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The forms of electron density structures in kinetic Alfvén wave turbulence are studied in connection with scintillation. The focus is on small scales $L
\sim 10^8-10^{10}$ cm where the Kinetic Alfvén wave (KAW) regime is active in the interstellar medium, principally within turbulent HII regions around bright stars. MHD turbulence converts to a KAW cascade, starting at 10 times the ion gyroradius and continuing to smaller scales. These scales are inferred to dominate scintillation in the theory of Boldyrev *et al.* [@boldyrev03a; @boldyrev03b; @boldyrev05; @boldyrev06]. From numerical solutions of a decaying kinetic Alfvén wave turbulence model, structure morphology reveals two types of localized structures, filaments and sheets, and shows that they arise in different regimes of resistive and diffusive damping. Minimal resistive damping yields localized current filaments that form out of Gaussian-distributed initial conditions. When resistive damping is large relative to diffusive damping, sheet-like structures form. In the filamentary regime, each filament is associated with a non-localized magnetic and density structure, circularly symmetric in cross section. Density and magnetic fields have Gaussian statistics (as inferred from Gaussian-valued kurtosis) while density gradients are strongly non-Gaussian, more so than current. This enhancement of non-Gaussian statistics in a derivative field is expected since gradient operations enhance small-scale fluctuations. The enhancement of density gradient kurtosis over current kurtosis is not obvious, yet it suggests that modest fluctuation levels in electron density may yield large scintillation events during pulsar signal propagation in the interstellar medium. In the sheet regime the same statistical observations hold, despite the absence of localized filamentary structures. Probability density functions are constructed from statistical ensembles in both regimes, showing clear formation of long, highly non-Gaussian tails.
author:
- 'K.W. Smith and P.W. Terry'
title: Damping of Electron Density Structures and Implications for Interstellar Scintillation
---
Introduction
============
Models of scintillation have a long history. Many [@lee75a; @lee75b; @sutton71] carry an implicit or explicit assumption of Gaussian statistics, applying to either the electron density field itself or its autocorrelation function (herein referred to as “Gaussian Models”). @lee75a is a representative approach. The statistics of the two-point correlation function of the index of refraction $\epsilon(\mathbf{r})$, $A(\rho) = \int{dz' \left< \epsilon(x,z) \epsilon(x+\rho,z') \right>}$ determines, among other effects, the scaling of pulsar signal width $\tau$ with dispersion measure $DM$. The index of refraction $\epsilon(\mathbf{r})$ is a function of electron density fluctuation $n(\mathbf{r})$. The quantity $A(0)$ enters the equations, representing the second moment of the index of refraction. If the distribution function of $\epsilon(\mathbf{r})$ has no second-order moment (as in a Lévy distribution) $A(0)$ is undefined. The assumption of Gaussian statistics leads to a scaling of $\tau \sim DM^2$, which contradicts observation for pulsars with $DM > 30$ cm$^{-3}$ pc. For these distant pulsars, $\tau \sim DM^4$ [@sutton71; @boldyrev03a; @boldyrev03b].
To explain the anomalous $DM^4$ scaling, @sutton71 argued that the pulsar signal encounters strongly scattering turbulent regions for longer lines of sight, essentially arguing that the statistics, as sampled by a pulsar signal, are nonstationary. Considering the pulse shape in time, @williamson72 [@williamson73; @williamson74] is unable to match observations with a Gaussian Model of scintillation unless the scattering region is confined to $1/4$ of the line of sight between the pulsar and Earth. These assumptions may have physical basis, since the ISM may not be statistically stationary, being composed of different regions with varying turbulence intensity [@boldyrev05].
The theory of @boldyrev03a [@boldyrev03b; @boldyrev05; @boldyrev06] takes a different approach to explain the anomalous $DM^4$ scaling by considering Lévy statistics for the density difference (defined below). Lévy distributions are characterized by long tails with no defined moments greater than first-order \[i.e., $A(0)$ is undefined for a Lévy distribution\]. The theory recovers the $\tau \sim DM^4$ relation with a statistically stationary electron density field. This theory also does not constrain the scattering region to a fraction of the line-of-sight distance.
The determinant quantity in the theory of Boldyrev *et al.* is the density difference, $\Delta n = n(\mathbf{x}_1, z) - n(\mathbf{x}_2, z)$. According to this model, if the distribution function of $\Delta n$ has a power-law decay as $|\Delta n| \rightarrow \infty$ and has no second moment, then it is possible to recover the $\tau \sim DM^4$ scaling [@boldyrev03b]. Assuming sufficiently smooth fluctuations, $\Delta n$ can be expressed in terms of the density gradient, $\mathbf{\sigma}(z)$: $n(\mathbf{x}_1) - n(\mathbf{x}_2)
\simeq \mathbf{\sigma}(z) \cdot (\mathbf{x}_1 - \mathbf{x}_2)$. Perhaps more directly, the density gradient enters the ray tracing equations \[Eqns. (7) in @boldyrev03a\], and is seen to be central to determining the resultant pulsar signal shape and width. This formulation of a scintillation theory does not require the distribution of $\Delta n$ to be Gaussian or to have a second-order moment.
The notion that the density difference is characterized by a Lévy distribution is a constraint on dynamical models for electron density fluctuations in the ISM. Consequently the question of how a Lévy distribution can arise in electron density fluctuations assumes considerable importance in understanding the ISM.
Previous work has laid the groundwork for answering this question. It has been established that electron density fluctuations associated with interstellar magnetic turbulence undergo a significant change in character near the scale $10 \rho_s$, where $\rho_s$ is the ion sound gyroradius [@ter01]. At larger scales, electron density is passively advected by the turbulent flow of an MHD cascade mediated by nonlinear shear Alfvén waves [@goldreich95]. At smaller scales the electron density becomes compressive and the turbulent energy is carried into a cascade mediated by kinetic Alfvén waves (KAW) [@ter01]. The KAW cascade brings electron density into equipartition with the magnetic field, allowing for a significant increase in amplitude. The conversion to a KAW cascade has been observed in numerical solutions of the gyrokinetic equations [@howes06], and is consistent with observations from solar wind turbulence [@harmon05; @bale05; @leamon98]. Importantly, it puts large amplitude electron density fluctuations (and large amplitude density gradients) at the gyroradius scale ($\sim 10^8-10^{10}$ cm), a desirable set of conditions for pulsar scintillation [@boldyrev06]. It is therefore appropriate to consider whether large-amplitude non-Gaussian structures can arise in KAW turbulence.
This question has been partially answered in a study of current filament formation in decaying KAW turbulence [@terry-smith07; @terry-smith08]. In numerical solutions to a two-field model with broadband Gaussian initial conditions large amplitude current filaments spontaneously arose. Each filament was associated with a large-amplitude electron density structure, circular in cross-section, that persisted in time. These electron density structures were not as localized as the corresponding current filaments, but were coherent and not mixed by surrounding turbulence. The observation of large amplitude current filaments is similar to the large-amplitude vortex filaments found in decaying 2D hydrodynamic turbulence [@mcw84]. Counterparts of such structures in 3D are predicted to be the dominant component for higher order structure functions [@she94].
@terry-smith07 proposed that a nonlinear refractive magnetic shear mechanism prevents the structures from mixing with turbulence. Radial shear in the azimuthally directed magnetic field associated with each large-amplitude current filament decreases the radial correlation length of the turbulent eddies and enhances the decorrelation rate. Eddies are unable to persist long enough to penetrate the shear boundary layer and disrupt the structure core. The structure persistence mechanism allows large-amplitude fluctuations to persist for many eddy-turnover times. As the turbulent decays these structures eventually dominate the statistics of the system. The spatial structure of the density field associated with localized circularly symmetric current filaments was shown from analytical theory [@terry-smith07] to yield a Lévy-distributed density gradient field. The kurtosis for the current field was significantly larger than the Gaussian-valued kurtosis of 3, indicating enhanced tails. The electron density and magnetic field kurtosis values were not significantly greater than 3. However, just as the current is non-Gaussian when the magnetic field is not, it is expected that numerical solutions should show non-Gaussian behavior for the density gradient. In the present paper, density gradient statistics are measured and found to be non-Gaussian. Rather than relying on kurtosis values alone, the probability density functions (PDFs) are computed from ensembles of numerical solutions, showing non-Gaussian PDFs for the density gradient field.
The previous studies of filament generation in KAW turbulence leave significant unanswered questions relating to structure morphology and its effect on scintillation. It is well established that MHD turbulence admits structures that are both filament-like and sheet-like. Can sheet-like structures arise in KAW turbulence? If so, what are the conditions or parameters favoring one type of structure versus the other? If sheet-like structures dominate in some circumstances, what are the statistics of the density gradient? Can they be sufficiently non-Gaussian to be compatible with pulsar scintillation scaling? It is desirable to consider such questions prior to calculation of rf wave scattering properties in the density gradient fields obtained from numerical solutions.
In this paper, we show that both current filaments and current sheet structures naturally arise in numerical solutions of a decaying KAW turbulence model. Each has a structure of the same type and at the same location in the electron density gradient. These structures become prevalent as the numerical solutions progress in time, and each is associated with highly non-Gaussian PDFs. Moreover, we show that small-scale current filaments and current sheets, along with their associated density structures, are highly sensitive to the magnitude of resistive damping and diffusive damping of density fluctuations. Current filaments persist provided that resistivity $\eta$ is small; similarly, electron density fluctuations and gradients are diminished by large diffusive damping in the electron continuity equation. The latter results from collisions assuming density fluctuations are subject to a Fick’s law for diffusion. The magnitude of the resistivity affects (1) whether current filaments can become large in amplitude, (2) their spatial scale, and (3) the preponderance of these filaments as compared to sheets. The magnitude of the diffusive damping parameter, $\mu$, similarly influences the amplitude of density gradients and, to a lesser degree, influences the extent to which electron density structures are non-localized.
In the ISM resistive and diffusive damping become important near resistive scales. However, it is well known that collisionless damping effects are also present [@lysak96; @bale05], and quite possibly dominate over collisional damping in larger scales near the ion Larmor radius. The collisional damping in the present work is understood as a heuristic approach that facilitates analysis of the effects of different damping regimes on the statistics of electron density fluctuations. By varying the ratio of resistive and diffusive damping we can, as suggested above, control the type of structure present in the turbulence. This allows us to isolate and study the statistics associated with each type of structure. It also allows us to assess and examine the type of environment conducive to formation of the structure. We consider regimes with large and small damping parameters, enabling us to explore damping effects on structure formation across a range from inertial to dissipative. Future work will address collisionless damping in greater detail.
Background Considerations for Structure Formation
-------------------------------------------------
The coherent structures observed in numerical solutions of decaying KAW turbulence, whether elongated sheets or localized filaments, are similar to structures observed in decaying MHD turbulence, as in @kinney95. In that work, the flow field initially gives rise to sheet-like structures. After selective decay of the velocity field energy, the system evolves into a state with sheets and filaments. During the merger of like-signed filaments, large-amplitude sheets arise, limited to the region between the merging filaments. These short-lived sheets exist in addition to the long-lived sheets not associated with the merger of filaments. In the two-field KAW system, however, there is no flow; the sheet and filament generation is due to a different mechanism, of which the filament generation has previously been discussed [@terry-smith07].
Other work [@biskamp89; @politano89] observed the spontaneous generation of current sheets and filaments in numerical solutions, with both Orszag-Tang vortex and randomized initial conditions. These 2D reduced MHD numerical solutions modeled the evolution of magnetic flux and vorticity with collisional dissipation coefficients $\eta$, the resistivity, and $\nu$, the kinematic viscosity. The magnetic Prandtl number, $\nu / \eta$, was set to unity. These systems are incompressible and not suitable for modeling the KAW system we consider here – they do however illustrate the ubiquity of current sheets and filaments, and serve as points of comparison. For Orszag-Tang-like initial conditions with large-scale flux tubes smooth in profile, current sheets are preferred at the interfaces between tubes. Tearing instabilities can give rise to filamentary current structures that persist for long times, but the large-scale and smoothness of flux tubes do not give rise to strong current filaments localized at the center of the tubes. To see this, consider a given flux tube, and model it as cylindrically symmetric and monotonically decreasing in $r$ with characteristic radial extent $a$,
$$\psi(r) = \psi_0 \left( 1 - (\frac{r}{a})^2 \right),$$
[ for $0 \leq r \leq a$. The current is localized at the center with magnitude]{}
$$J = - 4 \frac{\psi_0}{a^2}.$$
[ Thus flux tubes with large radial extent $a$ have a corresponding small current filament at their center. Hence, initial conditions dominated by a few large-scale flux tubes are not expected to have large amplitude current filaments at the flux tube centers, but favor current sheet formation and filaments associated with tearing instabilities in those sheet regions. At X points current sheet folding and filamentary structures can arise [see, e.g. @biskamp89 Fig. 10], but these regions are small in area compared to the quiescent flux-tube regions. Note that if, instead of Orszag-Tang-like initial conditions, the initial state is random, one expects some regions with flux tubes that have $a$ small, and therefore a sizable current filament at the center.]{}
Consider now the effect of comparatively large or small $\eta$. In the case of large $\eta$, the central region of a flux tube is smoothed by the collisional damping, thus having a strong suppressive effect on the amplitude of the current filament associated with such a flux tube. Large-amplitude current structures are localized to the interfaces between flux tubes. In the process of mergers between like-signed filaments (and repulsion between unlike-signed), large current sheets are generated at these interfaces, similar to the large-amplitude sheets generated in MHD turbulence during mergers [@kinney95]. For small $\eta$, relatively little suppression of isolated current filaments should occur; if these filaments are spatially separated owning to the buffer provided by their associated flux tube, they can be expected to survive a long time and only be disrupted upon the merger with another large-scale flux tube. Large $\eta$, then, allows current sheets to form at the boundaries between flux tubes while suppressing the spatially-separated current filaments at flux tube centers. Small $\eta$ allows interface sheets and spatially separated filaments to exist.
These simple arguments suggest that the evolution of the large-amplitude structures and their interaction with turbulence is thus strongly influenced by the damping parameters. As such, the magnitudes of the damping parameters are expected to affect the resultant pulsar scintillation scalings. The present paper considers the effect of variations of these damping parameters, $\eta$ and $\mu$. In the KAW model, the (unnormalized) resistivity takes the form $\eta = m_e \nu_e / n e^2$ and the density diffusion coefficient is $\mu =
\rho_e^2 \nu_e$, where $m_e$ is the electron mass, $\nu_e$ is the electron collision frequency, $n$ is the electron density, $e$ is the electron charge, and $\rho_e = v_{Te} / \omega_{ce}$, with $v_{Te}$ the electron thermal velocity, and $\omega_{ce}$ the electron gyrofrequency. The ratio of these terms, $c^2 \eta / 4\pi \mu = 2 / \beta$, where $\beta = 8 \pi n k T / B^2$ is the ratio of plasma to magnetic pressures. When we vary this ratio, as we will do in the numerical solutions presented here, we have in mind that we are representing regions of different $\beta$. However, as a practical matter in the numerical solutions, we must vary the damping parameters independently of the variation of $\beta$, since the kinetic Alfvén wave dynamics require a small $\beta$ to propagate. For the warm ionized medium, typical parameters are $T_e = 8000\;\mathrm{K}$, $n = 0.08\;\mathrm{cm}^{-3}$, $|B| =
1.4\;\mu\mathrm{G}$, $\delta B = 5.0\;\mu\mathrm{G}$ [@ferriere01]. With these parameters, the plasma $\beta$ formally ranges from $0.05-1.2$, spanning a range of plasma magnetization.
We present the results of numerical solutions of decaying KAW turbulence to ascertain the effect of different damping regimes on the statistics of the fields of interest, in particular the electron density and electron density gradient. In the $\eta \ll \mu$ regime (using normalized parameters), previous work [@crad91] had large-amplitude current filaments that were strongly localized with no discernible electron density structures ($\mu$ was large to preserve numerical stability). This regime is unable to preserve density structures or density gradients. The numerical solutions presented here have $\eta \sim \mu$ and $\eta \gg \mu$; in each limit the damping parameters are minimized so as to allow structure formation to occur, and are large enough to ensure numerical stability for the duration of each numerical solution. We investigate the statistics of both filaments and sheets in the context of scintillation in the warm ionized medium.
The paper is arranged as follows: section \[sec:KAW-model\] gives an overview of the KAW model and normalizations, its regime of validity, and its dispersion relation. Section \[sec:numerical-solution\] discusses the numerical method used and the field initializations. The negligible effect of initial cross-correlation between fields is discussed. Results for the two damping regimes are given in section \[sec:results\], where the type of structures that form, whether sheets-and-filaments or predominantly sheets, are seen to be dependent on the values of $\eta$ and $\mu$. PDFs from ensemble numerical solutions are presented in section \[sec:PDFs\], illustrating the strongly non-Gaussian statistics in the electron density gradient field for both the $\eta \sim \mu$ and $\eta \gg \mu$ regimes. This suggests that non-Gaussian electron-density gradients are robust to variation in $\eta$, as long as the overall damping in the continuity equation is not too large. Some discussions regarding the limitations of numerical approximation for this work and possible enhancements–particularly a model that addresses driven KAW turbulence–are given in concluding remarks.
Kinetic Alfvén Wave Model {#sec:KAW-model}
=========================
The kinetic Alfvén wave (KAW) model used in this paper is the same model used in theories of pulsar scintillation through the ISM [@terry-smith07; @terry-smith08] and in earlier work [@crad91]. It is a reduced, two-field, small-scale limit of a more general reduced three-field MHD system [@haz83; @rahman83; @fernandez97] that accounts for electron dynamics parallel to the magnetic field.
The 3-field model applies to large and small-scale fluctuations as compared to $\rho_s$, the ion gyroradius evaluated at the electron temperature. In large-scale strong turbulence magnetic and kinetic fluctuations are in equipartition, with electron density passively advected. In the limit of small spatial scales ($\leq 10 \rho_s$) the roles of kinetic and internal fluctuations are reversed – magnetic fluctuations are in equipartition with density fluctuations, and kinetic energy experiences a go-it-alone cascade without participating in the magnetic-internal energy interaction. The shear-Alfvén physics at large scale is supplanted by kinetic-Alfvén physics at small scale [@ter01].
In the Boldyrev *et al.* theory, the length scales that dominate scintillation for pulsars with $DM > 30 \;\mathrm{pc}\;\mathrm{cm}^{-3}$ are small, around $10^8-10^{10}\;\mathrm{cm}$. This motivates our focus on the small-scale regime of the more general 3-field system. The dominant interactions are between magnetic and internal fluctuations, via kinetic Alfvén waves. In these waves, electron density gradients along the magnetic field act on an inductive electric field in Ohm’s law. The electron continuity equation serves to close the system. The normalized equations are
$$\partial_t \psi = \nabla_{\|} n + \eta_0 J - \eta_2 \nabla^2 J,
\label{psi-eqn}$$
$$\partial_t n = - \nabla_{\|} J + \mu_0 \nabla^2 n - \mu_2 \nabla^2 \nabla^2 n,
\label{den-eqn}$$
$$\nabla_{\|} = \partial_z + \nabla\psi \times \mathrm{z} \cdot \nabla,$$
$$J = \nabla^2 \psi,$$
[ with $\psi = (C_s/c)e A_z/T_e$, the normalized parallel component of the vector potential and $n = (C_s/V_A)\tilde{n}/n_0$ the normalized electron density. The normalized resistivity is $\eta_0 = (c^2/4\pi V_A \rho_s)
\eta_{sp}$, with $\eta_{sp}$ the Spitzer resistivity, given in the introduction. The normalized diffusivity is $\mu = \rho_e^2 \nu_e /\rho_s
V_A$. The time and space normalizations are $\tau_A = \rho_s / V_A$ and $\rho_s = C_s / \Omega_i$. Here $C_s = (T_e/m_i)^{1/2}$ is the ion acoustic velocity, $V_A = B / (4\pi m_i n_0)^{1/2}$ is the Alfvén speed, and $\Omega_i
= eB/m_i c$ is the ion gyrofrequency. Electron density diffusion is presumed to follow Fick’s law; more detailed damping would necessarily consider kinetic effects and cyclotron resonances. The $\eta_2$ and $\mu_2$ terms (hyper-resistivity and hyper-diffusivity) are introduced to mitigate large-scale Fourier-mode damping by the linear diffusive terms. Throughout the remainder of the paper, we drop the $2$ subscript from $\eta_2$ and $\mu_2$ and refer to the hyper-dissipative terms as $\eta$ and $\mu$.]{}
Three ideal invariants exist: total energy $E = \int d^2x [(\nabla \psi)^2 +
\alpha n^2]$; flux $F = \int d^2x\; \psi^2$ and cross-correlation $H_c = \int
d^2 x\; n\psi$. Energy cascades to small scale (large $k$) while the flux and cross-correlation undergo an inverse cascade to large-scale (small $k$) [@fernandez97]. The inverse cascades require the initialized spectrum to peak at $k_0 \neq 0$ to allow for buildup of magnetic flux at large-scales for later times.
Linearizing the system yields a (dimensional) dispersion relation $\omega
= V_A k_z k_\perp \rho_s$. The mode combines perpendicular oscillation associated with a finite gyroradius with fluctuations along a mean field ($z$-direction). The oscillating quantities are magnetic field and density, out of phase by $\pi/2$ radians.
In the limit of strong mean field, quantities along the mean field ($z$-direction) equilibrate quickly, which allows $\partial / \partial z
\rightarrow 0$, or $k_z \rightarrow 0$. Kinetic Alfvén waves still propagate, as long as there are a broad range of scales that are excited, as in fully developed turbulence. As $k_z \rightarrow 0$, all gradients are localized to the plane perpendicular to the mean field. Presuming a large-scale fluctuation at characteristic wavenumber $\mathbf{k}_0$, smaller-scale fluctuations propagate linearly along this larger-scale fluctuation so long as their characteristic scale $k$ satisfies $k \gg k_0$. In this reduced, two dimensional system, the above dispersion relation is modified to be $\omega = V_A (\mathbf{b}_{k_0} \cdot \mathbf{k} / B) k \rho_s$ which is still Alfvénic but with respect to a perturbed large-scale amplitude perpendicular to the mean field. Relaxing the scale separation criterion yields $\omega \propto k^2$ for the general case.
Numerical Solution Method {#sec:numerical-solution}
=========================
We evolve Eqs. (\[psi-eqn\]) and (\[den-eqn\]) in a 2D periodic box, size $[2\pi] \times [2\pi]$ on a mesh of resolution $512 \times 512$. The $\psi$ and $n$ scalar fields are evolved in the Fourier domain, with the nonlinearities advanced pseudospectrally and with full $2/3$ dealiasing in each dimension [@orszag71]. The diffusive and resistive terms normally introduce stiffness into the equations; using an integrating factor removes any stability constraints stemming from these terms. Following the scheme outlined in @canuto90, we start with the semi-discrete formulation of Eqs. (\[psi-eqn\]) and (\[den-eqn\]):
$$\frac{d \psi_k}{d t} = -\eta k^4 \psi_k + \mathcal{F} \left[ \nabla_{\|} n \right]
\label{psi_discrete}$$
$$\frac{d n_k}{d t} = -\mu k^4 n_k - \mathcal{F} \left[ \nabla_{\|} J \right],
\label{den_discrete}$$
[ where $\mathcal{F}[\cdot]$ denotes the discrete Fourier transform. We do not explicitly expand the nonlinear terms as they will be integrated separately. The hyper-damping terms (proportional to $k^4$) are included above. Damping terms corresponding to the Laplacian operator (proportional to $k^2$) are not included in this section for clarity, but are trivial to incorporate. Equations \[psi\_discrete\] and \[den\_discrete\] can be put in the form ]{}
$$\frac{d}{dt} \left[e^{\eta k^4 t} \psi_k \right] = e^{\eta k^4 t} \mathcal{F} \left[ \nabla_{\|} n \right]$$
$$\frac{d}{dt} \left[e^{\mu k^4 t} n_k \right] = - e^{\mu k^4 t} \mathcal{F}
\left[ \nabla_{\|} J \right].$$
[ A second-order Runge-Kutta scheme for the $\psi_k$ difference equation is ]{}
$$\psi_k^{m+1/2} = e^{- \eta k^4 \Delta t/2} \left[
\psi_k^m + \Delta t/2
\mathcal{F}\left[ \nabla_{\|} n^m \right]
\right]$$
$$\psi_k^{m+1} = e^{- \eta k^4 \Delta t} \left[
\psi_k^{m+1/2} + \Delta t
\mathcal{F}\left[ \nabla_{\|} n^{m+1/2} \right]
\right]$$
[ with a similar form for the $n_k$ scheme.]{}
Initial conditions
------------------
The $\psi_k$ and $n_k$ fields are initialized such that the energy spectra are broad-band with a peak near $k_0 \sim 6-10$ and a power law spectrum for $k >
k_0$. The falloff in $k$ is predicted to be $k^{-2}$ for small-scale turbulence. @crad91 use $k^{-3}$, between the current-sheet limit of $k^{-4}$ and the kinetic-Alfvén wave strong-turbulence limit of $k^{-2}$. The numerical solutions considered here have either $k^{-2}$ or $k^{-3}$. The only qualitative difference between the two spectra is the scale at which structures initially form. The $k^{-2}$ spectra has more energy at smaller scales, leading to smaller characteristic structure size. After a few tens of Alfvén times these smaller-scale structures merge and the system resembles the initial $k^{-3}$ spectra.
The $n_k$ and $\psi_k$ phases can be either cross-correlated or uncorrelated. By cross-correlated we mean that the phase angle for each Fourier component of the $n_k$ and $\psi_k$ fields are equal. In general,
$$n_k = |A_k| e^{i\theta_1}, \qquad \psi_k = |B_k| e^{i \theta_2},$$
[ where $|A_k|$ and $|B_k|$ are the Fourier component’s amplitude, set according to the spectrum power-law. For cross-correlated initial conditions, $\theta_1 = \theta_2$ for all $\mathbf{k}$ at the initial time. For uncorrelated initialization, there is no phase relation between corresponding Fourier components of the $n_k$ and $\psi_k$ fields.]{}
@crad91 focused on the formation and longevity of current filaments in a turbulent KAW system. To preserve small-scale structure in the current filaments, these numerical solutions set $\eta = 0$ and had $\mu \sim 10^{-3}$, with a resolution of $128 \times 128$, corresponding to a $k_{max}$ of 44. Large-amplitude density structures that would have arisen were damped to preserve numerical stability up to an advective instability time of a few hundred Alfvén times, for the parameter values therein. The numerical solutions presented here explore a range of parameter values for $\eta$ and $\mu$. They make use of hyper-diffusivity and hyper-resistivity of appropriate strengths to preserve structures in $n$, $B$ and $J$. An advective instability is excited after $\sim 10^2$ Alfvén times if resistive damping is negligible. The $\eta = 0$ solutions–not presented here due to their poor resolution of small-scale structures–see large-amplitude current filaments arise, but they can be poorly resolved at this grid spacing. With no resistivity, the finite number of Fourier modes cannot resolve arbitrarily small structures without Gibbs phenomena resulting and distorting the current field.
We have found through experience that small hyper-resistivity and small hyper-diffusivity preserve large-amplitude density structures and their spatial correlation with the magnetic and current structures, while preventing the distortion resulting from poorly-resolved current sheets and filaments. They allow the numerical solutions to run for arbitrarily long times, and the effects of structure mergers become apparent. These occur on a longer timescale than the slowest eddy turnover times. The results presented here will consider two regimes of parameter values, the $\eta \approx \mu$ and $\eta \gg \mu$ regimes. The effect of cross-correlated and uncorrelated initial conditions will be addressed presently.
Results {#sec:results}
=======
It is of interest to examine whether cross-correlated or uncorrelated initial conditions affect the long-term behavior of the system. Two representative numerical solutions are presented here that reveal the system’s tendency to form spatially-correlated structures in electron density and current regardless of initial phase correlations. This study establishes the robustness of density structure formation in KAW turbulence and lends confidence that such structures should exist in the ISM under varying circumstances. The first numerical solution has cross-correlated initial conditions between the $n$ and $\psi$ fields; the second, uncorrelated. Damping parameters $\eta$ and $\mu$ are equal and large enough to ensure numerical stability while preserving structures in density, current and magnetic fields. These examples also serve to explore the intermediate $\eta / \mu$ regime.
The energy vs. time history for both numerical solutions are given in Figs.\[energy-time-correlated\] and \[energy-time-uncorrelated\]. Total energy is a monotonically decreasing function of time. The magnetic and internal energies remain in overall equipartition throughout the numerical solutions. Magnetic energy increases at the expense of internal energy and *vice versa*. This energy interchange is consistent with KAW dynamics and overall energy conservation in the absence of resistive or diffusive terms. The exchange is crucial in routinely producing large amplitude density fluctuations in this two-field model of nonlinearly interacting KAWs.
The total energy decay rates for the uncorrelated and correlated initial conditions in Figs. \[energy-time-correlated\] and \[energy-time-uncorrelated\] differ, with the latter decaying more strongly than the former. The damping parameters are identical for the two numerical solutions, and the decay-rate difference remains under varying randomization seeds. The magnitudes of the nonlinear terms during the span of a numerical solution in Eqs. (\[psi-eqn\]) and (\[den-eqn\]) for uncorrelated initial conditions are consistently larger than those of correlated initial conditions by a factor of 5. This difference lasts until 2500 Alfvén times, after which the decay rates are roughly equal in magnitude. The steeper energy decay during the run of numerical solutions with uncorrelated initial conditions (Fig. \[energy-time-uncorrelated\]) suggests that the enhancement of the uncorrelated nonlinearities transports energy to higher $k$ (smaller scale) more readily than the nonlinearities in the correlated case. Relatively more energy in higher $k$ enhances the energy decay rate as the linear damping terms dissipate more energy from the system. The initial configuration, whether correlated or uncorrelated, is seen to have an effect on the long-term energy evolution for these decaying numerical solutions. It will be shown below, however, that the correlation does not significantly affect the statistics of the resulting fields.
For cross-correlated initial conditions, we expect there to be a strong spatial relation between current, magnetic field and density structures through time. Figs. \[dendat-correlated-contour\] and \[bmag-correlated-contour\] show the $n$ and $|\mathrm{B}|$ contours at various times. For the latest time contour, the spatial structure alignment is evident. Further, in Fig.\[density-quiver-correlated\], the circular magnetic field structures (magnetic field direction and intensity indicated by arrow overlays) align with the large-amplitude density fluctuations. The correlation is evident once one notices that every positive-valued circular $n$ structure corresponds to counterclockwise-oriented magnetic field, and *vice versa*. Fig.\[density-quiver-correlated\] is at a normalized time of 5000 Alfvén times, defined in terms of the large $\mathrm{B}_0$. The system preserves the spatial structure correlation indefinitely, even after structure mergers.
The second representative numerical solution is one with uncorrelated initial conditions. Contour plots of density and $|\mathrm{B}|$ are given in Figs.\[dendat-uncorrelated-contour\] and \[bmag-uncorrelated-contour\], respectively. It is noteworthy that, similar to the cross-correlated initial conditions, spatially correlated density and magnetic field structures are discernible at the latest time contour.
In Fig. \[density-quiver-uncorrelated\] the circular density structures correspond to circular magnetic structures. Unlike Fig.\[density-quiver-correlated\] the positive density structures may correspond to clockwise or counterclockwise directed magnetic field structures. This serves to illustrate that, although the initial conditions have no phase relation between fields, after many Alfvén times circular density structures spatially correlate with magnetic field structures and persist for later times.
The kurtosis excess as a function of time, defined as $K(\Xi)=\left<\Xi^4\right>/\sigma_{\Xi}^4 - 3$, is shown in Figs.\[kurtosis-correlated\] and \[kurtosis-uncorrelated\] for correlated and uncorrelated initial conditions, respectively. Positive $K$ indicates a greater fraction of the distribution is in the tails as compared to a best-fit Gaussian. These figures indicate that the non-Gaussian statistics for the fields of interest are independent of initial correlation in the fields. In particular, the density gradients, $|\nabla n|$, are significantly non-Gaussian as compared to the current. Because scintillation is tied to density gradients, this situation is expected to favor the scaling inferred from pulsar signals.
The tendency of density structures to align with magnetic field structures regardless of initial conditions indicates that the initial conditions are representative of fully-developed turbulence. After a small number of Alfvén times the memory of the initial state is removed as the KAW interaction sets up a consistent phase relation between the fluctuations in the magnetic and density fields. Previous work [@terry-smith07] presented a mechanism whereby these spatially correlated structures can be preserved via shear in the periphery of the structures. The above figures indicate that this mechanism is at play even in cases where the initial phase relations are uncorrelated.
In the damping regime presented above, circularly symmetric structures in density, current and magnetic fields readily form and persist for many Alfvén times, until disrupted by mergers with other structures of similar amplitude. It is possible to define, for each circular structure, an effective separatrix that distinguishes it from surrounding turbulence and large-amplitude “sheets” that exist between structures. \[see, e.g., the magnetic field contours at later times in Fig. \[bmag-uncorrelated-contour\].\] The density field has significant gradients in both the regions surrounding the structure and within the structures themselves. The ability to separate these circular structures from the background sheets and turbulence is determined by the magnitudes – relative and absolute – of the damping parameters. Larger damping values erode the small-spatial-scale structures to a greater extent and, if large enough, disrupt the structure persistence mechanism that, for a fixed diameter, depends on a sufficiently large amplitude current filament to generate a sufficiently large radially sheared magnetic field.
The preceding results were for a damping regime where $\eta / \mu \sim 1$, an intermediate regime. Numerical solutions with $\mu = 0$ and $\eta$ small explore the regime where $\eta / \mu \rightarrow 0$. In this regime, which is opposite the regime used in Craddock et al., circularly symmetric current and magnetic structures are not as prevalent, rather, sheet-like structures dominate the large amplitude fluctuations. Current and magnetic field gradients are strongly damped, and the characteristic length scales in these fields are larger.
Contours of density for a numerical solution with $\mu = 0$ are shown in Fig.\[fig:density-zero-mu\]. Visual comparison with contours for runs with smaller damping parameters (Fig. \[dendat-uncorrelated-contour\], where $\eta = \mu$) indicate a preponderance of sheets in the $\mu = 0$ case, at the expense of circularly-symmetric structures as seen above. All damping is in $\eta$; any current filament that would otherwise form is unable to preserve its small-scale, large amplitude characteristics before being resistively damped. Inspection of the current and $|B|$ contours for the same numerical solution \[Figs. \[fig:current-zero-mu\] and \[fig:bmag-zero-mu\]\] reveal broader profiles and relatively few circular current and magnetic field structures with a well-defined separatrix as in the small $\eta$ case. Since there is no diffusive damping, gradients in electron density are able to persist, and electron density structures generally follow the same structures in the current and magnetic fields.
Kurtosis excess measurements for the $\mu = 0$ numerical solutions yield mean values consistent with the $\eta = \mu$ numerical solutions, as seen in Fig.\[fig:zero-mu-kurtosis\]. Magnetic field strength and electron density statistics are predominantly Gaussian, with current statistics and density gradient statistics each non-Gaussian. Perhaps not as remarkable in this case, the density gradient kurtosis excess is again seen to be greater than the current kurtosis excess – this is anticipated since the dominant damping of density gradients is turned off. With fewer filamentary current structures, however, the mechanism proposed in @terry-smith07 is not likely to be at play in this case, since few large-amplitude filamentary current structures exist. Sheets, evident in the density gradients in Fig.\[fig:dengrad-zero-mu\] and in the current in Fig. \[fig:current-zero-mu\] are the dominant large-amplitude structures and determine the extent to which the density gradients have non-Gaussian statistics. The current and density sheets are well correlated spatially. The largest sheets can extend through the entire domain, and evolve on a longer timescale than the turbulence. Sheets exist at the interface between large-scale flux tubes, and are regions of large magnetic shear, giving rise to reconnection events. With $\eta$ relatively large, the sheets evolve on timescales shorter than the structure persistence timescale associated with the long-lived flux tubes.
Sheets and filaments are the dominant large-amplitude, long timescale structures that arise in the KAW system. Filaments arise and persist as long as $\eta$ is small, with their amplitude and statistical influence diminished as $\eta$ increases. Sheets exist in both regimes, becoming the sole large-scale structure in the large $\eta$ regime. Density gradients are consistently non-Gaussian in both regimes as long as $\mu$ is small, although the density structures are different in both regimes. Density gradient sheets arise in the large $\eta$ regime and these density gradient sheets are large enough to yield non-Gaussian statistics.
Ensemble Statistics and PDFs {#sec:PDFs}
============================
To explicitly analyze the extent to which the decaying KAW system develops non-Gaussian statistics, ensemble runs were performed for both the $\eta / \mu
\sim 1$ and $\eta / \mu \ll 1$ regimes, and PDFs of the fields were generated.
For the $\eta / \mu \sim 1$ regime, 10 numerical solutions were evolved with identical parameters but for different randomization seeds. In this case $\eta
= \mu$ and both damping parameters have minimal values to ensure numerical stability. The fields were initially phase-uncorrelated. The density gradient ensemble PDF for two times in the solution results is shown in Fig.\[fig:dgx-PDF\]. Density gradients are Gaussian distributed initially. Many Alfvén times into the numerical solution the statistics are non-Gaussian with long tails. These PDFs are consistent with the time histories of density gradient kurtosis excess as shown above. The distribution tail extends beyond 15 standard deviations, almost 90 orders of magnitude above a Gaussian best-fit distribution. Similar behavior is seen in the current PDFs – initially Gaussian distributed tending to strongly non-Gaussian statistics with long tails for later times. Fig. \[fig:cur-PDF\] is the current PDF at an advanced time into the numerical solution. It is to be noted that the density gradient PDF has longer tails at higher amplitude than does the current PDF. One would expect these to be in rough agreement, since the underlying density and magnetic fields have comparable PDFs that remain Gaussian distributed throughout the numerical solution. The discrepancy between the density gradient and current PDFs suggests a process that enhances density derivatives above magnetic field derivatives. Future work is required to explore causes of this enhancement. This result is significant for pulsar scintillation, which is most sensitive to density gradients. Although interstellar turbulence is magnetic in nature, the KAW regime has the benefit of fluctuation equipartition between $n$ and $B$. The density gradient, however, is more non-Gaussian than the magnetic component, suggesting that this type of turbulence is specially endowed to produce the type of scintillation scaling observed with pulsar signals.
Ensemble runs for the $\eta / \mu \ll 1$ regime yield distributions similar to the $\eta / \mu \sim 1$ regime in all fields. The ensemble PDF for two times is shown in Fig. \[fig:dgx-PDF-zero-mu\]. The initial density gradient PDF is Gaussian distributed. For later times long tails are evident and consistent with the kurtosis excess measurements as presented above for the $\mu = 0$ case. The density gradient distribution has longer tails at higher amplitude than the current distribution; the overall distributions are similar to those for the $\eta / \mu \sim 1$ regime, despite the absence of filamentary structures and the presence of sheets. The strongly non-Gaussian statistics are insensitive to the damping regime, provided that the diffusion coefficient is small enough to allow density gradients to persist.
Discussion
==========
Using the normalizations for Eqs. (\[psi-eqn\]) and (\[den-eqn\]) and using $B=1.4 \mu$G, $n=0.08$ cm$^{-3}$ and $T_e=1$ eV, $\eta_{norm}$, the normalized Spitzer resistivity, is $2.4 \times 10^{-7}$ and $\mu_{norm}$, the normalized collisional diffusivity, is $1.9 \times 10^{-7}$. For a resolution of $512^2$, these damping values are unable to keep the system numerically stable. The threshold for stability requires the simulation $\eta$ to be greater than $5 \times 10^{-6}$, which is almost within an order of magnitude of the ISM value. The numerical solutions presented here, while motivated by the pulsar signal width scalings, more generally characterize the current and density gradient PDFs when the damping parameters are varied. We would expect the density gradients to be non-Gaussian when using parameters that correspond to the ISM. Future work will address the pulsar width scaling using electron density fields from the numerical solution.
The non-Gaussian distributions presented here are strongly tied to the fact that the system is decaying and that circular intermittent structures are preserved from nonlinear interaction. One can show that, in the KAW system, circularly symmetric structures (or filaments) are force free in Eqs.(\[psi-eqn\]) and (\[den-eqn\]), i.e., the nonlinearity is zero. Once a large-amplitude structure becomes sufficiently circularly symmetric and is able to preserve itself from background turbulence via the shear mechanism, that structure is expected to persist on long timescales relative to the turbulence. Structure mergers will lead to a time-asymptotic state with two oppositely-signed current structures and no turbulence. As structures merge, kurtosis excess increases until the system reaches a final two-filament state, which would have a strongly non-Gaussian distribution and large kurtosis excess.
If the system were driven, energy input at large scales would replenish large-amplitude fluctuations. New structures would arise from large amplitude regions whenever the radial magnetic field shear were large enough to preserve the structure from interaction with turbulence. One could define a structure-replenishing rate from the driving terms that would depend on the energy injection rate and scale of injection. The non-Gaussian measures for a driven system would be characterized by a competition between the creation of new structures through the injection of energy at large scales and the annihilation of structures by mergers or by erosion from continuously replenished small-scale turbulence. If erosion effects dominate, the kurtosis excess is maintained at Gaussian values, diminishing the PDF tails relative to a Lévy distribution. If replenishing effects dominate, however, the enhancement of the tails of the density gradient PDF may be observed in a driven system as it is observed in the present decaying system. We note that structure function scaling in hydrodynamic turbulence is consistent with the replenishing effects becoming more dominant relative to erosion effects as scales become smaller, i.e., the turbulence is more intermittent at smaller scales. The large range of scales in interstellar turbulence and the conversion of MHD fluctuations to kinetic Alfvén fluctuations at small scales both support the notion that the structures of the decaying system are relevant to interstellar turbulence at the scales of KAW excitations. This scenario is consistent with arguments suggested by @harmon05. They propose a turbulent cascade in the solar wind that injects energy into the KAW regime, counteracting Landau damping at scales near the ion Larmor radius. By doing so they can account for enhanced small-scale density fluctuations and observed scintillation effects in interplanetary scintillation.
We also observe that, although the numerical solutions presented here are decaying in time, the decay rate decreases in absolute value for later times (Figs. \[energy-time-correlated\] and \[energy-time-uncorrelated\]), approximating a steady-state configuration. The kurtosis excess (Figs.\[kurtosis-correlated\] and \[kurtosis-uncorrelated\]) for the density gradient field is statistically stationary after a brief startup period. Despite the decaying character of the numerical solutions, they suggest that the density gradient field would be non-Gaussian in the driven case.
The kurtosis excess – a measure of a field’s spatial intermittency – is itself intermittent in time. The large spikes in kurtosis excess correspond to rare events involving the merger of two large-amplitude structures, usually filaments. A large-amplitude short-lived sheet grows between the structures and persists throughout the merger, gaining amplitude in time until the point of merger. The kurtosis excess during this merger event is dominated by the single large-amplitude sheet between the merging structures. This would likely be the region of dominant scattering for scintillation, since a corresponding large-amplitude density gradient structure exists in this region as well. The temporal intermittency of kurtosis excess suggests that these mergers are rare and hence, of low probability. The heuristic picture of long undeviated Lévy flights punctuated by large angular deviations could apply to these merger sheets.
Conclusions
===========
Decaying kinetic Alfvén wave turbulence is shown to yield non-Gaussian electron density gradients, consistent with non-Gaussian distributed density gradients inferred from pulsar width scaling with distance to source. With small resistivity, large-amplitude current filaments form spontaneously from Gaussian initial conditions, and these filaments are spatially correlated with stable electron density structures. The electron density field, while Gaussian throughout the numerical solution, has gradients that are strongly non-Gaussian. Ensemble statistics for current and density gradient fields confirm the kurtosis measurements for individual runs. Density gradient statistics, when compared to current statistics, have more enhanced tails, even though both these fields are a single derivative away from electron density and magnetic field, respectively, which are in equipartition and Gaussian distributed throughout the numerical solution.
When all damping is placed in resistive diffusion ($\eta / \mu \rightarrow 0$ regime), filamentary structures give way to sheet-like structures in current, magnetic, electron density and density gradient fields. Kurtosis measurements remain similar to those for the small $\eta$ case, and the field PDFs also remain largely unchanged, despite the different large-amplitude structures at play.
The kind of structures that emerge, whether filaments or sheets, is a function of the damping parameters. With $\eta$ and $\mu$ minimal to preserve numerical stability and of comparable value, the decaying KAW system tends to form filamentary current structures with associated larger-scale magnetic and density structures, all generally circularly symmetric and long-lived. Each filament is associated with a flux tube and can be well separated from the surrounding turbulence. Sheets exist in this regime as well, and they are localized to the interface between flux tubes. With $\eta$ small and $\mu =
0$, the system is in a sheet-dominated regime. Both regimes have density gradients that are non-Gaussian with large kurtosis.
The effects on pulsar signal scintillation in each regime have yet to be ascertained directly. The conventional picture of a Lévy flight is a random walk with step sizes distributed according to a long-tailed distribution with no defined variance. This gives rise to long, uninterrupted flights punctuated by large scattering events. This is in contrast to a normally-distributed random walk with relatively uniform step sizes and small scattering events. The intermittent filaments that arise in the small $\eta$ and $\mu$ regime are suggestive of structures that could scatter pulsar signals through large angles, however the associated density structures are broadened in comparison to the current filament and would not give rise to as large a scattering event. Even broadened structures can yield Lévy distributed density gradients [@terry-smith07], but it is not clear how the Lévy flight picture can be applied to these broad density gradient structures. In the $\mu = 0$ regime, the large-aspect-ratio sheets may serve to provide the necessary scatterings through refraction and may map well onto the Lévy flight model.
An alternative possibility, suggested by the *temporal* intermittency of the kurtosis (itself a measure of a field’s *spatial* intermittency), is the encounter between the pulsar signal and a short-lived sheet that arises during the merger of two filamentary structures. These sheets are limited in extent and have very large amplitudes. At their greatest magnitude they are the dominant structure in the numerical solution. Their temporal intermittency distinguish them from the long-lived sheets surrounding them. It is possible that a pulsar signal would undergo large scattering when interacting with a merger sheet. This scattering would be a rare event, suggestive of a scenario that would give rise to a Lévy flight.
Acknowledgments
===============
We thank S. Boldyrev, S. Spangler and E. Zweibel for helpful discussions and comments.
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![Energy vs. time for cross-correlated initial conditions. Total energy is monotonically decreasing with time, and magnetic and internal energies remain in rough equipartition.[]{data-label="energy-time-correlated"}](energy-time-correlated.eps){width="\textwidth"}
![Energy vs. time for uncorrelated initial conditions. Total energy is monotonically decreasing with time, and magnetic and internal energies remain in rough equipartition.[]{data-label="energy-time-uncorrelated"}](energy-time-uncorrelated.eps){width="\textwidth"}
![Contours of $n$ for various times in a numerical solution with correlated initial conditions.[]{data-label="dendat-correlated-contour"}](dendat-correlated-contours.eps){width="\textwidth"}
![Contours of $|\mathrm{B}|$ for various times in a numerical solution with correlated initial conditions.[]{data-label="bmag-correlated-contour"}](bmag-correlated-contours.eps){width="\textwidth"}
![Contour plot of $n$ with $\mathbf{B}$ vectors overlaid. The positive, circularly-symmetric density structures correspond to counterclockwise-directed $\mathbf{B}$ structures; the opposite holds for negative circularly-symmetric density structures. These spatial correlations are to be expected for correlated initial conditions.[]{data-label="density-quiver-correlated"}](density-quiver-correlated-selected-hot.eps){width="\textwidth"}
![Contours of $n$ for various times in a numerical solution with uncorrelated initial conditions.[]{data-label="dendat-uncorrelated-contour"}](dendat-uncorrelated-contours.eps){width="\textwidth"}
![Contours of $|\mathrm{B}|$ for various times in a numerical solution with uncorrelated initial conditions.[]{data-label="bmag-uncorrelated-contour"}](bmag-uncorrelated-contours.eps){width="\textwidth"}
![Contour plot of $n$ with $\mathbf{B}$ vectors overlaid for a numerical solution with initially uncorrelated initial conditions. The positive, circularly-symmetric density structures correspond to magnetic field structures, although the sense (clockwise or counterclockwise) of the magnetic field structure does not correlate with the sign of the density structures. Circled in black are symmetric structures that display a high degree of spatial correlation. The circle gives an approximate indication of the separatrix for the structure.[]{data-label="density-quiver-uncorrelated"}](density-quiver-uncorrelated-selected-hot.eps){width="\textwidth"}
![Kurtosis excess for a numerical solution with phase-correlated initial conditions and $\eta / \mu = 1$.[]{data-label="kurtosis-correlated"}](correlated-kurtosis.eps){width="\textwidth"}
![Kurtosis excess for a numerical solution with phase-uncorrelated initial conditions and $\eta / \mu = 1$.[]{data-label="kurtosis-uncorrelated"}](uncorrelated-kurtosis.eps){width="\textwidth"}
![Electron density contour visualization with diffusive damping parameter $\mu = 0$ for various times.[]{data-label="fig:density-zero-mu"}](dendat-correlated-contours.eps){width="\textwidth"}
![Current density contour visualization with diffusive damping parameter $\mu = 0$ for various times.[]{data-label="fig:current-zero-mu"}](cur-correlated-contours.eps){width="\textwidth"}
![Magnitude of magnetic field contour visualization with diffusive damping parameter $\mu = 0$ for various times.[]{data-label="fig:bmag-zero-mu"}](bmag-correlated-contours.eps){width="\textwidth"}
![Kurtosis excess for a numerical solution with diffusive parameter $\mu
= 0$. Density gradient kurtosis remains greater than current kurtosis for the duration of the numerical solution.[]{data-label="fig:zero-mu-kurtosis"}](zero-diffusivity-kurtosis.eps){width="\textwidth"}
![Electron density gradient ($x$ direction) contour visualization with diffusive damping $\mu = 0$ for various times.[]{data-label="fig:dengrad-zero-mu"}](dengrad_x-correlated-contours.eps){width="\textwidth"}
![log-PDF of density gradients for an ensemble of numerical solutions with $\eta / \mu = 1$ at $t=0$ and $t=5000$. The density gradient field at $t=0$ is Gaussian distributed, while for $t=5000$ the gradients are enhanced in the tails, and deviate from a Gaussian. A best-fit Gaussian for each PDF is plotted for comparison.[]{data-label="fig:dgx-PDF"}](den-grad-pdf-2-times.eps){width="\textwidth"}
![log-PDF of current for an ensemble of numerical solutions with $\eta /
\mu = 1$ at $t=0$ and $t=5000$. The current at $t=0$ is Gaussian distributed. For $t=5000$ the current is non-Gaussian. Unlike the density gradient, the current is not enhanced in the tails of the PDF for later times relative to its initial Gaussian envelope.[]{data-label="fig:cur-PDF"}](current-pdf-2-times.eps){width="\textwidth"}
![Log-PDF of density gradient for an ensemble of numerical solutions with $\mu=0$ at $t=0$ and $t=5000$. The density gradient field at $t=0$ is Gaussian distributed, while for $t=5000$ the gradients are enhanced in the tails, and deviate from a Gaussian. A best-fit Gaussian for each PDF is plotted for comparison.[]{data-label="fig:dgx-PDF-zero-mu"}](den-grad-pdf-2-times-mu-0.eps){width="\textwidth"}
|
---
abstract: 'Using the multi-channel Transient Current Technique the currents induced by electron-hole pairs, produced by a focussed sub-nanosecond laser of 660 nm wavelength close to the Si-SiO$_2$ interface of $p^+n$ silicon strip sensors have been measured, and the charge-collection efficiency determined. The laser has been operated in burst mode, with bursts typically spaced by 1 ms, each consisting of 30 pulses separated by 50 ns. In a previous paper it has been reported that, depending on X-ray-radiation damage, biasing history and humidity, situations without charge losses, with hole losses, and with electron losses have been observed. In this paper we show for sensors before and after irradiation by X-rays to 1 MGy (SiO$_2$), how the charge losses change with the number of electron-hole pairs generated by each laser pulse, and the time interval between the laser pulses. This allows us to estimate how many additional charges in the accumulation layers at the Si-SiO$_2$ interface have to be trapped to significantly change the local electric field, as well as the time it takes that the accumulation layer and the electric field return to the steady-state situation. In addition, results are presented on the change of the pulse shape caused by the plasma effect for high charge densities deposited close to the Si-SiO$_2$ interface.'
address:
- '$^a$Institute for Experimental Physics, University of Hamburg, Hamburg, Germany'
- '$^b$DESY, Hamburg, Germany'
author:
- 'Thomas Poehlsen$^{a,}$'
- Julian Becker$^b$
- Eckhart Fretwurst$^a$
- Robert Klanner$^a$
- Jörn Schwandt$^a$
- Jiaguo Zhang$^a$
title: 'Study of the accumulation layer and charge losses at the Si-SiO$_2$ interface in p$^+$n-silicon strip sensors.'
---
Silicon strip sensors ,X-ray-radiation damage ,charge losses ,Si-SiO$_2$ interface ,accumulation layer ,plasma effect ,XFEL
Introduction {#chapter:introduction}
============
The high instantaneous intensity and the 4.5 MHz repetition rate of the European X-Ray Free-Electron Laser (XFEL) [@XFEL; @XFEL2; @Tschentscher:2011] pose new challenges for imaging X-ray detectors [@Graafsma:2009; @Klanner:2011]. The specific requirements for the detectors include a dynamic range of 0, 1 to more than 10$^4$ photons of typically 12.4 keV per pixel for an XFEL pulse duration of less than 100 fs, and a radiation tolerance for doses up to 1 GGy (SiO$_2$) for 3 years of operation.
One question is, if all charges are collected in the 220 ns between XFEL pulses for the high instantaneous charge-carriers densities, or if pile-up effects appear. In [@Becker:2010; @Becker:Thesis] the impact of the plasma effect [@Tove:1967], which occurs for high X-ray densities when the density of electron-hole ($eh$) pairs is large, typically of the order of the doping of the silicon crystal, has been studied. From these studies, it has been concluded that for 500 $\upmu $m thick sensors the operating voltage should be at least 500 V in order to assure the complete signal collection in-between XFEL pulses and a sufficient narrow point spread function for the measurement of the shape of narrow Bragg peaks. The present work concentrates on the collection of charges produced in the region below the Si-SiO$_2$ interface in segmented $p^+n$ sensors, where the potential, under certain biasing conditions, has a saddle point and the electric field is zero. The multi-channel Transient Current Technique (m-TCT) for charges produced by a sub-nanosecond laser light of 660 nm (absorption length in silicon of about 3.5 $\upmu $m at room temperature) is used for the studies.
The same study also allows a detailed investigation of the properties of the accumulation layer, which forms in segmented $p^+n$ sensors at the Si-SiO$_2$ interface, and of the close-by electric field. This is the main topic of this manuscript.
In [@Poehlsen:2012] it has been demonstrated that charge carriers produced close to the Si-SiO$_2$ interface can be lost, meaning, that they are not collected by an electrode of the sensor within $\lesssim 100$ ns. Depending on the biasing history and on environmental parameters like humidity, situations with losses of electrons, of holes and without losses have been observed. The different situations are related to the density of oxide charges, which strongly depends on the X-ray dose with which the sensor has been irradiated, and on the potential distribution on the surface of the sensor’s passivation layer, which changes when the biasing voltage is changed. Given the high surface resistivity of the passivation layer and its strong dependence on humidity, the time constants for reaching a steady-state of the surface potential can be as long as several days. As discussed in detail in [@Poehlsen:2012], the cause of the charge losses is the electric field which, for an electron-accumulation layer points away from the Si-SiO$_2$ interface, and for a hole-accumulation layer towards it. In this field charge carriers drift towards the accumulation layer, and are trapped for times longer than the integration times used in the analysis of the m-TCT data.
In [@Poehlsen1:2012] the time dependence of the electron and hole losses close to Si-SiO$_2$ interface for an non-irradiated silicon strip sensor and a sensor irradiated by X-rays to 1 MGy (SiO$_2$) have been investigated as function of biasing history and relative humidity.
In the present work we investigate how many additional charges have to be trapped in the accumulation layers to significantly change the collection of charges from the region close to the Si-SiO$_2$ interface and thus the local electric field, and the time dependence of returning to the steady-state conditions of the accumulation layers.
The work has been done within the AGIPD collaboration [@AGIPD; @AGIPD2] which is developing a large-area pixel-detector system for experimentation at the European XFEL and other X-ray sources.
Measurement techniques and analysis
===================================
Sensors under investigation {#sec:sensors}
---------------------------
The same DC-coupled $p^+n$ strip sensors produced by Hamamatsu [@Hamamatsu] as in [@Poehlsen:2012; @Poehlsen1:2012] were used for the investigations. Relevant sensor parameters are listed in Table \[tab:sensors3\], and a cross section of the sensor is shown in Figure \[fig:sensor3\]. The sensors are covered by a passivation layer with openings at the two ends of each strip for bonding. One sensor was investigated as produced, and another after irradiation to 1 MGy (SiO$_2$) with 12 keV photons and annealed for 60 minutes at 80$^\circ$C. The corresponding values for the oxide-charge density, $N_{ox}$, the integrated interface-trap density, $N_{it}$, and the surface-current density, $J_{surf}$, are listed in Table \[tab:irrad3\]. The values have been derived from measurements on MOS capacitors and gate-controlled diodes fabricated on the same wafer as the sensors [@Zhang:2011a; @Zhang:2011b; @Perrey:Thesis] and scaled to the measurement conditions.
![Schematic cross section of the DC-coupled $p^+n$ sensor, and definition of the $x$ and $y$ coordinates. The drawing is not to scale.[]{data-label="fig:sensor3"}](sensor1.png){width="10cm"}
Experimental setup {#sec:setup3}
------------------
To study the charge transport and charge collection in the sensor, the instantaneous currents induced in the electrodes by the moving charges were measured (Transient Current Technique - TCT [@Kraner:1993; @Becker:2010; @Becker:Thesis]). The multi-channel TCT setup, described in detail in [@Becker:Thesis], has been used for the measurements. The bias voltage was applied on the $n^+$ rear contact of the sensor. The current signal was read out on the rear contact and on 2 strips on the front side using Agilent 8496G attenuators, Femto HSA-X-2-40 current amplifiers and a Tektronix digital oscilloscope with $2.5$ GHz bandwidth (DPO 7254). The readout strips were grounded through the DC-coupled amplifiers ($\sim$ 50 $\Omega$ input impedance). The seven strips to the right and the seven strips to the left of the strips read out were connected to ground by 50 $\Omega$ resistors.
![Schematic of the pulse structure. The laser was operated in burst mode with 30 pulses per burst. Pulses inside a burst were separated by the time interval $t_1$, and the time interval between bursts was $t_2$.[]{data-label="fig:burst"}](pulse_structure.png){width="12cm"}
Electron-hole ($eh$) pairs were generated in the sensor close to its surface in-between the readout strips by red light from a laser focussed to an $rms$ of 3 $\upmu$m. The wavelength of the light was 660 nm, which has an absorption length in silicon at room temperature of approximately 3.5 $\upmu$m. We note that 1 keV X-rays have a similar absorption length in silicon. The number of generated $eh$ pairs was controlled by optical filters. For most of the measurements presented approximately 130 000 $eh$ pairs were generated, corresponding to 470 X-rays of 1 keV. The laser was used in burst mode with 30 pulses per burst. The pulse structure is shown in Figure \[fig:burst\]. The pulses in a burst were separated by $t_1 = 50$ ns, and the time interval between bursts $t_2 \approx 1$ ms, if not stated otherwise. To study the time dependence of the return to steady-state conditions after charges have been trapped, $t_1$ or $t_2$ were varied, with the other parameters fixed. For longer recovery times $t_2$ was varied between 500 ns and 10 ms and the signal from the first pulse of the burst was analysed. The recovery time $\Delta t$ is defined as the time interval between the pulse analysed and its preceding pulse. Hence $\Delta t = t_2$ if the first pulse is analysed. For short recovery times $t_2$ was set to 1 ms, $t_1$ varied between 50 and 500 ns, and the signal from pulse 30 analysed. In this case we have $\Delta t = t_1$. In this way two different measurements are available for $\Delta t = 500$ ns.
![Current transients for the first two pulses of the burst for strip $L$ (black solid line) and the rear contact (red dotted line) for 130 000 $eh$ pairs generated at $x_0 = 75$ $ \upmu$m for $t_1 = 50$ ns and $t_2 = 1$ ms. The vertical lines indicate the limits used to determine the base line and the signal. The results shown are for the non-irradiated sensor in conditions “dried @ 500 V” biased to 200 V.[]{data-label="fig:pulse1"}](integration_time_with_rear.png){width="14cm"}
Analysis method {#sec:analysis}
---------------
Figure \[fig:pulse1\] shows for the non-irradiated sensor biased to 200 V the current transients of the first two pulses of a pulse train measured at strip $L$ and at the rear contact for 130 000 $eh$ pairs generated at $x_0 = 75$ $ \upmu$m, half way between the readout strips $R$ and $NR$, as shown in Figure \[fig:sensor3\]. The red dotted line in Figure \[fig:pulse1\] shows the signal measured at the rear contact, and the black solid line the signal from strip $L$ at $x = 0$, which is 1.5 times the strip pitch away from the position where the $eh$ pairs were generated.
The signals are the sums of the currents induced by the holes, which drift to the $p^+$ strips, and the electrons, which drift to the $n^+$-rear contact. The holes are collected quickly, because the distance between the readout strips and the place where they were generated is small. The electron signals are significantly longer, as electrons have to traverse the entire sensor to reach the rear contact. The current transient on strip $L$ is the sum of a short negative signal from the holes drifting to strips $R$ and $NR$ and a slower positive signal from the electrons drifting to the rear contact. In the rear contact the holes as well as the electrons induce negative signals. The bi-polar signals, starting approximately 20 ns after the start of the signal pulse, are due to reflections from the amplifiers, which were connected to the electrodes by 2 m long cables. The noise and the reflection for the rear contact are significantly higher than for the signal from the readout strips. This is due to the higher capacitance of the rear contact and the bias-T used to decouple the high voltage.
In the analysis the induced charge for the i-th pulse in a burst, $Q_i$, is calculated off-line by integrating the current over the time interval $\delta t$ and subtracting the baseline current: $$Q_i = \int^{\tau_i + \delta t}_{\tau_i} (I - I_{baseline}) \cdot \mathrm{dt}
\quad \text{with} \quad
I_{baseline} = \frac{\int_{\tau_i - 8\,\text{ns}}^{\tau_i} I \cdot \mathrm{dt}}{8\,\text{ns}},
\quad \text{and} \quad
\tau_i = t_0 + (i-1) \cdot t_1.
\label{Q}$$ As indicated in Figure \[fig:pulse1\], $t_0$ is the time shortly before the first pulse starts and a value $\delta t = 16$ ns was chosen for the measurements with 130 000 $eh$ pairs. For the measurements in which the number of $eh$ pairs was varied between $10^5$ and $10^7$, $\delta t = 40$ ns had to be chosen in order to collect the entire charge.
The number of charge carriers lost is obtained from $Q^L$, the charge induced in strip $L$ in the following way: The integral of the hole signal is $Q_h^L = N_h \cdot q_0 \cdot (0 - \Phi _w^L(x_0))$. $N_h$ is the number of holes collected, $q_0$ the elementary charge, and the term in parenthesis the difference of the weighting potential $\Phi _w^L$ for the readout strip $L$ at the strips $R$ and $NR$ where the holes are collected ($\Phi _w^L(R) = \Phi _w^L(NR) = 0$), and $\Phi _w^L(x_0)$, the weighting potential at the position $x_0$, where the holes were generated. The charge induced by the electrons is $Q_e^L = N_e \cdot (-q_0) \cdot (0 - \Phi _w^L(x_0))$, where $N_e$ is the number of electrons collected at the rear contact. The total charge induced on strip $L$ is: $$Q^L = Q_e^L + Q_h^L = (N_e - N_h) \cdot q_0 \cdot \Phi _w^L(x_0).
\label{Q_sum}$$ If all holes and electrons are collected $N_e = N_h$ and $Q^L = 0$. For incomplete charge collection, assuming that there is negligible $eh$ recombination and only electrons or only holes are lost, the amount of charge lost is given by: $$Q_{lost} = (N_e - N_h) \cdot q_0 = Q^L/\Phi _w^L(x_0).
\label{Q_lost}$$ If more electrons than holes are collected $N_e > N_h$, $Q_{lost}$ is positive, and the number of holes lost is obtained from $N_h^{lost} = Q_{lost} / q_0$. In a similar way, for $N_e < N_h$ the number of electrons lost is $N_e^{lost} = -Q_{lost} / q_0$.
In this paper measurements from strip $L$ for light injected at the positions $x_0 = 40$ $\upmu$m and $x_0 = 75$ $\upmu$m are presented. For the corresponding weighting potentials $\Phi _w^L(x_0)$ values of 0.35 and 0.05 are used. For the analysis neither the signals from the rear contact nor from strip $R$ are used, but it has been verified that the corresponding signals agree with the results of the analysis from strip $L$. For more details on this method of determining the charge losses and on the way the values of the weighting potentials were obtained, we refer to [@Poehlsen:2012].
Results {#chapter:results}
=======
First, the three biasing and environmental conditions under which the measurements have been performed, are defined. Then, for the non-irradiated and for the irradiated (1 MGy) sensor in the three experimental conditions it is shown how the charge losses for 130 000 $eh$ pairs generated per pulse depend on the pulse number in the burst. Next, the time dependence of the recovery of the charge losses to the situation for the first pulse of the pulse train is investigated. Finally, for the irradiated sensor, the dependence of the charge losses on pulse number as function of the number of generated charge carriers in the range between $10^5$ and $10^7$ is shown. A discussion and qualitative explanations of the results are found in Section \[chapter:discussion\].
Measurement conditions {#sec:conditions}
----------------------
As discussed in detail in [@Poehlsen:2012; @Poehlsen1:2012], the observed charge losses depend on the X-ray-radiation damage and on the charge distributions inside and on top of the passivation layer. The latter changes when the biasing voltage is changed. After changing the biasing voltage, steady-state conditions are reached on top of the passivation layer after a time interval which, due to the dependence of the surface resistivity on humidity, strongly depends on the ambient relative humidity. In a dry atmosphere or in vacuum, this time can be as long as several days, whereas in a humid atmosphere, it can be as short as minutes. All measurements were performed at 200 V. The same biasing and environmental conditions were already used in [@Poehlsen:2012]:
- “humid”: Sensor biased to 200 V and kept in a humid atmosphere for $>$ 2 hours (relative humidity $>$ 60 $\%$), i.e. in steady-state conditions on top of the passivation layer,
- “dried @ 0 V”: Sensor stored at 0 V for a long time to reach steady-state conditions at 0 V, then kept in a dry atmosphere for $>$ 1 hour (relative humidity $<$ 5 $\%$), and then biased to 200 V for the measurements; thus the charge distribution on top of the passivation layer corresponds to the 0 V condition,
- “dried @ 500 V”: Sensor kept for $>$ 2 hours at 500 V in a humid atmosphere (relative humidity $>$ 60 $\%$) to reach steady-state conditions, then dried for $>$ 1 hour, and afterwards biased at 200 V in a dry atmosphere for the measurements.
![Fraction of charges lost as a function of pulse number in the burst for the non-irradiated sensor for $\sim$ 130 000 $eh$ pairs generated per pulse at $x_0 = 75$ $\upmu$m. In addition, some measurements with the laser at $x_0 = 40$ $\upmu$m are shown. The sensor was biased at 200 V. The nomenclature characterizing the different measurement conditions are explained in the text. Positive values correspond to hole losses and negative to electron losses.[]{data-label="fig:0Gy_pulse"}](pulse_0Gy.png){width="8cm"}
![Fraction of charges lost as a function of pulse number for the irradiated sensor (1MGy). The other conditions are the same as for Figure \[fig:0Gy\_pulse\].[]{data-label="fig:1MGy_pulse"}](pulse_1MGy.png){width="8cm"}
Charge losses as function of pulse number {#sec:pulse_number}
-----------------------------------------
Figures \[fig:0Gy\_pulse\] and \[fig:1MGy\_pulse\] show for the non-irradiated and the irradiated sensor for the three experimental conditions and $\sim$ 130 000 $eh$ pairs generated per pulse, the fraction of charges lost as function of the pulse number in the burst. The main results are summarised in Table \[tab:pulse\_number\]. The parameters of the burst mode were a time between the pulses $t_1=50$ ns, and a time between the bursts $t_2=1$ ms. As will be shown later the value $t_2=1$ ms is sufficient that the charge losses have recovered to the steady-state values before the first pulse of the following burst. In Figure \[fig:0Gy\_pulse\] the fractions of charges lost for the two laser positions, $x_0 = 40$ $\upmu $m and $x_0 = 75$ $\upmu $m, for the condition “dried @ 0 V” are shown. It can be seen, that the fluctuations for $x_0 = 40$ $\upmu $m are much smaller than for $x_0 = 75$ $\upmu $m. The reason is the difference in weighting potential, which is in the denominator in Equation (\[Q\_lost\]). It is 0.35 for $x_0 = 40$ $\upmu $m and 0.05 for $x_0 = 75$ $\upmu $m. However, $x_0 = 40$ $\upmu $m is only 15 $\upmu $m away from the center between the strips $R$ and $L$, and not for all conditions it can be assured, that no holes reach the readout strip $L$ by diffusion, which is assumed in the analysis. In [@Poehlsen:2012] it has been shown that there are situations where the diffusion of the holes is sufficiently small, so that the measurements at $x_0 = 40$ $\upmu $m give reliable results for the charge losses. This is the case for “dried @ 0 V”, and the results are compatible with the measurements at $x_0 = 75$ $\upmu $m. In the following, if no holes diffuse to strip $L$ the results for $x_0 = 40$ $\upmu $m are shown, else the results for $x_0 = 75$ $\upmu $m.
![Fraction of charges lost as a function of the recovery time $\Delta t$ for the non-irradiated sensor biased at 200 V for $\sim$ 130 000 $eh$ pairs generated. Left: Logarithmic time axis. Right: Linear time axis.[]{data-label="fig:0Gy_time"}](time_0Gy.png){width="15cm"}
Charge losses as function of recovery time {#sec:pulse_intensity}
------------------------------------------
![Fraction of charges lost as a function of the recovery time $\Delta t$ for the irradiated sensor biased at 200 V for $\sim$ 130 000 $eh$ pairs generated. Left: Logarithmic time axis. Right: Linear time axis.[]{data-label="fig:1MGy_time"}](time_1MGy.png){width="15cm"}
Figures \[fig:0Gy\_time\] and \[fig:1MGy\_time\] show the fraction of charges lost as a function of the recovery time $\Delta t$, defined in Section \[sec:setup3\], for the irradiated and non-irradiated sensor biased at 200 V and $\sim$ 130 000 $eh$ pairs generated. For the measurements at $\Delta t = 500$ ns there are two data points. As discussed in Section \[sec:setup3\], one is obtained from pulse number 30 for the laser timing $t_1 = 500$ ns and $t_2 = 1$ ms, the other from pulse number 1 for the timing $t_1 = 50$ ns and $t_2 = 500$ ns. It is seen that the values are compatible. Smooth transitions from the reduced charge losses at short recovery times to the larger steady-state losses, corresponding to the losses for the first pulse in Figures \[fig:0Gy\_pulse\] and \[fig:1MGy\_pulse\], are observed. In order to obtain a quantitative description of the measurements, they are fitted by the phenomenological function $$f_{lost}(\Delta t) = f_{lost}^\infty \left(1-e^{-(\Delta t/t_0)^{p}}\right),
\label{N_lost_fit}$$ with the free parameters, the steady-state fraction of charges lost, $f_{lost}^\infty$, the time constant, $t_0$, and the power in the exponent, $p$. The fit results are presented in Table \[tab:recovery\]. The discussion of the results is postponed to Section \[chapter:discussion\].
Effects of high charge densities {#sec:high_density}
--------------------------------
For the study of one consequence of the plasma effect, the increase of the pulse length, Figure \[fig:1Million\] shows the current transients of the first two pulses of the pulse train for the readout strip $L$ and the rear contact, for $10^5$, $3.6 \cdot 10^5$, $3.6 \cdot 10^6$ and $10^7$ $eh$ pairs produced at $x_0 = 75$ $ \upmu$m for the irradiated sensor biased to 200 V in the condition “dried @ 0 V”. We note that the condition “dried @ 0 V” corresponds to operation conditions typical for sensors.
Whereas the shapes of the signals from the rear contact (red dotted lines), which are mainly due to the electrons, are similar for $10^5$ and $3.6 \cdot 10^5$ $eh$ pairs, a significant change is observed for higher intensities. The signal peaks at $\sim 10$ ns, compared to $\sim 2$ ns, and the signal extends up to $\sim 35$ ns compared to $\lesssim 20$ ns. Also the signals from strip $L$ (black solid lines) change significantly. The short negative signals due to the holes moving to strips $R$ and $NR$ and the slower positive signals are very much reduced when normalised to the number of $eh$ pairs generated. The reason is that both electrons and holes are trapped in the $eh$ plasma, which dissolves by ambipolar diffusion, and the positive electron signal is to a good extent compensated by the negative signal induced by the holes moving towards strips $R$ and $NR$.
![Current transients for the first two pulses of the pulse train for strip $L$ (black solid line) and the rear contact (red dotted line) as function of the number of $eh$ pairs produced at $x_0 = 75$ $ \upmu$m for $t_1 = 50$ ns and $t_2 = 1$ ms. The vertical lines indicate the limits used to determine the base line ($t_0 - 8$ ns to $t_0$) and the signal ($t_0$ to $t_0 + 40$ ns). The measurements were made with the irradiated sensor in the condition “dried @ 0 V” biased to 200 V. Top left: $10^5$ $eh$ pairs. Top right: $3.6 \cdot 10^5$ $eh$ pairs. Bottom left: $3.6 \cdot 10^6$ $eh$ pairs. Bottom right: $10^7$ $eh$ pairs.[]{data-label="fig:1Million"}](_75um_OD25_1e5.png "fig:"){width="7.3cm"} ![Current transients for the first two pulses of the pulse train for strip $L$ (black solid line) and the rear contact (red dotted line) as function of the number of $eh$ pairs produced at $x_0 = 75$ $ \upmu$m for $t_1 = 50$ ns and $t_2 = 1$ ms. The vertical lines indicate the limits used to determine the base line ($t_0 - 8$ ns to $t_0$) and the signal ($t_0$ to $t_0 + 40$ ns). The measurements were made with the irradiated sensor in the condition “dried @ 0 V” biased to 200 V. Top left: $10^5$ $eh$ pairs. Top right: $3.6 \cdot 10^5$ $eh$ pairs. Bottom left: $3.6 \cdot 10^6$ $eh$ pairs. Bottom right: $10^7$ $eh$ pairs.[]{data-label="fig:1Million"}](_75um_OD20_4e5.png "fig:"){width="7.3cm"} ![Current transients for the first two pulses of the pulse train for strip $L$ (black solid line) and the rear contact (red dotted line) as function of the number of $eh$ pairs produced at $x_0 = 75$ $ \upmu$m for $t_1 = 50$ ns and $t_2 = 1$ ms. The vertical lines indicate the limits used to determine the base line ($t_0 - 8$ ns to $t_0$) and the signal ($t_0$ to $t_0 + 40$ ns). The measurements were made with the irradiated sensor in the condition “dried @ 0 V” biased to 200 V. Top left: $10^5$ $eh$ pairs. Top right: $3.6 \cdot 10^5$ $eh$ pairs. Bottom left: $3.6 \cdot 10^6$ $eh$ pairs. Bottom right: $10^7$ $eh$ pairs.[]{data-label="fig:1Million"}](_75um_OD10_4e6.png "fig:"){width="7.3cm"} ![Current transients for the first two pulses of the pulse train for strip $L$ (black solid line) and the rear contact (red dotted line) as function of the number of $eh$ pairs produced at $x_0 = 75$ $ \upmu$m for $t_1 = 50$ ns and $t_2 = 1$ ms. The vertical lines indicate the limits used to determine the base line ($t_0 - 8$ ns to $t_0$) and the signal ($t_0$ to $t_0 + 40$ ns). The measurements were made with the irradiated sensor in the condition “dried @ 0 V” biased to 200 V. Top left: $10^5$ $eh$ pairs. Top right: $3.6 \cdot 10^5$ $eh$ pairs. Bottom left: $3.6 \cdot 10^6$ $eh$ pairs. Bottom right: $10^7$ $eh$ pairs.[]{data-label="fig:1Million"}](_75um_OD05_10e6.png "fig:"){width="7.3cm"}
Next we have investigated for the irradiated sensor in the condition “dried @ 0 V”, where electron losses of $\sim 70$ % with little dependence on pulse number and recovery time had been observed, how the number of generated $eh$ pairs influences the charge losses as function of pulse number. The laser was used in burst mode with 30 pulses with the parameters $t_1 = 50$ ns and $t_2 = 1$ ms, and the number of $eh$ pairs generated at $x_0 = 40$ $\upmu$m was varied between $10^5$ and $10^7$. In order to take into account the increase of the pulse length due to the plasma effect, for this analysis the integration time $\delta t$ in Equation (\[Q\]) was increased to 40 ns, as indicated in Figure \[fig:1Million\].
![Fraction of electrons lost as function of the pulse number in the burst for the irradiated sensor biased at 200 V for the condition “dried at 0 V”. Between $10^5$ and $10^7$ $eh$ pairs per pulse were generated at $x_0 = 40$ $\upmu$m with pulse spacing $t_1=50$ ns and burst spacing $t_2=1$ ms.[]{data-label="fig:1MGy_intensity"}](pulse_intensity.png){width="8cm"}
Figure \[fig:1MGy\_intensity\] shows the fraction of electrons lost as function of the pulse number for different numbers of $eh$ pairs generated per pulse. For $10^5$ $eh$ pairs the number of electrons lost per pulse decreases with pulse number from $\sim$ 70 % to $\sim$ 60 % without reaching a constant value up to 30 pulses. This is similar to the data presented in Figure \[fig:1MGy\_pulse\]. For higher numbers of generated $eh$ pairs, the fraction of electrons lost for the first pulse decreases, and a strong further decrease is observed for the following pulses. The values at high pulse numbers also decrease with the numbers of generated $eh$ pairs. For $3.6 \cdot 10^5$ a saturation value of $\sim$ 30 % is obtained. For $10^7$ it is as small as $\sim$ 2 %. The observation that the fraction of electrons lost decreases with increasing number of generated $eh$ pairs already for the first pulse agrees with the expectation, that the local electric field changes already before the charges from that particular pulse are collected.
Discussion {#chapter:discussion}
==========
Plasma effect {#sec:discussion_plasma}
-------------
With respect to the questions, if there are pile-up effects due to charges trapped in the region below the Si-SiO$_2$ interface, we conclude from Figure \[fig:1Million\], that a significant lengthening of the current pulse occurs only when more than $3.6 \cdot 10^5$ $eh$ pairs are produced by the laser. For the AGIPD sensor the X-rays enter through the $n^+$ rear contact, and only $\sim$ 0.3 % of 12.4 keV X-rays (absorption length $\sim $ 250 $\upmu $m in silicon) interact in the $\sim $ 5 $\upmu $m close to the Si-SiO$_2$ interface, and thus $\sim $ $3.6 \cdot 10^4$ 12.4 keV X-rays are required to produce $3.6 \cdot 10^5$ $eh$ pairs there. We conclude, that the low-field region close to the Si-SiO$_2$ interface does not result in increased pulse lengths due to the plasma effect for the situation expected at the European XFEL, and that the conclusions of [@Becker:2010] remain valid.
![Simulated potential distribution for the non-irradiated sensor biased to 200 V with the biasing condition “humid”. See Figure \[fig:sensor3\] for the coordinate system.[]{data-label="fig:potential"}](potential_humi_0Gy.png){width="7.3cm"}
![Simulated potential distributions for the non-irradiated sensor biased to 200 V for different biasing conditions. Left: “dried at 0 V. Right: ”dried at 500 V". See Figure \[fig:sensor3\] for the coordinate system.[]{data-label="fig:potentials3"}](potential_dried0_0Gy.png "fig:"){width="7.3cm"} ![Simulated potential distributions for the non-irradiated sensor biased to 200 V for different biasing conditions. Left: “dried at 0 V. Right: ”dried at 500 V". See Figure \[fig:sensor3\] for the coordinate system.[]{data-label="fig:potentials3"}](potential_dried500V_0Gy.png "fig:"){width="7.3cm"}
Explanation of the charge losses {#sec:explain_losses}
--------------------------------
A detailed discussion and an explanation of the dependence of the charge losses on X-ray dose and biasing history has been presented in [@Poehlsen:2012]. It is briefly summarised here. Figure \[fig:potential\], taken from [@Poehlsen:2012], shows the simulated potential distribution for “humid”, the situation where no charge losses are observed. In the calculations a positive oxide-charge density of $10^{11}$ cm$^{-3}$ for a non-irradiated sensor has been assumed. At the Si-SiO$_2$ interface the potential has a parabolic shape in the $x$ direction, with a maximum value of $\sim $ 10 V in the center between the $p^+$ strips. In the $y$ direction the potential increases. Thus, for $eh$ pairs produced by the laser close to the interface, the electrons drift in the $y$ direction to the rear contact, the holes along the $x$ direction to the $p^+$ strips, and no charges are lost.
The simulated potential for the condition “dried @ 0 V”, where electrons are lost, is shown on the left of Figure \[fig:potentials3\], again taken from [@Poehlsen:2012]. For this condition the charge on top of the SiO$_2$ layer remains approximately zero, as it has been for the sensor at zero volt in steady-state conditions. The positive oxide charges cause an electron-accumulation layer at the Si-SiO$_2$ interface at a value of the potential of $\sim $ 29 V, a saddle point of the potential $\sim $ 5 $\upmu$m below the interface, and an electric field pointing from the SiO$_2$ into the silicon. Thus electrons drift towards the Si-SiO$_2$ interface where they are lost, i.e. not collected in the time interval during which the induced current is integrated.
The right side of Figure \[fig:potentials3\] shows the potential for the condition “dried @ 500 V”. In the humid steady-state condition at 500 V negative charges accumulate on top of the SiO$_2$. When the voltage is reduced to 200 V in dry conditions, the negative surface charges remain, overcompensate the positive oxide charges, produce a hole-accumulation layer at a potential value of $\sim $ 4 V at the Si-SiO$_2$ interface and an electric field which points from the silicon into the SiO$_2$. Thus holes drift to the Si-SiO$_2$ interface and are lost.
Explanation of the change of the charge losses {#sec:explain_change}
----------------------------------------------
Next we give a qualitative explanation of the change of the charge losses as function of pulse number and recovery time for the non-irradiated sensor.
As seen in Figures \[fig:0Gy\_pulse\] and \[fig:0Gy\_time\], if no charges are lost, i.e. all charges are collected before the next laser pulse arrives, the charge losses remain zero. This is expected, as the conditions do not change from pulse to pulse.
If, as for the situation “dried @ 500 V”, positive charges are trapped close to the interface, the value of the potential at the interface in-between the $p^+$ strips will increase and approach the no-charge-loss situation shown in Figure \[fig:potential\]. As summarised in Table \[tab:pulse\_number\], after $\sim $ 8 pulses of $\sim $ 130 000 $eh$ pairs spaced by 50 ns, the initial hole losses of $\sim 70$ % have decreased to a saturation value of $\sim 25$ %. We conclude that after $\sim $ 8 pulses, the additionally trapped charges move away from the position where they were produced in the 50 ns time interval between the laser pulses. For the recovery of the charge losses Figure \[fig:0Gy\_time\] shows a fast increase in the first few microseconds, followed by a much slower increase. The full recovery is reached at $\Delta t \approx 500$ $\upmu$s. We assume that the recovery is due to the diffusion of the excess holes over the potential barrier. Qualitatively the observations for the electron and for the hole losses are similar. The main difference is that the initial losses are $\sim 35$ % for electrons, compared to $\sim 70$ % for holes. Comparing the potential distributions shown in Figure \[fig:potentials3\], bigger charge losses are expected for holes than for electrons. The electric field responsible for hole trapping (right) is higher and extends over a larger region than the one for electron trapping (left). As in the case of hole trapping, trapped electrons change the potential towards the zero-loss situation. However, trapped electrons reduce the value of the potential at the interface, whereas trapped holes increase it.
Next we discuss the results for the irradiated sensor. The X-ray irradiation to 1 MGy increases the oxide-charge and interface-trap densities to an effective positive oxide-charge density of $\sim $ $2 \cdot 10^{12}$ cm$^{-2}$ [@Zhang:2011a]. In addition, the surface current increases by several orders of magnitude due to the interface states.
Figure \[fig:1MGy\_pulse\] shows, that for the condition “dried at 500 V” no charge losses are observed when $\sim $ 130 000 $eh$ pairs are generated per pulse. Apparently the negative surface charges on top of the passivation layer compensate the high effective oxide-charge density. For “humid” the density of negative surface charges is smaller, and does not fully compensate the positive oxide charges, and electron losses of $\sim 40$ % for the first pulse are observed. For “dried @ 0 V” the surface-charge density is essentially zero, resulting in even higher electron losses of $\sim 70$ % for the first pulse.
The electron losses as a function of pulse number for the irradiated sensor behave quite differently than for the non-irradiated sensor. For “humid” and 130 000 $eh$ pairs generated per pulse, the electron losses decrease to essentially zero after $\sim 15$ pulses, whereas for “dried @ 0 V” they hardly decrease and show no sign of saturation. As seen in Figure \[fig:1MGy\_intensity\], a much higher number of $eh$ pairs is required for the irradiated sensor in the condition “dried @ 0 V” to significantly change the electron losses. Also the shape of the recovery of the electron losses, shown in Figure \[fig:1MGy\_time\] for the irradiated sensor in conditions “humid”, is different. Whereas for the non-irradiated sensor an initial partial recovery with time constants of less than 1 $\upmu$s is followed by a slow full recovery until $\sim $ 200 $\upmu$s, the electron losses for the irradiated sensor recover with a single time constant of $\sim $ 6 $\upmu$s. We note, that in the discription by Equation (\[N\_lost\_fit\]), $p \approx 1$ we interprete as a single time constant, and $p<1$ we interprete as a recovery with both, slower and faster components (compare Figures \[fig:0Gy\_time\] and \[fig:1MGy\_time\]).
We finally comment, that we have made no attempt to simulate the dependence of the charge losses for the pulse structure used in the experiments. Given that it is a 3-D problem with charges spreading over large distances in-between the $p^+$ strips, a realistic simulation appeared out of reach.
Discussion of charge losses for high intensities {#sec:discuss_intensity}
-------------------------------------------------
To further study the dependence of the electron losses on pulse number for the irradiated sensor in the condition “dried @ 0 V”, the number of $eh$ pairs generated per pulse was varied between $10^5$ and $10^7$. The results are shown in Figure \[fig:1MGy\_intensity\]. It is observed that the fraction of electrons lost for the first pulse decreases from $\sim 70$ % for $10^5$ to $\sim 20$ % for $10^7$ $eh$ pairs. The explanation for this dependence is, that the charges deposited in a given pulse already change the local electric field, and thus already influence the charge collection for this first pulse. It is also observed that for $\gtrsim 3.6 \cdot 10^5$ $eh$ pairs generated, the electron losses saturate for higher pulse numbers. The saturation value decreases from $\sim 30$ % for $3.6 \cdot 10^5$ to $\sim 2$ % for $10^7$. We interpret this as evidence, that for the high radiation-induced effective oxide charge density and essentially zero negative charge on top of the SiO$_2$ layer, the maximum value of the potential at the Si-SiO$_2$ interface is high and many electrons have to be trapped to significantly reduce the electron losses. From the decrease of the charge losses for the first pulse with $eh$ intensity, we estimate that of the order of $10^6$ electrons have to be trapped locally in order to reduce the electron losses by about a factor 2. This number is significantly higher than for the irradiated sensor in conditions “humid”, where already electron losses of $\sim 10^5$ make a significant difference, or for the electron and hole losses for the non-irradiated sensor.
Summary
=======
Using the multi-channel Transient Current Technique, the currents induced by electron-hole pairs, produced by a focussed sub-nanosecond laser of 660 nm wavelength close to the Si-SiO$_2$ interface of $p^+n$-silicon strip sensors, have been measured, and charge-collection efficiencies determined. Sensors, before and after irradiation by 1 MGy (SiO$_2$) X-rays, have been investigated.
For high densities of electron-hole pairs deposited close to the Si-SiO$_2$ interface the plasma effect results in a significant increase in pulse length. However, the number of X-rays required to generate charge densities in this region so that these effects become significant are too high, to be of relevance for the AGIPD detector at the European XFEL.
As already reported previously, dependent on radiation dose and biasing history, not all electrons or holes are collected at the contacts of the sensors within the typical readout integration times of order $\lesssim 100$ ns, but are trapped close to the Si-SiO$_2$ interface. These lost charges result in a non-steady state of the accumulation layers and the nearby electric fields, which causes a reduction of the charge losses. The number of trapped charges required to significantly reduce further charge losses and possibly reach constant values, varies between $\sim$ $10^5$ and $\sim$ $10^6$ in the investigated cases. The recovery times to steady-state conditions depends on the X-ray dose with which the sensor had been irradiated.
Qualitative explanations of the findings have been given. Even if the results presented may be of limited practical relevance for the user of silicon sensors, they provide further insight into the complexities of the Si-SiO$_2$-interface region of segmented $p^+n$-silicon sensors.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was performed within the AGIPD Project which is partially supported by the European XFEL-Company. We would like to thank the AGIPD colleagues for the excellent collaboration. Support was also provided by the Helmholtz Alliance “Physics at the Terascale”, and the German Ministry of Science, BMBF, through the Forschungsschwerpunkt “Particle Physics with the CMS-Experiment”. J. Zhang was supported by the Marie Curie Initial Training Network “MC-PAD”.
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abstract: 'Temperature-dependent Brillouin scattering studies have been carried out on La$_{0.77}$Ca$_{0.23}$MnO$_3$ across the paramagnetic insulator - ferromagnetic metal (I-M) transition. The spectra show a surface Rayleigh wave (SRW) and a high velocity pseudo surface acoustic wave (HVPSAW) besides bulk acoustic waves (BAW). The Brillouin shifts associated with SRW and HVPSAW show blue-shifts, where as the frequencies of the BAW decrease below the I-M transition temperature (T$_C$) of 230 K. These results can be understood based on the temperature dependence of the elastic constants. We also observe a central peak whose width is maximum at T$_C$.'
author:
- 'Md. Motin Seikh'
- Chandrabhas Narayana
- 'L. Sudheendra'
- 'A. K. Sood'
- 'C.N.R. Rao'
title: 'Brillouin Scattering Studies of La$_{0.77}$Ca$_{0.23}$MnO$_3$ Across Metal-Insulator Transition'
---
Introduction
============
The study of the perovskite manganites, A$_{1-x}$B$_x$MnO$_3$ (ABMO) where A and B are trivalent and divalent ions, respectively, has attracted much attention in recent years due to their fascinating properties and technological potential [@jin94]. In particular, La$_{1-x}$Ca$_x$MnO$_3$ (LCMO) composition exhibit a transition from a paramagnetic insulating to a ferromagnetic metallic state for $0.2 \leq x \leq 0.5$ and show colossal magneto-resistance. In ABMO type manganites, the transition temperature as well as other properties are markedly affected by the average radius of the A-site cations, which in turn affects the band width of the [*e$_g$*]{} electrons due to the Mn$^{3+}$ ions. Jahn-Teller based electron-phonon coupling [@millis96; @roder96] as well as the double exchange [@zener51] play important roles in determining the properties of the manganites. There is experimental evidence for the presence of lattice distortions [@dai96; @billinge96; @radaelli96; @zhao96] which give rise to polarons which may be of lattice and/or of magnetic origin [@teresa97; @erwin96].
Brillouin scattering is a powerful probe to study the surface and bulk accoustic phonons as well as magnetic excitiations in opaque solids [@murugavel00]. The surface phonons can give rise to a surface Rayleigh wave (SRW), a pseudo surface acoustic wave (PSAW) and a high velocity pseudo surface acoustic wave (HVPSAW) [@carlotti92]. The Poynting vector of SRW lies parallel to the free surface and its particle displacement field decays exponentially within the medium. On the other hand, both PSAW and HVPSAW radiate energy into the bulk and get attenuated due to their decay into bulk phonons. Unlike the SRW, the PSAW propagates only along some specific directions of the surface of anisotropic media and its phase velocity is $\sim$ 40 % higher than the SRW velocities [@cunha95; @cunha98; @cunha01]. In comparison, the HVPSAW propagates in both anisotropic as well as isotropic media [@camley85] and has a phase velocity nearly twice that of regular SRW [@cunha95; @cunha98; @cunha01]. Both the PSAW and the HVPSAW have potential applications in surface accoustic wave devices due to their higher frequencies as compared to that of SRW [@cunha95; @cunha98; @cunha01].
We have carried out Brillouin scattering study of La$_{0.77}$Ca$_{0.23}$MnO$_3$ (LCMO23), which exhibits the paramagnetic insulator - ferromagnetic metal (I-M) transition at T$_C =$ 230 K. In addition to the SRW, the HVPSAW modes are observed for the first time in this manganite. The SRW and HVPSAW mode frequencies increase where as the frequencies of the bulk phonons decrease below T$_C$. This contrasting temperature-dependence can be understood in terms of the temperature-dependence of the elastic constants. A central peak, whose width is maximum at T$_C$, is also observed and we also examine the origin and behavior of the central peak in this paper.
Experimental Details
====================
Polycrystalline powders of LCMO23 were prepared by the solid-state reaction of stoichiometric amounts of lanthanum acetate, calcium carbonate and manganese dioxide. The materials were ground and heated at 1000 $^\circ$C for 60 hrs with two intermediate grindings. The sample was then further heated at 1200 $^\circ$C for 48 hrs. The polycrystalline powder was filled in a latex tube and pressed using a hydrostatic press at a pressure of 5 tons. The rod thus obtained was sintered at 1400 $^\circ$C for 24 hrs. The rod was then used to grow these crystals by the floating zone melting technique. The technique employs SC-M35HD double reflector infrared image furnace (Nichiden Machinery Ltd., Japan). The growth rate and rotation speeds were 10 mm/h and 30 rpm, respectively. The composition of the crystal was determined using Energy Dispersive Analysis of X-rays (EDAX) using LEICA S440I scanning electron microscope (SEM) of M/s LEICA, Japan using a Si-Li detector. The percentage of Mn$^{+4}$ was determined to be 23 $\pm$ 2 %. Figure 1 shows the resistivity and d.c. magnetization of the sample. The peak in the resistivity at T$_C$ = 230 K marks the I-M transition.
![\[fig1\] Temperature dependence of (a) Resistivity ($\rho/\rho_{300}$), where $\rho_{300}$ is the resistivity at 300 K (b) Magnetization of LCMO23.](fig1_lcmo.eps)
Brillouin spectra were recorded in the back-scattering geometry with an incident angle $\theta = 45^\circ$ with the surface normal using a JRS Scientific Instruments 3+3 pass tandem Febry-Perot interferometer equipped with a photo avalanche diode as a detector. The Nd:YAG single mode solid-state diode-pump frequency doubled laser (Model DPSS 532-400, Coherent Inc. USA) operating at 532 nm and power of $\sim$ 25 mW (focused to a diameter of $\sim$ 30 $\mu$m) was used to excite the spectra. The temperature experiments were carried out using a closed cycle helium cryostat (CTI Cryogenics, USA). The sample temperature was measured within an accuracy of $\pm$ 1 K. There is no laser damage of the sample surface as verified under a high magnification microscope. The typical time required per spectrum was 1.5 hrs.
The laser induced heating of the sample was estimated to be 66 K as measured from the anti-Stokes to Stokes intensity ratio of the 480 cm$^{-1}$ Raman band of the same sample recorded using similar experimental conditions. The data presented here have been corrected to take into account the laser heating.
Results and Discussion
======================
Figure 2 shows the Brillouin spectra recorded using two free spectral ranges. The spectra reveal four modes: 6.8 GHz (labeled as R$_1$), 15.3 GHz (R$_2$), 24.1 GHz (B$_1$) and 58.2 GHz (B$_2$). Figure 3 shows the linear dependence of the R$_1$ and R$_2$ mode frequencies on the magnitude of the wave vector parallel to the surface, $\vec{q}_\parallel$, thereby establishing them to be the surface modes. Upon rotating the crystal about the normal to the surface, the R$_1$ mode frequency shows an oscillatory behavior as a function of the rotation angle $\phi$, as expected for a SRW. The R$_2$ mode is associated with the HVPSAW, since its frequency is almost double than that of the R$_1$ mode with much broader line width, similar mode has been observed in GaAs along (1$\bar{1}$1) [@carlotti92] and discussed in the literature [@cunha95; @cunha98; @cunha01]. In addition, a weak but sharp peak is found at 8.6 GHz. This mode appears at certain orientations of the crystal and its frequency mode also shows a linear dependence on $\vec{q}$$_\parallel$, suggesting this to be the PSAW mode. The PSAW and HVPSAW modes appear in our spectra because the sample surface is not along any specific crystallographic direction. The peaks labeled B$_1$ and B$_2$ in Fig. 2 are associated with the bulk acoustic waves (BAW). The line widths of these modes are much higher due to the uncertainity in the wave-vector arising from the finite penetration depth of the incident radiation.
![\[fig2\]Room temperature spectrum of LCMO23 (a)recorded with free spectral range (FSR) 25 GHz (b) FSR 85 GHz. The solid lines through the data points are fits to the data, using two lorentzian peaks and an appropriate background on both Stokes and anti-Stokes side.](fig2_lcmo.eps)
![\[fig3\] The frequency dependence of both R and R$_\circ$ mode as a function of $\vec{q}_\parallel$. The solid line is a linear fit to the data points shown in solid circles and triangles. The slope of the line is the surface velocity.](fig3_lcmo.eps)
The temperature dependence of the changes in frequencies of all the four modes is shown in Fig. 4, where $\Delta \omega = \omega(T)- \omega(370 K)$. We see from Fig. 4(a) that the frequency of the SRW (R$_1$) mode increases below T$_C$. There is a gradual increase in the HVPSAW mode frequency as the temperature is lowered. The B$_1$ and B$_2$ modes, on the other hand, show a downward jump in frequency at T$_C$.
![\[fig4\] Percentage change in frequency as a function of temperature (a) R mode, (b) R$_\circ$ mode, (c) B$_1$ mode and (d) B$_2$ mode. Inset in (c) shows the magnetic field dependence of B$_1$ mode. The lines drawn are linear fits to act as guide to the eye.](fig4_lcmo.eps)
We now discuss the temperature dependence of the various modes. The SRW frequency can be calculated using the Greens function [@cummins01; @zhang98], which requires the values of the elastic constants. Since the orthorhombic distortions in LCMO23 are not large, we can assume it to be cubic and use the three elastic constants C$_{11}$, C$_{12}$ and C$_{44}$ to calculate the SRW frequency. Unfortunately, the elastic constants of LCMO are not available in the literature and we have, therefore, used the experimental values of elastic constants of La$_{0.835}$Sr$_{0.165}$MnO$_3$ (LSMO165) [@hazama00], measured as a function of temperature using ultrasonic attenuation. The elastic constant data are reproduced in Figure 5(a). The Greens function used here to calculate the SRW frequency of LSMO165 is for the (100) surface. The elastic constant values of LCMO23 should be similar to LSMO165, since the calculated SRW mode frequency for LSMO165 for $\theta = 45^{\circ}$ matches the value observed experimently in LCMO23 at 300 K.
 The temperature dependence of C$_{11}$, C$_{44}$, C$_{12}$ and C$_B$ for LSMO165 reproduced from Ref. \[19\]. (b) Percentage change in surface sound velocity of LCMO23 and LSMO165 (calculated) as a function of reduced temperature (T$_C -$T). (c) Percentage change in bulk sound velocity of LCMO23 of both B$_1$ (solid circles) and B$_2$ (open circles) modes and LSMO165 (calculated using C$_B$ values in (a) (solid curve)) as a function of reduced temperature (T$_C -$T).](fig5_lcmo.eps)
Figure 5(b) shows the relative change in the surfrace Rayleigh sound velocity as a function of the reduced temperature ($T - T_C$) for LCMO23 (experimental) as well as LSMO165 (calculated). It is seen that the there is a hardening of the SRW mode velocity below the T$_C$ and that the relative change in the SRW mode velocity is similar in LSMO165 and LCMO23. In La$_{1-x}$Sr$_x$MnO$_3$ (LSMO), hardening of bulk sound velocity is observed, for 0.11 $\leq x \leq$ 0.17, below the T$_C$ [@fujishiro97]. A 5 % hardening in the bulk sound velocity is found in the case of La$_{0.67}$Ca$_{0.33}$MnO$_3$ below the T$_C$ [@ramirez96]. The origin of the hardening of the sound velocity below T$_C$ should be the same in all these materials exhibiting an I-M transition. It is suggested [@lee97] that the mode frequency increases below T$_C$ due to the decrease in the electron-phonon coupling arising from the high mobility of carriers in the metallic state. Application of an external magnetic field further increases the sound velocity, since the magnetic field greatly enhances the mobility of the carriers [@lee97]. In the high temperature insulating phase, the electron-phonon coupling is larger, giving rise to a smaller frequency of the accoustic phonon. The temperature dependence of the HVPSAW mode frequency, also arises due to the increase in C$_{11}$ and C$_{44}$ across the I-M transition, just as in the case of the SRW mode.
We have carried out the experiments in the 90$^\circ$ geometry, in the case of the B$_1$ mode, to avoid strong reflection artefacts peaks, which appear around 27 and 33 GHz in the back-scattering geometry. The inset in Fig. 4(c) displays the magnetic field ($H$) dependence of the B$_1$ mode frequency showing an increase in the frequency with the magnetic field. In general, the magnetic exciation with frequency $\omega$ depends on $H$ as $\omega = \Delta + Dq^2 + \gamma H$, where $\Delta$ is the anisotropic spin wave energy gap, $D$ is spin stiffness coefficient, $q$ is wavevector transfer and $\gamma$ is the gyromagnetic ratio. This means, if the B$_1$ mode were a magnetic excitation, the slope (see inset of Fig. 4(c)) should be equal to $\gamma$. The observed slope is much smaller than the expected value of $\gamma$ (typically $\sim$ 2), suggesting that the B$_1$ mode is not a magnetic excitation. It is therefore a BAW with a strong magneto-elastic coupling. The mode B$_2$ does not show any dependence on H.
A $\sim$ 3 % softening is observed across the I-M transition temperature in the B$_1$ and B$_2$ modes (cf. Fig. 4(c) and (d)). This is in contrast to the behavior of the surface phonon and bulk sound velocities found in the ultrasonic measurements [@fujishiro97; @ramirez96; @lee97]. Taking into account the temperature dependence of elastic constants in Fig. 5(a) and considering that the bulk accoustic mode in a general direction can be expressed in terms of the bulk modulus, C$_B$ (= (C$_{11} +$ 2C$_{12}$)/3) [@hazama00], we have calculated the relative change in the sound velocity ($v$) of the LSMO165 using the equation $v = \sqrt{\frac{C_B}{\rho}}$. In Fig. 5(c), we have overlayed this plot on the relative changes in the sound velocity for B$_1$ and B$_2$ modes of LCMO23 as a function of reduced temperature ($T - T_C$). It is interesting that the percentage change in the velocities are similar in the experimentally observed data of LCMO23 and the calculated data of LSMO165. Since our experiments were carried out on an arbitrary cut crystal, the B$_1$ and B$_2$ modes should be related to the bulk modulus changes. The present study suggests that the decrease in C$_{12}$ and hence in C$_B$ below the T$_C$ is responsible for the temperature dependence of the B$_1$ and B$_2$ modes.
The intensity of the R$_2$ and B$_2$ modes do not show considerable changes across the I-M transtion. On the other hand, the intensity of the R$_1$ mode decreases through out the temperature range, making it difficult to measure it below 100 K. The B$_1$ mode also cannot be observed below 140 K. The linewidths of both the surface phonons do not change as a function of temperature, where as the linewidths of the bulk phonons increase with decreasing temperature as shown in Fig. 6. This can arise from the increase in the uncertainity in the wavevector transfer due to the reduced penetration depth of light with the increase in electronic conductivity [@sandercock82].
![\[fig6\] The temperature dependence of the full-width at half-maxima (FWHM) for the B$_1$ (open circles) and B$_2$ (solid circles) modes. The solid lines are linear fit to the data.](fig6_lcmo.eps)
In Fig. 7, we show the Brillouin spectra at various temperatures around T$_C$, in the 90$^\circ$ scattering geometry, revealing the presence of a central mode between 290 K and 180 K. We used three Lorentzian peaks, with an appropriate linear background, to fit the spectrum at each temperature. For temperatures below 180 K and above 290 K, we obtained a perfect fit (as shown in panels (a) and (b) of Fig. 7) with the elastically scattered mode giving an FWHM of 0.6 GHz over the entire temperature range of 80 to 370 K (except at low temperatures where the B$_1$ mode disappears, only one Lorentzian with a linear background was used to fit the spectra, as in the case of Fig. 7(e)). Between 290 K and 180 K, it was difficult to fit the spectra in this manner. Taking the FWHM of the elastically scattered mode as 0.6 GHz and adding an additional Lorentzian centered at $\omega = 0$, a good fit would be obtained as shown in (c) and (d) of Fig. 7. This suggests the presence of a central mode in the temperature 290 - 180 K range. Figure 8 shows the temperature dependence of both the FWHM and the intensity of the central mode, revealing that these quantities peak near the T$_C$. A similar observation of the central peak has been made in neutron scattering experiments across the paramagnetic to ferromagnetic transition [@lynn96; @dai00; @dai01; @bersuker90; @baca98] at $q \sim 0.1$ to 0.3 Å$^{-1}$ in La$_{1-x}$Ca$_x$MnO$_3$ ($x$ = 0.2, 0.25, 0.3 and 0.33), Nd$_{0.7}$Sr$_{0.3}$MnO$_3$ and Pr$_{0.63}$Sr$_{0.37}$MnO$_3$. The presence of the central peak in the manganite suggests the presence of competing magnetic states. In neutron diffraction, the spectral weight of the spin-wave excitation starts to decrease as T $\rightarrow$ T$_C$, while the weight for the spin-diffusion component increases rapidly. This spin-diffusion component dominates the fluctuation spectrum near T$_C$, in marked contrast to conventional ferromagnets. If one assumes the spin diffusion as the origin of the central peak in the present study, then the intrinsic width (FWHM or $\Gamma$) of the central peak is given by $\Gamma = \Lambda$q$^2$, where $\Lambda$ is the spin diffusivity. In the present study, $\Gamma/2 \pi = 21$ GHz at T$_c$ (see Fig. 8(a)) and $\vec{q}$ = $3.675\times10^{-3}$ Å$^{-1}$ leading to a value of spin diffusivity $\Lambda$ of 40.4 eV/Å$^2$. This value of $\Lambda$ is very much larger than that observed in the case of Nd$_{1-x}$Sr$_x$MnO$_3$ ($x = 0.3$), where $\Lambda = 26(2)$ meV/Å$^2$ [@baca98]. Thus the origin of central peak in the present case does not appear to be spin diffusion.
![\[fig7\] The Brillouin spectrum recorded at different temperatures showing the B$_1$ mode and the Rayleigh (elastic scattering) mode. The solid curve represents the fit to the data using a set of Lorentzian peaks and an appropriate linear background. The dotted curves represent the individual Lorentzians with the added linear background.](fig7_lcmo.eps)
![\[fig8\]Temperature dependence of the central mode (a)FWHM and (b)intensity as deduced from the fits shown in Fig. 7. The verticle dashed line represents the transition temperature (T$_C$). Inset in (a) shows the calculated linewidth $\Gamma_S = 2 \kappa$q$^2$ as described in the text.](fig8_lcmo.eps)
Entropy fluctuations with a width $\Gamma_S = 2 \kappa$q$^2$ [@sandercock82; @fabelinskii94], where $\kappa$ is the thermal diffusivity ($\kappa = \frac{k}{C \/\rho}$, where $k$ is the thermal conductivity, $C$ is the specific heat and $\rho$ is the density of the medium) can also be the origin of the quasielastic scattering. The value of $k$ at 300 K is 22.9 mWcm$^{-1}$K$^{-1}$ (for LCMO (x=0.25)) [@fujishiro01], while $C$ = 75.5 Jmol$^{-1}$K$^{-1}$ and $\rho$ = 6.2 gcm$^{-3}$ giving $\kappa = 1.1\times10^{-2}$ cm$^2$s$^{-1}$. The width $\Gamma$ turns out to be 3 GHz. Taking $\rho$(T), $C$(T) and $k$(T) for LCMO (x=0.25), the temperature dependence of $\Gamma_S$ is obtained as shown in the inset of Fig. 8. The temperature dependence of $\Gamma_S$ is in contrast to that observed in Fig. 8(a) showing that entropy fluctuations are not responsible for the central mode.
Entropy fluctuations can also be viewed as fluctuations in the density of the phonon $\vec{q}$$_1$, which scatters the light by the two-phonon difference process invloving emission of a phonon $\vec{q}$$_1$ with the simultaneous absorption of a phonon $\vec{q}$$_1 - \vec{q}$ from the same phonon branch [@sandercock82]. In the microscopic picture, scattering of light describes either the indirect [@landau34] or direct [@wehner72] coupling of light to phonon-density fluctuations, assuming at all times local thermodynamic equilibrium. Along with these, there is also the contribution to phonon density fluctuations from additional “dielectric fluctuations”, away from the local thermodynamic equilibrium [@coombs73]. In this case, the two phonon difference process is dominated by zone boundary phonons where the density of states is high, leading to a broad central peak extending beyond the Brillouin peaks. The line width of such a broad central peak is independent of q [@sandercock82]. The central peak will, therefore, have a narrow component with a linewidth of $\kappa q^2$ and a q-independent broad component from “dielectric fluctuations”. Since the central mode in our experiments, shown in Fig. 8, extends right upto the bulk modes, we suggest that it is associated with the q-independent “dielectric fluctuation”. It has been proposed that ‘defects’ can contribute to the central peak [@schwabl91]. Such defects can be the ferromagnetic insulating phase present before the I-M transition in LCMO23 [@huang00].
In conclusion, Brillouin scattering experiments on LCMO23 reveal, for the first time, the presence of a high velocity pseudo surface acoustic wave in manganites. The hardening of the SAW frequency below T$_C$ arises from the temperature dependence of C$_{11}$ and C$_{44}$, where as the softening of the bulk phonons below T$_C$ can be attributed to the decrease in C$_{12}$ with the decreasing temperature. The central peak is suggested to arise from non-equilibrium dielectric fluctuations Linked to the presence of a ferromagnetic insulating phase preceeding the I-M transition.
AKS thanks Prof. J. D. Comins for the computer program to calculate the surface mode spectra using Greens function. We would like thank Ms. G. Kavitha for implimenting the program and making the preliminary runs. Mr. Seikh would like to thank Council of Scientific and Industrial Research (CSIR), India for a research fellowship.
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abstract: 'The continuum spectrum of the planetary nebula Hf 2-2 close to the Balmer discontinuity is modeled in the context of the long standing problem of the abundance and temperature discrepancy found when analyzing optical recombination lines and collisionally excited forbidden lines in nebulae. Models are constructed using single and double Maxwell-Boltzmann distributions as well as $\kappa$-distributions for the energies of the free electrons. New results for the necessary continuum and line emission coefficients are presented calculated with $\kappa$-distributed energies. The best fit to the observed continuum spectrum is found to be a model comprising two components with dramatically different temperatures and with a Maxwell-Boltzmann distribution of electron energies. On the basis of a $\chi^2$ analysis, this model is strongly favored over a model with $\kappa$-distributed electron energies.'
author:
- |
P.J. Storey$^{1}$[^1], Taha Sochi$^{1}$[^2]\
$^{1}$University College London, Department of Physics and Astronomy, Gower Street, London, WC1E 6BT
date: 'Accepted XXX. Received XXX; in original form XXX'
title: 'The Continuum Emission Spectrum of [Hf 2-2]{} near the Balmer Limit and the ORL versus CEL abundance and temperature Discrepancy'
---
\[firstpage\]
atomic transition – atomic spectroscopy – Hf 2-2 – planetary nebulae – nebular physics – optical recombination lines – collisionally excited lines – Balmer continuum – abundance discrepancy – electron temperature – electron distribution – Maxwell-Boltzmann – kappa distribution.
Note: figures are colored in the online version.
Introduction {#Introduction}
============
The long standing problem in nebular physics related to the abundance and temperature discrepancy between the results obtained from optical recombination lines (ORL) and collisionally excited lines (CEL) has been visited by many researchers in the last few decades. Several explanations for the general trend of obtaining higher abundances and lower temperatures from ORL than from CEL of the same ions have been put forward. One explanation is a multi-component model of planetary nebulae in which low temperature, high metallicity components (clumps) produce the ORL and are embedded in a high temperature, relatively low metallicity, hydrogen-rich component that produces the CEL [@LiuSBDCB2000]. It has been suggested recently [@NichollsDS2012] that this discrepancy may be largely explained by the assumption of a different electron energy distribution, a $\kappa$-distribution, rather than the [Maxwell-Boltzmann]{} (MB) distribution which is traditionally accepted as the dominant distribution in the low density plasma found in the planetary nebulae.
The debate about the electron energy distribution in astronomical objects, including planetary nebulae, is relatively old and dates back to at least the 1940s when Hagihara [@Hagihara1944] suggested that the distribution of free electrons in gaseous assemblies deviates considerably from the MB distribution. This was refuted by @BohmA1947 who, on the basis of a detailed quantitative balance analysis, concluded that any deviation from the Maxwellian equilibrium distribution is very small. The essence of their argument is that for typical planetary nebulae conditions, the thermalization process of elastic collisions is by far the most frequent event and typically occurs once every second, while other processes that shift the system from its thermodynamic equilibrium, like inelastic scattering with other ions that leads to metastable excitation or recapture, occur at much larger time scales estimated to be months or even years. Bohm and Aller also indicated the significance of any possible deviation from a Maxwellian distribution on derived elemental abundances. However, nobody seems to have considered non-MB electron distributions as a possible solution for the ORL/CEL discrepancy problem until the recent proposal of @NichollsDS2012. In the light of this suggestion, it seems appropriate to ask whether there is any empirical evidence from observation for departures from MB energy distributions in nebulae.
A preliminary investigation of this possibility has been given recently by @SochiThesis2012 and @StoreySTemp2013 who proposed a method based on using atomic dielectronic recombination (DR) theoretical data in conjunction with observational line fluxes to sample the free electron distribution and compare to the theoretical distributions, i.e. [Maxwell-Boltzmann]{} and $\kappa$. For all the objects studied by @StoreySTemp2013 the limited observational data analyzed were found to be better described by a single MB distribution than a $\kappa$-distribution but the uncertainties were sufficient that a $\kappa$-distribution could not be conclusively excluded. A two-component model incorporating low temperature material was also found to not fit the data as well as a single MB model.
The current paper approaches this problem by modeling the magnitude of the Balmer discontinuity and nearby Balmer continuum of the planetary nebula [Hf 2-2]{}. This nebula is an extreme example of the ORL/CEL discrepancy trend since its abundance discrepancy factor (ADF), defined as the ratio of ORL abundance to CEL abundance, reaches an exceptionally high value between 68-84 [@Liu2003; @WessonLB2003; @ZhangLWSLD2004; @LiuLBL2004; @WessonL2004; @LiuBZBS2006; @WessonBLSED2008]. An exceptionally high temperature discrepancy between the CEL and ORL results, which differ by about an order of magnitude, has also been obtained for this nebula.
The shape of the Balmer continuum is principally determined by a convolution of the $n=2$ hydrogen recombination cross-section and the free electron energy distribution. The shape of the continuum therefore offers the possibility of gaining information about the free electron energy distribution. We shall see below that in the case of [Hf 2-2]{} the continuum shape is primarily determined by the free electron energy distribution. In section \[Hf2-2\] we outline the properties of [Hf 2-2]{} and in section \[Theory\] we deal with the theory describing the continuum processes and new results relating to recombination using a $\kappa$-distribution. The fitting procedure and the resulting fits to the spectral data with MB and $\kappa$ distributions are described in section \[Method\] and our conclusions in section \[Conclusions\].
[Hf 2-2]{} {#Hf2-2}
==========
[Hf 2-2]{} is a faint southern highly symmetric low density planetary nebula with a central cavity and interior disc-shaped multi-shell structure. Its spectrum includes carbon, nitrogen, oxygen, neon, sulphur and argon lines as well as helium. [Hf 2-2]{} seems to have a high abundance in He and C with a high C/O ratio and an exceptionally strong C[ [ii]{}]{} $\lambda$4267 Å signature. The nebula, which is in the galactic bulge at a distance of about 4.0-4.5 kpc, has a spectrally-varying close binary central system with a period of about 0.40 day [@Bond2000]. [Hf 2-2]{} has common spectral features with old novae like DQ Her [@CahnK1971; @Maciel1984; @Kaler1988; @Liu2003; @LiuBZBS2006; @SchaubBH2012], and therefore being a planetary nebula may be disputed.
This unusual nebula is marked by a number of exceptional features. [Hf 2-2]{} exhibits a very high ADF, possibly the highest recorded for a planetary nebula, with very strong ORL emission. [Hf 2-2]{} also has the very large C/O abundance ratio of about 19 [@PatriarchiP1994]. A third unusual feature is a very large Balmer jump associated with a rapid fall in the Balmer continuum intensity towards shorter wavelengths. An estimation based on the magnitude of the Balmer discontinuity indicates an electron temperature of about 780-1000 K in sharp contrast to the forbidden lines estimation of about 8800 K from an \[O[ [iii]{}]{}\] line [@Liu2003; @LiuLBL2004; @ZhangLWSLD2004; @LiuBZBS2006; @McnabbFLBS2012]. The deduced Balmer discontinuity temperature is one of the lowest observed for a planetary nebula. These findings led @LiuBZBS2006 to conclude that a multi-component model of cold hydrogen-deficient knots embedded into a metal-poor nebula may be inevitable to explain these exceptional observations.
With regard to ORL electron temperature derivations for [Hf 2-2]{}, @BastinThesis2006 obtained a mean electron temperature of $<600$ K from C[ [ii]{}]{} lines, while @Liu2006 derived an electron temperature of about 630 K and @McnabbFLBS2012 a temperature of about 3160 K from O[ [ii]{}]{}lines. As for the stellar temperature of this nebula, it is estimated by @LiuBZBS2006 between 50000–67000 K, while @SchaubBH2012 set the temperature of the two stars (assuming a binary system which seems to be largely accepted) in their model to 67000 K and 7500 K.
The observational data of [Hf 2-2]{} which is used in the present paper were obtained in 2001 by a Boller and Chivens long-slit spectrograph mounted on the ground-based 1.52 m European Southern Observatory telescope located in La Silla Chile. More details about this data set can be found in @LiuBZBS2006. The observed spectrum is shown in Figure \[ContFitSeg\], with the measured flux normalized to Balmer H11, an apparently unblended line close to the Balmer discontinuity.
![ The observed [Hf 2-2]{} spectrum (solid) and the continuum fit with a single [Maxwell-Boltzmann]{} distribution with $T=8800$ K (dashed). The fit was optimized on the two longer wavelength of the three wavelength segments (shown in the figure as horizontal lines): 3585-3645, 3780-3790 and 4180-4230 Å.[]{data-label="ContFitSeg"}](g/ContFitSeg)
Theory {#Theory}
======
To model the magnitude of the Balmer discontinuity and the shape of the Balmer continuum we consider the contributions to the continuum spectrum from recombination of H$^+$ and He$^+$, H 2-photon emission and the underlying scattered stellar continuum. Recombination of He$^{2+}$ is neglected due to the very low He$^{2+}$/H$^+$ fraction in [Hf 2-2]{} which is estimated to be about 0.002 [@ZhangLWSLD2004; @LiuBZBS2006].
The recombination of an atomic ion $X^+$ with an electron, $e^-$ of energy $E$ to a state $X^*$ of the recombined ion, $$X^+ + e^- \rightarrow X^* +h\nu,\label{genrecomb}$$ gives rise to continuous emission with emission coefficient (energy emitted per unit time per unit frequency and per unit particle densities) $$\gamma^*(\nu) = \frac{h^3 \alpha^3 c}{32\pi^2 m_e a_0^2} \frac{\omega^*}{\omega^+} \left(
\frac{h\nu}{R} \right)^3 \left( \frac{R}{E} \right)^{\frac{1}{2}} f(E)\ \sigma^*_{\nu} \label{gamma}$$ where, $\nu$ is the frequency of the emitted photon, $\omega^+$ and $\omega^*$ are the statistical weights of the initial and final states respectively, $f(E)$ is the free electron energy distribution function, $\sigma^*_{\nu}$ is the photoionization cross-section for the inverse process of Equation \[genrecomb\] and $R$ is the Rydberg energy constant. The other symbols have their usual meanings. In the present paper, the photoionization cross-sections for states of H and He are taken from @StoreyH91 and @HummerS98 respectively as described in more detail by @ErcolanoS06.
In this paper we consider two possible forms for $f(E)$, the Maxwell-Boltzmann distribution $$f_{_{\rm MB}}(E,T) =
\frac{2}{\left(kT\right)^{3/2}}\sqrt{\frac{E}{\pi}}e^{-\frac{E}{kT}}\label{ElDiEq2},$$ where $T$ is the electron temperature, and the $\kappa$-distribution [@Vasyliunas1968; @SummersT91] given by
[$$f_{\kappa}\left(E,
T_{\kappa}\right)=\frac{2\sqrt{E}}{\sqrt{\pi(kT_{\kappa})^3}}\frac{\Gamma\left(\kappa+1\right)}{(\kappa-\frac{3}{2})^{\frac{3}{2}}\Gamma\left(\kappa-\frac{1}{2}\right)}\left(1+\frac{E}{(\kappa-\frac{3}{2})
kT_{\kappa}}\right)^{-\left(\kappa+1\right)} \label{kappaEq}$$]{} where $\kappa$ is a parameter defining the distribution, $\Gamma$ is the gamma-function for the given arguments, and $T_{\kappa}$ is a characteristic temperature. Note that $f_{\kappa}$ becomes a MB distribution with $T_{\kappa} \rightarrow T$ as $\kappa \rightarrow \infty$.
The normalization of the continuum flux to H11 flux requires that the effective recombination coefficients for H11 should be calculated both with a Maxwell-Boltzmann distribution and a $\kappa$-distribution. The hydrogen line emissivities tabulated by @HummerS87 and @StoreyH95 were calculated assuming that the free electron energy distribution is described by a Maxwell-Boltzmann distribution for all physical processes between bound and continuum states involving free electrons. Here we use the techniques and computer codes described in the last two references and extend them to include a $\kappa$-distribution. At the electron number densities typical of photoionized nebulae ($10^2-10^5$ cm$^{-3}$) the populations of the low-lying states of H are determined primarily by recombination and radiative cascading and are relatively insensitive to the ambient electron density. Hence any error introduced into the calculation of collision rates between high-$n$ states caused by the use of a Maxwell-Boltzmann distribution rather than a $\kappa$-distribution should have minimal effect at nebular densities. We therefore make the approximation of computing the direct recombination coefficients to all the individual levels using a $\kappa$-distribution but retain a Maxwell-Boltzmann distribution for the energy and angular momentum changing collisions among the higher-$n$ states. This should provide a good approximation for the H11 emissivity.
For a general free electron energy distribution, the recombination coefficient to a state X$^*$ is given by
$$\alpha^*_{RC} = \frac{R^{\frac{5}{2}}}{\sqrt{2}c^2 m_e^{\frac{3}{2}}} \frac{\omega^*}{\omega^+}
\int_0^\infty \left( \frac{h\nu}{R} \right)^2 \left( \frac{R}{E} \right)^{\frac{1}{2}}
\sigma^*_{\nu}\ f(E)\ {\rm d}\left(\frac{E}{R} \right) \label{alphaEq}$$
On solving the collisional-radiative recombination problem for hydrogen we obtain the recombination coefficients to all levels and the effective recombination coefficients $\alpha_{\rm eff}(\lambda)$ for a transition of wavelength $\lambda$. We define a line emission coefficient $\epsilon(\lambda)$ as the energy emitted per unit volume per unit time for unit ion and electron density, so that $$\epsilon(\lambda) = \alpha_{\rm eff}(\lambda)\ \frac{hc}{\lambda}.$$
@NichollsDSKP2013 derive the following approximate expression for converting recombination coefficients calculated with a Maxwell-Boltzmann electron energy distribution to those applicable with a $\kappa$-distribution, $$x(\kappa) \equiv \frac{\alpha_{\kappa}(\lambda)}{\alpha_{\rm MB}(\lambda)} = \frac{\epsilon_{\kappa}(\lambda)}{\epsilon_{\rm MB}(\lambda)}
= \frac{(1-\frac{3}{2\kappa})\Gamma(\kappa+1)}{(\kappa-\frac{3}{2})^{\frac{3}{2}}\Gamma(\kappa-\frac{1}{2})}.\label{MBkappa}$$ To obtain this result, @NichollsDSKP2013 use the fact that the photoionization cross-section falls approximately as $(\nu_0/\nu)^3$ for $\nu > \nu_0$, where $\nu_0$ is the threshold frequency for photoionization. so that the integrand in Equation \[alphaEq\] contains a $1/\nu$ term. @NichollsDSKP2013 simplify the integral by moving the $1/\nu$ term outside of the integral, making it possible to carry out the integration analytically. In practice the integral is a convolution of the free-electron energy distribution with the $1/\nu$ term, which is energy dependent. At low temperatures the rapid fall in the electron energy distribution function with increasing energy means that it is a good approximation to neglect the energy dependence of the frequency term. At higher temperatures, both terms are essential in the calculation of the recombination coefficient. The expression for $x(\kappa)$ in Equation \[MBkappa\] is independent of temperature, density and transition. This approximation will fail at sufficiently high temperatures where the effects of the free-electron energy distribution and the frequency dependent term become comparable.
Using the approximate function $x(\kappa)$, we can express the results of our more complete collisional-radiative treatment for $\epsilon(\lambda)$ in terms of a correction factor $y(\lambda,T,\kappa)$ as follows
$$\epsilon_{\kappa}(\lambda) = \epsilon_{\rm MB}(\lambda)\ x(\kappa)\ y(\lambda,T,\kappa).
\label{epsEq}$$
The values of $\epsilon_{\rm MB}$ can be obtained from @HummerS87 and @StoreyH95. In Table \[H11MBkappa\] we tabulate values of $y(\lambda,T,\kappa)$ for H11 corresponding to various values of $\kappa$ at a range of temperatures. We also tabulate in the last line $x$ as a function of $\kappa$. Note that, in principle, $y$ also depends on electron density but in practice it is very insensitive to electron density for densities up to $10^5$ cm$^{-3}$, so we only tabulate values calculated at $10^3$ cm$^{-3}$.
------------- ------- ------- ------- ------- ------- ------- ------- --
logT\[K\]
50.0 20.0 10.0 7.0 5.0 3.0 2.0
2.6 0.994 0.994 0.996 0.997 0.999 1.004 1.017
2.7 0.994 0.995 0.997 0.999 1.002 1.011 1.031
2.8 0.995 0.996 0.999 1.001 1.005 1.017 1.044
2.9 0.995 0.997 1.001 1.004 1.009 1.024 1.057
3.0 0.996 0.998 1.002 1.007 1.013 1.031 1.071
3.1 0.996 0.999 1.004 1.009 1.016 1.038 1.084
3.2 0.996 1.000 1.005 1.011 1.020 1.045 1.099
3.3 0.996 1.000 1.007 1.013 1.023 1.053 1.115
3.4 0.996 1.000 1.008 1.016 1.027 1.062 1.133
3.5 0.995 1.001 1.010 1.019 1.032 1.072 1.154
3.6 0.995 1.001 1.012 1.022 1.037 1.083 1.177
3.7 0.995 1.001 1.014 1.026 1.043 1.096 1.203
3.8 0.995 1.003 1.017 1.031 1.051 1.112 1.235
3.9 0.995 1.004 1.020 1.036 1.059 1.128 1.269
4.0 0.994 1.004 1.022 1.040 1.066 1.144 1.306
4.1 0.995 1.006 1.027 1.047 1.076 1.165 1.350
4.2 0.995 1.008 1.032 1.054 1.087 1.187 1.398
4.3 0.995 1.010 1.036 1.060 1.097 1.210 1.450
$x(\kappa)$ 1.008 1.020 1.043 1.066 1.103 1.228 1.596
------------- ------- ------- ------- ------- ------- ------- ------- --
: Values of the correction factor $y(\lambda,T,\kappa)$, defined in Equation \[epsEq\], as a function of $\kappa$ and logT\[K\] for computing the emission coefficient, $\epsilon_{\kappa}(\lambda)$ for H11. In the last row of the table $x$, as defined in Equation \[MBkappa\], is given as a function of $\kappa$. \[H11MBkappa\]
As can be seen, the values from Equation \[MBkappa\] are a good approximation for large values of $\kappa$ but the ratio increases as $\kappa$ decreases. In addition, the values from Equation \[MBkappa\] are generally smaller than the more exact values, with the difference being largest for the smallest tabulated $\kappa$ and the highest temperatures, as expected.
Since our model of the recombination process with $\kappa$-distributed electron energies does not treat energy and angular momentum changing collisions correctly we cannot use the results to model the high Balmer lines as was done for example by @ZhangLWSLD2004 using the code originally authored by one of us (PJS). We therefore restrict the current model to calculation of the continuum processes only, normalized to the H11 flux.
Our model of the continuum also includes the hydrogen 2-photon emission. The emission coefficient for the hydrogen 2s-1s 2-photon emission is
$$\epsilon_{2q}(\nu) = \alpha({\rm 2s}) \frac{A_{2q}(\nu)}{A_{2q}+C_{{\rm 2s,2p}}\ N_p}\ h\nu,
\label{2q}$$
where $\alpha({\rm 2s})$ is the total recombination coefficient to the 2s state of H, $A_{2q}$ is the total 2s-1s spontaneous transition probability, $A_{2q}(\nu)$ is the probability per unit frequency, $C_{{\rm 2s,2p}}$ is the coefficient for proton collisional transitions between the 2s and 2p states and $N_p$ is the proton number density. We assume that the population of the 2p state is negligible at the relevant densities [@HummerS87], and we take $C_{{\rm 2s,2p}}$ from @Seaton55 and $A_{2q}$ and $A_{2q}(\nu)$ from @NussbaumerS84. Values of $\alpha({\rm
2s})$ for an MB distribution were provided by @HummerS87 and @StoreyH95. From our new calculation of the hydrogen collisional-radiative problem with a $\kappa$-distribution we obtain values of $\alpha({\rm 2s})$ as a function of $\kappa$ and temperature. These values can be related to those obtained with an MB distribution in the same way as in Equation \[epsEq\], via the function $x(\kappa)$ $$\alpha_{\kappa}({\rm 2s}) = \alpha_{\rm MB}({\rm 2s})\ x(\kappa)\ z({\rm 2s},T,\kappa) \label{zdef}$$ where we tabulate the correction factors $z({\rm 2s},T,\kappa)$ in Table \[2sMBkappa\].
----------- ------- ------- ------- ------- ------- ------- ------- --
logT\[K\]
50.0 20.0 10.0 7.0 5.0 3.0 2.0
2.6 0.994 0.994 0.995 0.995 0.996 1.000 1.009
2.7 0.994 0.994 0.996 0.997 0.999 1.005 1.020
2.8 0.994 0.995 0.997 0.999 1.001 1.009 1.029
2.9 0.995 0.996 0.998 1.000 1.004 1.014 1.038
3.0 0.995 0.996 0.999 1.002 1.006 1.018 1.047
3.1 0.995 0.996 1.000 1.003 1.007 1.022 1.054
3.2 0.994 0.996 1.000 1.003 1.009 1.025 1.062
3.3 0.994 0.996 1.001 1.004 1.011 1.029 1.070
3.4 0.994 0.996 1.001 1.005 1.013 1.033 1.079
3.5 0.994 0.997 1.002 1.007 1.015 1.038 1.089
3.6 0.994 0.998 1.004 1.009 1.018 1.045 1.101
3.7 0.995 0.999 1.006 1.012 1.022 1.052 1.114
3.8 0.998 1.001 1.010 1.017 1.028 1.062 1.131
3.9 0.997 1.002 1.011 1.019 1.032 1.069 1.147
4.0 0.994 1.000 1.010 1.019 1.033 1.075 1.162
4.1 0.994 1.000 1.011 1.022 1.038 1.085 1.182
4.2 0.995 1.002 1.014 1.026 1.044 1.096 1.207
4.3 0.994 1.001 1.015 1.028 1.048 1.107 1.230
----------- ------- ------- ------- ------- ------- ------- ------- --
: Values of the correction factor $z({\rm 2s},T,\kappa)$, defined in Equation \[zdef\], as a function of $\kappa$ and logT\[K\] for computing the total recombination coefficient to the 2s state of hydrogen with $\kappa$-distributed electron energies. \[2sMBkappa\]
Modeling the continuum {#Method}
======================
The observed spectrum of [Hf 2-2]{} with intensity normalized relative to H11 is shown in Figure \[ContFitSeg\]. The figure also shows the spectral segments that we use for fitting, chosen to be as free as possible from significant spectral lines. The wavelength ranges for the three segments are $\lambda=$ 3585-3645, 3780-3790 and 4180-4230 Å.
The continuum longward of the Balmer edge
-----------------------------------------
The observed continuum longward of the Balmer edge is comprised principally of hydrogen free-bound emission with principal quantum number $n>2$, helium free-bound emission, hydrogen 2-photon emission and scattered starlight from the central star of the nebula. The emissivity of the H 2-photon emission relative to H11 is relatively insensitive to the temperature of the emitting material and comprises up to 10% of the continuum in this spectral region. The contribution from $n>2$ H free-bound emission varies strongly with the temperature, being 3% just longward of the Balmer edge at $T=10^4$ K and falling rapidly as the temperature decreases. We model the remaining background contribution to the flux relative to H11 with an empirical power-law distribution
$$F(\lambda) = F_0\ \left( \frac{3647}{\lambda} \right)^{\beta} \label{power}$$
where $F_0$ is the contribution at the Balmer edge and wavelengths in Å are vacuum wavelengths.
The parameters $F_0$ and $\beta$ were obtained by fitting the model continuum to the two longer wavelength segments shown in Figure \[ContFitSeg\] by minimizing the rms deviations of the fit from the observed spectrum. In computing the model continuum we take $N({\rm He}^+)$/$N({\rm H}^+)$ = 0.103 [@LiuBZBS2006]. The same authors quote $N({\rm He}^{2+})$/$N({\rm H}^+)$ = 0.002, which means that He$^{2+}$ recombination makes a negligible contribution to the continuum, so we neglect this component. The derived continuum is very weakly sensitive to the electron number density through the 2-photon component (Equation \[2q\]) and we adopt $N_e= 1000$ cm$^{-3}$.
The best fit values of $F_0$ and $\beta$ were derived for two models. The first uses a MB distribution for the free electrons with a temperature of 8800 K chosen to reflect the forbidden line values derived by @LiuBZBS2006. In any model comprising two MB distributions, the low temperature component will contribute a negligible amount to the continuum longward of the Balmer edge, so it is appropriate to use the forbidden line temperature to determine the continuum component in this wavelength range.
As we shall see below, the best fit to the continuum using a single $\kappa$-distribution yields an electron temperature of a few thousand Kelvin and a small value of $\kappa$. So for our second model of the underlying continuum we adopt a $\kappa$-distribution with $\kappa=2$ and $T=3000$ K. The results of the two underlying continuum models are summarized in Table \[confit\]. The values of $F_0$ and $\beta$ are relatively insensitive to the assumptions about the electron energy distribution and its temperature. To illustrate the fit longward of the Balmer edge, we show in Figure \[ContFitSeg\] the full fit to the two longer wavelength segments of the continuum including all the processes described above, using the parameters (a) from Table \[confit\] and assuming a single MB distribution with $T=8800$ K.
$F_0$\[Å$^{-1}$\] $\beta$
--- ------------------- ---------
a 0.161 1.588
b 0.174 1.536
: Underlying continuum fitting parameters $F_0$ and $\beta$ for two scenarios (a) A single MB distribution with temperature 8800 K, and (b) a single $\kappa$-distribution with $\kappa=2$ and $T =3000$ K. \[confit\]
Fits with Maxwell-Boltzmann distributions
-----------------------------------------
It is clear from Figure \[ContFitSeg\] that the magnitude of the Balmer discontinuity and the slope of the Balmer continuum shortward of the discontinuity cannot be explained by emission from hydrogen and helium at the forbidden line temperature. This has already been discussed by @LiuBZBS2006 who estimated that the shape of the continuum implies that the emitting region has a temperature of $\approx 900$ K. In Figure \[SinglefreeMB\] we show the result of making a fit to all three spectral segments shown in Figure \[ContFitSeg\] using a single MB electron energy distribution in which the temperature is a free parameter. The background continuum is computed using the parameters from fit (a) in Table \[confit\]. The temperature of best fit is 1334 K and although the continuum model is improved compared to Figure \[ContFitSeg\], it matches neither the magnitude of the discontinuity nor the slope adequately.
![The observed [Hf 2-2]{} spectrum (solid) and the single-component Maxwell-Boltzmann fit (dashed) where the temperature is treated as a free parameter. The final optimized temperature is $T=1334$ K.[]{data-label="SinglefreeMB"}](g/SinglefreeMB)
As discussed earlier, one resolution of the ORL/CEL abundance and temperature discrepancy in nebulae that has been proposed is that there are relatively cold high-metallicity knots embedded in hotter lower abundance gas. We therefore attempt to match the continuum with two components having different temperatures. We set the temperature of one component equal to a typical forbidden line temperature for this object of 8800 K and allow the other to vary. The continuum emissivity has only weak density dependence so we choose representative electron densities for the two components of 1000 cm$^{-3}$ for the forbidden line region and 5000 cm$^{-3}$ for the cold material [@LiuBZBS2006]. The relative emission measure of the two components is then allowed to vary by making the volume ratio of the two components a free parameter. In Figure \[2MB1free\] we show the resulting fit, which is excellent. The optimum temperature of the cold component is 540 K and the fraction of the total volume occupied by the cold component is 0.00706. Hence the cold component has an emission measure, which is proportional to the product of the volume fraction and particle density squared, that is 15.0% of the total recombination line and continuum emission measure.
![The observed [Hf 2-2]{} spectrum (solid) and a two-component [Maxwell-Boltzmann]{} fit (dashed) with two temperatures: one is the forbidden line temperature which is fixed at $T=8800$ K and the other is free to vary. The ratio of the volumes of the two components is also treated as a free parameter.[]{data-label="2MB1free"}](g/2MB1free)
Fit with $\kappa$-distribution
------------------------------
An alternative possible resolution of the ORL/CEL problem is that there is no separation of the emitting material into high and low temperature components with significantly different abundances but rather a single medium in which the electron energy distribution is non-Maxwellian, with a $\kappa$-distribution being a possible candidate. The Balmer continuum close to the Balmer discontinuity provides a means of testing the validity of this proposal for the very low energy part of the distribution function. In Figures \[T100003kappa\] and \[T10003kappa\] we illustrate how the magnitude of the Balmer discontinuity and shape of the Balmer continuum change as $\kappa$ is varied for two different temperatures. Compared to a MB, $\kappa$-distributions have more particles at the lowest and highest energies and less at intermediate energies. At low energies the $\kappa$-distribution function thus falls more rapidly with increasing energy than a MB distribution. This can be seen in Figure \[T10003kappa\], for example, with the magnitude of the Balmer discontinuity increasing and the slope of the Balmer continuum becoming steeper as $\kappa$ decreases.
![Model continua with a $\kappa$-distribution of electron energies for a temperature $T=10000$ K; $\kappa=2$ (dashed blue line), $\kappa=5$ (dotted green line), and $\kappa=25$ (dot-dashed red line).[]{data-label="T100003kappa"}](g/T100003kappa)
![Model continua with a $\kappa$-distribution of electron energies for a temperature $T=1000$ K; $\kappa=2$ (dashed blue line), $\kappa=5$ (dotted green line), and $\kappa=25$ (dot-dashed red line).[]{data-label="T10003kappa"}](g/T10003kappa)
We now fit the chosen three spectral segments to a $\kappa$-distribution of which the temperature and value of $\kappa$ are free parameters and the final values that we obtained for these parameters are $\kappa=2.11$ and $T=4640$ K. Figure \[1kappa\] shows a plot of this fit. On visual inspection the best-fit $\kappa$ distribution where $\kappa$ and temperature are allowed to vary is significantly less good than the two-component MB fit. It should be remarked that the optimized $\kappa$ obtained from this fitting exercise is very different to values proposed by @NichollsDS2012 which usually fall in the range 10-20. For comparison, we include in Figure \[1kappa\] a fit carried out with a fixed value of $\kappa=10$. The temperature was allowed to vary and the best fit value was $1665$ K. In the next section we attempt to quantify the relative quality of these fits.
![Fits obtained by simultaneous optimization of $\kappa$ and $T$ (dashed red line) and by fixing $\kappa=10$ and allowing $T$ to vary (dotted blue line).[]{data-label="1kappa"}](g/1kappa)
Statistical Analysis {#Discussion}
--------------------
We wish to address the question of whether the fit to the continuum using a $\kappa$-distribution is significantly worse than the two-component MB fit. A visual inspection of Figures \[2MB1free\] and \[1kappa\] shows that while the-two-component fit provides a very good match to the three segments of the continuum used for fitting, the best $\kappa$-distribution fit underestimates the magnitude of the Balmer discontinuity and also underestimates the steepness of the decline at shorter wavelengths. We will use a comparison of $\chi^2$ values for these two fits to quantify the relative goodness of fit. To estimate $\chi^2$ we need an estimate of the uncertainties on the observational data, which by inspection and from instrumental considerations are not independent of wavelength, being much larger at the shortest wavelengths. We therefore compute separate rms deviations of the data from the two-component MB fit for each of the three wavelength segments. Since we use deviations from the two-component fit to define the data uncertainties it follows that the value of $\chi^2$ for the fit, summed over the three wavelength segments, will be given by $$\chi^2 = N_{df} \equiv N_{dp} - N_{fp} = 237,$$ where $N_{df}$ is the number of degrees of freedom, $N_{dp}$ is the number of data points in the three segments and $N_{fp}$ is the number of free parameters. We now use the estimates of the data uncertainties from the two-component MB fit to compute $\chi^2$ for the best fit $\kappa$-distribution and we find $\chi^2=433$. We wish to test the null hypothesis that the $\kappa$-distribution is an equally good fit to the data as the two-component MB fit. The following square-root transformation of $\chi^2$ [@AbSteg72],
$$t = \sqrt{2\chi^2} - \sqrt{2N_{df}},$$
is such that for large $N_{df}$, the transformed distribution is approximately normal with zero mean and unit standard deviation. For the best fit $\kappa$-distribution we find that $t=7.67$ standard deviations from the zero mean, indicating that the probability that the null hypothesis is true is vanishingly small.
Thus, a two-component [Maxwell-Boltzmann]{} model is highly favored compared to the single-component $\kappa$ model. The single-component MB model is even less likely to be correct than either of the two-parameter models based on the same $\chi^2$ analysis, giving $t=14.4$ standard deviations.
Conclusions {#Conclusions}
===========
The analysis of the Balmer continuum spectrum of [Hf 2-2]{} seems to indicate the signature of a two-component nebula with two different temperatures. Assuming that one component has a temperature of 8800 K, we find that the best fit to the data is obtained by adding a second component with temperature of 540 K and emission measure 15.0% of the total. We also modeled the continuum with a single component with $\kappa$-distributed free electron energies, but on the basis of a $\chi^2$ test we found this model to be significantly less likely to be correct. In the case of $\kappa$-distribution models, it is important to note that modeling the Balmer continuum only samples the free electron energy distribution at the lowest energies, below the mean energy, and therefore gives no information about departures from MB at energies above the mean energy which might affect the excitation of nebular forbidden lines, for example. The extreme nature of [Hf 2-2]{} and the similarity of some of its spectral features to old novae like DQ Her may also cast some doubt on the applicability of the conclusions reached to the properties of planetary nebulae in general.
Acknowledgment and Statement
============================
The work of PJS was supported in part by STFC (grant ST/J000892/1). During the final stages of writing this paper, our attention was drawn to a similar paper by Y. Zhang, X-W. Liu and B. Zhang (arXiv:1311.4974); however this did not have an impact on the methods or results reported in the present paper.
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[^1]: E-mail:[email protected] (PJS).
[^2]: E-mail: [email protected] (TMS). Corresponding author.
|
---
abstract: 'The generalized constrained search formalism is used to address the issues concerning density-to-potential mapping for excited states in time-independent density-functional theory. The multiplicity of potentials for any given density and the uniqueness in density-to-potential mapping are explained within the framework of unified constrained search formalism for excited-states due to Görling, Levy-Nagy, Samal-Harbola and Ayers-Levy. The extensions of Samal-Harbola criteria and it’s link to the generalized constrained search formalism are revealed in the context of existence and unique construction of the density-to-potential mapping. The close connections between the proposed criteria and the generalized adiabatic connection are further elaborated so as to keep the desired mapping intact at the strictly correlated regime. Exemplification of the unified constrained search formalism is done through model systems in order to demonstrate that the seemingly contradictory results reported so far are neither the true confirmation of lack of Hohenberg-Kohn theorem nor valid representation of violation of Gunnarsson-Lundqvist theorem for excited states. Hence the misleading interpretation of subtle differences between the ground and excited state density functional formalism are exemplified.'
author:
- Prasanjit Samal
- Subrata Jana
- 'Sourabh S. Chauhan'
title: 'Excited-State Density-Functional Theory Revisited: on the Uniqueness, Existence, and Construction of the Density-to-Potential Mapping'
---
introduction
============
Since its advent, density-functional theory (DFT) [@hk; @ks; @cs1; @cs2; @cs3; @cs4; @cs5; @py; @dg; @dj] is routinely applied for calculating the electronic, magnetic, spectroscopic and thermodynamic properties of atoms, molecules and materials in ground and excited states. In the last couple of decades, studying excited- states employing DFT has become the main research interest [@py; @dg; @dj; @gl1; @gl2; @zr; @vb; @at; @at2; @at3; @zls; @wh; @pl; @gok; @mlp; @anp; @gor1; @gor2; @gor3; @ln1; @ln2; @ln3; @pgg; @su; @fug; @anqc; @rvl; @cjcs; @pplb; @lps; @av; @jked; @gj; @thesis; @rvl; @cc; @rg; @dg; @kd; @sd; @gk; @cau; @est]. Thus one of the most natural approach to do excited-state DFT is to adopt the time-independent density functional formalism [@anp; @anqc; @an; @thesis] in which the individual excited-state energies are determined from the stationary states of the energy density functional. However, the question is whether there exists any such functional(s) for excited states analogous to the ground-state. Not only energy functionals but also the most fundamental and essential requirement for excited-state density functional theory ($e$DFT) is to establish the one-to-one mapping similar to the Hohenberg-Kohn theorem which is the main intent of the present work. Although the issue of density $\rho(\vec r)$ to potential $\hat v(\vec r)$ mapping for excited states has been addressed in the past [@sahni; @sahni1; @sahni2; @gb; @har1; @lpls], but the question still remains unanswered. So the current work will answer the critiques of density-to-potential mapping based on the generalized/unified constrained search(CS) due to Perdew-Levy(PL) [@pl], Görling [@gor1; @gor2; @gor3], Levy-Nagy(LN) [@ln1; @ln2; @ln3], Samal-Harbola(SH) [@shcpl; @shjp1; @shjp2; @shjp3] and Ayers-Levy-Nagy [@al; @aln; @pair].
In the present work, we will critically analyse and make furtherance to the $e$DFT ideas proposed by Samal and Harbola [@shjp2; @thesis]. According to it, (i) the CS approach can be extended to excited-state in the light of the stationary state formalism of Görling [@gor1; @gor2; @gor3] and variational $e$DFT formalism by Levy-Nagy [@ln1; @ln2; @ln3]; (ii) within the variational $e$DFT formalism, the construction of the Kohn-Sham(KS) system by comparing only the ground-state density is insufficient and can’t explain the existence of multiple potentials; (iii) the density-to-potential mapping in $e$DFT can be achieved through the following criteria: compare the ground states of the true and KS system energetically such that it can account for the most close resemblance of the densities in a least square sense. SH showed it by comparing the expectation value of the original ground-state KS Hamiltonian (obtained using the Harbola-Sahni [@hs] exact exchange potential) with that of the alternative KS systems. Finally, the kinetic energy of true and KS system need to be kept closest. This is also another way of comparing the ground states based on the differential virial theorem(DVT) [@hm]; (iv) the CS approach is capable of generating all the potentials for a given excited state density and at the same time uniquely establishes the density-to-potential mapping.
The work is organized as follows. In Sec.II, the generalized/unified CS $e$DFT will be briefly discussed from the prospective of density-to-potential mapping. It will be shown that there exist multitude of potentials for a given density. In Sec.III, furtherance of SH $e$DFT will be presented in order to show the existence and unique construction of the desired density-to-potential mapping. In this, we will show, how the proposed $e$DFT is also consistent with the generalized adiabatic connection(GAC) KS formalism [@gl1; @gor1; @gor2; @gor3; @gacj; @gacl; @gac1; @gac2; @gac3; @gac4; @gac5; @gac6; @gac7] and in principle applicable to (non-)coulombic densities. In Sec.IV, we will show the existence of multiple potentials for given ground or lowest excited states can never be ruled out even within Li. et al.[@lpls] demonstration of Gunnarsson and Lundqvist(GL) theorem [@gl1; @gl2]. However, based on the theories presented in Sec.II & III, these seemingly contradictory results will be explained in order to justify the non-violation of Hohenberg-Kohn(HK) [@hk] and GL theorems. Thus the density-to-potential mapping will be demonstrated within [@lpls] approach by making use of the unified $e$DFT for the two model systems (i.e. $1D$ quantum harmonic oscillator(QHO) with finite boundary and infinite well external potentials). For completeness, in Sec. V, same set of model systems will be used to exemplify density-to-potential mapping based on the CS formalism [@zp]. Finally, we will provide firm footing to density-to-potential mapping based on the proposed criteria of $e$DFT.
Unified Constrained-Search Formulation of DFT
=============================================
Although in principle the ground-state CS formalism [@cs1; @cs2; @cs3; @cs4; @cs5] has all the information about the excited-states, the desired density-to-potential mapping for individual excited-states are not so trivial and straightforward. To do so, series of attempts being made based on the original CS approach [@ln1; @ln2; @ln3; @pl; @gor1; @gor2; @gor3; @al; @anp; @aln; @mlp; @shcpl; @shjp1; @shjp2; @shjp3; @est; @pair]. In the recent past, the form of functional for ground state (both for degenerate and non-degenerate) has been extended [@shjp1; @shjp2; @shamim; @hem1; @hem2; @hem3] to study the excited states. Now we will briefly describe how the generalized CS formalism explains the existence of multiple potentials for any given fermionic density without hindering the density-to-potential mapping.
Let’s consider $N$ fermions trapped in a local external potential $\hat v_{\text{ext}}(\vec r)$, described by the Hamiltonian $$\hat H[\hat v;N] = \hat T + \hat V_{\text{ee}} + \sum_{i=1}^N\hat v_{\text{ext}}(\vec r_i),
\label{gcs1}$$ where $\hat T$ and $\hat V_{ee}$ are the kinetic and electron-electron interaction operators with the corresponding stationary states are given by $$\hat H[\hat v(\vec r), N] \Psi_k(\vec r) = E_k[\hat v(\vec r), N] \Psi_k(\vec r)~,
\label{gcs2}$$ where $\hat v_{\text{ext}}(\vec r) \equiv \hat v(\vec r)$. In Eq.(\[gcs2\]), $\Psi_k(\vec r) \equiv
\Psi_k[\hat v(\vec r), N] $ are the pure state $v-$representable stationary quantum states i.e. it is coming from the solution of the Schrödinger equation. But for $N-$representable densities (i.e. $\int
\rho(\vec r) d\vec r = N$) and therefore wavefunctions (i.e. $\int {\Psi[N]}^2 d\vec r = N$), similar to the HK universal functional there exists an analogous functional which is stationary w.r.t all the variations that do not change the density (i.e. $\delta_{\Psi \to \rho}$) and is given by $$Q^S[\rho;N] = \delta_{\Psi[N] \to \rho(\vec r)}\langle\Psi | \hat T + \hat V_{\text{ee}} | \Psi\rangle~.
\label{gcs3}$$ Now according to the Perdew and Levy extremum principle [@pl] and generalized CS formalism [@gor2; @shjp2; @al], the energy of the $k^{th}$ excited state is given by $$E_k = E[\rho_k;N] = Q^S[\rho_k;N] + \int\rho_k(\vec r) {v}_{\text{ext}}(\vec r) d\vec r .
\label{gcs4}$$ In Eq.(\[gcs4\]), the minimization occurs only over Görling’s stationary-state functional $Q^S[\rho_k]$ and the corresponding wavefunctions are given by $$\Psi_k^S = \Psi^S[\rho_k,N] = \arg\min_{\Psi[N] \to \rho_k}\langle\Psi[N]|\hat T + \hat V_{\text{ee}}|
\Psi[N]\rangle .
\label{swfc}$$ On the other hand, in the LN [@ln1; @ln2; @ln3] variational constrained minimization approach for excited-states leads to the $k^{th}$ stationary state energy $$\begin{aligned}
E_k[\rho,\rho_0] &=& \min_{\rho[\hat v] \to N}\Big\{\int \rho(\vec r) {v}_{\text{ext}}(\vec r)
d\vec r + F[\rho,\rho_0]\Big\} \nonumber\\
&=& \int \rho_k(\vec r) {v}_{\text{ext}}(\vec r) d\vec r + F[\rho_k,\rho_0]~,
\label{gcs5}\end{aligned}$$ where $\rho_0$ is the ground state density of the system under consideration. The LN energy density functional differs from the HKS ground-state and the stationary state $e$DFT functional due to the bifunctional $F[\rho,\rho_0]$, which is defined by $$\begin{aligned}
F_k[\rho,\rho_0] &=& \min_{\Psi[N] \to \rho, {\langle\Psi[N]|\Psi_j[\hat v;N]
\rangle=0, j < k}} \langle\Psi|\hat T + \hat V_{\text{ee}}|\Psi\rangle \nonumber \\
&=& F[\rho_k,\rho_0] ,
\label{gcs6}\end{aligned}$$ for the $k^{th}$ excited state. So the energy of the $k^{th}$ excited state can be re-expressed as $$\begin{aligned}
E_k[\rho,\rho_0] = \min_{\rho[\hat v] \to N} \Big\{\int \rho_k(\vec r) { v}_{\text{ext}}(\vec r) d\vec r
+ \nonumber\\ \min_{\Psi[N] \to \rho, {\langle\Psi[N]|\Psi_j[\hat v;N] \rangle=0, j < k}} \langle\Psi|
\hat T + \hat V_{\text{ee}}|\Psi\rangle\Big\}~
\label{gcs7}\end{aligned}$$ with the minimizing wavefunction denoted by $$\Psi_k^{LN}[\hat v;N] = \arg \min_{\Psi[N] \to \rho, {\langle\Psi[N]|\Psi_j[\hat v;N]
\rangle=0, j < k}} \langle\Psi|\hat T + \hat V_{\text{ee}}|\Psi\rangle .
\label{vwfc}$$ In the LN bifunctional $F[\rho,\rho_0]$, if $\rho = \rho_0$ then the functional reduces to HK universal functional and the same holds true for $Q^S[\rho]$. Also $F[\rho,\rho_0]$ is the generalization of the $e$DFT stationary state functional $Q^S[\rho]$ as described in the $\it Theorems 4,~5~\&~6$ of [@al]. These theorems are in fact an artifact of the orthogonality constraint. Since all the lower states $\Psi_j[\hat
v;N] (j < k)$ are determined from the external potential $\hat v_{\text{ext}}$ (which is a unique functional of ground state density $\rho_0$ according to HK [@hk] theorem), implies that the ground state density plays an important role in LN-formalism. So, in principle one can also write the excited-state density bifunctional as $F_k[\rho,{\hat v}_{\text{ext}}]$ instead of $F_k[\rho,\rho_0]$. If the electronic densities are $v-$representable then Eq.(\[gcs6\]) modifies to $$E_k[\hat v_{\text{ext}};N] = \int \rho_k(\vec r) {v}_{\text{ext}}(\vec r) d\vec r +
F_k[\rho,{\hat v}_{\text{ext}}].
\label{gcs8}$$ From the generalized CS energy functionals for any eigendensity given by the Eq.(\[gcs4\]) & Eq.(\[gcs8\]) there exist multiple potential functions [@shjp2; @thesis; @al]. In general, these generalized multiple local external potentials can be obtained through the Euler Lagrange equation $$\begin{aligned}
\frac{\delta}{\delta\rho}\Big[E_k - \mu\Big\{\int \rho_k(\vec r) d\vec r - N \Big\}\Big] = 0 \\
v_{\text{ext}}(\vec r) = \mu - \Big(\frac{\delta F[\rho,\rho_0]}{\delta\rho}\Big){\Big |}_{\rho = \rho_k} \\
{\text OR}~~
w_{\text{ext}}(\vec r) = \mu - \Big(\frac{\delta Q^S[\rho(\vec r)]}{\delta\rho}\Big){\Big |}_{\rho = \rho_k},
\label{gcs9}\end{aligned}$$ where $\mu = \left(\frac{\delta E[\rho_e]}{\delta\rho} \right)_N$. The actual potential is one of these which should be uniquely mapped to the given density as will be shown in the following sections. In particular, the local external potentials will be identical $v_{\text{ext}}(\vec r) = w_{\text{ext}}(\vec r)$ iff the density $\rho_k(\vec r)$ is pure state $v$-representable and $E_k$ is the corresponding eigen energy. Now to obtain the KS like equation for the generation of $\rho_k$ and to obtain $E_k$, one needs to first construct a non -interacting system with some external potential ${\hat v}'_{\text{ext}}$ such that it’s $m^{th}$ excited state density $\rho^{{\hat v}'_{\text{ext}}}_m(\vec r)$ (say) may be the same as $\rho_k(\vec r)$ of the original system ${\hat v}_{\text{ext}}$. In stationary-state $e$DFT [@gor2; @shjp2; @thesis], this is done by generalized adiabatic connection (GAC) [@gl1; @gor1; @gor3; @gacj; @gacl; @gac1; @gac2; @gac3; @gac4; @gac5; @gac6; @gac7]. Whereas, in LN variational $e$DFT [@ln1; @ln2; @ln3; @al], this is done by the constrained minimization of the expectation value $\langle\Psi[{\hat v}'_{\text{ext}},\rho^{{\hat v}'_{\text{ext}}}_m(\vec r)]|\hat T +
{\{{\hat V}_{\text{ee}}=0\}}|\Psi[{\hat v}'_{\text{ext}},\rho^{{\hat v}'_{\text{ext}}}_m(\vec r)]\rangle$, where $\Psi[{\hat v}'_{\text{ext}}, \rho^{{\hat v}'_{\text{ext}}}_m(\vec r)]$ gives the desire density of interest. Out of many such non-interacting $\Psi[{\hat v}'_{\text{ext}},\rho^{{\hat v}'_{\text{ext}}}_m(
\vec r)]$ s (different systems), the unique one is chosen whose ground-state density $\rho^{{\hat v}'_{
\text{ext}}}_0(\vec r)$(say) resembles with the ground-state density $\rho^{{\hat v}_{\text{ext}}}_0(\vec r)$ of the original system “most closely in a least-square sense"(i.e. the LN criterion). The matching of the ground-state densities actually matches the external potentials ${\hat v}'_{\text{ext}}$ and ${\hat v}_{
\text{ext}}$ according to the HK theorem [@hk]. But the difference occurs between the kinetic energies of the two systems. As matter of which, the discrepancy in the $\rho \Longleftrightarrow \hat v$ mapping arises because the LN criterion strictly depends upon the behavior of the bifunctional.
Proposed Constrained-Search Formulation of DFT
==============================================
The CS formulation described in the previous section implies that the content of the excited state functionals $Q^S[\rho_e]$ and $F[\rho_e,\rho_0]$ differs from the HK universal functional $F[\rho]$ except their stationarity with respect to variation in the external potential. Actually, only in the case of ground-state, all the three functionals are identical to one another and in general there exists a close link between Görling $Q^S[\rho_e]$ and Levy-Nagy $F[\rho_e,\rho_0]$ [@al]. So in the unified $e$DFT formalism, for a given excited-state eigendensity $\rho_e(\vec r)$, both $Q^S[\rho_e]$ and $F[\rho,\hat v_{ext}]$ are stationary about the corresponding $\hat v_{ext}$ which also holds for the desired excited-state $\Psi_k^S \equiv \Psi_k^{LN}$ [@shjp2; @thesis; @al]. Now due to the presence of orthogonality constraint in $F[\rho,\hat v_{ext}]$, several choices for the set of low lying states can be made to which $\Psi_k^{LN}$ will be orthogonal and for each choice, there may exists a generalized potential function $\hat w_{ext}$. So some extra deciding factors are required for setting up the $\rho \Longleftrightarrow \hat v$ mapping which is the intent of the current section.
Now resorting back to the work of Samal-Harbola [@shjp2], we would also like to re-emphasis that the direct or indirect comparison of ground states are not sufficient to establish the $\rho(\vec r)
\Longleftrightarrow {\hat v}_{ext}(\vec r)$ mapping or to construct the KS system for excited-states [@shcpl]. Given the discussions on unified CS $e$DFT in the previous section, we now present a consistent approach to address the density-to-potential mapping issues. Fundamentally rigorous and crucial tenets of the proposed DFT are: ([*i*]{}) There exist ways for mapping an excited-state density $\rho_e(\vec r)$ to the corresponding many-electron wavefunction $\Psi(\vec r)$ which in turn maps to the external potential $\hat v_{ext}(\vec r)$ through the $\rho$-stationary wavefunctions [@gor2; @shjp2; @al]. In this, the wavefunction depends upon the ground-state density $\rho_0$ implicitly. ([*ii*]{}) The KS system is to be defined through a comparison of the kinetic energy, ground-state density and variation of the energy w.r.t. symmetry of the excited-states.
The claim is, unified CS approach can provide the mapping from an excited-state density $\rho_e(\vec r)$ to many-body wavefunction. Stationary state formalism [@gor2; @shjp2] provides a straightforward method of mapping $\rho_e(\vec r) \Longleftrightarrow {\hat v}_{ext}(\vec r)$, just by making sure whether $\langle
\Psi_k|\hat T + \hat V_{\text{ee}}|\Psi_k\rangle$ is stationary or not, subject to the condition that $\Psi_k$ gives $\rho_e$. But [@gor2; @shjp2; @thesis; @al] shows that different $\Psi_k(\vec r)$s correspond to potentials ${\hat v}^k_{ext}(\vec r)$. The same problem also pervades through the variational $e$DFT approach as proposed by LN [@ln1; @ln2; @shjp2]. Thus unified CS gives, many different wavefunctions $\Psi_k(\vec r)$ and the corresponding external potential ${\hat v}^k_{ext}(\vec r)$ can be associated with a given density. Now if in addition to the excited-state density we also have the ground-state information $\rho_0$, then ${\hat v}_{ext}(\vec r)$ can be uniquely determined out of all possible multiple potentials ${\hat v}^k_{ext}(\vec r)$. Hence with the knowledge of $\rho_0$, it is quite trivial to select a particular $\Psi$ that belongs to a given $\left[\rho_e,\rho_0\right]$ combination by comparing ${\hat v}^k_{ext}(\vec r)$ with the actual ${\hat v}_{ext}(\vec r)$. Alternatively, one can think of it as finding $\Psi$ variationally for a $\left[\rho_e,{\hat v}_{ext}\right]$ combination. Its because the knowledge of $\rho_0$ and ${\hat v}_{ext}$ is equivalent. Now with the above information, the bifunctional $F[\rho_e,
\rho_0]$ can be redefined as $$F[\rho_e,\rho_0] \; = \;
\langle \Psi[\rho_e,\rho_{0}] | \hat{T} \;+\; \hat{V}_{ee} | \Psi[\rho_e,\rho_{0}]\rangle .
\label{cdpm1}$$ The above theoretical formulation is similar to that of LN [@ln1] but avoids the orthogonality constraint imposed by LN formalism. This is because, the densities for different excited states for a given ground-state density $\rho_0$ (that corresponds to a unique external potential ${\hat v}_{ext}$) can be found in following manner: take a density and search for $\Psi$ that makes $\langle\Psi|
\hat T + \hat V_{ee}|\Psi\rangle$ stationary and simultaneously make sure whether the corresponding potential ${\hat w}_{ext}$ $\Big(i.e.~ w_{ext} = -~\frac{\delta F[\rho,\rho_0]}{\delta\rho}{\Big|}_{\rho
= \rho_e} \Big)$ resembles the given $\rho_0$ ( or ${\hat v}_{ext}$); if not, search for another density and repeat the procedure until the correct $\rho$ is found. Thus it is clear that excited state orbitals $\Psi$ are now functional of $[\rho_e,\rho_0]$. So the correct density $\rho$ is excited state density of the potential and the $\Psi$ obtained in this method is also excited state wavefunction corresponding to that potential and density. After finding the correct density $\rho_e$, make a variation over it so that $(\rho_e \to \rho_e + \delta\rho)$ and again perform the CS to find $\Psi[\rho_e + \delta\rho;\rho_0]$. In this case, choose that $({\hat w}_{\text{ext}} + \delta
{\hat w}_{\text{ext}})$ which converges to ${\hat v}_{\text{ext}}$ as $\delta\rho \to 0$.
The above propositions for the excited-states in terms of their densities are quite reasonable, particularly because it’s development is parallel to that for the ground-state DFT. On the other hand, to construct a Kohn-Sham [@ks] system for a given density is not so trivial; and to carry out accurate calculations for excited-states, it is of prime importance to construct a KS system. Further, a KS system will be meaningful if the orbitals involve in an excitation match with the corresponding excitations in the true system. Samal-Harbola [@shjp2] have shown that the KS system constructed using the Levy-Nagy criterion fails in this regard. But using the form of the functional above a KS system can be defined for excited state. Actually, the state dependence of the excited-state exchange-correlation functional [@shjp1; @shamim; @hem1; @hem2; @hem3] leads to the discrepancies while one compares the ground-states either direct or indirect manner. But in principle, obtaining a KS system is plausible. Now by defining the non-interacting kinetic energy $T_s\left[\rho_e,\rho_0\right]$ and using it to further define the exchange-correlation functional as $$E_{xc}[\rho_e,\rho_0] = F[\rho_e,\rho_0] - E_{\text{H}}[\rho_e] - T_s[\rho_e,\rho_0],
\label{cdpm4}$$ solves the purpose. So the Euler equation for the excited-state densities becomes $${v}_{ext} = \mu - \Big\{\frac{\delta T_s\left[\rho_e,\rho_0\right]}{\delta\rho(\vec r)} + {V}_{\text
{H}}[\rho_e] + \frac{\delta E_{xc}\left[\rho_e,\rho_0\right]}{\delta\rho(\vec r)}\Big\}~,
\label{cdpm5}$$ which is equivalent to solving the KS equation $$\left\{- \frac{1}{2}\nabla^2 + {\hat v}_{s}(\vec r) \right\}\Psi_i(\vec r) = \varepsilon_i \Psi_i(\vec r)~,
\label{cdpm6}$$ where $$v_{s}(\vec r) = v_{ext}(\vec r) + \frac{\delta \Big\{F\left[\rho,\rho_0\right] - T_{s}\left[\rho,\rho_0
\right]\Big\}}{\delta\rho(\vec r)}{\Big|}_{\rho(\vec r) = \rho_e[v_{ext}(\vec r)]} ~.
\label{cdpm7}$$ In ground state DFT, one can easily find the $T_s[\rho_0]$ by minimizing the kinetic energy for a given density; here $T_s[\rho_0]$ for a given density is obtained by occupying the lowest energy orbitals for a non-interacting system. But in $e$DFT, to define $T_s\left[\rho_e,\rho_0\right]$ is not easy, as for the excited-states it is not clear which orbitals to occupy for a given density. Particularly because a density can be generated by many different configurations of the non-interacting systems. Levy-Nagy select one of these systems by comparing the ground-state density corresponding to the excited-state non-interacting system with the true ground-state density. However, LN criterion is not satisfactory as pointed out by Samal and Harbola [@shcpl]. The reason of this discrepancy is due to the inconsistency of the ground-state density of an excited state KS system with the true ground-state density. The ground -state density corresponding to the excited-state KS system is not same as the ground-state density of the true system. This means the desired state is not associated with ${\hat v}_{\text{ext}}(\vec r)$, rather it comes from a different local potential ${\hat v}'_{\text{ext}}(\vec r)$. To settle this inconsistency, KS system must be so chosen that it is energetically very close to the original system and it can be ensured through the following criterion. [*Criterion I: the non-interacting kinetic energy*]{} $T_s[\rho_e,\rho_0]$ [*obtained through the CS need to be very close to the actual*]{} $T[\rho_e,\rho_0]$, where $T_s[\rho_e,\rho_0]$ and $T[\rho_e,\rho_0]$ are defined as $$\begin{aligned}
T_s[\rho_e,\rho_0]&=&\min_{\Phi \to \rho_e} \langle\Phi|\hat T + \underbrace{\hat V_{\text{ee}}=0}|
\Phi\rangle\nonumber\\
T[\rho_e,\rho_0]&=&\min_{\Psi \to \rho_e}\langle\Psi|\hat T + \hat V_{\text{ee}} |\Psi\rangle .
\label{cdpm8} \end{aligned}$$ So defining $\Delta T = T - T_s$ smallest not only ensures that DFT exchange-correlation energy remains closer to the conventional quantum mechanical exchange-correlation energy but also keeps the structure of the KS potential appropriate for the desired excited-state which is shown below. Based on the DVT [@hm], it can be argued how for a given density $\rho_e$ one can have different exchange -correlation $\hat v_{xc}$ and external ${\hat v}_{\text{ext}}$ potentials. According to DVT, the exact expression for the gradient of the external potential (for interacting system) for a given excited-state density $\rho_e$ is $$\begin{aligned}
-\nabla {\hat v}_{\text{ext}} &=& -\frac{1}{4\rho_e(\vec r)}\nabla\nabla^2\rho_e(\vec r) +
\frac{1}{\rho_e(\vec r)}\vec Z(\vec r;\varGamma_1(\vec r;\vec r'))\nonumber\\
& + &\frac{2}{\rho_e(\vec r)}\int[\nabla {\hat u}(\vec r,\vec r')]\varGamma_2(\vec r,\vec r')d\vec r'~,
\label{dvt1}
\label{cdpm9}\end{aligned}$$ where ${\hat u} = \frac{1}{|\vec r - \vec r'|}$. This equation represents an exact relation between the gradient of the external potential ${\hat v}_{\text{ext}}$, the $e-e$ interaction potential ${\hat u}(\vec r,\vec r')$ and the density matrices $\rho(\vec r)$, $\varGamma_1(\vec r;\vec r')$ and $\varGamma_2(\vec r,\vec r')$. The vector field $\vec Z$ in Eq.(\[cdpm9\])is related to the kinetic-energy density tensor via $$Z_\alpha[\vec r;\varGamma_1(\vec r;\vec r')] = \Big[\frac{1}{4}\Big(\frac{\partial^2}
{\partial r'_\alpha\partial r''_\beta} + \frac{\partial^2}{\partial r'_\beta\partial r''_\alpha}
\Big)\varGamma_1(\vec r';\vec r'')\Big]_{\vec r'=\vec r''=\vec r}
\label{cdpm10}$$ So, $\vec Z$ can be called a ”local” functional of $\varGamma_1$. Similarly, for KS potential Eq.(\[cdpm9\]) reduces to $$\nabla {\hat v}_{\text{KS}} = -\frac{1}{4\rho_e(\vec r)}\nabla\nabla^2\rho_e(\vec r) + \frac{1}
{\rho_e(\vec r)}\vec Z_{\text{KS}}(\vec r;\varGamma_1(\vec r;\vec r')) .
\label{cdpm11}$$ As a given ground-state density $\rho_0$ fixes the external potential uniquely via HK theorem, which implies that $\rho$, $\varGamma_1$ and $\varGamma_2$ are also fixed from Eq.(\[cdpm9\]). Since the density matrices generated by some eigenfunction $\Psi$ of the Hamiltonian $\hat H$. So the fixed pair of excited-state and ground-state density i.e. $[\rho_e,\rho_0]$ may be arising from different configurations $-$ different configurations can be thought of as arising from different external potential or different exchange-correlation potential and this is due to the different $\varGamma_1$ and $\varGamma_2$ for a fixed $\rho_e$. Suppose a given density $\rho_e$ is generated through an $i^{th}$ KS system, then $$\nabla {\hat v}_{\text{KS}}^i = -\frac{1}{4\rho_e(\vec r)}\nabla\nabla^2\rho_e(\vec r) + \frac{1}
{\rho_e(\vec r)}\vec Z^i_{\text{KS}}(\vec r;\varGamma^i_{1 ({\text{KS}})}(\vec r;\vec r'))~.
\label{cdpm12}$$ If the density is generated through a $j^{th}$ external potential then $$\begin{aligned}
-\nabla {\hat v}_{\text{ext}}^j &=& -\frac{1}{4\rho_e(\vec r)}\nabla\nabla^2\rho(\vec r) + \frac{1}
{\rho(\vec r)}\vec Z^j(\vec r;\varGamma^j_1(\vec r;\vec r'))\nonumber\\
& + &\frac{2}{\rho_e(\vec r)}\int[\nabla u(\vec r,\vec r')]\varGamma^j_2(\vec r,\vec r')d\vec r'~.
\label{cdpm13}\end{aligned}$$ As a matter of which $$\begin{aligned}
- \nabla {\hat v}_{\text{xc}} = \frac{\vec Z_{\text{KS}}(\vec r;\varGamma_1(\vec r;\vec r')) -
\vec Z(\vec r;\varGamma_1(\vec r;\vec r'))}{\rho_e(\vec r)} \nonumber + \\
\frac{\int[\nabla {\hat u}(\vec r,\vec r')][\rho_e(\vec r) \rho_e(\vec r') - \varGamma_2(\vec r,
\vec r')] d\vec r'}{\rho_e(\vec r)}
\label{cdpm-1}\end{aligned}$$ becomes $$-\nabla {\hat v}_{xc}^{ij} = \frac{\vec Z^i_{\text{KS}}-\vec Z^j}{\rho(\vec r)} + \vec\varepsilon_
{\text{xc}}^j~,
\label{cdpm14}$$ where $\vec\varepsilon_{\text{xc}}^j$ is the field due to the Fermi-Coulomb hole of the $j^{th}$ system $[\varGamma_2^j]$ . So the kinetic energy difference between the true system and KS system is given by $$\Delta T = \frac{1}{2}\int\vec r.\Big\{\vec Z_{\text{KS}}\Big(\vec r;[\varGamma_{1({\text{KS}}}]
\Big) - \vec Z\Big(\vec r;[\varGamma_1]\Big)\Big\} d\vec r .
\label{cdpm15}$$ This difference should be kept the smallest for the true KS system so that it gives the KS system consistent with the original system. As a matter of which, we conclude that one way to establish the $\rho_e \Longleftrightarrow {\hat v}_{\text{ext}}$ mapping via the LN formalism [@ln1; @ln2; @ln3] is: if among the several potentials $-$ which have the same excited -state density, one can choose the correct KS potential by comparing the ground-state density i.e. keep that KS-potential whose ground-state density resembles with the true ground-state density. Keeping the ground-state density close we actually keep the external potential fixed via HK theorem. Thus LN criterion is exact for non-interacting system as there is no interaction, so the ground-state density match perfectly.
This proposal of LN for $\rho_e \Longleftrightarrow {\hat v}_{\text{ext}}$ mapping was carried by Samal and Harbola [@shjp2] but they argued in a slightly different way. They proposed that both for interacting and non-interacting case among all the multiple potentials, choose the correct KS potential whose ground-state density differ from the exact ground-state “most closely by least-square sense” which is done in the following manner. If $\rho_0(\vec r)$ is the exact ground state density and $\tilde{\rho_0}(\vec r)$ is that of the KS system (OR the alternate potentials ${\hat w}_{\text{ext}}$) then SH proposition can be further improved intuitively. [*Criterion II: the mean square distance between $\rho_0(\vec r)$ and $\tilde{\rho_0}(\vec r)$ should remain very close to zero*]{}. Thus $$\Delta[\rho_0(\vec r),\tilde{\rho_0}(\vec r)] = \min_{v[\tilde{\rho_0},\rho_e]} \left\{\int_
\infty\left|[\rho_0(\vec r) - \tilde{\rho_0}(\vec r)]\right|^2 d \vec r \right\}^\frac{1}{2} \geq 0~,
\label{cdpm16}$$ where the integration is carried out in the Sobolev space. This criterion is more appropriate in the context of $\rho_e \Longleftrightarrow {\hat v}_{\text{ext}}$ than the one proposed by [@shjp2; @aln]. The criterion as given in Eq.(\[cdpm16\]) will be fully satisfied if one makes use of the excited state functionals [@shjp1; @thesis; @shamim; @hem2; @hem3]. Otherwise it may fail in certain situations as pointed out by SH [@shjp2].
Instead of sticking to the [*Criterion I & II*]{}, one can even go beyond the same through [*Criterion III: compare the ground states of the true and alternate systems energetically*]{}. It can be done in the following manner in order to select the KS system for a given density. The alternative approach is to compare the ground-state expectation value of the KS system and the true system, instead of comparing their ground-state densities and kinetic energies. The procedure for comparing ground-state energy level is the following. First solve the exact DFT equation (say Harbola-Sahni [@hs] etc) for ground-state of the true system and obtain the ground-state of KS Hamiltonian $H_0$. If the expectation value of the ground state Hamiltonian of the true system is $\langle H_0 \rangle_{\text{true}}$ and that of the KS system is $\langle H_0 \rangle_{\text{KS}}$, then one need to choose that KS system whose $\langle H_0 \rangle_
{\text{KS}}$ $\simeq$ $\langle H_0 \rangle_{\text{true}}$. These criteria are well connected to the GAC-KS [@gl1; @gor1; @gor2; @gor3; @gacj; @gacl; @gac1; @gac2; @gac3; @gac4; @gac5; @gac6; @gac7] as discussed below.
Since GAC-KS in principle helps for the self-consistent treatment of excited states and could be considered as a plausible extension of HK theorem to the same. So now the furtherance of the propositions made by SH [@shjp2] as discussed previously will be justified within the GAC-KS. Indeed, relying on the principles of GAC-KS, unified CS formalism along with the SH criteria can also establish the density-to-potential mapping at the strictly correlated regime which will be shown below. In GAC, the $\lambda$ dependent Hamiltonian which is also used in the PL extremum principle is given by $$\hat H_\lambda[\hat v, N] = \hat T + \lambda \hat V_{ee} + \sum_{i=1}^N \hat v(\vec r_i) ,
\label{gaceq1}$$ with the corresponding equation of state $$\hat H_\lambda[\hat v,N] \Psi_\lambda[\hat v,N] = E_\lambda[\hat v,N] \Psi_\lambda[\hat v,N] ,
\label{gaceq2}$$ where $\lambda$ is the coupling constant with $0 \leq \lambda \leq 1$ allowing the electron-electron interaction to be triggered. Unlike the adiabatic connection (AC)-DFT, the external potential $\hat v(\vec r)$, is independent of $\lambda$. Analogous to the Levy-Lieb CS functionals, the GAC for the conjugate density functionals $F_\lambda[\rho]$ (density fixed AC) and $E_\lambda[\hat v]$ (potential fixed AC) are given by $$F_{\lambda = 1}[\rho] = F_{\lambda = 0}[\rho] + \int_0^1 \frac{dF_\lambda[\rho]}{d\lambda}~d\lambda ~,
\label{gaceq3}$$ $$E_{\lambda = 1}[\hat v] = E_{\lambda = 0}[\hat v] + \int_0^1 \frac{dE_\lambda[\hat v]}{d\lambda}~d\lambda ~.
\label{gaceq4}$$ Similar to Eq.(\[gaceq3\]) and (\[gaceq4\]), the excited-state functionals $T_\lambda[\rho,\rho_0]$, $Q^S_\lambda[\rho]$, $F_\lambda[\rho,\rho_0]$ and $E_\lambda[\rho,\rho_0]$ can be defined. Upon finding these $e$DFT functionals, one can define the GAC by starting at a $\rho$ stationary wavefunction for $\lambda = 1$ and then by gradually turning off ($\lambda = 0$) the electron-electron interaction. Thus the $\rho$-stationary wavefunctions for $0 \leq \lambda \leq 1$ will form the GAC in $e$DFT. Since the $\rho$-stationary wave functions for a given $\rho$ are numerable and the adiabatic connections do not overlap with each other, states $\Phi_i$ of non-interacting model systems equals to the $\rho$-stationary wave functions at $\lambda = 0$ (i.e.$\Phi_i = \Psi^S_{\lambda=0}[\rho]$) and can be assigned to real electronic states $\Psi_j = \Psi[\rho,\nu,\alpha=1]$ [@gor2]. These assigned model states are the eigenstates of the GAC-KS formalism. As discussed above, they are eigenstates of a Hamiltonian operator with local multiplicative potential. In this way, the GAC will define the path of going from a non- interacting system to an interacting system via a $\rho-$stationary path. Although for each of the interacting system, one can still end up with multiple non-interacting KS system. But with the criteria discussed previously it’s possible to select the appropriate ones. So once the $\rho \Longleftrightarrow
\hat v_{ext}$ for the interacting system is fixed, it do carries over to the KS system via GAC and vice versa. This shows how the proposed unified CS formalism not only establishes the density-to-potential mapping concretely but also constructs the KS system successfully. In the following sections we will exemplify what we have proposed so far through two model systems. This will be done in order address the critiques about density-to-potential mapping in $e$DFT.
DFT beyond the HK and GL Theorem
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The issue of non-uniqueness in the density-to-potential mapping is also persuaded [@lpls] in the context of GL theorem [@gl1; @gl2]. In [@lpls], it has been demonstrated for higher excited states of the considered $1D$ model system there is no equivalence of the GL/HK theorem. But the critical analysis of [@lpls] presented in this section will outline how the multiplicity of potentials still can’t be ruled out even in the case of ground as well as lowest excited states. So one need to go beyond [@lpls] approach in order to address the validity of HK & GL theorem for such state. In fact, relying on the principles of unified $e$DFT approach [@gor2; @shjp2; @thesis; @al] as discussed in Sec. II along with the proposed criteria of Sec. III, it will be shown here why the claim made in [@lpls] lacks merit to address the excited-state density-to-potential mapping. To validate the density-to-potential mapping (i.e. the analogue of HK/GL theorem) in [@lpls] proposed approach, we will consider as test cases: the examples of the $1D$ QHO with finite boundary and then the infinite potential well.
For clarity in understanding let’s first briefly discuss the theoretical formulation of [@lpls]. The Schrödinger equation of two non-interacting fermions subjected to local one dimensional potentials $v(x)$ and $w(x)$ s.t. $v(x) \neq w(x) + C$, where $C$ is a constant are given by $$\Big[-\frac{1}{2}\frac{d^2}{dx^2} + v(x)\Big]\Phi_i(x) = \varepsilon_i \Phi_i(x)~,
\label{gl1}$$ $$\Big[-\frac{1}{2}\frac{d^2}{dx^2} + w(x)\Big]\Psi_i(x) = \lambda_i \Psi_i(x)~.
\label{gl2}$$ Suppose that the eigenfunctions of the local potential $w(x)$ generates the ground/excited-state eigendensity of $v(x)$ as one of it’s eigendensity but with some arbitrary configuration which is either same or different from the original one. Then one possible way of achieving this is: the wavefunctions $\Psi(x)$ of the potential $w(x)$ can be associated to the wavefunctions $\Phi(x)$ of the potential $v(x)$ via the following unitary transformation i.e. $$\begin{aligned}
\begin{pmatrix}
\Psi_k(x) \\ \Psi_l(x)
\end{pmatrix}
&=&
\begin{pmatrix} \cos\theta(x) & \sin\theta(x) \\ -\sin\theta(x) & \cos\theta(x) \end{pmatrix}
\begin{pmatrix} \Phi_i(x) \\ \Phi_j(x) \end{pmatrix}\nonumber\\
&=&
\begin{pmatrix}
\Phi_i(x)\cos\theta(x)+\Phi_j(x)\sin\theta(x) \\-\Phi_i(x)\sin\theta(x)+\Phi_j(x)\cos\theta(x)
\end{pmatrix} ,
\label{gl3}\end{aligned}$$ As a matter of which the density preserving constraint will be satisfied and the ground/excited state density of two potentials remain invariant i.e. $$\rho(x) = |\Phi_i(x)|^2 + |\Phi_j(x)|^2 = |\Psi_k(x)|^2 + |\Psi_l(x)|^2 .
\label{gl4}$$ Now the potentials can be obtained from the Eqs.(\[gl1\]) and (\[gl2\]) by inverting the same $$\begin{aligned}
v(x) &=& \varepsilon_i + \frac{\ddot{\Phi_i}(x)}{2\Phi_i(x)} = \varepsilon_j + \frac{\ddot{\Phi_j}(x)}
{2\Phi_j(x)}\\
w(x) &=& \lambda_k + \frac{\ddot{\Psi_k}(x)}{2\Psi_k(x)} = \lambda_l + \frac{\ddot{\Psi_l}(x)}{2\Psi_l(x)} ~.
\label{gl5}\end{aligned}$$ Also from Eqs.(\[gl1\]) and (\[gl2\]), the difference between any two eigenvalues $\Delta$ and $\Delta'$ corresponding to the potentials $v(x)$ and $w(x)$ are given by $$\Delta = \varepsilon_j - \varepsilon_i = \frac{1}{2\Phi_i(x)\Phi_j(x)}\frac{d}{dx}[\Phi_j(x)
\dot{\Phi_i}(x) - \Phi_i(x)\dot{\Phi_j}(x)]~,
\label{gl6}$$ $$\Delta' = \lambda_k - \lambda_l = \frac{1}{2\Psi_k(x)\Psi_l(x)}\frac{d}{dx}[\Psi_l(x)\dot{\Psi_k}(x)
- \Psi_k(x)\dot{\Psi_l}(x)]~.
\label{gl7}$$ Now by plugging the values $\Psi_k(x)$ and $\Psi_l(x)$ from Eq.(\[gl3\]) back in Eq.(\[gl7\]), the rotation $\theta(x)$ can be obtained from the following $$\begin{array}{l}
\frac{d}{dx}[\dot{\theta}(x)\{\Phi^2_i(x)+\Phi^2_j(x)\}+\{\Phi_j(x)\dot{\Phi_i}(x)-\Phi_i(x)\dot
{\Phi_j}(x)\}]\\\\
=\Delta'[2\Phi_i(x)\Phi_j(x\cos2\theta(x)+\{\Phi^2_j(x)-\Phi^2_i(x)\}\sin2\theta(x)]
\label{gl8}
\end{array}$$ or $$\rho(x)\ddot{\theta}(x)+\dot{\rho}(x)\dot{\theta}(x)+f(\Phi_i(x),\Phi_j(x),\Delta,\Delta',\theta)=0~,
\label{gl9}$$ where $$\begin{aligned}
f=&2\Delta\Phi_i(x)\Phi_j(x)-\Delta'[\Phi_i(x)\Phi_j(x)\cos2\theta(x)\nonumber\\
&+\{\Phi^2_j(x)-\Phi^2_i(x)\}\sin2\theta(x)]~.
\label{gl10}\end{aligned}$$ The Eq.(\[gl9\]) is the central equation of [@lpls] theoretical framework which need to be solved numerically with proper initial conditions in order to obtain the alternate potential $w(x)$ for any given density and eigenvalue differences. In this work, the adopted numerical procedure to solve the above mentioned differential equation is very much accurate even at the boundary where obtaining appropriate structure and behavior of the multiple potentials and the corresponding wavefunctions are important and crucial.
Results: $1D$ Quantum Harmonic Oscillator
-----------------------------------------
The first model system for demonstrating the density-to-potential mapping is the $1D$ QHO defined by $$\begin{aligned}
v(x) = \frac{1}{2}\omega^2x^2~, {\mbox{where}}~~~-l\leq x \leq l ~.
\label{qhoeu1}\end{aligned}$$ So the wavefunctions and energy eigenvalues of the $n^{th}$ eigenstate are given by $$\begin{aligned}
\Phi_n(x)&=&\left(\frac{\omega}{\pi}\right)^\frac{1}{4}\frac{1}{\sqrt{2^n n!}}H_n(\sqrt{\omega}x)
\exp(-\frac{\omega x^2}{2})~,\label{eq28}\\
\varepsilon_n&=&(n+\frac{1}{2})\omega ~,
\label{qhoeq2}\end{aligned}$$ where $n=0,1,2.....$\
(atomic units are adopted i.e. $\hbar = 1$ and $m_e = 1$)
![Upper panel: Shows (red color) the ground state density of the 1D QHO and the corresponding transformed wavefunctions $\Psi_k$ (blue) and $\Psi_l$ (green) for $\Delta' = 10.0$. Lower panel: Shows the alternate potential associated with above wavefunctions and density.[]{data-label="qhofig0_0-1"}](qho_10_0-0all.eps){width="2.5in" height="3.0in"}
![The figure caption is same as Fig.\[qhofig0\_0-1\] but with $\Delta' = 46.0$.[]{data-label="qhofig0_0-2"}](qho_46_0-0all.eps){width="2.5in" height="3.0in"}
Fermions in The Ground State
----------------------------
Now consider two non-interacting fermions occupying the ground-state of the QHO i.e. $n = 0 = m$. So the eigenvalue difference for this state $\Delta = \varepsilon_0 - \varepsilon_0 = 0$ and the density is given by $$\rho(x) = 2\left(\frac{\omega}{\pi}\right)^\frac{1}{2}\exp(-\omega x^2)~.
\label{qhoeq3}$$ Thus the corresponding equation for rotation $\theta(x)$ can be obtained from the Eq.(\[gl9\]) and is given by $$\rho(x)\ddot{\theta}(x) + \dot{\rho}(x)\dot{\theta}(x) - \Delta'[2\left(\frac{\omega}{\pi}\right)^\frac{1}{2}
\exp(-\omega x^2)\cos2\theta(x)] = 0~.
\label{qhoeq4}$$ Now Eq.(\[qhoeq4\]) has to be solved with proper initial conditions. The initial conditions can be fixed by taking into consideration the symmetry of the differential Eq.(\[qhoeq4\]) and the normalization condition of the wavefunction. From Eq.(\[qhoeq4\]) it is clear that $\frac{d\theta}{dx}|_{(x=0)} = 0$ as both $\Phi(x)$ and $\rho(x)$ are symmetric about $x=0$. Now another condition is that $\Psi_k(x)$ and $\Psi_l(x)$ must also be normalized. So if we plot the renormalization $R$ $$\int^{l}_{-l} |\Psi_{k,l}(x)|^2 dx - 1 = R =0
\label{qhoeq5}$$ as a function of $\theta(x=0)$, then the points where $R = 0$ corresponds to the normalization of $\Psi_k(x)$ and $\Psi_l(x)$ [@lpls] and it will provide the initial condition on $\theta(x=0)$. After finding $\theta(x)$, the transformed set of normalized wavefunctions $\Psi_k(x)$ and $\Psi_l(x)$ is being obtained. Again using these wavefunctions the potential $w(x)$ can be determined from the Eq.(\[gl5\]). In Fig.\[qhofig0\_0-1\] and Fig.\[qhofig0\_0-2\], we have shown two different potentials which are obtained for the eigenvalue differences $\Delta' = 10.0$ and $\Delta' = 46.0$ respectively along with the corresponding wavefunctions. The important point of observation here is that the ground-state density $\rho^{QHO} = \rho_0[v(x)=v_{QHO}(x),N=2]$ now corresponds to some arbitrary excite-state having density $\rho_e[w(x) \neq v_{QHO}(x),N=2]$. As a matter of which, for the fixed $\rho_0^{QHO}$ and $\Delta^\prime$, the system gets transformed to some other system $w(x)$ for which $Q^S[\rho_e[w(x)]=\rho_0^{QHO}]$ and/or $F[\rho_e[w(x)]=\rho_0^{QHO},\tilde{\rho_0}]$ will be stationary. The corresponding stationary states are basically the transformed wavefunctions which are given by Eq.(\[gl2\]) $\Psi_l^S[w(x)] = \Psi_l[\rho_e,\tilde{\rho_0}]$. In this $\tilde{\rho_0}$ is the ground state density of the newly generated potential $w(x)$ and $\tilde{\rho_0} \neq \rho_0^{QHO}$. Now from the proposed criteria it follows that $\Delta T \neq 0$ , $\Delta[\rho_0(x),\tilde{\rho_0}(x)] > 0$ and new system is energetically far off from the original one. Hence the given ground-state density should be uniquely mapped to the original QHO potential $v(x)$ although there exist several multiple potentials $w(x)$. This result is consistent with the generalized/unified CS formalism [@gor2; @shjp2; @thesis; @al].
![The figure caption is same as Fig.\[qhofig0\_0-1\] but for the lowest excited state density being produced with $\Delta' = 15.0$.[]{data-label="qhofig0_1-1"}](qho_15_0-1all.eps){width="2.5in" height="3.0in"}
![The figure caption is same as Fig.\[qhofig0\_1-1\] but with $\Delta' = 35.0$.[]{data-label="qhofig0_1-2"}](qho_35_0-1all.eps){width="2.5in" height="3.0in"}
Fermions in The Lowest Excited State
------------------------------------
As the second example, we consider the lowest excited-state of the QHO. So the two non-interacting fermions are now occupying the $n=0$ and $m=1$ state. For this case, $\varepsilon_0 = \frac{1}{2}\omega$, $\varepsilon_1 = \frac{3}{2}\omega$ and $\Delta = \varepsilon_1 - \varepsilon_0 = \omega$ with the density $$\rho(x) = \left(\frac{\omega}{\pi}\right)^\frac{1}{2}\exp(-\omega x^2)(1 + 2\omega x^2),
\label{qhoeq6}$$ and the corresponding equation for rotation $\theta(x)$ is given by $$\begin{aligned}
\rho(x)\ddot{\theta}(x) + \dot{\rho}(x)\dot{\theta}(x) + 2\omega x\left(\frac{2\omega^2}{\pi}\right)^
\frac{1}{2}\exp(-\omega x^2)\nonumber\\
-\Delta'[2x\left(\frac{2\omega^2}{\pi}\right)^\frac{1}{2}\exp(-\omega x^2)\cos2\theta(x) + \nonumber\\
\frac{\omega}{\pi}\exp(-\omega x^2)\{2\omega x^2 - 1\}\sin2\theta(x)] = 0~.
\label{eq34}\end{aligned}$$ Since in this case $\Phi_0(x)$ is symmetric, $\Phi_1(x)$ is antisymmetric, so $\rho(x)$ symmetric around $x=0$. Thus Eq.(\[eq34\]) implies that $\theta(x)$ should be symmetric at $x=0$. The initial conditions on $\frac{d\theta}{dx}|_{(x=0)}$ is obtained from the behavior of the renormalization $R$ as a function of $\frac{d\theta}{dx}|_{(x=0)}$. Following the same procedure as before, in this case also we have obtained different potentials for the fixed lowest excited state density which are shown in the Fig.\[qhofig0\_1-1\] and Fig.\[qhofig0\_1-2\]. These two alternative potentials and the transformed wavefunctions correspond to two different eigenvalue differences $\Delta' = 16.0$ and $\Delta' = 35.0$. As described in the ground state case, in this case also the structure of the potential is different from the original 1D QHO as the potential should follow the structure of the wavefunctions. However, according to the unified CS $e$DFT, the results are never due to the violation of GL theorem. This is because the ground and lowest excited states of the newly found potential are quite different from that of the QHO. So following similar argument as in the previous case, now the lowest excited-state density of the QHO corresponds to some different eigendensity of the multiple potentials. Thus the multitude of potentials poses no issues for the validity of the GL theorem.
![The figure caption is same as Fig.\[qhofig0\_0-1\] but for one of the higher excited state density being produced with $\Delta' = 8.0$.[]{data-label="qhofig0_2-1"}](qho_8_0-2all.eps){width="2.5in" height="3.0in"}
![The figure caption is same as Fig.\[qhofig0\_2-1\] but with $\Delta' = 30.0$.[]{data-label="qhofig0_2-2"}](qho_30_0-2all.eps){width="2.5in" height="3.0in"}
![Upper panel: Shows the alternate wavefunctions $\Psi_k$ and $\Psi_l$ (green & red) resulting the ground state density of 1D potential well for $\Delta' = 200.0$. Lower panel: Shows the alternate potentials (green & red) and the density (magenta) associated with above wavefunctions.[]{data-label="pbfig2-1"}](well_1.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig2-1\] but with $\Delta' = 600.0$.[]{data-label="pbfig2-2"}](well_2.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig2-1\] but with $\Delta' = 1000.0$.[]{data-label="pbfig2-3"}](well_3.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig2-1\] but for the lowest excited state density being produced with $\Delta' = 200.0$.[]{data-label="pbfig3-1"}](well_4.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig3-1\] but with $\Delta' = 600.0$.[]{data-label="pbfig3-2"}](well_5.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig3-1\] but with $\Delta' = 1000.0$.[]{data-label="pbfig3-3"}](well_6.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig2-1\] but for one of the higher excited state density being produced with $\Delta' = 200.0$.[]{data-label="pbfig4-1"}](well_7.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig4-1\] but with $\Delta' = 600.0$.[]{data-label="pbfig4-2"}](well_8.eps){width="3.0in" height="3.5in"}
![The figure caption is same as Fig.\[pbfig4-1\] but with $\Delta' = 1000.0$.[]{data-label="pbfig4-3"}](well_9.eps){width="3.0in" height="3.5in"}
Fermions in Higher Excited States
---------------------------------
Here we consider one of the higher excited-state of 1D QHO (i.e. two non-interacting fermions are in the $n=0$ and $m=2$ states). For this case, the eigenvalue difference is $\Delta = \varepsilon_2 - \varepsilon_0
= 2\omega$ and the density corresponding to it is given by $$\rho(x) = \left(\frac{\omega}{\pi}\right)^{\frac{1}{2}}\exp(-\omega x^2)\{1 + (1 - 2\omega x^2)^2\}~.
\label{qho02-1}$$ Similarly, the corresponding equation for rotation $\theta(x)$ is given by $$\begin{aligned}
\rho(x)\ddot{\theta}(x) + \dot{\theta}(x)\dot{\theta}(x) + 4\omega\left(\frac{\omega}{2\pi}\right)^
{\frac{1}{2}}(2\omega x^2 -1)\exp(-\omega x^2) \nonumber\\
-\Delta'[\left(\frac{\omega}{2\pi}\right)^{\frac{1}{2}}(2\omega x^2 -1)\exp(-\omega x^2) \cos2\theta(x)
\nonumber\\
+ \left(\frac{\omega}{\pi}\right)^{\frac{1}{2}}\exp(-\omega x^2)\{\frac{1}{2}(2\omega x^2 - 1)^2 - 1\}
\sin2\theta(x)] = 0~.\nonumber\\
\label{qho02-2}\end{aligned}$$ Now by solving Eq.(\[qho02-2\]) for rotation $\theta(x)$ in analogous with the ground-state of the QHO and after taking care of the normalization of the transformed wavefunctions, the potential $w(x)$ is obtained for $\Delta' = 8.0,~30.0$. The potentials along with the wavefunctions are shown in Fig. \[qhofig0\_2-1\] & Fig.\[qhofig0\_2-2\]. Similar to ground and lowest excited-state, here too the given density is a different eigendensity of the new potentials. If it would have the same eigendensity of $w(x)$ then $w(x)$ should have been identical to the $v_{QHO}(x)$. But it is not the case. That’s why the generated potentials are completely different from the QHO.
Results: $1D$ Infinite Potential Well
-------------------------------------
As our second case study, we consider the model system same as that reported in [@lpls] (i.e. particles are trapped inside an $1D$ infinite potential well). For an infinite potential well with length varying from $0$ to $1$, the $n^{th}$ eigenfunction $\Phi_n(x)$ and the energy eigenvalue $\varepsilon_n$ are given by $$\Phi_n(x) = \sqrt{2}\sin (n\pi x) ~;~ \varepsilon_n = \frac{n^2\pi^2}{2}~,
\label{pbeq1}$$ where $n=1, 2, 3 ....$. The density $\rho(x)$ corresponding to the two potentials $v(x)$ and $w(x)$ is given by Eq.(\[gl4\]).
Fermions in The Ground State
----------------------------
For two spinless non-interacting particles in $n=1=m$ states, the energies of two states and the difference are $$\varepsilon_1 = \frac{\pi^2}{2} = \varepsilon_2 ~;~ \Delta = \varepsilon_2 - \varepsilon_1 = 0~.
\label{pbeq3}$$ The density corresponding to these states is $$\rho(x) = 4[\sin^2(\pi x)]~,
\label{pbeq5}$$ and the equation corresponding to Eq.(\[gl9\]) for the rotation $\theta(x)$ is $$\rho(x)\ddot{\theta}(x) + \dot{\rho}(x)\dot{\theta}(x) - \Delta'[4\sin^2\pi x \cos2\theta(x)] = 0~.
\label{pbeq6}$$ Since $\Phi_1(x)$ is symmetric and $\rho(x)$ is symmetric about $x=\frac{1}{2}$. Thus Eq.(\[pbeq6\]) indicates that $\theta(x)$ should be symmetric such that $\dot{\theta}(\frac{1}{2})=0$. With this initial condition and choosing any value of $\Delta'$ one can solve for $\theta(x)$ and subsequently obtain the $\Psi_k$ s. Now using these $\Psi_k$ s, the alternate potentials $w(x)$ will be obtained by using Eq. (\[gl5\]). Since the transformed wavefunction $\Psi_k(x)$ must also be normalized. This condition will be fulfilled by choosing the appropriate value of $\theta(\frac{1}{2})$ at which the $\Psi_k(x)$ should be normalized. Once $\Psi_k(x)$ is normalized then $\Psi_l(x)$ will also be normalized. Again by adopting the same procedure as that described in the case of $1D$ QHO, the alternative multiple potentials are obtained by making use of the following renormalization $R$ condition $$\int^1_0 |\Psi_k(x)|^2 dx - 1 = R = 0~.
\label{pbeq7}$$ All the wavefunctions, densities and multiple potentials are shown in the Figs.(\[pbfig2-1\] $to$ \[pbfig2-3\]). Here we have generated the multiple potentials for $\Delta' = 200.0,~600.0~\&~1000.0$ respectively. As is expected, the wavefunctions are totally different from the ground state of the $1D$ infinite well. Although the density remains to be the same in all the cases. But its not the ground state eigendensity of the multiple potentials. So this poses no issue for the HK theorem.
Fermions in The Lowest Excited State
------------------------------------
Now consider two fermions occupying the $n=1,~m=2$ (i.e. the lowest excited-state) eigenstates of the infinite potential well. Here too we have obtained several multiple potentials unlike [@lpls]. For this excited-state, the energy eigenvalues are $\varepsilon_1 = \frac{\pi^2}{2}$, $\varepsilon_2 =
2\pi^2$ with $\Delta = \frac{3\pi^2}{2}$. Hence the density arising from these two states is given by $$\rho(x) = 2[\sin^2(\pi x) + \sin^2(2\pi x)]~.
\label{pbeq8}$$ Similar to the previous examples, the equation for the rotation $\theta(x)$ is the following $$\begin{aligned}
\rho(x)\ddot{\theta}(x) + \dot{\rho}(x)\dot{\theta}(x) + 6\pi^2\sin(\pi x)\sin(2\pi x)\nonumber\\
-\Delta'[4\sin(\pi x)\sin(2\pi x)\cos2\theta(x)\nonumber\\
+ 2\{\sin^2(2\pi x) - \sin^2(\pi x)\}\sin2\theta(x)] = 0~.
\label{pbeq9}\end{aligned}$$ Here $\Phi_1(x)$ is symmetric, $\Phi_2(x)$ is antisymmetric and $\rho(x)$ symmetric about $x = \frac{1}
{2}$. Thus Eq.(\[pbeq9\]) predicts that $\theta(x)$ is antisymmetric such that $\theta(\frac{1}{2}) = 0$. In this case also normalization of both $\Psi_k(x)$ and $\Psi_l(x)$ are taken care and the proper $R$ (renormalization) values are obtained w.r.t. $\frac{d\theta}{dx}(\frac{1}{2})$. Quite interestingly, in this case also we have successfully generated multiple potentials for $\Delta' = 200.0,~600.0~\&~1000.0$. This is where [@lpls] failed to explain the validity of GL theorem. As expected, the potential follows the wavefunctions pattern. This is obvious at the boundary where the wavefunctions are perfectly vanishing, the potential shoots up to a very large positive value. The potentials along with wavefunctions are shown in the Figs.(\[pbfig3-1\] $to$ \[pbfig3-3\]). Following the same argument as in the case of the previous model system, the multiplicity of potentials obtained here are nothing to do with the GL theorem.
Fermions in Higher Excited States
---------------------------------
Now to complete our exploration on $1D$ well, we have considered here the second excited-state of it. This is the only excited-state for which [@lpls] reported multiple external potentials for various eigenvalue differences. We too generated multiple potentials and the corresponding wavefunctions for $\Delta' = 200.0,~600.0~\&~1000.0$ which are shown in the Figs.(\[pbfig4-1\] $to$ \[pbfig4-3\]). The results follow the trend similar to that of the ground and lowest excited-state. In all the cases, we have noticed that the potentials and the corresponding rotation angles can never attain flat structure at the boundary unlike [@lpls].
![(a) $\rho_e[n_1(2),n_2(1),n_4(1)]$ is the excited-state density of $1D$ potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$ is the ground-state density of potential $V_1$ whose excited-state configuration $[n_1(2),n_2(1),n_3(1)]$ results the same $\rho_e$. (b) $V_2[\rho_e]$ is the potential whose ground-state configuration results the same $\rho_e$ of (a) and is shown along with $V_1[\tilde{\rho}^{(1)}_0]$. (c) $\rho_e[n_1(2),n_3(1),n_4(1)]$ is the excited-state density of $1D$ potential well with ground-state $\rho_0$ and produced in an alternative configuration $[n_1(2),n_2(1),n_4(1)]~(V_1[\tilde{\rho}^{(1)}_0])$ besides the ground-state configuration leading to $V_2[\rho_e]$. (d) Shows all the alternative potentials of (c).[]{data-label="csfig1"}](exct_1exct_2.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e[n_1(1),n_2(2),n_4(1)]$ is the excited-state density of $1D$ potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$ and $\tilde{\rho}^{(2)}_0$ are the ground -state densities of $V_1$ and $V_2$ whose excited-state configurations $[n_1(2),n_2(1),n_3(1)]$ and $[n_1(2),n_2(1),n_4(1)]$ results the same $\rho_e$. (b) $V_3[\rho_e]$ is the potential whose ground-state configuration gives the same $\rho_e$ of (a) and is shown along with $V_1$, $V_2$. (c) $\rho_e[n_2(2),n_3(1),n_4(1)]$ is the excited-state density produced in alternative configuration $[n_1(2),n_2(1),n_3(1)]~(V_1[\tilde{\rho}^{(1)}_0])$, besides the ground-state configuration leading to $V_2[\rho_e]$. (d) Shows all the alternative potentials in (c).[]{data-label="csfig2"}](exct_3exct_4.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e[n_1(1),n_3(2),n_4(1)]$ is the excited-state density of $1D$ infinite potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$ and $\tilde{\rho}^{(2)}_0$ are the ground-state densities of $V_1$ and $V_2$, whose excited-state configurations $[n_1(2),n_2(1),n_3(1)]$ and $[n_1(2),n_3(1),n_4(1)]$ results the same $\rho_e$. (b) $V_3$ is the potential whose ground-state density is same as $\rho_e$ of (a) and is shown along with $V_1$, $V_2$. (c) $\rho_e[n_2(1),n_3(2),
n_4(1)]$ is the excited-state density produced via the alternative configurations $[n_1(2),n_2(1),
n_3(1)]~(V_1[\tilde{\rho}^{(1)}_0])$ besides the ground-state configuration leading to $V_2[\rho_e]$. (d) Shows all the alternative potentials of (c).[]{data-label="csfig3"}](exct_5exct_6.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e[n_1(1),n_2(1),n_4(2)]$ is the excited-state density of $1D$ infinite potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$, $\tilde{\rho}^{(2)}_0$ and $\tilde{\rho}^
{(3)}_0$ are the ground-state densities of $V_1$, $V_2$ and $V_3$ whose excited-state configurations $[n_1(2),n_2(1),n_3(1)]$,$[n_1(2),n_2(1),n_4(1)]$ and $[n_1(2),n_4(2)]$ results the same $\rho_e$. (b) $V_4$ is the potential whose ground-state density is same as $\rho_e$ of (a) and is shown along with $V_1$, $V_2$ and $V_3$. (c) $\rho_e[n_1(1),n_3(1),n_4(2)]$ is the excited-state density produced in the alternative configurations $[n_1(2),n_2(1),n_3(1)]~(V_1[\tilde{\rho}^{(1)}_0])$, $[n_1(2),n_2(1),n_4(1)]~(V_2[\tilde{\rho}^{(2)}_0])$ and $[n_1(2),n_3(1),n_4(1)]~(V_3[\tilde{\rho}
^{(3)}_0])$ besides the ground-state configuration leading to $V_4[\rho_e]$. (d) Shows all the alternative potentials of (c).[]{data-label="csfig4"}](exct_7exct_8.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e[n_2(1),n_3(1),n_4(2)]$ is the excited-state density of $1D$ infinite potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$ and $\tilde{\rho}^{(2)}_0$ are the ground state densities of $V_1$ and $V_2$, whose excited-state configurations $[n_1(2),n_2(1),n_3(1)]$ and $[n_1(2),n_2(1),n_4(1)]$ results the same $\rho_e$. (b) $V_3$ is the potential whose ground state density is same as $\rho_e$ of (a) and is shown along with $V_1$, $V_2$. (c) $\rho_e[n_1(2),
n_4(2)]$ is the excited-state density produced in alternative configurations $[n_1(2),n_2(1),n_3(1)]
~(V_1[\tilde{\rho}^{(1)}_0])$ and $[n_1(2),n_2(1),n_4(1)]~(V_2[\tilde{\rho}^{(2)}_0])$ besides the ground-state configuration leading to $V_3[\rho_e]$. (d) Shows all the alternative potentials of (c).[]{data-label="csfig5"}](exct_9exct_10.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e[n_2(2),n_3(2)]$ is the excited-state density of $1D$ infinite potential well with ground-state $\rho_0 $. $\tilde{\rho}^{(1)}_0$ and $\tilde{\rho}^{(2)}_0$ are the ground-state densities of $V_1$ and $V_2$, whose excited-state configurations $[n_1(2),n_2(1)
,n_4(1)]$ and $[n_1(2),n_4(2)]$ results the same $\rho_e$. (b) $V_3$ is the potential whose ground-state density is same as $\rho_e$ of (a) and is shown along with $V_1$, $V_2$. (c) $\rho_e[n_1(1),n_2(1),n_3(1),n_4(1)]$ is the excited-state density produced in alternative configurations $[n_1(2),n_2(1),n_3(1)]~(V_1[\tilde{\rho}^{(1)}_0])$ and $[n_1(2),n_2(1),n_4(1)]
~(V_2[\tilde{\rho}^{(2)}_0])$ besides the ground-state configuration leading to $V_3[\rho_e]$. (d) Shows all the alternative potentials of (c).[]{data-label="csfig6"}](exct_11exct_12.eps){width="3.3in" height="3.0in"}
![(a) $\rho_e^{(1)} [n=0, n=3]$(both half-filled) , $\rho_e^{(2)} [n=1, n=2]$ (both half filled) and $\rho_e^{(3)} [n=2, n=3]$(both half filled) are the excited-state densities of the potential $V$ produced as the ground state density of the potentials $V_1$, $V_2$ and $V_3$. (b) Shows all the four potentials $V$, $V_1$, $V_2$ and $V_3$ of (a). (c) $\rho_e [n=0, n=3]$ (both half filled) is the excited-state density of the potential $V$ produced in an alternative excited state configuration $[n=0, n=2]~(V_1)$. (d) Shows both the potentials of (c).[]{data-label="csfig7"}](qho_1qho_2.eps){width="3.0in" height="3.0in"}
To conclude this section, we would like to shed some light on the structure of the generated potentials at the boundary as its very important to be determined accurately. Since the wavefunctions die out towards the boundary. Thus the potentials obtained by the Schrödinger equation inversion (i.e. Eq.\[gl5\]) for specified eigenvalue differences will attain large positive value. Actually, in our approach we have gone way beyond [@lpls] to generate the accurate structure of the potential which is clear from the results. The important point to be noted is that the singularity of the potential plays the most crucial role if one directly solving the Schrödinger equation. But in getting the potential structure whether by inverting the Schrödinger equation or CS method solely depends on the wavefunction behavior in a given domain. So better access of the wavefunction’s behavior will by default lead to reliable potential structure.
Results within the CS formalism
===============================
In this section, we will discuss the results in connection with the density-to-potential mapping based on the CS-formalism discussed earlier. According to it, there exist multiple potentials for a given ground or excited state (eigen)density. But for the case of excited state density, when it is produced as some different excited-state of these multiple potentials (except the actual one) the corresponding ground-states are completely different from that of the original system. Similarly, one can produce potentials whose ground-state density may be same as the excited-state density of the original system. The results we have obtained for the systems of our study are fully consistent with the unified CS $e$DFT. The Zhao-Parr [@zp] CS method is being used to show the multiplicity of potentials for a given density.
To begin the CS exemplification (shown in Fig.\[csfig1\]), lets consider $four$ non-interacting particles in an $1D$ potential well, where two fermions are in $n=1$ state and one fermion each in $n=2$ and $n=4$ state. As a result, this gives some excited state density $\rho_e(x)$ associated with the above configuration which is shown in the Fig.\[csfig1\](a) and is given by $$\rho_e(x) = \rho_e^{\text{V}_0}(x) = 2|\Psi_1(x)|^2 + |\Psi_2(x)|^2 + |\Psi_4(x)|^2~,
\label{cseq1}$$ where $\Psi_i(x)$ s are the wavefunctions of the $1D$ potential well. In all our results shown in the figures (\[csfig1\]) to (\[csfig7\]), we have adopted notation $\rho(n_i(f_j))$, where $n_i$ denotes the quantum number of the eigenfunctions of the potential $V$ or $V_i$ ($i = 1, 2, 3, 4$) and $f_j$, the occupation. Using CS [@zp] the excited state density $\rho_e(x)$ given by Eq.(\[cseq1\]) is produced through another alternative potential $V_1$ (say) whose $n=1$ state is occupied with $2$ fermions (i.e. $f_1=2$) and $n=2, n=3$ with one fermion each (i.e. $f_2 = 1 = f_3$). Now the ground state density of the potential $V_1$ is different from that of the $V_0$ (i.e. particle in an infinite potential well) which is given by $\tilde{\rho}^{(1)}_0$ (Fig.\[csfig1\]a). As per our formalism, there can be many such multiple potentials having the given density as it’s eigendensity associated with some combination of eigenfunctions. So it is possible that one can also obtain second alternative potential $V_2$ (say) whose ground-state density is same as the above excited state density ($\rho_e(x)$) of the original system ($V_0$). In this way, we have studied six such excited states of the $1D$ potential well (Figs.\[csfig1\] to \[csfig6\]) and for each case we are able to produce symmetrically different multiple potentials for fix densities. Also in each case, we have produced the alternative potential whose ground-state density is nothing but the given excited-state density of the original configuration (i.e. $1D$ potential well).
As our final case study, we have considered the excited-states of the $1D$ QHO. This is also an interesting model system like the potential well. The results for this case, are shown in Fig.\[csfig7\]. Now consider the Fig.\[csfig7\]a, in this case we have produced three symmetrically different alternative potentials $V_1$, $V_2$ and $V_3$ (shown in Fig.\[csfig7\]b) whose ground-states densities (i.e. $\rho_0^{(1)}(x),
\rho_0^{(2)}(x)$ and $\rho_0^{(3)}(x)$)are same as the different excited-states densities (i.e. $\rho_e^{(1)}
(x), \rho_e^{(2)}(x)$ and $\rho_e^{(3)}(x)$) of the QHO potential $V(x)$. Here $\rho_e^{(1)}(x)$ corresponds to the configuration \[$n=0 (f_0=1), n=3 (f_3=1)$\]. Similarly, $\rho_e^{(2)}(x)$ and $\rho_e^{(3)}(x)$ are arising from the excited-state configurations \[$n=1 (f_1=1), n=2 (f_2=1)$\] and \[$n=2 (f_2=1), n=3 (f_3=1)$\] respectively. In Fig.\[csfig7\](d), we have produced a different potential $V_1$ whose excited-state density corresponding to the configuration \[$n=0 (f_0=1), n=2 (f_2=1)$\] is same as the excited-state density $\rho_e(x)$ (\[$n=0 (f_0=1), n=3 (f_3=1)$\]) of the original $1D$ QHO potential. Although we have produced so many potentials, but our criteria will only select the original potentials (i.e. the infinite potential well in the previous and QHO in the current study) for any given (i.e. either ground or excited-state) density. Thus establishes the excited-state $\rho(x) \Longleftrightarrow {\hat v}(x)$ mapping uniquely.
Discussions
===========
Now the conceptually basic questions of $e$DFT: what are the consequences as well as similarities and differences between the results of the CS formalism and that obtained in connection to the HK/GL theorem? Secondly, whether there arisen any critical scenario which is inconsistent with the HK and /or GL theorem(s)? This is because several multiple potentials are obtained for non-interacting fermions in the ground as well as lowest excited state. Not only that, [@gb; @lpls] have also claimed that for higher excited-states there is no analogue of HK theorem. So the seemingly contradictory results may give rise to the wrong conclusion about the validity of HK/GL theorem and non-existence of density-to- potential mapping for excited-states. However, the generalized/unified CS formalism overrules all these claims by showing that the ground-state density of a given symmetry (potential) can be the excited-state density of differing symmetry (potential). Now this excited-state will have a corresponding ground state which will be obviously quite different from the ground-state of the original system. As a matter of which there will exist a different potential according to HK theorem. This is also true for the excited -state density of the actual system: when it becomes either the ground-state or some arbitrary excited -state density of another potential. So the unified CS formalism justifies the non-violation of HK/GL theorem for such states.
Now based on the unified/generalized CS $e$DFT, one can very nicely interpret ours as well as [@lpls] results. Actually by keeping the excited/ground state density fix via a unitary transformation never guarantee the symmetries of the states involve will remain intact. This is because by changing the $\Delta'$ value and keeping either ground or the excited state density fix, we are forcing the system to change itself accordingly without hindering only the fixed density constraint. Since $\Delta'$ is not fixed. So in principle one can make several choices for $\Delta'$ and for each choice, the system will converge to different potentials (systems/configurations) which can give the desired density of ground/ excited-state of the original system (potential/configurations) as one of it’s eigendensity. Actually, the converged potentials are those for which the Görling and LN functionals are stationary and minimum respectively. So everything is again automatically fits into realm of generalized CS formalism and nothing really contradicting or posing issues for the $e$DFT formulations provided by [@gor1; @gor2; @gor3; @ln1; @ln2; @ln3; @shjp2; @thesis; @al; @aln]. Also the transformed quantum states leading to multitude of potential for a given density are energetically far off from the actual system and even the ground-states are also very different. Thus the generalized CS formalism proposed in this work along with the SH criteria can be considered as the most essential steps for establishing the $\rho(\vec r) \Longleftrightarrow {\hat v}_\text{ext}(\vec r)$ which further elaborated below.
Now the question is out of these existing multiple potentials in association with a fix density and $\Delta'$, which potential in principle should be picked in view of the $\rho(x) \Longleftrightarrow
{\hat v}(x)$? The criteria of selecting the exact potential out of all possibilities have already been discussed in Sec.III. First of all it is quite obvious from the Figs.(\[qhofig0\_0-1\] $to$ \[qhofig0\_2-2\]) and from Fig.\[pbfig2-1\] $to$ Fig.\[pbfig4-3\] that the ground-state densities of the generated alternate potentials are different from that of the original potential. This is also true even for the results of the CS formalism as shown in the Figs.(\[csfig1\] $to$ \[csfig7\]). So when we are fixing the excited-state density at the same time we should have taken care of the ground-state of the newly found system and the old one. Similarly, when several multiple potentials are generated for a given ground-state density, the same is not produced as the ground state eigendensity of the alternate potentials. So it’s quite obvious that there is no violation of the HK theorem. The criteria of taking care of the ground-states of the two system is given in Eq.(\[cdpm16\]). Additionally the kinetic energies of the two systems need to be kept closest, which we have pointed out on the basis of DVT. So in all the non-interacting model systems reported here, $\Delta T$ should have been zero. But the drastically differing structures of the transformed and original wavefunctions are nothing but the manifestation of non-vanishing difference of kinetic energies and thus leading to the multiple potentials. Furthermore, the most significant differences between the symmetries of the old and new systems implies that principally there exist discrepancies in the expectation values of the Hamiltonian w.r.t. the ground-states of various multiple potentials. This is what trivially follows from the reported results. Hence, the proposed criteria uniquely maps a given density of the $1D$ QHO/infinite well to a potential which is nothing but the $1D$ QHO/infinite well and discards rest of the multiple potentials.
Summary and CONCLUDING REMARKS
==============================
In this work, we have tried to obtain a consistent theory for $e$DFT based on the stationary state, variational and GAC formalism of modern DFT. We have provided a unified and general approach for dealing with excited-states which follows from previous attempts made by Perdew-Levy, Görling, Levy-Nagy-Ayers and in particular the work of Samal-Harbola in the recent past. In this current attempt, we have answered the questions raised about the validity of HK and GL theorems to excited-states. We have settled the issues by explaining why there exist multiple potentials not only for higher excited states but also for the ground as well as lowest excited state of given symmetry. In fact, the existing $e$DFT formalism allows the above possibility and at the same time keeps the uniqueness of density-to-potential mapping intact. So we have established in a rigorous fundamental footing the non-violation of the HK and GL theorem. Actually, the generalized CS approach gives us a strong basis in choosing a potential out of several multiple potentials for a fixed ground/excited state density. In our propositions, we have strictly defined the bi-density functionals for a fix pair of ground and excited-state densities in order to establish the density-to-potential mapping. Not only that, the theory also gives us a clear definition of excited-state KS systems through the comparison of kinetic and exchange-correlation energies w.r.t. the true system. It does takes care the stationarity and orthogonality of the quantum states. So everything fits quite naturally into the realm of modern DFT.
To conclude, we have demonstrated density-to-potential mapping for non-interacting fermions. For interacting case the GAC can be used to formulate all the theoretical and numerical contents in a similar way. We are working along this direction for strictly correlated fermions and the results will be reported in future. Finally, our conclusion is that nothing really reveals the manifestation of the failure or violation of the basic theorems and existing principles of modern DFT irrespective of the states under consideration. The method presented by Samal-Harbola and further progress being made here provides a most suitable framework and starting ground for the development of new density -functional methods for the self-consistent treatment of excited states. More realistically, the unified CS $e$DFT and further extensions to the SH criteria treat both the ground or excited states in an analogous manner. Hence, the present work endows the uniqueness of density-to-potential mapping for excited-states with a firm footing.
ACKNOWLEDGMENTS
===============
The authors thankfully acknowledge valuable discussions with Prof. Manoj K. Harbola and M. Hemanadhan.
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---
abstract: 'For a class of convection-diffusion equations with variable diffusivity, we construct third order accurate discontinuous Galerkin (DG) schemes on both one and two dimensional rectangular meshes. The DG method with an explicit time stepping can well be applied to nonlinear convection–diffusion equations. It is shown that under suitable time step restrictions, the scaling limiter proposed in \[Liu and Yu, SIAM J. Sci. Comput. 36(5): A2296–A2325, 2014\] when coupled with the present DG schemes preserves the solution bounds indicated by the initial data, i.e., the maximum principle, while maintaining uniform third order accuracy. These schemes can be extended to rectangular meshes in three dimension. The crucial for all model scenarios is that an effective test set can be identified to verify the desired bounds of numerical solutions. This is achieved mainly by taking advantage of the flexible form of the diffusive flux and the adaptable decomposition of weighted cell averages. Numerical results are presented to validate the numerical methods and demonstrate their effectiveness.'
address:
- 'Tsinghua University, Yau Mathematical Sciences Center, Beijing, China 100084'
- 'Iowa State University, Mathematics Department, Ames, IA 50011'
author:
- Hui Yu and Hailiang Liu
title: 'Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity'
---
Introduction
============
In this paper, we construct and analyze third order maximum-principle-satisfying (MPS) discontinuous Galerkin (DG) schemes for the following problem, $$\label{md}
\left\{
\begin{array}{ll}
M(x) \partial_t u + \nabla\cdot f(u) = \nabla \cdot(A\nabla u), & x\in \mathbb{R}^d, \; t>0,\\
u(0,x)=u_0 (x), & x\in \mathbb{R}^d.
\end{array}
\right.$$ This equation can be considered as a model for numerous physical problems. In this equation, $u=u(t, x)$ is a time-dependent unknown scalar function, $f(u)$ is the smooth vector flux, and $d$ is the spatial dimension. We assume that the diffusion tensor $A=A(x, u)$ is a symmetric and nonnegative definite matrix, and $M(x)$ is strictly positive scalar function. In this paper, we present our results and analysis for $d=1, 2$, extension to three dimension can be done as well.
Our concern for (\[md\]) arises from several typical scenarios. The first case with $f=0$ is a model for heat conduction in a non-uniform body with $M(x)=\frac{c(x)}{\rho(x)}$, where $c(x)$ is the specific heat and $\rho(x)$ the mass density. The heat flux is proportional to the temperature difference $-A \nabla u$, known as the Fourier law of heat conduction. Here $A$ measures the ability of the material to conduct heat, called the thermal conductivity. In the second case with $M=1$, we have the usual convection-diffusion equation, $$\label{conv_dif}
\partial_t u + \nabla \cdot f(u) = \nabla \cdot(A(x, u) \nabla u).$$ There are many interpretations and derivations from fluid dynamics and other application areas that motivate the convection-diffusion equation, such as Navier-Stokes equations and the porous medium equation. The third case is the Fokker-Planck equation, $$\begin{aligned}
\label{FP}
\partial_t \rho =\nabla \cdot(\nabla \rho +\nabla V(x) \rho),\end{aligned}$$ which can be rewritten in terms of $u=\rho e^{V(x)}$ as (\[md\]) with $A(x) = M(x)=e^{-V(x)}$, and $f=0$. This setting allows for many variants.
The above three types of problems can be formulated under the model class (\[md\]). One important solution to (\[md\]) is the one that is bounded by two constants dictated by the initial data, leading to the so-called Maximum Principle (MP). In other words, if $$c_1 = \min_{x} u_0, \quad c_2 = \max_{x} u_0,$$ then $u(t,x) \in [c_1, c_2]$ for any $x\in \mathbb{R}^d$ and $t>0$.
From the analytical viewpoint, the maximum principle is quite general but very significant due to its physical implications. From the numerical viewpoint, it is widely recognized that maximum principle provides a valuable tool in proving solvability results (existence and uniqueness of discrete solutions), enforcing numerical stability, and deriving convergence results (a priori error estimates) for the sequence of approximate solutions. Design of a high order scheme to preserve the maximum principle is known a challenging task. Our goal is to better understand how a high order DG scheme can be constructed for (\[md\]) to respect the MPS property. The main difficulty in the anisotropic case with a variable weight $M(x)$ is the derivation of suitable sufficient conditions so that the weighted cell average stays in $[c_1,c_2]$ during the time evolution. Such weighted cell average is essentially used for limiting the numerical solution into $[c_1, c_2]$, without destroying accuracy.
Related work
------------
Early discussion of the discrete maximum principle for the convection-diffusion equations includes the linear finite element solutions for parabolic equations [@Fu73], and recent developments [@FHK05; @FH06; @TW08; @VKH10; @FK12], as well as [@MH85] by the Petrov–Galerkin finite element method to solve convection dominated problems. However, they are under a different framework.
The present investigation involves the choice of numerical fluxes and monotonicity of weighted cell averages in terms of point values. In the largest sense, the origins of these ideas go all the way back to monotone schemes for hyperbolic conservation laws.
Indeed, for scalar conservation laws, i.e., (\[conv\_dif\]) with $A=0$, many first order classical schemes can be shown to be MPS (other names of this sort include bound-preserving, positivity preserving, or maximum-principle-preserving) since such low order accurate schemes are usually monotone. On the other hand, the Godunov Theorem states that a linear monotone scheme is at most first order accurate for the convection equation [@Le92]. To construct high order accurate MPS schemes for scalar convection, weak monotonicity in finite volume type schemes including DG methods was first used in [@ZS10; @ZS11; @ZLS12]. Here by weak monotonicity it means that each cell average is monotone with respect to point values in that cell, see e.g. [@Zh17]. The main idea in their work is to find sufficient conditions to preserve the desired bounds of cell averages by repeated convex combinations. A simple and efficient local MPS limiter can then be used to control the solution values at test points without affecting accuracy and conservation. Together with strong stability preserving (SSP) Runge–Kutta or multistep methods [@GKS11], which are convex combinations of several formal forward Euler steps, a high order accurate finite volume or DG scheme can be rendered MPS with the limiter.
For diffusion, a linear finite volume scheme can only be up to second order accurate in order to preserve the weak monotonicity, unless a non-conventional discretization is used in the scheme construction such as that in [@ZLS12]. For DG methods, in general only second order accuracy can be obtained to feature the MPS property; see [@ZZS13] for solving (\[conv\_dif\]) on triangular meshes.
The only DG method known to satisfy the weak monotonicity up to third order accuracy is the direct DG (DDG) method introduced in [@LY09; @LY10]. Indeed, the special method parameters in the DDG discretization allowed us to design in [@LY14] a third order MPS method for the linear Fokker-Planck equation (\[FP\]). A key idea in [@LY14] is the use of the non-logrithimic Landau formulation $$M\partial_t u= \nabla \cdot \left(M\nabla u\right) \quad\text{ with } M(x) = e^{-V(x)} \text{ and } u=\frac{\rho}{M},$$ so that the corresponding maximum principle on $\rho(t,x)$: $$c_1 e^{-V}\leq \rho(0, x) \leq c_2 e^{-V} \Longrightarrow c_1 e^{-V}\leq \rho(t, x) \leq c_2 e^{-V} \quad \forall t > 0,$$ reduces to $$c_1 \leq u(0, x) \leq c_2 \Longrightarrow c_1 \leq u(t, x) \leq c_2 \quad \forall t >0.$$ With this reformulation, one can show that each weighted cell average is monotone in terms of point values under appropriate CFL conditions. The result in [@LY14] is directly applicable to multi-dimensional diffusion on rectangular meshes. However, it gets subtle to ensure the MPS property on unstructured meshes; we refer to [@CHY16] for a third order such DDG method to solve diffusion equations on unstructured triangular meshes.
Another approach towards a positivity-preserving scheme with high order accuracy is to use the local DG (LDG) method [@BR97; @CS98], combined with some special positivity-preserving fluxes. Such an effort was first made in [@Zh17] for constructing high order accurate positivity-preserving DG schemes for compressible Navier–Stokes equations. Concerning further developments in this direction, we refer to [@SCS18; @SPZ18] for solving convection–diffusion problems. An MPS third-order LDG method using overlapping meshes has been recently proposed in [@DY19] for convection-diffusion equations.
One noteworthy alternative to enforce positivity in high order schemes is to take a convex combination of high order flux with a first order positivity-preserving one; the method has been applied to various high order schemes including finite difference, finite volume, and DG methods [@JX13; @XQX15; @YXQX16], while rigorous justification of accuracy for such methods seems difficult.
Present investigation
---------------------
The main distinction of our present investigation from the above mentioned works is the use of weighted cell averages for both ensuring the weak monotonicity and applying the scaling limiter, with special attention on difficulty caused by the anisotropic diffusivity.
The spatial discretization explored in this work is the DDG method introduced by Liu and Yan in [@LY09; @LY10]. Besides the usual advantages of a DG method (see e.g. [@HW07; @Ri08; @Sh09]), one main feature of the DDG method lies in numerical flux choices for the solution gradient, which involve second order derivatives evaluated crossing cell interfaces (see (\[flux\_DDG\_1Dp\]) below). With this choice, the obtained schemes are provably stable and optimally convergent as well as superconvergent for $\beta_1\not=0$ [@Liu15; @CLZ17]. Such method has also been successfully extended to various application problems, including Fokker-Planck type equations [@LY14; @LY15; @LW16; @LW17], and the three dimensional compressible Navier–Stokes equation [@CYLLL16].
Built upon the work [@LY14], we present third order DG methods for solving the initial value problem (\[md\]), with an application to nonlinear convection-diffusion equations of form (\[conv\_dif\]). Let us illustrate the main ideas via a simple one-dimensional equation subject to periodic boundary condition: $$M(x)\partial_t u = \partial_x(A(x)\partial_xu).$$ The computational cell is denoted by $I_j = [x_{j-\frac12},x_{j+\frac12}]$, where $x_{j+\frac12}$’s are the grid points. Let $\tau$ and $h$ be the time and space steps for a uniform mesh and $n$ the index for the time stepping. The update on $\langle u_h^n \rangle_j$, the weighted cell average of the numerical approximation $u_h^n$, is given by $$%\label{cell_1D}
\langle u^{n+1}_h\rangle_j = \langle u^n_h \rangle_j - \frac{\tau} {h} \left. A\widehat{\partial_x u_h^n}\right|_{\partial I_j}
\quad \text{ where } \langle u \rangle_j:= \frac{1}{h}\int_{I_j}M(x)u(x)\,dx.$$ Note that $\widehat{\partial_x u_h}$ is the approximation to $\partial_x u$ at $\partial I_j$, the cell interface of $I_j$, given by $$\begin{aligned}
\widehat{\partial_x u}& =\frac{\beta_0}{h}[u] +\{\partial_x u\} +\beta_1 h[\partial_x^2 u].\end{aligned}$$ As mentioned above, this form of the diffusive flux was originally introduced in [@LY09; @LY10] as part of the DDG method for solving the diffusion problem. If the flux parameters satisfy $$\begin{aligned}
\beta_0\geq 1 \quad\text{ and } \quad \frac{1}{8}\leq \beta_1\leq \frac{1}{4},\end{aligned}$$ then the procedure developed in [@LY14] can be extended to the present setting to conclude that, under a suitable CFL condition, the simple Euler forward will keep the cell average $ \bar u^n_j =\frac{\langle u_h^n\rangle_j}{\langle 1 \rangle_j} \in [c_1, c_2]$ in each time step, and the validity of the maximum principle when combined with a scaling limiter.
For a two dimensional problem with $A = \left(\begin{array}{cc} a & c \\ c & b \end{array}\right)$, the DDG method on shape-regular Cartesian meshes with $\kappa^{-1} \leq
\frac{\Delta x}{\Delta y}\leq \kappa$ can be rendered MPS if $$%\label{beta2d-}
\beta_0 \geq 1+\frac{\kappa|c| L(L-1)}{2 \min\{a, b\}} \quad\text{ and } \quad \frac{1}{8} \leq \beta_1 \leq \frac{1}{4}.$$ Here $L$ is the number of Gauss-Lobatto points used in the numerical evaluation of involved integrals. With $f(u)$ and $A = A(x,y,u)$, the MPS DDG schemes are analyzed in Section \[ssec2DNoninearCD\] where the parameter range and the CFL conditions are established.
The main conclusion is as follows: by applying the weighted MPS limiter introduced in [@LY14] to the DDG scheme designed here for , with the time discretization by an SSP Runge-Kutta method (see [@GKS09]), we obtain a third order accurate scheme solving (\[md\]) satisfying the strict maximum principle in the sense that the numerical solution never goes out of the range $[c_1, c_2]$ as indicated by the initial data.
Organization of the paper
--------------------------
The rest of the paper is organized as follows. In Section \[sec1D\], we design the numerical method for one dimensional problems. We first formulate the DDG scheme to solve the model heat equation, and prove the MPS property of the third order fully discretized DDG scheme, we then apply the result to show the MPS property for nonlinear convection-diffusion equations. Section \[sec2D\] is organized similarly for two dimensional problems on Cartesian meshes. In Section \[secLimiter\], we present the MPS limiter, with which the algorithm is complete. In Section \[secNum\], numerical tests for the DDG method are reported, including examples from the heat equation, porous media equation the Buckley-Leverett equation, and two dimensional diffusion with the anisotropic diffusion. Concluding remarks are given in Section \[secConclude\].
MPS schemes in one dimension {#sec1D}
============================
We first investigate the MPS property for third order DDG schemes to weighted diffusion equations, and show how to apply the scheme obtained to nonlinear convection-diffusion equations.
The diffusion equation
----------------------
We begin with the heat equation of the form $$\begin{aligned}
\label{Ma}
M(x)\partial_t u=\partial_x (A(x)\partial_x u),\end{aligned}$$ with $M(x)>0$ and $A(x)\geq 0$ on the spatial domain $\Omega$, subject to initial data $u_0(x)$ and the periodic boundary condition. It is known that the following maximum principle holds: $$\text{if } c_1 \leq u_0(x) \leq c_2 \;\; \forall x \in \Omega, \text{ then } c_1 \leq u(t, x) \leq c_2 \;\; \forall x \in \Omega, t>0.$$ In general, the problem can be either defined on a connected compact domain with proper boundary conditions, or it can involve the whole real line with solutions vanishing at the infinity. In our numerical scheme, we will always choose the spatial domain to be a connected interval. For simplicity, the periodic boundary conditions are applied.
We partition the domain $\Omega$ by regular cells such that $\Omega=\bigcup\limits_{j=1}^NI_j$ with $I_j = [x_{j-\frac12}, x_{j+\frac12}]$. Denote $h_j=x_{j+\frac12}-x_{j-\frac12}$ and $h=\max\limits_j h_j$. We seek numerical solutions in the discontinuous piecewise polynomial space, $$V_h=\{v \in L^2(\Omega) |\quad v|_{I_j}\in P^k(I_j), \quad j=1\cdots, N\}.$$ Here $P^k(I_j)$ is the space of $k$-th order polynomials on $I_j$. Note that the functions in $V_h$ can be double-valued at cell interfaces. Hence notations $v^-$ and $v^+$ are used for the left limit and right limit of $v$. The jump of these two values, $v^+-v^-$, is denoted by $[v]$, and the average by $\{v\}$.
Throughout this paper we adopt the DDG numerical flux of the form $$\begin{aligned}
\label{flux_DDG_1Dp}
\widehat{\partial_x v}& =\frac{\beta_0}{h_{j+\frac12}}[v] +\{\partial_x v\} +\beta_1 h_{j+\frac12}[\partial_x^2 v] \quad\text{ with } h_{j+\frac12} = \frac{h_j + h_{j+1}}{2}, \end{aligned}$$ when crossing the cell interface $x_{j+\frac12}$, and $(\beta_0, \beta_1)$ are in the range to be specified so that the underlying third order scheme can weakly satisfy the maximum-principle. The parameter range was first identified in [@LY14] for a third order DDG scheme to feature the MPS property for linear Fokker-Planck equations.
For model equation (\[Ma\]), we consider a $(k+1)$th-order DG scheme: to find $u_h\in V_h$ such that for any test function $v\in V_h$, $$\begin{aligned}
\int_{I_j} M(x) \partial_t u_h v\,dx & = -\int_{I_j} A(x) \partial_x u_h \partial_x v\,dx + \left. A \left[\widehat{\partial_x u_h}v +(u_h -\{u_h\})\partial_x v\right]\right|^{x_{j+\frac12}}_{x_{j-\frac12}},\end{aligned}$$ with the diffusive flux $\widehat{\partial_x u_h}$ as defined in (\[flux\_DDG\_1Dp\]). This DDG scheme with interface correction was proposed in [@LY10] for the diffusion problem, as an improved version of that in [@LY09]. Based on the DDG scheme in [@LY09], we will have $$\begin{aligned}
\int_{I_j} M(x) \partial_t u_h v\,dx & = -\int_{I_j} A(x) \partial_x u_h \partial_x v\,dx + \left. A \widehat{\partial_x u_h}v\right|^{x_{j+\frac12}}_{x_{j-\frac12}}.\end{aligned}$$
With the forward Euler time discretization, the weighted cell average for either of two DDG schemes evolves as $$\label{EF1Dv1}
\langle u^{n+1}_h \rangle_j= \langle u^n_h \rangle_j + \mu h\left. A \widehat{\partial_x u_h^n}\right|^{x_{j+\frac12}}_{x_{j-\frac12}},$$ where $\mu = \frac{\tau}{h^2}$ is the mesh ratio. For a concise presentation, a uniform mesh is assumed. Here and in what follows, we denote the time step length as $\tau$. Note that for periodic boundary conditions considered in this paper, we take $$u_{h,\frac12}^-=u^-_{h, N+\frac12}, \quad u_{h, N+\frac12}^+=u^+_{h, \frac12}$$ in the DDG numerical flux formula (\[flux\_DDG\_1Dp\]) when $j=\frac12, N+\frac12$.
We also use the notation $$\langle q(\xi) \rangle_j= \frac{1}{2}\int_{-1}^1M(x_j+\frac{h}{2}\xi)q(\xi)d\xi, \text{ for any } q(\xi) \text{ on }[-1, 1].$$ And the cell average $\bar u_j$ is $\bar u_j = \frac{\langle u_h \rangle_j}{ \langle 1 \rangle_j}$. For $j=1,\cdots, N$, we define $$\begin{aligned}
\label{dif1Dab}
a_j= \frac{\langle \xi-\xi^2 \rangle_j}{\langle 1-\xi \rangle_j}, \quad
b_j=\frac{\langle \xi +\xi^2 \rangle_j}{\langle 1+\xi\rangle_j}, \end{aligned}$$ and $$\begin{aligned}
\nonumber
\tilde{\omega}^1_j = &\frac{\langle \gamma-\xi(1+\gamma)+\xi^2\rangle_j }{2(1+\gamma)},\\ \label{c5ome}
\tilde{\omega}^2_j =&\frac{\langle 1-\xi^2\rangle_j }{1-\gamma^2}, \\ \nonumber
\tilde{\omega}^3_j =&\frac{\langle -\gamma +\xi(1-\gamma)+\xi^2\rangle_j }{2(1-\gamma)}\end{aligned}$$ with the weight $M(x)|_{I_j}=M(x_j+\frac{h}{2}\xi)$. We recall the following key result.
[@LY14 Lemma 3.3] $\tilde{\omega}^i_j >0$ for $i=1, 2,3$ if and only if $$\gamma \in (a_j, b_j),$$ where $a_j, b_j$ satisfy $-1<a_j<b_j<1$.
Here we only show $a_j< b_j$, which means that selection of $\gamma$ is always ensured. A direct calculation gives $$\begin{aligned}
b_j - a_j = 2\frac{\langle 1\rangle_j\langle\xi^2\rangle_j - \langle \xi\rangle_j^2}{\langle 1+\xi\rangle_j\langle 1-\xi \rangle_j} \geq 0,\end{aligned}$$ where the numerator can be reformulated as $$\begin{aligned}
& \int_{-1}^1\int_{-1}^1(\eta^2-\xi \eta)\tilde M(\xi) \tilde M(\eta)d\xi d\eta +\int_{-1}^1\int_{-1}^1(\xi^2-\eta \xi)\tilde M(\eta)\tilde M(\xi)d\eta d\xi\\
=& \int_{-1}^1\int_{-1}^1(\xi-\eta)^2 \tilde M(\xi)\tilde M(\eta)d\xi d\eta >0, \end{aligned}$$ where $\tilde M(\xi):= M(x_j+\frac{h}{2}\xi)$ has been used.
We thus have the following result.
\[thk2max\]($k=2$) The scheme (\[EF1Dv1\]) with $$\begin{aligned}
\label{betak2max}
\beta_0\geq 1
\quad \text{and} \quad
\frac{1}{8}\leq \beta_1 \leq \frac{1}{4} \end{aligned}$$ is maximum-principle-satisfying, namely, $c_1\leq \bar{u}^{n+1}_j \leq c_2 $ if $u_h^n(x)$ is in $[c_1, c_2]$ on $\{S_j\}_{j=1}^N$, where $$S_j = x_j +\frac{h}{2}\left\{-1, \gamma, 1\right\}$$ with $\gamma$ satisfying $$\begin{aligned}
\label{abmax}
a_j <\gamma < b_j\quad \text{and} \quad |\gamma|\leq 8\beta_1-1,\end{aligned}$$ under the CFL condition $
\mu \leq \mu_0,
$ where $\mu_0$ is given in below.
[*Step 1.*]{} Weighted integral decomposition. Define $$p_j(\xi)=u_h\left(x_j+\frac{h}{2}\xi\right) \text{ for } \xi\in [-1,1],$$ we see that in the case of ${p}_j(\xi)\in {P}^2[-1, 1]$, for any $\gamma \in (-1,1)$, the unique interpolation of $p_j$ at three points $\{-1, \gamma, 1\}$ gives the following $$\begin{aligned}
\label{pj}
{p}_j(\xi)=\frac{(\xi-1)(\xi -\gamma)}{2(1+\gamma)}{p}_j(-1)+\frac{(\xi -1)(\xi +1)}{(\gamma -1)(\gamma +1)}{p}_j(\gamma)+\frac{(\xi +1)(\xi -\gamma)}{2(1-\gamma)}{p}_j(1).\end{aligned}$$ This yields the following identity for the weighted average, $$\label{c5wp}
\langle u_h\rangle_j =\tilde{\omega}^1_j p_j(-1) + \tilde{\omega}^2_j p_j(\gamma) + \tilde{\omega}^3_j p_j(1),$$ where $\tilde{\omega}^i_j$ given in (\[c5ome\]) are ensured positive by Lemma 2.1.
[*Step 2.* ]{} Flux representation. A direct calculation gives $$\begin{aligned}
\label{c5fluxk2gamma}
h \left.\widehat{\partial_xu_h}\right|_{x_{j+\frac12}}
=& \alpha_3(-\gamma) p_{j+1}(-1)+\alpha_2(-\gamma)p_{j+1}(\gamma)+\alpha_1(-\gamma)p_{j+1}(1)\\
&- \left[\alpha_1(\gamma) p_j(-1)+\alpha_2(\gamma)p_j(\gamma)+\alpha_3(\gamma)p_j(1)\right],\nonumber\end{aligned}$$ where $$\begin{aligned}
\alpha_1(\gamma)= \frac{8\beta_1-1+\gamma}{2(1+\gamma)}, \quad
\alpha_2(\gamma)=2\frac{1-4\beta_1}{1-\gamma^2}, \quad
\alpha_3(\gamma)=\beta_0 +\frac{8\beta_1-3+\gamma}{2(1-\gamma)},\end{aligned}$$ are all positive due to (\[betak2max\]) and .
[*Step 3.*]{} Monotonicity under some CFL condition. We now substitute (\[c5wp\]) and (\[c5fluxk2gamma\]) into (\[EF1Dv1\]) to obtain $$\langle u_h^{n+1}\rangle_j=R_j^n(M(\cdot), \mu, h, A)$$ with $$\begin{aligned}
\label{c5avedeck2}
R_j^n(M(\cdot), \mu, h, A)= & \langle u^n_h \rangle_j + \mu \left(\left.Ah\widehat{\partial_xu_h^n}\right|_{x_{j+\frac12}} - \left.Ah \widehat{\partial_xu_h^n}\right|_{x_{j-\frac12}}\right) \\
=&\left[\tilde{\omega}_j^1-\mu \left(\alpha_3(-\gamma) A_{j-\frac12} +\alpha_1(\gamma) A_{j+\frac12} \right) \right]p_j(-1) \nonumber\\
&+\left[\tilde{\omega}_j^2 - \mu \left(\alpha_2(-\gamma) A_{j-\frac12} +\alpha_2(\gamma) A_{j+\frac12}\right)\right]p_j(\gamma)\nonumber\\
&+\left[\tilde{\omega}_j^3 - \mu \left(\alpha_1(-\gamma) A_{j-\frac12} +\alpha_3(\gamma) A_{j+\frac12} \right) \right]p_j(1) \nonumber\\
&+\mu A_{j+\frac12}\left[\alpha_3(-\gamma) p_{j+1}(-1)+\alpha_2(-\gamma)p_{j+1}(\gamma)+\alpha_1(-\gamma)p_{j+1}(1) \right]\nonumber \\
&+\mu A_{j-\frac12}\left[\alpha_1(\gamma) p_{j-1}(-1)+\alpha_2(\gamma)p_{j-1}(\gamma)+\alpha_3(\gamma)p_{j-1}(1) \right].\nonumber\end{aligned}$$ Here it is understood that $p_0(\xi):=p_N(\xi+|\Omega|)$ and $p_{N+1}(\xi+|\Omega|)=p_1(\xi)$ for incorporating the periodic boundary conditions. Note also that $\sum_{i}^3\alpha_i(\gamma)=\beta_0= \sum_{i}^3\alpha_i(-\gamma)$. Using the fact $ \tilde{\omega}_j^3(\gamma)=\tilde {\omega}_j^1(-\gamma)$ and formulas for $\alpha_2(\gamma)$, $\tilde{\omega}_j^2$, we see that if $\mu$ is chosen to be smaller than $\mu_0$ where $$\begin{aligned}
\label{CFL1Dk2}
\mu_0 =
\left(\max_{1\leq j\leq N} A(x_{j+\frac12}) \right)^{-1}
\min_{1\leq j\leq N}
\left\{ \frac{{\tilde\omega}_j^1(\pm \gamma)}{\alpha_3(\mp \gamma)
+\alpha_1(\pm \gamma) }, \frac{\langle 1-\xi^2\rangle_j}{4(1-4\beta_1)}
\right\},\end{aligned}$$ (\[c5avedeck2\]) is nondecreasing in the point values $p_j(\pm1), p_j(\gamma), p_{j\pm 1}(\pm1), p_{j\pm 1}(\gamma)$, hence when these values are replaced with the lower and upper bounds $c_1$ and $c_2$ respectively, we have $$c_1\sum_{i=1}^3\tilde{\omega}_j^i \leq \langle u_h^{n+1}\rangle_j \leq c_2 \sum_{i=1}^3\tilde {\omega}_j^i,$$ since the terms with $\alpha_i$’s are cancelled out. Moreover the sum of $\tilde{\omega}_j^i$ is $\langle 1\rangle_j$. Therefore $$c_1\langle 1\rangle_j \leq \langle u_h^{n+1}\rangle_j \leq c_2\langle 1\rangle_j \quad \Rightarrow \quad c_1 \leq \bar{u}_j^{n+1} \leq c_2.$$
For other types of boundary conditions, the boundary flux needs to be modified. A similar result to Theorem \[thk2max\] may be established as long as the PDE problem satisfies a maximum principle.
The use of $\gamma$ is essential for the success of our schemes, in particular when $M(x)$ is a function. For the special case $M(x)=1$, we have $\gamma\in (-\frac13, \frac13)$, hence $\gamma=0$, which corresponds to the usual Gauss quadrature point, is admissible. The CFL number given in may be optimized by carefully tuning $\gamma \in (a_j, b_j)$, but not in a linear fashion. Nevertheless, it was observed in [@LY14] that the larger $|\gamma|$ is, the better the scheme’s performance for the heat equation.
Application to nonlinear convection-diffusion equation
------------------------------------------------------
In this section we will demonstrate how to apply our MPS DG method in section 2.1 to the nonlinear convection-diffusion equation, $$\partial_t u +\partial_x f(u) =\partial_x (A(x, u)\partial_x u),$$ where $f(u)$ is a smooth function and diffusion coefficient $A(x, u)\geq 0$, subject to initial data $u(0, x)=u_0(x)$, and periodic boundary conditions. By applying the DG approximation, we obtain the following scheme. We seek $u_h\in V_h$ such that for any test function $v\in V_h$, $$\begin{aligned}
\label{semi1D}
\int_{I_j} \partial_t u_h v\,dx & = \int_{I_j} f(u_h)\partial_xv\,dx -\hat f(u_h^-, u_h^+) v\Big |^{x_{j+\frac12}}_{x_{j-\frac12}}\\ \notag
& \quad -\int_{I_j} A_h \partial_x u_h \partial_x v\,dx + \left.\{A_h\}\left[\widehat{\partial_x u_h}v +(u_h -\{u_h\})\partial_x v\right]\right|^{x_{j+\frac12}}_{x_{j-\frac12}}.\end{aligned}$$ For diffusion part, we adopt the DDG diffusive flux For convection, any monotone numerical flux can be used, i.e., $\hat f(u, v)$ is Lipschitz continuous, nondecreasing in $u$ and nonincreasing in $v$, consistent with $f(u)$ in the sense that $ \hat f(u, u)=f(u)$. For example, the global Lax-Friedrichs flux $$\hat f(u_h^-, u^{+}_h)=\frac{1}{2} (f(u^{-}_h) +f(u^{+}_h) -\sigma (u^{+}_h-u^{-}_h)),
\quad \sigma=\max_{u \in [c_1, c_2]}|f'(u)|.$$
We consider the first order Euler forward temporal discretization of (\[semi1D\]) to obtain $$\begin{aligned}
\label{fully_1D}
\int_{I_j} \frac{u_h^{n+1}-u_h^n}{\tau} v\,dx & = \int_{I_j} f(u^n_h)\partial_xv\,dx
- \hat f((u_h^n)^-, (u_h^n)^+) v\Big |^{x_{j+\frac12}}_{x_{j-\frac12}}\\ \notag
& \quad -\int_{I_j} A_h^n \partial_x u^n_h \partial_x v\,dx + \left.\{A_h^n\}\left[\widehat{\partial_x u^n_h}v +(u^n_h -\{u^n_h\})\partial_x v\right]\right|^{x_{j+\frac12}}_{x_{j-\frac12}}.\end{aligned}$$ By taking the test function $v = 1$ on $I_j$ and $0$ elsewhere, we obtain the evolutionary update for the cell average, $$\bar u^{n+1}_j= \bar u^n_j - \lambda \hat f^n \Big|^{x_{j+\frac12}}_{x_{j-\frac12}}
+ \mu h \{A_h^n\} \widehat{\partial_xu_h^n}
\Big|^{x_{j+\frac12}}_{x_{j-\frac12}},$$ where $\lambda = \frac{\tau}{h}$ and $\mu = \frac{\tau}{h^2}$ are the mesh ratios. Assuming that $ \bar u^n_j \in [c_1, c_2]$ for all $j$’s, we would like to derive some sufficient conditions such that $ \bar u^{n+1}_j \in [c_1, c_2] $ under certain restrictions on $\lambda$ and $\mu$.
For piecewise quadratic polynomials, the main result can be stated as follows.
\[thk2\_1D\]($k=2$) The scheme (\[fully\_1D\]) with $$\begin{aligned}
\beta_0\geq 1
\quad \text{and} \quad
\frac{1}{8}\leq \beta_1 \leq \frac{1}{4} \end{aligned}$$ is maximum-principle-satisfying; namely, $\bar{u}_{j}^{n+1}\in [c_1, c_2]$ if $u_h^n(x) \in [c_1, c_2]$ on the set $S_j$’s where $$S_j = x_j +\frac{h}{2}\left\{-1, \gamma, 1\right\}$$ with $\gamma$ satisfying $$\begin{aligned}
\label{ab}
-\frac{1}{3} <\gamma < \frac{1}{3} \quad \text{and} \quad |\gamma|\leq 8\beta_1-1, % \leq \gamma \leq 1+\frac{4\beta_1-1}{\beta_0-1/2},\end{aligned}$$ under the CFL condition $$\begin{aligned}
\lambda \leq \lambda_0, \quad \mu \leq \mu_0\end{aligned}$$ for some $\lambda_0$ and $\mu_0$ defined in (\[CFL1Dk2-\]) and (\[CFL1Dk2+\]), respectively.
We present the proof in four steps:\
[*Step 1.*]{} Split: we split the average $\bar u_j^n$ into two halves so that $$\bar u_j^{n+1}=\frac{1}{2} \mathcal{C}_j^n +\frac{1}{2}\mathcal{D}_j^n,$$ where the convection term is $$\label{cj}
\mathcal{C}_j=\bar u_j -2\lambda \hat f \Big |^{x_{j+\frac12}}_{x_{j-\frac12}}$$ and the diffusion term is $$\mathcal{D}_j =\bar u_j + 2\mu h \{A_h^n\} \widehat {\partial_xu_h}\Big |^{x_{j+\frac12}}_{x_{j-\frac12}}.$$ This split is for convenient presentation and may not lead to an optimal CFL condition.
[*Step 2.*]{} The integral decomposition. From (\[c5wp\]) it follows $$\bar u_j = \omega^1 p_j(-1)+\omega^2 p_j(\gamma)+\omega^3 p_j(1),$$ where $\omega^i=\tilde \omega^i, \langle u_h\rangle_j = \bar u_j$ for $M(x)\equiv1$, and $$\begin{aligned}
\omega^1 = \frac{1+3\gamma }{6(1+\gamma)},\; \;
\omega^2 = \frac{2}{3(1-\gamma^2)}, \;
\omega^3 = \frac{1-3\gamma}{6(1-\gamma)}.\end{aligned}$$ These coefficients are positive for $\gamma$ satisfying (\[ab\]).\
[*Step 3.* ]{} The convection term.
Using the cell average decomposition and the flux formula, we rewrite (\[cj\]) as $$\begin{aligned}
\mathcal{C}_j = & \bar u^n_j -2 \lambda\left(\left.\widehat{f}\right|_{x_{j+\frac12}} - \left. \widehat{f}\right|_{x_{j-\frac12}}\right) \\
=& \omega^3 p_j(1) - 2\lambda \hat f(p_j(1), p_{j+1}(-1)) + \omega^2 p_j(\gamma) + \omega^1 p_j(-1)
+ 2\lambda \hat f(p_{j-1}(1), p_j(-1)) \nonumber \\
= &: G(p_{j-1}(1), p_j(-1), p_j(\gamma), p_j(1), p_{j+1}(-1)). \nonumber\end{aligned}$$ For a monotone flux $\hat f(u, v)$ being Lipschitz continuous with Lipschitz constant $\mathcal{L}$, $G$ is non-increasing in all the four arguments provided the following condition is met, $$2 \lambda \mathcal{L} \leq \min\{ \omega^1, \omega^3\}.$$ Note that for the Lax-Friedrichs flux, $\mathcal{L} = \max\limits_{u\in[c_1,c_2]} |f'(u)|$. Moreover, the consistency of the flux $\hat f(u, u)=f(u)$ implies that $G(u, u, u, u)=u$. Hence we have $$\mathcal{C}_j \in [G(c_1, c_1, c_1, c_1, c_1), G(c_2, c_2, c_2, c_2, c_2)]=[c_1, c_2],$$ as long as the involved values are in $[c_1, c_2]$. It suffices to take $$\begin{aligned}
\label{CFL1Dk2-}
\lambda_0= \frac{1}{2\mathcal{L}} \min\{ \omega^1, \omega^3\}=\frac{1-3\gamma}{12\mathcal{L}(1-\gamma)},\end{aligned}$$ where we have used the fact that $\omega^1(\gamma)=\omega^3(-\gamma)$.
[*Step 4.*]{} The diffusion term. We apply the result in Theorem \[thk2max\] to the case with $M(x)\equiv1$ and $\mu$ replaced by $2\mu$ to conclude that $\mathcal{D}_j \in [c_1, c_2]$ if $u_h^n(x) \in [c_1, c_2]$ on $S_j$ and $\mu \leq \mu_0$ with $$\begin{aligned}
\label{CFL1Dk2+}
\mu_0 & = \frac{1}{2}\left(\max_j \{A_h^n\}_{j+\frac12} \right)^{-1}\min_{1\leq j\leq N}\left\{ \frac{{\omega}^1(\pm \gamma)}{\alpha_3(\mp \gamma)
+\alpha_1(\pm \gamma) }, \frac{1}{3(1-4\beta_1)}
\right\}\\ \notag
& =\frac{1}{12}\left(\max_{1\leq j\leq N} \{A_h^n\}_{j+\frac12}\right)^{-1}\min_{1\leq j\leq N}\left\{ \frac{1\pm 3\gamma}{\beta_0(1\pm \gamma)+8\beta_1-2}, \frac{2}{1-4\beta_1} \right\}.\end{aligned}$$
The above analysis can be readily carried over to (\[md\]) in one dimension, i.e., $$\begin{aligned}
\label{md+}
M(x)\partial_t u +\partial_x f(u) =\partial_x (A(x, u)\partial_x u). \end{aligned}$$ We summarize the result in the following.
\[thk2\_1D\_md\]($k=2$) The scheme when applied to with $$\begin{aligned}
\beta_0\geq 1
\quad \text{and} \quad
\frac{1}{8}\leq \beta_1 \leq \frac{1}{4} \end{aligned}$$ is maximum-principle-satisfying; namely, $\bar{u}_{j}^{n+1}\in [c_1, c_2]$ if $u_h^n(x) \in [c_1, c_2]$ on the set $S_j$’s where $$S_j = x_j +\frac{h}{2}\left\{-1, \gamma, 1\right\}$$ with $\gamma$ satisfying $$\begin{aligned}
\gamma \in (a_j, b_j) \; \text{as in \eqref{dif1Dab} and} \quad |\gamma|\leq 8\beta_1-1, % \leq \gamma \leq 1+\frac{4\beta_1-1}{\beta_0-1/2},\end{aligned}$$ under the CFL condition $$\begin{aligned}
\lambda \leq \lambda_0, \quad \mu \leq \mu_0\end{aligned}$$ for some $\lambda_0$ and $\mu_0$ defined in and , respectively.
The proof is entirely analogous to the proof of Theorem \[thk2\_1D\]. One only needs to replace $\omega^{i}$ by $\tilde\omega^{i}$ for $i=1, 2, 3$, and $\bar u_j$ by $\langle u_h\rangle_j$, respectively, so to obtain different CFL conditions. More precisely, (\[CFL1Dk2-\]) in Step 3 needs to be replaced by $$\begin{aligned}
\label{CFL1Dk2md_conv}
\lambda_0= \frac{1}{2\mathcal{L}} \min\{ \tilde\omega^1(\gamma), \tilde\omega^3(\gamma)\}= \frac{1}{2\mathcal{L}} \min\{ \tilde\omega^1(\pm\gamma)\}, \end{aligned}$$ and (\[CFL1Dk2\]) by a factor of $1/2$ gives $$\begin{aligned}
\label{CFL1Dk2md_diff}
\mu_0 = \frac{1}{2}
\left(\max_{1\leq j\leq N} A(x_{j+\frac12}) \right)^{-1}
\min_{1\leq j\leq N}
\left\{ \frac{{\tilde\omega}_j^1(\pm \gamma)}{\alpha_3(\mp \gamma)
+\alpha_1(\pm \gamma) }, \frac{\langle 1-\xi^2\rangle_j}{4(1-4\beta_1)}
\right\}.\end{aligned}$$
MPS schemes in two dimensions {#sec2D}
=============================
In this section, we design an MPS DG scheme to solve two dimensional problems on Cartesian meshes. Consider the model equation $$M(x,y)\partial_t u = \nabla\cdot(A\nabla u) \text{ with }
A = \left(\begin{array}{cc} a & c \\ c & b \end{array}\right), \quad \text{ for } (x, y)\in \Omega \subset \mathbb{R}^2, \quad t>0,$$ subject to the initial data $u_0(x, y)$ and periodic boundary conditions. Here $M(x, y) > 0$ is a given function, $a, b$ and $c$ are constant parameters so that $A$ is nonnegative definite. The domain $\Omega =I \times J$ is a rectangle given by two intervals $I$ and $J$ in $x$ and $y$ direction, respectively. Let $\cup_{i=1}^{N_x}\cup_{j=1}^{N_y} K_{ij} $ be a partition of the domain $\Omega$, with $K_{ij}=I_i\times J_j$, where $$I_{i}=[x_{i-\frac12}, x_{i+\frac12}], \quad J_j = [y_{j-\frac12}, y_{j+\frac12}].$$ The finite element space is defined as $$V_h=\{v\in L^2(\Omega), v\big|_{K_{ij}}\in Q^k(K_{ij}), i=1, \cdots, N_x, j=1,\cdots, N_y\}.$$ Here $Q^k(K_{ij})$ is the tensor product space of $P^k(I_i)$ and $P^k(J_j)$. Hence the DDG scheme can be formulated as follows: to find $u_h\in V_h$ such that for all $v\in V_h$, $$\begin{aligned}
\label{EFfully2D-}
\int_{K_{ij}} Mu_{h}^{n+1}v\,dxdy
= &\int_{K_{ij}} Mu_{h}^{n}v\,dxdy
- \tau \int_{K_{ij}} A\nabla u_h^n\cdot\nabla v\,dxdy \notag \\
&+ \tau \int_{\partial K_{ij}}A\widehat{\nabla u_h^n}\cdot \nu v\,ds
+ \tau \int_{\partial K_{ij}}A{\nabla v}\cdot \nu (u_h^n - \{u_h^n\})\,ds, \end{aligned}$$ where $\nu$ is the outward unit normal to the cell boundary $\partial K_{ij}$, and the numerical flux $$\label{df2}
\widehat{\nabla u_h}= (\widehat{ \partial_x u_h}, \widehat{\partial_y u_h})^\top$$ is defined as follows, $$\begin{aligned}
\label{beta2D}
\left.\widehat{\partial_x u_h}\right|_{(x_{i+\frac12},y)}& =\frac{\beta_{0}}{\Delta x}[u_h] +\{\partial_x u_h\} +\beta_{1} \Delta x [\partial_x^2 u_h], \;
\left.\widehat{\partial_y u_h}\right|_{(x_{i+\frac12},y)} =\{\partial_y u_h\}, \notag\\
\left.\widehat{\partial_y u_h}\right|_{(x,y_{j+\frac12})}& =\frac{\beta_{0}}{\Delta y}[u_h] +\{\partial_y u_h\} +\beta_{1} \Delta y [\partial_y^2 u_h], \;
\left.\widehat{\partial_x u_h}\right|_{(x,y_{j+\frac12})} =\{\partial_x u_h\}, \notag\end{aligned}$$ where $\beta_0, \beta_1$ are flux parameters to be determined to ensure the desired MPS property. Note that in $\widehat{\partial_y u_h}|_{(x_{i+\frac12},y)}$, jump terms do not show up since along interface $x = x_{i+\frac12}$ and $y \in [y_{j-\frac12}, y_{j+\frac12}]$, there is no jump of polynomials in $y$ direction. This argument applies to $\partial_x u_h|_{(x,y_{j+\frac12})}$ as well. Here for a concise expression of the numerical flux, a uniform mesh has been used, with $\Delta x = x_{i+\frac12}-x_{i-\frac12}$ and $\Delta y=y_{j+\frac12}-y_{j-\frac12}$.
To proceed, we recall some conventions similar to the one-dimensional case. The weighted cell average is defined as $$\langle u^{n+1}\rangle_{ij} = \frac{\int_{K_{ij}}Mu_h\,dxdy}{\Delta x \Delta y}
= \dashint_{J_j} \dashint_{I_i}Mu_h\,dxdy,$$ where $\dashint$ denotes the average integral. We also define the weighted interval average in $x$ and $y$, respectively, $$\langle \phi(\xi) \rangle_i(y)=\dashint_{-1}^1M\left(x_i+\frac{\Delta x}{2}\xi, y\right)\phi(\xi)d\xi, \quad
\langle \phi(\eta) \rangle_j(x)=\dashint_{-1}^1M\left(x, y_j+\frac{\Delta y}{2}\eta\right)\phi(\eta)d\eta.$$ The cell average $$\overline{u}_{ij} = \frac{\int_{K_{ij}}Mu_h\,dxdy}{\int_{K_{ij}}M\,dxdy}=\frac{\langle u_h\rangle_{ij}}{\langle 1\rangle_{ij}}$$ update can be obtained from (\[EFfully2D-\]) as $$\begin{aligned}
\label{EF2Dv1}
\langle u^{n+1}_h\rangle_{ij} =& \langle u^{n}\rangle_{ij}
+ \mu_x\Delta x \left.\dashint_{J_j}\big(a\widehat{\partial_x u_h^n} + c\widehat{\partial_y u_h^n}\big)\,dy\right|_{\partial I_i}
+ \mu_y\Delta y \left.\dashint_{I_i}\big(b\widehat{\partial_y u_h^n} + c\widehat{\partial_x u_h^n}\big)\,dx\right|_{\partial J_j},\end{aligned}$$ where $\mu_{x} = \frac{\tau }{(\Delta x)^2}$ and $\mu_{y} = \frac{\tau}{(\Delta y)^2}$. Let $\mu = \mu_x + \mu_y$ and decompose $\langle u^{n}\rangle_{ij}$ as $$\langle u^{n}\rangle_{ij} = \frac{\mu_x}{\mu} \langle u^{n}\rangle_{ij} + \frac{\mu_y}{\mu} \langle u^{n}\rangle_{ij},$$ so that (\[EF2Dv1\]) can be rewritten as $$\begin{aligned}
\label{EF2DH}
\langle u^{n+1}\rangle_{ij} =& \frac{\mu_x}{\mu}\dashint_{J_j}H_1(y)\,dy + \frac{\mu_y}{\mu}\dashint_{I_i} H_2(x)\,dx +B,\end{aligned}$$ where $$\begin{aligned}
& H_1(y) =\dashint_{I_i}M(x,y)u_{h}^n\,dx
+ \mu\Delta x\left. a\widehat{\partial_x u_h^n}\right|_{\partial I_i},\\
& H_2(x) =\dashint_{J_j}M(x,y)u_{h}^n\,dy
+ \mu\Delta y\left. b\widehat{\partial_y u_h^n} \right|_{\partial J_j},\\
& B = \frac{c\tau}{|K_{ij}|} \left[ \int_{J_j}\{\partial_y u_h\}dy |_{\partial I_i} +\int_{I_i}\{\partial_x u_h\}dx |_{\partial J_j} \right].\end{aligned}$$ Notice that $B$ can be expressed as a combination of point values of $u_h^n$ at four vertices of $K_{ij}$ as $$\begin{aligned}
B& =\frac{c\tau}{2\Delta x\Delta y}\big(2u_h^n(x_{i+\frac12}^-, y_{j+\frac12}^-) - 2u_h^n(x_{i+\frac12}^-, y_{j-\frac12}^+)
-2u_h^n(x_{i-\frac12}^+, y_{j+\frac12}^-) + 2u_h^n(x_{i-\frac12}^+, y_{j-\frac12}^+)\big) \\
& + \frac{c\tau}{2\Delta x\Delta y}\big(u_h^n(x_{i+\frac12}^+, y_{j+\frac12}^-) +u_h^n(x_{i+\frac12}^-, y_{j+\frac12}^+) - u_h^n(x_{i+\frac12}^-, y_{j-\frac12}^-)
-u_h^n(x_{i+\frac12}^+, y_{j-\frac12}^+)\big)\\
& + \frac{c\tau}{2\Delta x\Delta y}\big(u_h^n(x_{i-\frac12}^+, y_{j-\frac12}^-)+u_h^n(x_{i-\frac12}^-, y_{j-\frac12}^+) -u_h^n(x_{i-\frac12}^+, y_{j+\frac12}^+)
-u_h^n(x_{i-\frac12}^-, y_{j+\frac12}^-) \big).\end{aligned}$$ The two integrals in (\[EF2DH\]) can be approximated by the Gauss-Lobatto quadrature rule with sufficient accuracy. Let us assume that we use an $L-$point Gauss-Lobatto quadrature rule with $L\geq \frac{k+3}{2}$ points, which has accuracy of at least $O(h^{k+2})$. Let $$\begin{aligned}
&\hat S_{i}^x = \{x_{i-\frac12} = \hat x^1_i < \cdots < \hat x_i^{\sigma} < \cdots < \hat x^L_i=x_{i+\frac12}\}, \notag\\
&\hat S_{j}^y = \{y_{j-\frac12} = \hat y^1_j < \cdots < \hat y_j^{\sigma} < \cdots < \hat y^L_j=y_{j+\frac12}\}\end{aligned}$$ denote the quadrature points on $I_i$ and $J_j$, respectively, and $\hat\omega^{\sigma}$’s be the associated quadrature weights so that $$\sum_{\sigma=1}^L \hat \omega^{\sigma}=1.$$ Using the quadrature rule on the right-hand side of (\[EF2DH\]), we obtain the following scheme $$\label{c5EF2DHQ}
\langle u^{n+1}_h \rangle_{ij}= \frac{\mu_x}{\mu}\sum_{\sigma=1}^L \hat\omega^{\sigma}H_1(\hat y_j^\sigma) + \frac{\mu_y}{\mu}\sum_{\sigma=1}^L \hat\omega^{\sigma}H_2(\hat x_i^\sigma)+B.$$ Also set $$\begin{aligned}
S^x_i & = \{x_{i-\frac12}, x_i^{\gamma}, x_{i+\frac12}\} = x_i + \frac{\Delta x}{2}\{-1, \gamma^x, 1\}, \\
S^y_j & = \{y_{j-\frac12}, y_j^{\gamma}, y_{j+\frac12}\} = y_j + \frac{\Delta y}{2}\{-1, \gamma^y, 1\}\end{aligned}$$ to denote the test sets on $[x_{i-\frac12}, x_{i+\frac12}]$ and $[y_{j-\frac12}, y_{j+\frac12}]$, respectively, with $\gamma^x, \gamma^y$ satisfying $$\begin{aligned}
\label{c5angle2D}
& \frac{\langle \xi-\xi^2\rangle_i}{\langle 1- \xi \rangle_i} (y_j^\sigma) < \gamma^x < \frac{\langle \xi +\xi^2\rangle_i}{\langle 1+\xi \rangle_i} (y_j^\sigma), \quad |\gamma^x|\leq 8\beta_1-1,\\
& \frac{\langle \eta-\eta^2\rangle_j} {\langle 1- \eta \rangle_j} (x_i^\sigma) < \gamma^y < \frac{\langle \eta +\eta^2\rangle_j}{\langle 1+\eta \rangle_j} (x_i^\sigma), \quad |\gamma^y|\leq 8\beta_1-1.\nonumber\end{aligned}$$ We use $\otimes$ to denote the tensor product and define $$\label{c5set}
S_{ij}=(S_i^x\otimes\hat S^y_j)\cup (\hat S_i^x \otimes S_j^y).$$ The main result can now be stated in the following.
\[th2dk2\]($k=2$) Consider the two dimensional DDG scheme (\[EFfully2D-\]) on rectangular meshes. Assume the mesh is regularly shaped, i.e., $ \kappa^{-1} \leq\frac{\Delta x}{\Delta y} \leq \kappa
$, for some constant $\kappa > 0$, and the flux parameters $(\beta_0, \beta_1)$ satisfy $$\label{beta2dp}
\beta_0 \geq 1+\frac{\kappa|c|}{2 \hat \omega^1 \min\{a, b\}}.
\quad\text{ and } \quad
\frac{1}{8} \leq \beta_1 \leq \frac{1}{4}$$ If $u_h^n(x, y)\in [c_1, c_2]$ for all $(x, y)\in S_{ij}$, then there exists $\mu_0>0$ such that if $\mu \leq \mu_0$ the cell average $
\bar u^{n+1}_{ij} \in [c_1, c_2].
$ More precisely, we have $$\begin{aligned}
\label{CFL2d-}
\mu_0 = \min_{i,j}\underline\omega_{ij}
\min\left\{\frac{\hat\omega^1}{\hat\omega^1\max\{a,b\}\big(\beta_0 + \frac{8\beta_1-2}{1+\gamma}\big) + \kappa|c|},
\frac{1-\gamma^2}{4\max\{a,b\}(1-4\beta_1)}\right\},\end{aligned}$$ where $\underline\omega_{ij}>0$ is defined in equation , and $\gamma = \max\{|\gamma^x|, |\gamma^y|\} \leq 8\beta_1-1$.
The proof is relegated to Appendix A. We proceed to deal with nonlinear diffusion equations in the next subsection.
Application to nonlinear diffusion equations {#ssec2DNoninearCD}
--------------------------------------------
This section is devoted to application to nonlinear diffusion equations of the form $$\partial_t u = \nabla\cdot(A\nabla u) \text{ with }
A(x,y,u) = \left(\begin{array}{cc} a & c \\ c & b \end{array}\right) \quad \text{ for } (x, y)\in \Omega \subset \mathbb{R}^2, \quad t>0,$$ subject to initial data $u_0(x, y)$ and periodic boundary conditions, and $A(x,y,u)$ is nonnegative definite. This type of model arises in a wide range of applications. Hence the DDG scheme can be formulated as follows: to find $u_h\in V_h$ such that for all $v\in V_h$, $$\begin{aligned}
\label{EFfully2D}
\int_{K_{ij}} u_{h}^{n+1}v\,dxdy
= &\int_{K_{ij}} u_{h}^{n}v\,dxdy
- \tau \int_{K_{ij}} A_h^n \nabla u_h^n\cdot\nabla v\,dxdy \notag \\
&+ \tau \int_{\partial K_{ij}} \{A_h^n\}\widehat{\nabla u_h^n}\cdot \nu v\,ds
+ \tau \int_{\partial K_{ij}}\{A_h^n\} {\nabla v}\cdot \nu (u_h^n - \{u_h^n\})\,ds, \end{aligned}$$ where $\nu$ is the outward unit normal to the cell boundary $\partial K_{ij}$, the numerical flux $
\widehat{\nabla u_h}
$ is defined in (\[df2\]), and $A_h^n=A(x, y, u_h^n)$.
The cell average evolves according to $$\bar u^{n+1}_{ij} = \bar u^{n}_{ij}
+ \mu_x\Delta x \left.\dashint_{J_j}\big(\{a_h^n\}\widehat{\partial_x u_h^n} + \{c_h^n\}\widehat{\partial_y u_h^n}\big)\,dy\right|_{\partial I_i}
+ \mu_y\Delta y \left.\dashint_{I_i}\big(\{b_h^n\}\widehat{\partial_y u_h^n} + \{c_h^n\}\widehat{\partial_x u_h^n}\big)\,dx\right|_{\partial J_j}.$$ That is $$\begin{aligned}
\label{EF2Dv1+}
\bar u^{n+1}_{ij} = \frac{\mu_x}{\mu}\dashint_{J_j}H_1(y)\,dy + \frac{\mu_y}{\mu}\dashint_{I_i} H_2(x)\,dx +B, \end{aligned}$$ where $\mu_{x} = \frac{\tau }{(\Delta x)^2}$, $\mu_{y} = \frac{\tau}{(\Delta y)^2}$ and $\mu = \mu_x + \mu_y$, with $$\begin{aligned}
& H_1(y) =\dashint_{I_i}u_{h}^n\,dx
+ \mu\Delta x\left. \{a_h^n\}\widehat{\partial_x u_h^n}\right|_{\partial I_i},\\
& H_2(x) =\dashint_{J_j}u_{h}^n\,dy
+ \mu\Delta y\left. \{b_h^n\}\widehat{\partial_y u_h^n} \right|_{\partial J_j},\\
& B = \frac{\tau}{|K_{ij}|} \left[ \int_{J_j}\{c_h^n\} \{\partial_y u_h\}dy |_{\partial I_i} +\int_{I_i}\{c_h^n\}\{\partial_x u_h\}dx |_{\partial J_j} \right].\end{aligned}$$
The main result can be stated in the following.
\[th2dk2+\]($k=2$) Consider the two dimensional DDG scheme (\[EFfully2D\]) on rectangular meshes. Assume the mesh is regularly shaped, i.e., $ \kappa^{-1} \leq\frac{\Delta x}{\Delta y} \leq \kappa
$, for some constant $\kappa > 0$, and the flux parameters $(\beta_0, \beta_1)$ satisfy $$\label{beta2d}
\beta_0 \geq 1+\frac{2\kappa \|c\|_\infty L(L-1) }{(1-\gamma) \min\{a, b\}},
\quad\text{ and } \quad
\frac{1}{8} \leq \beta_1 \leq \frac{1}{4}$$ and $\gamma = \max\{|\gamma^x|, |\gamma^y|\} \leq 8\beta_1-1$, where $$\|c\|_\infty = \max_{(x,y) \in\Omega, u\in[c_1, c_2]}|c(x,y,u)|, \quad \min\{a, b\} = \min_{(x,y) \in \Omega, u\in[c_1, c_2]}\big\{a(x,y,u), b(x,y,u)\big\}.$$ If $u_h^n(x, y)\in [c_1, c_2]$ for all $(x, y)\in S_{ij}$, then there exists $\mu_0>0$ such that if $\mu \leq \mu_0$ the cell average $
\bar u^{n+1}_{ij} \in [c_1, c_2].
$ More precisely, we have $$\begin{aligned}
\label{CFL2d}
\mu_0 = \min\left\{\frac{1-3\gamma}{6\max\{a,b\}\big(\beta_0 + \frac{8\beta_1-2}{1+\gamma}\big)(1-\gamma) + 12\kappa\|c\|_\infty L(L-1)},
\frac{1}{6\max\{a,b\}(1-4\beta_1)}\right\},\end{aligned}$$ where $$\max\{a, b\} = \max_{(x,y) \in\Omega, u\in[c_1, c_2]}\big\{a(x,y,u), b(x,y,u)\big\}.$$
The proof is relegated to Appendix B.
Scaling limiter and the MPS algorithm {#secLimiter}
=====================================
Scaling limiter
---------------
The one dimensional result in Theorem \[thk2max\] and the two dimensional result in Theorem \[th2dk2\] tell us that for the DDG scheme with forward Euler time discretization, we need to modify $u_h^n$ such that it is in $[c_1, c_2]$ on the test set $S=S_j$ or $S_{ij}$. In one dimensional case, we can use the following scaling limiter $$\label{reconMax}
\tilde{u}_h(x) = \theta \left(u_h(x) - \bar{u}_j\right) + \bar{u}_j \quad
\text{ with } \theta = \min\left\{1, \left|\frac{\bar{u}- c_1}{\bar{u}_j - m_1}\right|,
\left|\frac{c_2-\bar{u}_j}{m_2- \bar{u}_j}\right| \right\},$$ where $$\label{zetaMax}
m_1 = \min_{x\in S_j}u_h(x), \qquad m_2 = \max_{x\in S_j}u_h(x).$$ In the two dimensional case, $$\label{recon2D}
\tilde{u}_h(x, y) = \theta \left(u_h(x, y)-\bar{u}_{ij}\right) + \bar{u}_{ij} \quad \text{ where }
\theta = \min\left\{1, \Big| \frac{\bar{u}_{ij}-c_1}{\bar{u}_{ij} - m_1} \Big|, \frac{c_2-\bar{u}_{ij}}{m_2-\bar{u}_{ij}}\Big| \right\},$$ where $$\label{c12}
m_1 = \min\limits_{(x, y)\in S_{ij}} u_h(x, y)\quad \text{and}\quad m_2=\max\limits_{(x, y)\in S_{ij}} u_h(x, y).$$ The modified polynomials are indeed in $[c_1, c_2]$ and preserve the cell average. Moreover, following [@LY14] it can be shown that the above scaling limiters do not destroy the accuracy. We summarize this for two dimensional case only.
If $\bar{u}_{ij} \in (c_1, c_2)$, then the modified polynomial (\[recon2D\]) is as accurate as $u_h(x,y)$ to approximate $u(t,x,y)$ for the same $t$ in the following sense: $$|u_h(x, y)-\tilde u_h(x, y)| \leq C_k \|u_h(\cdot, \cdot) - u(t, \cdot, \cdot)\|_{\infty},$$ where $C_k$ is a constant depending on the polynomial degree $k$ and the weight function $M(x,y)$.
Algorithm
---------
The fact that we only require $u_h^n$ be in the desired range $[c_1, c_2]$ at certain points in $\cup_{i,j} S_{ij}$ can be used to reduce the computational cost in a great deal.
Given the weighted $L^2$ projection $u_h^0$ computed from the initial data $u_0(x, y)$, the algorithm is stated below:
1. Initialization.
Obtain $u_h^0\in V_h$ using the standard piecewise $L^2$ projection $$\int_{I_{ij}} u_0(x, y)\phi dx dy =\int_{I_{ij}} u_h^0\phi dx dy \quad {\rm for} \quad \phi \in V_h.$$
2. Time evolution.\
For $n = 0, 1, 2, ...$,
1. Check the point values of $u_h^n$ on the test set $S_{ij}$. If one of them goes outside of $[c_1, c_2]$, reconstruct $\hat{u}_h^n$ using the formula (\[recon2D\]) and (\[c12\]) and set $u_h^n = \hat{u}_h^n$.
2. Use the scheme (\[EFfully2D\]) to compute $u_h^{n+1}$.
End
This algorithm is guaranteed to produce numerical solutions within the range with uniform third order accuracy for smooth exact solutions.
The algorithm with forward Euler time discretization can be extended to high order ODE solves, such as the strong stability preserving Runge-Kutta methods, since they are a convex linear combination of the forward Euler; see [@ZS10]. The desired MPS property can be ensured as long as the proper time step restrictions are respected.
Numerical tests {#secNum}
===============
In this section, we present the results of numerical tests using our third-order maximum-principle-preserving DDG schemes. The numerical integration is computed using the Gaussian quadrature rule. Since $M(x)$ and $A(x)$ can be complex functions, sufficient number of quadrature points will guarantee the desired order of accuracy and induce small numerical errors. Hence we take $16$ quadrature points in each cell through all the examples. As for the time stepping, we implement the SSP(3,3) scheme as in [@GKS09] for the strong-stability-preservation in time. The one-dimensional error is measured by the discrete norms: $$\begin{aligned}
e^h_p(t) =\left(\sum_{j=1}^{N_x}\|u_h(t,\cdot) - u(t,\cdot)\|^p_{L^p(I_j)}\right)^{\frac1p}, \text{ for } p = 1 \text{ or } 2, \end{aligned}$$ and $u(t,x)$ is taken as the exact solution or the reference solution given by greatly refined spacial discretization. If neither of them is available, we can compute the consecutive errors between $u_h(t,x)$ and $u_{\frac h2}(t,x)$, where the subindex indicates the mesh size $h$ and $\frac h2$. We introduce $e^h_\infty(t)$ to demonstrate the discrete MPS property at the test points in $S_j$: $$\begin{aligned}
\label{einf}
e^h_\infty(t) = \max_{x \in S_j, 0 \leq j \leq N} \{c_1 - u_h(t, x), u_h(t, x) - c_2\},\end{aligned}$$ If $e^h_\infty(t)>0$, then the discrete MPS property is violated.
One dimensional numerical tests
-------------------------------
In our numerical tests we choose scheme parameters as $$\beta_0 = 2, \; \beta_1 = 0.16, \; \gamma = 0.1.$$
Accuracy test {#accuracy-test .unnumbered}
-------------
We construct a linear problem of form to demonstrate the third order accuracy of our numerical schemes: $$\label{Eg1D7}
\left\{\begin{array}{ll}
M(x)\partial_t u = \partial_x\big(A(x) \partial_xu\big), &\quad x \in [1, 3], \; t>0, \\
u(0, x)=\sin (x^2-1), &\quad x \in [1, 3]
\end{array}\right.$$ with $M(x) = 4xe^{-x^2+1}, A(x) = \frac{e^{-x^2+1}}{x}$. The exact slution is given by $$u(t,x) = \exp(-t)\sin(x^2-1-t).$$ The boundary condition is imposed by using the exact solution. We take the final time $t=0.1$ with different mesh sizes $N_x = 16, 32, 64$ and $128$. Fig. \[Eg1D7L2error\] shows the logarithm of the $L^2$ error for $u_h$, denoted by circles. One can observe the third order of accuracy for $u_h$.
![ The accuracy test on . The figure shows the logarithm of the $L^2$ error with different number of meshes for $u_h$, denoted by circles. One can observe the third order of accuracy for $u_h$.[]{data-label="Eg1D7L2error"}](Eg1D7L2error.eps){width="\textwidth"}
Porous medium equation {#porous-medium-equation .unnumbered}
----------------------
The porous medium equation $$\label{pm}
\partial_t u =\partial_x^2 (u^m), \quad m>1$$ is known to admit the Barenblatt solution of the form $$B_m(t, x)=t^{-\alpha}\left[ 1-\frac{\alpha(m-1)}{2m}\frac{|x|^2}{t^{2\alpha}} \right]_+^{\frac{1}{m-1}}, \quad \text{ with }\alpha = \frac{1}{m+1},$$ which is compactly supported.
![The accuracy test on with $m=5$ at $t = 0.1$. The error $e_p^h$ is computed by removing the two nonsmooth corners. The figure shows the logarithm of the $L^2$ error with different number of meshes for $u$, denoted by circles. One can observe the third order of accuracy for $u_h$.[]{data-label="Eg1D2m5L2error"}](Eg1D2m5L2error.eps){width="\textwidth"}
Fig. \[Eg1D2m5L2error\] provides the accuracy test on with $m=5$ at $t = 0.1$. Removing the two non-smooth corners, one can observe the third-order accuracy for the numerical solutions of $u_h$.
We compute the numerical solution with initial data $B_m(1,x)$ subject to zero boundary conditions for $m=2$, up to final time $t=3$. From the numerical results in Fig. \[Eg1D2\](a) with the MPS limiter we see a sharp resolution of discontinuities, and keeping the solution strictly within the initial bounds everywhere for all time. Fig. \[Eg1D2\](b) shows a zoom-in at the nonsmooth corner for $x \in [-6, -4]$ where the solution $u$ is well simulated. In contrast, without MPS limiter, it brings in significant overshoots near the upper bound of the exact solution, as evidenced by oscillations already appearing at $t=1.0025$ in Fig. \[Eg1D2\](c). In addition, the scheme without the MPS limiter will blow up in a short time.
[0.32]{} ![The numerical solution to with $m = 2$ and $N_x = 200, \Delta t = 0.0001$. $\mu_0 \approx 3.66\times10^{-2}$. Fig. \[Eg1D2uh\_t3\] shows $u_h(t=3,x)$ (circles) with the MPS limiter against the exact solution (solid lines). Fig. \[Eg1D2nolimiter\] shows the numerical solution without the MPS limiter at $t = 1.0025$, zoomed in $[-1,1]$. This numerical solution blows up shortly. []{data-label="Eg1D2"}](Eg1D2uh_t3.eps "fig:"){width="\textwidth"}
[0.32]{} ![The numerical solution to with $m = 2$ and $N_x = 200, \Delta t = 0.0001$. $\mu_0 \approx 3.66\times10^{-2}$. Fig. \[Eg1D2uh\_t3\] shows $u_h(t=3,x)$ (circles) with the MPS limiter against the exact solution (solid lines). Fig. \[Eg1D2nolimiter\] shows the numerical solution without the MPS limiter at $t = 1.0025$, zoomed in $[-1,1]$. This numerical solution blows up shortly. []{data-label="Eg1D2"}](Eg1D2m2uhcorner_t3.eps "fig:"){width="\textwidth"}
[0.32]{} ![The numerical solution to with $m = 2$ and $N_x = 200, \Delta t = 0.0001$. $\mu_0 \approx 3.66\times10^{-2}$. Fig. \[Eg1D2uh\_t3\] shows $u_h(t=3,x)$ (circles) with the MPS limiter against the exact solution (solid lines). Fig. \[Eg1D2nolimiter\] shows the numerical solution without the MPS limiter at $t = 1.0025$, zoomed in $[-1,1]$. This numerical solution blows up shortly. []{data-label="Eg1D2"}](Eg1D2uhnolimiter_t0025.eps "fig:"){width="\textwidth"}
The Buckley-Leverett equation {#the-buckley-leverett-equation .unnumbered}
-----------------------------
The convection-diffusion Buckley-Leverett equation of the form $$\label{bl}
\partial_t u +\partial_x f(u) =\varepsilon \partial_x (\nu(u) \partial_x u),$$ is a model often used in reservoir simulations; see [@KT00]. Here $\varepsilon>0$ is a small parameter, $f$ has an s-shape: $$f(u)=\frac{u^2}{u^2+(1-u)^2},$$ and $$\nu(u)=4 u(1-u)1_{0\leq u\leq 1}.$$ We numerically solve (\[bl\]) with $\varepsilon=0.01$, subject to the following initial and boundary conditions $$\label{bldata}
u(0, x)=(1-3x)1_{0\leq x\leq 1/3}, \quad u(t, 0)=1, \quad u(t, 1)=0.$$ The exact solution is not available. With numerical convergence we demonstrate that our numerical scheme is capable of simulating the sharp corner of the solution that is moving in time. From the results in Fig. \[Eg1D3uh\] we observe the numerical convergence when the spacial mesh is refined. Moreover, the lower bound of $u_h(t,x)$ is well preserved around the corner point $x = 0.5$. We note that the numerical solution here is comparable to that obtained in [@KT00] by the second order central scheme.
![The numerical solution of problem and for $t = 0.2$ with $N_x = 36, 72, 144, 288$.[]{data-label="Eg1D3uh"}](Eg1D3uhNx.eps){width="40.00000%"}
Two dimensional numerical tests
-------------------------------
We consider the convection-diffusion equation: $$\label{aniCD}
\partial_t u + \nabla\cdot f(u) = \nabla\cdot(A\nabla u) \text{ for } (x, y) \in [-1, 1]\times[-1,1] \text{ and } t>0,$$ where $$\begin{aligned}
f(u) = u\vec v, \quad \vec v:=(0.01,0.01)^\top \end{aligned}$$ and $A$ is a symmetric, positive definite matrix. For such an equation, exact solutions can be found of the form $u(t,x,y)=a(t)\exp(-\xi^\top B(t) \xi)$ with $\xi=\boldsymbol{x}-\vec vt$, $\boldsymbol{x}:= (x,y)^\top$, provided $a'=-a{\rm tr} (AB)$ and $B'=-2B^\top AB$. Here $a(0)$ can be chosen small enough to ensure that the periodic boundary condition adopted is reasonable and accurate at a finite time. In fact, if we set $\sigma_0 = 0.01$ and a $2\times2$ matrix $\sigma(t)$ be such that $$\sigma(t) = \sigma_0^2{\rm{Id}} + 2At \quad \text{ with } \rm{Id} \text{ being the identity matrix.}$$ Then the function $$u(t,x,y) = \frac{\sigma_0^2}{|\det(\sigma)|^\frac12}\exp\left(-\frac{(\boldsymbol{x}-\vec vt)^T\sigma^{-1}(\boldsymbol{x}-\vec vt)}{2}\right), \quad \boldsymbol{x}:= (x,y)^\top$$ as given in [@CSG16], is a solution to equation .
In our numerical tests, we take three choices of the tensor $A$ as $$A =
\left(\begin{array}{cc}
1 & 0 \\ 0 & 1
\end{array}\right),
\left(\begin{array}{cc}
1 & 0 \\ 0 & 2
\end{array}\right)
\text{ or }
\left(\begin{array}{cc}
1 & 1 \\ 1 & 2
\end{array}\right),$$ which are usually denoted as isotropic, diagonally anisotropic and fully anisotropic diffusion tensors.
We begin to demonstrate the necessity of the MPS limiter using the first isotropic problem. Table \[MPSlimiter\] shows the minumum and maximum of the numerical solutions $u_h(t,x,y)$ using the MPS limiter over all the points in the test sets $S_{ij}$. They are well bounded in the interval $[0,1]$, satisfying the maximum principle. Table \[Nolimiter\] shows the error $e_{\infty}^h(t)$ in without the MPS limiter. One can observe that the numerical solutions $u_h(t,x,y)$ violates the maximum principle and the simulation will break down after some time. Larger $\beta_0$ helps to suppress the overshoot or undershoot, but cannot realize the bound preservation ideally.
$t$ $\min(u)$ $\max(u)$
------------------ ----------- -----------
0 0 1
$2\times10^{-6}$ 0 0.96111
$4\times10^{-6}$ 0 0.92156
$1\times10^{-5}$ 0 0.81327
$2\times10^{-5}$ 0 0.68236
: No limiter: $e_{\infty}^h(t)$[]{data-label="Nolimiter"}
$t$ $\beta_0 = 2$ $\beta_0=4$
------------------ --------------- -------------
0 1.705E-005 1.705E-005
$2\times10^{-6}$ 7.023E-004 4.168E-004
$4\times10^{-6}$ 1.339E-003 6.838E-004
$6\times10^{-6}$ 1.874E-003 6.387E-004
$8\times10^{-6}$ 2.403E-003 4.414E-004
: No limiter: $e_{\infty}^h(t)$[]{data-label="Nolimiter"}
In the first two cases, we take $\beta_0 = 2, \beta_1 = 0.16, \gamma = 0.1$ for both variables $x$ and $y$. For the fully anisotropic one, we take $\beta_0 = 4$. For smaller $\beta_0$, small oscillations develop in time and the scheme can become unstable. We observe a nice agreement of the numerical solution to the exact solution for all three cases of $A$.
[0.45]{} ![The contours of solutions to with the first choice of the matrix $A$ and $N_x = N_y = 200, \Delta t = 10^{-6}$. $\mu_0 \approx 4.62\times10^{-3}$. The final time $t = 0.0381$. Fig. \[Eg2DIso1ex\] shows the exact solution. Fig. \[Eg2DIso1\] shows the numerical solution with the MPS limiter. ](Eg2DIso1uex_t0_381.eps "fig:"){width="\textwidth"}
[0.45]{} ![The contours of solutions to with the first choice of the matrix $A$ and $N_x = N_y = 200, \Delta t = 10^{-6}$. $\mu_0 \approx 4.62\times10^{-3}$. The final time $t = 0.0381$. Fig. \[Eg2DIso1ex\] shows the exact solution. Fig. \[Eg2DIso1\] shows the numerical solution with the MPS limiter. ](Eg2DIso1uh_t0_381.eps "fig:"){width="\textwidth"}
[0.45]{} ![The contours of solutions to with the second choice of the matrix $A$ and $N_x = N_y = 200, \Delta t = 10^{-6}$. $\mu_0 \approx 4.62\times10^{-3}$. The final time $t = 0.03485$. Fig. \[Eg2DIso2ex\] shows the exact solution. Fig. \[Eg2DIso2\] shows the numerical solution with the MPS limiter. ](Eg2DIso2uex_t0_3485.eps "fig:"){width="\textwidth"}
[0.45]{} ![The contours of solutions to with the second choice of the matrix $A$ and $N_x = N_y = 200, \Delta t = 10^{-6}$. $\mu_0 \approx 4.62\times10^{-3}$. The final time $t = 0.03485$. Fig. \[Eg2DIso2ex\] shows the exact solution. Fig. \[Eg2DIso2\] shows the numerical solution with the MPS limiter. ](Eg2DIso2uh_t0_3485.eps "fig:"){width="\textwidth"}
For the equation with the third choice of the matrix $A$, we take care of the anisotropy by using a larger $\beta_0 = 4$, and adpative mesh size and time steps. In the beginning, the solution is highly concentrated around the origin and requires a good resolution. Therefore, for $t \in [0, 10^{-5}]$, we employ the discretization with $\Delta x = \Delta y = 0.005, \Delta t = 10^{-7}$. Afterwards, a coarser mesh with $\Delta x = \Delta y = 0.01, \Delta t = 10^{-6}$ is used. For time $t = 0.01$, Fig. \[Eg2DAni\](a-b) show the contours of the exact solution and the numerical solution with the MPS limiter. Fig. \[Eg2DAni\](c) shows a slice of the exact solution (circles) and the numerical solution (crosses) for $y = 0$. One can observe a good agreement of the two solutions.
[0.3]{} ![ with the third choice of the matrix $A$ and adaptive mesh size and time steps. $\mu_0 \approx 4.04\times10^{-3}$. In the beginning, the solution is highly concentrated around the origin and requires a good resolution. Therefore, for $t \in [0, 10^{-5}]$, we employ the discretization with $\Delta x = \Delta y = 0.005, \Delta t = 10^{-7}$. Afterwards, a coarser mesh with $\Delta x = \Delta y = 0.01, \Delta t = 10^{-6}$ is used. The final time $t = 0.01$. Fig. \[Eg2DAniuex\] shows the exact solution. Fig. \[Eg2DAniuh\] shows the numerical solution with the MPS limiter. Fig. \[Eg2DAniucut\] shows the good agreement between the exact solution (circles) and the numerical solution (crosses) for $y = 0$. []{data-label="Eg2DAni"}](Eg2DAniuex_t0_01.eps "fig:"){width="\textwidth"}
[0.3]{} ![ with the third choice of the matrix $A$ and adaptive mesh size and time steps. $\mu_0 \approx 4.04\times10^{-3}$. In the beginning, the solution is highly concentrated around the origin and requires a good resolution. Therefore, for $t \in [0, 10^{-5}]$, we employ the discretization with $\Delta x = \Delta y = 0.005, \Delta t = 10^{-7}$. Afterwards, a coarser mesh with $\Delta x = \Delta y = 0.01, \Delta t = 10^{-6}$ is used. The final time $t = 0.01$. Fig. \[Eg2DAniuex\] shows the exact solution. Fig. \[Eg2DAniuh\] shows the numerical solution with the MPS limiter. Fig. \[Eg2DAniucut\] shows the good agreement between the exact solution (circles) and the numerical solution (crosses) for $y = 0$. []{data-label="Eg2DAni"}](Eg2DAniuh_t0_01.eps "fig:"){width="\textwidth"}
[0.28]{} ![ with the third choice of the matrix $A$ and adaptive mesh size and time steps. $\mu_0 \approx 4.04\times10^{-3}$. In the beginning, the solution is highly concentrated around the origin and requires a good resolution. Therefore, for $t \in [0, 10^{-5}]$, we employ the discretization with $\Delta x = \Delta y = 0.005, \Delta t = 10^{-7}$. Afterwards, a coarser mesh with $\Delta x = \Delta y = 0.01, \Delta t = 10^{-6}$ is used. The final time $t = 0.01$. Fig. \[Eg2DAniuex\] shows the exact solution. Fig. \[Eg2DAniuh\] shows the numerical solution with the MPS limiter. Fig. \[Eg2DAniucut\] shows the good agreement between the exact solution (circles) and the numerical solution (crosses) for $y = 0$. []{data-label="Eg2DAni"}](Eg2DAniucut_t0_01.eps "fig:"){width="\textwidth"}
Concluding remarks {#secConclude}
==================
In this paper, we present third order accurate DDG schemes which can be proven maximum-principle-satisfying for a class of diffusion equations with variable diffusivity in terms of spatial variables and/or the unknown, in both one and two dimensional settings. Through careful theoretical analysis and numerical tests, we show that under suitable CFL conditions, with a simple scaling limiter involving little additional computational cost, the numerical schemes satisfy the strict maximum principle while maintaining uniform third order accuracy. The methodology extends to three dimensional rectangular meshes as well. The effectiveness of the maximum-principle-satisfying DG schemes has been demonstrated through extensive numerical examples.
The presence of variable diffusivity has led to slightly more restrictive CFL conditions in order to preserve the maximum principle. In two dimensional case, the CFL condition seems a main factor for the slow numerical time evolution. It would be interesting to extend the present result to implicit schemes so to improve the computational efficiency.
Proof of Theorem \[th2dk2\]
===========================
Upon regrouping, we decompose the right-hand side of (\[c5EF2DHQ\]) as $$\begin{aligned}
\label{EF2DHQ}
\langle u^{n+1}\rangle_{ij}
& = \frac{\mu_x}{\mu}\sum_{\sigma=2}^{L-1} \hat\omega^{\sigma}H_1(\hat y_j^\sigma) + \frac{\mu_y}{\mu}\sum_{\sigma=2}^{L-1} \hat\omega^{\sigma}H_2(\hat x_i^\sigma) \\ \notag
& \quad +\frac{\mu_x \hat \omega_1}{\mu} \big(H_1(y_{j-\frac12})+H_1(y_{j+\frac12}) \big) + \frac{\mu_y\hat \omega_1}{\mu} \big(H_2(x_{i-\frac12})+H_2(x_{i+\frac12}) \big) + B. \notag \end{aligned}$$ Here $\hat\omega_1 = \hat\omega_L =\frac{1}{L(L-1)}$ is used. From we see that $$H_1(\hat y_j^\sigma)=R_{i}(M(\cdot,\hat y_j^\sigma), \mu, \Delta x, a), \quad H_2(\hat x_i^\sigma)=R_j(M(\hat x_i^\sigma, \cdot), \mu, \Delta y, b)$$ for $1\leq \sigma \leq L$. Notice that the terms in $B$, induced from the nontrivial $c$, involve only polynomials values at four vertices of $K_{ij}$, we proceed to regroup and combine them with $H_1(y_{j\pm\frac12})$ and $H_2(x_{i\pm\frac12})$, respectively, in the following way: $$\begin{aligned}
B_1 & = \frac{c\tau}{2\Delta x\Delta y}\big(u_h(x_{i-\frac12}^-, y_{j-\frac12}^+) + u_h(x_{i-\frac12}^+, y_{j-\frac12}^+) - u_h(x_{i+\frac12}^-, y_{j-\frac12}^+) - u_h(x_{i+\frac12}^+, y_{j-\frac12}^+)\big) \; \text{with}\; H_1(y_{j-\frac12}), \\
B_2 & = \frac{c\tau}{2\Delta x\Delta y}\big(u_h(x_{i+\frac12}^-, y_{j+\frac12}^-) - u_h(x_{i-\frac12}^+, y_{j+\frac12}^-) + u_h(x_{i+\frac12}^+, y_{j+\frac12}^-) - u_h(x_{i-\frac12}^-, y_{j+\frac12}^-)\big) \; \text{with}\; H_1(y_{j+\frac12}), \\
B_3 & = \frac{c\tau}{2\Delta x\Delta y}\big(u_h(x_{i-\frac12}^+, y_{j-\frac12}^+) - u_h(x_{i-\frac12}^-, y_{j+\frac12}^-) + u_h(x_{i-\frac12}^+, y_{j-\frac12}^-) - u_h(x_{i-\frac12}^+, y_{j+\frac12}^+)\big) \; \text{with}\; H_2(x_{i-\frac12}), \\
B_4 & = \frac{c\tau}{2\Delta x\Delta y}\big(u_h(x_{i+\frac12}^-, y_{j-\frac12}^+) - u_h(x_{i+\frac12}^-, y_{j+\frac12}^-) + u_h(x_{i+\frac12}^-, y_{j-\frac12}^-) - u_h(x_{i+\frac12}^-, y_{j+\frac12}^+)\big) \; \text{with}\; H_2(x_{i+\frac12}). \end{aligned}$$ We shall also use the following notations. $$\begin{aligned}
\begin{array}{ll}
\tilde\omega_{i}^{x,1}(\gamma^x,y) = \frac{\langle \gamma^x-\xi(1+\gamma^x) + \xi^2\rangle_i(y)}{2(1+\gamma^x)},
&\tilde\omega_{j}^{y,1}(\gamma^y,x) = \frac{\langle \gamma^y-\eta(1+\gamma^y) + \eta^2\rangle_j(x)}{2(1+\gamma^y)},\\
\tilde\omega_{i}^{x,2}(\gamma^x,y) = \frac{\langle 1 - \xi^2\rangle_i(y)}{1-(\gamma^x)^2},
&\tilde\omega_{j}^{y,2}(\gamma^y,x) = \frac{\langle 1 - \eta^2\rangle_j(x)}{1-(\gamma^y)^2}, \\
\tilde\omega_{i}^{x,3}(\gamma^x,y) = \frac{\langle -\gamma^x+\xi(1-\gamma^x) + \xi^2\rangle_i(y)}{2(1-\gamma^x)},
&\tilde\omega_{j}^{y,3}(\gamma^y,x) = \frac{\langle -\gamma^y+\eta(1-\gamma^y) + \eta^2\rangle_j(x)}{2(1-\gamma^y)}.
\end{array}\end{aligned}$$ For the first group, we have $$\begin{aligned}
&H_1(y_{j-\frac12}) + \frac{\mu}{\mu_x\hat\omega_1}B_1 = R_i(M(\cdot, y_{j-\frac12}), \mu, \Delta x, a) + \frac{\mu}{\mu_x\hat\omega_1}B_1 \notag\\
=& \left[\tilde{\omega}_{i}^{x,1}(\gamma^x, y_{j-\frac12}) - \mu a \big(\alpha_3(-\gamma^x) +\alpha_1(\gamma^x)\big) + \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1} \right]u_h(x_{i-\frac12}^+, y_{j-\frac12}^+) \nonumber\\
&+\left[\tilde{\omega}_{i}^{x,2}(\gamma^x, y_{j-\frac12}) - \mu a \big(\alpha_2(-\gamma^x) +\alpha_2(\gamma^x)\big)\right]u_h(x_i^\gamma,y_{j-\frac12}^+)\nonumber\\
&+\left[\tilde{\omega}_{i}^{x,3}(\gamma^x, y_{j-\frac12}) - \mu a \big(\alpha_1(-\gamma^x) +\alpha_3(\gamma^x)\big) - \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right]u_h(x_{i+\frac12}^-, y_{j-\frac12}^+) \nonumber\\
&+\left(\mu a\alpha_3(-\gamma^x) - \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right)u_h(x_{i+\frac12}^+, y_{j-\frac12}^+)\nonumber \\
& +\mu a\left[\alpha_2(-\gamma^x)u_h(x_{i+1}^\gamma,y_{j-\frac12}^+)+\alpha_1(-\gamma^x)u_h(x_{i+\frac32}^-, y_{j-\frac12}^+) \right]\nonumber \\
&+\mu a\left[\alpha_1(\gamma^x) u_h(x_{i-\frac32}^+, y_{j-\frac12}^+)+\alpha_2(\gamma^x)u_h(x_{i-1}^\gamma,y_{j-\frac12}^+) \right]\nonumber \\
& + \left(\mu a\alpha_3(\gamma^x) + \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right)u_h(x_{i-\frac12}^-, y_{j-\frac12}^+),\end{aligned}$$ and the second group reduces to $$\begin{aligned}
&H_1(y_{j+\frac12}) + \frac{\mu}{\mu_x\hat\omega_1}B_2 = R_i(M(\cdot, y_{j+\frac12}), \mu, \Delta x, a) + \frac{\mu}{\mu_x\hat\omega_1}B_1 \notag\\
=& \left[\tilde{\omega}_{i}^{x,1}(\gamma^x, y_{j+\frac12}) - \mu a \big(\alpha_3(-\gamma^x) +\alpha_1(\gamma^x)\big) - \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1} \right]u_h(x_{i-\frac12}^+, y_{j+\frac12}^-) \nonumber\\
&+\left[\tilde{\omega}_{i}^{x,2}(\gamma^x, y_{j+\frac12}) - \mu a \big(\alpha_2(-\gamma^x) +\alpha_2(\gamma^x)\big)\right]u_h(x_i^\gamma,y_{j+\frac12}^-)\nonumber\\
&+\left[\tilde{\omega}_{i}^{x,3}(\gamma^x, y_{j+\frac12}) - \mu a \big(\alpha_1(-\gamma^x) +\alpha_3(\gamma^x)\big) + \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right]u_h(x_{i+\frac12}^-, y_{j+\frac12}^-) \nonumber\\
&+\left(\mu a\alpha_3(-\gamma^x) + \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right)u_h(x_{i+\frac12}^+, y_{j+\frac12}^-) \nonumber \\
& +\mu a\left[\alpha_2(-\gamma^x)u_h(x_{i+1}^\gamma,y_{j+\frac12}^-)+\alpha_1(-\gamma^x)u_h(x_{i+\frac32}^-, y_{j+\frac12}^-) \right]\nonumber \\
&+\mu a\left[\alpha_1(\gamma^x) u_h(x_{i-\frac32}^+, y_{j+\frac12}^-)+\alpha_2(\gamma^x)u_h(x_{i-1}^\gamma,y_{j+\frac12}^-) \right]\nonumber \\
& +
\left(\mu a\alpha_3(\gamma^x) - \frac{c\tau\mu}{2\Delta x\Delta y\mu_x\hat\omega_1}\right)u_h(x_{i-\frac12}^-, y_{j-\frac12}^+). \nonumber\end{aligned}$$ From the above two groups we see that all coefficients of solution values involved are nonnegative if
\[cfl1\] $$\begin{aligned}
& a\alpha_3(\pm\gamma^x) - \frac{\Delta x|c|}{2\Delta y\hat\omega_1} \geq 0, \\
& \mu \leq \min\left\{\frac{\tilde\omega_{i}^{x,1}(\pm\gamma^x, y_{j\pm\frac12})}{a\big(\alpha_1(\pm\gamma^x) + \alpha_3(\mp\gamma^x)\big) + \frac{\Delta x |c|}{\Delta y\hat\omega_1}},
\frac{\tilde{\omega}_{i}^{x,2}(\gamma^x, y_{j\pm \frac12})}{2\alpha_2(\gamma^x)}\right\}.\end{aligned}$$
Here we used $\tilde{\omega}_{i}^{x,3}(\gamma^x, y) = \tilde{\omega}_{i}^{x,1}(-\gamma^x, y)$ and $\alpha_2(\gamma^x) = \alpha_2(-\gamma^x)$.
Similarly, all coefficients of solution values in $$\begin{aligned}
H_2(x_{i-\frac12}) + \frac{\mu}{\mu_y\hat\omega_1}B_3 = R_j(M(x_{i-\frac12}, \cdot), \mu, \Delta y, b) + \frac{\mu}{\mu_y\hat\omega_1}B_3 \end{aligned}$$ and $$\begin{aligned}
H_2(x_{i+\frac12}) + \frac{\mu}{\mu_y\hat\omega_1}B_4 = R_j(M(x_{i+\frac12}, \cdot), \mu, \Delta y, b) + \frac{\mu}{\mu_y\hat\omega_1}B_4\end{aligned}$$ are also nonnegative if
\[cfl2\] $$\begin{aligned}
& b\alpha_3(\pm\gamma^y) - \frac{\Delta y|c|}{2\Delta x\hat\omega_1} \geq 0,\\
& \mu \leq \min\left\{\frac{\tilde\omega_{j}^{y,1}(\pm\gamma^y, x_{i\pm\frac12})}{b\big(\alpha_1(\pm\gamma^y) + \alpha_3(\mp\gamma^y)\big) + \frac{\Delta y |c|}{\Delta x\hat\omega_1}},
\frac{\tilde\omega_{j}^{y,2}(\gamma^y, x_{i\pm\frac12})}{2\alpha_2(\gamma^y)}\right\}.\end{aligned}$$
Here we used $\tilde{\omega}_{j}^{y,3}(\gamma^y, x) = \tilde{\omega}_{j}^{y,1}(-\gamma^y,x)$ and $\alpha_2(\gamma^y) = \alpha_2(-\gamma^y)$.
Since $\tilde\omega_i^{x,\sigma}, \tilde\omega_j^{y,\sigma}$ for $\sigma = 1, 2, 3$ only depend on $M(x,y)|_{K_{ij}}$ and $\gamma$, therefore bounded from below. We set such bound as $$\label{min_omega}
\underline\omega_{ij} = \min_{\gamma^x, \gamma^y, x\in \hat S_{i}^x, y\in \hat S_{j}^y}\{\tilde\omega_{i}^{x,1}(\pm\gamma^x, y),
\tilde\omega_{i}^{x,2}(\gamma^x,y),
\tilde\omega_{j}^{y,1}(\pm\gamma^y, x),
\tilde\omega_{j}^{y,2}(\gamma^y, x)\}.$$ Notice also that for $\gamma = \max\{|\gamma^x|, |\gamma^y|\}$, using $\beta_1\leq 1/4$, we have $$\alpha_1(\pm\gamma^x) + \alpha_3(\mp\gamma^x) = \beta_0 + \frac{8\beta_1-2}{1\pm\gamma^x} \leq \beta_0 + \frac{8\beta_1-2}{1-\gamma},$$ which also hods when $\gamma^x$ is replaced by $\gamma^y$. Hence both (\[cfl1\]b) and (\[cfl2\]b) are ensured by .
Observe that (\[cfl1\]a) and (\[cfl2\]a) are implied by $$\beta_0 +\min_{s\in [-\gamma, \gamma]}\frac{8\beta_1-3+s}{2(1-s)} \geq \frac{|c|\kappa}{2\hat \omega_1 \min\{a, b\}}.$$ Using $\beta_1 \leq 1/4$ we see that the minimum on the left hand side is $-1$, obtained at $s=\gamma$ when $\gamma = 8\beta_1-1$, hence this relation gives the lower bound in (\[beta2dp\]).
Proof of Theorem \[th2dk2+\]
=============================
For simplicity of presentation, in the following we consider $A=A(x, y)$ instead of $A(x, y, u)$, for which we only need to use $\{A_h\}$ on interfaces, so that $c=c(x, y)$, $a=a(x, y)$ and $b=b(x, y)$.
Since $u_h(x,y)$ is quadratic in terms of $x$ and $y$ respectively, we can use the formula (\[pj\]) to obtain $$\begin{aligned}
\dot p(\eta)=&\dot\omega^1(\eta)p(-1) + \dot\omega^2(\eta)(\gamma) + \dot\omega^3(\eta)p(1) \\
=& \frac{2\eta -1-\gamma}{2(1+\gamma)}{p}(-1)+\frac{2\eta }{(\gamma^2 -1)}{p}(\gamma)+\frac{2\eta +1-\gamma}{2(1-\gamma)}{p}(1).\end{aligned}$$ Here $\eta$ is the variable in the reference element $[-1,1]$, and $\dot\omega^\sigma(\eta) = \frac{d}{d\eta}\omega^\sigma(\eta)$. Then $$\begin{aligned}
\int_{J_j} c(x,y)\{\partial_y u_h(x, y)\}dy &= \{u_h(x, y_{j-\frac12}^+)\} \int_{-1}^1 c\left(x,y_j+\frac{\Delta y}{2}\eta \right)\dot\omega^1(\eta)\,d\eta \\
&\quad +\{u_h(x, y^\gamma_{j})\} \int_{-1}^1 c\left(x, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^2(\eta)\,d\eta \\
&\quad +\{u_h(x, y_{j+\frac12}^-)\} \int_{-1}^1 c\left(x, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^3(\eta)\,d\eta.\end{aligned}$$ Here the average $\{\cdot\}$ is taken with respect to $x^-$ and $x^+$. Using the quadrature rule on the right-hand side of (\[EF2Dv1+\]), we obtain the following scheme $$\label{c5EF2DHQ+}
\bar u^{n+1}_{ij}= \frac{\mu_x}{\mu}\sum_{\sigma=1}^L \hat\omega^{\sigma}H_1(\hat y_j^\sigma) + \frac{\mu_y}{\mu}\sum_{\sigma=1}^L \hat\omega^{\sigma}H_2(\hat x_i^\sigma)+B.$$ The test set now reduces to $$\begin{aligned}
& S^x_i = \{x_{i-\frac12}, x_i^{\gamma}, x_{i+\frac12}\} = x_i + \frac{\Delta x}{2}\{-1, \gamma, 1\}, \\
& S^y_j = \{y_{j-\frac12}, y_j^{\gamma}, y_{j+\frac12}\} = y_j + \frac{\Delta y}{2}\{-1, \gamma, 1\}\end{aligned}$$ with $\gamma$ satisfying $$\begin{aligned}
\label{c5angle2D+}
&|\gamma| \leq \frac13 \; \text{and} \; |\gamma|\leq 8\beta_1-1.\end{aligned}$$ It can be shown that there exists $L \geq \frac{2k+3}{2}$ and $\gamma$ satisfying (\[c5angle2D+\]) such that $x_i^\gamma \in \hat S^x_i$ and $y_j^\gamma\in \hat S^y_j$, the corresponding quadrature weight is denoted by $\hat \omega^*$. Upon regrouping, the right-hand side of (\[c5EF2DHQ+\]) may be decomposed as $$\begin{aligned}
\bar u^{n+1}_{ij}
= &\quad \frac{\mu_x}{\mu}\sum_{\sigma\neq 1, *, L} \hat\omega^{\sigma}H_1(\hat y_j^\sigma) + \frac{\mu_y}{\mu}\sum_{\sigma\neq1,*,L} \hat\omega^{\sigma}H_2(\hat x_i^\sigma) \\
& + \frac{\mu_x}{\mu} \big(\hat \omega^1 H_1(y_{j-\frac12})+ \hat \omega^* H_1(y_j^\gamma)+ \hat \omega^1 H_1(y_{j+\frac12}) \big) \\
& + \frac{\mu_y}{\mu} \big(\hat \omega^1 H_2(x_{i-\frac12})+ \hat \omega^* H_2(x_i^\gamma)+ \hat \omega^1 H_2(x_{i+\frac12}) \big) + B. \end{aligned}$$ Here $\hat\omega^1 = \hat\omega^L$ is used. The terms in $B$ will be regrouped correspondingly. More precisely, the first term involving integration on $J_j$ is regrouped in terms involving solution values at $y_{j-\frac12}^+, y_j^\gamma$ and $y_{j+\frac12}^-$, and combined with $H_1(y_{j-\frac12}), H_1(y_j^\gamma)$ and $H_1(y_{j+\frac12})$, respectively. We check upon the first group only: $$\begin{aligned}
&H_1(y_{j-\frac12}) + \frac{\mu}{\mu_x\hat\omega^1}B_1 = R_i(M(\cdot, y_{j-\frac12}), \mu, \Delta x, a) + \frac{\mu}{\mu_x\hat\omega^1}B_1 \\
=&\left[\omega^1 - \mu a \big(\alpha_3(-\gamma) +\alpha_1(\gamma)\big) - \frac{\mu\Delta x}{2\Delta y\hat\omega^1}
\int_{-1}^1c\left(x_{i-\frac12}, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^1(\eta)\,d\eta \right]u_h(x_{i-\frac12}^+, y_{j-\frac12}^+) \\
&+\left[\omega^2 - \mu a \big(\alpha_2(-\gamma) +\alpha_2(\gamma)\big)\right]u_h(x_i^\gamma,y_{j-\frac12}^+)\\
&+\left[\omega^3 - \mu a \big(\alpha_1(-\gamma) +\alpha_3(\gamma)\big) +\frac{\mu\Delta x}{2\Delta y\hat\omega^1}
\int_{-1}^1c\left(x_{i+\frac12}, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^1(\eta)\,d\eta \right]u_h(x_{i+\frac12}^-, y_{j-\frac12}^+) \\
&+\left(\mu a\alpha_3(-\gamma) +\frac{\mu\Delta x}{2\Delta y\hat\omega^1}\int_{-1}^1c\left(x_{i+\frac12}, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^1(\eta)\,d\eta
\right)u_h(x_{i+\frac12}^+, y_{j-\frac12}^+) \\
&+\mu a\left[\alpha_2(-\gamma)u_h(x_{i+1}^\gamma, y_{j-\frac12}^+)+\alpha_1(-\gamma)u_h(x_{i+\frac32}^-,y_{j-\frac12}^+) \right] \\
&+\mu a\left[\alpha_1(\gamma) u_h(x_{i-\frac32}^+, y_{j-\frac12}^+)+\alpha_2(\gamma)u_h(x_{i-1}^\gamma, y_{j-\frac12}^+) \right] \\
&+\left(\mu a\alpha_3(\gamma) -\frac{\mu\Delta x}{2\Delta y\hat\omega^1}\int_{-1}^1c\left(x_{i-\frac12}, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^1(\eta)\,d\eta \right)
u_h(x_{i-\frac12}^-, y_{j-\frac12}^+),\end{aligned}$$ From the three groups of $H_1(y_{j-\frac12}), H_1(y_j^\gamma)$ and $H_1(y_{j+\frac12})$, we see that all the coefficients of solutions values are nonnegative if $$\begin{aligned}
& a\alpha_3(\pm\gamma) - \frac{\Delta x}{2\Delta y \min\{\hat\omega_1, \hat \omega^*\}}
\max_{x \in S_i^x; \sigma}
\left|g_j^\sigma(x) \right| \geq 0,\\
& \mu \leq \min\left\{\frac{\min\{\omega^1,\omega^3\}}{a\big(\alpha_1(\pm\gamma) + \alpha_3(\mp\gamma)\big) + \frac{\Delta x}{2\Delta y \min\{\hat\omega_1, \hat \omega^*\} }\max_{x \in S_i^x; \sigma} \left|g_j^\sigma(x) \right|},
\frac{\omega^2}{2a\alpha_2(\gamma)}\right\},\end{aligned}$$ where $g_j^\sigma(x)=\int_{-1}^1 c\left(x, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^\sigma(\eta)\,d\eta$. The rest of terms in $B$ when combined with $H_2(y)$ for $y\in S_j^y$ led to the following conditions $$\begin{aligned}
& b \alpha_3(\pm\gamma) - \frac{\Delta y}{2\Delta x \min\{\hat\omega_1, \hat \omega^*\}}
\max_{y \in S_j^y; \sigma}
\left|\int_{-1}^1 c\left(x_i+\frac{\Delta x}{2}\xi, y\right)\dot\omega^\sigma(\xi)\,d\xi \right| \geq 0,\\
& \mu \leq \min\left\{\frac{\min\{\omega^1,\omega^3\}}{b\big(\alpha_1(\pm\gamma) + \alpha_3(\mp\gamma)\big) + \frac{\Delta y}{2\Delta x \min\{\hat\omega_1, \hat \omega^*\} }\max_{y \in S_j^y; \sigma} \left|g_i^\sigma(y)\right|},
\frac{\omega^2}{2a\alpha_2(\gamma)}\right\},\end{aligned}$$ where $g_i^\sigma(y)=\int_{-1}^1 c\left(x_i+\frac{\Delta x}{2}\xi, y)\right)\dot\omega^\sigma(\xi)\,d\xi$. Note that both $\left|\int_{-1}^1c\left(x, y_j+\frac{\Delta y}{2}\eta\right)\dot\omega^\sigma(\eta)\,d\eta\right|$ and $\left|\int_{-1}^1c\left(x_i+\frac{\Delta x}{2}\xi, y)\right)\dot\omega^\sigma(\xi)\,d\xi\right|$ are bounded from above by $$\begin{aligned}
\|c\|_\infty \left|\int_{-1}^1\dot\omega^\sigma(\eta) d\eta \right| \leq \frac{4\|c\|_\infty}{1-\gamma}.\end{aligned}$$ Using the fact that $\kappa^{-1} \leq \frac{\Delta x}{\Delta y}\leq \kappa$, $\omega^3 \leq \omega^1$, $\frac{1}{L(L-1)}=\hat \omega_1 < \hat \omega^*$, and $\alpha_3(\pm\gamma) \geq \beta_0 - 1$, as well as $$\alpha_1(\pm\gamma) + \alpha_3(\mp\gamma) \leq \beta_0 + \frac{8\beta_1 - 2}{1+\gamma}.$$ We see that all the above needed constraints are implied by the more severe constraints, including for $\beta_0$, and the CFL condition .
Acknowledgments {#acknowledgments .unnumbered}
===============
Liu’s research was partially supported by the National Science Foundation under Grant DMS1812666 and by NSF Grant RNMS (Ki-Net) 1107291.
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---
abstract: '> In this paper I describe how miscommunication problems are dealt with in the spoken language system DIALOGOS. The dialogue module of the system exploits dialogic expectations in a twofold way: to model what future user utterance might be about (predictions), and to account how the user’s next utterance may be related to previous ones in the ongoing interaction (pragmatic-based expectations). The analysis starts from the hypothesis that the occurrence of miscommunication is concomitant with two pragmatic phenomena: the deviation of the user from the expected behaviour and the generation of a conversational implicature. A preliminary evaluation of a large amount of interactions between subjects and DIALOGOS shows that the system performance is enhanced by the uses of both predictions and pragmatic-based expectations.'
author:
- |
Morena Danieli\
CSELT\
Centro Studi E Laboratori Telecomunicazioni S.p.A.\
Via G. Reiss Romoli 274\
I-10148 Torino, Italy\
E-Mail: [email protected]
title: 'On the use of expectations for detecting and repairing human-machine miscommunication'
---
The problem
===========
During the last few years it has been emerging that the success of spoken language systems is greatly improved by the contextual reasoning of dialogue modules. This tenet has spread through both the speech and the dialogue communities. Dialogue systems devoted to spoken language applications are able to detect partial communication breakdowns by other system modules, and that increases the robustness of human-machine interactions by speech.
During oral interactions with computers, communication problems often arise after the occurrence of errors during the recognition phase. Sometimes these errors cannot be solved by the semantic module: the utterances containing them are interpreted by the semantic analyser, but with an information content different from the speaker’s intended meaning. Detecting such miscommunications and repairing them through initialization of appropriate repair subdialogues is essential for the interaction to be successful.
Most of the research in this area has been devoted to providing the recognition and understanding modules with information generated on the basis of the dialogue context. They predict what the next user’s utterance will probably be about: throughout this paper I will refer to them with the name $''$predictions$''$. Sometimes they are passed down to the acoustic recognition level in order to decrease the huge number of lexical choices, sometimes they are used to help in deciding on multiple semantic interpretations. More often they are used during the contextual interpretation phase to accept or reject parser output.
Although predictions have proved useful, they only grasp one side of the miscommunication problem. Actually they are a means for reducing recognition errors, and their use allows the avoidance of one of the potential sources of miscommunication. However during spoken human-computer interactions, the detection of miscommunications may be outside the capabilities of the dialogue system, even though it uses predictions; on the contrary, the user is usually able to detect any speech errors. For example, in travel inquiry applications words belonging to the same class, such as proper names of place, may be highly confusable. When the dialogue prediction says that next user’s utterance is likely to be about a departure place, this does not exclude that the recognizer substitutes the actually uttered name with a phonetically similar one. Only the user is able to detect such kinds of errors. In this situation the dialogue system should identify the user’s detection of miscommunication and provide appropriate repairs.
In this paper I will argue that the dialogue module ability to detect user-initiated repairs is improved if the system is able to capture the pragmatic phenomena that accompany user’s detection of miscommunication. The paper offers an analysis of the pragmatic phenomena that occur when users detect miscommunication during task-oriented human-machine spoken dialogues. The account exploits both predictions and another notion of expectation that comes from the cognitive-based research area. This notion refers to conversants’ beliefs about the relation of future utterances with previous ones in a dialogue. A computational interpretation of this notion has been done in the model proposed by [@Roy95], where it is suggested that speech community predictions and cognitive based expectations are complementary notions. This paper claims that accounting for these two notions is useful for detecting and solving actual breakdowns in user-system communication. The working hypothesis is that in task-oriented dialogues miscommunication often generates conversational implicatures. I will show how they are dealt with by the dialogue module of DIALOGOS, an information inquiry spoken language system implemented by Cselt speech recognition and understanding group. By using the telephone, the system may be used to access the data base of the Italian public railway company. I will report dialogue examples and experimental data that show the effectiveness of the proposed analysis in task-oriented human-machine spoken dialogues.
The working hypothesis
======================
The conceptual background of this approach is inspired by the Gricean principles of conversation [@LC]. The user modelling of the dialogue module of DIALOGOS is based on the assumption that both the system and the user are active agents of the communicative process; in particular, it is assumed that both of them observe the Cooperation Principle (CP) in order to achieve the general goal of their linguistic interchange, i.e. to access a database to get the information that the user needs.
The system predictions are modelled on the basis of the CP. At each stage of the dialogue with the user the system expects that user’s reaction conforms with three of the original Gricean conversational maxims, re-interpreted in the context of human-machine communication in [@Norman]. For example, in this application domain, throughout the dialogue the system expects that:
- each user turn is not over or under-informative,
- each user turn communicates user’s $''$true’$''$ needs (i.e. to receive train timetables from place X to place Y, if the two places have been confirmed),
- is pertinent to the focus in hand.
However, as remarked earlier, in oral human-machine dialogue the communication process may be disturbed by several factors, the most usual of ones are recognition errors. The user’s detection of breakdowns and errors of the system has precise empirical consequences: these concern both user’s behaviour and her cognitive demand on the continuation of the interaction. In particular, these empirical consequences may be summarized as follows:
- the user utterance does not match dialogue predictions;
- the user asks the system to come back to the interpretation context where miscommunication occurred;
- both the user and the system should engage in a clarification subdialogue before continuing their interaction.
Since dialogue predictions are generated by taking into account the respect of the above listed maxims, and given that the system assumes that the user continues to respect the CP, each deviation of the user’s behaviour from dialogue predictions is ideally interpreted as a signal of the potential occurrence of miscommunication. By co-operating in the achievement of the goal of the conversation, the user intentionally violates the maxims when she detects that a misunderstanding occurred in previous turns. Accordingly to the conceptual background here adopted, the intentional violation of the conversational maxims gives place to a conversational implicature.
The working hypothesis is that the occurrence of a miscommunication goes along with two pragmatic counterparts: the first one is the deviation of the user’s behaviour from the dialogue predictions, the other one is the generation of a conversational implicature. the turn For being $''$conversational$''$ an implicature has to be inferable on the basis of the contextual knowledge. I suggest that in order to recognize conversational implicatures, the dialogue system should embody a model of user expectations that gives insights on how to relate user correction turns to the previous (normal) turns in an ongoing dialogue.
Examples of miscommunication
============================
Non-understanding
-----------------
The kind of miscommunications that occur during oral interactions of subjects with spoken language systems is usually caused by an actual breakdown in the flow of the dialogue. The most usual miscommunications may be labelled as $''$misunderstanding$''$ and $''$non-understanding$''$. The latter are usually less problematic.
[l]{}\
[**T1 S**]{}: Automatic Railway Information System. Please\
speak after the tone. Please give your point of\
departure and your destination.\
[**T2 U**]{}: Then ... What time .. I mean from \[NOISE\] eight\
*[$<$Recognition and understanding failure$>$]{}\
[**T3 S**]{}: Sorry, I did not understand. Please give your\
point of departure and your destination.\
[**T4 U**]{}: I want to leave from Milano in the evening.\
I ’d like to know departure \[NOISE\] from\
Milano to Roma.\
*[$<$confirmation=NO, departure-city=MILANO,]{}\
*[ arrival-city=ROMA, departure-time=EVENING$>$]{}\
[**T5 S**]{}: Do you want to travel from Milano to Roma\
leaving in the evening?\
[**T6 U**]{}: Exactly. Around eight p.m.\
*[ $<$confirmation=YES hour=EIGHT$>$ ]{}\
[**T7 S**]{}: Intercity 243 leaves from Milano Centrale at\
20 past 8 p.m.; it arrive at Roma Termini 6 a.m.\
\
****
In the dialogue reported in Figure \[D1ex\] non-understanding occurred in the turn T2: due to environment noise, the fragmentary utterance was not understood by the recognizer and no semantic representation was sent to the dialogue module by the parser. This symptom of actual breakdown causes the system to trigger an informative speech act of non-understanding and a requestive speech act for obtaining departure and arrival place names (T3).
In non-understanding the system completely fails to interpret a user’s utterance; that may happen either because a speech recognition error occurred or because the linguistic processor was not able to interpret it, for example if the semantic content of the sentence was out of its semantic coverage. Non-understanding is usually recognized by the dialogue system as soon as it happens. Within speech understanding systems non-understanding usually gives place to second-turn repairs [@Schegloff92]. For example, in the DIALOGOS system the dialogue module realizes that a non-understanding has occurred because it fails to receive any semantic interpretation for the user’s sentence uttered immediately before. In this case it enters a repair action by informing the user that her utterance was not understood and asking her to rephrase the sentence. At present the dialogue module does not know if the non-understanding was caused by a failure of the speech recognizer (because of out-of-vocabulary words, mispronouncing, unhearings, and so on) or by a failure of the parsing module. As a consequence, the user is addressed with a generic information of non-understanding.
Misunderstanding
----------------
Some kinds of failures in recognition and understanding cause actual human-machine misunderstanding. Let us consider the dialogue in Figure \[D2ex\]. The user utterance T2 has been recognized as $``$Arona$''$ instead of $``$Roma$''$ (in Italian they may be confusable). This causes a third-turn repair. In the turn T4 the user, instead of giving the departure place name, realizes that the sentence uttered in the turn T2 was not correctly recognized and she repeats the arrival city. In producing T3, the system generated a prediction about the departure city, but user’s turn T4 causes it to recover. This is done by reinterpreting T4 within the interpretation context initially generated for T2. Moreover the expectations generated in interpreting T4 are different from the ones generated for T2. In T4 the system realizes that a misunderstanding happened: it discards predictions about obtaining the departure city by refocusing the interpretation context on the possible confirmation of the arrival city.
[l]{}\
[**T1 S**]{}: Automatic Railway Information System. Please\
speak after the tone. Please give your point of\
departure and your destination.\
[**T2 U**]{}: I want to go to Roma.\
*[$<$ \*arrival-city=ARONA $>$]{}\
[**T3 S**]{}: To Arona. What is your point of departure?\
[**T4 U**]{}: I said that I was going to Roma.\
*[$<$ arrival-city=ROMA $>$]{}\
[**T5 S**]{}: Are you going to Roma?\
\
**
Identifying misunderstanding
============================
The example reported in Figure \[D2ex\] is an actual piece of dialogue (translated into English) between a subject and the DIALOGOS system. Let us now describe how the system deals with the misunderstanding. In the turn T3 the dialogue system generates a so-called $''$confirmation plus initiative$''$ speech act. This means that in a single dialogue turn the system tries to get both a confirmation of what it understood from the previous user turn, and other piece of information it needs to access the database. The selection of this speech act generates a large number of expectations. The most interesting for the present discussion are listed below. For brevity, in what follows I have considered only the predictions of statements with more informational content than simple confirmation and denial by the adverbs $''$yes$''$ and $''$no$''$.
1. A statement about the name of the departure city;
2. A statement about the name of the departure city and other required parameters, such as the date and the time of departure;
3. An explicit confirmation of the arrival, and the departure city;
4. An explicit denial of the understood arrival city and a request for another arrival city;
5. An explicit denial of the understood arrival city;
6. A statement including a new arrival city plus the departure city;
7. A statement including a new arrival city;
As we can observe, the expectations from (1) to (7) may be related to predictable lexical counterparts. In other words, they may be sent to the recognition level in the form of word class predictions, in order to be used for constraining the search space of the speech recognizer.
In addition, we can observe that at the dialogue level the realization of expectations (6) and (7) implies something more than the prediction of a lexical class. The linguistic realizations of (6) and (7) have the form of affirmative statements and the illocutionary force of disconfirmations. In both cases a conversational implicature is generated. The illocutionary goal of statements realizing (6) and (7) imply a precise cognitive demand by the user. Let us again consider Figure \[D2ex\]. T4 realizes dialogue expectation (7): the user asks the system to reinterpret T4 in the context of T2. Notice also that the user’s beliefs about the continuation of the interaction are slightly different for (6) and (7): in the case of (6) the focus of attention of both user’s and system are still on the same speech act, i.e. the disconfirmation of the arrival city and the proposal of a departure city. On the contrary, in (7) the user expects that [*before*]{} requesting a new parameter the system will be able to correct its previous misunderstanding. In other words, she demands a focus restriction in order to correct the misunderstanding. An appropriate system answer in situations like (6) and (7) requires that the system be able to grasp the illocutionary force of user statements in order to redirect the dialogue strategy.
Dealing with misunderstanding
=============================
Let us consider in Figure \[D3ex\] another excerpt from an interaction between DIALOGOS and a subject.
[l]{}\
[**T1 S**]{}: Automatic Railway Information System. Please\
speak after the tone. Please give your point of\
departure and your destination.\
[**T2 U**]{}: From Milano to Roma.\
*[ $<$ departure-city=MILANO \*arrival-city=ARONA $>$]{}\
[**T3 S**]{}: Do you want to travel from Milano to Arona?\
[**T4 U**]{}: From Milano to Roma.\
*[$<$departure-city=Milano arrival-city=ROMA $>$]{}\
[**T5 S**]{}: Do you want to arrive in Roma?\
\
**
In the turn T3, DIALOGOS prompts the user for confirmation with a Yes$/$No question. At this point the system linguistic predictions concern a confirmation or a negation speech act by the user. In T4 the user produces an affirmative sentence which has the illocutionary force of a disconfirmation. By analysing a large number of interactions between users and the DIALOGOS system, we actually realized that in this application context Italian speakers very often start a repair without an explicit negation word such as $``$No$''$: on the contrary, they repeat their previous utterances louder. Notice that in Italian repetitions mostly work for confirmation [@BAZZA], but in that case the intonational contour is unstressed.
As proposed above, in order to recognize the dialogic repetition of the turn T4 as a user-initiated repair, the dialogue system has to be able to take into account both the deviation of the user from the expected behaviour, and the illocutionary force of her utterance (i.e. the denial of the system interpretation of T2). This deviation is the signal of an actual misunderstanding and it is related to the presence of a conversational implicature.
The dialogue system initially interprets T4 with respect to its set of expectations. As soon as it realizes that the matched expectation realizes an implicature, it reinterprets T4 within the context of the utterance whose interpretation is now denied, i.e. the system traces back to the interpretation context of T2. The fact that T4 is now being reinterpreted in the context of T2 causes a refocusing of the dialogue context. In the present DIALOGOS version the refocusing mechanism is allowed through the maintenance of both a linguistic history and a context hierarchy (further details of this are reported in the section that describes the dialogue system).
All the examples discussed above concern third-turn repairs; however, repairs from misunderstanding may occur also in subsequent turns, as the example in Figure \[D4ex\] shows.
[l]{}\
[**T1 S**]{}: Automatic Railway Information System. Please\
speak after the tone. Tell me from where you\
want to leave and where you want to go.\
[**T2 U**]{}:... leaving from Pisa at eight \[NOISE\] this\
morning \[NOISE\] Firen-\
*[ $<$ \*departure-city=PISA-AEROPORTO]{}\
*[ departure-time=MORNING date=TODAY $>$]{}\
[**T3 S**]{}: From Pisa Aeroporto. Where are you going to?\
[**T4 U**]{}: To Firenze.\
$<$arrival-city=FIRENZE$>$\
[**T5 S**]{}: From Pisa Aeroporto to Firenze?\
[**T6 U**]{}: From Pisa ... Stazione Centrale to Firenze\
*[ $<$departure-city=PISA dep-station=CENTRALE ]{}\
*[ arrival-city=FIRENZE$>$]{}\
[**T7 S**]{}: From Pisa Centrale?\
[**T8 U**]{}: Yes\
*[$<$ confirmation=YES $>$ ]{}\
\
*****
In this example the sentence uttered in the turn T2 was badly recognized. For the sake of simplicity, let us concentrate on what happened about the names of the departure and arrival city. Due to disturbances over the phone line, the uttered arrival city name ($''$Firenze$''$) was not decoded. Moreover the phrase $''$from Pisa at eight$''$ (in Italian, $''$da Pisa alle otto$''$) was recognized as $''$Pisa Aeroporto$''$. The system expectations were not satisfied since the name of the arrival city was not acquired. Then the dialogue system decided to trigger a confirmation plus initiative speech act in order to obtain the missing parameter and to confirm departure. That choice resulted in the generation of T3. At this dialogue stage the set of expectations were the ones enumerated in the previous paragraph, although applied to the arrival city parameter. In T4 the user offered an arrival city. This matched the first expectation of our list. At this stage neither the user nor the system had grasped the inconsistency concerning departure. Since it had not obtained explicit confirmation for departure, the dialogue system addressed the user with the Yes$/$No question of turn T5. The contextual interpretation of the user utterance T6 detects the explicit confirmation of the arrival city and the misunderstanding that had occurred during the recognition of the departure city. The latter is refocused again in T7, and the user is addressed with a new Yes$/$No question.
Architecture of DIALOGOS
========================
DIALOGOS is a real-time task-oriented system composed of the following modules: the acoustical front-end, the linguistic processor, the dialogue manager and the message generator, and the text-to-speech synthesizer. Its vocabulary size is about 3,500 words including 2,983 place names. The acoustical front-end and the synthesizer are connected to the telephone network through a telephone interface, while the dialogue manager is connected to a Computer Information System to obtain information about Italian Railway time-tables. The acoustical front-end performs feature extraction and acoustic-phonetic decoding. The recognition module is based on a frame synchronous Viterbi decoding, where the acoustic matching is performed by a phonetic neural network. During the recognition, it makes use of language models that are class-based bigrams trained on 30K sentences. The training data were partially derived from the trial of a previous spoken language system applied to the same domain [@baggiacrim94].
The linguistic processor starts from the best-decoded sequence, and performs a multi-step robust partial parsing. In this strategy, partial solutions are accepted according to the linguistic knowledge [@ICASSP93]. At the end of the parsing stage a deep semantic representation for the user utterance is sent to the dialogue module, that will be described in the next subsection.
The dialogue module performs contextual interpretation and generates the answer which is sent to the text-to-speech synthesizer Eloquens [@Quazza93], that contains specific prosodic rules for the Italian language.
The dialogue system
-------------------
The dialogue strategy of DIALOGOS has been designed both to maintain the control of the cooperative interaction and to leave the expert user the freedom to guide the interaction. Referring to the classification reported in [@Smith95], the DIALOGOS system is able to support directive, suggestive and declarative initiative modes. It only exploits the directive mode in order to deal with repetitive problems at the recognition and understanding levels. The directive mode is implemented by automatic switching to the isolated word recognition modality.
The flow of a typical interaction with the system may be divided into two main stages: the acquisition of a set of parameters, used to select a reasonably small set of objects of interest to the user, and the presentation of the information related to the retrieved objects. These two stages may lead to the combination of several subdialogues, each of them with its own purpose. The system is able to move properly through nested subdialogues thanks to the maintenance of a dialogue history which relates the focus of the current utterance to the appropriate interpretation context. A detailed discussion of the dialogue strategy is given in [@Bet93].
The system performs contextual interpretation on the basis of the model of user-system interaction discussed above. The model has been implemented using a Transition Network formalism. Each node in the network represents a state in the dialogue: associated actions are executed when the interaction comes to that state. The actions are declared in a library of functions written in C. There are specific actions for each dialogue functionality, including contextual interpretation, updating of the context hierarchy (see below), and generation of dialogue predictions and expectations. A state transition is executed when the conditions associated with the arc are satisfied. Conditions are also library functions: they check both the current state of the dialogue model and the content of the user’s utterance, in order to direct the interaction into a new coherent state. Default transitions are associated with each node.
The contextual interpretation of user utterances takes into account the linguistic history and the global and local (active) focus [@Grosz], [@McCoy]. Each time a user utterance has to be interpreted, all the information useful for its interpretation (specifically, its semantic content and some surface information that may be used to solve references) are stored in a cycle-structure [@BaggiaGerb]. At each point of the dialogue, the linguistic history consists in the history of the previous cycle-structures. The interpretation of the utterance causes the creation of a local focus structure which is linked to the cycle-structure that has caused it [@Bet93]. The focus structures are hierarchically organized in a tree (that we name $''$context hierarchy$''$), whose root represents the global focus at the beginning of the dialogue. A new node in the tree, that is a new active focus, is created and linked in the hierarchy when the user operates a focus restriction or a focus shifting. The correct interpretation context of an utterance can be the active focus, if the utterance refers to the objects currently focused; when there is discrepancy between utterance focus and active focus, the hierarchy is climbed up for checking the semantic and pragmatic consistency between the current utterance and the previous ones. The first node where the consistency is verified is chosen as the utterance interpretation context.
Experimental Data
=================
Recently DIALOGOS has been tested in a large field trial. Five hundred subjects of different ages and levels of education called the system from all over Italy. They had never used a computerized telephone service before. Each of them made three calls to the system. The experimental material has been transcribed and it is currently being evaluated.
Subjects called the system from their own places (80%), from public telephone booths (10%), and by mobile phones used in noise environment (such as street, underground stations, and so on). They received a single page of printed instructions which contained a brief explanation of the service capabilities.
Each subject had to plan a trip from a city A to a city B in a certain date and time: they had to find out departure and arrival times of trains that satisfy their needs. The departure and the destination were specified in the scenarios, on the contrary the subjects had the freedom to choose the date and the time of the travel.
Some features of the dialogues collected in the test and already evaluated are shown in Table \[D5ex\]. The total number of evaluated dialogues, the number of continuous speech utterances, the number of isolated words, and the average number of continuous speech utterances per dialogue are reported.
--------- ------------ ------------ ----------------
No. of No. of No. of No. of
Dialogs Utterances Iso. Words Utt. per Dial.
923 9,124 442 9.8
--------- ------------ ------------ ----------------
: \[D5ex\]Corpus
A user-system interaction was considered successful when the user received all the information she needed. In particular, the transaction success is the measure of the success of the system in providing users with the train timetables they required. A definition of transaction success and related dialogue evaluation metrics is given in [@Danieli95]. The rate of transaction success for the 923 evaluated dialogues is 84%. We consider this result a promising starting point for further work within the conceptual framework described in this paper.
Conclusions
===========
This paper discusses the use of expectations for dealing with miscommunication in spoken dialogue systems. It has been suggested that the deviation of the user’s behaviour from predictions, along with the generation of a conversational implicature, are symptoms of actual miscommunication. The proposed analysis exploits two notions of expectations. The first one refers to predictions: they have been largerly used by the speech community to reduce speech and semantic interpretation errors. The second expectation notion gives insights on how to relate future user utterance to previous ones. The described approach has been implemented in the dialogue module of the DIALOGOS spoken language system.
The level of appropriateness of the system’s answer when miscommunication occurs is greatly enhanced when the spoken dialogue system is able to profit from both kinds of expectation. Some experimental data support this claim. Preliminary results (based on 923 dialogues with naive users) show that the rate of transaction success of the system is 84%.
Although this analysis is only a first step towards an adequate handling of miscommunication within our automatic speech recognition system, it has been shown to be useful in improving the overall performance of the system.
Acknowledgements
================
I would like to thank all the people that designed and implemented the current version of the DIALOGOS system: Dario Albesano, Paolo Baggia, Roberto Gemello, and Claudio Rullent. A special thanks to Elisabetta Gerbino who discussed and implemented with the author the preliminary version of the dialogue system. Carla Bazzanella and Elisabeth Maier, who carefully commented on an earlier draft of this paper, are gratefully acknowledged.
[X]{}
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abstract: |
The growing security threat of microarchitectural attacks underlines the importance of robust security sensors and detection mechanisms at the hardware level. While there are studies on runtime detection of cache attacks, a generic model to consider the broad range of existing and future attacks is missing. Unfortunately, previous approaches only consider either a single attack variant, e.g. *Prime+Probe*, or specific victim applications such as cryptographic implementations. Furthermore, the state-of-the art anomaly detection methods are based on coarse-grained statistical models, which are not successful to detect anomalies in a large-scale real world systems.
Thanks to the memory capability of advanced Recurrent Neural Networks (RNNs) algorithms, both short and long term dependencies can be learned more accurately. Therefore, we propose [*[FortuneTeller]{}*]{}, which for the first time leverages the superiority of RNNs to learn complex execution patterns and detects unseen microarchitectural attacks in real world systems. [*[FortuneTeller]{}*]{} models benign workload pattern from a microarchitectural standpoint in an unsupervised fashion, and then, it predicts how upcoming benign executions are supposed to behave. Potential attacks and malicious behaviors will be detected automatically, when there is a discrepancy between the predicted execution pattern and the runtime observation.
We implement [*[FortuneTeller]{}*]{} based on the available hardware performance counters on Intel processors and it is trained with 10 million samples obtained from benign applications. For the first time, the latest attacks such as Meltdown, Spectre, Rowhammer and Zombieload are detected with one trained model and without observing these attacks during the training. We show that [*[FortuneTeller]{}*]{} achieves the best false positive and false negative trade off compared to existing works under realistic workloads and target implementations with the highest F-score of $0.9970$.
author:
-
bibliography:
- 'main.bib'
title: |
[*[FortuneTeller]{}*]{}: Predicting Microarchitectural Attacks\
via Unsupervised Deep Learning
---
Introduction {#sec:intro}
============
In the past decade, we have witnessed the evolution of microarchitectural side-channel attacks [@yarom2014flush; @irazoqui2014wait; @evtyushkin2016jump; @gruss2016prefetch; @jang2016breaking], from being considered as a nuisance and largely dismissed by chip manufacturers to becoming frontpage news. The severity of the threat was demonstrated by the Spectre [@kocher2018spectre] and Meltdown [@lipp2018meltdown] attacks, which allow a user with minimum access right to easily read arbitrary locations in the memory by exploiting the transient effect of illegal instruction sequences. This was followed by a plethora of attacks [@maisuradze2018ret2spec; @van2018foreshadow; @schwarz2019zombieload; @minkin2019fallout] either extending the scope of the microarchitectural flaws or identifying new leakage sources. It is noteworthy that these critical vulnerabilities managed to stay hidden for decades. Only after years of experimentation, researchers managed to gain sufficient insight into, for the most part, the unpublished aspects of these platforms. This leads to the point that they could formulate fairly simple but very subtle attacks to recover internal secrets. Therefore, the natural question becomes: how can we discover dormant vulnerabilities and protect against such subtle attacks? A fundamental approach is to eliminate the leakage altogether by using formal analysis. However, given the tremendous level of complexity of modern computing platforms and lack of public documentation, formal analysis of the hardware seems impractical in the near future. What remains is the modus operandi: [**leaks are patched as they are discovered by researchers through inspection and statistical analysis**]{}.
Countermeasures for microarchitectural side-channel attacks focus on the operating system (OS) hardening [@LiuEtAl2016; @gruss2017kaslr], software synthesis [@rane2015raccoon; @cauligi2017fact] and analysis [@almeida2013formal; @weiser2018data; @wichelmann2018microwalk], and static [@irazoqui2016mascat] or dynamic [@Zhang2016cloudradar; @chiappetta2016real; @briongos2018cacheshield] detection of attacks. Static analysis is performed by evaluating the untrusted software against known malicious code patterns without running it on a target platform [@irazoqui2016mascat]. Alternatively, dynamic analysis aims to detect malicious behaviors in the system by analyzing the runtime footprint of the running processes [@chiappetta2016real]. Existing works on dynamic detection of microarchitectural attacks are based on collecting footprints from the hardware performance counters (HPCs) and limited modeling of malicious behaviors [@chiappetta2016real; @mushtaq2018nights; @Zhang2016cloudradar; @briongos2018cacheshield]. A crucial challenge for both detection techniques is the shortage of knowledge about new attack vectors. Therefore, modeling malicious behaviors for undiscovered attacks and accurately distinguishing them from benign activities are open problems. Moreover, microarchitectural attacks are in infancy, and supervised learning models, which are used as attack classifier [@mushtaq2018nights], are not reliable to detect known attacks due to the insufficient amount and imprecise labeling of the data. Hence, unsupervised methods are more promising to adapt the detection models to real world scenarios.
Anomaly-based attack detection, which has been also studied in other security applications [@shabtai2012andromaly; @feizollah2013study], aims to address the aforementioned challenge by only modeling the benign behaviors and detecting outliers. While there have been several efforts on anomaly-based detection of cache attacks [@briongos2018cacheshield; @chiappetta2016real], modern microarchitectures have a diverse set of components that suffers from side-channel attacks [@evtyushkin2016jump; @yarom2017cachebleed; @gruss2016prefetch; @moghimi2018memjam]. Thus, detection techniques will not be practical and usable, if they do not cover a broad range of both known and unseen attacks. This requires more advanced learning algorithms to comprehensively model the entire behavior of the microarchitecture. On the other hand, statistical methods for anomaly detection are not sufficient to analyze millions of events that are collected from a very complex system like the modern microarchiecture. A major limitation of the classical statistical learning methods is that they use a hand-picked set of features, which wastes the valuable information to characterize the benign execution patterns. As a result, these techniques fail at building a generic model for real-world systems.
The latest advancements in Deep Learning, especially in Recurrent Neural Networks (RNNs), shows that time dependent tasks such as language modeling [@sundermeyer2012lstm], speech recognition [@sak2014long] can be learned and upcoming sequences are predicted more efficiently by training millions of data samples. Similarly, computer programs are translated to processor instructions, and the corresponding microarchitectural events have time dependent behaviors. Modeling the sequential flow of these events for benign applications is extremely difficult by using logic and formal reasoning due to the complexity of the modern microarchitecture. We claim that these time dependent behaviors can be modeled in a large scale by observing sufficient number of benign execution flows. Since the long-term dependencies in the time domain can be learned with a high accuracy by training Long-short term memory (LSTM) and Gated Recurrent Unit (GRU) networks, the fingerprint of benign applications in a processor can also be learned efficiently. In addition, a challenging task of choosing the features of benign applications can be done automatically by LSTM/GRU networks in the training phase without any expert input.
**Our Contribution:** We propose [*[FortuneTeller]{}*]{} which is the first generic detection model/technique for microarchitectural attacks. [*[FortuneTeller]{}*]{} learns the benign behavior of hardware/software systems by observing microarchitectural events, and classifies any outlier that does not conform to the trained model as malicious behavior. [*[FortuneTeller]{}*]{} can detect unseen microarchitectural attacks, since it only requires training over benign execution patterns.
In summary, we propose [*[FortuneTeller]{}*]{} which:
- is a generic detection technique, that can be applied to detect attacks on other microarchitectures and execution environments.
- for the first time, can detect various attacks automatically, disregarding the victim application, including cryptogrpahic implementations, browser passwords, secret data in kernel environment, bit flips and so on.
- can detect attacks that were not observed during the training, or future attacks that may be introduced by the security community.
More specifically, we show:
1. different types of hardware performance counters can be used as the most optimum security sensor available on the commodity processors.
2. how to capture the system-wide low-level microarchitectural traces and learn noisy time-dependent sequences through advanced RNN algorithms by training a more advanced and generic model.
3. [*[FortuneTeller]{}*]{} performs better by comparing it to the state-of-the art detection techniques.
4. we can detect malicious behavior dynamically in an unsupervised manner including stealthy cache attacks (Flush+Flush), transient execution attacks (Meltdown, Spectre, Zombieload) and Rowhammer.
Outline:
--------
The rest of the paper is organized as follows: provides background information on microarchitecural attacks, performance counters and RNNs. Then, gives an overview of previous works. outlines the methodology and implementation of [*[FortuneTeller]{}*]{}. Also, information on our benign and attack dataset and performance counter selection are given. evaluates the results. The comparison with the prior works is given in . Finally, discusses the results and concludes our work.
Background {#sec:background}
==========
Microarchitectural Attacks
--------------------------
Modern computer architecture has a tremendously complex and optimized design. In order to improve the performance, several low-level features have been introduced such as speculative branches, out-of-order executions and shared last level cache (LLC). All these components are potential targets for microarchitectural attacks. Therefore, the following paragraphs give insight into microarchitectural attacks, which are examples of attacks that can be detected by [*[FortuneTeller]{}*]{}.
[***Flush+Reload (F+R)***]{} The LLC is shared among all cores in the processor. Flush+Reload attack [@yarom2014flush] aims to track accesses to specific cache lines by using the *clflush* instruction. First, adversary flushes the victim cache line. Then, the victim executes some instructions. Finally, the adversary reloads the same cache line and measures the access time. Flush+Reload attack is mostly used to recover cryptographic keys [@yarom2014recovering], which is applicable to perform attacks on systems with enabled memory deduplication such as cloud environments [@IrazoquiEtAl2015; @gulmezouglu2015faster].
[***Flush+Flush (F+F)***]{} Flush+Flush attack uses the *clflush* instruction to flush the specific cache lines [@GrussEtAl2016a]. Instead of measuring the time to access a cache line, the execution time of the *clflush* instruction is measured. This method is considered as a stealthy attack against detection methods, since the number of introduced cache misses is low by this attack. Flush+Flush attack is used to exploit the T-table implementation of AES and user’s keystrokes [@GrussEtAl2016a].
[***Prime+Probe (P+P)***]{} In a Prime+Probe attack, an adversary aims to fill an entire cache set, and then, measures the access time to the same cache set [@tromer2010efficient]. If a victim evicts any of the adversary’s cache line from the set, the adversary will observe access latency which leaks information about the victim’s memory access pattern. While it has a lower resolution compared to Flush+Reload and Flush+Flush, it has a broader applicability. Prime+Probe attack was applied in the cloud environment to steal secret keys [@ZhangEtAl2012; @inci2016cache; @irazoqui2015s], Javascript to detect the visited webpages [@OrenEtAl2015] and mobile phones to detect applications and user input [@lipp2016armageddon; @gulmezoglu2018undermining].
{width="85.00000%"}
Hardware Performance Counters (HPCs)
------------------------------------
[***Rowhammer***]{} DRAM cells have the possibility to leak charge over time. Rowhammer [@gruss2016rowhammer] triggers the leak by accessing neighboring rows repeatedly. This leads to bit flips, which enables adversaries with low access right to gain system privileges [@seaborn2015exploiting]. *clflush* instruction is also commonly used to increase repeated access to the DRAM by bypassing the cache [@aweke2016anvil].
[***Spectre***]{} Spectre attacks exploit speculative branches [@kocher2018spectre]. This attack is able to read memory addresses, which do not belong to the adversary by misusing the branch prediction. Therefore, sensitive data such as credentials stored in the browser can be leaked from the victim’s memory space. Spectre is also effective against the SGX environment to compromise the trusted execution [@van2018foreshadow; @chen2018sgxpectre].
[***Meltdown***]{} Meltdown attack focuses on out-of-order execution to read kernel memory addresses [@lipp2018meltdown]. The victim’s secret, which is loaded into the registers will be mapped to different cache lines. Flush+Reload is used to determine if a specific cache line has been accessed. An adversary with only user privileges can perform this attack to read the content of kernel address space. The same concept has also been applied to Intel SGX [@van2018foreshadow] to bypass the hardware supported memory isolation.
[***ZombieLoad***]{} Meltdown-style attacks can also specifically leak data from various microarchitectural resources such as store buffer [@minkin2019fallout], line fill buffer (LFB) [@schwarz2019zombieload] and load ports [@ridl]. ZombileLoad attack leaks data from memory load operations that are executed in other user processes and kernel context. Faulting/assisting loads that are executed by a malicious process can retrieve the stale data belonging to other security domains. This data that has been falsely forwarded from the shared resources may include secrets such as cryptographic keys or website URLs which can be transmitted over a covert channel such as the Flush+Reload technique.
*Hardware Performance Counters (HPCs)* store low-level hardware-related events in the CPU. These events are tracked as counters that are available through special purpose registers. There are various performance events available in processors. The counters are used to collect information about the system behavior while an application is running. They have been used by researchers to reverse-engineer the internal design choices in the processor [@maurice2015reverse], or to increase the performance of the software by analyzing the bottlenecks [@adhianto2010hpctoolkit]. These low-level counters are provided on all major architectures developed by ARM, Intel, AMD, and NVIDIA.
There are various tools to program and read performance counters. Intel PCM [@intelpcm] supports both core and offcore counters on Intel processors. The core counters give access to events within a single core of a processor, while the offcore counters profile events and activities across the cores and within the processor’s die. This includes some of the events related to the integrated memory controller and the Intel QuickPath Interconnect which is shared by all cores. Before using the performance counters, we need to program each of them to monitor a specific event. Afterwards, the counter state can be sampled. In this work, we only focused on core counters, since the offcore counters have a small variety.
Recurrent Neural Networks (RNNs)
--------------------------------
RNNs are a type of Artificial Netural Network algorithm, which is used to learn and predict the sequential data. RNNs are mostly applied to speech recognition and currently used by Apple’s Siri [@AppleSiri] and Google’s Voice Search [@chiu2018state]. The reason behind the integration of RNNs into real world applications is that it is the first algorithm to remember the temporal relations in the input through its internal memory. Therefore, RNNs are mostly preferred for tasks where sequential data is involved.
In a typical RNN structure, the information cycles through a loop. When the algorithm needs to make a decision, it uses the current input $x_t$ and hidden state $h_{t-1}$ where the learned features from the previous data samples are kept as shown in a. Basically, a RNN algorithm produces output based on the previous data samples, and provides the output as a feedback into the network. However, traditional RNN algorithms are not good at learning the long-term sequences because the amount of extracted information converges to zero with the increasing time steps. In other words, the gradient is vanished and the model stops learning after long sequences. In order to overcome this problem, two algorithms were introduced, as described below:
### Long-Short Term Memory
Long-Short Term Memory (LSTM) networks are modified RNNs, which essentially extends the internal memory to learn longer time sequences. LSTM networks consist of memory cell, input, forget and output gates as shown in b. The memory cell keeps the learned information from the previous sequences. While the cell state is modified by the forget gate, the output of the forget gate multiplies the specific positions in the input matrix by $0$ to forget and by $1$ to keep the information. The forget gate equation is as follow; $f_{t}=\sigma(W_f[h_{t-1},x_t]+b_t)$, where sigmoid function is applied to the weighted input and the previous hidden state. In the input gate, the useful input sections are determined to be fed into the cell state. The input gate equation is $i_t=\sigma(W_i[h_{t-1},x_t]+b_i)$, where sigmoid function is used as an activation function. This gate is combined with the input modulation gate to switch the cell state to forget memory. The activation function for input modulation gate is *tanh*. Finally, the output gate passes the output to the next hidden state by applying the equation $o_t=\sigma(W_o[h_{t-1},x_t]+b_o)$, where *tanh* is used as an activation function. Therefore, LSTM networks can select distinct features in the time sequence data more efficiently than RNNs, which enables learning the long-term temporal relations in the input.
Prior Works Learning Algorithm Approach Detected Attacks Target Implementations
-------------------------------------------- -------------------- ----------------- --------------------------------- ------------------------
Mushtaq et al. [@mushtaq2018nights] LDA/LR/SVM Supervised F+F, F+R Crypto
Zhang et al. [@Zhang2016cloudradar] DTW Semi-Supervised P+P, F+R Crypto/Hash
Chiappetta et al. [@chiappetta2016real] GS Unsupervised P+P, F+R Crypto
Briongos et al. [@briongos2018cacheshield] CPD Unsupervised F+F, F+R, P+P Crypto
F+F, F+R, P+P
**Rowhammer** **Benchmarks**
**Our Work** **LSTM/GRU** Unsupervised **Spectre Meltdown Zombieload** **Real-world Apps**
\[table:comparison\]
### Gated Recurrent Unit
Gated Recurrent Unit (GRU) is improved version of RNNs. GRU uses two gates called, update gate and reset gate. The update gate uses the following equation: $z_t=\sigma(W_zx_t+U_zh{t-1})$. Basically, both current input and the previous hidden state are multiplied with their own weights and added together. Then, a sigmoid activation function is applied to map the data between $0$ and $1$. The importance of the update gate is to determine the amount of the past information to be passed along to the future. Then, the reset gate is used to decide how much of the past information to forget. In order to calculate how much to forget, $r_t=\sigma(W_rx_t+U_rh_{t-1})$ equation is used, where the previous hidden state and current input are multiplied with their corresponding weights. Then, the results are summed and sigmoid function is applied. The output is passed to the current memory cell which stores the relevant information from the past. It is calculated as $h_{t}{'}=tanh(Wx_t+r_t \odot Uh_{t-1})$. The element-wise product between reset gate and weighted previous hidden layer state determines the information to be removed from previous time steps. Finally, the current information is calculated by the equation $h_t=z_t \odot h_{t-1}+(1-z_t) \odot h_t$. The purpose of this part is to use the information obtained from update gate and combine both reset and update gate information. Hence, while the relevant samples are learned by update gate, the redundant information such as noise is eliminated by reset gate.
In this work, the RNN algorithms are used in an unsupervised fashion where there is no need for separate validation dataset in the training phase. The validation error is calculated for each prediction in the next timestamp and the total validation error is given after each epoch.
Related Work {#sec:related_work}
============
Detecting Attacks using HPCs
----------------------------
Low-level performance monitoring events such as HPCs have been used as security sensors to detect malicious activities [@malone2011hardware; @herath2015these]. Similar to [@yuan2011security; @xia2012cfimon], *Numchecker* [@wang2013numchecker] and *Confirm* [@wang2015confirm] adopt these sensors to detect control flow violations, which are applied to *rootkits* and *firmware modifications*, respectively. In addition, classical ML algorithms such as support vector machines (SVMs) and k-nearest neighbors (KNNs) are adapted to naive heuristic-based techniques for multi-class classification [@bahador2014hpcmalhunter; @demme2013feasibility]. The latter explores neural network in a supervised fashion [@demme2013feasibility]. Tang et al. [@TangEtAl2014] train One-Class Support Vector Machine (OC-SVM) with benign system behavior and detect the malware in the system.
Despite the detection of malware and rootkits in the system, HPCs have also been used to detect microarchitectural attacks. Since our work focuses on microarchitectural attack detection, the features of prior approaches and our detection technique are compared in . Firstly, Chiappetta et al. [@chiappetta2016real] proposes to monitor HPCs and the data is analyzed by using Gaussian Sampling (GS) or probability density function (pdf) to detect the anomalies on cryptographic implementations dynamically. Later, Zhang et al. [@Zhang2016cloudradar] apply Dynamic Time Wrapping (DTW) to catch cryptographic implementation executions in the victim VMs. Then, the number of cache misses and hits in the attacker VMs are monitored during the execution of the sensitive operations. Briongos et al. [@briongos2018cacheshield] implement Change Point Detection (CPD) technique to determine the sudden changes in the time series data to detect F+F, F+R and P+P attacks. Finally, Mushtaq et al. [@mushtaq2018nights] detect the cache oriented microarchitectural attacks with supervised Linear Discriminant Analysis (LDA), Support Vector Machine (SVM) and Linear Regression (LR) technique under various system loads. We further compare the most related works with [*[FortuneTeller]{}*]{} in .
RNN Applications in Security
----------------------------
RNN algorithms are applied to other security domains to increase the efficiency of defensive technologies. For instance, Shin et al. [@shin2015recognizing] leverage RNNs to identify functions in the program binary. Once the model is trained with these function, the technique classifies the bytes to decide on whether it is the beginning of the function or not. Similarly, Pascanu et al. [@pascanu2015malware] apply RNNs to detect malware by training the APIs in an unsupervised way. The technique improves the true positive rate by 98% compared to previous studies. In another study, Melicher et al. [@melicher2016fast] introduce RNN-based technique to improve guessing attacks on password’s resistance. This study shows better accuracy than Markov models. Furthermore, Du et al. [@du2017deeplog] implement LSTM based anomaly detection to detect anomalies in the system. The LSTM model is trained with log data obtained from normal execution. Their results show that the traditional data mining techniques underperform LSTM model to detect the anomalies. Finally, Shen et al. [@shen2018tiresias] apply LSTM and GRU networks to predict the next security events with a precision of up to 0.93. These studies indicate that RNN based security applications are commonly used in other challenging environments.
[*[FortuneTeller]{}*]{} {#sec:methodology}
=======================
{width=".85\textwidth"}
Methodology
-----------
Our conceptual design for [*[FortuneTeller]{}*]{} consists of two phases as shown in : In the offline phase, [*[FortuneTeller]{}*]{} collects time sequence data from diverse set of benign applications by monitoring security sensors in the system. The collected data is used as the training data and it is fed into the RNN algorithm with a sliding window technique. The weights of the trained model are optimized by the algorithm itself since each data sample is also used as the validation. When there is no further improvement in the validation error, the training process stops. Once the RNN model is trained, it is ready to be used in a real time system.
In the online phase, the real-time sequences are captured from the same security sensors and given as input to RNN models. The prediction of the next measurement for each sensor is made by the pre-trained RNN models, dynamically. If the mean squared error (MSE) between the predicted value and the real time sensor measurement is consistently higher than a threshold, the anomaly flag is set. The online phase is the actual evaluation of [*[FortuneTeller]{}*]{} in a real world system.
Two separate detection models are trained with LSTM and GRU networks since they are known for their extraordinary capabilities in learning the long time sequences. Our purpose is to train an RNN-based detection model, which can predict the microarchitectural events in the next time steps with the minimal error. In our detection scenario, we consider a time series $X=\{\textbf{x}^{(1)},\textbf{x}^{(2)},\dots,\textbf{x}^{(n)}\}$, where each measurement $\textbf{x}^{(t)} \in R^m$ is an m-dimensional vector $\{x_{1}^{(t)},x_{2}^{(t)},\dots,x_{m}^{(t)}\}$ and each element corresponds to a sensor value at time $t$. As all temporal relations can not be discovered from millions of samples, a sliding window with a size of $W$ is used to partition the data into small chunks. Thereby, the input to RNN algorithm at time step $t$ is $\{x_{1}^{(t-W+1)},x_{2}^{(t-W+2)},\dots,x_{m}^{(t)}\}$, where the output is $\textbf{y}^{(t)}=\textbf{x}^{(t+1)}$. Note that, even though there is a fixed length sliding window in the problem formulation, the overall input size is not fixed. Finally, the trained model is saved to be used in real-world system.
To evaluate the trained model, test dataset is collected from benign applications and attack executions. The test dataset has the same structure with the training data, and is fed into the model to calculate the prediction error in the next time steps. The error at time step $t+1$ is $e^{(t+1)}$ which is equal to $1/m
\sum_{i=1}^{m}(y_{i}^{t+1}-x_{i}^{t+1})^2$. The model predicts the value of next measurement and then, the error for is summed up to one value.
To detect the anomalies in the system, a decision window $D$ and an anomaly threshold $\tau_{A}$ are used. If all the predictions in $D$ are higher than $\tau_{A}$, then the anomaly flag $F_A$ is set in the system in .
$$F_A=
\begin{cases}
1, & \forall e^{(t+1)} \in D \geq\tau_{A}\\
0, & otherwise
\end{cases}
\label{eq:anomaly}$$
The choice of $D$ directly determines the anomaly detection time. If $D$ is chosen as a small value, the attacks are discovered with a very small leakage. On the other hand, the false alarm risk increases in parallel, which is controlled by adjusting $\tau_{A}$. This trade-off is discussed further in .
Implementation
--------------
### Profiled Benchmarks and Attacks {#sec:dataset}
The main purpose of [*[FortuneTeller]{}*]{} is to train a generic model by profiling a diverse set of benign applications. Therefore, selecting benign applications is utmost importance. For the benign application dataset, benchmark tests in Phoronix benchmark suite [@phoronix] are profiled since the suite includes different type of applications such as cryptographic implementations, floating point and compression tests, web-server workloads etc. The complete list is given in Appendix, . It is important to note that some benchmark tests have multiple sub-tests and all the sub-tests are included in both training and test phases. In addition to CPU benchmarks, we evaluate our detection models against system, disk and memory test benchmarks. In order to increase the diversity, the daily applications such web browsing, video rendering, Apache server, MySQL database and Office applications with several parameters are profiled for real-world examples.
A subset of benign execution data is used to train our RNN models and then, the whole benign dataset is used to calculate the FPR (False Positive Rate) and TNR (True Negative Rate) of the models. In our work, FPs represents the benign applications which are classified as an attack/anomaly by the RNN model. If the benign application does not raise the alarm flag, it is considered as TNs.
For the attack executions we include traditional cache attack techniques such as F+F, F+R and P+P attacks. Different from previous works, these attacks are applied on arbitrary memory blocks to avoid any assumption on the target implementation. Spectre (v1 and v2) and Meltdown are implemented to read secrets such as passwords in a pre-determined memory location. In addition, two types of Rowhammer attacks namely, one-sided and double-sided, are applied to have bit flips. In order to test the efficiency of [*[FortuneTeller]{}*]{} we implemented a recent microarchitectural attack, Zombieload, to steal data across processes. For this purpose, a victim thread leaks pre-determined ASCII characters and the attacker reads the line-fill buffer to recover the secret. If the alarm flag is set during the execution of the attack, it is True Positive (TP). On the other hand, the undetected attack execution is represented by False Negative (FN).
### Performance Counter Selection {#sec:counters}
In our detection model, we leverage HPCs as security sensors. Although the number of available counters in a processor is more than 100, it is not feasible to monitor all counters concurrently. In an ideal system, we should be able to collect data from a diverse set of events to be able to train a generic model. However, due to the limited number of concurrently monitored events, we choose the most optimum subset of counters that give us information about common attacks. For this purpose, we perform a study of the best subset of performance counters.
In our experiments, we leverage Intel PCM tool [@intelpcm] to capture the system-wide traces. The set of counters in our experiments is chosen from *core* counters. The main reason to choose core counters is the high variety of the available counters such as branches, cache, TLB, etc. The number of core counters tested in the selection method is 36. The complete list is given in Appendix, .
In the data collection step, a subset of the counters is profiled concurrently, since the number of counters monitored in parallel is limited to four in Intel processors. For each subset, a separate dataset collected until all 36 counters are covered. The training data is collected from 30 different Phoronix benchmark tests [@phoronix] (1-30 in the ). In order to decide on the most suitable counters to detect microarchitectural anomalies, we collect a test dataset from 20 benchmark tests (1-56 in the ) and 6 microarchitectural attacks (174-179 in the ). The Zombieload attack is not included in the performance counter selection phase, since it was not released at that time. The sampling rate is chosen as 1 ms to have the minimal overhead in the system.
For every subset of counters one LSTM model is trained with a window size of $W=100$. The four dimensional data is given as an input to LSTM model and then, the final counters are selected based on their F-score given in Appendix, . It is observed that some counters have better accuracy than other counters for specific attacks. For instance, branch related counters have high correlation for Meltdown and Spectre attacks. However, the F-score is also around 0.3 because real-world applications also use the branches heavily. One of the important outcome of selection phase is that speculative branches are commonly integrated in the benign applications. Therefore, the counter selection shows that branch counters are not useful to detect speculative execution attacks. Thanks to our LSTM based counter selection technique, finding the most valuable counters is fully automated and the success rate of detecting anomalies with low FPR and FNR is increased significantly.
Even though it is allowed to choose up to 4 counters on the Intel server systems like Xeon, we selected 3 counters to profile for anomaly detection. The reason behind this is that in the desktop processors (Intel Core i5, i7) the programmable counters are limited to 3. The first selected counter is $L1\_Inst\_Miss$, which is more successful to detect Rowhammer, Spectre and Meltdown attacks with 14% FPR, where the F-score is 0.7979. As a second counter, $L1\_Inst\_Hit$ is chosen, since Flush+Flush and Flush+Reload attacks are detected with a high accuracy and the F-score is 0.8137. The reason behind the high F-score is that the *flush* instruction is heavily used in those attacks and the instruction cache usage also increases in parallel. Interestingly, Flush+Flush attack is known as a stealthy microarchitectural attack however, it is possible to detect it by monitoring instruction cache hit counter. The last selected counter is $LLC\_Miss$, which is successful to detect Rowhammer and Prime+Probe attacks with a high accuracy. These attacks cause frequent cache evictions in the LLC, which increases the number of anomalies in the $LLC\_Miss$ counter. These results show that the individual counters are not efficient to detect all the microarchitectural attacks. Therefore, there is a need for the integration of the aforementioned 3 counters to detect all the attacks with a high confidence rate.
Evaluation {#sec:results}
==========
In this section, we explain the experiments which are conducted to evaluate [*[FortuneTeller]{}*]{}. The experiments aim to answer the following research questions: 1) How does [*[FortuneTeller]{}*]{} perform in predicting the next performance counter values for benign applications with the increasing number of measurements()? What is the lowest possible FPR for server () and laptop environments ()? 3) How does the size of sliding window affect the performance of [*[FortuneTeller]{}*]{} ()? 4) How realistic is real time protection with [*[FortuneTeller]{}*]{} ()? 5) How much performance overhead is caused by [*[FortuneTeller]{}*]{} ()?
Experiment Setup
----------------
[*[FortuneTeller]{}*]{} is tested on two separate systems. The first system runs on an Intel Xeon E5-2640v3, which is a common processor used on server machines. It has 8 cores with 2.6 GHz base frequency and 20 MB LLC. The second device is used to illustrate a typical laptop/desktop machine, which is based on Intel(R) Core(TM) i7-8650U CPU with 1.90 GHz frequency. It has 8MB LLC and 2 cores in total.
Two types of RNN model namely, LSTM and GRU, are used to train [*[FortuneTeller]{}*]{}. The sliding window size, batch size and number of hidden LSTM/GRU layers are kept same in the training phase. Training of RNN models is done using the custom Keras [@chollet2015keras] scripts together with the Tensorflow [@abadi2016tensorflow] and GPU backend. The models are trained on a workstation with two Nvidia 1080Ti (Pascal) GPUs, a 20-core Intel i7-7900X CPU, and 64 GB of RAM.
RNN Model Training {#sec:RNN_training}
------------------
To detect the anomalies in the system, the first step is to learn the pattern of the benign applications. This is not an easy task, since the chosen benchmarks and real-world applications have complicated fingerprint in the microarchitectural level with the system noise. Moreover, the fingerprint at each execution is not identical and the execution of the application takes several seconds, which makes it difficult to learn long-term relations in the data. Therefore, the required number of measurements from each individual application plays a crucial role to train the [*[FortuneTeller]{}*]{} successfully. For this purpose, we choose 10 random benchmarks and a separate model for each of them is trained. The validation error obtained as a result of training is the critical metric to determine the capacity of the RNN algorithms as it indicates how well [*[FortuneTeller]{}*]{} guesses the next counter value. The first RNN model is trained with only 1 measurement and the number of measurements is increased gradually up to 44. It is observed that there is no further improvement in the validation error after 36 measurements for both LSTM and GRU networks in . Note that, the training data is scaled to \[0 1\] and the validation error is the average error of the 3 counters.
![Validation error with increasing number of measurements for Gnupg benchmark[]{data-label="fig:gnupg"}](gnupg_lstm_gru.eps){width="48.00000%"}
In , the prediction of $ICache.Hit$ counter value by using LSTM network is shown. The solid line represents the actual counter value while two other lines show the prediction values. When there is only one measurement to train the LSTM network, the prediction error is much higher. It means the trained model could not optimize the weights with small amount of data. On the other hand, once the number of measurements is increased to 36, the predictions are more consistent and close to actual counter value. The number of measurements directly affects the training time of the model. If the dataset is unnecessarily huge, the training time increases in linear. Therefore, it is decided to collect 36 measurements from each application in the training phase to achieve the best outcome from RNN algorithms in the real systems. With the accurate modeling of the benign behavior, the number of false alarms is reduced significantly. This is the main advantage of [*[FortuneTeller]{}*]{}, since the previous detection mechanisms apply a simple threshold technique to detect the anomalies when a counter value exceeds the threshold. In contrast, [*[FortuneTeller]{}*]{} can predict the sudden increases in the counter values and the correct classification can be made more efficiently than before.
![Prediction error in Gnupg for LSTM algorithm[]{data-label="fig:gnupg_prediction"}](gnupg_prediction.eps){width="48.00000%"}
Server Environment {#sec:server}
------------------
The first set of experiments is conducted in the server machine. Three core counters, $ICache.Miss$, $ICache.Hit$ and $LLC.Miss$ are monitored concurrently in the data collection process. The training dataset is collected with a sampling rate of 1 ms from $m=3$ core counters during the execution of benign applications. The dataset has 10 million samples (10,000 seconds) in total, collected from 67 randomly selected benchmark tests, 100 websites rendering in Google Chrome, Apache server/client benchmark, MySQL database and Office applications as listed in Appendix, . Note that the idle time frames between the executions are excluded from dataset to avoid the redundant information in the training phase. Firstly, LSTM model is trained with the collected dataset where the input size is $3\times10,000,000$. The sliding window size is selected as $W=100$, which means the total number of LSTM units equals to 100. The further details of window size analysis is given in . The training is stopped after 10 epochs since the validation error does not improve further. The validation error decreases to 0.0015. The training time for 10,000,000 samples takes approximately 4 days.
After the LSTM model is trained, a new dataset for the test phase is collected from counters by profiling 173 benign benchmark tests, 100 random websites, MySQL, Apache, Office applications and micro-architectural attacks. The length of the test data for each application would change, since our anomaly-based model has no assumption on the input length. Hence, the number of samples obtained from each application changes between 1000-20000. The number of samples for websites is around 1000, since the rendering process is extremely fast. However, some benchmarks have a longer execution time, which requires to collect data for a longer time. The remaining applications (Office, MySQL, Apache) are profiled for around 5 seconds.
Each application is monitored 50 times, and then, the test data is fed into the LSTM model to predict the counter values at the next time steps. Moreover, in order to make the test phase more realistic, the number of applications running concurrently is increased up to 5. The applications are chosen randomly from the test list in Appendix , and started at the same time. 100 measurements are collected from concurrently running applications. In total, 25,000,000 samples are collected for the test phase.
The prediction is made for all three counters at each time step (every 1 ms), and then, the mean squared error $e^{(t+1)}$ is computed between the actual and predicted counter values. $e^{(t+1)}$ in the prediction step is used to choose optimal decision window $D$ and $\tau_{A}$ to detect the anomalies in the system. If the prediction error is higher than the threshold $\tau_{A}$ for $D$ samples, the application is classified as an anomaly. The threshold and decision window are chosen as to equalize FNR and FPR. The trend between $\tau_{A}$ for $D$ is given in . For the lower $\tau_{A}$ values, the decision window is not applicable to detect the anomalies, since the benign applications and attack executions have higher error rates. Once the $\tau_{A}$ reaches $1.8 \times 10^6$, most of the attack executions are detected in $D=50$ samples. In other words, the microarchitectural attacks are caught in 50 ms by [*[FortuneTeller]{}*]{}. With the increasing $\tau_{A}$ values, the number of true positives begins decreasing, which yields to low detection rate.
![Threshold vs. Decision Window for benign applications (gray) and attack executions (red) using LSTM model in server scenario[]{data-label="fig:server_lstm"}](server_lstm.eps){width="48.00000%"}
The results show that P+P attack is the most difficult attack to be detected by the LSTM model in the server. This result is expected since P+P attack mostly focuses on specific cache sets and the cache miss ratio is smaller than other type of attacks. In addition, instruction cache is not heavily used by P+P attack, which makes the detection more difficult for [*[FortuneTeller]{}*]{}due to the lack of specific pattern. On the other hand, the highest TPR is obtained for Flush+Reload and Rowhammer attacks with 100% and 0% FNR. As these attacks increase the number of data cache misses and instruction hits through the extensive use of *clflush* instruction, the fluctuation in the counter values is higher than the other types of attacks and benign applications. The accuracy of the predicting the next values decreases when the variance is high in the counters, thus, the prediction error increases in parallel. Since the higher prediction rate is a strong indicator of the attack executions in the system, [*[FortuneTeller]{}*]{} detects them with a high accuracy. Note that, Zombieload is also detectable by the [*[FortuneTeller]{}*]{}, even though it was not included in the performance counter selection phase. This shows that [*[FortuneTeller]{}*]{} can detect the unseen microarchitectural attacks with the current trained models.
The ROC curves in indicate that LSTM networks have a better capability than GRU networks to detect the anomalies. The counter values are predicted with a higher error rate in GRU networks, which makes the anomaly detection harder. Some benign applications are always detected as anomaly by GRU, thus, the FPR is always high for different threshold values. The AUC (Area Under the Curve) for LSTM model is very close to perfect classifier with a value of 0.9840. On the other hand, the AUC for GRU model is 0.9125, which is significantly worse than LSTM model. There are several reasons behind the poor performance of GRU networks. The first reason is that GRU networks are not successful to learn the patterns of Apache server applications since there is a high fluctuation in the counter values. In addition, when the number of concurrently running applications increases, the false alarms increase drastically. On the other hand, LSTM networks are good at predicting the combination of patterns in the system. Therefore, the FPR is very small for LSTM model.
![ROC curve for LSTM and GRU models in server scenario[]{data-label="fig:server_ROC"}](ROC_Curve_Server.eps){width="29.70000%"}
Laptop Environment {#sec:laptop}
------------------
The experiments are repeated for the laptop environment to evaluate the usage of [*[FortuneTeller]{}*]{}. LSTM and GRU models are trained with 10 million samples, which is collected from benign applications. Since the laptops are mostly used for daily works, the counter values are relatively smaller than the server scenario. However, the applications stress the system more than the server scenario since the number of cores is lower.
![Threshold vs. Decision Window for benign applications (gray) and attack executions (red) using LSTM model in laptop scenario[]{data-label="fig:laptop_lstm"}](laptop_lstm.eps){width="48.00000%"}
When we analyze the relation between $D$ and $\tau_{A}$, we observe the same situation as in the server scenario. The lower $\tau_{A}$ values are not sufficient to differ the anomalies from benign executions. Therefore, we need to choose the optimal $\tau_{A}$ value slightly higher than the server scenario with a value of $3.8 \times 10^6$. The corresponding $D$ value is 60, which means that the anomalies are detected in 60 ms. The decision window is 10 ms bigger than server scenario however, the performance of [*[FortuneTeller]{}*]{} is better in laptop scenario. In , the ROC curves of LSTM and GRU models are compared. The AUC value of LSTM model is considerably higher than GRU model with a value of 0.9865. However, the AUC value for GRU is 0.8925. This shows that LSTM outperforms GRU model to predict the counter values of benign applications. This also concludes that FNR and FPR are lower for LSTM models.
Among the attack executions, Rowhammer attack can be detected with 100% success rate since the prediction error is very high. The other attacks have similar prediction errors, hence, [*[FortuneTeller]{}*]{} can detect the attacks with the same success rate. Since the computational power of laptop devices is low, the concurrent running applications have more noise on the counter values. Therefore, the prediction of the counter values is more difficult for RNN algorithms. While LSTM networks have small FPR for 4 and 5 applications running at the same time, GRU networks are not efficient to classify them as benign applications.
![ROC curve for LSTM and GRU models in laptop scenario[]{data-label="fig:laptop_ROC"}](ROC_Curve_Laptop.eps){width="30.00000%"}
The overall results show that LSTM works better than GRU networks for both laptop and server scenarios in . The first and second values represent the LSTM and GRU false alarm rates per second in percentages, respectively. In the server scenario, videos, MySQL and Office applications never give false positives. Websites running in Google Chrome have a small amount of false alarm. Therefore, the FPR and FNR are around 0.12% per second for LSTM network in server scenario overall. The main disadvantage of GRU networks is the poor performance in the prediction when the number of applications increases. The FPR and FNR are approximately 0.24%. This shows that the number of false alarms is twice more for GRU based [*[FortuneTeller]{}*]{}.
In laptop scenario, LSTM performs better, which is supported by the false alarm rate. The number of false alarms is lower than the server scenario for laptop devices with a value of 0.09%. On the other hand, the GRU networks are lack of ability to predict the counter values, thus, it is also reflected in false alarm rates. For every application, GRU has a higher false alarm rate than LSTM networks. Therefore, it is concluded that [*[FortuneTeller]{}*]{}should be trained with LSTM networks to have the better performance in both server and laptop scenarios.
------------ -------- -------- -------- --------
LSTM GRU LSTM GRU
Benchmarks 0.1400 0.1442 0.1202 0.4808
Websites 0.0550 0.0972 0.0278 0.6667
Videos 0.0000 0.0000 0.0000 0.4138
MySQL 0.0000 0.0000 0.0000 0.0000
Apache 0.0000 0.8333 0.0000 0.4030
Office 0.0000 0.0000 0.0000 0.0000
2 Apps 0.0000 0.0000 0.0000 0.5000
3 Apps 0.0000 0.1667 0.0000 0.5715
4 Apps 0.0750 0.2000 0.0500 0.6667
5 Apps 0.1250 0.2333 0.1000 0.8333
------------ -------- -------- -------- --------
: The False Alarm Rate in percentage per second for applications
\[table:accuracy\]
Varying size of Sliding Window {#sec:window_size}
------------------------------
We observed that the prediction results are affected by the size of the window. Therefore, we analyze the effect of sliding window size on anomaly detection with the data collected from core counters with 1 ms sampling rate in the server environment. 12 different window sizes are used to train LSTM and GRU models. The window size starts from 25 and increased by 25 at each step until reaching 300.
The changes in the validation error for both LSTM and GRU networks are depicted in . The overall GRU training error is higher than LSTM network for each window size. Both models reach the lowest error when the sliding window size is 100. Even though LSTM and GRU are designed to learn long sequences, it is recommended to choose the window size between 50-150. Since the best error is obtained with a window size of 100, all the models in the previous experiments are trained with this parameter. It is also important to note that the training time increases proportional to the size of the window.
![The validation error with varying size of sliding window[]{data-label="fig:lstm_gru_error"}](LSTM_GRU_error_comparison.eps "fig:"){width="47.00000%"}\[fig:LSTM\_GRU\_error\]
Time consumption for Testing {#sec:dynamic}
----------------------------
The dynamic detection of the anomalies also depends on the time consumption of predicting the next counter values. Therefore, the sampling rate should be chosen as close as to the timing consumption of predicting the next value. In our experiments, we observed that the prediction time is proportional to the size of the model. Since GRU has less number of cells in the architecture, the prediction of GRU is faster. While LSTM outputs prediction values for 3 counters in 2 ms, the prediction time for GRU is 1.7 ms. It shows that GRU is 15% faster than LSTM in the prediction phase. However, due to the high FPR of GRU networks [*[FortuneTeller]{}*]{} is trained with LSTM networks to detect anomalies in the real time system.
Performance Overhead {#sec:performance_overhead}
--------------------
The performance overhead of the proposed countermeasures is one of the most important concerns, since it affects all the applications running on the system. In this section, we evaluate the performance overhead for both server and laptop devices when core counters are used to collect data. The overhead amount is obtained with sampling rates of 1ms and 10$\mu$s. As it is expected the performance overhead increases in parallel with the sampling rate. In the server environment the overhead is around 7.7% when the sampling rate is chosen as 10$\mu$s. The overhead of individual tests fluctuates between 1% and 33% for benign applications. The performance of system and memory benchmarks is affected more than processor based benchmarks. On the other hand, when the sampling rate is decreased to 1ms, the performance overhead drops to 3.5%. The individual overheads change between 0.3% and 18%, which is more stable than the previous case.
In laptop scenario, the performance overhead is also calculated with the same benign benchmarks. The number of cores is smaller than the server scenario. Therefore, the performance monitoring unit only needs to read the counter values from 4 threads in parallel. On the other hand, since the system has lower features compared to the server machine, the overhead is increased when the sampling rate is increased .The overall performance degradation is 24.88% for 10$\mu$s. The overhead fluctuates heavily, which means that the applications suffer from the frequent interruptions to read the counter values. Once the sampling rate is decreased to 1 ms, the overhead drops to 1.6%, which is applicable in real-time systems. This overhead is also lower than the server scenario. Therefore, We preferred 1 ms sampling rate in our experiments.
Comparison of [*[FortuneTeller]{}*]{} with Prior Detection Methods {#sec:comparison}
==================================================================
There are several studies focused on microarchitectural attack detection as given in . While some works [@briongos2018cacheshield; @chiappetta2016real; @Zhang2016cloudradar] use unsupervised techniques, Mushtaq et al. [@mushtaq2018nights] benefits from supervised ML methods. All proposed methods claim that the false positive rate is very low in a real world scenarios. However, all these detection techniques are only applied for cryptographic implementations (AES, RSA, ECDSA etc.) and specific cache attacks (F+R, F+F, P+P). The performance of these techniques in real world scenarios (noisy environment, multiple concurrent processes) against transient execution attacks (Meltdown, Spectre, Zombieload etc.) and Rowhammer is questionable. In order to evaluate 4 proposed methods and [*[FortuneTeller]{}*]{}, we collected 6 million samples (4 million benign executions, 2 million attack executions) with 1ms sampling rate from 10 benign processes and 7 microarchitecture attacks by using **system-wide counters**. Note that each benign and attack execution is monitored 100 times in the server environment. The benign processes are chosen from diverse set of applications such as Apache, MySQL, browser and cryptographic implementations. The attacks cover cache-based, transient execution and Rowhammer attacks given in . The detection algorithms from previous works are rewritten in Matlab environment and tested with the collected data.
[***CPD from Briongos et al. [@briongos2018cacheshield]***]{} The first approach is Change Point Detection (CPD) which was implemented by Briongos et al. [@briongos2018cacheshield] to detect the anomalies in the victim process. The primary advantage of the method is to have the capability of self-learning by observing the number of cache misses. On the other hand, the assumption of having almost no LLC miss is a strong assumption, which is not applicable in real-world scenarios for system-wide profiling. Especially, when an application runs for the first time in the system, the number of cache misses increases drastically. This yields to high number of false positives at the beginning of the applications. Even though it is tried to eliminate the initial false positives by increasing the initial value of cache misses under attack ($\mu_{a}$), we still observe several false positives at the beginning. It is also difficult to monitor each PID in the system since there are hundreds of processes running at the same time.
For the evaluation of CPD method, we use the initial value of $\mu_{a} = 100$ and $\beta = 0.65$. When CPD method is applied to our dataset, we observe that the FPR is 3% and FNR is 10%. Therefore, the F-score is 0.9372. However, with the increasing number of concurrent processes, the false positive rate increases. This shows that CPD method is efficient for low system load however, it gives more false positives with increasing workload. The estimated detection time is around 300 ms for attack executions. The detection performance for Rowhammer and P+P attacks is poor since the number of cache misses is not high compared to benign processes. Therefore, these two attack types increase the FNR.
[***DTW from Zhang et al. [@Zhang2016cloudradar]***]{} The second detection method was proposed by Zhang et al. [@Zhang2016cloudradar], which benefits from Dynamic Time Warping (DTW) to detect the cryptographic implementations and then, the LLC hit and miss counters are monitored to detect the attacks. In the first step, DTW is used to compare the test data and the signature of cryptographic implementations obtained from branch instructions. Secondly, when the distance between test and target execution is very small, the LLC hit and LLC miss counters are monitored. If there is a sudden jump in these two counters, the anomaly flag is set. Again, this approach requires the PID of the monitored process.
In the evaluation of the method, we started with the application detection. Since the number of target applications is small in our dataset, DTW can detect them with 100% success rate in a noiseless environment. However, when there is a concurrent process running in the system, the DTW distance is always high. The reason behind this failure is that branch instructions are heavily affected by the other processes. Therefore, DTW is not suitable for real-world scenarios. Another drawback is that if the microarchitectural attack already started, the branch instructions is also affected, which prevents to detect the target application. Hence, the attack detection step never starts. When the target process is detected, the anomaly detection step begins. If another concurrent work starts running at the same time, the cache miss and hit counters start increasing, which increases the FPR extremely. Since there is only a simple threshold approach to detect the attacks and the proposed decision window (5 ms) is too small, the FPR raises. In these circumstances, the approach achieves 10% FPR. The attack detection is also not great since it is not possible to detect F+F attack with cache miss and hit counters. Thus, the the FNR increases in parallel which yields to 20% FNR. Overall, the detection technique has 0.8572 F-score.
[***PDF from Chiappetta et al. [@chiappetta2016real]***]{} In the third study, we evaluate the performance of normal distribution and probability density function, which is proposed by Chiappetta et al. [@chiappetta2016real]. The detection technique monitors five counters (total instructions, CPU cycles, L2 hits, L3 miss and L3 hits) to catch the anomalies in the cryptographic implementations. This technique is used in unsupervised manner by only learning the normal distribution of the attack execution (F+R) with its mean and variance in the system. After the normal distribution is calculated, the probability density function (pdf) of both attack and benign executions is calculated for each counter sample. Then, an optimal threshold ($\epsilon$) is chosen to separate the benign and attack processes. To evaluate the performance of the method, we collected a separate dataset with the aforementioned five counters. The results indicated that total instructions and L2 hits decrease the performance of detecting anomalies. On the other hand, L3 hits and miss counters overperform other counters. The main drawback of this method is that there is no learning and the decisions are made on only cache miss and variance values. Therefore, when there is a benign application with high variance and mean, it is more likely to be classified as an anomaly. Especially, Apache server benchmark and videos running in browsers give high FPR. It is also observed that the P+P, F+F and Rowhammer attacks are not detected with a high accuracy, which give 0.2145 and 0.3732 for FPR and FNR, respectively. The F-score of the detection technique is 0.7278.
[***OC-SVM from Mushtaq et al. [@mushtaq2018nights]***]{} The last method to compare is One Class Support Vector Machine (OC-SVM), which is used by Mushtaq et al. [@mushtaq2018nights] to detect the anomalies on cryptographic implementations. The scope is limited F+F and F+R attacks. The number of counters tested in [@mushtaq2018nights] is higher than three, which makes it impossible to monitor all of them concurrently. Therefore, we chose three counters (L1 miss, L3 hit and L3 total cache access), which give the highest F-score. Even though OC-SVM was used in a supervised way in [@mushtaq2018nights], we used it in unsupervised manner to maintain the consistency in the comparison. In the training phase, the model is trained with the 50% of the benign execution data. Then, the attack and benign dataset are tested with the trained model. The obtained confidence scores are used to find the optimal decision boundary to separate the benign and attack executions. The optimal decision boundary shows that the FPR and FNR are 0.2750 and 0.2778, respectively. The main problem is that OC-SVM is not sufficient to learn the diverse benign applications, which increases the FPR drastically. Moreover, Rowhammer and F+F attacks are not detected, which is the reason of higher FNR. Therefore, the F-score remains at 0.7240.
[***[*[FortuneTeller]{}*]{}***]{} Finally, we apply [*[FortuneTeller]{}*]{} to detect the anomalies in the system. Since the diversity of the benign executions is smaller in the comparison dataset, it is more easier to learn the patterns. It is also important to note that 50 measurements from each benign application is enough to reach the minimum prediction error. Once the LSTM model is trained with the benign applications, the attack executions and remaining benign application data are tested. The FPR and FNR remains at 0.2% and 0.4%, respectively. The F-score is 0.997 for the [*[FortuneTeller]{}*]{}.
**Technique** **F-score**
-------------------------------------------- --------------- -------------
Briongos et al. [@briongos2018cacheshield] CPD 0.9372
Zhang et al. [@Zhang2016cloudradar] DTW 0.8572
Chiappetta et al. [@chiappetta2016real] Normal Dist. 0.7278
Mushtaq et al. [@mushtaq2018nights] OC-SVM 0.7240
**Our work** **LSTM/GRU** **0.9970**
: Comparison of previous methods
\[tab:comp\_tech\]
The comparison results are summarized in . The lack of appropriate learning is significant in the wild. It is also obvious that even simple learning algorithm such as CPD can help to overperform other detection techniques. We also show that the detection accuracy increases by learning the sequential patterns of benign applications with the system-wide profiling. Therefore, it is significantly important to extract the fine-grained information from the hardware counters to achieve the low FPR and FNR. The common deficiencies of previous works are listed below:
- The detection methods focus on only cryptographic implementations, and the latest attacks such as Rowhammer, Spectre, Meltdown and Zombieload are not covered.
- There is no advanced learning technique applied in the detection methods. They mostly rely on the sudden changes in the counters, which increases the FPR heavily.
- The detection methods are either tested under no noise environment or the workload is not realistic. In addition, the FPR is not tested with a diverse set of applications.
Discussion {#sec:discussion}
==========
[***Bypassing [*[FortuneTeller]{}*]{}***]{} One of the questions about dynamic detection methods is that how an educated adversary can bypass the detection model? The common way is to put some delays between the attack steps to avoid increasing the counter values. For this purpose, we inserted different amounts of idle time frames between attack steps in Flush+Flush, Prime+Probe and Flush+Reload. We observed that the prediction errors in GRU and LSTM networks increases in parallel with the amount of sleep due to the high fluctuation. This shows that introducing delays between attack steps is not an efficient way to circumvent [*[FortuneTeller]{}*]{}. The reason behind this is the fluctuation in the time series data is not predicted well in the prediction phase. Therefore, we concluded that putting different amount of sleep between the attack steps is not enough to fool [*[FortuneTeller]{}*]{}. On the other hand, crafting adversarial examples is an efficient way to bypass Deep Learning based detection methods. For instance, Rosenberg et al [@rosenberg2017generic] shows that LSTM/GRU based malware detection techniques can be bypassed by carefully inserting additional API calls in between. Therefore, crafting adversarial code snippets to change the performance counters in the attack code may fool [*[FortuneTeller]{}*]{}. The main difficulty in this approach is that it is not possible to decrease the counter values by executing more instructions between attack steps. Therefore, applying adversarial examples on hardware counter values is not trivial.
[***Training Algorithm***]{} [*[FortuneTeller]{}*]{} investigates both available long-term dependency learning techniques. We observed that GRU performs worse than LSTM networks to predict the counter values in the next time steps. This is because of the lack of internal memory state, which keeps the relevant information from previous cells. This result is also supported with the high FPR and FNR of GRU networks. Since the prediction error increases for attack executions more than benign applications, the detection accuracy decreases. Therefore, we recommend to train LSTM networks for microarchitectural attack detection techniques.
[***Dynamic Detection***]{} The current implementation requires to have a GPU to train [*[FortuneTeller]{}*]{}, as GPU based training 40 times faster than CPU based training. The training is mostly done in an offline phase and it does not affect the dynamic detection. On the other hand, dynamic detection heavily depends on the matrix multiplication, since the trained model is loaded as a matrix in the system and the same matrix is multiplied with the current counter values. Hence, the required time to predict the next counter values is lower. In addition, we observed that the performance overhead is negligible for the matrix multiplication in the CPU systems. Therefore, [*[FortuneTeller]{}*]{} can be implemented in server/cloud/laptop environments, even though there is no GPU integrated in the system.
Conclusion {#sec:conclusion}
==========
This study presented [*[FortuneTeller]{}*]{}, which exploits the power of neural networks to overcome the limitations of the prior works, and further proposes a novel generic model to classify microarchitectural events. [*[FortuneTeller]{}*]{} is able to dynamically detect microarchitectural anomalies in the system through learning benign workload. In our study, we adopted two state-of-the-art RNN models: GRU and LSTM. We concluded that LSTM is more preferable compared to GRU for our use case. Further, the number of measurements and the sliding window size have a significant effect on the validation error in training phase, which makes it crucial to choose the optimal values to have better prediction results. [*[FortuneTeller]{}*]{} is applicable to both server and laptop environments with a high accuracy. In order to evaluate the performance of [*[FortuneTeller]{}*]{}, we used both benchmarks and real-world applications and achieved 0.12% and 0.09% FPRs for server and laptop environments, respectively. [*[FortuneTeller]{}*]{} is also tested against previous works in the realistic scenarios and it is concluded that, [*[FortuneTeller]{}*]{}overperforms other detection meahcnisms in the wild. While the performance overhead in laptop environment is less than server, [*[FortuneTeller]{}*]{} is still applicable in the real world systems with minimal overhead.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the National Science Foundation, under grants CNS-1618837 and CNS-1314770.
Appendix {#sec:appendix}
========
Tables for Performance Counters and Benchmarks
----------------------------------------------
Counter F-score
------------------------------------------------------- ------------
$Dtlb\_Load\_Misses.Miss\_Causes\_A\_Walk$ 0.5657
$Dtlb\_Load\_Misses.Walk\_Completed\_4K$ 0.5226
$Dtlb\_Load\_Misses.Walk\_Completed$ 0.5327
$Dtlb\_Load\_Misses.Walk\_STLB\_Hit\_4K$ 0.3627
$UOPS\_Issued\_Any$ 0.3663
$ICACHE.Hit$ **0.8137**
$ICACHE.Miss$ **0.7979**
$L1D\_Pend\_Miss.Pending$ 0.6818
$L1D\_Pend\_Miss.Request\_FB\_Full$ 0.6698
$L1D.Replacement$ 0.7523
$L2\_Rqsts\_Lat\_Cache.Miss$ 0.6244
$LLC\_Miss$ **0.8416**
$LLC\_Reference$ 0.6167
$IDQ.Mite\_UOPS$ 0.3383
$BR\_Inst\_Exec.Nontaken\_Cond.$ 0.2703
$BR\_Inst\_Exec.Taken\_Cond.$ 0.3390
$BR\_Inst\_Exec.Taken\_Direct\_Jmp$ 0.3455
$BR\_Inst\_Exec.Taken\_Indirect\_Jmp\_Non\_Call\_Ret$ 0.3137
$BR\_Inst\_Exec.Taken\_Indirect\_Near\_Return$ 0.2944
$BR\_Inst\_Exec.Taken\_Direct\_Near\_Call$ 0.3618
$BR\_Inst\_Exec.Taken\_Indirect\_Near\_Call$ 0.3592
$BR\_Inst\_Exec.All\_Cond.$ 0.2634
$BR\_Inst\_Exec.All\_Direct\_Jmp$ 0.3238
$BR\_Misp\_Exec.Nontaken\_Cond.$ 0.3648
$BR\_Misp\_Exec.Taken\_Cond.$ 0.4510
$BR\_Misp\_Exec.Taken\_Indirect\_Jmp\_Non\_Call\_Ret$ 0.4455
$BR\_Misp\_Exec.Taken\_Ret\_Near$ 0.3491
$BR\_Misp\_Exec.Taken\_Indirect\_Near\_Call$ 0.3553
$BR\_Misp\_Exec.All\_Branches$ 0.2700
$BR\_Inst\_Retired.Cond.$ 0.4623
$BR\_Inst\_Retired.Not\_Taken$ 0.4412
$BR\_Inst\_Retired.Far\_Branch$ 0.4608
$BR\_Misp\_Retired.All\_Branch$ 0.4615
$BR\_Misp\_Retired.Cond.$ 0.3786
$BR\_Misp\_Retired.All\_Branches\_Pebs$ 0.2111
$BR\_Misp\_Retired.Near\_Taken$ 0.2871
: Counter Selection for core counters
\[table:counters\]
=0.11cm
-- --------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- --
**System Tests & **Disk Tests & **Memory Tests & **Real-World & **Attacks\
1) Aobench& 41) Minion 1 & 81) Graphics 1 & 120) Apache & 153) Aio-stress & 165) Mbw & 174) Websites & 1) Flush+Flush\
2) Botan 1& 42) Minion 2 & 82) Graphics 2 & 121) Battery & 154) Blogbench 1 & 166) Ram 1 & 175) Videos & 2) Flush+Reload\
3) Botan 2& 43) Minion 3 & 83) Graphics 3 & 122) Compress & 155) Blogbench 2 & 167) Ram 2 & 176) MySQL & 3) Prime+Probe\
4) Botan 3& 44) Perl 1 & 84) Graphics 4 & 123) Git & 156) Compile & 168) Ram 3 & 177) Apache & 4) Meltdown\
5) Botan 4& 45) Perl 2 & 85) Graphics 5 & 124) Hint & 157) Dbench & 169) Ram 4 & 178) Office & 5) Spectre\
6) Botan 5& 46) Radiance 1 & 86) Graphics 6 & 125) Nginx & 158) Fio 1 & 170) Ram 5 & & 6) Rowhammer\
7) Bullet 1& 47) Radiance 2 & 87) Graphics 7 & 126) Optcarrot & 159) Fio 2 & 171) Stream & & 7) Zombieload\
8) Bullet 2& 48) Scimark 1 & 88) Hpcg & 127) Php 1 & 160) Iozone & 172) T-test & &\
9) Bullet 3& 49) Scimark 2 & 89) Luajit 1 & 128) Php 2 & 161) Postmark & 173) Tinymem & &\
10) Bullet 4& 50) Scimark 3 & 90) Luajit 2 & 129) Pybench & 162) Sqlite & & &\
11) Bullet 5& 51) Scimark 4 & 91) Luajit 3 & 130) Schbench & 163) Tiobench & & &\
12) Bullet 6& 52) Scimark 5 & 92) Luajit 4 & 131) Stress-ng 1 & 164) Unpack & & &\
13) Bullet 7& 53) Scimark 6 & 93) Luajit 5 & 132) Stress-ng 2 & & & &\
14) Cache 1& 54) Swet & 94) Luajit 6 & 133) Stress-ng 3 & & & &\
15) Cache 2& 55) Hackbench & 95) Mencoder & 134) Stress-ng 4 & & & &\
16) Cache 3& 56) M-queens & 96) Multichase 1 & 135) Stress-ng 5 & & & &\
17) Gzip& 57) Mrbayes & 97) Multichase 2 & 136) Stress-ng 6 & & & &\
18) Dcraw& 58) Npb 1 & 98) Multichase 3 & 137) Stress-ng 7 & & & &\
19) Encode& 59) Npb 2 & 99) Multichase 4 & 138) Stress-ng 8 & & & &\
20) Ffmpeg& 60) Npb 3 & 100) Multichase 5 & 139) Stress-ng 9 & & & &\
21) Fhourstones& 61) Npb 4 & 101) Polybench-c & 140) Stress-ng 10 & & & &\
22) Glibc 1& 62) Npb 5 & 102) Sample & 141) Stress-ng 11 & & & &\
23) Glibc 2& 63) Npb 6 & 103) Sudokut & 142) Stress-ng 12 & & & &\
24) Glibc 3& 64) Npb 7 & 104) C-ray & 143) Stress-ng 13 & & & &\
25) Glibc 4& 65) Povray & 105) Cloverleaf & 144) Stress-ng 14 & & & &\
26) Glibc 5& 66) Smallpt & 106) Dacapo 1 & 145) Stress-ng 15 & & & &\
27) Glibc 6& 67) Tachyon & 107) Dacapo 2 & 146) Stress-ng 16 & & & &\
28) Glibc 7& 68) Bork & 108) Dacapo 3 & 147) Sunflow & & & &\
29) Glibc 8& 69) Build Apache & 109) Dacapo 4 & 148) Sysbench 1 & & & &\
30) Gnupg& 70) Byte 1 & 110) Dacapobench 5 & 149) Sysbench 2 & & & &\
31) Java 1& 71) Byte 2 & 111) John 1 & 150) Tensorflow & & & &\
32) Java 2& 72) Byte 3 & 112) John 2 & 151) Tjbench & & & &\
33) Java 3& 73) Byte 4 & 113) John 3 & 152) Xsbench & & & &\
34) Java 4& 74) Clomp & 114) Mafft & & & & &\
35) Java 5& 75) Crafty & 115) N-queens & & & & &\
36) Java 6& 76) Dolfyn & 116) Openssl & & & & &\
37) Lzbenc 1& 77) Espeak & 117) Primesieve & & & & &\
38) Lzbench 2& 78) Fftw & 118) Stockfish & & & & &\
39) Lzbench 3& 79) Gcrypt & 119) Ttsiod & & & & &\
40) Lzbench 4& 80) Gmpbench & & & & & &\
**********
-- --------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- --
\[table:tests\]
|
---
author:
- 'Tim Fischer[^1], Sebastian Duffe and Götz S. Uhrig[^2]'
title: 'Microscopic Model for Bose-Einstein Condensation and Quasiparticle Decay'
---
Low-dimensional antiferromagnetic quantum spin systems display various fascinating properties, e.g., spin-Peierls transition [@bray75; @hase93a], appearance of a Haldane gap for integer spins [@halda83; @renar87], high-temperature superconductivity upon doping [@bedno86], and the Bose-Einstein condensation (BEC) in spin-dimer systems [@affle91; @shira97; @giama99; @garle07], where the latter one is characterized by a phase transition from a non-magnetic phase to a long-range antiferromagnetically ordered gapless phase at a critical magnetic field $H_{c1}$.
Another fascinating phenomenon recently observed in low-dimensional antiferromagnets is the decay of their elementary $S=1$ excitations, triplons [@schmi03c], at higher energies so that the triplons exist only in a restricted part of the Brillouin zone [@stone06; @masud06]. Theoretically as well, there is rising interest in the understanding and quantitative description of this phenomenon for gapped triplons [@kolez06; @zhito06a; @bibik07; @fisch10a] as well as for gapless magnons [@zhito99; @zheng06a; @chern09].
The description of quasiparticle decay faces an intrinsic difficulty. The merging of the long-lived, infinitely sharp elementary triplon with a multitriplon continuum requires to describe the resulting resonance and its edges precisely. This is still a challenge for numerical approaches such as exact diagonalization or dynamic density-matrix renormalization [@kuhne99a]. Diagrammatic approaches are able to capture the qualitative features but may encounter difficulties in the quantitative description in the regime of strong merging where the sharp mode dissolves completely in the continuum because this is a strong coupling phenomenon [@kolez06; @zhito06a]. Unitary transformations also face difficulties when modes of finite life-time occur [@fisch10a].
A crucial step in the understanding of both phenomena is to identify a suitable experimental system. The best studied candidate for the BEC in coupled spin-dimer systems is TlCuCl$_3$. Unfortunately, recent research suggests that the high field spectrum remains gapped [@sirke05; @johan05] in contrast to what is expected from a phase where a continuous symmetry is broken. This suggests the existence of anisotropies. A promising alternative for a BEC in a spin-dimer system is (CH$_3$)$_2$CHNH$_3$CuCl$_3$ (isopropylammonium trichlorocuprate(II), short: IPA-CuCl$_3$) where inelastic neutron scattering (INS) provides evidence for an almost exact realization of a BEC [@garle07; @zhelu07].
A suitable experimental system to study triplon decay in detail is searched for. The two-dimensional (2D) PHCC [@stone01; @stone06] is a candidate, but it involves eight different couplings so that a quantitative characterization is impossible to date. Due to its quasi one-dimensional (quasi 1D) structure, IPA-CuCl$_3$ is again a more promising candidate. This compound seems to realize the theoretically proposed situation for BEC in coupled spin ladders [@giama99].
But in spite of many years of intensive studies [@rober81; @manak97; @manak00b; @masud06; @manak07b; @garle07; @hong10a] no quantitative microscopic model for IPA-CuCl$_3$ is established. The present work aims at filling this gap. Theoretically, our study is based on continuous unitary transformations (CUTs) of models with quasiparticle decay [@fisch10a] and on high temperature series expansions for asymmetric spin ladders which are topologically equivalent to dimerized and frustrated spin chains [@buhle01a]. The experimental input used in INS data [@masud06] and magnetic susceptibility $\chi(T)$ data [@manak97]. We will illustrate why it is intrinsically difficult to determine the microscopic model.
Finally, we will compute the temperature and the magnetic field dependence of the lowest magnetic modes as well as the upper critical magnetic field $H_{c2}$, which induces full polarization. They all agree very well with experimental data [@garle07; @zhelu07; @manak08; @zhelu08; @nafra11] which supports the advocated model.
 Sketch of IPA-CuCl$_3$. Circles indicate Cu ions with $S=1/2$. The couplings $J_1$ and $J_4$ are ferromagnetic ($J_1,J_4<0$) while $J_2$ and $J_3$ are antiferromagnetic ($J_2,J_3>0$). Two spins linked by $J_3$ form a dimer.](fig1){width="0.87\columnwidth"}
Since the characterization of [IPA-CuCl$_3$ ]{} by Roberts *et al.* [@rober81] various spin models were discussed. Manaka *et al.* pointed out that the magnetic susceptibility of IPA-CuCl$_3$ can be explained by a ferro-antiferromagnetically alternating Heisenberg $S=1/2$ chain with ferromagnetic coupling twice as large as the antiferromagnetic coupling [@manak97]. According to Hida [@hida92] the magnetic ground state is thus given by a gapped Haldane state [@halda83].
The dispersions measured by INS [@masud06] and the crystal structure of IPA-CuCl$_3$ indicates that the system is quasi-2D. It is described by weakly coupled asymmetric spin $S=1/2$ Heisenberg ladders, see Fig. \[fig:ipa\], with
\[eq:ham\_def\] $$\begin{aligned}
\label{eq:hamiltonian}
H =& H_{\text{1D}} + H_{\perp}
\\
\label{eq:hamiltonian_ladder}
H_{\text{1D}} =& J_1 \sum_{r,s} \mathbf{S}_{1,r,s}\mathbf{S}_{2,r+1,s}
+ J_3 \sum_{r,s} \mathbf{S}_{1,r,s} \mathbf{S}_{2,r,s}
\notag \\
+& J_2 \sum_{r,s} \left(\mathbf{S}_{1,r,s} \mathbf{S}_{1,r+1,s}+
\mathbf{S}_{2,r,s} \mathbf{S}_{2,r+1,s} \right)
\\
\label{eq:hamiltonian_int}
H_{\perp} =& J_4\sum_{r,s} \mathbf{S}_{1,r,s} \mathbf{S}_{2,r+1,s+1}
\end{aligned}$$
with two ferromagnetic couplings $J_1$, $J_4<0$ and two antiferromagnetic couplings $J_2$, $J_3>0$. The dominant dimer coupling is $J_3$ so that we use the ratios $x=J_2/J_3$, $y=J_1/J_3$ and $z=J_4/J_3$. Let us first consider the ladders as isolated because the interladder coupling is small. The standard view of these ladders takes the $J_3$ bonds to form the rungs of the ladder. Then $J_1$ is a diagonal bond.
The key element of this model is the asymmetry of the spin ladders controlled by $J_1$. On the one hand, the presence of $J_1$ spoils the reflection symmetry about the center line of the ladder between the legs. This symmetry would imply a conserved parity such that the triplons on the dimers could be changed only by an even number [@knett01b; @schmi05b] so that no decay of a triplon into a pair of triplons could occur. Hence the very presence of $J_1$ opens an important decay channel for quasiparticle decay.
On the other hand, the two bonds $J_2$ and $J_1$ represent the coupling of adjacent dimers. Both contribute to the hopping of the triplons which is given in leading order by $2J_2-J_1$ [@uhrig96b] while the interaction of adjacent triplons is proportional to $2J_2+J_1$. With information only on the dispersion [@masud06] it is impossible to determine $J_1$ and $J_2$ separately. Hence, the same feature that induces the interesting quasiparticle decay makes it particularly difficult to establish a microscopic model.
The BEC occurring in TlCuCl$_3$ was successfully described by the bond-operator approach [@matsu02; @matsu04]. But this approach to spin-dimer systems is quantitatively reliable only as long as the interdimer couplings $J_\mathrm{inter}$ are significantly smaller than the dimer coupling $J_\mathrm{dimer}$: $|J_\mathrm{inter}| < J_\mathrm{dimer}/2$ [@norma11]. This limit requires $|J_i| < J_3/2$ for $i\in\{1,2,4\}$ for [IPA-CuCl$_3$ ]{}which does not hold [@manak97]. We will see below that $|J_1| \approx J_3$ provides very good fits.
Thus we apply self-similar CUTs (sCUTs) to isolated ladders [@mielk97b; @reisc04; @reisc06; @kehre06], modified to cope with decaying quasiparticles [@fisch10a]. We use an infinitesimal generator which decouples the subspaces with zero or one triplons from the remaining Hilbert space. We can still decouple the 1-triplon subspace from the 2-triplon subspace for the isolated ladder. The proliferating flow equations are truncated if the range of the corresponding process exceeds certain maximum extensions in real space [^3]. Thereby, the ladders are mapped to an effective model $$\label{eq:ham_1d_eff}
H_\text{1D,eff}=\sum_{h,l;\alpha} \omega_0(h)
t_{\alpha,h,l}^{\dag}t_{\alpha,h,l}^{\phantom{\dag}}$$ in terms of triplon creation $t_{\alpha,h,l}^{\dag}$ and annihilation operators $t_{\alpha,h,l}^{\phantom{\dag}}$ in momentum space, where $h$ is the wave vector component along the ladders, $l$ the one perpendicular to them, and $\alpha\in\left\{x,y,z\right\}$ the spin polarization. These operators are the Fourier transforms of the bond operators [@chubu89a; @sachd90] defined on the dimers in Fig. \[fig:ipa\].
The dispersion $\omega_0(h)$ depends only on $h$ because the CUT is applied to the isolated ladders which still have to be coupled. This coupling is achieved in leading order following the approach in Refs. [@uhrig04a; @uhrig05a]. The spin component $S^{\alpha}_{i,r,s}$ is taken as observable and transformed into the new basis by the CUT. Then it reads $S_{\text{eff},i,r,s}^{\alpha} := U^{\dag} S_{i,r,s}^{\alpha} U$ $$S_{\text{eff},i,r,s}^{\alpha}
= \sum_{\delta} a_{i,\delta} ( t_{\alpha,r+\delta,s}^{\dag} +
t_{\alpha,r+\delta,s}^{\phantom{\dag}} ) + \ldots \ ,$$ where the dots stand for normal-ordered higher terms in the real space triplon operators $t_{\alpha,r,s}^{\dag}$ ($t_{\alpha,r,s}$). Knowing $S_{\text{eff},i,r,s}^{\alpha}$ allows us in a second step to write down the effective interladder coupling $H_\text{int,eff}$ in real space $$\begin{aligned}
\nonumber
H_{\text{int,eff}} &=& J_4\sum_{r,s;\alpha}\sum_{\delta,\delta'}
a_{1,\delta}a_{2,\delta'} [ t_{\alpha,r,s}^{\dag}
(t_{\alpha,r+1+\left(\delta'-\delta\right),s+1}^{\dag}
\\
&& + t_{\alpha,r+1+\left(\delta'-\delta\right),s+1}^{\phantom{dag}})
+ \text{H.c.}].
\label{eq:ham_inter_eff}\end{aligned}$$ This neglects trilinear and higher contributions. The Fourier transform of $H_\text{int,eff}$ leads to $H_\text{eff}=H_\text{1D,eff}+H_\text{int,eff}$ amenable to a Bogoliubov diagonalization yielding
$$\begin{aligned}
\label{eq:hamilton-eff}
H_\text{eff} =& \sum_{h,l;\alpha} \omega(h,l)
b_{\alpha,h,l}^{\dag}b_{\alpha,h,l}^{\phantom{\dag}}
\\
\label{eq:dispersion}
\omega(h,l) =& \sqrt{\omega_0^2(h) + 4\omega_0(h) \lambda(h,l)}
\\
\notag
\lambda(h,l) =& - J_4
\sum_{\delta,\delta'}a_{1,\delta}a_{1,\delta'}\cos\left(2\pi\left[ h
\left(\delta + \delta' - 1 \right) - l \right]\right)
\end{aligned}$$
with bosonic operators $b_{\alpha,h,l}^{\dag}$ ($b_{\alpha,h,l}$). In the Bogoliubov diagonalization the hardcore property of the bosons is neglected. However this does not concern the large intraladder couplings, but only the small interladder couplings so that the approach is still very accurate [@exius10b]. The dispersion $\omega(h,l)$ makes a direct comparison with INS results possible.
$\ J_3$ \[meV\] $\ x=J_2/J_3\ $ $\ y=J_1/J_3\ $ $\ z=J_4/J_3$
------------------ ----------------- ----------------- ----------------
3.743 0.133 -2.0 -0.076
3.288 0.268 -1.4 -0.088
3.158 0.317 -1.2 -0.092
3.038 0.369 -1.0 -0.096
2.929 0.424 -0.8 -0.100
2.830 0.480 -0.6 -0.103
: \[tab\] Parameters for IPA-CuCl$_3$ compatible with INS [@masud06]
To determine the microscopic parameters we fix the value $y=J_1/J_3$ and fit $x=J_2/J_3$, $z=J_4/J_3$, and the energy scale $J_3$ to reproduce the experimental result (Eq. (2) in Ref. [@masud06]) $$\begin{aligned}
\nonumber
\omega(h,l)^2&=&a^2\cos^2(\pi h) +
[\Delta^2+4b^2\sin^2(\pi l)]\sin^2(\pi h)
\\
&&+ c^2\sin^2(2\pi h)\end{aligned}$$ with $a=4.08(9)$meV, $\Delta=1.17(1)$meV, $b=0.67(1)$meV and $c=2.15(9)$meV. Thus, we obtain the triples $(x, y, z)$ in Tab. \[tab\]. They all essentially imply the same dispersion, see Fig. \[fig:disp\]. Hence, on the basis of the the INS data, one cannot decide which of the triples applies to IPA-CuCl$_3$.
The quasiparticle decay occurs where the dispersion enters the 2-triplon continuum. It does not prevent to use the CUT for the isolated ladder since the realistic parameters turn out to be such that the triplons do not decay *without* the interladder coupling. A quantitative description of the decay is subject of ongoing research.
![\[fig:disp\] (Color online) Circles are INS data [@masud06]. (a) Dispersions $\omega(h,0)$ for $xyz$ triples in Tab. \[tab\]. The quasiparticle decay occurs where the dispersion enters the 2-triplon continuum. (b) Dispersion $\omega(0.5,l)$; all triples lead to coinciding curves.](fig2a "fig:"){width="0.90\columnwidth"}\
![\[fig:disp\] (Color online) Circles are INS data [@masud06]. (a) Dispersions $\omega(h,0)$ for $xyz$ triples in Tab. \[tab\]. The quasiparticle decay occurs where the dispersion enters the 2-triplon continuum. (b) Dispersion $\omega(0.5,l)$; all triples lead to coinciding curves.](fig2b "fig:"){width="0.90\columnwidth"}\
![\[fig:sus\] (Color online) Upper panel: Deviations of the experimental magnetic susceptibilities [@manak97] $\chi_\mathrm{B}$ and $\chi_\mathrm{C}$ in B and C direction relative to $\chi_\mathrm{A}$ for $g_\mathrm{A}=2.08$, $g_\mathrm{B}=2.06$, and $g_\mathrm{C}=2.25$, indicating anisotropies. Lower panel: Comparison of $\chi_\mathrm{A}(T)$ for various values $g_{\text{A}}$ with theoretical results obtained by Dlog-Padé approximated high temperature series expansions for the $xyz$ triples from Tab. \[tab\].](fig3a "fig:"){width="0.90\columnwidth"} ![\[fig:sus\] (Color online) Upper panel: Deviations of the experimental magnetic susceptibilities [@manak97] $\chi_\mathrm{B}$ and $\chi_\mathrm{C}$ in B and C direction relative to $\chi_\mathrm{A}$ for $g_\mathrm{A}=2.08$, $g_\mathrm{B}=2.06$, and $g_\mathrm{C}=2.25$, indicating anisotropies. Lower panel: Comparison of $\chi_\mathrm{A}(T)$ for various values $g_{\text{A}}$ with theoretical results obtained by Dlog-Padé approximated high temperature series expansions for the $xyz$ triples from Tab. \[tab\].](fig3b "fig:"){width="0.90\columnwidth"}
In complement to the INS we use the temperature dependence of the magnetic susceptibility $\chi(T)$ [@manak97]. Starting from the spin isotropic Hamiltonian the susceptibilities in different spatial direction have to be the same up to scaling proportional to the squares of the Landé $g$-factors. This means that $\chi_\mathrm{A} : \chi_\mathrm{B} : \chi_\mathrm{C}$ equals $g^2_\mathrm{A} : g^2_\mathrm{B} : g^2_\mathrm{C}$ where A, B, C indicate the directions normal to the corresponding surfaces of the crystal [@manak97]. Fig. \[fig:sus\]a displays that the three susceptibilities can be scaled to coincide for $g_\mathrm{A}=2.08$, $g_\mathrm{B}=2.06$, and $g_\mathrm{C}=2.25$ within about 3%. This choice of $g$-factors fulfills the experimental constraints [@manak97; @manak00b] $g_\mathrm{A}, g_\mathrm{B} \in [2.06,2.11]$ and $g_\mathrm{C}=2.25 -2.26$ best. We conclude that an spin isotropic Hamiltonian such as provides a very good description, although anisotropies, e.g., Dzyaloshinskii-Moriya terms, can be present with a relative size of a few percent. This agrees with findings from electron paramagnetic resonance [@manak00b].
Theoretically, we use the high temperature series expansion for the isolated asymmetric ladder [@buhle01a] providing series in $\beta=1/T$ up to order $\beta^{n+1}$ with $n=10$ denoted by $\chi_\text{1D}$. The 2D series $\chi_\text{2D}$ obeys the relation $\chi_\text{2D}^{-1}=\chi_\text{1D}^{-1}+J_4$ in interladder mean-field approximation, i.e., in leading order in $J_4$. We use standard Dlog-Padé approximation [@domb89] to deduce the full $\chi(T)$ from $\chi_\text{2D}$ and from the asymptotic behavior $\chi_\text{2D}(\beta) \propto \beta^0 \exp\left(-\Delta \beta \right)$ for $1/\beta \ll \Delta$. The result[^4] is plotted in Fig. \[fig:sus\] and compared to $\chi_\mathrm{m}$ measured in \[emu/g\] and converted according to $\chi(T) = {m_{\text{mol}} k_{\text{B}}}{\left(g \mu_{\text{B}}\right)^{-2}
N_{\text{A}}^{-1}} \chi_{\text{m}}(T)$. Here $m_{\text{mol}}$ is the molar mass of IPA-CuCl$_3$, $k_{\text{B}}$ the Boltzmann constant, $\mu_{\text{B}}$ the Bohr magneton and $N_{\text{A}}$ the Avogadro constant.
Fig. \[fig:sus\]b illustrates that theory and experiment agree indeed best for $g_\mathrm{A}=2.08$ and the triple of $y=-0.8$. As an asset, we stress that even without the value of $g_\mathrm{A}$, the *position* and the *shape* of the maximum of $\chi(T)$ fits best for the triple of $y=-0.8$ and one can deduce *deduce* that the $g_\mathrm{A}$-factor is around $2.08$. As a caveat, we stress the very weak dependence of $\chi(T)$ on $y$ in a triple tuned to the INS data. By assuming $g_{\text{A}}=2.08\pm 0.01$ we estimate the error of our analysis to be $x=0.42 \pm 0.06$, $y=-0.8 \pm 0.2$ and $z=-0.100 \pm 0.004$ implying $J_1= -2.3 \pm 0.6 \text{ meV}$, $J_2= 1.2 \pm 0.2\text{ meV}$, $J_3= 2.9 \pm 0.1 \text{ meV}$ and $J_4= -0.292 \pm 0.001\text{ meV}$. These values establish the microscopic model for IPA-CuCl$_3$. We highlight that the ferromagnetic coupling $J_1$ does not dominate over the antiferromagnetic coupling $J_3$ because $|y|\lessapprox 1$, in contrast to the previous purely 1D analysis [@manak97].
The derived microscopic model successfully passes three checks: The BEC is well-described, the upper critical field $H_{c2}$ agrees to experiment and the temperature dependence of the spin gap matches recent data.
First,we follow Refs. [@somme01; @matsu02; @matsu04] to describe the BEC and perform the local transformation
\[eq:magnetization\_trafo\] $$\begin{aligned}
{\left| \tilde{s}_{\mathbf{r}} \right>}&= u {\left| s_{\mathbf{r}} \right>} + v {\mathrm{e}}^{{\mathrm{i}}\mathbf{Q}_0 \mathbf{r}} \left(f {\left| {t}_{+,\mathbf{r}} \right>}+g
{\left| {t}_{-,\mathbf{r}} \right>}\right) \\
{\left| \tilde{t}_{+,\mathbf{r}} \right>}&= u \left(f {\left| {t}_{+,\mathbf{r}} \right>}+g
{\left| {t}_{-,\mathbf{r}} \right>}\right) - v {\mathrm{e}}^{{\mathrm{i}}\mathbf{Q}_0 \mathbf{r}} {\left| s_{\mathbf{r}} \right>} \\
{\left| \tilde{t}_{0,\mathbf{r}} \right>}&= {\left| {t}_{0,\mathbf{r}} \right>} \\
{\left| \tilde{t}_{-,\mathbf{r}} \right>}&= f {\left| {t}_{-,\mathbf{r}} \right>}-g
{\left| {t}_{+,\mathbf{r}} \right>}
\end{aligned}$$
in real space with $u=\cos(\theta)$, $v=\sin(\theta)$, $f=\cos(\varphi)$ and $g=\sin(\varphi)$, the position $\mathbf{r}=(r,s)$ and the wave vector $\mathbf{Q}_0=(\pi,0)$ of the minimum of the dispersion. The triplon states ${\left| {t}_{m} \right>}$ with $m\in\left\{-,0,+\right\}$ are given by ${\left| {t}_{-} \right>}=1/\sqrt{2}\left({\left| {t}_{x} \right>}-{\mathrm{i}}{\left| {t}_{y} \right>}\right)$, ${\left| {t}_{0} \right>}={\left| {t}_{z} \right>}$ and ${\left| {t}_{+} \right>}=1/\sqrt{2}\left({\left| {t}_{x} \right>}+{\mathrm{i}}{\left| {t}_{y} \right>}\right)$. The tensor product of all singlet states ${\left| s_{\mathbf{r}} \right>}$ is the vacuum ${\left| 0 \right>}$, so that the hardcore triplon creation operator with $m\in\{-,0,+\}$ is defined by $\tilde t_{m,\mathbf{r}}^\dag {\left| 0 \right>} :=
{\left| \tilde{t}_{m,\mathbf{r}} \right>}$ and the annihilation by $\tilde t_{m,\mathbf{r}} {\left| \tilde{t}_{m,\mathbf{r}} \right>} :={\left| 0 \right>}$ and so on. In this basis the magnetic field is described by the operator $-h(t_+^\dag t_+^{\phantom{\dag}} - t_-^\dag t_-^{\phantom{\dag}} )$. The two independent variables $\theta$ and $\varphi$ are varied to minimize the classical ground state energy. This choice also ensures that (i) all linear terms in the triplon operators vanish and (ii) a massless Goldstone mode appears as it has to be.
Previous work [@somme01; @matsu02; @matsu04] applied the transformation to the original spin model. This is not possible for [IPA-CuCl$_3$ ]{}because the dimers are too strongly coupled. Hence the CUT is mandatory and we apply the real space transformation to $H_\mathrm{1D,eff}+H_\mathrm{inter,eff}$ from Eqs. (\[eq:ham\_1d\_eff\],\[eq:ham\_inter\_eff\]) keeping the bilinear terms. Fourier transformation and Bogoliubov diagonalization finally provides the lowest lying modes. Their resulting gap energies are displayed in Fig. \[fig:gap\_H\]. *No* parameters are adjusted.
![\[fig:gap\_H\] (Color online) Gaps in IPA-CuCl$_3$ vs. the reduced magnetic field $gH/2$. Solid lines show theoretical results, see main text; symbols mark experimental data from Refs. [@zhelu07] (setup I & IV) and [@garle07] (setup II & III).](fig4){width="0.90\columnwidth"}
Second, the upper critical field $H_{c2}$ can be determined exactly for the spin model to be $H_{c2}=(2J_2+J_3)/(g\mu_{\text{B}}) \approx 2/g \cdot 45.8 \text{
T}$. After the transformation is applied to the dispersion obtained from CUT we obtain $H_{c2} \approx 2/g \cdot 45.1 \text{ T}$. The very good agreement of these two values strongly supports the approximations made. Additionally, the theoretical values also match the experimental result [@manak08] $H_{c2}=(43.9\pm 0.1) \text{
T} (2/g)$ within 4%. In view of the neglect of anisotropies and magnetoelastic effects, cf. Ref. [@johan05], this nice agreement lends independent support to the advocated microscopic model.
![\[fig:gap\_T\] (Color online) Spin gap in IPA-CuCl$_3$ vs. temperature $T$. Lines show theoretical results from CUTs and mean-field (solid) and from the nonlinear $\sigma$ model (dashed), respectively. Inset: Temperature dependence of the condensate fraction $s^2$.](fig5){width="0.90\columnwidth"}
Third, the temperature dependence of the gap $\Delta(T)$ supports that the low-lying excitations are hardcore triplons. We apply the mean-field approach in Refs. [@sachd90; @troye94; @ruegg95; @exius10a; @norma11] to $H_\mathrm{1D,eff}+H_\mathrm{inter,eff}$ from Eqs. (\[eq:ham\_1d\_eff\],\[eq:ham\_inter\_eff\]). In each nonlocal term ($t^\dag_{m,\mathbf{r}}t^{\phantom{\dag}}_{m,\mathbf{r}'}$ or $t^\dag_{m,\mathbf{r}}t^{{\dag}}_{-{m},\mathbf{r}'}$ or $t^{\phantom{\dag}}_{m,\mathbf{r}}t^{\phantom{\dag}}_{-{m},\mathbf{r}'}$ with $\mathbf{r}
\neq \mathbf{r}'$) all creation operators $t^\dag_{m,\mathbf{r}}$ are multiplied by the singlet annihilation $s_{\mathbf{r}}$ and the annihilation operators $t_{m,\mathbf{r}}$ by the singlet creation $s^\dag_{\mathbf{r}}$. Local terms remain unchanged because they do not change the local singlet number. Finally all singlet operators are replaced by the condensate value $s(T) = \left\langle
s^\dag \right\rangle = \left\langle s\right\rangle$ with $s\in \left[0,1\right]$. In a nutshell, a factor $s^2$ appears in front of each nonlocal term.
This implies a dependence of the dispersion on $s$ and hence on temperature [@troye94; @exius10a; @norma11], denoted by $\omega_{s(T)}(h,l)$. The self-consistent solution is found from the hardcore condition $1=\langle s^\dag_{\mathbf{r}} s_{\mathbf{r}}
+\sum_m t^\dag_{m,\mathbf{r}}t^{\phantom{\dag}}_{m,\mathbf{r}}\rangle$ leading to $s^2(T) = 1 - 3z/(1+3z)$ with $$z = \int_{-1/2}^{1/2} \operatorname{d\!}h
\int_{-1/2}^{1/2} \operatorname{d\!}l\ {\mathrm{e}}^{-\beta\omega_{s(T)}(h,l)} .$$ Figure \[fig:gap\_T\] compares the result (solid line) of this simple approximation to INS data [@zhelu08; @nafra11]. Up to $15$ K the experimental data is matched perfectly. We attribute the discrepancy at higher temperatures to the insufficient treatment of the hardcore constraint by the above approach (for $15$ K the condensate fraction $s^2$ is only $0.77$). Note that we only apply the mean-field theory to the dispersion obtained from CUT, not to the original spin model as done previously [@ruegg95; @norma11] because [IPA-CuCl$_3$ ]{}is not far enough in the dimer limit.
For comparison, we also include $\Delta(T)$ as derived from the nonlinear $\sigma$ model on 1-loop level [@senec93] in Fig. \[fig:gap\_T\] (dashed line). It is obtained from $$\label{eq:nlsm}
C=\int_{-1/2}^{1/2} \operatorname{d\!}h \int_{-1/2}^{1/2} \operatorname{d\!}l\
\frac{\coth\left(\beta\omega(h,l,T)/2 \right)}{\omega(h,l,T)}$$ with $\omega(h,l,T):=\sqrt{\omega^2(h,l) + \Delta^2(T) - \Delta^2(0)}$; the constant $C$ is determined for $T=0$. Interestingly, this approach describes the experimental data less accurately if the experimental dispersion at $T=0$ is used for $\omega(h,l)$, cf. Ref. [@nafra11]. We presume that the hardcore constraint is not accounted for sufficiently well by Eq. .
In summary, we showed that the available experimental evidence for [IPA-CuCl$_3$ ]{}is consistent with a quantitative model of weakly coupled asymmetric $S=1/2$ spin ladders with hardcore triplons as excitations. Such systems are of great current interest because they allow for the study of Bose-Einstein condensation of triplons and of the massless excitations above this condensate [@giama99; @garle07]. Additionally, they represent gapped quantum liquids known to display considerable quasiparticle decay [@kolez06; @zhito06a; @bibik07; @fisch10a].
Our high-precision analyses of inelastic neutron scattering data and of the temperature dependence of the magnetic susceptibility is based on advances in continuous unitary transformations [@fisch10a] and high temperature series expansions. The established quantitative model paves the way for further quantitative studies, both experimental and theoretical, of the decay of massive quasiparticles and of the condensation of hardcore bosons.
The latter is illustrated by the excellent agreement of the calculated gap energies as function of magnetic field. Additionally, the description of IPA-CuCl$_3$ by dispersive hardcore triplons is strongly supported by the agreement of the temperature dependence of the spin gap.
By this work, a quantitative model for IPA-CuCl$_3$ is established. Concomitantly, we exemplarily showed how CUT results in one dimension at zero temperature and zero magnetic field can be extended to render a quantitative description in two dimensions at finite temperature and finite magnetic field possible. We expect this approach to continue to be fruitful also for other systems.
We thank T. Lorenz, O.P. Sushkov, H. Manaka and A. Zheludev for insightful discussions and the latter two and B. Náfrádi for providing experimental data. This work was supported by the NRW Forschungsschule “Forschung mit Synchrotronstrahlung in den Nano- und Biowissenschaften” and by the DAAD.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: The truncation scheme used for the Hamiltonian is $(d_2,d_3,\ldots, d_8)=(10,8,8,5,5,3,3)$ and for the observables $(d_1,d_2, \ldots, d_6)=(10,10,8,8,6,6)$, where $d_j$ is the maximum extension for a process with $j$ creation and annihilation operators. Additionally, we keep only terms that create or annihilate at most $N=4$ triplons in the Hamiltonian and $N=3$ triplons in the observables, see also Ref. [@reisc06; @fisch10a].
[^4]: All theory curves rely on the \[7,4\] Dlog-Padé approximant in $u=\beta/(1+\beta)$. Data from other Dlog-Padé approximants, e.g., \[9,2\], agrees within line width except at very low temperatures.
|
---
abstract: '[GW170817/GRB170817A probably marks a double neutron star coalescence. Extended Emission $t_s\simeq (0.67\pm0.03)$s post-merger shows an estimated energy output ${\cal E}\simeq (3.5\pm1)\%M_\odot c^2$ determined by response curves to power-law signal injections, where $c$ is the velocity of light. It provides calorimetric evidence for a rotating black hole of $\sim 3M_\odot$, inheriting the angular momentum $J$ of the merged hyper-massive neutron star in the immediate aftermath of GW170817 following core-collapse about or prior to $t_s$. Core-collapse greatly increases the central energy reservoir to $E_J\lesssim 1M_\odot c^2$, accounting for ${\cal E}$ even at modest efficiencies in radiating gravitational waves through a non-axisymmetric thick torus. The associated multi-messenger output in ultra-relativistic outflows and sub-relativistic mass-ejecta is consistent with observational constraints from the GRB-afterglow emission of GRB170817A and accompanying kilonova.]{}'
author:
- 'Maurice H.P.M. van Putten'
- Massimo Della Valle
- Amir Levinson
title: 'Multi-messenger Extended Emission from the compact remnant in GW170817'
---
Introduction
============
GW170817 [@abb17a; @abb17b] is the first observation of a low-mass compact binary coalescence seen in a long duration ascending gravitational-wave chirp. By the accompanying GRB170817A identified by [*Fermi*]{}-GBM and INTEGRAL [@con17; @sav17; @gol17; @poz18; @kas17a] it represents the merger of a neutron star with another neutron star (NS-NS) or companion hole (NS-BH) with a chirp mass of about one solar mass. Potentially broad implications of the first has received considerable attention for our understanding of the origin of heavy elements [@kas17b; @sma17; @pia17; @dav17] and for entirely novel measurements of the Hubble constant [@gui17; @fre17]. By chirp mass, the nature of GW170817 is inconclusive in the absence of observing final coalescence at high gravitational-wave frequencies [@cou19]. For NS-NS coalescence, numerical simulations [e.g. @bai17] show gravitational radiation to effectively satisfy the canonical model signal of binary coalescence in a run-up to about 1kHz, beyond which the amplitude levels off and ultimately decays as the two stars merge into a single object at a maximal frequency $\sim$3kHz. In contrast, NS-BH mergers include tidal break-up [@lat76]. In a brief epoch of hyper-accretion, the black hole would be near-extremal with a remnant of NS debris to form a torus outside its Inner Most Stable Circular Orbit (ISCO). This process is marked by gravitational radiation switching off early on at a frequency 500-1500Hz [@val00; @fab09; @eti09; @fer10] and possibly quasi-normal mode oscillations at yet higher frequencies [e.g @yan18].
Here, we report on the energy output ${\cal E}$ in gravitational radiation post-merger, that appears as a descending chirp of Extended Emission marking spin-down of a compact remnant to binary coalescence at a Gaussian equivalent level of confidence of 4.2$\sigma$ [@van19]. We give a robust estimate of ${\cal E}$ by response curves determined by signal injection experiments in data of the LIGO detectors at Hanford (H1) and Livingston (L1). ${\cal E}$ introduces a new calorimetric constraint that may break the degeneracy of a NS or BH central engine.
${\cal E}$ reported here points to core-collapse of the merged NS produced in GW170817, inheriting its angular momentum $J$ while greatly increasing the associated spin-energy $E_J$ through collapse to a Kerr BH [@ker63].
After our injection experiments were initiated, we learned of an independent analysis of energy considerations by single-template injections, pointing qualitatively to similar energies without, however, identifying the origin of our Extended Emission [@oli19].
${\cal E}$ from pipeline response curves
========================================
We set out to determine response curves of our search pipeline by signal injections into LIGO data ([@val14]; Figs. 1-2), including whitening, butterfly filtering and image analysis of merged (H1,L1)-spectrograms (Appendix). Whitening is by normalizing the Fourier spectrum over an intermediate bandwidth of 2Hz, bringing about GW170817 more clearly than without whitening [@van19].
-0.35in
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We recall that GW170817 is observed as an [*ascending chirp*]{} signifying the merger of two compact stars with time-of-coalescence $t_c=1842.43$s followed by GRB170817A across a gap of 1.7s. In our injection experiments to LIGO data, we include a model DNS with the same chirp mass ${\cal M}_c=1.188M_\odot$ of GW170817 (Fig. 2) at time-of-coalescence about 1818s, producing two ascending chirps side-by-side (Fig. 3). A DNS is described by binary masses $M_1$, $M_2$, $\mu = M_1M_2/M$, $M=M_1+M_2$, at orbital separation $a$ and orbital frequency $\Omega \simeq c\sqrt{R_g/a^3}$ $(a>>R_g$), where $R_g=GM/c^2$ is the gravitational radius of the system, given the velocity of light $c$ and Newton’s constant $G$. This merger chirp has a quadrupole gravitational-wave frequency $f_{GW} = \pi^{-1}\Omega$, $$\begin{aligned}
f_{GW}(t)= A(t_c -t )^{-\frac{3}{8}}~~(t<t_c),
\label{EQN_fgw1}\end{aligned}$$ $[A]$=s$^{-5/8}$Hz, with strain $h(t) = ({4\mu}/{D})\left(M\Omega \right)^\frac{2}{3}$, $h(t) \simeq 1.7 \times 10^{-22} \left({M}/{3M_\odot}\right) \left({D}/{40\,\mbox{Mpc}}\right)^{-1} \left({f_{GW}}/{250\,\mbox{Hz}} \right)^\frac{2}{3}$ and $L_{GW} = ({32}/{5})\left({\cal M}_c\Omega\right)^{{10}/{3}}L_0$, where $L_0=c^5/G\simeq 200,000 M_\odot c^2$s$^{-1}$ [e.g. @fer10]. For GW170817, $A\simeq 138\,$s$^{-5/8}$Hz. Up to 260Hz in both H1 and L1, $L_{GW}$ reaches $1.35\times 10^{50}\mbox{erg\,s}^{-1} \simeq 7.5\times 10^{-5}M_\odot c^2\,\mbox{s}^{-1}$, i.e., $4\times10^{-10}L_0$. While small compared to $10^{-5}L_0$ of GW150914 at similar frequency, GW170817 produced the largest strain observed by its proximity of $D\simeq 40$Mpc. It emitted $E_0=0.43\%M_\odot c^2$ over 200-300Hz with $h=1.4-1.8\times 10^{-22}$ over $\Delta t \simeq 0.25\,$s across $74\,\mbox{km} < r < 97 \,\mbox{km}$ assuming $M_1=M_2$.
A merged (H1,L1)-spectrogram shows Extended Emission post-merger below 700Hz in the form of an exponential feature $$\begin{aligned}
f_{GW}(t)=(f_s-f_0)e^{-(t-t_s)/\tau_s}+f_0~~(t>t_s)
\label{EQN_fgw2}\end{aligned}$$ with the observed $\tau_s=3.01\pm0.2\,$s, $t_s=1843.1$s, $f_s=650\,$Hz and $f_0=98$Hz. For illustrative purposes, we note the isotropic equivalent strain $h=L_{GW}^{1/2}/(\Omega D)$ (in geometrical units, $c=G=1$) for the chirp mass of a small quadrupole mass-moment $\zeta=\delta m/M$ gives $h(t) \simeq 2.7 \times 10^{-23} \left({\zeta}/{3\%}\right) \left({D}/{40\,\mbox{Mpc}}\right)^{-1} \left({f_{GW}}/{650\,\mbox{Hz}} \right)^{{2}/{3}}$, $L_{GW} \simeq$ $2\times 10^{52} \left({\zeta}/{3\%}\right)^2 \left({10M}/{r} \right)^5 \mbox{erg~s}^{-1}$ $\simeq 1\% M_\odot c^2/$s$^{-1}$.
We next use phase coherent injections with frequency evolution (\[EQN\_fgw2\]) (Supplementary Data). No change in results below are found upon including phase incoherence by a Poisson distribution of random phase jumps over intermediate time scales, down to the duration $\tau = 0.5$s of our butterfly templates. $\tau=0.5$ appears intrinsic, as the Extended Emission feature tends to fade out as $\tau$ approaches 1s.
0.12in
The total energy output ${\cal E} =\int_0^T L_{GW}dt$ is computed numerically as sums ${\cal E}=E_0 K^{-1}\sum \nu_i^2h_i^2$ (samples at $t_i$, $i=1,2,..,n$) covering a post-merger interval of duration $T$, where $K=\sum \nu_{0,j}^2h_{0,j}^2$ (samples at $t_j$, $j=1,2,\cdots, m$) is a reference sum with energy $E_0$ over a duration $T_0$, where $\nu_i=f_{GW}(t_i)$ denotes gravitational-wave frequency. $E_0$ is conform quadrupole emission in the [same orientation]{} of the progenitor binary by conservation of orbital-to-spin angular momentum in transition to its remnant. Blind to any model in particular, we consider injections with power-law strain $h\propto f^\alpha$ with $T=7$s.
Fig. 3 shows the outcome of a signal-injection alongside GW170817EE after a calibration $C_h=0.7$ for observed-to-true strain due to non-ideal H1 and L1 detector orientations relative to GW170817. Extended to multiple injections, results show there is no interference with the merger signal or with one another.
Fig. 4 shows our estimated response curves $\chi({\cal E})$ for $h\propto f^\alpha$ ($0.1\le\alpha\le 1.0$). By $\hat{\chi}\simeq7.2$ of the Extended Emission to GW170817, we infer $$\begin{aligned}
{\cal E}\simeq (3.5\pm1)\%M_\odot c^2,
\label{EQN_E}\end{aligned}$$ For the descending chirp at hand (\[EQN\_fgw2\]), ${\cal E}$ mostly derives early on at high $f_{GW}$ with $L_{GW}\lesssim 1\%M_\odot c^2$s$^{-1}$ (Fig. 2).
Enhanced $E_J$ in collapse to a black hole
==========================================
${\cal E}$ in (\[EQN\_E\]) is a significant amount of energy, exceeding the merger output observed up to about 300Hz, emitted as a descending chirp over a secular time scale of seconds with $f_{GW}<700\,$Hz far below the characteristic frequency $c/R_S\simeq 30$kHz of the Schwarzschild radius $R_S=2R_g$. Important energies also appear in GRB170817A and mass-ejecta [e.g. @moo18a; @moo18b]. Of these, ${\cal E}$ and $f_{GW}$ will serve as primary observational constraints on the remnant, i.e., $E_J$ of a rapidly spinning merged NS or rotating BH.
0.2in
\[fig:chi\]
While a long-lived NS might be luminous in gravitational radiation through a baryon-loaded magnetosphere (Appendix), its spin-frequency $f_s=(1/2)f_{GW}$ inferred from our Extended Emission is less than one-fifth the break-up spin-frequency of about 2kHz. This modest initial spin limits $E_J$ to below $0.5\%M_\odot c^2$ and probably somewhat less based on more stringent limits [e.g. @hae09; @oli19]. However, $E_J$ greatly increases by core-collapse of the merged NS in the immediate aftermath of GW170817, here at time of core-collapse about or prior to $t_s=0.67(\pm0.03)$s post-merger (Fig. \[fig:chi\]), where the 30ms refers to our time-step in $t_s$. By the [@ker63] metric, $$\begin{aligned}
E_J = 2Mc^2\sin^2\left( \lambda/4\right) \lesssim 1M_\odot c^2 \left({M}/{3M_\odot}\right)\end{aligned}$$ in terms of $a/M=\sin\lambda$, $J=a\sin\lambda$. This potentially enormous energy reservoir amply accounts for ${\cal E}$ even at modest efficiency $\eta$, provided a mechanism is in place to tap and convert $E_J$ into gravitational radiation. Moderate frequencies $f_{GW}<700\,$Hz can be realized in catalytic conversion into quadrupole emission by a non-axisymmetric disk or torus, sufficiently wide or geometrically thick. Exhausting $E_J$, a descending chirp results due to expansion of the ISCO during black hole spin-down.
${\cal E}$ estimate from black hole spin-down
=============================================
To add some concreteness, we estimate $\eta$ in spin-down of an initially rapidly rotating BH, losing $J$ to matter in Alfvén waves through an inner torus magnetosphere [@van99; @van01]. By heating, a non-axisymmetric thick torus is expected to generate frequencies correlated to but below those of a thin torus about the ISCO [@cow02]. In geometrical units, an extended torus produces emission from an orbital radius $r\equiv zR_g$ at twice the local orbital frequency, i.e., $f_{GW}\simeq c\pi^{-1}\sqrt{R_g/r^3}$. Asymptotic scaling relations for large radii (modest $\eta$) [@van03] show $L_{GW} \sim 10^{52} \mbox{erg~s}^{-1}$ for a non-axisymmetric torus with mass ratio $\sigma = M_T/M \simeq 0.1$. Accompanying minor output is in MeV-neutrinos and $E_w\simeq \eta^2E_J$ in magnetic winds [@van02b; @van03] - [*most of $E_J$ is dissipated unseen in the event horizon, increasing area by Bekenstein-Hawking entropy [@van15].*]{} The observed 150Hz$ < f_{GW}<700\,$Hz indicates an effective radius of a quadrupole mass moment initially about three times the ISCO radius (Fig. 5), indicating a relatively thick torus. $f_{GW}$ decreases with $z$ with expansion of the ISCO during black hole spin-down.
By numerical integration of this spin-down process, catalytic conversion of $E_J$ gives (Fig. 5) $$\begin{aligned}
{\cal E} \simeq \left< \eta\right> E_{J} \simeq 3.6-4.3\% M_\odot c^2
\label{EQN_E2}\end{aligned}$$ for canonical values of initial $a/M$, depending somewhat on the start frequency $f_s=600-700$Hz, consistent with (\[EQN\_E\]) inferred from $\chi({\cal E})$.
The model estimate (\[EQN\_E2\]) uses effective values of disk mass $m$ and $K$ throughout. This does not readily predict $h(f_{GW})$ or the observed exponential feature (\[EQN\_fgw2\]), as $m$ and $K$ will be time-dependent and vary with $z$. Use of effective mean values is only for our present focus on total energy output.
While the nature of GW170817 by the chirp up to 300Hz is somewhat inconclusive [@cou19], ${\cal E}$ provides a novel calorimetric constraint on its remnant. ${\cal E}$ in (\[EQN\_E\]) challenges a hyper-massive NS [@oli19] yet is naturally accommodated by (\[EQN\_E2\]) in core-collapse to a Kerr BH.
In converting $E_J$, ${\cal E}$ is accompanied by MeV-neutrinos and magnetic winds [@van03] consistent with evidence for black hole spin-down in normalized light curves of long GRBs [@van12].
Multi-messenger Extended Emission
=================================
Starting with the merged NS from a DNS, a time-of-collapse about or prior to $t_s\simeq 0.67(\pm0.03)$s (Fig. 3) appears consistent - perhaps in mild tension - with the recently estimated time-of-collapse $0.98^{+0.31}_{-0.26}$s based on jet propagation times and mass of blue-ejecta [@gil19].
Sustained by Alfvén waves outwards over an inner torus magnetosphere, a torus developing a dynamo with magnetic field $B=O\left( 10^{16}\right)$G limited by dynamical stability over the lifetime of black hole spin [@van03] gives a characteristic time scale for the lifetime of rapid spin of the BH and hence of the BH-torus system, $$\begin{aligned}
T_s \simeq 1.5\,\mbox{s} \left( \frac{\sigma}{0.1}\right)^{-1} \left( \frac{z}{6}\right)^{4}\left( \frac{ M}{3M_\odot} \right),\end{aligned}$$ consistent with the duration $T_{90}$ (90% of gamma-ray counts over background) of GRB170817A. Over this secular time scale, the BH gently relaxes towards a nearly Schwarzschild BH as the ISCO expands. A relatively baryon-poor environment of the BH is ideally suited for it to also launch an ultra-relativistic baryon-poor jet within a baryon-rich disk or torus wind with [@van03] $$\begin{aligned}
E_j \simeq \frac{E_J}{4z^4}\simeq 5\times 10^{50}\mbox{erg}, E_w \simeq \eta^2 E_J \simeq 4\times 10^{51}\mbox{erg},
\label{EQN_MM}\end{aligned}$$ consistent with $E_j \sim 10^{49-50}$erg and $E_k = (1/2) M_{ej} v^2 \simeq 4.5\times 10^{51}$ in the relativistic ejecta of GRB170817A and $M_{ej}\simeq 5\%M_\odot$ of mass ejecta at mildly relativistic velocities $v\simeq 0.3c$ [@moo18a; @moo18b]. Emission terminates abruptly as the remnant torus collapses onto the black hole when $\Omega_H\simeq \Omega_T$ ($f_{GW} \simeq 10^2\,$Hz).
Conclusions
===========
${\cal E}\simeq (3.5\pm1)\%M_\odot c^2$ in Extended Emission measured by $\hat{\chi}({\cal E})$ through signal injections (Fig. 4) gives a powerful calorimetric constraint on the central engine. This outcome points to a Kerr BH formed in core-collapse of the merged NS in the immediate aftermath of GW170817.
At $f_{GW}<700$Hz, our ${\cal E}$ is consistent with post-merger bounds of LIGO [@abb17c between dashed lines $E_{gw}=0.01-0.1M_\odot c^2$ in Fig. 1] and [@oli19]. With $E_J$ of a Kerr BH, concerns of [@oli19] on detectability of Extended Emission are unfounded. Accurate time-integration of the complex scaling $L_{GW}\propto\left(f_{GW} [h/C_h]\right)^2$ highlights a need for measurement by signal injection, for which a one-frequency estimate of $h_{H1}$ alone [@van19] now appears inadequate.
Core-collapse greatly enhances $E_J$ in $J$ inherited from the merged NS up to about $1M_\odot c^2$ in a $\sim 3M_\odot$ BH. It amply accommodates ${\cal E}$ even at modest efficiencies in conversion to ${\cal E}$ over durations of seconds (Fig. 5). Accompanying minor emissions (\[EQN\_MM\]) in mass ejecta from the torus and ultra-high energy emission from the BH agree quantitatively with observational constraints on the associated kilonova and GRB170817A. GW170817 is too distant, however, to probe any MeV-neutrino emission [@bay12] its MeV-torus [@van03].
Conceivably, EE does not completely exhaust $E_J$, permitting low-luminosity latent emission including minor output in baryon-loaded disk winds and low-luminosity jets. While outside the present scope, such might be an alternative to the same from a long-lived NS remnant needed to account for ATo2017gfo [@ai18; @li18; @yu18; @pir19].
At improved sensitivity, LIGO-Virgo O3 observations may significantly improve on our ability to identify the nature of binary mergers involving a NS - including the tidal break-up in a NS-BH merger - and their remnants that might also be found in core-collapse supernovae and, possibly, accretion induced collapse of white dwarfs.
[**Acknowledgements.**]{} The authors thank the reviewer for a detailed reading and constructive comments. The first author gratefully thanks ACP, Aspen, Co, GWPop 2019 (PHY-1607611), and AEI, Hannover, where our signal injections were initiated in discussions with M. Alessandra Papa and B. Allen. We also thank A., V. Mukhanov and J. Kanner for constructive comments. We acknowledge use of the data set 10.7935/K5B8566F of the LIGO Laboratory and LIGO Scientific Collaboration, funded by the U.S. NSF, and support from NRF Korea (2015R1D1A1A01059793, 2016R1A5A1013277, 2018044640) and MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, MEXT Grant-in-Aid for Scientific Research on Innovative Areas 24103005, JSPS Core-to-Core Program, Advanced Research Networks, and ICRR.\
\
[**Supporting Data:**]{}\
\
[**WInjection.m**]{}, whitening and signal injection (Fig. 2), DOI 10.5281/zenodo.2613112\
[**EEE.m**]{}, estimated energy and efficiency of Extended Emission (Fig. 5), DOI 10.5281/zenodo.2613105\
Our broadband extended gravitational-wave emission (BEGE) pipeline aims for un-modeled ascending and descending chirps with a choice of intermediate time-scale of phase-coherence $0<\tau\lesssim1$s, expected from extreme transient events exhausting $E_J$ of their central engine in seconds:
- Butterfly filtering is matched filtering against a bank of time-symmetric chirp-like templates of intermediate duration $\tau$, densely covering a domain in $(f(t),\left| df(t)/dt\right|\ge\delta$) for some choice of $\delta>0$. Single detector spectrograms are extracted as scatter plots of correlations $\rho(t,f_c)$ between data segments (here, of 32 s duration) and time-symmetric chirp-like templates with central frequency $f_c$.
- To reduce noise in deep searches ($\kappa = 2$), spectrograms are merged by frequency coincidences ($\left| f_{c,H1}-f_{c,H2}\right| < \Delta f$) conform causality: $\Delta f$ is about $\left| df(t)/dt\right| \delta t$, where $\delta t=10$ms is the (maximal) signal propagation time between H1 and L1. We obtain satisfactory results with $\Delta f = 10$Hz (Fig. \[fig:ms\]).
- Candidate features (Fig. \[fig:chi\]) are evaluated by counting ‘hits:’ $\chi$($\rho>\kappa\sigma$) by H1&L1 over strips about a given family of curves - normalized to $\hat{\chi}$. For Extended Emission feature to GW170817, we use (\[EQN\_fgw2\]), giving $\hat{\chi}(t_s,f_s,f_0,\tau_s)$. The strip is of finite width ($\Delta f = 10$Hz, $\Delta t = 0.1$s), discretized with $\Delta t_s=0.030$s and, for background statistics, over $N = 16$ steps in each parameter gathered from 1956s of clean LIGO data in a scan over a total of 256M parameters ($N^3 = 4096$, $\Delta t_s=$30ms, [@van19]).
The merged NS produced by GW170817 may briefly emit GWs through a magnetosphere with field $B$, baryon-loaded with $M_b$ by dynamical mass ejecta and MeV-neutrino winds [e.g. @per14], by a quadrupole moment $\mu$ along its magnetic spin-axis misaligned with $J$ [@kal12], extending out to $l$ of its light cylinder. At Alfvén velocity $c_A = B/\sqrt{ 4\pi \rho}$, $B=B_{16}10^{16}$G with matter density $\rho$, $\mu$ greatly exceeds that of $B$ in vacuum [@hac17]. In geometrical units ($c=G=1$), the polar flux axis radiates like a rod with [@wal84] $L_{GW} = ({32}/{45}) \mu^2 \Omega^6 \simeq ({32}/{45}) (m \Omega)^2$ with $\mu = ml^2$. A star of mass $M$, radius $R$, Newtonian binding energy $W=M^2/(2R)\simeq 0.15$ generally satisfies $M >> W >> E_{J} >> E_{turb} >> E_{B}$ for turbulent motions $E_{turb}$ and $E_B = (1/6)B^2R^3$. Hence, $E_B = \left({E_{rot}}/{W}\right) \left({E_{turb}}/{E_{rot}}\right) \left({E_B}/{E_{turb}} \right) W \simeq 10^{-4} M$ for fiducial ratios of 0.1 for each factor with corresponding $B\simeq 4\times 10^{16}$G. $M_b$ enhances $m\simeq f_BE_B$ by $2\beta_A^{-2}$, $\beta_A=c_A/c$, where $f_B\simeq 0.5$ for a dipole field. Accordingly, $L_{GW} \simeq ({32}/{45}) \left(m\Omega\beta_A^{-2}\right)^2 \simeq 2\times 10^{52} \left( {B_{16}}/({\beta_A/0.1})\right)^4\left( {f_s}/{350\mbox{Hz}}\right)^2 \mbox{erg~s}^{-1}$ at rapid spin when $l$ is a few times $R$. Such burst will be short by canonical bounds on $E_J$ of a NS.
$E_J$ increases dramatically in continuing core-collapse to a BH. A numerical estimate of ${\cal E}$ derives from catalytic conversion of $E_{J} = 2M\sin^2(\lambda/2)\lesssim 0.29M$ at $a/M=\sin\lambda$ (non-extremal) at modest efficiency at orbital angular velocity $\Omega_T = \pi f_{GW}$ relative to $\Omega_H = \tan(\lambda/2)/(2M)$ of the BH. The estimated initial frequency of $\sim744\,$Hz at time-of-coalescence $t_c$ inferred from $t_s=0.67\,$s is below the orbital frequency at which the stars approach the ISCO of the system mass, about 1100Hz at $r\simeq16$ km. At this point, an equal mass DNS has $a/M = 0.72 < 1$ consistent with numerical simulations [e.g. @bai17], allowing collapse to a $\sim3M_\odot$ Kerr BH with $E_J \simeq 24\%M_\odot c^2$. For a torus radius $K$ times the ISCO radius, Fig. 5 shows the result of integration of the equations describing spin-down (Supplementary Data) with a ${\cal E} \simeq 3.6-4.3\%M_\odot$ at aforementioned canonical initial values $a/M$, subject to the observed gravitational-wave frequency $600\,\mbox{Hz} < f_{GW} < 700\,$Hz at $t_s=0.67$s post-merger - consistent with (\[EQN\_E\]).
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abstract: 'This work introduces a method to tune a sequence-based generative model for molecular *de novo* design that through augmented episodic likelihood can learn to generate structures with certain specified desirable properties. We demonstrate how this model can execute a range of tasks such as generating analogues to a query structure and generating compounds predicted to be active against a biological target. As a proof of principle, the model is first trained to generate molecules that do not contain sulphur. As a second example, the model is trained to generate analogues to the drug Celecoxib, a technique that could be used for scaffold hopping or library expansion starting from a single molecule. Finally, when tuning the model towards generating compounds predicted to be active against the dopamine receptor type 2, the model generates structures of which more than 95% are predicted to be active, including experimentally confirmed actives that have not been included in either the generative model nor the activity prediction model.'
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title: 'Molecular De-Novo Design through Deep Reinforcement Learning'
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Introduction
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Drug discovery is often described using the metaphor of finding a needle in a haystack. In this case, the haystack comprises on the order of $10^{60}-10^{100}$ synthetically feasible molecules [@Schneider2005], out of which we need to find a compound which satisfies the plethora of criteria such as bioactivity, drug metabolism and pharmacokinetic (DMPK) profile, synthetic accessibility, etc. The fraction of this space that we can synthesize and test at all - let alone efficiently - is negligible. By using algorithms to virtually design and assess molecules, *de novo* design offers ways to reduce the chemical space into something more manageable for the search of the needle.
Early *de novo* design algorithms [@Schneider2005] used structure based approaches to grow ligands to sterically and electronically fit the binding pocket of the target of interest [@Bohm1992; @Gillet1994]. A limitation of these methods is that the molecules created often possess poor DMPK properties and can be synthetically intractable. In contrast, the ligand based approach is to generate a large virtual library of chemical structures, and search this chemical space using a scoring function that typically takes into account several properties such as DMPK profiles, synthetic accessibility, bioactivity, and query structure similarity [@Reymond2013; @Hartenfeller2012]. One way to create such a virtual library is to use known chemical reactions alongside a set of available chemical building blocks, resulting in a large number of synthetically accessible structures [@Schneider2011]; another possibility is to use transformational rules based on the expertise of medicinal chemists to design analogues to a query structure. For example, Besnard *et al.* [@Besnard2012] applied a transformation rule approach to the design of novel dopamine receptor type 2 (DRD2) receptor active compounds with specific polypharmacological profiles and appropriate DMPK profiles for a central nervous system indication. Although using either transformation or reaction rules can reliably and effectively generate novel structures, they are limited by the inherent rigidness and scope of the predefined rules and reactions.
A third approach, known as inverse Quantitative Structure Activity Relationship (inverse QSAR), tackles the problem from a different angle: rather than first generating a virtual library and then using a QSAR model to score and search this library, inverse QSAR aims to map a favourable region in terms of predicted activity to the corresponding molecular structures [@Miyao2016; @Churchwell2004; @Wong2009]. This is not a trivial problem: first the solutions of molecular descriptors corresponding to the region need to be resolved using the QSAR model, and these then need be mapped back to the corresponding molecular structures. The fact that the molecular descriptors chosen need to be suitable both for building a forward predictive QSAR model as well as for translation back to molecular structure is one of the major obstacles for this type of approach.
The Recurrent Neural Network (RNN) is commonly used as a generative model for data of sequential nature, and have been used successfully for tasks such as natural language processing [@Mikolov2010] and music generation [@Eck2002]. Recently, there has been an increasing interest in using this type of generative model for the *de novo* design of molecules [@Segler2017; @Bombarelli2016; @Yu2016]. By using a data-driven method that attempts to learn the underlying probability distribution over a large set of chemical structures, the search over the chemical space can be reduced to only molecules seen as reasonable, without introducing the rigidity of rule based approaches. Segler *et al.* demonstrated that an RNN trained on the canonicalized SMILES representation of molecules can learn both the syntax of the language as well as the distribution in chemical space [@Segler2017]. They also show how further training of the model using a focused set of actives towards a biological target can produce a fine-tuned model which generates a high fraction of predicted actives.
In two recent studies, reinforcement learning (RL) [@Sutton1998] was used to fine tune pre-trained RNNs. Yu *et al.* [@Yu2016] use an adversarial network to estimate the expected return for state-action pairs sampled from the RNN, and by increasing the likelihood of highly rated pairs improves the generative network for tasks such as poem generation. Jaques *et al.* [@Jaques2016] use Deep Q-learning to improve a pre-trained generative RNN by introducing two ways to score the sequences generated: one is a measure of how well the sequences adhere to music theory, and one is the likelihood of sequences according to the initial pre-trained RNN. Using this concept of prior likelihood they reduce the risk of forgetting what was initially learnt by the RNN, compared to a reward based only on the adherence to music theory. The authors demonstrate significant improvements over both the initial RNN as well as an RL only approach. They later extend this method to several other tasks including the generation of chemical structures, and optimize toward molecular properties such as cLogP [@Leo1971] and QED drug-likeness [@Bickerton2012]. However, they report that the method is dependent on a reward function incorporating handwritten rules to penalize undesirable types of sequences, and even then can lead to exploitation of the reward resulting in unrealistically simple molecules that are more likely to satisfy the optimization requirements than more complex structures [@Jaques2016].
In this study we propose a policy based RL approach to tune RNNs for episodic tasks [@Sutton1998], in this case the task of generating molecules with given desirable properties. Through learning an augmented episodic likelihood which is a composite of prior likelihood [@Jaques2016] and a user defined scoring function, the method aims to fine-tune an RNN pre-trained on the ChEMBL database [@Gaulton2012] towards generating desirable compounds. Compared to maximum likelihood estimation finetuning [@Segler2017], this method can make use of negative as well as continuous scores, and may reduce the risk of catastrophic forgetting [@Goodfellow2013]. The method is applied to several different tasks of molecular *de novo* design, and an investigation was carried out to illustrate how the method affects the behaviour of the generative model on a mechanistic level.
Methods
=======
Recurrent Neural Networks
-------------------------
A recurrent neural network is an architecture of neural networks designed to make use of the symmetry across steps in sequential data while simultaneously at every step keeping track of the most salient information of previously seen steps, which may affect the interpretation of the current one [@Goodfellow2016]. It does so by introducing the concept of a *cell* (Figure \[fig:rnn\]). For any given step $t$, the $cell_{t}$ is a result of the previous $cell_{t-1}$ and the current input $x^{t-1}$. The content of $cell_t$ will determine both the output at the current step as well as influence the next cell state. The cell thus enables the network to have a memory of past events, which can be used when deciding how to interpret new data. These properties make an RNN particularly well suited for problems in the domain of natural language processing. In this setting, a sequence of words can be encoded into one-hot vectors the length of our vocabulary $X$. Two additional tokens, $GO$ and $EOS$, may be added to denote the beginning and end of the sequence respectively.
### Learning the data
Training an RNN for sequence modeling is typically done by maximum likelihood estimation of the next token $x^{t}$ in the target sequence given tokens for the previous steps (Figure \[fig:rnn\]). At every step the model will produce a probability distribution over what the next character is likely to be, and the aim is to maximize the likelihood assigned to the correct token: $$J(\Theta) = -\sum_{t=1}^T {\log P{(x^{t}\mid x^{t-1},...,x^{1})}}$$ The cost function $J(\Theta)$, often applied to a subset of all training examples known as a batch, is minimized with respect to the network parameters $\Theta$. Given a predicted log likelihood $\log
P$ of the target at step $t$, the gradient of the prediction with respect to $\Theta$ is used to make an update of $\Theta$. This method of fitting a neural network is called back-propagation. Due to the architecture of the RNN, changing the network parameters will not only affect the direct output at time $t$, but also affect the flow of information from the previous cell into the current one iteratively. This domino-like effect that the recurrence has on back-propagation gives rise to some particular problems, and back-propagation applied to RNNs is referred to as back-propagation through time (BPTT).
![ Depiction of maximum likelihood training of an RNN. $x^t$ are the target sequence tokens we are trying to learn by maximizing $P(x^t)$ for each step.[]{data-label="fig:rnn"}](ExternalFigures/article-figure0.pdf)
BPTT is dealing with gradients that through the chain-rule contains terms which are multiplied by themselves many times, and this can lead to a phenomenon known as exploding and vanishing gradients. If these terms are not unity, the gradients quickly become either very large or very small. In order to combat this issue, Hochreiter *et al.* introduced the Long-Short-Term Memory cell [@Hochreiter1997], which through a more controlled flow of information can decide what information to keep and what to discard. The Gated Recurrent Unit is a simplified implementation of the Long-Short-Term Memory architecture that achieves much of the same effect at a reduced computational cost [@Chung2014].
### Generating new samples {#subsec:generating samples}
Once an RNN has been trained on target sequences, it can then be used to generate new sequences that follow the conditional probability distributions learned from the training set. The first input - the $GO$ token - is given and at every timestep after we sample an output token $x^t$ from the predicted probability distribution $P(X^t)$ over our vocabulary $X$ and use $x^t$ as our next input. Once the $EOS$ token is sampled, the sequence is considered finished (Figure \[fig:rnn2\]).
![ Sequence generation by a trained RNN. Every timestep $t$ we sample the next token of the sequence $x^{t}$ from the probability distribution given by the RNN, which is then fed in as the next input.[]{data-label="fig:rnn2"}](ExternalFigures/article-figure1.pdf)
### Tokenizing and one-hot encoding SMILES
A SMILES [@SMILES] represents a molecule as a sequence of characters corresponding to atoms as well as special characters denoting opening and closure of rings and branches. The SMILES is, in most cases, tokenized based on a single character, except for atom types which comprise two characters such as “Cl” and “Br” and special environments denoted by square brackets (e.g \[nH\]), where they are considered as one token. This method of tokenization resulted in 86 tokens present in the training data. Figure \[fig:smiles\] exemplifies how a chemical structure is translated to both the SMILES and one-hot encoded representations.
There are many different ways to represent a single molecule using SMILES. Algorithms that always represent a certain molecule with the same SMILES are referred to as canonicalization algorithms [@Weininger1989]. However, different implementations of the algorithms can still produce different SMILES.
![ Depiction of a one-hot representation derived from the SMILES of a molecule. Here a reduced vocabulary is shown, while in practice a much larger vocabulary that covers all tokens present in the training data is used.[]{data-label="fig:smiles"}](ExternalFigures/article-figure2.pdf)
Reinforcement Learning
----------------------
Consider an Agent, that given a certain state $s\in\mathbb{S}$ has to choose which action $a\in\mathbb{A}(s)$ to take, where $\mathbb{S}$ is the set of possible states and $\mathbb{A}(s)$ is the set of possible actions for that state. The policy $\pi(a \mid s)$ of an Agent maps a state to the probability of each action taken therein. Many problems in reinforcement learning are framed as Markov decision processes, which means that the current state contains all information necessary to guide our choice of action, and that nothing more is gained by also knowing the history of past states. For most real problems, this is an approximation rather than a truth; however, we can generalize this concept to that of a partially observable Markov decision process, in which the Agent can interact with an incomplete representation of the environment. Let $r(a \mid s)$ be the reward which acts as a measurement of how good it was to take an action at a certain state, and the long-term return $G(a_t, S_t = \sum_{t}^T{r_t}$ as the cumulative rewards starting from $t$ collected up to time $T$. Since molecular desirability in general is only sensible for a completed SMILES, we will refer only to the return of a complete sequence.
What reinforcement learning concerns, given a set of actions taken from some states and the rewards thus received, is how to improve the Agent policy in such a way as to increase the expected return $\mathbb{E}[G]$. A task which has a clear endpoint at step $T$ is referred to as an episodic task [@Sutton1998], where $T$ corresponds to the length of the episode. Generating a SMILES is an example of an episodic task, which ends once the $EOS$ token is sampled.
The states and actions used to train the agent can be generated both by the agent itself or by some other means. If they are generated by the agent itself the learning is referred to as *on-policy*, and if they are generated by some other means the learning is referred to as *off-policy* [@Sutton1998].
There are two different approaches often used in reinforcement learning to obtain a policy: value based RL, and policy based RL [@Sutton1998]. In value based RL, the goal is to learn a value function that describes the expected return from a given state. Having learnt this function, a policy can be determined in such a way as to maximize the expected value of the state that a certain action will lead to. In policy based RL on the other hand, the goal is to directly learn a policy. For the problem addressed in this study, we believe that policy based methods is the natural choice for three reasons:
- [Policy based methods can learn explicitly an optimal stochastic policy [@Sutton1998], which is our goal.]{}
- [The method used starts with a prior sequence model. The goal is to finetune this model according to some specified scoring function. Since the prior model already constitutes a policy, learning a finetuned policy might require only small changes to the prior model.]{}
- [The episodes in this case are short and fast to sample, reducing the impact of the variance in the estimate of the gradients.]{}
In Section \[subsec:drd2\] the change in policy between the prior and the finetuned model is investigated, providing justification for the second point.
The Prior network
-----------------
Maximum likelihood estimation was employed to train the initial RNN composed of 3 layers with 1024 Gated Recurrent Units (forget bias 5) in each layer. The RNN was trained on the RDKit [@RDKit] canonical SMILES of 1.5 million structures from ChEMBL [@Gaulton2012] where the molecules were restrained to containing between 10 and 50 heavy atoms and elements $\in\{H, B, C, N, O, F, Si, P, S,
Cl, Br, I\}$. The model was trained with stochastic gradient descent for 50 000 steps using a batch size of 128, utilizing the Adam optimizer [@Kingma2014] ($\beta_1 = 0.9$, $\beta_2 = 0.999$, and $\epsilon = 10^{-8}$) with an initial learning rate of 0.001 and a 0.02 learning rate decay every 100 steps. Gradients were clipped to $[-3, 3]$. Tensorflow [@Tensorflow] was used to implement the Prior as well as the RL Agent.
The Agent network {#sec:agent}
-----------------
We now frame the problem of generating a SMILES representation of a molecule with specified desirable properties via an RNN as a partially observable Markov decision process, where the agent must make a decision of what character to choose next given the current cell state. We use the probability distributions learnt by the previously described prior model as our initial prior policy. We will refer to the network using the prior policy simply as the *Prior*, and the network whose policy has since been modified as the *Agent*. The Agent is thus also an RNN with the same architecture as the Prior. The task is episodic, starting with the first step of the RNN and ending when the $EOS$ token is sampled. The sequence of actions $A = {a_1, a_2,...,a_T}$ during this episode represents the SMILES generated and the product of the action probabilities $P(A) = \prod_{t = 1}^T{\pi(a_t \mid s_t)}$ represents the model likelihood of the sequence formed. Let $S(A)\in[-1,
1]$ be a scoring function that rates the desirability of the sequences formed using some arbitrary method. The goal now is to update the agent policy $\pi$ from the prior policy $\pi_{Prior}$ in such a way as to increase the expected score for the generated sequences. However, we would like our new policy to be anchored to the prior policy, which has learnt both the syntax of SMILES and distribution of molecular structure in ChEMBL [@Segler2017]. We therefore denote an augmented likelihood $\log P(A)_\mathbb{U}$ as a prior likelihood modulated by the desirability of a sequence: $$\log P(A)_\mathbb{U} = \log P(A)_{Prior} + \sigma S(A)$$ where $\sigma$ is a scalar coefficient. The return $G(A)$ of a sequence $A$ can in this case be seen as the agreement between the Agent likelihood $\log P(A)_\mathbb{A}$ and the augmented likelihood: $$G(A) = -[\log P(A)_\mathbb{U} - \log P(A)_\mathbb{A}]^2$$ The goal of the Agent is to learn a policy which maximizes the expected return, achieved by minimizing the cost function $L(\Theta) = -G$. The fact that we describe the target policy using the policy of the Prior and the scoring function enables us to formulate this cost function. In the appendix we show how this approach can be described using a REINFORCE [@Williams1992] algorithm with a final step reward of $r(t) = [\log P(A)_\mathbb{U} - \log P(A)_\mathbb{A}]^2 / \log P(A)_\mathbb{A}$. We believe this is a more natural approach to the problem than REINFORCE algorithms directly using rewards of $S(A)$ or $\log P(A)_{Prior} + \sigma S(A)$. In Section \[sec:sulphur\] we compare our approach to these methods. The Agent is trained in an on-policy fashion on batches of 128 generated sequences, making an update to $\pi$ after every batch has been generated and scored. A standard gradient descent optimizer with a learning rate of 0.0005 was used and gradients were clipped to $[-3, 3]$.
Figure \[fig:agent\] shows an illustration of how the Agent, initially identical to the Prior, is trained using reinforcement learning. The training shifts the probability distribution from that of the Prior towards a distribution modulated by the desirability of the structures. This method adopts a similar concept to Jaques *et al.* [@Jaques2016], while using a policy based RL method that introduces a novel cost function with the aim of addressing the need for handwritten rules and the issues of generating structures that are too simple.
In all the tasks investigated below, the scoring function is fixed during the training of the Agent. If instead the scoring function used is defined by a discriminator network whose task is to distinguish sequences generated by the Agent from ‘real’ SMILES (e.g. a set of actives), the method could be described as a type of Generative Adversarial Network [@Goodfellow2014], where the Agent and the discriminator would be jointly trained in a game where they both strive to beat the other. This is the approach taken by Yu *et al.* [@Yu2016] to finetune a pretrained sequence model for poem generation. Guimaraes *et al.* demonstrates how such a method can be combined with a fixed scoring function for molecular *de novo* design [@Guimaraes2017].
The DRD2 activity model {#sec:DRD2}
-----------------------
In one of our studies the objective of the Agent is to generate molecules that are predicted to be active against a biological target. The dopamine type 2 receptor DRD2 was chosen as the target, and corresponding bioactivity data was extracted from ExCAPE-DB [@Sun2017]. In this dataset there are 7218 actives (pIC50 $>$ 5) and 343204 inactives (pIC50 $<$ 5). A subset of 100 000 inactive compounds was randomly selected. In order to decrease the nearest neighbour similarity between the training and testing structures [@Sheridan2013; @Unterthiner2014; @Mayr2016], the actives were grouped in clusters based on their molecular similarity. The Jaccard [@Jaccard1901] index, for binary vectors also known as the Tanimoto similarity, based on the RDKit implementation of binary Extended Connectivity Molecular Fingerprints with a diameter of 6 (ECFP6 [@Rogers2010]) was used as a similarity measure and the actives were clustered using the Butina clustering algorithm [@Butina1999] in RDKit with a clustering cutoff of 0.4. In this algorithm, centroid molecules will be selected, and everything with a similarity higher than 0.4 to these centroids will be assigned to the same cluster. The centroids are chosen such as to maximize the number of molecules that are assigned to any cluster. The clusters were sorted by size and iteratively assigned to the test, validation, and training sets (assigned 4 clusters each iteration) to give a distribution of $\frac{1}{6}$, $\frac{1}{6}$, and $\frac{4}{6}$ of the clusters respectively. The inactive compounds, of which less than 0.5% were found to belong to any of the clusters formed by the actives, were split randomly into the three sets using the same ratios.
A support vector machine (SVM) classifier with a Gaussian kernel was built in Scikit-learn [@scikit-learn] on the training set as a predictive model for DRD2 activity. The optimal C and Gamma values utilized in the final model were obtained from a grid search for the highest ROC-AUC performance on the validation set.
Results and Discussion
======================
Structure generation by the Prior
---------------------------------
After the initial training, 94% of the sequences generated by the Prior as described in Section \[subsec:generating samples\] corresponded to valid molecular structures according to RDKit [@RDKit] parsing, out of which 90% are novel structures outside of the training set. A set of randomly chosen structures generated by the Prior, as well as by Agents trained in the subsequent examples, are shown in the appendix. The process of generating a SMILES by the Prior is illustrated in Figure \[fig:heatmap\]. For every token in the generated SMILES sequence, the conditional probability distribution over the vocabulary at this step according to the Prior is displayed. The sequence of distributions are depicted in Figure \[fig:heatmap\]. For the first step, when no information other than the initial GO token is present, the distribution is an approximation of the distribution of first tokens for the SMILES in the ChEMBL training set. In this case “O” was sampled, but “C”, “N”, and the halogens were all likely as well. Corresponding log likelihoods were -0.3 for “C”, -2.7 for “N”, -1.8 for “O”, and -5.0 for “F” and “Cl”.
A few (unsurprising) observations:
- [Once the aromatic “n” has been sampled, the model has come to expect a ring opening (i.e. a number), since aromatic moieties by definition are cyclic.]{}
- [Once an aromatic ring has been opened, the aromatic atoms “c”, “n”, “o”, and “s” become probable, until 5 or 6 steps later when the model thinks it is time to close the ring.]{}
- [The model has learnt the RDKit canonicalized SMILES format of increasing ring numbers, and expects the first ring to be numbered “1”. Ring numbers can be reused, as in the two first rings in this example. Only once “1” has been sampled does it expect a ring to be numbered “2” and so on.]{}
Learning to avoid sulphur {#sec:sulphur}
-------------------------
Model Prior Agent Action basis REINFORCE REINFORCE + Prior
-------------------------- ----------------- ----------------- ----------------- ----------------- -------------------
Fraction of valid SMILES $0.94 \pm 0.01$ $0.95 \pm 0.01$ $0.95 \pm 0.01$ $0.98 \pm 0.00$ $0.98 \pm 0.00$
Fraction without sulphur $0.66 \pm 0.01$ $0.98 \pm 0.00$ $0.92 \pm 0.02$ $0.98 \pm 0.00$ $0.92 \pm 0.01$
Average molecular weight $371 \pm 1.70$ $367 \pm 3.30$ $372 \pm 0.94$ $585 \pm 27.4$ $232 \pm 5.25$
Average cLogP $3.36 \pm 0.04$ $3.37 \pm 0.09$ $3.39 \pm 0.02$ $11.3 \pm 0.85$ $3.05 \pm 0.02$
Average NumRotBonds $5.39 \pm 0.04$ $5.41 \pm 0.07$ $6.08 \pm 0.04$ $30.0 \pm 2.17$ $2.8 \pm 0.11$
Average NumAromRings $2.26 \pm 0.02$ $2.26 \pm 0.02$ $2.09 \pm 0.02$ $0.57 \pm 0.04$ $2.11 \pm 0.02$
\[table:sulphur\]
Model Sampled SMILES
------------------- ------------------------------------------------------------------ -- -- -- --
CCOC(=O)C1=C(C)OC(N)=C(C\#N)C1c1ccccc1C(F)(F)F
Prior COC(=O)CC(C)=NNc1ccc(N(C)C)cc1\[N+\](=O)\[O-\]
Cc1ccccc1CNS(=O)(=O)c1ccc2c(c1)C(=O)C(=O)N2
CC(C)(C)NC(=O)c1ccc(OCc2ccccc2C(F)(F)F)nc1-c1ccccc1
Agent CC(=O)NCC1OC(=O)N2c3ccc(-c4cccnc4)cc3OCC12
OCCCNCc1cccc(-c2cccc(-c3nc4ccccc4\[nH\]3)c2OCCOc2ncc(Cl)cc2Br)c1
CCN1CC(C)(C)OC(=O)c2cc(-c3ccc(Cl)cc3)ccc21
Action level CCC(CC)C(=O)Nc1ccc2cnn(-c3ccc(C(C)=O)cc3)c2c1
CCCCN1C(=O)c2ccccc2NC1c1ccc(OC)cc1
CC1CCCCC12NC(=O)N(CC(=O)Nc1ccccc1C(=O)O)C2=O
REINFORCE CCCCCCCCCCCCCCCCCCCCCCCCCCCCNC(=O)OCCCCCC
CCCCCCCCCCCCCCCCCCCCCC1CCC(O)C1(CCC)CCCCCCCCCCCCCCC
Nc1ccccc1C(=O)Oc1ccccc1
REINFORCE + Prior O=c1cccccc1Oc1ccccc1
Nc1ccc(-c2ccccc2O)cc1
\[table:sulphursmiles\]
As a proof of principle the Agent was first trained to generate molecules which do not contain sulphur. The method described in Section \[sec:agent\] is compared with three other policy gradient based methods. The first alternative method is the same as the Agent method, with the only difference that the loss is defined on an action basis rather than on an episodic one, resulting in the cost function: $$J(\Theta) = [\sum_{t=0}^T{(\log \pi_{Prior}(a_t, s_t) - \log \pi_{\Theta}(a_t, s_t))} + \sigma S(A)]^2$$ We refer to this method as ‘Action basis’. The second alternative is a REINFORCE algorithm with a reward of $S(A)$ given at the last step. This method is similar to the one used by Silver *et al.* to train the policy network in AlphaGo [@Silver2016], as well as the method used by Yu *et al.* [@Yu2016]. We refer to this method as ‘REINFORCE’. The corresponding cost function can be written as: $$J(\Theta) = S(A)\sum_{t=0}^T \log \pi_{\Theta}(a_t, s_t)$$ A variation of this method that considers prior likelihood is defined by changing the reward from $S(A)$ to $S(A)+
\log P(A)_{Prior}$. This method is referred to as ‘REINFORCE + Prior’, with the cost function: $$J(\Theta) = [\log P(A)_{Prior} + \sigma S(A)]\sum_{t=0}^T \log \pi_{\Theta}(a_t, s_t)$$ Note that the last method by nature strives to generate only the putative sequence with the highest reward. In contrast to the Agent, the optimal policy for this method is not stochastic. This tendency could be restrained by introducing a regularizing policy entropy term. However, it was found that such regularization undermined the models ability to produce valid SMILES. This method is therefor dependent on only training sufficiently long for the model to reach a point where highly scored sequences are generated, without being settled in a local minima. The experiment aims to answer the following questions:
- [Can the models achieve the task of generating valid SMILES corresponding to structures that do not contain sulphur?]{}
- [Will the models exploit the reward function by converging on naïve solutions such as ’C’ if not imposed handwritten rules?]{}
- [Are the distributions of physical chemical properties for the generated structures similar to those of sulphur free structures generated by the Prior?]{}
The task is defined by the following scoring function: $$S(A) = \begin{cases}
\hphantom{-}1 & \text{if valid and no S} \\
\hphantom{-}0 & \text{if not valid} \\
-1 & \text{if contains S}
\end{cases}$$ All the models were trained for 1000 steps starting from the Prior and 12800 SMILES sequences were sampled from all the models as well as the Prior. A learning rate of 0.0005 was used for the Agent and Action basis methods, and 0.0001 for the two REINFORCE methods. The values of $\sigma$ used were 2 for the Agent and ’REINFORCE + Prior’, and 8 for ’Action basis’. To explore what effect the training has on the structures generated, relevant properties for non sulphur containing structures generated by both the Prior and the other models were compared. The molecular weight, cLogP, the number of rotatable bonds, and the number of aromatic rings were all calculated using RDKit. The experiment was repeated three times with different random seeds. The results are shown in Table \[table:sulphur\] and randomly selected SMILES generated by the Prior and the different models can be seen in Table \[table:sulphursmiles\]. For the ’REINFORCE’ method, where the sole aim is to generate valid SMILES that do not contain sulphur, the model quickly learns to exploit the reward funtion by generating sequences containing predominately ‘C‘. This is not surprising, since any sequence consisting only of this token always gets rewarded. For the ’REINFORCE + Prior’ method, the inclusion of the prior likelihood in the reward function means that this is no longer a viable strategy (the sequences would be given a low prior probability). The model instead tries to find the structure with the best combination of score and prior likelihood, but as is evident from the SMILES generated and the statistics shown in Table \[table:sulphur\], this results in small, simplistic structures being generated. Thus, both REINFORCE algorithms managed to achieve high scores according to the scoring function, but poorly represented the Prior. Both the Agent and the ’Action basis’ methods have explicitly specified target policies. For the ’Action basis’ method the policy is specified exactly on a stepwise level, while for the Agent the target policy is only specified to the likelihoods of entire sequences. Although the ’Action basis’ method generates structures that are more similar to the Prior than the two REINFORCE methods, it performed worse than the Agent despite the higher value of $\sigma$ used while also being slower to learn. This may be due to the less restricted target policy of the Agent, which could facilitate optimization. The Agent achieved the same fraction of sulphur free structures as the REINFORCE algorithms, while seemingly doing a much better job of representing the Prior. This is indicated by the similarity of the properties of the generated structures shown in Table \[table:sulphur\] as well as the SMILES themselves shown in Table \[table:sulphursmiles\].
Similarity guided structure generation
--------------------------------------
The second task investigated was that of generating structures similar to a query structure. The Jaccard index [@Jaccard1901] $J_{i, j}$ of the RDKit implementation of FCFP4 [@Rogers2010] fingerprints was used as a similarity measure between molecules $i$ and $j$. Compared to the DRD2 activity model (Section \[sec:DRD2\]), the feature invariant version of the fingerprints and the smaller diameter 4 was used in order to get a more fuzzy similarity measure. The scoring function was defined as: $$S(A) = -1 + 2 \times \frac{\min \{ J_{i, j}, k \}}{k}$$ This definition means that an increase in similarity is only rewarded up to the point of $k\in[0, 1]$, as well as scaling the reward from $-1$ (no overlap in the fingerprints between query and generated structure) to $1$ (at least $k$ degree of overlap). Celecoxib was chosen as our query structure, and we first investigated whether Celecoxib itself could be generated by using the high values of $k=1$ and $\sigma=15$. The Agent was trained for 1000 steps. After a 100 training steps the Agent starts to generate Celecoxib, and after 200 steps it predominately generates this structure (Figure \[fig:tanimoto\]).
![ Difference in learning dynamics for the Agents based on the canonical Prior, and those based on a reduced Prior where everything more similar than $J=0.5$ to Celecoxib have been removed.[]{data-label="fig:tanimoto"}](ExternalFigures/article-figure5.pdf)
Celecoxib itself as well as many other similar structures appear in the ChEMBL training set used to build the Prior. An interesting question is whether the Agent could succeed in generating Celecoxib when these structures are not part of the chemical space covered by the Prior. To investigate this, all structures with a similarity to Celecoxib higher than 0.5 (corresponding to 1804 molecules) were removed from the training set and a new reduced Prior was trained. The prior likelihood of Celecoxib for the canonical and reduced Priors was compared, as well as the ability of the models to generate structures similar to Celecoxib. As expected, the prior probability of Celecoxib decreased when similar compounds were removed from the training set from $\log_e P = -12.7$ to $\log_e P = -19.2$, representing a reduction in likelihood of a factor of 700. An Agent was then trained using the same hyperparameters as before, but on the reduced Prior. After 400 steps, the Agent again managed to find Celecoxib, albeit requiring more time to do so. After 1000 steps, Celecoxib was the most commonly generated structure (about a third of the generated structures), followed by demethylated Celecoxib (also a third) whose SMILES is more likely according to the Prior with $\log_e P = -15.2$ but has a lower similarity ($J = 0.87$), resulting in an augmented likelihood equal to that of Celecoxib.
These experiments demonstrate that the Agent can be optimized using fingerprint based Jaccard similarity as the objective, but making copies of the query structure is hardly useful. A more useful example is that of generating structures that are moderately to the query structure. The Agent was therefore trained for 3000 steps, starting from both the canonical as well as the reduced Prior, using $k = 0.7$ and $\sigma = 12$. The Agents based on the canonical Prior quickly converge to their targets, while the Agents based on the reduced Prior converged more slowly. For the Agent based on the reduced Prior where $k=1$, the fact that Celecoxib and demethylated Celecoxib are given similar augmented likelihoods means that the average similarity converges to around 0.9 rather than 1.0. For the Agent based on the reduced Prior where $k=0.7$, the lower prior likelihood of compounds similar to Celecoxib translates to a lower augmented likelihood, which lowers the average similarity of the structures generated by the Agent.
To explore whether this reduced prior likelihood could be offset with a higher value of $\sigma$, an Agent starting from the reduced Prior was trained using $\sigma=15$. Though taking slightly more time to converge than the Agent based on the canonical Prior, this Agent too could converge to the target similarity. The learning curves for the different model is shown in Figure \[fig:tanimoto\].
An illustration of how the type of structures generated by the Agent evolves during training is shown in Figure \[fig:celecoxib\]. For the Agent based on the reduced Prior with $k=0.7$ and $\sigma=15$, three structures were randomly sampled every 100 training steps from step 0 up to step 400. At first, the structures are not similar to Celecoxib. After 200 steps, some features from Celecoxib have started to emerge, and after 300 steps the model generates mostly close analogues of Celecoxib.
We have investigated how various factors affect the learning behaviour of the Agent. In real drug discovery applications, we might be more interested in finding structures with modest similarity to our query molecules rather than very close analogues. For example, one of the structures sampled after 200 steps shown in Figure \[fig:celecoxib\] displays a type of scaffold hopping where the sulphur functional group on one of the outer aromatic rings has been fused to the central pyrazole. The similarity to Celecoxib of this structure is $0.4$, which may be a more interesting solution for scaffold-hopping purposes. One can choose hyperparameters and similarity criterion tailored to the desired output. Other types of similarity measures such as pharmacophoric fingerprints [@Reutlinger2013], Tversky substructure similarity [@Senger2009], or presence/absence of certain pharmacophores could also be explored.
Target activity guided structure generation {#subsec:drd2}
-------------------------------------------
The third example, perhaps the one most interesting and relevant for drug discovery, is to optimize the Agent towards generating structures with predicted biological activity. This can be seen as a form of inverse QSAR, where the Agent is used to implicitly map high predicted probability of activity to molecular structure. DRD2 was chosen as the biological target. The clustering split of the DRD2 activity dataset as described in Section \[sec:DRD2\] resulted in 1405, 1287, and 4526 actives in the test, validation, and training sets respectively. The average similarity to the nearest neighbour in the training set for the test set compounds was 0.53. For a random split of actives in sets of the same sizes this similarity was 0.69, indicating that the clustering had significantly decreased training-test set similarity which mimics the hit finding practice in drug discovery to identify diverse hits to the training set. Most of the DRD2 actives are also included in the ChEMBL dataset used to train the Prior. To explore the effect of not having the known actives included in the Prior, a reduced Prior was trained on a reduced subset of the ChEMBL training set where all DRD2 actives had been removed.
Set Training Validation Test
----------- ---------- ------------ ------
Accuracy 1.00 0.98 0.98
ROC-AUC 1.00 0.99 1.00
Precision 1.00 0.96 0.97
Recall 1.00 0.73 0.82
: Performance of the DRD2 activity model
\[table:svm\_table\]
Model Prior Agent $\mathrm{Prior}^\dagger$ $\mathrm{Agent}^\dagger$
---------------------------------------------------------------------- ------- ------- -------------------------- --------------------------
Fraction valid SMILES 0.94 0.99 0.94 0.99
Fraction predicted actives 0.03 0.97 0.02 0.96
Fraction similar to train active 0.02 0.79 0.02 0.75
Fraction similar to test active 0.01 0.46 0.01 0.38
Fraction of test actives recovered ($\times 10^{-3}$) 13.5 126 2.85 72.6
Probability of generating a test set active ($\times 10^{-3}$) 0.17 40.2 0.05 15.0
$^{\dagger}$DRD2 actives witheld from the training of the Prior \[table:DRD2\]
The optimal hyperparameters found for the SVM activity model were $C=2^{7}, \gamma=2^{-6}$, resulting in a model whose performance is shown in Table \[table:svm\_table\]. The good performance in general can be explained by the apparent difference between actives and inactive compounds as seen during the clustering, and the better performance on the test set compared to the validation set could be due to slightly higher nearest neighbour similarity to the training actives (0.53 for test actives and 0.48 for validation actives).
The output of the DRD2 model for a given structure is an uncalibrated predicted probability of being active $P_{active}$. This value is used to formulate the following scoring function: $$S(A) = -1 + 2 \times P_{active}$$ The model was trained for 3000 steps using $\sigma = 7$. After training, the fraction of predicted actives according to the DRD2 model increased from 0.02 for structures generated by the reduced Prior to 0.96 for structures generated by the corresponding Agent network (Table \[table:DRD2\]). To see how well the structure-activity-relationship learnt by the activity model is transferred to the type of structures generated by the Agent RNN, the fraction of compounds with an ECFP6 Jaccard similarity greater than 0.4 to any active in the training and test sets was calculated.
In some cases, the model recovered exact matches from the training and test sets (c.f. Segler *et al.* [@Segler2017]). The fraction of recovered test actives recovered by the canonical and reduced Prior were 1.3% and 0.3% respectively. The Agent derived from the canonical Prior managed to recover 13% test actives; the Agent derived from the reduced Prior recovered 7%. For the Agent derived from the reduced Prior, where the DRD2 actives were excluded from the Prior training set, this means that the model has learnt to generate “novel” structures that have been seen neither by the DRD2 activity model nor the Prior, and are experimentally confirmed actives. We can formalize this observation by calculating the probability of a given generated sequence belonging to the set of test actives. For the canonical and reduced Priors, this probability was 0.17$\times 10^{-3}$ and 0.05$\times 10^{-3}$ respectively. Removing the actives from the Prior thus resulted in a threefold reduction in the probability of generating a structure from the set of test actives. For the Agents, the probabilities rose to 15.0$\times 10^{-3}$ and 40.2$\times 10^{-3}$ respectively, corresponding to an enrichment of a factor of 250 over the Prior models. Again the consequence of removing the actives from the Prior was a threefold reduction in the probability of generating a test set active: the difference between the two Priors is directly mirrored by their corresponding Agents. Apart from generating a higher fraction of structures that are predicted to be active, both Agents also generate a significantly higher fraction of valid SMILES (Table \[table:DRD2\]). Sequences that are not valid SMILES receive a score of -1, which means that the scoring function naturally encourages valid SMILES.
A few of the test set actives generated by the Agent based on the reduced Prior along with a few randomly selected generated structures are shown together with their predicted probability of activity in Figure \[fig:DRD2\_gen\]. Encouragingly, the recovered test set actives vary considerably in their structure, which would not have been the case had the Agent converged to generating only a certain type of very similar predicted active compounds.
Removing the known actives from the training set of the Prior resulted in an Agent which shows a decrease in all metrics measuring the overlap between the known actives and the structures generated, compared to the Agent derived from the canonical Prior. Interestingly, the fraction of predicted actives did not change significantly. This indicates that the Agent derived from the reduced Prior has managed to find a similar chemical space to that of the canonical Agent, with structures that are equally likely to be predicted as active, but are less similar to the known actives. Whether or not these compounds are active will be dependent on the accuracy of the target activity model. Ideally, any predictive model to be used in conjunction with the generative model should cover a broad chemical space within its domain of applicability, since it initially has to assess representative structures of the dataset used to build the Prior [@Segler2017].
Figure \[fig:heatmap2\] shows a comparison of the conditional probability distributions for the reduced Prior and its corresponding Agent when a molecule from the set of test actives is generated. It can be seen that the changes are not drastic with most of the trends learnt by the Prior being carried over to the Agent. However, a big change in the probability distribution even only at one step can have a large impact on the likelihood of the sequence and could significantly alter the type of structures generated.
Conclusion
==========
To summarize, we believe that an RNN operating on the SMILES representation of molecules is a promising method for molecular *de novo* design. It is a data-driven generative model that does not rely on pre-defined building blocks and rules, which makes it clearly differentiated from traditional methods. In this study we extend upon previous work [@Segler2017; @Bombarelli2016; @Yu2016; @Jaques2016] by introducing a reinforcement learning method which can be used to tune the RNN to generate structures with certain desirable properties through augmented episodic likelihood.
The model was tested on the task of generating sulphur free molecules as a proof of principle, and the method using augmented episodic likelihood was compared with traditional policy gradient methods. The results indicate that the Agent can find solutions reflecting the underlying probability distribution of the Prior, representing a significant improvement over both traditional REINFORCE [@Williams1992] algorithms as well as previously reported methods [@Jaques2016]. To evaluate if the model could be used to generate analogues to a query structure, the Agent was trained to generate structures similar to the drug Celecoxib. Even when all analogues of Celecoxib were removed from the Prior, the Agent could still locate the intended region of chemical space which was not part of the Prior. Further more, when trained towards generating predicted actives against the dopamine receptor type 2 (DRD2), the Agent generates structures of which more than 95% are predicted to be active, and could recover test set actives even in the case where they were not included in either the activity model nor the Prior. Our results indicate that the method could be a useful tool for drug discovery.
It is clear that the qualities of the Prior are reflected in the corresponding Agents it produces. An exhaustive study which explores how parameters such as training set size, model size, regularization [@Zaremba2014; @Wan2013], and training time would influence the quality and variety of structures generated by the Prior would be interesting. Another interesting avenue for future research might be that of token embeddings [@Bengio2003]. The method was found to be robust with respect to the hyperparameters $\sigma$ and the learning rate.
All of the aforementioned examples used single parameter based scoring functions. In a typical drug discovery project, multiple parameters such as target activity, DMPK profile, synthetic accessibility etc. all need to be taken into account simultaneously. Applying this type of multi-parametric scoring functions to the model is an area requiring further research.
Additional Files {#additional-files .unnumbered}
================
Additional file 1 — Equivalence to REINFORCE {#additional-file-1-equivalence-to-reinforce .unnumbered}
--------------------------------------------
Proof that the method used can be described as a REINFORCE type algorithm.
Additional file 2 — Generated structures {#additional-file-2-generated-structures .unnumbered}
----------------------------------------
Structures generated by the canonical Prior and different Agents.
Availability of data and materials {#availability-of-data-and-materials .unnumbered}
==================================
The source code and data supporting the conclusions of this article is available at https://github.com/MarcusOlivecrona/REINVENT, DOI:10.5281/zenodo.572576.
- [Project name: REINVENT]{}
- [Project home page: https://github.com/MarcusOlivecrona/REINVENT]{}
- [Archived version: http://doi.org/10.5281/zenodo.572576]{}
- [Operating system: Platform independent]{}
- [Programming language: Python]{}
- [Other requirements: Python2.7, Tensorflow, RDKit, Scikit-learn]{}
- [License: MIT]{}
Declarations {#declarations .unnumbered}
============
Ethics approval and consent to participate {#ethics-approval-and-consent-to-participate .unnumbered}
------------------------------------------
Not applicable.
Consent for publication {#consent-for-publication .unnumbered}
-----------------------
Not applicable.
List of abbreviations {#list-of-abbreviations .unnumbered}
---------------------
- [DMPK - Drug metabolism and pharmacokinetics]{}
- [DRD2 - Dopamine receptor D2]{}
- [QSAR - Quantitive structure activity relationship]{}
- [RNN - Recurrent neural network]{}
- [RL - Reinforcement Learning]{}
- [Log - Natural logarithm]{}
- [BPTT - Back-propagation through time]{}
- [$A$ - Sequence of tokens constituting a SMILES]{}
- [Prior - An RNN trained on SMILES from ChEMBL used as a starting point for the Agent]{}
- [Agent - An RNN derived from a Prior, trained using reinforcement learning]{}
- [$J$ - Jaccard index]{}
- [ECFP6 - Extended Molecular Fingerprints with diameter 6]{}
- [SVM - Support Vector Machine]{}
- [FCFP4 - Extended Molecular Fingerprints with diameter 4 and feature invariants]{}
Competing interests {#competing-interests .unnumbered}
-------------------
The authors declare that they have no competing interests.
Funding {#funding .unnumbered}
-------
MO, HC, and OE are employed by AstraZeneca. TB has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No 676434, “Big Data in Chemistry” (“BIGCHEM”, http://bigchem.eu). The article reflects only the authors’ view and neither the European Commission nor the Research Executive Agency (REA) are responsible for any use that may be made of the information it contains.
Author’s contributions {#authors-contributions .unnumbered}
----------------------
MO contributed concept and implementation. All authors co-designed experiments. All authors contributed to the interpretation of results. MO wrote the manuscript. HC, TB, and OE reviewed and edited the manuscript. All authors read and approved the final manuscript.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Thierry Kogej and Christian Tyrchan for general assistance and discussion, and Dominik Peters for his expertise in LaTeX.
|
---
abstract: 'If the threshold for $e^{-}e^{+}$ pair production depends on an angle between photon momenta, and if the $\gamma$-rays are collimated right [*in*]{} gamma-ray burst (GRB) source then another solution of the compactness problem is possible. The list of basic assumptions of the scenario describing the GRB with energy release $< 10^{49}$erg is adduced: the matter is about an alternative to the ultrarelativistic fireball if [*all*]{} long-duration GRBs are physically connected with core-collapse supernovae (SNe). The questions about radiation pressure and how the jet arises on account of even a small radiation field asymmetry in a compact GRB source of size $\lesssim 10^8$cm, and observational consequences of the compact model of GRBs are considered.'
---
[**Gravitational collapse as the source of gamma-ray bursts [^1]** ]{}\
\
Introduction: the root of the problem
=====================================
Gamma-ray bursts (GRBs) are the brief ($\sim$0.01-100s), intense flashes of $\gamma$-rays (mostly sub-MeV) with enormous electromagnetic energy release up to $\sim 10^{51} - 10^{53}$erg. The rapid temporal variability, $\delta T \lesssim 10$msec, observed in GRBs implies [*compact*]{} sources with a size smaller than $c\,\delta T \lesssim 3000$km.
But a problem immediately arises for distant GRB sources (e.g. \[1, 2\]): too large energy ($>10^{51}$ergs) is released in the observed (for the most GRBs) soft $\gamma$-rays ($<511$ keV and up to 1 MeV) in such a small volume for the sources at cosmological distances ($>1$Gpc). For a photon number density $n_{\gamma} \sim (10^{51}$erg$/(m_e c^2))/(c\,\delta T)^3
\sim 10^{57}/(3000\,$km$)^3 \sim 10^{32}$cm$^{-3}$ two $\gamma$-ray photons with a [*sum energy*]{} larger than $2 m_e c^2$ could interact with each other and produce electron positron pairs. The optical depth for pair creation is given approximately by $\tau_{e^+e^-}\sim n_{\gamma} r_{e}^2 (c\,\delta T) \sim
10^{16}$, where $r_e$ is the classical electron radius $e^2/(m_ec^2)$ (the cross-section for pair production is $\sim r_{e}^2$ or $\sim 10^{-25}$cm$^2$ at these semirelativistic energies). It is the essence of a so-called “compactness problem": the optical depth of the relatively low energy photons ($\sim 511$keV) must be so large that these photons could not be observed.
In the popular ultrarelativistic fireball GRB model \[3, 4\] in this definition a role of the high-energy photons is emphasized: the $\gamma$-ray photons with energies much larger than $m_e c^2$ (or $\gg 1$MeV) could interact with lower energy ($< 511$keV) “target” photons and produce $e^{-}e^{+}$ pairs. (e.g. \[4\]). In the ultrarelativistic fireball model it is supposed that the “heavy"/hard (or high-energy) photons [*must be present*]{} in all GRB spectra as high energy tails which contain a significant amount of energy. So, (see e.g. \[5\]) the optical depth [*of the high-energy photons*]{} ($\gg 1$MeV) would be so large that these photons could not be observed. In this theory the size of the region where the GRB prompt emission arises must be $\sim 10^{15} - 10^{17}$cm \[6\], if it is supposed that radiation (with 100MeV and 10GeV photons) is generated by ultrarelativistic jets moving with huge Lorentz factors $\sim $100-1000.
Below we will try to understand the [*observational soft*]{} (in the meaning of photon energies) GRB spectrum in a compact GRB model implies the GRB source with the size of $c\,\delta T \lesssim 10^8$cm, e. d. without involving huge kinematical motions of the radiating plasma, or without so enormous Lorentz factors. It concerns with another attempt of solving the compactness problem, namely, the dependence of the threshold for $e^{-}e^{+}$ pair production on the angle between photon momenta, a photon collimation [*in*]{} the source and the dependence of this collimation on GRB photon energy must be accounted for. Taking into consideration the compactness of the source and the fact that all long-duration GRBs are physically connected with core-collapse supernovae (SNe), it may be supposed that when observing these GRBs we directly observe the gravitational collapse of a massive and compact star core.
Typical GRB spectra and typical photon energies
===============================================
The GRB spectra are described in a review by Fishman and Meegan \[7\], see also the catalogue of the spectra by Preece et al. \[8\]. Typical observational GRB spectra turned out to be very diverse, but yet these are mainly soft (but not hard) $\gamma$-ray quanta. It has been known since the moment of GRBs discovery, when their spectra were presented in energy units: e.g., see a review by Mazets and Golenetsky \[9\], and many authors \[10-14\] point to the same again. Almost all GRBs have been detected in the energy range between 20 keV and 1 MeV. [*A few*]{} $\gamma$-ray quanta have been observed in GRBs above 100 MeV. In a review by Piran \[3\] also has paid attention to a puzzle of the origin of narrow distribution for the typical energy of the observed GRB radiation ($E_p < 511$keV, \[8\]). More, by 2000 it was clear that there were other two GRB classes: X-Ray Flashes (XRF) and X-Ray Rich Gamma Ray Bursts (XRR GRB) \[15, 16\]. These are GRBs either [*without*]{} (XRFs) or almost without (XRR GRB) $\gamma$-ray quanta.
Thus, for usual (mostly sub-MeV) GRBs, still there are too many lower energy $\gamma$-ray photons in a small volume $R^3$ with $R\sim c\,\delta T \lesssim 3000$ km. The observed fluxes give an estimate of a total GRB energy release to be of $\sim 10^{51}$ ergs in the form of just these [*low energy*]{} photons, or this “standard" estimation ($\sim 10^{51}$erg) was obtained from typical observational GRB spectra of just these, most frequently observed low-energy photons with the [*semirelativistic*]{} energies, up to 1 MeV, basically. (It is natural that the photon density was estimated using the simple assumption of spherical symmetry — see below.)
Nevertheless, if these theoretical (rather than observational) statements \[17\] on the possibility that [*as though*]{} (1) all GRB spectra have high energy tails and (2) the observed GRB spectra are non-thermal, are true indeed, the fireball theory \[4, 18\] with huge Lorentz factors is the only possible theoretical alternative for GRBs. It should be admitted though that the standard optically thin synchrotron shock emission model explains everything, except the observational spectra of GRBs themselves \[19\]. But for all that, it was left out of account that these (“target”) photons with $E_p < 511$keV are just the observed typical GRBs. So, it turns out that the main task, according to the standard fireball model, is not the explanation of this observed soft GRB spectrum in terms of photons’ energy (frequency), but the investigation of rare cases of release of hard quanta with energy of more than or $\sim 1$GeV.
As a result, the origin of the [*observed*]{} and substantially soft GRB spectra with a big number of photons $\lesssim 1$MeV remains not properly understood. It is especially incomprehensible against the background of conjurations about the huge gamma factor that is supposed to solve the compactness problem. But the question remains: why are mainly soft GRB spectra observed at ultrarelativistic motions of radiating plasma supposed in the fireball model? And what is more, as was noted above, sometimes the GRB spectra do not contain -ray quanta at all, as, for example, XRFs known already before 2000 \[20\]. Thus, when solving the compactness problem, we somehow imperceptibly incurred another problem of strong contradiction between the ultrarelativistic Lorentz factor $\Gamma \sim$ 100-1000 (in the fireball model with 100MeV and 10GeV photons) and observed soft ($\sim$ or $<1$MeV) -ray (GRB, XRR GRB) and X-ray (XRF) radiation of the most classical GRBs. Moreover, it is also important to point out here that the observed [*black-body*]{} prompt GRB radiation with a temperature $kT\sim 100$keV \[21, 22\] is inconsistent with the Lorentz factor $\approx 10^2 - 10^4$ for the reason that the mean observed temperature can easily exceed $kT$=1MeV in cosmological fireballs \[23\].
The threshold for $e^{-}e^{+}$ pair production
==============================================
The compactness problem was mentioned (before 1992, i.e. before the BATSE/EGRET mission) in connection with the famous burst of 1979 March 5 in the Large Magellanic Cloud. Already then a possibility of a [*photon collimation right in the source*]{} for explanation of observed soft spectra was not excluded \[1\] because the cross-section of electron-positron pair production $\sigma_{e^-e^+}$ (and annihilation also) depends not only on energy, but on the angle between momenta of colliding particles. For the first time in the paper by Aharonian and Ozernoy \[1\], and than later by Carrigan and Katz \[2\], a lot of interesting was said in connection with collimation of $\gamma$-rays leaving GRB source with high photon density in it.
It seems that just the collimation in GRB source solves the notorious compactness problem indeed. The paper by Carrigan and Katz \[2\] tells about modeling the observed GRB spectra allowing for the electron-positron pair production effects. These effects could produce effective collimation of the flux because of kinematics of the two-photon pair production: the opacity ($\tau_{e^-e^+}$) is also a sensitive function of the [*angular*]{} and [*spectral*]{} distribution of the radiation field [*in the source*]{}.
The argument proceeds as follows: [*two*]{} photons with energies $E_1$ and $E_2$, which are above the threshold energy ($E_1+E_2 > 2\cdot E_{th}$, $E_{th}= \sqrt {E_1 E_2}$) for electron-positron pair production $$E_{th}^2 = E_1 \cdot E_2 \geq 2(m_e c^2)^2 / (1-cos\theta_{12})$$ may produce a pair, where $2(m_e c^2)^2 = 2(511\,$keV$)^2$ and $\theta_{12}$ is the angle between the directions of the two $\gamma$-rays. The cross section for pair production reaches the maximum at a finite center-of-momentum photon energy: e.g. $E_1 + E_2 > 2\cdot E_{th} = 2\cdot$511keV for $\theta_{12} = 180\degr$, or $E_1 + E_2 > 2\cdot E_{th}\approx 2\cdot$700keV for $\theta_{12}\approx 90\degr$), or $E_1 + E_2 > 2\cdot E_{th}$ going to infinity ($\gg$1MeV) for $\theta_{12}\approx 0\degr$.
If the source photon spectrum is not sharply peaked, the relatively high-energy photons ($E > E_{th}$) will, therefore, form pairs predominantly with relatively low-energy photons ($E < E_{th}$). It means that the observed (or the emergent) GRB spectra will be soft, since the high-energy photons will be held by the threshold of pair production. Thus, because any [*reasonable*]{} source spectrum will contain much more low- or moderate-energy photons ($\lesssim 511$keV) than high-energy photons, the emergent spectrum will differ most markedly from the source spectrum at high photon energies ($E\gtrsim 1$MeV) at which it (the emergent spectrum) will be heavily depleted. In other words, the observed (emergent) spectrum becomes softer. Then, the $e^{-}e^{+}$ pairs eventually annihilate to produce two (infrequently 3) photons, but usually not one high- and one low-energy photon.
The result is that high-energy photons are preferentially removed from the observed spectrum. The observation of a measurable amount of quanta with $E > E_{th}= \sqrt {E_1 E_2}$ is not expected unless the optical depth $\tau_{e^-e^+}$ to pair production is equal to 1 or less, because the threshold for electron-positron pair production (1) is also a sensitive function of the angular distribution of the radiation field in the very source (see below). Thus, the observation of a considerable number of quanta with $E > 1$MeV due to the filter effect (1) is not expected, if only the optical depth for the $e^{-}e^{+}$ pair production is not proved $\lesssim 1$ indeed for various reasons, for example, because of anisotropy of the radiation field in the GRB source itself.
As is seen from the paper by Carrigan and Katz \[2\], in 1992 it was generally accepted that typical energies of most photons in observed GRB spectra are still rather small. Further in the peper, Carrigan and Katz adduce the estimates of distances to burst sources of such photons with the [*semirelativistic*]{} energies. The matter is that the problem of a compact source (in relation to the 1979 March 5 event in LMC) and a surprisingly big distance arises indeed, but not because of a problem with the release of “heavy" (100MeV, 1GeV, or more) ultrarelativistic photons which interfere with “light" ($\lesssim 1$MeV) target photons. The powerful 1979 March 5 event in LMC was observed without any super heavy photons in its spectrum. To make sure of it one should just look at the spectra of this burst published by Mazets and Golenetskii in their review \[9\].
To explain why the effect of the photon “$e^{-}e^{+}$ confinement" does not function in this GRB source (1979 March 5 event in LMC) different possibilities were discussed \[1, 2\]. In particular, the authors immediately point out to the angle dependence (1) of the threshold of the $e^{-}e^{+}$ production. A possible “loophole" exists if the source produces a [*strongly collimated*]{} beam of photons. Thus, [**the question is about an asymmetry of the radiation field in the source.**]{} In this case, even high-energy photons are below the threshold for the pair production if $\theta_{12}$ is small enough. The presence of such a “window" in the opacity for [*collimated*]{} photons suggests that in a region opaque to pair production much of the radiation may emerge through this window, in analogy to the great contribution of windows in the material opacity to radiation flow in the usual (Rosseland mean) approximation.
The use of the words “strongly collimated" in the (“old") paper \[2\] could be somewhat confusing. What means [*strongly*]{} indeed? At that time there were no observations of GRB spectra in the region of high energy $E$. Heavier photons with $E \sim 10$MeV (beyond the peak of $\sim 1$MeV) have been reliably observed only with EGRET/BATSE. In particular, from formula (1) for such photons an estimation of the collimation angle can be obtained (without any “target-photons"): $1 - cos\theta_{12} = 0.522245\,$MeV$^2/(10$MeV$\cdot 10$MeV$)\approx 0.005.$ It corresponds to $\theta_{12}$ less than $6^o$ only. It means that the quanta with energy $\sim 10$MeV leaving the source within a cone of $\sim 6^o$ opening angle do not give rise to pairs, and all [*softer*]{} radiation can be uncollimated at all. So the collision of 10MeV quanta with quanta of lower energy occurs at angles greater than $60^o$ ($0.522245\,$MeV$^2/(10\,$MeV$ \cdot 100\,$KeV$) \approx 0.5$), and softer quanta leaving the source within the cone of such opening angle do not prevent neither heavy nor (especially) light quanta to go freely to infinity.
Thus, formula (1) demands more or less strong collimation only for [*a small part*]{} of the heaviest quanta radiated by the source. If one looks at energetic spectra of typical GRBs (the same reference to Mazets and Golenetskii \[9\]) presented in $F($cm$^{-2}$s$^{-1}$KeV$^{-1})$ vs. $E$(KeV) — [*the number*]{} of photons per a time unit in an energy range unit per an area unit versus the photons energy, — then everything becomes clear. Only a small part or a small [*amount*]{} of quanta/photons observed beyond a threshold of $\approx 700$KeV can be collimated, but within a cone of $< 90^o$ opening angle.
At present, 6 degrees for 10MeV quanta would not be considered as a strongly collimated beam. Now such opening angles (of jets) are considered to be quite suitable in the “standard" or the most popular theory of fireballs. If one proceeds right away from an idea that it is necessary to release quanta with the energy up to 10MeV, then we would obtain at once a version of a collimated theory with the $\Gamma$ of $\sim 10$. But such a way in the standard fireball theory is a dead end also. The allowing for an initial collimation of GRB radiation can drastically change this model (see below) for the collimation arising [*directly in the source*]{} but not because of a huge $\Gamma$ of $\sim 1000$ what would be needed to solve the compactness problem if a ultra-relativistic jet is a GRB source indeed.
One way or another, the light flux is to lead to corresponding effects of radiation pressure upon the matter surrounding the source. And if in addition the radiation is collimated, then the arising of jets (at so enormous light flux) becomes an inevitable consequence of even a small asymmetry of the radiation field [*in the source*]{}.
But the question is if:
Is the jet a GRB source or not?
===============================
Indeed, perhaps one should take into account right away this angular dependence of the threshold of the pair $e^{-}e^{+}$ production (1) before the ultra relativistic limit, allowing for a possibility of a preferential (most probably by a magnetic field) direction in the burst source on the surface of a compact object – the GRB source. Can we do without the radiating and accelerated jet (in the model of fireball) up to a huge value of the Lorentz factor , but supposing that the source of GRB radiation is [*already*]{} collimated by the burst source itself (in a compact GRB model)?
The rather strong collimation of GRB -rays, reaching near-earth detectors, can be observably justified if, due to further accumulation of observational data about coincidence of GRBs and supernovae (SNe), it will turn out indeed that the GRBs could be the beginning of explosions of [*usual*]{} massive or core-collapse SNe \[24\]. At least, all results of photometrical and spectral observations of GRB host galaxies confirm the relation between GRB and evolution of a [*massive star*]{}, i.e., the close connection between GRB and relativistic collapse with SN explosion in the end of the star evolution \[25, 26, 27\]. The main conclusion resulting from the investigation of these galaxies is that the GRB hosts do not differ in anything from other galaxies with close value of redshifts $z$: neither in colors, nor in spectra, the massive star-forming rates \[27\], and the metallicities \[28\]. It means that these are generally starforming galaxies (“ordinary" for their redshifts) constituting the base of all deep surveys. In point of fact, this is the first result of [*the GRB optical identification*]{} with already known objects: GRBs are identified with ordinary (or the most numerous in the Universe at any $z$) galaxies up to $\approx 26$ stellar magnitudes. So, with allowing for the results of direct optical identifications this makes it possible to estimate directly from observations an average yearly rate of GRB events in every such galaxy by accounts of these galaxies for the number of galaxies brighter than 26th st. magn. It turns out to be equal to $ N_{GRB} \sim {\bf 10^{-8}} yr^{-1} galaxy^{-1}$. (But most probably this is only an upper estimate \[24\].) Allowing for the yearly rate of (massive) SN explosions $ N_{SN} \sim 10^{-3} - 10^{-2} yr^{-1} galaxy^{-1}$, the ratio of the number of GRBs, related with the collapse of massive stars (core-collapse SNe), to the number of such SNe is close to $ N_{GRB}/N_{SN} \sim 10^{-5} - 10^{-6}$. (This is also can be only the upper estimate for Ib/c type SNe \[24\].) Certainly, only the further increasing of the number of coincidences of GRBs and SNe ([*identifications of GRBs with Type Ib/c SNe*]{}) should finally tell us whether we have a core-collapse SN (spanning a large range of luminosities) [*in each*]{} GRB or whether the collapse of a massive star evolves following different paths according to the value of parameters as mass, angular momentum, and metallicity \[29\]. But here we proceed from the simplest assumption, which has been confirmed from 1998 by increasing number of observational facts, that [*all*]{} long-duration GRBs are related to explosions of massive SNe. Then the ratio $ N_{GRB}/N_{SN}$ should be interpreted as a very strict “$\gamma$-ray beaming" for a part quanta [*reaching an observer*]{}, when gamma-ray radiation (a part of it) of the GRB source propagates to very long distances within a very small solid angle
$$\Omega_{beam} = N_{GRB}/N_{SN} \sim (10^{-5} - 10^{-6})\cdot 4\pi.$$
Another possible interpretation of the so small value of $ N_{GRB}/N_{SN}$ — a relation to a rare class of some peculiar SNe — seems to be less possible (or hardly probable), since then GRBs would be related only to the $10^{-5}-10^{-6}$th part of all observed SNe in distant galaxies (up to 28th mag). These are already not simple peculiar SNe, with which the Paczyński’s hypernova is sometimes identified \[30\]. The peculiar SNe (hypernovae), such as 1997ef, 1998bw, 2002ap, turn out to be too numerous \[31\]. On the other hand, the more numerous are GRB/SN coincidences \[29\] of type of GRB030329/SN 2003dh, GRB060218/SN2006aj, or GRB/“red shoulder" in light curves, the more confident will be the idea that GRB radiation is collimated, but not related to a special class of SNe. The more so, that explosion geometry features (SN explosion can be axially symmetrical) make the attempts to select a class of “hypernovae" more complex \[33\] (see the end of their text). Now there are already other papers \[32\], pointing out to a possibility of collimated radiation from the GRB source (2).
Let us suppose that only [**the most collimated part**]{} of gamma radiation get to an observer, say, along a rotation axis of the collapsing core of a star with magnetic field. And if GRBs are so highly collimated, radiating only into a small fraction of the sky, then the energy of each event $E_{beam}$ must be much reduced, by several orders of magnitude in comparison at least with a (so called) “isotropic equivalent" $E_{iso}$, of a total GRB energy release ($E_{iso} \sim 10^{51} - 10^{52}$erg and up to $\sim 10^{53}$erg): $$E_{beam} = E_{iso} \Omega_{beam}/4\pi \sim 10^{45} - 10^{47}\,erg .$$
If it is just this case which is realized, and if the energy (3) of -rays propagating in the form of a narrow beam reaching an observer on Earth is only [*a part*]{} of the total radiated energy of the GRB source, then the other part (from $\sim 10^{47}$erg to $\sim 10^{49}$erg, see below) of its energy can be radiated in [*isotropic*]{} or almost isotropic way indeed. But at the spherical luminosity corresponding to a total GRB energy of, e.g., $\sim 10^{45} - 10^{47}$erg, no BATSE gamma-ray monitor detector, even the most sensitive one, would detect flux, corresponding to so low luminosity for objects at cosmological distances of $z \gtrsim 1$, and if the observer is outside the cone of the collimated component of radiation (2). I.e. (3) can be close to the lower estimate of the total radiated energy of GRB sources, corresponding to the flux measured within the solid angle (2), in which the most collimated component of the source radiation is propagating. (We always suppose that [*all*]{} long-duration GRBs are related to SNe.) So, there is a possibility at least to considerably [*reduce*]{} at once the total (bolometric) energy of GRB explosions.
Apparently, this question (what radiate: a central compact source or an extent jet?) is crucial for any GRB mechanism. If the GRB source radiation (mainly a hard component of the GRB spectrum) is collimated indeed, then we will have to return to the old idea: the radiation (GRB) arises [*on a surface*]{} of a compact object of the order of tens of kilometers(?). Further we will try to do without an (a priori) assumption that it is only the jet’s “end" which radiates. The jet arises for sure, but because of the strong pressure of the collimated radiation on the matter surrounding a compact (down to $10^7$cm and less) GRB source. Certainly, this jet accelerated by photons up to relativistic velocities will radiate also, but it would be already an afterglow, but not GRB itself.
The radiation pressure and origin of the jet
============================================
If the scenario: [*massive star*]{} —$>$ [*WR star*]{} —$>$ [*pre-SN = pre-GRB*]{} —$>$ [*the collapse of a massive star core*]{} with formation of a shell around WR is true, then it could be supposed that the reason for arising of a relativistic jet is the powerful light pressure of the collimated or non-isotropic prompt radiation of the GRB source onto the matter of the WR star envelope located immediately around the source itself — a collapsing core of this star.
For example, the radiation field arising around the compact source can be non-isotropic — axially symmetric due to magnetic field and effects of angular dependence (1) of the threshold of the $e^{-}e^{+}$ pair production. And only a part ($\sim 10\%$ or even $1\%$) of the total GRB energy ($\sim 10^{47} - 10^{49}$erg) may be the collimated radiation within the solid angle (2), which breaks through the dense envelope surrounding the collapsing core of the WR star and reaches the Earth. The main things now are: 1) [*the collimated flux*]{} of radiation from the source and 2) existence of [*dense*]{} gas (windy) environment pressed up by radiation from the GRB compact source embedded in it. This environment can be the most dense just near the source, if the density is close to $n = A r^{-2}$ (the WR law for stellar wind). Here the distance $r$ is measured from the WR star itself, and $A\sim 10^{34}$cm$^{-1}$ \[34\]. For the force of light pressure that can act on gas environment (plasma) around the GRB source (the WR star) we have $L_{GRB} \cdot (4\pi r^{2})^{-1} \cdot (\sigma_{T}/c)$, where $L_{GRB}$ is a so called isotropic [*luminosity*]{} equivalent of the source ($\sim 10^{50-51}$erg$\cdot$s$^{-1}$ and more), $r$ is a distance from the center (or from the source), $\sigma_{T} = 0.66 \cdot 10^{-24}$cm$^2$ is the Thomson cross-section, $c$ is the velocity of light. It is clear even without detailed calculation that near the WR core ($r\sim 10^9$cm) such a force can over and over exceed (by 12-13 orders) the light pressure force corresponding to [*the Eddington limit*]{} of luminosity ($\sim 10^{38}$erg$\cdot$s$^{-1}$ for 1$M_\odot$). The isotropic radiation with so huge luminosity $L_{GRB}\sim 10^{50-51}$erg$\cdot$s$^{-1}$ (or the light pressure) can also lead to fast acceleration (similar to an explosion) of environment adjacent to the source. But if we assume that the radiation of the GRB source is non-isotropic and a part of it is collimated or we have very strong beaming with the solid angle $\Omega_{beam} \sim (10^{-5} - 10^{-6})\cdot 4\pi$, then the forming of directed motion of relativistic/ultra-relativistic jets becomes inevitable, only because of so huge/enormous light pressure affecting the [*dense*]{} gas environment in the immediate vicinity of the source - collapsing stellar core.
We can estimate the size of the region [*within*]{} which such a jet can be accelerated by the radiation pressure up to relativistic velocities:\
1. If the photon flux producing the radiation pressure accelerating the matter at a distance $r$ from the center (near the GRB site) is equal to $L_{GRB}\cdot (4\pi r^{2})^{-1}$, then in the immediate vicinity from the GRB source (the collapsing core of WR star) such a flux can be enormous. It is [*inside*]{} this region where the jet originates and undergoes acceleration up to ultra relativistic velocities.\
2. To accelerate the matter up to velocity of at least $\sim 0.3c$, at the [*outer*]{} boundary of this region the photon flux must be at least not less than the Eddington flux $L_{Edd}\cdot (4\pi R_{*}^{2})^{-1}$. Here $L_{Edd}$ is the Eddington limit $\sim 10^{38}$erg$\cdot$s$^{-1}$ for 1$M_\odot$ and $R_{*}$ is the size of a compact object of $\sim 10^{6}$cm. (By definition: $L_{Edd}\cdot (4\pi R_{*}^{2})^{-1}$ is a flux [*stopping* ]{} the accretion onto a compact source — the falling of matter on the source at a parabolic velocity. For a neutron star it is equal to $\sim 0.3c$.)
From the condition that the photon flux $L_{GRB}\cdot (4\pi r^{2})^{-1}$ at distance $r$ is equal to $L_{Edd}\cdot (4\pi R_{*}^{2})^{-1}$ (or at least not less than this flux), and taking into account that the luminosity or rather its [*isotropic equivalent*]{} of the GRB radiation is $L_{GRB}\sim 10^{50-51}$erg$\cdot$s$^{-1}$, it is possible to obtain an estimate of the size of $\sim 10^{12}$ cm $\approx 14R_\odot$. At least, at this outer boundary the light pressure is still able to accelerate the initially stable matter up to sub-light velocities $\sim
0.3c$. And [*deeper*]{}, at less distances than $\sim 10^{12}$cm from the source, say, at $r\sim 10^{9}$cm (somewhere [*inside*]{} the region of the size less than the characteristic size of collapsing core of the massive star) the light accelerates the matter up to ultra relativistic velocities with the Lorentz factor of $\sim 10$ at $L_{GRB}\sim 10^{50}$erg$\cdot$s$^{-1}$. It can occur in a rather small volume of the typical size of $\lesssim R_\odot$. Thus, inside the region of a size of less (in any case) than $10-15 R_\odot$, a relativistic jet arises as a result of the strong light pressure onto the ambient medium.
Concluding remarks: the observational consequences
==================================================
[*The superluminal radio components:*]{} From the above-said it follows that the suggested compact GRB scenario allows also predicting the behavior of superluminal radio components which, e.g., have been observed for GRB 030329 \[37\]. If it is no considerable deceleration of the jet (bullet) with the Lorentz factor of order 10, hence we expect that the superluminal radio components related to the jet have the following properties:\
1) the radio component will move with the constant observed superluminal velocity;\
2) the characteristic observed velocity of the superluminal component is of the order of the Lorentz factor, i.e. of order $10\,c$.
Thus, it is undoubtedly that the GRB radiation is to be collimated in the compact model with GRB source $\sim 10^{8}-10^{6}$cm, but the collimation (2) concerns mainly only a small part of hard quanta. The pairs production threshold (1) for such quanta naturally and smoothly, according to the law $(1-cos\theta)^{1/2}$, rises with the decreasing of the angle $\theta$ between the direction at which the photon is radiated from the surface of the compact object and [*a selected*]{} direction (e.g. the magnetic field) on the surface. As a result, beside a soft component, the more and more hard part of the burst spectrum is passing through, and it is possible to suggest non-isotropic (axially symmetrical) field of radiation around the source.
[*The non-collimated XRFs and SNe:*]{} E.g. the XRFs can be not collimated at all or slightly collimated (XRR GRB), but with the low total bolometric energy of $\sim 10^{47}$erg. Since most probably these are actually the explosions of massive SNe at distances of 100Mpc \[35, 36\], they can be observed much more frequently than it is predicted by the standard fireball GRB model. One should try to find early spectral and photometrical SN features. Then, in general, the observational problem of XRF/XRR/GRB identification becomes a special section in the study of cosmological SNe. (It will be recalled that the GRB030329/SN2003dh was a XRR GRB but not a classical GRB and XRF/GRB060218/SN2008aj was the X-ray flash \[29, 32\].)
As to normal classical GRBs and especially those ones with many heavy quanta in spectra, it is possible to obtain directly from formula (1) a kinematical estimate of the limit collimation of this $\gamma$-radiation, which, in turn, independently agrees with the observational ratio (2) of the yearly rates $N_{GRB}/N_{SN} \sim 10^{-5} - 10^{-6}$. If the matter concerns quanta with $E\sim 100$MeV \[7\] of distant and the most distant GRBs, then from $1 - cos\theta_{12}\approx 0.5$MeV$^2/(100$MeV$\cdot100$MeV$) = 0.5\cdot 10^{-4}$ it follows that the radiation of such GRBs turns out to be the most collimated. Such photons must be radiated in the cone of an opening of $\approx 0.5^o$ and be detected in the spectra of the rather distant GRBs with $z\sim 1$ and farther because of geometrical factor only.
[*The Amati law:*]{} Thus, a natural consequence of our compact model of the GRB source is the fact that distant bursts ($z\gtrsim 1$) turn out to be harder ones, while close “GRBs" ($z \sim 0.1$) look like XRF and XRR GRBs with predominance of soft X-ray quanta in their spectra (though the factor $1+z$ also works). Naturally, the effects of observational selection due to finite sensitivity of GRB detectors should be also taken into account. For example, the soft spectral component of the distant (classical) GRBs is “cut" out by the detector sensitivity threshold. And the isotropic X-ray burst, simultaneous with the GRB, can be simply not seen in distant (classical) GRBs because of the low total/bolometric luminosity of the source in the compact GRB model ($< 10^{49}$erg). Actually, XRF and XRR GRBs have lower values of $E_{iso}$ (so called isotropic equivalent), than GRBs \[16, 32\]. It is an important [*observational*]{} result of BeppoSAX and HETE-2 missions. We mean the detection of obvious XRFs and XRR GRBs first by BeppoSAX \[16\] and then by HETE-2. In our compact model of GRB source it (the Amati law) can be a “simple" consequence of formula (1) + collimation (most probably) by magnetic field on the surface of the compact object.
[*The yearly rate of core-collapse SNe and Fast X-ray Trasients:*]{} In the scenario of jet formation, which was discussed in this paper an isotropic X-ray, optical and radio emission of GRB [*afterglow*]{} is possible. At that an initial assumption was just a possibility of the GRB collimation (2), which follows from the comparison of the rates of GRBs and SN explosions in distant galaxies. It means that the close relation between GRBs and SNe was taken as the basic assumption: [*all*]{} long GRBs are always accompanied by SN explosions, which are sometimes observed, and sometimes not \[24\]. In other words, the long GRB is [*the beginning*]{} of a massive star collapse or the beginning of SN explosion, and GRBs must always be accompanied by SN explosions (of Ib/c type or of [*other*]{} types of massive SNe). Then in any case the total energy release at the burst in -rays can be [*not more*]{} than the total energy released by any SN ($ <$ or $\sim
10^{49}$erg) in all [*electromagnetic*]{} waves. But with so “low" total energy of the GRB explosion ($\lesssim 10^{49}$erg) the only possibility to see GRB at cosmological distances ($z\gtrsim 1$) is the detection of at least the most collimated part of this energy ($1-10$%) leaving the source within the solid angle of $\Omega_{beam} \sim (10^{-5} - 10^{-6})\cdot 4\pi$. The rest can be inaccessible for GRB [*detectors*]{} with a limit sensitivity of $\sim 10^{-7}$erg$ \cdot $s$^{-1}\cdot $cm$^{-2}$. Certainly, it does not concern the 10 000 times more sensitive X-ray [*telescopes*]{} which were used to make sky surveys with the ArielV, HEAO-1, Einstein satellites \[15\]. For the limit sensitivity of $\sim 10^{-11}$erg$\cdot $s$^{-1}\cdot $cm$^{-2}$ in the band of $0.2-3.5$keV the X-ray observatory (Einstein) recorded [*Fast X-ray Trasients*]{} (unidentified with anything) at a rate of $\sim 10^6 yr^{-1}$ all over the sky. It agrees well with an average rate of the massive SNe explosions in distant galaxies, but for the present, GRB-detectors see only $\sim 10^{-4}$ part of this huge number of the distant SN explosions as GRBs.
It is natural that at the total/bolometric energy of “GRB" $\sim 10^{47} - 10^{49}$erg and at the GRB energy (3) released in the narrow cone (2), “the fireball" also looks in quite a different way. As to the compactness problem solved by the fireball model for GRB energies of $10^{52} - 10^{53}$erg, there is no such a problem for “-burst" energies $\sim 10^{47} - 10^{49}$erg. In any case, allowing for the low -ray collimation from the surface of the compact object – GRB/XRR/XRF source, which is necessary for the angular dependence of $e^-e^+$ pair production (1), this problem is solved under quite different physical conditions in the GRB-source than that supposed by Piran \[18\]. In the scenario: [*massive star*]{} —$>$ [*WR*]{} —$>$ [*pre-SN $=$ pre-GRB*]{}, in which only a small part of the most collimated radiation with the collimation (2) goes to infinity and, correspondingly, with the total energy of $10^4 - 10^6$ times less than in the standard theory, the source can actually be of a size $\lesssim 10^8$cm. It means that at the energies of up to $\sim 10^{49}$erg the old (“naive") estimate of the source size resulting directly from the time variability of GRB can be quite true. Thus, the point can be that the burst energy is much less than in the standard fireball model.
[*The strong polarization of the GRB radiation:*]{} But in such a model \[38\] the compact source must always have some radiating [*surface*]{} (but not an event horizon) and, respectively, always occupy some finite volume. Such an object can have both a strong regular magnetic field and a nonuniformly-raidating surface connected with it. The radiation field arising around the source could be anisotropic, e.g., axially symmetric due to the local magnetic field. In particular, non-uniform radiation at the source surface (e.g. polar caps) could lead to efficient collimation or anisotropy of the radiation field \[2\], due to the influence of the angular dependence (1) for the e- e+ pair-creation threshold. Such anisotropy could be associated with the transport of radiation in a medium with a strong ($\sim 10^{14}-10^{16}$G) magnetic field , when the absorption coefficient for photons polarized orthogonal to the magnetic field is very small \[39, 40\]. In this case, the observation of strong linear polarization of the GRB radiation should be another consequence of our compact GRB model.
We always suppose that [*all*]{} long-duration GRBs are related to core-collapse SNe, or the rate of GRB-SNe is the rate of all massive star deaths. Thus, in the compact model of GRBs, the formation of massive ($\gtrsim 3M_\odot$) and compact remnants of the core-collapse SNe (with the massive progenitor stars $> 30-40\,M_\odot$) can be always accompanied by the GRB (or XRF) phenomenon. But the observations should finally tell us whether we have a SN in each GRB or whether the collapde of a massive star evolves following different paths. Whatever the answer may be the fundamental point of the connection is that GRBs may serve as a guideline to better understand the mechanism, and possibly solve the long standing problem of the core-collapse SN explosion, since in the GRBs we have additional information related to the core-collapse \[41\].
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[^1]: to be published in Proceedings of the Conference “Problems of Practical Cosmology", see http://ppc08.astro.spbu.ru/index.html
|
---
abstract: 'Despite widespread interest, a detailed understanding of the dynamics of proton transfer at interfaces is lacking. Here we use *ab initio* molecular dynamics to unravel the connection between interfacial water structure and proton transfer for the widely studied and experimentally well-characterized water-ZnO(10$\bar{1}$0) interface. We find that upon going from a single layer of adsorbed water to a liquid multilayer changes in the structure are accompanied by a dramatic increase in the proton transfer rate at the surface. We show how hydrogen bonding and rather specific hydrogen bond fluctuations at the interface are responsible for the change in the structure and proton transfer dynamics. The implications of this for the chemical reactivity and for the modelling of complex wet oxide interfaces in general are also discussed.'
author:
- Gabriele Tocci
- Angelos Michaelides
bibliography:
- 'water\_on\_zno\_bib.bib'
title: 'Solvent Induced Proton Hopping at a Water-Oxide Interface'
---
Proton transfer in water is a process of central importance to a number of fields in science and technology. Consider for example proton conduction across polymeric membranes used in fuel cells [@nature_fuel_cell_review] or through protein channels in cells [@proton_transfer_water_chain_protein_2]. Proton transfer reactions are also key to many processes in catalysis such as the production of hydrogen from methanol or biomass [@nature_reforming_biomass; @science_norsk_zno_cu_methanol], or water formation [@water_pt_angelos]. Whilst it is notoriously difficult to characterize proton transfer under industrial or biological conditions, considerable insight and understanding has been gained by examining well-defined model systems. One such model system is the example of the solvated proton in pure liquid water [@quantum_nature_proton_Tuckerman; @marx_review; @quantum_nature_proton_Marx_h3o+; @tuckerman_prl09]. Another model system is water adsorbed on atomically flat solid surfaces. Indeed, whereas traditionally most work on well defined water/solid interfaces has focused on structure characterization (e.g. Ref. [@javi_review] and references therein), increasingly the focus is turning to proton transfer and related properties such as surface acidity and water dissociation [@salmeron_nilsson_jacs_wat_copper; @PRL_bonn; @sulpizi_jctc; @sprik_tio2_jctc; @wat_gan_serra; @wat_al2o3; @wat_inp_gap_wood].
Of the various water-solid interfaces that have been examined, water on ZnO$(10\bar{1}0)$ plays a central role in heterogeneous catalysis [@science_norsk_zno_cu_methanol; @MeyerPRLCuZnO; @meyer_wat_zno04] and light harvesting [@ZnO_dye]. It is also a well-defined system that has been the focus of a number of studies under ultra high vacuum (UHV) conditions [@meyer_wat_zno04; @dulub_prl04] which have hinted at potentially interesting dynamical behavior. Specifically, Meyer *et al.* found that at monolayer (ML) coverage one out of every two water molecules is dissociated, forming a so-called partially dissociated (PD) overlayer [@meyer_wat_zno04]. Subsequently they found that this PD overlayer could coexist with an overlayer of intact molecular (M) water [@dulub_prl04]. Moreover they suggested that the two states may rapidly interchange such that an average configuration, intermediate between the two, is at times observed in their scanning tunneling microscopy images. These findings prompted a number of follow up studies which focussed on the structure of water on the surface or on the level of dissociation [@woll_pccp; @meyer_06pccp; @raymand_2011; @holtaus_jctc; @todorova]. This previous work indicates that water on ZnO$(10\bar{1}0)$ might be a highly suitable system for investigating proton hopping in interfacial water. However, the key issue of how proton hopping occurs in this system and how it relates to the aqueous water environment is still not understood. Indeed, this is true for most water/solid interfaces where major gaps in our understanding of the mechanisms of proton motion at interfaces remain.
This work focuses on understanding proton transfer at the liquid water ZnO(10$\bar{1}$0) interface. Although techniques for characterization of well-defined aqueous interfaces have emerged (e.g. Refs. [@salmeron_nilsson_jacs_wat_copper; @PRL_bonn]), probing the microscopic nature of proton transfer at interfaces remains a formidable challenge for experiment. On the other hand, *ab initio* molecular dynamics (AIMD), as we use here, has reached such a state of maturity that it is now possible to simulate bond making and bond breaking events at complex solid-liquid interfaces (see e.g. Refs. [@Galli_water_sic; @wat_gan_serra; @sprik_tio2_jctc; @wat_nacl_pccp]). Here, we find that upon going from a water ML – characteristic of UHV – to a liquid film (LF) – characteristic of ambient conditions – changes in the structure and in the proton transfer dynamics of interfacial water are observed. Although moderate alterations in the structure of the contact layer are found, the proton transfer rate increases more than tenfold. Analysis reveals that H-bond fluctuations induced by the liquid are responsible for the structural change and for the substantial increase in proton transfer. This effect is unlikely to be specific to water on ZnO, implying that proton transfer may be significantly faster under aqueous conditions than at the low coverages typical of UHV-style studies. This fast proton transfer may also affect the chemical activity of a surface, being particularly relevant to heterogeneous catalysis under wet conditions [@hu_solvent; @nature_zeol; @neurock_zeol].
The work reported here was carried out within the framework of density functional theory (DFT). Full details of the computational set-up can be found in the Supporting Information [@SI]. However, in brief, the key features of the simulations are that we used the PBE [@pbe] exchange-correlation functional and the CP2K code [@cp2kvond]. The surface model is made of $6\times 3$ primitive surface unit cells and a 3 bilayer slab. There is one water molecule per primitive cell at ML coverage, whereas the LF is comprised of 144 molecules, resulting in a $\approx 2$ nm thick overlayer. The AIMD simulations are performed in the canonical ensemble close to room temperature. We performed extensive tests on the set-up to explore the sensitivity of our results to issues such as basis set and exchange-correlation functional, including functionals that account for exact exchange and van der Waals forces [@SI]. Overall we find that compared to other interfacial water systems this one is rather benign and none of our main conclusions are affected by the specific details of the DFT set-up. In particular, although the importance of van der Waals dispersion forces between water molecules and water on surfaces is being increasingly recognised (see e.g. Refs. [@jiri_rev; @bis_prl_ice; @Galli_water_vdw; @javi_vdw_prl]) they do not have a significant impact on the dynamics of this system [@SI].
![Spatial probability distribution function of the O and H atoms projected on ZnO$(10\bar{1}0)$ for (a) the water monolayer and (b) the contact layer of the liquid film. Gray, red and white spheres are Zn O, and H atoms, respectively. The topmost Zn and O surface atoms are shown using larger spheres. In (a) a H$_2$O and a OH that are connected via a H-bond are circled in red and black, respectively. (c) top and (d) side view of the partially dissociated water dimer, which is the basic building block of the $(2\times 1)$ overlayer structure. Snapshots of the liquid film showing water in a new type of structure enclosed in a blue oval (e) and partially dissociated dimer structure enclosed in a green oval (f).[]{data-label="fig:spat_dist"}](LF_ML_pdf_structure){width="50.00000%"}
Let us first consider the adsorption of water on ZnO$(10\bar{1}0)$ at ML coverage. Fig. \[fig:spat\_dist\](a) shows the spatial probability distribution function of the O and H atoms adsorbed on the surface at ML coverage. This illustrates the average structure of the overlayer projected onto the surface. Only the PD structure is observed, and it has a similar structure (bond lengths differ by $<0.05$ [Å]{}) to the zero temperature geometry optimized structure. Figs. \[fig:spat\_dist\](c) and (d) show snapshots of the PD state in top and side views, respectively. The OHs and the H$_2$Os sit in the trenches and are covalently bound to the surface-Zn atoms. A H-bond is formed between the surface-Os and the H$_2$Os and also between the surface OHs and the dissociated water. In addition, the H$_2$Os donate a H-bond to the neighbouring OHs and lie essentially flat on the surface, whereas the OHs are tilted up and point away from the surface.
A snapshot of the liquid water film is illustrated in Fig. \[fig:str\](a) and in Fig. \[fig:str\](b) the planar averaged density profile as a function of distance from the surface is reported. The density profile shows a pronounced layering, as previously reported for water on various substrates [@wat_inp_gap_wood; @wat_nacl; @Galli_water_graphene; @fenter04_prog_surf_sci; @PhysRevLett_wat_mica]. For convenience we discuss the density profile in terms of the regions observed, and label them from 0 to 3. Region 0 shows up as a small peak close to the surface and this corresponds to the chemisorbed Hs. These are the Hs that bond to the surface as a result of dissociation of some of the H$_2$Os. The large peak of $\approx 3.2$ g/cm$^3$ in region 1 at about 2.0 [Å]{} corresponds to a mixture of OHs and H$_2$Os in immediate contact with the surface. The second peak in region 1 of about 0.7 g/cm$^3$ also arises from a mixture of OHs and H$_2$Os, that however sit on top of the surface-O atom. Between regions 1 and 2 there is a depletion of H$_2$Os, then in region 2 the oscillations are damped until in region 3 the density decay, characteristic of the liquid-vacuum interface, is observed [@wat_kuo_sci].
![(a) Snapshot of a liquid water film on ZnO$(10\bar{1}0)$(a). (b) Planar averaged density profile as a function of the distance from the surface, where different regions are identified and labelled from 0 to 3. In (b) the zero in the distance is the average height of the top surface ZnO layer and the density reported is the planar averaged density of adsorbed species. In (a) the top four surface layers are shown and the water overlayer is colored according to the regions shown in the density profile (b). Regions (going from 0 to 3) correspond to chemisorbed H atoms, H$_2$Os/OHs adsorbed on the surface, mainly bulk-like liquid water, and water in the liquid vapor interface. []{data-label="fig:str"}](density_final_bright){width="50.00000%"}
The structure of the contact layer in the LF differs from the ML in a number of ways (*c.f*. Figs. \[fig:spat\_dist\](a) and (b)). First, although there are remnants of the ($2\times 1$) structure (see green ovals in Figs. \[fig:spat\_dist\](b) and (f)), the symmetry present at ML coverage is now broken. Second, the proton distribution is more delocalized in the contact layer of the LF than in the ML. Third, and most notably, the coverage in the LF has increased to $1.16\pm 0.03$, with excess waters sitting in a new configuration on top of a surface-O (circled in red in Fig. \[fig:spat\_dist\](e)). At this new site adsorption can happen either molecularly or dissociatively and in either case the adsorbate accepts a H-bond from a H$_2$O sitting on the top-Zn site. Analysis of the overlayer reveals that H-bonding with the liquid above stabilizes the excess H$_2$O at the top-O site which gives rise to the higher coverage [@note_diff_ads_energy]. Despite the structural change between the ML and the contact layer of the LF, we did not observe any exchange of water. Further, the level of dissociation is not altered in the two cases. This can be seen in Figs. \[fig:ads\_Hs\](a) and (b) where the trajectory of the percentage of adsorbed H atoms is reported for the two systems. At an average of $50\%$ dissociation in the case of the ML and $55 \pm 5 \%$ for the contact layer of the LF the difference is not significant [@disagreement_reaxff].
![Time evolution of the percentage of H atoms adsorbed on the surface for (a) water monolayer and (b) liquid water overlayer. (c) Proton hopping frequency $\nu(\tau)$ as a function of the residence time $\tau$ for the liquid film (black) and the monolayer (red). The inset is a $\log$-$\log$ plot of the total number of hops as a function of $\tau$, obtained as $\int_0^\tau \nu(\tau^{\prime}) d \tau^{\prime} $. The full 35 ps of analysis are shown in the inset. \[fig:ads\_Hs\]](water_zno_tau_freq_hydrox_new){width="50.00000%"}
Whilst the changes in the structure between the ML and LF are interesting and important, remarkable differences in the proton transfer dynamics are observed. This is partly shown by the fluctuations in the percentage of adsorbed H atoms, which represent proton transfer events to and from the surface (Fig. 3). Clearly by comparing Figs. \[fig:ads\_Hs\](a) and (b) it can be seen that the fluctuations are much more pronounced in the LF than in the ML. However, proton transfer to and from the surface is only part of the story as proton hopping between the H$_2$Os and OHs is also observed in the contact layer of the LF. Indeed this is already clear by looking at the proton distribution within the green ovals in Fig. \[fig:spat\_dist\](b). In the analysis reported in Fig. \[fig:ads\_Hs\](c) all events are included and the hopping of each proton is monitored. Specifically, we plot the hopping frequency ($\nu=$ number of hops$/$(time$\times$sites)) against $\tau$. $\tau$ is defined as the time a proton takes to return to the O it was initially bonded to, and therefore measures the lifetime of a proton transfer event, with larger values of $\tau$ corresponding to longer lived events. Fig. \[fig:ads\_Hs\](c) thus reveals that proton transfer is more frequent in the LF than in the ML. Specifically, in the LF there are more events at all values of $\tau$, with a maximum in the frequency distribution of about 0.02$/$(ps$\times$site) at $\tau \approx 20$ fs. In contrast in the ML the $\nu$ distribution never reaches values larger than 0.005$/$(ps$\times$site). The $\approx 20$ fs lifetime of the hopping events observed here is similar to the timescale of interconversion between Zundel-like and Eigen-like complexes in liquid water ($< 100$ fs) obtained from femtosecond spectroscopy. It is also in the same ballpark as other theoretical estimates of proton transfer lifetimes obtained from work on proton transport in liquid water or on other water/solid interfaces [@tuckerman_prl09; @fs_spectroscopy_liq_wat; @jacs_akimov_2013]. The total number of hops (inset in Fig. \[fig:ads\_Hs\](c)) is $\approx 0.4/$site but about $10/$site in the LF. While only proton hopping between the overlayer and the surface is observed in the ML, in the LF $\approx 1/4$ of the hops are within the contact layer with the remaining $3/4$ of all hops being to and from the surface. Proton hopping events are also longer lived in the LF than in the ML. This is demonstrated by the long tail in the frequency distribution of the LF and more clearly by the inset in Fig. \[fig:ads\_Hs\](c), which shows that the longest hopping events are only about 0.2 ps in the ML but as long as $\approx 4$ ps in the LF. Events with a lifetime of the order of the picosecond are characteristic of Grotthus-like diffusion [@marx_review] in liquid water or in other water/solid interfaces [@jacs_akimov_2013; @wat_inp_gap_wood; @wat_inp_gap_wood_jpcc; @wat_gan_serra], which are however not observed here, although may occur at longer timescales than we can simulate [@raymand_2011].
To gain further insights in the two systems we plot in Fig. \[fig:pdf\_delta\] free energy ($\Delta F$) surfaces $\Delta F$ for the various distinct proton transfer events considered here. The free energy surfaces have been obtained in a standard manner from $\Delta F= -k_B T \log P($O-O,$\delta)$. The probability distribution $P($O-O,$\delta)$ is a function of O-O distances and of $\delta$, the position of the H with respect to the two Os. With reference to Fig. \[fig:pdf\_delta\] $\delta_{1\textrm{-}2}$ is defined as the difference in the distances between H and two oxygens, O$_1$ and O$_2$, i.e. $\delta_{1\textrm{-}2}=$ O$_{1}$-H $-$ O$_{2}$-H. There are some clear differences between the free energy maps of the ML and the LF. First, the single minimum in Fig. \[fig:pdf\_delta\](a) shows that in the ML protons do not hop between adsorbed H$_2$O and OHs. In contrast in the LF two clear minima are identified revealing that hopping between adsorbed H$_2$Os and OHs occurs readily. The approximate free energy barrier is $\approx 100$ meV. Secondly, proton hopping to and from the surface happens both in the ML (Fig. \[fig:pdf\_delta\](b) and in the LF (Fig.\[fig:pdf\_delta\](d)), but the free energy barrier is noticeably lower in the LF ($\approx 70$ meV) than it is for the ML ($\approx 160$ meV).
In order to understand why the hopping frequency increases so much upon going from ML to multilayer we have examined the time dependence of the H-bonding network at the interface. This reveals an intimate connection between the local H-bonding environment of a molecule and its proclivity towards proton transfer. From the AIMD trajectory we see this connection between H-bonding environment and the hopping of individual protons and we demonstrate in Fig. \[fig:barrier\_ML\_constr\](a) that this holds on average for the entirety of all water-to-surface proton hopping events. Specifically, Fig. \[fig:barrier\_ML\_constr\](a) shows the mean length of the O-H bonds that break in a proton transfer event ($\langle$O$_{\mathrm{w}}$-H$\rangle$) as a function of time. We find that this is correlated with $\langle$O$_{\mathrm{w}}$-O$_{\mathrm{d}}\rangle$, the mean distance between O$_{\mathrm{w}}$ and O$_{\mathrm{d}}$, where O$_{\mathrm{d}}$ is the O of the nearest molecule donating a H-bond to O$_{\mathrm{w}}$. At time t$\,<0$ water is intact at a distance $\langle$O$_{\mathrm{w}}$-H$\rangle \approx 1.0$ [Å]{}. Just before $t=0$, the point at which the $\langle$O$_{\mathrm{w}}$-H$\rangle$ bond breaks, there is a sharp increase in the $\langle$O$_{\mathrm{w}}$-H$\rangle$ distance and then it levels off at $\approx$ 1.4 [Å]{}, about 200 fs after dissociation. Accompanying these changes in the $\langle$O$_{\mathrm{w}}$-H$\rangle$ distance are changes in $\langle$O$_{\mathrm{w}}$-O$_{\mathrm{d}}\rangle$ distances. Crucially about $150$ fs before proton transfer there is a net decrease in the intermolecular separation that shortens $\langle$O$_{\mathrm{w}}$-O$_{\mathrm{d}}\rangle$ from about 3.1 to 2.9 [Å]{}. It can be seen clearly that this change in intermolecular separation occurs before the $\langle$O$_{\mathrm{w}}$-H$\rangle$ bonds start to break revealing that rearrangement in H-bonding is required prior to proton transfer. Similar behaviour has recently been reported for the liquid water/InP(001) interface [@wat_inp_gap_wood]. Further, O-H bond lengthening due to the presence of additional water was reported for water on Al$_2$O$_3$ [@wat_al2o3]. Here, we illustrate that an increase in the O-H bond length occurs before the O-O distance decreases. Not only are the two distances correlated but it is the decrease in the O-O distance which produces an increase in the O-H distance.
![Analysis of the role of H-bond fluctuations on proton transfer. (a) Average O-O distance and average O-H distance as a function of time for all proton transfer events to the surface. The O-O distance plotted (black line) is the distance between the O of the molecule involved in the proton transfer event (O$_{\mathrm{w}}$) and the O of the nearest molecule from which it accepts a H-bond (O$_{\mathrm{d}}$). The O-H distance (red line) is the distance between O$_{\mathrm{d}}$ and the H that is involved in proton transfer. The black and red vertical lines indicate the approximate moment where there is a significant change in the $\langle$O$_{\mathrm{w}}$-O$_{\mathrm{d}}\rangle$ distances, respectively. The insets show snapshots of specific molecules before and after dissociation. (b) Activation energy (E$_a$) for water dissociation at ML coverage as a function of the O$_{\mathrm{w}}$-O$_{\mathrm{d}}$ distance (calculated using VASP [@vasp1; @vasp2], see [@SI]). \[fig:barrier\_ML\_constr\]](h2o_zno_dist_deprotonation){width="50.00000%"}
Through a careful series of additional calculations in which an individual proton transfer event was examined we established that the proton transfer barrier depends critically on the intermolecular distance. As shown in Fig. \[fig:barrier\_ML\_constr\](b) for relatively large distances of 3.4 [Å]{} there is a small $\approx 10$ meV barrier. As the O$_{\mathrm{w}}$-O$_{\mathrm{d}}$ distance decreases, so too does the barrier until at $ \approx 3.1$ [Å]{} where there is no barrier and the intact water state is not stable. Given that fluctuations in the H-bond distances are more pronounced in the LF than in the ML and lead at times to relatively short O$_{\mathrm{w}}$-O$_{\mathrm{d}}$ separations, it is this that causes the more frequent proton transfer. An estimate of the H-bond distance fluctuations is obtained by computing the root mean square displacement the O-O distances in the contact layer, which gives 0.43 [Å]{} in the LF compared to the much smaller value of 0.15 [Å]{} in the ML. This increase is also the reason why hopping does not occur between neighbouring H$_2$Os and OHs on the ML while it does in the LF. H-bond distance fluctuations are also responsible for a proportion of events having a long lifetime (with $\tau \approx 1$ ps, see inset of Fig. \[fig:ads\_Hs\](c)), although actual hydrogen bond forming and breaking and not just fluctuations in the distance may participate in this case. While we never observe H-bond forming or breaking in the ML, the H-bond lifetime is of the order of the picosecond in the LF and this correlates well with the long lived proton transfer events.
We have shown that there are clear differences in the properties of water in contact with ZnO$(10\bar{1}0)$ upon going from UHV-like to more ambient-like conditions. Changes in the adsorption structure upon increasing the coverage above 1 ML have previously been predicted for a number of substrates including ZnO$(10\bar{1}0)$ [@wat_tio2; @wat_tio2_reply; @wat_al2o3; @raymand_2011; @salmeron_nilsson_jacs_wat_copper]. The specific observation here is that the liquid water film leads to a $\approx 16\%$ increase in the water coverage and a breaking of the $(2\times 1)$ periodicity observed at ML. This arises because of H-bonding between the molecules in the contact layer and the molecules above it. It should be possible to verify this increased capacity for water adsorption using a technique such as *in situ* surface X-ray diffraction.
We have demonstrated that there is a substantial increase in the proton transfer rate in the contact layer of the LF and that this is caused by H-bond fluctuations that lower the proton transfer barrier. A H-bond induced lowering of the dissociation barrier upon increasing the water coverage has been discussed before [@water_diss_ru_mich; @meyer_wat_zno04; @wat_mgo_odel; @salmeron_nilsson_jacs_wat_copper; @xiaoliang_wat_mgo; @wat_al2o3; @wat_inp_gap_wood_jpcc]. Here, however, we have demonstrated that the barriers to dissociation and recombination are lowered in general because of the presence of the liquid. As in the case of liquid water and water on other substrates we show (Fig. \[fig:pdf\_delta\]) that there is a strong dependence between the proton transfer barrier and the distance between the Os on either side of the hopping proton [@quantum_nature_proton_Tuckerman; @marx_review; @xin_wat_metal; @wat_gan_serra]. However, we have also identified a connection between the molecule involved in the proton transfer and the molecules in its first solvation shell (Fig. \[fig:barrier\_ML\_constr\]). This observation is somewhat similar to the structural diffusion of the excess proton in liquid water [@marx_review; @quantum_nature_proton_Marx_h3o+]. The key difference between the two is that concerted H-bond breaking and making is required for proton diffusion in liquid water [@tuckerman_prl09], while only fluctuations in the H-bond distance are needed for proton transfer (but not diffusion) to occur. Because fluctuations of the solvent provide the mechanism for the increased proton transfer rate, a similar effect is expected also on other substrates, e.g. on reactive metal surfaces upon which water dissociates [@water_diss_ru_mich; @salmeron_nilsson_jacs_wat_copper].
Finally, since the barrier to proton transfer is sensitive to changes in specific H-bond distances it is likely that implicit solvent models will be inadequate for this class of system as they do not account for H-bond fluctuations. A solvent induced increase in the proton transfer rate may also affect the chemical activity of the substrate and therefore have important consequences for heterogeneous catalysis under wet conditions [@hu_solvent; @nature_zeol; @neurock_zeol; @science_FeO_Merte]. Given that the O-O distance correlates with the barrier height and that H-bond distances of adsorbed H$_2$Os/OHs are related to the lattice constant of the substrate, it might be possible to tailor the proton hopping rate through e.g. strain or doping of the substrate. In conclusion, we have reported on a detailed AIMD study of water on ZnO. In so doing we have tried to bridge the gap between studies of proton transfer in liquid water and low coverage UHV-style work. This has revealed a substantial increase in the rate of proton transfer upon increasing the coverage from a monolayer to a liquid multilayer. We have tracked down the enhanced proton transfer rate to specific solvent induced fluctuations in the H-bond network, which yield configurations with relatively short intermolecular distances wherein the barrier to proton transfer is lowered. These findings are potentially relevant to the modelling of wet interfaces in general and to heterogeneous catalysis.
Supplementary Information
=========================
Further tests and computational details. This material is available free of charge via the Internet at DOI:[10.1021/jz402646c](http://pubs.acs.org/doi/abs/10.1021/jz402646c#10.1021/jz402646c)
Acknowledgments
===============
We are grateful for support from the FP7 Marie Curie Actions of the European Commission, via the Initial Training Network SMALL (MCITN-238804). A. M. is supported by the European Research Council and the Royal Society through a Wolfson Research Merit Award. We are also grateful for computational resources to the London Centre for Nanotechnology and to the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC (EP/F067496), for access to HECToR.
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---
author:
- 'Benedict H. Gross'
title: On the periods of abelian varieties
---
Abstract
========
In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker’s limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.
Looking for a thesis
====================
In my third year of graduate school at Harvard I was still looking for a thesis topic. John Tate had suggested a problem on $p$-adic Galois representations, but I couldn’t see how to make any progress on it. Fortunately for me, Neal Koblitz and David Rohrlich had arrived at Cambridge as BP assistant professors, and I started to talk to them about their work. Rohrlich showed me how the periods of eigen-differentials on the Fermat curve $F(d)$ of exponent $d$ could be explicitly calculated, using the values of Euler’s gamma function at rational arguments with denominator $d$. At the time, I was reading André Weil’s book “Elliptic Functions according to Eisenstein and Kronecker” [@W]. Weil ended with a proof of the Chowla-Selberg formula, which yields an expression, using the values of the gamma function at rational arguments with denominator $d$, for the periods of an elliptic curve with complex multiplication by an order in the imaginary quadratic field of discriminant $-d$.
The similarity of these two expressions led me to wonder if there was any connection between them. In this direction, I was able to identify some products of gamma values at rational arguments $a/d$, which were the periods of certain differential forms of degree $n = \phi(d)/2$ on a factor $J$ of the Jacobian of the Fermat curve $F(d)$, and could also be viewed (via the Chowla-Selberg formula) as the periods of forms of the same degree on the product abelian variety $A^n$, where $A$ was an elliptic curve with complex multiplication by the imaginary quadratic field of discriminant $-d$. If I could relate the periods of these forms on the two abelian varieties geometrically, it would give an independent proof of the implication of the Chowla-Selberg formula for elliptic periods. (Their original proof used techniques from analytic number theory, including Kronecker’s limit formula.) At the time I didn’t see how this identification of periods could be made, but thought it might be a good topic to investigate further.
After I found the relation between the periods of higher degree forms, I came across a paper [@W2] that Weil had recently written, which indicated that he was thinking about similar questions (and was way ahead of me). Since I heard that he was coming to Harvard to speak at Lars Ahlfors’ $70^{th}$ birthday conference, I wrote Weil a note asking if we could get together. He suggested that I meet him at the University guest house, and we ended up taking a long walk through the back streets of Cambridge. Weil described his own years as a graduate student in Rome, where had spoken with Vito Volterra about period integrals.
In fact, periods were the subject of Weil’s talk at the Ahlfors conference the next day. He remained skeptical of William V.D. Hodge’s famous conjecture describing the cohomology classes of algebraic cycles, and suggested that it might be time to search for a counter-example. David Mumford had found some interesting candidates in the Hodge ring of abelian varieties with complex multiplication, and Weil observed that these Hodge classes actually existed on a continuous family of abelian varieties, whose members had endomorphisms by an order in an imaginary quadratic field [@W3].
About half-way into this talk, with the single-minded focus that only a graduate student could possess, I realized that the techniques Weil was using could be modified slightly to give a geometric proof of the Chowla-Selberg formula for elliptic integrals. The continuous family of abelian varieties he highlighted (over a base which would now be called a Shimura variety associated to a group of unitary similitudes) had one fiber isomorphic to the factor $J$ of the Fermat Jacobian and another fiber isomorphic to the product $A^n$ of elliptic curves with complex multiplication. I could guess that the integrals of a differential form of higher degree on this family were constant (the form was horizontal for the Gauss-Manin connection), and that would give a geometric proof of the period identity.
I spent the next few weeks learning the algebraic geometry that was necessary to write this all up, and sent the first draft of my argument in a letter to Weil (which eventually became the paper [@G]). He responded with a letter of encouragement, containing some suggestions for further work. The summer after Weil’s lecture at Harvard, I attended a conference on automorphic forms and $L$-functions held in Corvallis, Oregon. All of the experts in the field had gathered there, and once they realized this, quite sensibly decided to lecture to each other. We graduate students present were frequently lost, and turned to Jean-Pierre Serre for help. After one of our remedial sessions, Serre mentioned that he had heard from Weil about my work on periods. He asked if I would like to meet Pierre Deligne, who was also thinking about these questions. After a quick introduction, the three of us sat down to talk.
I began by saying that I had found a new proof of the period implication of the Chowla-Selberg formula, using some techniques from algebraic geometry. Deligne immediately asked, in all seriousness, if I had proved the Hodge conjecture. I replied that I would be delighted to hear that I had done so, as I was still looking for a thesis topic (and felt that a proof of the Hodge conjecture would probably be sufficient). He then asked me to explain what I had actually done. After about fifteen minutes I had gone through the argument above, and we all agreed that I hadn’t proved the Hodge conjecture. But Deligne thought that my argument could be extended to yield something in that direction. A few weeks later, he sent me a handwritten note of three pages outlining his proof of a fundamental theorem: that all Hodge cycles on abelian varieties are absolutely Hodge [@D]. So that was what I had been doing! (For an illuminating general discussion of the Hodge conjecture, see [@D2].)
About the same time, Serre asked me a question about Hecke characters for imaginary quadratic fields, and referred me to his paper with Tate [@ST], which gives an elegant treatment of the algebraic Hecke characters associated to abelian varieties with complex multiplication. I started to talk with Tate about it, and one thing led to another. So I ended up writing my thesis on the arithmetic of elliptic curves with complex multiplication [@G2], rather than on their periods.
In this expository paper, I will try to pull the two topics together. I will begin by reviewing the original analytic proof of the Chowla-Selberg formula. I will then introduce the elliptic curves $A = A(p)$ in my thesis, with complex multiplication by the integers of $k =\Q(\sqrt{-p})$. These curves are defined over the Hilbert class field $H$ of $k$, and are isogenous over $H$ to all of their Galois conjugates [@G2]. I will show how the Chowla-Selberg formula gives information on the periods of $A$ at the complex places of $H$, and will reinterpret that result in terms of the Faltings height of $A$ [@F]. I will then describe a beautiful conjecture of Pierre Colmez [@C], which gives a formula for the Faltings height of a general abelian variety with complex multiplication by the ring of integers of a CM field in terms of logarithmic derivatives of Artin $L$-functions at $s=0$, and will summarize the recent progress that has been made in that direction.
I will end by explaining Deligne’s construction of a motive of rank $2$ and weight $n$ with complex multiplication by an imaginary quadratic field $k$ from an abelian variety of dimension $n$ with endomorphisms by $k$ (which is an abstraction of the geometric argument in my Chowla-Selberg paper). I will work this construction out for the abelian variety $B = B(p) = \operatorname{Res}_{H/k} A(p)$ of dimension $h(-p)$ as well as for a factor $C$ of the Jacobian of the Fermat curve of exponent $p$, which has dimension $(p-1)/2$. The comparison of these two motives (they differ only by a Tate twist) yields a Hodge class in the middle cohomology of the product variety $B\times C$. In the simplest non-trivial case, when $p = 7$, $B$ is the elliptic curve $A(7) = X_0(49)$ with affine equation $$y^2 + xy = x^3 - x^2 - 2x - 1$$ and $C$ is the Jacobian of the hyperelliptic curve of genus $3$ with affine equation $$z^2 - z = t^7.$$ In this case, the abelian variety $B \times C$ has dimension $4$ and has a Hodge class of type $(2,2)$. Is there a codimension $2$ cycle on $B \times C$ which is defined over $\Q$ and has this class in cohomology?
The Chowla-Selberg formula
==========================
Let $k$ be an imaginary quadratic field with discriminant $-d$, ring of integers $\O$, class number $h$, and unit group of order $w$. Let $\mathfrak a_i$ be ideals of $\O$ which represent the distinct ideal classes. We fix an embedding of $k$ into $\bC$, so the ideals $\mathfrak a_i$ give lattices and the quotients $\bC/\mathfrak a_i$ correspond to the $h$ distinct complex elliptic curves with complex multiplication by $\O$. Let $\Delta$ be the function on lattices in $\bC$ corresponding the the usual cusp form of weight $12$, and let $\Gamma(x)$ be Euler’s Gamma function. Then the Chowla- Selberg formula is the equality [@CS] $$\prod \Delta(\mathfrak a_i ) \Delta(\mathfrak a_i^{-1}) = (2\pi/d)^{12h} \prod\Gamma(a/d)^{6w\epsilon(a)}$$ where the first product is taken over the distinct ideal classes (the product $\Delta(\mathfrak a) \Delta(\mathfrak a^{-1})$ depends only on the ideal class of $\mathfrak a$) and the second product is taken over the elements $0 < a < d$ which are relatively prime to $d$. Finally $$\epsilon: (\Z/d\Z)^* \rightarrow \{\pm1\}$$ is the quadratic character which describes the splitting of primes in $k$, by quadratic reciprocity.
The analytic proof involves the computation of the logarithmic derivative of the zeta function of $k$ at the point $s = 0$ in two different ways. Recall that the zeta function of $k$ is the Dirichlet series given by the sum over all non-zero ideals $\mathfrak a$ of the ring $\O$ of integers of $k$ $$\zeta_k(s) = \sum (N\mathfrak a)^{-s}$$ where the norm $N \mathfrak a$ of an ideal is equal to its index in the ring $\O$. This series converges in the half-plane where the real part of $s$ is greater than one. It has a meromorphic continuation to the entire complex plane, with a simple pole at $s = 1$ and no other singularities.
First, we may write the zeta function of $k$ as a sum of $h$ partial zeta funtions $\zeta_k(\mathfrak a_i,s)$, which are defined by taking the partial sum over the ideals $\mathfrak a$ of $\O$ in the same class as $\mathfrak a_i$. Kronecker’s limit formula [@W pg 73] gives the first two terms in the Taylor expansion of $\zeta_k(\mathfrak a_i,s)$ at the point $s= 0$ $$\zeta_k(\mathfrak a_i,s) = -\frac{1}{w} - \frac{1}{12w}\log(\Delta(\mathfrak a_i)\Delta(\mathfrak a_i^{-1}))s + O(s^2)$$ Hence $$\zeta_k(s) = -\frac{h}{w} - \frac{1}{12w} \sum \log(\Delta(\mathfrak a_i)\Delta(\mathfrak a_i^{-1}))s +O(s^2)$$ and $$d\log\zeta_k(s)|_{s=0} = \frac{1}{12h}\sum \log(\Delta(\mathfrak a_i)\Delta(\mathfrak a_i^{-1})).$$
On the other hand, we may also write the zeta function of $k$ as the product of the Riemann zeta function $\zeta(s)= \ \sum n^{-s}$ and the Dirichlet $L$ function $L(\epsilon,s) = \sum \epsilon(n) n^{-s}$ associated the the character $\epsilon$: $$\zeta_k(s) = \zeta(s) L(\epsilon,s).$$ This identity was obtained by Dirichlet when the real part of $s$ is greater than $1$, by identifying the terms in the Euler product, using quadratic reciprocity. It then holds for all $s$ by analytic continuation.
From the product, it follows that $$d\log\zeta_k(s) = d\log \zeta(s) + d\log L(\epsilon,s).$$ These logarithmic derivatives at $s = 0$ can be calculated from Lerch’s expansion of the Hurwitz zeta function (with $0 < x \leq 1$) [@W pg 59-60]: $$H(x,s) = \sum_{n = 0}^{\infty}\frac{1}{(n+x)^s} = (1/2 - x) + \log (\Gamma(x)/\sqrt{2\pi})s + O(s^2).$$ Taking $x = 1$, we find that $\zeta(0) = -1/2$ and $d\log \zeta(s)|_{s = 0} = \log(2\pi).$ On the other hand, summing over $0 < a < d$ with $a$ prime to $d$ we find $$L(\epsilon, s) = d^{-s} \sum \epsilon(a)H(a/d,s).$$ Since $\sum \epsilon(a) = 0$, we find $L(\epsilon,0) = -\sum \epsilon(a) (a/d)$ and $\zeta_k(0) = 1/2\sum \epsilon(a)(a/d))$. Comparing the last formula with the formula for $\zeta_k(0)$ obtained by summing the partial zeta functions gives Dirichlet’s famous class number formula $$h = -(w/2) \sum_{0 < a < d} \epsilon(a) (a/d).$$
We also obtain the following formula for the logarithmic derivative $$d\log L(\epsilon,s)|_{s=0} = (w/2h)\sum_{0 < a < d} \epsilon (a) \log \Gamma(a/d) - \log d.$$ Adding this to the logarithmic derivative of $\zeta(s)$ at $s = 0$, we get a second expression for the logarithmic derivative of $\zeta_k(s)$: $$d\log \zeta_k(s)|_{s=0} = \log (2\pi) - \log d + (w/2h) \sum_{0 < a < d} \epsilon(a) \log \Gamma(a/d).$$ Comparing this with the one obtained via Kronecker’s limit formula, multiplying both sides by $12h$, and exponentiating gives the Chowla-Selberg formula: $$\prod \Delta(\mathfrak a_i ) \Delta(\mathfrak a_i^{-1}) = (2\pi/d)^{12h} \prod\Gamma(a/d)^{6w\epsilon(a)}$$
Elliptic periods
================
In this section, I will describe what the Chowla-Selberg formula yields on the periods of elliptic curves with complex multiplication in the simplest case, when the discriminant $d$ of the imaginary quadratic field is a prime. Let $p$ be a prime number with $p \equiv 3~ (\rm mod~4)$. Let $k = \Q(\sqrt{-p})$ be the imaginary quadratic field of discriminant $-p$ and let $\O$ be the ring of integers of $k$. We will also assume that $p > 3$, so that the group of units $\O^* = \langle \pm 1 \rangle$ and $w = 2$. In this simple case, the character $\epsilon(a) = (a|p)$ is just the quadratic residue symbol and the Chowla- Selberg formula states: $$\prod_{i = 1}^h \Delta(\mathfrak a_i ) \Delta(\mathfrak a_i^{-1}) = (2\pi/p)^{12h} \prod_{a = 1}^{p-1}\Gamma(a/p)^{12\epsilon(a)}.$$
To interpret this as a result on elliptic periods, we will introduce the elliptic curves $A(p)$. Let $H$ be the Hilbert class field of the imaginary quadratic field $k$, which is an abelian extension of $k$ with Galois group isomorphic to the ideal class group of $\O$. Elliptic curves $A$ with complex multiplication by $\O$ over $H$ are determined up to isomorphism by two invariants [@G2 §9]: an algebraic Hecke character $$\chi_A: I_H^* \rightarrow k^*$$ on the idèles of $H$ whose restriction to the principal idèles $H^*$ is given by the norm, and the modular invariant $j(A) = j(\tau) \in H$. Here $\tau$ is a point in the upper half-plane which is a root of an integral quadratic polynomial $ax^2 + bx + c$ with discriminant $b^2 -4ac = -p$. The character $\chi_A$ determines the isogeny class of $A$ over $H$ and the invariant $j(A)$ determines the isomorphism class of $A$ over $\overline{\Q}$.
The elliptic curve $A = A(p)$ has invariant $j(A) = j((1 + \sqrt{-p})/2)$, which generates the subfield $F = H \cap \R$, and character $\chi_A$ of conductor $(\sqrt{-p})$. On idèles $b = (b_v)$ where the local components at places $v$ dividing $p$ are units which are congruent to $1$, let $\mathfrak b$ be the corresponding fractional ideal of $H$, and let $\mathfrak a = N_{H/k}(\mathfrak b)$. Then $\mathfrak a$ is principal, and $\chi_A(b)$ is the unique generator of $\mathfrak a$ which is a square modulo $\sqrt{-p}$. Since the character $\chi_A$ is equivariant for the action of $\operatorname{Gal}(H/\Q)$ on both $I_H$ and $k^*$, the curve $A = A(p)$ is isogenous to all of its conjugates over $H$ [@G2 §11]. The curve $A$ descends to its field of moduli $F = \Q(j(A))$, where it defines an isogeny class containing two isomorphism classes. We specify an isomorphism class by insisting that $A$ has a minimal Weierstrass model over the ring of integers of $F$ with discriminant $\Delta= -p^3$ [@G3]. The Néron differential $\omega$ determined by this model is well-defined up to sign, and the Chowla-Selberg formula gives a formula for the product of its periods over the complex places of $H$: $$\prod_{v | \infty}~~ \int_{A_v(H_v)} |\omega_v \wedge \overline{\omega_v}| = (2\pi/p)^h \prod_{a = 1}^{p-1} \Gamma(a/p)^{\epsilon(a)}.$$ Taking logarithms of both sides and dividing by $h$, we find the equivalent formula $$\frac{1}{h} \log~ (\prod_{v | \infty}~~ \int_{A_v(H_v)} |\omega_v \wedge \overline{\omega_v}| )= \log 2\pi - \log p + \frac{1}{h} \sum \chi(a) \log \Gamma(a/p) = d\log \zeta_k(s)|_{s=0}.$$
We now derive the formula for the product of periods from the Chowla-Selberg formula. Since $H = \Q(\sqrt{-p}, j(A))$, we have a unique embedding $v_1: H \rightarrow \bC$ which maps $\sqrt{-p}$ to a complex number with positive imaginary part, and $j(A)$ to the real number $j((1 + \sqrt{-p})/2)$. For each class $\mathfrak a$ let $\sigma_{\mathfrak a}$ be the corresponding element in the Galois group of $H/k$, using the Artin reciprocity law, and let $v_{\mathfrak a}: H \rightarrow \bC$ be the embedding defined by $v_{\mathfrak a} = v_1 \circ \sigma^{-1}_{\mathfrak a}$. Then the complex period lattice of $\omega_{v_{\mathfrak a}}$ has the form $\Omega_{\mathfrak a}\cdot \mathfrak a$, for a non-zero complex number $\Omega_{\mathfrak a}$. The period integral at the place $v = v_{\mathfrak a}$ is then given by $$\int_{A_v(H_v)} |\omega_v \wedge \overline{\omega_v}| = \Omega_{\mathfrak a}\cdot\overline{\Omega_{\mathfrak a}} \cdot N(\mathfrak a) \sqrt p.$$ On the other hand, since $\Delta$ is a modular form of weight $12$, we have $\Delta(\Omega_{\mathfrak a} \cdot \mathfrak a) = \Omega_{\mathfrak a}^{-12} \Delta(\mathfrak a)$, and the value of $\Delta$ on the period lattice of $\omega_{v_{\mathfrak a}}$ is given by $\sigma^{-1}_{\mathfrak a}(\Delta(\omega)) = \sigma^{-1}_{\mathfrak a}(-p^3) = -p^3.$ This gives the formula $-p^3 \cdot\Omega_{\mathfrak a}^{12}=\Delta(\mathfrak a).$ Since $\overline{\Delta(\mathfrak a)} = \Delta(\overline{\mathfrak a}) = N(\mathfrak a)^{-12} \Delta(\mathfrak a^{-1})$ we conclude that $$\Omega_{\mathfrak a}^{12} \cdot \overline{\Omega_{\mathfrak a}}^{12} \cdot N(\mathfrak a)^{12} \cdot p^6= \Delta(\mathfrak a) \Delta(\mathfrak a^{-1})$$
Now take the product over all embeddings and use the formula for $\prod \Delta(\mathfrak a ) \Delta(\mathfrak a^{-1})$ given by Chowla and Selberg. We find that $$\prod_{v | \infty}~~ \int_{A_v(H_v)} |\omega_v \wedge \overline{\omega_v}|^{12} = (2\pi/p)^{12h} \prod_{a = 1}^{p-1} \Gamma(a/p)^{12\epsilon(a)}.$$ Since both sides are positive real numbers, we may take the $12^{th}$ roots to obtain the stated formula on elliptic periods. For generalizations to elliptic curves with complex multiplication by non-maximal orders, see [@K], [@NT].
We will need another result on the periods of regular differentials $\omega$ on $A$ over $H$, when integrated over rational $1$-cycles on the curve at the completions $H_v$ [@G2 Thm 21.2.2]. Define an equivalence relation $a \sim b$ on non-zero complex numbers if the ratio $a/b$ lies in $k^*$. Then we have $$\prod_{v |\infty} ~~\int_{\gamma_v} \omega_v ~~\sim~~(2 \pi i)^{-m}\prod_{\epsilon(a) = +1} \Gamma(a/p)$$ where $\gamma_v$ is any non-trivial $1$-cycle in the rational homology of $A$ over the complex completion $H_v$ and $$m = \sum_{\epsilon(a) = +1} a/p = \frac{p-1}{4} - \frac{h}{2}$$ The product of $\Gamma$ values and the above sum is taken over the quadratic residues in $(\Z/p\Z)^*$, and $a$ is the unique representative of the class which lies between $1$ and $p$.
Let $B = B(p) = \operatorname{Res}_{F/\Q} A(p)$. Then $B$ is an abelian variety of dimension $h$ over $\Q$ which has complex multiplication over $k$ by a CM field $E$ which contains $k$ [@G2 §15]. The field $E$ is generated over $k$ by certain $h^{th}$ roots of ideals which are $h^{th}$ powers. Let $\omega_B$ be a non-zero regular differential of degree $h$ on $B$ over $k$. Then $\omega_B$ is unique up to scaling by $k^*$ and it follows from the above that the non-zero periods are given up to equivalence by $$\int_{\gamma_B} \omega_B ~ \sim ~ (2 \pi i)^{-m}\prod_{\epsilon(a) = +1} \Gamma(a/p)$$ where $\gamma_B$ is any $h$-cycle in the rational homology of $B$.
Colmez’s conjecture for the Faltings height
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The formula we obtained for the product of the periods of $(1,1)$ forms $\int_{A_v(H_v)} |\omega_v \wedge \overline{\omega_v}|$, using the values of the Gamma function at rational numbers with denominator $p$, can be used to compute the Faltings height of the abelian variety $A = A(p)$ over $\overline{\Q}$. To define this height, we let $\alpha = \sqrt{-p}$ in $H$ and pass to the quadratic extension $K = H(\sqrt\alpha)$ where the elliptic curve $A$ has good reduction everywhere. Let $\mathscr A$ be the Néron model of $A$ over $K$ and let $\Omega(\mathscr A)$ be the projective $\O_K$-module of Néron differentials. In this special case, the projective module $\Omega(\mathscr A)$ is free, and generated by $\omega' = \sqrt\alpha \cdot \omega$. The Faltings height $h(A)$ of $A$ over $\overline{\Q}$ is then defined as $$h(A) = \frac{-1}{[K:\Q]} \sum_w \log\int_{A_w(\bC)} |\omega'_w \wedge \overline{\omega'_w}|$$ where the sum is taken over the complex places $w$ of $K$.
Since there are two complex places $w$ of $K$ above each complex place $v$ of $H$, and $$\int_{A_w(\bC)} |\omega'_w \wedge \overline{\omega'_w}| = \sqrt p \int_{A_v(\bC)} |\omega_v \wedge \overline{\omega_v}|$$ we obtain the formula $$h(A) = \frac{-1}{2h} \sum_v \log\int_{A_v(\bC)} |\omega_v \wedge \overline{\omega_v}| ~~-~~ \frac{1}{4}\log p$$ Combining this with the Chowla-Selberg formula for $A(p)$, we find that $$h(A) = -\frac{1}{2} d\log \zeta_k(s)|_{s=0} -\frac{1}{4} \log p = -\frac{1}{2}d\log L(\epsilon,s)_{s = 0} - \frac{1}{4}\log p -\frac{1}{2}\log(2\pi) .$$
While looking for a product formula for periods, analogous to the classical product formula for algebraic numbers, Pierre Colmez was led to a beautiful generalization of the above result. He conjectures a precise formula, expressing the Faltings height of an abelian variety with complex multiplication in terms of the logarithmic derivatives of Artin $L$-functions at $s = 0$ [@C]. The amazing idea of relating the periods of abelian varieties with complex multiplication by an abelian field to the logarithmic derivatives of Dirichlet $L$-functions at $s = 0$ is due to Anderson [@A]. Colmez was able to extend this work and establish his product formula in the abelian case (with an assist from Obus [@O] at the prime $2$). Substantial progress on the general Colmez conjecture was recently made by Andreatta, Goren, Howard, and Madapusi-Pera [@AGHM], and by Yuan and Zhang [@YZ]. These authors establish an average version of the conjectural formula, which I will describe below.
Let $E$ be a CM field of degree $2g$, with ring of integers $\O_E$ and totally real subfield $E^+$. Let $B$ be an abelian variety of dimension $g$ with complex multiplication by $\O_E$, defined over $\overline{\Q}$. Then $B$ is defined over a finite extension $K$ of $\Q$, which is contained in $\overline{\Q}$. We will assume that $K$ is large enough so that all endomorphisms of $B$ are defined over $K$, $K$ contains the normal closure of $E$, and the abelian variety $B$ has everywhere good reduction over $K$ (cf. [@ST]). The CM type $\Phi$ of $B$ is defined as the set of embeddings $E \rightarrow K$ which result from the diagonalization of the action of $\O_E$ on the $g$ dimensional tangent space $\operatorname{Lie}(B/K)$. If $c$ is complex conjugation on $E$, then the union $\Phi \cup \Phi\circ c$ is the set of all embeddings of $E$ into $K$. Hence there are $2^g$ possible CM types $\Phi$, for each CM field $E$ of degree $2g$. The automorphism group of the normal closure of $E$ acts on the finite set of CM types for $E$ by composition.
Let $\mathscr B$ be the Néron model of $B$ over the ring of integers $\O_K$ of $K$, and let $\Omega = \det(\operatorname{Lie}(\mathscr B)^{\vee})$ be the projective $\O_K$ module (of rank $1$) of Néron differentials. Choose a non-zero element $\omega \in \Omega$, and define the Faltings height $h(B)$ of $B$ by the fomula $$[K: \Q] \cdot h(B) = -\sum_v \log\int_{B_v(\bC)} |\omega_v \wedge \overline{\omega_v}| + \log\#(\Omega/\O_K\omega)$$ where the sum is taken over the complex places $v$ of $K$. Faltings shows that the height is independent of the choice of differential $\omega$ and the field of definition $K$ of $B$, and Colmez shows that it depends only on the CM type $\Phi$ of $B$ (in fact, it depends only on the Galois orbit of the CM type), so we can denote it $h(E,\Phi)$. His conjecture gives a precise formula for the height $h(E, \Phi)$ in terms of logarithmic derivatives of Artin $L$-series at $s = 0$. For $g = 1$ this is what we obtained from the Chowla-Selberg formula, and Tonghai Yang [@Y] has resolved the case when $g = 2$. The average version of the Colmez conjecture, which was recently proved, is the simpler statement that $$\frac{1}{2^g} \sum_{\Phi} h(E,\Phi) = -\frac{1}{2} d \log L(V,s)_{s = 0} - \frac{1}{4} \log f(V) - \frac{g}{2} \log 2\pi$$ where the sum is taken over all the possible CM types for $E$, $V$ is the $g$ dimensional representation of the Galois group of $\Q$ induced from the non-trivial quadratic character of $E/E^+$, $L(V,s)$ is its Artin $L$-function, and $f(V) = (-1)^g\operatorname{disc}E/\operatorname{disc}E^+$ is its conductor.
A generalization of Colmez’s conjecture to the logarithmic derivatives of Artin $L$-functions at all negative integers was proposed by Maillot and Roessler [@MR]. There has also been recent progress on the conjecture that I made with Deligne in [@G], giving the periods of varieties acted on by automorphisms of finite order in terms of rational values of the $\Gamma$ function [@Fr] [@MR2].
Deligne’s motive
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In this section, we recall Deligne’s construction of a motive of rank $2$ and weight $n$ from an abelian variety $A$ of dimension $n$ with complex multiplication by an imaginary quadratic field $k$. This construction uses two facts. First, the higher de Rham, Betti, and étale cohomology groups of an abelian variety $A$ are given by the exterior powers of the first cohomology: $H^n(A) = \wedge^nH^1(A)$. Second, the exterior powers of a $k$ vector space $V$ embed in a natural way as direct factors of its exterior powers over $\Q$.
Let $A$ be an abelian variety of dimension $n$ over $\Q$, which has endomorphisms by the imaginary quadratic field $k$ over the extension field $k$. By this we mean that there is a homomorphism $$k \rightarrow \operatorname{End}_k(A) \otimes \Q.$$ These endomorphisms act linearly on the $k$ vector space $\operatorname{Lie}(A/k)$, which decomposes as $u$ copies of the identity embedding of $k$ and $v$ copies of the conjugate embedding, with $u + v = n$. Deligne constructs a subspace of dimension $2$ in the middle cohomology group $H^n(A)$ of $A$, in all cohomology theories (Betti, deRham, Hodge, and $\ell$-adic). This should be a motive $M = M(A)$ in the sense of Grothendieck; for us it will suffice that we can define its cohomological realizations. In that sense, the definition of $M$ is given by $$M = \wedge^n_kH^1(A) \subset \wedge^nH^1(A) = H^n(A).$$ For example, suppose $V = H_B^1(A)$ is Betti cohomology, which is a vector space of dimension $2n$ over $\Q$ with an action of $k$. Then the exterior power $\wedge^n_kV = M_B$ (which has dimension $1$ over $k$ and dimension $2$ over $\Q$) embeds as a direct factor of the $\Q$-vector space $\wedge^nV = H_B^n(A)$: it is the subspace of $\wedge^nV$ on which elements $\alpha$ in the group $k^*$ act by the two characters $\alpha^n$ and $\overline{\alpha}^n$.
The motive $M$ has weight $n$ and rank $2$ over $\Q$ and has complex multiplication by $k$ over the extension field $k$. The Hodge numbers of $M_B \otimes \bC$ are $(u,v) + (v,u)$. Indeed, the two characters $\alpha^n$ and $\overline{\alpha}^n$of $k^*$ appear in bi-degrees $(u,v)$ and $(v,u)$ respectively. This shows that the Hodge decomposition is algebraic, and occurs over $k \subset \bC$. The periods of $M$ are the integrals of a deRham class $\omega_M$ of type $(u,v)$ over $k$ over rational classes in the dual of the one dimensional $k$-vector space $M_B$. These define an equivalence class in $\bC^*/k^*$, which determines the Hodge structure of $M$.
The $\ell$-adic realization $M_{\ell}$ of $M$ is the induced representation of the character $$\rho_{M,\ell}: \operatorname{Gal}(\overline{k}/k) \rightarrow (k\otimes\Q_{\ell})^*$$ which comes from the determinant of the Galois represention on the $k \otimes \Q_{\ell}$-vector space $H^1_{\ell}(A)$. All of these $\ell$-adic characters should come from a single algebraic Hecke character on the idèles $I_k$ of $k$: $$\psi_M: I_k \rightarrow k^*$$ which is equivariant for complex conjugation and has algebraic part $z \rightarrow z^u\overline{z}^v$. The process of passing from an algebraic Hecke character to a compatible family of $\ell$-adic characters is described below.
We will now work out the realizations of the motive $M$ for the abelian variety $B = B(p) = \operatorname{Res}_{F/\Q}A(p)$ of dimension $n = h(-p)$ over $\Q$. In this case, the action on $\operatorname{Lie}(B/k)$ is by $n$ copies of the standard embedding of $k$, so the Hodge numbers of $M_B \otimes \bC$ are $(n,0) + (0,n)$. The periods of $M_B$ are given by the integrals of the non-zero regular differential $\omega_B$ defined over $k$ over classes $\gamma$ in the rational homology: $$\int_{\gamma_B} \omega_B ~ \sim ~ (2 \pi i)^{-m}\prod_{\chi(a) = +1} \Gamma(a/p),$$ where $m = \sum_{\epsilon(a) = +1} \langle a/p \rangle = (p-1)/4 - h/2$
We can determine the $\ell$-adic character $\rho_{M,\ell}$ as follows. The abelian variety $B$ has complex multiplication over $k$ by the CM field $E$ of degree $2h$ . This determines an algebraic Hecke character $\psi_{M(B)}: I_k^* \rightarrow E^*$ whose algebraic part is the standard embedding of $k^*$ into $E^*$ [@G2 §8]. The representation of $\operatorname{Gal}(\overline{\Q}/k)$ on $H^1_{\ell}(B)$ can be described as follows [@ST]. Let $f_{\ell}$ be the embedding $(k \otimes \Q_{\ell})^* \rightarrow (E \otimes \Q_{\ell})^*$ and let $$\psi_{\ell}: I_k \rightarrow (E \otimes \Q_{\ell})^*$$ be defined by $\psi_{\ell}(a) = \psi_{M(B)}(a) \cdot f_{\ell}(a_{\ell})^{-1}$. Then $\psi_{\ell}$ is trivial on $k^*$. Since $\psi_{\ell}$ is continuous and its image is totally disconnected, it is also trivial on the connected component of the idèle class group, and the group of connected components of the idèle class group is isomorpic to the abelianized Galois group of $k$ (via the inverse of the Artin reciprocity law). This gives a homomorphism $\psi_{\ell}: \operatorname{Gal}(\overline{k}/k) \rightarrow (E \otimes \Q_{\ell})^*$, and the representation of this Galois group on $H^1_{\ell}(B)$ decomposes as the direct sum of the $2h$ characters obtained by the different embeddings of $(E \otimes \Q_{\ell})$ into $\overline{\Q_{\ell}}$. Its determinant is therefore given by the composition of $\psi_{\ell}$ with the norm $N$ from $(E \otimes \Q_{\ell})^*$ to $(k \otimes \Q_{\ell})^*$.
It follows that the $\ell$-adic character $\rho_{M,\ell}$ corresponds to the algebraic Hecke character $$\psi_M = N \circ \psi_B: I_k \rightarrow k^*$$ which is equivariant for complex conjugation. The Hecke character $\psi_M = N \circ \psi$ has algebraic part given by the map $\alpha \rightarrow \alpha^h$ and conductor $(\sqrt{-p})$. For an ideal $\frak a$ which is prime to $p$, the value $\psi_M(\frak a) = N \circ \psi_B(\frak a)$ is the unique generator $\beta$ of the ideal $(\frak a)^h$ in $k$ which is congruent to a square $(\mod \sqrt{-p})$. This follows from the fact that $\psi_B(\frak a)$ is the unique $h^{th}$ root of $\beta$ in the CM field $E$.
The induced $2$-dimensional $\ell$-adic representations $M_{\ell}$ of $\operatorname{Gal}(\overline{\Q}/\Q)$ correspond to a holomorphic modular form of weight $h + 1$ for the group $\Gamma_0(p^2)$, which is a newform with integral Fourier coefficients.
A factor of the Fermat Jacobian
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We will now compute Deligne’s motive $M = M(J)$ for a factor $C$ of the Jacobian of the Fermat curve of exponent $p$. This factor is the Jacobian of the quotient curve $X = X(r,s,t)$ with affine equation [@GR] $$y^p = x^r(1-x)^s,$$ where $(r,s,t)$ is a triple of integers with $0< r,s,t < p$ and $r+s+t = p$. Like the Fermat curve, the curve $X$ is an abelian cover of $\mathbb P^1$, which is ramified only at $\{0,1,\infty\}$. The map from the Fermat curve $u^p + v^p = 1$ to the curve $X$ is given by $(u,v) \rightarrow (x,y) = (u^p,u^rv^s)$.
For a rational number $x$ with $n \leq x < n+1$ we let $\langle x \rangle = x - n$, so $0 \leq \langle x \rangle < 1$. The genus of $X$ is $n = (p-1)/2$ and for $a \in (\Z/p\Z)^*$ the differentials $$\omega_a = x^{\langle ar/p\rangle - 1}(1-x)^{\langle as/p \rangle - 1}dx$$ give a basis for the first de Rham cohomology over $\Q$. The curve $X$ has an automorphism of order $p$ over $K = \Q(\mu_p)$, given by $(x,y) \rightarrow (x,\zeta.y)$, where $\zeta$ is a primitive $p^{th}$ root of unity. The differential $\omega_a$ is an eigenvector with eigenvalue $\zeta^a$. Moreover, $\omega_a$ is of the first kind if and only if [@G §2] [@W pg 815] $$\langle ar/p \rangle + \langle as/p \rangle + \langle at/p \rangle = 1.$$ The periods of $\omega_a$ over any $1$-cycle $\gamma$ in the rational homology of $X$ have the form [@G2 Appendix] [@W pg 815] $$\int_{\gamma} \omega_a = I(\gamma)^{\sigma_a}B(\langle ar/p\rangle, \langle as/p \rangle)$$ where $I(\gamma)$ is an element in $K = \Q(\mu_p)$ which depends on the rational cycle $\gamma$ and $\sigma_a$ is the automorphism of $K$ that maps $\zeta$ to $\zeta^a$. Finally, $B(u,v)$ is Euler’s beta function, defined by the integral $$B(u,v) = \int_0^1 x^{u-1} (1-x)^{v-1} dx.$$
The differentials $\omega_a$ correspond to eigenforms of degree one on the Jacobian $C = C(r,s,t)$ of the curve $X$. This abelian variety has dimension $n =(p-1)/2$; over the extension field $K = \Q(\mu_p)$ it has complex multiplication by the ring $\Z[\mu_p]$ of integers in $K$. The CM type of $C$ consists of the elements $$\Phi = \{a \in (\Z/p\Z)^*: \langle ar/p \rangle+ \langle as/p \rangle + \langle at/p \rangle = 1\}.$$ We will henceforth assume that $p \equiv 3~ (\mod ~4)$ and that $ p > 3$, and will consider $C$ as an abelian variety of dimension $n$ over $\Q$ which has endomorphisms by the imaginary quadratic field $k = \Q(\sqrt{-p})$ over $k$. Indeed, $k$ is the unique quadratic subfield of $K = \Q(\mu_p)$ in this case. The subgroup of the Galois group $(\Z/p\Z)^* = \operatorname{Gal}(K/\Q)$ which fixes $k$ consists of the squares $$\operatorname{Gal}(K/k) = \{a \in (\Z/p\Z)^*: \epsilon(a) = +1\}.$$
Our objective is to compute the various realizations of Deligne’s rank $2$ motive $M = M(C)$, and compare them to the motive $M(B)$ studied in the previous section. The Hodge numbers of $M_B \otimes \bC$ are $(u,v) + (v,u)$, where $$u = \#\{a \in \Phi: \epsilon(a) = +1\}$$ $$v = \#\{a \in \Phi: \epsilon(a) = -1\}$$ Indeed, the CM type $\Phi$ gives the action of $K$ on $\operatorname{Lie}(C/K)$, so the action of the subfield $k$ is by $u$ copies of the standard embedding and $v$ copies of the conjugate embedding. Clearly, $u + v = n$. Using Dirichlet’s class number formula, one can show [@G Lemma 4] that $u - v = h.\epsilon(r,s,t) $ with $\epsilon(r,s,t) = \epsilon(r) + \epsilon(s) + \epsilon(t)$.
In [@G Thm 2] I computed the periods of the form $\omega_C$ of type $(u,v)$ on $C$ from the periods of the $1$-forms $\omega_a$. All the cycles of degree $n = (p-1)/2$ come from products of $1$-cycles, and $$\int_{\gamma_1.\gamma_2\ldots \gamma_n}\omega_C = \prod_{\epsilon(a) = +1} B(\langle ar/p\rangle, \langle as/p \rangle) \det((I(\gamma_i)^{\sigma_a})).$$ The latter determinant lies in $k$, as applying an element of the Galois group of $K/k$ to the determinant permutes the columns. Since this Galois group is abelian of odd order, this permutation is even and the determinant is unchanged. Hence $$\int_{\gamma} \omega_C~\sim~ \prod_{\epsilon(a) = +1} B(\langle ar/p\rangle, \langle as/p \rangle).$$ Using Euler’s relation between the beta and gamma functions, and his functional equation for the gamma function: $$B(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y) ~~~~~\Gamma(x)\Gamma(1-x) = \pi/\sin(\pi x)$$ we find after some calculation that this product of beta values is equivalent to $$(2 \pi i)^{-n} \prod_{\epsilon(a) = +1}\Gamma(ar/p)\Gamma(as/p)\Gamma(at/p)$$ where we recall that $n = (p-1)/2$.
If $\epsilon(r) = +1$ we have $$\prod_{\epsilon(a) = +1} \Gamma(ar/p) ~\sim~\prod_{\epsilon(a) = +1} \Gamma(a/p)$$ whereas if $\epsilon(r) = -1$ we have $$\prod_{\epsilon(a) = +1} \Gamma(ar/p) ~\sim~ (2\pi i)^n~/~\prod_{\epsilon(a) = +1} \Gamma(a/p).$$ Since the same holds for $s$ and $t$, when $\epsilon(r,s,t) = +1$ we find the simple formula $$\int_{\gamma} \omega_C ~\sim~\prod_{\epsilon(a) = +1} \Gamma(a/p).$$ This gives the periods of $M(C)$ when $\epsilon(r,s,t) = +1$. In this case, $u - v = h$ and $u + v = n = (p-1)/2$. Hence $v = (p-1)/4 - h/2$ and $$\int_{\gamma} \omega_C \sim \int_{\gamma} \omega_B \cdot (2\pi i)^v$$ Hence the Betti, de Rham, and Hodge realizations of $M(C)$ are Tate twists of the corresponding realizations of $M(B)$.
Next, we compute the $\ell$-adic realization of $M(C)$. The representation of $\operatorname{Gal}(\overline{\Q}/k)$ on $H^1_{\ell}(C/k)$ is induced from an abelian representation of $\operatorname{Gal}(\overline{\Q}/K)$ corresponding to the Hecke character $\psi_C= \psi(r,s,t)$ given by Jacobi sums [@GR §1 §3]. This algebraic Hecke character is an Galois equivariant homorphism on the idèles of $K$ $$\psi_C:I_K \rightarrow K^*$$ whose algebraic part is the map of tori $K^* \rightarrow K^*$ determined by the CM type $\Phi(r,s,t)$ of $C$.
The determinant of the induced representation is then given by the transfer of the inducing character, as the sign of the permutation representation of the odd abelian group $\operatorname{Gal}(K/k)$ is trivial. By class field theory, the transfer of the inducing character is given by the restriction of the algebraic Hecke character $\psi_C$ to the idèles of the subfield $k$ [@S2 Ch XIII]. This restriction gives a Galois equivariant homorphism $$\psi_{M(C)}: I_k \rightarrow k^*$$ whose algebraic part is the homomorphism $z \rightarrow z^u\overline{z}^v$. Since the conductor of the Jacobi sum Hecke character $\psi_C$ is either $(1-\zeta)$ or $(1-\zeta)^2$, its restriction $\psi_{M(C)}$ to $I_k$ has conductor dividing $(\sqrt{-p})$. Since $u + v = (p-1)/2$ is odd, the conductor is equal to $(\sqrt{-p})$. A bit more analysis shows that when $\epsilon (r,s,t) = +1$, we have $$\psi_{M(C)} = \psi_{M(B)} \cdot N^v$$ in $\operatorname{Hom}(I_k,k^*)$, where $N$ is the idèlic norm. Since the $2$-dimensional representation of $\operatorname{Gal}(\overline{\Q}/\Q)$ on $H_{\ell}(M(C))$ is induced from this character of $\operatorname{Gal}(\overline{\Q}/K)$, we can also identify the $\ell$-adic realization of $M(C)$ with a Tate twist of the $\ell$-adic realization of $M(B)$. Summing up, we have shown
Assume that $\epsilon(r,s,t) = \epsilon(r) + \epsilon(s) + \epsilon(t) = +1$. Then $u - v = h$ and the Deligne motives $M(B)$ and $M(C)$ of rank $2$ associated to the abelian varieties $B = B(p)$ and $C = C(r,s,t)$ differ by a Tate twist $$M(C)(m) = M(B)$$ with $m = \sum_{\epsilon(a) = +1} \langle a/p\rangle = (p-1)/4 - h/2$.
A similar analysis yields the identity $M(C)(m) = M(B)$ when $\epsilon(r,s,t) = -1$.
A Hodge class
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In this section, we assume that $\epsilon(r,s,t) = \pm1$. We recall that $C = C(r,s,t)$ is the Jacobian of the curve $X(r,s,t)$ with equation $y^p = x^r(1-x)^s$ and $B = \operatorname{Res}_{F/\Q}A(p)$. Let $d = (p-1)/2 + h = \dim(B \times C)$. Note that $d$ is even, as both $(p-1)/2$ and $h$ are odd.
The rank $4$ motive $M(B) \otimes M(C)(d/2)$, occurs as a submodule of $H^d(B \times C)(d/2)$ by the Kunneth decomposition. It has Hodge numbers $(h,-h) + ((0,0) + (0,0) + (-h,h)$. Since the rank $2$ motives $M(B)$ and $M(C)$ are both symplectic and differ by a Tate twist, it follows that the tensor product $(M(B) \otimes M(C)(d/2)$ contains the rank $2$ Artin motive $\Q + \Q(\epsilon)$. In particular, there is a Hodge class in the middle cohomology of $B \times C$ which is defined over $\Q$.
Are these Hodge classes algebraic, or do they give counter-examples to the Hodge conjecture? For example, when $p \equiv 7 ~(\mod~8)$ the triple $(r,s,t) = (1,1,p-2)$ has $\epsilon (r,s,t) = +1$. In this case, the curve $X = X(r,s,t)$ is hyperelliptic, with affine equation $$z^2 - z = t^p$$ Is there an algebraic cycle of codimension $d/2 = (p-1)/4 + h/2$ which is defined over $\Q$ and gives this Hodge class in the middle cohomology of the abelian variety $B \times C$?
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Benedict H. Gross\
Leverett Professor of Mathematics, Emeritus\
Harvard University\
Harvard Department of Mathematics\
1 Oxford St.\
Cambridge, MA 02138\
USA\
[email protected]
|
---
abstract: 'In this paper, we propose and analyze a gradient flow based Kohn-Sham density functional theory. First, we prove that the critical point of the gradient flow based model can be a local minimizer of the Kohn-Sham total energy. Then we apply a midpoint scheme to carry out the temporal discretization. It is shown that the critical point of the Kohn-Sham energy can be well-approximated by the scheme. In particular, based on the midpoint scheme, we design an orthogonality preserving iteration scheme to minimize the Kohn-Sham energy and show that the orthogonality preserving iteration scheme produces approximations that are orthogonal and convergent to a local minimizer under reasonable assumptions. Finally, we report numerical experiments that support our theory.'
author:
- Xiaoying Dai
- Qiao Wang
- 'Aihui Zhou[^1]'
bibliography:
- 'reference.bib'
title: 'Gradient flow based discretized Kohn-Sham density functional theory [^2]'
---
0.2cm
[**Keywords.**]{} density functional theory, dynamical system, eigenvalue problem, energy minimization, gradient flow, orthogonality preserving 0.2cm
[**AMS subject classifications.**]{}
Introduction
============
Kohn-Sham density functional theory (DFT) is the most widely used model in electronic structure calculations [@Kohn_Sham3]. We see that to solve the Kohn-Sham equation, which is a nonlinear eigenvalue problem, some self-consistent field (SCF) iterations are demanded [@dai2011finite; @dai2008three; @Ele_Str3]. However, the convergence of SCF iterations is not guaranteed, especially for large scale systems with small band gaps, for which the performance of the SCF iterations is unpredictable [@dai2017conjugate; @SCF2]. It has been shown by numerical experiments that the SCF iterations usually converge for systems with larger gap between the occupied orbitals and the remainder [@SCF1]. We understand that there are a number of works trying to illustrate this phenomenon and see that SCF iterations do converge if the gap is uniformly large enough locally or globally [@cai2017eigenvector; @liu2014convergence; @liu2015analysis; @SCF1].
In order to obtain approximations of the Kohn-Sham DFT that are convergent, in recent two decades, people pay much attention to study the direct energy minimization model. Instead of solving the Kohn-Sham equation, people minimize the Kohn-Sham total energy under an orthogonality constraint [@dai2017conjugate; @Opt1; @liu2014convergence; @liu2015analysis; @Opt3; @Dyn2; @Opt4; @Opt6; @Str_Pre2; @SCF2; @BaiZhengjian]. It is shown in [@Opt7] that a monotonic optimization approach may produce a locally convergent approximations. We observe that the iterations based on the optimization should be carefully carried out due to the orthogonality constraint, for which the existing methods are indeed either retraction (see, e.g., [@dai2017conjugate; @SCF2]) or manifold path optimization approaches (see, e.g., [@dai2017conjugate; @Str_Pre2; @SCF2]). We see that some backtracking should be applied in a monotonic optimization method, due to not only theory but also practice.
In this paper, we introduce and analyze a gradient flow based discretized Kohn-Sham DFT for electronic structure calculations. First, we prove that our gradient flow based discretized Kohn-Sham DFT preserves orthogonality and models the ground state well. We then propose a midpoint scheme to carry out the temporal discretization, which is of orthogonality preserving, too. We mention that our numerical scheme avoids a retraction process and does not need any backtracking. Based on the midpoint scheme, finally, we design and analyze an orthogonality preserving iteration scheme for solving the discretized Kohn-Sham energy. It is shown by theory and numerics that our orthogonality preserving iteration scheme is convergent provided some reasonable assumption.
For illustration, we provide Figure \[Table: Differences\] to show the differences among the three approaches. In the midpoint scheme of the gradient flow based model (blue dashed line with square symbol endpoint), the auxiliary point of midpoint scheme of the dynamical system is inside the manifold. In the manifold path method (black solid line with circle symbol endpoint), the path is on the manifold and the energy is decreasing when the iteration is moving along the path. In the retraction method (red solid line with triangle symbol endpoint), the auxiliary point is in the tangent space and outside the manifold.
[c|c|c|c]{} &
---------------
Gradient Flow
Based Model
---------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
--------------
Minimization
Model
--------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
------------
Eigenvalue
Model
------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
\
-------------
Orthogona
-lization
requirement
-------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
& No &
--------------------------
Yes(retraction method),
no(manifold path method)
--------------------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&Yes\
-----------
Auxiliary
points
location
-----------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
-----------
Inside
manifold
(midpoint
scheme)
-----------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
------------------------
Outside manifold
(retraction method),
on manifold
(manifold path method)
------------------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
----------
On
manifold
----------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
\
------------
Energy
decreasing
------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&Yes & Yes &May not\
-------------
Convergence
result
assumptions
-------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
------------
Local
uniqueness
of
minimizer
------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
------------
Local
uniqueness
of
minimizer
------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
&
-----------------
Eigenvalue gap
large enough
(depending on
discretization)
-----------------
: Comparison of three models for Kohn-Sham DFT[]{data-label="Table: Differences"}
We observe that there are some existing works on the gradient flow methods of eigenvalue problems. We refer to [@Oja1; @Dyn1; @Oja] and references cited therein for linear eigenvalue problems and [@Dyn3] for the ground state of Bose-Einstein condensate (which requires the smallest eigenvalue and its associated eigenfunction only). We point out that our gradient flow based model is different from the gradient flow model proposed in [@Opt7] for the Kohn-Sham DFT, in which the numerical scheme is either the retraction approach or the manifold path approach.
We organize the rest of the paper as follows. In section 2, we introduce some necessary notation and the Kohn-Sham DFT models. Then we come up with our gradient flow based model and prove its local convergence and convergence rate of the asymptotic behaviours in section 3. In section 4, we propose a midpoint scheme to realize temporal discretization of the gradient flow based model and investigate the relevant properties including preserving orthogonality automatically, updating inside the manifold as well as the local convergence. Based on the midpoint scheme, in section 5, we design and analyze an orthogonality preserving iteration scheme for solving the discretized Kohn-Sham energy. In section 6, we provide numerical experiments that support our theory. Finally, we present some concluding remarks.
Preliminaries
=============
In this section, we introduce some basic notation and the Kohn-Sham models.
Basic notation
--------------
We apply the standard $\textup{L}^2$-inner product $(\cdot, \cdot)_{\textup{L}^2(\mathbb{R}^3)}$, which is defined as $$(u, v)_{\textup{L}^2(\mathbb{R}^3)} = \displaystyle\int_{\mathbb{R}^3}u(x)v(x)\textup{d}x,$$ denote $\textup{L}^2$-norm $\Vert \cdot\Vert_{\textup{L}^2(\mathbb{R}^3)}$ by $\Vert u\Vert_{\textup{L}^2(\mathbb{R}^3)} = (u, u)_{\textup{L}^2(\mathbb{R}^3)}^{\frac12}$, and $\textup{L}^1$-norm $\Vert \cdot\Vert_{\textup{L}^1(\mathbb{R}^3)}$ by $$\Vert u\Vert_{\textup{L}^1(\mathbb{R}^3)} = \displaystyle\int_{\mathbb{R}^3}|u(x)|\textup{d}x.$$ We define $\textup{H}^1$-norm $\Vert \cdot\Vert_{\textup{H}^1(\mathbb{R}^3)}$ as $$\Vert u\Vert^2_{\textup{H}^1(\mathbb{R}^3)} = \|u\|^2_{\textup{L}^2(\mathbb{R}^3)}+\|\nabla u\|^2_{\textup{L}^2(\mathbb{R}^3)}$$ and use Sobolev space $\textup{H}^1(\mathbb{R}^3)$ $$\textup{H}^1(\mathbb{R}^3) = \{u\in\textup{L}^2(\mathbb{R}^3):\Vert u \Vert_{\textup{H}^1(\mathbb{R}^3)} < +\infty\}.$$
Let $\Psi = (\psi_1, \psi_2, \ldots, \psi_N)\in\big(\textup{H}^1(\mathbb{R}^3)\big)^N$ and $\Phi = (\varphi_1, \varphi_2, \ldots, \varphi_N)$ $\in \big(\textup{H}^1(\mathbb{R}^3)\big)^N$ . Define product matrix $$\Psi\odot\Phi = (\psi_i\varphi_j)_{i,j = 1}^N\in \big(\textup{L}^1(\mathbb{R}^3)\big)^{N\times N}$$ and inner product matrix $$\langle\Psi^\top\Phi\rangle = \big( (\psi_i, \varphi_j)_{\textup{L}^2(\mathbb{R}^3)} \big)_{i,j = 1}^N\in\mathbb{R}^{N\times N}.$$ For $\mathcal{F} = (\mathcal{F}_1, \mathcal{F}_2, \ldots, \mathcal{F}_N)\in \big( (\textup{H}^{1}(\mathbb{R}^3))^N \big)' = \big( \textup{H}^{-1}(\mathbb{R}^3) \big)^N$, we set $$\langle \mathcal{F}, \Psi\rangle = \big( \langle \mathcal{F}_i, \psi_j\rangle \big)_{i,j = 1}^N\in\mathbb{R}^{N\times N}.$$ We then introduce the Stiefel manifold defined as $$\mathcal{M}^N = \big\{U \in \big(\textup{H}^1(\mathbb{R}^3) \big)^N:\langle U^\top U\rangle = I_N \big\}.$$ For $U \in \big(\textup{H}^1(\mathbb{R}^3) \big)^N$ and any matrix $P \in \mathbb{R}^{N\times N}$, we denote $$UP = \Big( \sum\limits_{j = 1}^N p_{j1}u_j, \sum\limits_{j = 1}^N p_{j2}u_j, \ldots, \sum\limits_{j = 1}^N p_{jN}u_j \Big).$$ We see that $$U\in\mathcal{M}^N \Leftrightarrow UP\in\mathcal{M}^N, \quad \forall P \in\mathcal{O}^{N},$$ where $$\mathcal{O}^{N}=\{P\in\mathbb{R}^{N\times N}:P^\top P = I_N\}.$$
We define an equivalent relation “$\sim$” on $\mathcal{M}^N$ as $$U \sim \hat U\Leftrightarrow \exists P\in \mathcal{O}^{N},~\hat U = UP,$$ and get a Grassmann manifold, which is a quotient of $\mathcal{M}^N$ $$\mathcal{G}^N = \mathcal{M}^N\slash\sim.$$ We introduce an equivalent class of $U\in \mathcal{M}^N$ by $$[U] = \{UP: P\in\mathcal{O}^{N}\},$$ an inner product as $$(U, \hat U) = \textup{tr}\big(\langle U^\top \hat U \rangle \big)$$ together with an associated norm $$||| U ||| = (U, U)^{1\slash2}$$ on $\big(\textup{H}^1(\mathbb{R}^3) \big)^N$.
Give a finite-dimensional space $V_{N_g}\subset \textup{H}^1(\mathbb{R}^3)$ spanned by $\phi_1, \phi_2, \ldots, \phi_{N_g}$. We denote $\Phi = (\phi_1, \phi_2, \ldots, \phi_{N_g})$. We see that for any $U\in (V_{N_g})^N$, there exists $C\in\mathbb{R}^{N_g\times N}$ such that $$\label{Equ: spacialDiscrete}
U = \Phi C = \Big(\sum\limits_{j = 1}^{N_g} c_{j1}\phi_j, \sum\limits_{j = 1}^{N_g} c_{j2}\phi_j, \ldots, \sum\limits_{j = 1}^{N_g} c_{jN}\phi_j\Big).$$ We define a closed $\delta$-neighborhood of $U$ by $$B(U, \delta) = \{\hat U\in (V_{N_g})^N :\textup{dist}(U, \hat U)\leqslant \delta\}$$ where $$\textup{dist}(U, \hat U) = ||| U - \hat U |||,$$ and for $U\in(V_{N_g})^N\bigcap\mathcal{M}^N$ introduce a closed $\delta$-neighborhood of $[U]$ on $\mathcal{G}^N$ by $$B([U], \delta) = \{[\hat U]\in \mathcal{G}^N: \hat U \in (V_{N_g})^N\bigcap\mathcal{M}^N, \textup{dist}([U],[\hat U])\leqslant \delta\},$$ where $$\textup{dist}([U],[\hat U]) = \inf\limits_{P\in\mathcal{O}^{N}} ||| U - \hat U P |||.$$
For simplicity, we use notation $$\{U, W\} = UW^\top - WU^\top, \:\forall U, W \in(V_{N_g})^N$$ where $UW^\top$ and $WU^\top$ denote operators on $(V_{N_g})^N$: $$\begin{split}
(UW^\top) V &= U \langle W^\top V\rangle,\\
(WU^\top) V &= W \langle U^\top V\rangle,
\end{split}$$ for any $V\in(V_{N_g})^N$.
Obviously $$\{U, W\} + \{W, U\} = 0, \:\forall U, W \in (V_{N_g})^N.$$ Namely $\{U, W\}$ is skew-symmetric.
Kohn-Sham models
----------------
The energy based Kohn-Sham DFT model for a system of $N$ electron orbitals with external potential contributed by M nuclei of charges is the following constrained optimization problem on the Stiefel manifold $$\label{Opt: Kohn-Sham Energy}
\begin{array}{cc}
\inf\limits_{U\in (\textup{H}^1(\mathbb{R}^3))^N} &E(U)\\
\textup{s.t.} &~U\in\mathcal{M}^N
\end{array}$$ where $E(U)$ is the Kohn-Sham energy $$\label{Equ: Kohn-Sham Energy}
\begin{split}
E(U) =& \frac12\sum\limits_{i = 1}^N f_i \int_{\mathbb{R}^3}|\nabla u_i(r)|^2 \textup{d}r + \int_{\mathbb{R}^3}V_{ext}(r)\rho_{_U}(r)\textup{d}r\\
&+ \frac12\int_{\mathbb{R}^3} \int_{\mathbb{R}^3} \frac{\rho_{_U}(r)\rho_{_U}(r')}{|r - r'|}\textup{d}r \textup{d}r' + E_{xc}\big(\rho_{_U}\big).
\end{split}$$ Here $u_i\in\textup{H}^1(\mathbb{R}^3)(i = 1, 2, \ldots, N)$ are Kohn-Sham orbitals, $$\rho_{_U}(r) = \sum\limits_{i = 1}^N f_i |u_i(r)|^2 = \textup{tr}(U\odot U F)$$ is the associated electron density with $f_i$ being the occupation number of the $i$-th orbital and $F = \textup{diag}(f_1, f_2, \ldots, f_N)$. $V_{ext}(r)$ is the external potential generated by the nuclei: for full potential calculations, $$V_{ext}(r) = -\sum\limits_{I = 1}^{M}\frac{Z_I}{|r - R_I|},$$ $Z_I$ and $R_I$ are the nuclei charge and position of the $I$-th nuclei respectively; while for pseudo potential approximations, the formula for the energy is still (\[Equ: Kohn-Sham Energy\]) (see, e.g., [@dai2017conjugate]). The fourth term in is the exchange-correlation energy, to which some approximations, such as LDA(Local Density Approximation), GGA(General Gradient Approximation) and so on[@Ele_Str2; @LDA], should be applied. We assume that $E(U)$ is bounded from below with orthogonality constraint of $U$, which is of physics. For simplicity, we consider the case of $F = 2I_N$.
We see that for any $ U\in\mathcal{M}^N$ and all $P\in\mathcal{O}^N$, there hold $$\rho_{_{UP}} = \textup{tr}((UP)\odot UP F) = 2\textup{tr}(U\odot UP P^\top) = 2\textup{tr}(U\odot U) = \rho_{_U}$$ and $$E(UP) = E(U).$$ Instead we consider an optimization problem on $\mathcal{G}^N$ $$\label{Equ: EnergyGrassmann}
\begin{array}{cc}
\inf\limits_{U\in (\textup{H}^1(\mathbb{R}^3))^N} &E(U)\\
\textup{s.t.} &~[U]\in\mathcal{G}^N
\end{array}$$ and define level set $$\mathcal{L}_{E} = \{[U]\in \mathcal{G}^N : E(U)\leqslant E\}.$$ To introduce the gradient on $\mathcal{G}^N$, we suppose $$E_{xc}\big(\rho_{_U}\big) = \int_{\mathbb{R}^3} \varepsilon_{xc}\big(\rho_{_U}\big)(r)\rho_{_U}(r)\textup{d}r$$ and assume that the exchange-correlation energy is differentiable and the exchange-correlation potential $$v_{xc}(\rho) = \frac{\delta\big(\rho \varepsilon_{xc}(\rho) \big)}{\delta\rho}.$$ We may write the gradient of $E(U)$ as $$\nabla E(U) = (E_{u_1}, E_{u_2}, \ldots, E_{u_N}) \in \big(\textup{H}^{-1}(\mathbb{R}^3)\big)^N,$$ where $E_{u_i}\in\textup{H}^{-1}(\mathbb{R}^3)$ is defined by $$\label{Equ: Derivative}
\begin{split}
\langle E_{u_i}, v\rangle =& 4\bigg( \frac12(\nabla u_i, \nabla v)_{\textup{L}_2} + (V_{ext}~u_i, v)_{\textup{L}_2}\\
&+ \Big(\int_{\mathbb{R}^3}\frac{\rho_{_U}(r')}{|r - r'|}\textup{d}r~u_i, v\Big)_{\textup{L}_2} + \Big(v_{xc}\big(\rho_{_U}\big)~u_i, v\Big)_{\textup{L}_2} \bigg), \forall v\in\textup{H}^1(\mathbb{R}^3).
\end{split}$$ Obviously $$\label{Equ: GradientSym}
\big\langle\nabla E(U), U\big\rangle = \big\langle\nabla E(U), U\big\rangle^\top,\forall U\in\big(\textup{H}^1(\mathbb{R}^3)\big)^N.$$ We see from [@Grassmann1] that the gradient on Grassmann manifold $\mathcal{G}^N$of $E(U)$ at $[U]$ is $$\label{Equ: GrassmannGradientPrm}
\nabla_{G}E(U) = \nabla E(U) - U\big\langle\nabla E(U), U\big\rangle^\top, ~\forall U\in \mathcal{M}^N.$$ To propose a gradient flow based model preserving orthogonality, we need to extend the domain of $\nabla_{G}E(U)$ from $\mathcal{M}^N$ to $\big(\textup{H}^{1}(\mathbb{R}^3)\big)^N$. We then define extended gradient $\nabla_{G}E(U):\big(\textup{H}^{1}(\mathbb{R}^3)\big)^N \longrightarrow \big(\textup{H}^{-1}(\mathbb{R}^3)\big)^N$ as follows $$\label{Equ:GrassmannGradient}
\nabla_{G}E(U) = \nabla E(U)\langle U^\top U\rangle - U \big\langle\nabla E(U), U\big\rangle^\top, ~\forall U\in \big(\textup{H}^{1}(\mathbb{R}^3)\big)^N.$$ Note that is consistent with for $[U]\in\mathcal{G}^N$ since $\langle U^\top U\rangle = I_N$.
We see from [@dai2017conjugate; @Grassmann1] that the tangent space on $\mathcal{G}^N$ is $$\mathcal{T}_{[U]}\mathcal{G}^N = \big\{ W\in \big(\textup{H}^1(\mathbb{R}^3)\big)^N:\langle W^\top U \rangle= 0 \big\}$$ and the Hessian of $E(U)$ on $\mathcal{G}^N$ is $$\textup{Hess}_{G}E(U)[V, W] = \textup{tr}\big(\langle V^\top \nabla^2E(U)W\rangle\big) - \textup{tr}\big(\langle V^\top W\rangle \langle U^\top \nabla E(U)\rangle\big),\:\forall V, W \in \mathcal{T}_{[U]}\mathcal{G}^N.$$
If $U\in (V_{N_g})^N$, then we may view $\nabla E(U)\in (V_{N_g})^N$ in the sense of isomorphism and $$\big\langle\nabla E(U), V\big\rangle = \big\langle\big(\nabla E(U)\big)^\top V\big\rangle,~\forall V\in(V_{N_g})^N.$$ As a result, $\nabla_{G}E(U)\in (V_{N_g})^N$ and we may write $$\label{Equ: GrassmannGradientDis}
\nabla_{G}E(U)
= \mathcal{A}_{U} U, ~\forall U\in(V_{N_g})^N,
$$ where $$\mathcal{A}_{U} = \{\nabla E(U), U\}.$$
Gradient flow based model {#Sec: GrdFlw}
=========================
In this section, we propose and analyze a gradient flow based model.
The model
---------
Different from the Kohn-Sham equation and the Kohn-Sham energy minimization model, we propose a gradient flow based model of Kohn-Sham DFT as follows: $$\label{Equ: GradientFlowOp}
\left\{
\begin{array}{l}
\displaystyle\frac{\textup{d}U}{\textup{d}t} = -\nabla_G E(U), \quad 0< t <\infty\\
U(0) = U_{0},
\end{array}
\right.$$ where $U(t)\in (V_{N_g})^N$ and $U_{0}\in\mathcal{M}^N$. We see that is different from the standard gradient flow model presented in [@Opt7], which applies the $\nabla_G E(U)$ in rather than . We point out that whether the solution of keeps on the Stiefel manifold is unclear. However, we see from Proposition \[Prop: GradientFlowOp\] that our new $\nabla_G E(U)$ defined by guarantees that the solution keeps on the Stiefel manifold. Namely, is an orthogonality preserving model whenever the initial is orthogonal.
\[Lemma: ZeroTrace\] If $A, B\in \mathbb{R}^{N\times N}$ and $$A^\top = A, \quad B^\top = -B,$$ then $$\textup{tr}(AB) = 0.$$
We see that $$\textup{tr}(AB) = \textup{tr}(AB)^\top = \textup{tr}(B^\top A^\top) = -\textup{tr}(BA) = -\textup{tr}(AB),$$ which indicates $$\textup{tr}(AB) = 0.$$
\[Prop: GradientFlowOp\] The solution of satisfies ${U(t)}\in\mathcal{M}^N$. Moreover, there holds $$\frac{\textup{d}E\big( U(t) \big)}{\textup{d}t} = -\Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big|^2\leqslant 0, \quad 0 < t < \infty.$$
A direct calculation shows that $$\begin{split}
\frac{\textup{d}}{\textup{d}t}\big\langle U(t)^\top U(t)\big\rangle &= \bigg\langle\Big(\frac{\textup{d}}{\textup{d}t}U(t)\Big)^\top U(t)\bigg\rangle + \Big\langle U(t)^\top \frac{\textup{d}}{\textup{d}t}U(t)\Big\rangle\\
&= \Big(\big\langle U(t)^\top \mathcal{A}_{U(t)} U(t)\big\rangle \Big) - \Big(\big\langle U(t)^\top \mathcal{A}_{U(t)} U(t)\big\rangle \Big) = 0,
\end{split}$$ which indicates $$\big\langle{U(t)}^\top U(t)\big\rangle = I_N$$ due to $\langle{U_0}^\top U_0\rangle = I_N$. Consequently, we see from Lemma \[Lemma: ZeroTrace\] that $$\begin{split}
&\Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big|^2 - \textup{tr}\Big\langle\nabla E\big(U(t)\big)^\top \nabla_{G} E\big(U(t)\big)\Big\rangle\\
=&\textup{tr}\bigg(\Big\langle \nabla E\big(U(t)\big)^\top U(t)\Big\rangle \Big\langle U(t)^\top \nabla_{G} E\big(U(t)\big)\Big\rangle\bigg) = 0.
\end{split}$$ As a result, $$\frac{\textup{d}E\big( U(t) \big)}{\textup{d}t} = \frac{\delta E}{\delta U}\cdot \frac{\textup{d}U}{\textup{d}t}= -\textup{tr}\Big\langle\nabla E\big(U(t)\big)^\top \nabla_{G} E\big(U(t)\big)\Big\rangle = -\Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big|^2\leqslant 0.$$
Critical points
---------------
We denote Lagrange function of $$\mathcal{L}(U, \Lambda) = E(U) - \frac12\big(\langle U^\top U\rangle - I_N \big)\Lambda$$ for $U \in (V_{N_g})^N$ and $\Lambda\in\mathbb{R}^{N\times N}$, then the corresponding first-order necessary condition is as follows $$\begin{aligned}
\nabla_U \mathcal{L}(U, \Lambda) &\equiv& \nabla E(U) - U\Lambda = 0,\label{Equ: EigenGrad}\\
\nabla_{\Lambda} \mathcal{L}(U, \Lambda) &\equiv& \frac12\big(I_N - \langle U^\top U\rangle \big) = 0.\label{Equ: EigenOrtho}
\end{aligned}$$
We call $[U]$ a critical point of if $$\nabla_{G} E(U) = 0.$$ Obviously, for such a critical point, we have $$\nabla E(U) = U \langle U^\top \nabla E(U)\rangle,$$ which suggests $$\begin{aligned}
\nabla_U \mathcal{L}\big(U, \langle U^\top \nabla E(U)\rangle\big) = 0,\\
\nabla_{\Lambda} \mathcal{L}\big(U, \langle U^\top \nabla E(U)\rangle\big) = 0.
\end{aligned}$$ Thus we see that such a critical point may be a local minimizer.
As $t\to\infty$, we know that energy $E\big( U(t) \big)$ decreases monotonically, thus $\lim\limits_{t\rightarrow\infty}$ $E\big( U(t) \big)$ exists provided that $E\big( U(t) \big)$ is bounded from below. The following statement tells us the asymptotical behavior of the extended gradient flow(c.f. [@Lyapunov]).
\[Theo: Inf ConvergeOp\] If $U(t)$ is a solution of , then $$\begin{split}
&\liminf\limits_{t\rightarrow\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big| = 0.
\end{split}$$
We see from Proposition \[Prop: GradientFlowOp\] that $$\begin{split}
&\int_{0}^{+\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big|^2 \textup{d} t = -\int_{0}^{+\infty} \frac{\textup{d} E(U)}{\textup{d}t} \textup{d}t\\
=& E\big(U(0)\big) - \lim\limits_{t\rightarrow \infty} E\big(U(t)\big) < +\infty.
\end{split}$$ Since $\Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big|^2$ is nonnegative function, we have $$\liminf\limits_{t\rightarrow\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big| = 0.$$
Suppose that the local minimizer $[U^*]$ is the unique critical point of in $B([U^*], \delta_1)$. For a fixed constant $\delta_2 \in(0, \delta_1]$, we define $$E_0 = \min \{ E([\tilde U])~|~[\tilde U] \in \overline{B([U^*], \delta_1)\backslash B([U^*], \delta_2)}\}.$$ Here and hereafter, we assume that as an operator from $\big(V_{N_g}\big)^N$ to $\big(V_{N_g}\big)^N$, $\nabla E$ is continuous in $B(U^*, \delta_1)$.
\[Theo: ContConv\] If the initial value satisfies $$E(U_0)\leqslant \frac{E_0 + E(U^*)}{2}\equiv E_1,$$ then $$\begin{aligned}
\lim\limits_{t\rightarrow\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big| = 0,\\
\lim\limits_{t\rightarrow\infty} E\big(U(t)\big) = E(U^*),\\
\lim\limits_{t\rightarrow\infty}\textup{dist}([U(t)], [U^*]) = 0.
\end{aligned}$$
We obtain from Theorem \[Theo: Inf ConvergeOp\] that there exists a sequence $\{\tau_k\}_{i = 1}^{\infty}$ so that $\lim\limits_{k\rightarrow\infty}\tau_k = +\infty$ and $\lim\limits_{k\rightarrow\infty} \big|\big|\big| \nabla_{G} E\big(U(\tau_k)\big) \big|\big|\big| = 0$. The uniqueness of critical point in $B([U^*], \delta_1)$ implies $E_0> E([U^*])$. Due to $E([U(t)]) \leqslant E_1, \forall t\geqslant 0, $ we have $$[U(\tau_k)]\in B([U^*], \delta_2)\bigcap \mathcal{L}_{E_1},$$ where $\mathcal{L}_{E_1}=\{[U]\in\mathcal{G}^N: E(U)\le E_1\}$ is the level set. Since set $$\mathcal{S}= \{\tilde U\in (V_{N_g})^N:[\tilde U]\in B([U^*], \delta)\bigcap \mathcal{L}_{E_1}\}$$ is compact, there exist a subsequence $\{U(\tau_{k_l})\}$ and $\hat U\in\mathcal{S}$ that $\lim\limits_{l\rightarrow\infty}U(\tau_{k_l}) = \hat U$. Since $\nabla E$ is continuous, then $\nabla_{G} E$ is also continuous, so $\nabla_{G} E(\hat U) = 0$. By the uniqueness of critical point in $B([U^*], \delta_1)$ again, we get $[\hat U] = [U^*]$ and $$\lim\limits_{t\rightarrow\infty} E\big(U(t)\big) = \lim\limits_{l\rightarrow\infty} E\big(U(\tau_{k_l})\big) = E(U^*).$$ We claim that $\lim\limits_{t\rightarrow\infty} \textup{dist}([U(t)], [U^*]) = 0$. Otherwise, there exists a subsequence $\{U(\tau_p)\}$ that for some fixed $\hat\delta > 0$, $\textup{dist}([U(\tau_p)], [U^*]) \geqslant \hat\delta$. Since $\mathcal{S}$ is compact, there exist a subsequence $\{U(\tau_{p_q})\}$ and $\bar U\in\mathcal{S}$ that $\lim\limits_{q\rightarrow\infty}U(\tau_{p_q}) = \bar U$. Therefore $$E(\bar U) = \lim\limits_{q\rightarrow\infty} E\big(U(\tau_{p_q})\big) = E([U^*]),$$ and $[\bar U] = [U^*]$, which contradicts the assumption $\textup{dist}([U(\tau_p)], [U^*]) \geqslant \hat\delta$.
Clearly, there exists $P(t)\in\mathcal{O}^N$ that $$|||U(t)P(t) - U^*|||= \textup{dist}([U(t)], [U^*]),$$ then $$\lim\limits_{t\rightarrow\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)\big) \Big|\Big|\Big| = \lim\limits_{t\rightarrow\infty} \Big|\Big|\Big| \nabla_{G} E\big(U(t)P(t)\big) \Big|\Big|\Big| = \Big|\Big|\Big| \nabla_{G} E\big(U^*\big) \Big|\Big|\Big| = 0.$$
Indeed, we may have some convergence rate.
If $E(U_0)\leqslant E_1$ and $$\label{Equ: Hess}
\textup{Hess}_G E(U)[D, D] \geqslant \sigma ||| D |||^2 \quad \forall [U]\in B([U^*], \delta_3),\: \forall D \in \mathcal{T}_{[U]}\mathcal{G}^N\bigcap(V_{N_g})^N$$ for some $\delta_3\in(0,\delta_1]$ and $\sigma > 0$, then there exists $\hat{T} > 0$ such that $$\begin{aligned}
&\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|\leqslant e^{-\sigma (t - \hat T)},\\
&E\big(U(t)\big) - E(U^*) \leqslant \frac{1}{2\sigma} e^{-2\sigma (t - \hat T)}
\end{aligned}$$ for all $t \geqslant {\hat T}$.
We see that $$\begin{split}
&\frac12\frac{\textup{d}}{\textup{d}t}\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2 = \textup{tr} \bigg( \Big\langle\nabla_{G} E\big(U(t)\big)^\top \frac{\textup{d}}{\textup{d}t} \nabla_{G} E\big(U(t)\big) \Big\rangle\bigg)\\
=&\textup{tr} \bigg( \Big\langle\nabla_{G} E\big(U(t)\big)^\top \nabla^2 E\big(U(t)\big) \frac{\textup{d}}{\textup{d}t} U(t) \Big\rangle \bigg) \\
&- \textup{tr} \bigg( \Big\langle \nabla_{G} E\big(U(t)\big)^\top \frac{\textup{d}}{\textup{d}t}U(t)\Big\rangle \Big\langle U(t)^\top\nabla E\big(U(t)\big) \Big\rangle\bigg)\\
&-\textup{tr} \bigg( \Big\langle \nabla_{G} E\big(U(t)\big)^\top U(t)\Big\rangle \frac{\textup{d}}{\textup{d}t}\bigg(\Big\langle U(t)^\top\nabla E\big(U(t)\big) \Big\rangle\bigg)\bigg),
\end{split}$$ which together with Lemma \[Lemma: ZeroTrace\] leads to $$\frac12\frac{\textup{d}}{\textup{d}t}\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2
= -\textup{Hess}_G E\big(U(t)\big)\big[\nabla_{G} E\big(U(t)\big), \nabla_{G} E\big(U(t)\big)\big].$$ Note that Theorem \[Theo: ContConv\] implies that there exists $\hat T > 0$ such that $$U(t)\in B([U^*], \delta_3), \quad \forall t\geqslant \hat T.$$ Hence, we obtain from (\[Equ: Hess\]) that $$\frac{\textup{d}}{\textup{d}t}\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2 \leqslant - 2\sigma \Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2,\quad t\geqslant \hat T.$$ Using Grönwall’s inequality we arrive at $$\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2 \leqslant e^{-2\sigma(t-\hat T)},\quad t\geqslant \hat T.$$ Therefore, for all $t\ge \hat T$, there hold $$\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|\leqslant e^{-\sigma (t - \hat T)}$$ and $$E\big(U(t)\big) - E(U^*) = \int_t^{+\infty}\Big|\Big|\Big|\nabla_{G} E\big(U(t)\big)\Big|\Big|\Big|^2 \textup{d}t\leqslant \frac{1}{2\sigma} e^{-2\sigma (t - \hat T)}.$$
We understand that (\[Equ: Hess\]) has been already applied in [@dai2017conjugate; @Opt7]. We observe that $\sigma$ in is related to the gap between the $(N+1)$-th eigenvalue and the $N$-th eigenvalue of the Kohn-Sham equation.
Temporal discretization {#Sec: Exact}
=======================
We may apply various temporal discretization approaches to solve . In this section, we propose and analyze a midpoint point scheme. Our analysis shows that the midpoint point scheme is quite efficient and recommended.
A midpoint scheme
-----------------
Let $\{t_n:n = 0, 1, 2 \cdots \}\subset[0, +\infty)$ be discrete points such that $$0 = t_0 < t_1 < t_2 < \cdots < t_n < \cdots,$$ and $\lim\limits_{n\rightarrow+\infty} t_n = +\infty$. Set $$\Delta t_n = t_{n + 1} - t_n,$$ and consider a midpoint scheme as follows $$\label{Equ: MidpointOp}
\frac{U_{n + 1} - U_{n}}{\Delta t_n} = -\nabla_G E(U_{n + 1\slash2}),$$ where $U_{n + 1\slash2} = (U_{n + 1} + U_{n})\slash{2}$. Equivalently $$\label{Equ: Midpoint}
\frac{U_{n + 1} - U_{n}}{\Delta t_n} = - \mathcal{A}_{U_{n + 1\slash2}}U_{n + 1\slash2}.$$ Our midpoint scheme is an implicit method and we will propose and analyze a practical scheme to solve in the next section.
First, we investigate the existence of the solution of in a neighborhood of $U^*$, which requires that $\nabla E(U)$ is Lipschitz continuous locally $$|||\nabla E(U_1) - \nabla E(U_2) ||| \leqslant L_0 ||| U_1 - U_2|||,~\forall U_1, U_2\in B(U^*, \delta_1),$$ which is true for LDA when $\rho > 0$. However, it is still open whether $\rho > 0$ [@Positivity].
\[Lemma: Implicit Function TheoryOp\] There exist such $\delta_a, \delta_b, \delta^* > 0$ and a unique function $g: B(U^*, \delta_a) \times [-\delta^*, \delta^*]\rightarrow B(U^*, \delta_b)$ which satisfies $$\label{Equ: Implicit FunctionOp}
\begin{split}
g(U, s) - U = -s\nabla_G E\Big(\frac{g(U, s) + U}{2}\Big)
\end{split}$$ for some $\delta_a, \delta_b$ and $\delta^* >0$.
We define $\mathcal{H}$ on $(V_{N_g})^N\times(V_{N_g})^N\times \mathbb{R}$ by $$\mathcal{H}(X, Y, t) := Y - X + t\nabla_G E\Big(\frac{Y + X}{2}\Big).$$ Obviously, $\mathcal{H}(U^*, U^*, 0) = 0$ and $\frac{\partial}{\partial Y}\mathcal{H}(X, Y, t)$ exists. Since $$\frac{\partial}{\partial Y}\mathcal{H}(U^*, U^*, 0) = I,$$ we see from implicit function theory that there exists a unique function $g: B(U^*, \delta_a) \times [-\delta^*, \delta^*]\rightarrow B(U^*, \delta_b)$ which satisfies $\mathcal{H}(U, g(U, s), s) = 0$ for some $\delta_a, \delta_b, \delta^* > 0$. Thus we complete the proof.
Due to Lemma \[Lemma: Implicit Function TheoryOp\], we see that $U_{n + 1} = g(U_n, \Delta t_n)$ is the solution of . Then we arrive at the following Algorithm \[Alg: Midpoint\] and refer to Theorem \[Theo: MainConvergeOp\] for the choice of $\delta_T$.
\[Alg: Midpoint\]
Given $\varepsilon > 0$, $\delta_T > 0$, initial data $U_0\in (V_{N_g})^N\bigcap\mathcal{M}^N$, calculate gradient $\nabla_{G} E(U_0)$, let $n = 0$
We will see from Proposition \[Prop: OrthoOp\] that the approximations produced by midpoint scheme are orthogonality preserving, which is significant in electronic structure calculations, for instance. The following lemmas are helpful in our analysis.
$(I + s\mathcal{A}_{U})^{-1}$ exists for all $s\in\mathbb{R}$ and $U\in(V_{N_g})^N$.
Since $\mathcal{A}_{U}$ is skew-symmetric, the corresponding eigenvalues are pure imaginary numbers. As a result, the eigenvalues of $\big(I + s\mathcal{A}_{U}\big)$ belongs to the set $$\{1 + \mu_j\imath:\mu_j\in\mathbb{R}\}$$ where $\imath$ is the imaginary unit that $\imath^2 = -1$, which implies $\big(I + s\mathcal{A}_{U}\big)$ is invertible.
\[Lemma: OrthoOp\] If $U\in(V_{N_g})^N\bigcap\mathcal{M}^N$, then $$\hat U \equiv \big(2(I + s\mathcal{A}_{\tilde U})^{-1}U - U\big)\in (V_{N_g})^N\bigcap\mathcal{M}^N$$ for all $s\in\mathbb{R}$ and $\tilde U\in(V_{N_g})^N$.
A simple calculation shows that $$\begin{split}
\hat U &= 2(I + s\mathcal{A}_{\tilde U})^{-1}U - U = (I + s\mathcal{A}_{\tilde U})^{-1}\big(2I - (I + s\mathcal{A}_{\tilde U})\big)U\\
&= (I + s\mathcal{A}_{\tilde U})^{-1}(I - s\mathcal{A}_{\tilde U})U.
\end{split}$$ We have $$\begin{split}
\langle \hat U^\top \hat U \rangle &= \langle U^\top (I + s\mathcal{A}_{\tilde U}) (I - s\mathcal{A}_{\tilde U})^{-1} (I + s\mathcal{A}_{\tilde U})^{-1}(I - s\mathcal{A}_{\tilde U})U \rangle = \langle U^\top U \rangle = I_N
\end{split}$$ and complete the proof.
\[Prop: OrthoOp\] If $U_n$ is obtained from Algorithm \[Alg: Midpoint\], then $U_n\in$$(V_{N_g})^N\bigcap\mathcal{M}^N$ for all $n\in\mathbb{N}$.
We split into two equations $$\label{Equ: Midpoint IntermediateOp}
\begin{split}
\frac{U_{n + 1\slash2} - U_{n}}{{\Delta t_n}\slash{2}} &= - \mathcal{A}_{U_{n + 1\slash2}}U_{n + 1\slash2},\\
\frac{U_{n + 1} - U_{n + 1\slash2}}{{\Delta t_n}\slash{2}} &= - \mathcal{A}_{U_{n + 1\slash2}}U_{n + 1\slash2},
\end{split}$$ and obtain $$\label{Equ: Midpoint Intermediate1Op}
\begin{split}
U_{n + 1\slash2} &= \Big(I + \frac{\Delta t_n}{2}\mathcal{A}_{U_{n + 1\slash2}}\Big)^{-1}U_n,\\
U_{n + 1} &= 2\Big(I + \frac{\Delta t_n}{2}\mathcal{A}_{U_{n + 1\slash2}}\Big)^{-1}U_n - U_n.
\end{split}$$ Therefore, we arrive at the conclusion from Lemma \[Lemma: OrthoOp\].
We see from that the midpoint scheme of gradient flow based method may be reviewed as a mixed scheme of an implicit Euler method of a temporal step ${\Delta t_n}\slash{2}$ and an explicit Euler method of the temporal step ${\Delta t_n}\slash{2}$ provided an auxiliary point. We will see an crucial difference between our midpoint scheme of gradient flow based method and the retraction optimization method afterwards.
\[Lemma: Inner PropertyOp\] If $U\in(V_{N_g})^N\bigcap\mathcal{M}^N$, then spectrum $\sigma\big(\langle {\bar U}^\top {\bar U} \rangle\big)$ of $\langle {\bar U}^\top {\bar U}\rangle$ satisfies $$\sigma\big(\langle {\bar U}^\top {\bar U}\rangle\big)\subset [0, 1],$$ where $$\bar U\equiv (I + s\mathcal{A}_{\tilde U})^{-1}U$$ for all $s\in\mathbb{R}$ and $\tilde U\in(V_{N_g})^N$.
For any eigenvalue $\lambda_j \in \sigma\big(\langle {\bar U}^\top {\bar U} \rangle\big)$, we have $$\label{Equ: Midpoint Pf1Op}
\begin{split}
&0 \leqslant \lambda_j \leqslant \Vert \langle {\bar U}^\top {\bar U} \rangle \Vert_2 = \Vert \langle U^\top \big(I - s^2(\mathcal{A}_{\tilde U})^2\big)^{-1} U\rangle \Vert_2\\
\leqslant& \big\Vert \big(I - s^2(\mathcal{A}_{\tilde U})^2\big)^{-1}\big\Vert \Vert \langle U^\top U \rangle \Vert_2 = \big\Vert \big(I - s^2(\mathcal{A}_{\tilde U})^2\big)^{-1}\big\Vert.
\end{split}$$ Note that $\mathcal{A}_{\tilde U}$ is skew-symmetric, which implies its eigenvalues are pure imaginary numbers. We obtain $$\label{Equ: Midpoint Pf2Op}
\begin{split}
&\Vert \big(I - s^2(\mathcal{A}_{\tilde U})^2\big)^{-1}\Vert\\
=& \max\big\{(1 + s^2\mu_j^2)^{-1}:\mu_j {\imath} \in \sigma\big(\mathcal{A}_{\tilde U}\big), \mu_j \in \mathbb{R}\big\} \leqslant 1,
\end{split}$$ where $\imath$ is the imaginary unit satisfying $\imath^2 = -1$. This completes the proof.
Combining and Lemma \[Lemma: Inner PropertyOp\], we arrive at
\[Prop: Inner PropertyOp\] If $U_n$ is obtained from Algorithm \[Alg: Midpoint\], then spectrum\
$\sigma\big(\langle {U_{n + 1\slash2}}^\top {U_{n + 1\slash2}} \rangle\big)$ of $\langle {U_{n + 1\slash2}}^\top {U_{n + 1\slash2}} \rangle$ satisfies $$\sigma\big(\langle {U_{n + 1\slash2}}^\top {U_{n + 1\slash2}} \rangle\big)\subset[0, 1].$$
Since $$\sigma\big(\langle {U}^\top {U}\rangle\big) = \{1\},$$ for all $U\in \mathcal{M}^N$, we see from Proposition \[Prop: Inner PropertyOp\] that for the midpoint scheme of the gradient flow based model, the auxiliary updating points are inside the Stiefel manifold. Nevertheless, we understand from Lemma 3.2 in [@dai2017conjugate] that the auxiliary points for the retraction optimization method are outside the Stiefel manifold. In fact, since $U_{n}\in\mathcal{M}^N$ and $\langle {D_{n}}^\top U_{n} \rangle= 0$, we have $$\langle{\tilde U_n}^\top {\tilde U_n}\rangle = I_N + (\Delta t_n)^2 \langle {D_{n}}^\top D_{n}\rangle,$$ for $\tilde U_{n} = U_{n} + \Delta t_n D_{n}$ and obtain [@dai2017conjugate]
\[Prop: SVDOp\] Suppose $U_{n}\in\mathcal{M}^N$, $\langle {D_{n}}^\top U_{n}\rangle = 0$, and $\tilde U_{n} = U_{n} + \Delta t_n D_{n}$ is the auxiliary point of retraction optimization method, then $$\sigma\big( \langle {\tilde U_{n}}^\top {\tilde U_n} \rangle \big) \subset \Big[1, 1 + (\Delta t_n)^2 \Vert \langle {D_{n}}^\top D_{n}\rangle \Vert_2 \Big].$$
Convergence
-----------
Now we investigate the convergence of the midpoint scheme. First we show that the energy decreases for small time step. In this section, we always assume that $\nabla E$ is local Lipschitz continuous in the neighborhood of a local minimizer $U^*\in(V_{N_g})^N\bigcap\mathcal{M}^N$ as follows $$\label{Equ: Local LipschitzOp}
\begin{split}
||| \nabla E(U_i) - \nabla E(U_j) ||| \leqslant L ||| U_i - U_j |||, \quad\forall U_i, U_j\in B\big(U^*, \max\{\delta_a, \delta_b\}\big)
\end{split}$$
\[Lemma: Energy decreaseOp\] There holds $$\label{Equ: GradientGrassLip}
\begin{split}
||| \nabla_{G}E(U_i) - \nabla_{G}E(U_j) ||| \leqslant L_1 ||| U_i - U_j |||, \quad \forall U_i, U_j\in B\big(U^*, \max\{\delta_a, \delta_b\}\big),
\end{split}$$ where $L_1 = 2\alpha\big( 2||| \nabla E(U^*) ||| + 2L \max\{\delta_a, \delta_b\} + \alpha L\big)$. Moreover, there exists a upper bound $\delta_s$ of $s$ that $$\label{Equ: Energy DifOp}
\begin{split}
E(U) - E\big(g(U, s)\big) \geqslant \frac{s}{4N}\Big|\Big|\Big| \nabla_G E\Big(\frac{g(U,s) + E(U)}{2}\Big)\Big|\Big|\Big|^2,\\ \forall U \in B(U^*, \delta_a)\bigcap\mathcal{M}^N, \forall s \in[0, \delta_s],
\end{split}$$ where $\delta_a, \delta_b$ are defined in Lemma \[Lemma: Implicit Function TheoryOp\].
First, we have that $||| \nabla E(U) |||$ is bounded over $B(U^*, \max\{\delta_a, \delta_b\})$ since $$\label{Equ: localBoundedOp}
\begin{split}
& ||| \nabla E(U_i) ||| \leqslant ||| \nabla E(U^*) ||| + ||| \nabla E(U_i) - \nabla E(U^*) ||| \\
\leqslant& ||| \nabla E(U^*) ||| + L ||| U_i - U^* ||| \leqslant ||| \nabla E(U^*) ||| + L \max\{\delta_a, \delta_b\},
\end{split}$$ which together with and leads to $$\label{Equ: Lip2Op}
\begin{split}
&||| \nabla E(U_i) \langle U_i^\top U_i \rangle - \nabla E(U_j) \langle U_j^\top U_j \rangle |||\\
\leqslant& \big|\big|\big| \nabla E(U_i) \big(\langle U_i^\top U_i \rangle - \langle U_j^\top U_j \rangle \big) \big|\big|\big|
+ ||| \big(\nabla E(U_i) -\nabla E(U_j)\big) \langle U_j^\top U_j\rangle ||| \\
\leqslant& \big( ||| \nabla E(U^*) ||| + L \max\{\delta_a, \delta_b\}\big) \big(||| U_i ||| + ||| U_j |||\big) ||| U_i - U_j |||\\
&+ L ||| U_j |||^2 ||| U_i - U_j |||\\
\leqslant& \alpha \big( 2||| \nabla E(U^*) ||| + 2L \max\{\delta_a, \delta_b\} + \alpha L\big)||| U_i - U_j |||,
\end{split}$$ and $$\label{Equ: Lip3Op}
\begin{split}
&||| U_i \langle U_i^\top \nabla E(U_i)\rangle - U_j \langle U_j^\top \nabla E(U_j)\rangle ||| \\
\leqslant& \big|\big|\big| U_i \big(\langle U_i^\top \nabla E(U_i)\rangle - \langle U_j^\top \nabla E(U_j)\rangle\big) \big|\big|\big|
+ ||| (U_i - U_j) \langle U_j^\top \nabla E(U_j)\rangle ||| \\
\leqslant& ||| U_i ||| \big(||| \nabla E(U_i) ||| + L||| U_j |||\big) ||| U_i - U_j ||| + ||| U_j|||\cdot||| \nabla E(U_j)|||\cdot ||| U_i - U_j |||\\
\leqslant& \alpha\big( 2||| \nabla E(U^*) ||| + 2L \max\{\delta_a, \delta_b\} + \alpha L\big)||| U_i - U_j |||,
\end{split}$$ where $\alpha = \max\big\{ ||| U ||| : U\in B\big(U^*, \max\{\delta_a, \delta_b\}\big)\big\}$.
Due to the triangle inequality $$\label{Equ: Lip1Op}
\begin{split}
&||| \nabla E(U_i) - \nabla E(U_j)|||\\
\leqslant& ||| \nabla E(U_i) \langle U_i^\top U_i \rangle - \nabla E(U_j) \langle U_j^\top U_j \rangle ||| + ||| U_i \langle U_i^\top \nabla E(U_i)\rangle - U_j \langle U_j^\top \nabla E(U_j)\rangle |||,
\end{split}$$ we obtain from and that $$\begin{split}
&||| \nabla E(U_i) - \nabla E(U_j)||| \leqslant L_1 ||| U_i - U_j |||.
\end{split}$$
Now we are going to prove the remainder. For given $s\in[0, \delta^*]$, Lemma \[Lemma: Implicit Function TheoryOp\] tells us that $g(U, s)$ exists uniquely. Then we define $S(t) = t g(U, s) + (1 - t) U$ for $t\in[0,1]$, and see that $E\big( S(t) \big)$ is differentiable in (0,1). We understand that there exists a $\xi\in (0,1)$ such that $$\begin{split}
&E(g(U, s)) - E(U) = E(S(1)) - E(S(0)) = \textup{tr} \big\langle \nabla E\big(S(\xi)\big)^\top \frac{\textup{d}}{\textup{d}t}S(\xi)\big\rangle\\
=& \textup{tr}\big\langle \nabla E\big(S(\xi)\big)^\top \big(g(U, s) - U\big) \big\rangle
=-s\;\textup{tr}\Big\langle\nabla E\big(S(\xi)\big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\Big\rangle.
\end{split}$$ We divide the left part into two terms and obtain $$\label{Equ: EstimateOp}
\begin{split}
E\big(g(U, s)\big) &- E(U) =-s\;\textup{tr}\left\langle\nabla E\Big(S\big(\frac12\big)\Big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\right\rangle\\
&+ s\;\textup{tr}\left\langle \Big(\nabla E\Big(S\big(\frac12\big)\Big) - \nabla E\big(S(\xi)\big)\Big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\right\rangle.
\end{split}$$
For the first term, we see that $$\begin{split}
&\textup{tr}\left\langle\nabla E\Big(S\big(\frac12\big)\Big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\right\rangle = -\frac12\textup{tr} (\mathcal{A}_{S(\frac12)})^2\\
=&\frac12\textup{tr} (\mathcal{A}_{S(\frac12)})^*(\mathcal{A}_{S(\frac12)}) =\frac12\big\Vert \mathcal{A}_{S(\frac12)} \big\Vert^2.
\end{split}$$ Due to Proposition \[Prop: Inner PropertyOp\], we have $$\begin{split}
&\Big|\Big|\Big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\Big|\Big|\Big| \leqslant \Vert \mathcal{A}_{S(\frac12)}\Vert \cdot \Big|\Big|\Big| S\big(\frac12\big) \Big|\Big|\Big| \leqslant \sqrt{N} \big\Vert \mathcal{A}_{S(\frac12)}\big\Vert,
\end{split}$$ thus $$\label{Equ: Tmp1Op}
\begin{split}
&\textup{tr}\left\langle\nabla E\Big(S\big(\frac12\big)\Big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\right\rangle \geqslant \frac{1}{2N} \Big|\Big|\Big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\Big|\Big|\Big|^2.
\end{split}$$
For the second term of the last line in , since $$||| S(t) - U^*||| = \big|\big|\big| t\big(g(U, s) - U^*\big) + (1 - t)(U - U^*) \big|\big|\big|\leqslant \max\{\delta_a, \delta_b\}, \forall t \in[0,1],$$ by local Lipschitz continuity of $\nabla E$, we get $$\label{Equ: Tmp2Op}
\begin{split}
&\textup{tr}\left\langle\Big(\nabla E\Big(S\big(\frac12\big)\Big) - \nabla E\big(S(\xi)\big)\Big)^\top \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\right\rangle\\
\leqslant& L \big|\big|\big| S\big(\frac12\big) - S(\xi) \big|\big|\big| \cdot \big|\big|\big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big) \big|\big|\big| \leqslant \frac{sL}{2} \big|\big|\big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big)\big|\big|\big|^2.
\end{split}$$
Combining with and , we have $$\begin{split}
&E(U) - E\big(g(U, s)\big) \geqslant s\big(\frac{1}{2N} - \frac{sL}{2}\big)\Big|\Big|\Big| \nabla_G E\Big(\frac{g(U,s) + E(U)}{2}\Big)\Big|\Big|\Big|^2
\end{split}$$ and reach the conclusion when $\delta_s = \min\big\{{1}\slash(2NL), \delta^*\big\}$.
We define a mapping $$\hat g: B\big([U^*], \delta_a\big)\times[0, \delta^*] \rightarrow B\big([U^*], \delta_b\big)$$ as follows $$\hat g\big([U], s\big) = \big[g\big(\arg\min\limits_{\tilde U\in [U]} ||| \tilde U - U^* |||, s\big)\big].$$ and we always assume that the local minimizer $[U^*]\in\mathcal{G}^N$ is the unique critical point of in $B\big([U^*], \delta_c\big)$ for some $\delta_c\in (0,\delta_1]$ from now on.
\[Lemma: Self-mappingOp\] There holds $$\hat g\Big(B\big([U^*], \delta_e\big)\bigcap\mathcal{L}_{E_e}\times[0, \delta_T]\Big) \subset B\big([U^*], \delta_e\big)\bigcap\mathcal{L}_{E_e}$$ for some $\delta_e > 0$, $E_e\in\mathbb{R}$, $\delta_T \in [0, \delta_s]$ where $\delta_s$ is defined in Lemma \[Lemma: Energy decreaseOp\].
We use the notation in Lemma \[Lemma: Implicit Function TheoryOp\] and Lemma \[Lemma: Energy decreaseOp\]. Set $\delta_e = \min\big\{\delta_a, \frac12\delta_c\big\}$ and $$E_{c} = \min \big\{ E(\tilde U) : [\tilde U] \in \overline{B([U^*], \delta_c)\backslash B([U^*], \delta_e)} \big\}.$$ We observe that $[\tilde U]\in B\big([U^*], \delta_e\big)$ if $E(\tilde U) \leqslant \frac{E_c + E (U^*)}{2}\equiv E_e$ and $[\tilde U]\in B\big([C^*], \delta_c\big)$.
For $[U]\in B\big([U^*], \delta_e\big)$ and $s\in[0, \delta_s]$, we observe that there exists a $\tilde U\in [U]$ such that $||| \tilde U - U^* ||| = \textup{dist}\big([U], [U^*]\big)\leqslant \delta_e$. For simplicity, we still use $U$ to denote $\tilde U$. We obtain from Lemma \[Lemma: Energy decreaseOp\] that $g(U, s) \in B(U^*, \delta_b)$ and $E\big(g(U, s)\big) \leqslant E(U) \leqslant E_e$ for any fixed $s\in[0,\delta_s]$.
Due to $$\begin{split}
&\textup{dist}\big([g(U, s)], [U]\big) \leqslant ||| g(U,s) - U ||| \leqslant s \Big|\Big|\Big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big) \Big|\Big|\Big|,
\end{split}$$ and $$\begin{split}
&\Big|\Big|\Big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big) \Big|\Big|\Big| = \Big|\Big|\Big| \mathcal{A}_{S(\frac12)}S\big(\frac12\big) - \mathcal{A}_{U^*}U^* \Big|\Big|\Big| \leqslant L_1\Big|\Big|\Big| S\big(\frac12\big) - U^* \Big|\Big|\Big|\leqslant L_1 \max\{\delta_a, \delta_b\},
\end{split}$$ we obtain $$\hat g\big([U], s\big) \in B\big([U^*], \delta_c\big), ~~\forall s \in [0,\delta_T],$$ where $$\delta_T =\left\{
\begin{array}{ll}
\min\big\{\frac{\delta_c - \delta_e}{L_1 \max\{\delta_a, \delta_b\}}, \delta_s\big\} & \delta_b > \delta_e,\\
\delta_s & \delta_b \leqslant \delta_e.
\end{array}
\right.$$ Since $E\Big(g\big([U], s\big)\Big) \leqslant E_e$, by definition of $E_e$, we have $\hat g\big([U], s\big) \in B\big([U^*], \delta_e\big)$.
Since $$g(UP, s) = g(U, s)P,~ \forall P\in\mathcal{O}^{N},$$ we may directly solve to get a representative of $\hat g(U, s)$ with respect to any representative $U$ of $[U]$.
Consequently we arrive at the convergence of the midpoint scheme of the gradient flow based model of Kohn-Sham DFT.
\[Theo: MainConvergeOp\] If $[U_{0}]\in B\big([U^*], \delta_e\big)$ and $\sup\{\Delta t_n : n \in \mathbb{N}\} \leqslant \delta_T$, then the sequence $\{U_n\}$ produced by Algorithm \[Alg: Midpoint\] satisfies $$\begin{aligned}
\lim\limits_{n\rightarrow\infty} |||\nabla_{G} E(U_{n})||| = 0,\\
\lim\limits_{n\rightarrow\infty} E(U_{n}) = E(U^*),\\
\lim\limits_{n\rightarrow\infty} \textup{dist}([U_{n}], [U^*]) = 0,\label{Equ: DistanceConv}
\end{aligned}$$ where $\delta_e$ and $\delta_T$ are defined in Lemma \[Lemma: Self-mappingOp\].
We see from Lemma \[Lemma: Energy decreaseOp\] that $E(U_{n + 1}) \leqslant E(U_{n})$. Since $B\big([U^*], \delta_e\big)\bigcap\mathcal{L}_{E_e}$ is compact, we obtain from \[Lemma: Self-mappingOp\] that $\{ E\big([U_{n}]\big)\}_{n = 0}^{\infty}$ is bounded below. So $\lim\limits_{n\rightarrow\infty} E\big([U_{n}]\big)$ exists. Note that implies $$\begin{split}
&E(U_{n}) - E(U_{n + 1}) \geqslant \frac{\Delta t_n}{4N} |||\nabla_G E(U_{n + 1\slash2}) |||^2,
\end{split}$$ we have $$\begin{split}
&\sum\limits_{n = 0}^\infty \frac{\Delta t_n}{4N}||| \nabla_G E(U_{n + 1\slash2}) |||^2 \leqslant E(U_{0}) - \lim\limits_{n\rightarrow\infty} E(U_{n}) < +\infty,
\end{split}$$ which together with $\sum\limits_{n = 0}^{\infty} \Delta t_n = +\infty$ leads to $$\inf\big\{ ||| \nabla_G E(U_{k + 1\slash2}) ||| : k\in\mathbb{N}, k\geqslant n\big\} = 0,~\forall n \in\mathbb{N}.$$ Therefore $$\begin{split}
&\liminf\limits_{n\rightarrow\infty} ||| \nabla_G E(U_{n + 1\slash2}) ||| = 0.
\end{split}$$ Consequently, there exists a subsequence $\{U_{n_{k + 1\slash2}}\}_{k = 0}^{\infty}$ such that $$\lim\limits_{k\rightarrow\infty} ||| U_{n_{k + 1}} - U_{n_{k}} ||| \leqslant \delta_T \lim\limits_{k\rightarrow\infty} ||| \nabla_G E(U_{n_{k + 1\slash2}}) ||| = 0.$$ Note that $$\mathcal{\hat S}\equiv \big\{U\in(V_{N_g})^N:[U]\in B\big([U^*], \delta_e\big)\bigcap \mathcal{L}_{E_e}\big\}$$ is compact, we have a subsequence of $\{U_{n_{k}}\}_{k = 0}^{\infty}$, for simplicity, we write as $\{U_{n_{k}}\}_{k = 0}^{\infty}$, satisfying $$\lim\limits_{k\rightarrow\infty} U_{n_k} = \bar U$$ for some $\bar U \in\mathcal{\hat S}$. Then $$\lim\limits_{k\rightarrow\infty} U_{n_{k + 1\slash 2}} = \lim\limits_{k\rightarrow\infty} U_{n_k} + \frac{U_{n_{k + 1}} - U_{n_k}}{2} = \bar U.$$ According to the Proposition \[Prop: GradientFlowOp\] and Lemma \[Lemma: Energy decreaseOp\], we have $$\nabla_{G} E(\bar U) = \mathcal{A}_{\bar U}\bar U = 0.$$ This means $\liminf\limits_{n\rightarrow\infty} ||| \nabla_{G} E(U_{n})||| = 0$.
Lemma \[Lemma: Self-mappingOp\] tells us that $[\bar U]\in B\big([U^*], \delta_e\big)\bigcap\mathcal{L}_{E_e}\subset B\big([U^*], \delta_c\big)$. Due to the uniqueness of the critical point in $B\big([C^*], \delta_c\big)$, we have $[\bar U] = [U^*]$ and $$\lim\limits_{n\rightarrow\infty}E\big(U_{n}\big) = \lim\limits_{k\rightarrow\infty} E\big(U_{n_k}\big) = E(U^*).$$
Next we show that $\lim\limits_{n\to\infty}\textup{dist}\big([U_{n}], [U^*]\big) = 0$. If it is not true, then there exists a subsequence $\{U_{n_l}\}_{l = 0}^{\infty}$ and $\check\delta > 0$ that $\textup{dist}\big([U_{n_l}], [U^*]\big)\geqslant \check\delta$. Since $\mathcal{\hat S}$ is compact, there exists a subsequence of $\{U_{n_l}\}_{l = 0}^{\infty}$, for simplicity again written as $\{U_{n_l}\}_{l = 0}^{\infty}$, which satisfies $\lim\limits_{l\rightarrow\infty} U_{n_l} = \check U$ for some $\check U\in\mathcal{\hat S}$. Thus we have $$E(\check U) = \lim\limits_{l\rightarrow\infty} E(U_{n_l}) = E(U^*).$$ Again by the uniqueness of local minimizer in $B\big([U^*], \delta_e\big)$, we obtain $[\check U] = [U^*]$, which contradicts the assumption $\textup{dist}([U_{n_l}], [U^*]) \geqslant \check\delta$.
Clearly there exists $P_n\in\mathcal{O}^N$ that $$|||U_nP_n - U^*|||= \textup{dist}([U_n], [U^*]),$$ then $$\lim\limits_{n\rightarrow\infty} \big|\big|\big| \nabla_{G} E(U_n) \big|\big|\big| = \lim\limits_{n\rightarrow\infty} \big|\big|\big| \nabla_{G} E(U_nP_n) \big|\big|\big| = \big|\big|\big| \nabla_{G} E(U^*) \big|\big|\big| = 0.$$ This completes the proof.
Theorem \[Theo: MainConvergeOp\] shows that the approximations produced by Algorithm \[Alg: Midpoint\] converge to the unique local minimizer under some mild assumptions, in which no uniform gap between the required and nonrequired eigenvalues, or namely *uniformly well posed* (UWP) property in [@UWP_Bris; @liu2014convergence; @liu2015analysis; @UWP_Yang], is needed.
Convergence rate
----------------
We are able to have some convergence rate of the approximations obtained from Algorithm \[Alg: Midpoint\].
\[Lemma: HessAppr\] For $U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$ and $\tau\in(0, \delta_T]$, set $$\begin{split}
U_+ = \Big(I + \frac{\tau}{2}\mathcal{A}_{U_+}\Big)^{-1}U,\\
U_- = \Big(I - \frac{\tau}{2}\mathcal{A}_{U_-}\Big)^{-1}U.\\
\end{split}$$ If holds true, then there exists some $\delta_{r_1} > 0$ such that $$\begin{split}
\textup{tr}\Big( \big\langle \big(U_{+} - U_{-}\big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-})\big) \big\rangle\Big) \geqslant \frac{\sigma}{2} ||| U_{+} - U_{-}|||^2
\end{split}$$ for all $\tau\in(0,\delta_{r_1}]$ and $U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$, where $\delta_{r_1}\in(0,\delta_T]$ is a positive constant, $\delta_T$ is defined in Theorem \[Theo: MainConvergeOp\] and $\delta_a$ is defined in Lemma \[Lemma: Implicit Function TheoryOp\].
Note that $$\label{Equ: Explain0}
\begin{split}
&\textup{tr}\Big( \big\langle \big(U_{+} - U_{-}\big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-})\big) \big\rangle\Big)\\
=&\frac{\tau}{2}\textup{tr}\Big( \big\langle \big(\nabla_G E(U_{+}) + \nabla_G E(U_{-}) \big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-}) \big) \big\rangle \Big).
\end{split}$$ Since $$\begin{split}
&\lim\limits_{\tau \to 0}\frac{\textup{tr}\Big( \big\langle \big( \nabla_G E(U_{+}) \big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-}) \big) \big\rangle \Big)}{\tau}\\
=& \textup{Hess}_G E(U)[\nabla E_G(U), \nabla E_G(U)] \geqslant \sigma ||| \nabla E_G(U) |||^2 = \lim\limits_{\tau \to 0} \frac{\sigma|||U_{+} - U_{-}|||^2}{\tau^2},
\end{split}$$ we have $$\label{Equ: Explain1}
\begin{split}
\lim\limits_{\tau\to 0} \tau \frac{\textup{tr} \Big( \big\langle \big( \nabla_G E(U_{+}) \big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-}) \big) \big\rangle \Big) } {|||U_{+} - U_{-}|||^2} \geqslant \sigma.
\end{split}$$ Similarly, $$\label{Equ: Explain2}
\begin{split}
\lim\limits_{\tau\to 0} \tau \frac{\textup{tr} \Big( \big\langle \big( \nabla_G E(U_{-}) \big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-}) \big) \big\rangle \Big) } {|||U_{+} - U_{-}|||^2} \geqslant \sigma.
\end{split}$$ Therefore, we see from , and that $$\begin{split}
\lim\limits_{\tau\to 0} \frac{\textup{tr}\Big( \big\langle \big(U_{+} - U_{-}\big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-})\big) \big\rangle\Big)}{|||U_{+} - U_{-}|||^2}\geqslant\sigma.
\end{split}$$ Then we know for any $U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$, there exists a $\delta_U > 0$ that $$\begin{split}
\frac{\textup{tr}\Big( \big\langle \big(U_{+} - U_{-}\big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-})\big) \big\rangle\Big)}{|||U_{+} - U_{-}|||^2} > \frac{\sigma}{2}
\end{split}$$ for all $\tau\in(0, \delta_U]$. We denote $$\begin{split}
\mathcal{C}_U := \bigg\{ V: \frac{\textup{tr}\Big( \big\langle \big(V_{+} - V_{-}\big)^\top \big(\nabla_G E(V_{+}) - \nabla_G E(V_{-})\big) \big\rangle\Big)}{|||V_{+} - V_{-}|||^2}> \frac{\sigma}{2}, \\
\forall \tau\in(0, \delta_U]\bigg\},
\end{split}$$ then we have $\mathcal{C}_U\neq\varnothing$ since $U\in\mathcal{C}_U$. Due to $$B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N\subset\bigcup\limits_{U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N} \mathcal{C}_U$$ and the compactness of $B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$, we know there exist finite sets $\mathcal{C}_{U_{(l)}}$ that $$B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N\subset\bigcup\limits_{l = 1}^\ell \mathcal{C}_{U_{(l)}}.$$ Set $$\delta_{r_1} = \min\{\delta_{U_{(1)}}, \delta_{U_{(2)}}, \ldots,\delta_{U_{(\ell)}}, \delta_T \},$$ and we complete the proof.
\[Lemma: GradientConvEst\] For $U\in B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N$ and $\tau\in(0, \delta_T]$, if we set $$\begin{split}
U_+ &= \Big(I + \frac{\tau}{2}\mathcal{A}_{U_+}\Big)^{-1}U,\\
\bar {U}_{+} &= 2U_+ - U,\\
\end{split}$$ then there exists some $\delta_{r_2} > 0$ that satisfies $$E(U) - E(\bar{U}_{+}) \leqslant \frac{\tau(L + 3)}{2} ||| \nabla_G E(U_{+})|||^2,$$ for all $\tau\in(0,\delta_{r_2}]$ and $U\in B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N$, where $\delta_{r_2}\in(0,\delta_T]$ is a positive constant, $\delta_T$ is defined in Theorem \[Theo: MainConvergeOp\] and $\delta_a$, $\delta_b$ and $L$ are defined in Lemma \[Lemma: Energy decreaseOp\].
We see from and that $$E(U) - E(\bar{U}_{+}) \leqslant \tau \textup{tr}\langle \nabla E(U_{+})^\top \nabla_G E(U_{+})\rangle + \frac{\tau L}{2} ||| \nabla_G E(U_{+})|||^2.$$ Note that $$\begin{split}
\lim\limits_{\tau \to 0} \frac{\textup{tr}\langle \nabla E(U_{+})^\top \nabla_G E(U_{+})\rangle}{|||\nabla_G E(U_{+})|||^2} =
\frac{\textup{tr}\langle \nabla E(U)^\top \nabla_G E(U)\rangle}{|||\nabla_G E(U)|||^2} = 1.
\end{split}$$ Then we see that for any $U\in B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N$, there exists a $\hat\delta_U > 0$ that $$\begin{split}
\frac{\textup{tr}\langle \nabla E(U_{+})^\top \nabla_G E(U_{+})\rangle}{|||\nabla_G E(U_{+})|||^2} < \frac{3}{2}
\end{split}$$ for all $\tau\in(0, \hat\delta_U]$. We denote $$\begin{split}
\hat{\mathcal{C}}_U := \bigg\{ V: \frac{\textup{tr}\langle \nabla E(V_{+})^\top \nabla_G E(V_{+})\rangle}{|||\nabla_G E(V_{+})|||^2} < \frac{3}{2}, \forall \tau\in(0, \hat\delta_U]\bigg\},
\end{split}$$ then we have $\hat{\mathcal{C}}_U\neq\varnothing$ since $U\in\hat{\mathcal{C}}_U$. Due to $$B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N\subset\bigcup\limits_{U\in B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N} \hat{\mathcal{C}}_U$$ and the compactness of $B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N$, we note that there exist finite sets $\hat{\mathcal{C}}_{U_{(l)}}$ that $$B(U^*, \min\{\delta_a, \delta_b\})\bigcap\mathcal{M}^N\subset\bigcup\limits_{l = 1}^{\hat\ell} \hat{\mathcal{C}}_{U_{(l)}}.$$ Set $$\delta_{r_2} = \min\{\hat\delta_{U_{(1)}}, \hat\delta_{U_{(2)}}, \ldots,\hat\delta_{U_{(\hat\ell)}}, \delta_T \},$$ and we arrive at the conclusion.
\[Theo: ConvRate\] Suppose Hessian coercivity holds true as . If $[U_{0}]\in B\big([U^*], \delta_e\big)$ and $\Delta t_n = \tau \leqslant \delta_{r_1}, \forall n\geqslant N_0$, then the sequence $\{U_n\}$ produced by Algorithm \[Alg: Midpoint\] satisfies $$\begin{split}
|||\nabla_G E(U_{n})||| \leqslant \Big(1 + \frac{L_1\tau}{2}\Big) \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{(n - N_0 + 1)\slash 2} |||\nabla_G E(U_{N_0 - 1\slash2})|||, \\
\forall n \geqslant N_0,
\end{split}$$ where $N_0$ is a positive integer, $\delta_e$ and $\delta_T$ are defined in Lemma \[Lemma: Self-mappingOp\], $L_1$ is defined in and $\delta_{r_1}$ is defined in Lemma \[Lemma: HessAppr\].
Moreover, if $\Delta t_n = \tau \leqslant \min\{\delta_{r_1}, \delta_{r_2}\}$, $\forall n\geqslant N_1$, then $$\begin{split}
&E(U_n) - E(U^*)\\
\leqslant& \frac{(L + 3)(4 + \tau^2L_1^2 + 2\sigma \tau)}{8\sigma} \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{n - N_1 + 1} |||\nabla_G E(U_{N_1 - 1\slash2})|||^2, \forall n \geqslant N_1,
\end{split}$$ where $N_1\geqslant N_0$ is a positive integer, $L$ is defined in and $\delta_{r_2}$ is defined in Lemma \[Lemma: GradientConvEst\].
Due to , there exist $N_0\in\mathbb{N}$ that $U_n\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$, $\forall n \geqslant N_0$ and $N_1\geqslant N_0$ that $U_n\in B(U^*, \min\{\delta_3, \delta_a, \delta_b\})\bigcap\mathcal{M}^N$, $\forall n \geqslant N_1$.
We observe that $$\label{Equ: Rate1}
\begin{split}
&\tau||\nabla_G E(U_{n + 1\slash2})|||^2 - \tau|||\nabla_G E(U_{n - 1\slash2})|||^2\\
=& \tau \textup{tr}\Big( \big\langle \big( \nabla_G E(U_{n + 1\slash2}) + \nabla_G E(U_{n - 1\slash2}) \big)^\top\big(\nabla_G E(U_{n + 1\slash2}) - \nabla_G E(U_{n - 1\slash2}) \big) \big\rangle \Big)\\
=& -2\textup{tr}\Big(\big\langle \big(U_{n + 1\slash2} - U_{n - 1\slash2}\big)^\top \big( \nabla_G E(U_{n + 1\slash2}) - \nabla_G E(U_{n - 1\slash2}) \big) \big\rangle \Big).
\end{split}$$ And the parallelogram identity yields $$\begin{split}
&4||| U_{n + 1\slash2} - U_{n - 1\slash2} |||^2 = \tau^2 ||| \nabla_G E(U_{n + 1\slash2}) + \nabla_G E(U_{n - 1\slash2}) |||^2\\
=& 2\tau^2||| \nabla_G E(U_{n + 1\slash2})|||^2 + 2\tau^2||| \nabla_G E(U_{n + 1\slash2})|||^2\\
& -\tau^2||| \nabla_G E(U_{n + 1\slash2}) - \nabla_G E(U_{n - 1\slash2}) |||^2,
\end{split}$$ which together with $$\begin{split}
&|||\nabla_G E(U_{n + 1\slash2}) - \nabla_G E(U_{n - 1\slash2}) |||^2 \leqslant L_1^2 |||U_{n + 1\slash2} - U_{n - 1\slash2} |||^2
\end{split}$$ leads to $$\label{Equ: Rate2}
\begin{split}
&||| U_{n + 1\slash2} - U_{n - 1\slash2} |||^2 \geqslant \frac{2\tau^2}{4 + \tau^2L_1^2} \Big( |||\nabla_G E(U_{n + 1\slash2})|||^2 + |||\nabla_G E(U_{n - 1\slash2})|||^2 \Big).
\end{split}$$
Thus we obtain from Lemma \[Lemma: HessAppr\], and that $$\begin{split}
&\tau|||\nabla_G E(U_{n + 1\slash2})|||^2 - \tau|||\nabla_G E(U_{n - 1\slash2})|||^2\\
\leqslant& - \frac{2\sigma\tau^2}{4 + \tau^2L_1^2} \Big( |||\nabla_G E(U_{n + 1\slash2})|||^2 + |||\nabla_G E(U_{n - 1\slash2})|||^2 \Big), ~\forall n \geqslant N_0.
\end{split}$$ Namely, we have $$\begin{split}
\Big(1 + \frac{2\sigma \tau}{4 + \tau^2L_1^2} \Big) |||\nabla_G E(U_{n + 1\slash2})|||^2 \leqslant& \Big( 1 - \frac{2\sigma \tau}{4 + \tau^2L_1^2} \Big) |||\nabla_G E(U_{n - 1\slash2})|||^2, \forall n \geqslant N_0,
\end{split}$$ or $$|||\nabla_G E(U_{n + 1\slash2})||| \leqslant \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{1\slash2} |||\nabla_G E(U_{n - 1\slash2})|||, \forall n \geqslant N_0.$$ Therefore, $$\label{Equ: GradientConvEst}
|||\nabla_G E(U_{n + 1\slash2})||| \leqslant \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{(n - N_0 + 1)\slash 2} |||\nabla_G E(U_{N_0 - 1\slash2})|||, \forall n \geqslant N_0$$ and $$\begin{split}
&|||\nabla_G E(U_{n})||| \leqslant |||\nabla_G E(U_{n + 1\slash2})||| + |||\nabla_G E(U_{n}) - \nabla_G E(U_{n + 1\slash2}) |||\\
\leqslant& |||\nabla_G E(U_{n + 1\slash2})||| + L_1 ||| U_{n} - U_{n + 1\slash2} |||\\
\leqslant& \Big(1 + \frac{L_1\tau}{2}\Big)|||\nabla_G E(U_{n + 1\slash2})|||\\
\leqslant& \Big(1 + \frac{L_1\tau}{2}\Big) \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{(n - N_0 + 1)\slash 2} |||\nabla_G E(U_{N_0 - 1\slash2})|||, \forall n \geqslant N_0.
\end{split}$$
Finally, we obtain from Lemma \[Lemma: GradientConvEst\] that $$\begin{split}
&E(U_n) - E(U_{n + 1}) \leqslant \frac{\tau (L + 3)}{2} ||| \nabla_G E(U_{n + 1\slash2}) |||^2 \\
\leqslant& \frac{\tau (L + 3)}{2} \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{n - N_1 + 1} |||\nabla_G E(U_{N_1 - 1\slash2})|||^2, \forall n \geqslant N_1.
\end{split}$$ Consequently, $$\begin{split}
&E(U_n) - E(U^*)\\
\leqslant& \frac{(L + 3)(4 + \tau^2L_1^2 + 2\sigma \tau)}{8\sigma} \Big(\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\Big)^{n - N_1 + 1} |||\nabla_G E(U_{N_1 - 1\slash2})|||^2, \forall n \geqslant N_1.
\end{split}$$ This completes the proof.
Note that $$\begin{split}
&\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}
= 1 - \frac{4\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\\
=&1 - \frac{4\sigma}{4\slash \tau + \tau L_1^2 + 2\sigma }
\geqslant 1 - \frac{4\sigma}{4L_1 + 2\sigma }
\end{split}$$ where the equality holds if and only if $\tau = {2}\slash{L_1}$. As a result, Algorithm \[Alg: Midpoint\] possesses the optimal convergence rate if $$\label{Equ: OptimalRate}
\tau = \left\{
\begin{array}{ll}
\min\{\delta_{r_1}, \delta_{r_2}\}, &\quad \displaystyle\frac{2}{L_1} > \min\{\delta_{r_1}, \delta_{r_2}\},\\
\displaystyle\frac{2}{L_1}, &\quad \displaystyle\frac{2}{L_1} \leqslant \min\{\delta_{r_1}, \delta_{r_2}\}.
\end{array}
\right.$$ Moreover, if $U_{k + 1\slash2} \neq U_{k - 1\slash2}$ for some $k \geqslant N_0$, then $L_1\geqslant {\sigma}\slash{2}$. Notice that $$\frac{4 + \tau^2L_1^2 - 2\sigma \tau}{4 + \tau^2L_1^2 + 2\sigma \tau}\geqslant 1 - \frac{4\sigma}{4L_1 + 2\sigma }\geqslant 0$$ where the last equality holds when $L_1 = {\sigma}\slash{2}$. Then we see that convergence rate can approach $0$ given proper assumptions in theory.
An orthogonality preserving iteration
=====================================
We understand that the convergence of SCF iteration of nonlinear eigenvalue models can neither be predicted by theory nor by numerics for those systems in large scale with small energy gap. In this section, we propose and analyze an orthogonality preserving iteration scheme based on the gradient flow based model, which is indeed a practical version of the midpoint scheme proposed in section \[Sec: Exact\]. In implementation of Algorithm \[Alg: Midpoint\], we are not able to get the exact $U_{n+1}$ of . Some approximation should be taken into account in solving , which then produces the orthogonality preserving iteration scheme that will be proved to be convergent.
An iteration
------------
With the gradient flow based approach, in this subsection, we are able to design a convergent orthogonality preserving iteration scheme for solving the Kohn-Sham equation. We recall and split midpoint scheme (\[Equ: MidpointOp\]) into two equations $$\label{Equ: MidpointSplitOp}
\begin{split}
\frac{U_{n + 1\slash2} - U_{n}}{{\Delta t_n}\slash{2}} &= - \nabla_{G} E(U_{n + 1\slash2}),\\
\frac{U_{n + 1} - U_{n + 1\slash2}}{{\Delta t_n}\slash{2}} &= - \nabla_{G} E(U_{n + 1\slash2}),
\end{split}$$ and provide partition $$0 = t_0 < t_1 < t_2 < \cdots < t_n < \cdots,$$ where $\lim\limits_{n\rightarrow+\infty} t_n = +\infty$ and $\Delta t_n = t_{n + 1} - t_n$.
We may solve the first equation of approximatively and then update the approximation using $U_{n + 1} = 2U_{n + 1\slash2} - U_n$. Consequently, we obtain Algorithm \[Alg: Self-consistent\].
\[Alg: Self-consistent\]
Given $\varepsilon > 0$, $\tilde\delta_T > 0$, initial data $U_0\in(V_{N_g})\bigcap\mathcal{M}^N$, calculate gradient $\nabla_{G} E(U_0)$, let $n = 0$
Although Algorithm \[Alg: Self-consistent\] involves time step $\Delta t_n$, we can regard the time step as a parameter and then Algorithm \[Alg: Self-consistent\] becomes a nonlinear operator iteration.
We refer to Theorem \[Theo: inexactMainConvergeOp\] for the choice of $\tilde\delta_T$ in Algorithm \[Alg: Self-consistent\]. Due to the low-rank structure in $ \Big( I + s \mathcal{A}_{U} \Big)^{-1}$, we may apply Sherman-Morrison-Woodbury formula [@dai2017conjugate; @Str_Pre2] to obtain $$\begin{split}
\Big( I + s \mathcal{A}_{U} \Big)^{-1}\tilde U =~& \tilde U + s[\nabla E(U)\quad U]\\
\cdot\Bigg(I_{2N} + s \bigg[&
\begin{array}{cc}
\langle U^\top \nabla E(U) \rangle & -\langle U^\top U\rangle\\
\langle\big(\nabla E(U)\big)^\top \nabla E(U) \rangle & -\langle U^\top \nabla E(U)\rangle
\end{array}
\bigg]\Bigg)^{-1}
\bigg[
\begin{array}{c}
\langle U^\top \tilde U\rangle\\
\big\langle \big(\nabla E(U)\big)^\top \tilde U\big\rangle
\end{array}
\bigg].
\end{split}$$ We observe that the computational complexity from $U_n$ to $U_{n + 1}$ of Algorithm \[Alg: Self-consistent\] is mainly determined by $\langle U^\top U\rangle$, $\langle U^\top \nabla E(U) \rangle$ and $\big\langle\big(\nabla E(U)\big)^\top \nabla E(U) \big\rangle$. If $\nabla E$ is a dense operator, the computational complexity of Algorithm \[Alg: Self-consistent\] is $\mathcal{O}(NN_g^2)$; otherwise, if $\nabla E$ is sparse, generated by finite element bases for example, the computational complexity can be reduced to $\mathcal{O}(N^2N_g)$.
Similar to section \[Sec: Exact\], we have
If $U_n$ is obtained from Algorithm \[Alg: Self-consistent\], then $U_n\in(V_{N_g})^N\bigcap\mathcal{M}^N$ for all $n\in\mathbb{N}$.
By the mathematical induction, we obtain that the auxiliary updating points are inside the Stiefel manifold, too.
\[Prop: inexactInnerProperty\] If $U_n$ is obtained from Algorithm \[Alg: Self-consistent\], then spectrum\
$\sigma\big(\langle {U_{n + 1\slash2}^{(k)}}^\top {U_{n + 1\slash2}^{(k)}}\rangle\big)$ of $\langle {U_{n + 1\slash2}^{(k)}}^\top {U_{n + 1\slash2}^{(k)}}\rangle$ satisfies $$\sigma\big(\langle {U_{n + 1\slash2}^{(k)}}^\top {U_{n + 1\slash2}^{(k)}}\rangle \big)\subset[0, 1],$$ for any $p_n\in\mathbb{N}_+$ and $k = 1, 2, \ldots, p_n$.
Convergence
-----------
Now we prove the convergence of the orthogonality preserving iteration scheme. First we prove a useful lemma.
\[Lemma: inexactMidAppOp\] If $U_{n + 1\slash2}^{(k)}$ is defined in Algorithm \[Alg: Self-consistent\] for any $p\in\mathbb{N}_+$ and $$\begin{split}
||\mathcal{A}_{U_i} - \mathcal{A}_{U_j} || \leqslant \hat L ||| U_i - U_j ||| \qquad \forall U_i, U_j\in B(U_{N + 1\slash2}, \delta_r),
\end{split}$$ then there exists a upper bound $\delta_z$ for $\Delta t_n$ that $$||| U_{n + 1\slash2}^{(k)} - U_{n + 1\slash2}||| \leqslant C\Delta t_n$$ and $$U_{n + 1\slash2}^{(k)} \in B(U_{n + 1\slash2}, \delta_r)$$ for all $\Delta t_n \in [0, \delta_z]$ and $k = 1, 2, \ldots, p$, where $U_{n + 1\slash2}$ is the solution of and $C$ is a constant.
We prove the lemma by mathematical induction. Set $$\delta_z =
\left\{
\begin{array}{l}
\min\Big\{\frac{2}{\hat L\sqrt{N}}, \frac{2\delta_r}{||| \nabla_G E(U_{n + 1\slash2}) |||}, \delta^*\Big\}, \quad ||| \nabla_G E(U_{n + 1\slash2}) ||| > 0,\\
\min\Big\{\frac{2}{\hat L\sqrt{N}}, \delta^*\Big\}, \quad ||| \nabla_G E(U_{n + 1\slash2}) ||| = 0.\\
\end{array}
\right.$$ Clearly, the claim holds when $k = 0$ because $$||| U_{n + 1\slash2}^{(0)} - U_{n + 1\slash2} ||| = ||| U_n - U_{n + 1\slash2}||| = \frac{||| \nabla_G E(U_{n + 1\slash2}) |||}{2} \Delta t_n.$$
Suppose the claim holds for $k - 1$. Since $U_{n + 1\slash2}$ is the solution of , we have $$\begin{split}
&U_{n + 1\slash2}^{(k)} - U_{n + 1\slash2}\\
=&\Big( I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}^{(k - 1)}} \Big)^{-1}U_{n} - \Big( I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}} \Big)^{-1}U_{n}\\
=&\frac{\Delta t_n}{2}\Big( I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}^{(k - 1)}} \Big)^{-1} \Big(\mathcal{A}_{U_{n + 1\slash2}} - \mathcal{A}_{U_{n + 1\slash2}^{(k - 1)}}\Big) \Big( I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}} \Big)^{-1}U_{n}.
\end{split}$$ Note that $\mathcal{A}_{U}$ is skew-symmetric. We obtain $$\label{Equ: inexactMidpoint Pf2Op}
\begin{split}
&\bigg\Vert \Big(I + \frac{\Delta t_n}{2}\mathcal{A}_{U} \Big)^{-1}\bigg\Vert = \bigg\Vert \Big(I - \frac{1}{4}(\Delta t_n \mathcal{A}_{U})^2\Big)^{-1}\bigg\Vert^{\frac12} \leqslant 1, \quad\forall U\in (V_{N_g})^N
\end{split}$$ and hence $$\begin{split}
&||| U_{n + 1\slash2}^{(k)} - U_{n + 1\slash2} |||\\
\leqslant& \frac{\Delta t_n}{2} \bigg\Vert\Big(I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}^{(k - 1)}} \Big)^{-1}\bigg\Vert \bigg\Vert\mathcal{A}_{U_{n + 1\slash2}} - \mathcal{A}_{U_{n + 1\slash2}^{(k - 1)}}\bigg\Vert \bigg\Vert\Big(I + \frac{\Delta t_n}{2} \mathcal{A}_{U_{n + 1\slash2}} \Big)^{-1}\bigg\Vert ||| U_n |||\\
\leqslant&\frac{\Delta t_n\hat L\sqrt{N}}{2} ||| U_{n + 1\slash2}^{(k - 1)} - U_{n + 1\slash2} ||| \leqslant \frac{\hat L\sqrt{N}\delta_r}{2} \Delta t_n.
\end{split}$$ By $\Delta t_n \in [0, \delta_z]$, we see $U_{n + 1\slash2}^{(k)} \in B(U_{n + 1\slash2}, \delta_r)$ and the claim also holds for $k$. Thus we confirm that $U_{n + 1\slash2}^{(k)} \in B(U_{n + 1\slash2}, \delta_r)$ and $$||| U_{n + 1\slash2}^{(k)} - U_{n + 1\slash2} ||| \leqslant C\Delta t_n$$ for all $k = 0, 1, \ldots, p$, where $C$ is some constant.
Similar to the midpoint scheme in Lemma \[Lemma: Energy decreaseOp\], we prove the local energy descending property for Algorithm \[Alg: Self-consistent\]. We introduce a mapping $h_p$ from $(U, s)\in(V_{N_g})^N\times\mathbb{R}$ to $h_p(U, s)\in(V_{N_g})^N$ as follows: $$h_p(U, s) = 2\bar U^{(p)} - U,$$ where $\bar U^{(p)}$ is recursively defined by $$\begin{split}
\bar U^{(k)} &= \Big( I + \frac{s}{2} \mathcal{A}_{\bar U^{(k - 1)}} \Big)^{-1} U,~k = p, p - 1, \ldots, 1,\\
\bar U^{(0)} &= U.
\end{split}$$ In this section, we always assume that $\nabla E$ is local Lipschitz continuous in the neighborhood of a local minimizer $U^*\in(V_{N_g})\bigcap\mathcal{M}^N$: $$\label{Equ: inexactLocalLipschitzOp}
\begin{split}
||| \nabla E(U_i) - \nabla E(U_j))||| \leqslant \tilde L ||| U_i - U_j |||, \forall U_i, U_j\in B(U^*, \tilde\delta_L),
\end{split}$$ where $\tilde\delta_L > \max\{\delta_a, \delta_b\}$.
\[Lemma: inexactEnergyDecreaseOp\] There holds $$\label{Equ: Local Lipschitz2Op}
\begin{split}
||| \nabla_G E(U_i) - \nabla_G E(U_j)||| \leqslant \tilde L_1 ||| U_i - U_j|||, \quad\Vert \mathcal{A}_{U_i} - \mathcal{A}_{U_j}\Vert \leqslant \tilde L_1 ||| U_i - U_j|||, \\
\forall U_i, U_j\in B(U^*, \tilde\delta_L).
\end{split}$$ Moreover, there exists a upper bound $\tilde\delta_s$ for $s$ that $$\begin{split}
E(U) - E\big(h_p(U, s)\big) \geqslant \frac{s}{4N}\Big|\Big|\Big| \nabla_G E\Big(\frac{h_p(U,s) + E(U)}{2}\Big)\Big|\Big|\Big|^2,\\
\forall U \in B(U^*, \delta_a)\bigcap\mathcal{M}^N, \forall s \in[0, \tilde\delta_s].
\end{split}$$ Meanwhile $h_p(U, s)\in B(U^*, \tilde\delta_L)$.
We only need to prove that $h_p(U, s)\in B(U^*, \tilde\delta_L)$. Set $$S(t) = tg(U, s) + (1 - t)U,~t\in[0, 1].$$ We obtain from Lemma \[Lemma: Energy decreaseOp\] that for $U\in B(U^*, \delta_a)$ and $s\in[0,\delta_s]$, there holds $$\begin{split}
\Big|\Big|\Big| S\big(\frac12\big) - U^* \Big|\Big|\Big| \leqslant& \frac12 ||| g(U, s) - U^*||| + \frac12||| U - U^*||| \leqslant \frac12\delta_b + \frac12\delta_a,
\end{split}$$ which implies $$S\big(\frac12\big)\in B(U^*, \frac12\delta_b + \frac12\delta_a)\subset B(U^*, \tilde\delta_L).$$ Note that Lemma \[Lemma: inexactMidAppOp\] implies $$\bar U^{(p)}(s)\in B\Big(S\big(\frac12\big), \frac12(\tilde\delta_L - \delta_b) \Big) \subset B(U^*, \tilde\delta_L)$$ provided $$s\in
\left\{
\begin{array}{l}
\bigg[0, \min\Big\{\frac{2}{\hat L_1\sqrt{N}}, \frac{\tilde\delta_L - \delta_b}{||| \nabla_G E(S(\frac12)) |||}, \delta^*\Big\}\bigg], \quad \big|\big|\big| \nabla_G E\big(S\big(\frac12\big)\big) \big|\big|\big| > 0,\\[0.3cm]
\bigg[0, \min\Big\{\frac{2}{\tilde L_1\sqrt{N}}, \delta^*\Big\}\bigg], \quad \big|\big|\big| \nabla_G E\big(S\big(\frac12\big)\big) \big|\big|\big| = 0.
\end{array}
\right.$$ While there holds $$\big|\big|\big| \nabla_G E\big(S\big(\frac12\big)\big) \big|\big|\big| = \big|\big|\big| \nabla_G E\big(S\big(\frac12\big)\big) - \nabla_G E\big(U^*\big)\big|\big|\big| \leqslant \tilde L_1||| S\big(\frac12\big) - U^* ||| \leqslant \frac12\tilde L_1(\delta_a + \delta_b),$$ we have $$\bar U^{(p)}(s)\in B\Big(S\big(\frac12\big), \frac12(\tilde\delta_L - \delta_b) \Big)$$ as long as $$s\in \bigg[0, \min\Big\{\frac{2}{\tilde L_1\sqrt{N}}, \frac{2(\tilde\delta_L - \delta_b)}{\tilde L_1(\delta_a + \delta_b)}, \delta^*\Big\}\bigg].$$ Therefore, we get $$\begin{split}
&||| h_p(U, s) - U^*||| \leqslant ||| h_p(U, s) - g(U, s)||| + ||| g(U, s) - U^*|||\\
\leqslant& 2\big|\big|\big| \bar U^{(p)} - S\big(\frac12\big)\big|\big|\big| + \delta_b \leqslant \tilde\delta_L.
\end{split}$$
Similarly, we have $$\begin{split}
&E(U) - E(h_p(U, s)) \geqslant s\Big(\frac{1}{2N} - \frac{s\tilde L}{2}\Big)\Big|\Big|\Big| \nabla_G E\Big(\frac{h_p(U,s) + E(U)}{2}\Big)\Big|\Big|\Big|^2.
\end{split}$$ All the above results hold when $s\in[0,\tilde\delta_s]$, where $$\tilde\delta_s = \min\Big\{\frac{2}{\tilde L_1\sqrt{N}}, \frac{2(\tilde\delta_L - \delta_b)}{\tilde L_1(\delta_a + \delta_b)}, \frac1{2N\tilde L},\delta^*\Big\}.$$
We can define a mapping $$\hat h_p: B\big([U^*], \delta_a\big)\times[0, \tilde\delta_s] \rightarrow B\big([U^*], \tilde\delta_L\big)$$ such that $$\hat h_p\big([U], s\big) = \Big[h_p\big(\arg\min\limits_{\tilde U\in [U]} ||| \tilde U - U^* |||, s\big)\Big].$$
\[Lemma: inexactSelf-mappingOp\] There holds $$\hat h_p\Big(B\big([U^*], \tilde\delta_e\big)\bigcap\mathcal{L}_{\tilde E_e}\times[0, \tilde\delta_T] \Big)\subset B([U^*], \tilde\delta_e)\bigcap\mathcal{L}_{\tilde E_e}$$ for some $\tilde\delta_e > 0$, $\tilde E_e\in\mathbb{R}$, $\tilde\delta_T \in [0, \tilde\delta_s]$ where $\tilde\delta_s$ is defined in Lemma \[Lemma: inexactEnergyDecreaseOp\].
Then comparing with the midpoint scheme case in Theorem \[Theo: MainConvergeOp\], we arrive at the following convergence result. Since the proof is similar, we omit the details.
\[Theo: inexactMainConvergeOp\] If $[U_{0}]\in B\big([U^*], \tilde\delta_e\big)$ and $\sup\{\Delta t_n : n \in \mathbb{N}\} \leqslant \tilde\delta_T$, then for any $p_n\in\mathbb{N}_+$, the sequence $\{U_n\}$ produced by Algorithm \[Alg: Self-consistent\] satisfies $$\begin{aligned}
\lim\limits_{n\rightarrow\infty} ||| \nabla_{G} E(U_{n}) ||| = 0,\\
\lim\limits_{n\rightarrow\infty} E(U_{n}) = E(U^*),\\
\lim\limits_{n\rightarrow\infty} \textup{dist}\big([U_{n}], [U^*]\big) = 0,
\end{aligned}$$ where $\tilde\delta_e$, $\tilde\delta_T$ are defined in Lemma \[Lemma: inexactSelf-mappingOp\].
Finally, we turn to the convergence rate of the approximations produced by Algorithm \[Alg: Self-consistent\].
\[Lemma: inexactHessAppr\] For $U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$ and $\tau\in(0, \tilde\delta_T]$, set $$\begin{split}
U_+ = \Big(I + \frac{\tau}{2}\mathcal{A}_{U_+}\Big)^{-1}U,\\
U_- = \Big(I - \frac{\tau}{2}\mathcal{A}_{U_-}\Big)^{-1}U.\\
\end{split}$$ If holds true, then there exists some $\tilde\delta_{r_1} > 0$ such that $$\begin{split}
\textup{tr}\Big( \big\langle \big(U_{+} - U_{-}\big)^\top \big(\nabla_G E(U_{+}) - \nabla_G E(U_{-})\big) \big\rangle\Big) \geqslant \frac{\sigma}{2} ||| U_{+} - U_{-}|||^2
\end{split}$$ for all $\tau\in(0,\tilde\delta_{r_1}]$ and $U\in B(U^*, \min\{\delta_3, \delta_a\})\bigcap\mathcal{M}^N$, where $\tilde\delta_{r_1}\in(0,\tilde\delta_T]$ is a positive constant, $\tilde\delta_T$ is defined in Theorem \[Theo: inexactMainConvergeOp\] and $\delta_a$ is defined in Lemma \[Lemma: Implicit Function TheoryOp\].
\[Lemma: inexactGradientConvEst\] For $U\in B(U^*, \tilde\delta_L)\bigcap\mathcal{M}^N$ and $\tau\in(0, \tilde\delta_T]$, if $$\begin{split}
U_+ &= \Big(I + \frac{\tau}{2}\mathcal{A}_{U_+}\Big)^{-1}U,\\
\bar{U}_{+} &= 2U_+ - U,\\
\end{split}$$ then there exists some $\tilde\delta_{r_2} > 0$ that satisfies $$E(U) - E(\bar{U}_{+}) \leqslant \frac{\tau(\tilde L + 3)}{2} ||| \nabla_G E(U_{+})|||^2,$$ for all $\tau\in(0,\tilde\delta_{r_2}]$ and $U\in B(U^*, \tilde\delta_L\})\bigcap\mathcal{M}^N$, where $\tilde\delta_{r_2}\in(0,\tilde\delta_T]$ is a positive constant, $\tilde\delta_T$ is defined in Theorem \[Theo: inexactMainConvergeOp\] and $\tilde\delta_L$ and $\tilde L$ are defined in Lemma \[Lemma: inexactEnergyDecreaseOp\].
\[Theo: inexactConvRate\] Suppose Hessian coercivity holds true as . If $[U_{0}]\in B\big([U^*], \tilde\delta_e\big)$ and $\Delta t_n = \tau \leqslant \tilde\delta_{r_1}, \forall n\geqslant \tilde N_0$, then the sequence $\{U_n\}$ produced by Algorithm \[Alg: Self-consistent\] satisfies $$\begin{split}
|||\nabla_G E(U_{n})||| \leqslant \Big(1 + \frac{\tilde L_1\tau}{2}\Big)\Big(\frac{4 + \tau^2\tilde L_1^2 - 2\sigma \tau}{4 + \tau^2\tilde L_1^2 + 2\sigma \tau}\Big)^{(n - \tilde N_0 + 1)\slash 2} |||\nabla_G E(U_{\tilde N_0 - 1\slash2})||| , \\
\forall n \geqslant \tilde N_0,
\end{split}$$ where $\tilde N_0$ is a positive integer, $\tilde \delta_e$ and $\tilde \delta_T$ are defined in Lemma \[Lemma: inexactSelf-mappingOp\], $\tilde L_1$ is defined in and $\tilde \delta_{r_1}$ is defined in Lemma \[Lemma: inexactHessAppr\].
Moreover, if $\Delta t_n = \tau \leqslant \min\{\tilde \delta_{r_1}, \tilde \delta_{r_2}\}$, $\forall n\geqslant \tilde N_1$, then $$\begin{split}
&E(U_n) - E(U^*)\\
\leqslant& \frac{(\tilde L + 3)(4 + \tau^2\tilde L_1^2 + 2\sigma \tau)}{8\sigma} |||\nabla_G E(U_{N_1 - 1\slash2})|||^2 \Big(\frac{4 + \tau^2\tilde L_1^2 - 2\sigma \tau}{4 + \tau^2\tilde L_1^2 + 2\sigma \tau}\Big)^{n - \tilde N_1 + 1}, \forall n \geqslant \tilde N_1,
\end{split}$$ where $\tilde N_1\geqslant \tilde N_0$ is a positive integer, $\tilde L$ is defined in and $\tilde \delta_{r_2}$ is defined in Lemma \[Lemma: inexactGradientConvEst\].
Similarly, Algorithm \[Alg: Self-consistent\] reaches the optimal convergence rate when $$\tau = \left\{
\begin{array}{ll}
\min\{\delta_{r_1}, \delta_{r_2}\}, &\quad \displaystyle\frac{2}{\tilde L_1} > \min\{\delta_{r_1}, \delta_{r_2}\},\\
\displaystyle\frac{2}{\tilde L_1}, &\quad \displaystyle\frac{2}{\tilde L_1} \leqslant \min\{\delta_{r_1}, \delta_{r_2}\}.
\end{array}
\right.$$ Furthermore, if $U_{k + 1\slash2} \neq U_{k - 1\slash2}$ for some $k \geqslant \tilde N_0$, then we have $\tilde L_1\geqslant{\sigma}\slash{2}$ and convergence rate of the approximations produced by Algorithm \[Alg: Self-consistent\] can approach $0$ given proper assumptions in theory.
Compared with Algorithm \[Alg: Midpoint\], Algorithm \[Alg: Self-consistent\] is computable. In particular, Algorithm \[Alg: Self-consistent\] does not require a large band gap and Theorem \[Theo: inexactConvRate\] tells the convergence rate of the orthogonality preserving iterations.
Numerical experiments
=====================
Our code of the orthogonality preserving iterations of the gradient flow based model is developed based on by PHG toolbox[@PHG]. We adopt quadratic finite elements in the spacial discretization. For the exchange-correlation potential, we choose the local density approximation(LDA) in [@LDA]: $$v_{xc}(\rho) = \varepsilon_{xc}(\rho) + \rho\frac{\delta\varepsilon_{xc}(\rho)}{\delta\rho},$$ where $\varepsilon_{xc}(\rho) = \varepsilon_x(\rho) + \varepsilon_c(\rho)$ with $$\varepsilon_x(\rho) = -\frac34 (\frac{3}{\pi})^{1\slash3}\rho^{1\slash3}$$ and $$\varepsilon_c(\rho) =
\left\{
\begin{array}{l}
-0.1423/(1 + 1.0529\sqrt{r_s} + 0.3334r_s) \:\textup{if}\: r_s \geqslant 1,\\
0.0311\ln r_s -0.048 + 0.0020 r_s \ln r_s -0.0116 r_s \:\textup{if}\: r_s < 1,
\end{array}
\right.$$ here $r_s = \big({3}\slash(4\pi\rho)\big)^{1\slash 3}$. We see from Theorem \[Theo: inexactMainConvergeOp\] that the approximations produced by Algorithm \[Alg: Self-consistent\] is convergent given proper $\tilde\delta_t$ and $\tilde\delta_T$. In implementation of Algorithm \[Alg: Self-consistent\], we apply some self-adapted time step sizes and some acceleration techniques.
We give four examples whose molecular structures can be found in Figure \[Fig: Structure\].
*Example* 1. Consider the gradient flow model for lithium hydride(LiH) with orbits number $N = 2$ on a fixed tetrahedral finite element mesh over $[-56, 55]\times [-54, 53]\times[-54, 53] \subset\mathbb{R}^3$ from an adaptive refinement finite element method[@AFEM] with degrees of freedom $N_g = 10971$(see Figure \[Fig: LiHPlus\]). We see from Figure \[Fig: LiHPlus\] that the approximations of electron density between the two nuclei converge. Figure \[Fig: LiH\] shows the energy and the gradient convergence curve. We see that the energy approximations converge monotonically and the approximations of the gradient oscillate to zero.
Moreover, the approximated energy of the ground state of LiH we obtain is\
$-7.990787295248$ a.u., which closes to the experimental value $-8.0705$ a.u. in [@LiH1] and also consistent with the numerical result $-8.044572$ a.u. [@dai2008three] and other numerical results in [@LiH1; @LiH2]. The minor ground state energy difference results from spacial discretization, boundary condition approximation and precision of LDA model of exchange-correlation term.
*Example* 2. For methane(CH~4~) whose orbits number $N = 5$, we compute the gradient flow model on a fixed tetrahedral finite element mesh on $[-56, 55]\times [-54, 53]\times[-54, 53]\subset\mathbb{R}^3$ from an adaptive refinement finite element method[@AFEM] with degrees of freedom $N_g = 17267$(see Figure \[Fig: CH4Plus\]). We see from Figure \[Fig: CH4Plus\] that the approximations of electron density converge to a regular tetrahedron shape. We learn form Figure \[Fig: CH4\] that both the approximations of energy and the gradient converge well.
*Example* 3. We choose a fixed tetrahedral finite element mesh on $[-56, 55]\times [-54, 53]\times[-54, 53]\subset\mathbb{R}^3$ from an adaptive refinement finite element method[@AFEM] with degrees of freedom $N_g = 16531$ and apply the gradient flow based model to compute the ground state of ethyne(C~2~H~2~) with orbits number $N = 7$(see Figure \[Fig: C2H2Plus\]). We observe from Figure \[Fig: C2H2Plus\] that the approximations of electron density converge. And similar to the examples above, the convergence curve of the approximated energy and the approximated gradient behaves as expected in Figure \[Fig: C2H2\].
*Example* 4. We apply the gradient flow based model to compute the ground state of benzene(C~6~H~6~) with orbits number $N = 21$ on a fixed tetrahedral finite element mesh on $[-56, 55]\times [-54, 53]\times[-54, 53]\subset\mathbb{R}^3$ generated by an adaptive refinement finite element method[@AFEM] with degrees of freedom $N_g = 20541$(see Figure \[Fig: C6H6Plus\]). We see from Figure \[Fig: C6H6Plus\] that the approximations of electron density are convergent. We understand from Figure \[Fig: C6H6\] that the approximations of energy converges monotonically and the lower limit of the norm of the gradient approximations converge to zero.
*Examples* 1-4 indicate that our orthogonality preserving iterations of the gradient flow based model (Algorithm \[Alg: Self-consistent\]) work well in ground state calculations.
Concluding remarks
==================
In this paper, we have proposed and analyzed a gradient flow based model of Kohn-Sham DFT, which is an alternative way to solve Kohn-Sham DFT apart from the existing eigenvalue model with SCF iterations and the energy minimization model with optimization approaches. First we have established a continuous dynamical system based on the extended gradient flow and proven that the solution remains on the Stiefel manifold, and then we have proven the local convergence of the dynamical system. Apart from that, local convergence rate can be further estimated if the Hessian is coercive locally. Second, we have come up with a midpoint scheme to discretize the dynamical system in the temporal direction and proven that it preserves orthogonality. We should mention that the auxiliary updating points of the midpoint scheme distribute inside the Stiefel manifold while those of retraction optimization methods distribute outside the Stiefel manifold. Compared with manifold path optimization methods diminishing energy locally [@Str_Pre2], our midpoint scheme is a global approximation of the gradient on the step size interval. We also have proven the local convergence and estimated the convergence rate of the midpoint scheme under mild assumptions. In particular, based on the midpoint scheme, we have then proposed and analyzed an orthogonality preserving iteration scheme for the Kohn-Sham model and proven that the scheme is convergent under mild assumptions and the corresponding convergence rate can be estimated. Without annoying orthogonality preserving strategy and backtracking in optimization model and divergence of small gap systems in SCF iterations of nonlinear eigenvalue model, the gradient flow based model of Kohn-Sham DFT is promising. It is worthwhile to look into the relationship between our orthogonality preserving scheme from the gradient flow based model and the conventional self-consistent field iteration from the nonlinear eigenvalue model. Moreover, our gradient flow based model can be extended to other models in electronic structure calculations such as Hartree-Fock type models. In this paper, we have mainly discussed the midpoint scheme to discretize the gradient flow based model. We may study other orthogonality preserving discretizations in temporal, such as the leapfrog scheme. Finally, we should mention that it is very useful if the convergence of the approximations of the gradient flow based model can be speed up, which is indeed our on-going work.
[^1]: LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China ([email protected], [email protected], [email protected]).
[^2]: This work was partially supported by the National Natural Science Foundation of China under grant 91730302 and 11671389 and the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences under grant QYZDJ-SSW-SYS010.
|
---
abstract: 'The aim of this study is to investigate bursts-related EEG signals in a focal hand dystonia patient. Despite of considering time domain and frequency domain techniques as mutually exclusive analysis, in this contribution we have taken advantage from both of them: particularly, in the frequency domain, coherence was used to identify the most likely frequency bands of interaction between brain and muscles; then, in the time domain, cross-correlation was exploited to verify the physiological reliability of such a relationship in terms of signal transmission delay from the centre to the periphery. Our preliminary results suggest - in line with recent literature - that activity in the high $\beta$ band (around 30 Hz) could represent an electroencephalographic correlate for the pathological electromyographic bursts affecting the focal hand dystonia condition. Even though a future study on a larger sample is needed to statistically support these preliminary findings, this contribution allows to think of new kinds of rehabilitation from focal hand dystonia that could target the actual electroencephalographic correlate of the pathology, i.e. phenotypically expressed by bursts, with the consequence of a relevant functional improvement.'
author:
- |
Giulia Cisotto, Umberto Michieli, Leonardo Badia\
Dept. of Information Engineering, University of Padova\
via Gradenigo 6B, 35131 Padova, Italy\
email: [ {giulia.cisotto, leonardo.badia}@dei.unipd.it ]{}
bibliography:
- 'CMC.bib'
title: A coherence study on EEG and EMG signals
---
Introduction
============
Materials and Methods
=====================
One FHD patient and one healthy subject (HS) were involved in this pilot study. The EEG was recorded from one monopolar EEG channel placed on C3, the standard location of the International 10-20 System over the left hemisphere corresponding to the brain region related to the functioning of the (dominant) right-hand. The EMG was recorded from one bipolar channel placed on the *abductor pollicis brevis*, the intrinsic hand muscle responsible for the abduction of the thumb. Both signals were sampled with a sampling frequency of 1 kHz and quantized at 16 bit. In the experiment, the participants were sitting quietly on a comfortable chair in front of a screen placed 1 meter apart from them, on a table. They were simply required to rest with opened eyes for about 3 minutes with their limbs laying on the table in front of them. At a first glance, it was possible to assess a clear difference between the two EMG signals: in the FHD patient, the amplitude of the signal assumed values up to $\pm$ 200 V, while its power spectral density (PSD) took significant values in the frequency band $[5$, $200]$ Hz. However, in case of HS, the amplitude of the EMG signal did not exceed $\pm$ 20 V, with a significant PSD extended from $5$ to $50$ Hz.
In the offline analysis, signals were preprocessed to limit their frequency range in the frequency band of interest. Specifically, the EEG was filtered through an elliptic filter of order $24$ with a passband of $[4,45]$ Hz. The EMG was processed by a high-pass elliptic filter of order $11$ with cut-off frequency at $5$ Hz. A series of notch filters of order $14$ were used with cut-off frequencies at $50$ Hz and subsequent harmonics up to $350$ Hz were put to reduce the effect of the mains.
Then, CMC as well as cross-correlation have been computed between the EEG (otherwise labelled as signal $x[m]$) and the EMG (otherwise labelled as signal $y[m]$) signals, in order to assess the quantitative relationship between them, both in the frequency domain and in the time domain.
Frequency domain analysis: cortico-muscular coherence
-----------------------------------------------------
The coherence of two discrete-time signals $x[m]$ and $y[m]$, regarded as stochastic processes, is given by: $${\rm Coh}_{xy}(f) \triangleq \frac{\mathcal{P}_{xy}(f)}{\sqrt{\lvert \mathcal{P}_x(f) \rvert} \cdot \sqrt{ \lvert \mathcal{P}_y(f) \rvert}},$$ where $\mathcal{P}_{x}(f)$ is the PSD of $x[m]$ and $\displaystyle \mathcal{P}_{xy}(f)=\frac{1}{n} \sum_{i=1}^{n} X_i(f)Y_i^*(f)$ is the cross-power spectral density (CPSD) between $x[m]$ and $y[m]$.
In order to provide a statistically significant result, a confidence level $CL$ of $95$ $\%$, i.e. a critical level of $\alpha=0.05$, was obtained from the following formula [@Mima1999]: $$CL = 1-(1-\alpha)^{\frac{1}{N-1}},$$ where $N$ is the number of signal segments used to estimate the coherence value.
The PSD, the CPSD and the coherence values were estimated via the Fast Fourier Transform (FFT)-based Welch’s method: specifically, the signal length was set to $L = 1024$ samples ($1.024$ s) and the number of FFT points to $1024$ samples. Border effects were mitigated by a Hann sliding windowing with overlap of $50$ $\%$ [@Mitra2007].
Time domain analysis: cross-correlation function
------------------------------------------------
Generally speaking, given two discrete-time signals $x[m]$ and $y[m]$, their cross-correlation function is defined as: $${\tt r}_{xy}[n] \triangleq \sum_{m=-\infty}^{+\infty} x^*[m] y[n+m].$$ Cross-correlation is particularly useful to evaluate the similarity between two signals as a function of the time shift $n$ (expressed in number of samples) of the second signal behind the first one.
In the present analysis, the absolute value of the correlation between the EEG and the EMG signals computed at its maximum and normalized by the square root of the product of the signals energies $E_x$ and $E_y$ was evaluated. Therefore, the quantity: $$\displaystyle {\tt r}_{max}=\frac{\max (r_{xy}[n])}{\sqrt{E_x E_y}}.$$ was considered as a measure of similarity between the two signals.
Moreover, the lag $n$ which the maximum was found at was taken into account as a measure of the transmission delay from the brain to the muscle, i.e. the time taken for a motor command to travel from its origin in the central nervous system to the target effector at the periphery.
Particularly, $71$ pairs of EEG and EMG signals were extracted from the whole EEG and EMG recordings of the FHD patient. They were selected empirically as examples of bursty EMG activity (with their corresponding EEG). The duration of these signals was variable ($0.70 \pm 0.66$ s): all of them were included in the analysis to keep into account the variability of the burst events.
![Absolute value of CMC for the healthy participant ($CL=0.067$ with $N=44$ and $\alpha=0.05$).[]{data-label="fig:coerenza_sano_totale"}](coerenza_sano.pdf){width="50.00000%"}
![Absolute value of CMC for the pathological subject ($CL=0.008$ with $N=393$ and $\alpha=0.05$).[]{data-label="fig:coerenza_patologico_totale"}](coerenza_patologico.pdf){width="50.00000%"}
To support the physiological meaning of the EEG-EMG coherent components, the cross-correlation function was computed between the narrow-band EEG signals filtered in the high $\beta$ band, i.e. between $26$ and $31$ Hz, and the EMG signal limited to $250$ Hz by a band-pass filter with frequency band $[5, 250]$ Hz.
Results
=======
EEG-EMG coherence
-----------------
In this section the results are shown in regards to the CMC measure for both the HS and the FHD patient. In the case of the healthy participant, the CMC spectrum could be seen in \[fig:coerenza\_sano\_totale\]. It has to be noted that peaks above the confidence level can be observed in the frequency range between $5$ and $20$ Hz, only, with a particularly strong coherence at $20$ Hz.
On the other hand, the CMC spectrum of the FHD patient is reported in \[fig:coerenza\_patologico\_totale\]. It can be observed that a larger frequency band contribute to the coherence between EEG and EMG signals. It is also important to highlight the presence of peaks in the upper side of the spectrum, i.e. $[20, 45]$ Hz. This is probably due to the larger bandwidth of the pathological EMG of the patient, as mentioned in the previous section.
In order to confirm our hypothesis, we selected a portion of the whole recorded data where bursts mostly affected the EMG signal and evaluated the CMC in this particular case. As a further support, we selected another portion of EMG signal where healthy-like activity could be observed and computed CMC as well. Two typical examples of both situations are reported next. \[fig:125\_coerenza\] shows the coherence result when comparing two chunks of the EEG and EMG signals for the healthy-like case. Here, two main peaks can be seen at the frequencies of $8$ Hz and $18$ Hz, but no significant coherence values at frequencies higher than $30$ Hz.
![CMC, in absolute value, between chunks of healthy-like EEG and EMG for the pathological subject ($CL=0.776$ with $N=3$ and $\alpha = 0.05$).[]{data-label="fig:125_coerenza"}](1_2-5_coerenza.pdf){width="50.00000%"}
On the contrary, \[fig:117120\_coerenza\] reports the coherence spectrum in case of bursts-affected chunks. Significantly, the figure shows that coherence values at low frequencies are heavily reduced, whereas some peaks around $20$ and $35$ Hz appeared, hence the hypothesis that higher frequencies components are related to bursty EMG activity in the FHD patient could actually be supported.
![CMC, in absolute value, between chunks of bursts-affected EEG and EMG for the pathological subject ($CL=0.451$ with $N=6$ and $\alpha = 0.05$).[]{data-label="fig:117120_coerenza"}](117_120_coerenza.pdf){width="50.00000%"}
EEG-EMG cross-correlation
-------------------------
As mentioned above, cross-correlation of EEG and EMG was computed to investigate the physiological reliability on the relationship between the high $\beta$ band EEG component with the EMG.
\[fig:isto\_maxnorm\] reports the empirical distribution of the maximum values of the cross-correlation function found from the $71$ pairs of EEG and EMG signals. Mean value was found of $0.683$, with variance of $0.0293$. This result certainly shows a strong connection between the narrow-band EEG and the EMG, as indicated by the high average value.
Finally, \[fig:isto\_lag\] displays the empirical distribution of the lag where the maximum value of the cross-correlation function of the $71$ pairs of EEG and EMG signals was found. Mean value occurred at $-11.65$ ms. As neural impulses propagate at a speed of about $100$ m/s, the transmission of signals from the brain to the hand muscles could be estimated of about $10$ ms, which is in line with the results we achieved.
The standard deviation is considerably high (it was found to be about $100$ ms) but this could be explained because of the limited size of the data sample. Indeed, we expect that the tendency observed in this study could be further confirmed (with a reduced standard deviation), with an increased sample size.
Discussion
==========
Conclusions
===========
The aim of this study is to investigate bursts-related EEG signals in a focal hand dystonia patient. Despite of considering time domain and frequency domain techniques as mutually exclusive analysis, in this contribution we have taken advantage from both of them: particularly, in the frequency domain, cortico-muscular coherence was used to identify the most likely frequency bands of interaction between brain and muscles; then, in the time domain, cross-correlation was exploited to verify the physiological reliability of such a relationship in terms of signal transmission delay from the centre to the periphery. The most interesting result suggested that the high $\beta$ band activity in the EEG could be responsible for the bursty activity observed in the EMG. Even though a future study on a larger sample is needed to statistically support these preliminary findings, this contribution allows to think of new kinds of rehabilitation interventions for focal hand dystonia patients that could target the actual EEG correlate of the pathology with consequence improvement of the motor functions.
|
---
abstract: |
Consider a nonlinear regression model : $y_{i}=g\left( \mathbf{x}_{i},\mathbf{\theta}\right) +e_{i},$ $i=1,...,n,$ where the $\mathbf{x}_{i}$ are random predictors $\mathbf{x}_{i}$ and $\mathbf{\theta}$ is the unknown parameter vector ranging in a set $\mathbf{\Theta\subset}R^{p}\mathbf{.}$ All known results on the consistency of the least squares estimator and in general of M estimators assume that either $\mathbf{\Theta}$ is compact or $g$ is bounded, which excludes frequently employed models such as the Michaelis-Menten, logistic growth and exponential decay models. In this article we deal with the so-called *separable* models, where $p=p_{1}+p_{2},$ $\mathbf{\theta=}\left( \mathbf{\alpha,\beta}\right) $ with $\mathbf{\alpha\in}A\subset R^{p_{1}},$ $\mathbf{\beta\in}B\subset R^{p_{2},}$and $g$ has the form $g\left( \mathbf{x,\theta}\right) =\mathbf{\beta}^{T}\mathbf{h}\left( \mathbf{x,\alpha}\right) $ where $\mathbf{h}$ is a function with values in $R^{p_{2}}.$ We prove the strong consistency of M estimators under very general assumptions, assuming that $\mathbf{h}$ is a bounded function of $\mathbf{\alpha,}$ which includes the three models mentioned above.
Key words and phrases: Nonlinear regression, separable models, consistency, robust estimation.
author:
- 'María Victoria Fasano$^{1}$ ([email protected])'
- |
Ricardo Maronna$^{1}$ ([email protected])\
$^{1}$Departamento de Matemática, Facultad de Ciencias Exactas,\
Universidad de La Plata, C.C. 172, La Plata 1900, Argentina.
date:
title: Consistency of M estimates for separable nonlinear regression models
---
Introduction
============
Consider i.i.d. observations $\left( \mathbf{x}_{i},y_{i}\right) ,$ $\ i=1,...,n,$ given by the nonlinear model with random predictors:$$y_{i}=g\left( \mathbf{x}_{i},\mathbf{\theta}_{0}\right) +e_{i},\ \label{nonlinmod}$$ where $\mathbf{x}_{i}\in R^{q}$ and $e_{i}$ are independent, and the unknown parameter vector $\mathbf{\theta}_{0}$ ranges in a set $\Theta\subset R^{p}.$ An important case, usually called *separable*, are models where $p=p_{1}+p_{2}$ and $\mathbf{\theta}_{0}\mathbf{=}\left( \mathbf{\alpha}_{0}\mathbf{,\beta}_{0}\right) $ with $\mathbf{\alpha}_{0}\mathbf{\in
}A\subset R^{p_{1}}$ and $\mathbf{\beta}_{0}\mathbf{\in}B\subset R^{p_{2}},$ and $g$ of the form$$g\left( \mathbf{x,\theta}\right) =g\left( \mathbf{x,\alpha,\beta}\right)
=\sum_{j=1}^{p_{2}}\beta_{j}h_{j}\left( \mathbf{x,\alpha}\right) ,
\label{lincompo}$$ where $h_{j}$ $(j=1,...,p_{2})$ are functions of $X\times R^{p_{2}}\rightarrow
R.$ Usually $B$ is the whole of $R^{p_{2}}$ or an unbounded subset of it. Examples are the Michaelis-Menten model, with$$p_{1}=p_{2}=q=1,\ x\geq0,~\alpha,\beta>0,~h\left( x,\alpha\right) =\frac
{x}{x+\alpha}, \label{Michael}$$ the logistic growth model, with$$q=1,\ p2=1,\ \ p_{2}=1,\ x\geq0,\ \alpha_{j}>0,\ \beta>0,\ h\left(
\mathbf{x,\alpha}\right) =\frac{e^{\alpha_{2}x}}{1+\alpha_{1}\left(
e^{\alpha_{2}x}-1\right) }, \label{logis}$$ the exponential decay model, with $$~q=1,~p_{2}=p_{1}+1,\ ~x\geq0,~\alpha_{j}<0,~\beta_{j}\geq0,~g\left(
\mathbf{x,\alpha,\beta}\right) =\beta_{0}+\sum_{j=1}^{p_{1}}\beta
_{j}e^{\alpha_{j}x}, \label{expdecay}$$ and the exponential growth model, like (\[expdecay\]) but with $\alpha
_{j}>0.$
The classical least squares estimate (LSE) is given by$$\widehat{\mathbf{\theta}}=\arg\min_{\mathbf{\theta\in}\Theta}\sum_{i=1}^{n}\left( y_{i}-g\left( \mathbf{x}_{i},\mathbf{\theta}\right) \right)
^{2}.$$
The consistency of the LSE assuming $\mathrm{E}\left( e_{i}\right) =0$ and $\mathrm{Var}\left( e_{i}\right) =\sigma^{2}<\infty$ has been proved by several authors under the assumption of a compact $\Theta$; in particular Amemiya (1983), Jennrich (1969) and Johansen (1984). Wu (1981) assumes that $\Theta$ is a finite set.
Richardson and Bhattacharyya (1986) do not require the compactness of $\Theta,$ but they assume $g\left( \mathbf{x,\theta}\right) $ to be a bounded function of $\mathbf{\theta,}$ which excludes most separable models.
Shao (1992) showed the consistency of the LSE without requiring the compacity of $\Theta$ nor the boundedness of $g,$ but requires assumptions on $g$ that exclude the simplest separable models. For example, in the case $g\left(
x,\mathbf{\theta}\right) =\beta e^{\alpha x},$ for any $x_{0}>0$ one can make $g\left( x_{0},\mathbf{\theta}\right) =$constant with $\alpha\rightarrow
-\infty$ and $\beta\rightarrow0.$ This fact violates both Condition 1 and Condition 2 in page 427 of his paper.
The well-known fact that the LSE is sensitive to outliers has led to the development of *robust estimates* that are simultaneously highly efficient for normal errors and resistant to perturbations of the model. One of the most important families of robust estimates are the *M-estimates* proposed by Huber (1973) for the linear model. For nonlinear models they are defined by$$\mathbf{\hat{\theta}}_{n}\mathbf{=}\arg\min_{\mathbf{\theta}\in\Theta}\sum_{i=1}^{n}\rho\left( \frac{y_{i}-g\left( \mathbf{x}_{i},\mathbf{\theta
}\right) }{\widehat{\sigma}}\right) , \label{defMestimaSig}$$ where $\rho$ is a loss function whose properties will be described in the next section and $\widehat{\sigma}$ is an estimate of the error’s scale. However, at this stage of our research we deal with the simpler case of known $\sigma.$ Then it may be assumed without loss of generality that $\sigma=1$ and therefore we shall deal with estimates of the form$$\mathbf{\hat{\theta}}_{n}\mathbf{=}\arg\min_{\mathbf{\theta}\in\Theta}\sum_{i=1}^{n}\rho\left( y_{i}-g\left( \mathbf{x}_{i},\mathbf{\theta
}\right) \right) . \label{defMestima}$$
All published results on the consistency of robust estimates for nonlinear models require the compacity of $\Theta.$ Oberhofer (1982) deals with the $L_{1}$ estimator. Vainer and Kukush (1998) and Liese and Vajda (2003, 2004) deal with M estimates. The latter deal with $O\left( n^{-1/2}\right) $ consistency and asymptotic normality of M estimates in more general models. Stromberg (1995) proved the consistency of the Least Median of Squares estimate (Rousseeuw, 1984), and Čížek (2005) dealt with the consistency and asymptotic normality of the Least Trimmed Squares estimate. Fasano et al. (2012) study the functionals related to M estimators in linear and nonlinear regression; in the latter case, they also assume a compact $\mathbf{\Theta.}$
In this article we will prove the consistency of M estimates for separable models without assuming the compactness of $\Theta$, but assuming the boundedness of the $h_{j}$s$\mathbf{;}$ this case includes the exponential decay, logistic growth and Michaelis-Menten models. It can thus be considered as a generalization of (Richardson and Bhattacharyya, 1986).
The assumptions
===============
It will be henceforth assumed that $\rho$ is a $\rho
$–function in the sense of (Maronna et al, 2006). i.e., $\rho\left( u\right) $ is a continuous nondecreasing function of $|u|,$ such that $\rho\left( 0\right) =0$ and that if $\rho(u)<\sup_{u}\rho(u)$ and $0\leq u<v$ then $\rho(u)<\rho(v).$ We shall consider two cases: unbounded $\rho$ and bounded $\rho.$ The first includes convex function, in particular the LSE with $\rho\left( x\right) =x^{2}$ and the well-known Huber function$$\rho_{k}(x)=\left\{
\begin{array}
[c]{ccc}x^{2} & \mathrm{if} & \left\vert x\right\vert \leq k\\
2k\left\vert x\right\vert -k^{2} & \mathrm{if} & \left\vert x\right\vert >k
\end{array}
\right. \label{dfhubrho}$$ and the second includes the bisquare function $\rho\left( x\right)
=\min\left\{ 1-\left( 1-\left( x/k\right) ^{2}\right) ^{3},1\right\} $, where $k$ is in both cases a constant that controls the estimator’s efficiency.
Let $\mathbf{h}\left( \mathbf{x,\alpha}\right) =\left( h_{1}\left(
\mathbf{x,\alpha}\right) ,...,h_{p_{2}}\left( \mathbf{x,\alpha}\right)
\right) ^{\prime}$ where in general $\mathbf{a}^{\prime}$ denotes the transpose of $\mathbf{a.}$The necessary assumptions are:
A
: $B$ is a closed set such that $t\mathbf{\beta\in}B$ for all $\mathbf{\beta\in}B$ and $t>0.$
B
: $\sup_{\mathbf{\alpha\in}A}\mathrm{E}|\rho\left( y-\mathbf{\beta
}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right) |<\infty$ for all $\mathbf{\ \beta\in}B.$
C
: The function $\mathrm{E}\rho\left( e-t\right) $ –where $e$ denotes any copy of $e_{i}$– has a unique minimum at $t=0.$ Put $\lambda
_{0}=\mathrm{E}\rho\left( e\right) .$
D
: $\mathbf{h}$ is continuous in $\mathbf{\alpha}$ a.s. and$$\mathbf{\alpha}\not =\mathbf{\alpha}_{0}\Rightarrow\sup_{\mathbf{\beta}\in
B}\mathrm{P}\left\{ \mathbf{\beta}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha
}\right) =\mathbf{\beta}_{0}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}_{0}\right) \right\} <1\text{ } \label{CL1}$$
E
: Let $S=\sup_{t}\rho\left( t\right) $ (which may be infinite). Then$$\delta=:\sup_{\mathbf{\beta\not =0,}\text{ }\mathbf{\alpha}\in A}\mathbf{P}\left( \mathbf{\beta}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha
}\right) =0\right) <1-\frac{\lambda_{0}}{S}. \label{CL2}$$
F
: Call $\mathcal{U}$ the family of all open neighborhoods of $\mathbf{\alpha}_{0}.$ Then $$\sup_{\mathbf{\beta}}\inf_{U\in\mathcal{U}}\sup_{\mathbf{\alpha\notin}U}\mathrm{P}\left\{ \mathbf{\beta}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha
}\right) =\mathbf{\beta}_{0}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}_{0}\right) \right\} <1.$$
G
: $\mathbf{h}$ is bounded as a function of $\mathbf{\alpha,}$ i.e., $\sup_{\mathbf{\alpha}\in A}\left\Vert \mathbf{h}\left( \mathbf{x,\alpha
}\right) \right\Vert <\infty\ \mathrm{a.s.}$
We now comment on the assumptions.
For (A) to hold in examples (\[Michael\])-(\[logis\])-(\[expdecay\]) we must enlarge the range of $\beta_{j}$s to $\beta_{j}\geq0.$ However, to ensure the validity of (D) and (F), it will be assumed that the elements of the true vector $\mathbf{\beta}_{0}$ are all positive.
If $\rho$ is bounded, (B) holds without further conditions. Sufficient conditions for Huber’s $\rho$ and for the LSE are finite moments of $e$ and of $\mathbf{h}\left( \mathbf{x,\alpha}\right) ,$ of orders one and two, respectively.
A sufficient condition for (C) is that the distribution of $e$ has an even density $f\left( u\right) $ that is nonincreasing for $u\geq0$ and is decreasing in a neighborhood of $u=0$ (see Lemma 3.1 of Yohai (1987)). If $\rho$ is strictly convex with a derivative $\psi,$ then a sufficient condition is $\mathrm{E}\psi\left( e\right) =0,$ which for the LSE reduces to $\mathrm{E}e=0.$
Assumption (D) is required for ensure uniqueness of solutions. For examples (\[Michael\])-(\[logis\]) it is very easy to verify. For (\[expdecay\]) it follows from the well-known linear independence of exponentials.
If $S=\infty,$ (E) just means that $\delta<1$ (since $\lambda_{0}<\infty$ by (B)). Otherwise it puts a bound on $\delta.$ In our examples we have $\delta=0,$ since $\mathbf{\beta}^{\prime}\mathbf{h}>0$ if $\mathbf{\beta}$ has a single nonnull (positive) element.
Assumption (F) is required in the case of non-compact $A,$ to prevent the estimator $\widehat{\mathbf{\alpha}}$ from escaping to the border. In our examples the border for the $\alpha_{j}$s is either zero of infinity, and (F) is easily verified by a detailed but elementary calculation (taking into account the remark above that all elements of $\mathbf{\beta}_{0}$ are positive). For example, in (\[Michael\]) it suffices to consider neighborhoods of the form $\left( \alpha_{0}/K,K\alpha_{0}\right) $ with $K$ sufficiently large.
Finally, (G) is easily verified for models (\[Michael\])-(\[logis\])-(\[expdecay\]).
The results
===========
For separable ** models ** the M-estimate is given by $$\mathbf{\hat{\theta}}_{n}=\left( \widehat{\mathbf{\alpha}}_{n},\widehat
{\mathbf{\beta}}_{n}\right) \mathbf{=}\arg\min_{\mathbf{\alpha\in}A,\text{
}\mathbf{\beta}\in B}\frac{1}{n}\sum_{i=1}^{n}\rho\left( y_{i}-\mathbf{\beta
}^{\prime}\mathbf{h}\left( x_{i},\mathbf{\alpha}\right) \right) .$$
We now state our main result.
Assume model (\[lincompo\]) with conditions A-B-C-D-E-F-G. Then the M estimate $\left( \widehat{\mathbf{\alpha}}_{n},\widehat{\mathbf{\beta}}_{n}\right) $ is strongly consistent for $\mathbf{\theta}_{0}.$
We shall first need an auxiliary result, based on a proof in (Bianco and Yohai, 1996).
Assume model (\[lincompo\]) with conditions A-B-C-D-E and $A$ *compact*. Then $\left\Vert \mathbf{\hat{\beta}}_{n}\right\Vert $ is ultimately bounded with probability one.
**Proof of the Lemma:** Put$$\lambda\left( \mathbf{\alpha,\beta}\right) =\mathrm{E}\rho\left(
y-\mathbf{\beta}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right)
.$$
It follows from (C) that $\mathbf{\lambda(\alpha},\mathbf{\beta)}$ attains its minimum only when $\mathbf{\beta}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha
}\right) =\mathbf{\beta}_{0}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}_{0}\right) $ a.s. and by (\[CL1\]) this happens when $\left(
\mathbf{\alpha},\mathbf{\beta}\right) =\left( \mathbf{\alpha}_{0},\mathbf{\beta}_{0}\right) .$ Therefore $$\left( \mathbf{\alpha},\mathbf{\beta}\right) \neq\left( \mathbf{\alpha}_{0},\mathbf{\beta}_{0}\right) \Rightarrow\lambda\left( \mathbf{\alpha
},\mathbf{\beta}\right) >\lambda\left( \mathbf{\alpha}_{0},\mathbf{\beta
}_{0}\right) =\lambda_{0}. \label{unique}$$
Let $\Gamma=\left\{ \mathbf{\gamma}\in B:\left\Vert \mathbf{\gamma
}\right\Vert =1\right\} .$ Then we may write $\mathbf{\beta}=t\mathbf{\gamma
}$ with $t=\left\Vert \mathbf{\beta}\right\Vert \in R_{+}$ and $\mathbf{\gamma
}\in\Gamma.$
We divide the proof into two cases.
**Case I: bounded** $\rho:$ Assume that $S=\sup_{u}\rho\left( u\right)
<\infty.$ To simplify notation it will be assumed without loss of generality that $S=1.$ For each $\left( \mathbf{\alpha},\mathbf{\gamma}\right) \in
A\times\Gamma$ we have$$\lim_{t\rightarrow\infty}\mathrm{E}\rho\left( \mathbf{y-}t\mathbf{\gamma
}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right) \geq
1-\delta>\lambda_{0},$$ where $\delta$ is defined in (\[CL2\]). Let $$\xi=1-\delta-\lambda_{0}>0,\ \ \varepsilon=\frac{\xi}{4}<\frac{1-\delta}{4}.$$
Since (\[CL2\]) implies that $\mathrm{P}\left( \left\vert \mathbf{\gamma
}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right\vert >0\right)
\geq1-\delta$ for $\mathbf{\gamma\in}\Gamma,$ then for each $\left(
\mathbf{\alpha},\mathbf{\gamma}\right) \in A\times\Gamma$ there are positive $a,b$ such that$$\mathrm{P}\left( \left\vert y\right\vert \leq a,\left\vert \mathbf{\gamma
}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right\vert \geq
b\right) \geq1-\delta-\varepsilon. \label{CL3}$$
Then by (\[CL3\]) there exists $T>0$ such that $t>T$ implies$$\mathrm{E}\inf_{t>T}\rho\left( \mathbf{y-}t\mathbf{\gamma}^{\prime}\mathbf{h}\left( \mathbf{x,\alpha}\right) \right) >1-\delta-2\varepsilon.
\label{CL4}$$
Therefore (\[CL4\]) implies that for each $\left( \mathbf{\alpha
},\mathbf{\gamma}\right) \in A\times\Gamma$ there exist a neighborhood $U\left( \mathbf{\alpha},\mathbf{\gamma}\right) \subset A\times\Gamma$ and $T\left( \mathbf{\alpha},\mathbf{\gamma}\right) \in R_{+}$ such that $$\mathrm{E}\inf_{\left( \mathbf{\alpha}_{1}\mathbf{,\gamma}_{1}\right) \in
U\left( \mathbf{\alpha,\gamma}\right) }\inf_{\ t>T\left( \mathbf{\alpha
,\gamma}\right) }\rho\left( \mathbf{y-}t\mathbf{\gamma}_{1}^{\prime
}\mathbf{h}\left( \mathbf{x,\alpha}_{1}\right) \right) >1-\delta
-2\varepsilon=\lambda_{0}+\frac{\xi}{2}. \label{CL5}$$
The neighborhoods $\{U\left( \mathbf{\alpha,\gamma}\right) :\mathbf{\alpha
\in}A,\mathbf{\gamma}\in\Gamma\}$ are a covering of the compact set $A\times\Gamma,$ and therefore there exists a finite subcovering thereof: $\left\{ U_{j}=U\left( \alpha_{j},\mathbf{\gamma}_{j}\right) \right\}
_{j=1}^{N}$. Let $T_{0}=\max_{j}T\left( \alpha_{j},\mathbf{\gamma}_{j}\right) .$
We shall show that $\lim\sup_{n\rightarrow\infty}\left\Vert \mathbf{\hat
{\beta}}_{n}\right\Vert \leq T_{0}$ a.s. Put for brevity$$\lambda_{n}\left( \mathbf{\alpha,\beta}\right) =\frac{1}{n}\sum_{i=1}^{n}\rho\left( y_{i}-\mathbf{\beta}^{\prime}\mathbf{h}\left( x_{i},\mathbf{\alpha}\right) \right) .$$ Then$$\begin{aligned}
\inf_{\left\Vert \mathbf{\beta}\right\Vert >T_{0}}\inf_{\mathbf{\alpha}\in
A}\lambda_{n}\left( \mathbf{\alpha,\beta}\right) & \geq\frac{1}{n}\sum_{i=1}^{n}\inf_{\mathbf{\alpha}\in A,\mathbf{\gamma}\in\Gamma}\inf_{t>T_{0}}\rho\left( y_{i}-t\mathbf{\gamma}^{\prime}\mathbf{h}\left(
x_{i},\mathbf{\alpha}\right) \right) \\
& =\min_{j=1,...,N}\frac{1}{n}\sum_{i=1}^{n}\inf_{\left( \mathbf{\alpha
,\gamma}\right) \in U_{j}}\inf_{t>T_{0}}\rho\left( y_{i}-t\mathbf{\gamma
}^{\prime}\mathbf{h}\left( x_{i},\mathbf{\alpha}\right) \right) ,\end{aligned}$$ and therefore (\[CL5\]) and the Law of Large Numbers imply$$\lim\inf_{n\rightarrow\infty}\inf_{\left\Vert \mathbf{\beta}\right\Vert
>T_{0}}\inf_{\mathbf{\alpha}\in A}\lambda_{n}\left( \mathbf{\alpha,\beta
}\right) \geq\lambda_{0}+\frac{\xi}{2}~\mathrm{a.s.},$$ while$$\lambda_{n}\left( \widehat{\mathbf{\alpha}}_{n}\mathbf{,}\widehat
{\mathbf{\beta}}_{n}\right) =\inf_{\beta\in B}\inf_{\mathbf{\alpha}\in
A}\lambda_{n}\left( \mathbf{\alpha,\beta}\right) \leq\lambda_{n}\left(
\mathbf{\alpha}_{0}\mathbf{,\beta}_{0}\right) \rightarrow\lambda
_{0}~\mathrm{a.s.}$$ which shows that ultimately $\left\Vert \mathbf{\hat{\beta}}_{n}\right\Vert
\leq T_{0}$ with probability one.
**Case II: unbounded** $\rho:$ Here an analogous but simpler procedure shows the existence of $T_{0}$ and neighborhoods $U\left( \mathbf{\alpha
,\gamma}\right) $ such that the left-hand member of (\[CL5\]) is larger than $2\lambda_{0},$ and the rest of the proof is similar.$\blacksquare$
**Proof of the Theorem:** If $A$ is not compact, we employ the same approach as in (Richardson and Bhattacharyya, 1986): the Čech-Stone compactification yields a compact set $\widetilde{A}\supset A$ such that each bounded continuous function on $A$ has a unique continuous extension to $\widetilde{A}.$ We have to ensure that (B), (D) and (E) continue to hold for $\mathbf{\alpha\in}\widetilde{A}.$ Since each element of $\widetilde{A}$ is the limit of a sequence of elements of $A,$ (B) and (E) are immediate; and (D) follows from assumption (F). Therefore we can apply the Lemma to conclude that $\left( \widehat{\mathbf{\alpha}}_{n}\mathbf{,}\widehat{\mathbf{\beta}}_{n}\right) $ remains ultimately in a compact a.s. The Theorem then follows from Theorem 1 of Huber (1967).$\blacksquare$
Acknowledgements:
=================
**** This research was partially supported by grants PID 5505 from CONICET and PICTs 21407 and 00899 from ANPCYT, Argentina.
**References**
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Fasano, M.V., Maronna, R.A., Sued, M., Yohai, V.J., 2012. Continuity and differentiability of regression M functionals. Bernouilli (to appear).
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Liese, F. Vajda, I., 2004. A general asymptotic theory of M-estimators II. Math. Meth. Statist., 13, 82-95.
Maronna, R.A., Martin, R.D., Yohai, V.J., 2006. Robust Statistics: Theory and Methods, John Wiley and Sons, New York.
Oberhofer, W., 1982. The consistency of nonlinear regression minimizing the $L_{1}$ norm. Ann. Statist., 10, 316-319.
Richardson, G.D., Bhattacharyya, B.B., 1986. Consistent estimators in nonlinear regression for a noncompact parameter space. Ann. Statist., 14, **** 1591-1596.
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Yohai, V. J., 1987. High breakdown-point and high efficiency estimates for regression. Ann. Statist., 15, 642-656.
|
---
abstract: 'Simulations of structure formation in the Universe predict accretion shock waves at the boundaries of the large-scale structures as sheets, filaments, and clusters of galaxies. If magnetic fields are present at these shocks, particle acceleration should take place, and could contribute to the observed cosmic rays of high energies. When the radio plasma of an old invisible radio lobe is dragged into such a shock wave, the relativistic electron population will be reaccelerated and the plasma becomes radio-luminous again. Such tracers of the accretion shock waves are observed at the boundaries of some clusters of galaxies: the so-called cluster radio relics, which are large regions of diffuse radio emission, without any parent galaxy nearby. The observed properties of the cluster radio relics are naturally explained by accretion shock waves. Radio relics therefore give the first evidence for the existence of accretion shocks of the large-scale structure formation and they allow investigations of the shock properties.'
author:
- 'T.A. Ensslin'
- 'P.L. Biermann'
- 'U. Klein'
- 'S. Kohle'
title: 'Shock Waves of the Large-Scale Structure Formation in the Universe'
---
Shock Waves of the Large-Scale Structure Formation
====================================================
The large-scale structure of the Universe, seen in the structured galaxy distribution, is still forming. Matter is flowing out of the cosmic voids onto sheets and filaments. Within the filaments the matter flows to the density cusps frequently located at the intersection points of filaments: the clusters of galaxies. Whenever the flow passes from one structure into another, its velocity suddenly changes and several Mpc sized shock waves occur. At these shock waves the kinetic energy of the gravitationally accelerated gas is dissipated, mainly thermalized to temperatures of a few $0.1$ keV in filaments and several keV in clusters of galaxies. The shock velocity at filaments is expected to be of the order of several $100$ km s$^{-1}$, and the accretion shocks at clusters $1000-2000$ km s$^{-1}$. From simulation of structure formation rough values of the cluster accretion shock radius and velocity can be given in terms of the cluster temperature as an indicator of the gravitational potential (Kang et al. 1997): $$\begin{aligned}
\label{eq:rs}
r_{\rm s} &=& 4.24\, h_{50}^{-1} {\rm Mpc} \,( kT_{\rm
obs}/{6.06\,{\rm keV}} )^{1/2} (1+z)^{-3/2}\\
\label{eq:Vs}
V_{\rm s, predicted} &=& 1750 \, {\rm km\, s^{-1}}\, ({kT_{\rm
obs}}/{6.06\,{\rm keV}} )^{1/2}\,\,.\end{aligned}$$ The dissipated accretion power per shock surface is: $$\label{eq:Qflow}
Q_{\rm flow} \approx \frac{1}{2}\,n_{\rm e}\,m_{\rm p}\,V_{\rm s}^3
\approx 4\cdot 10^{44} \;\, \frac{\rm erg\,s^{-1}}{\rm Mpc^2}\,\,\frac{n_{\rm
e}}{10^{-5}\,{\rm cm^{-3}}}\,\left( \frac{kT_{\rm obs}}{6.06\,{\rm keV}}
\right)^{3/2}\,.$$
Particle Acceleration at Cluster Accretion Shocks
==================================================
If magnetic fields are present at the location of the accretion shocks, charged particles are accelerated. The acceleration of protons is mainly limited by photo-meson production to $< 10^{20}$ eV, but could contribute to the UHECR spectrum below this energy (Kang et al. 1997). Relativistic electrons lose their energy by IC scattering of MWB photons and synchrotron losses. Especially the synchrotron radio emission should be a sensitive tracer of shock waves. The best detectability is given if a strongly magnetized plasma is present with a preaccelerated population of electrons. This is the case whenever an old remnant radio lobe of a former radio galaxy is dragged into such a shock. Since the Universe was sufficiently polluted by radio plasma at the epoch of quasar activity (En[ß]{}lin et al. 1997, 1998b), such shocks should be traceable today. We argue that this is the case for cluster radio relics.
Cluster Radio Relics
=====================
- Peripherally located sources of irregular extended radio emission.
- Steep radio spectra $\alpha = 1.0-1.5$.
- Typical radio luminosities of $10^{41...42}$ erg s$^{-1}$.
- Radio relics are believed to be the remnants of radio lobes of radio galaxies, where the former active galaxy has become inactive or has moved away.
- But cluster radio relics have frequently no nearby possible parent galaxy within an electron cooling-time travel distance.
- Further: The radio spectra of cluster radio relics do usually not show an apparent cutoff, as the spectra of old remnants of radio galaxies do.
- Thus the electron population is (re)accelerated there.
- Nine known examples: 0038-096 in A85, 0917+75 in A786, 1140+203 in A1367, 1253+275 in Coma, 1712+64 in A2255, 1706+78 in A2256, 2006-56 and 2010-57 in A3667, 1401-33 in S753
A Simple Model
===============
The orientation of the magnetic fields of the infalling plasma is assumed to be distributed randomly. The plasma is compressed at the shock by the compression ratio $R$. This determines the spectral index of the electron population. The high energy cutoff of the downstream electron population is a function of the distance to the shock plane. High resolution radio observations should therefore reveal a flatter (steeper) spectrum at the outer (inner) edge of the relic, which is closer (more distant) to the accretion shock. This is indeed observed for the relics in Coma and A3667. The overall radio spectrum is a superposition of the spectra at different distances and is composed by a flat low frequency spectrum, then a spectral index steeper by $0.5$ due to the superposition of different cutoffs, and a final cutoff. The observed spectral indices of relics belong to the steeper region, expect in the case of the relic in A2256, which is not due to a peripheral accretion shock but due to a massive cluster merger. The shock compression ratio can be calculated[^1] from the observed radio spectrum: $$\label{eq:R}
R = ({\alpha+1})/({\alpha-0.5})\,\, ,$$ The ratio of break- to cutoff-frequency allows (if observed) a rough estimate of the electron diffusion coefficient (En[ß]{}lin et al. 1998a): $$\label{eq:kappa2}
\kappa_2 \approx {V_{\rm s}\,D}\,{R}^{-1}\, \sqrt{{\nu_{\rm
break}}/{\nu_{\rm cut}}} < 3 \cdot 10^{30} \,{\rm cm^2\,s^{-1}} \,\,.$$ The thickness $D \approx 0.1$ Mpc, and the other parameters are taken for the relic 1253+275 in Coma. The thickness divided by the post shock velocity $V_{\rm s}/R$ gives the age of the relic after it passed the shock. It should be similar to the cooling time of electrons visible at the spectral break. Both ages agree roughly for all known relics to be of the order of $\approx 10^8$ yr. The shock efficiency which is necessary in order to power the radio emission is $0.1...5\%$.
Radio Polarization
===================
The magnetic fields of the old radio lobe get compressed in the shock and therefore aligned with the shock plane. If the shock is observed inclined by a viewing angle $\delta$ the projected fields are perpendicular to the projected shock plane. For relics with observed radio polarization (Coma, A2256, A786) this is true. If the magnetic fields are not dynamical important, the radio polarization of fields, which were originally randomly oriented, can be estimated (En[ß]{}lin et al. 1998a). Since an independent very rough estimate of the viewing angle $\delta$ is available from Eq. \[eq:rs\] and the projected observed radius, the expected correlation between polarization and expected viewing angle can be checked (Fig. \[fig:pol\]).
Probing Large-Scale Flows
===========================
Cluster radio relics allow to measure properties of the large-scale flows. An extrapolation of the cluster gas density by the usual $\beta$-profile divided by the compression ration gives an estimate of the density of the infalling gas: $n_{\rm e} \approx0.5...1.0\cdot
10^{-5}\,{\rm cm^{-3}}$. The temperature jump of the shock is also given by the compression ratio. Assuming that the temperature inside the shock is half of the central cluster temperature, which is a typical observed decrease, one gets for the temperature of the infalling gas $0.5 ... 1$ keV. Also the predicted shock velocities (Eq. \[eq:Vs\]) can be checked: Using the observed parameters of the shock the velocity $V_{\rm s}$ of the infalling matter can be estimated. This is consistent with the prediced one ($V_{\rm
s,predicted}$) from cosmological structure formation (Fig. \[fig:vs\]).
Conclusions
============
- Giant shock waves of the cosmological large-scale motion of the on-going structure formation are places of particle acceleration.
- The conditions for particle acceleration are ideal if an old radio lobe is dragged into such a shock wave.
- [*First observational evidence for these shock waves is presented*]{}: Cluster radio relics trace accretion shock fronts where the gas of large-scale filaments flows into clusters of galaxies
- Observed properties of cluster radio relics can be explained if they are understood as remnant radio plasma within cluster accretion shocks.
- Properties of the large-scale flows can be measured.
- The long lasting outstanding problem of the energy supply of cluster radio relics is solved by the accretion shock theory.
[**References**]{}\
Biermann P.L., 1993, A&A [ 271]{}, 649\
Jokipii J.R., 1987, ApJ 313, 842\
En[ß]{}lin T.A., Biermann P.L., Kronberg P.P., Wu X.-P., 1997, ApJ 477, 560\
En[ß]{}lin T.A., Biermann P.L., Klein U., Kohle S., 1998a, A&A 332, 395\
En[ß]{}lin T.A., Wang Y., Nath B.B., Biermann P.L., 1998b, A&A 333, L47\
Giovannini G., Feretti L., Stanghellini C., 1991, A&A [ 252]{}, 528\
Kang H., Rachen J.P., Biermann P.L., 1997, MNRAS 286, 257\
Roettiger K., Burns J.O., Pinkney J., 1995, ApJ 453, 634\
[^1]: Since particle acceleration is most efficient for oblique shocks (Jokipii 1987), patches with magnetic fields oriented parallel to the shock plane will dominate the acceleration. The spatial diffusion coefficient, which determines the efficiency of the acceleration process, is only poorly understood in such complicated circumstances. Thus, we assume a momentum independent coefficient, because of the success of this simplification in the similarly complicated case of SNR CR-acceleration (Biermann 1993).
|
---
abstract: 'There are several models for the effective thermal conductivity of two-phase composite materials in terms of the conductivity of the solid and the disperse material. In this paper, we generalise three models of Maxwell type (namely, the classical Maxwell model and two generalisations of it obtained from effective medium theory and differential effective medium theory) so that the resulting effective thermal conductivity accounts for radiative heat transfer within gas voids. In the high-temperature regime, radiative transfer within voids strongly influences the thermal conductivity of the bulk material. Indeed, the utility of these models over classical Maxwell-type models is seen in the high-temperature regime, where they predict that the effective thermal conductivity of the composite material levels off to a constant value (as a function of temperature) at very high temperatures, provided that the material is not too porous, in agreement with experiments. This behaviour is in contrast to models which neglect radiative transfer within the pores, or lumped parameter models, as such models do not resolve the radiative transfer independently from other physical phenomena. Our results may be of particular use for industrial and scientific applications involving heat transfer within porous composite materials taking place in the high-temperature regime.'
author:
- |
Kristian B. Kiradjiev,$^{a}$ Svenn Anton Halvorsen,$^b$ Robert A. Van Gorder,$^{a*}$\
and Sam D. Howison$^a$\
$^a$ Mathematical Institute, University of Oxford\
Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road\
Oxford OX2 6GG United Kingdom\
$^b$ Teknova AS, Tordenskjolds gate 9, 5th floor\
NO-4612 Kristiansand S, Norway\
$^* $Email: [email protected]
title: 'Maxwell-type models for the effective thermal conductivity of a porous material with radiative transfer in the voids'
---
*Keywords*: effective thermal conductivity; porous media; radiation; Maxwell model; effective medium theory; differential effective medium theory
Introduction {#sec:1}
============
Estimating thermal properties of bulk or composite materials, such as their effective thermal conductivity, is crucial for understanding of heat transfer in them. This on its own is of great significance for various industries which process raw materials, often from a granular or particulate state. There are various models, such as lumped parameter models, which condense the effects of different phenomena such as solid conduction, radiation, and conduction through contact points, into equivalent parameters. See [@Kunii1960] and [@Kandula2011], for example, for several expressions for effective thermal conductivity. While useful, such lumped parameter models may obscure the contribution from each phenomenon, and often lack a rigorous foundation. For this reason, it is desirable to derive such models from first principles.
Common models for calculating the effective thermal conductivity of a solid composite material include the Maxwell model (based on the pioneering work of Maxwell [@Maxwell1873], where an expression for the effective thermal conductivity was derived via far-field perturbations to solutions of the steady heat equation), as well as variants thereof, including the effective medium theory (EMT) model and the differential effective medium theory (DEMT) model. EMT uses a similar approach to that used in deriving the Maxwell model, and can be applied to many other physical properties (see [@Landauer1952], for instance, for the electrical resistance problem). In [@Carson2005; @Xu2016], there is a comparison between the Maxwell model and EMT, and those works outline how certain bounds on the thermal conductivities can be obtained. The multipole expansion method [@Mogilevskaya2012] gives similar results. In applying the DEMT, one incrementally adds one of the materials to the composite, and considers the effect of an infinitesimal change in the composite material composition on the effective thermal conductivity, obtaining a differential equation for the effective thermal conductivity in terms of the volume fraction of inclusions; see [@Bauer1993; @Ordonez-Miranda2010]. Reviews of many current models and methodologies, including those outlined above, can be found in [@Carson2002; @Karayacoubian2006; @Pietrak2015].
Recent work has involved the application of Maxwell-like models to the study of composite materials [@xu2016reconstruction], including polymer composites [@kim2017two; @zhai2018effective]. Such models have recently proven useful in understanding nanoflake thermal annealing [@bernal2017thermally], and in the understanding of effective thermal conductivity in a variety of materials, such as for a wood cell modelled as a constituent element of briquette chips [@sova2018effective], polyethylene nanocomposites [@zabihi2017effective], phase-change materials [@abujas2016performance] and composites [@wang2017experimental], fiber-reinforced concrete [@liu2017theoretical], alumina-graphene hybrid filled epoxy composites [@akhtar2017alumina], metal-graphene composites [@wejrzanowski2016thermal], transparent and flexible polymers containing fillers [@ngo2016thermal], and composite materials for LED heat sink applications [@terentyeva2017analyzing]. Maxwell-like models have also motivated theoretical methods for upscaling the thermal conduction equation in periodic composite materials [@mathieu2017method]. The thermal conductivity of composites made up of metallic and non-metallic microparticles or nanoparticles embedded in a solid matrix has been considered in [@ordonez2014thermal], following on from a model for the effective thermal conductivity of metal-nonmetal particulate composites, obtained in [@ordonez2012model]. The role of pore shape on the thermal conductivity of porous media has been studied using the Bruggeman differential effective medium theory [@ordonez2012effect].
All of these models were originally derived for composite materials with heat conduction as the only mode of heat transfer. However, radiation can play a strong role in heat transfer in porous media when there is a non-negligible volume fraction of gas voids to solid material. For a review of radiative heat transfer theory, see [@Modest1993]. Examples of applications where radiation within the gas voids can influence the effective thermal conductivity of composite materials include multilayer thermal insulation systems [@spinnler2004studies]; porous partially stabilised zirconia [@hsu1992measurements]; monolithic organic aerogels [@lu1992thermal]; ultralight metal foams [@zhao2004thermal] and open-celled metal alloy foams [@zhao2004temperature]; and Earth materials within the mantle [@schatz1972thermal]. Note that many such applications involve high-temperature regimes. The effect of identical approximately spherical pores and anisometric cylindrical pores on the thermal conductivity of alumina, graphite, and nickel has been investigated separately in [@francl1954thermal], and pore orientation was shown to affect the value of the effective thermal conductivity for a given porosity, particularly above 500 C. Experimental results for the thermal conductivity of a range of porous materials were observed in [@luikov1968thermal], and for large temperatures thermal conductivities were shown to level off in some cases, rather than to increase without bound. In general, due to the high-temperature regime required in many industries, it is very difficult to conduct experiments, which supports the need for good models for predicting the effective thermal conductivity of composite materials. Some experimental methods for determining this can be seen in [@Kosowska-Golachowska2014], while a review of experimental methods for characterising thermal contact resistance is given in [@xian2017experimental]. Loeb [@loeb1954thermal] obtained a variety of formulae for thermal conductivity of porous media, which involve the conductivity of the solid material, the emissivity of the surface of the pores, and the size, shape, and distribution of the pores. Loeb [@loeb1954thermal] was able to show that materials can be prepared having different thermal conductivities in different directions. A number of correlations between effective thermal conductivity and the packing structure in beds of spherical particles exist in the literature; see, for instance, [@van2010review], for a review on this topic. A homogenization approach, making use of the separation of length scales, was used in [@Allaire2014], where the authors derive an effective thermal conductivity tensor and study the second-order corrections to the temperature field.
In this paper, we generalise effective thermal conductivity models of Maxwell type to include radiation in the gas pores between densely-packed solid particles, justifying the use of an effective radiative conductivity, often as employed in lumped parameter models to account for radiation. In particular, we consider the Maxwell, EMT, and DEMT models, and extend them to the case when we have a solid matrix of material with multiple gas voids, a reasonable assumption for modelling densely-packed particle beds. We allow for thermal conduction in the solid phase and radiation in the gas phase, neglecting thermal conduction in the gas, because its thermal conductivity is relatively small. Similarly, we ignore any convective heat flux in the gas, on the basis that the heat capacity of the gas is also relatively small.
The remainder of the paper is organised as follows. In Section 2, we review the Maxwell, EMT, and DEMT models for the effective thermal conductivity of a solid matrix of material with multiple inclusions of a second material. In Section 3, we extend the Maxwell model to include radiation in the gas phase, deriving a new effective thermal conductivity. In Sections 4 and 5, respectively, we derive EMT and DEMT models for effective thermal conductivity, accounting for radiative transfer in the gas phase. In Section 6, we compare and discuss the results of these three models. We conclude in Section 7.
Effective thermal conductivity models of Maxwell type {#sec:2}
=====================================================
Maxwell [@Maxwell1873] models the composite material as a continuous matrix of constant conductivity $k_{1}$ containing multiple spherical inclusions of radius $a$ (not necessarily in a regular array) of constant conductivity $k_{2}$ and applies an external temperature field with gradient of magnitude $T'_{\infty}$, which drives heat through the medium (see Figure \[fig:7\]). Further, he assumes that the sizes of the particle inclusions are small relative to the inter-particle distances, so that the thermal disturbances to the temperature field due to each inclusion can be considered independently.
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
Consider a single spherical inclusion. To obtain the perturbed temperature field $T_{1}$ and $T_{2}$ outside and inside the sphere, respectively, we solve Laplace’s equation with continuity of temperature and conductive flux across the interface (ignoring interfacial resistance), and a matching condition with the far-field temperature. Taking spherical coordinates ($r,\varphi,\theta$) with origin at the centre of the sphere, and $\theta=0$ parallel to the imposed temperature gradient, we have $$\begin{aligned}
\nabla^2 T_{1}&=0 \qquad &&\text{ for } \qquad r> a, \label{eq:81}\\
\nabla^2 T_{2}&=0 \qquad &&\text{ for } \qquad r< a, \label{eq:80}\\
T_{1}&=T_{2} \qquad &&\text{ on } \qquad r=a, \label{eq:83}\\
k_{1}\dfrac{\partial T_{1}}{\partial r}&=k_{2}\dfrac{\partial T_{2}}{\partial r} \qquad &&\text{ on } \qquad r=a, \label{eq:91}\\
T_{1} &\to T'_{\infty}r\cos{\theta} \qquad &&\text{ as } \qquad r\to \infty. \label{eq:70}\end{aligned}$$ This is solved to obtain $$\begin{aligned}
T_{1}=\left(T'_{\infty}r+\frac{C_{1}(a,k_{1},k_{2})}{r^2}\right) \cos{\theta}, \label{eq:308}\\
T_{2}=C_{2}(a,k_{1},k_{2})r\cos{\theta},
\label{eq:310}\end{aligned}$$ where $$\begin{aligned}
C_{1}(a,k_{1},k_{2})=\frac{k_{1}-k_{2}}{2k_{1}+k_{2}}a^3T'_{\infty}, \label{eq:309}\\
C_{2}(a,k_{1},k_{2})=\frac{3k_{1}}{2k_{1}+k_{2}}T'_{\infty}.
\label{eq:311}\end{aligned}$$ Here, $C_{1}(a,k_{1},k_{2})$ determines the far-field behaviour induced by the spherical inclusion $a$. Maxwell next proposed to consider a large sphere of radius $A$ in the matrix material with $n$ spherical inclusions of the second material inside it (see Figure \[fig:7\]). By the assumption that the spherical inclusions are far from each other so that they do not interact, applying the superposition principle, we get that the coefficient, $C_{n}(a,k_{1},k_{2})$, for the far-field behaviour, induced by the $n$ spheres, is simply $$C_{n}(a,k_{1},k_{2})=nC_{1}(a,k_{1},k_{2})=\frac{k_{1}-k_{2}}{2k_{1}+k_{2}}\phi A^3 T'_{\infty},
\label{eq:241}$$ where in the last expression $\phi=n a^3/A^3$ is the volume fraction of the spherical inclusions.
The crux of the method lies in considering the large sphere, which has numerous spherical inclusions inside, as a continuous medium with effective thermal conductivity $k_{\mathit{eff}}$ (see Figure \[fig:7\]). Using , the far-field perturbation coefficient $C_{1}(A,k_{1},k_{\mathit{eff}})$ due to a single spherical inclusion with a thermal conductivity $k_{\mathit{eff}}$ is $$C_{1}(A,k_{1},k_{\mathit{eff}})=\frac{k_{1}-k_{\mathit{eff}}}{2k_{1}+k_{\mathit{eff}}}A^3 T'_{\infty}.
\label{eq:251}$$ Since the far-field perturbation is the same both ways, we have $$C_{n}(a,k_{1},k_{2})=C_{1}(A,k_{1},k_{\mathit{eff}}),
\label{eq:261}$$ and, therefore, rearranging for $k_{\mathit{eff}}$, we obtain $$k_{\mathit{eff}}=\frac{2k_{1}+k_{2}+2\phi(k_{2}-k_{1})}{2k_{1}+k_{2}-\phi(k_{2}-k_{1})}k_{1}=k_{1}+\frac{3 k_{1} (k_{2}-k_{1})}{2k_{1}+k_{2}}\phi + O(\phi^2).
\label{eq:3}$$ This approach gives a good estimate of the effective thermal conductivity in the dilute porosity limit [@Pietrak2015]. Note that Maxwell’s result is consistent with the lower and upper bounds for the effective thermal conductivity of an isotropic medium, derived by other means in [@Hashin1962].
Another approach is that of Effective Medium Theory (EMT), which is used in many other situations involving effective properties such as conductivities, polarisation, and the like. It is derived in a similar way to Maxwell’s model, but, crucially, it does not assume an ambient medium solely composed of the matrix material extending to the far-field, where we look for temperature perturbations. However, the assumption for a dilute porosity limit still holds. As one can check in the end, the two formulae agree up to $O(\phi)$ and differ at $O(\phi^2)$ for small $\phi$.
In reviewing the effective medium theory model, we follow the approach outlined in [@Xu2016]. Unlike Maxwell’s model, we begin by treating the composite medium, consisting of spherical inclusions of a different material sufficiently far apart, as a single material with effective conductivity $k_{\mathit{eff}}$. The temperature field of this medium is determined solely by the prescribed temperature gradient and has magnitude $$T_{1}=T'_{\infty} r\cos \theta.
\label{eq:301}$$ Now, as in Maxwell’s model, suppose we pick a spherical region of radius $A$ at random (Figure \[fig:11\]), remove the spherical inclusions (of radius $a$) from it (suppose they are $n$ in number), and replace them with particles of the same conductivity as the matrix material, resulting in a sphere of conductivity $k_{1}$ (Figure \[fig:11\]). The resulting far-field temperature obtained for a single sphere of radius $A$ immersed in a matrix of thermal conductivity $k_{\mathit{eff}}$ for $r \gg A$ reads $$T_{1}=\left(T'_{\infty}r+\frac{C_{1}(A,k_{\mathit{eff}},k_{1})}{r^2}\right) \cos{\theta},
\label{eq:312}$$ with $C_{1}(A,k_{\mathit{eff}},k_{1})$ defined as in , noting that the order of the arguments is different this time, because, for this particular calculation, the matrix material has conductivity $k_{\mathit{eff}}$ and the sphere has conductivity $k_{1}$.
{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
We now want to return the original spherical inclusions within the sphere of radius $A$. We first need to vacate $n$ spherical holes of conductivity $k_{1}$, and then replace them with spheres of the second material (Figure \[fig:11\]). Again, we assume sparsity of the spherical inclusions so that the interaction between them is negligible. The respective contributions to the far-field temperature coefficient are given by $$\begin{aligned}
C_{n}(a,k_{\mathit{eff}},k_{1})=nC_{1}(a,k_{\mathit{eff}},k_{1})&=\frac{k_{\mathit{eff}}-k_{1}}{2k_{\mathit{eff}}+k_{1}}\phi A^3T'_{\infty}, \label{eq:321}\\
C_{n}(a,k_{\mathit{eff}},k_{2})=nC_{1}(a,k_{\mathit{eff}},k_{2})&=\frac{k_{\mathit{eff}}-k_{2}}{2k_{\mathit{eff}}+k_{2}}\phi A^3T'_{\infty}. \label{eq:331}\end{aligned}$$ Combining -, we obtain the net far-field temperature $$\begin{aligned}
T_{1} = \left\lbrace \vphantom{\frac{C_{1}(A,k_{\mathit{eff}},k_{1})-C_{n}(a,k_{\mathit{eff}},k_{1})+C_{n}(a,k_{\mathit{eff}},k_{2})}{r^2}}T'_{\infty}r +\frac{C_{1}(A,k_{\mathit{eff}},k_{1})-C_{n}(a,k_{\mathit{eff}},k_{1})+C_{n}(a,k_{\mathit{eff}},k_{2})}{r^2} \right\rbrace\cos{\theta} .
\label{eq:341}
\end{aligned}$$ Since we effectively arrive at the initial configuration of homogenised medium with the prescribed temperature gradient, then $$C_{1}(A,k_{\mathit{eff}},k_{1})-C_{n}(a,k_{\mathit{eff}},k_{1})+C_{n}(a,k_{\mathit{eff}},k_{2})=0,
\label{eq:351}$$ which gives an implicit expression for $k_{\mathit{eff}}$: $$\frac{k_{\mathit{eff}}-k_{1}}{2k_{\mathit{eff}}+k_{1}}(1-\phi)+\frac{k_{\mathit{eff}}-k_{2}}{2k_{\mathit{eff}}+k_{2}}\phi=0.
\label{eq:361}$$ This expression is symmetric in $k_{1}$ and $k_{2}$, provided $\phi \leftrightarrow 1-\phi$, as the model can be applied when either of the materials is dilute in the other one. Equation has a unique positive root $$\begin{aligned}
k_{\mathit{eff}} & = \frac{1}{4}\left\lbrace \vphantom{\sqrt{\phi(k_{2})^2}} 3\phi (k_{2}-k_{1})+(2k_{1}-k_{2}) \right. \\
& \qquad \left. +\sqrt{(3\phi (k_{2}-k_{1})+(2k_{1}-k_{2}))^2+8k_{2}k_{1}}\right\rbrace \\ & = k_{1}+\frac{3 k_{1} (k_{2}-k_{1})}{2k_{1}+k_{2}}\phi + O(\phi^2),
\label{eq:5}
\end{aligned}$$ for $\phi \ll 1$, which agrees with Maxwell’s result up to $O(\phi)$. A third model, the Differential Effective Medium Theory (DEMT), motivated by the early work of Bruggeman [@bruggeman1935dielectric], considers media of various particle volume fractions [@Bauer1993; @Ordonez-Miranda2010] but with a range of particle sizes present in the composite medium. In this approach, one incrementally adds one of the materials to the composite, and considers the effect on the effective thermal conductivity, obtaining a differential equation for it in terms of the porosity.
In reviewing the differential effective medium theory approach, we follow [@Ordonez-Miranda2010]. We assume that the effective thermal conductivity is given as a function of the particle volume fraction, $\phi$, resulting in the ansatz $$k_{\mathit{eff}}(\phi)=k_{1}(1+b(k_{1},k_{2})\phi+c(k_{1},k_{2})\phi^2+\cdots).
\label{eq:410}$$ Here $b(k_{1},k_{2})$ determines the behaviour of $k_{\mathit{eff}}$ in the dilute limit $\phi \ll 1$, while $c(k_{1},k_{2})$ is the second-order correction which partially accounts for the particle interactions. We remove composite material of volume $\Delta V$, and replace it with the same volume of particle inclusions. Treating the homogeneous material as a new matrix, we may express the new effective conductivity as $$k_{\mathit{eff}}(\phi+\mathrm{d}\phi) =k_{\mathit{eff}}(\phi)\left(1+b(k_{\mathit{eff}}(\phi),k_{2})\frac{\mathrm{d}V}{V} +\cdots\right),
\label{eq:411}$$ where $\mathrm{d}\phi=(\mathrm{d}V-\mathrm{d}V_{p})/V$ is the net increase in the particle volume fraction and $\mathrm{d}V_{p}$ is the volume of the particles removed. Assuming that $\mathrm{d}V_{p}/V_{p}=\mathrm{d}V/V$ on average, we have $\mathrm{d}V/V=\mathrm{d}\phi/(1-\phi)$. Hence, in the limit $\mathrm{d}\phi\to 0$, we obtain $$\dfrac{\mathrm{d}k_{\mathit{eff}}}{\mathrm{d}\phi}=\frac{k_{\mathit{eff}}}{1-\phi}b(k_{\mathit{eff}},k_{2}),
\label{eq:423}$$ and integrating this differential equation with the condition $k_{\mathit{eff}}(0)=k_{1}$, we find $$\int_{k_{1}}^{k_{\mathit{eff}}}\frac{\mathrm{d}s}{sb(s,k_{2})}=-\log(1-\phi).
\label{eq:444}$$ From Maxwell’s result , we find $$b(k_{1},k_{2})=\frac{3(k_{2}-k_{1})}{2k_{1}+k_{2}}.
\label{eq:451}$$ After performing the integration in , we obtain the effective thermal conductivity implicitly as $$\left(\frac{k_{\mathit{eff}}-k_{2}}{k_{1}-k_{2}}\right)^3\frac{k_{1}}{k_{\mathit{eff}}}=(1-\phi)^3.
\label{eq:461}$$ This result was obtained (and the cubic equation solved explicitly) in [@kamiuto1990examination] (see equations (2) and (3) of [@kamiuto1990examination]) and [@kamiuto1993combined] (see equation (10) of [@kamiuto1993combined]). In order for the manipulations in the derivation of this model to be valid (in particular, to be able to consider incremental changes of the particle volume fractions), particles of a range of sizes are assumed to be present in the composite material, which is usually the case in many industrial processes. We note that if we expand $k_{\mathit{eff}}$ in powers of $\phi$ for small $\phi$, we again obtain agreement with Maxwell’s and EMT models up to $O(\phi)$.
As we saw, all three models agree up to $O(\phi)$ in the limit $\phi \to 0$. Maxwell’s model is a classical result, which has a surprising accuracy beyond $O(\phi)$ when compared to standard multiple-scales approximations for ordered media [@Bruna2015]. The effective medium theory model has an intrinsic symmetry in its constituent materials, which can be used to give estimates of the effective conductivity of composite materials, in which the inclusions are in a continuous rather than discrete phase. The differential effective medium theory model is applicable when there is a gradient in particle size in the composite material, so can be used for heterogeneous materials, whereas normally Maxwell’s and EMT models assume particle inclusions of a uniform size distribution.
We now extend these three models to the case of a solid matrix of material with multiple gas voids. The assumption of a solid matrix of material with multiple gas voids, a reasonable assumption for modelling densely packed particle beds, which commonly arise in applications in this area. As noted above, we consider conductive heat transfer only via the solid phase. [^1] We do, however, include radiation in the gas phase.
Maxwell model with radiation {#sec:3}
============================
First, we extend the Maxwell model. We consider the case of spherical pores, filled with gas of negligible thermal conductivity.
Similarly to Section \[sec:2\], we begin by looking at perturbations to the far-field temperature induced by a single small spherical gas void $V$ of radius $a$ (see Figure \[fig:4\]) when we apply an external temperature field of constant magnitude $T'_{\infty}$. The energy flux within the void is assumed to be entirely radiative. Noting that the temperature must be measured on the absolute scale, by the Stefan–Boltzmann law, the flux emitted per unit area at a point $\mathbf{r}$ on the void surface $\Sigma$ is $\epsilon\sigma T^4(\mathbf{r})$, where $\epsilon$ is the emissivity of the surface (assumed to be a gray body) and $\sigma$ is the Stefan–Boltzmann constant. The incident flux from points elsewhere on the surface is given by [@Modest1993] $$\begin{split}
\int_\Sigma \epsilon\sigma T^4(\mathbf{r}')
\frac{ \cos \gamma(\mathbf{r},\mathbf{r}')
\cos \gamma'(\mathbf{r},\mathbf{r}') \,
\mathrm{d}S'}{\pi|\mathbf{r}-\mathbf{r}'|^2}\\
=\int_\Sigma \epsilon\sigma T^4(\mathbf{r}') \,
\mathrm{d}F(\mathbf{r},\mathbf{r}'), \label{eq:53}
\end{split}$$ where $\gamma$ is the angle between the normal at $\mathbf{r}$ and the line of sight from $\mathbf{r}$ to $\mathbf{r}'$, $\gamma'$ being defined similarly. The term $$\mathrm{d}F(\mathbf{r},\mathbf{r}')
=\frac{ \cos \gamma(\mathbf{r},\mathbf{r}')
\cos \gamma'(\mathbf{r},\mathbf{r}') \,
\mathrm{d}S'}{\pi|\mathbf{r}-\mathbf{r}'|^2} \label{eq:55}$$ is known as the view factor. It is a geometrical property of the domain and satisfies $\int_\Sigma \mathrm{d}F(\mathbf{r},\mathbf{r}') = 1$ for all $\mathbf{r}$. For a sphere of radius $a$, simple trigonometry shows that the view factor is $\mathrm{d}S'/(4\pi a^2)$.
![Geometry of a spherical void in the Maxwell model with radiation.[]{data-label="fig:4"}](Fig2){width="0.25\linewidth"}
Denoting the solid-matrix-material conductivity simply by $k$, we now need to solve the following problem: $$\begin{aligned}
\nabla^2 T&=0 \qquad &&\text{ for } \qquad r > a, \label{eq:8}\\
k\dfrac{\partial T}{\partial r}&=\epsilon \sigma T^4-\int_{\Sigma} \frac{\epsilon \sigma T^4}{4\pi a^2} \, \mathrm{d}S \qquad &&\text{ on } \qquad r=a, \label{eq:9}\\
T &\to T'_{\infty} r \cos{\theta} \qquad &&\text{ as } \qquad r\to \infty. \label{eq:10}\end{aligned}$$
As the voids are assumed to be small, the temperature variation across them is also small. Hence, we expand $T$ around some reference temperature $\hat{T}_{0}$, which is taken to be the temperature on the ‘equator’ of the void $\theta=\pi/2$, exploiting the fact that $$\delta=\frac{aT'_{\infty}}{\hat{T}_{0}}\ll 1.
\label{eq:11}$$ We expand $T$ for $r\geq a$ as $$T \sim \hat{T}_{0}+\delta \hat{T}_{1} + O(\delta^2).
\label{eq:12}$$ The $O(1)$ terms cancel and so $$\delta k \dfrac{\partial \hat{T}_{1}}{\partial r}=\epsilon \sigma \hat{T}_{0}^4+4 \delta \epsilon \sigma \hat{T}_{0}^3\hat{T}_{1}-\int_{\Sigma} \epsilon \sigma (\hat{T}_{0}^4+4 \delta \hat{T}_{0}^3\hat{T}_{1})\mathrm{d}F \text{ on } r=a. \label{eq:13}$$ The first term on the left-hand side cancels with the first term in the integral since, by definition, $$\int_{\Sigma} \mathrm{d}F = 1.
\label{eq:14}$$ This simplifies to $$\frac{k}{4 \epsilon \sigma \hat{T}_{0}^3}\dfrac{\partial \hat{T}_{1}}{\partial r}=\hat{T}_{1}-\int_{\Sigma} \hat{T}_{1}\mathrm{d}F \qquad \text{ on } \qquad r=a.
\label{eq:15}$$ We further note that the integral term in this equation vanishes identically: all points on the void surface have, to $O(\delta)$, the same incident flux (but different radiative fluxes). We readily find that $$\hat{T}_{1}(r,\theta)=\left(\frac{C_{1}}{r^2}+C_{2}r\right)\cos{\theta},
\label{eq:16}$$ where, upon using and , we have $$\begin{aligned}
C_{1}(a,k,k_{r})&=\frac{a^2 \hat{T}_{0}(k-k_{r})}{2k+k_{r}}=\frac{a^2 \hat{T}_{0}(\Lambda-1)}{2\Lambda+1}, \label{eq:17}\\
C_{2}(a,k,k_{r})&=\frac{\hat{T}_{0}}{a}, \label{eq:18}\end{aligned}$$ where $k_{r}=4 \epsilon \sigma \hat{T}_{0}^3 a$ is the effective radiative conductivity for the spherical inclusions, and $\Lambda=k/k_{r}$ plays the role of a conduction-to-radiation-ratio parameter. We note that this expression for $k_{r}$ can be obtained by linearising the Stefan-Boltzmann law for radiative heat flux and comparing it with a Fourier heat flux with an effective thermal conductivity $k_{r}$. Before we proceed, we note that with - is the unique solution for $\hat{T}_{1}$. The proof is non-standard, and we record it in Appendix A.
Having found $C_{1}$ in , we repeat the analysis from Section \[sec:2\] to obtain that the coefficient for the far-field behaviour due to $n$ spheres is $$n\delta C_{1}(a,k,k_{r})=\frac{k-k_{r}}{2k+k_{r}}\phi A^3 T'_{\infty},
\label{eq:24}$$ which has the same functional form as with $k_{2}$ replaced with $k_{r}$ (we note that the factor of $\delta$ comes from the expansion ). Thus, we find that the effective thermal conductivity is given by $$k_{\mathit{eff}}=\frac{2k+k_{r}+2\phi(k_{r}-k)}{2k+k_{r}-\phi(k_{r}-k)}k=\frac{2\Lambda+1+2\phi(1-\Lambda)}{2\Lambda+1-\phi(1-\Lambda)}k.
\label{eq:27}$$ We again note the immediate relation to , with $k_2$ replaced with $k_r$. This comes as no surprise given that we have linearised the radiative heat flux in assuming a small gas void in a uniform temperature gradient. Furthermore, note that if $\phi=0$, then $k_{\mathit{eff}}=k$, which is exactly as expected since this corresponds to the case when there is only the matrix material present. We also note that if $\Lambda=k/k_{r} \gg 1$ (the case where the radiative conductivity is much smaller than the solid one), then we obtain the asymptotic value for the effective thermal conductivity as $$k_{\mathit{eff}}=\frac{2(1-\phi)}{2+\phi}k.
\label{eq:28}$$ Similarly, if $\Lambda \ll 1$, then $$k_{\mathit{eff}}=\frac{1+2\phi}{1-\phi}k.
\label{eq:29}$$ These asymptotic scalings suggest that the effective thermal conductivity under the Maxwell model levels off to the constant value in the high-temperature regime.
Effective medium theory with radiation {#sec:4}
======================================
Having already calculated the relevant coefficients for the far-field behaviour due to the spherical inclusions with radiation in Section \[sec:3\], we straightforwardly generalise the EMT model presented in Section \[sec:2\] to obtain $$\begin{aligned}
k_{\mathit{eff}} & = \frac{1}{4}\left\lbrace\vphantom{\sqrt{(3\phi (k_{r}-k)+(2k-k_{r}))^2}} 3\phi (k_{r}-k)+(2k-k_{r})\right.\\
& \qquad \left. +\sqrt{(3\phi (k_{r}-k)+(2k-k_{r}))^2+8k_{r}k}\right\rbrace \\
& = \frac{k}{4\Lambda}\left\lbrace\vphantom{\sqrt{(3\phi (k_{r}-k)+(2k-k_{r}))^2}} 3\phi (1-\Lambda)+(2\Lambda-1)\right.\\
& \qquad \left. +\sqrt{(3\phi (1-\Lambda)+(2\Lambda-1))^2+8\Lambda}\right\rbrace .
\label{eq:37}
\end{aligned}$$ Note again the expected similarities between and , with $k_r$ replacing $k_2$.
Asymptotic bounds can be obtained in either the large or small $k_r$ limits. If $k_{r} \ll 1$, for example (i.e., $\Lambda \gg 1$), then $$k_{\mathit{eff}} \sim \left(1-\frac{3}{2}\phi\right)k.
\label{eq:38}$$ This is valid for $\phi < 2/3$, which lies in our assumed region of dilute-porosity limit. If $k_{r} \gg 1$ (i.e, $\Lambda \ll 1$), then $$k_{\mathit{eff}}=\frac{k}{1-3\phi},
\label{eq:39}$$ which is valid for $\phi < 1/3$. We thus observe that the topology undergoes a percolation threshold. If we consider when radiation is large, then we see that, for porosity $0 \leq \phi < 1/3$, the conductivity scales with $k$. Thus, for small enough porosity, the effective thermal conductivity under the EMT model levels off to the constant value given in as temperature increases, as was also true of the Maxwell model.
Differential effective medium theory with radiation {#sec:5}
===================================================
We use the analysis in Section \[sec:2\] to generalise the differential effective medium theory and include radiation in the gas phase. This time, expanding our generalised Maxwell model result (which gives the solution in the dilute limit) for small $\phi$, we find $$b(k,k_{r})=\frac{3(k_{r}-k)}{2k+k_{r}}.
\label{eq:45}$$ After performing the integration in the analogue of , we obtain $$\left(\frac{k_{\mathit{eff}}-k_{r}}{k-k_{r}}\right)^3\frac{k}{k_{\mathit{eff}}}=(1-\phi)^3,
\label{eq:46}$$ or $$\left(\frac{\Lambda k_{\mathit{eff}}/k-1}{\Lambda-1}\right)^3\frac{k}{k_{\mathit{eff}}}=(1-\phi)^3,
\label{eq:4612}$$ which again is with $k_{2}$ replaced by $k_{r}$.
{width="0.38\linewidth"} {width="0.38\linewidth"}
We remark that this method relies on being able to incrementally change the medium, which is primarily applicable provided that there is a variety of gas void sizes. This is frequently the case in real-world applications and industrial processes.
As was done for previous models, asymptotic bounds may be obtained in the small or large $k_r$ limits. If $k_r \ll 1$ (i.e., $\Lambda \gg 1$), then we have $$k_{\mathit{eff}}=(1-\phi)^{3/2}k,
\label{eq:47}$$ while if $k_r \gg 1$ (i.e., $\Lambda \ll 1$), then we have $$k_{\mathit{eff}}=\frac{k}{(1-\phi)^3}.
\label{eq:48}$$ The DEMT theory can be seen as a perturbation of the dilute limit, and hence is valid for $\phi \ll 1$. Thus, for reasonable values of porosity, the effective thermal conductivity under the DEMT model will level off to a constant value in the high-temperature regime, as was true of the Maxwell and EMT models.
Results and discussion {#sec:6}
======================
We compare numerically the three models for the effective thermal conductivity due to radiation within the gas voids. We choose parameter values corresponding to those used in [@Kosowska-Golachowska2014], in order to calibrate our model to obtain physically meaningful results. We apply the models obtained in earlier sections to a solid matrix made of anthracite with gas pores inside. We use an anthracite conductivity of $30 \text{W}\text{m}^{-1}\text{K}^{-1}$, $\epsilon=1$, porosity of $\phi=0.1$, and average radii of gas inclusions either $0.01$m or $0.05$m. In Figure \[fig:14\], we compare results from the three models for the two different void radii by plotting the dependence of the effective thermal conductivity on temperature. The first thing to notice is that all three models give effective thermal conductivities which exhibit similar behaviours. The material with smaller gas pores results in greater agreement between the three effective thermal conductivities for a wider range of temperatures, as the radiation effect is less pronounced for smaller gas voids.
All three models predict a saturation in the effective thermal conductivity at high temperatures. This is in contrast to many lumped parameter models, which often predict unbounded growth. The difference is partially due to the fact that there is a solid matrix with gas voids inside, therefore the solid is the rate-limiting factor. Now, comparing this observation with what is seen in the one-dimensional case, which corresponds to gas and solid blocks connected in series, we also find that $k_{\mathit{eff}}=1/(\phi/k_{r}+(1-\phi)/k)\to k/(1-\phi)$ when the temperature $T$ becomes very large. This is an interesting result, which says that if we have a porous solid material with gas bubbles inside (of small void fraction), the effective conductivity will eventually saturate with increasing temperature. This is in contrast to what one might expect if we have a composite material consisting of solid particles dispersed in a gaseous matrix, when the effective conductivity can be highly dependent on temperature (potentially unbounded).
For our given parameters, we also evaluate the conduction-to-radiation parameter to be $\Lambda = k/k_r \approx 0.5$ for radius of $0.01$m and $T=3000$ C, which shows that even for high-temperature regimes (such as those in industrial processes), solid conduction is comparable with radiative effects. This means that radiation does not become dominant. Of course, for voids of a larger radius, $\Lambda$ decreases. For lower temperatures of $T=1000$ C, we have that $\Lambda \approx 10$, while for higher temperatures of $T=5000$ C, we have that $\Lambda \approx 0.1$. Therefore, in these regimes we expect our asymptotic limits , , , , , and may hold. Indeed, according to we should have $k_{\mathit{eff}} \sim 40\text{W}\text{m}^{-1}\text{K}^{-1}$, and this is roughly the saturation value for the effective thermal conductivity (under the Maxwell model with radiation) we observe in Figure \[fig:14\].
Conclusions {#sec:7}
===========
We have considered Maxwell’s model, effective medium theory (EMT), and differential effective medium theory (DEMT) for the effective thermal conductivity of a porous material with small gas-filled voids in which radiation effects within the voids are taken into account. The corresponding expressions giving the relationship between the effective thermal conductivity and temperature, and parametrically depending on the thermal conductivity of the solid and the porosity of the material, are derived. The formulas for the effective thermal conductivity obtained under each model compare naturally with their original counterparts, if one were to use $k_{r}$ in place of the conductivity of the inclusions $k_{2}$, as is intuitively expected, justifying the use of the effective radiative conductivity in these models when considering the case of a solid matrix with gaseous inclusions. Furthermore, the standard formulas without radiation are obtained when the void fraction vanishes, providing another consistency check of our results. Our results justify the use of an effective radiative conductivity, obtained from linearising the Stefan-Boltzmann law.
One interesting finding was that the predicted effective thermal conductivity saturates with increasing temperature; as the temperature increases, the rate of increase of the conductivity of the composite material slows down as the temperature increases, provided the porosity of the bulk material was not too large. Recall that in [@luikov1968thermal] there were a number of experimental results and scaling laws for the thermal conductivity of a range of porous materials, and at high temperatures thermal conductivities were observed to level off for some materials, rather than to increase without bound. (The effective thermal conductivities may have levelled off eventually for other materials, but the range of temperatures was limited in some cases, and was usually below what we considered in Figure \[fig:14\].) Such asymptotically bounded effective thermal conductivity is in contrast to what is observed in certain lumped parameter models, as those models often more crudely approximate the underlying physics. That this saturation occurs has important implications for a variety of industrial processes and may give insight into material design.
Acknowledgments {#acknowledgments .unnumbered}
===============
This publication is based on work supported by the EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with Teknova, where it is part of the research project Electrical Conditions and their Process Interactions in High Temperature Metallurgical Reactors (ElMet), with financial support from The Research Council of Norway and the companies Alcoa, Elkem, and Eramet Norway. K. Kiradjiev thanks Teknova for financial support and the opportunity to work on-site during parts of this project. The authors thank C. M. Rooney and M. Sparta for helpful discussions.
Uniqueness of first-order correction, $\hat{T}_{1}$
===================================================
We prove that the first-order correction $\hat{T}_{1}$ to the temperature field in Maxwell’s model with radiation is unique.
Suppose that there are two solutions for $\hat{T}_{1}$, say, $\hat{T}_{11}$ and $\hat{T}_{12}$. Let $\tilde{T}=\hat{T}_{11}-\hat{T}_{12}$. Then, $$\begin{aligned}
\nabla^2\tilde{T}&=0 \qquad &&\text{ for } \qquad r > a, \label{eq:19}\\
\frac{k}{4 \epsilon \sigma \hat{T}_{0}^3}\dfrac{\partial \tilde{T}}{\partial r}&=\tilde{T}-\int_{\Sigma} \tilde{T} \mathrm{d}F \qquad &&\text{ on } \qquad r=a, \label{eq:20}\\
\tilde{T} &\to 0 \qquad &&\text{ as } \qquad r\to \infty, \label{eq:21}\\
\nabla \tilde{T} &= O(1/r^2) \qquad &&\text{ as } \qquad r\to \infty. \label{eq:22}\end{aligned}$$ Using the Divergence theorem, consider the following: $$\begin{aligned}
0&\leq \int_{\mathbb{R}\setminus V} k|\nabla\tilde{T}|^2\mathrm{d}V=\int_{\mathbb{R}\setminus V} k|\nabla\tilde{T}|^2\mathrm{d}V+\int_{\mathbb{R}\setminus V} k\tilde{T}\nabla^2 \tilde{T} \mathrm{d}V\\ &=\int_{\mathbb{R}\setminus V}\nabla\cdot (k\tilde{T}\nabla\tilde{T})\mathrm{d}V
=\int_{\Sigma} k\tilde{T}\nabla\tilde{T}\cdot(-\mathbf{n})\mathrm{d}S \\ &=-\int_{\Sigma} k\tilde{T}\dfrac{\partial \tilde{T}}{\partial r}\mathrm{d}S=4\epsilon\sigma \hat{T}_{0}^3\int_{\Sigma}\tilde{T}\left(-\tilde{T}+\int_{\Sigma} \tilde{T} \mathrm{d}F\right)\mathrm{d}S\\
&=4 \epsilon\sigma \hat{T}_{0}^3\left(-\int_{\Sigma}\tilde{T}^2 \mathrm{d}S+\frac{1}{4\pi a^2}\int_{\Sigma}\tilde{T} \mathrm{d}S\int_{\Sigma}\tilde{T} \mathrm{d}S\right)\leq 0,
\end{aligned}
\label{eq:125}$$ where $V$ is the region $\{r \leq a \}$, $\mathbf{n}$ is the outwards-pointing unit normal to $\Sigma$, and the last inequality follows from the Cauchy-Schwarz inequality, *viz*., $$\left(\int_{\Sigma}\tilde{T} \mathrm{d}S\right)^2\leq \int_{\Sigma} \mathrm{d}S\int_{\Sigma}\tilde{T}^2 \mathrm{d}S,
\label{eq:126}$$ with $\int_{\Sigma} \mathrm{d}S=4\pi a^2$ being the measure (in this case, surface area) of the sphere. Again because of the Cauchy-Schwarz inequality, equality is possible only when $\tilde{T}$ and $1$ are linearly dependent, i.e., when $\tilde{T}$ is a constant. Using , we obtain $\tilde{T}=0$. Therefore, the solution for $\hat{T}_{1}$ is unique.
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[^1]: The extension to include gas conductivity is straightforward but lengthy.
|
---
abstract: 'It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.'
author:
- |
Robert S. Lubarsky\
Dept. of Mathematical Sciences\
Florida Atlantic University\
Boca Raton, FL 33431\
[email protected]
title: On the Cauchy Completeness of the Constructive Cauchy Reals
---
Introduction
============
Are the reals Cauchy complete? This is, for every Cauchy sequence of real numbers, is there a real number which is its limit?
This sounds as though the answer should be “of course". After all, the reals are defined pretty much to make this true. The reason to move from the rationals to the reals is exactly to “fill in the holes" that the rationals have. So however you define ${\mathbb R}$, you’d think its Cauchy completeness would be immediate. At the very least, this property would be a litmus test for any proffered definition.
In fact, for the two most common notions of real number, Dedekind and Cauchy real, this is indeed the case, under classical logic. First off, classically Cauchy and Dedekind reals are equivalent anyway. Then, taking a real as an equivalence class of Cauchy sequences, given any Cauchy sequence of reals, a canonical representative can be chosen from each real, and a limit real can be built from them by a kind of diagonalization, all pretty easily.
Constructively, though, this whole procedure breaks down. Starting even at the beginning, Cauchy and Dedekind reals are no longer equivalent notions (see [@FH] or [@LR]). While the Dedekind reals are complete, working with the Cauchy reals, it’s not clear that a representative can be chosen from each equivalence class, or, even if you could, that a limit could be built from them by any means.
It is the purpose of this paper to show that, indeed, such constructions are not in general possible, answering a question of Martin Escardo and Alex Simpson [@ES].
While the original motivation of this work was to show the final theorem, that the Cauchy reals are not provably Cauchy complete, it is instructive to lay out the framework and examine the related questions so laid bare. A Cauchy real is understood as an equivalence class of Cauchy sequences of rationals. When working with a Cauchy sequence, one usually needs to know not only that the sequence converges, but also how fast. In classical set theory, this is definable from the sequence itself, and so is not problematic. The same cannot be said for other contexts. For instance, in recursion theory, the complexity of the convergence rate might be important. In our context, constructive (a.k.a. intuitionistic) set theory, the standard way to define a modulus of convergence just doesn’t work. Certainly given a Cauchy sequence $X(n)$ and positive rational $\epsilon$, there is an integer $N$ such that for $m, n > N \; \mid X(n) - X(m) \mid <
\epsilon$: that’s the definition of a Cauchy sequence. A modulus of convergence is a function $f$ such that for any such $\epsilon$ $f$ on $\epsilon$ returns such an $N$. Classically this is easy: let $f$ return the least such $N$. Intuitionistically that won’t work. And there’s no obvious alternative. So a real is taken to be an equivalence class of pairs $\langle X, f \rangle$, where $X$ is a Cauchy sequence and $f$ a modulus of convergence.
One immediate source of confusion here is identifying reals with sequence-modulus pairs. A real is an equivalence class of such pairs, and it is not obvious how a representative can be chosen constructively from each real; in fact, this cannot in general be done, as we shall see. This distinction has not always be made though. For instance, as observed by Fred Richman, in [@TvD], the Cauchy completeness of the reals was stated as a theorem, but what was proved was the Cauchy completeness of sequence-modulus pairs. To be precise, what was shown was that, given a countable sequence, with its own modulus of convergence, of sequence-modulus pairs, then there is a limit sequence, with modulus. For that matter, it is not hard (and left to the reader) that, even if the given sequence does not come equipped with its own modulus, it still has a Cauchy sequence as a limit, although we will have to punt on the limit having a modulus. But neither of those two observations is the Cauchy completeness of the reals.
Nonetheless, these observations open up the topic about what kinds of behavior in the limit one can expect given certain input data. There are two, independent parameters. Does the outside Cauchy sequence have a modulus of convergence? And are its individual members sequence-modulus pairs, or merely naked sequences? Notice that, while the first question is yes-no, the second has a middle option: the sequences have moduli, but not uniformly. Perhaps each entry in the big sequence is simply a Cauchy sequence of rationals, and it is hypothesized to have a modulus somewhere, with no information about the modulus given. These possibilities are all summarized in the following table.
Entries in the outside Cauchy sequence are:
------------- --------------- --------------------- --------------------
seq-mod pairs seqs that have mods seqs that may not
somewhere have mods anywhere
Outside seq
has a mod
Outside seq
doesn’t
have a mod
------------- --------------- --------------------- --------------------
\
Not to put the cart before the horse, is it even possible to have a Cauchy sequence with no modulus of convergence, or with only non-uniform moduli? It has already been observed that the obvious classical definition of such does not work intuitionistically, but it still remains to be shown that no such definition is possible.
The goal of this paper is to prove the negative results as much as possible, that any given hypothesis does not show that there is a limit Cauchy sequence, or in certain cases that there is no limit with a modulus.
The positive results are all easy enough and so are left as exercises. They are:
1. Every Cauchy sequence with modulus of sequence-modulus pairs has a limit sequence with modulus.
2. Every Cauchy sequence of sequence-modulus pairs has a limit sequence.
3. Every Cauchy sequence with modulus of Cauchy sequences has a limit sequence.
In tabular form, the positive results are:
Entries in the outside Cauchy sequence are:\
------------- ----------------- --------------------- --------------------
seq-mod pairs seqs that have mods seqs that may not
somewhere have mods anywhere
Outside seq There is a There is a There is a
has a mod limit with mod. limit sequence. limit sequence.
Outside seq
doesn’t There is a
have a mod limit sequence.
------------- ----------------- --------------------- --------------------
\
Regarding the first and last columns, the negative results are that these are the positive results cited above are the best possible. In detail:\
[**Theorem \[thm1\]**]{} [*IZF$_{Ref}$ does not prove that every Cauchy sequence has a modulus of convergence. It follows that IZF$_{Ref}$ does not prove that every Cauchy sequence of sequence-modulus pairs converges to a Cauchy sequence with a modulus of convergence.*]{}\
[**Theorem \[thm2\]**]{} [*IZF$_{Ref}$ does not prove that every Cauchy sequence of Cauchy sequences converges to a Cauchy sequence.*]{}\
[**Theorem \[thm3\]**]{} [*IZF$_{Ref}$ does not prove that every Cauchy sequence with a modulus of convergence of Cauchy sequences converges to a Cauchy sequence with a modulus of convergence.*]{}\
The middle column is discussed briefly in the questions in the last section of this paper.
In tabular form, these negative results are:
Entries in the outside Cauchy sequence are:
------------- --------------- --------------------- ---------------------
seq-mod pairs seqs that have mods seqs that may not
somewhere have mods anywhere
Outside seq See questions, Limit may not have
has a mod section 7. a mod. thm \[thm3\]
Outside seq Limit may not There may not
doesn’t have a mod. See questions, even be a limit.
have a mod thm \[thm1\] section 7. thm \[thm2\]
------------- --------------- --------------------- ---------------------
\
Then there is the major negative result, the original and ultimate motivation of this work:\
[**Theorem \[thm4\]**]{} [*IZF$_{Ref}$ does not prove that every Cauchy sequence of reals has a limit.*]{}\
Recalling that a real is here taken as an equivalence class of sequence-modulus pairs, to prove this result it would suffice to construct a Cauchy sequence (perhaps itself without modulus) of reals, with no sequence-modulus pair as a limit. We will do a tad better, constructing a Cauchy sequence, with modulus, of reals, with no Cauchy sequence, even without modulus, as a limit.
At this point a word about the meta-theory is in order. The results here are stated as non-theorems of IZF$_{Ref}$, which is the variant of IZF in which the Collection schema is replaced by the Reflection schema. The point is that these independence results are not meant to be based on a weakness of the underlying set theory. Hence the set theory taken is the strongest version of the intuitionistic theories commonly considered. The results would remain valid if IZF were augmented by yet stronger hypotheses, such as large cardinals. Of course, these remarks do not apply if IZF is augmented by whatever choice principle would be enough to build the sequences and moduli here claimed not to exist. Clearly Dependent Choice is strong enough for everything at issue here: choosing representatives from equivalence classes, building moduli, constructing Cauchy sequences. The question what weaker choice principle/s would suffice is addressed in [@LRi].
Regarding the methodology, counter-examples are constructed in each case. These examples could be presented as either topological or Kripke models. While certain relations among topological and Kripke models are known, it is not clear to the author that the natural models in the cases before us are really the same. While the family resemblance is unmistakable (compare sections 2 and 3 below), it is, for example, at best nonobvious where the non-standard integers in the Kripke models are hidden in the topological ones. A better understanding of the relations among Kripke and topological models would be a worthwhile project for some other time. For now, we would like to present the reader with adequate information without being long-winded. Hence all of the constructions will be presented as topological models, since there is better technology for dealing with them. In particular, there is already a meta-theorem (see [@G]) that the (full) model over any Heyting algebra models IZF (easily, IZF$_{Ref}$ too). So we will never have to prove that our topological models satisfy IZF$_{Ref}$. In contrast, we know of no such meta-theorem that would apply to the Kripke models in question. In the simplest case, the first theorem, the Kripke model will also be given, so the reader can see what’s going on there. But even a cursory glance at that argument should make it clear why the author does not want to repeat the proofs of IZF$_{Ref}$ and all the auxiliary lemmas, and the reader likely does not want to read them, three more times.
One last word about notation/terminology. For p an open set in a topological space and $\phi$ a formula in set theory (possibly with parameters from the topological model), “$p \subseteq \|
\phi \|", ``p \Vdash \phi$", and “$p$ forces $\phi$" all mean the same thing. Also, “WLOG" stands for “without loss of generality".
Not every Cauchy sequence has a modulus of convergence
======================================================
\[thm1\] IZF$_{Ref}$ does not prove that every Cauchy sequence has a modulus of convergence. It follows that IZF$_{Ref}$ does not prove that every Cauchy sequence of sequence-modulus pairs converges to a Cauchy sequence with a modulus of convergence.
The second assertion follows immediately from the first: given a Cauchy sequence $X(n)$, for each $n$ let $X_n$ be the constant sequence $X(n)$ paired with some modulus of convergence independent of $n$. Sending $X$ to the sequence $\langle X_n \mid
n \in {\bf N} \rangle$ embeds the Cauchy reals into Cauchy sequences of sequence-modulus pairs. If provably every one of the latter had a modulus, so would each of the former.
To prove the first assertion, we will build a topological model with a specific Cauchy sequence $Z(n)$ of rationals with no modulus of convergence.
The topological space $T$ consists of all Cauchy sequences of rationals.
A basic open set is given by $(p, I)$, where $p$ is a finite sequence of rationals and $I$ is an open interval. A Cauchy sequence $X$ is in (the open set determined by) $(p, I)$ if $p
\subseteq X$, rng($X \backslash p) \subseteq I$, and lim($X$) $\in
I$. (Notice that, under this representation, the whole space $T$ is given by ($\emptyset, {\mathbb R}$), and the empty set is given by $(p, \emptyset)$ for any $p$.)
For this to generate a topology, it suffices to show that the basic open sets are closed under intersection. Given $(p, I)$ and $(q, J)$, if $p$ and $q$ are not compatible (i.e. neither is an extension of the other), then $(p, I) \cap (q, J) = \emptyset$. Otherwise WLOG let $q \supseteq p$. If rng($q \backslash p) \not
\subseteq I$ then again $(p, I) \cap (q, J) = \emptyset$. Otherwise $(p, I) \cap (q, J) = (p \cup q, I \cap J) = (q, I \cap
J)$.
Let $M$ be the Heyting-valued models based on $T$, as describes in e.g. [@G]. Briefly, a set in $M$ is a collection of objects of the form $\langle \sigma, (p, I) \rangle$, where $\sigma$ inductively is a set in $M$. It is shown in [@G] that $M
\models$ IZF$_{Coll}$ (assuming IZF$_{Coll}$ in the meta-theory). Similarly, assuming IZF$_{Ref}$ in the meta-theory yields $M$ $\models$ IZF$_{Ref}$.
We are interested in the term $\{ \langle {\bar p}, (p, I) \rangle
\mid (p, I)$ is an open set$\}$. (Here ${\bar p}$ is the canonical name for $p$. Each set in $V$ has a canonical name in $M$ by choosing $(p, I)$ to be ($\emptyset, {\mathbb R}$) hereditarily: ${\bar x} = \{ \langle {\bar y}, (\emptyset, {\mathbb R}) \rangle
\mid y \in x \}$.) We will call this term $Z$.
$\|$ Z is a Cauchy sequence $\|$ = T.
To see that $\|Z$ is total$\| = T$, let $N$ be an integer. (Note that each integer in $M$ can be identified locally with an integer in $V$. For notational ease, we will identify integers in $M$ and $V$.) Let $p$ be any sequence of rationals of length $> N$. Then $(p, {\mathbb R}) \subseteq \| Z(N) = p(N) \| \subseteq \| N \in$ dom($Z) \|$. As $T$ is covered by the open sets of that form, $T$ $\subseteq \| N \in$ dom($Z) \|$. That $Z$ is a function is similarly easy.
As for $Z$ being Cauchy, again let $N$ be an integer and $X$ be in $T$. Since $X$ is Cauchy, there is an integer $M$ such that beyond $M$ $X$ stays within an interval $I$ of size 1/(2$N$). Of course, $X$’s limit might be an endpoint of $I$. So let $J$ extend $I$ on either side and still have length less that 1/$N$. Then $X \in (X
\upharpoonright M, J) \subseteq \| \forall m, n > M \mid Z(m) -
Z(n) \mid \; \leq 1/N \|$, making $Z$ “Cauchy for 1/$N$", to coin a phrase.
In order to complete the theorem, we need only prove the following
$\|$Z has no modulus of convergence$\|$ = T.
Suppose $(p, I) \subseteq \| f$ is a modulus of convergence for $Z\|$. WLOG $I$ is a finite interval. Let $n$ be such that 1/$n$ is less than the length of $I$, and let $\epsilon$ be (length($I$) - 1/$n$)/2.
Let $(q, J) \subseteq (p, I)$ force a value $m$ for $f(n)$. WLOG length($q$) $> m$, as $q$ could be so extended. If $J=I$, then $(q, J)$ could be extended simply by extending $q$ with two values a distance greater than 1/$n$ apart, thereby forcing $f$ not to be a modulus of convergence. So $J \subset I$. That means either inf $J$ $>$ inf $I$ or sup $J$ $<$ sup $I$. WLOG assume the latter. Let mid $J$ be the midpoint of $J$, $q_0$ be $q$ extended by mid $J$, and $J_0$ be (mid $J$, sup $J$). Then ($q_0, J_0) \subseteq
(q, J)$, and therefore ($q_0, J_0) \subseteq \|f(n) = m\|$.
What ($q_1, J_1$) is depends:
CASE I: There is an open set $K$ containing sup($J_0$) such that ($q_0, K) \subseteq \|f(n) = m\|$. Then let $j_{max}$ be the sup of the right-hand endpoints (i.e. sups) of all such $K$’s. Let $q_1$ be $q_0$ extended by sup $J_0$.
: ($q_1$, (sup $J_0, j_{max})) \subseteq \|f(n) =
m\|$.
Let $j \in$ (sup $J_0, j_{max}$). By hypothesis, there is a $K$ such that ($q_1$, (sup $J_0, j)) \subseteq (q_1, K) \subseteq
\|f(n) = m\|$. Since ($q_1$, (sup $J_0, j_{max}$)) is the union of the various ($q_1$, (sup $J_0, j$))’s over such $j$’s, the claim follows.
Of course, $J_1$ will be (sup $J_0, j_{max}$).
CASE II: Not Case I. Then extend by the midpoint again. That is, $q_1$ is $q_0$ extended by mid $J_0$, and $J_1$ is (mid $J_0$, sup $J_0$). Also in this case, ($q_1, J_1) \Vdash f(n) = m$.
Clearly we would like to continue this construction. The only thing that might be a problem is if the right-hand endpoint of some $J_k$ equals (or goes beyond!) sup $I$, as we need to stay beneath $(p, I)$. In fact, as soon as sup($J_k) >$ sup($I$) - $\epsilon$ (if ever), extend $q_k$ by something within $\epsilon$ of sup $I$, and continue the construction with left and right reversed. That is, instead of going right, we now go left. This is called “turning around".
What happens next depends.
CASE A: We turned around, and after finitely many more steps, some $J_k$ has its inf under inf($I$) + $\epsilon$. Then extend $q_k$ by something within $\epsilon$ of inf $I$. This explicitly blows $f$ being a modulus of convergence for $Z$.
CASE B: Not Case A. So past a certain point (either the stage at which we turned around, or, if none, from the beginning) we’re marching monotonically toward one of $I$’s endpoints, but will always stay at least $\epsilon$ away. WLOG suppose we didn’t turn around. Then the construction will continue for infinitely many stages. The $q_k$’s so produced will in the limit be a (monotonic and bounded, hence) Cauchy sequence $X$. Furthermore, lim $X$ is the limit of the sup($J_k$)’s. Finally, $X \in (p, I)$. Hence there is an open set $(q', K)$ with $X \in (q', K) \subseteq
\|f(n) = m'\|$, for some $m'$. Let $k$ be such that sup($J_k$) $\in K$, and $q_k \supseteq q'$. Consider ($q_k, J_k$). Note that ($q_k, J_k \cap K$) extends both ($q_k, J_k$) and $(q', K)$, hence forces both $f(n) = m$ and $f(n) = m'$, which means that $m=m'$.
Therefore, at this stage in the construction, we are in Case I. By the construction, $J_{k+1}$ = (sup $J_k, j_{max}$), where $j_{max}
\geq$ sup $K >$ lim $X$ = lim$_k$ (sup($J_k$)) $\geq$ sup($J_{k+1}$) = $j_{max}$, a contradiction.
Same theorem, Kripke model version
==================================
[**Theorem \[thm1\]**]{} [*IZF$_{Ref}$ does not prove that every Cauchy sequence has a modulus of convergence.*]{}
To repeat the justification given in the introduction, even though the theorem proved in this section is exactly the same as in the previous, this argument is being given for methodological considerations. The prior construction is topological, the coming one is a Kripke model, and it is not clear (at least to the author) how one could convert one to the other, either mechanically via a meta-theorem or with some human insight. Hence to help develop the technology of Kripke models, this alternate proof is presented.
Construction of the Model
-------------------------
Let $M_{0} \prec M_{1} \prec$ ... be an $\omega$-sequence of models of ZF set theory and of elementary embeddings among them, as indicated, such that the sequence from $M_{n}$ on is definable in $M_{n}$, and such that each thinks that the next has non-standard integers. Notice that this is easy to define (mod getting a model of ZF in the first place): an iterated ultrapower using any non-principal ultrafilter on $\omega$ will do. We will ambiguously use the symbol f to stand for any of the elementary embeddings inherent in the M$_{n}$-sequence.
The Kripke model $M$ will have underlying partial order a non-rooted tree; the bottom node (level 0) will have continuum (in the sense of $M_0$) many nodes, and the branching at a node of level $n$ will be of size continuum in the sense of $M_{n+1}$. (We will eventually name each node by associating a Cauchy sequence to it. Some motivation will be presented during this section, and the final association will be at the end of this section.) Satisfaction at a node will be indicated with the symbol $\models$. There is a ground Kripke model, which, at each node of level $n$, has a copy of $M_{n}$. The transition functions (from a node to a following node) are the elementary embeddings given with the original sequence of models (and therefore will be notated by $f$ again). Note that by the elementarity of the extensions, this Kripke model is a model of classical ZF. More importantly, the model restricted to any node of level $n$ is definable in $M_{n}$, because the original $M$-sequence was so definable.
The final model $M$ will be an extension of the ground model that will be described like a forcing extension. That is, $M$ will consist of (equivalence classes of) the terms from the ground model. The terms are defined at each node separately, inductively on the ordinals in that model. At any stage $\alpha$, a term of stage $\alpha$ is a set $\sigma$ of the form $\{ \langle
\sigma_{j}, (p_{j}, I_{j}) \rangle \mid j \in J \}$, where $J$ is some index set, each $\sigma_{i}$ is a term of stage $< \alpha$, each $p_{j}$ is a finite function from ${\mathbb N}$ to ${\mathbb
Q}$, and each $I_{j}$ is an open rational interval on the real line. Note that all sets from the ground model have canonical names, by choosing each $p_{j}$ to be the empty function and $I_j$ to be the whole real line, hereditarily.
Notice also that the definition of the terms given above will be interpreted differently at each node of the ground Kripke model, as the ${\mathbb N}$ and ${\mathbb Q}$ change from node to node. However, any term at a node gets sent by the transition function $f$ to a corresponding term at any given later node. The definitions given later, such as the forcing relation $\Vdash$, are all interpretable in each $M_{n}$, and coherently so, via the elementary embeddings.
As a condition, each finite function $p$ is saying “the Cauchy sequence includes me", and each interval $I$ is saying “future rationals in the Cauchy sequence have to come from me". For each node of level $n$ there will be an associated Cauchy sequence $r$ (in the sense of $M_{n}$) such that at that node the true $p$’s and $I$’s will be those compatible with $r$ (or, perhaps, those with which $r$ is compatible, as the reader will). You might reasonably think that compatibility means “$p \subset r$ and rng($r \backslash p) \subseteq I$": roughly, “$r$ extends $p$, and anything in $r$ beyond $p$ comes from $I$". But that’s not quite right. Consider the Cauchy sequence $r(n) = 1/n \; (n \geq$ 1). rng$(r) \subseteq$ (0, 2), but in a non-standard extension, $r$’s pattern could change at a non-standard integer; at that point, it would be too late for $r$ to change by a standard amount, but it could change by an infinitesimal amount. So the range of $r$ could include (infinitely small) negative numbers, which are outside of (0, 2). Hence we have the following
A condition (p, I) and a Cauchy sequence r are if p $\subseteq$ r, rng(r$\backslash$p) $\subseteq$ I, and lim(r) $\in$ I.
(p, I) is compatible with a finite function q if p $\subseteq$ q and rng(q$\backslash$p) $\subseteq$ I.
Given this notion of compatibility, speaking intuitively here, a term $\sigma$ can be thought of as being interpretable (with notation $\sigma^{r}$) inductively in $M_{n}$ as $\{
\sigma_{j}^{r} \mid \langle \sigma_{j}, p_j, I_{j} \rangle \in
\sigma$ and $r$ is compatible with ($p_j, I_j) \}$. (This notion is hidden in the more formal development below, where we define and then mod out by =$_M$.)
Our next medium-term goal is to define the primitive relations at each node, =$_{M}$ and $\in_{M}$ (the subscript being used to prevent confusion with equality and membership of the ambient models $M_{n}$). In order to do this, we need first to develop our space’s topology.
(q, J) $\leq$ (p, I) ((q, J) (p, I)) if q $\supseteq$ p, J $\subseteq$ I, and rng(q$\backslash$p) $\subset$ I.
[*C*]{} = $\{ (p_j, I_j) \mid j \in J \}$ (p, I) if each (p$_j$, I$_j$) extends (p, I) and each Cauchy sequence r compatible with (p, I) is compatible with some (p$_j$, I$_j$).
$\leq$ induces a notion of compatibility of conditions (having a common extension). We say that a typical member $\langle \sigma,$ (p, I) $\rangle$ of a term is with (q, J) if (p, I) and (q, J) are compatible.
We need some basic facts about this p.o., starting with the fact that it is a p.o.
1. $\leq$ is reflexive, transitive, and anti-symmetric.
2. If (p, I) and (q, J) are each compatible with a Cauchy sequence r, then they are compatible with each other.
3. If (p, I) and (q, J) are compatible, then their glb in the p.o. is (p$\cup$q, I$\cap$J).
4. {(p, I)} covers (p, I).
5. A cover of a cover is a cover. That is, if [*C*]{} covers (p, I), and, for each (p$_j$, I$_j$) $\in$ [*C*]{}, [*C*]{}$_j$ covers (p$_j$, I$_j$), then $\bigcup_j$[*C*]{}$_j$ covers (p, I).
6. If [*C*]{} covers (p, I) and (q, J) $\leq$ (p, I), then (q, J) is covered by [*C*]{} $\wedge$ (q, J) =$_{def}$ {(p$_j\cup$q, I$_j\cap$J) $\mid$ (q, J) is compatible with (p$_j$, I$_j$) $\in$ [*C*]{}}.
Left to the reader.
Now we are in a position to define =$_{M}$ and $\in_{M}$. This will be done via a forcing relation $\Vdash$.
$(p, I) \Vdash \sigma =_{M} \tau$ and $(p, I)
\Vdash \sigma \in_{M} \tau$ are defined inductively on $\sigma$ and $\tau$, simultaneously for all $(p, I)$:
$(p, I) \Vdash \sigma =_{M} \tau$ iff for all $\langle \sigma_{j}, (p_{j}, I_j) \rangle \in \sigma$ compatible with (p, I) $(p \cup p_j, I \cap I_{j}) \Vdash \sigma_{j} \in_{M} \tau$ and vice versa, and
$(p, I) \Vdash \sigma \in_{M} \tau$ iff there is a cover [*C*]{} of (p, I) such that for all $(p_j, I_j) \in {\it C}$ there is a $\langle \tau_{k}, (p_{k}, I_k) \rangle \in \tau$ such that $(p_j,
I_j) \leq (p_k, I_k)$ and $(p_j, I_j) \Vdash \sigma =_{M}
\tau_{k}$.
(We will later extend this forcing relation to all formulas.)
At a node (with associated real r), for any two terms $\sigma$ and $\tau$, $\sigma =_{M} \tau$ iff, for some (p, I) compatible with r, (p, I) $\Vdash \sigma =_{M} \tau$.
Also, $\sigma \in_{M} \tau$ iff for some (p, I) compatible with r, (p, I) $\Vdash \sigma \in_{M} \tau$.
Thus we have a first-order structure at each node.
The transition functions are the same as before. That is, if $\sigma$ is an object at a node, then it’s a term, meaning in particular it’s a set in some $M_{n}$. Any later node has for its universe the terms from some $M_{m}, m \geq n$. With $f$ the elementary embedding from $M_{n}$ to $M_{m}$, $f$ can also serve as the transition function between the given nodes. These transition functions satisfy the coherence conditions necessary for a Kripke model.
To have a Kripke model, $f$ must also respect =$_{M}$ and $\in_{M}$, meaning that $f$ must be an =$_{M}$- and $\in_{M}$-homomorphism (i.e. $\sigma =_{M} \tau \; \rightarrow \;
f(\sigma$) =$_{M} f(\tau$), and similarly for $\in_{M}$). In order for these to be true, we need an additional restriction on the model. By way of motivation, one requirement is, intuitively speaking, that the sets $\sigma$ can’t shrink as we go to later nodes. That is, once $\sigma_{j}$ gets into $\sigma$ at some node, it can’t be thrown out at a later node. $\sigma_{j}$ gets into $\sigma$ because $r$ is compatible with $(p_j, I_{j})$ (where $\langle \sigma_j, (p_j, I_j) \rangle \in \sigma$). So we need to guarantee that if $r$ and $(p, I)$ are compatible and $r'$ is associated to any extending node then $r'$ and $(p, I)$ are compatible for any condition $(p, I)$. This holds exactly when $r'$ extends $r$ and all of the entries in $r'\backslash r$ are infinitesimally close to lim($r$). This happens, for instance, when $r' = f(r)$. Other such examples would be $f(r)$ truncated at some non-standard place and arbitrarily extended by any Cauchy sequence through the reals with standard part lim($r$); in fact, all such $r'$ have that form. We henceforth take this as an additional condition on the construction: once $r$ is associated to a node, then for any $r'$ associated to an extending node, rng($r'\backslash r$) must consist only of rationals infinitely close to lim($r$).
f is an $=_{M}$ and $\in_{M}$-homomorphism.
If $\sigma =_{M} \tau$ then let $(p, I)$ compatible with $r$ witness as much. At any later node, $(p, I) = f((p, I)) = (f(p),
f(I)) \Vdash f(\sigma) =_{M} f(\tau)$. Also, the associated real $r'$ would still be compatible with $(p, I)$. So the same $(p, I)$ would witness $f(\sigma) =_{M} f(\tau)$ at that node. Similarly for $\in_{M}$.
We can now conclude that we have a Kripke model.
\[equalitylemma\] This Kripke model satisfies the equality axioms:
1. $\forall x \; x=x$
2. $\forall x, y \; x=y \rightarrow y=x$
3. $\forall x, y, z \; x=y \wedge y=z \rightarrow x=z$
4. $\forall x, y, z \; x=y \wedge x \in z \rightarrow y \in z$
5. $\forall x, y, z \; x=y \wedge z \in x \rightarrow z \in y.$
1: It is easy to show with a simultaneous induction that, for all $(p, I)$ and $\sigma$, $(p, I)$ $\Vdash \sigma =_{M} \sigma$, and, for all $\langle \sigma_{j}, (p_j, I_j) \rangle \in \sigma$ compatible with $(p, I), (p \cup p_j, I \cap I_{j}) \Vdash
\sigma_{i} \in_{M} \sigma$.
2: Trivial because the definition of $(p, I)$ $\Vdash \sigma =_{M}
\tau$ is itself symmetric.
3: For this and the subsequent parts, we need some lemmas.
If (p’, I’) $\leq$ (p, I) $\Vdash \sigma =_{M}
\tau$ then (p’, I’) $\Vdash \sigma =_{M} \tau$, and similarly for $\in_{M}$.
By induction on $\sigma$ and $\tau$.
If (p, I) $\Vdash \rho =_{M} \sigma$ and (p, I) $\Vdash
\sigma =_{M} \tau$ then (p, I) $\Vdash \rho =_{M} \tau$.
Again, by induction on terms.
Returning to proving property 3, the hypothesis is that for some $(p, I)$ and $(q, J)$ each compatible with $r$, $(p, I) \Vdash
\rho =_{M} \sigma$ and $(q, J) \Vdash \sigma =_{M} \tau$. By the first lemma, $(p \cup q, I \cap J) \Vdash \rho =_{M} \sigma,
\sigma =_{M} \tau$, and so by the second, $(p \cup q, I \cap J)
\Vdash \rho =_{M} \tau$. Also, $(p \cup q, I \cap J)$ is compatible with $r$.
4: Let $(p, I) \Vdash \rho =_{M} \sigma$ and $(q, J) \Vdash \rho
\in_{M} \tau$. We will show that $(p \cup q, I \cap J) \Vdash
\sigma \in_{M} \tau$. Let [*C*]{} be a cover of $(q, J)$ witnessing $(q, J) \Vdash \rho \in_{M} \tau$. We will show that $(p \cup q, I \cap J) \wedge {\it C} = (p, I) \wedge {\it C}$ is a cover of $(p \cup q, I \cap J)$ witnessing $(p \cup q, I \cap J)
\Vdash \sigma \in_{M} \tau$. Let $(q_i, J_i) \in$ [*C*]{} and $\langle \tau_k, p_k, I_k \rangle$ be the corresponding member of $\tau$. By the first lemma, $(p \cup q_i, I \cap J_{i}) \Vdash
\rho =_{M} \sigma$, and so by the second, $(p \cup q_i, I \cap
JS_{i}) \Vdash \sigma =_{M} \tau_{k}$.
5: Similar, and left to the reader.
With this lemma in hand, we can now mod out by =$_{M}$, so that the symbol “=" is interpreted as actual set-theoretic equality. We will henceforth drop the subscript $_{M}$ from = and $\in$, although we will not distinguish notationally between a term $\sigma$ and the model element it represents, $\sigma$’s equivalence class.
At this point, we need to finish specifying the model in detail. What remains to be done is to associate a Cauchy sequence to each node. At the bottom level, assign each Cauchy sequence from $M_0$ to exactly one node. Inductively, suppose we chose have the sequence $r$ at a node with ground model $M_{n}$. There are continuum-in-the-sense-of-$M_{n+1}$-many immediate successor nodes. Associate each possible candidate $r'$ in $M_{n+1}$ with exactly one such node. (As a reminder, that means each member of rng($r'$$\backslash$$r$) is infinitely close to lim($r$).)
By way of notation, a node will be named by its associated sequence. Hence “$r$ $\models \phi$" means $\phi$ holds at the node with sequence $r$.
Note that, at any node of level $n$, the choice of $r$’s from that node on is definable in $M_{n}$. This means that the evaluation of terms (at and beyond the given node) can be carried out over $M_{n}$, and so the Kripke model (from the given node on) can be defined over $M_{n}$, truth predicate and all.
The Forcing Relation
--------------------
Which $(p, I)$’s count as true determines the interpretation of all terms, and hence of truth in the end model. We need to get a handle on this. As with forcing, we need a relation $(p, I) \Vdash
\phi$ which supports a truth lemma. Note that, by elementarity, it doesn’t matter in which classical model $M_n$ or at what node in the ground Kripke model $\Vdash$ is being interpreted (as long as the parameters are in the interpreting model, of course).
(p, I) $\Vdash \phi$ is defined inductively on $\phi$:
(p, I) $\Vdash \sigma =_{M} \tau$ iff for all $\langle \sigma_{j}, (p_{j}, I_j) \rangle \in \sigma$ compatible with (p, I) $(p \cup p_j, I \cap I_{j}) \Vdash \sigma_{j} \in_{M} \tau$ and vice versa
(p, I) $\Vdash \sigma \in_{M} \tau$ iff there is a cover [*C*]{} of (p, I) such that for all $(p_j, I_j) \in {\it C}$ there is a $\langle \tau_{k}, (p_{k}, I_k) \rangle \in \tau$ such that $(p_j,
I_j) \leq (p_k, I_k)$ and $(p_j, I_j) \Vdash \sigma =_{M}
\tau_{k}$.
(p, I) $\Vdash \phi \wedge \psi$ iff (p, I) $\Vdash \phi$ and (p, I) $\Vdash
\psi$
(p, I) $\Vdash \phi \vee \psi$ iff there is a cover [*C*]{} of (p, I) such that, for each (p$_j$, I$_j$) $\in$ [*C*]{}, (p$_j$, I$_j$) $\Vdash \phi$ or (p$_j$, I$_j$) $\Vdash \psi$
(p, I) $\Vdash \phi \rightarrow \psi$ iff for all (q, J) $\leq$ (p, I) if (q, J) $\Vdash \phi$ then (q, J) $\Vdash \psi$
(p, I) $\Vdash \exists x \; \phi(x)$ iff there is a cover [*C*]{} of (p, I) such that, for each (p$_j$, I$_j$) $\in$ [*C*]{}, there is a $\sigma$ such that (p$_j$, I$_j$) $\Vdash
\phi(\sigma)$
(p, I) $\Vdash \forall x \; \phi(x)$ iff for all $\sigma$ (p, I) $\Vdash \phi(\sigma)$
\[helpful lemma\]
1. If (q, J) $\leq$ (p, I) $\Vdash \phi$ then (q, J) $\Vdash \phi$.
2. If [*C*]{} covers (p, I), and (p$_j$, I$_{j}$) $\Vdash \phi$ for all (p$_j$, I$_{j}$) $\in$ [*C*]{}, then (p, I) $\Vdash \phi$.
3. (p, I) $\Vdash \phi$ iff for all r compatible with (p, I) there is a (q, J) compatible with r such that (q, J) $\Vdash \phi$.
4. Truth Lemma: For any node r, r $\models \phi$ iff (p, I) $\Vdash \phi$ for some (p, I) compatible with r.
1\. A trivial induction, using of course the earlier lemmas about $\leq$ and covers.
2\. Easy induction. The one case to watch out for is $\rightarrow$, where you need to invoke the previous part of this lemma.
3\. Trivial, using 2.
4\. By induction on $\phi$, in detail for a change.
In all cases, the right-to-left direction (“forced implies true") is pretty easy, by induction. (Note that only the $\rightarrow$ case needs the left-to-right direction in this induction.) Hence in the following we show only left-to-right (“if true at a node then forced").
=: This is exactly the definition of =.
$\in$: This is exactly the definition of $\in$.
$\wedge$: If $r \models \phi \wedge \psi$, then $r \models \phi$ and $r \models \psi$. Inductively let $(p, I) \Vdash \phi$ and $(q, J) \Vdash \psi$, where $(p, I)$ and $(q, J)$ are each compatible with $r$. That means that $(p, I)$ and $(q, J)$ are compatible with each other, and $(p \cup q, I \cap J)$ suffices.
$\vee$: If $r \models \phi \vee \psi$, then WLOG $r \models \phi$ . Inductively let $(p, I)$ $\Vdash \phi$, $(p, I)$ compatible with $r$. {$(p, I)$} suffices.
$\rightarrow$: Suppose to the contrary $r$ $\models \phi
\rightarrow \psi$ but no $(p, I)$ compatible with $r$ forces such. Work in the node $f(r)$. (Recall that $f$ is the universal symbol for the various transition functions in sight. What we mean more specifically is that if $r$ $\in M_n$, i.e. if $r$ is a node from level $n$, then $f(r)$ is the image of $r$ in $M_{n+1}$, i.e. in the Kripke structure on level $n+1$.) Let $(p, I)$ be compatible with $f(r)$ and $p$ have non-standard (in the sense of $M_n$) length (equivalently, $I$ has infinitesimal length). Since $(p,
I)$ $\not\Vdash \phi \rightarrow \psi$ there is a $(q, J)$ $\leq$ $(p, I)$ such that $(q, J)$ $\Vdash \phi$ but $(q, J)$ $\not\Vdash
\psi$. By the previous part of this lemma, there is an $r'$ compatible with $(q, J)$ such that no condition compatible with $r'$ forces $\psi$. At the node $r'$, by induction, $r'$ $\not\models \psi$, even though $r'$ $\models \phi$ (since $r'$ is compatible with $(p, I)$ $\Vdash \phi$). This contradicts the assumption on $r$ (i.e. that $r$ $\models \phi \rightarrow \psi$), since $r'$ extends $r$ (as nodes).
$\exists$: If $r$ $\models \exists x \; \phi(x)$ then let $\sigma$ be such that $r$ $\models \phi(\sigma)$. Inductively there is a $(p, I)$ compatible with $r$ such that $(p, I)$ $\Vdash
\phi(\sigma)$. {$(p, I)$} suffices.
$\forall$: Suppose to the contrary $r$ $\models \forall x \;
\phi(x)$ but no $(p, I)$ compatible with $r$ forces such. As with $\rightarrow$, let $(p, I)$ non-standard be compatible with $f(r)$. Since $(p, I)$ $\not\Vdash \forall x \; \phi(x)$ there is a $\sigma$ such that $(p, I)$ $\not\Vdash \phi(\sigma)$. By the previous part of this lemma, there is an $r'$ compatible with $(p,
I)$ such that, for all $(q, J)$ compatible with $r', (q, J)
\not\Vdash \phi(\sigma)$. By induction, that means that $r'
\not\models \phi(\sigma)$. This contradicts the assumption on $r$ (i.e. that $r$ $\models \forall x \; \phi(x)$), since $r'$ extends $r$ (as nodes).
The Final Proofs
----------------
Using $\Vdash$, we can now prove
This Kripke model satisfies IZF$_{Ref}$.
Note that, as a Kripke model, the axioms of intuitionistic logic are satisfied, by general theorems about Kripke models.
- Empty Set: The interpretation of the term $\emptyset$ will do.
- Infinity: The canonical name for $\omega$ will do. (Recall that the canonical name $\bar{x}$ of any set $x \in V$ is defined inductively as $\{ \langle \bar{y}, (\emptyset, \mathbb{R})
\rangle \mid y \in x \}.)$
- Pairing: Given $\sigma$ and $\tau$, $\{ \langle \sigma,
(\emptyset, {\mathbb R}) \rangle , \langle \tau, (\emptyset,
{\mathbb R}) \rangle \}$ will do.
- Union: Given $\sigma$, $\{ \langle \tau, J \cap J_i \rangle
\mid$ for some $\sigma_i, \; \langle \tau, J \rangle \in \sigma_i$ and $\langle \sigma_i, J_i \rangle \in \sigma \}$ will do.
- Extensionality: We need to show that $\forall x \; \forall y
\; [\forall z \; (z \in x \leftrightarrow z \in y) \rightarrow x =
y]$. So let $\sigma$ and $\tau$ be any terms at a node $r$ such that $r \models ``\forall z \; (z \in \sigma \leftrightarrow z \in
\tau)"$. We must show that $r \models ``\sigma = \tau"$. By the Truth Lemma, let $r \in J \Vdash ``\forall z \; (z \in \sigma
\leftrightarrow z \in \tau)"$; i.e. for all $r' \in J, \rho$ there is a $J'$ containing $r'$ such that $J \cap J' \Vdash \rho \in
\sigma \leftrightarrow \rho \in \tau$. We claim that $J \Vdash
``\sigma = \tau"$, which again by the Truth Lemma suffices. To this end, let $\langle \sigma_i, J_i \rangle$ be in $\sigma$; we need to show that $J \cap J_i \Vdash \sigma_i \in \tau$. Let $r'$ be an arbitrary member of $J \cap J_i$ and $\rho$ be $\sigma_i$. By the choice of $J$, let $J'$ containing $r'$ be such that $J
\cap J' \Vdash \sigma_i \in \sigma \leftrightarrow \sigma_i \in
\tau$; in particular, $J \cap J' \Vdash \sigma_i \in \sigma
\rightarrow \sigma_i \in \tau$. It has already been observed in \[equalitylemma\], part 1, that $J \cap J' \cap J_i \Vdash
\sigma_i \in \sigma$, so $J \cap J' \cap J_i \Vdash \sigma_i \in
\tau$. By going through each $r'$ in $J \cap J_i$ and using \[helpful lemma\], part 3, we can conclude that $J \cap J_i
\Vdash \sigma_i \in \tau$, as desired. The other direction ($``\tau \subseteq \sigma"$) is analogous.
- Set Induction (Schema): Suppose $r$ $\models ``\forall x \;
((\forall y \in x \; \phi(y)) \rightarrow \phi(x))"$; by the Truth Lemma, let $J$ containing $r$ force as much. We must show $r$ $\models ``\forall x \; \phi(x)"$. Suppose not. Using the definition of satisfaction in Kripke models, there is an $r'$ extending (i.e. infinitesimally close to) $r$(hence in $J$) and a $\sigma$ such that $r'$ $\not \models \phi(\sigma)$. By elementarity, there is such an $r'$ in $M_n$, where $n$ is the level of $r$. Let $\sigma$ be such a term of minimal $V$-rank among all $r'$s in $J$. Fix such an $r'$. By the Truth Lemma (and the choice of $J$), $r'$ $\models ``(\forall y \in \sigma \;
\phi(y)) \rightarrow \phi(\sigma)"$. We claim that $r'$ $\models
``\forall y \in \sigma \; \phi(y)"$. If not, then for some $r''$ extending $r'$ (hence in $J$) and $\tau, r'' \models \tau \in
\sigma$ and $r'' \not \models \phi(\tau)$. Unraveling the interpretation of $\in$, this choice of $\tau$ can be substituted by a term $\tau$ of lower $V$-rank than $\sigma$. By elementarity, such a $\tau$ would exist in $M_n$, in violation of the choice of $\sigma$, which proves the claim. Hence $r'$ $\models
\phi(\sigma)$, again violating the choice of $\sigma$. This contradiction shows that $r \models ``\forall x \; \phi(x)"$.
- Separation (Schema): Let $\phi(x)$ be a formula and $\sigma$ a term. Then $\{ \langle \sigma_i, J \cap J_i \rangle \mid \langle
\sigma_i, J_i \rangle \in \sigma$ and $J \Vdash \phi(\sigma_i) \}$ will do.
- Power Set: A term $\hat{\sigma}$ is a canonical subset of $\sigma$ if for all $\langle \sigma_i, \hat{J_i} \rangle \in
\hat{\sigma}$ there is a $J_i \supseteq \hat{J_i}$ such that $\langle \sigma_i, {J_i} \rangle \in \sigma$. $\{ \langle
\hat{\sigma}, (\emptyset, {\mathbb R}) \rangle \mid \hat{\sigma}$ is a canonical subset of $\sigma \}$ is a set (in $M_n$), and will do.
- Reflection (Schema): Recall that the statement of Reflection is that for every formula $\phi(x)$ (with free variable $x$ and unmentioned parameters) and set $z$ there is a transitive set $Z$ containing $z$ such that $Z$ reflects the truth of $\phi(x)$ in $V$ for all $x \in Z$. So to this end, let $\phi(x)$ be a formula and $\sigma$ be a set at a node $r$ of level $n$ (in the tree which is this Kripke model’s partial order). Let $k$ be such that the truth of $\phi(x)$ at node $r$ and beyond is $\Sigma_{k}$ definable in $M_n$. In $M_{n}$, let $X$ be a set containing $\sigma$, $r$, and $\phi$’s parameters such that $X \prec_{k}
M_{n}$. Let $\tau$ be $\{ \langle \rho, (\emptyset, {\mathbb R})
\rangle \mid \rho \in X$ is a term}. $\tau$ will do.
We are interested in the canonical term $\{ \langle {\bar p}, (p,
I) \rangle \mid $ $p$ is a finite function from ${\mathbb N}$ to ${\mathbb Q}$ and $I$ is a non-empty, open interval from the reals with rational endpoints$\}$, where ${\bar p}$ is the canonical name for $p$. We will call this term $Z$. Note that at node $r$ $Z$ gets interpreted as $r$.
For all nodes r, r $\models$ “Z is a Cauchy sequence".
To see that $\bot \models$ “$Z$ is total", suppose $r$ $\models$ “$N$ is an integer". Then ($\langle N, r(N) \rangle, {\mathbb
R})$ is compatible with $r$ and forces “$Z(N) = r(N)$". That $Z$ is a function is similarly easy.
As for $Z$ being Cauchy, again let $r$ $\models$ “$N$ is an integer". Since $r$ is Cauchy, there is an integer $M$ such that beyond $M$ $r$ stays within an interval $I$ of size $1/(2N)$. Of course, future nodes might be indexed by Cauchy sequences $s$ extending $r$ that go outside of $I$, but only by an infinitesimal amount. So let $J$ extend $I$ on either side and still have length less that $1/N$. Then ($r \upharpoonright M, J$) is compatible with $r$, and forces that $Z$ beyond $M$ stay in $J$, making $Z$ “Cauchy for $1/N$", to coin a phrase.
In order to complete the theorem, we need only prove the following
For all nodes r, r $\models$ “Z has no modulus of convergence."
Suppose $r \models ``f$ is a modulus of convergence for $Z$." Let $(p, I)$ compatible with $r$ force as much. WLOG $I$ is a finite interval. Let $n$ be such that $1/n$ is less than the length of $I$, and let $\epsilon$ be (length($I) - 1/n)/2$.
Let $(q, J) \leq (p, I)$ force a value $m$ for $f(n)$. WLOG length$(q) > m$, as $q$ could be so extended. If $J=I$, then $(q,
J)$ could be extended simply by extending $q$ with two values a distance greater than $1/n$ apart, thereby forcing $f$ not to be a modulus of convergence. So $J \subset I$. That means either inf $J$ $>$ inf $I$ or sup $J <$ sup $I$. WLOG assume the latter. Let mid $J$ be the midpoint of $J, \; q_0$ be $q$ extended by mid $J$, and $J_0$ be (mid $J$, sup $J$). Then $(q_0, J_0) \leq (q, J)$, and therefore $(q_0, J_0) \Vdash f(n) = m$.
What $(q_1, J_1$) is depends:
CASE I: There is an open set $K$ containing sup($J_0$) such that $(q_0, K) \Vdash f(n) = m$. Then let $j_{max}$ be the sup of the right-hand endpoints (i.e. sups) of all such $K$’s. Let $q_1$ be $q_0$ extended by sup $J_0$.
: $(q_1, (\sup J_0, j_{max})) \Vdash f(n) = m.$
Let $r$ be any Cauchy sequence compatible with $(q_1$, (sup J$_0,
j_{max}$)). Since lim $r < j_{max}$, $r$ (that is, rng($r
\backslash q_0$)) is actually bounded below $j_{max}$. By the definition of $j_{max}$, there is an open $K$ containing sup $J_0$ such that $r$ is bounded by sup $K$. As $r$ is bounded below by sup $J_0$, $r$ (again, rng($r \backslash q_0$)) is contained within $K$. As ($q_0, K) \Vdash f(n) = m, \; r \models f(n) = m$.
Of course, $J_1$ will be (sup $J_0, j_{max}$).
CASE II: Not Case I. Then extend by the midpoint again. That is, $q_1$ is $q_0$ extended by mid $J_0$, and $J_1$ is (mid $J_0$, sup $J_0$). Also in this case, $(q_1, J_1) \Vdash f(n) = m$.
Clearly we would like to continue this construction. The only thing that might be a problem is if the right-hand endpoint of some $J_k$ equals (or goes beyond!) sup $I$, as we need to stay beneath $(p, I)$. In fact, as soon as sup($J_k) >$ sup$(I) -
\epsilon$ (if ever), extend $q_k$ by something within $\epsilon$ of sup $I$, and continue the construction with left and right reversed. That is, instead of going right, we now go left. This is called “turning around".
What happens next depends.
CASE A: We turned around, and after finitely many more steps, some $J_k$ has its inf within inf($I) + \epsilon$. Then extend $q_k$ by something within $\epsilon$ of inf $I$. This explicitly blows $f$ being a modulus of convergence for $Z$.
CASE B: Not Case A. So past a certain point (either the stage at which we turned around, or, if none, from the beginning) we’re marching monotonically toward one of $I$’s endpoints, but will always stay at least $\epsilon$ away. WLOG suppose we didn’t turn around. Then the construction will continue for infinitely many stages. The $q_k$’s so produced will in the limit be a (monotonic and bounded, hence) Cauchy sequence $r$. Furthermore, lim $r$ is the limit of the sup($J_k$)’s. Finally, $r$ is compatible with $(p, I)$. Hence $r \models ``f$ is total", and so $r \models f(n)
= m'$, for some $m'$. Let some condition compatible with $r$ force as much. This condition will have the form $(q', K)$, where lim $r
\in K$. Let $k$ be such that sup($J_k) \in K$, and $q_k \supseteq
q'$. Consider $(q_k, J_k$). Note that $(q_k, J_k \cap K)$ extends both $(q_k, J_k$) and $(q', K)$, hence forces both $f(n) = m$ and $f(n) = m'$, which means that $m=m'$.
Therefore, at this stage in the construction, we are in Case I. By the construction, $J_{k+1} = (\sup J_k, j_{max}$), where $j_{max}
\geq$ sup $K >$ lim $r$ = lim$_k (\sup(J_k)) \geq \sup(J_{k+1}) =
j_{max}$, a contradiction.
Not every Cauchy sequence of Cauchy sequences converges
=======================================================
\[thm2\] IZF$_{Ref}$ does not prove that every Cauchy sequence of Cauchy sequences converges to a Cauchy sequence.
The statement of the theorem itself needs some elaboration. The distance $d(x_{0n}, x_{1n})$ between two Cauchy sequences $x_{0n}$ and $x_{1n}$ is the sequence $\mid x_{0n} - x_{1n} \mid$. $x_{0n}
< x_{1n}$ if there are $m, N \in {\mathbb N}$ such that for all $k
> N \; x_{0k} + 1/m < x_{1k}$. A rational number $r$ can be identified with the constant Cauchy sequence $x_{n} = r. \; x_n$ = 0 if $\forall m \; \exists N \; \forall k > N \; \mid x_k \mid <
1/m$. $x_{0n}$ and $x_{1n}$ are equal (as reals, equivalent as Cauchy sequences if you will) if $d(x_{0n}, x_{1n})$ = 0. With these definitions in place, we can talk about Cauchy sequences of Cauchy sequences, and limits of such. The theorem is then that it is consistent with IZF$_{Ref}$ to have a convergent sequences of Cauchy sequences with no limit.
Note that we are not talking about reals! A real number would be an equivalence class of Cauchy sequences (omitting, for the moment, considerations of moduli of convergence). It would be weaker to claim that the sequence of reals represented by the constructed sequence of sequences has no limit. After all, given a sequence of reals, it’s not clear that there is a way to choose a Cauchy sequence from each real. We are claiming here that even if your task is made easier by being handed a Cauchy sequence from each real, it may still not be possible to get a “diagonalizing", i.e. limit, Cauchy sequence.
The Topological Space and Model
-------------------------------
Let $T$ be the space of Cauchy sequences of Cauchy sequences. By way of notation, if $X$ is a member of $T$, then $X_j$ will be the $j^{th}$ Cauchy sequence in $X$; as a Cauchy sequence of rationals, $X_j$ will have values $X_j(0), X_j$(1), etc. Still notationally, if $X_n \in T$, then the $j^{th}$ sequence in $X_n$ is $X_{nj}$. In the classical meta-universe, the Cauchy sequence $X_j$ has a limit, lim($X_j$); in addition, the sequence $X$ has a limit, which will be written as lim($X$).
A basic open set $p$ is given by a finite sequence $\langle (p_j,
I_j) \mid j<n_p \rangle$ of basic open sets from the space of the previous theorem (i.e. $p_j$ is a finite sequence of rationals and $I_j$ is an open interval), plus an open interval $I_p$. $X \in p$ if $X_j \in (p_j, I_j)$ for each $j < n_p$, if lim($X_j$) $\in
I_p$ for each $j \geq n_p$, and lim($X$) $\in I$. Note that $q
\subseteq p$ ($q$ $p$) if $n_q \geq n_p, (q_j,
K_j) \subseteq (p_j, I_j)$ for $j < n_p, K_j \subseteq I_p$ for $j
\geq n_p$, and $I_q \subseteq I_p$.
$p$ and $q$ are compatible (where WLOG $n_p \leq n_q$) if, for $j
< n_p \; (p_j, I_j$) and ($q_j, K_j$) are compatible, for $n_p
\leq j < n_q \; K_j \cap I_p \not = \emptyset$, and $I_q \cap I_p
\not = \emptyset$. In this case, $p \cap q$ is not the basic open set you’d think it is, but rather a union of such. The problem is that for $n_p \leq j < n_q$ it would be too much to take the $j^{th}$ component to be $(q_j, K_j \cap I_p$), because that would leave out all extensions of $q_j$ with entries from $K_j
\backslash I_p$ before they finally settle down to $K_j \cap I_p$. So $p \cap q$ will instead be covered by basic open sets in which the $j^{th}$ component will be ($r_j, K_j \cap I_p$), where ($r_j,
K_j) \subseteq (q_j, K_j$). (So the given basic open sets form not a basis for the topology, but rather a sub-basis.)
As always, the sets in the induced Heyting-valued model $M$ are of the form $\{\langle \sigma_{k}, p_{k} \rangle \mid k \in K \}$, where $K$ is some index set, each $\sigma_{k}$ is a set inductively, and each $p_{k}$ is an open set. Note that all sets from the ground model have canonical names, by choosing each $p_{k}$ to be $T$ (i.e. $n_p$ = 0 and $I_p = {\mathbb R}$), hereditarily. $M$ satisfies IZF$_{Ref}$.
The Extensions $\leq_j$ and $\leq_\infty$
-----------------------------------------
In the final proof, we will need the following notions.
$\leq_j$: q $\leq_j$ p for some $j <
n_p$ if q and p satisfy all of the clauses of q extending p except possibly for the condition on the $j^{th}$ component: (q$_j$, K$_j$) need not be a subset of $(p_j, I_j)$, although we will still insist that (q$_j$, I$_j$) be a subset of $(p_j, I_j)$.
More concretely, $q_j$ comes from $p_j$ by extending with elements from $I_j$; it’s just that we’re no longer promising to keep to $I_j$ in the future. Notice that $\leq_j$ is not transitive; the transitive closure of $\leq_j$ will be notated as $\leq_j^*$.
$\leq_\infty$: q $\leq_\infty$ p if q and p satisfy all of the clauses of q extending p except possibly for the last, meaning that I$_q$ need not be a subset of I$_p$.
$\leq_\infty^*$ is the transitive closure of $\leq_\infty$.
Suppose q $\subseteq$ p, q $\subseteq \|$f(n)=m$\|$ for some particular m and n, and $j < n_p$. Then for all x $\in$ I$_j$ there is an r $\subseteq$ p, r $\leq_j^*$ q such that r $\subseteq \|$f(n)=m$\|$ and x $\in$ L$_j$, where (r$_j$, L$_j$) is r’s $j^{th}$ component.
If $x \in K_j$, then we are done: let $r$ be $q$. So assume WLOG that $x \geq \sup(K_j$). The inspiration for this construction is the construction of the previous theorem. The main difference is that not only do we have ($q_j, K_j$) to contend with, we also have all of $q$’s other components around. Hence the notion of a $j$-extension: we will do the last theorem’s construction on the $j^{th}$ coordinates, and leave all the others alone.
First off, we would like to show that $q$ has a $j$-extension $q'
\subseteq p$ also forcing $f(n)=m$ such that sup($K_j) \in K'_j$. Toward this end, let $X \in T$ be a member of (the open set determined by) $q$ except that lim($X_j$) = sup($K_j$). $X$ is in $p$, so there is some $r \subseteq p$ such that $X \in r$ and $r$ forces a value for $f(n)$, say $m'. \; q$ and $r$ are compatible though: apart from the $j^{th}$ component, $X$ is in both, and the only thing happening in the $j^{th}$ component is that, in $r$, sup($K_j$) $\in L_j$, meaning that $K_j$ and $L_j$ overlap. So any common extension of both $q$ and $r$ would have to force $f(n)=m$ and $f(n)=m'$; since $p$ already forces that $f$ is a function, $m=m'$. Using $r$, it is easy to construct the desired $q'$: take the $j^{th}$ component from $r$, and let each other component be the intersection of the corresponding components from $r$ and $q$.
If there is such a $q'$ such that $x \in K'_j$, then we are done. Else we would like to mimic the last theorem’s construction by having in our next condition the interval part of the $j^{th}$ component be (sup($K_j$), $j_{max}$) (for a suitably defined $j_{max}$). The problem is, $q$ has all these other components around. For any real $y < j_{max}$ we could find a $j$-extension of $q$ with (sup($K_j$), $y$) in the $j^{th}$ component, but not necessarily for $y = j_{max}$ itself.
To this end, consider all such $q'$ as above. Each $q'$ can be extended (to say $q''$) by restricting the interval in the $j^{th}$ component to (sup($K_j$), sup($K'_j$)). Let $q_1$ be such a $q''$ where that interval is at least half as big as possible (i.e. among all such $q''$, where of course sup($K'_j$) has to be bounded by sup($I_j$)).
Continue this construction so that $q_n$ is defined from $q_{n-1}$ just as $q_1$ was defined from q. WLOG dovetail this construction with extending all other components so that after infinitely many steps we would have produced an $X \in T$. (This remark needs a word of justification about the $j^{th}$ components. By the definition of $j$-extension alone, it is not clear that a sequence of $j$-extending conditions $q_0 \geq_j q_1 \geq_j$ ... converges to a point in $T$. In our case, though, by the construction itself, the various $K_{nj}$’s are monotonically increasing and bounded, hence the $X_j$ so determined is Cauchy.)
If at some finite stage we have covered $x$, then we are done. If not, then sup($X_j$) = sup$_n$(sup($K_{nj}$)) $\leq x \in I_j$, so that $X \in p$. So there is some $r \subseteq p$ with $X \in r$ such that $r$ forces a value for $f(n)$, say $m'$. Let $\epsilon$ be sup($L_j) - \sup(X_j$). Eventually in the construction, $K_{nj}$ will be contained within $\epsilon$ of sup($X_j$). With $r$ as the witness, at the next stage $K_{(n+1)j}$ would go beyond sup($X_j$), which is a contradiction. Hence this case is not possible, and at some finite stage we must have covered $x$, as desired.
We have a similar lemma for $\infty$-extensions.
Suppose q $\subseteq$ p and q $\subseteq \|$f(n)=m$\|$ for some particular m and n. Then for all x $\in$ I$_p$ there is an r $\subseteq$ p, r $\leq_\infty^*$ q such that r $\subseteq \|$f(n)=m$\|$ and x $\in$ I$_r$.
Similar to the above.
Observe that the same arguments work for preserving finitely many values of $f$ simultaneously.
The Final Proof
---------------
We are interested in the canonical term $\{ \langle {\bar p_j}, p
\rangle \mid p$ is an open set$\}$, where ${\bar p_j}$ is the canonical name for the sequence $\langle p_j \mid j<n_p \rangle$ from $p$. We will call this term $Z$. It should be clear that $T =
\| Z$ is a Cauchy sequence of Cauchy sequences$\|$. Hence we need only prove
T = $\|$Z does not have a limit$\|$.
Suppose $p \subseteq \| f$ is a Cauchy sequence$\|$. It suffices to show that for some $q \subseteq p, \; q \subseteq \| f \not =$ lim$(Z)\|$.
If $p$ can ever be extended to force infinitely many values for $f$ simultaneously, then do so, and further extend (it suffices here to extend merely the last component) to force $Z$ away from $f$’s limit. This suffices for the theorem.
If this is not possible, then the construction will be to build one or two sequences of open sets, $p_k$ and possibly $r_k$, indexed by natural numbers $k$. It is to be understood even though not again mentioned that the construction below is to be dovetailed with a countable sequence of moves designed to produce a single member of $T$ in the end (i.e. each individual component must shrink to a real as in the previous theorem, the $n_{p_k}$’s must be unbounded as $k$ goes through ${\mathbb N}$, and the last components $I_{p_k}$ must shrink to something of length 0).
First, let $p_0$ be built by extending $p$ by cutting $I_p$ to its bottom third, and let $L$ be some point in $I_p$’s top half. If $p_0$ can be extended (to $p_1$) so that $f$ is forced to have an additional value (that is, beyond what has already been forced) in $I_p$’s top half, then do so. Else proceed as follows. First extend $p_0$ to force an additional value for $f$, necessarily in $I_p$’s bottom half. Then by the second lemma above, $\infty$-extend that latter condition, to $q$ say, preserving the finitely many values of $f$ already determined, and getting $L$ into $I_q$. Typically $n_q > n_p$, so let $\bar{q}$ be such that $n_{\bar{q}} = n_q$, if $j < n_p$ then $\bar{q}$’s $j^{th}$ component is the same as $p$’s, if $n_p \leq j < n_q$ then $\bar{q}$’s $j^{th}$ component is ($\emptyset, I_p$), and $I_{\bar{q}} = I_p$. Note that $q \subseteq \bar{q} \subseteq p$, so we can apply the first lemma above to $q$ and $\bar{q}$. Starting from $q$, iteratively on $j$ from $n_p$ up to $n_q$, $j$-extend to get $L$ into the interval part of the $j^{th}$ component, while preserving the finitely many values of $f$ already determined. Call the last condition so obtained $r_0$. Finally, $\infty$-extend $r_0$ to get the last component to be a subset of $I_{p_0}$, while still preserving $f$ of course. Let this latter condition be $p_1$.
Stages $k > 0$ will be similar. To start, if possible, extend $p_k$ to force an additional value for $f$ in $I_p$’s top half. Call this condition $p_{k+1}$.
If that is not possible, first extend $p_k$ to force a new value for $f$, necessarily in $I_p$’s bottom half. Then $\infty$-extend (to $q$ say) to get $L$ into the last component $I_q$. After that, $j$-extend for each $j$ from $n_{r_i}$ to $n_q$ to get $L$ in those components, where $i$ is the greatest integer less than $k$ such that $r_i$ is defined. (It bears mentioning that $r_h$ is defined if and only if at stage $h$ we are in this case.) If need be, shrink those components to be subsets of $I_{r_i}$, for the purpose of getting $r_k \subseteq r_i$ (once we define $r_k$). That last condition will be $r_k$. Next, $\infty$-extend $r_k$ to get the last component to be a subset $I_{p_k}$. This final condition is $p_{k+1}$.
This completes the construction.
If the second option happens only finitely often, let $k$ be greater than the last stage where it happens. Then not only does $p_k$ force lim($Z$) to be in $I_p$’s bottom third, as all $p_i$’s do actually, but also $p_k$ is respected in the rest of the construction: for $i > k$, $p_i \subseteq p_k$. Let $l \geq k$ be such that $6/l <$ length($I_p$) (i.e. the distance between $I_p$’s top half and bottom third is greater than $1/l$). Recall that $p
\subseteq \| f$ is a Cauchy sequence$\|$; that is, $p \subseteq \|
\forall \epsilon > 0 \; \exists N \; \forall m, n \geq N \; |f(m)
- f(n)| < \epsilon\|$. Since $1/l >$ 0, $p \subseteq \| \exists N
\; \forall m, n \geq N \; |f(m) - f(n)| < 1/l\|$. That means there is a cover $C$ of $p$ such that each $q \in C$ forces a particular value for $N$. Let $S$ be $\bigcap_{j \geq k} p_j$, and let $q \in
C$ contain $S$. Similarly, let ${\hat q}$ containing $S$ force a value for $f(N)$. $q \wedge {\hat q} \wedge p_k$ is non-empty because it contains $S$, and $q \wedge {\hat q} \wedge p_k$ forces by the construction that $f(N)$ is in $I_p$’s top half, by the choice of $q$ that lim($f$) is away from $I_p$’s bottom third, and by choice of $k$ that lim($Z$) is in $I_p$’s bottom third. In short, $q \wedge {\hat q} \wedge p_k \subseteq \| f \not =$ lim($Z)\|$.
Otherwise the second option happens infinitely often. Then we have an infinite descending sequence of open sets $r_k$, and a similar argument works. Let $S$ be $\bigcap_j r_j$, where the intersection is taken only over those $j$’s for which $r_j$ is defined. Let $k$ be such that $r_k \subseteq \|$lim($Z) - $midpoint($I_p) <
\epsilon \|$, for some fixed $\epsilon >$ 0. Let $q$ containing $S$ be such that, for a fixed value of $N$, $q \subseteq \|
\forall m, n \geq N \; |f(m) - f(n)| < \epsilon\|$. Let ${\hat q}$ force a particular value for $f(N)$, necessarily in $I_p$’s bottom half. Again, $q \wedge {\hat q} \wedge p_k \subseteq \| f \not =$ lim($Z)\|$.
The given Cauchy sequence has a modulus, but the limit doesn’t
==============================================================
\[thm3\] IZF$_{Ref}$ does not prove that every Cauchy sequence with a modulus of convergence of Cauchy sequences converges to a Cauchy sequence with a modulus of convergence.
c is a convergence function for a Cauchy sequence $\langle X_j
\mid j \in \bf{N} \rangle$ if c is a decreasing sequence of positive rationals; for all n, if j, k $\geq$ n then $\mid X_j - X_k \mid \leq c(n)$; and lim(c(n)) = 0.
Notice that convergence functions and moduli of convergence are easily convertible to each other: if $c$ is the former, then $d(n)$ := the least $m$ such that $c(m) \leq 2^{-n}$ is the latter; and if $d$ is the latter, then $c(n) := 2^{-m}$, where $m$ is the greatest integer such that max($m, d(m)) \leq n$, is the former. Therefore the current construction will be of a Cauchy sequence $\langle X_j \mid j \in {\bf N} \rangle$ with a convergence function but no limit. Without loss of generality, the convergence function in question can be taken to be $c(n) =
2^{-n}$.
Let the topological space $T$ be $\{ \langle X_j \mid j \in {\bf
N} \rangle \mid \langle X_j \mid j \in \bf{N} \rangle$ is a Cauchy sequence of Cauchy sequences with convergence function $2^{-n}$ }. As in the previous section, for $X \in T, \; X_j$ will be the $j^{th}$ Cauchy sequence in $X$’s first component. The real number represented by $X_j$, i.e. $X_j$’s limit, will be written as lim($X_j$). In the classical meta-universe, the limit of the sequence $\langle X_j \mid j \in \bf{N} \rangle$ will be written as lim($X$).
$T$ is a subset of the space from the previous section, and the topology of $T$ is to be the subspace topology. That is, a basic open set $p$ is given again by a finite sequence $\langle (p_j,
I_j) \mid j<n_p \rangle$ and an open interval $I_p$. $X \in p$ if, again, $X_j \in (p_j, I_j)$ for each $j<n_p$, lim($X_j$) $\in I_p$ for each $j\geq n_p$, and lim($X$) $\in I_p$. $p$ and $q$ are compatible under the same conditions as before, and $p \cap q$ is covered by basic open sets, just as in the last theorem; the convergence function causes no extra trouble.
Note that $q \subseteq p \; (q$ $p$) if all of the same conditions from the last section hold: $n_q \geq n_p$, ($q_j, K_j) \subseteq (p_j, I_j)$ for $j < n_p, \; K_j \subseteq
I_p$ for $j \geq n_p$, and $I_q \subseteq I_p$.
In the following, we will need to deal with basic open sets in canonical form. The issue is the following. Suppose, in $p$, $I_0$ = (0, 1) and $I_1$ = (0, 10). Then $X_1$ could certainly contain elements from (0, 10). However, when it comes to taking limits, $X_1$ has 2 as an upper bound, because of $I_0$ and the convergence function $2^{-n}$, but this is not reflected in $I_1$.
p is in canonical form if, for $j < k < n_p,
\mid$sup(I$_j$) - sup(I$_k)\mid \leq 2^{-j}$, and also $\mid$sup(I$_j$) - sup(I$_p)\mid \leq 2^{-j}$.
The value of canonical form is that, if for $j < n_p$ lim($X_j) =
\sup (I_j)$ and if lim($X$) = sup($I_p$), then, although $X \not
\in p$, $X$ could still be in $T$.
Every open set is covered by open sets in canonical form.
Let $X \in p$ open. If, in $q \subseteq p$, $J_k$ is an interval with midpoint lim($X_k$) and radius independent of $k$, and $I_q$ an interval with midpoint lim($X$) and the same radius, then $q$ will be canonical. We will construct such a $q$ containing $X$.
By way of choosing the appropriate radius, as well as $n_q$, let $\delta$ be half the distance from lim($X$) to the closer of $I_p$’s endpoints. Let $N \geq n_p$ be such that for all $k \geq
N$ lim($X_k$) is within $\delta$ of lim($X$). Let $r \leq \delta$ be such that for all $k < N$ (lim($X_k) - r$, lim($X_k) + r$) $\subseteq I_k$. Let $n_q \geq N$ be such that for all $k \geq
n_q$ lim($X_k$) is within $r$ of lim($X$). For $k < n_q$ let $J_k$ be the neighborhood with center lim($X_k$) and radius $r$, and let $q_k$ be an initial segment of $X_k$ long enough so that beyond it $X_k$ stays within $J_k$. Let $I_q$ be the neighborhood with center lim($X$) and radius $r$. This $q$ suffices.
As always, the sets in the induced Heyting-valued model $M$ are of the form $\{\langle \sigma_{k}, p_{k} \rangle \mid k \in K \}$, where $K$ is some index set, each $\sigma_{k}$ is a set inductively, and each $p_{k}$ is an open set. Note that all sets from the ground model have canonical names, by choosing each $p_{k}$ to be $T$ (i.e. $n_p$ = 0 and $I_p = {\mathbb R}$), hereditarily. $M$ satisfies IZF$_{Ref}$.
We are interested in the canonical term $\{ \langle {\bar p_j}, p
\rangle \mid p$ is an open set$\}$, where ${\bar p_j}$ is the canonical name for the sequences $\langle p_j \mid j<n_p \rangle$ from $p$. We will call this term $Z$. It should be clear that $T$ = $\|$Z is a Cauchy sequence of Cauchy sequences with convergence function $2^{-n} \|$. Hence we need only prove
T = $\|$No Cauchy sequence equal to lim(Z) has a modulus of convergence$\|$.
Suppose $p \subseteq \| f$ is a modulus of convergence for a Cauchy sequence $g \|$, $p$ in canonical form. It suffices to show that for some $q \subseteq p$, $q \subseteq \| g \not =$ lim($Z)\|$.
Let $\epsilon <$ (length $I_p$)/2. Let $q \subseteq p$ in canonical form force “$f(\epsilon) = N$"; WLOG $n_q > N$. We can also assume (by extending again if necessary) that $q$ forces a value for $g(N)$; WLOG $g(N) \leq$ midpoint($I_p$). Let $X \in p$ be on the boundary of $q$; that is, $X_k$ extends $q_k$ ($k <
n_q$), $X_k$ beyond length($q_k$) is a sequence through $J_k$ with limit $\sup(J_k$), and $X$ beyond $n_q$ is a sequence through $I_q$ with limit $\sup(I_q$) (more precisely, $\langle \lim(X_k)
\mid k \geq n_q \rangle$ is such a sequence).
(Technical aside: By the canonicity of $q$’s form, $X \in T$. But why should $X$ be in $p$? This could fail only if $\sup(J_k) =
\sup(I_k$) or if $\sup(I_q) = \sup(I_p$). The latter case would actually be good. The point of the current argument is to get a condition $r$ (forcing the things $q$ forces) such that $I_r$ contains points greater than $g(N) + \epsilon$, which would fall in our lap if $\sup(I_q) = \sup(I_p$). If $\sup(I_q) < \sup(I_p$) and $\sup(J_k) = \sup(I_k$), then $X_k$ must be chosen so that $\lim(X_k$) is slightly less than this sup. Could this interfere with 2$^{-n}$ being a convergence function for $X$? No, by the canonicity of $p$. If $l$ is another index such that $\sup(J_l) =
\sup(I_l$), then by letting $\lim(X_l$) be shy of this sup by the same amount as for $k$ the convergence function 2$^{-n}$ is respected (for these two indices). If $\sup(J_l) < \sup(I_l$), then what to do depends on whether $\sup(J_k$) and $\sup(J_l$) are strictly less than 2$^{-min(k,l)}$ apart or exactly that far apart. In the former case, there’s some wiggle room in the $k^{th}$ slot for $\lim(X_k$) to be less than $\sup(J_k$). In the latter, $\sup(J_l$) must be $\sup(J_k$) either increased or decreased by 2$^{-min(k,l)}$. The first option is not possible, by the canonicity of $p$, as $\sup(J_l) < \sup(I_l$). In the second option, having $\lim(X_k$) be less than $\sup(J_k$) brings $\lim(X_k$) and $\lim(X_l$) even closer together. Similar considerations apply to comparing $\lim(X_k$) and $\lim_k(X_k) =
\sup(I_q$).)
Let $q_1$ in canonical form containing $X$ force values for $f(\epsilon$) and $g(f(\epsilon$)). Since $q_1$ and $q$ are compatible, they force the same such values. WLOG $q_1$ is such that $\sup(I_{q_1}$) is big (that is, $\sup(I_{q_1}) - \sup(I_q$) is at least half as big as possible). Continuing inductively, define $q_{n+1}$ from $q_n$ as $q_1$ was defined from $q$. Continue until $I_{q_n}$ contains points greater than $g(N) +
\epsilon$. This is guaranteed to happen, because, if not, the infinite sequence $q_n$ will converge to a point $X$ in $p$. Some neighborhood $r$ of $X$ forcing values for $f(\epsilon$) and $g(f(\epsilon$)) will contain some $q_n$, witnessing that $q_{n+1}$ would have been chosen with larger last component than it was, as in the previous proofs.
Once the desired $q_n$ is reached, shrink $I_{q_n}$ to be strictly above $g(N) + \epsilon$. Call this new condition $r$. $r$ forces “$\lim(Z) > g(N) + \epsilon$", and $r$ also forces “$g \leq g(N)
+ \epsilon$". So $r$ forces “$g \not = \lim(Z$)", as desired.
The reals are not Cauchy complete
=================================
\[thm4\] IZF$_{Ref}$ does not prove that every Cauchy sequence of reals has a limit.
As stated in the introduction, what we will actually prove will be what seems to be the hardest version: there is a Cauchy sequence, with its own modulus of convergence, of real numbers, with no Cauchy sequence as a limit, even without a modulus of convergence. Other versions are possible, such as changing what does and doesn’t have a modulus. After all of the preceding proofs, and after the following one, it should not be too hard for the reader to achieve any desired tweaking of this version.
Let $T$ consist of all Cauchy sequences of Cauchy sequences, all with a fixed convergence function of 2$^{-n}$. An open set $p$ is given by a finite sequence $\langle (p_j, I_j) \mid j < n_p
\rangle$ as well as an interval $I_p$, with the usual meaning to $X \in p$.
Recall from the previous section:
$p$ is in canonical form if, for each $j < k < n_p,$ $\mid
\sup(I_j) - \sup (I_k) \mid \leq 2^{-j}$. Also, $\mid \sup(I_j) -
\sup (I_p) \mid \leq 2^{-j}$.
Also from the last section:
Every open set is covered by sets in canonical form.
Henceforth when choosing open sets we will always assume they are in canonical form.
$p$ and $q$ are similar, $p \sim q$, if $n_p = n_q, I_p = I_q, I_k
= J_k$, and length($p_k$) = length($q_k$). So $p$ and $q$ have the same form, and can differ only and arbitrarily on the rationals chosen for their components.
If moreover $p_k = q_k$ for each $k \in J$ then we say that $p$ and $q$ are $J$-similar, $p \sim_J q$.
If $p \sim q$, this induces a homeomorphism on the topological space $T$, and therefore on the term structure. (To put it informally, wherever you see $p_k$, or an initial segment or extension thereof, replace it (or the corresponding part) with $q_k$, and vice versa. This applies equally well to members of $T$, open sets, and (hereditarily) terms.)
If $p, q$, and $r$ are open sets, $\sigma$ is a term, and $q$ and $r$ are similar, then the image of $p$ under the induced homeomorphism is notated by $p_{qr}$ and that of $\sigma$ by $\sigma_{qr}$.
p $\Vdash \phi(\vec{\sigma})$ iff p$_{qr} \Vdash \phi(\vec{\sigma}_{qr})$.
A straightforward induction.
$\sigma$ has support $J$ if for all $p\sim_J q$ $\bot \Vdash
\sigma = \sigma_{pq}$. $\sigma$ has finite support if $\sigma$ has support $J$ for some finite set $J$.
The final model $M$ is the collection of all terms with hereditarily finite support.
As always, let $Z$ be the canonical term. Note that $Z$ is not in the symmetric submodel! However, each individual member of $Z$, $Z_j$, is, with support $\{ j \}$. Also, so is $\langle [Z_j] \mid
j \in {\bf N} \rangle$, which we will call $[Z]$, with support $\emptyset$. (Here, for $Y$ a Cauchy sequence, $[Y]$ is the equivalence class of Cauchy sequences with the same limit as $Y$, i.e. the real number of which $Y$ is a representative.) That’s because no finite change in $Z_j$ affects \[$Z_j$\]. (Notice that even though each member of \[$Z_j$\] has support $\{ j \}$, \[$Z_j$\]’s support is still empty.) It will ultimately be this sequence \[$Z$\] that will interest us. But first:
M $\models$ IZF$_{Ref}$.
As far as the author is aware, symmetric submodels have been studied only in the context of classical set theory, not intuitionistic, and, moreover, the only topological models in the literature are full models, in which the terms of any given model are all possible terms built on the space in question, not submodels. Nonetheless, the same proof that the full model satisfies IZF (easily, IZF$_{Ref}$) applies almost unchanged to the case at hand. To keep the author honest without trying the patience of the reader, only the toughest axiom, Separation, will be sketched.
To this end, suppose the term $\sigma$ and formula $\phi$ have (combined) support $J$ (where the support of a formula is the support of its parameters, which are hidden in the notation used). The obvious candidate for a term for the appropriate subset of $\sigma$ is $\{ \langle \sigma_i, p \cap p_i \rangle \mid \langle
\sigma_i, p_i \rangle \in \sigma \wedge p \Vdash
\phi(\sigma_i)\}$, which will be called Sep$_{\sigma, \phi}$. We will show that this term has support $J$.
To this end, let $q \sim_J r$. We need to show that $\bot \Vdash$ Sep$_{\sigma, \phi}$ = (Sep$_{\sigma, \phi})_{qr}$. In one direction, any member of (Sep$_{\sigma, \phi})_{qr}$ is of the form $\langle \sigma_i, p \cap p_i \rangle_{qr}$, where $\langle
\sigma_i, p \cap p_i \rangle \in$ Sep$_{\sigma, \phi}$, i.e. $\langle \sigma_i, p_i \rangle \in \sigma$ and $p \Vdash
\phi(\sigma_i)$. We need to show that $(p \cap p_i)_{qr} \Vdash
(\sigma_i)_{qr} \in$ Sep$_{\sigma, \phi}$. Since $\bot \Vdash
\sigma = \sigma_{qr}$ and $(p_i)_{qr} \Vdash (\sigma_i)_{qr} \in
\sigma_{qr}, (p_i)_{qr} \Vdash (\sigma_i)_{qr} \in \sigma$. In addition, by the lemma above, $p_{qr} \Vdash
\phi_{qr}((\sigma_i)_{qr})$ (where $\phi_{qr}$ is the result of taking $\phi$ and applying the homeomorphism to its parameters). Since $\phi$’s parameters have support $J$, $\bot \Vdash \phi_{qr}
= \phi$, and $p_{qr} \Vdash \phi((\sigma_i)_{qr})$. Summarizing, $(p \cap p_i)_{qr} \Vdash (\sigma_i)_{qr} \in \sigma \wedge
\phi((\sigma_i)_{qr})$, so $(p \cap p_i)_{qr} \Vdash
(\sigma_i)_{qr} \in $ Sep$_{\sigma, \phi}$, as was to be shown.
The other direction is similar.
So there was no harm in taking the symmetric submodel. The benefit of having done so is the following
Extension Lemma: Suppose q, r $\subseteq$ p, q $\subseteq \| f(n) = m \|$ and for $j \in$ J (q$_j$, K$_j$) = (r$_j$, L$_j$) (i.e. q and r agree on f’s support). Then r has an extension forcing f(n) = m.
Take a sequence of refinements of $q$ converging to a point $X$ on $q$’s boundary, as follows. Consider $j < n_r, j \not \in J$. If $K_j \cap L_j$ is non-empty, then just work within the latter set. Else either $\sup(K_j) < \inf(L_j$), in which case let $\lim(X_j)
= \sup(K_j$), or $\inf(K_j) > \sup(L_j$), in which case let $\lim(X_j) = \inf(K_j$). (In what follows, we will consider only the first of those two cases.) Similarly for $I_q$ and $I_r$. As usual, since $X \in p, \; X$ has a neighborhood forcing a value for $f(n)$; since $X$ is on $q$’s boundary, any such neighborhood has to force the same value for $f(n)$ that $q$ did. Let $q_1$ be such a neighborhood where, for $j$ the smallest integer not in $J$, $\sup((K_1)_j) - \sup(K_j$) is at least half as big as possible.
To continue this construction, consider what would happen if $\sup(K_1)_j < \inf(L_j$). We would like to take another point $X$, this time on the boundary of $q_1$, with $\lim(X_j) =
\sup(K_1)_j$. The only possible obstruction is that ($q_1)_j$ might have entries far enough away from $\sup(K_1)_j$ so that the constraint of the convergence function would prevent there being such an $X$. In this case, change ($q_1)_j$ so that this is no longer an obstruction. Since $j \not \in J$, the new condition is $J$-similar to the old, and so will still force the same value for $f(n)$.
Repeat this construction, making such that each of the finitely many components $j < n_r$, $j \not \in J$ and the final component get paid attention infinitely often (meaning $\sup((K_{n+1})_j) -
\sup((K_n)_j)$ is at least half as big as possible). This produces a sequence $q_n$. Eventually $q_n$ will be compatible with $r$. If not, let $X$ be the limit of the $q_n$’s. If $X \not \in r$ then, for some component $j$, $\lim(X_j) < \inf(L_j$). $X$ has a neighborhood, say $q_{\infty}$, forcing $f(n) = m$. At some large enough stage at which $j$ gets paid attention, the existence of $q_{\infty}$ would have made the $j^{th}$ component of the next $q_n$ contain $\lim(X_j$), a contradiction.
With the Extension Lemma in hand, the rest of the proof is easy. It should be clear that \[$Z$\] has convergence function 2$^{-n}$. So it remains only to show
$\|$ \[Z\] has no limit $\|$ = T.
Suppose $p \subseteq \| f$ is a Cauchy sequence $\|$. It suffices to find a $q \subseteq p$ such that $q \subseteq \| f \not =
\lim([Z]) \|$.
By the Extension Lemma, all of $f$’s values are determined by $f$’s finite support $J$. So $f$ cannot be a limit for \[$Z$\], as any such limit has to be affected by infinitely many components.
Questions
=========
There is a variant of the questions considered nestled between the individual Cauchy sequences of the big Cauchy sequence being adorned with a modulus of convergence and not. It could be that each such sequence has a modulus of convergence, but the sequence is not paired with any modulus in the big sequence. Looked at differently, perhaps the big sequence is one of Cauchy sequences with moduli of convergence but not uniformly. Certainly this extra information would not weaken any of the positive results. Would it weaken any of the negative theorems though? Presumably not: knowing that each of the individual sequences has a modulus doesn’t seem to help to build a limit sequence or a modulus for such, if there’s no way you can get your hands on them. Still, in the course of trying to prove this some technical difficulties were encountered, so the questions remain open.
The negative results here open up other hierarchies. Starting with the rationals, one could consider equivalence classes of Cauchy sequence with moduli of convergence. By the last theorem, that may not be Cauchy complete. So equivalences classes can be taken of sequences of those. This process can be continued, presumably into the transfinite. Is there a useful structure theorem here? All of this can be viewed as taking place inside of the Dedekind reals, which are Cauchy complete. There is a smallest Cauchy complete set of reals, namely the intersection of all such sets. As pointed out to me by the referee, this could be a proper subset of the Dedekind reals, since that is the case in the topological model of [@FH]. Naturally enough, the same is also the case in the Kripke model of [@LR]. Is there any interesting structure between the Cauchy completion of the rationals and the Dedekind reals? What about the corresponding questions for other notions of reals, such as simply Cauchy sequences sans moduli?
As indicated in the introduction, the first two models, one topological and the other Kripke, are essentially, even if not substantially, different. What is the relation between the two?
In the presence of Countable Choice, all of the positive results you could want here are easily provable (e.g. every Cauchy sequence has a modulus of convergence, the reals are Cauchy complete, etc.). Countable Choice itself, though, is a stronger principle than necessary for this, since, as pointed out to me by Fred Richman, these positive results are true under classical logic, but classical logic does not imply Countable Choice. Are there extant, weaker choice principles that would suffice instead? Can the exact amount of choice necessary be specified? These questions will start to be addressed in the forthcoming [@LRi], but there is certainly more that can be done than is even attempted there.
[99]{} M.P. Fourman and J.M.E. Hyland, Sheaf models for analysis, in M.P. Fourman, C.J. Mulvey, and D.S. Scott (eds.), Applications of Sheaves, Lecture Notes in Mathematics Vol. 753 (Springer-Verlag, Berlin Heidelberg New York, 1979), p. 280-301 M. Escardo and A. Simpson, A universal characterization of the closed Euclidean interval, Sixteenth Annual IEEE Symposium on Logic in Computer Science, 2001, p.115-125 R.J. Grayson, Heyting-valued models for intuitionistic set theory, in M.P. Fourman, C.J. Mulvey, and D.S. Scott (eds.), Applications of Sheaves, Lecture Notes in Mathematics Vol. 753 (Springer-Verlag, Berlin Heidelberg New York, 1979), p. 402-414 R. Lubarsky and M. Rathjen, On the constructive Dedekind reals, submitted for publication R. Lubarsky and F. Richman, Representing real numbers without choice, in preparation S. MacLane and I. Moerdijk, Sheaves in Geometry and Logic (Springer-Verlag, New York, 1992) A.S. Troelstra and D. van Dalen, Constructivism in Mathematics, Vol. 1 (North Holland, Amsterdam New York Oxford Todyo, 1988)
|
---
author:
- 'Atanu Bhatta, Prashanth Raman and Nemani V Suryanarayana'
title: Scalar Blocks as Gravitational Wilson Networks
---
Introduction
============
The correlation function of a set of primary operators in a $d$-dimensional CFT can be decomposed into its partial waves. For example, the correlation function of four scalar primary operators can be decomposed as \_1(x\_1) [O]{}\_2(x\_2) [O]{}\_3(x\_3) [O]{}\_4(x\_4) = \_[[O]{}]{} C\_[12 [O]{}]{} C\^[O]{}\_[ 34]{} W\^[(d)]{}\_[, l]{} (\_i, x\_i) where $C_{12 {\cal O}}$ are the OPE coefficients and the partial wave $W^{(d)}_{\Delta, l} (x_i)$ is W\^[(d)]{}\_[, l]{} (\_i, x\_i) = ()\^[ (\_1 - \_2)]{} ()\^[ (\_3 - \_4)]{} (x\_[12]{}\^2)\^[-(\_1+\_2)]{} (x\_[34]{}\^2)\^[-(\_3+\_4)]{} G\_[, l]{} (u,v) The pre-factor is determined by the conformal invariance and the function $G_{\Delta, l} (u,v)$ – referred to as the conformal block – depends only on the conformally invariant cross-ratios $u, v$. A lot is known about these conformal partial waves/blocks. For instance, a general expression for conformal partial waves (CPW) of four-point scalar correlators is given in [@Dolan:2011dv] (see also [@Dolan:2000ut; @Dolan:2003hv]). Written in terms of the complex coordinates $z, \bar z$ where $u= z \, \bar z$ and $v = (1-z) \, (1- \bar z)$, closed form expressions are known for all even $d$ for scalar CPW [@Dolan:2011dv; @SimmonsDuffin:2012uy]. Also closed form expressions for scalar conformal blocks for particular choice $z = \bar z$ are known for all dimensions [@ElShowk:2012ht; @Hogervorst:2013kva]. Powerful recursion relations between blocks in even $d$ are found in [@SimmonsDuffin:2012uy]. A different choice of parametrising the cross-ratios through $z = x \, e^{i \,\theta}$ and $\bar z = x \, e^{-i \, \theta}$ was also advocated in [@Dolan:2011dv; @Hogervorst:2013sma].
Since AdS/CFT provides a natural avenue to answer questions in CFT$_d$ in terms of $AdS_{d+1}$ gravity (and vice versa) it is natural to ask how to compute the conformal partial waves of a given correlation function of primary operators in a CFT holographically. To achieve this two distinct prescriptions have been proposed so far in the literature:
1. [**Geodesic Witten Diagrams**]{} [@Hijano:2015zsa]: This prescription is based on the second order Einstein-Hilbert formulation of gravity in which the conformal partial waves are given by the so called geodesic Witten diagrams. This has been generalised further in [@Hijano:2015qja; @Nishida:2016vds; @Dyer:2017zef; @Belavin:2017atm; @Tamaoka:2017jce; @Kraus:2017ezw; @Anand:2017dav; @Nishida:2018opl].
2. [**Gravitational Open Wilson Networks**]{} [@Bhatta:2016hpz; @Besken:2016ooo]: This prescription is suitable for the first order Hilbert-Palatini formulation of the bulk theory in which the conformal partial waves are given by appropriate gravitational open Wilson networks (OWN). These are studied and generalised for 2d CFTs in [@Fitzpatrick:2016mtp; @Besken:2017fsj; @Hikida:2017ehf; @Hikida:2018dxe].
In this paper we restrict ourselves to the second prescription, and provide further computational methods for its implementation in general dimensions. Before proceeding further let us review some essential aspects of this construction (see [@Bhatta:2016hpz] for more details).
In the first-order Hilbert-Palatini formulation of $AdS_{d+1}$ gravity [@MacDowell:1977jt; @Freidel:2005ak] the basic fields are the vielbeins $e^a$ and the spin-connections $\omega^{ab}$. They are conveniently combined into a 1-form gauge field $A$ in the adjoint of the $so(1, d+1)$ algebra as: \[gaugeconnection\] A = e\^a M\_[0a]{} + \^[ab]{} M\_[ab]{} where $\{ M_{0a}, \, M_{ab}\}$ are the generators of $so(1, d+1)$ with $a,b = 1, \cdots, d+1$. In this theory we consider a set of gauge covariant Wilson Network operators. In particular,
- One starts with an open, directed and trivalent graph (such as in Fig. (1) ) whose every line (internal as well as external) carries a representation label of the (Euclidean) conformal algebra $so(1,d+1)$.
- The representations of interest are those non-unitary infinite dimensional irreps which are obtained by appropriate Wick rotation of the corresponding UIR of the associated Lorentzian conformal algebra $so(2,d)$ of the CFT$_d$. Such an irrep can be labeled by $(\Delta; l_1, \cdots, l_{[d/2]})$ where $\Delta$ is the conformal weight and $l_i$ label which irrep the primary transforms in, under the boundary rotation group $so(d)$.
.5cm
\[scale = .5\] (0,0) circle (5 cm); (0,0) circle (5 cm); (-4,0) – (-3.33,0); (-3.33,0) – (-2.83,0); (-4.93,0.93) – (-4.43,0.43); (-4.43,0.43) – (-4,0); (-4.93,0.93) circle (2pt); (-4.93,-0.93) – (-4.43,-0.43); (-4.43,-0.43) – (-4,0); (-4.93,-0.93) circle (2pt); (-4,0) circle (4pt); (-2.83,0) – (-2.83,0.465); (-2.83,0.465)–(-2.83,0.93); (-2.83,0) – (-2.83,-0.465); (-2.83,-0.465) – (-2.83,-0.93); (-2.83,0) circle (4pt); (2.83,0) – (3.415,0); (4,0) – (3.415,0); (4,0) – (4.465,0.465); (4.93,0.93) – (4,0); (4.93,0.93) circle (2pt); (4,0) – (4.465,-0.465); (4.93,-0.93) – (4.465,-0.465); (4.93,-0.93) circle (2pt); (4,0) circle (4pt); (2.83, 0.93) – (2.83,0.465); (2.83, 0.465) – (2.83,-2.4); (2.83,-2.4) – (3.615,-2.4); (3.615,-2.4) – (4.4,-2.4); (2.83,0) circle (4pt); (4.4,-2.4) circle (2pt); (0,4) – (0,3.45); (0,2.83) – (0,3.45); (0.93,4.93) – (0.43,4.43); (0,4) – (0.43, 4.43); (0.93,4.93) circle (2pt); (-0.93,4.93) – (-0.43,4.43); (0,4)–(-0.43,4.43) ; (-0.93,4.93) circle (2pt); (0,4) circle (4pt); (-0.93,2.83) – (-0.43,2.83); (-0.43,2.83) – (0.43,2.83); (0.43,2.83) – (0.93,2.83); (0,2.83) circle (4pt); (0,-2.83) – (0,-3.415); (0,-4) – (0,-3.415); (0,-4) – (-0.465,-4.465); (0.93,-4.93) – (0.465,-4.465); (0,-4) – (0.465,-4.465); (0.93,-4.93) circle (2pt); (-0.93,-4.93) – (-0.465,-4.465); (-0.93,-4.93) circle (2pt); (0,-4) circle (4pt); (-0.93,-2.83) – (-0.465,-2.83); (-0.465,-2.83) – (0.465,-2.83); (0.465,-2.83) – (0.93,-2.83); (0,-2.83) circle (4pt); (-2.83,2.83) – (-3.25,3.25); (-3.54,3.54) – (-3.25,3.25); (-3.54,3.54) circle (2pt); (-2.83,1.83) – (-2.83,2.33); (-2.83,2.33) – (-2.83,2.83); (-2.83,2.83) – (-2.33,2.83); (-2.33,2.83) – (-1.83,2.83); (-2.83,2.83) circle (4pt); (-2.83,1.83) – (-2.83,0.93); (-1.83,2.83) – (-0.93,2.83); (3.54,3.54) – (3.185,3.185); (2.83,2.83) – (3.185,3.185); (3.54,3.54) circle (2pt); (2.83,2.83) – (2.83,2.33); (2.83,2.33) – (2.83,1.83); (1.83,2.83) – (2.33,2.83); (2.33,2.83) – (2.83,2.83); (2.83,2.83) circle (4pt); (2.83,1.83) – (2.83,0.93); (1.83,2.83) – (0.93,2.83); (-2.83,-2.83) – (-3.185,-3.185); (-3.54,-3.54) – (-3.185,-3.185); (-3.54,-3.54) circle (2pt); (-2.83,-1.83) – (-2.83,-2.33); (-2.83,-2.33) – (-2.83,-2.83); (-2.83,-2.83) – (-2.33,-2.83); (-2.33,-2.83) – (-1.83,-2.83); (-2.83,-2.83) circle (4pt); (-2.83,-1.83) – (-2.83,-0.93); (-1.83,-2.83) – (-0.93,-2.83); (0.93,-2.83) – (2.83,-2.83); (2.83,-2.83) – (2.83,-3.49); (2.83,-3.49) – (2.83,-4.15); (2.83,-4.15) circle (2pt) ; at (3.4,-4.7) [$ \mathcal{O}_{R_n}(x_n)$]{}; at (5.8,-2.4) [$ \mathcal{O}_{R_1}(x_1)$]{}; at (6.3,-0.96) [$ \mathcal{O}_{R_2}(x_2)$]{}; (5.5,0.93) arc (10:20:5.5 cm); (1.35,-5.4) arc (-76:-85:6 cm); at (3.415,0.35) [$ R$]{}; at (2.3,0.5) [$R'$]{}; at (3.4,2.33) [$R''$]{}; at (-0.7,-3.415) [$R^{(k)}$]{};
- Next, one associates an open Wilson line (OWL) operator W\_y\^x(R, C) = P [exp]{} \[\_y\^x A\] for 1-form $A$ in (\[gaugeconnection\]) to the line labelled by the irrep R, connecting the points $x$ and $y$ in the graph.
- At every trivalent vertex where three lines carrying representation labels $(R_1, R_2, R_3)$ join – one glues the corresponding OWLs with the appropriate Clebsch-Gordan coefficients to make the vertex gauge invariant.
- One projects each of the external lines onto [*Cap States*]{} [@Verlinde:2015qfa; @Miyaji:2015fia; @Nakayama:2015mva] – a set of states in the conformal module $R$ labelling that leg that also provides a finite-dimensional irrep of the $so(d+1)$ subalgebra whose generators are $M_{ab}$ used in (\[gaugeconnection\]).
- One evaluates these OWNs for the gauge connection $A$ that corresponds to the Euclidean Poincare $AdS_{d+1}$. Such a gauge connection has to satisfy the flatness condition F:= dA + A A =0.
- Finally one takes the external legs to the boundary and reads out the leading component of the OWN - and these compute the relevant conformal partial waves.
This leading component of the OWN satisfies the conformal Ward identities and conformal Casimir equations expected of the partial waves of a correlator of primaries that are inserted at the points on the boundary to which the end points of the external legs of the OWN approach.
In short, the basic ingredients needed to compute our OWNs are (i) Wilson lines, (ii) CG coefficient and (iii) the cap states. These were found for $d=2$ in [@Bhatta:2016hpz] for the most general case. When the external legs were taken to the boundary the computation reduced to simple Feynman-like rules that require the knowledge of what we called [*legs*]{} (more precisely the conformal wave functions) and the CG coefficients. The explicit computations using these rules to find the global conformal blocks of correlators of primary operators (with any conformal dimension and spin) was demonstrated explicitly for $d=2$ in [@Bhatta:2016hpz] (see also [@Besken:2016ooo]).
Even though the general prescription for computing the partial waves of correlators of any set of primaries (in arbitrary representations of the rotation group of the boundary theory) in general CFT$_d$ using OWNs was laid down in [@Bhatta:2016hpz], the actual computations in higher dimensions could not be carried through as some of the necessary ingredients were missing. In this work we would like to report some progress in this direction. In particular, we will demonstrate how to implement our prescription explicitly for the scalar CPW $W^{(d)}_{\Delta, 0} (\Delta_i, x_i)$ in any CFT$_d$. Our results include a simplification of the computation of OWN[*s*]{} using the concept of [*OPE modules*]{} - which are close analogues of the OPE blocks that were studied in the literature [@Czech:2016xec; @Boer:2016pqk]. With this simplification we compute the scalar 4-point blocks in general dimension and show that our prescription reproduces the known answers [@Dolan:2011dv]. Remarkably, our results are naturally given in Gegenbauer polynomial basis [@Dolan:2011dv; @Hogervorst:2013sma]. Further, we show that there is a non-trivial recursion relation that emerges from our prescription which relates the scalar blocks in $d+2$ dimensions to those of $d$ dimensions. This relation reproduces the one in [@SimmonsDuffin:2012uy] in the context of even $d$, and provides an analogue for the odd $d$ cases.
The rest of the paper is organised as follows: The section \[sec2\] contains the construction of the modules and the conformal wave functions required for the computation of scalar blocks. We also introduce the concept of OPE module here and use it to carry out the computation of the 4-point scalar blocks in general dimensions. The section \[sec3\] contains details of how our answers match with several known results in $d \le 4$. In section \[sec4\] we derive recursion relations between different dimensions. In section \[sec5\] we include a couple of generalisations: most general bulk analysis in $d=1$, more general bulk geometries in $d=2$. We provide a discussion of our results and open questions in section \[sec6\]. The appendices contain some relevant mathematical results used in the text.
Scalar OWN in General Dimensions {#sec2}
================================
In this section we would like to provide details on how to explicitly compute the OWN[*s*]{} in $AdS_{d+1}$ spaces, with all lines (both external and internal) carrying scalar representations.
Collecting the Ingredients
--------------------------
As has been alluded to in the introduction the basic ingredients are Wilson lines, cap states and CG coefficients. We start with collecting these ingredients first. .5cm .5cm We will be evaluating the OWN in the background of the Euclidean $AdS_{d+1}$ geometry with ${\mathbb R}^d$ boundary ([*i.e*]{}, Poincare $AdS_{d+1}$) with the metric: \[eadsdplus1\] l\^[-2]{} ds\^2\_[AdS\_[d+1]{}]{} = d \^2 + e\^[2 ]{} dx\^i dx\^i. For this, working with the frame: e\^i = l e\^ dx\^i, i = 1, , d, e\^[d+1]{} = l dthe Wilson line reduces to W\_y\^x(R, C) = P [exp]{} \[\_y\^x A\] = g(x) g\^[-1]{} (y) as was shown in [@Bhatta:2016hpz], with g(x) = e\^[- M\_[0,d+1]{}]{} e\^[- x\_a (M\_[0,a]{}+M\_[a,d+1]{})]{} g\_0 , where the algebra generators are taken in the representation $R$ of $so(1, d+1)$. Using the standard identification of $so(1, d+1)$ generators as the conformal generators of ${\mathbb R}^d$: \[algdefs\] D=-M\_[0,d+1]{}, P\_= M\_[0]{} + M\_[,d+1]{}, K\_= -M\_[0]{} + M\_[,d+1]{}, [and]{} M\_ where $\alpha, \beta =1, \cdots, d$, the coset element $g(x)$ reads: \[generaldg\] g(x) = e\^[ D]{} e\^[- x\^a P\_a]{} g\_0. This gives us the Wilson lines. .5cm .5cm To project the external legs of the OWN operator we seek states, in the representation space $R$ carried by that external leg, that transform in a (finite dimensional) irrep of the subalgebra $so(d+1)$ with generators $\{M_{\alpha\beta}, M_{\alpha, d+1}\}$ [@Nakayama:2015mva]. In particular, for the scalar cap this finite dimensional representation is the trivial one, that is, annihilated by $\{M_{\alpha\beta}, M_{\alpha, d+1}\}$. Let us now construct these states. In terms of the generators in (\[algdefs\]) the $so(1, d+1)$ algebra reads &=& - (\_ P\_- \_ P\_), \[M\_, K\_\] = - (\_ K\_- \_ K\_), \[P\_, K\_\] &=& -2 M\_ -2 \_ D, \[D, P\_\] = P\_, \[D, K\_\] = -K\_ , \[M\_, M\_\] &=& \_ M\_ + \_ M\_ - \_ M\_ - \_ M\_ . We work with irreps $R$ of $so(1, d+1)$ that become UIR of $so(2,d)$ obtained by a Wick rotation. This implies the following reality conditions M\_[0,d+1]{}\^= M\_[0,d+1]{}, M\_[0]{}\^= - M\_[0,]{}, M\_[, d+1]{}\^= M\_[, d+1]{}, M\_\^= - M\_ . In terms of the generators in (\[algdefs\]) these mean: \[reality\] D\^= D, P\_\^= K\_, M\_\^= - M\_. Then the scalar cap state $|\Delta \rangle\!\rangle$ is defined to be a state in the scalar module $(\Delta, l_i =0)$ that satisfies the conditions: M\_ |= (P\_+ K\_) |= 0. We can construct it as a linear combination of states in the module over the scalar primary (lowest weight) state $|\Delta \rangle$ which satisfies \[scalarcapeqn\] D |= |, M\_ |= K\_ |=0. Rest of the basis states of the module take the form $|\Delta, k_i \rangle = {\cal N}_{\vec k} \, P_1^{k_1} \cdots P_d^{k_d} |\Delta \rangle$. The solution to the scalar cap state equation (\[scalarcapeqn\]) was provided first in [@Nakayama:2015mva] (see also [@Verlinde:2015qfa; @Miyaji:2015fia] for $d=2$ case). We rederive it here for completeness. For this note that the cap state has to be a singlet under $so(d)$ and therefore can only depend on $P_\alpha P^\alpha$. So write \[eq:scalar\_cap\] |= \_[n=0]{}\^C\_n (, d) (P\_P\^)\^n | , and impose $(P_\alpha + K_\alpha) |\Delta \rangle\!\rangle = 0$ to determine the coefficients $C_n$. Carrying out this straightforward exercise gives \[scalarcapcns\] C\_n (, d) = With these (\[eq:scalar\_cap\]) can be seen to be equivalent to the one in [@Nakayama:2015mva] using the definition of the Bessel function of first kind $J_\alpha (x)$. We will need the dual (conjugate under (\[reality\])) of this cap state which is given by: \[hcscalarcap\] | = \_[n=0]{}\^C\_n (, d) | (K\_K\^)\^n with the same $C_n$ as in (\[scalarcapcns\]).[^1]
In fact one can obtain more general cap states. For instance, in the case of $d=2$, we [@Bhatta:2016hpz] provided expressions for cap states in the module over the primary state $|h, \bar h \rangle$ that transform under $(j,m)$ representation of $so(3)$ algebra. In other dimensions one should seek caps that transform under arbitrary finite dimensional irreps of $so(d+1)$ – to be used in computing the OWN[*s*]{} with primaries that are not just scalars (see (\[vectorcap\]) for the vector cap state – provided for illustration). We however will not pursue this further here. .5cm .5cm The last ingredient in the computation of the OWN expectation values is the Clebsch-Gordan coefficients (CGC) of the gauge algebra $so(1,d+1)$. Some of these are known – see for instance [@kerimov1984]. Those are however not in a form that lends itself readily to our purposes. So here we propose a method to derive them using the 3-point functions.
For this first recall that the CG coefficients are defined as the invariant tensors in the product of three representations. That is, the CGC that appear in the tensor product decomposition $R_1 \otimes R_2 \rightarrow R_3$ satisfy: \[cgcinv\] R\_1\[g(x)\]\_[[**m**]{}\_1 [**m**]{}\_1’]{} R\_2\[g(x)\]\_[[**m**]{}\_2 [**m**]{}\_2’]{} C\^[R\_1, R\_2; R\_3]{}\_[[**m**]{}\_1’ ,[**m**]{}\_2’; [**m**]{}\_3’]{} R\_3\[g(x)\^[-1]{}\]\_[[**m**]{}\_3’ [**m**]{}\_3]{} = C\^[R\_1, R\_2; R\_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3]{} where $R_i[g(x)]_{{\bf m}_i {\bf m}_i'}$ is used to denote the matrix elements of $g(x)$ in the representation $R_i$, whose basis elements are collectively labelled by ${\bf m}_i$. In terms of the algebra elements $M_{AB}$ with $A, B = 0, 1, \cdots, d+1$, this eq. reads: \[cgcrecursion\] R\_1\[M\_[AB]{}\]\_[[**m**]{}\_1 [**m**]{}\_1’]{} C\^[R\_1, R\_2; R\_3]{}\_[[**m**]{}\_1’ ,[**m**]{}\_2; [**m**]{}\_3]{} + R\_2\[M\_[AB]{}\]\_[[**m**]{}\_2 [**m**]{}\_2’]{} C\^[R\_1, R\_2; R\_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2’; [**m**]{}\_3]{} = C\^[R\_1, R\_2; R\_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3’]{} R\_3\[M\_[AB]{}\]\_[[**m**]{}\_3’ [**m**]{}\_3]{} which is the recursion relation that determines the CGC. Now we argue that this is equivalent to the conformal Ward identity of the 3-point function of primary operators corresponding to the irreps $(R_1, R_2, R_3)$. The prescription of [@Bhatta:2016hpz] for the 3-point function of scalar primaries is to extract the leading term, [*i.e,*]{} the coefficient of $e^{-\rho (\Delta_1 + \Delta_2 + \Delta_3)}$ term – in the boundary limit of \[own3ptfn\] \_1| g(x\_1) |\_1, [**m**]{}\_1 \_2| g(x\_2) |\_2, [**m**]{}\_2 C\^[\_1, \_2; \_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3]{} \_3, [**m**]{}\_3 | g\^[-1]{}(x\_3) |\_3 We now show that this quantity satisfies the conformal Ward identity. To see this we note the following identities [@Bhatta:2016hpz]: \[gmmig\] g(x) M\_[AB]{} &=& l\^\_[AB]{} (x) \_g(x) + M\_[bc]{} g(x) M\_[AB]{} g\^[-1]{} (x) &=& -l\^\_[AB]{} (x) \_g\^[-1]{} (x) + g\^[-1]{} (x) M\_[bc]{} where the $l^\mu_{AB} (x)$ are the components of the Killing vector of the background geometry (\[eadsdplus1\]) carrying the indices of the corresponding $so(1, d+1)$ algebra generator $M_{AB} \in \{M_{0a}, M_{ab}\}$ of the left hand side. Next we consider: && \_1| g(x\_1) M\_[AB]{} |\_1, [**m**]{}\_1 \_2| g(x\_2) |\_2, [**m**]{}\_2 C\^[\_1, \_2; \_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3]{} \_3, [**m**]{}\_3 | g\^[-1]{}(x\_3) |\_3 && + \_1| g(x\_1) |\_1, [**m**]{}\_1 \_2| g(x\_2) M\_[AB]{} |\_2, [**m**]{}\_2 C\^[\_1, \_2; \_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3]{} \_3, [**m**]{}\_3 | g\^[-1]{}(x\_3) |\_3 && && - \_1| g(x\_1) |\_1, [**m**]{}\_1 \_2| g(x\_2) |\_2, [**m**]{}\_2 C\^[\_1, \_2; \_3]{}\_[[**m**]{}\_1 ,[**m**]{}\_2; [**m**]{}\_3]{} \_3, [**m**]{}\_3 | M\_[AB]{} g\^[-1]{}(x\_3) |\_3 which vanishes identically as a consequence of the recursion relation (\[cgcrecursion\]) for the CGC. On the other hand using the identities (\[gmmig\]) above and the fact that the scalar cap is killed by $M_{ab}$’[*s*]{} we see that the OWN for the 3-point function (\[own3ptfn\]) is invariant under simultaneous transformation of the three bulk points $(x_1, x_2, x_3)$ under any $AdS_{d+1}$ isometry. This in turn implies the conformal Ward identity in the limit of the external points $x_i$ approaching the boundary. It is of course true that the Ward identity completely determines the coordinate dependence of the 3-point function. Therefore, the question of finding the CGC is translated into finding expressions for the quantities $\langle\! \langle \Delta | g(x) |\Delta, {\bf m}\rangle$ and $\langle \Delta, {\bf m} | g^{-1}(x) |\Delta \rangle\!\rangle$ in the large radius limit, and then amputating them from the corresponding 3-point function (Fig. 2).[^2]
\[scale=0.6\] (-4,0) circle (3cm); (-4,0) circle (3cm); (-1,0) – (-2,0); (-2,0) – (-4,0); (-1,0) circle (2pt); (scissors) at (-3,0) ; (-6.12,-2.12) – (-5.6,-1.6); (-4,0) – (-5.6,-1.6); (scissors) at (-5.06,1.06) ; (-6.12,-2.12) circle (2pt); (-6.12,2.12) – (-5.6,1.6); (-4,0) – (-5.6,1.6); (scissors) at (-5.06,-1.06) ; (-6.12,2.12) circle (2pt); (-4,0) circle (4pt); (1,0.05) – (1.5,0.05); (1,-0.05) – (1.5,-0.05); (2.5,0) circle (10pt); (2.5,0) – (2.25,.25); (2.5,0) – (2.25,-.25); (2.5,0) – (2.85,0); at (-6.6,-2.7) [$\mathcal{O}_{\Delta_2}(x_2)$]{}; at (-6.6,2.6) [$\mathcal{O}_{\Delta_1}(x_1)$]{}; at (0,0) [$\mathcal{O}_{\Delta}(x)$]{};
.5cm
Processing the Ingredients
--------------------------
To proceed further we need the explicit expressions for the [**in-going**]{} legs $\langle\! \langle \Delta | g(x) |\Delta, {\bf m}\rangle$ and the [**out-going**]{} legs $\langle \Delta, {\bf m} | g^{-1}(x) |\Delta \rangle\!\rangle$ which are matrix elements of $g(x)$ and $g^{-1}(x)$ between the cap states $|\Delta \rangle \! \rangle$ and normalised basis elements $|\Delta, {\bf m} \rangle$ of the scalar module. So we turn to finding a suitable orthonormal basis for the module over a scalar primary $|\Delta \rangle$ next. .5cm .5cm The descendent states take the form $|\Delta, \{k_1, k_2, \dots ,k_d\} \rangle \sim \prod_{i=1}^{d}P_i^{k_i} |\Delta \rangle$. These states are eigenstates of the dilatation operator $D$ with eigenvalue $\Delta + \sum_{i=1}^d k_i$. States with different eigenvalues of $D$ are orthogonal. The set of states with a given conformal weight form a reducible representation of the rotation algebra $so(d)$ – which can be decomposed into a sum of irreps of $so(d)$. Then states belonging to different irreps will also be orthogonal. Therefore, a more suitable basis to work with would be in terms of the hyperspherical harmonics of the boundary $so(d)$ rotation algebra, $ ( P^2 )^s M^l_{{\bf m}} ({\bf P}) |\Delta\rangle $ where ${\bf m}$ denotes $(m_{d-2}, \cdots, m_2, m_1)$, whose conformal dimension is $\Delta + l + 2s$. In the rest of the paper we follow the conventions of [@WenAvery; @Junker] for hyperspherical functions.[^3] We define orthonormal states in this basis as follows[^4] ( P\^2 )\^s M\^l\_[[**m**]{}]{} ([**P**]{}) |&:=& A\_[l,s]{} | ; { l,[**m**]{},s}\
| (K\^2)\^s [M\^l\_[[**m**]{}]{}]{}\^([**K**]{}) &:=& A\^\*\_[l,s]{} ; { l,[**m**]{},s} | with ; { l’,[**m**]{}’,s’} | ; { l,[**m**]{},s} = \_[ll’]{} \_[[**m**]{} [**m**]{}’]{} \_[ss’]{} To find the normalisation $A_{l,s}$ let us start with the following observation \[obsiden\] | e\^[[**y**]{} ]{} e\^[[**x**]{} ]{} |= . On the left hand side of the above identity we expand the plane waves $e^{{\bf x} \cdot {\bf P}}$ in terms of spherical waves:[^5] \[swe\] e\^[[**x**]{} ]{} = \_[l=0]{}\^(2 l + d - 2)(d-4)!! j\_[l]{}\^[d]{}(x P) C\_[l]{}\^() where $j_l^{d}(x)$ is the spherical Bessel function and $C_l^{\mu} (z)$ is the Gegenbauer polynomials as defined below C\_[l]{}\^(x) = \_[k=0]{}\^[\[l/2\]]{} (-1)\^k x\^[l-2k]{} and j\_[l]{}\^[d]{}(x) = \_[s=0]{}\^ . One can also write Gegenbauer polynomials in terms of hyperspherical harmonics using the well known identity \[gptosh\] \_[[**m**]{}]{} Y \_[l; [**m**]{}]{}\^[\*]{}(\_[x]{}) Y\_[l;[**m**]{}]{}(\_[y]{})= C\_[l]{}\^() Substituting these into the we get: e\^[[**x**]{} ]{} = 4 a \^ \_[l=0]{}\^\_[s=0]{}\^\_[[[**m**]{}]{}]{} M\_[[**m**]{}]{}\^[l \*]{}([**x**]{})M\_[[**m**]{}]{}\^[l]{}([**P**]{})(P\^[2]{})\^[s]{} where $M_{{\bf m}}^{l}({\bf x}) = x^l \, Y_{l;{\bf m}}(\Omega_{x}) $ and a =
, &\
, &
Similarly e\^[[**y**]{} ]{} = 4 a \^ \_[l=0]{}\^\_[s=0]{}\^\_[[[**m**]{}]{}]{} M\_[[**m**]{}]{}\^[l \*]{}([**y**]{})M\_[[**m**]{}]{}\^[l]{}([**K**]{})(K\^[2]{})\^[s]{} . Therefore the left hand side of takes the following form \[lhs\] | e\^[[**y**]{} ]{} e\^[[**x**]{} ]{} |= \_[l=0]{}\^\_[s=0]{}\^(x\^2)\^s (y\^2)\^[s]{} \_[[**m**]{}]{} M\^[l]{}\_[[**m**]{}]{} ([**y**]{}) M\^[l \*]{}\_[[**m**]{}]{} ([**x**]{}) |A\_[l,s]{}|\^2 ()\^2\
Next we want to expand the [*rhs*]{} of (\[obsiden\]) in the same basis. For this we first write = with $t = x \, y$ and $\xi = t^{-1} {\bf x} \cdot {\bf y}$. We would now like to expand this quantity in terms of Gegenbauer polynomials $C_n^{\mu}(x)$. Luckily this exercise was done in [@Cohl2013] which reads[^6] \[cohlformula\] = \_[k=0]{}\^C\_k\^() t\^k \_2F\_1(+k, -; +k+1; t\^2) However, we are interested in expanding the [*lhs*]{} of (\[cohlformula\]) in $d$-dimensional hyperspherical harmonics in ${\bf x}$ which requires us to choose $\mu = (d-2)/2$. Using the series representation of the hypergeometric function: \_2F\_1(+k, -; +k+1; t\^2) = \_[n=0]{}\^ \
and using the identity we finally arrive at \[rhs\] = \_[l,s=0]{}\^ [(x\^2)]{}\^[l+2s]{} [(y\^2)]{}\^[l+2s]{} \_[[**m**]{}]{}M\^[l]{}\_[[**m**]{}]{} ([**y**]{}) M\^[l \*]{}\_[[**m**]{}]{} ([**x**]{})\
Comparing with , we get[^7] |A\_[l,s]{} |\^2 = Having found an orthonormal basis for the scalar module we would like to now compute the legs (conformal wave functions) as described in the beginning of this section.
.5cm For this we start with $g(x) = e^{\rho D} e^{- {\bf x \cdot P}}$. Then
&& |g(x)|; {l, [**m**]{}, s} && =\_[n=0]{}\^ (-1)\^n C\_n | (K\^2)\^n e\^[D]{} e\^[- [**x P**]{}]{} |; {l, [**m**]{}, s} &&= \_[n=0]{}\^ A\_[0,n]{}\^\*,{0,0,n}| e\^[D]{} e\^[- [**x P**]{}]{} ([P\^2]{})\^s M\^l\_[**m**]{} ([**P**]{}) | &&= \_[n=0]{}\^ (-1)\^n C\_n A\_[0,n]{}\^\*\_[l’=0]{}\^ \_[s’=0]{}\^ \_[[**m**]{}’]{} M\^[l’ \*]{}\_[[**m**]{}’]{}(-[**x**]{}) && ,{0,0,n}| e\^[D]{} (P\^2)\^[s+s’]{} M\^[l’]{}\_[[**m**]{}’]{}([**P**]{})M\^[l]{}\_[m]{}([**P**]{}) | Now using the identity for the hyperspherical harmonics \[eq:sp\_harmonics\_decomp\] M\^[l]{}\_[**m**]{}([**P**]{}) M\^[l’]{}\_[**m’**]{}([**P**]{}) = \_[L]{} \_[**n**]{}
l & l’ & L\
[**m**]{} & [**m’**]{} & [**n**]{}
(P\^2)\^ M\^L\_[**n**]{}([**P**]{}) where $ \big[ \begin{smallmatrix} l & l' & L \\ {\bf m} & {\bf m'} & {\bf n} \end{smallmatrix}\big] $ is $so(d)$ CG coefficients, we find $$\begin{aligned}
&\langle\!\langle \Delta |g(x)|\Delta ; \{l, {\bf m}, s\}\rangle \cr
&= \frac{4a\pi^{\frac{d}{2}}}{A_{l,s}} \sum_{n=0}^{\infty} (-1)^n C_n A^*_{0,n}\sum_{l'=0}^{\infty} \sum_{s'=0}^{\infty} \frac{(x^2)^{s'}}{s'! \, 2^{l'+2s'}\,\Gamma(l'+s'+d/2)} \sum_{\bf m'}
M^{l' *}_{\bf m'}(-{\bf x})\,\, e^{\rho(\Delta + l+l'+ 2(s+s'))} \cr
&\qquad \qquad \qquad \qquad \qquad \qquad \times \sum_{L} \sum_{\bf n}
\begin{bmatrix}
l & l' & L \\
{\bf m} & {\bf m'} & {\bf n}
\end{bmatrix}
\langle \Delta;\{0,0,n\}|(P^2)^{s+s'+ (l+l'-L)/2} M^L_{\bf n}({\bf P})|\Delta \rangle \cr
&= \frac{4a\pi^{\frac{d}{2}}}{A_{l,s}} \sum_{n=0}^{\infty} (-1)^n C_n A^*_{0,n}\sum_{l'=0}^{\infty} \sum_{s'=0}^{\infty} \frac{(x^2)^{s'}}{s'! \, 2^{l'+2s'}\,\Gamma(l'+s'+d/2)} \sum_{\bf m'}
M^{l' *}_{\bf m'}(-{\bf x})\,\, e^{\rho(\Delta + l+l'+ 2(s+s'))} \cr
&\qquad \qquad \qquad \qquad \qquad \qquad \times \sum_{L} \sum_{\bf n}
\begin{bmatrix}
l & l' & L \\
{\bf m} & {\bf m'} & {\bf n}
\end{bmatrix}
\, A_{L,s+s'+\frac{l+l'-L}{2}} \, \delta_{L0} \, \delta_{{\bf n}0} \, \delta_{n\left(s+s'+\frac{l+l'-L}{2} \right)}\nonumber
\end{aligned}$$ Carrying out the summation over $L$ and ${\bf n}$ we find $$\begin{aligned}
&\langle\!\langle \Delta |g(x)|\Delta ; \{l, {\bf m}, s\}\rangle \cr
&= \frac{4a\pi^{\frac{d}{2}}}{A_{l,s}} \sum_{n=0}^{\infty} (-1)^n C_n A^*_{0,n}\sum_{l'=0}^{\infty} \sum_{s'=0}^{\infty} \frac{(x^2)^{s'}}{s'! \, 2^{l'+2s'}\,\Gamma(l'+s'+d/2)}
M^{l}_{\bf m}(-{\bf x})\,\, e^{\rho(\Delta + l+l'+ 2(s+s'))} \cr
&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \times \delta_{ll'} \, A_{0,s+s'+\frac{l+l'}{2}} \, \delta_{n\left(s+s'+\frac{l+l'}{2} \right)}
\end{aligned}$$ where we have used \_[**m’**]{}
l & l’ & 0\
[**m**]{} & [**m’**]{} & 0
M\^[l’ \*]{}\_[**m’**]{}([**x**]{}) = \_[ll’]{} M\^[l]{}\_[**m**]{}([**x**]{}) . Therefore && |g(x)|; {l, [**m**]{}, s} && && = \_[n=0]{}\^ (-1)\^n C\_n A\^\*\_[0,n]{} \_[s’=0]{}\^ M\^[l ]{}\_[**m**]{}(-[**x**]{}) e\^[(+ 2(l+s+s’))]{} A\_[0,s+s’+l]{} \_[n(s+s’+l )]{} && &&= e\^ (x\^2)\^[-l-s]{} M\^[l]{}\_[**m**]{}(-[**x**]{}) \_[n=0]{}\^ (-1)\^n C\_n |A\_[0,n]{}|\^2 && &&=e\^[-]{} M\^[l ]{}\_[**m**]{}(-[**x**]{})(-1)\^[s+l]{} (e\^[2]{})\^[+l+s]{}(l+d/2)\_s ()\_[l+s]{} && \_2F\_1 (+l+s, l+s+d/2; l+d/2; -e\^[2]{}x\^2 ) Now we want to take $\rho \to \infty $ limit. We rewrite the hypergeometric function in the above expression using the identity \_2F\_1( a, b; c; z) = (1-z)\^[-a]{} \_2F\_1( a,c- b; c; ) as && \_2F\_1( +l+s, l+s+d/2; l+d/2; -e\^[2]{}x\^2) &&= (1+e\^[2]{}x\^2)\^[--l-s]{} \_2F\_1(+l+s, -s; l+; ) In the $\rho \to \infty $ limit the argument of the hypergeometric function tends to unity. As the following identity holds \_2F\_1 (-n, b; c; 1 ) = = , to the leading order in $e^{\rho}$ the in-going leg becomes |g(x)|; {l, [**m**]{}, s} e\^[-]{} (-1)\^[s+l]{} M\^[l]{}\_[**m**]{}(-[**x**]{}) (x\^2)\^[--l-s]{} ()\_[l+s]{} (d/2--s)\_s + where dots are subleading terms in $\rho \rightarrow \infty$ limit. Finally we use $ (-x)_n = (-1)^n(x-n+1)_n$ and $ (-1)^l M^{l}_{\bf m} ({\bf x}) = M^{l}_{\bf m} (-{\bf x}) $ to get \[finalincoming\] \_ e\^|g(x)|; {l, [**m**]{}, s}= 4 a \^[d/2]{} ()\_[l+s]{} (-)\_s (x\^2)\^[--l-s]{} M\^[l]{}\_[**m**]{}([**x**]{})\
.5cm For this we start with $g^{-1}(y) = e^{{\bf y\cdot P}} e^{-\rho D}$, and compute && ; {l,[**m**]{},s}|g\^[-1]{}(y)| &&= \_[n=0]{}\^ (-1)\^n C\_n e\^[-(+2n)]{} ; {l,[**m**]{},s}|e\^[[**yP**]{}]{}(P\^2)\^n| &&= 4a\^[d/2]{} \_[n=0]{}\^ (-1)\^n C\_n e\^[-(+2n)]{} \_[l’=0]{}\^ \_[s’=0]{}\^ \_[**m’**]{} M\^[l’ \*]{}\_[ **m’**]{}([**y**]{}) && A\_[l’,s’+n]{} \_[ll’]{} \_[**mm’**]{} \_[s(s’+n)]{} &&= 4a\^[d/2]{} \_[n=0]{}\^ (-1)\^n C\_n A\_[l,s]{} e\^[-(+2n)]{} M\^[l \*]{}\_[**m**]{}([**y**]{}) &&=e\^[-]{} A\_[l,s]{} \_[n=0]{}\^ (-1)\^n C\_n e\^[-2n]{} M\^[l \*]{}\_[**m**]{}([**y**]{}) As $\rho \to \infty $, to the leading order only the $n=0$ term contributes, so that we have the result \[finaloutgoing\] \_ e\^ ; {l,[**m**]{},s}|g\^[-1]{}(y)|= A\_[l,s]{} M\^[l \*]{}\_[**m**]{}([**y**]{}) The results of these rather lengthy, albeit straightforward exercises are (\[finalincoming\], \[finaloutgoing\]). These two sets of functions (\[finalincoming\]) and (\[finaloutgoing\]) provide a representation and its conjugate representation respectively of the conformal algebra $so(1,d+1)$, on which the conformal generators $\{D, M_{\alpha\beta}, P_\alpha, K_\alpha\}$ act through their differential operator representations on scalar primaries with dimension $\Delta$. One can use these to derive matrix representations of the conformal generators and therefore, can be more appropriately called the [*conformal wave functions*]{}.
Finally let us quickly carry out a check on our conformal wave functions, namely, that when they are used in our OWN prescription they have to reproduce the appropriate two-point function for the scalar primaries. According to our prescription the two-point function can be obtained as \_ ([**x**]{}) \_ ([**y**]{}) &&= \_ e\^[2]{}| g(x) g\^[-1]{}(y) | && = \_ e\^[2]{}\_[l=0]{}\^ \_[s=0]{}\^\_[**m**]{} |g(x)|; {l, [**m**]{}, s}; {l,[**m**]{},s}|g\^[-1]{}(y)|
(0,0) circle (3cm); (0,0) circle (3cm); (-3,0) – (-2,0); (-2,0) – (-1,0); (-3,0) circle (2pt); (1,0) – (2,0); (2,0) – (3,0); (3,0) circle (2pt); at (0.5,0) [$\langle l,{\bf m},s |$ ]{}; at (-0.5,0) [$ | l,{\bf m},s \rangle $ ]{}; at (-5,0) [ $ \sum_{l, {\bf m}, s}$]{};
As $\rho \to \infty$ the above diagram evaluates to && \_[l=0]{}\^ \_[s=0]{}\^\_[m=-l]{}\^[l]{} |g(x)|; {l, m, s}; {l,m,s}|g\^[-1]{}(y)| &&= e\^[-2]{} (x\^2)\^[-]{}(4 a \^[d/2]{})\^2 \_[l=0]{}\^ \_[s=0]{}\^ ()\^[2s]{} (x\^2)\^[-l]{}\_[**m**]{} M\^[l]{}\_[**m**]{}([**x**]{}) M\^[l \*]{}\_[**m**]{}([**y**]{}) Finally using and comparing with we obtain \_ ([**x**]{}) \_ ([**y**]{}) &=& 4 a\^2 \^[d/2]{} (x\^2)\^[-]{} (1-2 + )\^[-]{} &&=4 a\^2 \^[d/2]{} |[**x-y**]{}|\^[-2]{} This is the expected result for two-point function (up to an overall constant factor - which can be gotten rid of by multiplying the cap states by appropriate overall factors).
Introducing OPE module
----------------------
Finally we need to amputate the legs (\[finalincoming\], \[finaloutgoing\]) we have found in the previous subsection from the correlation function of three scalar primaries to find the CGC we need. The explicit expressions adapted to our method are given in appendix \[appA\]. However, to compute, for example, the 4-point conformal partial waves we need CGC[*s*]{} that are already connected to two legs at a time – which is obtained easily by starting with an appropriate 3-point function and amputating only one leg. This object depends on the boundary coordinates where two of the primaries are inserted, and carries labels of basis vectors of the module of the third primary. This is a close cousin of the so called OPE block [@Czech:2016xec; @Boer:2016pqk], which we call the [*OPE module*]{}.
These OPE modules can be characterised by two types of identities. To spell them out let us label the representations of the conformal algebra $so(1, d+1)$ of interest by $(\Delta, {\bf l})$ where $\Delta$ is the conformal dimension and ${\bf l}$ represents all the independent Casimirs of the representation. States in such a representation $R$ can be labelled by $(\Delta, {\bf l}; {\bf m}, s)$ where ${\bf m}$ is again a collective index of magnetic quantum numbers. It turns out there are two types of these OPE modules which we denote by ${\cal B}^{(\Delta_1, {\bf l}_1; {\bf x}_1), (\Delta_2, {\bf l}_2; {\bf x}_2)}_{(\Delta_3, {\bf l}_3; {\bf m}_3, s_3)}$ and ${\cal B}_{(\Delta_1, {\bf l}_1; {\bf x}_1), (\Delta_2, {\bf l}_2; {\bf x}_2)}^{(\Delta_3, {\bf l}_3; {\bf m}_3, s_3)}$. Then these OPE modules are supposed to satisfy the Ward identities: ([L]{}\_[x\_1]{}\[M\_[AB]{}\] + [L]{}\_[x\_2]{}\[M\_[AB]{}\]) [B]{}\^[(\_1, [**l**]{}\_1; [**x**]{}\_1), (\_2, [**l**]{}\_2; [**x**]{}\_2)]{}\_[(, [**l**]{}; [**m**]{}, s)]{} &=& [[M]{}\_[(, [**l**]{}; [**m**]{}, s)]{}]{}\^[(, [**l**]{}; [**m**]{}’, s’)]{} \[M\_[AB]{}\] [B]{}\^[(\_1, [**l**]{}\_1; [**x**]{}\_1), (\_2, [**l**]{}\_2; [**x**]{}\_2)]{}\_[(, [**l**]{}; [**m**]{}’, s’)]{} ([L]{}\_[x\_1]{}\[M\_[AB]{}\] + [L]{}\_[x\_2]{}\[M\_[AB]{}\]) [B]{}\_[(\_1, [**l**]{}\_1; [**x**]{}\_1), (\_2, [**l**]{}\_2; [**x**]{}\_2)]{}\^[(, [**l**]{}; [**m**]{}, s)]{} &=& - [B]{}\_[(\_1, [**l**]{}\_1; [**x**]{}\_1), (\_2, [**l**]{}\_2; [**x**]{}\_2)]{}\^[(, [**l**]{}; [**m**]{}’, s’)]{} [[M]{}\_[(, [**l**]{}; [**m**]{}’, s’)]{}]{}\^[(, [**l**]{}; [**m**]{}, s)]{} \[M\_[AB]{}\]\
where we denote the differential operator representation and the matrix representation of the conformal generator $M_{AB}$ by ${\cal L}[M_{AB}]$ and ${\cal M} [M_{AB}]$ respectively. From these identities it is very easy to see that both types of OPE modules satisfy the corresponding conformal Casimir equations.
For the scalar blocks of interest here, the two types of OPE modules can be obtained by amputating either an in-going (\[finalincoming\]), or an out-going leg (\[finaloutgoing\]) from the appropriate 3-point functions: $\langle {\cal O}_{\Delta_1} (x_1) {\cal O}_{\Delta_2} (x_2) {\cal O}_{\Delta} (x) \rangle$, $\langle {\cal O}_{\Delta} (x) {\cal O}_{\Delta_3} (x_3) {\cal O}_{\Delta_4} (x_4)\rangle$. See Fig. (4) for a pictorial representation of this procedure.
(-4,0) circle (3cm); (-4,0) circle (3cm); (-4,0) – (-2,0); (-2,0) – (-1,0); (-1,0) circle (2pt); (scissors) at (-3,0) ; (-6.12,-2.12) – (-5,-1); (-5,-1) – (-4,0); (-6.12,-2.12) circle (2pt); (-6.12,2.12) – (-5,1); (-5,1) – (-4,0); (-6.12,2.12) circle (2pt); (-4,0) circle (4pt); (1,0.05) – (1.5,0.05); (1,-0.05) – (1.5,-0.05); (5,0) circle (3cm); (5,0) circle (3cm); (2.88,2.12) – (4,1); (4,1) – (5,0); (2.88,-2.12) – (4,-1); (4,-1) – (5,0); (5,0) circle (4pt); (5,0) – (5.15,0); (5.15,0) – (5.65,0); (2.88,2.12) circle (2pt); (2.88,-2.12) circle (2pt); at (-6.7,-2.7) [$\mathcal{O}_{\Delta_2}(x_2)$]{}; at (-6.6,2.6) [$\mathcal{O}_{\Delta_1}(x_1)$]{}; at (0,0) [$\mathcal{O}_{\Delta}(x)$]{}; at (-3,-0.5) [$amputation$ ]{}; at (6.8,0) [$|\Delta, s, l, {\bf m}\rangle$]{}; at (2.4,2.6) [$\mathcal{O}_{\Delta_1}(x_1)$]{}; at (2.3,-2.7) [$\mathcal{O}_{\Delta_2}(x_2)$]{};
Finally the method to obtain the 4-point conformal partial wave using the OWN prescription reduces to taking two types of OPE modules defined above and contracting the module indices.
Computing the 4-point CPW
-------------------------
Having equipped ourselves with all the ingredients needed, we now turn to compute four-point conformal blocks for scalar primaries of conformal weights $\Delta_i$ for $i = 1,2,3,4$. For simplicity we take the operator insertion points to be at ${\bf x_1} \to \infty,\, {\bf x_2} \to {\bf u} ,\, {\bf x_3} \to {\bf x}$ and $ {\bf x_4} \to {\bf 0}$ with ${\bf u} \cdot {\bf u} =1$. As elucidated above this four-point conformal block can be computed using two specific OPE modules.
One of the OPE modules we need can be extracted from the three-point function, with the operator insertions at $(\infty,\, {\bf u},\, {\bf y} ) $ by amputating the out-going leg anchored at the boundary-point ${\bf y}$. The corresponding OPE module is shown in the figure 5 below.
\[scale=0.6\] (-1,2) – (1,3) – (1,-2) – (-1,-3); (2,0) – (1,0.5); (0,1) – (1,0.5); (2,0) – (1,-0.5); (0,-1) – (1,-0.5); (0,1) circle (2pt); (0,-1) circle (2pt); (2.15,0) – (3,0); (2,0) circle (4pt); (2,0) – (2.15,0); at (0,1.5) [$\small \mathcal{O}_{\Delta_2} ({\bf u})$]{}; at (0,-1.5) [$\small \mathcal{O}_{\Delta_1} (\infty)$]{}; at (0,-3) [$\rho \to \infty$ ]{}; at (4,0) [$|\Delta;\, l, { \bf m}, s \rangle$]{};
The three-point function takes the form \_[\_1]{}() \_[\_2]{} ([**u**]{}) \_ ([**y**]{}) &=& \_[z ]{} (z\^2)\^[\_1]{} \_[\_1]{}([**z**]{}) \_[\_2]{} ([**u**]{}) \_ ([**y**]{}) = which can be expanded in terms of hyperspherical harmonics using as && \_[\_1]{}() \_[\_2]{} ([**u**]{}) \_ ([**y**]{}) && &&= (4\^[d/2]{}) \_[l=0]{}\^ \_[s=0]{}\^ (y\^2)\^s \_[**m**]{} M\^[l]{}\_[[**m**]{}]{} ([**u**]{}) M\^[l \*]{}\_[[**m**]{}]{} ([**y**]{}) Amputation of the out-going leg (\[finaloutgoing\]) ending at ${\bf y}$ from the above expression gives \[leftopemodule\] \^ ( )\_[l+s]{} ( - )\_s M\^l\_[[**m**]{}]{} ([**u**]{}) where $\Delta_{ij} \equiv \Delta_i - \Delta_j$. Similarly to find the other OPE module we start with the three-point function && \_([**y**]{}) \_[\_3]{} ([**x**]{}) \_[\_4]{} ([**0**]{}) = (y\^2)\^ (x\^2)\^ Expanding this in hyperspherical harmonics gives && \_([**y**]{}) \_[\_3]{} ([**x**]{}) \_[\_4]{} ([**0**]{}) && &&= (4\^[d/2]{}) (x\^2)\^ \_[l, s=0]{}\^ (x\^2)\^s (y\^2)\^[-l-s]{} \_[**m**]{} M\^[l]{}\_[[**m**]{}]{} ([**y**]{}) M\^[l \*]{}\_[[**m**]{}]{} ([**x**]{}) &&
\[scale=0.6\] (-1,3) – (1,2) – (1,-3) – (-1,-2); (-2,0) – (-1,0.5); (-1,0.5) – (0,1); (0,1) circle (2pt); (-2,0) – (-1,-0.5); (-1,-0.5) – (0,-1); (0,-1) circle (2pt); (-2,0) circle (4pt); (-2,0) – (-2.15,0); (-2.15,0) – (-3,0); at (-4,0) [$\langle \Delta; l, {\bf m}, s |$]{}; at (0,1.5) [$\small \mathcal{O}_{\Delta_3} ({\bf x})$]{}; at (0,-1.5) [$\small \mathcal{O}_{\Delta_4} ({\bf 0})$]{}; at (0,-3) [$\rho \to \infty$ ]{};
Now amputating the in-going leg (\[finalincoming\]) starting from ${\bf y}$, we obtain \[rightopemodule\] && (x\^2)\^ \^ ( )\_[l+s]{} ( - )\_s (x\^2)\^s M\^[l \*]{}\_[[**m**]{}]{} ([**x**]{}) && Finally we glue the OPE modules (\[leftopemodule\]) and (\[rightopemodule\]) to compute the four-point conformal partial wave (see figure 7).
\[scale=0.8\] (-4,0) circle (3cm); (-4,0) circle (3cm); (4,0) circle (3cm); (4,0) circle (3cm); (-0.25,0.05) – (0.25,0.05); (-0.25,-0.05) – (0.25,-0.05); (-6.25,2) – (-5.875,1); (-5.875,1) – (-5.5,0); (-6.25,-2) – (-5.875,-1); (-5.875,-1) – (-5.5,0); (-5.5,0) circle (4pt); (-5.5,0) – (-5.45,0); (-5.45,0) – (-5,0); (-6.25,2) circle (2pt); (-6.25,-2) circle (2pt); (-2.5,0) – (-2.125,1); (-1.75,2) – (-2.125,1); (-2.5,0) – (-2.125,-1); (-1.75,-2) – (-2.125,-1); (-2.5,0) circle (4pt); (-2.5,0) – (-2.55,0); (-2.55,0) – (-3,0); (-1.75,2) circle (2pt); (-1.75,-2) circle (2pt); (5.5,0) – (5.875,1); (6.25,2) – (5.875,1); (5.5,0) – (5.875,-1); (6.25,-2) – (5.875,-1); (5.5,0) circle (4pt); (2.5,0) – (4,0); (4,0) – (5.5,0); (6.25,2) circle (2pt); (6.25,-2) circle (2pt); (1.75,2) – (2.125,1); (2.125,1) – (2.5,0); (1.75,-2) – (2.125,-1); (2.125,-1) – (2.5,0); (2.5,0) circle (4pt); (1.75,2) circle (2pt); (1.75,-2) circle (2pt); at (-3.5,0) [$\langle l, {\bf m},s |$ ]{}; at (-4.5,0) [$| l, {\bf m},s \rangle$ ]{}; at (-9,0) [ $ \sum_{l, {\bf m},s }$]{}; at (-6.6,2.6) [$\mathcal{O}_{\Delta_1} (x_1)$]{}; at (-6.6,-2.6) [$\mathcal{O}_{\Delta_2} (x_2)$]{}; at (-1.4,2.6) [$\mathcal{O}_{\Delta_4} (x_4)$]{}; at (-1.4,-2.7) [$\mathcal{O}_{\Delta_3} (x_3)$]{}; at (1.4,2.6) [$\mathcal{O}_{\Delta_1} (x_1)$]{}; at (1.4,-2.7) [$\mathcal{O}_{\Delta_2} (x_2)$]{}; at (6.6,-2.7) [$\mathcal{O}_{\Delta_3} (x_3)$]{}; at (6.6,2.6) [$\mathcal{O}_{\Delta_4} (x_4)$]{}; at (4,0.3) [$\mathcal{O}_{\Delta}$]{};
.5cm
Thus our prescription for the corresponding four-point conformal partial wave gives W\^[(d)]{}\_[, 0]{} (\_i, [**x**]{}) &=& (x\^2)\^ \_[l,s]{} && ( - )\_s ( - )\_s (x\^2)\^s \_[[**m**]{}]{} [M\^l\_[[**m**]{}]{}]{}\^([**x**]{}) M\^l\_[[**m**]{}]{} ([**u**]{})\
Using we express this in terms of Gegenbauer polynomials \[ourblock\] W\^[(d)]{}\_[, 0]{} (\_i, [**x**]{}) &=& (x\^2)\^ \_[l,s]{} && ( - )\_s ( - )\_s x\^[l+2s]{} C\_[l]{}\^()\
This is our final result for the scalar conformal partial wave. Even though we assumed $d \ge 2$, we will see in the next section this result also holds for $d=1$. Notice that as advertised in the introduction our answer is naturally given in terms of Gegenbauer polynomials.
A result for the same quantity already exists in the literature in terms of the cross ratios [@Dolan:2011dv]. In appendix \[appB\] we show our answer agrees with their result.
In principle one can put together the conformal wave functions of section 2, and the CGC of appendix \[appA\] suitably to generate the scalar CPW[*s*]{} of any higher-point scalar correlators as well (as was done for $d=2$ case in [@Bhatta:2016hpz]).
Recovery of Results in $d \le 4$ {#sec3}
================================
In this section we want to recover the known results for four-point scalar conformal partial waves in $d = 1, \cdots, 4$ from our answer (\[ourblock\]). For this we find it convenient to express our answer in different variables. Writing ${\bf x \cdot u} = x \, \cos \theta$, we define \[zbzdefs\] z = x e\^[i]{}, |z = x e\^[-i]{} . In terms of these variables $(z, \bar z)$ the four-point CPW (\[ourblock\]) takes the form \[ourblockzbz\] W\^[(d)]{}\_[, 0]{} (\_i; z, |z) &&= (z|z)\^ \_[l,s]{} &&( - )\_s ( - )\_s (z|z)\^[s+]{} C\_[l]{}\^() .5cm Substituting $d=4$ in (\[ourblockzbz\]) and manipulating further we find W\^[(4)]{}\_[, 0]{} (\_i, z, |z) &=& (z |z)\^[(- \_3-\_4)]{} \_[l,s=0]{}\^((+ \_[34]{})+l+s) ((- \_[12]{})+l+s) && ((- \_[12]{})+s-1)((+ \_[34]{})+s-1) && (z\^[l+s+1]{} [|z]{}\^s - z\^s [|z ]{}\^[l+s+1]{}) &=& () (-1) () (-1) (z |z)\^[(- \_3-\_4)]{} && && where $\alpha = \tfrac{1}{2}(\Delta- \Delta_{12})$ and $\beta = \tfrac{1}{2}(\Delta+ \Delta_{34})$. The details of the calculation of how to go from the first to the second expression are relegated to appendix \[appC\]. Our answer perfectly matches with the known results [@Dolan:2011dv]. .5cm When $d=3$ the Gegenbauer polynomials used to express the answer (\[ourblockzbz\]) become the Legendre polynomials, [*i.e.,*]{} $C^{1/2}_l (\cos\theta) = P_l(\cos\theta)$. Therefore, our answer reads W\^[(3)]{}\_[, 0]{}(\_i; z, |z) &&= (z|z)\^ \_[l,s=0]{}\^ &&( - )\_s ( - )\_s (z|z)\^[s+]{} P\_l() We are not aware of any closed form for this case. There exists a conjectured formula by [@Hogervorst:2016hal] where the $d=3$ four-point block is written in terms of 2$d$ blocks. We have checked that our answer also agrees with [@Hogervorst:2016hal] for large ranges of $l$ and $s$. .5cm To recover the answer for $d=2$ we have to take the $d\rightarrow 2$ limit of (\[ourblockzbz\]). We find W\^[(2)]{}\_[, 0]{}(\_i; z, |z) &=& (z|z)\^ \_[l,s=0]{}\^ ( )\_s ( )\_s (z|z)\^[s+]{} (l) &=& (z|z)\^ \_[l,s=0]{}\^ (z\^[l+s]{} |z\^s + z\^s |z\^[l+s]{}) where we have used the following identity \_[0]{} C\^\_l() = T\_l () = (l) where $T_l$ are Chebyshev polynomials of the first kind. Finally performing the summations we recover the familiar answer for scalar CPW in two dimensions W\^[(2)]{}\_[, 0]{} (\_i; z, |z) &=& (z|z)\^[(- \_3-\_4)]{} \_2F\_1 \_2F\_1 && .3cm This case corresponds to $\frac{d-2}{2} = -1/2$ and the corresponding Gegenbauer polynomials take the following form: C\^[-]{}\_l () = \_[l,0]{} - \_[l,1]{} + (l-2) P\_[l-1]{}() Further, in this case all the positions of the operators are simply real numbers. In particular, the unit vector ${\bf u}$ becomes either $1$ or $-1$. Without loss of generality we take ${\bf u}=1$. Then the argument of the Gegenbauer polynomials in , $ \hat{ {\bf x} }\cdot {\bf u}$ also becomes $\pm 1$ depending on the sign of ${\bf x}$. For both the cases the Gegenbauer polynomial simplifies to C\^[(-1/2)]{}\_l (1) = \_[l0]{} \_[l1]{} = \_[l0]{} - [sign]{} ([**x**]{}) \_[l1]{} Then the expression for 4-point CPW splits into two parts as follows \[onedblockone\] W\^[(1)]{}\_[, 0]{}(\_i, x) &=& (x\^2)\^ where $\alpha = \tfrac{1}{2}(\Delta- \Delta_{12})$ and $\beta = \tfrac{1}{2}(\Delta+ \Delta_{34})$ as before. Now using the following identities for Pochhammer symbols &&(A )\_s ( A + )\_s = ( 2A)\_[2s]{} , (A)\_[s+1]{} (A + )\_s = ( 2A)\_[2s+1]{} && for $A \in \{\alpha, \beta, \Delta \}$, and s! ()\_[s]{} = , s! ()\_[s+1]{} = we can show that the expression (\[onedblockone\]) can be written as a single sum, which can be carried out to yield the answer \[1d4ptblock\] W\^[(1)]{}\_[, 0]{}(x) &=& x\^[- \_3-\_4]{} \_2F\_1 ( 2, 2; 2; x ) where $x = |{\bf x}|$. This expression agrees with the known result [@Qiao:2017xif; @Gross:2017aos] for the $d=1$ case.
Seed Blocks and Recursion Relations {#sec4}
===================================
There exist in the literature some powerful recursion relations that enable one to compute the CPW in a given dimension in terms of those in lower dimensions [@SimmonsDuffin:2012uy; @Penedones:2015aga; @Hogervorst:2016hal]. For instance, one such recursion relation among the even dimensional CPW[*s*]{} was given in [@SimmonsDuffin:2012uy]. In this section we give a different (and simpler) proof of this relation using our answer, and provide a counterpart of such a relation among the odd dimensional CPW[*s*]{}. For this we begin by extracting the conformal block from the CPW via the relation: $W^{(d)}_{\Delta, 0} (\Delta_i, {\bf x}):= (x^2)^{-\frac{1}{2}( \Delta_3+\Delta_4)} G^{\mu}_{\Delta}(\alpha,\beta; {\bf x})$ where $\alpha = \tfrac{1}{2}(\Delta- \Delta_{12}),\,
\beta =\tfrac{1}{2}(\Delta+ \Delta_{34})$ and $\mu = \frac{d-2}{2}$. Then from (\[ourblock\]) we have: \[defG\] G\^\_(,; [**x**]{}) = && (x\^2)\^ \_[l,s=0]{}\^ (1+) x\^[l+2s]{} C\_[l]{}\^() . Differentiating with respect to $\cos\theta$ and using the identity C\_[l]{}\^(z) = 2 C\_[l-1]{}\^[+1]{}(z) (\[defG\]) becomes &=& (x\^2)\^ \_[l=1]{}\^\_[s=0]{}\^ (1+) x\^[l+2s]{} 2 C\_[l-1]{}\^[+1]{}() &=& (x\^2)\^ \_[l, s=0]{}\^ (1+) x\^[l+2s]{} C\_[l]{}\^[+1]{}()\
&=& G\^[+1]{}\_[+1]{}(+1,+1; [**x**]{}) where, in going from the first to the second line we have replaced $l \rightarrow l+1$ and used the identity: $(\alpha)_{n+1}=\alpha \, (\alpha+1)_n$. By applying this relation repeatedly (say $k$ times) we arrive at: \[recident\] G\^\_(,; [**x**]{}) &=& ()\^k G\^[-k]{}\_[-k]{} (-k ,-k; [**x**]{}) &=& ()\^k G\^[-k]{}\_[-k]{} (-k ,-k; [**x**]{}) where we have used $(\alpha-k)_k \, (\alpha)_{-k} =1$. Since $\mu \rightarrow \mu+1$ corresponds to $d \rightarrow d+2$ the equation (\[recident\]) says that we can get all even (odd) dimensional conformal blocks starting from, say the 2$d$ (3$d$) blocks. Writing $d=2k+2 + 2 \gamma$ where $\gamma = 0$ for even $d$ and $\gamma = 1/2$ for odd $d$, we can recast this result as G\^[k+]{}\_(,; [**x**]{})=()\^[k]{} G\^\_[- k]{} (- k ,- k; [**x**]{}) where using (\[zbzdefs\]) we have defined =(z-|[z]{}) . This result for the case of $\gamma = 0$ (relating different even dimensional blocks) is the one found in [@SimmonsDuffin:2012uy] – whereas the case of $\gamma = 1/2$ is its odd dimensional counterpart.
Some Odds and Ends {#sec5}
==================
In this section we present a couple of additional results that are a selection of possible generalisations in various directions of the cases considered so far. One of the limitations is the restriction to scalar operators (both in the external and the internal legs). The cases of $d=1$ and $d=2$ are the simplest to address in this regard. The $d=2$ case was solved completely in [@Bhatta:2016hpz]. The $d=1$ case can also be treated in full generality, which we present here.
Complete $d=1$ analysis
-----------------------
First we would like to compute the cap state for 1$d$ case and then the 1$d$ global blocks. We begin with the infinite dimensional matrix representations [@Jackiw:1990ka] of global conformal algebra $sl(2,\bf{R})$ for CFT$_1$: L\_[1]{}|h,n= |h,n-1, && L\_[-1]{}|h,n= |h,n+1 , L\_[0]{}|h,n&=& (h+n) |h,nwhere $D = L_0$, $P = L_{-1}$ and $K= L_1$. The bulk is the ${\mathbb H}^2$ space whose tangent space rotation group is $SO(2)$. Therefore, the cap state $|h, \theta \rangle\!\rangle$ transforms as a 1-dimensional irrep of $SO(2)$: (L\_[1]{}-L\_[-1]{})|h, = |h, The parameter $\theta$, a purely imaginary number, is related to the spin of the general bulk field – we will elaborate further on this shortly. This equation can be solved for $|h, \theta \rangle \! \rangle$ as a linear combination of states in the module: |h, = \_[n=0]{}\^ C\_[n]{}|h,n , writing $C_{n}= \sqrt{\frac{\Gamma(2h)}{n! \, \Gamma(2h+n)}} ~ f_{n}$ with the $f_n$ satisfying the recursion relation f\_[n+1]{}= f\_[n]{}+ n (2h+n-1) f\_[n-1]{} . It is not difficult to see that the $f_n$ are generated by $G(x)=\sum_{n=0}^{\infty}\frac{x^{n}}{n!} \, f_{n}$ where G(x)=(1-x)\^[-h-]{}(1+x)\^[-h+]{} . We find the coefficient of $\frac{x^n}{n!}$ in $G(x)$ to be: f\_[n]{}=(-1)\^[n]{} (h-)\_[n]{} \_[2]{}F\_[1]{}(-n, h+, -h+-n+1;-1) . Having obtained the expression for the most general cap state in $d=1$, we can repeat the rest of the exercises carried out in section 2 on these caps. Working with the coset element $g(x) = e^{\rho L_{0}} \, e^{-x L_{-1}}$ we can extract the leading terms in the large-$\rho$ limit of $\langle\!\langle h, \theta | g(x)|h,k\rangle$ and $\langle h,k | g^{-1}(y)|h, \theta \rangle\! \rangle$. With some further analysis we find the following simple answers in the $\rho\rightarrow \infty $ limit: \[1dlegs\] \_ e\^[h]{} h, | g(x)|h,k &= &(-1)\^[-h- ]{} x\^[-2h-k]{} \_ e\^[h]{} h,k | g\^[-1]{}(y)|h, &=& y\^[k]{} Notice that even though the general cap states depend on the spin-parameter $\theta$ the final expressions (\[1dlegs\]) for the legs have essentially no dependence on it. For example, putting the legs together and performing the sum over $k$ gives \_ e\^[2h]{} h, | g(x) g\^[-1]{}(y)|h, = (-1)\^[-h- ]{} x\^[-2h]{} \_[k=0]{}\^ ()\^k = (-1)\^[-h- ]{}\
A comparison of the $d=1$ legs here with the holomorphic part of the $d=2$ case of [@Bhatta:2016hpz] enables us to immediately write down the 1$d$ blocks by starting with the holomorphic parts of $d=2$ blocks and replacing $h \rightarrow \Delta$ and $z \rightarrow |x|$. It is evident that this will give rise to the 4-point block found above (\[1d4ptblock\]), and higher-point ones to match with [@Qiao:2017xif; @Gross:2017aos]. .5cm .5cm To better understand the role of $\theta$ we must first look at the linearised bulk equations satisfied by the the legs $\langle h,k | g^{-1}(x)| h, \theta \rangle\!\rangle$ and $\langle\!\langle h, \theta | g(x)|h,k\rangle$. To this end we first list the following identities [@Bhatta:2016hpz] satisfied by $g^{-1}(x)$ && L\_[0]{} g\^[-1]{}(x) = (-\_ +x \_[x]{}) g\^[-1]{}(x), L\_[-1]{} g\^[-1]{}(x) = -\_[x]{}g\^[-1]{}(x) ,&& L\_[1]{} g\^[-1]{}(x)= (2 x \_ -x\^[2]{} \_[x]{} +e\^[-]{} \_[x]{}) g\^[-1]{}(x) + g\^[-1]{}(x) e\^[-]{} (L\_[1]{}-L\_[-1]{}) . Using these we can easily compute the action of the $sl(2,\bf{R})$ Casimir operator $C_{2}$ on $g^{-1}(x)$ C\_[2]{}&=& 2 L\_[0]{}\^[2]{} - L\_[1]{} L\_[-1]{} -L\_[-1]{} L\_[1]{} C\_[2]{} g\^[-1]{}(x)& =& 2 (\^[2]{}\_+\_ +e\^[-2 ]{} \_[x]{}\^[2]{}) g\^[-1]{}(x) +e\^[-]{} \_[x]{} g\^[-1]{}(x) (L\_[1]{}-L\_[-1]{}) Thus we see that the legs $\langle h,k | g^{-1}(x)| h, \theta \rangle\!\rangle$ satisfy the second order PDE: \[firstpde\] (\_\^[2]{} +\_ + e\^[-2 ]{} \_[x]{}\^[2]{}+ e\^[-]{} \_[x]{}) h,k | g\^[-1]{}(x)| h, = (-1)h,k |g\^[-1]{}(x)| h, . It is not difficult to see that the other legs $\langle\!\langle h, \theta | g(x)|h,k\rangle$ also satisfy the same equation. We would now like to interpret this equation as that of a bulk local field in the background $AdS_{2}$ geometry with metric $ds^2_{{\mathbb H}^2} = d\rho^2 + e^{2\rho} \, dx^2$.
Since the boundary isometry group is just ${\mathbb Z}_{2}$ we would expect the boundary conformal primary operators to be characterised by a scaling dimension $\Delta$ and a parity $\pm 1$. But any general bulk local field in two dimensions (once one trades off the spacetime indices for the tangent space ones) has to have only two parameters: the mass and the spin on which the bulk covariant derivative acts as D\_(x) = \_(x) + \^[ab]{}\_L\_[ab]{} (x) where $L_{ab}$ is the tangent space rotation generator in the representation of $\psi (x)$. Redefining the coordinates $z = e^{-\rho} +i x \, , ~ \bar{z} = e^{-\rho} - i x $ the metric of $AdS_2$ becomes $ds^{2} = \frac{4 dz d\bar{z}}{(z+\bar{z})^2}$. For this geometry we have the following non-zero vielbeins, spin-connections and Christoffel connections: e\^[+]{} =, e\^[-]{} = , [\^[+]{}]{}\_[+]{}== - [\^[-]{}]{}\_[-]{}, \_[z z]{}\^[z]{}= = \_[|[z]{} |[z]{}]{}\^[|[z]{}]{} Since the tangent space is just ${\mathbb R}^2$, there is only one rotation generator $L_{+-}$, and we can take the field $\psi (x)$ to be an eigenstate of it with eigenvalue $i \, \theta$. Then it is easy to show that such a field satisfies the following \[secondpde\] (- m\^2) (x) &=& (\_\^[2]{} +\_ + e\^[-2 ]{} \_[x]{}\^[2]{}+ e\^[-]{} \_[x]{}) (x) -(m\^2 + \^2) (x) =0 Comparing (\[firstpde\]) and (\[secondpde\]) we make the following identifications: (-1)= m\^[2]{}+ . Therefore, we conclude that, when it is available, the parameter $\theta$ represents the spin of the bulk field.[^8]
OWN[*s*]{} in more general $AdS_3$ geometries
---------------------------------------------
In [@Bhatta:2016hpz] we provided the computation of CPW of vacuum correlators in CFT$_2$ of primaries in general representations of the conformal algebra using the Wilson network prescription. Here we extend this result to include CPW of correlators in any (heavy) state, and to thermal correlators. This involves computing the OWN in appropriate locally $AdS_3$ geometries. Recall that in Fefferman-Graham gauge the most general solution to $AdS_3$ gravity [@Banados:1998gg] is l\^[-2]{} ds\^2 &=& d\^2 + (dx\_1\^2 + dx\_2\^2) (e\^[2]{} + T(z) |T(|[z]{}) e\^[-2]{}) && + (T(z) + |T(|[z]{})) (dx\_1\^2 - dx\_2\^2) + 2i (T(z) - |T(|[z]{})) dx\_1 dx\_2 When $- \infty < x_i < \infty$ – it is a Euclidean locally $AdS_3$ geometry with boundary ${\mathbb R}^2$. For constant values of $T, \bar T \ge 0$ these are interpreted as BTZ black holes. When $-1/4 < T, \bar T <0 $ these represent heavy CFT states. We restrict to the constant $T, \bar T$ cases from now on. The relevant coset element is g(x) &=& e\^[ (L\_0 + |L\_0)]{} e\^[-z (L\_[-1]{} - T L\_1) ]{} e\^[-|z (|L\_[-1]{} - |[T]{} |L\_1) ]{} One can carry out the rest of the computations following [@Bhatta:2016hpz]. We find that the expressions for legs in the $\rho \rightarrow \infty$ are: \_ e\^[(h + |h)]{} h, |h; j, m | g(x)|h, |h; k,|k & =& ( )\^[2h]{} ( )\^[2|h]{} && ()\^k ()\^[|k]{} && \_ e\^[(h + |h)]{} h, |h; k, |k | g\^[-1]{}(x)|h, |h; j,-j = Putting them together for the 2-point function yields: \_[(h, |h)]{}(x\_1) [O]{}\_[(h, |h)]{}(x\_2) \_[([T]{}, |[ T]{})]{} = ()\^[2h]{} ()\^[2 |h]{} . which is the well known two-point function of a thermal CFT [@Gubser:1997cm; @Datta:2011za] (see also [@Chen:2016uvu] and more recently [@Castro:2018srf]). The higher-point blocks can also be computed for these geometries [@ourunpub].
Discussion {#sec6}
==========
In this paper we have continued to develop further our prescription [@Bhatta:2016hpz] to compute the conformal partial waves of CFT correlation functions using the gravitational open Wilson network operators in the holographic dual gravity theories. In particular, we have demonstrated how to use gravitational Open Wilson Networks to compute 4-point scalar partial waves (both external and the exchanged operators being scalars) in any dimension. Our result for the scalar CPW are naturally given in the Gegenbauer polynomial basis. We have compared our results with the known answers wherever available and found complete agreement. Our methods also lead to a simpler proof of the recursion relation of [@SimmonsDuffin:2012uy] in even dimensional CFT[*s*]{}, and lead to analogous recursion relations for odd dimensions.
The CPW for correlation functions of any primary with any exchanged operator has already been achieved in $d=2$ case in [@Bhatta:2016hpz] and here in (in the simpler case of) $d=1$. It remains to generalise these computational techniques to obtain the CPWs for arbitrary representations in $d \ge 3$.[^9] This involves finding the relevant cap states, and from there the relevant legs (conformal wave functions), and OPE modules etc. This work is in progress [@BRS3] and we hope to report on it in the near future. So far we have found the caps states for vectors, rank-2 antisymmetric and symmetric traceless tensor representations, and working on finding others. For those who may be interested, we present here the expressions of the cap states for the vector representation of the tangent space rotation algebra $so(d+1)$. This is constructed as a linear combination of the basis elements of the conformal module over a vector primary state and it is given by: \[vectorcap\] |\_, &=& \_[n=0]{}\^(P\_P\^)\^n \_[=1]{}\^d A\^[(n)]{}\_(, d) |\_, + \_[n=0]{}\^(P\_P\^)\^n B\^[(n)]{} (, d) P\_ \_[=1]{}\^d P\_|\_, |\_, d+1 &=& \_[n=0]{}\^C\^[(n)]{} (, d) (P\_P\^)\^n \_[=1]{}\^d P\_|\_, with $A^{(n)} = C^{(n)} (\Delta_\phi-1)$ and $B^{(n)} = \frac{ C^{(n)}}{2}+ 2(n+1)C^{(n+1)} \left(\Delta_\phi + n - \frac{d}{2} +2 \right)$. .5cm Of course one would like to see if our method gives answers in forms more amenable to potential applications, such as in the bootstrap approach towards the classification of CFT[*s*]{}. Since our answers are in Gegenbauer polynomial basis it is possible that they may be found more suitable – as working with this basis is much simpler (as we have seen in section 4, for example).
An interesting set of future directions should include exploring the role of Weight shifting operators [@Costa:2011dw; @Karateev:2017jgd; @Costa:2018mcg] in our formalism.
It may be of interest to compute objects similar to our OWN[*s*]{} in both flat and de Sitter gravity theories. Such diagrams could provide a basis of partial waves for S-matrices for scattering problems in these spaces.
Another possible generalisation should involve inclusion of boundaries and defects to the CFT [@McAvity:1995zd; @Nakayama:2016cim; @Gadde:2016fbj] in the formalism considered.
We hope that this program will naturally lend itself to answering dynamical questions as well in CFT[*s*]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank ICTS, Bengaluru for hospitality during the final stages of this work, and the participants of the workshop “AdS/CFT @ 20 and Beyond" for discussions. AB would like to thank IoP, Bhubaneswar for hospitality while the work was in progress.
CGC required for scalar CPW {#appA}
===========================
Here we record results of CGC of the representations considered in the text for the algebra $so(1, d+1)$. These are needed to compute the CPW for higher-point functions from OWNs. Before we present the detailed derivation of the CGC from the 3-point functions we collect a few facts about the irreducible representations of $so(d)$ which we will use in the extraction of the Clebsch-Gordan coefficients.
A finite dimensional irreducible representation of $so(d)$ is uniquely defined by its highest weight $[\mu_{1},\mu_{2},...\mu_{k}]$ with &&\_1 \_2 \_[k-1]{} |\_[k]{}| &&\_1 \_2 \_[k-1]{} \_[k]{} 0 The components $\mu_{i}$ are either simultaneously integers (tensorial representations) or half-integers (spinorial representations). We only consider symmetric traceless representations of $so(d)$ as these are the only relevant ones for the scalar CGC of $so(d+1,1)$. These could be represented on the Hilbert space $H$ of square integrable function on $S^{d-1}$. The Hilbert space can be decomposed into an orthogonal sum of subspaces $H^{l}$ of homogenous polynomials of degree $l$ in $d$ variables. We introduce a complete orthonormal basis ${|l, {\bf M} \rangle}$ on $H^{l}$, where ${\bf M} =(m_{d-2},m_{d-3},...,m_2,m_1)$ label these basis states provided they fulfil: l=m\_[d-1]{} m\_[d-2]{}m\_2 |m\_1| m\_1 m\_i \_[>0]{} i2 The dimension of the space $H$ is $d_{l}= (2l+d-2) \frac{(l+d-3)!}{l!(d-2)!}$ - the number of independent components of a general symmetric traceless tensor of rank $l$ in $d$ dimensions. The matrix elements of the representation $D^{l}$ read: D\^[l]{}\_[[**M**]{} [**M**]{}’]{}(g) = l, [**M**]{}|D\^[l]{}(g) |l,[**M**]{}’In particular, D\^[l]{}\_[[**M**]{} [**0**]{}]{}(g) = N\^[d]{}\_[l [**M**]{}]{} \_[k=1]{}\^[d-2]{} C\_[m\_[k+1]{}-m\_[k]{}]{}\^[m\_[k]{}+k/2]{} (\_[k+1]{}) \^[m\_k]{}(\_[k+1]{}) e\^[i m\_[1]{} \_[1]{}]{} where $N^{d}_{l\, {\bf M}}$ is the normalisation w.r.t the Haar measure on $so(d)$, $C_{\lambda}^{n}(z)$ are the Gegenbauer polynomials. The angles $0 \leq \Phi_{1}\leq 2 \pi$ and $0 \leq \Phi_{i}\leq \pi$ for $i\neq 1$ can be identified with the Euler angles of a rotation $g$ which maps the north pole $a=(0,\cdots,0,1) \in {\mathbb R}^d$ to an arbitrary point on $S^{d-1}$. Then the hyperspherical harmonics on $S^{d-1}$ are defined as follows: |e= D\^[l]{}(g) |a, Y\_[l [**M**]{}]{}(e) = e| l, [**M**]{} , a| l,[**M**]{}= \_[[**M**]{} [**0**]{}]{} where $V_{d}= \frac{2 \pi^{d/2}}{\Gamma(d/2)}$ is the volume of unit $S^{d-1}$ sphere. Therefore, we get Y\_[l [**M**]{}]{}(e) &= & D\^[l]{}\_[[**M**]{} [**0**]{}]{}(g) We finally list the following properties of hyperspherical harmonics which can be easily derived using the definitions given above:
1. $ Y^{*}_{l\, {\bf M}}(e) = (-1)^{m_1} Y_{l \, \bar{{\bf M}}}(e) $ where $\bar{{\bf M}}=(m_{d-2},\cdots,m_2,-m_1)$.
2. $
Y_{l_1\, {\bf M}_1}(e)Y_{l_2\, {\bf M}_2}(e) = \sum_{l_3, {\bf M}_3} \left( {l_1 \, l_2 \, l_3}\atop{{\bf M}_1\, {\bf M}_2 \,{\bf M}_3} \right) \left( {l_1 \, l_2 \, l_3}\atop{{\bf 0} \, {\bf 0} \,{\bf 0}} \right) Y^{*}_{l_3\,{\bf M}_3}(e)$
3. $\left( {l_1 \, l_2 \, l_3}\atop{0 \, 0 \,0}\right) = 0$ unless $l_1+l_2+l_3$ is an even integer and $l_3= |l_1-l_2|,\cdots,l_1+l_2$.
4. $\left( {l \, l' \, 0}\atop{{\bf M} \, {\bf M}' \,0}\right) = \frac{(-1)^{l-m1}}{\sqrt{d_{l}}}\delta_{l \, l'}\delta_{{\bf M}\,{\bf M}'}$
5. $ \sum_{\{{\bf m}_i\}} (-1)^{({\bf m}_2)_1} \,\left({l_1\,l_3 \,L_2}\atop{{\bf m}_1\, {\bf m}_3\,{\bf M}_2}\right)\left({l_1\,l_2 \,L_3}\atop{\bar{{\bf m}}_1\, {\bf m}_2\,{\bf M}_3}\right)\left({l_2\,l_3 \,L_1}\atop{\bar{{\bf m}}_2 \,\bar{{\bf m}}_3\,{\bf M}_1}\right)= (-1)^{l_2+L_2-L_3} \,\left({ L_3\,L_1\,L_2}\atop{\bar{{\bf M}}_3 \,\bar{{\bf M}}_1 \,{\bf M}_2 } \right)\, \left\{\begin{matrix}
L_1 & L_2 & L_3 \\
l_1 & l_2 & l_3
\end{matrix}
\right\} $
.5cm .5cm We extract CG coefficients for $so(1,d+1)$ for three scalars from the three-point function for scalar primary operators amputating the out-going and in-going legs we had found earlier. The out-going and in-going legs takes the following forms respectively \_ e\^ |g(x)|; {l,[**m**]{}, s}&=& M\^[l]{}\_[[**m**]{}]{}([**x**]{}) (x\^2)\^[--l-s]{} && &=& \^[1/2]{} (x\^2)\^[--l-s]{} M\^[l\^]{}\_[[**m**]{}]{}([**x**]{}) && and \_ e\^ ; {l,[**m**]{},s}|g\^[-1]{}(y)|&=& M\^[l\^]{}\_[[**m**]{}]{}([**y**]{}) && &=&\^[1/2]{} (y\^2)\^[s]{} M\^[l\^]{}\_[[**m**]{}]{}([**y**]{}) && The 3-point function of the scalar primary operators with conformal dimensions $\Delta_1, \Delta_2$ and $\Delta_3$ can be expanded as && (4 \^[ d/2]{})\^3 \_[i=1]{}\^[3]{} \_[l\_i=0]{}\^ \_[s\_i=0]{}\^ \_[[**m**]{}\_i]{} && && (x\^2)\^[-\_1-l\_1-l\_3-s\_1-s\_3]{} (y\^2)\^[-\_[23]{}/2-l\_2+s\_1-s\_2]{} (z\^2)\^[s\_2+s\_3]{} && M\^[l\_1]{}\_[[**m**]{}\_1]{}([**x**]{}) M\^[l\_3]{}\_[[**m**]{}\_3]{}([**x**]{}) M\^[l\_1\^]{}\_[[**m**]{}\_1]{}([**y**]{}) M\^[l\_2]{}\_[[**m**]{}\_2]{}([**y**]{})M\^[l\_2\^]{}\_[[**m**]{}\_2]{}([**z**]{}) M\^[l\_3\^]{}\_[[**m**]{}\_3]{}([**z**]{}) && where \_[12]{} \_1 +\_2 -\_3, \_[23]{} \_2 +\_3 -\_1, \_[31]{} \_3 +\_1 -\_2 We use the following identities: \[eq:sp\_harmonics\_decomp\] M\^[l]{}\_[[**m**]{}]{}([**x**]{}) M\^[l’]{}\_[[**m**]{}’]{}([**x**]{}) = \_[L,M]{} ( [l l’ L]{} ) ( [l l’ L]{} ) (x\^[2]{})\^ M\^[L\^]{}\_[[**M**]{}]{}([**x**]{}) M\^[l\^]{}\_[[**m**]{}]{}(**[x]{}) =(-1)\^[m\_[1]{}]{} M\^[l]{}\_[ |[m]{} ]{} (**[ x]{}) where, $\bar{{\bf m}}=(m_{n-2},...,m_{2},-m_1)$ to rewrite the product of spherical harmonics in the summand as && M\^[l\_1]{}\_[[**m**]{}\_1]{}([**x**]{}) M\^[l\_3]{}\_[[**m**]{}\_3]{}([**x**]{}) M\^[l\_1\^]{}\_[[**m**]{}\_1]{}([**y**]{}) M\^[l\_2]{}\_[[**m**]{}\_2]{}([**y**]{})M\^[l\_2\^]{}\_[[**m**]{}\_2]{}([**z**]{})M\^[l\_3\^]{}\_[[**m**]{}\_3]{}([**z**]{}) &=& (-1)\^[m\_1+m\_2+m\_3]{}M\^[l\_1]{}\_[[**m**]{}\_1]{}([**x**]{}) M\^[l\_3]{}\_[[**m**]{}\_3]{}([**x**]{}) M\^[l\_1]{}\_[|[[**m**]{}\_1]{}]{}([**y**]{})M\^[l\_2]{}\_[[**m**]{}\_2]{}([**y**]{})M\^[l\_2]{}\_[|[[**m**]{}\_2]{}]{}([**z**]{}) M\^[l\_3]{}\_[|[[**m**]{}\_3]{}]{}([**z**]{}) &=& (-1)\^[m\_2]{} \_[i=1]{}\^[3]{} \_[L\_i,M\_i]{} (-1)\^[M\_2]{} ([l\_1l\_3 L\_2]{})([l\_1l\_2 L\_3]{})([l\_2l\_3 L\_1]{}) && ([l\_1l\_3 L\_2]{})([l\_1l\_2 L\_3]{})([l\_2l\_3 L\_1]{}) && (x\^2)\^ (y\^2)\^ (z\^2)\^ M\^[L\_2]{}\_[[**M**]{}\_2]{} ([**x**]{}) M\^[L\_3]{}\_[[**M**]{}\_3]{} ([**y**]{}) M\^[L\_1]{}\_[[**M**]{}\_1]{} ([**z**]{}) Inserting the above relation in the summand and performing the $\{m_i\}$ summations we get && \_[{[**m**]{}\_i}]{} (-1)\^[m\_2]{} ([l\_1l\_3 L\_2]{})([l\_1l\_2 L\_3]{})([l\_2l\_3 L\_1]{}) && && = (-1)\^[l\_2+L\_2-L\_3]{} ([ L\_3L\_1L\_2]{} ) {****
l\_2 & l\_1 & L\_3\
L\_2 & L\_1 & l\_3
} && && = (-1)\^[l\_2+L\_2-L\_3]{} ([ L\_3L\_1L\_2]{} ) {
L\_1 & L\_2 & L\_3\
l\_1 & l\_2 & l\_3
} Now the three-point function takes the form &&(4 \^[ d/2]{})\^3 \_[i=1]{}\^[3]{} \_[l\_i=0]{}\^ \_[s\_i=0]{}\^ \_[[**m**]{}\_i]{} && \_[i=1]{}\^[3]{} \_[L\_[i]{}, [**M**]{}\_[i]{}]{} (x\^2)\^[-\_1-l\_1-l\_3-s\_1-s\_3+]{} (y\^2)\^[-\_[23]{}/2-l\_2+s\_1-s\_2+]{} (z\^2)\^[s\_2+s\_3+]{} && (-1)\^[l\_2+L\_2-L\_3+M\_2]{} M\^[L\_2]{}\_[[**M**]{}\_2]{} ([**x**]{}) M\^[L\_3]{}\_[[**M**]{}\_3]{} ([**y**]{}) M\^[L\_1]{}\_[[**M**]{}\_1]{} ([**z**]{}) && ([l\_1l\_3 L\_2]{})([l\_1l\_2 L\_3]{})([l\_2l\_3 L\_1]{})([ L\_3L\_1L\_2]{} ) {
L\_1 & L\_2 & L\_3\
l\_1 & l\_2 & l\_3
} which can also be written as &&(4 \^[ d/2]{})\^3 \_[i=1]{}\^[3]{} \_[l\_i=0]{}\^ \_[s\_i=0]{}\^ \_[[**m**]{}\_i]{} && \_[i=1]{}\^[3]{} \_[L\_[i]{}, M\_[i]{}]{} (x\^2)\^[-\_1-l\_1-l\_3-s\_1-s\_3+]{} (y\^2)\^[-\_[23]{}/2-l\_2+s\_1-s\_2+]{} (z\^2)\^[s\_2+s\_3+]{} && (-1)\^[l\_2+L\_2-L\_3]{} M\^[L\_2]{}\_[|[[**M**]{}]{}\_2]{} ([**x**]{}) M\^[L\^\_3]{}\_[|[[**M**]{}]{}\_3]{} ([**y**]{}) M\^[L\^\_1]{}\_[|[[**M**]{}]{}\_1]{} ([**z**]{}) && ([l\_1l\_3 L\_2]{})([l\_1l\_2 L\_3]{})([l\_2l\_3 L\_1]{})([ L\_3L\_1L\_2]{} ) {
L\_1 & L\_2 & L\_3\
l\_1 & l\_2 & l\_3
} Note that the $so(d)$ $3\,j$- coefficients $\left({l \,l' \,L}\atop{0\,0\,0}\right)$ is non-vanishing only when $l+l'-L$ is even integer. This suggests to change the following variables as l\_1+l\_3 = 2K\_2 +L\_2, l\_1+l\_2 = 2K\_3 +L\_3, l\_2+l\_3 = 2K\_1 +L\_1 i.e. l\_1 &=& + K\_2+K\_3-K\_1 l\_2 &=& + K\_3+K\_1-K\_2 l\_3 &=& + K\_1+K\_2-K\_3 Then the powers of $x^2,\, y^2,\, z^2$ becomes (excluding the powers within the spherical harmonics) (x\^2)\^[-\_1-L\_2-K\_2-s\_1-s\_3]{} (y\^2)\^[--+s\_1-s\_2-K\_1+K\_2]{} (z\^2)\^[s\_2+s\_3+K\_1]{} respectively. Comparing with the legs we want to amputate from the three-point function we make the following change of variables in the summand K\_2+s\_1+s\_3 &=& S\_2\
K\_1+s\_2+s\_3 &=& S\_1\
--+s\_1-s\_2-K\_1+K\_2 &=& S\_3 The last one of the above relations impose the following selection rule (\_2 +L\_1+2S\_1) + (\_3+L\_3+2S\_3) = (\_1+L\_2+2S\_2) So $S_3$ is not an independent variable and $s_3$ is undetermined in terms of new variables. We call it $s_3 =S$. In terms of the new variables the three-point function becomes && && \_[i=1]{}\^[3]{} \_[L\_i=0]{}\^ \_[S\_i=0]{}\^ \_[M\_i]{} (\_2+L\_1+2S\_1+\_3+L\_3+2S\_3-\_1-L\_2-2S\_2) && && \_[K\_3=0]{}\^\_[K\_1=0]{}\^[S\_1]{}\_[K\_2=0]{}\^[S\_2]{}\_[S=0]{}\^[(S\_2-K\_2, S\_1-K\_1)]{} && && && && (-1)\^[+K\_3+K\_1-K\_2]{} && && ([ + K\_2+K\_3-K\_1, + K\_1+K\_2-K\_3 ,L\_2 ]{} ) && && ([ + K\_2+K\_3-K\_1, + K\_3+K\_1-K\_2,L\_3]{} ) && && ([ + K\_3+K\_1-K\_2, + K\_1+K\_2-K\_3,L\_1]{} ) && && {
L\_1 & L\_2 & L\_3\
+ K\_2+K\_3-K\_1 & + K\_3+K\_1-K\_2 & + K\_1+K\_2-K\_3
} && && ([ L\_3L\_1L\_2]{} ) (x\^2)\^[-\_1-L\_2-S\_2]{} (y\^2)\^[S\_1]{} (z\^2)\^[S\_3]{} M\^[L\_2\^[ \*]{}]{}\_[-[**M**]{}\_2]{} ([**x**]{}) M\^[L\_3]{}\_[-[**M**]{}\_3]{} ([**y**]{}) M\^[L\_1]{}\_[-[**M**]{}\_1]{} ([**z**]{}) where we have arranged the order as well as the limits of the summations appropriately. According to our prescription the three-point function can be recovered as && \_[i=1]{}\^[3]{} \_[L\_i=0]{}\^ \_[S\_i=0]{}\^ \_[[**M**]{}\_i]{} \_1 |g(x)|\_1 ; {L\_2,[**M**]{}\_2, S\_2} \_2; {L\_1,[**M**]{}\_1,S\_1}|g\^[-1]{}(y)|\_2 && \_3; {L\_3,[**M**]{}\_3,S\_3}|g\^[-1]{}(z)|\_3 C\^[(\_2,S\_1), (\_3,S\_3); (\_1,S\_2)]{}\_[(L\_1,[**M**]{}\_1), (L\_3,[**M**]{}\_3); (L\_2,[**M**]{}\_2)]{} where $C^{(\Delta_2,S_1),\, (\Delta_3,S_3);\, (\Delta_1,S_2)}_{(L_1,{\bf M}_1),\, (L_3,{\bf M}_3);\, (L_2,{\bf M}_2)}$ is $so(1,d+1)$ CG coefficient. Comparing above with the three-point function we write && && C\^[(\_2,S\_1), (\_3,S\_3); (\_1,S\_2)]{}\_[(L\_1,[**M**]{}\_1), (L\_3,[**M**]{}\_3); (L\_2,[**M**]{}\_2)]{} && && = && \^[1/2]{} \^[1/2]{} && && \^[1/2]{} && \_[K\_3=0]{}\^\_[K\_1=0]{}\^[S\_1]{}\_[K\_2=0]{}\^[S\_2]{}\_[S=0]{}\^[min(S\_2-K\_2, S\_1-K\_1)]{} (-1)\^[+K\_3+K\_1-K\_2]{} && && && && && && && ([ + K\_2+K\_3-K\_1, + K\_1+K\_2-K\_3 ,L\_2 ]{} ) && && ([ + K\_2+K\_3-K\_1, + K\_3+K\_1-K\_2,L\_3]{} ) && && ([ + K\_3+K\_1-K\_2, + K\_1+K\_2-K\_3,L\_1]{} ) && && {
L\_1 & L\_2 & L\_3\
+ K\_2+K\_3-K\_1 & + K\_3+K\_1-K\_2 & + K\_1+K\_2-K\_3
} && && ([ L\_3L\_1L\_2]{} )
Manipulation of the $d$-dimensional result {#appB}
==========================================
The four-point block of a scalar correlation function in general dimensions in our method takes the following form: (x\^2)\^[(- \_3-\_4)]{} && \_[l,s]{} (+l+s) (+l+s) (+s- )(+s-) && (x\^2)\^s [C]{}\^\_l ([**x**]{} ) where $\mu = \frac{d-2}{2}$. One of the questions we have to address is how our computations match with those known in the literature. There is a famous expression for the conformal blocks in any dimension in terms the cross ratios $u, v$ as found by Dolan and Osborn. We now prove the following identity towards establishing the equivalence between our answers and theirs. \_[l,s=0]{}\^ (z |z)\^[s+ ]{} C\_l\^() is equal to \_[r,q=0]{}\^ (z |z)\^r (z + |z - z |z)\^q To establish this we first note the following identities/definitions: (z |z)\^ C\^\_l() := \_[k=0]{}\^[\[l/2\]]{} (-1)\^k (z+ |z)\^[l-2k]{} (z |z)\^[k]{} (z + |z - z |z)\^q = \_[p=0]{}\^q (-1)\^p ( [q p]{} ) (z+ |z)\^[q-p]{} (z |z)\^[p]{} Using the double sum identity: \_[q=0]{}\^\_[p=0]{}\^q a\_[p, q-p]{} = \_[m=0]{}\^\_[n=0]{}\^a\_[n,m]{} = \_[l=0]{}\^\_[k=0]{}\^[\[l/2\]]{} a\_[k, l-2k]{} the first expression can be written as \_[s=0]{}\^ \_[m,n=0]{}\^ (-1)\^n (z |z)\^[n+s]{} (z+ |z)\^[m]{}\
The second of the expressions can be manipulated to: \_[r=0]{}\^ \_[m,p=0]{}\^ (-1)\^p (z |z)\^[r+p]{} (z+|z)\^[m]{} In the next step we extract the coefficients of $(z \bar z)^q (z+\bar z)^m$ in both these expressions. For this in the first expression we change $n\rightarrow p, ~ s\rightarrow q-p$ and in the second we change $p\rightarrow p, ~ r \rightarrow q-p$. Then in both the expressions the indices $q$ and $m$ run freely over all non-negative integers and the index $p$ runs over $0, 1, \cdots, q$. The corresponding coefficient for the first expression is: \[conjlhs\] \_[p=0]{}\^q (-1)\^p and for the second expression is: \[conjrhs\] \_[p=0]{}\^q (-1)\^p Now the final step is to compare these two expressions (\[conjlhs\]) and (\[conjrhs\]) for arbitrary integers $\{d \ge 1, q \ge 0, m \ge 0\}$. We conjecture that these expressions are identical. We have verified this claim for various special cases exactly, and for large subsets of the integer parameters $\{d \ge 1, q \ge 0, m \ge 0\}$ using Mathematica.
Details of CPW computation in $d=4$ {#appC}
===================================
To establish the result for four-point scalar CPW in $d=4$ we start by expanding the answer in power series. z \_2F\_1(,,, z) \_2F\_1(-1,-1,-2, |z) &=& \_[m=0]{}\^\_[n=0]{}\^ && We now divide the [*rhs*]{} into three terms with $m+1>n$, $m+1<n$ and $m+1=n$. The piece coming from terms with $m+1 =n$ are real and therefore cancel with the corresponding terms from the complex conjugate combination. The remaining parts are obtained by considering the restricted sums \_[n=0]{}\^\_[m=n]{}\^+ \_[m=0]{}\^\_[n=m+2]{}\^ Let us consider the conjugate term next: |z \_2F\_1(,,, |z) \_2F\_1(-1,-1,-2, z) &=& \_[m=0]{}\^\_[n=0]{}\^ && again we split this into three types of terms as above and drop the term that is real. Then we can split the rest into two types of terms by writing the sum as before in two parts: \_[n=0]{}\^\_[m=n]{}\^+ \_[m=0]{}\^\_[n=m+2]{}\^ Noticing that the first sum in the first term and the second sum in the second have more $z$’s than $\bar z$’s we would like to combine them. In these two we introduce two new variables $m=n+p$ and $n=m+2+q$ to replace $m$ and $n$ respectively. Combing these we have: && \_[n=0]{}\^\_[p=0]{}\^ &&-\_[m=0]{}\^\_[q=0]{}\^ In the second term we can replace $m \rightarrow m-1$ and still sum over the new $m$ from $0$ to $\infty$ as there will be term $(m-1)!$ in the denominator which kills the $m=0$ term. Then && \_[n=0]{}\^\_[p=0]{}\^ &&-\_[m=0]{}\^\_[q=0]{}\^ Now we change dummy variables $n \rightarrow s$, $p\rightarrow l$ in the first term and $m \rightarrow s$ and $q \rightarrow l$ in the second term and combine terms to write this as: && \_[l=0]{}\^\_[s=0]{}\^ && Using - = This can be seen to be: && \_[l=0]{}\^\_[s=0]{}\^ which is precisely the first term in our OWN computation of the block. The remaining two terms are simply conjugates of what we have dealt with so far and therefore are going to reproduce the second term in our OWN computation.
[10]{}
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[^1]: This scalar cap in the $d=2$ case can be seen to be equivalent to that with $h=\bar h$ cap used in [@Bhatta:2016hpz] (see also [@Miyaji:2015fia] and more recently [@Castro:2018srf] for a different perspective).
[^2]: Expressions of CGC for the scalar module obtained using this procedure can be found in appendix \[appA\].
[^3]: It turns out that this choice is responsible for giving the CPW[*s*]{} as a sum over contributions of given spin $l$, namely the Gegenbauer polynomial basis.
[^4]: Note that the $so(d)$ symmetry dictates that the normalisation of these states do not depend on [**m**]{}.
[^5]: Even though this formal expansion looks odd as it apparently depends not only on $P$ whose square is ${\bf P} \cdot {\bf P}$, but also appears in the denominator of the argument of the Gegenbauer polynomial – we will shortly see that this is not a problem once interpreted correctly.
[^6]: This is a remarkable generalisation of how the Gegenbauer Polynomials $C_k^\mu(x)$ are defined through its generating function when $\Delta = \mu$.
[^7]: While this work was in progress [@Terashima:2017gmc] appeared where the same result was obtained in a different context.
[^8]: Amusingly the same equation (\[secondpde\]) arises for a complex scalar in $AdS_2$ minimally coupled to a background electric field preserving its isometries, and with strength $\theta$.
[^9]: The generalisation of geodesic Witten diagrams to include some of the other representations was achieved in [@Nishida:2016vds; @Dyer:2017zef; @Tamaoka:2017jce; @Nishida:2018opl].
|
---
abstract: |
@Hilberg90 supposed that finite-order excess entropy of a random human text is proportional to the square root of the text length. Assuming that Hilberg’s hypothesis is true, we derive Guiraud’s law, which states that the number of word types in a text is greater than proportional to the square root of the text length. Our derivation is based on some mathematical conjecture in coding theory and on several experiments suggesting that words can be defined approximately as the nonterminals of the shortest context-free grammar for the text. Such operational definition of words can be applied even to texts deprived of spaces, which do not allow for Mandelbrot’s “intermittent silence” explanation of Zipf’s and Guiraud’s laws. In contrast to Mandelbrot’s, our model assumes some probabilistic long-memory effects in human narration and might be capable of explaining Menzerath’s law.\
**Keywords:** excess entropy, grammar-based compression, Guiraud’s law, Zipf’s law
author:
- |
[Ł]{}ukasz Dbowski\
Institute of Computer Science\
Polish Academy of Sciences\
[[email protected]]([email protected])
bibliography:
- 'books.bib'
- 'ql.bib'
- 'nlp.bib'
- 'ai.bib'
- 'neuro.bib'
- 'mine.bib'
- 'tcs.bib'
title: |
On Hilberg’s Law and Its Links\
with Guiraud’s Law
---
Introduction {#SecIntroduction}
============
Over a decade ago, @Hilberg90 reinterpreted @Shannon50’s ([-@Shannon50]) well-known experimental data and formulated a novel hypothesis concerning the entropy of human language. The hypothesis states that block entropy $H(n)$ of a text drawn from natural language production, except for disputable constant and linear terms, is proportional to the square root of the text length $n$ measured in phonemes (or letters), $$\begin{aligned}
\label{HilbergH}
H(n)\approx h_0 + h_\mu n^{\mu} + h n,\end{aligned}$$ where $\mu\approx 1/2$. For brevity, we call relation (\[HilbergH\]) Hilberg’s law. Hilberg’s publication appeared in a technical journal of telecommunications. It was popularized among natural scientists by Ebeling [@EbelingNicolis91; @EbelingPoschel94] and stimulated some discussions [@BialekNemenmanTishby00; @CrutchfieldFeldman01; @Shalizi01; @Debowski01b; @Debowski02].
In this article, we shall discuss some interaction between Hilberg’s law and the better known Guiraud’s and Zipf’s laws. Empirical Guiraud’s law [@Guiraud54] states that the number of orthographic word types $V$ in a text behaves like $$\begin{aligned}
\label{Guiraud}
V\propto N^\rho,\end{aligned}$$ where $\rho<1$ is constant and $N$ is the length of the text measured in orthographic word tokens. On the other hand, Zipf’s-Mandelbrot’s law [@Zipf35; @Zipf49; @Mandelbrot54] states that any text obeys relation $$\begin{aligned}
\label{Zipf}
c(w)\propto \frac{1}{r(w)^{B}},\end{aligned}$$ where $B>1$ is constant, frequency $c(w)$ is the count of word $w$ in the text, and rank $r(w)$ is the position of word $w$ in the list of words sorted in descending order by $c(w)$.
We do not know to what extent Hilberg’s law is valid. Formula (\[HilbergH\]) presupposes some stationary probabilistic model of the entire natural language production, which is a highly hypothetical entity itself. Nevertheless, we would like to argue that some form of Guiraud’s law can be deduced from equation (\[HilbergH\]). Strictly speaking, assuming that Hilberg’s law is true for all $n$, we shall only infer some lower bound for the growth of the vocabulary size. Despite that restriction, we think that our explanation of Guiraud’s law can be more linguistically plausible than the famous joint derivation of Guiraud’s and Zipf’s laws provided by @Mandelbrot53. The latter derivation is known also as “intermittent silence” explanation [@Miller57; @Li98].
Hilberg’s law concerns the probabilistic distribution of arbitrary phoneme or letter strings, i.e. the law constrains the distribution of all human texts. On the other hand, both Guiraud’s and Zipf’s laws concern the distribution of individual words in texts. Saying that Guiraud’s law can be deduced from Hilberg’s law, we presuppose some procedure which transforms the distribution of phoneme strings (i.e.texts) into the corresponding distribution of words. In some naive approach, we could assume that the text is a string of phonemes or spaces and the words are the space-to-space strings of phonemes. In fact, “intermittent silence” explanation assumes that the text is a string of probabilistically independent random tokens taking the values of spaces and phonemes. Given this assumption and the space-to-space definition of word, Mandelbrot deduced Zipf’s law, and hence Guiraud’s law can be deduced as well [@Kornai02].
Unfortunately, “intermittent silence” explanation cannot be applied to natural language. We know that the occurrences of phonemes in the language production exhibit some strong probabilistic dependence and there are no definite spaces between the words in human speech [@Jelinek97]. If we want to derive Zipf’s law from the distribution of mere phoneme strings, we must use some definition of word tokens which could be applied to the text deprived of spaces and which would match empirically the definition of word tokens given by spelling conventions or by semantic considerations.
Some well-defined tokenization of the space-deprived text into word-like strings can be given by grammar-based text compression [@KiefferYang00]. In grammar-based compression, the text is represented as a special context-free grammar, called an admissible grammar. That class of context-free grammars should not be confused with phrase structure grammars: The nonterminals of admissible grammars correspond to fixed strings of phonemes rather than to part-of-speech classes. Each admissible grammar gives some tokenization of the text into hierarchically structured word-like strings being the nonterminal tokens. It was empirically confirmed that for the grammars which approximate the shortest admissible grammar for a human text, the nonterminals usually correspond to the orthographic words [@DeMarcken96; @NevillManning96].
We will show that the expected number of nonterminal types for the shortest admissible grammar cannot be less than proportional to so called finite-order excess entropy of the random text. It is some mathematical result based on a line of theorems and one unproved conjecture. On the other hand, if Hilberg’s hypothesis is true then the finite-order excess entropy of the text is roughly proportional to the square root of the text length. The close empirical correspondence between the nonterminals and the orthographic words allows us to claim that Hilberg’s law implies some lower bound for the vocabulary growth, i.e. some form of Guiraud’s law.
The rest of this article fills in the details of the deductions and empirical observations mentioned in the previous paragraphs:
- In section \[SecHilberg\], we introduce the definitions of stationary distribution, block entropy, excess entropy, and infinitary distributions. We sketch the history of Hilberg’s law and the general research of block entropy for natural language production.
- In section \[SecGrammars\], we introduce the concepts of admissible and irreducible grammars. We also discuss some empirical evidence that the shortest admissible grammar matches largely the linguistic tokenization for the human text.
- In section \[SecCodes\], we relate block entropy to the expected length of irreducible grammar-based codes. Assuming some mathematical conjecture, we show that the expected total length of the non-initial productions of the shortest grammar cannot be less than finite-order excess entropy.
- In section \[SecGuiraud\], we discuss Guiraud’s law in detail and we argue that Hilberg’s law explains it better than the assumption of “intermittent silence”. Some arguments for Hilberg’s law explanation are: (i) non-randomness of texts, (ii) empirical detectability of word boundaries and internal structures, (iii) possibility of explaining Menzerath’s law, and (iv) significant variation of word frequencies across different texts.
Excess entropy and Hilberg’s law {#SecHilberg}
================================
Let us imagine some infinite sequence of characters, e.g.$$\begin{aligned}
\label{SomeRoses}
{\tt the\_rose\_is
\_a\_hose\_is\_a\_rose\_is\_a\_hose\_is\_a\_rose\_is\_a\_hose...}\,\, ,\end{aligned}$$ where subsequence ${\tt \_a\_rose\_is\_a\_hose\_is}$ is repeated infinitely to fix our imagination. For such an (infinite) sequence we can compute the relative frequency of any (finite) string which appears in that sequence.
For example, let us define probability $\mathbf{P}({\tt rose})$ as the relative frequency of string ${\tt rose}$ in the infinite sequence (\[SomeRoses\]). We shall do it in two steps. Let $a_i$ stand for the $i$th character of (\[SomeRoses\]), i.e. $a_{1}={\tt t}$, $a_{2}={\tt h}$, $a_{3}={\tt e}$, $a_{4}={\tt \_}$, $a_{5}={\tt r}$ etc. We will write the finite substrings as $a_{m:n}:=(a_{m},a_{m+1},...,a_{n})$. The relative frequency $\mathbf{P}({\tt rose};n)$ of string ${\tt rose}$ in string $a_{1:n}$ is the number of all positions $a_i$, $1\le i\le n$, where string ${\tt
rose}$ starts divided by $n$. For any equality relation $\phi$ let us define ${\left[\mkern-3mu\left[\,\phi\,\right]\mkern-3mu\right]}=1$ if $\phi$ is true and ${\left[\mkern-3mu\left[\,\phi\,\right]\mkern-3mu\right]}=0$ if $\phi$ is false. Thus, $\mathbf{P}({\tt rose};k)$ can be expressed as $$\begin{aligned}
\mathbf{P}({\tt rose};k):=\frac{1}{k} \sum_{i=1}^k
{\left[\mkern-3mu\left[\,a_{i:i+3}={\tt rose}\,\right]\mkern-3mu\right]},\end{aligned}$$ where $:=$ means definition. We have $\mathbf{P}({\tt
rose};1)=0$, $\mathbf{P}({\tt rose};5)=1/5$, $\mathbf{P}({\tt
rose};10)=1/10$, $\mathbf{P}({\tt rose};30)=2/30$ and so on.
Let us define probability $\mathbf{P}({\tt rose})$ as the limit of relative frequencies of string ${\tt rose}$ in the initial substrings of (\[SomeRoses\]). So we will write $$\begin{aligned}
\label{PRose}
\mathbf{P}({\tt rose}):=\lim_{n\rightarrow\infty} \mathbf{P}({\tt rose};n). \end{aligned}$$ Every $20$th character in sequence (\[SomeRoses\]) is a position where string ${\tt rose}$ starts, so $\mathbf{P}({\tt rose})=1/20$. Analogically, we can define probability $\mathbf{P}(v)$ for any string $v$, $$\begin{aligned}
\label{PMises}
\mathbf{P}(v):=\lim_{k\rightarrow\infty} \frac{1}{k}
\sum_{i=1}^k {\left[\mkern-3mu\left[\,a_{i:i+{\operatorname{len}v}-1}=v\,\right]\mkern-3mu\right]},\end{aligned}$$ where ${\operatorname{len}v}$ is the number of characters in $v$. Hence, for (\[SomeRoses\]) we obtain not only $\mathbf{P}({\tt
t})=0$ (there are no ${\tt t}$’s), $\mathbf{P}({\tt s})=1/5$ (two in ten characters are ${\tt s}$), and $\mathbf{P}({\tt e})=1/10$ but also $\mathbf{P}({\tt e\_is\_a})=1/10$, $\mathbf{P}({\tt
a\_rose\_is\_a\_hose})=1/20$, and $\mathbf{P}({\tt
a\_rose\_is\_a\_rose})=0$.
Now let us take some general sequence $(a_{1},a_{2},a_{3},...)$. Let $\mathbb{V}$ be the finite set of characters that appear in that sequence. Let $\mathbb{V}^+$ be the set of all finite strings formed by concatenating the characters in $\mathbb{V}$. For any sequence $(a_{1},a_{2},a_{3},...)$ such that limit (\[PMises\]) exists for each string $v\in\mathbb{V}^+$, probability function $\mathbf{P}$ satisfies relations $$\begin{aligned}
\label{PConstistency}
0\le \mathbf{P}(v)\le 1,\quad
\sum_{a\in\mathbb{V}} \mathbf{P}(a)=1,\quad
\sum_{a\in\mathbb{V}} \mathbf{P}(av)= \mathbf{P}(v)=
\sum_{a\in\mathbb{V}} \mathbf{P}(va).\end{aligned}$$ We will call any function $\mathbf{P}$ satisfying conditions (\[PConstistency\]) for all $v\in\mathbb{V}^+$ a stationary distribution.[^1] It is an open question whether for any stationary distribution $\mathbf{P}$ exists such $(a_{1},a_{2},a_{3},...)$ that we have (\[PMises\]) for all $v\in\mathbb{V}^+$.
Let $\mathbb{V}^n$ be the set of all $n$-character long strings. We define block entropy $H(n)$ of any stationary distribution $\mathbf{P}$ as the entropy of strings of length $n$, $$\begin{aligned}
\label{DefEntropy}
H(n):=
-\sum_{v\in\mathbb{V}^n} \mathbf{P}(v) \log_2 \mathbf{P}(v)
.\end{aligned}$$ We also put $H(0):=0$ for algebraic convenience.
For any stationary distribution $\mathbf{P}$ block entropy $H(n)$ is a nonnegative, growing, and concave function of $n$ [@CrutchfieldFeldman01], i.e., $$\begin{aligned}
\label{Hproperties}
H(n)\ge 0,\quad
H'(n)\ge 0,\quad
H''(n)\le 0,\end{aligned}$$ where $$\begin{aligned}
H'(n):=H(n)-H(n-1),\quad
H''(n):=H(n)-2H(n-1)+H(n-2).\end{aligned}$$ Because of inequalities (\[Hproperties\]), we can define entropy rate as $$\begin{aligned}
\label{DefRate}
h:=\lim_{n\rightarrow\infty}H(n)/n=\lim_{n\rightarrow\infty}H'(n)\ge 0. \end{aligned}$$
If entropy rate satisfies $h>0$ then $H(n)$ grows almost linearly against the string length $n$ for very long strings, $H(n)\approx h
n$. We can ask how fast $H(n)$ approaches $hn$. The departure of $H(n)$ from the linear growth is known as excess entropy.
Finite-order excess entropies $E(n)$ are some functions of $H(n)$ and $H(2n)$, $$\begin{aligned}
\label{DefEn}
E(n)&:=2H(n)-H(2n)=-\sum_{k=2}^n (k-1)H''(k)-
\sum_{k=n+1}^{2n} (2n-k+1)H''(k). \end{aligned}$$ So defined functions are nonnegative and growing, i.e., $E(n)\ge
E(n-1)\ge 0$. @CrutchfieldFeldman01 proved that (total) excess entropy $E$ can be defined equivalently as $$\begin{aligned}
E:=\lim_{n\rightarrow\infty}E(n)=
\lim_{n\rightarrow\infty}[H(n)-hn].\end{aligned}$$ We also have inequality $$\begin{aligned}
\label{Control}
E=-\sum_{k=2}^\infty (k-1)H''(k)\ge -\sum_{k=2}^\infty H''(k)=H(1)-h.\end{aligned}$$
Let $vu$ be the concatenation of strings $v$ and $u$. We will say that stationary distribution $\mathbf{P}$ is an IID distribution if $$\begin{aligned}
\label{PRandom}
\mathbf{P}(vu)=\mathbf{P}(v)\mathbf{P}(u)\end{aligned}$$ for all strings $v,u\in\mathbb{V}^+$. (IID stands for independent identically distributed random variables.) Distributions $\mathbf{P}$ can be IID even for some quite ordered underlying sequences $(a_{1},a_{2},a_{3},...)$. For instance, $\mathbf{P}$ given through (\[PMises\]) is IID for the sequence of digits of consecutive natural numbers $(a_{1},a_{2},a_{3},...)=
(1,2,3,4,5,6,7,8,9,1,0,1,1,...)$, which is called Champernowne sequence [@LiVitanyi93]. Anyway, we do not expect that we could obtain IID distribution $\mathbf{P}$ if we substituted some collection of human texts for sequence $(a_{1},a_{2},a_{3},...)$.
For any IID distribution $\mathbf{P}$ we have $H(n)=nH(1)$ so $h=H(1)$ and $E=0$. Conversely, if $H(1)-h>0$ or $E>0$, then distribution $\mathbf{P}$ cannot be IID. For the extreme departures from the IID case, we have $h=0$ or $E=\infty$. Stationary distributions exhibiting $h=0$ are called deterministic while the distributions obeying $E=\infty$ are called infinitary [@CrutchfieldFeldman01]. In appendix \[SecMath\], we present some properties of infinitary distributions which could be important for their possible applications in quantitative and computational linguistics but which are not so relevant for the main reasoning of this article.
Let us assume that we could obtain some definite stationary distribution $\mathbf{P}$ through formula (\[PMises\]) if we substituted the infinite concatenation of some human texts for $(a_{1},a_{2},a_{3},...)$. We will call such an infinite sequence $(a_{1},a_{2},a_{3},...)$ natural language production. Research in the hypothetical stationary distribution of natural language production has attracted many scientists. The first one to work in this area was @Shannon50. He tried to estimate block entropy using the guessing method and assuming some correspondence between particular instances of English texts and the hypothetical random English language production. Shannon published some estimates of $H(n)$ for strings of $n$ consecutive letters, where $n\le 100$.
Shannon was not convinced of any particular asymptotics of block entropy $H(n)$ for the natural language production [@Hilberg90] but the later researchers in quantitative linguistics tried to model $H(n)$ by some simple formulae. For example, @HoffmanPiotrovskij79 proposed a model of exponential convergence, $$\begin{aligned}
\label{HMMH}
H(n)/n=(h_0-h)\exp{\left[\, -n/n_0 \,\right]}+h.\end{aligned}$$ @Petrova73 fitted model (\[HMMH\]) to French language data and obtained $1/n_0\in (0.24, 0.33)$.
On the other hand, @Hilberg90 replotted the original plot of $H(n)$ vs. $n$ by @Shannon50 into a log-log scale and observed that a simple square-root dependence fits all the data points, $$\begin{aligned}
\label{HilbergHori}
H(n)\propto n^{\mu},
\quad
\mu\approx 1/2,
\quad
n\le 100.\end{aligned}$$ For our convenience, we will call Hilberg’s law an algebraic relation which is slightly more general than Hilberg’s original hypothesis (\[HilbergHori\]). We will say that Hilberg’s law holds for any stationary distribution $\mathbf{P}$ if only relation (\[HilbergH\]) holds with $\mu\approx 1/2$ and $h_\mu>0$ for any $n$. For such definition, Hilberg’s law is independent of any hypothesis on the particular value of entropy rate $h$ and the constant term $h_0$.
While Shannon estimated block entropy using the guessing method, Ebeling and his collaborators tried to estimate the asymptotics of $H(n)$ by counting $n$-tuples in the samples of various symbolic sequences. Using improved entropy estimators, the researchers fitted the general formula (\[HilbergH\]) with $\mu\approx 1/2$ for natural language texts and $\mu\approx 1/4$ for classical music transcripts. For English and German texts $H(n)$ could be safely estimated for $n\le 30$ characters with $h_0\approx 0$, $h_{\mu}\approx 3.1$ bits and $h\approx 0.4$ bits [@EbelingNicolis92; @EbelingPoschel94]. In contrast, Shannon’s guessing data, reinterpreted by @Hilberg90, suggest that equation (\[HilbergH\]) can be extrapolated at least for $n\le 100$.
It is important to note that the estimation of block entropy $H(n)$ based on the naive estimation of probabilities $\mathbf{P}(v)$ for all strings $v$ of length $n$ is expensive in the input data. In order to estimate the value of $H(n)$, we need a sample of length about $2^{H(n)}$ [@HerzelSchmittEbeling94]. If we try to make shortcuts, we assume some particular properties of the unknown stationary distribution $\mathbf{P}$. Even @Shannon50’s ([-@Shannon50]) guessing method need not give the reliable estimates of $H(n)$ for the language production if the probabilistic language model internalized by the experimental subjects differs from the model estimated from the corpus [@BodHayJannedy03; @Hug97].
Let us note that for the block entropy of formula (\[HilbergH\]), finite-order excess entropies are $$\begin{aligned}
\label{HilbergE}
E(n)\approx h_0 + (2-2^\mu)h_{\mu}n^{\mu}.\end{aligned}$$ If relations (\[HilbergE\]) hold with $0<\mu<1$ for any $n$ then the total excess entropy is $E=\infty$. Hence, every stationary distribution exhibiting Hilberg’s law is infinitary.
At the moment, we have no clear idea how one could verify if Hilberg’s law holds for the hypothetical stationary distribution of the language production. Nevertheless, we can provide some mixed inductive and deductive arguments that Hilberg’s law implies some phenomena that can be observed in human language.
Words and the shortest grammars {#SecGrammars}
===============================
In the following sections, we shall argue that Hilberg’s law can explain some quantitative laws concerning the distribution of word types in the language production. Nevertheless, before we can speak of any distribution of words in a finite string of phonemes or letters, we need to delimit the word tokens themselves. If the words are some objective entities of the language, there should be some method for identifying the boundaries between the words in a sufficiently long string of phoneme or letter tokens even if we delete the spaces between words and ignore the lexicon.
Let us take some text deprived of spaces, e.g. $$\begin{aligned}
\label{Ex}
v={\tt shouldawoodchuckchuckifawoodchuckcouldchuckwood}. \end{aligned}$$ We can express our knowledge of word tokens describing string $v$ by means of a two-level context-free grammar $$\begin{aligned}
\label{IdealEx}
\begin{array}{rl}
G&={\left\{
\begin{array}{ccc}
\multicolumn{2}{c}{
b_0\mapsto b_5\,b_1\,b_7\, b_6\, b_2\, b_1\,b_7\, b_4\, b_6\, b_3,} &
b_1\mapsto{\tt a}, \\
b_2\mapsto{\tt if}, &
b_3\mapsto {\tt wood}, &
b_4\mapsto {\tt could}, \\
b_5\mapsto {\tt should}, &
b_6\mapsto {\tt chuck}, &
b_7\mapsto {\tt woodchuck}
\end{array}
\right\}}
.
\end{array}\end{aligned}$$ Symbols $b_i$ are called nonterminals. For each $b_i$ there is some production rule $(b_i\mapsto g_i)\in G$. On the other hand, the typewriter-typed symbols, which have no productions rules in the grammar, will be called terminals. Nonterminal $b_0$ is called the initial symbol of the grammar. If we recursively substitute productions $g_i$ for all nonterminals $b_i$ where $(b_i\mapsto
g_i)\in G$, then $b_0$ expands into string $v$ with the requested tokenization into the words. Namely, $$\begin{aligned}
v={\tt {\, \overline{should} \,}{\, \overline{a} \,}{\, \overline{woodchuck} \,}{\, \overline{chuck} \,}
{\, \overline{if} \,}{\, \overline{a} \,}{\, \overline{woodchuck} \,}{\, \overline{could} \,}{\, \overline{chuck} \,}{\, \overline{wood} \,}
},\end{aligned}$$ where notation ${\, \overline{g} \,}$ means that $G$ contains rule $b_i\mapsto g$ for some $i\neq 0$ [@DeMarcken96].
Of course, if we were not given any previous knowledge of English lexicon, we could propose other tokenizations for text (\[Ex\]). For instance, $$\begin{aligned}
\label{RecCodeEx}
\begin{array}{rl}
G&={\left\{
\begin{array}{ccc}
\multicolumn{2}{c}{b_0\mapsto {\tt sh} b_1 b_4 b_2
{\tt if} b_4 {\tt c} b_1 b_2 b_3,} &
b_1\mapsto {\tt ould}, \\
b_2\mapsto {\tt chuck}, &
b_3\mapsto {\tt wood}, &
b_4\mapsto {\tt a} b_3 b_2
\end{array}
\right\}}
\end{array}\end{aligned}$$ yields $$\begin{aligned}
v={\tt
sh{\, \overline{ould} \,}{\, \overline{a{\, \overline{wood} \,}{\, \overline{chuck} \,}} \,}{\, \overline{chuck} \,}
if{\, \overline{a{\, \overline{wood} \,}{\, \overline{chuck} \,}} \,}c{\, \overline{ould} \,}{\, \overline{chuck} \,}{\, \overline{wood} \,}
}.\end{aligned}$$ In the extreme, we could define $b_0$ as the entire string $v$ or each $b_i$, $i\neq 0$, as a single letter. Since we ignore English lexicon, we need some purely formal criterion for deciding what grammars $G$ are good for arbitrary strings $v$ and what are not.
Let us state some formal definitions. Context-free grammar $G$ will be called a grammar (more precisely, admissible grammar) for string $v$ [cf. @KiefferYang00] if:
1. For each nonterminal $b_i$ there is exactly one production $g_i$ such that $(b_i\mapsto g_i)\in G$.
2. Nonterminal $b_0$ expands into $v$ if we recursively substitute productions $g_i$ for all $b_i$.
The set of all admissible grammars for $v$ will be denoted by $F(v)$. Each grammar $G\in F(v)$ is allowed to produce only one derivation, which is the finite text $v$ itself. In contrast, context-free grammars producing a single infinite derivation are known as L-systems.
Some a priori criterion for deciding which admissible grammars approximate the correct tokenizations of texts makes use of the principle of minimum description length [@Rissanen78; @LehmanShelat02]. Define the length ${\operatorname{len}g_i}$ of production $g_i$ as the total number of its terminal and nonterminal symbols, e.g. ${\operatorname{len}{\tt sh} b_1 b_4 b_2 {\tt if} b_4
{\tt c} b_1 b_2 b_3}=12$ and ${\operatorname{len}{\tt a} b_3 b_2}=3$. According to the principle of minimum description length, the best grammar for string $v$ is grammar $G^{\operatorname{MDL}}(v)$ having the minimal length, $$\begin{aligned}
\label{GMDLDefinition}
G^{\operatorname{MDL}}(v)&:=\mathop{\arg \min}_{G\in F(v)} {\operatorname{len}G},\end{aligned}$$ where the length of a grammar is the total length of all its productions, $$\begin{aligned}
\label{Glength}
{\operatorname{len}G}:=\sum_{(b_i\mapsto g_i)\in G} {\operatorname{len}g_i}.\end{aligned}$$ Strictly speaking, there can be more than one grammar having the minimal length, so object $G^{\operatorname{MDL}}(v)$ is slightly indeterminate.
Grammar $G^{\operatorname{MDL}}(v)$ usually cannot be computed in a reasonable amount of time but there is a multitude of heuristic algorithms which compute grammars whose lengths approximate ${\operatorname{len}G^{\operatorname{MDL}}(v)}$ [@Lehman02; @LehmanShelat02]. Various algorithms for computing the approximations of $G^{\operatorname{MDL}}(v)$ usually perform some kind of local search on set $F(v)$ and output so called irreducible grammars. Grammar $G$ is called irreducible [@KiefferYang00 section 3.2] if:
1. Each nonterminal expands recursively into a different string of terminals.
2. Each nonterminal except for $b_0$ appears at least twice in productions $g_i$.
3. There is no string $y$ of ${\operatorname{len}y}\ge 2$ which appears more than once in productions $g_i$.
It can be shown that there is an irreducible grammar for $v$ whose length equals $\min_{G\in F(v)} {\operatorname{len}G}$. Hence, we can assume that $G^{\operatorname{MDL}}(v)$ is irreducible.
Various algorithms for computing the irreducible approximations of $G^{\operatorname{MDL}}(v)$ have been tested empirically on natural language data. @Wolff80, @NevillManning96, and @DeMarcken96 reported that those algorithms return quite sound representations of English texts. The nonterminals of some irreducible approximations of $G^{\operatorname{MDL}}(v)$ can be interpreted as syllables, morphemes, words, and fixed phrases. Some of the heuristic algorithms identify the correct boundaries of about $90\%$ of orthographic words in the Brown corpus, in a text deprived of spaces, capitalization, and punctuation [@DeMarcken96]. Here is an example of the computed tokenization given by @DeMarcken96:
$
\begin{array}[c]{c}
{\tt
{\, \overline{ {\, \overline{f{\, \overline{or} \,}} \,} {\, \overline{ {\, \overline{t{\, \overline{he} \,}} \,} {\, \overline{
{\, \overline{{\, \overline{p{\, \overline{ur} \,}} \,}{\, \overline{{\, \overline{{\, \overline{po} \,}s} \,}e} \,}} \,} {\, \overline{of} \,} } \,}} \,}} \,}
{\, \overline{{\, \overline{{\, \overline{ma{\, \overline{in} \,}} \,}{\, \overline{ta{\, \overline{in} \,}} \,}} \,}{\, \overline{{\, \overline{in} \,}g} \,}} \,}
{\, \overline{{\, \overline{{\, \overline{in} \,}{\, \overline{t{\, \overline{er} \,}} \,}} \,}{\, \overline{{\, \overline{n{\, \overline{a{\, \overline{t{\, \overline{i{\, \overline{on} \,}} \,}} \,}} \,}} \,}{\, \overline{al} \,}} \,}} \,}
}
\\
{\tt
{\, \overline{{\, \overline{pe} \,}{\, \overline{a{\, \overline{ce} \,}} \,}} \,} {\, \overline{{\, \overline{an} \,}d} \,}
{\, \overline{{\, \overline{{\, \overline{p{\, \overline{ro} \,}} \,}{\, \overline{{\, \overline{mo} \,}t} \,}} \,}{\, \overline{{\, \overline{in} \,}g} \,}} \,}
{\, \overline{t{\, \overline{he} \,}} \,}
{\, \overline{{\, \overline{adv{\, \overline{a{\, \overline{n{\, \overline{ce} \,}} \,}} \,}} \,}{\, \overline{{\, \overline{{\, \overline{me} \,}n} \,}t} \,}} \,} {\, \overline{
{\, \overline{of} \,} {\, \overline{a{\, \overline{ll} \,}} \,} } \,}
}
\\
{\tt
{\, \overline{{\, \overline{pe} \,}{\, \overline{op} \,}{\, \overline{le} \,}} \,} {\, \overline{{\, \overline{ {\, \overline{t{\, \overline{he} \,}} \,} {\, \overline{
{\, \overline{{\, \overline{un} \,}{\, \overline{it} \,}{\, \overline{ed} \,}} \,} {\, \overline{{\, \overline{{\, \overline{st{\, \overline{at} \,}} \,}e} \,}s} \,} } \,}} \,}
{\, \overline{ {\, \overline{of} \,} {\, \overline{a{\, \overline{me} \,}{\, \overline{r{\, \overline{ic} \,}} \,}a} \,} } \,}} \,}
{\, \overline{{\, \overline{{\, \overline{jo} \,}{\, \overline{in} \,}} \,}{\, \overline{ed} \,}} \,}
}
\\
{\tt
{\, \overline{in} \,} {\, \overline{f{\, \overline{o{\, \overline{un} \,}d} \,}} \,} {\, \overline{{\, \overline{in} \,}g} \,} {\, \overline{ {\, \overline{t{\, \overline{he} \,}} \,}
{\, \overline{ {\, \overline{{\, \overline{un} \,}{\, \overline{it} \,}{\, \overline{ed} \,}} \,}
{\, \overline{{\, \overline{n{\, \overline{a{\, \overline{t{\, \overline{i{\, \overline{on} \,}} \,}} \,}} \,}} \,}s} \,} } \,}} \,}
}
.
\end{array}
$
The results of the automatic tokenization are especially impressive for strongly isolating languages, such as English and Chinese [@DeMarcken96]. The same algorithms need not be so effective for highly inflective languages, where numerous orthographic alternations occur within the morphological stems (e.g. for Polish). The pursuit for better tokenization algorithms cannot be separated from the quest for the data compression algorithms which identify the inflectional paradigms [@Goldsmith01] or the abstract phrase syntax structures [@NowakPlotkinJansen00].
The shortest grammar and excess entropy {#SecCodes}
=======================================
Let us denote the set of the non-initial rules of grammar $G$ as $G_0:=G\setminus {\left\{ b_0\mapsto g_0 \right\}}$, where $A\setminus B$ is the difference of sets $A$ and $B$. We will call $G_0$ the vocabulary of $G$. The length of the vocabulary is defined as $$\begin{aligned}
{\operatorname{len}G_0}:=\sum_{(b_i\mapsto g_i)\in G_0}
{\operatorname{len}g_i}={\operatorname{len}G} -{\operatorname{len}g_0}.\end{aligned}$$ We use notation ${\operatorname{len}G_0^{\operatorname{MDL}}(v)}:={\operatorname{len}G^{\operatorname{MDL}}(v)} -{\operatorname{len}g_0^{\operatorname{MDL}}(v)}$ respectively.
If the average length of the word-like productions $g_i$, $i\neq 0$, does not depend significantly on the text then we may suppose that $G_0^{\operatorname{MDL}}(v)$ is proportional to the number of word types in text $v$. In fact, we can observe an analog of Guiraud’s law (\[Guiraud\]). If we look at the data published by @NevillManning96 [figure 3.12 (b), p. 69], we can observe empirical proportionality $$\begin{aligned}
\label{NevillVocabulary}
{\operatorname{len}G_0^{\operatorname{SEQUITUR}}(v)}\propto ({\operatorname{len}v})^\alpha, \end{aligned}$$ where $1/2<\alpha<1$ and $G_0^{\operatorname{SEQUITUR}}(v)$ is some approximation of $G_0^{\operatorname{MDL}}(v)$ computed by the algorithm called $\operatorname{SEQUITUR}$. In this section, we would like to present some general theoretical result. We shall relate the length of $G_0^{\operatorname{MDL}}(v)$ to the finite-order excess entropy. It is well known that there are intimate relations between block entropy and the expected lengths of some codes used in data compression. In particular, @KiefferYang00 discuss the concept of grammar-based codes, which represent strings $v\in\mathbb{V}^+$ as uniquely decodable binary strings $C(v)\in{\left\{ 0,1 \right\}}^+$ by the mediation of the admissible grammars.
Let $F=\bigcup_{v\in\mathbb{V}^+} F(v)$ be the set of admissible grammars for all strings. Function $C:\mathbb{V}^+\rightarrow{\left\{ 0,1 \right\}}^+$ is called a grammar-based code if $$\begin{aligned}
\label{GrammarBased}
C(v)=\mathbf{B}(G^C(v)), \end{aligned}$$ where grammar transform $G^C$ computes grammar $G^C(v)\in F(v)$ and grammar encoder $\mathbf{B}$ represents any grammar $G\in F$ as a unique binary string $\mathbf{B}(G)\in{\left\{ 0,1 \right\}}^+$. Let us introduce the expected length of code $C$ for the strings of length $n$ drawn from stationary distribution $\mathbf{P}$, $$\begin{aligned}
\label{ExpectedMDL}
H^C(n)&:=
\sum_{v\in\mathbb{V}^n} \mathbf{P}(v)\cdot {\operatorname{len}C(v)}.\end{aligned}$$ Code $C$ is called universal (more precisely, weakly minimax universal) if $$\begin{aligned}
\label{ChannelInequality}
H^C(n) &\ge H(n),
\\
\label{RateEquality}
\lim_{n\rightarrow\infty}H^C(n)/n &=\lim_{n\rightarrow\infty}H(n)/n\end{aligned}$$ for any stationary distribution $\mathbf{P}$. See @CoverThomas91 [sections 5.1–6 and 12.10] for a general background in information and coding theory.
Additionally, let us call $C$ an irreducible code if for each input string $v\in\mathbb{V}^+$, grammar $G^C(v)$ is irreducible. @KiefferYang00 [theorem 8] prove the following result:
\[theoEncoder\] There exists such grammar encoder $\mathbf{B}$ that any irreducible code of form (\[GrammarBased\]) is weakly minimax universal.
It is a very strong and profound theorem. In particular, code $\operatorname{MDL}(v):=\mathbf{B}(G^{\operatorname{MDL}}(v))$ is universal since the shortest grammar $G^{\operatorname{MDL}}(v)$ is irreducible. Theorem \[theoEncoder\] can be used to prove universality of the modified $\operatorname{SEQUITUR}$ code by @NevillManning96 [@KiefferYang00 section 6.2]. Universality of the famous Lempel-Ziv code, however, is proved differently since it is not an irreducible code and it uses a different grammar encoder [@CoverThomas91 section 12.10].
It has been checked empirically that codes whose grammars are shorter usually enjoy shorter lengths. For instance, @Grassberger02 compressed 135 GB of English text and obtained compression rates (in bits per character) ${\operatorname{len}\operatorname{LZ}(v)}/{\operatorname{len}v} \approx 2.6$ for Lempel-Ziv code $\operatorname{LZ}$ and ${\operatorname{len}\operatorname{NSRPS}(v)}/{\operatorname{len}v} \approx 1.8$ for some heuristic irreducible code $\operatorname{NSRPS}$. Other researchers reported comparable results [@DeMarcken96].
By analogy to definition (\[DefEn\]) of finite-order excess entropy $E(n)$, let us introduce the expected excess code length $$\begin{aligned}
\label{DefThn}
E^C(n)&:= 2H^C(n)-H^C(2n)
\nonumber
\\
&\,\,= \sum_{v,u\in\mathbb{V}^n} \mathbf{P}(vu)
{\left[\, {\operatorname{len}C(v)} + {\operatorname{len}C(u)} - {\operatorname{len}C(vu)} \,\right]}
.\end{aligned}$$
\[theoDiffThEbar\] For any weakly minimax universal code $C$ inequality $$\begin{aligned}
\label{DiffThEbar}
E^C(n)\ge E(n)\end{aligned}$$ is true for infinitely many $n$. (See appendix \[SecProof\] for the proof.)
Inequality (\[DiffThEbar\]) is valid in particular for $C=\operatorname{MDL}$ or for any irreducible code.
Now, we shall link the expected excess code length $E^{\operatorname{MDL}}(n)$ with the length of $\operatorname{MDL}$ vocabulary. Let $L^m(v):={\operatorname{len}G^{\operatorname{MDL}}(v)}$ be the length of the shortest grammar and $L_0^m(v):={\operatorname{len}G_0^{\operatorname{MDL}}(v)}$ be the length of its vocabulary. Define $L^{>1}(v)$ as the maximal length of a string which appears in string $v$ at least twice.
\[theoVocabulary\] We have inequalities $$\begin{aligned}
\label{ReZero}
L^m(v)&\le {\operatorname{len}v},
\\
\label{ReOne}
L^m(v),L^m(u)&\le L^m(vu) + L^{>1}(vu),
\\
\label{ReBoth}
0 \le L^m(v)+L^m(u)-L^m(vu)&\le L_0^m(vu) +
L^{>1}(vu).
\end{aligned}$$ (See appendix \[SecProof\] for the proof.)
Inequality (\[ReBoth\]) states that the vocabulary length for the shortest grammar cannot be roughly less than the excess length of the shortest grammar. In a slightly heuristic reasoning, we shall argue that the excess length of the shortest grammar multiplied by a slowly growing function cannot be less than the excess length of code $\operatorname{MDL}$. In order to do it we need some pretty strong symmetrical bound for the length of code $\operatorname{MDL}$ in terms of the length of the shortest grammar.
It is known that function $\mathbf{B}$ of Theorem \[theoEncoder\] satisfies $\operatorname{len}\mathbf{B}(G)\le
\gamma({\operatorname{len}G})$, where $\gamma(n):= n\cdot (c +\log n)$ for some constant $c$ [@KiefferYang00 section 4]. The following symmetrical bound for code $\operatorname{MDL}$ seems probable:
\[conjBound\] There is inequality $$\begin{aligned}
{\left| {\operatorname{len}\operatorname{MDL}(v)}-\gamma(L^m(v)) \right|}\le f_2(L^m(v)),
\end{aligned}$$ where $\gamma(n):= n\cdot f_1(n)$ and functions $f_i\ge 0$ satisfy $0\le f_i(n+1)-f_i(n)\le c_i/n$ for some constants $c_i$.
Now we can give a bound for the excess length of code $\operatorname{MDL}$ in terms of the excess length of the shortest grammar.
\[theoIfBound\] If Conjecture \[conjBound\] is true then $$\begin{aligned}
&{\operatorname{len}\operatorname{MDL}(v)}
+ {\operatorname{len}\operatorname{MDL}(u)}
- {\operatorname{len}\operatorname{MDL}(vu)}
\nonumber
\\
\label{IfBound}
&\quad\le
{\left[\, L^m(v)+L^m(u)-L^m(vu) + d_1 \,\right]}{\left[\, d_2 + c_1 \log {\operatorname{len}vu}
+c_1 \frac{L^{>1}(vu)}{L^m(vu)} \,\right]}
+ c_1 L^{>1}(vu),\end{aligned}$$ where $d_1=3c_2/c_1$ and $d_2=\max (f_1(1),f_2(1)c_1/c_2)$. (See appendix \[SecProof\] for the proof.)
Recall that $H^{\operatorname{MDL}}(n)/n=\sum_{v\in\mathbb{V}^n} \mathbf{P}(v)\cdot
L^m(v)/{\operatorname{len}v}$ approaches entropy rate $h$ for $n\rightarrow\infty$ by Theorem \[theoEncoder\]. We may speculate that $h>0$ for the language production. Let us assume a stronger statement, namely, that $$\begin{aligned}
{\operatorname{len}v} \le d_3 L^m(v)\end{aligned}$$ for some constant $d_3$ and (almost) every human text $v$. On the other hand, notice that $L^{>1}(v)\le {\operatorname{len}v}$ follows by definition of $L^{>1}(v)$. By these two inequalities, we have $L^{>1}(vu)/L^m(vu)\le
d_3$. Combining the latter with (\[IfBound\]) and (\[ReBoth\]) gives $$\begin{aligned}
{\operatorname{len}\operatorname{MDL}(v)}
&+ {\operatorname{len}\operatorname{MDL}(u)}
- {\operatorname{len}\operatorname{MDL}(vu)}
\nonumber
\\
\label{ThenBound}
&\le
{\left[\, L_0^m(vu) +L^{>1}(vu) +d_1 \,\right]}{\left[\, d_4 + c_1 \log
{\operatorname{len}vu} \,\right]}, \end{aligned}$$ where $d_4:=d_2+ c_1(d_3+1)$. Averaging (\[ThenBound\]) with $\mathbf{P}(vu)$ for $v,u\in\mathbb{V}^n$, we obtain $$\begin{aligned}
\label{SolMDLbar}
{\left[\, L_0^m[2n] + L^{>1}[2n]+ d_1 \,\right]}{\left[\, d_4 + c_1
\log (2n) \,\right]} \ge E^{\operatorname{MDL}}(n),
\end{aligned}$$ where $$\begin{aligned}
\label{ExpectedGMDL}
L_0^m[n]:=
\sum_{v\in\mathbb{V}^n} \mathbf{P}(v)\cdot {\operatorname{len}L_0^m(v)},
\quad
L^{>1}[n]:=
\sum_{v\in\mathbb{V}^n} \mathbf{P}(v)\cdot {\operatorname{len}L^{>1}(v)}.\end{aligned}$$
By inequality (\[SolMDLbar\]) and Theorem \[theoDiffThEbar\], we also have $$\begin{aligned}
\label{DiffGMDLEbar}
{\left[\, L_0^m[2n] + L^{>1}[2n]+ d_1 \,\right]}
{\left[\, d_4 + c_1\log (2n) \,\right]}\ge E(n)\end{aligned}$$ for infinitely many $n$. In particular, if stationary distribution $\mathbf{P}$ obeys Hilberg’s law (\[HilbergH\]) then inequality $$\begin{aligned}
\label{GMDLHilberg}
L_0^m[n] + L^{>1}[n] \ge {\mathop{\text{const}}}\cdot n^{\mu}/\log n\end{aligned}$$ holds for infinitely many $n$ by equation (\[HilbergE\]).
Hilberg’s law and Guiraud’s law {#SecGuiraud}
===============================
In this section, we would like to make the final step in deriving Guiraud’s law from relation (\[GMDLHilberg\]). First, let us have a closer look at Guiraud’s and Zipf’s laws. It is widely-known that if Zipf’s law (\[Zipf\]) holds with the same $B$ for all $N$ then Guiraud’s law (\[Guiraud\]) is satisfied with $\rho=1/B$ for large $N$, cf. @Kornai02 [section 3.2] or @FerrerSole01b.
In fact, the number of word types $V$ and the number of word tokens $N$ can be computed given the word frequencies, $$\begin{aligned}
V&=\sum_{w:\,c(w)>0} 1, \quad
N=\sum_{w:\,c(w)>0} c(w),\end{aligned}$$ so any relation between $V$ and $N$ is a function of the exact distribution of frequencies $c(w)$. The converse is not true. In general, frequency $c(w)$ cannot be computed given only $w$, $V$, and $N$ since different texts usually have different keywords. Still, we may seek for hypothetical derivations of formula (\[Zipf\]) given formula (\[Guiraud\]) and some additional assumptions.
One could ask if Guiraud’s law or Zipf’s law do hold with the same $\rho$ or $B$ for texts of various size and origin. The answer is complex. For instance, @Kornai02 [section 2.5] discusses Guiraud’s law extensively and according to the plot in his article value $\rho\approx 0.75$ holds perfectly for samples of sizes $N\in{\left[\, 1.4\cdot 10^5,1.8\cdot 10^7 \,\right]}$ drawn from San Jose Mercury News corpus. Such value of $\rho$ would correspond to $B\approx 1.33$ if formula (\[Zipf\]) with constant $B$ held for all word ranks. Nevertheless, if we investigate the rank-frequency plot for so large collections of texts, we encounter a different regularity.
@FerrerSole01b discovered that parameter $B$ in formula (\[Zipf\]) depends on word rank $r(w)$. For multi-author corpora there are two regimes where $B$ is almost constant. Namely, we have $$\begin{aligned}
\label{TwoRegimes}
B=
\begin{cases}
B_1, & 0\le r(w)\le R_1,
\\
B_2, & R_1\le r(w),
\end{cases}\end{aligned}$$ where $B_2<B_1\approx 1$. Let us note that for sufficiently short text collections (those with $V < R_1$) only one of two regimes can be observed. For single-author corpora and $r(w)\ge R_1$, we have an exponential decay of $c(w)$ rather than a power-law.
In another case of some multi-author collection of English texts counting $1.8\cdot 10^8$ word tokens, @MontemurroZanette02 reported $B_1\approx 1$, $B_2\approx 2.3$ and $R_1\approx 6000$. The investigated collection is only 10 times larger than SJMN corpus surveyed by Kornai. If formula (\[Zipf\]) with constant $B\approx
2.3$ held for all word ranks then we would have Guiraud’s law (\[Guiraud\]) with $\rho\approx 0.43$. Anyway, if there are two regimes of $B$, like in (\[TwoRegimes\]), then we could obtain Guiraud’s law (\[Guiraud\]) with $\rho\approx 0.75$ for all $N$ if also parameter $R_1$ depends on the text length $N$. Until we have more experimental data on the dependence between $N$ and $R_1$, we can be only sure that there is inequality $$\begin{aligned}
\label{GuiraudH}
V\ge {\mathop{\text{const}}}\cdot N^{0.43}.\end{aligned}$$
Let $V(v)$ be the number of orthographic word types in text $v$ and $N(v)$—the number of orthographic word tokens therein. If we assume that the mean length of the word tokens in text $v$ does not change substantially with $v$ then text length $N(v)$ measured in orthographic words is proportional to text length ${\operatorname{len}v}$ measured in phonemes or letters, $$\begin{aligned}
\label{Npropto}
N(v)\propto {\operatorname{len}v}.\end{aligned}$$
In view of section \[SecGrammars\], we may suppose that the number of orthographic word types $V(v)$ is proportional to the number of the production rules in the shortest grammar $G^{\operatorname{MDL}}(v)$, cf.@NevillManning96 [figure 3.12 (c) vs. (a), p. 69]. If the mean length of the non-initial productions does not change substantially against $v$ then the number of the rules is proportional to length $L_0^{m}(v)$ of the vocabulary of the shortest grammar $G^{\operatorname{MDL}}(v)$, cf. @NevillManning96 [figure 3.12 (a) vs. (b), p.69]. Resuming, we would have proportionality $$\begin{aligned}
\label{Vpropto}
V(v)\propto L_0^{m}(v).\end{aligned}$$ Assuming relations (\[Npropto\]) and (\[Vpropto\]), we can restate Guiraud’s law (\[GuiraudH\]) as $$\begin{aligned}
\label{GMDLGuiraudH}
L_0^{m}(v)\ge {\mathop{\text{const}}}\cdot ({\operatorname{len}v})^{0.43},\end{aligned}$$ which resembles relation (\[NevillVocabulary\]) reported by @NevillManning96. Except for the effects of averaging and the negligible length $L^{>1}(v)$ of the longest substring appearing more than once, inequality (\[GMDLGuiraudH\]) is implied by inequality (\[GMDLHilberg\]) with the very rough estimate $\mu\approx 1/2$ done by Hilberg. We could say that Hilberg’s law can be some explanation of Guiraud’s law. Let us discuss the plausibility of such explanation.
Zipf’s law is often understood as a specific algebraic relationship between the counts and ranks of various objects—not necessarily words. In such generalization, Zipf’s law is observed also out of the linguistic domain, e.g. in income distribution [@Pareto97]. We do not know if one can find a general explanation of Zipf’s law both in linguistic and non-linguistic contexts. Explaining Zipf’s law in the purely linguistic context seems somehow easier. One needs “only” to assign some reasonable relative frequency $\mathbf{P}(v)$ to every string $v$ of phonemes and then to define how any finite string $v$ should be cut into words. The existence or nonexistence of relation (\[Zipf\]) should follow by pure mathematical deduction from these two assumptions.
That idea inspired @Mandelbrot53 to formulate some classical explanation of Zipf’s law. His assumptions are:
1. Stationary distribution $\mathbf{P}$ is an IID distribution, i.e. it satisfies (\[PRandom\]).
2. Set $\mathbb{V}$ of atomic symbols is the set of phonemes and spaces. The word tokens in any text are defined as the space-to-space strings of phonemes.
Given these assumptions Mandelbrot derived Zipf’s law for space-to-space words and hence Guiraud’s law can be inferred as well. In fact, Mandelbrot did not discuss Guiraud’s law but, as we have said, Zipf’s law does imply Guiraud’s law automatically. Mandelbrot’s explanation assuming the existence of “intermittent silences” was quoted or rediscovered by many researchers, e.g. by @Belevitch56, @Miller57, @BellClearyWitten90 and @Li92. There is some historical summary of that literature done by @Li98.
Although Mandelbrot’s explanation of Zipf’s and Guiraud’s laws earned some popularity among natural scientists, we should stress that both of its assumptions are false with respect to the intended application to natural language. First, we would object to modeling human language production by an IID distribution. Second, Mandelbrot’s definition of word is biased by the spelling conventions of the most popular alphabetic scripts which use blank spaces to separate words. No regular “intermittent silences” appear in the spoken versions of the corresponding ethnic languages [@Jelinek97]. That phenomenon is a challenge for automatic speech recognition and it motivated some interest in the shortest admissible grammars as a means for restoring the boundaries between the words [@DeMarcken96].
In this article, we present another explanation of Guiraud’s law. Our assumptions are:
1. Stationary distribution $\mathbf{P}$ exhibits Hilberg’s law (\[HilbergH\]) for all $n$.
2. We may assume that $\mathbb{V}$ is a set of phonemes only. The word tokens in any text are defined as the nonterminal tokens of the shortest admissible grammar.
We think that the derivation of Guiraud’s law based on Hilberg’s law is better linguistically justified than the classical explanation by Mandelbrot. There are several reasons for that claim:
1. The new explanation assumes that human narration exhibits strong probabilistic dependence, it is not a IID distribution. In appendix \[SecMath\], we recall that no infinitary distribution $\mathbf{P}$ can be modeled by a stationary hidden Markov chain with a finite number of hidden states. This fact can have some important implications for computational linguistics [@Jelinek97].
2. The new explanation does not assume the pre-existence of spaces between the words in the natural language production. Children can learn the correct tokenization of speech into the words even if they do not know yet what the words are.
3. Space-to-space words for the IID distributions do not have any definite internal structure. It is no longer true for the new explanation. The nonterminals of the shortest grammar exhibit the internal structure of recursive rule productions. Such nonterminals have well-defined parts. Without any change of the model, we can speak not only of Guiraud’s and Zipf’s laws for the words but we can also discuss laws which relate words to their elements. Some example of the latter is Menzerath’s law, which states that the longer the word is the shorter its constituents are [@Menzerath28; @Altmann80]. By means of the grammar-based codes one can define the structure of word-like objects and investigate many quantitative linguistic laws not only for the language production but also for any other stationary distributions.
4. Stationary distribution is called ergodic (roughly) if the relative frequency of any fixed word does not vary significantly across different texts. By some theorem, every IID distribution is ergodic [@Debowski05 chapter 4]. Nevertheless, empirical studies do not corroborate Mandelbrot’s assumption that language production $\mathbf{P}$ is ergodic. The mere existence of concept “the keywords of the text” reflects the fact that different texts use different vocabularies systematically. Words, once they appear in some text, tend to reappear. Let us stress that some significant variation of the word frequencies *can* be modelled by non-ergodic stationary distributions. Many non-ergodic stationary distributions are infinitary [@Debowski05 chapters 4 and 5], see also appendix \[SecMath\]. It is an interesting question whether Hilberg’s law (\[HilbergH\]) implies non-ergodicity of stationary distribution $\mathbf{P}$. Some further discussion of Hilberg’s law and non-ergodic distributions could give us insight where to seek general quantitative laws in the intertext variability of language. Any such laws would be of great importance to computational linguistics as well.
Conclusions {#SecConclusions}
===========
In this article, we have discussed some implications of @Hilberg90’s ([-@Hilberg90]) hypothesis on the entropy of natural language production. That hypothesis states that finite-order excess entropy $E(n)$ of the $n$-letter strings is proportional to the square root of $n$. So far, the proportionality has been roughly verified only for $n\le 50$. On the other hand, we have argued that Hilberg’s hypothesis, when extrapolated to $n$ of the text length magnitude, provides a better explanation of Guiraud’s law than the classical explanation based on the existence of “intermittent silences” [@Mandelbrot53].
The new explanation is based on two points. First, we observe that the tokenization of a text into orthographic words and their morphemes matches largely the production rules of the shortest admissible grammar for the text. Second, we use some partially heuristic, but largely deductive, mathematical reasoning to argue that the length of the non-initial production rules of the shortest grammar cannot be less than finite-order excess entropy.
In the future research, the rough match of the linguistically-motivated tokenizations and the tokenizations given by the shortest grammars should be surveyed as one of the fundamental problems of quantitative linguistics. One should survey Zipf’s, Guiraud’s, and Menzerath’s laws for the nonterminals of the admissible grammars and the orthographic words simultaneously across a large range of text sizes and languages. Proportionalities (\[Npropto\]) and (\[Vpropto\]) should be verified as well.
It seems that the existence of a rich formal structure in the natural language production is reflected by its high total excess entropy $E$ rather than by simply positive entropy gain $H(1)-h$. We think that the further discussion of Hilberg’s hypothesis can improve the quality of statistical language models both in quantitative and computational linguistics, see appendix \[SecMath\] and our doctoral dissertation [@Debowski05].
Since the shortest admissible grammars reproduce also the internal structure of words, the behavior of excess entropy might be linked not only with Guiraud’s and Zipf’s laws but also with Menzerath’s law. The shortest grammars can be used as the *definition* of words and their constituents in any symbolic string [@NevillManning96]. Adopting such a definition, empirical researchers can survey the form of Guiraud’s, Zipf’s, and Menzerath’s laws also in the non-linguistic symbolic data (such as DNA). Last but not least, mathematicians can prove some rigorous theorems.
Proofs {#SecProof}
======
[\[theoDiffThEbar\]]{} For any function $f$ we have identity $$\begin{aligned}
\sum_{k=0}^{m-1}
{\left[\, 2f(2^k n)-f(2^{k+1} n) \,\right]}\cdot\frac{1}{2^{k+1}}=
f(n)-\frac{f(2^m n)}{2^m n}\cdot n\end{aligned}$$ for each finite $m$. Hence, if (\[RateEquality\]) is true then we obtain $$\begin{aligned}
H(n)-hn &= \sum_{k=0}^\infty
{\left[\, 2H(2^k n)-H(2^{k+1} n) \,\right]}\cdot\frac{1}{2^{k+1}}=
\sum_{k=0}^\infty \frac{E(2^k n)}{2^{k+1}}
,
\\
H^C(n)-hn &= \sum_{k=0}^\infty
{\left[\, 2H^C(2^k n)-H^C(2^{k+1} n) \,\right]}\cdot\frac{1}{2^{k+1}}=
\sum_{k=0}^\infty \frac{E^C(2^k n)}{2^{k+1}}
.\end{aligned}$$ Because of inequality (\[ChannelInequality\]), we have $H(n)-hn\le H^C(n)-hn$ so $$\begin{aligned}
\label{ThEbarSeries}
\sum_{k=0}^\infty \frac{E(2^k n)}{2^{k+1}}\le
\sum_{k=0}^\infty \frac{E^C(2^k n)}{2^{k+1}}.\end{aligned}$$ If we put $n=2^p M$ with any $p$ and some fixed $M$ then (\[ThEbarSeries\]) yields $$\begin{aligned}
\label{ThEbarSeries2}
\sum_{k=p}^\infty \frac{E^C(2^k M)-E(2^k M)}{2^{k+1}}\ge
0.\end{aligned}$$
Assume that $E^C(2^k M)-E(2^k M)\ge 0$ holds only for finitely many $k$. Then we would have $E^C(2^k M)-E(2^k M)< 0$ for all $k\ge p$ and some $p$. Hence, we would have $$\begin{aligned}
\label{ThEbarSeries3}
\sum_{k=p}^\infty \frac{E^C(2^k M)-E(2^k M)}{2^{k+1}}<
0.\end{aligned}$$ Since (\[ThEbarSeries3\]) stays in contradiction with (\[ThEbarSeries2\]), our assumption that $E^C(2^k M)-E(2^k M)\ge 0$ only for finitely many $k$ was false. We must have $E^C(2^k
M)-E(2^k M)\ge 0$ for infinitely many $k$, and this is exactly inequality (\[DiffThEbar\]) which we were to prove.
[\[theoVocabulary\]]{} In order to prove (\[ReZero\]), notice that $G={\left\{ b_0\mapsto
v \right\}}$ is a grammar for $v$. Its length satisfies ${\operatorname{len}v}={\operatorname{len}G}\le {\operatorname{len}G^{\operatorname{MDL}}(v)}$ by (\[Glength\]) and (\[GMDLDefinition\]).
Now, let us prove (\[ReOne\]) and (\[ReBoth\]). Since vocabulary $G_0^{\operatorname{MDL}}(vu)$ cannot beat vocabularies $G_0^{\operatorname{MDL}}(v)$ and $G_0^{\operatorname{MDL}}(u)$ in the efficient representation of any strings $v$ and $u$ respectively, we observe inequalities $$\begin{aligned}
\label{Left}
{\operatorname{len}G^{\operatorname{MDL}}(v)}
&\le
{\operatorname{len}g_L} +{\operatorname{len}G_0^{\operatorname{MDL}}(vu)},
\\
\label{Right}
{\operatorname{len}G^{\operatorname{MDL}}(u)}
&\le
{\operatorname{len}g_R} +{\operatorname{len}G_0^{\operatorname{MDL}}(vu)},\end{aligned}$$ where $G_0^{\operatorname{MDL}}(vu)\cup {\left\{ b_0\mapsto g_L \right\}}$ and $G_0^{\operatorname{MDL}}(vu)\cup {\left\{ b_0\mapsto g_R \right\}}$ are some grammars for $v$ and $u$ respectively. Analogically, $$\begin{aligned}
\label{Inverse}
{\operatorname{len}G^{\operatorname{MDL}}(vu)}
&\le {\operatorname{len}G^{\operatorname{MDL}}(v)} + {\operatorname{len}G^{\operatorname{MDL}}(u)} \end{aligned}$$ since $G_0^{\operatorname{MDL}}(v)\cup G_0^{\operatorname{MDL}}(u)\cup {\left\{ b_0\mapsto
g_0^{\operatorname{MDL}}(v) g_0^{\operatorname{MDL}}(u) \right\}}$ is a grammar for $vu$.
Assume that $g_L$ and $g_R$ are obtained by splitting the initial production $g_0^{\operatorname{MDL}}(vu)$ into two parts and recursively expanding the nonterminal at the border if necessary. That is, we have either $g_Lg_R=g_0^{\operatorname{MDL}}(vu)$ or $g_L=y_Lx_L$, $g_R=x_Ry_R$, and $g_0^{\operatorname{MDL}}(vu)=y_Lb_iy_R$, where nonterminal $b_i$ expands recursively into string $x_Lx_R\in\mathbb{V}^+$. Grammar $G^{\operatorname{MDL}}(vu)$ is irreducible so we must have ${\operatorname{len}x_Lx_R}\le
L^{>1}(vu)$, where $L^{>1}(vu)$ is the maximal length of a string which appears in string $vu$ at least twice. Thus, $$\begin{aligned}
\label{Sum}
{\left| {\operatorname{len}g_L}+{\operatorname{len}g_R}- {\operatorname{len}g_0^{\operatorname{MDL}}(vu)} \right|}\le L^{>1}(vu).\end{aligned}$$ By (\[Sum\]), adding (\[Left\]) and (\[Right\]) yields $$\begin{aligned}
{\operatorname{len}G^{\operatorname{MDL}}(v)} + {\operatorname{len}G^{\operatorname{MDL}}(u)}
&\le {\operatorname{len}g_0^{\operatorname{MDL}}(vu)} + 2{\operatorname{len}G_0^{\operatorname{MDL}}(vu)} + L^{>1}(vu)
\nonumber
\\
\label{Both}
&= {\operatorname{len}G^{\operatorname{MDL}}(vu)} + {\operatorname{len}G_0^{\operatorname{MDL}}(vu)} + L^{>1}(vu).\end{aligned}$$ In fact, we can rewrite (\[Both\]) and (\[Inverse\]) as (\[ReBoth\]). By (\[Sum\]), we also have ${\operatorname{len}g_L},{\operatorname{len}g_R}\le {\operatorname{len}g_0^{\operatorname{MDL}}(vu)}+ L^{>1}(vu)$. Inserting these two inequalities into (\[Left\]) and (\[Right\]) respectively yields (\[ReOne\]).
[\[theoIfBound\]]{} According to Conjecture \[conjBound\], we have $$\begin{aligned}
\label{BoundI}
&{\operatorname{len}\operatorname{MDL}(v)}
+ {\operatorname{len}\operatorname{MDL}(u)}
- {\operatorname{len}\operatorname{MDL}(vu)}
\nonumber
\\
&\quad\le
\gamma(L^m(v))+ \gamma(L^m(u))-\gamma(L^m(vu))
+ f_2(L^m(v))+f_2(L^m(u))+f_2(L^m(vu))
\end{aligned}$$ By $0\le f_i(n+1)-f_i(n)\le c_i/n$ and (\[ReOne\]), there is $$\begin{aligned}
\label{BoundL}
f_i(n)&\le f_i(1)+\sum_{k=2}^n c_i/k<f_i(1) +c_i\log n,
\\
f_i(L^m(v))
&\le
f_i(L^m(vu)) + c_i L^{>1}(vu)/L^m(vu).
\end{aligned}$$ Hence by (\[ReZero\]), $$\begin{aligned}
\label{BoundG}
\gamma(L^m(v))&+ \gamma(L^m(u))-\gamma(L^m(vu))
\nonumber
\\
&\le
{\left[\, L^m(v)+L^m(u)-L^m(vu) \,\right]} f_1(L^m(vu))
+ c_1 {\left[\, L^m(v)+L^m(u) \,\right]}\frac{L^{>1}(vu)}{L^m(vu)}
\nonumber
\\
&=
{\left[\, L^m(v)+L^m(u)-L^m(vu) \,\right]}
{\left[\, f_1(L^m(vu)) + c_1\frac{L^{>1}(vu)}{L^m(vu)} \,\right]}
+ c_1 L^{>1}(vu)
\nonumber
\\
&\le
{\left[\, L^m(v)+L^m(u)-L^m(vu) \,\right]}
{\left[\, f_1({\operatorname{len}vu}) + c_1 \frac{L^{>1}(vu)}{L^m(vu)} \,\right]}
+ c_1 L^{>1}(vu)
.
\end{aligned}$$ On the other hand, $$\begin{aligned}
\label{BoundF}
f_2(L^m(v))+f_2(L^m(u))+f_2(L^m(vu))
&\le
3f_2(L^m(vu)) + 2c_2 L^{>1}(vu)/L^m(vu)
\nonumber
\\
&\le
3{\left[\, f_2({\operatorname{len}vu}) + c_2 \frac{L^{>1}(vu)}{L^m(vu)} \,\right]}
.
\end{aligned}$$ Inserting (\[BoundG\]), (\[BoundF\]), and (\[BoundL\]) into (\[BoundI\]) we obtain (\[IfBound\]).
Some properties of infinitary distributions {#SecMath}
===========================================
Infinitary distributions seem to be a new interesting class of the stochastic models for human narration. The mathematics of excess entropy is just being developed, cf.@Debowski05 for an overview. Our program is to bring together some advanced results of mathematics (measure-theoretic probability theory, coding theory) and some quantitative linguistic intuitions. We can give a linguistic interpretation to some mathematical theorems and a formal language to express some vague hypotheses about the obscure nature of probabilistic language models.
We would like to mention four facts about infinitary distributions which can be important for quantitative and computational linguistics in the view of Hilberg’s hypothesis. These are:
1. There are infinitary distributions which are not deterministic stationary distributions. That is, total excess entropy $E=\infty$ does not imply entropy rate $h=0$.
2. All stationary distributions which consist in a random description of some infinite random object must be infinitary and nonergodic [@Debowski05 chapter 5].
Hence, we may suppose that $E=\infty$ holds for the stationary distribution of the language production because almost every human text refers systematically to a different and potentially infinite fictitious world.
3. For some infinitary distributions, value $\mathbf{P}(v)$ can be computed for every string $v$ by some finite procedure, cf. @Berthe94 and @Gramss94.
4. No infinitary distribution can be represented by a finite-state hidden Markov model (HMM), cf. @CrutchfieldFeldman01, @Upper97, @CoverThomas91 [section 2.8, data processing inequality].
In spite of their inadequacy as the models of infinitary distributions, finite-state HMMs are the standard heuristic models of natural language engineering. It happens so only for the necessity of the effective search for the most probable hidden states. Some well-known applications of HMMs are automatic speech recognizers [@Jelinek97] and trigram part-of-speech taggers [@ManningSchutze99; @Debowski04]. It was observed that the error rate of trigram taggers decreases as a negative power of the size of the training data. When we increase the training data size ten times, the error rate diminishes only by half [@Megyesi01]. In fact, such power-law decay of the error rate can be also some consequence of Hilberg’s law [@BialekNemenmanTishby00].
The lack of space disallows us to exactly explain the terminology and the reasons for the mathematical facts mentioned above. We will try to popularize some ideas of our thesis among the linguistic audience in the next articles.
[^1]: Stationary distributions are the distributions of stationary stochastic processes [@Upper97]. For simplicity, we avoid the mathematical terms of stochastic processes, random variables and probabilistic spaces [@Billingsley79; @Kallenberg97]. Since we do not need these notions to present the core reasonings, we ignore them to make the article as elementary as possible.
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[**Nuclear multifragmentation and fission: similarity and differences**]{}
V. Karnaukhov$^{1,}$[^1], H. Oeschler$^2$, S. Avdeyev$^1$, V. Rodionov$^1$,\
V. Kirakosyan$^1$, A. Simonenko$^1$, P. Rukoyatkin$^1$, A. Budzanowski$^3$,\
W. Karcz$^3$, I. Skwirczynska$^3$, B. Czech$^3$, L. Chulkov$^4$, E. Kuzmin$^4$,\
E. Norbeck$^5$, A. Botvina$^6$
[ $^1$Joint Institute for Nuclear Research, 141980 Dubna, Russia\
$^2$Institut für Kernphysik, Darmstadt University of Technology, 64289 Darmstadt, Germany\
$^3$H. Niewodniczanski Institute of Nuclear Physics, 31-342 Cracow,Poland\
$^4$Kurchatov Institute, 123182 Moscow, Russia\
$^5$University of Iowa, Iowa City, IA 52242, USA\
$^6$Institute for Nuclear Research, 117312 Moscow, Russia\
]{}
Thermal multifragmentation of hot nuclei is interpreted as the nuclear [*liquid–fog*]{} phase transition deep inside the spinodal region. The experimental data for p(8.1GeV) + Au collisions are analyzed. It is concluded that the decay process of hot nuclei is characterized by [*two size parameters*]{}: transition state and freeze-out volumes. The similarity between dynamics of fragmentation and ordinary fission is discussed. The IMF emission time is related to the mean rupture time at the multi-scission point, which corresponds to the kinetic freeze-out configuration.
Introduction
============
The study of the highly excited nuclei is one of the challenging topics of nuclear physics, giving access to the nuclear equation of state for temperatures below ${\it T}_{c}$ – the critical temperature for the [*liquid-gas*]{} phase transition. The main decay mode of hot nuclei is a copious emission of intermediate mass fragments (IMF), which are heavier than $\alpha$–particles but lighter than fission fragments. An effective way to produce hot nuclei is via heavy-ion collisions. But in this case the heating of nuclei is accompanied by compression, strong rotation, and shape distortion, which influence the decay properties of excited nuclei. One gains simplicity, and the picture becomes clearer, when light relativistic projectiles (protons, antiprotons, pions) are used. In this case, fragments are emitted by only one source – the slowly moving target spectator. Its excitation energy is almost entirely thermal. Light relativistic projectiles provide a unique possibility for investigating [*thermal multifragmentation*]{}. The decay properties of hot nuclei are well described by statistical models of multifragmentation (SMM and MMMC [@1; @2]). This is an indication, that the system is thermally equilibrated or close to that. For the case of peripheral heavy-ion collisions the partition of the excited system is also governed by heating.
The van der Walls equation can be used with nuclear matter because of the similarity of the nucleon-nucleon force to the force between molecules in a classical gas [@3; @4; @5]. In both cases there exists a region in the PVT diagram corresponding to a mixture of liquid and gas phases. This region can contain unstable, homogeneous matter for short times. In a classical gas this can be achieved by cooling through the critical point. In the nuclear case this can be achieved by a sudden expansion of the liquid phase at a temperature well below the critical temperature. The separation of the homogeneous matter into a mixture of stable liquid and gas is called spinodal decomposition. One can imagine that a hot nucleus (at ${\it T}$ = 7–10 MeV) expands due to thermal pressure and enters the unstable region. Due to density fluctuations, a homogeneous system is converted into a mixed phase consisting of droplets (IMF) and nuclear gas interspersed between the fragments. Thus the final state of this transition is a [*nuclear fog*]{} [@5]. Note that classical fog is unstable, it condensates finally into bulk liquid. The charged nuclear fog is stable in this respect. But it explodes due to Coulomb repulsion and is detected as multifragmentation. It is more appropriate to associate the spinodal decomposition with the [*liquid-fog*]{} phase transition in a nuclear system rather than with the [*liquid-gas*]{} transition [@6; @8]. This scenario is supported by a number of observations; some of them are the following:\
(a) the density of the system at break-up is much lower than the normal one ${\rho}_{0}$ [@8];\
(b) the mean life-time of the fragmenting system is very small ($\approx$ 50 [*fm/c*]{}) [@9];\
(c) the break-up temperature is significantly lower than $T_{c}$, the critical temperature for the [*liquid-gas*]{} phase transition [@6; @7].\
In this paper we concentrate on the dynamics of thermal multifragmentation and its similarity to ordinary fission. This similarity was noted first by Lopez and Randrup in their statistical theory of multifragmentation [@10; @11]. First of all, there are two characteristic volumes (or configurations) for both processes. Secondly, the time scale characterizations for fragmentation and fission are similar with respect to their ingredients.\
Experimental data have been obtained using the $4\pi$-device FASA installed at the external beam of the Nuclotron (Dubna). At present, the setup consists of twenty five [*dE-E*]{} telescopes surrounded by a fragment multiplicity detector, which is composed of 58 thin CsI(Tl) scintillation counters.\
Two characteristic volumes in thermal multifragmentation
========================================================
Traditionally, in statistical models [@1; @2], multifragmentation is characterized by just one size\
parameter – the freeze-out volume, ${\it V}_{f}$. There are a number of papers with experimental estimations of this characteristic volume, but the values obtained deviate significantly. A mean freeze-out volume $\sim$ 7${\it V}_{0}$ (${\it V}_{0}$ = volume at normal density) was found in $\it ref.$ [@12] from the average relative velocities of the IMFs for $^4$He(14.6 MeV) + Au collisions. In paper [@13] the nuclear caloric curves were considered within the Fermi-gas model to extract average nuclear densities for different systems. It was found that ${\it V}_{f}$ $\approx$ 2.5${\it V}_{0}$ for the medium and heavy masses. In $\it ref.$ [@14] the mean IMF kinetic energies were analyzed for Au + Au collisions at 35$\cdot$A MeV. The freeze-out volume was found to be $\sim$ 3${\it V}_{0}$. The average source density for the fragmentation in the 8.0 GeV/$\it c$ ${\pi}^{-}$ + Au interaction was estimated to be $\approx$ (0.25–0.30)${\rho}_{0}$, at E\*/A$\sim$5 MeV from the moving-source-fit Coulomb parameters [@15; @16].\
In our paper [@8], the data on the charge distribution and kinetic energy spectra of IMFs produced in [*p*]{}(8.1GeV)+Au collisions were analyzed using the combined INC +Exp+SMM model. The events with IMF multiplicity [*M*]{}$\ge$2 were selected. The results obtained are shown at [*fig.*]{}1. It was shown that one should use two volume (or density) parameters to describe the process of multifragmentation, not just one as in the traditional approach. The first, ${V}_{t}$ = (2.6${\pm}$0.3)${\it V}_{0}$ (or ${\rho}_{t}$ ${\approx}$ 0.38${\rho}_{0}$), corresponds to the stage of pre-fragment formation. Strong interaction between pre-fragments is still significant at this stage.
![Proposed spinodal region for the nuclear system. The experimental points were obtained by the FASA collaboration. The arrow line shows the way from the starting point at [*T*]{}=0 and normal nuclear density ${\rho}_{0}$ to the break-up point at ${\rho}_{t}$ and to the kinetic freeze-out at the mean density ${\rho}_{f}$. Critical temperature is estimated in [@7].[]{data-label="1"}](pic1.eps "fig:"){width="8.1cm"}\
The second one, ${\it V}_{f}$=(5${\pm}$1)${\it V}_{0}$, is the kinetic freeze-out volume. It is determined by comparing the measured fragment energy spectra with the model predictions using multi-body Coulomb trajectories. The calculations have been started with placing all charged particles of a given decay channel inside the freeze-out volume ${\it V}_{f}$. In this configuration the fragments are already well separated from each other, they are interacting via the Coulomb force only. Actually, the system at freeze-out belongs to the mixed phase sector of the phase diagram, with a mean density that is five times less than normal nuclear density. The first characteristic volume, ${\it V}_{t}$, was obtained by analyzing the IMF charge distributions, [*Y*]{}(Z), within the SMM model with a [*free*]{} size parameter [*k*]{}. For simplicity, the dependence of the charge distribution on the critical temperature ${\it T}_{c}$ was neglected in this analysis.\
Recently, we performed a more sophisticated consideration of [*Y*]{}(Z) with two free parameters, ${V}_{t}$ and ${\it T}_{c}$. A comparison of the data with the calculations was done for the range 3 $<$ Z $<$ 9, in which minimal systematic errors were expected. [*Figure* ]{}2 shows ${\chi}^{2}$ for comparison of the measured and calculated IMF charge distributions as a function of ${\it V}_{t}$ /${\it V}_{0}$ for different values of the critical temperature. The minimum value of ${\chi}^{2}$ decreases with increasing ${\it T}_{c}$ in the range from 15 to 19 MeV and saturates after that. The corresponding value of ${\it V}_{t}$/${\it V}_{0}$ increases from 2.4 to 2.9. The measured IMF charge distribution is well reproduced by SMM with ${\it V}_{t}$ in the range (2.5–3.0)${\it V}_{0}$, which is close to the value obtained in [@8]. As for the kinetic freeze-out volume, the present value coincides with the one given in [@8], but its uncertainty is only half as much because the estimated systematic error is less:\
${V}_{f}$ = (5.0 ${\pm}$0.5)${\it V}_{0}$.\
Our previous conclusion about the value of the critical temperature [@6; @7] is also confirmed: ${\it T}_{c}$ exceeds 15 MeV. Note, this value is twice as large as estimated in \[17,18\] with the Fisher droplet model. This contradiction is waiting for further efforts to clarify the point.
![Value of ${\chi}^{2}$ as a function of ${\it V}_{t}/{\it V}_{0}$ for comparison of the measured and model predicted IMF charge distributions. The calculations with the INC\*+SMM combined model was performed under the assumption of two free parameters: ${\it V}_{t}$ – the effective volume at the stage of pre-fragment formation, and ${\it T}_{c}$ – the critical temperature for the liquid-gas phase transition.[]{data-label="2"}](pic2.eps "fig:"){width="7.6cm"}\
Comparison of multifragmentation and fission dynamics
=====================================================
The occurrence of two characteristic volumes for multifragmentation has a transparent meaning. The first volume, ${\it V}_{t}$, corresponds to the fragment formation stage at the top of the fragmentation barrier. Here, the properly extended hot target spectator transforms into closely packed pre-fragments. The final channel of disintegration is completed during the evolution of the system up to the moment, when receding and interacting pre-fragments become completely separated at ${\it V}_{f}$. This is just as in ordinary fission. The saddle point (which has a rather compact shape) resembles the final channel of fission having already a fairly well defined mass asymmetry. Nuclear interaction between fission pre-fragments ceases after the descent of the system from the top of the barrier to the scission point. In the papers by Lopez and Randrup [@10; @11] the similarity of the two processes was used to develop a theory of multifragmentation based on a generalization of the transition-state approximation first suggested by Bohr and Wheeler in 1939. The transition states are located at the top of the barrier or close to it. The phase space properties of the transition states are decisive for the further fate of the system, for specifying the final channel. No size parameters are used in the theory.\
![Upper: qualitative presentation of the potential energy of the hot nucleus (with excitation energy ${\it E}_{0}^{*}$) as a function of the system radius. The ground state energy of the system corresponds to [*E*]{}=0. [*B*]{} is the fragmentation barrier, [*Q*]{} is the released energy. Bottom: schematic view of the multifragmentation process and its time scale: ${\it t}_{1}$ – thermalization time, ${\it t}_{2}$ – time of the expansion driven by thermal pressure, ${\it t}_{3}$ – the mean time of the multi-scission point (with dispersion, which is measured as fragment emission time, ${\tau}_{em}$ ).[]{data-label="3"}](pic3.eps "fig:"){width="6.5cm"}\
Being conceptually similar to the approach of [*ref.*]{} [@10; @11], the statistical multifragmentation model (SMM) uses a size parameter that can be determined by fitting to data. The size parameter obtained from the IMF charge distribution can hardly be called a freeze-out volume. This is the “transition state volume”, corresponding to the top of the fragmentation barrier (see [*fig.*]{}3). The freeze-out volume, ${\it V}_{f}$, corresponds to the multi-scission point, when fragments became completely separated and start to be accelerated in the common electric field. In the statistical model (SMM) used, the yield of a given final channel is proportional to the corresponding statistical weight. This means that the nuclear interaction between pre-fragments is neglected when the system volume is ${\it V}_{t}$ and that this approach can be viewed as a simplified transition-state approximation. Nevertheless, the SMM well describes the IMF charge (mass) distributions for thermally driven multifragmentation. Note once again that in the traditional application of the SMM, only one size parameter is used. The shortcoming of such a simplification of the model is obvious now.\
The evidence for the existence of two characteristic multifragmentation volumes changes the\
understanding the time scale of the process (see the bottom of [*fig.*]{}3). One can imagine the following ingredients of the time scale: ${\it t}_{1}$ – the mean thermalizataion time of the excited target spectator, ${\it t}_{2}$ – the mean time of the expansion to reach the transition state, ${\it t}_{3}$ – the mean time up to the multi-scission point.\
The system configuration on the way to the scission point contains several pre–fragments connected by necks. Their random rupture is characterized by the mean time, ${\tau}_{n}$, which seems to be a decisive ingredient of the fragment emission time: ${\tau}_{em}$ ${\approx}$ ${\tau}_{n}$. Formally ${\tau}_{em}$ may be understood as the standard deviation of ${t}_{3}$:\
${\tau}_{em}$ =$(<{t}^{2}_{3}>-<{t}_{3}>^{2})^{1/2}$.\
In the earlier papers, the emission time was related to the mean time of density fluctuations in the system at the stage of fragment formation, at [*t*]{} ${\approx}$ $t_{2}$ [@19].\
What are the expected values of these characteristic times? Thermalization or energy relaxation time after the intranuclear cascade, ${\it t}_{1}$, is model estimated to be (10–20) [*fm/c*]{} [@20; @21]. The Expanding Emitting Source model (EES) predicts $<$${\it t}_{2}$ – ${\it t}_{1}$$>$ ${\approx}$ 70 [*fm/c*]{} for [*p*]{}(8.1GeV)+Au collisions [@22]. The model calculation in [@23] results in estimation of ${\it t}_{3}$ to be (150–200) [*fm/c*]{}. The only measured temporal characteristic is a fragment emission time, ${\tau}_{em}$, which is found in number of papers to be ${\approx}$ 50 [*fm/c*]{} ([*e.g.*]{} see [@9]). It would be very important to find a way to measure the value of ${\it t}_{3}$.\
Note, that in the case of ordinary fission ${\it t}_{2}$ is specified by the fission width ${\Gamma}_{f}$, which corresponds to a mean time of about ${10}^{-19} s$ (or $3.3 {\cdot} {10}^{4}$ [*fm/c*]{}) for an excitation energy of around 100 MeV \[24\]. The value $<$${\it t}_{3}$ – ${\it
t}_{2}$$>$ is model estimated in a number of papers to be about 1000 [*fm/c*]{} [@25]. A mean neck rupture time, considered as a Rayleigh instability, is estimated in [@26] to be:
$${\tau}_{n}=
[1.5{\cdot}{({R_n}/{\it fm})}^3]^{1/2}{\cdot}{10}^{-22} s$$
Generally, the values of ${\tau}_{n}$ are found to be 200–300 [*fm/c*]{} for fission.\
Using (1) for the estimation of the mean time for the rupture of the multi-neck configuration in fragmentation, one gets ${\tau}_{n}$ between 40 and 115 [*fm/c*]{} under the assumption of a neck radius $R_n$ between 1 and 2 [*fm*]{}. This estimation is in qualitative agreement with the measured values of the fragment emission time ${\tau}_{em}$. Evidently, the multifragmentation process is much faster than high energy fission. [*Table*]{} 1 summarizes these estimates.\
[*Table*]{} 1 Characteristic times (in [*fm/c*]{}) for fragmentation and fission; the models used are given in brackets; experimental values are marked (Exp).
${\it t}_1$ ${\it t}_2 $ $<$ ${\it t}_{3}$ – ${\it t}_{2}$ $>$ ${\sigma}({\it t}_{3})$
--------------- ------------- ---------------------- --------------------------------------- -------------------------
Fragmentation 20 80 150 $50{\pm}10$
(UU) (EES) (QMD) (Exp)
Fission $3 {\cdot} {10}^{4}$ $ 2 {\cdot} {10}^{3}$ 200
(Exp) (LD) (RI)
As for the spatial characteristics, the relative elongation of the very heavy systems (Z$>$99) at the fission scission point is similar to that for the multi-scission point of medium heavy nuclei. For the fission of the lighter nuclei, (Po–Ac), the scission elongation is larger [@26].\
A few words about the experimental possibility for finding the total time scale for fragmentation, [*i.e.*]{} the mean value of ${\it t}_{3}$. It can be done by the analysis of the fragment-fragment correlation function with respect to relative angle. But, in contrast to the usual IMF-IMF correlation, one of the detected fragments should be the particle ejected during the thermalization time ${\it t}_{1}$. For the light relativistic projectiles, it may be the pre-equilibrium IMF’s; for heavy-ion induced multifragmentation, a projectile residual (PR) may be used as the trigger related to the initial collision. In the last case the PR–TIMF correlation function should be measured, where TIMF is the intermediate mass fragment from the disintegration of the hot target spectator created via the partial fusion.
**Conclusion**
==============
Thermal multifragmentation of hot nuclei is interpreted as the nuclear [*liquid–fog*]{} phase transition inside the spinodal region. Experimental evidence is presented for the existence of two characteristic volumes for the process: transition state and kinetic freeze-out volumes. This is similar to that for ordinary fission. The dynamics is similar also for the two processes, but multifragmentation is much faster than high energy fission. The IMF emission time is related to the mean rupture time at the multi-scission point, which corresponds to the freeze-out configuration.\
The authors are grateful to A. Hrynkiewicz, A.I. Malakhov, A.G. Olchevsky for support and to I.N. Mishustin and W. Trautmann for illuminating discussions. The research was supported in part by the Russian Foundation for Basic Research, Grant ¹ 06-02-16068, the Grant of the Polish Plenipotentiary to JINR, Bundesministerium für Forschung und Technologie, Contract [*No*]{} 06DA453.
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[^1]: Email address: [email protected]
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---
abstract: |
This paper is concerned with nonlinear filtering of the coefficients in asset price models with stochastic volatility. More specifically, we assume that the asset price process $
S=(S_{t})_{t\geq0}
$ is given by $$dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t},$$ where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and $\theta=(\theta_{t})_{t\geq0}$ is a cádlág strong Markov process. The random process $\theta$ is unobservable. We assume also that the asset price $S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<\ldots.$ This is an appropriate assumption when modelling high frequency financial data (e.g., tick-by-tick stock prices).
In the above setting the problem of estimation of $\theta$ can be approached as a special nonlinear filtering problem with measurements generated by a multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While quite natural, this problem does not fit into the standard diffusion or simple point process filtering frameworks and requires more technical tools. We derive a closed form optimal recursive Bayesian filter for $\theta_{t}$ , based on the observations of $(\tau_{k},\log
S_{\tau_{k}})_{k\geq1}$. It turns out that the filter is given by a recursive system that involves only deterministic Kolmogorov-type equations, which should make the numerical implementation relatively easy.
address:
- 'Caltech, M/C 228-77, 1200 E. California Blvd. Pasadena, CA 91125, USA.'
- 'Department of Electrical Engineering-Systems, Tel Aviv University, 69978 Tel Aviv, Israel'
- 'Department of Mathematics, University of Southern California, Los Angeles, CA 90089-1113, USA '
author:
- Jakša Cvitanić
- Robert Liptser
- Boris Rozovskii
title: A filtering approach to tracking volatility from prices observed at random times
---
[^1] [^2]
**Introduction** {#Sec-1}
================
In the classical Black-Scholes model for financial markets, the stock price $S_{t}$ is modelled as a Geometric Brownian motion, that is, with diffusion coefficient equal to $\sigma S_{t}$, where “volatility” $\sigma$ is assumed to be constant. The volatility parameter is the most important one when it comes to option pricing; consequently, many researchers have generalized the constant volatility model to so-called stochastic volatility models, where $\sigma_{t}$ is itself random and time dependent. There are two basic classes of models: complete and incomplete. In complete models, the volatility is assumed to be a functional of the stock price; in incomplete models, it is driven by some other source of noise that is possibly correlated with the original Brownian motion. In this paper we study a particular incomplete model in which the volatility process is independent of the driving Brownian motion process. This has the economic interpretation of the volatility being influenced by market, political, financial, and other factors that are independent of the “systematic risk” (the Brownian motion process) associated with the particular stock price under study. It is also close in spirit to the way traders think about volatility – as a parameter that changes with time and whose future value in a given period of interest has to be estimated/predicted. They need an estimate of the volatility to decide how they will trade in financial markets, especially derivatives markets. In fact, the notion of volatility is so important to traders that they even quote option prices in volatility units rather than in dollars (or some other currency). Investment banks also depend on modelling future volatility in order to price custom-made financial products, whose payoff depends on the future path of the underlying stock price. Recently, new contracts have been developed that directly trade the volatility itself (volatility swaps, for example). We plan to address the issue of pricing options within the framework of our model in future research.
Estimating volatility from observed stock prices is not a trivial task in either complete or incomplete models, in part because the prices are observed at discrete, possibly random time points. Since volatility itself is not observed, it is natural to apply filtering methods to estimate the volatility process from historical stock price observations. Nevertheless, this has only recently been investigated in continuous-time models, in particular by Frey and Runggaldier [@FR]. See Runggaldier [@R] for an up-to-date survey. See also Elliott et al [@EHJ] for a discrete-time approach with equally spaced observations, Gallant and Tauchen [@GT] for an approximating algorithm in continuous time, Malliavin and Mancino [@MM] for a nonparametric approach, as well as Fouque et al. [@FPS], Rogers and Zane [@RZ], and Kallianpur and Xiang [@KX] for still other approaches. There is also a rich econometrics, time-series literature on ARCH-GARCH models of stochastic volatility, that presents an alternative way to model and estimate volatility; see Gourieroux [@Gou] for a survey.
Our paper was prompted by Frey and Runggaldier [@FR]. Like that paper, we assume that the asset price process $S=(S_{t})_{t\geq0}$ is given by $$dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t},$$ where $B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function, and $\theta=(\theta_{t})_{t\geq0}$ is a cádlág strong Markov process. The “volatility” process $\theta$ is unobservable, while the asset price $S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<\ldots$ This assumption is designed to reflect the discrete nature of high frequency financial data such as tick-by-tick stock prices. The random time moments $\tau_{k}$ can be interpreted as instances at which a large trade occurs or at which a market maker updates his quotes in reaction to new information (see Frey [@F] ). Hence, it is natural to assume that $(\tau_{k})_{k\geq1}$ might also be correlated with $\theta.$
In the above setting the problem of volatility estimation can be regarded as a special nonlinear filtering problem.
Frey and Runggaldier [@FR] derive a Kallianpur-Striebel type formula (see e.g. [@KalStr]) for the optimal mean-square filter for $\theta_{t}$ based on the observations of $S_{\tau_{1}},S_{\tau_{2}},...$ for all $\tau_{k}\leq t$ and investigate Markov Chain approximations for this formula. We extend this result in that we derive the exact filtering equations for $\theta_{t}$ that allow us to compute the conditional distribution of $\theta_{t}$ given $S_{\tau_{1}\wedge t}$, $S_{\tau_{2}\wedge t}$,…. Moreover, our framework includes general random times of observations, not just doubly stochastic Poisson processes.
We remark that while being natural, the Frey and Runggaldier model adopted in this paper does not quite fit into the standard diffusion or simple point process filtering frameworks (cf. [@LSII], [@KrZa], [@Roz1]) and requires more technical tools. In particular, the general filtering theory for diffusion processes requires that the diffusion coefficient of the observation process does not depend on the state process, while in our case the presence of $\theta_{t}$ in the diffusion coefficient is crucial. The standard" filtering theory for point processes is also not applicable in the present setting since the observation process $(\tau_{i},S_{\tau_{i}})_{i\geq1}$ is a multivariate process (see also Remark \[r1\]).
It turns out that the resulting filtering equations are simpler than their counterparts in the case of continuous observations. In the latter case, the nonlinear filters are described by infinite dimensional stochastic differential equations. For example, if $\theta_{t}$ is a diffusion process, the filtering equations (e.g., Kushner filter or Zakai filter) are given by stochastic partial differential equations (see, e.g., [@Roz1]). In contrast, in our setting, the filtering equation can be reduced to a recursive system of linked *deterministic* equations of Kolmogorov’s type. Therefore, the numerical implementation of the filter is much simpler (see the follow up paper [@CRZ]).
We describe the model in Section 2, state the main results and examples in Section 3, provide the proofs in Section 4, and present more detailed examples in Section 5.
**Mathematical model** {#Sec-2}
======================
Risky asset and observation times
---------------------------------
Let us fix a probability space $(\Omega,\mathcal{F},\mathsf{P})$ equipped with a filtration $\mathbf{F}=(\mathcal{F}_{t})_{t\geq0}$ that satisfies the usual conditions (see, e.g. [@LSMar]). All random processes considered in the paper are assumed to be defined on $(\Omega,\mathcal{F},\mathsf{P})$ and adapted to $\mathbf{F}$.
It is assumed that there is a risky asset with the price process $S=(S_{t})_{t\geq0}$ given by the Itô equation $$\label{2.1b}
dS_{t}=r(\theta_{t})S_{t}dt+v(\theta_{t})S_{t}dB_{t},$$ where $B=(B_{t})_{t\geq0}$ is a standard Brownian motion and $\theta =(\theta_{t})_{t\geq0}$ is a cádlág Markov jump-diffusion process in $\mathbb{R}$ with the generator $\mathcal{L}$. To simplify the discussion, it is assumed that $r(x)$ and $v(x)$ are measurable bounded functions on $\mathbb{R}$, the initial condition $S_{0}$ is constant, and $v(x)$ and $S_{0}$ are positive.
The process $(\theta_{t})_{t\geq0}$ is referred to as the *volatility process*. It is unobservable, and the only observable quantities are the values of the log-price process $X_{t}=\log S_{t}$ taken at stopping times $(\tau_{k})_{k\geq0}$, so that $\tau_{0}=0,\tau_{k}<\tau_{k+1}$ if $\tau _{k}<\infty,$ and $\tau_{k}$ $\uparrow\infty$ as $k\uparrow\infty.$
In accordance with , the log-price process is given by $$X_{t}=\int_{0}^{t}\Big(r(\theta_{s})-\frac{1}{2}v^{2}(\theta_{s}
)\Big)ds+\int_{0}^{t}v(\theta_{s})dB_{s}.$$ For notational convenience, set $X_{k}:=X_{\tau_{k}}.$ Thus, the observations are given by the sequence $(\tau_{k},X_{k})_{k\geq0}$.
\[rem-1\] [(Note on the reading sequence.) The reader interested primarily in applying our results to real data can focus her attention on Example \[examplechain\], which appears to be the most practical model to work with. That example provides self-contained formulas for estimating the conditional [(]{}filtering[)]{} distribution of the volatility process. We report on the numerical results related to this example in the follow-up paper [@CRZ].]{}
Clearly, the observation process $(\tau_{k},X_{k})_{k\geq0}$ is a multivariate (marked) point process (see, e.g. [@JS], [@Last]) with the counting measure $$\mu(dt,dy)=\sum_{k\geq1}\mathbf{I}_{\{ \tau_{k}<\infty\} }
\delta_{{\{\tau_{k},X_{k}\}}}(t,y)dtdy,$$ where $\delta_{{\{\tau_{k},X_{k}\}}}$ is the Dirac delta-function on $\mathbb{R}_{+}\times\mathbb{R}$.
We introduce two filtrations related to $(\tau_{k},X_{k})_{k\ge0}$: $(\mathcal{G}(n))_{n\ge0}$ and $(\mathcal{G}_{t})_{t\ge0}$, where
- $\mathcal{G}(n):=\sigma\{(\tau_{k},X_{k})_{k\leq n}\}$,
- $\mathcal{G}_{t}:=\sigma(\mu([0,r]\times\Gamma):r\leq
s,\Gamma\in \mathcal{B}(\mathbb{R})),$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R.}$
It is a standard fact (see III.3.31 in [@JS]) that $$\mathcal{G}_{\tau_{k}}=\mathcal{G}(k),\ k=0,1\ldots$$ and $\{\tau_{k}\}$ is a system of stopping times with respect to $(\mathcal{G}_{t})_{t\geq0}$.
\[r1\] [Although $\mathcal{G}_{\tau_{k}}$ contains all the relevant information carried by the observations obtained up to time $\tau_{k},$ the filtration $\big(\mathcal{G}_{t}\big)_{t\geq 0}$ provides additional information between the observation times. To elucidate this point on a more intuitive level, we note that the length of the time elapsed between $\tau_{k}$ and $\tau_{k+1}$carries additional information about the state of $\theta_{t}$ after $\tau_{k}.$ Specifically, if the frequency of observations is proportional to the stock’s volatility $v(\theta_{t})$, $t\in[\hskip-.015in[\tau_{k},\tau_{k+1}]\hskip-.015in]$ , the larger values of $t-\tau_{k}$ might indicate lower values of $v(\theta_{t})$.]{}
Volatility process
------------------
A more precise description of the volatility process is in order now. Let $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ and $(\mathbb{R}_{+}\times
\mathbb{R},\mathcal{B}(\mathbb{R}_{+})\otimes\mathcal{B}(\mathbb{R}))$ be measurable spaces with Borel $\sigma$-algebras. The volatility process $\theta=(\theta_{t})_{t\geq0}$ is defined by the Itô equation $$d\theta_{t}=b(t,\theta_{t})dt+\sigma(t,\theta_{t})dW_{t}+\int_{\mathbb{R}
}u(\theta_{t-},x)(\mu^{\theta}-\nu^{\theta})(dt,dx),
\label{2.1}$$ where $W_{t}$ is a standard Wiener process and $\mu^{\theta}=\mu^{\theta}(dt,dx)$ is a Poisson measure on $\big(
\mathbb{R}_{+}\times\mathbb{R},\mathcal{B}(\mathbb{R}_{+})
\otimes\mathcal{B}(\mathbb{R})\big) $ with the compensator $\nu^{\theta}(dt,dx)=K(dx)dt$, where $K(dx)$ is a $\sigma-$finite non-negative measure on $\big(
\mathbb{R},\mathcal{B}(\mathbb{R})\big) $. We assume that $E\theta_{0}^{2}<\infty$, the functions $b(t,z),\sigma(t,z),$ and $u(z,x)$ are Lipschitz continuous in $z$ uniformly with respect to other variables, and $$|b(t,z)|+|\sigma(t,z)|^{2}+\int_{\mathbb{R}}|u(z,x)|^{2}K(dx)\leq
C(1+|z|^{2}).$$ It is well known that under these assumptions possesses a unique strong solution adapted to $\mathbf{F}$, and $E\theta_{t}^{2}<\infty$ for any $t\geq0$.
The generator $\mathcal{L}$ of the volatility process is given by $$\begin{gathered}
\mathcal{L}f(x):=b(t,x)f^{\prime}(x)+\frac{1}{2}\sigma^{2}(t,x)f^{\prime
\prime}(x)\label{2.10a}\\
+\int_{\mathbb{R}}\Big(f(x+u(x,y))-f(x)-f^{\prime}
(x)u(x,y)\Big)K(dy).\nonumber\end{gathered}$$
Before proceeding with the assumptions and main results we shall introduce additional notation. Set $$m(s,t)=\int_{s}^{t}\left(
r(\theta_{u})-\frac{1}{2}v^{2}(\theta_{u})\right)du,$$ and $$\sigma^{2}(s,t)=\int_{s}^{t}v^{2}(\theta_{u})du.$$ For simplicity, it is assumed that $v^{2}(s,t)$ is bounded away from zero. Let us denote by $\rho_{s,t}(y)$ the density function of the normal distribution with mean $m(s,t)$ and the variance $\sigma^{2}(s,t)$: $$\rho_{s,t}(y):=\frac{1}{\sqrt{2\pi}\sigma(s,t)}e^{-\frac{(y-m(s,t))^{2}
}{2\sigma^{2}(s,t)}}.
\label{rho}$$ Clearly, $\rho$ is the conditional density of the stock’s log-increments $X_{t}-X_{s}$ given $\theta$.
Let $\mathcal{F}^{\theta}=(\mathcal{F}_{t}^{\theta})_{t\geq0}$ be the right-continuous filtration generated by $(\theta_{t})_{t\geq0}$ and augmented by $\mathsf{P}$-zero sets from $\mathcal{F}$. Denote by $G_{k}^{\theta}$ the conditional distribution of $\tau_{k+1}$with respect to[^3] $\mathcal{F}^{\theta}\vee\mathcal{G}(k).$ That is, $G_{k}^{\theta}$ is the distribution of the time of the next observation, given previous history, and given $\theta$: $$G_{k}^{\theta}(dt) =\mathsf{P}\big(\tau_{k+1}\in
dt|\mathcal{F}^{\theta}\vee\mathcal{G}(k)\big).
\label{Gktheta}$$ Without loss of generality we can and will assume that $G_{k}^{\theta}(dt) $ is the regular version of the right hand side of (\[Gktheta\]).
Let $N=(N_{t})_{t\geq0}$ be the counting process with interarrival times: $\tau_0=0$, $(\tau_{k}-\tau_{k-1})_{k\geq1},$ that is $$N_{t}=\sum_{k\geq1}I(\tau_{k}\leq t).$$
Assumptions
-----------
The following assumptions will be in force throughout the paper:
A.0: For every $\mathcal{G}$-predictable and a.s. finite stopping time $S$, $$\mathsf{P}(N_{S}-N_{S-}\neq0 |\mathcal{G}_{S-})=0 \ {\rm or} \ 1.$$
A.1: The Brownian motion $B$ is independent of $\big(\theta,N\big)$.
A.2: For every $k$, there exists a $\mathcal{G}(k)$-measurable integrable random measure $\Phi_{k}$ on $\mathcal{B}(\mathbb{R}_{+})$, so that, for almost all $\omega \in\Omega,$ $\Phi_{k}\big([
0,\tau_{k}(\omega)]\big) =0$ and $G_{k}^{\theta}$ is absolutely continuous with respect to $\Phi_{k}$ $.$
Denote by $\phi(\tau_{k},t)=\phi(\theta,\tau_{k},t)$ the Radon-Nikodym derivative of $G_{k}^{\theta}(dt)$ with respect to $\Phi_{k}(dt),$ i.e. for almost every $\omega,$ $$\phi(\tau_{k},t):=\frac{dG_{k}^{\theta}\big((\tau _{k},t]\big)}
{d\Phi_{k}\big((\tau_{k},t]\big)}.
\label{fika}$$
Assumption A.0 is not essential for the derivation of the filter. However, under this assumption the structure of the optimal filter is simpler, and in the practical examples important for this paper, this assumption holds anyway. In particular, A.0is verified if the conditional distribution $
G_{k}^{\theta}=\mathsf{P}\big(\tau_{k+1}\leq
t|\mathcal{F}^{\theta} \vee\mathcal{G}(k)\big)
$ is absolutely continuous with respect to the Lebesgue measure[^4] or if the arrival times $\tau_{k}$ are non-random.
The following two simple but important examples illustrate the assumption A.2.
\[ex:cox copy(1)\] Let ($\tau_{k})_{k\geq0}$ be the jump times of a doubly stochastic Poisson process (Cox process) with the intensity $n(\theta_{t}).$ In this case, $$\mathsf{P}\big(\tau_{k+1}\leq
t|\mathcal{F}^{\theta}\vee\mathcal{G}(k)\big )=
\begin{cases}
1-e^{-\int_{\tau_{k}}^{t}n(\theta_{s})ds} & ,t\geq\tau_{k}\\
0 & ,\text{otherwise}.
\end{cases}$$ Then, one can take $\Phi_{k}(ds)=ds$ and $\phi(\tau _{k},s)=n(\theta_{t})\exp\big(-\int_{\tau_{k}}^{s}n(\theta _{u})du\big)$. If $n(\theta_{t})=n$ is a constant, one could also choose $$\Phi_{k}(ds)=n\exp\big\{n(\tau_{k}-s)\big\}ds \quad\text{and}\quad
\phi(\tau_{k},s)=1.$$
\[ex:constep0\] If the filtering is based on non-random observation times $\tau_{k}$ (e.g., $\tau_{k}=kh$ where $h$ is a fixed time step) then a natural choice would be $$\Phi_{k}(ds)=\delta_{\{\tau_{k+1}\}}(s)ds \ \text{and} \ \phi (\tau_{k},s)=1.$$
For practical purposes, $\Phi_{k}(ds)$ must be known or easily computable as soon as the the observations $(
\tau_{i},X_{i})_{i\leq k}$ become available. In contrast, the Radon-Nikodym density $\phi(\tau_{k}) $ is, in general, a function of the volatility process and is subject to estimation.
We note that A.2 could be weakened slightly by replacing $G_{k}^{\theta}$ by a regular version of the conditional distribution of $\tau_{k+1}$with respect to $\mathcal{F}_{\tau_{k+1}-}^{\theta}\vee\mathcal{G}(k).$ The latter assumption would make the proof a little bit more involved and we leave it to the interested reader.
**Main results and introductory examples** {#Sec-3}
==========================================
Main result {#sec-3.1ab}
-----------
For a measurable function $f$ on $\mathbb{R}$ with $E|f(\theta_{t})|<\infty,$ define the conditional expectation estimator $\pi_{t}(f)$ by $$\pi_{t}(f):=E\big(f(\theta_{t})|\mathcal{G}_{t}\big)=\int_{\mathbb{R}}
f(z)\pi_{t}(dz),
\label{eq:p0}$$ where $\pi_{t}(dz):=d\mathsf{P}(\theta_{t}\leq z|\mathcal{G}_{t})$ is the filtering distribution. (Note that we omit the argument $\theta_{t}$ of $f$ in the estimator $\pi_{t}(f)$). In the spirit of the Bayesian approach, it is assumed that the a priori distribution $$\pi_{0}(dx)=\mathsf{P}\big(\theta_{0}\in dx\big)$$ is given.
Let $\sigma\{\theta_{\tau_{k}}\}$ be the $\sigma$-algebra generated by $\theta_{\tau_{k}}$. For $t>\tau_{k}$, let us define the following *structure functions*:
$$\psi_{k}(f;t,y,\theta_{\tau_{k}}):=E\Big(f(\theta_{t})\rho_{{\tau_{k},t}
}(y-X_{k})\phi(\tau_{k},t)\big|\sigma\big\{\theta_{\tau_{k}}\big\}\vee
\mathcal{G}(k)\Big),
\label{nado}$$
and its integral with respect to $y$ $$\overline{\psi}_{k}(f;t,\theta_{\tau_{k}}):=\int_{\mathbb{R}}\psi_{k}\big(
f;t,y,\theta_{\tau_{k}}\big)
dy=E\Big(f(\theta_{t})\phi(\tau_{k}
,t)\big|\sigma\big\{\theta_{\tau_{k}}\big\}\vee\mathcal{G}(k)\Big),
\label{nado1}$$ where $\rho$ and $\phi$ are given by (\[rho\]) and (\[fika\]), respectively. If $f\equiv1$, the argument $f$ in $\psi$ and $\bar{\psi}$ is replaced by $1.$
Write $$\Phi_{k}(\{\tau_{k+1}\}):=\int_{0}^{\infty}I(t=\tau_{k+1})\Phi_{k}(dt),$$ i.e. $\Phi_{k}(\{\tau_{k+1}\})$ is the jump of $\Phi_{k}(dt)$ at $\tau_{k+1}$.
Finally, for $t\geq\tau_{k}$ and a bounded function $f$, define $$\mathcal{M}_{k}(f;t,\pi_{t})
:=\frac{\pi_{\tau_{k}}( \bar{\psi}_{k}(f;t)) -\pi_{t-}(f)\pi_{\tau_{k}}
(\bar{\psi}_{k}(1;t))}{\int_{t}^{\infty}\pi_{\tau_{k} }(
\bar{\psi}_{k}(1;s))\Phi_{k}(ds)},$$ whenever the denominator is not zero, and $\mathcal{M}_{k}(f;t,\pi_{t})=0$ if the denominator is zero.
The main result of this paper is as follows:
\[mainthm\] Assume A.0-A.2. Then for every measurable bounded function $f$ in the domain of the generator $\mathcal{L}$ such that $\int_{0} ^{t}E|\mathcal{L}f(\theta_{s})|ds<\infty$ for any $t\geq0,$ the following system of equations holds[:]{}
[1)]{} For every $k=0,1\ldots,$
$$\pi_{\tau_{k+1}}(f)=\frac{\pi_{\tau_{k}}(\psi_{k}(f;t,y))}{\pi_{\tau_{k}}
(\psi_{k}(1;t,y))}_{\big\{\substack{t=\tau_{k+1}\\y=X_{k+1}}
\big\}}-\mathcal{M}_{k}(f;t,\pi_{t})
_{_{\{t=\tau_{k+1}\}}}\cdot\Phi(\{\tau_{k+1}\}).
\label{eq:jump}$$
[2)]{} For every $k=0,1\ldots$ and $t\in]\hskip-1.5pt]\tau_{k},\tau _{k+1}[\hskip-1.5pt[$, $$d\pi_{t}(f)=\pi_{t}(\mathcal{L}f)dt-\mathcal{M}_{k}(f;t,\pi_{t})\Phi_{k}(dt).
\label{eq:cont}$$
**Remarks**
-----------
- 1. Equations (\[eq:jump\]), (\[eq:cont\]) form a closed system of equations for the filter $\pi_{t}(f)$. It is often convenient and customary (see e.g. [@Roz1], [@Roz2] and the references therein) to write a differential equation for a measure-valued process $H_{t}(dx)$ in its variational form, i.e. as the related system of equations for $H_{t}(f)$ for all $f$ from a sufficiently rich class of test functions belonging to the domain of the operator $\mathcal{L}.$ In our setting, such a reduction to the variational form is a necessity, since in some cases the filtering measure $\pi_{s}(dx)=\mathsf{P}(\theta_{s}\in dx|\mathcal{G}_{s})
$may not belong to the domain of $\mathcal{L}$. However, in the important examples discussed below, there is no need to resort to the variational form. The interested reader who is unaccustomed to the variational approach might benefit from looking first into the examples at the end of this section and in Section \[Sec-Ex\], where the filtering equations are written as equations for posterior distributions.
- 2. The system (\[eq:jump\]) simplifies considerably if $$\label{m0}
\mathcal{M}_{k}(f;t,\pi_{t})
_{_{\{t=\tau_{k+1}\}}}\cdot\Phi(\{\tau_{k+1}\})=0, \ \text{\rm for all }k.$$ Obviously, (\[m0\]) holds if for all $k,$ $\Phi_{k}(dt)$ is continuous at $t=\tau_{k+1}$ as in the case when $N_{t}$ is a Cox process. In fact, (\[m0\]) holds true in many other interesting cases, even when $\Phi_{k}(dt)$ has jumps at all $\tau_{k+1}$, as in the case of fixed observation intervals (see Example \[ex:constep\] below). We note then that the following *separation principle* holds.
\[cor:sp\] Assume [(\[m0\])]{}. Then the filtering at the observation times $\{\tau_{k}\}_{k\geq1}$ does not require filtering between them; it is done by the Bayes type recursion[:]{} $$\pi_{\tau_{k+1}}(f)=\frac{\pi_{\tau_{k}}(\psi_{k}(f;t,y))}{\pi_{\tau_{k}}
(\psi_{k}(1;t,y))}_{\big\{\substack{t=\tau_{k+1}\\y=X_{k+1}}\big\}}.$$
- 3. Note that for high-frequency observations, even if condition (\[m0\]) is not met, for all practical purposes, it may suffice to compute the volatility estimates only at the observation times. In that case, one would only use the relatively simple recursion formula (\[eq:jump\]), and disregard equation (\[eq:cont\]).
- 4. Clearly, the structure functions“ $\psi$ and $\bar {\psi}$ are of paramount importance for computing the posterior distribution of the volatility process. We would like to stress that these do not involve the observations and could be pre-computed off-line” using just the *a priori* distribution. Then, on-line", when the observations become available, one needs only to plug in the obtained measurements $(\tau_{k},X_{k}),$ and to compute $\pi_{t}(f)$ by recursion$.$ This feature is important for developing efficient numerical algorithms.
- 5. Note also that for almost every $\omega\in\Omega,$ filtering equation (\[eq:cont\]) is a *linear deterministic* equation of Kolmogorov’s type, rather than a *nonlinear stochastic* partial differential equation. The latter is typical of the nonlinear filtering of diffusion processes. The well-posedness and the regularity properties of equation (\[eq:cont\]) are well researched in the literature on second order parabolic deterministic integro-differential equations (see e.g. [@LM], [@MP], [@SK] and the references therein).
\[examplechain\] (*Volatility as a Markov Chain*.) Let us now assume that the counting process is a Cox process with intensity $n(\theta_{t})$, and take $\phi(\tau_{k},s)=n(\theta_{t})e^{-\int_{\tau_{k}
}^{s}n(\theta_{u})du}$ and $\Phi_{k}(ds)=ds.$ Also assume $\theta=(\theta_{t})_{t\leq T}$ is a homogeneous Markov jump process taking values in the finite alphabet $\{a_{1},\ldots,a_{M}\}$ with the intensity matrix $\Lambda=\|\lambda(a_{i},a_{j})\|$ and the initial distribution $p_{q}=\mathsf{P}(\theta_{0}=a_{q}),\ q=1,\ldots,M$. (This is one of the two models of the state process discussed in [@FR].) In this case, $$\mathcal{L}f(\theta_{s})=\sum_{j}\lambda(\theta _{s},a_{j})f(a_{j}).$$ Denote by $\theta_{t\text{ }}^{j}$ the process $\theta_{t\text{
}}$ starting from $a_{j}$, and $$\begin{gathered}
p_{ji}(t):=\mathsf{P}\big(\theta_{t}=a_{i}|\theta_{0}=a_{j}\big), \quad
\pi_{j}(t)=\mathsf{P}\big(
\theta_{t}=a_{j}\big|\mathcal{G}_{t}\big),
\\
r_{ji}\left( t,z\right)
:=E\big(e^{-\int_{0}^{t}n(\theta_{u}^{j})du}
\rho_{_{0,t}}^{j}(z)|\theta_{t}^{j}=a_{i}\big),\end{gathered}$$ where $\rho_{_{0,t}}^{j}(z)$ is obtained by substituting $\theta_{s}^{j}$ for $\theta_{s}$ in $\rho_{_{0,t}}(z).$ It follows from Theorem \[mainthm\] (for details see Example \[exs1\] ), with $f(\theta_{t})
:=I_{\{\theta_{t}=a_{i}\}},$ that $$\pi_{i}(\tau_{k})=\frac{n(a_{i})
\sum_{j}r_{ji}(\tau _{k}-\tau_{k-1},X_{k}-X_{k-1})
p_{ji}(\tau_{k}-\tau _{k-1})
\pi_{j}(\tau_{k-1})}{\sum_{i,j}n(a_{i})r_{ji}(
\tau_{k}-\tau_{k-1},X_{k}-X_{k-1})p_{ji}(\tau
_{k}-\tau_{k-1})\pi_{j}(\tau_{k-1})}.
\label{chainTk}$$ This recursion can be easily computed, once one computes off-line the values $r_{ij}$. This example is also treated in more detail in Section \[Sec-Ex\].
**Proofs** {#Section3}
==========
In the proof of the main result we want to show that $$d\pi_{t}(f) =\pi_{t}(\mathcal{L}f)dt +dM_{t},$$ where $M_{t}$ is a martingale, and then we find a (integral) martingale representation of $M_{t}$ with respect to the measure $\mu-\nu$, where $\nu$ is a compensator of $\mu$. We first find the compensator.
${\mbox{\boldmath $(\mathcal{G}_{t})$}}$-compensator of ${\mbox{\boldmath $\mu$}}$ {#subsec-3}
----------------------------------------------------------------------------------
Denote by $\mathcal{P}(\mathcal{G})$ be the predictable $\sigma$-algebra on $\Omega\times[0,\infty)$ with respect to $\mathcal{G}$ and and set $$\widetilde{\mathcal{P}}(\mathcal{G})=\mathcal{P}(\mathcal{G})\otimes
\mathcal{B}(\mathbb{R}).$$
A nonnegative random measure $\nu(dt,dy)$ on $\widetilde{\mathcal{P} }(\mathcal{G})$ is called a $\widetilde{\mathcal{P}}(\mathcal{G})$ -compensator of $\mu$ if for any $\widetilde{\mathcal{P}}(\mathcal{G})$-measurable, nonnegative function $\varphi(t,y)=\varphi(\omega,t,y)$, $$\begin{split}
\mathrm{(i)}\quad &
\int_{0}^{t}\int_{\mathbb{R}}\varphi(s,y)\nu(ds,dy)
\ \text{is ${\mathcal{P}}(\mathcal{G})$-measurable}\\
\mathrm{(ii)}\quad & E\int_{0}^{\infty}\int_{\mathbb{R}}\varphi
(t,y)\mu(dt,dy)=E\int_{0}^{\infty}\int_{
\mathbb{R}}\varphi(t,y)\nu(dt,dy).
\end{split}$$ Let $
G_{k}(ds,dx)=G_{k}(\omega,ds,dx)
$ be a regular version of the conditional distribution of $
\big(
\tau_{k+1} ,X_{k+1}\big) $ given $\mathcal{G}\left( k\right)
$ ( it is assumed that $G_{k}([0,\tau_{k}],dx)=0$): $$\begin{aligned}
\label{GGG}
\mathsf{G}_{k}(dt,dy)=d\mathsf{P}\big(\tau_{k+1}\le
t,X_{k+1}\le y|\mathcal{G}(k)\big).\end{aligned}$$ Denote $G_{k}(ds)=G_{k}(dt,\mathbb{R}),$ that is, $G_{k}(t)=\mathsf{P}(\tau_{k+1}\leq
t~|~\mathcal{G}(k))$ (with probability one).
By Theorem III.1.33 [@JS] (see also Proposition 3.4.1 in [@LSMar]), $${\nu}(dt,dy)=\sum_{k\geq0}I_{]\hskip-1.5pt]\tau_{k},\tau_{k+1}]\hskip-1.5pt]}
(t)\frac{G_{k}(dt,dy)}{G_{k}([t,\infty),\mathbb{R})},$$ We now derive a representation, suitable for the filtering purposes, of the $\widetilde{\mathcal{P}}(\mathcal{G})$-compensator $\nu$ in terms of the structure functions (\[nado\]), (\[nado1\]), and the posterior distribution of $\theta\ $.
\[lem-4.1\] The $\widetilde{\mathcal{P}}(\mathcal{G})-$compensator $\nu$ admits the following version[:]{} $$\nu(dt,dy)=\sum_{k\geq0}I_{]\hskip-1.5pt]\tau_{k},\tau_{k+1}]\hskip-1.5pt]}
(t)\frac{\pi_{\tau_{k}}(\psi_{k}(1;t,y))}{\int_{t}^{\infty}\pi_{\tau_{k}
}(\overline{\psi}_{k}(1;s))\Phi_{k}(ds)}\Phi_{k}(dt)dy.
\label{eq:nu}$$
By A.1 for $t>\tau_{k}$, with probability 1,
$$\begin{split}
& \mathsf{P}\big(\tau_{k+1}\leq t,X_{k+1}\leq
y|\mathcal{F}^{\theta}\vee\mathcal{G}(k)\big)
\\
&=E\Big(\mathsf{P}\big(\tau_{k+1}\leq t,X_{k+1}\leq
y|\mathcal{F}^{\theta }\vee\mathcal{G}(k)
\vee\sigma(\tau_{k+1})
\big)\big|\mathcal{F}^{\theta}\vee\mathcal{G}(k)\Big)
\\
& =E\Big(I_{(\tau_{k+1}\leq t)}\mathsf{P}\big(
X_{k+1}\leq y|\mathcal{F}^{\theta}\vee\mathcal{G}\left( k\right)
\vee \sigma\left( \tau_{k+1}\right) \big)
|\mathcal{F}^{\theta}\vee\mathcal{G}(k)\Big)
\\
&=E\left(I_{(\tau_{k+1}\leq t)}\int_{-\infty}^{y}\rho _{\tau_{k},\tau_{k+1}}(
z-X_{k})dz|\mathcal{F}^{\theta}
\vee\mathcal{G}(k)\right)
\\
& =\int_{\tau_{k}}^{t}\int_{-\infty}^{y}\rho_{\tau_{k},s}(z-X_{k}
)dzG_{k}^{\theta}(ds) ,
\end{split}$$
where we recall that $G_{k}^{\theta}$ is a regular version of the conditional distribution of $\tau_{k+1}$with respect to $\mathcal{F}^{\theta} \vee\mathcal{G}(k).$ Thus, by A.2, for $t>\tau_{k}$, with probability 1,
$$\begin{gathered}
\label{eq:sep}
\mathsf{P}\big(\tau_{k+1}\leq t,X_{k+1}\leq
y|\mathcal{F}^{\theta}
\vee\mathcal{G}(k)\big)
\nonumber\\
=\int_{\tau_{k}}^{t}\int_{-\infty}^{y}\rho_{\tau_{k},s}(
z-X_{k})\phi(\tau_{k},s) dz\Phi_{k}(ds).\end{gathered}$$
By (\[nado\]), using notation (\[eq:p0\]), we see that $$E\big(E\big[\phi(\tau_{k},s)\rho_{\tau_{k},s}(z-X_{k})|\sigma\{\theta
_{\tau_{k}}\}\vee\mathcal{G}(k)
\big]|\mathcal{G}(k)
\big)=\pi_{\tau_{k}}(\psi_{k}(1;s,z)).$$ This, together with (\[eq:sep\]), yields, recalling definition (\[GGG\]), $$G_{k}\big(ds,dz)=\pi_{\tau_{k}}(\psi_{k}(1;s,z))\Phi_{k}(ds)dz.$$ In the same way, for $t>\tau_{k}$, with probability 1,
$$G_{k}\big([t,\infty],\mathbb{R})=\int_{t}^{\infty}\pi_{\tau_{k}}
(\overline{\psi}_{k}(1;s))\Phi_{k}(ds).
\label{eq:denom}$$
This completes the proof.
\[re:den0\] [If the right hand of (\[eq:denom\]) is zero, then $
\mathsf{P}\big(\tau_{k+1}\geq t|\mathcal{G}(k)\big)=0.
$ Hence, $I_{]\hskip-1.5pt]\tau_{k},\tau_{k+1}]\hskip-1.5pt]}(t)=0$ with probability 1 and, by the $\ 0/0=0$ convention, the corresponding term in (\[eq:nu\]) is zero.]{}
Semimartingale representation of the optimal filter {#sec-4}
---------------------------------------------------
In this section we will prove the following result.
\[theo-4.1\] For any bounded function $f$ from the domain of the operator $\mathcal{L}$ such that $\int_{0}^{t}E|\mathcal{L}f(\theta_{s})|ds<\infty$ for all $t<\infty$, the differential of the optimal filter $\pi_{s}(f)$ is given by equation $$\begin{aligned}
\label{filteq}
d\pi_{s}(f) & =\pi_{s}(\mathcal{L}f)ds
\\
&
+\int_{\mathbb{R}}\Big(\sum_{k\geq0}I_{]\hskip-1.5pt]\tau_{k},\tau
_{k+1}]\hskip-1.5pt]}(s)\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}{\pi_{\tau_{k}
}(\psi_{k}(1;s,y))}-\pi_{s-}(f)\Big)(\mu-\nu)(ds,dy).\nonumber\end{aligned}$$
It suffices to verify the statement for twice continuously differentiable functions $f$ with $f,f^{\prime}f^{\prime\prime}$ bounded. By Itô’s formula,
$$\begin{split}
&
f(\theta_{t})=f(\theta_{0})+\int_{0}^{t}\mathcal{L}f(\theta_{s})ds+\int
_{0}^{t}f^{\prime}(\theta_{s})\sigma(\theta_{s})dW_{s}\\
&
+\int_{0}^{t}\int_{\mathbb{R}}f^{\prime}(\theta_{s-})u(\theta_{s-}
,x)(\mu^{\theta}-\nu^{\theta})(ds,dx).
\end{split}$$ Denote $$L_{t}=\int_{0}^{t}f^{\prime}(\theta_{s})\sigma(\theta_{s})dW_{s}+\int_{0}
^{t}\int_{\mathbb{R}}f^{\prime}(\theta_{s-})u(\theta_{s-},x)(\mu^{\theta}
-\nu^{\theta})(ds,dx).$$ Then, we have $$\begin{aligned}
\pi_{t}(f) & =E\big(f(\theta_{0})|\mathcal{G}_{t}\big)\\
& +E\Bigg(\int_{0}^{t}\mathcal{L}f(\theta_{s})ds\Big|\mathcal{G}
_{t}\Bigg)+E\big(L_{t}|\mathcal{G}_{t}\big).\end{aligned}$$ Set $$\begin{split}
M_{t} & =\big\{E\big(f(\theta_{0})|\mathcal{G}_{t}\big)-\pi_{0}(f)\big\}\\
&
\quad+\Bigg\{E\Bigg(\int_{0}^{t}\mathcal{L}f(\theta_{s})ds\Big|\mathcal{G}
_{t}\Bigg)-\int_{0}^{t}\pi_{s}\big(\mathcal{L}f\big)ds\Bigg\}+E\big(L_{t}
|\mathcal{G}_{t}\big).
\end{split}$$ Obviously, the process $E\big(f(\theta_{0})|\mathcal{G}_{t}\big)-\pi_{0}(f)$ is a $\mathcal{G}_{t}$-martingale. Process $L_{t}$ is a $\mathcal{F}_{t} $-martingale. Since $\mathcal{G}_{t}\subseteq\mathcal{F}_{t}$, for $t>t^{\prime},$ $$E\big(E(L_{t}|\mathcal{G}_{t})|\mathcal{G}_{t^{\prime}}\big)=E\big(E(L_{t}
|\mathcal{F}_{t^{\prime}})|\mathcal{G}_{t^{\prime}}\big)=E(L_{t^{\prime}
}|\mathcal{G}_{t^{\prime}}).$$ Consequently, $E(L_{t}|\mathcal{G}_{t})$ is a martingale too.
Finally, $E\big(\int_{0}^{t}\mathcal{L}f(\theta_{s})ds|\mathcal{G}_{t}\big)-\int
_{0}^{t}\pi_{s}\big((\mathcal{L}f)\big)ds$ is also a $\mathcal{G}_{t} $-martingale. Indeed, for $t>s>t^{\prime},$ we have $E\big(\pi_{s}
\big(\mathcal{L}f)\big|\mathcal{G}_{t^{\prime}}\big)=E\big(\mathcal{L}
f(\theta_{s})|\mathcal{G}_{t^{\prime}}\big)$ which yields $$\begin{aligned}
& E\left[
E\Bigg(\int_{0}^{t}\mathcal{L}f(\theta_{s})ds\Big|\mathcal{G}
_{t}\Bigg)-\int_{0}^{t}\pi_{s}(\mathcal{L}f)ds\Bigg|\mathcal{G}_{t^{\prime}
}\right]
\\
& \quad=E\Bigg(\int_{0}^{t^{\prime}}\mathcal{L}f(\theta_{s}
)ds\Big|\mathcal{G}_{t^{\prime}}\Bigg)-\int_{0}^{t^{\prime}}\pi_{s}
(\mathcal{L}f)ds.\end{aligned}$$
Thus, $M_{t}$ is a $\mathcal{G}_{t}$-martingale. In particular, this means that $\pi_{t}(f)$ is a $\mathcal{G}$ - semimartingale with paths in the Skorokhod space $\mathbb{D}_{[0,\infty)}(\mathbb{R})$, so that $\pi_{t}(f)$ is a right continuous process with limits from the left. By the Martingale Representation Theorem ( see e.g. Theorem 1 and Problem 1.c in Ch.4, §8. in [@LSMar]), $$M_{t}=\int_{0}^{t}\int_{\mathbb{R}}H(s,y)(\mu-\nu)(ds,dy).$$ It is a standard fact that $
\mathsf{P}(N_{S}-N_{S-}\neq0
|\mathcal{G}_{S-})={\nu}(\{S\},\mathbb{R}_{+}).
$ Hence, due to assumption A.0, by Theorem 4.10.1 from [@LSMar] (see formulae (10.6) and (10.15)), $$H(t,y)=\mathsf{M}_{\mu}^{\mathsf{P}}\big(\triangle
M|\widetilde{\mathcal{P} }(\mathcal{G})\big)(t,y),$$ where $\triangle M_{t}=M_{t}-M_{t-}$ and the conditional expectation $\mathsf{M}_{\mu}^{\mathsf{P}}\big(g|\widetilde{\mathcal{P}}(\mathcal{G}
)\big)$ is defined by the following relation (see, e.g. [@LSMar], Ch. 2, §2 and Ch. 10, §1): for any $\widetilde{\mathcal{P}}(\mathcal{G} )$-measurable bounded and compactly supported function $\varphi(t,y),$ $$E\int_{0}^{\infty}\int_{\mathbb{R}}\varphi(t,y)g_{t}\mu(dt,dy)
=E\int_{0}^{\infty}\int_{\mathbb{R}}\varphi(t,y)\mathsf{M}_{\mu}^{\mathsf{P}
}\big(g\big|\widetilde{\mathcal{P}}(\mathcal{G})\big)(t,y)\nu(dt,dy).$$ By Lemma 4.10.2, [@LSMar], $$\mathsf{M}_{\mu}^{P}\big(\pi_{t}(f)\big|\widetilde{\mathcal{P}}(\mathcal{G}
)\big)(t,y)=\mathsf{M}_{\mu}^{P}\big(f\big|\widetilde{\mathcal{P}}
(\mathcal{G})\big)(t,y).
\label{4.10a}$$ Since, $\pi_{t-}(f)$ is $\widetilde{\mathcal{P}}(\mathcal{G})$-measurable (which implies $\mathsf{M}_{\mu}^{\mathsf{P}}(\pi_{-}(f)|\widetilde
{\mathcal{P}}(\mathcal{G}))(t,y)=\pi_{t-}(f)$ ), by (\[4.10a\]), $$\begin{gathered}
\mathsf{M}_{\mu}^{\mathsf{P}}\big(\triangle
M\big|\widetilde{\mathcal{P}
}(\mathcal{G})\big)(t,y)
=\mathsf{M}_{\mu}^{\mathsf{P}}\big(\pi_{t}(f)-\pi_{t-}(f)\big|\widetilde
{\mathcal{P}}(\mathcal{G})\big)(t,y)\nonumber\\
\\
=\mathsf{M}_{\mu}^{\mathsf{P}}\big(f\big|\widetilde{\mathcal{P}
}(\mathcal{G})\big)(t,y)-\pi_{t-}(f).\end{gathered}$$ To complete the proof one needs to show that $$\mathsf{M}_{\mu}^{\mathsf{P}}\big(f(\theta_{.})
\big|\widetilde
{\mathcal{P}}(\mathcal{G})\big)(s,y)=\sum_{k\geq0}I_{]\hskip-1.5pt]\tau
_{k},\tau_{k+1}]\hskip-1.5pt]}(s)\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}
{\pi_{\tau_{k}}(\psi_{k}(1;s,y))}.
\label{gg}$$ To prove (\[gg\]), it suffices to demonstrate that for any $\widetilde {\mathcal{P}}(\mathcal{G})$-measurable bounded and compactly supported function $\varphi(t,y),$ $$\begin{gathered}
E\sum_{k\geq0}\int_{(\tau_{k},\tau_{k+1}]\cap(\tau_{k},\infty)}\int
_{\mathbb{R}}\varphi(t,y)\frac{\pi_{\tau_{k}}(\psi_{k}(f;t,y))}{\pi_{\tau_{k}
}(\psi_{k}(1;t,y))}\nu(dt,dy)
\nonumber\\
=E\int_{0}^{\infty}\int_{\mathbb{R}}
\varphi(t,y)f(\theta_{t})\mu(dt,dy).
\label{show}\end{gathered}$$
By monotone class arguments, we can assume that $\varphi(t,x)
=v(t)g(x)$, where $v(t)$ is a $\mathcal{P}(\mathcal{G})$-measurable process and $g(x)$ is a continuous function on $\mathbb{R}$. By Lemma III.1.39 [@JS], since ${v} $[$(t)$ is $\mathcal{P}(\mathcal{G})-$measurable, it must be of the form $$v(t)=v_{0}+\sum_{k\geq1}^{\infty}v_{k}(t)I_{]\hskip-1.5pt]\tau_{k},\tau_{k+1}]
\hskip-1.5pt]}(t),
\label{4.11b}$$ where $v_{0}$ is a constant and $v_{k}\left( t\right) $ are $\mathcal{G} \left( k\right) \otimes\mathcal{B}(\mathbb{R}_{+})$-measurable functions. ]{}
Owing to (\[4.11b\]) and Lemma \[lem-4.1\], in order to prove (\[show\]), it suffices to verify the equality $$\begin{gathered}
E\left[
\int_{(\tau_{k},\tau_{k+1}]\cap(\tau_{k},\infty)}\int_{\mathbb{R}
}g(y)v_{k}(t)\frac{\pi_{\tau_{k}}(\psi_{k}(f;t,y))}{\pi_{\tau_{k}}(\psi
_{k}(1;t,y))}\Phi_{k}(dt)dy\right]
\nonumber\\
\nonumber\\
=E\left[
v_{k}(\tau_{k+1})g(X_{k+1})f(\theta_{\tau_{k+1}})1_{\{
\tau_{k+1}<\infty\}}\right],
\label{enough}\end{gathered}$$
The next step follows the ideas of Theorem III.1.33 [@JS]. We have $$\begin{aligned}
& E\left[
v_{k}(\tau_{k+1})g(X_{k+1})f(\theta_{\tau_{k+1}})1_{\{
\tau_{k+1}<\infty\}}\right]
\\
& =E\left[E\big(
v_{k}(\tau_{k+1})g(X_{k+1})f(\theta_{\tau_{k+1} })1_{\{
\tau_{k+1}<\infty\}}|\mathcal{G}(k)
\vee\mathcal{F}^{\theta}\big)\right]
\\
&=E\left(
\int_{(\tau_{k},\infty)}\int_{\mathbb{R}}v_{k}(s)g(y)E\left[
f(\theta_{s})G_{k}^{\theta}\left( ds,dy\right)
|\mathcal{G}\left( k\right) \right]\right),\end{aligned}$$ where, as before, $G_{k}^{\theta}$ $(ds,dy)$ is a regular version of the conditional distribution of $\big(
\tau_{k+1},X_{k+1}\big)$ with respect to $\mathcal{F}^{\theta}\vee\mathcal{G}(k).$
By Fubini Theorem, and recalling notation (\[GGG\]), $$\begin{aligned}
& E\left(
\int_{(\tau_{k},\infty)}\int_{\mathbb{R}}v_{k}(s)g(y)E\left[
f(\theta_{s})G_{k}^{\theta}(ds,dy)
|\mathcal{G}(k)
\right]\right)
\\
&=E\left(
\int_{(\tau_{k},\infty)}\int_{\mathbb{R}}v_{k}(s)g(y)\frac
{E\left[f(\theta_{s})G_{k}^{\theta}(ds,dy)
|\mathcal{G}(k)\right]}{G_{k}\left(\left[s,\infty\right];\mathbb{R}\right)
}\int_{[s,\infty]}G_{k}(du,\mathbb{R})\right)
\\
& =E\left(
\int_{\tau_{k}}^{\tau_{k+1}}\int_{\mathbb{R}}v_{k}(s)g(y)\frac
{E\left[f(\theta_{s})G_{k}^{\theta}(ds,dy)
|\mathcal{G}(k)\right]}{G_{k}\left(\left[
s,\infty\right];\mathbb{R}\right) }\right).\end{aligned}$$ By (\[eq:sep\]), $$G_{k}^{\theta}(ds,dy)
=\rho_{{\tau_{k},s}}(z-X_{k})\phi(\tau _{k},s)\Phi_{k}(ds)dy.$$
Hence, for $s>\tau_{k}$, $$\begin{aligned}
& E\left[f(\theta_{s})G_{k}^{\theta}(ds,dy)
|\mathcal{G}(k\right]
\\
& =E\Big(E\left(f(\theta_{s})\rho_{{\tau_{k},s}}(y-X_{k})\phi(\tau
_{k},s)|\sigma\{\theta_{\tau_{k}}\}
\vee\mathcal{G}(k)\right)\big|\mathcal{G}(k)\Big)\Phi_{k}(ds)dy
\\
&=\pi_{\tau_{k}}(\psi_{k}(f;s,y))dy\Phi_{k}(ds).\end{aligned}$$
This, together with (\[eq:denom\]), yields
$$\begin{aligned}
&
E\left(\int_{\tau_{k}}^{\tau_{k+1}}\int_{\mathbb{R}}v_{k}(s)g(y)\frac
{E\left[f(\theta_{s})G_{k}^{\theta}(ds,dy)
|\mathcal{G}(k)\right]}{G_{k}\left(\left[
s,\infty\right];\mathbb{R}\right)}\right)
\\
&=E\left(
\int_{\tau_{k}}^{\tau_{k+1}}\int_{\mathbb{R}}v_{k}(s)g(y)\frac
{\pi_{\tau_{k}}(\psi_{k}(f;s,y))dy}{\int_{s}^{\infty}\pi_{\tau_{k}}
\left(\bar{\psi}(1;t)\right)\Phi_{k}(dt)}\Phi_{k}(ds)\right) ,\end{aligned}$$
so that (\[enough\]) is satisfied, and the proof follows.
Proof of Theorem \[mainthm\] {#sec-6}
----------------------------
In this section we show that Theorem \[mainthm\] follows from Lemma \[lem-4.1\] and Theorem \[theo-4.1\].
Firstly, we note that the stochastic integral in the right hand side of (\[filteq\]) can be written as the difference of the integrals with respect to $\mu$ and $\nu.$ Indeed, since $f$ is bounded, this follows from [@JS], Proposition II.1.28.
By applying Lemma \[lem-4.1\] and integrating over $y$ one gets that for $t\in]\hskip-1.5pt]\tau_{k},\tau_{k+1}]\hskip-1.5pt],$
$$\begin{aligned}
&
\int_{\mathbb{R\times}(\tau_{k},t]}\Big(\frac{\pi_{\tau_{k}}(\psi
_{k}(f;s,y))}{\pi_{\tau_{k}}(\psi_{k}(1;s,y))}-\pi_{s-}(f)\Big)\nu(ds,dy)\\
& =\int_{(\tau_{k},t]}\frac{\pi_{\tau_{k}}\left(
\bar{\psi}_{k}(f;s)\right) -\pi_{s-}(f)\pi_{\tau_{k}}\left(
\bar{\psi}_{k}(1;s)\right)}{\int
_{s}^{\infty}\pi_{\tau_{_{k}}}\left(\bar{\psi}_{k}(1;u)\right)
\Phi _{k}(du)}\Phi_{k}(ds).\end{aligned}$$
This equation verifies that follows from the semimartingale representation , for $t$ between the consecutive observation times.
For the jump part , we note that $$\int_{0}^{t}\int_{\mathbb{R}}\pi_{s-}(f)\mu(ds,dy)=
\sum_{\tau_{k+1}\leq t}\pi_{(
\tau_{k+1})-}(f)$$ and $$\int_{0}^{t}\int_{\mathbb{R}}\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}{\pi
_{\tau_{k}}(\psi_{k}(1;s,y))}\mu(ds,dy)
=\sum_{\tau_{k+1}\leq
t}\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}{\pi_{\tau_{k}}(\psi_{k}(1;s,y))}
_{\big\{\substack{s=\tau_{k+1}\\y=X_{k+1}}\big\}}.$$ Now, (\[filteq\]) can be rewritten as follows: $$\label{closeq1}
\begin{aligned}
\pi_{t}(f)&=\pi_{0}(f)
+\int_{0}^{t}\pi
_{s}\big(\mathcal{L}f\big)ds
\\
& +\sum_{\tau_{k+1}\leq t}\left(
\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}
{\pi_{\tau_{k}}(\psi_{k}(1;s,y))}
_{\big\{\substack{s=\tau_{k+1}
\\y=X_{k+1}}\big\}} -\pi_{(\tau_{k+1})-}(f)
\right)
\\
&
-\sum_{k\geq0}\int_{(\tau_{k},t\wedge\tau_{k+1}]}\mathcal{M}_{k}(f;s,\pi_{s})
\Phi_{k}(ds).
\end{aligned}$$ Suppose $t\in]\hskip-1.5pt]\tau_{k},\tau_{k+1}[\hskip-1.5pt[.$ Then, $$\begin{aligned}
\pi_{t}(f)&=\pi_{\tau_{k}}(f)
\\
&+\int_{\tau_{k}}^{t}\pi_{s}\big(\mathcal{L}f)ds
-\int_{\tau_{k} }^{t}\mathcal{M}_{k}(f;s,\pi_{s})
\Phi_{k}(ds).\end{aligned}$$ It follows that $$\begin{aligned}
& \pi_{(\tau_{k+1})-}(f)
\\
&=\pi_{\tau_{k}}(f)
+\int_{\tau_{k}}^{\tau_{k+1}}\pi _{s}\big(\mathcal{L}f\big)
ds-\int_{\tau_{k}}^{(\tau _{k+1})-}\mathcal{M}_{k}(f;s,\pi_{s})\Phi_{k}(ds).\end{aligned}$$ Therefore, from (\[closeq1\]), $$\pi_{\tau_{k+1}}(f)
=\frac{\pi_{\tau_{k}}(\psi_{k}(f;s,y))}
{\pi_{\tau_{k}}(\psi_{k}(1;s,y))}
_{\big\{\substack{s=\tau_{k+1}
\\y=X_{k+1}}\big\}}
-\mathcal{M}_{k}(f;t,\pi_{t})
_{\{t=\tau_{k+1}\}} \Phi(\{\tau_{k+1}\}).$$
This completes the proof.
**Examples** {#Sec-Ex}
============
In this Section we consider some important special cases of Theorem \[mainthm\].
\[exs1\] (*Markov chain volatility and Cox process arrivals.*) Recall the setting of Example \[examplechain\] and its notation $r_{ij}$, $\pi_{j}(t),$ and $\theta^{j}$. It follows from Example \[ex:cox copy(1)\] that in this case $\Phi_{k}(\{ \tau_{k+1}\})=0$ for all $k$’s. Hence the second term in the right hand side of equation (\[eq:jump\]) is zero. By (\[nado\]), for $f(\theta
_{t})=\mathbf{1}_{\{\theta_{t}=a_{i}\}}$ and $t>\tau_{k}$, $$\psi_{k}(f;t,y,\theta_{\tau_{k}})=n(a_{i})\left[
E\big(I_{\{\theta_{t}=a_{i}\}}e^{-\int_{s}^{t}n(\theta_{u}
)du}\rho_{_{s,t}}(y-x)|\theta_{s}\big)\right]_{\big\{\substack
{s=\tau_{k}\\x=X_{k}}\big\}}.$$ Thus, owing to the homogeneity of $\theta_{t},$ for $t$ $>\tau_{k},$ $$\begin{aligned}
&\qquad \pi_{\tau_{k}}(\psi_{k}(f;t,y))
\\
&=\sum_{j}n(a_{i})E\Big(I_{\{\theta_{t}=a_{i}\}}e^{-\int
_{s}^{t}n(\theta_{u})du}\rho_{s,t}(y-x)\big|\theta_{s}=a_{j}
\Big)_{\big\{\substack{s=\tau_{k}\\x=X_{k}}\big\}}\pi_{j}(\tau_{k})
\\
&=\sum_{j}n(a_{i})E\Big(I_{\{ \theta_{t-s}^{j}=a_{i}\}}
e^{-\int_{0}^{t-s}n(\theta_{u})du}\rho_{0,t-s}^{j}
(y-x)\Big)_{\big\{\substack{s=\tau_{k}\\x=X_{k}}\big\}}\pi_{j}(\tau_{k})
\\
&=\sum_{j}n(a_{i})E\Big[I_{\{\theta_{t-s}^{j}=a_{i}\}}
E\Big(e^{-\int_{0}^{t-s}n(\theta_{u})du}\rho_{0,t-s}^{j}(y-x)\big|\theta
_{t-s}^{j}\Big)\Big]_{\hskip
-.05in\big\{\substack{s=\tau_{k}\\x=X_{k}}\big\}}
\pi_{j}(\tau_{k})
\\
&=\sum_{j}n(a_{i})r_{ji}(t-\tau_{k},y-X_{k})
p_{ji}(t-\tau_{k})\pi_{j}(\tau_{k}).
\end{aligned}$$ Similar formula holds for the denominator of the first term of the right hand side of the equation. Now equation follows from .
Repeating the previous calculations and using the notation $$\bar{r}_{ji}(t)
:=E\big(e^{-\int_{0}^{t}n(\theta_{u}^{j})du}|\theta_{t}^{j}=a_{i}\big),$$ it is readily checked that, for $t>\tau_{k}$, $$\pi_{\tau_{k}}\big(\bar{\psi}_{k}(\mathbf{1}_{\{\theta_{t}=a_{i}
\}};t)\big)=n(a_{i})
\sum_{j}\pi_{j}(\tau_{k})\bar{r}_{ji}(t-\tau_{k})p_{ji}(t-\tau_{k})$$ and $$\pi_{\tau_{k}}\big(\bar{\psi}_{k}(1,t)\big)=\sum_{i,j}\pi_{j}(\tau
_{k})n(a_{i})\bar{r}_{ji}(t-\tau_{k})p_{ji}(t-\tau _{k})$$ which are needed in computing . It is easily verified that in the setting of this example, equation reduces to the following: $$d\pi_{i}(t)=\sum_{j}\lambda(a_{j},a_{i})
\pi_{j}(t)dt+\bar {D}(\tau_{k},t)
\pi_{i}(t)dt+D_{i}(\tau_{k},t)dt,
\label{eq:ko2}$$ where $$\begin{aligned}
D_{i}(\tau_{k},t)&=-\frac{n(a_{i})
\sum _{j}\bar{r}_{ji}(t-\tau_{k})p_{ji}(t-\tau_{k})
\pi_{j}(\tau _{k})}{\int_{t}^{\infty}\sum_{i,j}n(a_{i})\bar{r}_{ji}
(s-\tau_{k})p_{ji}(s-\tau_{k})\pi_{j}(\tau_{k})ds}
\\
\bar{D}(\tau_{k},t)&=\frac{\sum_{l,j}n(a_{l})\bar{r}_{jl}(t-\tau_{k})p_{jl}
(t-\tau_{k})\pi_{j}(\tau_{k})}{\int_{t}^{\infty}\sum_{i,j}n(a_{i})
\bar{r}_{ji}(s-\tau _{k})p_{ji}(s-\tau_{k})
\pi_{j}(\tau_{k})ds}.\end{aligned}$$ Note that equation (\[eq:ko2\]) is considered for a fixed $\omega$ and $t>\tau_{k}(\omega).$ Therefore, $\tau_{k}$ and $\pi_{\cdot }(\tau_{k})$ should be viewed as known quantities.
\[Poisson\](*Poisson arrivals.*) Suppose that the interarrival times between the observations are exponential with constant intensity $n(\theta)\equiv\lambda$. In other words, $N_{t}$ is Poisson process with constant parameter $\lambda.$ In this case, the volatility process $\theta$ is independent of $N_{t}.$ Then, on the interval $\tau_{k}<t<\tau_{k+1},$ equation (\[eq:ko2\]) reduces to $$\begin{aligned}
\label{eq:po}
d\pi_{i}(t)&=\sum_{j}\lambda(a_{j},a_{i})
\pi_{j}
(t)dt
\nonumber\\
& -\lambda\Big(\sum_{j}p_{ji}(t-\tau_{k})
\pi_{j}(\tau_{k})-\pi _{i}(t)\Big)dt.\end{aligned}$$ On the other hand, owing to the independence of $N$ and $\theta,$ it is readily checked that on the interval $\tau_{k}<t<\tau_{k+1},$ $$\pi_{i}(t)={\ }\sum_{j}p_{ji}(t-\tau_{k})
\pi_{j}(\tau_{k}).$$ Therefore, the filtering equation (\[eq:po\]) [is simply the forward Kolmogorov equation for ]{}$\theta.$
A similar effect appears also in the following example.
\[ex:constep\](Fixed observation intervals.) Assume for simplicity that the Markov process $\theta_{t}$ is homogeneous. Also assume that $\tau_{k}=kh,$ where $h$ is a fixed time step. Notice that $$\mathcal{G}_{t}=\mathcal{G}(k) \ \text{for any $t\in
[\hskip-.015in[\tau_{k},\tau_{k+1}[\hskip-.015in[$}.$$ Denote by $P(t,x,dy)$ the transition probability kernel of the process $\theta_{t}$, given that $\theta_{0}=x$, and let $T_{t}$ denote the associated transition operator.
In accordance with Example \[ex:constep0\], one can take $\phi (\tau_{k},t)\equiv1$ and $\Phi_{k}(dt)=\delta_{\{\tau_{k+1}\}}(t)dt.$ Thus, we get $$\begin{aligned}
\psi_{k}(f;t,y,\theta_{\tau_{k}})&=E\left[
f(\theta _{t})\rho_{{\tau_{k},t}}(y-X_{k})\big|\sigma\{
\theta_{\tau_{k}}\} \vee\mathcal{G}(k)\right],
\\
\bar{\psi}_{k}(f;t,\theta_{\tau_{k}})
&=T_{t-\tau_{k} }f(\theta_{\tau_{k}})
:=\int f(y)\mathsf{P}(t-\tau_{k} ,\theta_{\tau_{k}},dy).\end{aligned}$$ Since $\Phi_{k}(dt)=0$ on $[\hskip-.015in[\tau_{k},\tau
_{k+1}[\hskip-.015in[$, is reduced to the forward Kolmogorov equation $$\frac{\partial_{t}}{\partial t}\pi_{t}(f)=\pi_{t}(\mathcal{L}f)$$ subject to the initial condition $\pi_{\tau_{k}}(f).$ The unique solution of this equation is given by $\pi_{t}(f)=\pi_{\tau_{k}}(T_{t-\tau_{k}}f)$, $t<\tau_{k+1}$. Hence, $$\pi_{\tau_{k+1}-}(f)=\pi_{\tau_{k}}(T_{h}f)$$ Since $\phi(\tau_{k},t)\equiv1,$ the denominator of $\mathcal{M}_{k}$ is equal to 1 when $t=\tau_{{k+1}}$. This together with the formula $\Phi(\{\tau_{k+1}\})=1$ yields $$\mathcal{M}_{k}(f;t,\pi_{t})_{t=\tau_{k+1}}\Phi(\{\tau_{k+1}\})=\pi_{\tau_{k}}
\big(T_{h}f\big)-\pi_{\tau_{k+1}-}(f).
\label{eq:dopjump}$$ Owing to (\[eq:dopjump\]), we get $
\mathcal{M}_{k}(f;t,\pi_{t})_{t=\tau_{k+1}}\Phi
(\{\tau_{k+1}\})=0.
$
This yields the following recursion formula: $$\begin{aligned}
\pi_{\tau_{k+1}}(f)& =\frac{\pi_{\tau_{k}}(\psi _{k}(f;t,y))}{\pi_{\tau_{k}}(
\psi_{k}(1;t,y))}_{\substack{t=\tau_{k+1}\\y=X_{\tau_{k+1}}}}
\\
& =\frac{\int_{\mathbb{R}}E\big(
f(\theta_{t-\tau_{k}})\rho_{0{,t-}\tau
_{k}}(y-z)|\theta_{0}=z\big)\pi_{\tau_{k}}(dz)}
{\int_{\mathbb{R}}E\big(
\rho_{0{,t-}\tau_{k}}(y-z)|\theta_{0}=z\big)
\pi_{\tau_{k}}\left(
dz\right)}_{\substack{t=\tau_{k+1}\\y=X_{\tau_{k+1}}}}.\end{aligned}$$
**Acknowledgement:** We are grateful to the anonymous Associate Editor and the referee for their constructive suggestions, especially regarding a simplified presentation of the results. We are very much indebted to Remigijus Mikulevicius for many important suggestions, and to Ilya Zaliapin, whose numerical experiments helped to discover an error in a preprint version of the paper.
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[^1]: The research of J. Cvitanić was supported in part by the National Science Foundation, under Grant NSF-DMS-00-99549 and 04-03575.
[^2]: The research of B.L. Rozovskii was supported in part by the Army Research Office and the Office of Naval Research under the grants DAAD19-02-1-0374 and N0014-03-0027.
[^3]: Here and below $\mathcal{F}^{1}\vee\mathcal{F}^{2}$ stands for the $\sigma$-algebra generated by the $\sigma-$algebras $\mathcal{F}^{1}\ $and $\mathcal{F}^{2}.$
[^4]: More generally, it holds if the compensator of the counting process $N_{t}$ is a continuous process.
|
---
abstract: 'This talk is devoted to the statistical analysis of the new catalogue of Chemically Peculiar stars compiled from papers, where chemical abundances of those stars were given. The catalogue contains chemical abundances and physical parameters of 428 stars based on high-resolution spectroscopy data. Spearman’s rank correlation test was applied for 416 CP (108 HgMn, 188 ApBp and 120 AmFm) stars and the correlation between chemical abundances and different physical parameters (effective temperature, surface gravity and rotational velocity) was checked. From dozens interesting cases we secluded four cases: the Mn peculiarities in HgMn stars, the Ca correlation with respect to effective temperature in AmFm stars, the case of helium and iron in ApBp stars. We applied also Anderson-Darling (AD) test on ApBp stars to check if multiplicity is a determinant parameter for abundance peculiarities.'
author:
- 'S. Ghazaryan$^{1}$$^{\ast}$, G. Alecian$^{2}$, A. A. Hakobyan$^{1}$'
title: Statistical analysis of the new catalogue of CP stars
---
\[firstpage\]
***Keywords:** Chemically Peculiar stars - abundances – catalogues – individual stars: HgMn, ApBp and AmFm.*
Introduction
============
Historically, main sequence Chemically Peculiar (CP) stars were divided in 4 groups by Preston (1974) – AmFm, ApBp, HgMn and $\lambda{Boo}$ stars. Later, two groups of CP stars were added to the known ones - He-weak and He-rich stars. All CP stars have different physical properties. The effective temperatures of all CP stars are in the range of 7000-30000K, some of them are non-magnetic, some- magnetic. More than 50 percent of CP stars are binaries or belongs to the multiple systems. But all those stars have one generality – in the atmospheres of those stars we see peculiarities of different chemical element’ abundances. AmFm stars are non-magnetic and they are characterized by underabundances of calcium and scandium, and high overabundances of iron and nickel (see Kunzli & North 1998; Gebran, Monier & Richard 2008, for example). HgMn stars are non-magnetic as well; they are characterized by strong overabundances of mercury and manganese (Hg overabundance riches up to 107) (see Catanzaro, Leone & Leto 2003; Dolk, Wahlgren & Hubrig 2003, Alecian et al. 2009, etc.). In contrary, ApBp stars are magnetic and characterized by silicon, chromium, europium and strontium overabundances (see Leckrone 1981; Ryabchikova et al. 1999; Kochukhov et al. 2006, etc.). In this research we studied 3 groups of CP stars – AmFm, ApBp and HgMn stars.
{width="100.00000%"}
The new catalogue of CP stars
=============================
Our new catalogue of CP stars consists in 428 stars, for which all chemical abundances were determined through high resolution spectroscopy techniques. 108 stars are identified as HgMn stars (see Ghazaryan & Alecian 2016), 128 stars as AmFm and 188 as ApBp stars. The types of 11 stars are uncertain and one star is a known horizontal-branch star (Feige 86). The catalogue contains physical parameters such as effective temperature, gravity, rotational velocity, and chemical abundances with their error measurements. If for a given element the abundances of different ions were given, we took the mean of them for the abundance value, and the error bars were recalculated by the mean square of the errors as in Ghazaryan & Alecian (2016). Compiled abundances with their errors are shown in Fig. 1. The detailed information on our catalogue is given in Ghazaryan et al. (2018).
[|c|\*[9]{}[c|]{}]{}&&&\
$Elements$&$\rho$&$p$&$N$&$\rho$&$p$&$N$&$\rho$&$p$&$N$\
He & **-0.40** & **0.001** & **65** & -0.10 & 0.450 & 65 & 0.12 & 0.355 & 64\
C & -0.23 & 0.105 & 51 & 0.00 & 0.982 & 51 & 0.03 & 0.841 & 50\
O & 0.23 & 0.156 & 38 & -0.28 & 0.091 & 38 & 0.15 & 0.391 & 37\
Mg & **-0.39** & **0.001** & **69** & -0.07 & 0.580 & 69 & 0.09 & 0.489 & 68\
Al & **-0.42** & **0.005** & **44** & 0.24 & 0.112 & 44 & **0.32** & **0.037** & **44**\
Si & 0.01 & 0.929 & 74 & -0.15 & 0.192 & 74 & -0.01 & 0.913 & 73\
S & **-0.61** & **0.000** & **52** & 0.05 & 0.743 & 52 & -0.03 & 0.813 & 51\
Ti & 0.24 & 0.064 & 62 & 0.11 & 0.403 & 62 & 0.10 & 0.448 & 61\
Cr & **-0.28** & **0.010** & **86** & -0.02 & 0.828 & 86 & **0.22** & **0.041** & **85**\
Mn & **0.48** & **0.000** & **70** & 0.06 & 0.650 & 70 & **0.26** & **0.032** & **69**\
Fe & 0.07 & 0.488 & 90 & 0.06 & 0.561 & 90 & -0.11 & 0.313 & 89\
Ni & -0.05 & 0.730 & 48 & -0.01 & 0.945 & 48 & 0.11 & 0.473 & 47\
Cu & 0.32 & 0.106 & 26 & & & & 0.22 & 0.289 & 26\
Zn & -0.18 & 0.339 & 29 & 0.00 & 0.999 & 29 & 0.12 & 0.533 & 29\
Ga & 0.30 & 0.083 & 34 & 0.23 & 0.184 & 34 & 0.00 & 0.997 & 33\
Sr & **-0.49** & **0.001** & **45** & **0.33** & **0.027** & **45** & 0.11 & 0.492 & 44\
Y & -0.17 & 0.220 & 55 & 0.20 & 0.134 & 55 & 0.19 & 0.161 & 54\
Zr & 0.25 & 0.133 & 37 & 0.25 & 0.136 & 37 & **0.38** & **0.021** & **36**\
Xe & **0.41** & **0.032** & **28** & -0.04 & 0.823 & 28 & 0.11 & 0.568 & 28\
Hg & **-0.22** & **0.045** & **86** & **0.27** & **0.011** & **86** & -0.01 & 0.944 & 85\
Statistical analysis
====================
The abundance anomalies may be explained by theoretical models, including atomic diffusion, and for that reason it is interesting to know the correlation between abundances and physical parameters, such as effective temperature, surface gravity and rotational velocity. To check that correlation we applied Spearman’s rank test (see Spearman 1904) between abundances and mentioned physical parameters. The statistical analysis was done for each element measured in more than 11 stars. In Table 1 as an example we show statistical results for HgMn stars. As you see, we found significant correlation with respect to effective temperature for Mg, Al, S, Cr, Mn, Sr, Xe, Hg. Possible correlations suggested by Ghazaryan & Alecian (2016) for Ni, Ti, and Si are not confirmed in this study. We found also significant correlation with respect to gravity for strontium and confirmed it for mercury suggested in our previous paper. Considering rotation velocity as parameter, correlations have been found for aluminum, chromium, manganese, and zirconium.
Such types of tables were created for ApBp and AmFm stars as well. Dozens correlations between abundances and fundamental parameters were found for those type of stars too. Four noteworthy cases are secluded and explained in detail in our paper (see Ghazaryan et al. 2018).
We have applied AD test on single CP stars and those being in binary systems and do not find any relation between abundance anomalies and multiplicity in all three CP type stars, possibly because of the lack of data.
Conclusions
===========
We present a unique catalogue of 428 Chemically Peculiar stars observed by spectroscopy during the last decades. Our new compilation of the main physical parameters and the abundances of elements from helium to uranium for 108 HgMn, 188 ApBp, 120 AmFm stars, plus 12 other peculiar stars (including 1 horizontal-branch star) confirms the increase of overabundances for heavy elements with atomic number (see Smith 1996) and the large scatter of the abundances anomalies. This scatter is not only due to the heterogeneity of the data, or abundance determination errors, but it is real. The applied statistical tests proofs that there is no any correlation between abundance anomalies and multiplicity of HgMn, ApBp and AmFm stars. We have found a significant number of correlations with the effective temperatures, but also some (fewer) with gravity and rotational velocity. We discuss also four noteworthy cases, but this does not mean that there are only four cases that deserve discussion. We are convinced that considering as a whole, the abundance measurements in CP stars will lead to interesting understanding of the physical processes in play in the atmospheres of those stars. In the near future we plan to extend our database to other categories of CP stars (such as stars with helium anomalies), and we have no doubt that a major extension of such a database will be achieved from the final GAIA catalogue.
Acknowledgements {#acknowledgements .unnumbered}
================
Alecian G., Gebran M., Auvergne M., et al. 2009, A&A, 506, 69\
Asplund M., Grevesse N., Sauval A. J., Scott P. 2009, ARA&A, 47, 481\
Catanzaro G., Leone F., Leto P. 2003, A&A, 407, 669\
Dolk L., Wahlgren G. M., Hubrig S. 2003, A&A, 402, 299\
Gebran M., Monier R., Richard O. 2008, A&A, 479, 189\
Ghazaryan S., Alecian G. 2016, MNRAS, 460, 1912\
Ghazaryan S., Alecian G., Hakobyan A. A. 2018, MNRAS, 480, 2953\
Kochukhov O., Tsymbal V., Ryabchikova T., et al. 2006, A&A, 460, 831\
Kunzli M., North P. 1998, A&A, 330, 651\
Leckrone D. S. 1981, ApJ, 250, 687\
Preston G. W. 1974, ARA&A, 12, 257\
Ryabchikova T., Piskunov N., Savanov I., et al. 1999, A&A, 343, 229\
Smith K. C. 1996, Ap$\&$SS, 237, 77\
Spearman C. 1904, Am. J. Psychol., 15, 72\
|
---
abstract: 'Under many [*in vitro*]{} conditions, some small viruses spontaneously encapsidate a single stranded (ss) RNA into a protein shell called the capsid. While viral RNAs are found to be compact and highly branched because of long distance base-pairing between nucleotides, recent experiments reveal that in a head-to-head competition between a ssRNA with no secondary or higher order structure and a viral RNA, the capsid proteins preferentially encapsulate the linear polymer! In this paper, we study the impact of genome stiffness on the encapsidation free energy of the complex of RNA and capsid proteins. We show that an increase in effective chain stiffness because of base-pairing could be the reason why under certain conditions linear chains have an advantage over branched chains when it comes to encapsidation efficiency. While branching makes the genome more compact, RNA base-pairing increases the effective Kuhn length of the RNA molecule, which could result in an increase of the free energy of RNA confinement, that is, the work required to encapsidate RNA, and thus less efficient packaging.'
author:
- Siyu Li
- 'Gonca Erdemci-Tandogan'
- Paul van der Schoot
- Roya Zandi
bibliography:
- 'bibfile.bib'
title: 'The effect of RNA stiffness on the self-assembly of virus particles'
---
introduction
============
Ribonucleic acid (RNA) is one of the molecules of life, which plays a central role in the cell as information carriers, enzymes, gene regulators, et cetera. It is made out of four elementary building nucleotides, being A(denine), G(uanine), C(ytosine) and U(racil) [@Higgs2000]. As shown by Crick and Watson, purines (A,G) pair with complementary pyrimidines (C,U), leading primarily to the pairs CG and AU. There exist also so-called wobble pairs of GU. Single stranded RNA is quite flexible with a Kuhn length of, depending on the ionic strength of the solution, one or two $nm$ [@Chen2011], and can form double helical stems (A helices) with a Kuhn length of about 140 $nm$ [@Kebbekus1995; @Abels2005]. So, double stranded RNA is stiffer than double stranded DNA, which has a Kuhn length of 100 $nm$, noting that the Kuhn length is twice the persistence length of a so-valled wormlike chain.
The pairing of bases over long distances along the backbone gives rise to the secondary or folded structure of RNA. Pairing of bases can be represented by so-called arch diagrams. Nested arches represent helices, while crossings give rise to the so-called pseudoknots [@Nussinov]. The nested pairings can be described quantitatively by recursion relations [@McCaskill1990; @Zuker; @Vienna], which exactly sum all possible pairings without pseudoknots. From a geometrical point of view, the generated structures can be viewed as branched polymers. The size of an ideal, Gaussian linear polymer scales as the number of “segments” to the power $\nu=1/2$, while ideal branched ones have a scaling exponent $\nu=1/4$ [@Schwab2009]. Note that there is no excluded volume interaction between monomers of an ideal chain. For self-avoiding chains the scaling exponents are $\nu=3/5$ and $\nu=1/2$ for the linear and branched polymers, respectively [@Grosberg97; @Schwab2009]. However, because of its *tertiary* structures that include pseudoknots, RNAs are significantly more compact than branched polymers. Indeed, several numerical studies and surveys have found the exponent $\nu=1/3$ to be small for RNA, reflecting this more compact structure[@Fang2011; @Ben-Shaul2015].
Many small viruses encapsidate a single stranded RNA into a protein shell called the capsid. Under appropriate physico-chemical conditions of acidity and ionic strength, this process is spontaneous and the virus can readily assemble [*in vitro*]{} from solutions containing protein subunits and RNA [@Cornelissen2007; @Ren2006; @Bogdan; @Anze2; @Zlotnick; @Sun2007; @Nature2016]. Note that in the absence of genome, capsids do not form at physiological pH and salt concentrations. Many spherical viruses adopt structures with icosahedral symmetry [@Fejer:10; @Rapaport:04a], which imposes a constraint on the number of subunits in capsids. The structural index $T$, introduced by Casper and Klug, defines the number of protein subunits in viral shells, which is 60 times the $T$ number. Note that $T=1,3,4,7,\ldots$ can assume only certain “magic” integer numbers [@Wagner2015956; @Luque:2010a; @Chen:2007b; @Stefan].
Quite interestingly, virus protein subunits are able to co-assemble with a wide variety of negatively charged cargos, including non-cognate RNAs of different length and sequence, synthetic polyanions, and negatively charged nanoparticles [@Sun2007; @Kusters2015; @Zandi2016]. It is now widely accepted that electrostatic interactions between the positive charges on the coat protein tails and negative charges on the cargo is the main driving force for the spontaneous assembly of simple viruses in solution [@Cornelissen2007; @Ren2006; @Bogdan; @Anze2; @Zlotnick; @Hsiang-Ku; @Venky2016]. Still, several recent self-assembly experimental studies reveal the importance of non-electrostatic interactions, associated with specific structures of the genome, for the selection of one RNA over another by the capsid proteins[@Patel2014].
The self-assembly studies of Comas-Garcia [*et al.*]{} [@Comas] reveal in particular the importance of RNA topology. They carried out a number of experiments in which a solution of the capsid proteins of cowpea chlorotic mottle virus(CCMV) were mixed with equal amount of RNA1 of Brome Mosaic virus (BMV) and RNA1 of Cowpea Chlorotic Mottle Virus (CCMV). In this head-to-head competition, the amount of coat protein (CP) of CCMV was selected such that it could only encapsidate one of the genomes. Quite unexpectedly, the RNA1 of CCMV (the cognate RNA) lost to RNA1 of BMV, [*i.e*]{}, only RNA1 of BMV was encapsidated by CCMV CPs. These experiments emphasize the impact of RNA structure on the assembly of viral shells, as RNA1 of BMV has a more compact structure than that of CCMV [@Gonca2014].
Following these experiments a number of simulation studies, using quenched (fixed) branched polymers as a model for RNA, have shown that the optimal length of encapsidated RNA increases when accounting for its secondary structure [@elife; @Ben-Shaul2015]. Mean-field calculations using annealed (equilibrium) branched polymers as model RNAs have also shown that the length of encapsidated polymer increases as the propensity to form larger numbers of branched points increases [@Gonca2014; @Gonca2016; @Li2017]. More importantly, these calculations show that a higher level of branching considerably increases the depth of the free-energy gain associated with the encapsulation of RNA by a positively charged shell. This implies that the efficiency of genome packaging goes up with increasing the level of branching, so with increasing compact secondary structure of the genome.
In fact, it was shown in Refs. [@Yoffe2008; @Bruinsma2016] that while RNA molecules of the same nucleotide length and composition might have similar amounts of base pairing, non-viral RNAs have significantly less compact structures than viral ones. The compactness of viral RNAs has been associated with the presence of a larger fraction of higher-order junctions or branch points in their secondary structure [@Yoffe2008; @Li-tai; @Gopal2014]. Figure \[structure\](a) and (b) illustrate the secondary structures of CCMV RNA and those of a randomly sequenced RNA with the same length. The structures are obtained through the Vienna RNA software package [@Vienna]. As shown in the figure, CCMV RNA has considerably larger number of branched points than non-viral RNA of the same length.
Above-mentioned theoretical and experimental studies indicate that in a head-to-head competition between two different types of RNAs, the RNA with a larger number of branching junctions or branch points should have a competitive edge over others [@Gonca2014; @Gonca2016; @Li2017]. A naive physical explanation is that branching causes RNA molecules to become more compact than structureless linear polymers of similar chain length, which are then easier to accommodate in the limited space provided by the cavity of a capsid. According to these theories and simulations, a linear chain should definitely “loose” to a branched one of the same number of monomers when competing head-to-head for a limited number of capsid proteins.
To probe the effect of RNA structure and test the above theories on the self-assembly of virions more systematically, Beren *et al.* [@Beren2017] recently performed a set of [*in vitro*]{} packaging experiments with polyU, an RNA molecule that has no folded secondary structure. They examined whether RNA topology, i.e., the secondary structure or level of branching, allows the viral RNA to be exclusively packaged by its cognate capsid proteins. More specifically, they studied the competition between CCMV viral RNA with polyU of equal number of nucleotides for virus capsid proteins. They find that CCMV CPs are capable of packaging polyU RNAs and, quite interestingly, polyU outcompetes the native CCMV RNA in a head-to-head competition for the capsid proteins. These findings are in sharp contrast with the previous experimental, theoretical, simulation and scaling studies noted above, which suggest that the branching and compactness of RNA must lead to a more efficient capsid assembly. That being said, the scaling theory of Ref. [@Paul:13a] already hints at the subtle interplay between Kuhn length, solvent quality and linear charge density dictating the free energy gain of encapsulation.
To explain these intriguing experimental findings, we employ a mean-field density functional theory and study the impact of RNA branching, while allowing for differences in Kuhn length. We further consider that double helical sequences have a larger linear charge density than non-hybridized sequences along the chain. In all previous theoretical and simulation studies related to the impact of RNA topology on virus assembly, the focus has been on the importance of the degree of branching, ignoring the impact of base-pairing on the RNA Kuhn length and linear charge density.
As noted above, the Kuhn length of single stranded RNA under physiological conditions of monovalent salt is between one and two $nm$ depending on the ionic strength [@Chen2011], while that of a double stranded RNA is about 140 $nm$ [@Kebbekus1995; @Abels2005]. The average duplex length of viral RNA is about six nucleotide pairs [@Fang2011], which corresponds to about five $nm$. This value is much smaller than the persistence length of double stranded RNA[@Yoffe2008], suggesting that viral RNA can be modeled as a flexible polymer with an average Kuhn length of about six paired nucleotides. There are of course also loop sequences that in our model act as end, hinge and branching points, but how this translates into an effective Kuhn length for the entire branched chain representation of the RNA is unclear. Plausibly, the effective Kuhn length of the internally hybridized chain should be larger than that of the equivalent unstructured non-hybridized chain. Furthermore, another major difference between the linear and branched (base-paired) ssRNA structures seems to be the linear charge density, which doubles for the latter on account of base pairing (hybredization).
In this paper, we vary the degree of branching as well as the effective Kuhn length and linear charge density of a model RNA, and study their impact on the optimal length of encapsulated genome by capsid proteins. We find that as we increase the chain stiffness or Kuhn length the free energy of encapsulation of RNA becomes less negative than that of a linear chain, at least under certain conditions. Hence, a larger Kuhn length, associated with base-pairing, might decrease the efficiency of packaging of RNA compared to a linear polymer. In contrast, our results indicate that increasing the linear charge density improves the efficiency of packaging of both linear and branched polymers. Thus base-pairing has two competing effects: it makes the chain stiffer, which increases the work required to encapsidate the chain, but at the same time it increases the linear charge density that lowers the encapsidation free energy and augment the packaging efficiency. These results are consistent with the experiments of Beren [*et al.*]{} [@Beren2017], in which the linear RNA, PolyU, outcompetes the cognate RNA of CCMV when they are both in solution with a limited amount of capsid proteins of CCMV, that is, sufficient to encapsidate either PolyU or CCMV RNA but not both.
The remainder of this paper is organized as follows. In the next section, we introduce the model and present the equations that we will employ later. In Section III, we present our results and discuss the impact of the Kuhn length on the capsid stability and optimal length of encapsidated genome in Section IV. Finally, in Section V, we present our conclusion and summarize our findings.
Model
=====
To obtain the free energy associated with a genome trapped inside a spherical capsid, we consider RNA as a generic flexible branched polyelectrolyte that interacts with positive charges residing on the inner surface of the capsid. We focus on the case of annealed branched polymers as the degree of branching of RNAs, a statistical quantity, can be modified by its interaction with the positive charges on the capid proteins [@McPherson]. Within mean-field theory, the free energy of a negatively charged chain in a salt solution confined inside a positively charged spherical shell can be written as [@Borukhov; @Gonca2014; @Gonca2016; @adsorption2015; @Erdemci2016; @Venky2016; @Li2017] $$\begin{gathered}
\label{free_energy}
\beta F = \!\!\int\!\! {\mathrm{d}^3}{{\mathbf{r}}}\Big[
\tfrac{\ell^2}{6} |{\nabla\Psi({\bf{r}})}|^2
+\frac{1}{2}\upsilon \Psi^4 ({\mathbf{r}})+W\big[\Psi({\bf{r}}) \big]\\
-\tfrac{ 1}{8 \pi \lambda_B} |{\nabla \beta e \Phi({\bf{r}})}|^2
-2\mu\cosh\big[\beta e \Phi({\bf{r}})\big]
+ \beta \tau \Phi({\bf{r}})\Psi^2({\bf{r}})
\Big]\\
+ \int\!\! {\mathrm{d}^2}r \Big[ \beta \sigma \, \Phi({\bf{r}}) \Big].\end{gathered}$$ with $\beta$ the inverse of temperature in units of energy, $v$ the effective excluded volume per monomer, $\lambda_B=e^2 \beta/4 \pi \epsilon$ the Bjerrum length, $e$ the elementary charge, $\mu$ the number density of monovalent salt ions, and $\tau$ the charge of the statistical Kuhn segment of the chain. The dielectric permittivity of the medium $\epsilon$ is assumed to be constant [@Janssen2014]. The quantity $\ell$, the Kuhn length of the polymer, is defined as an effective stiffness averaged over the entire sequence along the genome. Further, the fields $\Psi(r)$ and $\Phi(r)$ describe the square root of the monomer density field and the electrostatic potential, respectively, and the term $W[\Psi]$ corresponds to the free energy density of an annealed branched polymer as described in Eq. \[W\_branched\] below.
As discussed in the Introduction, the secondary structure of the RNA molecules contain considerable numbers of junctions of single-stranded loops from which three or more duplexes exit. This makes RNA act effectively as a flexible branched polymer in solution. While the Kuhn length for a single stranded, non self-hybridized ssRNA is a few nanometers and that for a double stranded RNA is about 140 nanometers, the Kuhn length of viral RNA is not well determined, as we discussed above. In the absence of exact measurements, we employ an average or effective value for $\ell$, which presumably will be larger if the number of consecutive base pairs (duplexes) between single stranded segments or stem loops along the RNA is larger. Further, we consider the limit of long chains consisting of a very large number of segments $N \rightarrow \infty$ for our confined chains, where $N$ denotes the number of segments. In this formal limit, we employ the ground-state dominance approximation implicit in Eq. (1), as it has proven to be accurate provided $N \gg 1$, i.e., for very long chains [@deGennes1979]. We specify below the connection between the number of segments and the number of nucleotides that make up the RNA, differentiating between self-hybridized and non self-hybridized RNAs.
The first term in Eq. is the entropic cost of deviation from a uniform chain density and the second term describes the influence of excluded volume interactions. The last two lines of Eq. are associated with the electrostatic interactions between the chain segments, the capsid and the salt ions at the level of Poisson-Boltzmann theory [@Siber2008; @Borukhov; @Shafir]. The term $W[\Psi]$ represents the free energy density associated with the annealed branching of the polymer [@Lubensky; @Nguyen-Bruinsma; @Lee-Nguyen; @Elleuch], $$\begin{aligned}
\label{W_branched}
W[\Psi]&= -\frac{1}{\sqrt{\ell^3}}(f_e\Psi+\frac{\ell^3}{6} f_b \Psi^3),\end{aligned}$$ where $f_e$ and $f_b$ are the fugacities of the end and branched points of the annealed polymer, respectively [@adsorption2015]. Note that the stem-loop or hair-pin configurations of RNA are counted as end points. The quantity $\frac{1}{\sqrt{\ell^3}} f_e\Psi$ indicates the density of end points and $\frac{\sqrt{\ell^3}}{6} f_b \Psi^3$ the density of branch points. The number of end $N_e$ and branched points $N_b$ are related to the fugacities $f_e$ and $f_b$, respectively, and can be written as $$\begin{aligned}
\label{NeNb}
N_e =- \beta f_e \frac{\partial{F}}{\partial{f_e}} \qquad {\rm and} \qquad N_b =- \beta f_b \frac{\partial{F}}{\partial{f_b}}.\end{aligned}$$ There are two additional constraints in the problem. Note first that the total number of monomers (Kuhn lengths) inside the capsid is fixed [@deGennes; @Hone], $$\label{constraint}
N = \int {\mathrm{d}^3}{\bf{r}} \; \Psi^2 ({\bf{r}}).$$ We impose this constraint through a Lagrange multiplier, $\mathcal{E}$, introduced below. Second, there is a relation between the number of the end and branched points, $$\label{branch_constraint}
N_e = N_b+2,$$ as there is only a single polymer in the cavity that by construction has no closed loops as it has to mimic the secondary structure of an RNA. The polymer is linear if $f_b=0$, and the number of branched points increases with . For our calculations, we vary $f_b$ and find $f_e$ through Eqs. and . Thus, $f_e$ is not a free parameter.
Varying the free energy functional with respect to the monomer density field $\Psi(r)$ and the electrostatic potential $\Phi(r)$, subject to the constraint that the total number of monomers inside the capsid is constant [@Hone], we obtain two self-consistent non-linear differential equations, which couple the monomer density with the electrostatic potential in the interior of the capsid. The resulting equations are
\[eq:diff\] $$\begin{aligned}
&\frac{\ell^2}6\nabla^2\Psi=-\mathcal{E}\Psi({\mathbf r})+ \beta\tau \Phi({\mathbf r})\Psi({\mathbf r})+\upsilon\Psi^3+\frac{1}{2}\frac{\partial{W}}{\partial{\Psi}} \\
% &\hspace{2cm} -\frac{f_e}{2\sqrt{\ell^3}}-\frac{\sqrt{\ell^3}}{4}f_b\Psi^2,\label{eq:branchdiff_a}\\
&\tfrac{\beta e^2}{4\pi \lambda_B} \nabla^2\Phi_{in}({\mathbf r}) = 2\mu e\sinh \beta e\Phi_{in}({\mathbf r}) \! - \! \tau \Psi^2({\mathbf r}) \\
&\tfrac{\beta e^2}{4\pi \lambda_B} \nabla^2\Phi_{out}({\mathbf r}) = 2\mu e\sinh \beta e\Phi_{out}({\mathbf r}) \! \label{eq:diff_b}\end{aligned}$$
with $\mathcal{E}$ the earlier mentioned Lagrange multiplier enforcing the fixed number of monomers in the cavity. The boundary conditions for the electrostatic potential inside and outside of the spherical shell of radius $R$ are,
\[eq:BCEL\] $$\begin{aligned}
%\left\{
%\begin{array}{ll}
{\hat n\cdot}{\nabla \Phi_{in}}\mid_{r=R}-{\hat n\cdot}{\nabla \Phi_{out}}\mid_{r=R}&= {{4\pi\lambda_B}}{\sigma}/\beta e^2\label{eq:BCEL_a}\\
\Phi_{in}({\bf r})\mid_{r=R}&=\Phi_{out}({\bf r})\mid_{r=R}\label{eq:BCEL_b}\\
\Phi_{out}(\bf {r})\mid_{r=\infty}&=0. %\end{array}
%\right.\end{aligned}$$
The boundary condition (BC) for the electrostatic potential is obtained by minimizing the free energy assuming the surface charge density $\sigma$ is fixed. The concentration of the polymer outside of the capsid is assumed to be zero. The BC for the inside monomer density field $\Psi$ is of Neumann type (${\hat n\cdot}{\nabla \Psi}|_s=0$) that can be obtained from the energy minimization [@Hone] but our findings are robust and our conclusion do not change if we impose the Dirichlet boundary condition $\Psi(r)\mid_{r=R}=0$. The former represent a neutral surface, whilst the latter a repelling surface.[@deGennes1979]
![Genome density profile as a function of distance from the capsid center for a linear polymer with $l=1 nm$ (solid line), $l=2 nm$ (dashed line) and $l=4 nm$ (dotted line). Other parameters used correspond to a $T=3$ virus: the total capsid charges on capsid $Q_c = 1800 e$, the strength of excluded volume interaction $\upsilon =0.05 nm^3$, the fugacity $f_b =0$, the quantity $\mu$ corresponds to a salt concentration of $100 mM$, the capsid radius $R =12 nm$, the temperature $T=300 K$ and total number of nucleotides for all three cases equals 1000. []{data-label="profile"}](profile.pdf){width="45.00000%"}
Results
=======
We solved the coupled equations given in Eqs. for the $\Psi$ and $\Phi$ fields, subject to the boundary conditions in Eqs. through a finite element method (FEM). The polymer density profiles $\Psi^2$ as a function of the distance from the center of the shell, $r$, are shown in Fig. \[profile\] for different values of the RNA stiffness $\ell$ and a fixed number of nucleotides, presuming the RNA not to have any secondary structure. Note that for simplicity we assume that a linear chain with $\ell=1$ $nm$ contains one nucleotide and carries one negative charge, so $\tau=-e$. $\ell=2$ $nm$ has two nucleotides with two negative charges and so on. Thus in our figures the numerical value of $\ell$ also indicates the number of nucleotides in one Kuhn length for linear chains. For the three plots in Fig. \[profile\], the total number of nucleotides is calculated using Eq. \[constraint\] and is equal to 1000. It is worth mentioning that Eq. \[constraint\] gives us the total number of Kuhn lengths $N$ and we multiply it by $\ell$ the number of nucleotides along one Kuhn length to obtain the total number of nucleotides.
As illustrated in the figure, the polymer density becomes larger at the wall as the Kuhn length decreases, even though the linear charge density is fixed. In all plots for Fig. \[profile\] we assumed that the excluded volume is kept constant. Arguably, the excluded volume parameter $\upsilon$ depends on $\ell$, and usually it is assumed that $\upsilon \propto \ell^3$[@deGennes1979]. As we will discuss in Sect. \[discussion\], our conclusions about the role of stiffness in the encapsidation free energy are robust and should not sensitively depend on the strength of the excluded volume interaction.
![Encapsidation free energy of a linear polymer as a function of number of nucleotides for $\ell = 1 nm$ (solid line), $\ell = 2 nm$ (dashed line) and $\ell = 4 nm$ (dotted line). As the stiffness $\ell$ increases, the optimal number of nucleotides moves towards shorter chains. The quantity $\tau$ indicates the number of negative charges in one Kuhn segment. Other parameters used are the total number of charges on the capsid $Q_c = 1800$, the excluded volume parameter $\upsilon =0.05 nm^3$, the quantity $\mu$ corresponds to a salt concentration of $100 mM$, the radius of the cavity of the capsid $R=12 nm$ and the absolute temperature $T=300 K$.[]{data-label="Flin"}](linfixedvolnew.pdf){width="45.00000%"}
To investigate the packaging efficiency of a linear chain as a function of its stiffness, we obtained the free energy of the encapsidation of the linear polymer model as a function of number of nucleotides for different values of $\ell$, as illustrated in Fig. \[Flin\]. The figure shows that the optimal number of nucleotides trapped in the shell increases as $\ell$ decreases. We emphasize again that since we assumed that the size of a single nucleotide is about one $nm$, the numerical value of $\ell$ represents the number of nucleotides within one Kuhn length. This implies that the number of nucleotides and hence the number of charges per Kuhn segment should increase as the Kuhn length increases. For example, in our parametrization $\ell = 4 nm$ represents four nucleotides (resulting in $\tau = - 4 e$). We observe the same behavior for the free energy of branched polymers, that is, increasing $\ell$ causes the optimal length of genome to move towards shorter chains. Obviously the stiffness value $\ell$ is larger for the RNAs whose average number of base pairs in the duplex segments is larger.
The concept of the number of nucleotides per Kuhn length is trickier to implement for the branched polymers taken as model for self-hybridized ssRNA. For example, a branched polymer with the Kuhn length $\ell = 1 nm$ represents in our model description two nucleotides and a charge of $\tau = - 2 e$. When the average number of base pairs is about 8 in duplex segments of an ssRNA, we consider the Kuhn length is about eight $nm$, but the number of nucleotides and number of charges per Kuhn length $\tau$ will be 16. Thus, in our prescription of the self-hybridized ssRNA the number of nucleotides is twice the value of $\ell$ within a Kuhn length as a result of base pairing. We also examined the impact of the fugacity on the optimal number of nucleotides. There is a direct relation between the fugacity and the number of branched points: As the fugacity increases the number of branched points of RNA increases too, see [@Gonca2014; @Gonca2016; @Li2017]. Figure \[bradiffb\] illustrates that the optimal number of nucleotides increases and the encapsidation free energy becomes more negative, indicating a more stable complex, as the fugacity of branching and hence the number of branch points increases. The solid line in the figure shows the free energy of a linear polymer. For the case shown in the figure, the Kuhn length of the linear chain is $\ell=1 nm$ but that for the branched polymers $\ell=4 nm$, corresponding to four base-paired nucleotides. The number of charges within one Kuhn length then is $\tau=-8 e$.
Figure \[bradiffb\] reveals that the free energy of the linear chain is lower than that of the branched one in certain regions of parameter space. For example, for a branched polymer with fugacity $f_b=0.1$, $\ell=4 nm$ and $\tau=-8 e$ (dotted line), the encapsidation free energy of a linear chain with $\ell=1 nm$ and $\tau = -e$ is always lower than that of the branched polymer, and thus, in a head-to-head competition with a limited number of proteins, the linear chain will be the one that is preferentially encapsidated by capsid proteins. This shows that the work of compaction of linear chains could be lower than that of a branched polymer, depending on the stiffness and the degree of branching of the polymers involved. Note that for a fixed $\ell$ while the number of branch points ($f_b$) increases, at some point, the branched polymers outcompetes the linear polymer for binding to capsid proteins, as is illustrated in the figure.
![ As the fugacity $f_b$ (and hence the number of branched points) increases, the optimal number of nucleotides moves towards longer chains. Other parameters are $Q_c = 1800e$, $\upsilon = 0.05 nm^3$, the quantity $\mu$ corresponds to a salt concentration of $100 mM$, $R=12 nm$ and $T=300 K$.[]{data-label="bradiffb"}](branchedfbchange.pdf){width="45.00000%"}
We next studied the free energy of a branched polymer with a fixed fugacity for different values of the stiffness $\ell$. As illustrated in Fig. \[linwin\] for a fugacity $f_b=0.1$, the linear chain (solid) “looses" to a branched one when four nucleotides have formed two base pairs with $\ell=2 nm$ and $\tau=-4 e$ (dashed line). However, the figure shows that as $\ell$ increases, for $\ell=4 nm$ and $8 nm$ (dotted and dotted-dashed lines), their encapsidation free energies become larger than that of the linear chain, indicating that in a head-to-head competition the linear polymer will be encapsidated. Thus, if the average number of nucleotides in duplex segments increases, it becomes energetically more costly to confine RNA inside the capsid.
![ Other parameters used are $Q_c = 1800 e$, $\upsilon =0.05 nm^3$, the quantity $\mu$ corresponds to a salt concentration of $100 mM$, $R=12 nm$ and $T = 300 K$.[]{data-label="linwin"}](linearwins.pdf){width="45.00000%"}
Discussion
==========
![[]{data-label="chargedensity"}]({lbchangetau}.pdf){width="45.00000%"}
Recent experiments emphasized on the crucial role of the RNA topology in the efficiency of virus assembly. As noted in the introduction, Comas-Garcia [*et al.*]{} [@Comas] have shown that CCMV capsid proteins exclusively encapsidate BMV RNA in the presence of the cognate CCMV RNA under conditions where there is a limited number of capsid proteins in solution. The simulations and analytical studies performed in Refs. [@Gonca2014; @Gonca2016; @Li2017; @elife; @GoncaPRL2017] are consistent with these results: the viral RNA with a larger degree of branching has a competitive edge over the other viral RNAs or non-viral randomly branched RNAs, keeping all other chain quantities equal.
Indeed, all mean-field theories, numerical calculations and simulations up to now have indicated that the encapsidation free energy of both annealed and quenched branched polymers is significantly lower than that of linear polymers. This suggests that if there are equal amounts of linear and branched polymers in a solution, but there are sufficient capsid proteins to encapsulate exclusively half of the genomes in solutions, only the branched polymer is encapsidated by capsid proteins. Nevertheless, according to a series of more recent experiments by Beren *et al.* [@Beren2017] in a head-to head competition between a linear (polyU) chain and CCMV RNA of equal length, surprisingly, and in contrast to theoretical predictions, the linear chain outcompetes the cognate RNA.
While previous theoretical studies have focused on the scaling behavior of linear and branched flexible polymers [@Vanderschoot2009; @Gonca2014; @Gonca2016; @Li2017; @Siber2008; @SiberZandi2010; @GoncaPRL2017], in this paper we study the impact of the stiffness or Kuhn length on the encapsidation of RNA by capsid proteins. In general the duplexed segments of viral RNA contain on average about five to six base-pairs [@Fang2011]. Note that some studies show that viral RNAs must have between 60 and 70 per cent of their nucleotides in duplexes, so the linear charge density is almost a factor of two larger and the effective chain length about twice shorter [@BORODAVKA2016]. We argue that while the base pairing on the one hand makes the RNA more compact, on the other hand it increases the effective Kuhn length or the statistical length of the polymer unit. This leads to an increase in the work of compaction of the flexible chain by capsid proteins, which is directly related to the encapsidation free energy of the polymer as plotted in Fig. \[linwin\]. We emphasize again that the findings of this paper is not in contradiction with the previous studies: The more strongly branched a polymer is, the more competitive it becomes to be encapsidated by capsid proteins. However, in this work we show that because of base-pairing, the RNA also becomes stiffer and under appropriate conditions can no longer outcompete the linear polymer for binding to capsid proteins. Since branching due to base-pairing causes both the stiffness and the linear charge density of an otherwise linear polymer to increase, one might wonder which effect, higher charge density or larger stiffness, makes the viral RNA less competitive than a linear polymer. Figure \[chargedensity\] distinguishes the effect of stiffness and charge density. The dashed lines in the figure correspond to linear polymers with $\ell=1 nm$ but different numbers of charges per Kuhn segment $\tau=-e,-4e,-10e$. In the plots, the longer the dashes are, the higher the charge density is. As illustrated in the figure, the encapsidation free energy becomes lower as the charge density increases. The charge density has the same impact on the encapsidation free energy of branched polymers. Figure \[chargedensity\] shows that as the charge density of branched polymer increases (dotted lines), their free energy decreases. The more distance between the dots, the higher the charge density of the branched polymer. Quite interestingly, the figure shows that the effect of stiffness overshadows the impact of charge density. A branched polymer with the stiffness of $\ell=2 nm$ and charge density of $\tau=-4e$ or $-10e$ has a higher free energy than a linear polymer with the stiffness of $\ell=1 nm$ but the charge density of $\tau=-4e$. These examples do not correspond to “real” RNA as it is not possible to increase the number of charges to more than $2e$ per base pair, but they clarify that base-pairing has three competing effects. First, it makes RNA stiffer, which increases the work of encapsidation but, second, in parallel gives rise to the branching effect and, third, a higher charge density, which both lowers the encapsidation free energy and enhances the packaging efficiency of RNA by capsid proteins.
Another important point to consider, is the change in the excluded volume interaction that must somehow be connected with the variation in the Kuhn length. We repeated the calculations done for Fig. \[linwin\], but considered the excluded volume effect, which approximately goes as $\ell^3$ [@deGennes1979]. We found that our conclusion is robust and that the excluded volume interaction only slightly modifies the boundary in the parameter space where the linear polymers are able to outcompete the branched ones. The results of this study can explain the intriguing findings of the experiments of Beren *et al.* [@Beren2017] in which the unstructured polyU RNA is preferentially packaged and outcompetes native RNA CCMV, despite the fact that viral RNAs have more branch points and as such have a more compact structure. Last but not least we note that the interaction of RNA with capsid proteins could modify the preferred curvature of proteins and result into the capsid of different sizes and $T$ numbers as demonstrated in [@Beren2017]. However, since very little is known about this effect, in this paper we exclusively focused on the impact of RNA stiffness resulting from its base pairing in the RNA encapsidation free energy.
Conclusions
===========
Results of our field theory calculations have shown that competition between different forms of RNA for encapsulation by virus coat proteins is a complex function of the degree of branching, effective stiffness of the polymer, linear charge density and excluded volume interactions. The conclusion of previous works that the more branched an RNA is on account of its secondary, base-paired structure, the larger the competitive edge it has to be encapsulated in the presence of coat proteins needs to be refined. Under appropriate conditions of linear charge density and effective chain stiffness, we find that a linear chain may in fact outcompete even the native RNA of a virus, as was recently also shown experimentally. Of course, our conclusions are based on coarse-grained model in which the RNA binding domains of the coat proteins are represented by a smooth, uniformly charged wall. In future work we intend to more realistically model these polycationic tails that form a complex with the polynucleotide. Of particular interest here is the impact of excluded volume interactions between these tails and the polynucleotide.\
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Jef Wagner for useful discussions. This work was supported by the National Science Foundation through Grant No. DMR -1719550 (R.Z.).
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abstract: 'Hyperspectral image is a substitution of more than a hundred images, called bands, of the same region. They are taken at juxtaposed frequencies. The reference image of the region is called Ground Truth map (GT). the problematic is how to find the good bands to classify the pixels of regions; because the bands can be not only redundant, but a source of confusion, and decreasing so the accuracy of classification. Some methods use Mutual Information (MI) and threshold, to select relevant bands. Recently there’s an algorithm selection based on mutual information, using bandwidth rejection and a threshold to control and eliminate redundancy. The band top ranking the MI is selected, and if its neighbors have sensibly the same MI with the GT, they will be considered redundant and so discarded. This is the most inconvenient of this method, because this avoids the advantage of hyperspectral images:: some precious information can be discarded. In this paper we’ll make difference between useful and useless redundancy. A band contains useful redundancy if it contributes to decreasing error probability. According to this scheme, we introduce new algorithm using also mutual information, but it retains only the bands minimizing the error probability of classification. To control redundancy, we introduce a complementary threshold. So the good band candidate must contribute to decrease the last error probability augmented by the threshold. This process is a wrapper strategy; it gets high performance of classification accuracy but it is expensive than filter strategy.'
author:
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title: 'Band Selection and Classification of Hyperspectral Images using Mutual Information: An algorithm based on minimizing the error probability using the inequality of Fano'
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Hyperspectral images, classification, feature selection, error probability, redundancy.
Introduction
============
In the feature classification domain, the choice of data affects widely the results. For the Hyperspectral image, the bands don’t all contain the information; some bands are irrelevant like those affected by various atmospheric effects, see Figure.4, and decrease the classification accuracy. And there exist redundant bands to complicate the learning system and product incorrect prediction \[14\]. Even the bands contain enough information about the scene they may can’t predict the classes correctly if the dimension of space images, see Figure.3, is so large that needs many cases to detect the relationship between the bands and the scene (Hughes phenomenon) \[10\]. We can reduce the dimensionality of hyperspectral images by selecting only the relevant bands (feature selection or subset selection methodology), or extracting, from the original bands, new bands containing the maximal information about the classes, using any functions, logical or numerical (feature extraction methodology) \[11\]\[9\]. Here we focus on the feature selection using mutual information. Hyperspectral images have three advantages regarding the multispectral images \[6\], see Figure.1\
\
**First:** the hyperspectral image contains more than a hundred images but the multispectral contains three at ten images.\
**Second:** hyperspectral image has a spectral resolution (the central wavelength divided by de width of spectral band) about a hundred, but that of multispectral is about ten.\
**Third:** the bands of a hyperspectral image is regularly spaced, those of multispectral image is large and irregularly spaced.\
**Comment:** when we reduce hyperspectral images dimensionality, we must save the precision and high discrimination of substances given by hyperspectral image.\
![Precision an dicrimination added by hyperspectral images[]{data-label="fig_sim"}](figure1){width="3in"}
In this paper we use the Hyperspectral image AVIRIS 92AV3C (Airborne Visible Infrared Imaging Spectrometer). \[2\]. It contains 220 images taken on the region “Indiana Pine” at “north-western Indiana”, USA \[1\]. The 220 called bands are taken between 0.4µm and 2.5µm. Each band has 145 lines and 145 columns. The ground truth map is also provided, but only 10366 pixels are labeled fro 1 to 16. Each label indicates one from 16 classes. The zeros indicate pixels how are not classified yet, Figure.2.\
![The Ground Truth map of AVIRIS 92AV3C and the 16 classes []{data-label="fig_sim"}](figure2){width="3in"}
The hyperspectral image AVIRIS 92AV3C contains numbers between 955 and 9406. Each pixel of the ground truth map has a set of 220 numbers (measures) along the hyperspectral image. This numbers (measures) represent the reflectance of the pixel in each band. So the pixel is shown as vector off 220 components. Figure.3 shows the vector pixel’s notion \[7\]. So reducing dimensionality means selecting only the dimensions caring a lot of information regarding the classes.
![The notion of pixel vector []{data-label="fig_sim"}](figure3){width="3in"}
We can also note that not all classes are carrier of information. In Figure. 4, for example, we can show the effects of atmospheric affects on bands: 155, 220 and other bands. This hyperspectral image presents the problematic of dimensionality reduction.
Mutual information based selection
==================================
In this paragraph we inspect a recent method called band selection scheme using mutual information, and a rejection bandwidth algorithm to eliminate redundancy \[3\]. \[7\].
Definition of mutual information
--------------------------------
This is a measure of exchanged information between tow ensembles of random variables A and B : $$I(A,B)=\sum\;log_2\;p(A,B)\;\frac{p(A,B)}{p(A).p(B)}$$
Considering the ground truth map, and bands as ensembles of random variables, we calculate their interdependence.
Geo \[3\] uses also the average of bands 170 to 210, to product an estimated ground truth map, and use it instead of the real truth map. Their curves are similar. Figure 4.
Bands selection using mutual information
----------------------------------------
From the mutual information curve, we can make threshold, and we retain the bands that have mutual information value above threshold. But the adjacent bands are possibly redundant. Geo\[3\] propose an algorithm to eliminate redundancy. It’s described in \[3\] as following: “Let *B*$_{m}$ be the band that maximizes the mutual information. And N the number of *B*$_{m}$ neighboring bands. We define: $$d(n)=\Delta (MI(n)-MI(n-1))$$ If *max*$_{n}$d(n) is down to a $threshold$ only, *B*$_{m}$ is retained”, i.e. its N neighbors will be discarded, because they my be redundante . Let X be the number of bands to be selected. At some point in the selection process, let S be the set of selected bands, and let R be the set of remaining bands. We initialize the process with SS=Ø and R=[1,2,,,220]{}.
Select *band index*$_{s}$ $S$=*argmax*$_{s}$ MI(s) Neighbours set *N*=$\{n|n=S-(B+1),…,S,…,S+B $}
$SS\gets \textit{SS} \cup \textit{S}$ and $R\gets \textit{R} \setminus\textit{SS}\setminus\textit{N}$ $SS\gets \textit{SS} \cup \textit{S}$ and $R\gets \textit{R} \setminus\textit{SS}$
For more details refer to \[3\].
Discussion and critics of method
--------------------------------
This algorithm is applied at mutual information calculated with the estimated ground truth map Figure.4. Like at \[3\] 50The most inconvenient of this method is how it measures redundancy: small values of d(n)=Δ(MI(n)-MI(n-1)) doesn’t necessary an expression of redundancy. It’s seed at \[3\] that the neighboring bands are possibly redundant. So with this method, the advantage (the precision viewed at section I) of hyperspectral images regarding the multispectral images is avoid, because the precious information can be avoided.
Partial conclusion
------------------
In this section we inspect the effectiveness of mutual information based selection for hyperspectral images. In the next step we use also the mutual information.
One inconvenient if this filter approach is the time made to adjust manually the parameters. It can be expensive. But the most inconvenient is t hat reduce redundancy by eliminating the precision given by the notion of hyperspectral images. We propose now an algorithm avoiding only useless redundancy. We apply this rule: “If a band decreases the error probability, it will be retained even if it contains redundant information”.
![Mutual information of AVIRIS with the Ground Truth map (solid line) and with the ground apporoximated by averaging bands 170 to 210 (dashed line) .[]{data-label="fig_sim"}](figure4){width="3in"}
The mesure of error probability
===============================
Inequality of Fano
------------------
Here we inspect the inequality of Fano \[8\]: $$\;\frac{H(C/X)-1}{Log_2(N_c)}\leq\;P_e\leq\frac{H(C/X)}{Log_2}\;$$with : $$\;\frac{H(C/X)-1}{Log_2(N_c)}=\frac{H(C)-I(C;X)-1}{Log_2(N_c)}\;$$ and : $$P_e\leq\frac{H(C)-I(C;X)}{Log_2}=\frac{H(C/X)}{Log_2}\;$$
The expression of conditional entropy *H(C/X)* is calculated between the ground truth map (i.e. the classes C) and the subset of bands candidates X. Nc is the number of classes. So when the features X have a higher value of mutual information with the ground truth map, (is more near to the ground truth map), the error probability will be lower. But it’s difficult to compute this conjoint mutual information *I(C;X)*, regarding the high dimensionality \[14\].This propriety makes Mutual Information a good criterion to measure resemblance between too bands, like it’s exploited in section II. Furthermore, we will interest at case of one feature candidate X.\
Corollary: for one feature X, as X approaches the ground truth map, the interval *P*$_{e}$ is very small.
Algorithm based on inequality of Fano
--------------------------------------
Our idea is based on this observation: the band that has higher value of Mutual Information with the ground truth map can be a good approximation of it. So we note that the subset of selected bands are the good ones, if thy can generate an estimated reference map, sensibly equal the ground truth map. It’s clearly that’s an Incremental Wrapper-based Subset Selection (IWSS) approach\[16\] \[13\].
Our process of band selection will be as following: we order the bands according to value of its mutual information with the ground truth map. Then we initialize the selected bands ensemble with the band that has highest value of MI. At a point of process, we build an approximated reference map *C*$_{est}$ with the already selected bands, and we put it instead of $X$ for computing the error probability (*P*$_{e}$); the latest band added (at those already selected) must make *P*$_{e}$ decreasing, if else it will be discarded from the ensemble retained. Then we introduce a complementary threshold *T*$_{h}$to control redundancy. So the band to be selected must make error probability less than ( *P*$_{e}$ - *T*$_{h}$) , where *P*$_{e}$ is calculated before adding it. The algorithm following shows more detail of the process:
*Let SS be the ensemble of bands already selected and S the band candidate to be selected. *Build*$_{estimated}C()$ is a procedure to construct the estimated reference map. *P*$_{e}$ is initialized with a value [*P*$_{e}^*$ ]{} . X the number of bands to be selected, $SS$ is empty and $R={1..220}$.*
Select *band index*$_{s}$ $S$=*argmax*$_{s}$ MI(s) $SS\gets \textit{SS} \cup \textit{S}$ and $R\gets \textit{R} \setminus\textit{S}$ *C*$_{est}$= *Build*$_{estimated}C(SS)$ $$Pe=\frac{H(C/C_{est})}{Log_2}\ - \frac{H(C/C_{est})-1}{Log_2(N_c)};\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ $Pe\gets Pe^*$ $SS\gets \textit{SS} \setminus \textit{S}$
Results and analysis
---------------------
We apply this algorithm on the hyperspectral image AVIRIS 92AV3C \[1\], in the same conditions of section 2. 3.
The procedure to construct the estimated reference map *C*$_{est}$ is the same SVM classifier used for classification.
Table. I shows the results obtained for several thresholds. We can see the effectiveness selection bands of our algorithm, and the important effect of avoiding redundancy.
[lllllll]{} ${\mathrm Bands } $& & & & $Th$\
${\mathrm retained }$ & 0.00 & 0.001 & 0.008 &0.015 & 0.020 & 0.030\
10 & 55.43 & 55.43 & 55.58 & 53.09 & 60.06 & 71.62\
18 & 59.09 & 59.09 & 64.41 & 73.70 & 82.62 & 90.00\
20 & 63.08 & 63.08 & 68.50 & 76.15 & 84.36 & -\
25 & 66.02 & 66.12 & 74.62 & 84.41 & 89.06 & -\
27 & 69.47 & 69.47 & 76.00 & 86.73 & 91.70 & -\
30 & 73.54 & 73.54 & 79.04 & 88.68 & - & -\
35 & 76.06 & 76.06 & 81.38 & 92.36 & - & -\
40 & 78.96 & 79.41 & 86.48 & - & - & -\
45 & 80.58 & 80.60 & 89.09 & - & - & -\
50 & 81.63 & 81.20 & 91.14 & - & - & -\
53 & 82.27 & 81.22 & 92.67 & - & - & -\
60 & 86.13 & 86.23 & - & - & - & -\
70 & 86.97 & 87.55 & - & - & - & -\
80 & 89.11 & 89.42 & - & - & - & -\
90 & 90.55 & 90.92 & - & - & - & -\
100 & 92.50 & 93.18 & - & - & - & -\
102 & 92.62 & 93.34 & - & - & - & -\
114 & 93.34 & - & - & - & - & -\
Figure.5 shows more detail of the accuracy curves, versus number of bands retained, for several thresholds. This covers all behaviors of the algorithm:\
**First:** For the highest threshold values (0.1, 0.05, 0.03 and 0.02) we obtain a hard selection: a few more informative bands are selected; the accuracy of classification is 90% with less than 20 bands selected.\
**Second:** For the medium threshold values (0.015, 0.012, 0.010, 0.008, 0.006), some redundancy is allowed, in order to made increasing the classification accuracy.\
**Tired:** For the small threshold values (0.001 and 0), the redundancy allowed becomes useless, we have the same accuracy with more bands.\
**Finally:** for the negative thresholds, for example -0.01, we allow all bands to be selected, and we have no action of the algorithm. This corresponds at selection bay ordering bands on mutual information . The performance is low.\
We can not here that \[15\] uses two axioms to characterize feature selection. Sufficiency axiom: the subset selected feature must be able to reproduce the training simples without losing information. The necessity axiom “simplest among different alternatives is preferred for prediction”. In the algorithm proposed, reducing error probability between the truth map and the estimated minimize the information loosed for the samples training and also the predicate ones.
We note also that we can use the number of features selected like condition to stop the search; so we can obtain an hybrid approach filter-wrapper\[16\].
{width="3.5in"}
**Partial conclusion:** the algorithm proposed is a very good method to reduce dimensionality of hyperspectral images.
We illustrate in Figure .6, the Ground Truth map originally displayed, like at Figure .1, and the scene classified with our method, for threshold as 0.03, so 18 bands selected.\
Table II indicates the classification accuracy of each class, for several thresholds.\
-------------------- ---------------------- ------- ------- ------- ------- ------- -------
${\mathrm Class }$ $Total$ $Th$
${\mathrm pixels }$ 0.00 0.001 0.008 0.015 0.020 0.030
1 : 54 86.96 82.61 86.96 83.96 78.26 86.96
2 : 1434 91.07 89.40 89.54 89.12 88.01 83.96
3 : 834 89.93 90.89 89.69 86.09 83.69 81.53
4 : 234 96.32 83.76 86.32 87.18 87.18 86.32
5 : 597 95.93 95.53 94.34 95.93 95.93 95.53
6 : 747 98.60 98.60 98.32 98.60 98.32 98.32
7 : 26 84.62 84.62 84.62 84.62 84.62 84.62
8 : 489 98.37 98.37 98.78 97.96 98.78 98.78
9 : 20 100 100 100 100 100 100
10: 968 92.15 92.98 91.32 91.74 90.91 89.05
11: 2468 93.84 94.17 92.54 92.71 91.90 91.25
12: 614 91.21 93.49 92.83 92.18 88.93 87.30
13: 212 98.06 98.06 98.06 98.06 98.06 98.06
14: 1294 97.53 97.86 97.22 97.84 97.99 97.53
15: 390 79.52 77.71 75.90 74.10 78.92 64.46
16: 95 93.48 93.48 93.48 93.48 93.48 93.48
-------------------- ---------------------- ------- ------- ------- ------- ------- -------
: Accuracy of classification(%) of each class for numerous thresholds ($Th$)
{width="3.5in"}
**First :**we can not the effectiveness of this algorithm for particularly the classes with a few number of pixels, for example class number 9.
**Second:** we can not that 18 bands (i.e. threshold 0.03) are sufficient to detect materials contained in the region. It’s also shown in Figure .6
**Tired:** one of important comment is that most of class accuracy change lately when the threshold changes between 0.03 and 0.015
Conclusion
===========
In this paper we presented the necessity to reduce the number of bands, in classification of Hyperspectral images. Then we have comment results of a filtering redundancy mutual information based scheme. We carried out their effectiveness to select bands able to classify the pixels of ground truth. And also we have carried out their inconvenient as: the elimination of precision by discarding neighboring bands having sensibly the same mutual information with the ground truth map. We introduce an algorithm also based on mutual information and using a measure of error probability (inequality of Fano). To choice a band, it must contributes to reduce error probability. A complementary threshold is added to avoid redundancy. So each band retained has to contribute to reduce error probability by a step equal to threshold even if it caries a redundant information. We can tell that we conserve the useful redundancy by adjusting the complementary threshold. This algorithm is a feature selection methodology. But it’s a wrapper approach, because we use the classifier to make the estimated reference map. This is a limitation that must be avoided by searching another procedure to estimate reference map more rapidly, in order to implement it in a real time applications. This scheme is very interesting to investigate and improve, considering its performance.
[1]{}
D. Landgrebe, “On information extraction principles for hyperspectral data: A white paper,” Purdue University, West Lafayette, IN, Technical Report, School of Electrical and Computer Engineering, 1997. Téléchargeable ici : http://dynamo.ecn.purdue.edu/ landgreb/whitepaper.pdf.
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abstract: 'We investigate the large-scale circulation (LSC) of turbulent Rayleigh-Bénard convection in a large box of aspect ratio $\Gamma =32$ for Rayleigh numbers up to $Ra=10^9$ and at a fixed Prandtl number $Pr=1$. A conditional averaging technique allows us to extract statistics of the LSC even though the number and the orientation of the structures vary throughout the domain. We find that various properties of the LSC obtained here, such as the wall-shear stress distribution, the boundary layer thicknesses and the wind Reynolds number, do not differ significantly from results in confined domains ($\Gamma \approx 1$). This is remarkable given that the size of the structures (as measured by the width of a single convection roll) more than doubles at the highest $Ra$ as the confinement is removed. An extrapolation towards the critical shear Reynolds number of $Re_s^{\textrm{crit}} \approx 420$, at which the boundary layer (BL) typically becomes turbulent, predicts that the transition to the ultimate regime is expected at $Ra_{\textrm{crit}} \approx \mathcal{O}(10^{15})$ in unconfined geometries. This result is in line with the Göttingen experimental observations. Furthermore, we confirm that the local heat transport close to the wall is highest in the plume impacting region, where the thermal BL is thinnest, and lowest in the plume emitting region, where the thermal BL is thickest. This trend, however, weakens with increasing $Ra$.'
author:
- |
Alexander Blass, Roberto Verzicco,\
Detlef Lohse, Richard J.A.M. Stevens,
- 'Dominik Krug ,'
bibliography:
- 'literature\_turbulence.bib'
title: 'Flow organization in laterally unconfined Rayleigh-Bénard turbulence'
---
Introduction
============
Rayleigh-Bénard (RB) convection [@ahl09; @loh10; @chi12; @xia13] is the flow in a box heated from below and cooled from above. Such buoyancy driven flow is the paradigmatic example for natural convection which often occurs in nature, e.g. in the atmosphere. For that case, a large-scale horizontal flow organization is observed in satellite pictures of weather patterns. Other examples include the thermohaline circulation in the oceans [@rah00], the large-scale flow patterns that are formed in the outer core of the Earth [@gla99b], where reversals of the large-scale convection roll are of prime importance, convection in gaseous giant planets [@bus94] and in the outer layer of the Sun [@mie00]. Thus, the problem is of interest in a wide range of scientific disciplines, including geophysics, oceanography, climatology, and astrophysics.
For a given aspect ratio and given geometry, the dynamics in RB convection are determined by the Rayleigh number $Ra=\beta g\Delta H^3 /(\kappa \nu)$ and the Prandtl number $Pr=\nu/\kappa$. Here, $\beta$ is the thermal expansion coefficient, $g$ the gravitational acceleration, $\Delta$ the temperature difference between the horizontal plates, which are separated by a distance $H$, and $\nu$ and $\kappa$ are the kinematic viscosity and thermal diffusivity, respectively. The dimensionless heat transfer, i.e. the Nusselt number $Nu$, along with the Reynolds number $Re$ are the most important response parameters of the system.
For sufficiently high $Ra$, the flow becomes turbulent, which means that there are vigorous temperature and velocity fluctuations. Nevertheless, a large-scale circulation (LSC) develops in the domain such that, in addition to the thermal boundary layer (BL), a thin kinetic BL is formed to accomodate the no-slip boundary condition near both the bottom and top plates. Properties of the LSC and the nature of the BLs are highly relevant to the theoretical description of the problem. In particular, the unifying theory of thermal convection [@gro00; @gro01; @gro11; @ste13] states that the transition from the classical to the ultimate regime takes place when the kinetic BLs become turbulent. This transition is shear based and driven by the large-scale wind, underlying the importance of the LSC to the overall flow behavior.
So far, the LSC and BL properties have mainly been studied in small aspect ratio cells, typically for $\Gamma=1/2$ and $\Gamma=1$. Various studies have shown that the BLs indeed follow the laminar Prandtl-Blasius (PB) type predictions in the classical regime [@ahl09; @zho10; @zho10b; @ste12; @shi15; @shi17]. Previous studies by, for example [@wag12] and [@sch16], have used results from direct numerical simulations (DNS) in aspect ratio $\Gamma=1$ cells to study the properties of the BLs in detail. [@wag12] showed that an extrapolation of their data gives that for $Pr=0.786$ the critical shear Reynolds number of $420$ is reached at $Ra\approx1.2\times10^{14}$. Despite the wealth of studies in low aspect-ratio domains, many natural instances of thermal convection take place in very large aspect ratio systems, as mentioned above. Previous research has demonstrated that several flow properties are significantly different in such unconfined geometries. @har03 and @har08 performed DNS at $Ra=\mathcal{O}(10^7)$ and $\Gamma =20$. They observed large-scale structures by investigating the advective heat transport and found the most energetic wavelength of the LSC at $4H-7H$. Recently, DNS by [@ste18] for $\Gamma =128$ and $Ra=\mathcal{O}(10^7-10^9)$ also reported ‘superstructures’ with wavelengths of $6-7$ times the distance between the plates. Similar findings were made by @pan18 over a wide range of Prandtl numbers $0.005\leq Pr \leq 70$ and $Ra$ up to $10^7$. It was shown that the signatures of the LSC can be observed close to the wall, which [@par04] described as clustering of thermal plumes originating in the BL and assembling the LSC. @kru20 showed that the presence of the LSC leads to a pronounced peak in the coherence spectrum of temperature and wall-normal velocity. Based on DNS at $\Gamma=32$ and $Ra=\mathcal{O}(10^5-10^9)$, they determined that the wavelength of this peak shifts from $\hat{l}/H \approx 4$ to $\hat{l}/H \approx 7$ as $Ra$ is increased.
[@ste18] have shown that in periodic domains, the heat transport is maximum for $\Gamma=1$ and reduces with increasing aspect ratio up to $\Gamma \approx 4$ when the large-scale value is obtained. They also found that fluctuation-based Reynolds numbers depend on the aspect ratio of the cell. However, other than the structure size, it is mostly unclear how the large-scale flow organization and BL properties are affected by different geometries. Not only is the size of the LSC more than 2 times larger without confinement (note that $\hat{l}$ measures the size of two counter-rotating rolls combined), but also other effects, such as corner vortices, are absent in periodic domains. Therefore one would expect differences in wind properties and BL dynamics. It is the goal of this paper to investigate these differences. Doing so comes with significant practical difficulty due to the random orientation of a multitude of structures that are present in a large box. To overcome this, we adopt the conditional averaging technique that was devised in [@ber20] to reliably extract LSC features even under these circumstances. Details on this procedure are provided in section §\[section\_results\] after a short description of the dataset in §\[section\_method\]. Finally, in §\[shear\_nusselt\] and §\[section\_BL\] we present results on how superstructures affect the flow properties in comparison to the flow formed in a cylindrical $\Gamma=1$ domain [@wag12] and summarize our findings in §\[section\_conclusions\].
Numerical method {#section_method}
================
The data used in this manuscript have previously been presented by [@ste18] and [@kru20]. A summary of the most relevant quantities for this study can be found in Table \[tab:stats\]; note that there and elsewhere we use the free-fall velocity $V_{ff} = \sqrt{g\beta H\Delta}$ as a reference scale. In the following, we briefly report details on the numerical method for completeness. We carried out periodic RB simulations by numerically solving the three-dimensional incompressible Navier-Stokes equations within the Boussinesq approximation. They read: $$\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \bcdot \bnabla \boldsymbol{u} =-\bnabla P
+ \nu
\nabla^2\boldsymbol{u}+\beta g \theta \hat{z},
\label{eqn:Navier}$$ $$\bnabla \bcdot \boldsymbol{u} =0,
\label{eqn:div}$$ $$\frac{\partial \theta}{\partial t} + \boldsymbol{u} \bcdot \bnabla \theta =
\kappa
\nabla ^2 \theta.
\label{eqn:temp}$$ Here, $\boldsymbol{u}$ is the velocity vector, $\theta$ the temperature, and the kinematic pressure is denoted by $P$. The coordinate system is oriented such that the unit vector $\hat{z}$ points up in the wall-normal direction, while the horizontal directions are denoted by $x$ and $y$. We solve (\[eqn:Navier\]) - (\[eqn:temp\]) using AFiD, the second-order finite difference code developed by Verzicco and coworkers [@ver96; @poe15c]. We use periodic boundary conditions and a uniform mesh in the horizontal direction and a clipped Chebyshev-type clustering towards the plates in the wall-normal direction. For validations of the code against other experimental and simulation data in the context of RB we refer to @ver96 [@ver97; @ver03; @ste10; @koo18].
The aspect ratio of our domain is $\Gamma =L/H=32$, where $L$ is the length of the two horizontal directions of the periodic domain. The used numerical resolution ensures that all important flow scales are properly resolved [@shi10; @ste10].
In this manuscript, we define the decomposition of instantaneous quantities into their mean and fluctuations such that $\psi (x,y,z,t)= \Psi (z) + \psi ' (x,y,z,t)$, where $\Psi = \langle \psi (x,y,z,t) \rangle _{x,y,t}$ is the temporal and horizontal average and $\psi '$ the fluctuations with respect to this mean.
![\[fig:spectra\] (a) Premultiplied temperature power spectra $k \Phi _{\theta \theta}$ for $Ra=10^5;10^7;10^9$. The blue line indicates the cut-off wavenumber $k_{\textrm{cut}} = 2/H$ used for the low-pass filtering. The dashed black lines indicate alternative cut-offs ($k_{\textrm{cut}} = 1.8/H $ and $k_{\textrm{cut}} = 2.5/H$) considered in panel (c). The white plusses are located at $k=0.57/\lambda _\theta ^*$ and $z=0.85 \lambda _\theta ^*$ (with $\lambda _\theta ^* =H/(2Nu)$) in all cases, which corresponds to the location of the inner peak [@kru20] (b) Coherence spectra of temperature and wall-normal velocity at mid-height, figure adopted from [@kru20]. The black line illustrates the choice of $k_{\textrm{cut}} = 2/H$ and the legend of figure \[fig:counts\]a applies for the $Ra$-trend. (c) Snapshot of temperature fluctuations for $Ra=10^7$ at mid-height. The black lines show contours of $\theta_L ' =0$ evaluated for different choices of $k_{\textrm{cut}}$.](./Figures/spectra){width="\textwidth"}
![\[fig:method\] Illustration of the conditional averaging method based on simulation data for $Ra=10^7$. (a) Temperature fluctuation field at mid-height and corresponding distance field (right). The black lines correspond to the zero-crossings $\theta _L' =0$ relative to which the distance $d^*$ is defined (see blow-up in panel b). Note that by definition isolines $\theta _L '=0$ correspond to contours of $d=0$ in the distance field. (b) Illustration of the distance definition; for every point $d^*$ is equal to the radius of the smallest circle around that point which touches a $\theta _L' =0$ contour. (c) Illustration of the decomposition of the horizontal velocity $v$ into the parallel $v_p$ and the normal $v_n$ component to the gradient vector $d$. The color scheme in (b) and (c) indicates the $d$-field as in (a).](./Figures/method){width="\textwidth"}
[cccccc]{} $Ra$ & $N_x \times N_y \times N_z$ & $Nu$ & $\hat{l}/H$ & $v_{RMS}/V_{ff}$ & $\lambda _\theta ^*/H$\
\
$ 1\times10^{5} $ & $2048 \times 2048 \times 64$ & 4.35 & 4.4 & 0.2172 & 0.115\
$ 4\times10^{5} $ & $2048 \times 2048 \times 64$ & 6.48 & 4.5 & 0.2214 & 0.077\
$ 1\times10^{6} $ & $3072 \times 3072 \times 96$ & 8.34 & 4.9 & 0.2198 & 0.060\
$ 4\times10^{6} $ & $3072 \times 3072 \times 96$ & 12.27 & 5.4 & 0.2152 & 0.041\
$ 1\times10^{7} $ & $4096 \times 4096 \times 128$ & 15.85 & 5.9 & 0.2107 & 0.032\
$ 1\times10^{8} $ & $6144 \times 6144 \times 192$ & 30.94 & 6.3 & 0.1968 & 0.016\
$ 1\times10^{9} $ & $12288 \times 12288 \times 384$ & 61.83 & 6.6 & 0.1805 & 0.008\
\[1pt\]
[\[tab:stats\]]{}
Conditional Averaging {#section_results}
=====================
Extracting features of the LSC in large aspect ratio cells poses a significant challenge. The reason is that there are multiple large-scale structures of varying sizes, orientation, and inter-connectivity at any given time. It is therefore not possible to extract properties of the LSC by using methods that rely on tracking a single or a fixed small number of convection cells, which have been proven to be successful in analyzing the flow in small [@sun08; @wag12] to intermediate [@ree08] aspect-ratio domains. To overcome this issue, we use a conditional averaging technique developed in [@ber20], where this framework was employed to study the modulation of small-scale turbulence by the large flow scales. This approach is based on the observation of [@kru20] that the premultiplied temperature power spectra $k\Phi_{\theta\theta}$ (shown in figure \[fig:spectra\]a) is dominated by two very distinct contributions. One is due to the ‘superstructures’ whose size (relative to $H$) increases with increasing $Ra$ and typically corresponds to wavenumbers $kH \approx 1-1.5$. The other contribution relates to a ‘near-wall peak’ with significantly smaller structures whose size scales with the thickness of the BL [@kru20]. This implies that this peak shifts to larger $k$ (scaled with $H$) as the BLs get thinner at higher $Ra$. Hence, there is a clear spectral gap between superstructures and small-scale turbulence, which widens with increasing $Ra$, as can readily be seen from figure \[fig:spectra\]a. This figure also demonstrates that a spectral cut-off $k_{\textrm{cut}} = 2/H $ is a good choice to separate superstructure contributions from the other scales over the full $Ra$ range $10^5\leq Ra\leq 10^9$ considered here.
The choice for $k_{\textrm{cut}} = 2/H $ is further supported by considering the spectral coherence $$\gamma ^2 _{\theta w}(k)=\frac{|\Phi_{\theta w}(k)|^2}{\Phi_{\theta \theta}(k)\Phi_{w w}(k)},$$ where $\Phi_{w w}$ and $\Phi_{\theta w}$ are the velocity power spectrum and the co-spectrum of $\theta$ and $w$, respectively. The coherence $\gamma^2$ may be interpreted as a measure of the correlation *per scale*. The results at $z = 0.5H$ in figure \[fig:spectra\]b indicate that there is an almost perfect correlation between $\theta'$ and $w'$ at the superstructure scale. Almost no energy resides at the scales corresponding to the high-wavenumber peak in $\gamma ^2 _{\theta w}$ [see @kru20], such that the coherence there is of little practical consequence. The threshold $k_{\textrm{cut}} = 2/H$ effectively delimits the large-scale peak in $\gamma^2_{\theta w}$ towards larger $k$ for all $Ra$ considered, such that this value indeed appears to be a solid choice to distinguish the large-scale convection rolls from the remaining turbulence. To confirm this, we overlay a snapshot of $\theta'$ with zero-crossings of the low-pass filtered signal (with cut-off wavenumber $k_{\textrm{cut}}$) $\theta'_L$ in figure \[fig:spectra\]c. These contours reliably trace the visible structures in the temperature field. Furthermore, it becomes clear that slightly different choices for $k_{\textrm{cut}}$ do not influence the contours significantly. This is consistent with the fact that only limited energy resides at the scales around $k\approx 2/H$, such that $\theta'_L$ only changes minimally when $k_{\textrm{cut}}$ is varied within that range. In the following, we adopt $k_{\textrm{cut}} = 2/H$ to obtain $\theta'_L$ except when we study the effect of the choice for $k_{\textrm{cut}}$.
We use $\theta'_L$ evaluated at mid-height to map the horizontal field onto a new horizontal coordinate $d$. To obtain this coordinate, first the distance $d^*$ to the nearest zero-crossing in $\theta'_L$ is determined for each point in the plane. This can be achieved efficiently using a nearest-neighbor search. Then the sign of $d$ is determined by the sign of $\theta'_L$, such that $d$ is given by $$d=\operatorname{sgn}(\theta '_L){d^*}.$$ All results presented here are with reference to the lower hot plate. Hence $d<0$ and $d>0$ correspond to plume impacting and plume emitting regions, respectively. The averaging procedure is illustrated in figure \[fig:method\]a,b. Another important aspect is a suitable decomposition of the horizontal velocity component $v$. Figure \[fig:method\]c shows how we decompose $v$ into one component ($v_p$) parallel the local gradient $\nabla d$, and another component ($v_n$) normal to it. This ensures that $v_p$ is oriented normal to the zero-crossings in $\theta'_L$ for small $|d|$, where the wind is strongest. However, at larger $|d|$, the orientation may vary from a simple interface normal, which accounts for curvature in the contours. It should be noted that the $d$-field is determined at mid-height and consequently applied to determine the conditional average at all $z$-positions. This is justified since @kru20 showed that there is a strong *spatial* coherence of the large scales in the vertical direction. Therefore, the resulting zero-contours would almost be congruent if $\theta'_L$ was evaluated at other heights. The time-averaged conditional average is obtained by averaging over points of constant $d$, while we make use of the symmetry around the mid-plane to increase the statistical convergence. Mathematically, the conditioned averaging results in a triple decomposition according to $\psi (x,y,z,t) = \Psi (z) + \overline{\psi}(z,d)+ \widetilde{\psi}(x,y,z,t)$, where the overline indicates conditional and temporal averaging.
Applying the outlined method to our RB dataset results in a representative large-scale structure like the one depicted in figure \[fig:lsc\] for $Ra= 10^7$. In general, we find $\overline{\theta}<0$ with predominantly downward flow for $d<0$, while lateral flow towards increasing $d$ dominates in the vicinity of $d = 0$. In the plume emitting region $d>0$ the conditioned temperature $\overline{\theta}$ is positive and the flow upward. In interpreting the results it is important to keep in mind that the averaging is ‘sharpest’ close to the conditioning location ($d = 0$) and ‘smears out’ towards larger $|d|$ as the size of individual structures varies. We normalize $d$ with $\hat{l}$ to enable a comparison of results across $Ra$. Based on the location of the peak in $\gamma^2$, [@kru20] found that the superstructure size is $\hat{l}=5.9 H$ at $Ra= 10^7$. As indicated, the conditionally averaged flow field in figure \[fig:lsc\] corresponds to approximately half this size.
![\[fig:counts\] (a) PDF of the normalized distance parameter $d/\hat{l}$. (b) Sample velocity profile to illustrate the slope method ($\lambda$) and the level method ($\ell$) used to determine the instantaneous BL thicknesses.](./Figures/slope_counts){width="\textwidth"}
We present the probability density function (PDF) of the distance parameter $d$ in figure \[fig:counts\]a. The data collapse to a reasonable degree, indicating that there are no significant differences in how the LSC structures vary in time and space across the considered range of $Ra$. Visible deviations are at least in part related also to uncertainties in determining $\hat{l}$ via fitting the peak of the $\gamma^2$-curve.
The LSC is carried by $v_p$, which is also supported by the fact that the velocity component normal to the gradient $\nabla d$ averages to zero, i.e. $\overline{v}_n \approx 0 $, for all $d$. The determination of the viscous BL thickness is therefore based on $v_p$ only. We use the ‘slope method’ to determine the viscous ($\lambda _u$) and thermal ($\lambda_\theta$) BL thickness. Both are determined locally in space and time and are based on instantaneous wall-normal profiles of $\theta$ and $v_p$, respectively. As sketched in figure 4b, $\lambda$ is given by the location at which linear extrapolation using the wall-gradient reaches the level of the respective quantity. Here the ‘level’ (e.g. $u_L$ for velocity) is defined as the local maximum within a search interval above the plate. In agreement with [@wag12] we find that the results for both thermal and viscous BL do not significantly depend on the search region when it is larger than $4 \lambda_\theta ^*$. Therefore, we have adopted this search region in all our analyses.
In figure \[fig:T\_vp\]a we present the conditionally averaged temperature $ \overline{\theta}$ as a function of $z/H$ at three different locations of $d/\hat{l}$. Consistent with the conditioning on zero-crossings in $\theta'_L =0$, we find that $\overline{\theta}\approx 0$ for all $z$ at $d = 0$. In the plume impacting ($d/\hat{l} = -0.25$) and emitting ($d/\hat{l} = 0.25$) regions, $\overline{\theta}$ is respectively negative and positive throughout. On both sides, $\overline{\theta}$ attains nearly constant values in the bulk, the magnitude of which is decreasing significantly with increasing $Ra$.
Profiles for the mean wind velocity $\overline{v}_p(z) $ at $d = 0$ are shown in figure \[fig:T\_vp\]b,c. These figures show that the viscous BL becomes thinner with increasing $Ra$, while the decay from the velocity maximum to $0$ at $z/H = 0.5$ is almost linear for all cases. We note that of all presented results the wind profile is most sensitive to the choice of the threshold $k_{\textrm{cut}}$. The reason is that the obtained wind profile depends on both the contour location and orientation. To provide a sense for the variations associated with the choice of $k_{\textrm{cut}}$, we compare the present result at $Ra = 10^7$ to what is obtained using alternative choices ($k_{\textrm{cut}} = 1.8/H$ and $k_{\textrm{cut}} = 2.5/H$) in the inset of figure \[fig:T\_vp\]b. This plot shows that results within the BL are virtually insensitive to the choice of $k_{\textrm{cut}}$ while the differences in the bulk consistently remain below 5%. In panel (c) of figure \[fig:T\_vp\] we re-plot the data from figure \[fig:T\_vp\]b normalized with the BL thickness $\overline{\lambda}_u(d=0)$ and the velocity maximum $\overline{v}_p^{\textrm{max}}$. The figure shows that the velocity profiles for the different $Ra$ collapse reasonably well for $z\lessapprox \overline{\lambda}_u$. A comparison to the experimental data by [@sun08], which were recorded in the center of a slender box with $\Gamma =1$ and $Pr = 4.3$, reveals that, although the overall shape of the profiles is similar, there are considerable differences in the near-wall region. With their precise origin unknown, these discrepancies could be related to the differences in $Pr$ and $\Gamma$ but also to experimental uncertainties.
![\[fig:T\_vp\] (a) Conditioned temperature $\overline{\theta}/\Delta$ at $d=0$ and in the plume impacting $(d/\hat{l}=-0.25)$ and in the plume emitting region $(d/\hat{l}=0.25)$ for various $Ra$, see legend in (c). (b) Wind velocity $\overline{v}_p/V_{ff}$ at $d=0$ versus $z/H$ at the same $Ra$. The inset shows the sensitivity of the results to different choices of $k_{\textrm{cut}}$ in the range $1.8\leq k_{\textrm{cut}} H \leq 2.5$ (same range used in figure \[fig:spectra\]) for $Ra=10^7$. (c) Mean wind velocity normalized by its maximum value for various $Ra$ (see legend). The dashed and dotted black lines in (c) represent experimental data from [@sun08] at $\Gamma =1$ for $Ra=1.25 \times 10^9$ and $Ra=1.07 \times 10^{10}$, respectively.](./Figures/meanvalues2){width="\textwidth"}
![\[fig:timescale\] (a) Timescale $\mathcal{T}$ versus $Ra$ using different methods. The datasets are: The time needed to circulate the flow along a streamline, which passes through $ z^*/H$ at $d=0$ (red circles), see figure \[fig:lsc\]; the timescale calculated with the geometric method of [@pan18] (blue squares). We also show the [@pan18] data itself, which were calculate for the smaller $Pr=0.7$ (black diamonds). (b) Average velocity $v_{\textrm{wind}}$ determined along the streamline chosen in (a), normalized with $v_{RMS}$. (c) Comparison between the length of the streamline and the circumference $\pi (0.25 \hat{l}+0.5 H)$ of the ellipse (geometric method), both used to calculate the respective timescale in (a).](./Figures/timescale_vel){width="\textwidth"}
Another interesting question that we can address based on our results concerns the evolution timescale $\mathcal{T}$ of the LSC. We estimate $\mathcal{T}$ as the time it takes a fluid parcel to complete a full cycle in the convection roll. To do this we compute the streamline that passes through the location $z^*/H$ of the velocity maximum $\overline{v}_p(z^*/H)=\overline{v}_p^{\max}$ at $d=0$ as shown in figure \[fig:lsc\]. The integrated travel time $\mathcal{T}$ along this averaged streamline as a function of $Ra$ is presented in figure \[fig:timescale\]a. We find $\mathcal{T}/T_{ff} \gg 1$, i.e., the typical timescale of the LSC dynamics is much longer than the free-fall time $T_{ff} = \sqrt{H / (\beta g \Delta)}$. Up to $Ra =10^7$ the timescale $\mathcal{T}$ grows approximately according to $\mathcal{T}/T_{ff}
\sim Ra^{0.14}$, but the trend flattens out at $Ra$ beyond that value.
To compare our results to other estimates in the literature, we also adopt the method used by @pan18 to estimate $\mathcal{T}$. These authors assumed the LSC to be an ellipse and set the effective velocity to $\frac{1}{3} \, v_{RMS}$, where the prefactor $1/3$ is purely empirical. The results for the ‘geometric method’ are compared to the corresponding results by @pan18 in figure \[fig:timescale\]a. Results are consistent between the two methods in terms of the order of magnitude. However, the actual values, especially at lower $Ra$, differ significantly, and also the trends do not fully agree. The streamline approach allows us to determine the average convection velocity along the streamline $v_{\textrm{wind}}\equiv \mathcal{L}/\mathcal{T}$, where $\mathcal{L}$ is the length of the streamline. Figure \[fig:timescale\]b show that $v_\textrm{wind}$ is indeed proportional to $v_{RMS}$ with $v_\textrm{wind} \approx 0.45 \, v_{RMS}$ in the considered $Ra$ number regime. In figure \[fig:timescale\]c, we present $\mathcal{L}$ along with the ellipsoidal estimate used in @pan18. From this, it appears that an ellipse does not very well represent the streamline geometry. Further, it becomes clear that it is the difference in the length-scale estimate that leads to the different scaling behaviors for $\mathcal{T}$ in figure \[fig:timescale\]a.
 Normalized shear stress $\overline{\tau} _w$ as a function of $d/\hat{l}$ and (b) mean shear stress $\langle \overline{\tau}_w \rangle _J$ and maximum shear stress $\overline{\tau}_w ^{\max}$ versus $Ra$. The filled symbols show data of the present study ($\Gamma=32$ periodic domain), while the open symbols represent the data of [@wag12] for $\Gamma=1$ with a cylindrical domain. The blue symbols show the maximum shear stress and the red symbols the mean shear stress over the interval $J=\{d/\hat{l} \; | \; d/\hat{l} \in[-0.2 : 0.15] \}$.](./Figures/taucombined_new){width="\textwidth"}
Wall shear stress and heat transport {#shear_nusselt}
====================================
The shear stress $\overline{\tau}_w $ at the plate surface is defined through $$\overline{\tau} _w/\rho = - \nu
\langle \partial _z \overline{v}_p \rangle _t.$$ Here $\partial _z$ is the spatial derivative in wall-normal direction. In figure \[fig:tau\_w\]a we show that the normalized shear stress $ \overline{\tau}_w / \overline{\tau}_w ^{\max}$ as a function of the normalized distance $d/\hat{l}$ is nearly independent of $Ra$. Similar to findings in smaller cells [@wag12], the curves are asymmetric with the maximum ($d/\hat{l} \approx -0.05$) shifted towards the plume impacting region. The value of $ \overline{\tau}_w / \overline{\tau}_w ^{\max}$ drops to about 0.25 in both the plume impacting ($d/\hat{l} = -0.25$) and the plume emitting region ($d/\hat{l} = 0.25$).
We use the good collapse of the $ \overline{\tau}_w / \overline{\tau}_w ^{\max}$ profiles across the full range of $Ra$ considere to separate regions with significant shear from those with little to no lateral mean flow. We define the ‘wind’ region based on the approximate criterion $ \overline{\tau}_w / \overline{\tau}_w ^{\max} \gtrapprox 0.5$, which leads to the interval $J=\{d/\hat{l} \; | \; d/\hat{l} \in[-0.2 : 0.15] \}$ that is indicated by the blue shading in figure \[fig:tau\_w\]a. We use the average over this interval to evaluate the wind properties and indicate this by $\langle \rangle_J$. In figure \[fig:tau\_w\]b the data for mean $\langle \overline{\tau}_w\rangle_J$ and for maximum $\overline{\tau}_w ^{\max}$ wall shear stress are compiled for the full range of $Ra$ considered. Both quantities are seen to increase significantly as $Ra$ increases. Around $Ra=1\times 10^6\textup{--}4\times 10^6$ we can see a transition point at which the slope steepens. For lower $Ra$ the scaling of $\langle \overline{\tau} _w \rangle_{J}$ is much flatter. A fit to the data for $Ra\geq 4\times 10^6$ gives $${\overline{\tau}_w /\rho \over V_{ff}^2} \sim Ra^{0.24},$$for both $\langle \overline{\tau}_w\rangle_J$ and $\overline{\tau}_w ^{\max}$. Overall, we find that the shear stress at the wall due to the turbulent thermal superstructures (in the periodic $\Gamma=32$ domain with $Pr=1$) compares well with the shear stress in a cylindrical $\Gamma =1$ domain by [@wag12] with $Pr=0.786$. Most importantly, the scaling with $Ra$ is the same for both cases. The actual shear stress seems to be somewhat higher in the cylindrical aspect ratio $\Gamma=1$ domain than in the periodic domain in which the flow is unconfined. In part this difference may be related to the difference in $Pr$. Besides that, as we will show in the next section, the shear Reynolds number is slightly lower for the periodic domain than in the confined domain.
 Local heat flux $\overline{Nu}$ at the wall normalized by the global heat flux $Nu$ as function of the normalized spatial variable $d/ \hat{l}$. (b) Values in the impacting ($ -0.3 \leq d/ \hat{l} \leq -0.2$) and emitting ($0.2 \leq d/ \hat{l} \leq 0.3$) range as a function of $Ra$.](./Figures/nucombined){width="\textwidth"}
![\[fig:heattransport\] Large-scale turbulent heat transport term $\widetilde{(\theta w)} / ( Nu \kappa \Delta/H)$ evaluated in (a) the plume impacting region, (b) for small $|d|$ around zero, and (c) in the plume emitting region.](./Figures/heattransport){width="\textwidth"}
Next, we consider the local heat flux at the plate surface, given by $$\overline{Nu}(d) \equiv - {H \over\Delta }\partial _z \overline{\theta}(d),$$ which is plotted in figure \[fig:Nu\]a for the full range of $Ra$. In all cases $\overline{Nu}/Nu$ is higher than one on the plume impacting side ($d<0$). This is consistent with the impacting cold plume increasing the temperature gradient in the BL locally. The fluid subsequently heats up while it is advected along the plate towards increasing $d$ by the LSC. As a consequence, the wall gradient is reduced and $\overline{Nu}$ decreases approximately linearly with increasing $d/\hat{l}$, which is consistent with observations by @ree08 and @wag12. This leads to the ratio $\overline{Nu}/Nu$ dropping below 1 for $d>0$. For increasing $Ra$, the local heat flux becomes progressively more uniform across the full range of $d$. To quantify this, we plot the mean local heat fluxes in the plume impacting and emitting regions, respectively, in figure \[fig:Nu\]b. The former is decreasing while the latter is increasing with increasing $Ra$, bringing the two sides closer. Again, and in both cases, a change of slope is visible in the range of $Ra=1\times 10^6 \textup{--} 4\times 10^6$. In this context it is interesting to note that in a recent study on two-dimensional RB convection [@zhu18c] it was found that at significantly higher $Ra \gtrapprox 10^{11}$ the heat transport in the plume emitting range dominated, reversing the current situation. If we extrapolate the trend for $Ra\geq 4\times 10^6$ in our data, we can estimate that a similar reversal may occur at $Ra \approx \mathcal{O}(10^{12}\textup{--}10^{13})$, see figure \[fig:Nu\]b.
A possible mechanism that might explain this behavior is increased turbulent (or convective) mixing, which can counteract the diffusive growth of the temperature BLs. To check this hypothesis, we plot the heat transport term $\widetilde{(\theta w)} \equiv \overline{w \theta} - \overline{w} \overline{\theta}$ in figure \[fig:heattransport\]. The normalization in the figure is with respect to the total heat flux $Nu$, the plotted quantity reflects the fraction of $Nu$ carried by $\widetilde{(\theta w)}$. It is obvious from these results that the convective transport contributes significantly, even within the BL height $\langle\overline{\lambda}_\theta \rangle_J$. Moreover, this relative contribution is independent of $Ra$ (except for the lowest value considered) in the plume impacting region (see figure \[fig:heattransport\]a). However, figure \[fig:heattransport\]b shows that already around $d = 0 $ the convective transport in the BL increases with increasing $Ra$. This trend is much more pronounced in the plume emitting region $d>0.2$, see figure \[fig:heattransport\]c. Hence, convective transport in the BL plays an increasingly larger role for $d\geq 0$ with increasing $Ra$. Its effect is to mix out the near-wall region, thereby increasing the temperature gradient at the wall. It is conceivable that the increased convective transport in the near-wall region (provided the trend persists) eventually leads to a reversal of the $\overline{Nu} (d)$ trend observed at moderate $Ra$ in figure \[fig:Nu\]a.
Thermal and viscous boundary layers {#section_BL}
===================================
![\[fig:lambda\] (a) Thermal BL thickness $\overline{\lambda} _\theta$ normalized by the estimated thermal BL thickness $\lambda _\theta ^*$ and (b) viscous BL thickness $\overline{\lambda} _u$ normalized by the mean viscous BL thickness in the interval $d/ \hat{l} \in J$ versus normalized distance $d/ \hat{l}$. The color indicates the Rayleigh number, see legend.](./Figures/lambda){width="\textwidth"}
Next, we study how the BL thicknesses $\lambda_\theta$ and $\lambda_u$ vary along the LSC. In figure \[fig:lambda\]a we present $\overline{\lambda}_\theta$, normalized by $\lambda_\theta ^*$. As expected from figure \[fig:Nu\], $\overline{\lambda}_\theta$ is generally smaller in the plume impacting region and then increases along the LSC. However, unlike $\overline{Nu}$, $\overline{\lambda}_\theta$ is not determined by the gradient alone but also depends on the temperature level (see figure \[fig:counts\]b) such that differences arise. Specifically, $\overline{\lambda}_\theta/\lambda_\theta ^*$ is rather insensitive for $Ra \geq 4\times 10^6$ in the plume impacting region ($d/\hat{l}<-0.1$). Furthermore, for $Ra \geq 10^7$, the growth of the thermal BL with $d/\hat{l}$ comes to an almost complete stop beyond $d= 0$, which is entirely consistent with the conclusions drawn in the discussion on $\widetilde{(\theta w)}$ above. Finally, we note that $\overline{\lambda}_\theta$ is generally larger than the estimate $\lambda_\theta ^*$, which agrees with previous observations by [@wag12].
 Mean BL thicknesses versus $Ra$ for the present data at $\Gamma = 32$ (filled symbols) and those of @wag12 with $Gamma = 1$ (open symbols). (b) Most probable BL ratio $\overline{\Lambda} ^{MP} = \lambda_\theta / \lambda_u $ versus normalized distance $d/ \hat{l}$ for various $Ra$.](./Figures/lambdaratio){width="\textwidth"}
There is no obvious choice for the normalization of the viscous BL thickness and we therefore present $\overline{\lambda} _u$ normalized with its mean value $\langle \overline{\lambda} _u \rangle_{J}$ in figure \[fig:lambda\]. Overall these curves for $\overline{\lambda} _u$ exhibit a similar trend as we observed previously for $\overline{\lambda} _\theta$. The values of $\overline{\lambda} _u$ are smaller in the plume impacting region ($d<0$) and the variation with $Ra$ is limited. Also for $\overline{\lambda} _u/ \langle \overline{\lambda} _u \rangle_{J}$ the growth with increasing $d$ is less pronounced the higher $Ra$ and the curves almost collapse for $d>0$ at $Ra\geq 10^7$.
Figure \[fig:lambda\_tu\]a shows $\langle \overline{\lambda}_\theta \rangle_{J}$ and $\langle \overline{\lambda}_u \rangle_{J}$ as a function of $Ra$. For the thermal BL thickness, the scaling appears to be rather constant over the full range and from fitting $4 \times 10^6 \leq Ra \leq 10^9$ we obtain $$\langle \overline{\lambda}_\theta \rangle_{J}/H \sim Ra^{-0.30}.$$ The reduction of the viscous BL thickness $\langle \overline{\lambda}_u \rangle_{J}$ with $Ra$ is significantly slower than for the thermal BL thickness $\langle \overline{\lambda}_\theta \rangle_{J}$. For low $Ra$, $\langle \overline{\lambda}_u \rangle_{J} < \langle \overline{\lambda}_\theta \rangle_{J}$. However, due to the different scaling of the two BL thicknesses, $\langle \overline{\lambda}_u \rangle_{J} > \langle \overline{\lambda}_\theta \rangle_{J}$ for $Ra \approx 4\times 10^6$. Comparing the periodic $\Gamma =32$ data with the confined $\Gamma =1$ case reported in [@wag12], we note that the results for $\langle \overline{\lambda}_\theta \rangle$ agree closely between the two geometries. The scaling trends for $\langle \overline{\lambda}_u \rangle$ also appear to be alike in both cases. However, the viscous BL is significantly thinner in the smaller box. This situation is similar, and obviously also related to, the findings we reported for the comparison of the wall-shear stress in figure \[fig:tau\_w\]b.
We further computed the instantaneous BL ratio $\Lambda = \lambda_\theta / \lambda _u$ for which results are presented in figure \[fig:lambda\_tu\]b. Since the statistics of $\Lambda$ were found to be quite susceptible to outliers, we decided to report the most probable value $\overline{\Lambda}^{MP}$ as this provides a more robust measure than the mean. The Prandtl-Blasius BL theory for the flow over a flat plate suggests that $\Lambda = 1$ for $Pr= 1$. The figure shows that $\overline{\Lambda}^{MP}$ is almost constant as function of $d/\hat{l}$. However, unexpectedly, $\overline{\Lambda}^{MP}$ turns out to depend on $Ra$. For $Ra =10^5$, $\overline{\Lambda}^{MP}\approx 2$, which is larger than the theoretical prediction, but similar to the ratio of the means reported in figure \[fig:lambda\_tu\]a. $\overline{\Lambda}^{MP}$ decreases with $Ra$ and approaches the predicted value of $1$ for $Ra=10^9$. We note that, although this $Ra$ dependence is not expected, it was also observed by e.g. [@wag12].
![\[fig:lambda\_ell\] Comparison of mean BL thicknesses versus $Ra$ using the slope method and the location of the respective temperature and velocity levels (level method).](./Figures/lambda_ell){width="70.00000%"}
When interpreting results for the BL thicknesses, it should be kept in mind that different definitions exist in the literature [@pui07; @zho10; @zho10b; @sch12; @zho13; @pui13; @sch14; @shi15; @shi17b; @chi19]. We note that values may depend on the boundary layer definition that is employed. To get at least a sense for which of the observations transfer to other possible BL definitions, we compare the results for $\lambda$ (the slope method) to those obtained by the location of the temperature and velocity levels $(\overline{\ell})$ (level method, see figure \[fig:counts\]b) in figure \[fig:lambda\_ell\]. We note that the scalings versus $Ra$ are very similar, albeit not exactly the same, for both definitions of the BL thickness. However, the offset between $\overline{\lambda}$ and $\overline{\ell}$ is not the same for velocity and temperature. As a consequence, there is no crossover between $\overline{\ell_\theta}$ and $\overline{\ell_u}$ within the range of $Ra$ considered.
![\[fig:re\_res\] (a) Wind Reynolds number $Re_\textrm{wind}$ versus $Ra$ obtained in a periodic $\Gamma=32$ domain compared to the corresponding values obtained by [@wag12] for a cylindrical $\Gamma =1$ domain. We also show the predictions from the unifying theory [@gro00; @gro01] using the updated prefactors [@ste13]. (b) $Re_s$ versus $Ra$ with estimations for $Ra_\textrm{crit}$. In both panels, we have fitted our own datapoints only from $Ra=4\times 10^6$ onwards to achieve consistent comparisons with the data by [@wag12], where only data from $Ra=3\times 10^6$ on is available.](./Figures/Re_Res){width="\textwidth"}
In figure \[fig:re\_res\] we compare the wind Reynolds number, which we determined as follows, $$Re_{\textrm{wind}} = \langle \overline{u}_L \rangle_{J} \, H / \nu ,$$ with the results of [@wag12]. The figure shows that our $Re_{\textrm{wind}}$ obtained in a periodic $\Gamma=32$ domain with $Pr=1$ agree surprisingly well with the results from [@wag12] obtained in a cylindrical $\Gamma=1$ sample with $Pr=0.786$. The obtained $Re$ values obtained by [@wag12] are slightly higher than our values. We note that the lower $Pr$ results in slightly higher $Re_\textrm{wind}$. This means that the main finding in this context is that $Re_\textrm{wind}$ in the turbulent superstructures is almost the same, perhaps slightly lower, than in a confined $\Gamma=1$ sample [@wag12]. We note that the predictions for the wind Reynolds number obtained from the unifying theory for thermal convection [@gro00; @gro01] are in good agreement with the data. The unifying theory, using the updated constants found by [@ste13], namely predicts that for $Pr=1$ the wind Reynolds number scales as $Re_{GL}=0.395\times Ra^{0.439}$, while the data for $Ra \geq 4\times10^6$ are well approximated by $Re_{\textrm{wind}}=0.22 \times Ra^{0.470}$. To estimate when the BLs become turbulent we calculate the shear Reynolds number $$Re_s= { \left [ \overline{u}_L \times \lambda _u^{MP} \right ] ^{\max} \over 2 \nu}.$$ We expect the BL to become turbulent and the ultimate regime to set in [@gro00; @gro11] at a critical shear Reynolds number of $Re_s^\textrm{crit} \approx 420$ [@ll87]. A fit to our data gives $$Re_s=0.09\times Ra^{0.243},$$ from which we can extrapolate that $Re_s^\textrm{crit}=420$ is reached at $Ra_\textrm{crit} \approx 1.3\times 10^{15}$. Of course, this estimate comes with a significant error bar as our data for $\Gamma =32$ is still far away from the expected critical $Ra$ number. Nevertheless, it agrees well with the result from [@wag12], who find $Ra_\textrm{crit}\approx 1.2\times 10^{14}$ for a cylindrical $\Gamma=1$ cell and the results from [@sun08] who find from experiments that $Ra_\textrm{crit}\approx 2\times 10^{13}$. We emphasize that all these estimates are consistent with the observation of the onset of the ultimate regime at $Ra_{*}\approx 2\times 10^{13}$ in the Göttingen experiments [@he12; @he15]. As is explained by [@ahl17] also measurements of the shear Reynolds number in low $Pr$ number simulations by [@sch16] support the observation of the ultimate regime in the Göttingen experiments.
Conclusions {#section_conclusions}
===========
We have used a conditional averaging technique to investigate the properties of the LSC and the boundary layers in $\Gamma =32$ RB convection for unit Prandtl number and Rayleigh numbers up to $Ra=10^9$. The resulting quasi-two-dimensional representation of the LSC allowed us to analyze the wind properties as well as wall shear and local heat transfer. We found the distribution of the wall shear stress $\overline{\tau} _w$ to be asymmetric. The maximum of $\overline{\tau} _w$ is located closer to the plume impacting side and its value increases as $\overline{\tau} _w ^{\max}
/ (\rho \beta g H \Delta )
\sim Ra^{0.24}$ with increasing $Ra$. The local heat transfer at the wall, represented by the conditioned Nusselt number $\overline{Nu}$, has its highest values in the plume impacting zone at all $Ra$ considered here. Going from the plume impacting towards the plume emitting region, $\overline{Nu}$ is seen to decrease consistently as is expected from the fluid near the hot wall heating up. However, as $Ra$ is increased, the differences in $\overline{Nu}$ even out more and more. For the plume emitting side in particular, we were able to connect this trend to increased advective transport in the wall-normal direction at higher $Ra$. When extrapolating the trends for $\overline{Nu}$ to $Ra$ higher than those available here, our results appear consistent with @zhu18c. These authors observed a reversal of the $\overline{Nu}$-distribution in 2D RB turbulence above $Ra \gtrapprox 10^{11}$ with higher values of the heat transport in the emitting region.
Further, we examined the thermal and the viscous BLs. At low $Ra$, both increase along $d$ in an approximately linear fashion, whereas flat plate boundary layer theory would suggest a growth proportional $\sqrt{d}$ [@ll87]. As $Ra$ increases, and especially for $d>0$, the growth becomes successively weaker and stops entirely beyond $Ra \gtrapprox10^8$. Again, this is likely a consequence of the increased convective mixing in this region. For increasing $Ra$, both $\overline{\lambda}_{\theta}$ and $\overline{\lambda}_u$ become thinner, with $\overline{\lambda} _\theta$ showing an effective scaling of $\langle \overline{\lambda} _{\theta} \rangle _J /H \sim Ra^{-0.3}$. At $Ra\gtrapprox 4\times 10^6$ we observed a crossover point where the thermal BL becomes smaller than the viscous BL. It should be noted that the crossover appears specific to the definition of $\overline{\lambda}$ since a similar behavior was not observed when an alternative definition ($\overline{\ell}$, based on the location of the level) was employed. Nevertheless, the scaling behavior of $\overline{\lambda}$ and $\overline{\ell}$ was seen to be very similar. When calculating instantaneous BL ratios, a convergence to $\overline{\Lambda}^{MP} \rightarrow 1$ for high enough $Ra$ can be observed as predicted by the PB theory for laminar BLs. As pointed out in @shi14, the PB limit only strictly applies to wall parallel flow and the ratio is expected to be higher if the flow approaches the plate at an angle. This incidence angle is higher at smaller $\Gamma$ which can explain why at comparable $Ra$ the BL ratios reported in @wag12 are slightly higher than what is found here.
We expected to find significant differences in the LSC statistics obtained in a confined $\Gamma =1$ system and a large $\Gamma =32$ system. However, surprisingly, we find that the thermal BL thickness $\langle \overline{\lambda}_{\theta} \rangle _J$ obtained for both cases agrees very well. It turns out that the viscous BL thickness $\langle \overline{\lambda}_u \rangle _J$ is significantly larger for the periodic $\Gamma =32$ case than in a $\Gamma=1$ cylinder. However, the wall shear and its scaling with $Ra$ are similar in both cases. Here we find that in a periodic $\Gamma=32$ domain, the shear Reynolds number scales as $Re_s \sim Ra^{0.243}$. This is a bit lower than the corresponding result for $\Gamma=1$, although one needs to keep in mind the slight difference in $Pr$ ($Pr = 0.786$ at $\Gamma = 1$ vs. $Pr = 1$ for $\Gamma=32$) is responsible for part of the observed difference. An extrapolation towards the critical shear Reynolds number of $Re_s^{\textrm{crit}} \approx 420$ when the laminar-type BL becomes turbulent predicts that the transition to the ultimate regime is expected at $Ra_{\textrm{crit}} \approx \mathcal{O}(10^{15})$. This is slightly higher than the corresponding result for a $\Gamma=1$ cylinder, i.e. $Ra_{\textrm{crit}} \approx \mathcal{O}(10^{14})$, by [@wag12]. However, it should be noted that considering inherent uncertainties and differences in $Pr$, the results for $\Gamma=32$ the observed transition to the ultimate regime in the Göttingen experiments [@he12; @he15] and previous measurements of the shear Reynolds number [@wag12; @sch16]. So surprisingly, we find that in essentially unconfined very large aspect ratio systems, in which the resulting structure size is significantly larger, the differences in terms of $Re_{\textrm{wind}}$ or $Re_s$ with respect to the $\Gamma=1$ cylindrical case are marginal.
Acknowledgments {#acknowledgments .unnumbered}
===============
We greatly appreciate valuable discussions with Olga Shishkina. This work is supported by NWO, the University of Twente Max-Planck Center for Complex Fluid Dynamics, the German Science Foundation (DFG) via program SSP 1881, and the ERC (the European Research Council) Starting Grant No. 804283 UltimateRB. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. ([[www.gauss-centre.eu](www.gauss-centre.eu)]{}) for funding this project by providing computing time on the GCS Supercomputer SuperMUC-NG at Leibniz Supercomputing Centre ([[www.lrz.de](www.lrz.de)]{}).
Declaration of Interests {#declaration-of-interests .unnumbered}
========================
The authors report no conflict of interest.
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---
abstract: 'In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.'
author:
- |
**Umberto Guarnotta, Salvatore A. Marano[^1]\
\
\
\
\
**Dumitru Motreanu\
\
\
****
title: '**On a singular Robin problem with convection terms**'
---
**Keywords:** Robin problem, quasilinear elliptic equation, gradient dependence, singular term.
**AMS Subject Classification:** 36J60, 35J62, 35J92.
Introduction
============
Let $\Omega\subseteq{\mathbb{R}}^N$ ($N\geq 3$) be a bounded domain with a $ C^2$-boundary $\partial \Omega$ and let $f:\Omega\times {\mathbb{R}}\times{\mathbb{R}}^N\to [0,+\infty)$, $g:\Omega\times (0,+\infty)\to [0,+\infty)$ be two Carathéodory functions. In this paper, we study existence and uniqueness of solutions to the following Robin problem: $$\label{problem} \tag{${\rm P}$}
\left\{
\begin{array}{ll}
- {\rm \, div} a(\nabla u)=f(x,u,\nabla u) + g(x,u)\;\; &\mbox{in}\;\;\Omega, \\
u > 0\;\; &\mbox{in}\;\;\Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}}+\beta |u|^{p-2}u= 0\;\; &\mbox{on}\;\;\partial \Omega,
\end{array}
\right.$$ where $a:{\mathbb{R}}^N\to{\mathbb{R}}^N$ denotes a continuous strictly monotone map having suitable properties, which basically stem from Liebermann’s nonlinear regularity theory [@Li] and Pucci-Serrin’s maximum principle [@PS]; see Section \[S2\] for details. Moreover, $\beta>0$, $1<p<+\infty$, while $\frac{\partial}{\partial \nu_a}$ denotes the co-normal derivative associated with $a$.
This problem gathers together several hopefully interesting technical features, namely:
- The involved differential operator appears in a general form that includes non-homogeneous cases.
- $f$ depends on the solution and its gradient. So, the reaction exhibits nonlinear convection terms.
- $g$ can be singular at zero, i.e., $\displaystyle{\lim_{s\to 0^+}}g(x,s)=+\infty$.
- Robin boundary conditions are imposed instead of (much more frequent) Dirichlet ones.
All these things have been extensively investigated, although separately. For instance, both differential operator and Robin conditions already appear in [@GMP] where, however, the problem has a fully variational structure, whilst [@PW] falls inside non-variational settings. The paper [@FarMotPug] addresses the presence of convection terms; see also [@MMM; @MW; @ZLM], which exhibit more general contexts. Last but not least, singular problems were considered especially after the seminal works of Crandall-Rabinowitz-Tartar [@CRT] and Lazer-McKenna [@LM]. Among recent contributions on this subject, we mention [@GP2; @PapWink]. Finally, [@LMZ] treats a $p$-Laplacian Dirichlet problem whose right-hand side has the same form as that in . It represented the starting point of our research.
Several issues arise when passing from Dirichlet to Robin boundary conditions. Accordingly, here, we try to develop some useful tools in this direction, including the localization of solutions to an auxiliary variational problem inside an opportune sublevel of its energy functional, constructed for preserving some compactness and semicontinuity properties (cf. Section \[S3\]).
Our main result, Theorem \[existence\], establishes the existence of a regular solution to chiefly via sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as Schaefer’s fixed point theorem. Uniqueness is also addressed, but only when $ p=2$ (vide Section \[S4\]).
Usually, linear problems possess only one solution, whereas multiplicity is encountered in nonlinear phenomena. Hence, it might be of interest to seek hypotheses on $f$ and $g$ that yield uniqueness even if $p \neq 2$. As far as we know, this is still an open problem.
Preliminaries {#S2}
=============
Let $X$ be a set and let $C\subseteq X$. We denote by $\chi_C$ the characteristic function of $C$. If $C\neq\emptyset$ and $\Gamma:C\to C$ then $${\rm Fix}(\Gamma):=\{ x\in C: x=\Gamma(x)\}$$ is the fixed point set of $\Gamma$. The following result, usually called Schaefer’s theorem [@GP p. 827] or Leray-Schauder’s alternative principle, will play a basic role in the sequel.
\[schaefer\] Let $X$ be a Banach space, let $C \subseteq X$ be nonempty convex, and let $\Gamma: C\to C$ be continuous. Suppose $\Gamma$ maps bounded sets into relatively compact sets. Then either $\{ x\in C:x = t\,\Gamma(x)\;\mbox{for some}\; t\in (0,1)\} $ turns out unbounded or ${\rm Fix}(\Gamma)\neq\emptyset$.
Given a partially ordered set $(X,\leq)$, we say that $X$ is downward directed when for every $x_1,x_2\in X$ there exists $x\in X$ such that $x\leq x_i$, $i=1,2$. The notion of upward directed set is analogous.
If $Y$ is a real function space on a set $\Omega\subseteq{\mathbb{R}}^N$ and $u,v\in Y$, then $u\leq v$ means $u(x)\leq v(x)$ for almost every $x\in\Omega$. Moreover, $Y_+ :=\{ u\in Y: u\geq 0\}$, $\Omega(u\leq v):=\{x\in\Omega: u(x)\leq v(x)\}$, etc.
Let $X,Y$ be two metric spaces and let $\mathscr{S}:X\to 2^Y$. The multifunction $\mathscr{S}$ is called lower semicontinuous when for every $x_n\to x$ in $X$, $y\in\mathscr{S}(x)$ there exists a sequence $\{y_n\}\subseteq Y$ having the following properties: $y_n\to y$ in $Y$; $y_n\in\mathscr{S}(x_n)$ for all $n\in{\mathbb{N}}$.
Finally, if $X$ is a Banach space and $J\in C^1(X)$, then $${\rm Crit}(J):=\{ x \in X:J'(x) = 0\}$$ is the critical set of $J$.
The monograph [@CLM] represents a general reference on these topics.
Given any $s>1$, the symbol $s'$ will indicate the conjugate exponent of $s$, namely $s':=\frac{s}{s-1}$.
Henceforth, for $1<p<+\infty$, $\beta>0$, $\Omega$ as in the Introduction, and $u:\overline{\Omega} \to{\mathbb{R}}$ appropriate, the notation below will be adopted: $$\|u\|_\infty :=\operatorname*{ess \, sup}_{x \in \Omega} |u(x)|\, ;\quad \|u\|_{C^1(\overline{\Omega})}:=\|u\|_\infty + \|\nabla u\|_\infty\, ;$$ $$\|u\|_p:=\left( \int_{\Omega} |u|^p dx \right)^{\frac{1}{p}}\, ;\quad \|u\|_{p,\partial \Omega} := \left( \int_{\partial \Omega}
|u|^p d\sigma \right)^{\frac{1}{p}}\, ;$$ $$\|u\|_{1,p}:=\left( \|u\|_p^p+\|\nabla u\|_p^p\right)^\frac{1}{p}\, ;\quad \|u\|_{\beta,1,p}:=\left( \beta\|u\|_{p,\partial\Omega}^p+\|\nabla u\|_p^p \right)^\frac{1}{p}\, .$$ Here, $\sigma$ denotes the $(N-1)$-dimensional Hausdorff measure on $\partial\Omega$. If $\nu (x)$ is the outward unit normal vector to $\partial\Omega$ at its point $x$ then $\frac{\partial}{\partial\nu_a}$ stands for the co-normal derivative associated with $a$, defined extending the map $u\mapsto\langle a(\nabla u),\nu\rangle$ from $C^1(\overline{\Omega})$ to $W^{1,p}(\Omega)$.
The trace inequality ensures that $\|u\|_{p,\partial \Omega}$ makes sense whenever $ u \in W^{1,p}(\Omega) $; see for instance [@E] or [@KJF].
It is known [@FMP] that $${\rm int}(C^1(\overline{\Omega})_+)=\left\{u \in C^1(\overline{\Omega}): u(x) > 0\;\forall\, x\in\overline{\Omega} \right\}.$$
If $\beta>0$, then $\| \cdot \|_{\beta,1,p}$ is a norm on $W^{1,p}(\Omega)$ equivalent to $\|\cdot \|_{1,p}$. In particular, there exists $c_1=c_1(p,\beta,\Omega)\in (0,1) $ such that $$\label{equivnorm}
c_1\|u\|_{1,p}\leq\|u\|_{\beta,1,p}\leq\frac{1}{c_1}\|u\|_{1,p}\quad\forall\, u \in W^{1,p}(\Omega)\, .$$ For the proof we refer to [@PW].
Let $\omega\in C^1(0,+\infty)$ satisfy $$C_1 \leq\frac{t \omega'(t)}{\omega(t)} \leq C_2\, , \quad C_3 t^{p-1} \leq \omega(t) \leq C_4 (1+t^{p-1})$$ in $(0,+\infty)$, with $C_i$ suitable positive constants. We say that the operator $a:{\mathbb{R}}^N\to{\mathbb{R}}^N$ fulfills assumption $\underline{\rm{H(a)}}$ when:
- $a(\xi) = a_0(|\xi|)\xi $ for all $\xi \in {\mathbb{R}}^N $, where $ a_0:(0,+\infty)\to(0,+\infty)$ is $C^1$, $t \mapsto ta_0(t)$ turns out strictly increasing, and $$\lim_{t \to 0^+} ta_0(t) = 0, \quad \lim_{t \to 0^+} \frac{ta_0'(t)}{a_0(t)} > -1.$$
- $ \displaystyle{|Da(\xi)| \leq C_5 \frac{\omega(|\xi|)}{|\xi|}} $ in ${\mathbb{R}}^N \setminus \{0\}$.
- $\displaystyle{\langle Da(\xi)y,y\rangle\geq\frac{\omega(|\xi|)}{|\xi|} |y|^2} $ for every $y,\xi\in{\mathbb{R}}^N$, $\xi \neq 0$.
Various differential operators comply with ${\rm H(a)}$. Three classical examples are listed below.
- The so-called $p$-Laplacian: $\Delta_p u:={\rm div}\left( |\nabla u|^{p-2} \nabla u\right)$, which stems from $a_0(t):=t^{p-2}$.
- The $(p,q)$-Laplacian: $\Delta_p u+\Delta_q u$, where $1< q< p< +\infty$. In this case, $a_0(t):=t^{p-2}+t^{q-2}$.
- The generalized $p$-mean curvature operator: $$u \mapsto{\rm div}\left[ (1+|\nabla u|^2)^{\frac{p-2}{2}} \nabla u \right],$$ corresponding to $a_0(t):=(1+t^2)^{\frac{p-2}{2}}$.
Finally, define $$G_0(t):=\int_{0}^{t} s a_0(s){\rm d}s\;\;\forall\, t\in{\mathbb{R}}\quad\mbox{as well as}\quad G(\xi):=G_0(|\xi|)\;\;\forall\,\xi\in{\mathbb{R}}^N.$$
\[opestimate\] Under hypothesis $ {\rm H(a)} $, there exists $ c_2 \in (0,1) $ such that $$|a(\xi)|\leq\frac{1}{c_2}(1+|\xi|^{p-1})\quad\mbox{and}\quad c_2 |\xi|^p\leq \langle a(\xi),\xi\rangle\leq\frac{1}{c_2}(1+|\xi|^p)$$ for all $\xi\in{\mathbb{R}}^N$. In particular, $$c_2 |\xi|^p \leq G(\xi) \leq \frac{1}{c_2}(1+|\xi|^p)\,,\;\;\xi \in {\mathbb{R}}^N.$$
See [@GMP Lemmas 2.1–2.2] or [@PW Lemma 2.2 and Corollary 2.3].
Existence {#S3}
=========
Throughout this section, the convection term $f$ and the singularity $g$ will fulfill the assumptions below where, to avoid unnecessary technicalities, ‘for all $ x $’ takes the place of ‘for almost all $ x $’.\
$\underline{\rm{H(f)}}$ $f: \Omega \times {\mathbb{R}}\times{\mathbb{R}}^N \to [0,+\infty)$ is a Carathéodory function. Moreover, to every $M > 0$ there correspond $c_M, d_M > 0$ such that $$f(x,s,\xi)\leq c_M + d_M |s|^{p-1}\quad \forall\, (x,s,\xi) \in\Omega\times{\mathbb{R}}\times{\mathbb{R}}^N\;\;\text{with}\;\; |\xi|\leq M.$$ $ \underline{\rm{H(g)}}$ $g: \Omega \times (0,+\infty) \to [0,+\infty) $ is a Carathéodory function having the properties:
- $g(x,\cdot)$ turns out nonincreasing on $(0,1]$ whatever $x \in\Omega$, and $ g(\cdot,1)\not\equiv 0$.
- There exist $c,d>0$ such that $$g(x,s)\leq c+d s^{p-1} \quad \forall\, (x,s) \in\Omega \times (1,+\infty).$$
- With appropriate $\theta\in\rm{int}(C^1(\overline{\Omega})_+)$ and ${\varepsilon}_0 > 0$, the map $x\mapsto g(x,{\varepsilon}\theta(x)) $ belongs to $ L^{p'}(\Omega) $ for any ${\varepsilon}\in (0,{\varepsilon}_0)$.
The paper [@LMZ] contains meaningful examples of functions $g$ that satisfy H(g).
Fix $w\in C^1(\overline{\Omega})$. We first focus on the singular problem (without convection terms) $$\label{auxprob} \tag{${\rm P}_w$}
\left\{
\begin{array}{ll}
- {\rm div} \, a(\nabla u) = f(x,u,\nabla w) + g(x,u)\;\; &\mbox{in}\;\;\Omega, \\
u > 0 \;\; &\mbox{in}\;\; \Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}} + \beta |u|^{p-2}u= 0 \;\; &\mbox{on}\;\; \partial \Omega.
\end{array}
\right.$$
$u\in W^{1,p}(\Omega)$ is called a subsolution to when $$\label{subsol}
\int_{\Omega} \langle a(\nabla u),\nabla v \rangle{\rm d}x
+ \beta\int_{\partial \Omega} |u|^{p-2}uv{\rm d}\sigma
\leq \int_{\Omega}[f(\cdot,u,\nabla w)+g(\cdot,u)]v{\rm d}x$$ for all $v\in W^{1,p}(\Omega)_+ $. The set of subsolutions will be denoted by $ \underline{U}_w $.\
We say that $u\in W^{1,p}(\Omega)$ is a supersolution to if $$\label{super}
\int_{\Omega} \langle a(\nabla u),\nabla v \rangle {\rm d}x + \beta\int_{\partial \Omega} |u|^{p-2}uv {\rm d}\sigma
\geq \int_{\Omega} [f(\cdot,u,\nabla w)+g(\cdot,u)]v {\rm d}x$$ for every $v \in W^{1,p}(\Omega)_+ $, and indicate with $ \overline{U}_w $ the supersolution set.\
Finally, $u\in W^{1,p}(\Omega)$ is called a solution of provided $$\int_{\Omega} \langle a(\nabla u),\nabla v \rangle {\rm d}x + \beta\int_{\partial \Omega} |u|^{p-2}uv {\rm d}\sigma = \int_{\Omega}[f(\cdot,u,\nabla w)+g(\cdot,u)]v {\rm d}x$$ for all $ v \in W^{1,p}(\Omega)_+ $. The corresponding solution set will be denoted by $ U_w $. Obviously, $U_w =\overline{U}_w\cap \underline{U}_w $.
\[supersol\] If $ u_1,u_2 \in\overline{U}_w $ (resp. $u_1,u_2\in\underline{U}_w$), then $\min\{u_1,u_2\}\in\overline{U}_w$ (resp. $\max\{ u_1,u_2\} \in\underline{U}_w$). In particular, the set $\overline{U}_w $ (resp. $ \underline{U}_w $) is downward (resp. upward) directed.
This proof is patterned after that of [@LMZ Lemma 10] (see also [@CLM]). Thus, we only sketch it. Pick $u_1,u_2\in\overline{U}_w$, set $u:=\min\{u_1,u_2\} $, and define, for every $t\in{\mathbb{R}}$, $$\eta_{\varepsilon}(t):= \left\{
\begin{array}{ll}
0 \quad &\mbox{when} \quad t < 0, \\
\frac{t}{{\varepsilon}}\quad &\mbox{if} \quad 0 \leq t \leq {\varepsilon}, \\
1 \quad &\mbox{for} \quad t > {\varepsilon},
\end{array}
\right.$$ where ${\varepsilon}>0$. Further, to shorten notation, write $\bar\eta_{\varepsilon}(x):=\eta_{\varepsilon}(u_2(x)-u_1(x))$. Evidently, both $\bar\eta_{\varepsilon}\in W^{1,p}(\Omega)_+$ and $$\nabla\bar\eta_{\varepsilon}=\eta'_{\varepsilon}(u_2-u_1)\,\nabla(u_2-u_1).$$ Let $\hat v\in C^1(\overline{\Omega})_+$. Since $u_i$ fulfills , one has $$\int_{\Omega}\langle a(\nabla u_i),\nabla v\rangle{\rm d}x + \beta \int_{\partial \Omega} |u_i|^{p-2}u_i v {\rm d}\sigma
\geq\int_{\Omega} [f(\cdot,u_i,\nabla w)+g(\cdot,u_i)]v {\rm d}x$$ whatever $v\in W^{1,p}(\Omega)_+ $. Choosing $v:=\bar\eta_{\varepsilon}\,\hat v$ when $ i = 1 $, $v:= (1-\bar\eta_{\varepsilon})\hat v$ if $ i = 2 $, and adding term by term produces $$\label{termbyterm}
\begin{split}
&\int_{\Omega} \langle a(\nabla u_1) - a(\nabla u_2),\nabla (u_2-u_1) \rangle \eta_{\varepsilon}'(u_2-u_1)\hat v {\rm d}x \\
&+ \int_{\Omega}\langle a(\nabla u_1), \nabla \hat v \rangle\,\bar\eta_{\varepsilon}{\rm d}x
+ \int_{\Omega} \langle a(\nabla u_2),\nabla \hat v \rangle (1-\bar\eta_{\varepsilon}) {\rm d}x \\
&+\beta\left( \int_{\partial \Omega} |u_1|^{p-2}u_1\bar\eta_{\varepsilon}\hat v {\rm d}\sigma
+ \int_{\partial \Omega} |u_2|^{p-2}u_2 (1-\bar\eta_{\varepsilon})\hat v {\rm d}\sigma \right) \\
&\geq \int_{\Omega} [f(\cdot,u_1,\nabla w)+g(\cdot,u_1)]\bar\eta_{\varepsilon}\hat v {\rm d}x\\
&+ \int_{\Omega} [f(\cdot,u_2,\nabla w)+g(\cdot,u_2)](1-\bar\eta_{\varepsilon})\hat v {\rm d}x.
\end{split}$$ The strict monotonicity of $a$, combined with $\eta_{\varepsilon}'(u_2-u_1)\hat v\geq 0$, lead to $$\int_{\Omega}\langle a(\nabla u_1)-a(\nabla u_2),\nabla (u_2-u_1)\rangle\eta_{\varepsilon}'(u_2-u_1)\hat v{\rm d}x \leq 0.$$ For almost every $x\in\Omega$ we have $$\nabla u(x) = \left\{
\begin{array}{ll}
\nabla u_1(x)\;\; &\mbox{if}\; u_1(x)<u_2(x), \\
\nabla u_2(x)\;\; &\mbox{otherwise,}
\end{array}
\right.$$ as well as $$\lim_{{\varepsilon}\to 0^+}\bar\eta_{\varepsilon}(x)= \chi_{\Omega(u_1<u_2)}(x).$$ Hence, letting ${\varepsilon}\to 0^+$ and using the dominated convergence theorem, inequality becomes $$\int_{\Omega}\langle a(\nabla u),\nabla\hat v\rangle {\rm d}x+\beta\int_{\partial \Omega}|u|^{p-2}u\hat v {\rm d}\sigma
\geq\int_{\Omega}[f(\cdot,u,\nabla w)+g(\cdot,u)]\hat v {\rm d}x;$$ see [@LMZ Lemma 10] for more details. Since $\hat v\in C^1(\overline{\Omega})_+$ was arbitrary, by density one arrives at $u\in\overline{U}_w$.
\[subsollemma\] Let ${\rm H(f)}$ and ${\rm H(g)}$ be satisfied. Then there exists a subsolution $\underline{u}\in{\rm int}(C^1(\overline{\Omega})_+)$ to independent of $w$ and such that $\|\underline{u}\|_\infty \leq 1$.
Given any $\delta>0$, consider the problem $$\label{subprob}
\left\{
\begin{array}{ll}
- {\rm div} \, a(\nabla u) = \tilde{g}(x,u) \;\; &\mbox{in}\;\;\Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}} + \beta |u|^{p-2}u= 0\;\; &\mbox{on}\;\;\partial \Omega,
\end{array}
\right.$$ where $\tilde{g}(x,s):=\min\{g(x,s),\delta\}$, $(x,s)\in\Omega\times (0,+\infty)$. Standard arguments yield a nontrivial solution $ \underline{u} \in W^{1,p}(\Omega) $ to , because $\tilde{g}$ is bounded. Testing with $-\underline{u}^- $ we get $$-\int_{\Omega}\langle a(\nabla \underline{u}),\nabla\underline{u}^- \rangle {\rm d}x
-\beta \int_{\Omega}|\underline{u}|^{p-2} \underline{u} \underline{u}^- {\rm d}\sigma =
-\int_{\Omega} \tilde{g}(x,\underline{u}) \underline{u}^- {\rm d}x \leq 0,$$ whence, by Proposition \[opestimate\], $$c_2\|\underline{u}^-\|_{\beta,1,p}^p
\leq\int_{\Omega}\langle a(\nabla \underline{u}^-), \nabla \underline{u}^- \rangle {\rm d}x
+ \beta\int_{\Omega} (\underline{u}^-)^{p} {\rm d}\sigma \leq 0.$$ Therefore, $\underline{u}\geq 0$. Regularity up to the boundary [@Li] and strong maximum principle [@PS] then force $\underline{u}\in{\rm int}(C^1(\overline{\Omega})_+)$. Using the maximum principle one next has $$\label{subsupnorm}
\|\underline{u}\|_\infty \leq 1$$ once $\delta$ is small enough. Let $\theta$ and ${\varepsilon}_0$ be as in ${\rm (g_3)}$. Since $\underline{u},\theta\in{\rm int}(C^1(\overline{\Omega})_+) $, there exists ${\varepsilon}\in (0,{\varepsilon}_0)$ such that $\underline{u} - {\varepsilon}\theta \in {\rm
int}(C^1(\overline{\Omega})_+)$. Via ${\rm (g_1)}$, , and ${\rm (g_3)} $, we thus infer $$\label{subsolsummability}
0 \leq g(\cdot,\underline{u}) \leq g(\cdot,{\varepsilon}\theta)\in L^{p'}(\Omega).$$ The conclusion is achieved by verifying that $\underline{u}\in\underline{U}_w$ for any $w\in C^1(\overline{\Omega})$. Pick such a $w$, test with $v \in W^{1,p}(\Omega)_+ $, and recall the definition of $ \tilde{g} $, to arrive at $$\begin{split}
&\int_{\Omega} \langle a(\nabla \underline{u}),\nabla v\rangle{\rm d}x+\beta\int_{\partial \Omega}\underline{u}^{p-1}v {\rm d}\sigma
=\int_{\Omega}\tilde{g}(\cdot,\underline{u})v {\rm d}x \\
&\leq\int_{\Omega} g(\cdot,\underline{u})v{\rm d}x\leq\int_{\Omega} [f(\cdot,u,\nabla w) + g(\cdot,\underline{u})] v {\rm d}x,
\end{split}$$ as desired.
This proof shows that the subsolution $\underline{u}$ constructed in Lemma \[subsollemma\] enjoys the further property: $$\label{subformula}
\int_{\Omega} \langle a(\nabla\underline{u}),\nabla v\rangle{\rm d}x
+\beta \int_{\partial \Omega}|\underline{u}|^{p-2}\underline{u}v {\rm d}\sigma
\leq\int_{\Omega} g(\cdot,\underline{u})v{\rm d}x\;\; \forall\, v \in W^{1,p}(\Omega)_+.$$
Given $w\in C^1(\overline{\Omega})$, consider the truncated problem $$\label{truncprob}
\left\{
\begin{array}{ll}
- {\rm div} \, a(\nabla u) =\hat{f}(x,u)+\hat{g}(x,u)\;\; &\mbox{in}\;\;\Omega, \\
u > 0\;\; &\mbox{in}\;\; \Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}} + \beta |u|^{p-2}u= 0\;\;&\mbox{on}\;\;\partial \Omega,
\end{array}
\right.$$ where $$\label{1f}
\hat{f}(x,s):= \left\{
\begin{array}{ll}
f(x,\underline{u}(x),\nabla w(x))\;\; &\mbox{if}\;\;s \leq \underline{u}(x), \\
f(x,s,\nabla w(x))\;\; &\mbox{otherwise,}
\end{array}
\right.$$ $$\label{1}
\hat{g}(x,s):= \left\{
\begin{array}{ll}
g(x,\underline{u}(x))\;\; &\mbox{if}\;\; s \leq \underline{u}(x), \\
g(x,s)\;\; &\mbox{otherwise.}
\end{array}
\right.$$ The energy functional corresponding to writes $$\begin{split}
\mathscr{E}_w(u):=\frac{1}{p} \int_{\Omega} G(\nabla u){\rm d}x+\frac{\beta}{p} \int_{\partial \Omega} |u|^p {\rm d}\sigma
- \int_{\Omega}\hat{F}(\cdot,u){\rm d}x-\int_{\Omega} \hat{G}(\cdot,u){\rm d}x
\end{split}$$ for all $ u \in W^{1,p}(\Omega) $, with $$\hat{F}(x,s):=\int_{0}^{s} \hat{f}(x,t) {\rm d}t, \quad \hat{G}(x,s):=\int_{0}^{s} \hat{g}(x,t) {\rm d}t.$$ Hypotheses ${\rm H(f)}$–${\rm H(g)}$ ensure that $\mathscr{E}_w$ is of class $ C^1 $ and weakly sequentially lower semicontinuous; see, e.g., [@GMP Lemma 3.1]. Under the additional condition $$\label{condition}
d_M+d< c_1^p c_2 \quad \forall\, M >0,$$ it turns out also coercive, as the next lemma shows.
\[estlemma\] Let $\mathscr{B}$ be a nonempty bounded set in $C^1(\overline{\Omega})$. If ${\rm H(f)}$, ${\rm H(g)}$, and hold true then there exist $\alpha_1 \in (0,1)$, $\alpha_2 > 0$ such that $$\mathscr{E}_w(u) \geq\frac{\alpha_1}{p}\|u\|_{1,p}^p - \alpha_2(1+\|u\|_{1,p})\quad\forall\, (u,w)\in W^{1,p}(\Omega)\times\mathscr{B}.$$
Put $\hat{M}:=\displaystyle{\sup_{w \in \mathscr{B}}}\|w\|_{C^1(\overline{\Omega})}$. By –, Proposition \[opestimate\] entails $$\begin{split}
\mathscr{E}_w(u) &\geq\frac{c_2}{p} \|\nabla u\|_p^p+\frac{\beta}{p}\|u\|_{p,\partial \Omega}^p
-\int_{\Omega} [f(\cdot,\underline{u},\nabla w)+ g(\cdot,\underline{u})]\underline{u} {\rm d}x \\
&- \int_{\Omega(u>\underline{u})}\left(\int_{\underline{u}}^{u} f(\cdot,t,\nabla w) {\rm d}t \right){\rm d}x
-\int_{\Omega(u>\underline{u})} \left(\int_{\underline{u}}^{u} g(\cdot,t) {\rm d}t \right) {\rm d}x.
\end{split}$$ Hypothesis ${\rm H(f)} $ along with Hölder’s inequality imply $$\begin{split}
\int_{\Omega(u>\underline{u})}\left( \int_{\underline{u}}^{u} f(\cdot,t,\nabla w){\rm d}t \right){\rm d}x
&\leq\int_{\Omega(u>\underline{u})}\left( \int_{0}^{u} f(\cdot,t,\nabla w){\rm d}t \right){\rm d}x \\
&\leq c_{\hat{M}}|\Omega|^{\frac{1}{p'}} \|u\|_p + \frac{d_{\hat{M}}}{p} \|u\|_p^p \\
&\leq c_{\hat{M}} |\Omega|^{\frac{1}{p'}} \|u\|_{1,p} +
\frac{d_{\hat{M}}}{p} \|u\|_{1,p}^p.
\end{split}$$ Exploiting , ${\rm (g_2)}$, and Hölder’s inequality again, we have $$\begin{split}
&\int_{\Omega(u>\underline{u})} \left(\int_{\underline{u}}^{u} g(\cdot,t) {\rm d}t \right) {\rm d}x \\
&\leq\int_{\Omega(u>\underline{u})}\left( \int_{\underline{u}}^{1} g(\cdot,t) {\rm d}t \right) {\rm d}x
+ \int_{\Omega(u>1)}\left( \int_{1}^{u} g(\cdot,t) {\rm d}t \right) {\rm d}x \\
&\leq \int_{\Omega(u>\underline{u})} g(\cdot,\underline{u}) {\rm d}x
+\int_{\Omega(u>1)}\left( \int_1^{u} (c+dt^{p-1}) {\rm d}t \right) {\rm d}x \\
&\leq \int_{\Omega} g(\cdot,\underline{u}) {\rm d}x+ c |\Omega|^{\frac{1}{p'}}\|u\|_p + \frac{d}{p} \|u\|_p^p \\
& \leq\int_{\Omega} g(\cdot,\underline{u}) {\rm d}x+ c |\Omega|^{\frac{1}{p'}} \|u\|_{1,p} + \frac{d}{p} \|u\|_{1,p}^p.
\end{split}$$ Hence, through we easily arrive at $$\begin{split}
\mathscr{E}_w(u) &\geq\frac{c_2}{p}\|u\|_{\beta,1,p}^p-\frac{d_{\hat{M}}+d}{p} \|u\|_{1,p}^p
- (c_{\hat{M}} + c) |\Omega|^{\frac{1}{p'}}\|u\|_p - K \\
&\geq\frac{c_1^p c_2 - d_{\hat{M}} - d}{p} \|u\|_{1,p}^p- (c_{\hat{M}} + c) |\Omega|^{\frac{1}{p'}} \|u\|_{1,p} - K\\
&\geq\frac{c_1^p c_2 - d_{\hat{M}} - d}{p}\|u\|_{1,p}^p-\max\{(c_{\hat{M}}+c) |\Omega|^{\frac{1}{p'}},K\} (1+\|u\|_{1,p}),
\end{split}$$ where $$\begin{split}
K &:=\int_{\Omega} [f(\cdot,\underline{u},\nabla w)]+g(\cdot,\underline{u})]\underline{u} {\rm d}x
+\int_{\Omega} g(\cdot,\underline{u}) {\rm d}x \\
&\leq\int_{\Omega} (c_{\hat{M}} + d_{\hat{M}}){\rm d}x + 2 \int_{\Omega} g(\cdot,{\varepsilon}\theta) {\rm d}x
\leq (c_{\hat{M}} + d_{\hat{M}}) |\Omega| + 2 \|g(\cdot, {\varepsilon}\theta)\|_{p'} |\Omega|^{\frac{1}{p}}
\end{split}$$ due to ${\rm H(f)}$ and –. Now, the conclusion follows from .
\[regularity\] A standard application of Moser’s iteration technique [@Le] shows that any solution to lies in $L^\infty(\Omega)$. By Liebermann’s regularity theory [@Li], it actually is Hölder continuous up to the boundary.
\[criticalprop\] Let ${\rm H(f)}$, ${\rm H(g)}$, and be satisfied. Then $$\emptyset \neq {\rm Crit}(\mathscr{E}_w) \subseteq U_w \cap \{ u \in C^1(\overline{\Omega}): u \geq \underline{u}\}.$$
Since $\mathscr{E}_w$ is coercive (cf. Lemma \[estlemma\]), the Weierstrass-Tonelli theorem produces ${\rm Crit}(\mathscr{E}_w)\neq\emptyset$. Pick any $ u \in {\rm Crit}(\mathscr{E}_w) $, test with $ (\underline{u}-u)^+ $, and exploit –, besides , to achieve $$\begin{split}
&\int_{\Omega} \langle a(\nabla u),\nabla(\underline{u}-u)^+ \rangle {\rm d}x + \beta \int_{\partial \Omega} |u|^{p-2}u(\underline{u}-u)^+ {\rm d}\sigma \\
&= \int_{\Omega} [\hat{f}(\cdot,u) + \hat{g}(\cdot,u)](\underline{u}-u)^+ {\rm d}x \\
&\geq\int_{\Omega}\hat{g}(\cdot,u)(\underline{u}-u)^+{\rm d}x
=\int_{\Omega} g(\cdot,\underline{u})(\underline{u}-u)^+{\rm d}x \\
&\geq \int_{\Omega} \langle a(\nabla\underline{u}),\nabla(\underline{u}-u)^+ \rangle {\rm d}x
+ \beta\int_{\partial \Omega}|\underline{u}|^{p-2}\underline{u}(\underline{u}-u)^+ {\rm d}\sigma.
\end{split}$$ Rearranging terms we get $$\int_{\Omega} \langle a(\nabla \underline{u}) - a(\nabla
u),\nabla(\underline{u}-u)^+ \rangle {\rm d}x + \beta \int_{\partial
\Omega} (|\underline{u}|^{p-2}\underline{u} -
|u|^{p-2}u)(\underline{u}-u)^+ {\rm d}\sigma \leq 0.$$ The strict monotonicity of $a$, combined with [@P Lemma A.0.5], entail $$\nabla(\underline{u}-u)^+ =0\;\;\text{in}\;\; \Omega,\quad(\underline{u}-u)^+ =0\;\;\text{on}\;\; \partial \Omega.$$ So, $\|(\underline{u}-u)^+\|_{\beta,1,p} = 0$, which means $u\geq \underline{u}$. Finally, by – one has $u \in U_w $, while $u\in C^1(\overline{\Omega})$ according to Remark \[regularity\].
For every $w\in C^1(\overline{\Omega})$ we define $$\mathscr{S}(w):=\{ u\in C^1(\overline{\Omega}): u \in U_w,\, u \geq \underline{u},\, \mathscr{E}_w(u) < 1 \}.$$
\[Scompact\] Under assumptions ${\rm H(f)}$, ${\rm H(g)}$, and , the multifunction $\mathscr{S}:C^1(\overline{\Omega})\to 2^{C^1(\overline{\Omega})}$ takes nonempty values and maps bounded sets into relatively compact sets.
If $ w\in C^1(\overline{\Omega})$, then there exists $ \hat u_w\in {\rm Crit}(\mathscr{E}_w) $ such that $$\hat u_w\in C^1(\overline{\Omega}),\quad\hat u_w\geq\underline{u},\quad\mathscr{E}_w(\hat u_w) = \inf_{W^{1,p}(\Omega)} \mathscr{E}_w \leq
\mathscr{E}_w(0) = 0 < 1;$$ cf. the proof of Lemma \[criticalprop\]. Hence, $\mathscr{S}(w)\neq\emptyset$, because $\hat u_w \in \mathscr{S}(w)$. Let $\mathscr{B}\subseteq C^1(\overline{\Omega}) $ nonempty bounded. From Lemma \[estlemma\] it follows $$\frac{\alpha_1}{p}\|u\|_{1,p}^p-\alpha_2(1+\|u\|_{1,p})\leq\mathscr{E}_w(u)<1\;\;\forall\, u\in\mathscr{S}(w),\;w\in \mathscr{B},$$ whence $\mathscr{S}(\mathscr{B})$ turns out bounded in $W^{1,p}(\Omega)$. By nonlinear regularity theory [@Li], the same holds when $C^{1,\alpha}(\overline{\Omega})$, with suitable $ \alpha\in (0,1)$, replaces $W^{1,p}(\Omega)$. Recalling that $C^{1,\alpha}(\overline{\Omega})\hookrightarrow C^1(\overline{\Omega})$ compactly yields the conclusion.
To see that $\mathscr{S}$ is lower semicontinuous, we shall employ the next technical lemma.
\[reclemma\] Let $\alpha,\beta,\gamma > 0$, let $1< p <+\infty$, and let $\{a_k\}\subseteq [0,+\infty)$ satisfy the recursive relation $$\label{reclaw}
\alpha a_k^p\leq\beta a_k + \gamma a_{k-1}^p\;\;\forall\, k \in {\mathbb{N}}.$$ If $\gamma<\alpha$, then the sequence $\{a_k\}$ is bounded.
Using the obvious inequality $$a_k \leq T + T^{1-p} a_k^p,\quad T > 0,$$ becomes $$\left(\alpha-\beta T^{1-p}\right) a_k^p\leq\beta T+\gamma a_{k-1}^p\;\;\forall\, k \in {\mathbb{N}}.$$ Since $\sigma:= 1/p < 1$, this entails $$\left(\alpha-\beta T^{1-p}\right)^\sigma a_k \leq\left(\beta T+\gamma a_{k-1}^p\right)^\sigma\leq (\beta T)^\sigma +
\gamma^\sigma a_{k-1}$$ or, equivalently, $$\label{recfinal}
a_k \leq\left(\frac{\beta T}{\alpha-\beta T^{1-p}}\right)^\sigma
+\left(\frac{\gamma}{\alpha-\beta T^{1-p}}\right)^\sigma a_{k-1},\quad k \in {\mathbb{N}},$$ provided $T>0 $ is large enough. Choosing $T>\left( \frac{\beta}{\alpha - \gamma} \right)^{\frac{1}{p-1}}$, the coefficient of $ a_{k-1} $ turns out strictly less than 1. A standard computation based on completes the proof.
\[Slsc\] Suppose ${\rm H(f)}$–${\rm H(g)}$ hold and, moreover, $$\label{reccond}
d_M+d< \frac{c_1^p c_2}{p}\quad\forall\, M>0.$$ Then the multifunction $\mathscr{S}:C^1(\overline{\Omega})\to 2^{C^1(\overline{\Omega})}$ is lower semicontinuous.
The proof is patterned after that of [@LMZ Lemma 20]. So, some details will be omitted. Let $$\label{convwn}
w_n \to w\;\;\mbox{in}\;\; C^1(\overline{\Omega}).$$ We claim that to each $\tilde u\in\mathscr{S}(w)$ there corresponds a sequence $\{ u_n\}\subseteq C^1(\overline{\Omega})$ enjoying the following properties: $$u_n \in \mathscr{S}(w_n),\;\; n\in{\mathbb{N}};\quad u_n\to\tilde u\;\;\text{in}\;\; C^1(\overline{\Omega}).$$ Fix $\tilde u\in\mathscr{S}(w)$. For every $n\in{\mathbb{N}}$, consider the auxiliary problem $$\label{vwnprob} \tag{${\rm P}_{\tilde u,w_n}$}
\left\{
\begin{array}{ll}
- {\rm div}\, a(\nabla u) = f(x,\tilde u,\nabla w_n) + \hat{g}(x,\tilde u)\;\;&\mbox{in}\;\;\Omega, \\
u > 0\;\; &\mbox{in}\;\;\Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}}+\beta |u|^{p-2}u= 0\;\;&\mbox{on}\;\;\partial \Omega,
\end{array}
\right.$$ with $\hat{g}(x,s)$ given by . One has $\hat{g}(x,\tilde u)=g(x,\tilde u)$, because $\tilde u\in\mathscr{S}(w)$, while the associated energy functional writes $$\begin{split}
\mathscr{E}_{\tilde u,w_n}(u)&:=\frac{1}{p} \int_{\Omega} G(\nabla u){\rm d}x+\beta \int_{\partial \Omega}|u|^p {\rm d}\sigma\\
&-\int_{\Omega} f(x,\tilde u,\nabla w_n)u{\rm d}x-\int_{\Omega}\hat{g}(x,\tilde u)u{\rm d}x,\;\; u\in W^{1,p}(\Omega).
\end{split}$$ Since $\mathscr{E}_{\tilde u,w_n}$ turns out strictly convex, the same argument exploited to show Lemma \[criticalprop\] yields here a unique solution $u_n^0\in{\rm int(C^1(\overline{\Omega})_+)}$ of such that $$\label{convEn}
\mathscr{E}_{\tilde u,w_n}(u_n^0)\leq 0.$$ Via –, reasoning as in Lemmas \[estlemma\] and \[Scompact\] (but for $\mathscr{E}_{\tilde u,w} $ instead of $ \mathscr{E}_w $ and $\mathscr{B}:=\{w_n:n \in {\mathbb{N}}\}$), we deduce that $\{u_n^0\}\subseteq C^1 (\overline{\Omega})$ is relatively compact. Consequently, $u_n^0\to u^0$ in $C^1(\overline{\Omega})$, where a subsequence is considered when necessary. By again and Lebesgue’s dominated convergence theorem, $u^0 $ solves problem $({\rm P}_{\tilde u,w})$. Thus, a fortiori, $u^0 =\tilde u$, because $({\rm P}_{\tilde u,w})$ possesses one solution at most. An induction procedure provides now a sequence $\{u_n^k\}$ such that $u_n^k$ solves problem $({\rm P}_{u_n^{k-1},w_n})$, the inequality $\mathscr{E}_{u_n^{k-1},w_n}(u_n^k)\leq 0$ holds, and $$\label{nindex}
\lim_{n\to+\infty }u_n^k=\tilde u\;\;\mbox{in}\;\; C^1(\overline{\Omega})\;\;\mbox{for all}\;\; k\in{\mathbb{N}}.$$ : $\{u_n^k\}_{k \in {\mathbb{N}}}\subseteq C^1(\overline{\Omega})$ is relatively compact.\
In fact, recalling , pick $M=\displaystyle{\sup_{n\in{\mathbb{N}}}} \|w_n\|_{C^1(\overline{\Omega})}$. Through Hölder’s and Young’s inequalities, besides , we obtain $$\label{recprinc}
\frac{1}{p} \int_{\Omega} G(\nabla u_n^k) {\rm d}x
+\frac{\beta}{p}\int_{\partial \Omega} |u_n^k|^p {\rm d}\sigma\geq\frac{c_1^p c_2}{p}\|u_n^k\|_{1,p}^p,$$ $$\label{recf}
\begin{split}
&\int_\Omega f(\cdot,u_n^{k-1},\nabla w_n) u_n^k {\rm d}x
\leq c_M |\Omega|^{\frac{1}{p'}}\|u_n^k\|_p + d_M \int_\Omega |u_n^{k-1}|^{p-1}|u_n^k| {\rm d}x \\
&\leq c_M |\Omega|^{\frac{1}{p'}} \|u_n^k\|_p + d_M \left(\frac{1}{p'} \|u_n^{k-1}\|_p^p + \frac{1}{p} \|u_n^k\|_p^p \right),
\end{split}$$ as well as $$\label{recg}
\begin{split}
\int_\Omega & \hat{g}(\cdot,u_n^{k-1}) u_n^k {\rm d}x\\
&=\int_{\Omega(u_n^{k-1}\leq 1)} \hat{g}(\cdot,u_n^{k-1}) u_n^k {\rm d}x
+\int_{\Omega(u_n^{k-1}> 1)} \hat{g}(\cdot,u_n^{k-1}) u_n^k {\rm d}x \\
&\leq\int_{\Omega(u_n^{k-1}\leq 1)} g(\cdot,\underline{u}) u_n^k {\rm d}x+\int_{\Omega(u_n^{k-1}> 1)} g(\cdot,u_n^{k-1}) u_n^k {\rm d}x \\
&\leq(\| g(\cdot,\underline{u})\|_{p'}+c|\Omega|^{\frac{1}{p'}}) \|u_n^k\|_p
+d\int_\Omega |u_n^{k-1}|^{p-1}|u_n^k| {\rm d}x \\
&\leq(\| g(\cdot,\underline{u})\|_{p'}+c|\Omega|^{\frac{1}{p'}})\|u_n^k\|_p
+d\left(\frac{1}{p'} \|u_n^{k-1}\|_p^p + \frac{1}{p} \|u_n^k\|_p^p \right).
\end{split}$$ Since $\mathscr{E}_{u_n^{k-1},w_n}(u_n^k) \leq 0$, estimates – entail $$\label{recgeneral}
\begin{split}
&\frac{c_1^p c_2 - d_M - d}{p} \|u_n^k\|_{1,p}^p\\
&\leq\left(\| g(\cdot,\underline{u})\|_{p'}+(c_M + c)|\Omega|^{\frac{1}{p'}}\right)\|u_n^k\|_{1,p}
+\frac{d_M + d}{p'} \|u_n^{k-1}\|_{1,p}^p
\end{split}$$ for all $k\in{\mathbb{N}}$. Thanks to , Lemma \[reclemma\] applies, and the sequence $\{u_n^k\}_{k \in {\mathbb{N}}}$ turns out bounded in $W^{1,p}(\Omega)$. Standard arguments involving regularity up to the boundary (cf. the proof of Lemma \[Scompact\]) yield the claim.
We may thus assume there exists $\{u_n\}\subseteq C^1(\overline{\Omega})$ fulfilling $$\label{kindex}
\lim_{k \to \infty} u_n^k = u_n\;\,\mbox{in}\;\; C^1(\overline{\Omega})$$ whatever $n\in{\mathbb{N}}$. By and Lebesgue’s dominated convergence theorem one has $u_n\in U_{w_n}$. Moreover, as in the proof of Lemma \[criticalprop\], $u_n \geq \underline{u}$. Due to and , the double limit lemma [@GP Proposition A.2.35] gives $$\label{doublelimit}
u_n\to\tilde u\;\;\mbox{in}\;\; C^1(\overline{\Omega}).$$ Thus, it remains to show that $\mathscr{E}_{w_n}(u_n) < 1$. From we easily infer $\mathscr{E}_{w_n}(\tilde u)\to\mathscr{E}_{w}(\tilde u)$. Since $\mathscr{E}_{w_n}$ is of class $C^1$, via and one arrives at $$\lim_{n\to+\infty}\left(\mathscr{E}_{w_n}(u_n) - \mathscr{E}_{w}(\tilde u)\right)=0,$$ namely $\mathscr{E}_{w_n}(u_n)\to\mathscr{E}_{w}(\tilde u)$. This completes the proof, because $\tilde u\in\mathscr{S}(w)$, whence $\mathscr{E}_{w}(\tilde u)<1$.
\[comparison\] Under ${\rm H(f)}$, ${\rm H(g)}$, and , the set $\mathscr{S}(w)$, $w\in C^1(\overline{\Omega})$, is downward directed.
Let $u_1,u_2\in \mathscr{S}(w)$ and let $\hat{u}:=\min\{u_1,u_2\}$. By Lemma \[supersol\] we have $\hat u\in
\overline{U}_w$. Consider the problem $$\label{compareprob}
\left\{
\begin{array}{ll}
- {\rm div} \, a(\nabla u) = h(x,u)\;\; & \mbox{in}\;\;\Omega, \\
u > 0\;\; & \mbox{in}\;\;\Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}}+\beta |u|^{p-2}u= 0\;\; & \mbox{on}\;\;\partial \Omega,
\end{array}
\right.$$ where $$h(x,s) = \left\{
\begin{array}{ll}
f(x,\underline{u}(x),\nabla w(x)) + g(x,\underline{u}(x))\;\; & \mbox{for}\; s \leq \underline{u}(x), \\
f(x,s,\nabla w(x)) + g(x,s)\;\; & \mbox{if}\; \underline{u}(x) < s < \hat{u}(x), \\
f(x,\hat u(x),\nabla w(x)) + g(x,\hat u(x))\;\; & \mbox{when} \; s \geq\hat{u}(x).
\end{array}
\right.$$ The associated energy functional writes $$\tilde{\mathscr{E}}_w(u):=\frac{1}{p}\int_{\Omega} G(\nabla u){\rm d}x+\beta\int_{\partial \Omega} |u|^p {\rm d}x - \int_{\Omega}{\rm d}x \int_0^{u} h(\cdot,t){\rm d}t,\; u \in W^{1,p}(\Omega).$$ Arguing as in Lemma \[Scompact\] produces a solution $\tilde{u}\in C^1(\overline{\Omega})$ to such that $\tilde{\mathscr{E}}_w(\tilde{u})\leq 0$. Next, adapt the proof of Lemma \[criticalprop\] and exploit the fact that $\hat{u}$ is a supersolution of to achieve $\underline{u}\leq\tilde{u}\leq\hat{u}$. Consequently, $\tilde{u}\in U_w$ and $$\mathscr{E}_w(\tilde{u})=\tilde{\mathscr{E}}_w(\tilde{u})\leq 0<1.$$ This forces $\tilde{u}\in \mathscr{S}(w)$, besides $\tilde{u}\leq\min\{u_1,u_2\}$.
\[wellposed\] If ${\rm H(f)}$, ${\rm H(g)}$, and hold true then for every $w\in C^1(\overline{\Omega})$ the set $\mathscr{S}(w)$ possesses absolute minimum.
Fix $w \in C^1(\overline{\Omega})$. We already know (see Lemma \[comparison\]) that $\mathscr{S}(w)$ turns out downward directed. If $\mathscr{C} \subseteq \mathscr{S}(w)$ is a chain in $\mathscr{S}(w)$ then there exists a sequence $\{u_n\}
\subseteq\mathscr{S}(w) $ satisfying $$\lim_{n \to \infty} u_n = \inf \mathscr{C}.$$ On account of Lemma \[Scompact\] and up to subsequences, one has $u_n\to\hat u$ in $C^1(\overline{\Omega})$. Thus, $\hat u =\inf\mathscr{C}$. By Zorn’s Lemma, $\mathscr{S}(w)$ admits a minimal element $u_w$. It remains to show that $u_w=\min \mathscr{S}(w)$. Pick any $u\in\mathscr{S}(w)$. Through Lemma \[comparison\] we get $\tilde{u}\in\mathscr{S}(w)$ such that $\tilde{u} \leq\min\{u_w,u\}$. The minimality of $u_w$ entails $u_w=\tilde{u}$. Therefore, $u_w\leq u$, as desired.
This proof is patterned after the one in [@LMZ Theorem 23].
Lemma \[wellposed\] allows to consider the function $\Gamma: C^1(\overline{\Omega})\to C^1(\overline{\Omega})$ given by $$\Gamma(w):=\min\mathscr{S}(w)\quad\forall\, w\in C^1(\overline{\Omega}).$$
\[propgamma\] Under assumptions ${\rm H(f)}$, ${\rm H(g)}$, and , $\Gamma$ is continuous and maps bounded sets into relatively compact sets.
It is analogous to that of [@LMZ Lemma 24]. So, we will omit details. Let $\mathscr{B}\subseteq C^1(\overline{\Omega})$ be bounded. Since $\Gamma(\mathscr{B})\subseteq\mathscr{S}(\mathscr{B})$ and $\mathscr{S}(\mathscr{B})$ turns out relatively compact (cf. Lemma \[Scompact\]), $\Gamma(\mathscr{B})$ enjoys the same property. Next, suppose $ w_n\to w$ in $C^1(\overline{\Omega})$. Setting $u_n :=\Gamma (w_n)$, one evidently has $u_n \to u$ in $C^1(\overline{\Omega})$, where a subsequence is considered when necessary. The function $u$ complies with $u\geq\underline{u} $ and $\mathscr{E}_w(u) < 1$ (see the proof of Lemma \[Slsc\]). Via the Lebesgue dominated convergence theorem, from $u_n\in U_{w_n}$ it follows $u\in U_w$. Plugging all together, we get $u \in \mathscr{S}(w)$. It remains to verify that $u =\Gamma(w)$. Lemma \[Slsc\] provides a sequence $\{v_n\}\subseteq C^1(\overline{\Omega})$ fulfilling both $v_n\in\mathscr{S}(w_n)$ for all $n\in{\mathbb{N}}$ and $v_n\to\Gamma(w)$ in $ C^1(\overline{\Omega})$. The choice of $\Gamma$ entails $u_n=\Gamma(w_n) \leq v_n$, besides $\Gamma(w)\leq u$. Letting $n \to+\infty$ we thus arrive at $$\Gamma(w)\leq u=\lim_{n\to+\infty} u_n\leq\lim_{n \to+\infty} v_n=\Gamma(w),$$ i.e., $u=\Gamma(w)$, which completes the proof.
To establish our main result, the stronger version below of H(f) will be employed. $\underline{{\rm H'(f)}}$ $ f:\Omega \times{\mathbb{R}}\times{\mathbb{R}}^N\to [0,+\infty) $ is a Carathéodory function such that $$f(x,s,\xi)\leq c_3 + c_4 |s|^{p-1} + c_5 |\xi|^{p-1} \quad\forall\, (x,s,\xi) \in \Omega \times {\mathbb{R}}\times {\mathbb{R}}^N,$$ with appropriate $c_3,c_4,c_5>0$. Condition is substituted by $$\label{condition2}
c_4 +(2p-1)c_5+ d< c_1^p c_2\, .$$
Assumption ${\rm H'(f)}$ clearly implies $ {\rm H(f)}$, with $ c_M:=c_3+c_5 M^{p-1}$ and $d_M:=c_4$. Likewise, forces while reads as $$\label{reccond2}
c_4+d<\frac{c_1^p c_2}{p}\, .$$
\[existence\] Let ${\rm H'(f)}$, ${\rm H(g)}$, and – be satisfied. Then problem possesses a solution $u\in{\rm int}(C^1(\overline{\Omega})_+) $. The set of solutions to is compact in $C^1(\overline{\Omega})$.
Define $$\Lambda(\Gamma):=\{ u \in C^1(\overline{\Omega}): u=\tau\,\Gamma(u)\;\mbox{for some}\;\tau\in (0,1)\}.$$ : $\Lambda(\Gamma)$ is bounded in $W^{1,p}(\Omega)$.\
To see this, pick any $u\in\Lambda(\Gamma)$. Since $\frac{u}{\tau}=\Gamma(u)\in\mathscr{S}(u)$, one has $\mathscr{E}_u \left( \frac{u}{\tau} \right)<1$. Assumption ${\rm H'(f)}$, combined with Young’s and Hölder’s inequalities, produces $$\begin{split}
\int_{\Omega \left(\frac{u}{\tau}>\underline{u}\right)}\left(\int_{\underline{u}}^{\frac{u}{\tau}} f(\cdot,t,\nabla u) {\rm d}t \right) {\rm d}x
&\leq\int_{\Omega}\left( \int_{0}^{\frac{u}{\tau}} (c_3 + c_4 t^{p-1}+ c_5|\nabla u|^{p-1}) {\rm d}t \right) {\rm d}x \\
&\leq c_3 \left\|\frac{u}{\tau}\right\|_1 + \frac{c_4}{p} \left\|\frac{u}{\tau}\right\|_p^p+ c_5 \int_{\Omega} |\nabla u|^{p-1} \left|\frac{u}{\tau}\right| {\rm d}x \\
&\leq c_3 |\Omega|^{\frac{1}{p'}}\left\|\frac{u}{\tau}\right\|_p+\frac{c_4}{p} \left\|\frac{u}{\tau}\right\|_p^p+c_5 \left(\frac{\left\|\frac{u}{\tau}\right\|_p^p}{p}+\frac{\|\nabla u\|_p^p}{p'} \right) \\
&\leq c_3|\Omega|^{\frac{1}{p'}}\left\|\frac{u}{\tau}\right\|_{1,p}+\frac{c_4+c_5}{p}\left\|\frac{u}{\tau}\right\|_{1,p}^p + \frac{c_5}{p'}\|u\|_{1,p}^p.
\end{split}$$ Analogously, on account of , $$\begin{split}
\int_\Omega f(\cdot,\underline{u},\nabla u)\underline{u} {\rm d}x
&\leq\int_\Omega\left( c_3 \underline{u}+c_4 \underline{u}^p+c_5 |\nabla u|^{p-1}\right)\underline{u} {\rm d}x \\
&\leq \left( c_3 + c_4 + \frac{c_5}{p} \right) |\Omega| + \frac{c_5}{p'} \|\nabla u\|_p^p \\
&\leq \left( c_3 + c_4 + \frac{c_5}{p} \right) |\Omega| + \frac{c_5}{p'} \|u\|_{1,p}^p. \\
\end{split}$$ Reasoning as in Lemma \[estlemma\] and recalling that $\tau\in (0,1)$, we thus achieve $$\begin{split}
1&>\mathscr{E}_u\left( \frac{u}{\tau} \right)\\
&\geq\frac{c_1^p c_2 - c_4 - (2p-1)c_5-d}{p}\left\|\frac{u}{\tau}\right\|_{1,p}^p
-(c_3 + c) |\Omega|^{\frac{1}{p'}}\left\| \frac{u}{\tau} \right\|_{1,p} - K',
\end{split}$$ where $$K':=\left( c_3+c_4+\frac{c_5}{p} \right)|\Omega|+2 \|g(\cdot,{\varepsilon}\theta)\|_{p'} |\Omega|^{\frac{1}{p}}.$$ Thanks to , the above inequalities force $$\| u\|_{1,p}\leq\left\| \frac{u}{\tau} \right\|_{1,p} \leq K^*,$$ with $K^*>0$ independent of $u$ and $\tau$. Thus, the claim is proved.\
By regularity [@Li], the set $\Lambda(\Gamma)$ turns out bounded in $C^1(\overline{\Omega})$. Hence, due to Lemma \[propgamma\], Theorem \[schaefer\] applies, which entails ${\rm Fix}(\Gamma) \neq \emptyset$. Let $u\in {\rm Fix}(\Gamma) $. From $u =\Gamma(u)\in\mathscr{S}(u)$ we deduce both $u\geq\underline{u}$ and $u\in U_u$. Accordingly, $$\hat{f}(\cdot,u)=f(\cdot,u,\nabla u),\quad\hat{g}(\cdot,u) = g(\cdot,u),$$ namely the function $u$ solves problem . Further, $u\in{\rm int}(C^1(\overline{\Omega})_+)$ because of the strong maximum principle.
Finally, arguing as in Lemma \[regularity\] ensures that each solution to lies in $C^{1,\alpha}(\overline{\Omega})$. Since $C^{1,\alpha}(\overline{\Omega})\hookrightarrow C^1(\overline{\Omega})$ compactly and the solution set of is closed in $C^1(\overline{\Omega})$, the conclusion follows.
The same techniques can be applied for finding solutions to the Neumann problem $$\left\{
\begin{array}{ll}
- {\rm \, div} a(\nabla u) + |u|^{p-2}u = f(x,u,\nabla u) + g(x,u)\;\;&\mbox{in}\;\; \Omega, \\
u > 0\;\;&\mbox{in}\;\;\Omega, \\
\displaystyle{\frac{\partial u}{\partial \nu_a}} = 0\;\;&\mbox{on}\;\;\partial \Omega.
\end{array}
\right.$$ In fact, it is enough to replace the norm $\|\cdot\|_{\beta,1,p}$ with the standard one $\|\cdot\|_{1,p}$.
Uniqueness (for $p=2$) {#S4}
======================
Throughout this section, $p=2$, the operator $a$ fulfills H(a), while the nonlinearities $f$ and $g$ comply with H(f) and H(g), respectively. The following further conditions will be posited:
- There exists $c_6\in (0,1]$ such that $$\langle a(\xi)-a(\eta),\xi-\eta\rangle\geq c_6 |\xi - \eta|^2\quad\forall\,\xi,\eta \in {\mathbb{R}}^N.$$
- With appropriate $c_7,c_8>0$ one has $$\label{effeone}
[f(x,s,\xi)-f(x,t,\xi)](s-t)\leq c_7|s-t|^2$$ $$\label{effetwo}
|f(x,t,\xi) - f(x,t,\eta)| \leq c_8 |\xi-\eta|$$ in $\Omega\times{\mathbb{R}}\times{\mathbb{R}}^N$.
- There is $c_9>0$ such that $$\label{gone}
[g(x,s) - g(x,t)](s-t)\leq c_9 |s-t|^2\;\;\forall\, x \in\Omega,\; s,t \in [1,+\infty).$$ Moreover, $$\label{gtwo}
g(x,s)\leq g(x,1)\;\;\mbox{in}\;\;\Omega\times (1,+\infty).$$
The parametric $(2,q)$-Laplacian $\Delta+\mu\Delta_q$, where $1<q< 2$, $\mu\geq 0$, satisfies ${\rm H(a)}$ and $({\rm a}_4)$; cf. [@P Lemma A.0.5].
Under the above assumptions, problem admits a unique solution provided $$\label{uniqcond}
c_7 + c_1 c_8 + c_9 < c_1^2 c_6.$$
Suppose $u,v$ solve , test with $u-v$, and subtract to arrive at $$\label{weakcomp}
\begin{split}
&\int_{\Omega}\langle a(\nabla u)-a(\nabla v),\nabla(u-v)\rangle{\rm d}x+\beta\int_{\partial \Omega} |u-v|^2 {\rm d}\sigma\\
&= \int_{\Omega} [f(\cdot,u,\nabla u) - f(\cdot,v,\nabla v)](u-v){\rm d}x\\
&+ \int_{\Omega} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x.
\end{split}$$ The left-hand side of can easily be estimated from below via $({\rm a}_4)$ as follows: $$\label{aestimate}
\int_{\Omega}\langle a(\nabla u)-a(\nabla v),\nabla(u-v)\rangle {\rm d}x+\beta\int_{\partial \Omega}|u-v|^2 {\rm d}\sigma
\geq c_6 \|u-v\|_{\beta,1,2}^2.$$ Using – and Hölder’s inequality we get $$\label{festimate}
\begin{split}
\int_{\Omega} & [f(\cdot,u,\nabla u) - f(\cdot,v,\nabla v)](u-v) {\rm d}x \\
&=\int_{\Omega} [f(\cdot,u,\nabla u) - f(\cdot,v,\nabla u)](u-v) {\rm d}x \\
&\phantom{pppppp}+\int_{\Omega}[f(\cdot,v,\nabla u) - f(\cdot,v,\nabla v)](u-v) {\rm d}x \\
&\leq c_7\int_{\Omega} |u-v|^2 {\rm d}x + c_8\int_{\Omega} |\nabla u - \nabla v| |u-v| {\rm d}x \\
&\leq c_7 \|u-v\|_2^2 + c_8 \|\nabla(u-v)\|_2\|u-v\|_2 \\
&\leq \frac{c_7}{c_1^2} \|u-v\|_{\beta,1,2}^2+ \frac{c_8}{c_1} \|u-v\|_{\beta,1,2}^2.
\end{split}$$ Observe now that $$\label{gestimate}
\begin{split}
&\int_{\Omega} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&= \int_{\Omega(\max\{u,v\} \leq 1)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&+ \int_{\Omega(\min\{u,v\} > 1)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&+ \int_{\Omega(u \leq 1 < v)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&+ \int_{\Omega(v \leq 1 < u)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x.
\end{split}$$ By hypothesis $({\rm g}_1)$ in H(g) one has $$\label{integral1}
\int_{\Omega(\max\{u,v\} \leq 1)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \leq 0.$$ Inequality entails $$\label{integral2}
\begin{split}
&\int_{\Omega(\min\{u,v\} > 1)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&\leq c_9 \|u-v\|_2^2 \leq \frac{c_9}{c_1^2} \|u-v\|_{\beta,1,2}^2.
\end{split}$$ Thanks to $({\rm g}_1)$ again and we obtain $$\label{integral3}
\begin{split}
&\int_{\Omega(u \leq 1 < v)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x \\
&\leq \int_{\Omega(u \leq 1 < v)} [g(\cdot,1) - g(\cdot,v)](u-v) {\rm d}x \leq 0.
\end{split}$$ Likewise, $$\label{integral4}
\int_{\Omega(v \leq 1<u)} [g(\cdot,u) - g(\cdot,v)](u-v) {\rm d}x\leq 0.$$ Plugging – into and – into yields $$c_6 \|u-v\|_{\beta,1,2}^2 \leq\left(\frac{c_7}{c_1^2} + \frac{c_8}{c_1} + \frac{c_9}{c_1^2}\right) \|u-v\|_{\beta,1,2}^2.$$ On account of , this directly leads to $u=v$, as desired.
The conditions that guarantee existence or uniqueness, namely , , and , represent a balance between data (growth or variation of reaction terms) and structure (driving operator and domain) of the problem .
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is performed within PTR 2018–2020 - Linea di intervento 2: ‘Metodi Variazionali ed Equazioni Differenziali’ of the University of Catania and partly funded by Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2017 ‘Nonlinear Differential Problems via Variational, Topological and Set-valued Methods’ (Grant Number 2017AYM8XW)
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[^1]: Corresponding Author
|
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abstract: |
We propose a construction of Skyrme fields from holonomy of the spin connection of gravitational instantons. The procedure is implemented for Atiyah–Hitchin and Taub–NUT instantons.
The skyrmion resulting from the Taub–NUT is given explicitly on the space of orbits of a left translation inside the whole isometry group. The domain of the Taub–NUT skyrmion is a trivial circle bundle over the Poincare disc. The position of the skyrmion depends on the Taub–NUT mass parameter, and its topological charge is equal to two.
author:
- |
Maciej Dunajski[^1]\
Department of Applied Mathematics and Theoretical Physics,\
University of Cambridge,\
Wilberforce Road, Cambridge CB3 0WA, UK.
title: '-70pt'
---
Introduction
============
The Standard Model of elementary particles is a quantum gauge theory with a non–abelian gauge group $SU(3)\times SU(2) \times U(1)$. This model is at some level fundamental, and provides a complete field theory of interacting quarks. Thus in principle it should describe protons and neutrons. On the other hand the Lagrangian underlying the Standard Model leads to non-linear field equations, which has so far made it impossible to obtain exact results about the bound states describing the particles of the theory.
An alternative is to look for theories which ignore the internal structure of particles, and instead give an effective, low energy description of baryons. The Skyrme model [@Skyrme_paper] is an example of such theory, where baryons arise as solitons. The topological degree of these solitons is identified with the baryon number in a mechanism which naturally leads to a topological baryon number conservation.
In a recent paper [@AMS11] Atiyah, Manton and Schroers (AMS) proposed a far reaching generalisation of the Skyrme model which involves several topological invariants, and aims to give a geometrical and topological interpretations to the electric charge, the baryon number and the lepton number. In the AMS model static particles are described in terms of gravitational instantons - Riemannian four manifolds which satisfy the Einstein equations and whose curvature is concentrated in a finite region of a space-time [@GH].
The electrically charged particles correspond to non–compact asymptotically locally flat (ALF) instantons $(M, g)$ – complete four–dimensional Riemannian manifolds which solve the Einstein equations (possibly with cosmological constant) and approach $S^1$ bundle over $S^2$ at infinity. The first Chern class of the asymptotic $U(1)$ fibration gives the electric charge. Neutral particles correspond to compact instantons. In all cases the baryon number has a topological origin and is identified with the signature of $M$.
The AMS model is inspired by the Atiyah–Manton approach [@AM89] to the Skyrme model of baryons, where a static Skyrme field $U:\R^3\rightarrow SU(2)$ with the boundary condition $U(\bf{x})\rightarrow {\bf 1} $ as $|{\bf x}|\rightarrow \infty$ arises from a holonomy of a Yang–Mills instanton on $\R^4$ along one of the directions. Thus the physical three–space $\R^3$ is regarded as the space of orbits of a one–parameter group of conformal isometries of $\R^4$. AMS use this as a motivation for their model, but in the AMS approach the three–space is (for the electrically charged particles) the base space of asymptotic circle fibrations.
The idea behind the present paper is to use the AMS model as a motivation for relating particles to gravitational instantons, but then to proceed in a way analogous to the Atiyah–Manton construction to recover a skyrmion from a gravitational instanton. Thus, in our case, the three–space ${\mathcal B}$ will arise as a quotient of $M$ by a certain $S^1$ action. The $SU(2)$ instanton holonomy will be replaced by a holonomy of a spin connection on $\spp_+$, where $TM\otimes \C\cong
\spp_+\otimes\spp_-$, and $\spp_\pm$ are rank two complex vector bundles over $M$ (see e. g. [@Dunajski_book]).
In the next Section we shall reinterpret the $\mathfrak{su}(2)$ spin connection on $\spp_+$ as the potential for a self–dual Yang–Mills field on the gravitational instanton background. In Section \[section3\] we shall compute the holonomy of this potential along the orbits of an $SO(2)$ left–translation inside the whole isometry group of $(M, g)$, using the Atiyah–Hitchin and Taub–NUT instantons as examples. This will give rise to a skyrmion on the space of orbits ${\mathcal B}$ of $SO(2)$ in $M$. We shall find the expression for the topological charge density, and compute this charge for the Taub–NUT skyrmion. In Section \[section4\] we shall construct the Riemannian metric $h_{\mathcal B}$ on the three–dimensional domain ${\mathcal B}$ of the skyrmion. In the case of Taub–NUT skyrmion this metric is complete and describes a trivial circle fibration over the upper half–plane, with circular fibres of non–constant radius: $$h_{\mathcal B}=g_{\HH^2}+R^2d\psi^2,\quad\mbox{where}\quad
R^2=\frac{1}{(\mu r+1)^2\sin{\theta}^2+\cos{\theta}^2}.$$ Here the constant parameter $\mu$ is the inverse mass in the Taub–NUT space, and $y=r\sin{\theta}>0, x=r\cos{\theta}$ are coordinates on $\HH^2$ with the hyperbolic metric $g_{\HH^2}=y^{-2}(dx^2+dy^2)$. In these coordinates the skyrmion is given by \[skyrmion\_introduction\] U=. The skyrmion (\[skyrmion\_introduction\]) is localised on the imaginary axis, around the point $(0,5/(4\mu))$. We should again, at this point, emphasise the difference between our construction and the AMS approach. The ‘physical’ three–space in [@AMS11] admits an isometric $SO(3)$ action, whereas the three–dimensional Riemannian manifold $({\mathcal B}, h_{\mathcal B})$ which supports the skyrmion admits only one isometry in the Taub–NUT case. This is because the generator of the left translation used in the construction of the quotient belongs to a two–dimensional abelian subalgebra inside the full Lie algebra of the isometry group $U(2)$. Thus one Killing vector of the Taub–NUT space descends down to the quotient. In the Atiyah–Hitchin case the isometry algebra $SO(3)$ does not contain two–dimensional abelian sub-algebras, and the quotient space ${\mathcal B}$ does not admit any Killing vectors, or conformal Killing vectors.
Spin connection as gauge potential {#section2}
==================================
Properties of a single particle are invariant with respect to ordinary rotations in three-space. Thus the corresponding instanton should admit $SO(3)$ or its double cover $SU(2)$ as the group of isometries, i. e. the metric should take the form \[metric\] g=f\^2 dr\^2+(a\_1 \_1)\^2+(a\_2\_2)\^2+(a\_3\_3)\^2, where $\eta_i$ are the left invariant one–forms on $SU(2)$ such that $$d\eta_1=\eta_2\wedge\eta_3, \quad d\eta_2=\eta_3\wedge\eta_1,
\quad \quad d\eta_3=\eta_1\wedge\eta_2$$ and $(a_1, a_2, a_3, f)$ are functions of $r$. There is no loss of generality in this diagonal ansatz, as the induced metric can always be diagonalised on a surface of constant $r$, and then the Einstein equations imply [@tod_cohomogeneity] that the non–diagonal components are fixed (to zero) in the ‘evolution’ in $r$. The diffeomorphism freedom can be used to set $f=-a_2/r$.
In the AMS setup the Atiyah–Hitchin manifold [@AH] is a model for the proton and the self–dual Taub NUT manifold corresponds to the electron. Although the spin connection $\gamma_-$ of (\[metric\]) does not vanish in the invariant frame (\[metric\]), the curvature of $(\spp_-, \gamma_-)$ is zero as both AH and Taub–NUT metrics have self–dual Riemann curvature. Thus we shall consider the connection $\gamma=\gamma_+$ on $\spp_+$. It is best calculated using the self–dual two–forms [@Dunajski_book] $$\begin{aligned}
\label{two_forms}
\Sigma_i&=&{\bf e}_0\wedge {\bf e}_i+\frac{1}{2}\varepsilon_{ijk} {\bf e}_j\wedge {\bf e}_k,
\quad \mbox{where}\quad i, j, k =1, \dots, 3\quad\mbox{and}\\
{\bf e}_0&=&fdr,\quad {\bf e}_1=a_1\eta_1,\quad {\bf e}_2=a_2\eta_2,\quad
{\bf e}_3=a_3\eta_3.\nonumber\end{aligned}$$ The spin connection coefficients $\gamma_{ij}$ are skew–symmetric and are determined from the relations $
d\Sigma_i+\gamma_{ij} \wedge \Sigma_j=0.
$ We find \[connection\] P\_1=f\_1(r)\_1, P\_2=f\_2(r)\_2, P\_3=f\_3(r)\_3, \_[ij]{}=\_[ijk]{} P\_k, where the functions $f_i(r)$ depend on the coefficients $a_i(r)$ and their derivatives.
- The Taub–NUT metric is a unique non–flat complete self–dual Einstein metric with isometric $SU(2)$ action such that the generic orbit is three–dimensional, and the $SU(2)$ action rotates the anti–self–dual two–forms. In this case $$a_1=a_2=r\sqrt{\epsilon+\frac{m}{r}}, \quad
a_3=m\sqrt{\epsilon+\frac{m}{r}}^{-1},$$ where $\epsilon$ and $m$ are constants. At $r=0$ the three-sphere of constant $r$ collapses to a point – an example of a NUT singularity. The SD spin connection coefficients[^2] give the Pope–Yuille instanton [@PY] (see [@cherkis] for a discussion on more general Yang-Mills instantons on self-dual ALF spaces) \[taub\_nut\] f\_1=f\_2=-, f\_3=.
- The Atiyah–Hitchin metric is a unique (up to taking a double covering) complete self–dual Einstein metric with isometric $SO(3)$ (rather than $SU(2)$) action such that the generic orbit is three–dimensional, and the action rotates the anti–self–dual two–forms [@AH]. It is a metric on a moduli space of 2-monopoles with fixed centre. The $SO(3)$ isometric action can be traced back to the 2-monopole configuration, where the rotation group acts on the pair of unoriented spectral lines, and $r$ is (a function of) an angle between these lines. The coordinate $r$ parametrises the orbits of $SO(3)$ in the moduli space. In this paper we shall use the double cover of the moduli space of centered 2-monopoles, and, following [@AMS11], still call it the Atiyah–Hitchin (or AH) manifold.
In the case of the Atiyah–Hitchin metric we shall only need the asymptotic formulae. The coordinate $r$ ranges between $\pi$ and $\infty$, and at $r=\pi$ the $SO(3)$ orbits collapses to a two–sphere (a bolt). For large $r$ $$a_1=a_2=r\sqrt{1-\frac{2}{r}} +O(e^{-r}),
\quad a_3=-2\sqrt{1-\frac{2}{r}}^{-1} +O(e^{-r}),$$ which leads to the asymptotic expressions \[AHlarge\_r\] f\_1=f\_2=-, f\_3=. For $r$ close to $\pi$ we have $$\begin{aligned}
a_1&=&2(r-\pi)+ O((r-\pi)^2) ,\quad a_2=\pi+\frac{1}{2}(r-\pi)
+O((r-\pi)^2),\\
a_3&=&-\pi+\frac{1}{2}(r-\pi) + O((r-\pi)^2), \end{aligned}$$ which gives \[AHsmall\_r\] f\_1=-3, f\_2=, f\_3=.
To make contact with the ‘skyrmions from instantons’ ansatz of [@AM89] we need to reinterpret the self–dual spin connection $\gamma$ as $\mathfrak{su}(2)$–valued gauge field $A$. This is done [@charap; @kor] by setting \[gauge\_field\] A=P\_1\_1+P\_2\_2+P\_3\_3, where the matrices ${\bf t}_i$ generate the Lie algebra $\mathfrak{su}(2)$ with the commutation relations $[{\bf t}_i, {\bf t}_j]=-(1/2)\varepsilon_{ijk} {\bf t}_k$, and the one–forms $P_j$ are given by (\[connection\]). Topology of the Yang–Mills field is determined by the topology of the gravitational instanton (see [@dunajski_2] for a related construction where topology of an abelian vortex is determined by the topology of the underlying background surface). The topological charge of the Yang–Mills instanton is in general fractional, despite the action being finite. The relation between the Yang–Mills, and Einstein curvatures is easily expressed using the $SO(3)$ representation spaces: $$\begin{aligned}
A&=&P_i \otimes {\bf t}_i=\frac{1}{2}\varepsilon_{ijk}\gamma_{jk} \otimes
{\bf t}_i\\
F&=& dA+A\wedge A=\frac{1}{2}\varepsilon_{ijk}R_{jk}\otimes {\bf t}_i,
\quad\mbox{where}\;\;R_{ij}=d\gamma_{ij}
+\gamma_{ik}\wedge\gamma_{kj}\end{aligned}$$ is the Riemann curvature two–form of the gravitational instanton metric. Therefore $F$ is a self–dual Yang–Mills field $$F=*F,$$ where $*$ is the Hodge operator on the gravitational instanton background.
The two–form $R_{ij}$ can be decomposed in terms of the Ricci tensor, Weyl tensor and Ricci scalar as $
R_{ij}=W_{ijk}\Sigma_k+\Phi_{ijk}\Omega_k,
$ where $\Sigma_k$ and $\Omega_k$ are basis of SD (see \[two\_forms\]) and ASD two–forms respectively. The coefficients $\Phi_{ijk}$ have nine components corresponding to the trace–free Ricci tensor. The Bianchi identity $R_{ij}\wedge\Sigma_j=0$ gives $W_{ijj}=0$, so $W_{ijk}$ can be further decomposed into a self–dual Weyl tensor (with five independent components), and the totally skew part $\Lambda\varepsilon_{ijk}$, where $\Lambda$ is a multiple of the Ricci scalar. In the self–dual vacuum case we have $\Phi_{ijk}=0, \Lambda=0$. Using the identities $$\Sigma_i\wedge\Sigma_j=2\delta_{ij}\mbox{vol}, \quad
\mbox{Tr}({\bf t}_i{\bf t}_j)=-\frac{1}{2}\delta_{ij}, \quad
\varepsilon_{ijk}\varepsilon_{kpq}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}$$ we find $ \mbox{Tr}(F\wedge F)=-(1/2)|W|^2\mbox{vol}$, where $|W|^2=W_{ijk}W^{ijk}$. In the case considered in this paper, where $P_j$ are given by the one–forms (\[connection\]) we find (with $\cdot =d/dr$) $$\begin{aligned}
F&=&(\dot{f}_1 dr\wedge\eta_1+(f_1-f_2f_3)\;\eta_2\wedge \eta_3)\otimes {\bf t}_1\\
&+&(\dot{f}_2 dr\wedge\eta_2+(f_2-f_1f_3)\;\eta_3\wedge \eta_1)\otimes {\bf t}_2\\
&+&(\dot{f}_3 dr\wedge\eta_3+(f_3-f_1f_2)\;\eta_1\wedge \eta_2)
\otimes {\bf t}_3.\end{aligned}$$ The instanton number is $k=-c_2$, where the Chern number is given by $$c_2=-\frac{1}{8\pi^2}\int_M \mbox{Tr}(F\wedge F).$$ In our case $$\mbox{Tr}(F\wedge F)=\frac{d}{dr}\Big(f_1 f_2 f_3-\frac{1}{2}(f_1^2+f_2^2+f_3^2)\Big)
dr\wedge \eta_1\wedge \eta_2\wedge \eta_3,$$ and integration by parts gives $k_{TN}=1$ and $k_{AH}=2.$ In evaluating the $r$–integrals we took into account that the radial direction is oppositely oriented in the AH and the Taub–NUT cases. This is a consequence of a fact (carefully discussed in [@AMS11] ) that the Taub–NUT metric is self-dual for the orientation which in the limit $\epsilon\rightarrow 0$ gives the standard orientation on $\C^2$. The Atiyah–Hitchin manifold on the other hand is self–dual for the orientation opposite to the complex orientations given by the underlying hyper–Kähler structure.
Skyrmions from spin connection holonomy {#section3}
=======================================
The Skyrme model is a non–linear theory of pions in three space dimensions [@Skyrme_paper]. The model does not involve quarks, and is to be regarded as a low energy, effective theory of QCD. A static skyrmion is a map $$U:\R^3\rightarrow SU(2)$$ satisfying the boundary conditions $U\rightarrow{\bf 1}$ as $|{\bf x}|\rightarrow \infty$. The boundary conditions imply that $U$ extends to a one–point compactification $S^3=\R^3\cup \{\infty\}$, and thus $U$ is partially classified by its integer topological degree taking values in $\pi_3(S^3)$. In the Skyrme model this topological degree is identified with the baryon number, which by continuity is conserved under time evolution. The non–linear field equations resulting from the Skyrme Lagrangian are not integrable and no explicit solutions are known. In contrast to other soliton models, the Bogomolny bound is not saturated in the Skyrme case, and the energy is always greater than the baryon number.
A good approximation of skyrmions is given by holonomy of $SU(2)$ instantons in $\R^4$, computed along straight lines in one fixed direction [@AM89]. Choosing the lines to be parallel to the $s=x^4$ axis gives the Atiyah–Manton ansatz $$U({\bf x})={\mathcal P}\exp{\Big(\int_{-\infty}^{\infty}
A_4({\bf x}, s)ds\Big)}$$ where $A_4$ is a component of the Yang–Mills instanton on $\R^4$. The end points of each line should be identified with the north–pole of the four sphere compactification of $\R^4$. The boundary conditions at ${\bf x}\rightarrow \infty$ are then satisfied as small circles on $S^4$ corresponding to straight lines shrink to a point (Figure 1). Moreover the instanton number of $A$ is equal to the baryon number of the resulting skyrmion.
{width="10cm" height="5cm"}
[[**Figure 1.**]{} [*Boundary conditions from the instanton ansatz.*]{}]{}
In this section we shall adapt the Atiyah–Manton construction to gravitational instantons. While the underlying principle still applies, and holonomy of the spin connection along certain geodesics gives rise to a scalar group–valued field, the properties of the resulting skyrmion are very different. In particular it is not defined on $\R^3$ which affects the boundary conditions. The topological degree can still be found, but we shall not interpret it as the baryon number as this is given by the signature of the underlying gravitational instanton. In particular the baryon number is zero for the Taub–NUT skyrmion, but the Skyrme topological degree is not.
Let $K=K^a \p/\p x^a, a=0, \dots, 3$ be a vector field generating a one–parameter group of transformations of $M$ with orbits $\Gamma$. The holonomy of the gauge field $A$ along $\Gamma$ arises from a solution to an ordinary differential equation $K^aD_a \Psi=0$, where $D_a=\p_a+A_a$, and $\Psi=\Psi(s, x^j)$ takes its value in the Lie group $SU(2)$. Here $K=\p/\p s$ and $x^j$ are the coordinates on the space of orbits ${\mathcal B}$ of $K$ in $M$. If the trajectories $\Gamma$ are non–compact, then one imposes the initial condition $\Psi(0, x^j)={\bf 1}$ at $s=-\infty$, and sets the Skyrme field to be $U(x^j)=\lim_{s\rightarrow \infty}\Psi(s, x^j)$. This gives \[integral\] U=[P]{}, where ${\mathcal P}$ denotes the $s$–ordering. In the case of $\Gamma$ being a circle one breaks it up into an interval.
We now have to choose the curves $\Gamma$ along which the holonomy is to be calculated. We shall need an explicit parametrisation of the one–forms $\eta_i$ in the metric (\[metric\]) \[forms\_eta\] \_2+i\_1=e\^[-i]{}(d+i d), \_3=d+d, where to cover $SU(2)=S^3$ in the Taub–NUT case we require the ranges $$0\leq\theta\leq\pi, \quad 0\leq\phi\leq 2\pi, \quad
0\leq\psi\leq 4\pi$$ so that $\int \eta_1\wedge \eta_2\wedge \eta_3=-16\pi^2.$ In the AH case take $0\leq\psi\leq 2\pi$, and make an identification $(\theta, \phi, \psi)\cong (\pi-\theta, \phi+\pi, -\psi)$ so that $\int \eta_1\wedge \eta_2\wedge \eta_3=-4\pi^2.$
One natural choice for $\Gamma$ is the family of the asymptotic circles with the $\psi$ coordinate varying, but this gives a trivial result as the resulting Skyrme field depends only on $r$, is Abelian, and its topological charge vanishes. More generally, the holonomy should be calculated along the curves which are orbits of a Killing vector (or at least a conformal Killing vector) as otherwise the space of orbits $\mathcal{B}$ of $K$ in $M$ does not admit a metric even up to scale. However a metric on $\mathcal{B}$ is necessary to compute the energy of the skyrmion. This rules out the asymptotic circles in the AH case.
We shall instead pick a left translation $SO(2)$ inside the isometry group $SO(3)$ (or its double cover $SU(2)$) of (\[metric\]). Without lose of generality we can always choose the Euler angles in (\[forms\_eta\]) so that the generator of this left translation is the right invariant vector field $K=\p/\p\phi$. The $S^1$ fibres of ${\mathcal B}$ have no points in common and the resulting skyrmion can only be defined up to conjugation. However the preferred gauge has been fixed by choosing the $SO(3)$ or $SU(2)$ invariant frame (\[two\_forms\]) in which the spin connection components are proportional to the left–invariant one forms, and the coefficients only depend on the radial coordinate. This procedure is analogous to the one used by Atiyah and Sutcliffe [@AS05]. The Yang–Mills connection resulting from our procedure is given in the radial gauge $A_r=0$ as $$A=f_1(r)\eta_1\otimes {\bf t}_1+f_2(r)\eta_2\otimes {\bf t}_2+f_3(r)\eta_3\otimes {\bf t}_3.$$ The residual gauge freedom $A\rightarrow \rho A \rho^{-1} - d\rho\, \rho^{-1}$, where $\rho=\rho(\theta, \psi, \phi)\in SU(2)$ can either be fixed by demanding regularity of the point $r=0$ (which is singled out as the fixed point of the isometry $K$) or by imposing the symmetry requirement $${\mathcal L}_{R_i} A=0,\quad i=1, 2, 3$$ where the right–invariant Killing vector fields $R_i$ generate left–translations and thus preserve the left–invariant one–forms $\eta_i$ i.e. ${\mathcal L}_{R_i}\eta_j=0$ and ${\mathcal L}$ denotes the Lie derivative.
The anti–symmetric matrix $\nabla_a K_b$ has rank four at $r=0$, where the norm of $K$ vanishes. Thus the Killing vector has an isolated fixed point $r=0$, which is an anti–NUT in the terminology of [@GH]. Therefore some care needs to be taken when constructing the metric on the three–dimensional domain of the skyrmion - this will be done in Section \[section4\].
Restricting the left–invariant forms $\eta_j$ to the $\phi$–circles gives $$\eta_j={{n}_j}\;d\phi,$$ where the unit vector ${\bf{n}}$ is given in the [*unusual*]{} spherical polar coordinates $(\psi, \theta)$ by $${\bf{n}}=(\cos{\psi}\sin{\theta}, \sin{\psi}\sin{\theta}, \cos{\theta}).$$ The integral (\[integral\]) can be performed explicitly, as the component $A_\phi$ of the gauge field (\[gauge\_field\]) does not depend on $\phi$. This yields \[final\_skyrme\] U(r, , )=, where the Pauli matrices $\tau_j$ are related to the generators of $\mathfrak{su}(2)$ by ${\bf t}_j=(i/2)\tau_j$. The topological charge of the skyrmion is[^3] \[charge\] B=-\_ ((U\^[-1]{}dU)\^3) $$=-\frac{1}{2\pi}\int
({\frac{d{f}_1}{dr}f_2f_3\;{{n}_1}^2
+{f_1}\frac{d{f}_2}{dr}f_3\;{n}_2^2+{f_1}{f}_2\frac{d{f}_3}{dr}\;{{n}_3}^2})
\;\frac{\sin{(\pi\kappa)}^2}{\kappa^2}\sin{\theta}\; dr\; d\theta \;d\psi,
$$ where $
\kappa=\sqrt{{f_1}^2{n}_1^2+{f_2}^2{{n}_2}^2+{f_3}^2{n}_3^2}.
$
Let us first consider the Taub–NUT case, where $f_j$ are given by (\[taub\_nut\]). The field $U$ does not satisfy the boundary conditions usually expected from a skyrmion, as $U(0)={\bf 1}$, and $f_j\rightarrow (-1, -1, 1)$ as $r\rightarrow \infty$. It nevertheless gives rise to a well defined constant group element at $\infty$, as there $(f_1{n}_1, f_2 {n}_2,
f_3{n}_3)$ tends to a unit vector and, setting ${\bf k}=(-{n}_1, -{n}_2, {n}_3)$, we get $$\lim_{r\rightarrow\infty} U=\cos{(-\pi)}{\bf 1}+i({\bf k}\cdot{\bf\tau})\sin{(-\pi)}=
-{\bf 1}.$$ The functions $f_1$ and $f_2$ monotonically decrease and $f_3$ monotonically increases. Thus - as the angle $\psi$ varies between $0$ and $4\pi$ - each element of the target space except $U=-{\bf 1}$ has exactly two pre-images in the space of orbits on $\p/\p\phi$. Thus the topological charge of the Taub–NUT skyrmion is $$B_{TN}=2.$$ This is confirmed by evaluating the integral (\[charge\]). We stress that the value of this topological charge is intimately related to the period of the $\psi$ coordinate. The density does not depend on $\psi$, so of the range is $\psi\in [0, k\pi]$, then $B_{TN}=k/2$. In the Atiyah–Hitchin case $f_j\rightarrow (-1, -1, 1)$ when $r\rightarrow \infty$, and the Skyrme field tends to a constant group element $-{\bf 1}$ at infinity. We do not expect the skyrmion to have a constant value at $r=\pi$, as this corresponds to a bolt two–surface in the AH manifold. Formulae (\[AHsmall\_r\]) give $$U(r=\pi, \psi, \theta)=\exp{(3i\pi\cos{\psi}\sin{\theta}\;\tau_1)}.$$ This skyrmion has a constant direction in $\mathfrak{su}(2)$ at the surface of the bolt, with magnitude varying along its boundary.
The degree of the Atiyah–Hitchin skyrmion is still well defined, but we have so far failed in calculating it by direct integration. Instead we shall use the method of counting pre-images. Following [@AH; @GM] we consider the parametrisation of the radial functions $a_i(r)$ by elliptic integrals. Set $$\begin{aligned}
r&=&2 K(\sin(\beta/2)),\\
w_1&=&-r\frac{d r}{d\beta}\sin{\beta}-\frac{1}{2}r^2(1+\cos{\beta}),\\
w_2&=&-r\frac{d r}{d\beta}\sin{\beta},\\
w_3&=&-r\frac{d r}{d\beta}\sin{\beta}+\frac{1}{2}r^2(1-\cos{\beta})\end{aligned}$$ where $K$ is the elliptic integral $$K(k)=\int_0^{\pi/2}\frac{1}{\sqrt{1-k^2\sin{\tau}^2}}d\tau$$ so that when $\beta\in[0, \pi)$, then $r(\beta)\in [\pi, \infty)$ is a monotonically increasing function. The radial functions in the AH metric are $$a_1=\sqrt{\frac{w_2w_3}{w_1}}, \quad a_2=\sqrt{\frac{w_1w_3}{w_2}},\quad
a_3=-\sqrt{\frac{w_1w_2}{w_3}},$$ and we find the spin connection coefficients (\[connection\]) to be $$\begin{aligned}
f_1&=&\frac{1}{2}\Big(\frac{a_3}{a_2}+\frac{a_2}{a_3}- \frac{{a_1}^2}{a_2a_3} \Big)-\frac{r}{a_2} \frac{d a_1}{dr},\\
f_2&=&\frac{1}{2}\Big(\frac{a_3}{a_1}+\frac{a_1}{a_3}-\frac{{a_2}^2}{a_1a_3}\Big)-\frac{r}{a_2} \frac{d a_2}{dr},\\
f_3&=&\frac{1}{2}\Big(\frac{a_1}{a_2}+\frac{a_2}{a_1}-\frac{{a_3}^2}{a_1a_2}\Big)-\frac{r}{a_2} \frac{d a_3}{dr}.\end{aligned}$$ The graphs of $r(\beta), a_i(\beta), f_i(\beta)$ are shown on Figure 2 (the graphs provided in [@GM] give $a_i$ as functions of $r$.).
{width="6cm" height="8cm"} {width="6cm" height="8cm"}
[ [**Figure 2.**]{} [*Radial functions $a_1, a_2, a_3$ and $r$ (1st graph) and the spin connection coefficients (2nd graph) as functions of $\beta\in [0, \pi)$.*]{}]{}
The functions $f_1$ and $f_3$ are monotonically increasing, and the function $f_2$ appears to have a local minimum near $\beta=3.0635$. Thus each point on $SU(2)$ except $U=-{\bf 1}$ has exactly one pre-image in ${\mathcal B}$, and the topological charge of the Atiyah–Hitchin skyrmion is $B_{AH}=1$.
Geometry of the Taub–NUT skyrmion {#section4}
=================================
In both AH and Taub–NUT cases the resulting skyrmion is defined on the space of orbits ${\mathcal B}$ of the isometry $\phi\rightarrow\phi+{const}$ in $(M, g)$. This space has a natural conformal metric induced by (\[metric\]). To find it, perform the standard Kaluza–Klein reduction on $\phi$, simply by completing the square. Set $$\Omega^2={a_1}^2\cos{\psi}^2\sin{\theta}^2+{a_2}^2\sin{\psi}^2\sin{\theta}^2+
{a_3}^2\cos{\theta}^2.$$ Then $$g=h+\Omega^2(d\phi+\Omega^{-2}\omega)^2,$$ where $(h, \omega)$ are a metric and a one form respectively on the space of orbits of $\p/\p\phi$ given by $$h=f^2 dr^2+ ({a_1}^2\sin{\psi}^2+{a_2}^2\cos{\psi}^2) d\theta^2 +{a_3}^2d\psi^2
-\Omega^{-2}\omega^2,$$ $$\omega=({a_2}^2-{a_1}^2)\sin{\psi}\cos{\psi}\sin{\theta} d\theta
+{a_3}^2\cos{\theta}d\psi.$$ The metric $h$ is only defined up to scale on the space of orbits, and we can choose this scale freely.
One choice of the conformal factor which takes into account the range of the angular coordinate $\theta$ in the Taub–NUT case is[^4] $$h_{\mathcal B}=\frac{1}{{a_1}^2\sin{\theta}^2}\;h=g_{\HH^2}+R^2\;d\psi^2$$ This metric is defined on a trivial circle bundle over the hyperbolic plane. The vector $\p/\p\psi$ is an isometry of $h_{\mathcal B}$ which is a consequence of the fact that the right translations $\psi\rightarrow \psi+const$ of the Taub–NUT space commute[^5] with the left translations $\phi\rightarrow \phi+const$. In the upper half plane model where $x=r\cos{\theta}, y=r\sin{\theta}$ the hyperbolic metric $g_{\HH^2}$ and the varying radius $2R$ (as $\psi$ is between $0$ and $4\pi$) of the $S^1$ fibres are given by $$g_{\HH^2}=\frac{dx^2+dy^2}{y^2},
\quad R^2=\frac{1}{(a_1/a_3)^2\sin{\theta}^2+\cos{\theta}^2}=
\frac{x^2+y^2}{y^2({\mu}\sqrt{x^2+y^2}+1)^2+x^2},$$ where $\mu=\epsilon/m$. The radius of the circles tends to two on the real line boundary of the upper half–plane, and shrinks away from the boundary. The metric is complete, as the radius does not vanish anywhere on ${\mathcal B}$. The density of the resulting skyrmion (\[final\_skyrme\]) attains its maximum at the $y$–axis in the upper half–plane model, where it is given by $$\frac{\pi\mu y(\mu y+2)}{(\mu y+1)^2}
\sin{\Big(\frac{\pi \mu y}{\mu y+1}\Big)}^2.$$ The location of the maximum is a root of the transcendental equation $\tan{\hat{y}}=\hat{y}^{-1}(\pi^2-\hat{y}^2)$, where $\mu y=\pi/\hat{y}-1$. The approximate solution is $y=5/(4\mu)$, and the resulting maximal density is approximately $1.22$ for all values of the parameter $\mu$. The skyrmion density is independent on $\psi$.
{width="5cm" height="5cm"} {width="8cm" height="8cm"}
[[**Figure 3a.**]{} [*Skyrmion density on the upper half plane.*]{}]{}
In the disc model of the hyperbolic space, the boundary of ${\mathcal{B}}$ is a flat torus. Let the map $\DD\rightarrow \HH^2$ be given by $$x+iy=\frac{z-i}{iz-1},$$ where $|z|<1$. The radius of fibers of $S^1\rightarrow
{\mathcal B}\rightarrow \DD$ is discontinuous at the point $z=-i$ corresponding to $y=\infty$ on the boundary (Figure 3b).
{width="5cm" height="5cm"}
[[**Figure 3b.**]{} [*Circle fibration over the Poincare disc, with shrinking fibers.*]{}]{}
The Ricci scalar of $h_{\mathcal B}$ is also discontinuous on the boundary, and equals $-2$ if $|z|=1$ and $z\neq -i$, and $-6$ at $z=-i$ (Figure 4). At the centre of the disc the Ricci scalar depends on $\mu$ and is given by $-2(3\mu^2+3\mu+1)/(\mu+1)^2$. This does not cause a problem as the point $z=-i$ is not a part of the manifold ${\mathcal B}$.
{width="6cm" height="8cm"} {width="6cm" height="8cm"}
[[**Figure 4.**]{} [*Density plots of the Ricci scalar (Fig. 4a) of $({\mathcal B}, h_{\mathcal B})$ and the radii of the circles (Fig. 4b) in the fibration $S^1\rightarrow \DD$ for $\mu=1$.*]{}]{}
The density of the Taub–NUT skyrmion peaks on the diameter joining $z=-i$ and $z=i$. The skyrmion is located on this diameter at $z=i(4\mu-5)/(4\mu+5)$. (Figure 5). At the centre of the disc the skyrmion varies along the fibres according to $$U=\exp{(i\pi\mu/(\mu+1)(\cos{\psi}\tau_1+\sin{\psi}\tau_2)}).$$ The point $r=0$ corresponds to the point $z=i$ on the boundary of the disc, where $U={\bf 1}$. Note that this point has been removed from the domain of the skyrmion, as it is the fixed point (in four dimensions) of the isometry used to obtain the quotient ${\mathcal B}$. The boundary conditions $r\rightarrow \infty$ translate to $U=-{\bf 1}$ at the point $z=-i$ for all $\psi$.
{width="15cm" height="15cm"}
[[**Figure 5.**]{} [*Density of the Taub–NUT skyrmion in the Poincare disc model with $z=u+iv, \mu=4$*]{}.]{}
Further remarks
===============
We have constructed $SU(2)$–valued Skyrme fields from a holonomy of a non–flat $\mathfrak{su}(2)$ spin connection on $\spp_+$ corresponding to ALF gravitational instantons. The holonomy is calculated along orbits of an isometry generating a one–parameter subgroup $SO(2)$ of the full isometry group. In case of Taub–NUT, the Skyrme field carries a non–zero topological charge. This rules out the interpretation of the charge as the baryon number – which vanishes – but opens up a possibility of assigning other integral charges to particles in the AMS model. A lepton number is an obvious candidate[^6], as no proposal of its topological interpretation has been put forward in [@AMS11].
A computation of the Skyrme field can be carried over for other gravitational instantons. To do it for the Fubini–Study metric on $\CP^2$, one needs to express it in the Bouchiat–Gibbons form [@BG] adapted to the $SO(3)$ (rather than $U(2)\subset SU(3))$ action. This, with $r\in [0, \pi/2]$, leads to a connection (\[connection\]) with $$\begin{aligned}
f_1&=&\frac{2\sin{(r+\pi/2)}\cos{r}+\cos{r}^2}{\sin{(r+\pi/2)}},\\
f_2&=&\frac{2\cos{(r/2+\pi/4)}^2+2\sin{r}\cos{(r/2+\pi/4)}^2-\cos{r}^2}
{2\sin{r}\cos{(r/2+\pi/4)}},\\
f_3&=&-\frac{2\cos{(r/2+\pi/4)}^2+2\sin{r}\sin{(r/2+\pi/4)}^2+\cos{r}^2-2}
{2\sin{r} \sin{(r/2+\pi/2)}}.\end{aligned}$$ The instanton number of the corresponding gauge field is fractional $k_{\CP^2}=9/2$, which reflects the fact that $\CP^2$ does not admit a spin structure, and the gauge field resulting from the spin connection is not globally defined. The resulting Skyrme field behaves similarly to the AH case, as $(f_1, f_2, f_3)$ equals $(3, 0, 0)$ at $r=0$, and $(0, 0, 0)$ at $r=\pi/2$ . While the number and the position of nuts and bolts might depend on the choice of the isometry – $\CP^2$ has three nuts for $\p/\p\phi$ and a nut and a bolt for $\p/\p\psi$ – the total number of nuts and bolts is constrained by a topological equality [@GH] $$\sum \mbox{nut}+\sum\chi(\mbox{bolt})=\chi(M)$$ and the RHS is equal to $3$ for $\CP^2$.
We can also compute the Skyrme field for the Euclidean Schwarzschild metric. This gravitational instanton does not appear in [@AMS11], as its curvature is not self–dual. It is nevertheless possible that it can be used as a model for the neutrino in place of $S^4$ - the case for self–duality of the underlying Riemannian manifolds was not overwhelmingly strong in [@AMS11]. The topology of the Euclidean Schwarzschild is $\R^2\times S^2$, with boundary $S^1\times S^2$. Now there are two Yang–Mills fields constructed out of self–dual and anti–self–dual spin connections, and the Euler and Pontriagin numbers are linear combinations of the two instanton numbers. The Schwarzschild manifold has signature zero, which is compatible with AMS interpretation. The metric is asymptotically flat, and the asymptotic circle fibration is trivial. Thus there is no associated electric charge.
### Acknowledgements {#acknowledgements .unnumbered}
I thank Gary Gibbons, Nick Manton and Prim Plansangkate for useful discussions. I also thank the anonymous referees for their comments which led to several improvements in the manuscript.
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[^1]: Email: [email protected]
[^2]: For comparison, computing the ASD connection on $\spp_-$ would also give (\[connection\]), but this time with $f_1=f_2=f_3=1$. The Maurer–Cartan equations on $SU(2)$ then imply that the curvature of this connection vanishes, and so the metric has self–dual Riemannian curvature.
[^3]: In the special case of the usual hedgehog ansatz where $f_1=f_2=f_3=F(r)$, and $0\leq\psi\leq 2\pi$ this formula reduces to the known expression $$B=-2\int_{r_0}^\infty \sin{(\pi F)}^2\;\frac{d F}{dr}dr.$$
[^4]: This conformal metric also admits a Weyl connection such that the Einstein–Weyl equations hold on ${\mathcal B}$. This is true for any conformal structure on the space of orbits of an isometry in a Riemannian manifold with self–dual Weyl curvature – see e.g. [@Dunajski_book]
[^5]: This would not be the case for the AH metric, where the domain of the resulting skyrmion does not admit any isometries.
[^6]: It may be that some combination of the Skyrme charge, and the Euler number and the signature corresponds to the lepton number. An identification of the lepton number with $(B-1)$ is consistent with the AMS proposal, but possibly too naive.
|
---
abstract: 'We characterize a general solution to the vacuum Einstein equations which admits isolated horizons. We show it is a non-linear superposition – in precise sense – of the Schwarzschild metric with a certain free data set propagating tangentially to the horizon. This proves Ashtekar’s conjecture about the structure of spacetime near the isolated horizon. The same superposition method applied to the Kerr metric gives another class of vacuum solutions admitting isolated horizons. More generally, a vacuum spacetime admitting any null, non expanding, shear free surface is characterized. The results are applied to show that, generically, the non-rotating isolated horizon does not admit a Killing vector field and a spacetime is not spherically symmetric near a symmetric horizon.'
---
Space-Times Admitting Isolated Horizons
Jerzy Lewandowski
*Instytut Fizyki Teoretycznej, Uniwersytet Warszawski,*
*ul. Hoża 69, 00-681 Warszawa, Poland, [email protected]*
The quantum geometry considerations applied to black hole entropy [@qentropy] led Ashtekar et al to a new approach to black hole mechanics. The idea is to consider a null surface which locally has the properties of the Schwarzschild horizon, but is not necessarily infinitely extendible, so the spacetime metric in a neighborhood is not necessarily that of Schwarzschild. Such a surface was called a non rotating isolated horizon (NRIH). The number of degrees of freedom describing a spacetime admitting a NRIH is much larger than that describing a static black hole (see below). In a series of works the laws of the black hole thermodynamics and mechanics were extended to this case [@phase; @mech].
In this letter we completely characterize a general solution to the Einstein vacuum equations which admits an isolated horizon and, in particular, a NRIH. For that purpose, we use Friedrich’s characteristic Cauchy problem defined on null surfaces [@Friedrich] (The idea of constructing solutions to the Einstein’s equations starting with data defined on a null surface was first formulated by Newman [@Newman]). The null Cauchy problem formulation gives rise to our superposition method: Given a local solution to the Einstein vacuum equations and the data it defines on a null surface, a new solution can be constructed from the null surface data and certain new data freely defined on a transversal null surface. We show that a general solution which admits a NRIH is given by the superposition of the data defined by the Schwarzschild metric on the horizon and the data defined freely on a transversal null surface. This result is then applied to prove that a generic NRIH does not admit a Killing vector field. Even though there are vector fields defined on the horizon which Lie annihilate the metric tensor [@Racz], none of them, generically, can be extended to a neighborhood. The statement concerns the null vector fields as well as the space like vectors generating symmetries of the internal geometry induced on the 2 dimensional cross sections of the NRIH.
We also characterize a general solution to the Einstein vacuum equations which admits a null non expanding surface. An interesting subclass of space-times is obtained by superposing, in our sense, the data defined by the Kerr metric on its horizon with the data freely defined on a transversal null surface. By analogy to the non rotating case, the resulting null surface equipped with the data corresponding to those of the Kerr metric may be thought of as a rotating isolated horizon.
Another way to extend our results is to admit matter fields in spacetime. In particular the Maxwell field fits the null surfaces formulations of the Cauchy problem very well. We use Newman-Penrose spin connection and curvature coefficients in the notation of [@exact]. All our considerations and results will be [*local*]{} in the following sense: Given two null 3-surfaces $N_0$ and $N_1$ intersecting in the their future, by [*locally*]{} we mean ‘in the past part of a suitable neighborhood of a $N_0\cap N_1$ bounded by the incoming parts of the surfaces’.
[**Isolated horizons: definitions.**]{} Consider a null 3-submanifold $N_0$ of a 4-dimensional spacetime $M$ diffeomorphic to $$\label{times}
S_2 \times [v_0,v_1],$$ where 2-spheres $S_2$ can be identified with space-like cross sections and the intervals $[v_0, v_1]$ lie along the null generators of $N_0$. We say that $N_0$ is an isolated horizon if the intrinsic, degenerate metric tensor induced in $N_0$ is annihilated by the Lie derivative with respect to any vector field $$l = -o^A o^{A'}$$ tangent to the null generators of $N_0$. In other words, $l$ is non expanding and shear free, $$\rho=\sigma=0.$$ An isolated horizon $N_0$ equipped with a foliation by space-like 2-cross sections is called non rotating isolated horizon (NRIH) whenever a transversal, future oriented null vector field $$n\ =\ -\iota^A\iota^{A'},$$ defined on $N_0$ by the gradient of a function $v$ labeling the leaves of the foliation[^1] satisfies the following conditions on $N_0$:
i\) $n$ is shear free, and its expansion is a negative function of $v$, $$\lambda = 0,\ \mu = f(v) < 0;$$ ii) moreover, it is assumed that the Newman-Penrose spin-coefficient $\pi$ vanishes $$\pi = 0;$$ iii) the Ricci tensor component $R_{\mu\nu}m^\mu\bar{m}^\nu$ is a function, say $K$, of the function $v$ only; $$R_{\mu\nu}m^\mu\bar{m}^\nu = K(v),$$ where $m$ is a null, complex valued vector field tangent to the slices $v=const$ normalized by $m^\mu {\bar m}_\mu=1$;
$iv$) The vector field $ k^\mu = G^{\mu \nu}l_\nu$, where $G_{\mu\nu}$ is the Einstein tensor, is causal, $k^\mu k_\mu \le 0.$
The vanishing of the shear and of the expansion of $l$ is rescaling invariant. We normalize $l$ such that $$l^\mu n_\mu = -1.$$ A NRIH will be denoted by $(N_0, [(l,n)])$ where the bracket indicates, that the vector fields $(l,n)$ are defined up to the foliation preserving transformations $v \mapsto v'(v)$.
[**Space-times admitting NRIH.**]{}\
Suppose now, that $(N_0, [(l, n)])$, is a NRIH and the Einstein vacuum equations hold in the past of a neighborhood of $N_0$. To characterize (locally) a general solution we need to introduce another null surface, $N_1$ say. Let $N_1$ be a surface generated by finite segments of the incoming null geodesics which intersect $N_0$ at $v=v_1$ (see (\[times\]): we are assuming that the cartesian product corresponds to the foliation and the variable $v$) and are parallel to the vector field $n$ at the intersection points. Thus the intersection, $$N_0 \cap N_1 =: S$$ is the cross section $v=v_1$ of $N_0$. Locally (see above for the definition of ‘locally’), the metric tensor is uniquely characterized (up to diffeomorphisms) by Friedrich’s reduced data: $$\begin{aligned}
{\rm on}\ \ S&:& m, {\rm Re}\rho,\
{\rm Re}\mu, \ \sigma,\ \lambda, \pi,\label{reducedS}\\
{\rm on}\ \ N_0&:&\ \ \Psi_0,\label{reduced0} \\
{\rm on}\ \ N_1&:&\ \ \Psi_4,\label{reduced1}\end{aligned}$$ where $m$ is a complex valued vector field tangent to $S$. The resulting solution is given by a null frame which satisfies the following gauge conditions, $$\label{gaugeM}
\nu = \gamma =\tau=\pi -\alpha-{\bar \beta}= \mu -{\bar \mu} =0,$$ locally in the spacetime, and $$\label{gaugeN}
\epsilon = 0 \ {\rm on}\ \ \ N_0.$$ Conversely, given submanifolds $N_0\cup N_1$ of a time oriented 4-manifold $M$, the triple $(M,N_0,N_1)$ being diffeomorphic (by the time orientation preserving diffeomorphism) to the one above, every freely chosen data (\[reducedS\], \[reduced0\], \[reduced1\]) corresponds to a unique solution to the vacuum Einstein equations. Let $(N_0, [(l,n)])$ be a NRIH . To calculate Friedrich’s data, we need to satisfy the gauge conditions (\[gaugeN\]), (\[gaugeM\]). Since $N_0$ is non-diverging and shear-free, we can choose on $N_0$ a normalized complex vector field $m$ tangent to the foliation, such that $m$ is Lie constant along the null generators of $N_0$. This implies the vanishing of $\epsilon - \bar{\epsilon}$. From the generalized ‘0th law’ [@mech] we know that, if we parameterize the foliation of $N_0$ by a function $v'$ such that $$\mu' = {\rm const} \ \ {\rm on}\ \ N_0,$$ then owing to the vacuum Einstein’s equations $$\epsilon' + \bar{\epsilon'}= {\rm const}\ \ {\rm on} \ \ N_0.$$ The geometric meaning of this law is that another function v := [exp]{}(2v’) defines an affine parameter along the null generators of $N_0$. Therefore, if we use the pair $(l,n)$ corresponding to the function $v$, then $$l^\mu l_{\nu;\mu}\ =\ 0,\ {\rm hence}\ \ \epsilon = 0, \ \ {\rm on}\ \ N_0.$$ (Incidently, in this normalization, $l^\mu n_{\nu;\mu}\ =\ 0 $ due to $\pi = 0$). The vanishing of $\pi -\alpha -\bar{\beta},\, \mu
-\bar{\mu}$ is automatically ensured on $N_0$ by the pullback of $n$ on $N_0$ being $dv$. Finally, the gauge conditions (\[gaugeM\]) can be satisfied locally in $M$ by appropriate rotations of a null frame along the incoming geodesics, not affecting the data already fixed on $N_0$.
Now, we can consider the reduced data of the horizon. It follows directly from the definition, that $$\label{necessary}
{\rm on}\ S:\ \sigma =\lambda = {\rm Re}(\rho)= \pi = 0, \
\mu = {\rm const}<0\
\ {\rm and \ \ on}\ N_0:\ \Psi_0\ =\ 0.\$$ We also know [@mech] that the property (iii) in the definition of NRIH implies that the 2-metric tensor induced on $S$ is spherically symmetric. The above conditions are necessary for the reduced data to define a NRIH.
Conversely, suppose that reduced data (\[reducedS\]), (\[reduced0\]) satisfy the conditions (\[necessary\]) and that they define a spherically symmetric 2-metric on the slice $S$. Then, it follows from the Einstein vacuum equations that, locally, $N_0$ is a NRIH. To summarize, locally, $N_0$ is a NRIH if and only if the vacuum space-time is given by the reduced data (\[reducedS\]), (\[reduced0\]), (\[reduced1\]) such that (\[necessary\]) holds and the vector field $m$ defines on the slice $S$ a homogeneous 2-metric tensor. The degrees of freedom are: $i)$ the radius $r_0$ of the 2-metric of $S$, and $ii)$ a complex valued function $\Psi_4$ freely defined on $N_1$. The constant $\mu_{|S}$ can be rescaled to be any fixed $\mu_0<0$.
[**Non existence of Killing vector fields for NRIH.**]{} Let us apply now our very result to the issue of the existence of Killing vectors. The usual way one addresses that problem is writing the Killing equation and trying to solve it. Another way is to look for invariant objects and see if those have a common symmetry[^2]. (Perhaps the first way is a little better to prove the existence whereas the second way may be more useful to disprove it.) We will apply the second one. As it was indicated in [@phase], a null surface admits [*at most one*]{} structure of NRIH. Moreover, let as fix a number $\mu_0<0$ and use the rescaling freedom to fix the null vector fields $(l,n)$ representing the NRIH structure $[(l,n)]$, such that $$\mu = \mu_0, \ \ {\rm on}\ \ N_0.$$ There is exactly one pair $(l,n)$ on $N_0$ which satisfies the NRIH properties and the normalization of $\mu$. Every isometry of spacetime preserving $N_0$, preserves the value of $\mu$. Therefore, it necessarily preserves the vector fields $l$ and $n$. Hence, the potential local isometry preserves also the function $$|\Psi_4|^2\ =\ |C_{\nu\mu\alpha\beta}n^\nu m^\mu n^\alpha m^\beta|^2,$$ where $C$ is the Weyl tensor. Let us use the above isometry invariant to see whether $N_0$ admits a tangential null Killing vector field. On $N_0$, the (would be) Killing vector is of the form $$\ksi \ =\ b_0 l$$ where $b_0$ is a function. The following should be true $$0\ =\ b_0 l^\mu(|\Psi_4|^2)_{;\mu}\ =\ -8b_0 \epsilon |\Psi_4|^2 \ =\
-4 b_0 ({\rm surface\ \ gravity}) |\Psi_4|^2,$$ the second equality being the consequence of the Einstein equations and the Bianchi identities. Since the surface gravity is not zero, this contradicts the existence of a null Killing vector field on the horizon unless $$\Psi_4\ =\ 0.$$ The general formula for a possible Killing vector field tangential to $N_0$ is $$\ksi\ =\ b_0 l + K$$ where $K$ is tangent to the leaves of the foliation and together with $b_0$ is subject to the following restrictions. Since the isometry generated by $\ksi$ has to preserve the foliation of $N_0$, and the flow generated by $l$ already does, the function $b_0$ is constant on each leaf of the foliation. Since the symmetry has to preserve the vector field $l$, $b_0$ is constant on $N_0$ and $K$ commutes with $l$. Finally, because the symmetry should preserve the internal degenerate metric tensor on $N_0$, $K$ on each leaf is a Killing vector field. On the other hand, the equation $$\ksi(|\Psi_4|^2)\ = \ 0,$$ implies $$\label{sym}
b_0\ =\ {1\over 8\epsilon} K({\rm ln}|\Psi_4|^2).$$ For a generic $\Psi_4$ defined on the cross section $S$ of $N_0$, the right hand side of (\[sym\]) is not constant on $S$ for any Killing vector field of $S$.[^3] So, generically, there is no Killing vector field in a past neighborhood of a NRIH which is tangent to $N_0$. (Sufficient conditions for the existence of a Killing symmetry of an isolated horizon will be derived in a forthcoming paper.)
[**General isolated horizons.**]{}\
A solution admitting the general isolated horizon can also be characterized using the reduced data. One can easily check that, whenever $$\label{geniso}
\sigma\ =\ \rho\ =0, \ \ {\rm on}\ \ S, \ \ {\rm and}\ \ \Psi_0\ =\ 0,
{\rm on}\ \ N_0,$$ in the reduced data set (\[reducedS\]), (\[reduced0\], (\[reduced1\]), then the corresponding solution satisfies $\sigma =
\rho = \Psi_0 = 0$ on $N_0$, hence $N_0$ is an isolated horizon. Of course the above data is also necessarily an isolated horizon data.
Therefore:
[*$N_0$ is an isolated horizon in Einstein’s vacuum space-time, if and only if it is locally given by the reduced data (\[reducedS\], \[reduced0\], \[reduced1\]) and the conditions (\[geniso\]), the remaining data ${\rm Re}\mu,\, \lambda,\, \pi$ on $S$ and $\Psi_4$ on $N_1$ being arbitrary.*]{}
[**The superposition method.**]{} There is one feature of the characteristic Cauchy problem of [@Friedrich] we would like to emphasize more strongly here because of its relevance to the generalization of BH mechanics. Given a reduced data (\[reducedS\]), (\[reduced0\]), (\[reduced1\]) one can evolve it, in particular, along the surface $N_0$. The data determines at each point of $N_0$ a vacuum solution: a null 4-frame, the spin connection and the Weyl tensor. Remarkably, the evolution of every field along the null generators of $N_0$ is independent of $\Psi_4$ except the evolution of $\Psi_4$ itself. We tend to think of this construction as a non-linear superposition of a vacuum solution given near $N_0$ with the contribution coming from data $\Delta \Psi_4$ given on $N_1$ and evolved tangentially to $N_0$. If we know a spacetime whose Newman-Penrose coefficients on $N_0$ we particularly like, but $\Psi_4$ is not relevant for us, by varying $\Psi_4$ on a transversal null surface $N_1$ we obtain a large family of solutions each of which has the desired properties on $N_0$. For example, let us take the Schwarzschild metric as the preferred solution, $N_0$ being a part of its horizon. The family of solutions obtained by the superposition with $\Psi_4$ coming in tangentially to $N_0$, is [*exactly the set of general vacuum solutions admitting a NRIH which we have derived in this paper*]{}. For every member of this family, on $N_0$, the 4-metric tensor, and all, except $\Psi_4$, Newman-Penrose coefficients are the same, as those of Schwarzschild. Ashtekar and collaborators wrote the laws of BH mechanics of Schwarzshild horizon purely in terms of the spin and curvature coefficients on $N_0$, not involving $\Psi_4$. That is why the laws hold automatically for a general NRIH [@mech].
[**Kerr like isolated horizons.**]{} The superposition can be well applied to the Kerr metric. Consider reduced data given by the following recipe:
a\) take a reduced data for Kerr, such that $N_0$ is an isolated horizon, and $N_1$ is an arbitrary transversal null surface;
b\) Keep $\Psi_0 $ on $N_0$, and ${\rm Re}\rho,{\rm
Re}\mu,\sigma,\lambda, \pi$ on the intersection $S$, but use an arbitrary function for $\Psi_4$ on $N_1$.
The resulting solution will be described by the same 4-metric tensor and the Newman-Penrose coefficients on $N_0$, except $\Psi_4$, as the original Kerr metric.
Following this example and the definition of NRIH, we propose to define a [*Kerr like isolated horizon*]{} to be a null surface $N_0$ equipped with an induced (degenerate) intrinsic metric tensor and the Newman-Penrose spin connection coefficients of the Kerr solution. This will determine the Weyl tensor spin coefficients except $\Psi_4$. Therefore, if we formulate the laws of rotating BH exclusively in terms of this data on the horizon, the same laws will hold for every metric tensor admitting the Kerr like isolated horizon. Since an analogous Schwarzschild like horizon would be exactly a NRIH, the above definition is a natural step toward defining a rotating case.
[**NRIH in the Einstein-Maxwell case.**]{} In a non-vacuum case, the conditions imposed on a NRIH imply restrictions on the stress energy tensor of the matter. They are [@phase] $$\Phi_{00}\ =\ \Phi_{01}\ =\ \Phi_{02}\ =\ 0\ =\ \delta(\Phi_{11} +
{1\over 8}R)$$ the last equation being the condition $iii)$ in the definition of NRIH. Those conditions are met by an electro magnetic field such that $$\Phi_{0}=0,$$ and $|\Phi_1|^2$ is constant on the leaves of the foliation of $N_0$.
If we assume the Einstein-Maxwell equations to hold on $N_0\cup N_1$, by looking at the Newman-Penrose version of the Maxwell equations, it is easy to complete the vacuum free data with suitable data for the electro-magnetic field. Indeed, for $\Phi_0$ given on $N_0$, $\Phi_1$ defined on $S$ and $\Phi_2$ defined on $N_1$, the Einstein-Maxwell equations determine the metric tensor, connection, curvature and electro magnetic field on the null surfaces $N_0$ and $N_1$, as well as their rates of change in the transversal directions. Then, $N_0$ is a NRIH if and only if the Einstein-Maxwell data is given by the reduced data (\[reducedS\]-\[reduced1\]) of the vacuum NRIH case, and $$\Phi_0\ = \ 0, \ {\rm on}\ \ N_0,\ |\Phi_1| \ = \ {\rm const}, \ \
{\rm on} S,$$ $\Phi_2$ being arbitrary on $ N_1$.
The electro magnetic field affects only the evolution of the gravitational data in the direction transversal to $N_0\cup N_1$. In this case the existence/uniqueness statements can be found in Friedrich’s contribution in [@Friedrich3].
[**Acknowledgments.**]{} I am most grateful to Abhay Ashtekar for the introduction to his idea of the isolated horizons and for numerous discussions, and to Helmut Friedrich for drawing my attention to his characteristic Cauchy problem which was crucial for this paper. I have also benefited from conversations with Alan Rendall, Ted Newman, Istvan Racz, Pawel Nurowski, Bob Wald, Christopher Beetle and Stephen Fairhurst. Finally, I would like to thank MPI for Gravitational Physics in Potsdam-Golm and the organizers of the workshop Strong Gravitational Fields held in Santa Barbara for their hospitality. This research was supported in part by Albert Einstein MPI, University of Santa Barbara, and the Polish Committee for Scientific Research under grant no. 2 P03B 060 17.
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Kramer D, Stephani H, MacCallum M, Herlt E 1980 [*Exact solutions of Einstein’s field equations*]{}, (VEB Deutscher Verlag der Wissenschaften, Berlin) and Erratum!
Drerulle N and Piran T (eds.) 1983 [*Gravitational Radiation*]{} North-Holland Amsterdam
[^1]: That is, for every vector $X$ tangent to $N_0$, we have $X^a
n_{a }= X^a v_{,a}$.
[^2]: Scalar invariants can be defined on $M$ or even on the bundle of null directions, see Nurowski [*et al*]{} [@Nurek]
[^3]: If it were constant, on the other hand, then necessarily $b_0=0$ provided the orbits of $K$ in the 2-sphere are closed.
|
---
abstract: 'Differential cross sections for the production of $Z$ bosons or off-shell photons $\gamma^*$ in association with jets are measured in proton-antiproton collisions at center-of-mass energy $\sqrt{s}=1.96$ TeV using the full data set collected with the Collider Detector at Fermilab in Tevatron Run II, and corresponding to 9.6 fb$^{-1}$ of integrated luminosity. Results include first measurements at CDF of differential cross sections in events with a boson and three or more jets, the inclusive cross section for production of $Z/\gamma^*$ and four or more jets, and cross sections as functions of various angular observables in lower jet-multiplicity final states. Measured cross sections are compared to several theoretical predictions.'
bibliography:
- 'prdzjets.bib'
title: 'Measurement of differential production cross section for bosons in association with jets in [$p\bar{p}$]{} collisions at $\sqrt{s}=1.96$ TeV\'
---
\[sec:Intro\]Introduction
=========================
Studies of the production of jets in association with a boson, henceforth referred to as processes, are central topics in hadron collider physics. Differential cross section measurements provide stringent tests for perturbative quantum chromodynamics (QCD) predictions [@Gross:1973ju]. In addition, production is a background to many rare standard model (SM) processes, such as Higgs-boson production, and searches for non-SM physics. Dedicated measurements can help to improve the theoretical modeling of production.
Differential cross sections have been previously measured in proton-antiproton collisions by the CDF [@Aaltonen:2007ae] and D0 [@Abazov:2008ez; @*Abazov:2009av; @*Abazov:2009pp] collaborations as functions of several variables, including the jet transverse momentum, the jet rapidity, and various angular observables. These measurements are in qualitative agreement with predictions from perturbative QCD at the next-to-leading order (NLO) expansion in the strong-interaction coupling, but are limited by the small number of events with high multiplicity of jets. Recently, measurements have also been published by the ATLAS [@Aad:2013ysa; @*Aad:2011qv] and CMS [@Chatrchyan:2011ne; @*Chatrchyan:2013tna; @*Khachatryan:2014zya] collaborations in proton-proton collisions at the LHC, since the understanding of these SM processes is essential in the search for non-SM physics at the LHC.
In this article, measurements of differential cross sections for production are presented, using the full data sample of proton-antiproton collisions collected with the CDF II detector in Run II of the Tevatron Collider, which corresponds to 9.6 fb$^{-1}$ of integrated luminosity. The results include differential cross sections as functions of jet transverse momentum, [$p_{\textrm{T}}$]{}, and rapidity, $y$ [^1], extended for the first time at CDF to the final state; the total cross section as a function of jet multiplicity up to four jets; and several differential distributions for events with a boson and at least one or two jets. Measurements are compared to NLO [@Campbell:2002tg; @Berger:2008sj] and approximate next-to-next-to-leading order (NNLO) perturbative QCD predictions [@Rubin:2010xp], to NLO QCD predictions including NLO electroweak corrections [@Denner:2011vu], and to distributions from various Monte Carlo (MC) generators that use parton showers interfaced with fixed-order calculations [@Mangano:2002ea; @Alioli:2010qp].
This paper is organized as follows: Section \[sec:CDF\] contains a brief description of the CDF II detector. The data sample and the event selection are presented in Sec. \[sec:Data\]. The MC samples used across the analysis are listed in Sec. \[sec:MCsamples\]. The estimation of the background contributions is described in Sec. \[sec:Backgrounds\]. The unfolding procedure is explained in Sec. \[sec:unf\]. The systematic uncertainties are addressed in Sec. \[sec:sys\]. The theoretical predictions are described in Sec. \[sec:pred\]. The measured differential cross sections are shown and discussed in Sec. \[sec:results\]. Section \[sec:conclusion\] summarizes the results.
\[sec:CDF\] The CDF II Detector
===============================
The CDF II detector, described in detail in Ref. [@Abulencia:2005ix], is composed of a tracking system embedded in a $1.4$ T magnetic field, surrounded by electromagnetic and hadronic calorimeters and muon spectrometers. The CDF experiment uses a cylindrical coordinate system in which the $z$ axis lies along the proton beam direction, $\phi$ is the azimuthal angle, and $\theta$ is the polar angle, which is often expressed as pseudorapidity $\eta =
-\ln [\tan(\theta/2)]$. The tracking system includes a silicon microstrip detector [@Sill:2000zz] covering a pseudorapidity range of $|\eta|<2$, which provides precise three-dimensional reconstruction of charged-particle trajectories (tracks). The silicon detector is surrounded by a $3.1$ m long open-cell drift chamber [@Affolder:2003ep], which covers a pseudorapidity range $|\eta|<1$, providing efficient pattern recognition and accurate measurement of the momentum of charged particles. The calorimeter system is arranged in a projective-tower geometry and measures energies of photons and electrons in the $|\eta|<3.6$ range. The electromagnetic calorimeter [@Balka:1987ty; @Hahn:1987tx] is a lead-scintillator sampling calorimeter, which also contains proportional chambers at a depth corresponding approximately to the maximum intensity of electron showers. The hadronic calorimeter [@Bertolucci:1987zn] is an iron-scintillator sampling calorimeter. The muon detectors [@Ascoli:1987av], located outside the calorimeters, consist of drift chambers and scintillation counters covering a pseudorapidity range of $|\eta|<1.0$. Finally, the luminosity is computed from the rate of inelastic [$p\bar{p}$]{} collisions determined by the Cherenkov counters [@Elias:1999qg] located close to the beam pipe.
\[sec:Data\]Data Sample and Event Selection
===========================================
The data sample consists of and + jets candidate events, which have been collected using a three-level online event selection system (trigger) [@Winer:2001gj] between February 2002 and September 2011. In the electron channel, the trigger requires a central () electromagnetic calorimeter cluster with GeV matched to a charged particle with [GeV/$c$]{}. In the analysis, events are selected by requiring two central electrons with GeV and reconstructed invariant mass in the range [GeV/$c^2$]{}. Details on the electron identification requirements are given in Ref. [@Abulencia:2005ix]. In the muon channel, the trigger requires a signal in the muon detectors associated with a charged particle reconstructed in the drift chamber with and [GeV/$c$]{}. In the analysis, events are selected by requiring two reconstructed muons of opposite electric charge with and [GeV/$c$]{}, and reconstructed invariant mass in the range [GeV/$c^2$]{}. Quality requirements are applied to the tracks in order to reject misidentified muons, and all the muon candidates are required to to be associated with an energy deposit in the calorimeter consistent with a minimum ionizing particle. More details on the muon reconstruction and identification can be found in Ref. [@Abulencia:2005ix].
In addition to a [$Z$]{} boson candidate, one or more jets with [GeV/$c$]{} and rapidity are required. Jets are reconstructed using the midpoint algorithm [@Abulencia:2005yg] in a cone of radius [^2]. Calorimeter towers are clustered if the energy deposits correspond to a transverse energy larger than 0.1 GeV [^3] and used as seeds if larger than 1 GeV. Towers associated with reconstructed electrons and muons are excluded. A split-merge procedure is used, which merges a pair of cones if the fraction of the softer cone’s transverse momentum shared with the harder cone is above a given threshold; otherwise the shared calorimeter towers are assigned to the cone to which they are closer. The split-merge threshold is set to 0.75. Jet four-momenta are evaluated by adding the four-momenta of the towers according to the E-scheme, $p^{\mu}_{\textrm{jet}} = \sum{p^{\mu}_{\textrm{towers}}}$, described in Ref. [@Blazey:2000qt]. With such a recombination scheme, jets are in general massive, and in order to study the jet kinematic properties, the variables [$p_{\textrm{T}}$]{} and $y$ are used, which account for the difference between $E$ and $p$ due to the jet mass. Since the jet transverse momentum measured by the calorimeter, [$p_{\textrm{T,cal}}$]{}, is affected by instrumental effects, an average correction [@Bhatti:2005ai] is applied to [$p_{\textrm{T,cal}}$]{}. These effects, mainly due to the noncompensating nature of the calorimeter and the presence of inactive material, are of the order of 30% for [$p_{\textrm{T,cal}}$]{} around 40 [GeV/$c$]{} and reduce to about 11% for high [$p_{\textrm{T,cal}}$]{} jets. A further correction is applied to account for the energy contributions to jets from multiple [$p\bar{p}$]{} interactions, but no modification is made to account for underlying-event contributions or fragmentation effects. The requirement of [GeV/$c$]{} is applied to the corrected jet transverse momentum. Events are selected if the leptons are separated from the selected jets by $\Delta{R}_{\ell-\textrm{jet}} \geqslant
0.7$ [^4].
\[sec:MCsamples\]Monte Carlo Simulation
=======================================
Samples of [$Z/\gamma^* \rightarrow e^+e^- + \textrm{jets}$]{}, [$Z/\gamma^* \rightarrow \mu^+\mu^- + \textrm{jets}$]{}, and [$Z/\gamma^* \rightarrow \tau^+\tau^- + \textrm{jets}$]{} events are generated using [[alpgen]{}]{} v2.14 [@Mangano:2002ea] interfaced to [[pythia]{}]{} 6.4.25 [@Sjostrand:2006za] for the parton shower, with CTEQ5L parton distribution functions (PDF) [@Lai:1999wy] and using the set of *tuning* parameters denoted as Tune Perugia 2011 [@Skands:2010ak]. The MLM matching procedure [@Alwall:2007fs] is applied to avoid double-counting of processes between the matrix-element calculations and the parton-shower algorithm of [[pythia]{}]{}. In addition, samples of , associated production of [$W$]{} and [$Z$]{} bosons ($WW$, $WZ$, $ZZ$), and inclusive production are generated using [[pythia]{}]{} v6.2 with the same PDF set and Tune A [@Affolder:2001xt]. All the samples are passed through a full CDF II detector simulation based on <span style="font-variant:small-caps;">geant</span> [@Brun:1987ma], where the <span style="font-variant:small-caps;">gflash</span> [@Grindhammer:1989zg] package is used for parametrization of the energy deposition in the calorimeters, and corrected to account for differences between data and simulation in the trigger selection and lepton identification efficiencies. The electron [$E_{\textrm{T}}$]{} and the muon [$p_{\textrm{T}}$]{} scale and resolution are corrected to match the dilepton invariant mass distributions $M_{\ell\ell}$ observed in the data in the region [GeV/$c^2$]{}. Simulated samples are also reweighted with respect to the number of multiple [$p\bar{p}$]{} interactions in the same bunch crossing so as to have the same instantaneous luminosity profile of the data. The MC samples are used to determine background contributions and derive the unfolding correction factors described in Sec. \[sec:unf\].
\[sec:Backgrounds\]Background Contributions
===========================================
The selected sample of data events is expected to include events from various background processes. The largest background contributions come from pair production of [$W$]{} and [$Z$]{} bosons, $WW$, $WZ$, $ZZ$, and top-antitop quarks, ; a smaller contribution comes from [$Z/\gamma^* \rightarrow \tau^+\tau^- + \textrm{jets}$]{} events. Inclusive jets and events contribute to the background if one or more jets are misidentified as electrons or muons. Various strategies are used to estimate the background contributions. In the channel, a data-driven method is used to estimate the inclusive jets and background contribution. First, the probability for a jet to pass the electron selection requirements is evaluated using an inclusive jet data sample. This is denoted as *fake* rate and is parametrized as a function of the jet transverse energy. The fake rate is applied to jets from a sample of events with one reconstructed electron: for each event, all the possible electron-jet combinations are considered as candidates, the jet transverse energy is corrected to match on average the corresponding electron energy, and all the electron-jet pairs that fulfill the selection requirements are weighted with the corresponding fake rate associated with the jet, and used to estimate the background rate for each observed distribution.
In the muon channel, the and inclusive jets processes constitute a source of background if a track inside a jet is identified as a muon. To estimate this background contribution, events containing muon pairs are reconstructed following the analysis selection but requiring the charge of the two muons to have the same electric charge.
The other background contributions, originating from , associated production of [$W$]{} and [$Z$]{} bosons ($WW$, $WZ$, $ZZ$), and + jets, are estimated with simulated samples. The sample is normalized according to the approximate NNLO cross section [@oai:arXiv.org:0807.2794], the $WW$, $WZ$ and $ZZ$ samples are normalized according to the NLO cross sections [@oai:arXiv.org:hep-ph/9905386], and the + jets sample is normalized according to the [$Z$]{} inclusive NNLO cross section [@Abulencia:2005ix]. The total background varies from about $2\%$ to $6\%$ depending on jet multiplicity as shown in Table \[tab:backg\], which reports the sample composition per jet-multiplicity bin in the electron and muon channels.
\[tab:backg\]
[lcccc]{} &\
Backgrounds & $\geqslant 1$ jet & $\geqslant 2$ jets & $\geqslant 3$ jets & $\geqslant 4$ jets\
QCD, $W$ + jets & $ 25.9 \pm 3.9$ & $ 4.0 \pm 0.6$ & $0.6 \pm 0.1$ & $\leqslant 0.1$\
$WW$, $ZZ$, $ZW$ & $ 119 \pm 36$ & $ 43 \pm 13$ & $4.2 \pm 1.3$ & $ 0.3 \pm 0.1$\
& $ 45 \pm 13$ & $ 25.4 \pm 7.6$ & $2.9 \pm 0.9$ & $ 0.2 \pm 0.1$\
+ jets & $ 7.2 \pm 2.2$ & $ 0.5 \pm 0.1$ & $<0.1$ & $<0.1$\
Total background & $ 197 \pm 38 $ & $ 73 \pm 15 $ & $ 7.8 \pm 1.5$ & $ 0.7 \pm 0.1$\
Data & $ 12910 $ & $ 1451 $ & $ 137 $ & $ 13 $\
\
[lcccc]{} &\
Backgrounds & $\geqslant 1$ jet & $\geqslant 2$ jets & $\geqslant 3$ jets & $\geqslant 4$ jets\
QCD, $W$ + jets & $ 51 \pm 51$ & $ 18 \pm 18$ & $3 \pm 3$ & $ 1 \pm 1$\
$WW$, $ZZ$, $ZW$ & $ 190 \pm 57$ & $ 69 \pm 21$ & $6.7 \pm 2.0$ & $ 0.5 \pm 0.2$\
& $ 68 \pm 21$ & $ 38 \pm 12$ & $4.5 \pm 1.3$ & $ 0.5 \pm 0.1$\
+ jets & $ 9.4 \pm 2.8$ & $ 1.2 \pm 0.3$ & $\leqslant 0.1$& $< 0.1$\
Total background & $ 318 \pm 79 $ & $ 126 \pm 30 $ & $ 14.3 \pm 3.8$ & $ 2.0 \pm 1.0$\
Data & $ 19578$ & $ 2247 $ & $ 196$ & $ 13$\
\
Figure \[fig:invmass\] shows the invariant mass distribution for
events in the electron and muon decay channels. The region outside the mass range used in the analysis contains a larger fraction of background processes. Table \[tab:backg\_sidebands\] shows the comparison between data and
\[tab:backg\_sidebands\]
[lcccc]{} & &\
Backgrounds & $40 \leqslant M_{ee} < 66\quad$ & $\quad116 < M_{ee} \leqslant 145\quad$ & $\quad40 \leqslant M_{\mu\mu} < 66\quad$ & $\quad116 < M_{\mu\mu} \leqslant 145$\
QCD, $W$ + jets & $ 15.9 \pm 2.4$ & $ 2.9 \pm 0.4$ & $ 37 \pm 37$ & $ 8 \pm 8$\
$WW$, $ZZ$, $ZW$ & $ 5.2 \pm 1.6$ & $ 3.2 \pm 1.0$ & $ 7.5 \pm 2.3$ & $ 4.6 \pm 1.4$\
& $ 19.7 \pm 5.9$ & $ 15.6 \pm 4.7$ & $ 30.1 \pm 9.0$ & $ 22.4 \pm 6.7$\
+ jets & $ 10.9 \pm 3.3$ & $ 0.3 \pm 0.1$ & $ 17.5 \pm 5.2$ & $ 0.3 \pm 0.1$\
Total background & $ 51.7 \pm 7.3$ & $ 21.9 \pm 4.8$ & $ 92 \pm 39$ & $ 35 \pm 11$\
+ jets ([[alpgen]{}]{}) & $238.6 \pm 6.5$ & $196.7 \pm 5.6$ & $ 335.4 \pm 7.2$ & $ 289.0 \pm 6.4$\
Total prediction & $290.3 \pm 9.8$ & $218.6 \pm 7.3$ & $ 428 \pm 39$ & $ 324 \pm 12$\
Data & $312$ & $226$ & $ 486$ & $ 334$\
signal plus background prediction for events in the low- and high-mass regions $40 \leqslant M_{\ell\ell} <
66$ [GeV/$c^2$]{} and $116 < M_{\ell\ell} \leqslant 145$ [GeV/$c^2$]{}, respectively. The good agreement between data and expectation supports the method used to estimate the sample composition.
\[sec:unf\]Unfolding
====================
Measured cross sections need to be corrected for detector effects in order to be compared to the theoretical predictions. The comparison between data and predictions is performed at the particle level, which refers to experimental signatures reconstructed from quasi-stable (lifetime greater than 10 ps) and color-confined final-state particles including hadronization and underlying-event contributions, but not the contribution of multiple [$p\bar{p}$]{} interactions in the same bunch crossing [@Buttar:2008jx]. Detector-level cross sections are calculated by subtracting the estimated background from the observed events and dividing by the integrated luminosity. Measured cross sections are unfolded from detector level to particle level with a bin-by-bin procedure. For each bin of a measured observable $\alpha$, the [[alpgen+pythia]{}]{} + jets and + jets MC samples are used to evaluate the unfolding factors, which are defined as $U_{\alpha}=\frac{d\sigma^{\textrm{MC}}_{\textrm{p}}}{d\alpha}/\frac{d\sigma^{\textrm{MC}}_{\textrm{d}}}{d\alpha}$, where $\sigma^{\textrm{MC}}_{\textrm{p}}$ and $\sigma^{\textrm{MC}}_{\textrm{d}}$ are the simulated particle-level and detector-level cross sections, respectively. Measured particle level cross sections are evaluated as $\frac{d\sigma_{\textrm{p}}}{d\alpha} = \frac{d\sigma_{\textrm{d}}}{d\alpha}
\cdot U_{\alpha}$, where $\sigma_{\textrm{d}}$ is the detector-level measured cross section. The simulated samples used for the unfolding are validated by comparing measured and predicted cross sections at detector level. The unfolding factors account for reconstruction efficiency, particle detection, and jet reconstruction in the calorimeter. Unfolding factors are typically around 2.5 (1.7) in value and vary between 2.3 (1.6) at low [$p_{\textrm{T}}$]{} and 3 (2) at high [$p_{\textrm{T}}$]{} for the () channel.
At particle level, radiated photons are recombined with leptons following a scheme similar to that used in Ref. [@Denner:2011vu]. A photon and a lepton from decays are recombined when $\Delta{R}_{\gamma-\ell} \leqslant 0.1$. If both charged leptons in the final state are close to a photon, the photon is recombined with the lepton with the smallest $\Delta{R}_{\gamma-\ell}$. Photons that are not recombined to leptons are included in the list of particles for the jet clustering. With such a definition, photons are clustered into jets at the particle level, and + $\gamma$ production is included in the definition of . The contribution of the + $\gamma$ process to the cross section is at the percent level, and taken into account in the [[pythia]{}]{} simulation through photon initial- (ISR) and final-state radiation (FSR).
Reconstruction of experimental signatures and kinematic requirements applied at particle level establish the measurement definition. Requirements applied at the detector level are also applied to jets and leptons at the particle level so as to reduce the uncertainty of the extrapolation of the measured cross section. Jets are reconstructed at particle level in the simulated sample with the midpoint algorithm in a cone of radius $R=0.7$, the split-merge threshold set to 0.75, and using as seeds particles with ${\ensuremath{p_{\textrm{T}}}}\geqslant 1$ [GeV/$c$]{}. The measured cross sections are defined in the kinematic region [GeV/$c^2$]{}, , [GeV/$c$]{} ($\ell=e,~\mu$), [GeV/$c$]{}, , and .
\[sec:sys\]Systematic Uncertainties
===================================
All significant sources of systematic uncertainties are studied. The main systematic uncertainty of the + jets measurement is due to the jet-energy-scale correction. The jet-energy scale is varied according to Ref. [@Bhatti:2005ai]. Three sources of systematic uncertainty are considered: the absolute jet-energy scale, multiple [$p\bar{p}$]{} interactions, and the $\eta$-dependent calorimeter response. The absolute jet-energy scale uncertainty depends on the response of the calorimeter to individual particles and on the accuracy of the simulated model for the particle multiplicity and [$p_{\textrm{T}}$]{} spectrum inside a jet. This uncertainty significantly affects observables involving high-[$p_{\textrm{T}}$]{} jets and high jet multiplicity. The jet-energy uncertainty related to multiple [$p\bar{p}$]{} interactions arises from inefficiency in the reconstruction of multiple interaction vertices, and mainly affects jets with low [$p_{\textrm{T}}$]{} and high rapidity, and events with high jet multiplicity. The $\eta$-dependent uncertainty accounts for residual discrepancies between data and simulation after the calorimeter response is corrected for the dependence on $\eta$.
Trigger efficiency and lepton identification uncertainties are of the order of $1\%$ and give small contributions to the total uncertainty.
A $30\%$ uncertainty is applied to the MC backgrounds yield estimation, to account for missing higher-order corrections on the cross-section normalizations [@Aaltonen:2007ae]. In the channel, a $15\%$ uncertainty is assigned to the data-driven QCD and background yield estimation, to account for the statistical and systematic uncertainty of the fake-rate parametrization. In the channel a $100\%$ uncertainty is applied to the subtraction of QCD and background, which accounts for any difference between the observed same-charge yield and the expected opposite-charge background contribution. The impact of both sources to the uncertainties of the measured cross sections is less than $2\%$. The primary vertex acceptance is estimated by fitting the beam luminosity as a function of $z$ using minimum bias data, the uncertainty on the primary vertex acceptance is approximately $1\%$. Finally, the luminosity estimation has an uncertainty of $5.8\%$ which is applied to the measurements [@Klimenko:2003if]. As examples, systematic uncertainties as functions of inclusive jet [$p_{\textrm{T}}$]{} in the channel and inclusive jet rapidity in the channel are shown in Fig. \[fig:Total\_Sys\_1J\],
the corresponding systematic uncertainties as functions of inclusive jet [$p_{\textrm{T}}$]{} in the channel and inclusive jet rapidity in the channel have similar trends.
\[sec:pred\]Theoretical Predictions
===================================
Measured differential cross sections are compared to several theoretical predictions such as NLO perturbative QCD calculations evaluated with [[mcfm]{}]{} [@Campbell:2002tg] and [[blackhat+sherpa]{}]{} [@Berger:2008sj], approximate NNLO [[loopsim+mcfm]{}]{} predictions [@Rubin:2010xp], perturbative NLO QCD predictions including NLO electroweak corrections [@Denner:2011vu], and to generators based on LO matrix element (ME) supplemented by parton showers (PS), like [[alpgen+pythia]{}]{} [@Mangano:2002ea; @Sjostrand:2006za], and NLO generators interfaced to PS as [[powheg+pythia]{}]{} [@Alioli:2010qp]. For the [[loopsim+mcfm]{}]{} predictions, the notation $\overline{\textrm{n}}^p$N$^q$LO introduced in Ref. [@Rubin:2010xp] is used, which denotes an approximation to the N$^{p+q}$LO result in which the $q$ lowest loop contributions are evaluated exactly, whereas the $p$ highest loop contributions are evaluated with the [[loopsim]{}]{} approximation; according to such a notation, the approximate NNLO [[loopsim+mcfm]{}]{} predictions are denoted with [$\overline{\textrm{n}}$NLO]{}. The NLO [[mcfm]{}]{} predictions are available for final states from production in association with one or more, and two or more jets, [[loopsim+mcfm]{}]{} only for the final state, NLO [[blackhat+sherpa]{}]{} for jet multiplicity up to , and [[powheg+pythia]{}]{} predictions are available for all jet multiplicities but have NLO accuracy only for . The [[alpgen]{}]{} LO calculation is available for jet multiplicities up to + 6 jets but, for the current comparison, the calculation is restricted to up to . Electroweak corrections at NLO are available for the final state. Table \[tab:theory\_predictions\] lists the theoretical predictions which are compared to measured cross sections.
\[tab:theory\_predictions\]
[lllll]{} Prediction & QCD order & EW order & Parton shower & Jets multiplicity\
[[mcfm]{}]{} & LO/NLO & LO & no & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1$ and $2$ jets\
[[blackhat+sherpa]{}]{} & LO/NLO & LO & no & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1, 2$, and 3 jets\
[[loopsim+mcfm]{}]{} & [$\overline{\textrm{n}}$NLO]{} & LO & no & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1$ jet\
[[nlo qcd $\otimes$ nlo ew]{}]{} & NLO & NLO & no & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1$ jet\
[[alpgen+pythia]{}]{} & LO & LO & yes & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1,2,3$, and $4$ jets\
[[powheg+pythia]{}]{} & NLO & LO & yes & ${\mbox{\ensuremath{Z/\gamma^*}}}{} + \geqslant 1,2,3$, and $4$ jets\
The input parameters of the various predictions are chosen to be homogeneous in order to emphasize the difference between the theoretical models. The MSTW2008 [@Martin:2009iq] PDF sets are used as the default choice in all the predictions. The LO PDF set and one-loop order for the running of the strong-interaction coupling constant $\alpha_s$ are used for the LO [[mcfm]{}]{} and [[blackhat+sherpa]{}]{} predictions; the NLO PDF set and two-loop order for the running of $\alpha_s$ for [[powheg]{}]{}, [[alpgen]{}]{}, NLO [[mcfm]{}]{}, and NLO [[blackhat]{}]{} predictions; the NNLO PDF set and three-loop order for the running of $\alpha_s$ for the [$\overline{\textrm{n}}$NLO]{} [[loopsim]{}]{} prediction. The contribution to the NLO [[mcfm]{}]{} prediction uncertainty due to the PDF is estimated with the MSTW2008NLO PDF set at the 68% confidence level (CL), by using the Hessian method [@Pumplin:2001ct]. There are 20 eigenvectors and a pair of uncertainty PDF associated with each eigenvector. The pair of PDF corresponds to positive and negative 68% CL excursions along the eigenvector. The PDF contribution to the prediction uncertainty is the quadrature sum of prediction uncertainties from each uncertainty PDF. The impact of different PDF sets is studied in [[mcfm]{}]{}, [[alpgen]{}]{} and [[powheg]{}]{}. The variation in the predictions with CTEQ6.6 [@Nadolsky:2008zw], NNPDF2.1 [@Ball:2011mu], CT10 [@Lai:2010vv], and MRST2001 [@Martin:2001es] PDF sets is of the same order of the MSTW2008NLO uncertainty. The LHAPDF 5.8.6 library [@Whalley:2005nh] is used to access PDF sets, except in [[alpgen]{}]{}, where PDF sets are provided within the MC program.
The nominal choice [@Berger:2010vm; @*Berger:2009ep; @Bauer:2009km] for the functional form of the renormalization and factorization scales is $\mu_0=\hat{H}_T/2=\frac{1}{2}\big(\sum_j p_{T}^{j} +
p_{T}^{\ell^+} + p_{T}^{\ell^-}\big)$ [^5], where the index $j$ runs over the partons in the final state. An exception to this default choice is the [[alpgen]{}]{} prediction, which uses $\mu_0=\sqrt{m_{Z}^{2} + \sum_j p_{T}^{j}}$; the difference with respect to $\mu_0=\hat{H}_T/2$ was found to be negligible [@Camarda:2012yha]. The factorization and renormalization scales are varied simultaneously between half and twice the nominal value $\mu_0$, and the corresponding variations in the cross sections are considered as an uncertainty of the prediction. This is the largest uncertainty associated with the theoretical models, except for the [[alpgen+pythia]{}]{} prediction, where the largest uncertainty is associated with the variation of the renormalization scale using the Catani, Krauss, Kuhn, Webber (CKKW) scale-setting procedure [@Catani:2001cc]. In the [[alpgen]{}]{} prediction, the value of the QCD scale, $\Lambda_{QCD}$, and the running order of the strong-interaction coupling constant in the CKKW scale-setting procedure, $\alpha_{s}^{\textrm{CKKW}}$, are set to $\Lambda_{QCD}=0.26$ and one loop, respectively [@Cooper:2011gk]. These settings match the corresponding values of $\Lambda_{QCD}$ and the running order of $\alpha_s$ for ISR and FSR of the [[pythia]{}]{} Tune Perugia 2011. The variation of the CKKW renormalization scale is introduced together with an opposite variation of $\Lambda_{QCD}$ in the [[pythia]{}]{} tune. Simultaneous variations of the renormalization and factorization scales for the matrix element generation in [[alpgen]{}]{} were found to be smaller than the variation of the CKKW scale [@Camarda:2012yha]. The differences with respect to the previously used Tune A and Tune DW [@Albrow:2006rt; @*Field:2006gq] are studied, with the $\alpha_s$-matched setup of Tune Perugia 2011 providing a better modeling of the shape and normalization of the differential cross sections. In the case of Tune A and Tune DW, the running of $\alpha_{s}^{\textrm{CKKW}}$ in [[alpgen]{}]{} and $\Lambda_{QCD}$ in [[pythia]{}]{} is determined by the PDF set, which is CTEQ5L in both to avoid mismatch. The [[powheg]{}]{} calculation is performed with the weighted events option, and the Born suppression factor for the reweight is set to $10$ [GeV/$c$]{}, following Ref. [@Alioli:2010qp]. Further studies on the impact of different choices of the functional form of the renormalization and factorization scales have been performed in Ref. [@Camarda:2012yha].
In the LO and NLO [[mcfm]{}]{} predictions, jets are clustered with the native [[mcfm]{}]{} *cone* algorithm with $R=0.7$. This is a seedless cone algorithm that follows the jet clustering outlined in Ref. [@Blazey:2000qt]. The split-merge threshold is set to 0.75, and the maximum $\Delta{R}$ separation $R_{sep}$ for two partons to be clustered in the same jet [@Ellis:1992qq], is set to [@Aaltonen:2007ae]. For the [[loopsim+mcfm]{}]{} prediction the minimum jet [$p_{\textrm{T}}$]{} for the generation is set to $1$ [GeV/$c$]{}, and the jet clustering is performed with the fastjet [@Cacciari:2011ma] interface to the SISCone [@Salam:2007xv] jet algorithm with cone radius $R=0.7$ and a split-merge threshold of 0.75. The same parameters and setup for the jet clustering are used in the [[blackhat+sherpa]{}]{} calculation, and the predictions are provided by the [[blackhat]{}]{} authors.
A recently developed MC program allows the calculation of both NLO electroweak and NLO QCD corrections to the cross sections [@Denner:2011vu]. In such a prediction, the QCD and electroweak part of the NLO corrections are combined with a factorization ansatz: NLO QCD and electroweak corrections to the LO cross section are evaluated independently and multiplied. Such a combined prediction is referred to as [[nlo qcd $\otimes$ nlo ew]{}]{}. The prediction is evaluated with the configuration described in Ref. [@Denner:2011vu], except for the renormalization and factorization scales, which are set to $\mu_0=\hat{H}_T/2$, and the predictions are provided by the authors.
Fixed-order perturbative QCD predictions need to be corrected for nonperturbative QCD effects in order to compare them with the measured cross sections, including the underlying event associated with multiparton interactions, beam remnants, and hadronization. Another important effect that is not accounted for in the perturbative QCD predictions and needs to be evaluated is the quantum electrodynamics (QED) photon radiation from leptons and quarks. Both ISR and FSR are considered, with the main effect coming from FSR. The inclusion of QED radiation also corrects the cross sections for the contribution of + $\gamma$ production, which enters the definition of the particle level used in this measurement. The nonperturbative QCD effects and the QED radiation are estimated with the MC simulation based on the $\alpha_s$-matched Perugia 2011 configuration of [[alpgen+pythia]{}]{}, where [[pythia]{}]{} handles the simulation of these effects. To evaluate the corrections, parton-level and particle-level [[alpgen+pythia]{}]{} cross sections are defined: parton-level cross sections are calculated with QED radiation, hadronization, and multiparton interactions disabled in the [[pythia]{}]{} simulation, whereas these effects are simulated for the particle-level cross sections. Kinematic requirements on leptons and jets and jet-clustering parameters for the parton and particle levels are the same as those used for the measured cross sections, and photons are recombined to leptons in $\Delta{R}=0.1$ if radiated photons are present in the final state. The corrections are obtained by evaluating the ratio of the particle-level cross sections over the parton-level cross sections, bin-by-bin for the various measured variables. Figure \[fig:CC\_Pt1Y1\_P2H\] shows the parton-to-particle
corrections as functions of inclusive jet [$p_{\textrm{T}}$]{} and inclusive jet rapidity for events, with the contributions from QED ISR and FSR radiation, hadronization, and underlying event. The corrections have a moderate dependence on jet multiplicity, as shown in Fig. \[fig:CC\_NJ\_Incl\_P2H\]. Figure \[fig:CC\_Pt1Y1\_TUNE\] shows
![Parton-to-particle corrections as a function of jet multiplicity. The relative contributions of QED radiation, hadronization, and underlying event are shown. \[fig:CC\_NJ\_Incl\_P2H\]](CC_NJ_Incl_P2H.pdf){width="\figsize"}
the parton-to-particle corrections evaluated with various tunes of the underlying-event and hadronization model in [[pythia]{}]{}, namely Tune A [@Affolder:2001xt], Tune DW [@Albrow:2006rt; @*Field:2006gq], Tune Perugia 2011 [@Skands:2010ak], and Tune Z1 [@Field:2011iq], and with the [[alpgen+pythia]{}]{} or [[powheg+pythia]{}]{} simulations. The corrections are generally below $10\%$, and independent of the [[pythia]{}]{} MC tune and of the underlying matrix-element generator.
The cross sections are measured using the midpoint algorithm for the reconstruction of the jets in the final state. The midpoint algorithm belongs to the class of iterative cone algorithms. Though they present several experimental advantages, iterative cone algorithms are not infrared and collinear safe, which means that the number of hard jets found by such jet algorithms is sensitive to a collinear splitting or to the addition of a soft emission. In particular the midpoint jet algorithm used in this measurement is infrared unsafe, as divergences appear in a fixed-order calculation for configurations with three hard particles close in phase space plus a soft one, as discussed in Refs. [@Salam:2009jx; @Salam:2007xv]. In order to compare the measured cross sections with a fixed-order prediction, an infrared and collinear safe jet algorithm that is as similar as possible to the midpoint algorithm, is used in the prediction. This is the SISCone algorithm with the same split-merge threshold of 0.75 and the same jet radius $R=0.7$ of the midpoint algorithm used for the measured cross sections. The additional uncertainty coming from the use of different jet algorithms between data and theory is estimated by comparing the particle-level cross sections for the two jet algorithms. Figure \[fig:CC\_Pt1\_Incl\_JETALG\] shows the cross section ratios
of midpoint and SISCone jet algorithms for inclusive jet [$p_{\textrm{T}}$]{} and rapidity in the final state. The difference at parton level between SISCone and midpoint is between $2\%$ and $3\%$. Larger differences between midpoint and SISCone are observed if the underlying event is simulated; however, they do not affect the comparison with fixed-order predictions. Figure \[fig:CC\_NJ\_Incl\_JETALG\] shows the same
![Ratio of differential cross sections evaluated with the midpoint and with the SISCone jet algorithms, as a function of jet multiplicity in . \[fig:CC\_NJ\_Incl\_JETALG\]](CC_NJ_Incl_JALGSIS.pdf){width="\figsize"}
comparison as a function of jet multiplicity. The difference at parton level between midpoint and SISCone is always below $3\%$ and generally uniform.
\[sec:results\]Results
======================
The differential cross sections of production in [$p\bar{p}$]{} collisions are measured independently in the and decay channels and combined using the best linear unbiased estimate (BLUE) method [@Lyons:1988rp]. The BLUE algorithm returns a weighted average of the measurements taking into account different types of uncertainty and their correlations. Systematic uncertainties related to trigger efficiencies, lepton reconstruction efficiencies, and QCD and background estimation are considered uncorrelated between the two channels; all other contributions are treated as fully correlated.
Inclusive cross sections are measured for number of jets $N_{\textrm{jets}} \geqslant 1,2,3$, and $4$, various differential cross sections are measured in the , , and final states. Table \[tab:results\] summarizes the measured cross sections.
\[tab:results\]
[ll]{} Final state & Measured quantity (Fig.)\
& Inclusive cross section for $N_{\textrm{jets}} \geqslant 1,2,3$, and $4$ (\[fig:CC\_NJ\_Incl\_APLB\])\
& Leading jet [$p_{\textrm{T}}$]{} (\[fig:CC\_Pt1\_Lead\_APBEL\]), inclusive jet [$p_{\textrm{T}}$]{} (\[fig:CC\_Pt1\_Incl\_APML\],\[fig:CC\_Pt1\_Incl\_MCFM\]), inclusive jet $y$ (\[fig:CC\_Y1\_Incl\_APBL\],\[fig:CC\_Y1\_Incl\_MCFM\]), ${\ensuremath{p_{\textrm{T}}}}^Z$ (\[fig:CC\_ZPt1\_APMEL\]), $\Delta{\phi}_{Z,\textrm{jet}}$ (\[fig:CC\_Zj\_DPhi\_LPA\]), [$H_{\textrm{T}}^{\textrm{jet}}$]{} (\[fig:CC\_Htj1\_MAPL\])\
& $2\textit{nd}$ leading jet [$p_{\textrm{T}}$]{} (\[fig:CC\_Pt2\_Lead\]), inclusive-jet $y$ (\[fig:CC\_Y2\_Incl\]), $M_{\mathit{jj}}$ (\[fig:CC\_DJ\_Mass\_AM\]), dijet $\Delta{R}$ (\[fig:CC\_DJ\_DR\_AM\]), dijet $\Delta{\phi}$ (\[fig:CC\_DJ\_DPhi\_AM\]), dijet $\Delta{y}$ (\[fig:CC\_DJ\_DY\_AM\]), $\theta_{Z,{\mathit{jj}}}$ (\[fig:CC\_Zjj\_Theta\_AM\])\
& $3$rd leading jet [$p_{\textrm{T}}$]{} (\[fig:CC\_Pt3\_Y3\_BH\] a), inclusive-jet $y$ (\[fig:CC\_Pt3\_Y3\_BH\] b)\
Cross section for the production of a boson in association with $N$ or more jets\[sec:Znjet\_results\]
------------------------------------------------------------------------------------------------------
The production cross sections are measured for $N_{\textrm{jets}}$ up to four and compared to LO and NLO perturbative QCD [[blackhat+sherpa]{}]{}, LO-ME+PS [[alpgen+pythia]{}]{}, and NLO+PS [[powheg+pythia]{}]{} predictions. The cross section is compared also to the [$\overline{\textrm{n}}$NLO]{} [[loopsim+mcfm]{}]{} prediction. Figure \[fig:CC\_NJ\_Incl\_APLB\] shows the inclusive cross
{width="\figsizestar"}
section as a function of jet multiplicity for + $\geqslant$ 1, 2, 3 and 4 jets. The measured cross section is in general good agreement with all the predictions. The blue dashed bands show the theoretical uncertainty associated with the variation of the renormalization and factorization scales, except for the [[alpgen+pythia]{}]{} prediction, where the band shows the uncertainty associated with the variation of the CKKW renormalization scale. The [[alpgen+pythia]{}]{} LO-ME+PS prediction provides a good model of the measured cross sections, but has large theoretical uncertainty at higher jet multiplicities. The [[blackhat+sherpa]{}]{} NLO perturbative QCD prediction shows a reduced scale dependence with respect to the [[alpgen+pythia]{}]{} LO-ME+PS prediction. The [[powheg+pythia]{}]{} NLO+PS prediction has NLO accuracy only for , but it can be compared to data in all the measured jet multiplicities, where a general good agreement is observed. The [[loopsim+mcfm]{}]{} [$\overline{\textrm{n}}$NLO]{} prediction is currently available only for , where it shows a very good agreement with the measured cross section and a reduced scale-variation uncertainty at the level of $5\%$.
The [[blackhat+sherpa]{}]{} NLO perturbative QCD calculation appears to be approximately $30\%$ lower than data, with the difference covered by the scale-variation uncertainty. Such a difference is not observed in the comparison with LO-ME+PS [[alpgen+pythia]{}]{} and NLO+PS [[powheg+pythia]{}]{} predictions, in agreement with recent measurements using the anti-$k_t$ jet algorithm [@Aad:2013ysa; @*Aad:2011qv], which do not show any difference with the NLO predictions at high jet multiplicities. The reason of this difference has been found to be related to the different $\Delta
R$ angular reach [@Salam:2009jx] between the SISCone and anti-$k_t$ algorithms, and how it is influenced by additional radiation between two hard particles [@Camarda:2012yha]. The difference between data or LO-ME+PS with respect to the NLO prediction in the final state is explained with the presence of higher-order QCD radiation, which reduces the angular reach of the SISCone algorithm and increases the cross section in this particular configuration.
Cross section for the production of a boson in association with one or more jets\[sec:Z1jet\_results\]
------------------------------------------------------------------------------------------------------
Figures \[fig:CC\_Pt1\_Lead\_APBEL\] and \[fig:CC\_Pt1\_Incl\_APML\] show the leading-jet and inclusive-jet cross sections differential in [$p_{\textrm{T}}$]{} for events.
{width="\figsizestar"}
{width="\figsizestar"}
All the theoretical predictions are in reasonable agreement with the measured cross sections. The NLO electroweak corrections give a $5\%$ negative contribution in the last and leading jet [$p_{\textrm{T}}$]{} bin, due to the large Sudakov logarithms that appear in the virtual part of the calculation [@Denner:2011vu]. The scale-variation uncertainty is quite independent of the jet [$p_{\textrm{T}}$]{} and of the order of $4\%-6\%$ for the [$\overline{\textrm{n}}$NLO]{} [[loopsim]{}]{} prediction. Figure \[fig:CC\_Pt1\_Incl\_MCFM\] shows variations in the
{width="\figsizestar"}
[[mcfm]{}]{} prediction with different values of the strong-interaction coupling constant at the $Z$ boson mass, $\alpha_s(M_Z)$, factorization scale, PDF sets, and choice of the functional form of the factorization and renormalization scales.
Figure \[fig:CC\_Y1\_Incl\_APBL\] shows the inclusive-jet cross
{width="\figsizestar"}
sections differential in rapidity for events. All predictions correctly model this quantity. In the high-rapidity region the measured cross section is higher than predictions; however, the difference is covered by the uncertainty due to the contribution of multiple [$p\bar{p}$]{} interaction. The [$\overline{\textrm{n}}$NLO]{} [[loopsim+mcfm]{}]{} prediction has the lowest scale-variation theoretical uncertainty, which is of the order of $4\%-6\%$, and the PDF uncertainty is between $2\%$ and $4\%$. In the high-rapidity region the [[alpgen]{}]{} prediction is lower than other theoretical models; however, the difference with data is covered by the large CKKW renormalization scale-variation uncertainty of this prediction. Figure \[fig:CC\_Y1\_Incl\_MCFM\] shows variations in the
{width="\figsizestar"}
[[mcfm]{}]{} prediction with different values of $\alpha_s(M_Z)$, factorization scale, PDF sets, and choice of the functional form of the factorization and renormalization scales.
Figure \[fig:CC\_ZPt1\_APMEL\] shows the production cross section
{width="\figsizestar"}
differential in ${\ensuremath{p_{\textrm{T}}}}({\mbox{\ensuremath{Z/\gamma^*}}}{})$ for the final state. The perturbative QCD fixed-order calculations [[mcfm]{}]{} and [[loopsim+mcfm]{}]{} fail in describing the region below the $30$ [GeV/$c$]{} jet [$p_{\textrm{T}}$]{} threshold, where multiple-jet emission and nonperturbative QCD corrections are significant. The low [$p_{\textrm{T}}$]{} region is better described by the [[alpgen+pythia]{}]{} and [[powheg+pythia]{}]{} predictions, which include parton shower radiation, and in which the nonperturbative QCD corrections are applied as part of the [[pythia]{}]{} MC event evolution. In the intermediate [$p_{\textrm{T}}$]{} region, the ratios of the data over the NLO [[mcfm]{}]{}, NLO+PS [[powheg+pythia]{}]{} and [$\overline{\textrm{n}}$NLO]{} [[loopsim+mcfm]{}]{} predictions show a slightly concave shape, which is covered by the scale-variation uncertainty. The NLO electroweak corrections related to the large Sudakov logarithms are negative and of the order of $5\%$ in the last [$p_{\textrm{T}}$]{} bin.
Figure \[fig:CC\_Zj\_DPhi\_LPA\] shows the differential cross section
{width="\figsizestar"}
as a function of the -leading jet $\Delta{\phi}$ variable in events. The [[alpgen+pythia]{}]{} prediction shows good agreement with the measured cross section in the region $\Delta{\phi} \geqslant \pi / 2$. In the region $\Delta{\phi} < \pi / 2$ the [[alpgen+pythia]{}]{} prediction is lower than the data, with the difference covered by the scale-variation uncertainty. The [[powheg+pythia]{}]{} prediction has very good agreement with the data over all of the -jet $\Delta{\phi}$ spectrum, and is affected by smaller scale-variation uncertainty. The difference between the [[alpgen+pythia]{}]{} and [[powheg+pythia]{}]{} predictions is comparable to the experimental systematic uncertainty, which is dominated by the uncertainty from the contribution of multiple [$p\bar{p}$]{} interactions. Hence, the measured cross section cannot be used to distinguish between the two models. The NLO [[mcfm]{}]{} prediction fails to describe the region $\Delta{\phi} < \pi / 2$ because it does not include the + 3 jets configuration, whereas [$\overline{\textrm{n}}$NLO]{} [[loopsim+mcfm]{}]{}, which includes the + 3 jets with only LO accuracy, predicts a rate approximately 2–3 times smaller than the rate observed in data in this region.
Some observables have larger NLO-to-LO K-factors, defined as the ratio of the NLO prediction over the LO prediction, and are expected to have significant corrections at higher order than NLO [@Rubin:2010xp]. The most remarkable example is the [$H_{\textrm{T}}^{\textrm{jet}}$]{}, defined as ${\ensuremath{H_{\textrm{T}}^{\textrm{jet}}}}= \sum{{\ensuremath{p_{\textrm{T}}^{\textrm{jet}}}}}$, in events. Figure \[fig:CC\_Htj1\_MAPL\] shows the measured
{width="\figsizestar"}
cross section as a function of [$H_{\textrm{T}}^{\textrm{jet}}$]{} compared to the available theoretical predictions. The NLO [[mcfm]{}]{} prediction fails to describe the shape of the [$H_{\textrm{T}}^{\textrm{jet}}$]{} distribution, in particular it underestimates the measured cross section in the high [$H_{\textrm{T}}^{\textrm{jet}}$]{} region, where the NLO-to-LO K-factor is greater than approximately two and a larger NLO scale-variation uncertainty is observed. The LO-ME+PS [[alpgen+pythia]{}]{} prediction is in good agreement with data, but suffers for the large LO scale uncertainty. The [[powheg+pythia]{}]{} prediction also is in good agreement with data, but is still affected by the larger NLO scale-variation uncertainty in the high [$p_{\textrm{T}}$]{} tail. The [$\overline{\textrm{n}}$NLO]{} [[loopsim+mcfm]{}]{} prediction provides a good modeling of the data distribution, and shows a significantly reduced scale-variation uncertainty.
Cross section for the production of a boson in association with two or more jets \[sec:Z2jet\_results\]
-------------------------------------------------------------------------------------------------------
Figures \[fig:CC\_Pt2\_Lead\] to \[fig:CC\_Zjj\_Theta\_AM\] show measured differential cross sections in the final state. Figures \[fig:CC\_Pt2\_Lead\] and \[fig:CC\_Y2\_Incl\] show the
{width="\figsizestar"}
{width="\figsizestar"}
measured cross section as a function of the $2\textit{nd}$ leading jet [$p_{\textrm{T}}$]{} and inclusive jet rapidity compared to [[alpgen+pythia]{}]{} and [[blackhat+sherpa]{}]{} predictions. Measured distributions are in good agreement with the theoretical predictions. Figure \[fig:CC\_DJ\_Mass\_AM\] shows the measured cross
{width="\figsizestar"}
section as a function of the dijet mass, $M_{\mathit{jj}}$. The cross section in the first bin is overestimated by the [[mcfm]{}]{} prediction, but correctly described by the [[alpgen+pythia]{}]{} prediction. In the $M_{\mathit{jj}}$ region above approximately 160 [GeV/$c^2$]{}, the measured cross sections are $10\%-20\%$ higher than both predictions. However, the systematic uncertainty, mainly due to the jet-energy scale, is as large as the observed difference. Figure \[fig:CC\_DJ\_DR\_AM\] shows the measured cross section as a
{width="\figsizestar"}
function of the dijet $\Delta{R}$ compared to [[alpgen+pythia]{}]{} and [[mcfm]{}]{} predictions. Some differences between data and theory are observed at high $\Delta{R}$, where the measured cross section is approximately $50\%$ higher than the theoretical predictions. The dijet $\Delta{\phi}$ and $\Delta{y}$ differential cross sections also are measured, and the results are shown in Figs. \[fig:CC\_DJ\_DPhi\_AM\] and \[fig:CC\_DJ\_DY\_AM\]. The dijet $\Delta{\phi}$ appears reasonably
{width="\figsizestar"}
{width="\figsizestar"}
modeled by the [[alpgen+pythia]{}]{} and [[mcfm]{}]{} predictions, whereas the dijet $\Delta{y}$ shows a shape difference, which reaches $50\%$ at $\Delta{y} = 3-3.6$, and is related to the observed difference between data and theory at $\Delta{R} \gtrsim 4$. This region is affected by large experimental uncertainties, mainly due to the pile-up subtraction, and large theoretical uncertainty. Figure \[fig:CC\_Zjj\_Theta\_AM\]
{width="\figsizestar"}
shows the measured cross section as a function of the dihedral angle $\theta_{Z,{\mathit{jj}}}$ between the decay plane and the jet-jet plane [^6]. The measured cross section is in good agreement with the [[alpgen+pythia]{}]{} and [[mcfm]{}]{} predictions.
Cross section for the production of a boson in association with three or more jets \[sec:Z3jet\_results\]
---------------------------------------------------------------------------------------------------------
Figure \[fig:CC\_Pt3\_Y3\_BH\] shows the differential cross sections as
a functions of $3\textit{rd}$ leading jet [$p_{\textrm{T}}$]{} and inclusive jet rapidity in events with a reconstructed decay and at least three jets. The NLO [[blackhat+sherpa]{}]{} prediction is approximately $30\%$ lower than the measured cross sections for + $\geqslant 3$ jets events, but data and predictions are still compatible within the approximately $25\%$ scale-variation uncertainty and the $15\%$ systematic uncertainty, dominated by the jet-energy scale. Apart from the difference in the normalization, the shape of the measured differential cross sections is in good agreement with the NLO [[blackhat+sherpa]{}]{} prediction.
\[sec:conclusion\]Summary and Conclusions
=========================================
The analysis of the full proton-antiproton collisions sample collected with the CDF II detector in Run II of the Tevatron, corresponding to $9.6$ fb$^{-1}$ integrated luminosity, allows for precise measurements of inclusive and differential cross sections, which constitute an important legacy of the Tevatron physics program. The cross sections are measured using the decay channels and in the kinematic region , , , , , and , with jets reconstructed using the midpoint algorithm in a radius $R=0.7$. The measured cross sections are unfolded to the particle level and the decay channels combined. Results are compared with the most recent theoretical predictions, which properly model the measured differential cross sections in , 2, and 3 jets final states. The main experimental uncertainty is related to the jet-energy scale, whereas the largest uncertainty of the theoretical predictions is generally associated with the variation of the renormalization and factorization scales. Among perturbative QCD predictions, [[loopsim+mcfm]{}]{} shows the lowest scale-variation uncertainty and, therefore, gives the most accurate cross-section prediction for the final state. The [[mcfm]{}]{} and [[blackhat+sherpa]{}]{} fixed-order NLO predictions are in reasonable agreement with the data in the , 2, and 3 jets final states. The [[alpgen+pythia]{}]{} prediction provides a good modeling of differential distributions for all jets multiplicities. The [[powheg+pythia]{}]{} prediction, due to the NLO accuracy of the matrix elements and to the inclusion of nonperturbative QCD effects, provides precise modeling of final states both in the low- and high-[$p_{\textrm{T}}$]{} kinematic regions. The effect of NLO electroweak virtual corrections to the + jet production is studied and included in the comparison with the measured cross sections: in the high [$p_{\textrm{T}}$]{} kinematic region, corrections are of the order of $5\%$, which is comparable with the accuracy of predictions at higher order than NLO. The large theoretical uncertainty associated with the variation of the renormalization and factorization scales suggests that the inclusion of higher order QCD corrections, by mean of exact or approximate calculations, will improve the theoretical modeling of processes.
The understanding of associated production of vector bosons and jets is fundamental in searches for non-SM physics, and the results presented in this paper support the modeling of currently employed in Higgs-boson measurements and searches for physics beyond the standard model.
We thank the Fermilab staff and the technical staffs of the participating institutions for their vital contributions. This work was supported by the U.S. Department of Energy and National Science Foundation; the Italian Istituto Nazionale di Fisica Nucleare; the Ministry of Education, Culture, Sports, Science and Technology of Japan; the Natural Sciences and Engineering Research Council of Canada; the National Science Council of the Republic of China; the Swiss National Science Foundation; the A.P. Sloan Foundation; the Bundesministerium für Bildung und Forschung, Germany; the Korean World Class University Program, the National Research Foundation of Korea; the Science and Technology Facilities Council and the Royal Society, UK; the Russian Foundation for Basic Research; the Ministerio de Ciencia e Innovación, and Programa Consolider-Ingenio 2010, Spain; the Slovak R&D Agency; the Academy of Finland; the Australian Research Council (ARC); and the EU community Marie Curie Fellowship contract 302103.
[^1]: The rapidity is defined as $y=\frac{1}{2}\ln(\frac{E+p_Z}{E-p_Z})$; the transverse momentum and energy are defined by ${\ensuremath{p_{\textrm{T}}}}= p
\sin{\theta}$ and ${\ensuremath{E_{\textrm{T}}}}= E \sin{\theta}$
[^2]: The jet cone radius $R$ is defined as $R =
\sqrt{\eta^{2}+\phi^{2}}$
[^3]: The transverse energy is evaluated using the position of the tower with respect to the primary interaction vertex.
[^4]: $\Delta{R}$ is defined as $\Delta{R} =
\sqrt{\Delta{y}^{2}+\Delta{\phi}^{2}}$
[^5]: In [[blackhat]{}]{} and [[powheg]{}]{} predictions, the alternative definition $\mu_0=\hat{H'}_{\textrm{T}}/2=\frac{1}{2}\big(\sum_j p_{\textrm{T}}^{j} + E_{\textrm{T}}^{Z}\big)$ with $E_{\textrm{T}}^{Z} = \sqrt{M_{Z}^{2} + p_{\textrm{T},Z}^{2}}$ is used, where the index $j$ runs over the partons in the final state.
[^6]: $\theta_{Z,{\mathit{jj}}}$ is defined as $\theta_{Z,{\mathit{jj}}} =
\arccos{ \frac{(\vec{\ell_1} \times \vec{\ell_2}) \cdot (\vec{j_1} \times
\vec{j_2})} {|\vec{\ell_1} \times \vec{\ell_2}||\vec{j_1} \times
\vec{j_2}|}}$, where $\vec{\ell}$ and $\vec{j}$ are the momentum three-vectors of leptons and jets.
|
---
abstract: |
The commuting graph of a finite non-commutative semigroup $S$, denoted ${\mathcal{G}}(S)$, is a simple graph whose vertices are the non-central elements of $S$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Let ${\mathcal{I}}(X)$ be the symmetric inverse semigroup of partial injective transformations on a finite set $X$. The semigroup ${\mathcal{I}}(X)$ has the symmetric group $\operatorname{Sym}(X)$ of permutations on $X$ as its group of units. In 1989, Burns and Goldsmith determined the clique number of the commuting graph of $\operatorname{Sym}(X)$. In 2008, Iranmanesh and Jafarzadeh found an upper bound of the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$, and in 2011, Dolzan and Oblak claimed that this upper bound is in fact the exact value.
The goal of this paper is to begin the study of the commuting graph of the symmetric inverse semigroup ${\mathcal{I}}(X)$. We calculate the clique number of ${\mathcal{G}}({\mathcal{I}}(X))$, the diameters of the commuting graphs of the proper ideals of ${\mathcal{I}}(X)$, and the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ when $|X|$ is even or a power of an odd prime. We show that when $|X|$ is odd and divisible by at least two primes, then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is either $4$ or $5$. In the process, we obtain several results about semigroups, such as a description of all commutative subsemigroups of ${\mathcal{I}}(X)$ of maximum order, and analogous results for commutative inverse and commutative nilpotent subsemigroups of ${\mathcal{I}}(X)$. The paper closes with a number of problems for experts in combinatorics and in group or semigroup theory.
*$2010$ Mathematics Subject Classification*. 05C25, 20M20, 20M14, 20M18.
*Keywords*: Commuting graphs of semigroups; symmetric inverse semigroup; commutative semigroups; inverse semigroups; nilpotent semigroups; clique number; diameter.
author:
- |
João Araújo\
[Universidade Aberta, R. Escola Politécnica, 147]{}\
[1269-001 Lisboa, Portugal]{}\
[&]{}\
[Centro de Álgebra, Universidade de Lisboa]{}\
[1649-003 Lisboa, Portugal, [email protected]]{}\
\
Wolfram Bentz\
[Centro de Álgebra, Universidade de Lisboa]{}\
[1649-003 Lisboa, Portugal, [email protected]]{}\
\
- |
Janusz Konieczny\
[Department of Mathematics, University of Mary Washington]{}\
[Fredericksburg, Virginia 22401, USA, [email protected]]{}
title: The Commuting Graph of the Symmetric Inverse Semigroup
---
Introduction
============
The commuting graph of a finite non-abelian group $G$ is a simple graph whose vertices are all non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if $xy=yx$. Commuting graphs of various groups have been studied in terms of their properties (such as connectivity or diameter), for example in [@BaBu03; @Bu06; @IrJa08; @Se01]. They have also been used as a tool to prove group theoretic results, for example in [@Be83; @RaSe01; @RaSe02].
For the particular case of the commuting graph of the finite symmetric group $\operatorname{Sym}(X)$, it has been proved [@IrJa08] that its diameter is $\infty$ when $|X|$ or $|X|-1$ is a prime, and is at most $5$ otherwise. It has been claimed [@DoOb11] that if neither $|X|$ nor $|X|-1$ is a prime, then the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$ is exactly $5$. The claim is correct but the proof contains a gap (see the end of Section 6). The clique number of ${\mathcal{G}}(\operatorname{Sym}(X))$ follows from the classification of the maximum order abelian subgroups of $\operatorname{Sym}(X)$ [@BuGo89; @KoPr89]. In addition, there is a very interesting conjecture (which is still open, as far as we know) that there exists a common upper bound of the diameters of the (connected) commuting graphs of finite groups.
The concept of the commuting graph carries over to semigroups. Suppose $S$ is a finite non-commutative semigroup with center $Z(S)=\{a\in S:ab=ba\mbox{ for all $b\in S$}\}$. The *commuting graph* of $S$, denoted ${\mathcal{G}}(S)$, is the simple graph (that is, an undirected graph with no multiple edges or loops) whose vertices are the elements of $S-Z(S)$ and whose edges are the sets $\{a,b\}$ such that $a$ and $b$ are distinct vertices with $ab=ba$.
In 2011, Kinyon and the first and third author [@ArKiKo11] initiated the study of the commuting graphs of (non-group) semigroups. They calculated the diameters of the ideals of the semigroup $T(X)$ of full transformations on a finite set $X$ [@ArKiKo11 Theorems 2.17 and 2.22], and for every natural number $n$, constructed a semigroup of diameter $n$ [@ArKiKo11 Theorem 4.1]. (The latter result shows that the aforementioned conjecture on the diameters of finite groups does not hold for semigroups.) Finally, the study of the commuting graphs of semigroups led to the solution of a longstanding open problem in semigroup theory [@ArKiKo11 Proposition 5.3].
The goal of this paper is to extend to the finite symmetric inverse semigroups part of the research already carried out for the finite symmetric groups. The *symmetric inverse semigroup* ${\mathcal{I}}(X)$ on a set $X$ is the semigroup whose elements are the partial injective transformations on $X$ (one-to-one functions whose domain and image are included in $X$) and whose multiplication is the composition of functions. We will write functions on the right ($xf$ rather than $f(x))$) and compose from left to right ($x(fg)$ rather than $f(g(x))$. The semigroup ${\mathcal{I}}(X)$ is universal for the class of inverse semigroups since every inverse semigroup can be embedded in some ${\mathcal{I}}(X)$ [@Ho95 Theorem 5.1.7]. This is analogous to the fact that every group can be embedded in some symmetric group $\operatorname{Sym}(X)$ of permutations on $X$. We note that ${\mathcal{I}}(X)$ contains an identity (the transformation that fixes every element of $X$) and a zero (the transformation whose domain and image are empty). The class of inverse semigroups is arguably the second most important class of semigroups, after groups, because inverse semigroups have applications and provide motivation in other areas of study, for example, differential geometry and physics [@La98; @Pa99].
Various subsemigroups of the finite symmetric inverse semigroup ${\mathcal{I}}(X)$ have been studied. One line of research in this area has been the determination of subsemigroups of ${\mathcal{I}}(X)$ of a given type that are either maximal (with respect to inclusion) or largest (with respect to order). (See, for example, [@AnFeMi07; @GaKo94; @Ya99; @Ya05].)
In 1989, Burns and Goldsmith [@BuGo89] obtained a complete classification of the abelian subgroups of maximum order of the symmetric group $\operatorname{Sym}(X)$, where $X$ is a finite set. These abelian subgroups are of three different types depending on the value of $n$ modulo $3$, where $n=|X|$. We extend this result to the commutative subsemigroups of ${\mathcal{I}}(X)$ of maximum order (Theorem \[tgen\]). We also determine the maximum order commutative inverse subsemigroups of ${\mathcal{I}}(X)$ (Theorem \[tinv\]) and the maximum order commutative nilpotent subsemigroups of ${\mathcal{I}}(X)$ (Theorem \[tnil\]). As a corollary of Theorem \[tgen\], we obtain the clique number of the commuting graph of ${\mathcal{I}}(X)$ (Corollary \[ccli\]).
We also find the diameters of the commuting graphs of the proper ideals of ${\mathcal{I}}(X)$ (Theorem \[tpro\]), the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ when $n=|X|$ is even (Theorem \[tdie\]) and when $n$ is a power of an odd prime (Theorem \[tpow\]), and establish that the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $4$ or $5$ when $n$ is odd and divisible by at least two distinct primes (Proposition \[pdio\]). The diameter results extend to ${\mathcal{G}}({\mathcal{I}}(X))$ the results obtained for ${\mathcal{G}}(\operatorname{Sym}(X))$ by Iranmanesh and Jafarzadeh [@IrJa08] and Dolzan and Oblak [@DoOb11]. (However, see our discussion at the end of Section \[scdg\] regarding a problem with Dolzan and Oblak’s proof.) We conclude the paper with some problems that we believe will be of interest for mathematicians working in combinatorics and semigroup or group theory (Section \[spro\]).
The concept of the commuting graph of a transformation semigroup is central for associative algebras since, in a sense, the study of associativity is the study of commuting transformations and centralizers [@ArKo12]. This paper builds upon the results on centralizers of transformations in general and of partial injective transformations in particular [@AnArKo11; @ArKo03; @ArKo04; @Ko99; @Ko02; @Ko03; @Ko04; @Ko10; @KoLi98; @Li96].
Throughout this paper, we fix a finite set $X$ and reserve $n$ to denote the cardinality of $X$. To simplify the language, we will sometimes say “semigroup in ${\mathcal{I}}(X)$” to mean “subsemigroup of ${\mathcal{I}}(X)$.” We will denote the identity in ${\mathcal{I}}(X)$ by $1$ and the zero in ${\mathcal{I}}(X)$ by $0$.
Commuting Elements of ${\mathcal{I}}(X)$
========================================
In this section, we collect some results about commuting transformations in ${\mathcal{I}}(X)$ that will be needed in the subsequent sections.
Let $S$ be a semigroup with zero. An element $a\in S$ is called a *nilpotent* if $a^p=0$ for some positive integer $p$; the smallest such $p$ is called the *index* of $a$. We say that $S$ is a *nilpotent semigroup* if every element of $S$ is a nilpotent. A special type of a nilpotent semigroup is a *null semigroup* in which $ab=0$ for all $a,b\in S$. Note that every nonzero nilpotent in a null semigroup has index $2$. We say that $S$ is a *null monoid* if it contains an identity $1$ and $ab=0$ for all $a,b\in S$ such that $a,b\ne1$. Clearly, all null semigroups and all null monoids are commutative.
For ${\alpha}\in {\mathcal{I}}(X)$, we denote by $\operatorname{dom}({\alpha})$ and $\operatorname{im}({\alpha})$ the domain and image of ${\alpha}$, respectively. The $\emph{rank}$ of ${\alpha}$ is the cardinality of $\operatorname{im}({\alpha})$ (which is the same as the cardinality of $\operatorname{dom}({\alpha})$ since ${\alpha}$ is injective). The union $\operatorname{span}({\alpha})=\operatorname{dom}({\alpha})\cup\operatorname{im}({\alpha})$ will be called the *span* of ${\alpha}$.
Let ${\alpha},{\beta}\in{\mathcal{I}}(X)$. We say that ${\beta}$ is *contained* in ${\alpha}$ (or ${\alpha}$ *contains* ${\beta}$) if $\operatorname{dom}({\beta})\subseteq\operatorname{dom}({\alpha})$ and $x{\beta}=x{\alpha}$ for all $x\in\operatorname{dom}({\beta})$. We say that ${\alpha}$ and ${\beta}$ in ${\mathcal{I}}(X)$ are *completely disjoint* if $\operatorname{span}({\alpha})\cap\operatorname{span}({\beta})=\emptyset$. Let $M=\{{\gamma}_1,\ldots,{\gamma}_k\}$ be a set of pairwise completely disjoint elements of ${\mathcal{I}}(X)$. The *join* of the elements of $M$, denoted ${\gamma}_1{\sqcup}\cdots{\sqcup}{\gamma}_k$, is the element ${\alpha}$ of ${\mathcal{I}}(X)$ whose domain is $\operatorname{dom}({\gamma}_1)\cup\ldots\cup\operatorname{dom}({\gamma}_k)$ and whose values are defined by $x{\alpha}=x{\gamma}_i$, where ${\gamma}_i$ is the (unique) element of $M$ such that $x\in\operatorname{dom}({\gamma}_i)$. If $M=\emptyset$, we define the join to be $0$. Let $x_0,x_1,\ldots,x_k$ be pairwise distinct elements of $X$.
- A *cycle* of length $k$ ($k\geq1$), written $(x_0\,x_1\ldots\, x_{k-1})$, is an element $\rho\in{\mathcal{I}}(X)$ with $\operatorname{dom}(\rho)=\{x_0,x_1,\ldots,x_{k-1}\}$, $x_i\rho=x_{i+1}$ for all $0\leq i<k-1$, and $x_{k-1}\rho=x_0$.
- A *chain* of length $k$ ($k\geq1$), written $[x_0\,x_1\ldots\, x_k]$, is an element $\tau\in{\mathcal{I}}(X)$ with $\operatorname{dom}(\tau)=\{x_0,x_1,\ldots,x_{k-1}\}$ and $x_i\tau=x_{i+1}$ for all $0\leq i\leq k-1$.
The following decomposition result is given in [@Li96 Theorem 3.2].
\[pdec\] Let ${\alpha}\in{\mathcal{I}}(X)$ with ${\alpha}\ne0$. Then there exist unique sets ${\Gamma}=\{\rho_1,\ldots,\rho_k\}$ of cycles and ${\Omega}=\{\tau_1,\ldots,\tau_m\}$ of chains such that the transformations in ${\Gamma}\cup{\Omega}$ are pairwise completely disjoint and ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k{\sqcup}\tau_1{\sqcup}\cdots{\sqcup}\tau_m$.
Let ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k{\sqcup}\tau_1{\sqcup}\cdots{\sqcup}\tau_m$ as in Proposition \[pdec\]. Note that every $\rho_i$ and every $\tau_j$ is contained in ${\alpha}$. Moreover, for every integer $p>0$, ${\alpha}^p=\rho_1^p{\sqcup}\cdots{\sqcup}\rho_k^p{\sqcup}\tau_1^p{\sqcup}\cdots{\sqcup}\tau_m^p$. For example, if $${\alpha}=\begin{pmatrix}1&2&3&4&5&6&7&8\\2&3&4&1&6&7&8&-\end{pmatrix}=(1\,2\,3\,4){\sqcup}[5\,6\,7\,8]\in{\mathcal{I}}(\{1,2,\ldots,8\}),$$ then ${\alpha}^2=(1\,3){\sqcup}(2\,4){\sqcup}[5\,7]{\sqcup}[6\,8]$, ${\alpha}^3=(1\,4\,3\,2){\sqcup}[5\,8]$, and ${\alpha}^4=(1){\sqcup}(2){\sqcup}(3){\sqcup}(4)$.
Let ${\alpha}\in{\mathcal{I}}(X)$. Then:
- ${\alpha}\in\operatorname{Sym}(X)$ if and only if ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k$ is a join of cycles and $\cup_{i=1}^k \operatorname{dom}(\rho_i)=X$. The join ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k$ is equivalent to the cycle decomposition of ${\alpha}$ in group theory. Note that a cycle $(x_0\,x_1\ldots\, x_{t-1})$ differs from the corresponding cycle in $\operatorname{Sym}(X)$ in that the former is undefined for every $x\in X-\{x_0,x_1,\ldots,x_{t-1}\}$, while the latter fixes every such $x$.
- ${\alpha}$ is a nilpotent if and only if ${\alpha}=\tau_1{\sqcup}\cdots{\sqcup}\tau_m$ is a join of chains; and ${\alpha}^2=0$ if and only if ${\alpha}=[x_1\,y_1]{\sqcup}\cdots{\sqcup}[x_m,y_m]$ is a join of chains of length $1$, where we agree that ${\alpha}=0$ if $m=0$.
The following proposition has been proved in [@Li96 Theorem 10.1].
\[pcen\] Let ${\alpha},{\beta}\in{\mathcal{I}}(X)$. Then ${\alpha}{\beta}={\beta}{\alpha}$ if and only if the following conditions are satisfied:
- If $\rho=(x_0\,x_1\ldots\, x_{k-1})$ is a cycle in ${\alpha}$ such that some $x_i\in\operatorname{dom}({\beta})$, then every $x_j\in\operatorname{dom}({\beta})$ and there exists a cycle $\rho'=(y_0\,y_1\ldots\,y_{k-1})$ in ${\alpha}$ (of the same length as $\rho$) such that $$x_0{\beta}=y_j,\,\,x_1{\beta}=y_{j+1},\ldots,x_{k-1}{\beta}=y_{j+k-1},$$ where $j\in\{0,1,\ldots,k-1\}$ and the subscripts on the $y_i$s are calculated modulo $k$;
- If $\tau=[x_0\,x_1\ldots\, x_k]$ is a chain in ${\alpha}$ such that some $x_i\in\operatorname{dom}({\beta})$, then there are $p\in\{0,1,\ldots,k\}$ and a chain $\tau'=[y_0\,y_1\ldots\, y_m]$ in ${\alpha}$, with $m\geq p$, such that $\operatorname{dom}({\beta})\cap\{x_0,x_1,\ldots,x_k\}=\{x_0,x_1,\ldots,x_p\}$ and $$x_0{\beta}=y_{m-p},\,\,x_1{\beta}=y_{m-p+1},\ldots,x_p{\beta}=y_m;$$
- If $x\not\in\operatorname{span}({\alpha})$ and $x\in\operatorname{dom}({\beta})$, then either $x{\beta}\not\in\operatorname{span}({\alpha})$ or there exists a chain $\tau'=[y_0\,y_1\ldots\, y_m]$ in ${\alpha}$ such that $x{\beta}=y_m$.
The way to remember Proposition \[pcen\] is that ${\alpha}{\beta}={\beta}{\alpha}$ if and only if ${\beta}$ maps cycles in ${\alpha}$ onto cycles in ${\alpha}$ of the same length, and it maps initial segments of chains in ${\alpha}$ onto terminal segments of chains in ${\alpha}$.
An element ${\varepsilon}\in{\mathcal{I}}(X)$ is an idempotent (${\varepsilon}{\varepsilon}={\varepsilon}$) if and only if ${\varepsilon}=(x_1){\sqcup}(x_2){\sqcup}\cdots{\sqcup}(x_k)$ is a join of cycles of length $1$; and ${\sigma}\in{\mathcal{I}}(X)$ is a permutation on $X$ if and only if $\operatorname{dom}({\sigma})=X$ and ${\sigma}$ is a join of cycles. For a function $f:A\to B$ and $A_0\subseteq A$, we denote by $f|_{A_0}$ the restriction of $f$ to $A_0$.
The following lemma will be important in our inductive arguments in Sections \[sinv\] and \[sgen\].
\[lres1\] Suppose ${\gamma}\in{\mathcal{I}}(X)$ is either an idempotent such that ${\gamma}\notin\{0,1\}$, or a permutation on $X$ such that not all cycles in ${\gamma}$ have the same length. Then there is a partition $\{A,B\}$ of $X$ such that ${\beta}|_A\in{\mathcal{I}}(A)$ and ${\beta}|_B\in{\mathcal{I}}(B)$ for all ${\beta}\in{\mathcal{I}}(X)$ such that ${\gamma}{\beta}={\beta}{\gamma}$.
Suppose ${\gamma}=(x_1){\sqcup}(x_2){\sqcup}\cdots{\sqcup}(x_k)\in{\mathcal{I}}(X)$ is an idempotent such that ${\gamma}\notin\{0,1\}$. Let $A=\operatorname{dom}({\gamma})=\{x_1,x_2,\ldots,x_k\}$ and $B=X-A$. Then $A\ne\emptyset$ (since ${\gamma}\ne0$), $B\ne\emptyset$ (since ${\gamma}\ne1$), and $A\cap B=\emptyset$. Note that $B=X-\operatorname{span}({\gamma})$. Let ${\beta}\in{\mathcal{I}}(X)$ be such that ${\gamma}{\beta}={\beta}{\gamma}$. Let $x_i\in A$ and $y\in B$ be such that $x_i,y\in\operatorname{dom}({\beta})$. Then $x_i{\beta}=x_j\in A$ by (1) of Proposition \[pcen\], and $y{\beta}\in B$ by (3) of Proposition \[pcen\]. Hence ${\beta}|_A\in{\mathcal{I}}(A)$ and ${\beta}|_B\in{\mathcal{I}}(B)$.
Suppose ${\gamma}\in{\mathcal{I}}(X)$ is a permutation on $X$ such that not all cycles in ${\gamma}$ have the same length. Select any cycle $\rho$ in ${\gamma}$ and let $k$ be the length of $\rho$. Let $$A=\{x\in X:\mbox{$x\in\operatorname{span}(\rho')$ for some cycle $\rho'$ in ${\gamma}$ of length $k$}\}$$ and let $B=X-A$. Then $A\ne\emptyset$ (since $\rho$ is a cycle in ${\gamma}$ of length $k$), $B\ne\emptyset$ (since not all cycles in ${\gamma}$ have length $k$), and $A\cap B=\emptyset$. Let ${\beta}\in{\mathcal{I}}(X)$ be such that ${\gamma}{\beta}={\beta}{\gamma}$. Let $x\in A$ and $y\in B$ be such that $x,y\in\operatorname{dom}({\beta})$. Then $x{\beta}\in A$ and $y{\beta}\in B$ by (1) of Proposition \[pcen\]. Hence ${\beta}|_A\in{\mathcal{I}}(A)$ and ${\beta}|_B\in{\mathcal{I}}(B)$.
It is straightforward to prove the following lemma.
\[lres2\] Let $\{A,B\}$ be a partition of $X$. Suppose ${\alpha},{\beta}\in{\mathcal{I}}(X)$ are such that ${\alpha}|_A,{\beta}|_A\in{\mathcal{I}}(A)$ and ${\alpha}|_B,{\beta}|_B\in{\mathcal{I}}(B)$. Then:
- $({\alpha}{\beta})|_A=({\alpha}|_A)({\beta}|_A)$ and $({\alpha}{\beta})|_B=({\alpha}|_B)({\beta}|_B)$.
- ${\alpha}{\beta}={\beta}{\alpha}$ if and only if $({\alpha}|_A)({\beta}|_A)=({\beta}|_A)({\alpha}|_A)$ and $({\alpha}|_B)({\beta}|_B)=({\beta}|_B)({\alpha}|_B)$.
We conclude this section with a lemma that is an immediate consequence of the definition of commutativity.
\[lclo\] For all ${\alpha},{\beta}\in{\mathcal{I}}(X)$, if ${\alpha}{\beta}={\beta}{\alpha}$, then $(\operatorname{im}{\alpha}){\beta}\subseteq\operatorname{im}({\alpha})$ and $(\operatorname{dom}({\alpha})){\beta}^{-1}\subseteq\operatorname{dom}({\alpha})$.
The Largest Commutative Inverse Semigroup in ${\mathcal{I}}(X)$ {#sinv}
===============================================================
In this section, we will prove that the maximum order of a commutative inverse subsemigroup of ${\mathcal{I}}(X)$ is $2^n$, and that the semilattice $E({\mathcal{I}}(X))$ of idempotents is the unique commutative inverse subsemigroup of ${\mathcal{I}}(X)$ of the maximum order (Theorem \[tinv\]).
An element $a$ of a semigroup $S$ is called *regular* if $a=axa$ for some $x\in S$. If all elements of $S$ are regular, we say that $S$ is a *regular semigroup*. An element $a'\in S$ is called an *inverse* of $a\in S$ if $a=aa'a$ and $a'=a'aa'$. Since regular elements are precisely those that have inverses (if $a=axa$ then $a'=xax$ is an inverse of $a$), we may define a regular semigroup as a semigroup in which each element has an inverse [@Ho95 p. 51]. The most extensively studied subclass of the regular semigroups has been the class of inverse semigroups (see [@Pe84] and [@Ho95 Chapter 5]). A semigroup $S$ is called an *inverse semigroup* if every element of $S$ has exactly one inverse [@Pe84 Definition II.1.1]. An alternative definition is that $S$ is an inverse semigroup if it is a regular semigroup and its idempotents (elements $e\in S$ such that $ee=e$) commute [@Ho95 Theorem 5.1.1].
A *semilattice* is a commutative semigroup consisting entirely of idempotents. A semilattice can also be defined as a partially ordered set $(S,\leq)$ such that the greatest lower bound $a\wedge b$ exists for all $a,b\in S$. Indeed, if $S$ is a semilattice, then $(S,\leq)$, where $\leq$ is a relation on $S$ defined by $a\leq b$ if $a=ab$, is a poset with $a\wedge b=ab$ for all $a,b\in S$. Conversely, if $(S,\leq)$ is a poset such that $a\wedge b$ exists for all $a,b\in S$, then $S$ with multiplication $ab=a\wedge b$ is a semilattice. (See [@Ho95 Proposition 1.3.2].) For a semigroup $S$, denote by $E(S)$ the set of idempotents of $S$. The set $E({\mathcal{I}}(X))$ is a semilattice, which, viewed as a poset, is isomorphic to the poset $(\mathcal{P}(X),\subseteq)$ of the power set $\mathcal{P}(X)$ under inclusion.
For semigroups $S$ and $T$, we will write $S\cong T$ to mean that $S$ is isomorphic to $T$.
\[lres3\] Let $S$ be a commutative semigroup in ${\mathcal{I}}(X)$. Suppose there is a partition $\{A,B\}$ of $X$ such that ${\alpha}|_A,{\beta}|_A\in{\mathcal{I}}(A)$ and ${\alpha}|_B,{\beta}|_B\in{\mathcal{I}}(B)$ for all ${\alpha},{\beta}\in S$. Let $S_{\!{\mbox{\tiny $A$}}}=\{{\alpha}|_{A}:{\alpha}\in S\}$ and $S_{\!{\mbox{\tiny $B$}}}=\{{\alpha}|_{B}:{\alpha}\in S\}$. Then:
- $S_{\!{\mbox{\tiny $A$}}}$ is a commutative semigroup in ${\mathcal{I}}(A)$ and $S_{\!{\mbox{\tiny $B$}}}$ is a commutative semigroup in ${\mathcal{I}}(B)$.
- If $S$ is an inverse semigroup, then $S_{\!{\mbox{\tiny $A$}}}$ and $S_{\!{\mbox{\tiny $B$}}}$ are inverse semigroups.
- If $S$ is a maximal commutative semigroup in ${\mathcal{I}}(X)$, then $S\cong S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$.
To prove (1), first note that $S_{\!{\mbox{\tiny $A$}}}$ is a subset of ${\mathcal{I}}(A)$. It is closed under multiplication since for all ${\alpha},{\beta}\in S$, we have ${\alpha}{\beta}\in S$, and so, by Lemma \[lres2\], $({\alpha}|_A)({\beta}|_A)=({\alpha}{\beta})|_A\in S_{{\mbox{\tiny $A$}}}$. Finally, $S_{\!{\mbox{\tiny $A$}}}$ is commutative by Lemma \[lres2\] and the fact that $S$ is commutative. The proof for $S_{\!{\mbox{\tiny $B$}}}$ is the same.
To prove (2), suppose that $S$ is an inverse semigroup. Let ${\alpha}|_A\in S_{\!{\mbox{\tiny $A$}}}$, where ${\alpha}\in S$. Since $S$ is a regular semigroup, there exists ${\beta}\in S$ such that ${\alpha}={\alpha}{\beta}{\alpha}$. Then ${\beta}|_A\in S_{\!{\mbox{\tiny $A$}}}$ and, by Lemma \[lres2\], ${\alpha}|_A=({\alpha}{\beta}{\alpha})|_A=({\alpha}|_A)({\beta}|_A)({\alpha}|_A)$. Thus ${\alpha}|_A$ is a regular element of $S_{\!{\mbox{\tiny $A$}}}$, and so $S_{\!{\mbox{\tiny $A$}}}$ is a regular semigroup. Hence $S_{\!{\mbox{\tiny $A$}}}$ is an inverse semigroup since it is a subsemigroup of ${\mathcal{I}}(A)$ and the idempotents in ${\mathcal{I}}(A)$ commute. The proof for $S_{\!{\mbox{\tiny $B$}}}$ is the same.
To prove (3), suppose that $S$ is a maximal commutative semigroup in ${\mathcal{I}}(X)$. Define a function $\phi:S\to S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$ by ${\alpha}\phi=({\alpha}|_A,{\alpha}|_B)$. Then $\phi$ is a homomorphism since for all ${\alpha},{\beta}\in S$, $$({\alpha}{\beta})\phi=(({\alpha}{\beta})|_A,({\alpha}{\beta})|_B)=(({\alpha}|_A)({\beta}|_A),({\alpha}|_B)({\beta}|_B))=({\alpha}|_A,{\alpha}|_B)({\beta}|_A,{\beta}|_B)=({\alpha}\phi)({\beta}\phi).$$ Further, for all ${\alpha},{\beta}\in S$, $({\alpha}|_A,{\alpha}|_B)=({\beta}|_A,{\beta}|_B)$ implies ${\alpha}={\beta}$ (since $\{A,B\}$ is a partition of $X$). Thus $\phi$ is one-to-one. Let $({\sigma},\mu)\in S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$. Then ${\sigma}={\alpha}|_A$ and $\mu={\beta}|_B$ for some ${\alpha},{\beta}\in S$. Define ${\gamma}\in{\mathcal{I}}(X)$ by ${\gamma}|_A={\alpha}|_A$ and ${\gamma}|_B={\beta}|_B$. Let ${\delta}\in S$. Then ${\alpha}{\delta}={\delta}{\alpha}$ and ${\beta}{\delta}={\delta}{\beta}$, and so, by Lemma \[lres2\], $({\gamma}|_A)({\delta}|_A)=({\alpha}|_A)({\delta}|_A)=({\delta}|_A)({\alpha}|_A)=({\delta}|_A)({\gamma}|_A)$ and $({\gamma}|_B)({\delta}|_B)=({\beta}|_B)({\delta}|_B)=({\delta}|_B)({\beta}|_B)=({\delta}|_B)({\gamma}|_B$). Hence ${\gamma}{\delta}={\delta}{\gamma}$, which implies that ${\gamma}\in S$ since $S$ is a maximal commutative semigroup in ${\mathcal{I}}(X)$. Thus ${\gamma}\phi=({\gamma}|_A,{\gamma}|_B)=({\alpha}|_A,{\beta}|_B)=({\sigma},\mu)$, and so $\phi$ is onto.
A subgroup $G$ of $\operatorname{Sym}(X)$ is called *semiregular* if the identity is the only element of $G$ that fixes any point of $X$ [@Wi64]. It is easy to see that $G$ is semiregular if and only if for every ${\sigma}\in G$, all cycles in ${\sigma}$ have the same length. If $G$ is a semiregular subgroup of $\operatorname{Sym}(X)$ with $n=|X|$, then the order of $G$ divides $n$ [@Wi64 Proposition 4.2], and so $|G|\leq n$.
We can now prove our main theorem in this section.
\[tinv\] Let $X$ be a finite set with $n\geq1$ elements. Then:
- If $S$ is a commutative inverse subsemigroup of ${\mathcal{I}}(X)$, then $|S|\leq 2^n$.
- The semilattice $E({\mathcal{I}}(X))$ is the unique commutative inverse subsemigroup of ${\mathcal{I}}(X)$ of order $2^n$.
We will prove (1) and (2) simultaneously by induction on $n$. The statements are certainly true for $n=1$. Let $n\geq2$ and suppose that (1) and (2) are true for every symmetric inverse semigroup on a set with cardinality less than $n$.
Let $S$ be a maximal commutative inverse semigroup in ${\mathcal{I}}(X)$. Let $G=S\cap\operatorname{Sym}(X)$ and $T=S-G$. If $G$ is a semiregular subgroup of $\operatorname{Sym}(X)$ and $T=\{0\}$, then $|S|=|G|+1\leq n+1<2^n$ (since $n\geq2$).
Suppose $G$ is not semiregular or $T\ne\{0\}$. In the former case, $G$ (and so $S$) contains a permutation ${\sigma}$ such that not all cycles of ${\sigma}$ are of the same length. Suppose $T\ne\{0\}$. Let $0\ne{\alpha}\in T$ and let ${\alpha}'$ be the inverse of ${\alpha}$ in $S$. Then ${\alpha}={\alpha}{\alpha}'{\alpha}$ and ${\varepsilon}={\alpha}{\alpha}'$ is an idempotent. Note that ${\varepsilon}\ne1$ (since ${\alpha}\notin\operatorname{Sym}(X)$) and ${\varepsilon}\ne0$ (since ${\alpha}={\varepsilon}{\alpha}$ and ${\alpha}\ne0$).
Thus, in either case, by Lemmas \[lres1\] and \[lres3\], there is a partition $\{A,B\}$ of $X$ such that $S\cong S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$, where $S_{\!{\mbox{\tiny $A$}}}$ is a commutative inverse semigroup in ${\mathcal{I}}(A)$ and $S_{\!{\mbox{\tiny $B$}}}$ is a commutative inverse semigroup in ${\mathcal{I}}(B)$. Let $k=|A|$ and $m=|B|$. Then $1\leq k,m<n$ with $k+m=n$, and so, by the inductive hypothesis, $|S|=|S_{\!{\mbox{\tiny $A$}}}|\cdot|S_{\!{\mbox{\tiny $B$}}}|\leq 2^k\cdot2^m=2^{k+m}=2^n$.
Suppose that $S\ne E({\mathcal{I}}(X))$. Then, since $S$ is a maximal commutative inverse semigroup in ${\mathcal{I}}(X)$, $S$ is not included in $E({\mathcal{I}}(X))$, and so it is not a semilattice. It follows that $S_{\!{\mbox{\tiny $A$}}}\ne E({\mathcal{I}}(A))$ or $S_{\!{\mbox{\tiny $B$}}}\ne E({\mathcal{I}}(B))$ (since $S\cong S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$ and the direct product of two semilattices is a semilattice). We may assume that $S_{\!{\mbox{\tiny $A$}}}\ne E({\mathcal{I}}(A))$. By the inductive hypothesis again, ${\mathcal{I}}(A)<2^k$, and so $|S|=|S_{\!{\mbox{\tiny $A$}}}|\cdot|S_{\!{\mbox{\tiny $B$}}}|<2^k\cdot2^m=2^{k+m}=2^n$.
We have proved that $|S|\leq 2^n$ and if $S\ne E({\mathcal{I}}(X))$ then $|S|<2^n$. Statements (1) and (2) follow.
The Largest Commutative Nilpotent Semigroups in ${\mathcal{I}}(X)$ {#snil}
==================================================================
In this section, we consider nilpotent semigroups in ${\mathcal{I}}(X)$, that is, the semigroups whose every element is a nilpotent. We determine the maximum order of a commutative nilpotent semigroup in ${\mathcal{I}}(X)$, and describe the commutative nilpotent semigroups in ${\mathcal{I}}(X)$ of the maximum order (Theorem \[tnil\]).
\[dnul\] [Let $X$ be a set with $n\geq2$ elements and let $\{K,L\}$ be a partition of $X$. Denote by $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ the subset of ${\mathcal{I}}(X)$ consisting of all nilpotents of the form $[x_1\,y_1]{\sqcup}\cdots{\sqcup}[x_r\,y_r]$, where $x_i\in K$, $y_i\in L$, and $0\leq r\leq\min\{|K|,|L|\}$. ]{}
For example, let $n=4$, $X=\{1,2,3,4\}$, $K=\{1,2\}$, and $L=\{3,4\}$. Then $$S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}=\{0,[1\,3],[1\,4],[2\,3],[2\,4],[1\,3]{\sqcup}[2\,4],[1\,4]{\sqcup}[2\,3]\}.$$
\[lmax\] Any set $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ from [Definition \[dnul\]]{.nodecor} is a null semigroup of order $\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!$, where $m=\min\{|K|,|L|\}$.
Let ${\alpha},{\beta}\in S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ and suppose $x\in\operatorname{dom}({\alpha})$. Then $x{\alpha}\notin\operatorname{dom}({\beta})$ (since $x{\alpha}\in L$), and so $x\notin\operatorname{dom}({\alpha}{\beta})$. It follows that ${\alpha}{\beta}=0$.
Let $m=\min\{|K|,|L|\}$. Suppose $m=|K|$, so $|L|=n-m$. Let ${\alpha}=[x_1\,y_1]{\sqcup}\cdots{\sqcup}[x_r\,y_r]$ be a transformation in $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ of rank $r$. Then, clearly, $0\leq r\leq m$. The domain of ${\alpha}$ can be selected in $\binom{m}{r}$ ways, the image in $\binom{n-m}{r}$ ways, and the domain can be mapped to the image in $r!$ ways. It follows that $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ contains $\binom{m}{r}\binom{n-m}{r}r!$ transformations of rank $r$, and so $|S|=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!$. The result is also true when $m=|L|$ since $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ has the same order as $S_{\!{\mbox{\tiny $L$}}\!,{\mbox{\tiny $K$}}}$.
\[dbal\]
A null semigroup $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ from [Definition \[dnul\]]{.nodecor} such that $|K|={\left\lfloor\frac{n}{2}\right\rfloor}$ and $L=n-{\left\lfloor\frac{n}{2}\right\rfloor}$, or vice versa, will be called a *balanced null semigroup*. By Lemma \[lmax\], any balanced null semigroup has order $$\label{e1snil}
{\lambda}_n=\sum_{r=0}^{{\left\lfloor\frac{n}{2}\right\rfloor}}\binom{{\left\lfloor\frac{n}{2}\right\rfloor}}{r}\binom{n-{\left\lfloor\frac{n}{2}\right\rfloor}}{r}r!.$$ If $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ is a balanced null semigroup, then the monoid $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}\cup\{1\}$ will be called a *balanced null monoid*.
Note that ${\lambda}_n$ from (\[e1snil\]) is also defined for $n=1$, and that ${\lambda}_1=1$ is the order of the trivial nilpotent semigroup $S=\{0\}$.
Our objective is to prove that the maximum order of a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ is ${\lambda}_n$, and that, if $n\notin\{1,3\}$, the balance null semigroups $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ are the only commutative nilpotent subsemigroups of ${\mathcal{I}}(X)$ of order ${\lambda}_n$ (Theorem \[tnil\]). We will need some combinatorial lemmas, which we present now.
\[lcom1\] For every $n\geq4$, ${\lambda}_n={\lambda}_{n-1}+{\left\lfloor\frac{n}{2}\right\rfloor}{\lambda}_{n-2}$.
Let $m={\left\lfloor\frac{n}{2}\right\rfloor}$. Consider a balanced null semigroup $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$, where $|K|=n-m$ and $|L|=m$. Then ${\lambda}_n=|S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}|$. Fix $x\in K$. Then $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}=S_1\cup S_2$, where $S_1=\{{\alpha}\in S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}:x\notin\operatorname{dom}({\alpha})\}$ and $S_2=\{{\alpha}\in S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}:x\in\operatorname{dom}({\alpha})\}$. Then $S_1=S_{\!{\mbox{\tiny $K$}}-\{x\}\!,{\mbox{\tiny $L$}}}\subseteq{\mathcal{I}}(X-\{x\})$ with $|K-\{x\}|=\left\lfloor\frac{n-1}{2}\right\rfloor$ and $|L|=(n-1)-\left\lfloor\frac{n-1}{2}\right\rfloor$. Thus $|S_1|={\lambda}_{n-1}$.
Let ${\alpha}\in S_2$. Then ${\alpha}=[x\,y]{\sqcup}{\beta}$, where $y\in L$ and ${\beta}\in S_{\!{\mbox{\tiny $K$}}-\{x\}\!,{\mbox{\tiny $L$}}-\{y\}}\subseteq{\mathcal{I}}(X-\{x,y\})$ with $|K-\{x\}|=(n-2)-\left\lfloor\frac{n-2}{2}\right\rfloor$ and $|L|=\left\lfloor\frac{n-2}{2}\right\rfloor$. For a fixed $y\in L$, the mapping ${\alpha}=[x\,y]{\sqcup}{\beta}\to{\beta}$ is a bijection from $\{{\alpha}\in S_2:x{\alpha}=y\}$ to $S_{\!{\mbox{\tiny $K$}}-\{x\}\!,{\mbox{\tiny $L$}}-\{y\}}$. Thus, since there are $|L|=m$ choices for $y$, we have $|S_2|=m|S_{\!{\mbox{\tiny $K$}}-\{x\}\!,{\mbox{\tiny $L$}}-\{y\}}|=m{\lambda}_{n-2}$. Hence $${\lambda}_n=|S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}|=|S_1|+|S_2|={\lambda}_{n-1}+{\left\lfloor\frac{n}{2}\right\rfloor}{\lambda}_{n-2}$$ since $m={\left\lfloor\frac{n}{2}\right\rfloor}$.
\[lcom2\] Let $a,b$ be integers such that $1\leq a,b\leq n$, $a<{\left\lfloor\frac{n}{2}\right\rfloor}$, and $b=n-a$. Then $$\sum_{r=0}^{a}\binom{a}{r}\binom{b}{r}r!<\sum_{r=0}^{a+1}\binom{a+1}{r}\binom{b-1}{r}r!.$$
Since $a<{\left\lfloor\frac{n}{2}\right\rfloor}$, and $b=n-a$, we have $a<b$ and hence $a+1\leq b$. Let $0\leq r\leq a$. Then
$$\begin{aligned}
-b\leq -a-1 &\Rightarrow-br\leq -ar-r\notag \\
&\Rightarrow ba+b-br\leq ba+b-ar-r\notag \\
&\Rightarrow b(a+1-r)\leq (b-r)(a+1)\notag \\
&\Rightarrow \frac{b}{b-r}\leq \frac{(a+1)}{(a+1-r)}\notag \\
&\Rightarrow \frac{b}{(b-r)(a-r)!(b-r-1)!}\leq \frac{(a+1)}{(a+1-r)(a-r)!(b-r-1)!}\notag \\
&\Rightarrow \frac{b}{(b-r)!(a-r)!}\leq \frac{(a+1)}{(a+1-r)!(b-r-1)!}\notag \\
&\Rightarrow \frac{a!(b-1)!b}{(b-r)!(a-r)!}\leq \frac{a!(b-1)!(a+1)}{(a+1-r)!(b-r-1)!}\notag \\
&\Rightarrow \frac{a!b!}{r!r!(b-r)!(a-r)!}\leq \frac{(b-1)!(a+1)!}{r!r!(a+1-r)!(b-r-1)!}\notag \\
&\Rightarrow \frac{a!}{r!(a-r)!}\frac{b!}{r!(b-r)!}\leq \frac{(a+1)!}{r!(a+1-r)!}\frac{(b-1)!}{r!(b-r-1)!}\notag \\
&\Rightarrow \binom{a}{r}\binom{b}{r}\leq \binom{a+1}{r}\binom{b-1}{r}\notag \\
&\Rightarrow \binom{a}{r}\binom{b}{r}r!\leq \binom{a+1}{r}\binom{b-1}{r}r!\notag\end{aligned}$$
Hence $\sum_{r=0}^{a} \binom{a}{r}\binom{b}{r}r!\leq \sum_{r=0}^{a}\binom{a+1}{r}\binom{b-1}{r}r!$, and so $\sum_{r=0}^{a} \binom{a}{r}\binom{b}{r}r!< \sum_{r=0}^{a+1}\binom{a+1}{r}\binom{b-1}{r}r!$.
\[lcom3\] Let $n>10$. Then:
- ${\lambda}_n+1>2({\lambda}_{n-1}+1)$.
- For every positive integer $k$ such that $k\geq10$ and $n-k\geq10$, $${\lambda}_n+1>({\lambda}_k+1)({\lambda}_{n-k}+1).$$
To prove (1), fix $a\in X$ and consider a partition $\{A,B\}$ of $X-\{a\}$ such that $|A|=\left\lfloor\frac{n-1}2\right\rfloor$ and $|B|=(n-1)-|A|$. Note that ${\lambda}_{n}=|S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}|$ and ${\lambda}_{n-1}=|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|$. We will consider two cases. [**Case 1.**]{} $n$ is even. In this case $|B|= |A|+1$, hence for every ${\alpha}\in S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$, we can select an element $y_{\alpha}\in B-\operatorname{im}({\alpha})$. Then the mapping $\phi:S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}\to S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}$ defined by ${\alpha}\phi={\alpha}{\sqcup}[a\,y_{\alpha}]$ is one-to-one with $\operatorname{im}(\phi)\subseteq S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}-S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$. Since $n>10$, we can select $y_1,y_2\in B$ such that $y_1,y_2\ne y_{\alpha}$ where ${\alpha}=0$. Then $[a\,y_1],[a\,y_2]\in S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}-(S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}\cup\operatorname{im}(\phi))$, which implies $${\lambda}_n=|S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}|\geq|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|+|\operatorname{im}(\phi)|+|\{[a\,y_1],[a\,y_2]\}|={\lambda}_{n-1}+{\lambda}_{n-1}+2>2{\lambda}_{n-1}+1.$$
[**Case 2.**]{} $n$ is odd. Let $m=|A|=|B|=\frac{n-1}{2}$. By direct calculations, ${\lambda}_{11}=4051$ and $2{\lambda}_{10}+1=3093$. So (1) is true for $n=11$. Suppose $n\geq13$ and note that $m\geq6$. Denote by $J_{m-2}$ the set of transformations of $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ of rank at most $m-2$ and note that $$|J_{m-2}|={\lambda}_{n-1}-m!-\binom{m}{1}^2(m-1)!={\lambda}_{n-1}-(m+1)m!$$ (since $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ has $m!$ transformations of rank $m$, and $\binom{m}{1}^2(m-1)!$ transformations of rank $m-1$). For every ${\alpha}\in J_{m-2}$, select two distinct elements $y_{\alpha},z_{\alpha}\in B-\operatorname{im}({\alpha})$ (possible since $|B|=m$ and $\operatorname{rank}({\alpha})\leq m-2$). Then the mappings $\phi,\psi:J_{m-2}\to S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}$ defined by ${\alpha}\phi={\alpha}{\sqcup}[a\,y_{\alpha}]$ and ${\alpha}\psi={\alpha}{\sqcup}[a\,z_{\alpha}]$ are one-to-one with $\operatorname{im}(\phi)\cup\operatorname{im}(\psi)\subseteq S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}-S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ and $\operatorname{im}(\phi)\cap\operatorname{im}(\psi)=\emptyset$. Therefore, $$\begin{aligned}
\lambda_n&=|S_{\!{\mbox{\tiny $A$}}\cup\{a\}\!,{\mbox{\tiny $B$}}}|\geq|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|+|\operatorname{im}(\phi)|+|\operatorname{im}(\psi)|\notag\\
&={\lambda}_{n-1}+2(\lambda_{n-1}-(m+1)m!)=2{\lambda}_{n-1}-2(m+1)m!+{\lambda}_{n-1}\notag\\
&>2{\lambda}_{n-1}-2(m+1)m!+\left(\binom{m}{0}^2m!+\binom{m}{1}^2 (m-1)!+\binom{m}{2}^2 (m-2)!\right)\notag\\
&=2{\lambda}_{n-1}-2(m+1)m!+\left(m!+m \cdot m!+\frac{m(m-1)}{4}m!\right)\notag\\
&=2{\lambda}_{n-1}+m!\left(-2m-2+1+m+\frac{m(m-1)}{4}\right)\notag\\
&=2\lambda_{n-1}+\frac{m!}{4}\left(m^2-5m-4\right)>2{\lambda}_{n-1}+1,\notag\end{aligned}$$ where the first strong inequality follows from the fact that ${\lambda}_{n-1}=|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|$, $m\geq6$, and the expression $\binom{m}{0}^2m!+\binom{m}{1}^2 (m-1)!+\binom{m}{2}^2(m-2)!$ only counts the transformations in $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ of ranks $m$, $m-1$, and $m-2$; and the last strong inequality inequality follows from the fact that for $m\geq6$, $\frac{m!}{4}\geq180$ and $m^2-5m-4\ge 2$.
To prove (2), suppose $k\geq10$ and $n-k\geq10$. We may assume that $k\leq n-k$. Consider a partition $\{A,B,C,D\}$ of $X$ such that $$|A|=\left\lfloor\frac{n-k}2\right\rfloor,\,\, |B|=(n-k)-|A|,\,\, |D|=\left\lfloor\frac{k}2\right\rfloor,\,\,|C|=k-|D|.$$ Then, either $|A|+|C|={\left\lfloor\frac{n}{2}\right\rfloor}$ or $|B|+|D|={\left\lfloor\frac{n}{2}\right\rfloor}$, and so ${\lambda}_n=|S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}|$, ${\lambda}_{n-k}=|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|$ and ${\lambda}_k=|S_{{\mbox{\tiny $C$}},{\mbox{\tiny $D$}}}|$.
Let $S$ be the subsemigroup of $S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}$ consisting of all ${\alpha}$ such that ${\alpha}|_A\in S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ and ${\alpha}|_C\in S_{{\mbox{\tiny $C$}},{\mbox{\tiny $D$}}}$. We can construct a bijection between $S$ and $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}\times S_{{\mbox{\tiny $C$}},{\mbox{\tiny $D$}}}$ as in the proof of Lemma \[lres3\], hence $|S|={\lambda}_{n-k}{\lambda}_k$. Since the inequality in (2) is equivalent to ${\lambda}_n>{\lambda}_k{\lambda}_{n-k}+{\lambda}_k+{\lambda}_{n-k}$, it suffices to construct more then $2\lambda_{n-k}\ge\lambda_{n-k}+\lambda_{k}$ elements of $S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}-S$. We will consider two cases. [**Case 1.**]{} $n-k$ is odd. In this case $|B|=|A|+1$, so for each ${\alpha}\in S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$, we can select an element $b_{\alpha}\in B-\operatorname{im}({\alpha})$. Now, for any pair $(c,{\alpha})\in C\times S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$, let ${\alpha}_c={\alpha}{\sqcup}[c\,b_{\alpha}]$. It is clear that ${\alpha}_c\in S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}-S$ and that the mapping $({\alpha},c)\to{\alpha}_c$ is one-to-one. Since $k\geq10$, we have $|C|=k-\left\lfloor\frac{k}2\right\rfloor\ge 5$. Thus, we have constructed $|C|\cdot|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|\geq5\lambda_{n-k}>2\lambda_{n-k}$ elements in $S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}-S$. [**Case 2.**]{} $n-k$ is even. Let $m=\frac{n-k}{2}$. Note that for any ${\alpha}\in S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ of rank smaller then $m$, we can find $b_{\alpha}\in B-\operatorname{im}({\alpha})$ and define ${\alpha}_c$ as in Case 1. This construction yields $|C|({\lambda}_{n-k}-m!)\ge5({\lambda}_{n-k}-m!)$ distinct elements of $S_{\!{\mbox{\tiny $A$}}\cup{\mbox{\tiny $C$}},{\mbox{\tiny $B$}}\cup{\mbox{\tiny $D$}}}-S$. Since $m\geq5$, we have $${\lambda}_{n-k}=|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|>\binom{m}{0}\binom{m}{0}m!+\binom{m}{1}\binom{m}{1}(m-1)!> 2m!,$$ where the first inequality follows from the fact that $\binom{m}{0}\binom{m}{0}m!+\binom{m}{1}\binom{m}{1}(m-1)!$ only counts the elements of $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ of rank $m$ and $m-1$. Thus $3{\lambda}_{n-k}>6m!$, and so $$|C|({\lambda}_{n-k}-m!)\ge5({\lambda}_{n-k}-m!)=3\lambda_{n-k}+2\lambda_{n-k}-5m!>6m!+2\lambda_{n-k}-5m!>2\lambda_{n-k}.$$ The result follows.
\[lcom4\] If $n\geq6$, then ${\lambda}_n>2{\lambda}_{n-1}$.
If $n>10$, then ${\lambda}_n>2{\lambda}_{n-1}+1>2{\lambda}_{n-1}$ by Lemma \[lcom3\]. If $6\leq n\leq10$, then the result can be checked by direct calculations: $$\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\hline
$n$&2&3&4&5&6&7&8&9&10\\\hline
${\lambda}_n$&2&3&7&13&34&73&209&501&1546\\\hline
$2{\lambda}_{n-1}$&2&4&6&14&26&68&146&418&1002\\\hline
\end{tabular}$$
We begin the proof of Theorem \[tnil\] with introducing the following notation.
\[nabc\] [ Let $S$ be any commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$. We define the following subset $C=C(S)$ of $X$: $$\label{e1nabc}
C=\{c\in X:c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\beta})\mbox{ for some ${\alpha},{\beta}\in S$}\}.$$ For a fixed $c\in C$, we define $$\begin{aligned}
A_c&=\{a\in X:a{\alpha}=c\mbox{ for some ${\alpha}\in S$}\},\notag\\
B_c&=\{b\in X:c{\alpha}=b\mbox{ for some ${\alpha}\in S$}\}.\notag\end{aligned}$$ Note that $A_c$ and $B_c$ are not empty (by the definition of $C$) and that $A_c\cap B_c=\emptyset$. (Indeed, if $a\in A_c\cap B_c$, then $a{\alpha}=c$ and $c{\beta}=a$ for some ${\alpha},{\beta}\in S$, that is, ${\alpha}=[\ldots a\,c\ldots]{\sqcup}\cdots$ and ${\beta}=[\ldots c\,a\ldots]{\sqcup}\cdots$. It then follows from Proposition \[pcen\] that ${\alpha}{\beta}\ne{\beta}{\alpha}$, which is a contradiction.) ]{}
In the following lemmas, $S$ is a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ and $C$ is the subset of $S$ defined by (\[e1nabc\]). Our immediate objective is to obtain certain bounds on $|A_c|$ and $|B_c|$ (see Lemma \[lbou\]).
\[lqabc\] Let $c\in C$, $a\in A_c$, and $b\in B_c$. Then:
- There is a unique $q=q(c,a,b)\in C$ such that for all ${\alpha}\in S$, if $a{\alpha}=c$, then $q{\alpha}=b$.
- For all ${\beta}\in S$, if $c{\beta}=b$, then $a{\beta}=q$, where $q=q(c,a,b)$ is the unique element from [(1)]{}.
To prove (1), suppose ${\alpha}\in S$ with $a{\alpha}=c$, that is, ${\alpha}=[\ldots a\,c\ldots]{\sqcup}\cdots$. Since $b\in B_c$, $c{\beta}=b$ for some ${\beta}\in S$. Since $c\in\operatorname{dom}({\beta})$, Proposition \[pcen\] implies that $a\in\operatorname{dom}({\beta})$. Let $q=a{\beta}$. Then $q{\alpha}=(a{\beta}){\alpha}=(a{\alpha}){\beta}=c{\beta}=b$. Let ${\alpha}'\in S$ be such that $a{\alpha}'=c$. By the foregoing argument, there exists $q'\in X$ such that $a{\beta}=q'$ and $q'{\alpha}'=b$. But then $q=a{\beta}=q'$, so $q$ is unique. Moreover, $q\in C$ since $q\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\beta})$.
To prove (2), suppose ${\beta}\in S$ with $c{\beta}=b$. Since $a\in A_c$, $a{\alpha}=c$ for some ${\alpha}\in S$. But then, by the proof of (1), $a{\beta}=q$.
\[ldif1\] Let $c\in C$, $a,a_1,a_2\in A_c$, and $b,b_1,b_2\in B_c$. Then:
- If $q(c,a,b_1)=q(c,a,b_2)$, then $b_1=b_2$.
- If $q(c,a_1,b)=q(c,a_2,b)$, then $a_1=a_2$.
To prove (1), let $q=q(c,a,b_1)=q(c,a,b_2)$. Since $a\in A_c$, there is ${\alpha}\in S$ such that $a{\alpha}=c$. But then, by Lemma \[lqabc\], $b_1=q{\alpha}=b_2$. The proof of (2) is similar.
We can now prove the lemma concerning the sizes of $A_c$ and $B_c$.
\[lbou\] Suppose $C\ne\emptyset$. Then, there exists $c \in C$ such that one of the following conditions holds:
- $|A_c|\geq2$ and $|B_c|\geq2$;
- $|A_c|=1$ and $|B_c|\leq{\left\lfloor\frac{n}{2}\right\rfloor}$; or
- $|B_c|=1$ and $|A_c|\leq{\left\lfloor\frac{n}{2}\right\rfloor}$.
Suppose to the contrary that for every $c\in C$, none of (a)–(c) holds. Let $c\in C$. Then, since (a) does not hold for $c$, $|A_c|=1$ or $|B_c|=1$.
Suppose $|A_c|=1$, say $A_c=\{a\}$. Then, since (b) does not hold for $c$, $|B_c|>{\left\lfloor\frac{n}{2}\right\rfloor}$. Let $b\in B_c$. We claim that $b\notin C$. Suppose to the contrary that $b\in C$. Construct elements $b_0,b_1,b_2,\ldots$ in $C\cap B_c$ as follows. Set $b_0=b$. Suppose $b_i\in C\cap B_c$ has been constructed ($i\geq0$). Let $b_{i+1}$ be any element of $B_c$ such that $b_{i+1}{\gamma}_i=b_i$ for some ${\gamma}_i\in S$. Then $b_{i+1}\in C$ as $b_{i+1} \in \operatorname{dom}({\gamma}_i) \cap B_c=\operatorname{dom}({\gamma}_i)\cap \{c\}S$. If such an element $b_{i+1}$ does not exist, stop the construction. Note that the construction must stop after finitely many steps. (Indeed, otherwise, since $X$ is finite, we would have $b_k=b_j$ with $k>j\geq0$. But then $b_j{\gamma}=b_k{\gamma}=b_j$ for ${\gamma}={\gamma}_{k-1}{\gamma}_{k-2}\cdots{\gamma}_j\in S$, which is impossible since $S$ consists of nilpotents.) Thus, there exists $i\geq0$ such that $b_i\in C\cap B_c$ and no element of $B_c$ is mapped to $b_i$ by some transformation in $S$.
Let $b'=b_i$ and note that $A_{b'}\subseteq X-B_c$. Since $A_c=\{a\}$, $a{\alpha}=c$ for some ${\alpha}\in S$. Since $b'\in B_c$, $c{\beta}=b'$ for some ${\beta}\in S$. Let $q=q(c,a,b')$. Then, by Lemma \[lqabc\], $q{\alpha}=b'$, and so $\{c,q\}\subseteq A_{b'}$. If $c\ne q$, then $|A_{b'}|\geq2$. Suppose $c=q$. Then $a({\alpha}{\alpha})=c{\alpha}=q{\alpha}=b'$, and so $\{c,a\}\subseteq A_{b'}$. But $a\ne c$ (since $a{\alpha}=c$ and ${\alpha}$ is a nilpotent), and we again have $|A_{b'}|\geq2$. On the other hand, since $A_{b'}\subseteq X-B_c$ and $|B_c|>{\left\lfloor\frac{n}{2}\right\rfloor}$, we have $|A_{b'}|\leq{\left\lfloor\frac{n}{2}\right\rfloor}$. But $b'\in C$ with $2\leq|A_{b'}|\leq{\left\lfloor\frac{n}{2}\right\rfloor}$ contradicts our assumption (see the first sentence of the proof).
The claim has been proved. Hence, no element of $B_c$ is in $C$, that is, $C\subseteq X-B_c$. Now, by Lemma \[lqabc\], for each $b_i \in B_c$, there exists $q_i=q(c,a,b_i)\in C$ such that $a\in A_{q_i}$ and $b_i\in B_{q_i}$. Moreover, by Lemma \[ldif1\], $q_i\ne q_j$ if $i\ne j$. But this is a contradiction since $|B_c|>{\left\lfloor\frac{n}{2}\right\rfloor}>|X-B_c|\geq|C|$.
If $|B_c|=1$, we obtain a contradiction in a similar way. This concludes the proof.
We continue the proof of Theorem \[tnil\] by considering two cases. First, we suppose that $S$ is a commutative semigroup of nilpotents such that $C=\emptyset$, that is, there is no $c\in X$ such that $c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\beta})$ for some ${\alpha},{\beta}\in S$. Note that this implies that each nonzero element of $S$ is a nilpotent of index $2$.
\[pcase1\] Let $X$ be a set with $n\geq2$ elements and let $m={\left\lfloor\frac{n}{2}\right\rfloor}$. Let $S$ be a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ with $C=\emptyset$. Suppose $S\ne S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ for every balanced null semigroup $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ (see [Definition \[dbal\]]{}). Then $|S|<\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!$.
Let $A=\{x\in X:x\in\operatorname{dom}({\alpha})\mbox{ for some ${\alpha}\in S$}\}$ and $B=X-A$. Since $C=\emptyset$, we have $A\cap\{y\in X:y\in\operatorname{im}({\beta})\mbox{ for some ${\beta}\in S$}\}=\emptyset$, and so $S\subseteq S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$.
Suppose $|A|=m$. Then $S\ne S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ by the assumption, and so, by Lemma \[lmax\], $|S|<|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!$. Suppose $|A|<m$. Let $a=|A|$ and $b=|B|=n-a$. By Lemma \[lmax\] again, $$|S|\leq|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|=\sum_{r=0}^{a}\binom{a}{r}\binom{b}{r}r!.$$ Applying Lemma \[lcom2\] $m-a$ times, we obtain $$|S|\leq\sum_{r=0}^{a}\binom{a}{r}\binom{b}{r}r!<\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!.$$ Suppose $|A|>m$. Consider the semigroup $S'=\{{\alpha}^{-1}:{\alpha}\in S\}$ and note that $S'$ is a nilpotent commutative semigroup with $C=C(S')=\emptyset$ and the corresponding set $A'$ included in the original set $B$. Since $|A'|\leq m$, $|S|=|S'|<\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!$ by the foregoing argument.
Second, we suppose that $S$ is a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ such that $C\ne\emptyset$. Note that this is possible only if $n\geq3$. Fix $c\in C$ that satisfies one of the conditions (1)–(3) from Lemma \[lbou\]. Our objective is to prove that for all $n\geq3$, $$\label{eqsl1}
|S|\leq{\lambda}_n=\sum_{r=0}^{{\left\lfloor\frac{n}{2}\right\rfloor}}\binom{{\left\lfloor\frac{n}{2}\right\rfloor}}{r}\binom{n-{\left\lfloor\frac{n}{2}\right\rfloor}}{r}r!.$$
We will proceed by strong induction on $n=|X|$. Let $n=3$. Then the maximal commutative nilpotent semigroups in ${\mathcal{I}}(X)$ are the balanced null semigroups $\{0,[i\,j],[i\,k]\}$ and $\{0,[i\,k],[j\,k]\}$, and the cyclic semigroups $\{0,[i\,j\,k],[i\,k]\}$, where $i,j,k$ are fixed, pairwise distinct, elements of $X$. Thus (\[eqsl1\]) is true for $n=3$. [**Inductive Hypothesis.**]{} Let $n\geq4$ and suppose that (\[eqsl1\]) is true whenever $3\leq|X|<n$. Consider the following subset of $S$: $$\label{eqsl2}
S_c=\{{\alpha}\in S:c\in\operatorname{span}({\alpha})\}.$$ Then $S-S_c$ is a commutative nilpotent subsemigroup of ${\mathcal{I}}(X-\{c\})$. If there is no $d\in X-\{c\}$ such that $d\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\beta})$ for some ${\alpha},{\beta}\in S-S_c$, then $|S-S_c|\leq {\lambda}_{n-1}$ by Proposition \[pcase1\]. If such a $d\in X-\{c\}$ exists, then $|S-S_c|\leq{\lambda}_{n-1}$ by the inductive hypothesis. Thus, at any rate, $$\label{eqsl3}
|S-S_c|\leq{\lambda}_{n-1}.$$
We now want to find a suitable upper bound for the size of $S_c$ (Lemma \[lbound\]). To this end, we will map $S_c$ onto a commutative subset $S_c^*$ of ${\mathcal{I}}(X-\{c\})$ and analyze the preimages of the elements of $S_c^*$.
\[dscs\]
For ${\alpha}\in S_c$ with $c\in\operatorname{im}({\alpha})$, let $U_{\alpha}$ be the smallest subset of $X$ containing $c{\alpha}^{-1}$ and closed under all transformations ${\gamma}^{-1}$ and ${\alpha}{\delta}{\alpha}^{-1}$, where ${\gamma},{\delta}\in S_c$.
For ${\alpha}\in S_c$ with $c\in\operatorname{dom}({\alpha})$, let $D_{\alpha}$ be the smallest subset of $X$ containing $c$ and closed under all transformations ${\gamma}^{-1}$ and ${\alpha}{\delta}{\alpha}^{-1}$, where ${\gamma},{\delta}\in S_c$.
For ${\alpha}\in S_c$, define ${\alpha^*}\in{\mathcal{I}}(X-\{c\})$ as follows: $${\alpha^*}=
\left\{\begin{array}{ll}
{\alpha}|_{X-U_{\alpha}} & \mbox{if $c\in\operatorname{im}({\alpha})-\operatorname{dom}({\alpha})$,}\\
{\alpha}|_{X-D_{\alpha}} & \mbox{if $c\in\operatorname{dom}({\alpha})-\operatorname{im}({\alpha})$,}\\
{\alpha}|_{X-(D_{\alpha}\cup\, U_{\alpha})} & \mbox{if $c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\alpha})$.}
\end{array}\right.$$ Let $S_c^*=\{{\alpha^*}:{\alpha}\in S_c\}$ and note that $S_c^*$ is a subset of ${\mathcal{I}}(X-\{c\})$.
We will need the following lemma about the sets $U_{\alpha}$ and $D_{\alpha}$.
\[lpre\] Let ${\alpha},{\beta}\in S_c$. Then:
- If $c\in\operatorname{im}({\alpha})$, then $U_{\alpha}\subseteq\operatorname{dom}({\alpha})$. Moreover, if $c\in\operatorname{im}({\beta})$ and $c{\alpha}^{-1}=c{\beta}^{-1}$, then $U_{\alpha}=U_{\beta}$ and $x{\alpha}=x{\beta}$ for all $x\in U_{\alpha}$.
- If $c\in\operatorname{dom}({\alpha})$, then $D_{\alpha}\subseteq\operatorname{dom}({\alpha})$. Moreover, if $c\in\operatorname{dom}({\beta})$ and $c{\alpha}=c{\beta}$, then $D_{\alpha}=D_{\beta}$ and $x{\alpha}=x{\beta}$ for all $x\in D_{\alpha}$.
- If $c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\alpha})$, then $D_{\alpha}=U_{\alpha}$. Moreover, if $c\in\operatorname{im}({\beta})$ and $c{\alpha}^{-1}=c{\beta}^{-1}$, then $c\in\operatorname{dom}({\beta})$, $U_{\alpha}=U_{\beta}=D_{\beta}$, and $x{\alpha}=x{\beta}$ for all $x\in U_{\alpha}$. If $c\in\operatorname{dom}({\beta})$ and $c{\alpha}=c{\beta}$, then $c\in\operatorname{im}({\beta})$, $U_{\alpha}=U_{\beta}=D_{\beta}$, and $x{\alpha}=x{\beta}$ for all $x\in U_{\alpha}$.
To prove (1), suppose $c\in\operatorname{im}({\alpha})$ and let $a=c{\alpha}^{-1}$. Then clearly $a\in\operatorname{dom}({\alpha})$. By Lemma \[lclo\], $\operatorname{dom}({\alpha})$ is closed under ${\gamma}^{-1}$ for all ${\gamma}\in S_c$. Let $x\in\operatorname{dom}({\alpha})$ and ${\delta}\in S_c$ be such that $x({\alpha}{\delta}{\alpha}^{-1})$ is defined. Since $x{\alpha}\in\operatorname{im}({\alpha})$, we have $(x{\alpha}){\delta}\in\operatorname{im}({\alpha})$ by Lemma \[lclo\], and so $x({\alpha}{\delta}{\alpha}^{-1})=((x{\alpha}){\delta}){\alpha}^{-1}\in\operatorname{dom}({\alpha})$. Thus $\operatorname{dom}({\alpha})$ is also closed under ${\alpha}{\delta}{\alpha}^{-1}$ for all ${\delta}\in S_c$. It follows that $U_{\alpha}\subseteq\operatorname{dom}({\alpha})$.
Suppose $c\in\operatorname{im}({\beta})$ and $c{\alpha}^{-1}=c{\beta}^{-1}$. Let $a=c{\alpha}^{-1}=c{\beta}^{-1}$. Let $x\in U_{\beta}$. We will prove that $x\in U_{\alpha}$ and $x{\alpha}=x{\beta}$ by induction on the minimum number of steps needed to generate $x$ from $a$.
If $x=a$, then $x\in U_{\alpha}$ and $x{\alpha}=x{\beta}$ since $x=a=c{\beta}^{-1}=c{\alpha}^{-1}$. Suppose $x=y{\gamma}^{-1}$ for some $y\in U_{\beta}$ and ${\gamma}\in S_c$. Then $y\in U_{\alpha}$ and $y{\alpha}=y{\beta}$ by the inductive hypothesis. Then $x=y{\gamma}^{-1}\in U_{\alpha}$ by the definition of $U_{\alpha}$. Further, $y\in C$ (since $y\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\gamma})$), $x\in A_y$ (since $x{\gamma}=y$), and $y{\alpha}\in B_y$. Since we also have $y{\beta}=y{\alpha}$, Lemma \[lqabc\] implies $$x{\alpha}=q(y,x,y{\alpha})=q(y,x,y{\beta})=x{\beta}.$$ Finally, suppose $x=y({\beta}{\delta}{\beta}^{-1})$ for some $y\in U_{\beta}$ and ${\delta}\in S_c$. Then $y\in U_{\alpha}$ and $y{\alpha}=y{\beta}$ by the inductive hypothesis. Let $p=y({\alpha}{\delta})$. Then $y{\alpha}\in C$ (since $y{\alpha}\in\operatorname{dom}({\delta})\cap\operatorname{im}({\alpha}))$), $y\in A_{y{\alpha}}$, and $p\in B_{y{\alpha}}$ (since $(y{\alpha}){\delta}=p$). Again, since $y{\beta}=y{\alpha}$, Lemma \[lqabc\] implies $$p{\alpha}^{-1}=q(y{\alpha},y,p)=q(y{\beta},y,p)=p{\beta}^{-1}.$$ Then $x=y({\beta}{\delta}{\beta}^{-1})=(y({\alpha}{\delta})){\beta}^{-1}=p{\beta}^{-1}=p{\alpha}^{-1}=y({\alpha}{\delta}{\alpha}^{-1})$. It follows that $x\in U_{\alpha}$ and $x{\alpha}=p=x{\beta}$. We have proved that $U_{\beta}\subseteq U_{\alpha}$ and $x{\alpha}=x{\beta}$ for all $x\in U_{\beta}$. By symmetry, $U_{\alpha}\subseteq U_{\beta}$ and $x{\alpha}=x{\beta}$ for all $x\in U_{\alpha}$. We have proved (1). The proof of (2) is similar.
To prove (3), suppose $c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\alpha})$, say $c{\alpha}=b$ and $a{\alpha}=c$. Then $a=c{\alpha}^{-1}\in D_{\alpha}$ and $c=a{\alpha}{\alpha}{\alpha}^{-1}\in U_\alpha$. Hence $U_{\alpha}=D_{\alpha}$ by the definitions of $U_{\alpha}$ and $D_{\alpha}$. The remaining claims in (3) follow from (1) and (2).
\[lsccom\] Any two transformations in $S_c^*$ commute.
Let ${\alpha},{\beta}\in S_c$. We want to prove that ${\alpha^*}{\beta^*}={\beta^*}{\alpha^*}$. Let $x\in X-\{c\}$. Since ${\alpha}{\beta}={\beta}{\alpha}$, both ${\alpha}{\beta}$ and ${\beta}{\alpha}$ are either defined at $x$ or undefined at $x$. In the latter case, both ${\alpha^*}{\beta^*}$ and ${\beta^*}{\alpha^*}$ are undefined at $x$.
So suppose that $x({\alpha}{\beta})=x({\beta}{\alpha})$ exists. If both ${\alpha^*}{\beta^*}$ and ${\beta^*}{\alpha^*}$ are defined at $x$, then $x({\alpha^*}{\beta^*})=x({\alpha}{\beta})=x({\beta}{\alpha})=x({\beta^*}{\alpha^*})$. Hence, it suffices to show that $$x({\alpha^*}{\beta^*})\mbox{ is undefined}{\Leftrightarrow}x({\beta^*}{\alpha^*})\mbox{ is undefined}.$$ By symmetry, we may suppose that that $x({\alpha^*}{\beta^*})$ is undefined. We consider two possible cases. [**Case 1.**]{} $x{\alpha^*}$ is undefined. Since we are working under the assumption that $x({\alpha}{\beta})$ exists (and so $x{\alpha}$ exists), it follows from Definition \[dscs\] and Lemma \[lpre\] that $x\in K$, where $K=U_{\alpha}$ or $K=D_{\alpha}$. Since $x({\alpha}{\beta})$ exists, it is in $\operatorname{im}({\alpha})$ by Lemma \[lclo\], so $x({\alpha}{\beta}{\alpha}^{-1})$ exists. Hence $x({\alpha}{\beta}{\alpha}^{-1})\in K$ by the definitions of $U_{\alpha}$ and $D_{\alpha}$. We have $x({\alpha}{\beta}{\alpha}^{-1}) =(x{\beta}{\alpha}){\alpha}^{-1}=x{\beta}$, and so $x{\beta}\in K$. Thus $(x{\beta}){\alpha^*}$ is undefined, and so $x({\beta^*}{\alpha^*})$ is undefined. [**Case 2.**]{} $x{\alpha^*}$ is defined and $(x{\alpha^*}){\beta^*}$ is undefined. This can only happen when $x{\alpha^*}=x{\alpha}$ is in $K$, where $K=U_\beta$ or $K=D_{\beta}$. By the definitions of $U_{\beta}$ and $D_{\beta}$, $x=(x{\alpha}){\alpha}^{-1}\in K$ as well. But then $x{\beta^*}$ is undefined, and hence $x({\beta^*}{\alpha^*})$ is also undefined.
\[lemp\] Let ${\alpha}\in S_c$. Then:
- If $c\in\operatorname{im}({\alpha})$, then $\operatorname{span}({\alpha^*})\cap B_c=\emptyset$.
- If $c\in\operatorname{dom}({\alpha})$, then $\operatorname{span}({\alpha^*})\cap A_c=\emptyset$.
To prove (1), let $c\in\operatorname{im}({\alpha})$ and $b\in B_c$, that is, $c{\gamma}=b$ for some ${\gamma}\in S_c$. Note that $b\in\operatorname{im}({\alpha})$ by Lemma \[lclo\]. Then, since $c{\alpha}^{-1}\in U_{\alpha}$, we have $b{\alpha}^{-1}=(c{\alpha}^{-1})({\alpha}{\gamma}{\alpha}^{-1})\in U_{\alpha}$. Thus $b{\alpha}^{-1}\notin\operatorname{dom}({\alpha^*})$, and so $b\notin\operatorname{im}({\alpha^*})$. If $b\notin\operatorname{dom}({\alpha})$, then clearly $b\notin\operatorname{dom}({\alpha^*})$. Suppose $b\in\operatorname{dom}({\alpha})$. We have already established that $b{\alpha}^{-1}\in U_{\alpha}$. Thus $b=(b{\alpha}^{-1})({\alpha}{\alpha}{\alpha}^{-1})\in U_{\alpha}$, and so $b\notin\operatorname{dom}({\alpha^*})$. We have proved (1). The proof of (2) is similar.
We can now obtain an upper bound for the size of $S_c$.
\[lbound\] Let $p=|A_c|$ and $t=|B_c|$. Then $$|S_c|\leq(p-1){\lambda}_{n-t-1}+(t-1){\lambda}_{n-p-1}+2{\lambda}_{n-p-t-1}.$$
Let $A=A_c$, $B=B_c$, and consider the following subsets of $S_c^*$: $$\begin{aligned}
F_A&=\{{\alpha^*}\in S_c^*:\operatorname{span}({\alpha^*})\cap A\ne\emptyset\},\notag\\
F_B&=\{{\alpha^*}\in S_c^*:\operatorname{span}({\alpha^*})\cap B\ne\emptyset\},\notag\\
F_0&=\{{\alpha^*}\in S_c^*:\operatorname{span}({\alpha^*})\cap(A\cup B)=\emptyset\}.\notag\end{aligned}$$ Suppose ${\alpha^*}\in F_A$. Then, by Lemma \[lemp\], $c\in\operatorname{im}({\alpha})-\operatorname{dom}({\alpha})$ and $\operatorname{span}({\alpha^*})\cap B=\emptyset$. Hence ${\alpha^*}\in{\mathcal{I}}(X-(B\cup\{c\}))$. Similarly, if ${\alpha^*}\in F_B$, then $c\in\operatorname{dom}({\alpha})-\operatorname{im}({\alpha})$, $\operatorname{span}({\alpha^*})\cap A=\emptyset$, and ${\alpha^*}\in{\mathcal{I}}(X-(A\cup\{c\}))$. If ${\alpha^*}\in F_0$, then clearly ${\alpha^*}\in{\mathcal{I}}(X-(A\cup B\cup\{c\}))$. Thus, $S_c^*=F_A\cup F_B\cup F_0$ and the sets $F_A$, $F_B$, and $F_0$ are pairwise disjoint.
By Lemma \[lsccom\], $F_A$, $F_B$, and $F_0$ are sets of commuting transformations (as subsets of $S_c^*$). Let $F$ be any subset of $S_c^*$ and denote by $\langle F\rangle$ the semigroup generated by $F$. Then $\langle F\rangle$ is clearly commutative. Suppose to the contrary that $\langle F\rangle$ is not a nilpotent semigroup. Then it contains a nonzero idempotent, say ${\varepsilon}={\alpha^*}_1\cdots{\alpha^*}_k$, where ${\alpha^*}_i\in F$. Let $x\in X$ be any element fixed by ${\varepsilon}$. Then $x({\alpha^*}_1\cdots{\alpha^*}_k)=x$, and so $x({\alpha}_1\cdots{\alpha}_k)=x$ since each ${\alpha^*}_i$ is a restriction of ${\alpha}_i$. But this is a contradiction since ${\alpha}_1\cdots{\alpha}_k$ is a nilpotent as an element of $S$. Thus $\langle F\rangle$ is a nilpotent semigroup.
Hence, by Proposition \[pcase1\] and the inductive hypothesis applied to $\langle F_A\cup F_0\rangle\subseteq {\mathcal{I}}(X-(B\cup\{c\}))$, $\langle F_B\cup F_0\rangle\subseteq {\mathcal{I}}(X-(A\cup\{c\}))$, and $\langle F_0\rangle\subseteq{\mathcal{I}}(X-(A\cup B\cup\{c\}))$), we have $$\label{e1lbound}
|F_A|+|F_0|\leq{\lambda}_{n-t-1},\,\,|F_B|+|F_0|\leq{\lambda}_{n-p-1},\,\,|F_0|\leq{\lambda}_{n-p-t-1}.$$
Suppose ${\alpha^*}\in F_A$. Then $c\in\operatorname{im}({\alpha})-\operatorname{dom}({\alpha})$, and so $a{\alpha}=c$ for some $a\in X$. Note that $a\in A$. Fix $a_0\in\operatorname{span}({\alpha^*})\cap A$. Suppose to the contrary that $a_0=a$. Then $a_0\notin\operatorname{dom}({\alpha^*})$ since $a_0=a=c{\alpha}^{-1}\in U_{\alpha}$ and ${\alpha^*}={\alpha}|_{X-U_{\alpha}}$. Hence $a_0\in\operatorname{im}({\alpha^*})$, that is, $x{\alpha^*}=a_0=a$ for some $x\in\operatorname{dom}({\alpha^*})$. But this is a contradiction since $x=a{\alpha}^{-1}\in U_{\alpha}$, and so $x\notin\operatorname{dom}({\alpha^*})$. We have proved that $a_0\ne a$. Suppose ${\alpha^*}={\beta^*}$. By the foregoing argument, there is $a'\in A$ such that $a'{\beta}=c$ and $a'\ne a_0$. Moreover, if $a=a'$, then ${\alpha}={\beta}$ by Lemma \[lpre\].
It follows that any ${\alpha^*}\in F_A$ has at most $p-1$ preimages under the mapping $^*$ (which correspond to the number of elements from the set $A-\{a_0\}$ that ${\alpha}$ can map to $c$ if ${\alpha^*}\in F_A$). By similar arguments, any ${\alpha}\in F_B$ has at most $t-1$ preimages under $^*$, and any ${\alpha^*}\in F_0$ has at most $p+t$ preimages under $^*$. These considerations about the number of preimages that an element of $S_c^*$ can have, together with (\[e1lbound\]), give $$\begin{aligned}
|S_c|&\leq(p-1)|F_A|+(t-1)|F_B|+(p+t)|F_0|\notag\\
&=(p-1)(|F_A|+|F_0|)+(t-1)(|F_B|+|F_0|)+2|F_0|\notag\\
&\leq (p-1){\lambda}_{n-t-1}+(t-1){\lambda}_{n-p-1}+2{\lambda}_{n-p-t-1},\notag\end{aligned}$$ which completes the proof.
The following proposition will finish our inductive proof of (\[eqsl1\]). The proposition is stronger than what we need in this section, but we will also use it in the proof of the general case.
\[pcase2\] Let $X$ be a set with $n\geq4$. Let $S$ be a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ with $C\ne\emptyset$. Then:
- If $n\leq7$, then $|S|<{\lambda}_n$.
- If $n\geq8$, then $|S|<{\lambda}_n-n$.
We have checked that (1) is true by direct calculations using GRAPE [@So06], which is a package for GAP [@Scel92]. For $n\in\{4,5,6,7\}$, we have calculated the orders of the maximal commutative nilpotent semigroups and the number of semigroups of each order. The following table contains the maximum order of a commutative nilpotent semigroup (row 2) and the number of commutative nilpotent semigroups of the maximum order.
$$\begin{tabular}{|c|c|c|c|c|}\hline
$n$&4&5&6&7\\\hline
Max order&7&13&34&73\\\hline
No of sgps of max order&6&20&20&70\\\hline
\end{tabular}$$ The numbers in the second row of the table are ${\lambda}_4$, ${\lambda}_5$, ${\lambda}_6$, and ${\lambda}_7$ (see the table in Lemma \[lcom4\]). The numbers in the third row are $\binom{4}{2}$, $2\binom{5}{2}$, $\binom{6}{3}$, and $2\binom{7}{3}$. This means that the commutative nilpotent semigroups of the maximum order are the balanced null semigroups $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ since, for $m={\left\lfloor\frac{n}{2}\right\rfloor}$, there are $\binom{n}{m}$ such semigroups if $n$ is even, and $2\binom{n}{m}$ such semigroups if $n$ is odd (see the proof of Theorem \[tgen\]). Since $C=\emptyset$ for each semigroup $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ (balanced or not), (1) follows.
To prove (2), suppose $n\geq8$. Let $c\in C$ be an element that satisfies one of the conditions (1)–(3) from Lemma \[lbou\]. Let $p=|A_c|$ and $t=|B_c|$. By (\[eqsl3\]) and Lemma \[lbound\], $$\label{e1}
|S|=|S-S_c|+|S_c|\leq{\lambda}_{n-1}+(p-1){\lambda}_{n-t-1}+(t-1){\lambda}_{n-p-1}+2{\lambda}_{n-p-t-1}.$$ We consider four possible cases. [**Case 1.**]{} $p\geq2$ and $t\geq2$. By (\[e1\]), $$\begin{aligned}
|S|&\leq{\lambda}_{n-1}+(p-1){\lambda}_{n-t-1}+(t-1){\lambda}_{n-p-1}+ 2{\lambda}_{n-p-t-1}\notag\\
\label{e11}&\leq{\lambda}_{n-1}+(p-1){\lambda}_{n-3}+(t-1){\lambda}_{n-3}+2{\lambda}_{n-5}\\
\label{e21}&\leq{\lambda}_{n-1}+(n-3){\lambda}_{n-3}+2{\lambda}_{n-5},\end{aligned}$$ where (\[e11\]) follows from $n-t-1,n-p-1\leq n-3$ and $n-p-t-1\leq n-5$, and (\[e21\]) from $p+t\leq n-1$ (so $p+t-2\leq n-3$). For $n=8$ and $n=9$, ${\lambda}_{n-1}+(n-3){\lambda}_{n-3}+2{\lambda}_{n-5}<{\lambda}_n-n$ by direct calculations: $$\begin{tabular}{|c|c|c|}\hline
$n$&8&9\\\hline
${\lambda}_n-n$&201&492\\\hline
${\lambda}_{n-1}+(n-3){\lambda}_{n-3}+2{\lambda}_{n-5}$&144&427\\\hline
\end{tabular}$$ For $n\geq10$, ${\lambda}_{n-5}>n$ (see the table in Lemma \[lcom4\]), and so $$\begin{aligned}
|S|&\leq{\lambda}_{n-1}+(n-3){\lambda}_{n-3}+2{\lambda}_{n-5}\notag\\
\label{e2a1}&<{\lambda}_{n-1}+(n-3){\lambda}_{n-3}+3{\lambda}_{n-5}-n\\
\label{e31}&<{\lambda}_{n-1}+(n-3){\lambda}_{n-3}+\mbox{$\frac{3}{4}$}{\lambda}_{n-3}-n\\
&<{\lambda}_{n-1}+(n-2){\lambda}_{n-3}-n\notag\\
\label{e41}&\leq {\lambda}_{n-1}+\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}$}{\lambda}_{n-2}-n\\
\label{e51}&={\lambda}_n-n,\end{aligned}$$ where (\[e2a1\]) follows from ${\lambda}_{n-5}>n$ when $n\geq10$ (see Lemma \[lcom4\] and the table in its proof), (\[e31\]) from ${\lambda}_{n-3}>4{\lambda}_{n-5}$ when $n\geq10$ (see Lemma \[lcom4\] and the table in its proof), (\[e41\]) from ${\lambda}_{n-2}>2{\lambda}_{n-3}>\frac{n-2}{{\left\lfloor\frac{n}{2}\right\rfloor}}{\lambda}_{n-3}$ when $n\geq8$ (see Lemma \[lcom4\]), and (\[e51\]) from Lemma \[lcom1\]. [**Case 2.**]{} $p=1$ and $t=1$. Then, by (\[e1\]), $$\begin{aligned}
|S|&\leq{\lambda}_{n-1}+2{\lambda}_{n-3}\notag\\
\label{e12}&<{\lambda}_{n-1}+3{\lambda}_{n-3}-n\\
\label{e22}&<{\lambda}_{n-1}+\mbox{$\frac{3}{2}$}{\lambda}_{n-2}-n\\
&<{\lambda}_{n-1}+\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}$}{\lambda}_{n-2}-n\notag\\
\label{e32}&={\lambda}_n-n,\end{aligned}$$ where (\[e12\]) follows from ${\lambda}_{n-3}>n$ when $n\geq8$, (\[e22\]) from ${\lambda}_{n-2}>2{\lambda}_{n-3}$ when $n\geq8$ (see Lemma \[lcom4\]), and (\[e32\]) from Lemma \[lcom1\]. [**Case 3.**]{} $p=1$ and $2 \le t\leq{\left\lfloor\frac{n}{2}\right\rfloor}$. Again by (\[e1\]), $$\begin{aligned}
|S|&\leq{\lambda}_{n-1}+(t-1){\lambda}_{n-2}+2{\lambda}_{n-t-2}\notag\\
&\leq{\lambda}_{n-1}+(\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}-1$}){\lambda}_{n-2}+2{\lambda}_{n-4}\notag\\
\label{e13}&\leq{\lambda}_{n-1}+(\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}-1$}){\lambda}_{n-2}+3{\lambda}_{n-4}-n+1\\
\label{e23}&<{\lambda}_{n-1}+(\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}-1$}){\lambda}_{n-2}+\mbox{$\frac34$}{\lambda}_{n-2}-n+1\\
&={\lambda}_{n-1}+\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}$}{\lambda}_{n-2}-\mbox{$\frac14$}{\lambda}_{n-2}-n+1\notag\\
\label{e33}&<{\lambda}_{n-1}+\mbox{${\left\lfloor\frac{n}{2}\right\rfloor}$}{\lambda}_{n-2}-n\\
\label{e43}&={\lambda}_n-n,\end{aligned}$$ where (\[e13\]) follows from ${\lambda}_{n-4}\geq n-1$ when $n\geq8$ (see Lemma \[lcom4\] and the table in its proof), (\[e23\]) from ${\lambda}_{n-2}>4{\lambda}_{n-4}$ when $n\geq8$ (see Lemma \[lcom4\] and the table in its proof), (\[e33\]) from $\frac14{\lambda}_{n-2}>1$ for $n\geq6$, and (\[e43\]) from Lemma \[lcom1\]. [**Case 4.**]{} $2\le p\leq{\left\lfloor\frac{n}{2}\right\rfloor}$ and $t=1$. This case is symmetric to Case 3.
The inductive proof of (\[eqsl1\]) is complete. As a bonus, we have Proposition \[pcase2\]. We can now prove the main theorem of this section.
\[tnil\] Let $X$ be a set with $n\geq1$ elements and let $m={\left\lfloor\frac{n}{2}\right\rfloor}$. Then:
- The maximum cardinality of a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$ is $${\lambda}_n=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!.$$
- If $n\notin\{1,3\}$, then the only commutative nilpotent subsemigroups of ${\mathcal{I}}(X)$ of order ${\lambda}_n$ are the balanced null semigroups $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$.
Let $S$ be a commutative nilpotent subsemigroup of ${\mathcal{I}}(X)$. If $n=1$, then $S=\{0\}$, so $|S|=1={\lambda}_1$. Let $n\geq2$. If $S=S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ is a balanced null semigroup, then $|S|={\lambda}_n$ by Lemma \[lmax\]. Suppose $S$ is not one of the balanced null semigroups $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$. If $C=\emptyset$, then $|S|<{\lambda}_n$ by Proposition \[pcase1\]. Suppose $C\ne\emptyset$. If $n=3$, then $S=\langle[i\,j\,k]\rangle=\{0,[i\,j\,k],[i\,k]\}$, where $i,j,k$ are pairwise distinct elements of $X$, so $|S|=3={\lambda}_3$. If $n\geq4$, then $|S|<{\lambda}_n$ by Proposition \[pcase2\]. The result follows.
The Largest Commutative Semigroups in ${\mathcal{I}}(X)$ {#sgen}
========================================================
In this section, we determine the maximum order of a commutative subsemigroup of ${\mathcal{I}}(X)$, and describe the commutative subsemigroups of ${\mathcal{I}}(X)$ of the maximum order (Theorem \[tgen\]).
\[leix1\] Let $X$ be a set with $n<10$ elements. Suppose $S$ is a commutative subsemigroup of ${\mathcal{I}}(X)$ such that $S\ne E({\mathcal{I}}(X))$, where $E({\mathcal{I}}(X))$ is the semilattice of idempotents of ${\mathcal{I}}(X)$. Then $|S|<2^n$.
The lemma is vacuously true when $n=1$. It is also true when $n=2$ since then the only maximal commutative subsemigroups of ${\mathcal{I}}(X)$ other than $E({\mathcal{I}}(X))$ are $\operatorname{Sym}(X)\cup\{0\}$ and $\{0,1,[i\,j]\}$, where $i$ and $j$ are distinct elements of $X$. Let $n\geq3$ and suppose, as the inductive hypothesis, that the result is true whenever $|X|<n$. Let $G=S\cap\operatorname{Sym}(X)$ and $T=S-G$.
Suppose $G$ is a semiregular subgroup of $\operatorname{Sym}(X)$ and $T$ is a nilpotent semigroup. Then $|G|\leq n$ (since $G$ is semiregular) and $|T|\leq{\lambda}_n$ (by Theorem \[tnil\]). Thus $|S|\leq {\lambda}_n+n<2^n$, where the latter inequality follows from the table below. $$\begin{tabular}{|c|c|c|c|c|c|c|c|}\hline
$n$&3&4&5&6&7&8&9\\\hline
${\lambda}_n+n$&6&11&18&40&80&217&510\\\hline
$2^n$&8&16&32&64&128&256&512\\\hline
\end{tabular}$$
Suppose $G$ is not a semiregular group or $T$ is not a nilpotent semigroup. Then, by Lemmas \[lres1\] and \[lres3\], there is a partition $\{A,B\}$ of $X$ such that $S\cong S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$, where $S_{\!{\mbox{\tiny $A$}}}$ is a commutative subsemigroup of ${\mathcal{I}}(A)$ and $S_{\!{\mbox{\tiny $B$}}}$ is a commutative subsemigroup of ${\mathcal{I}}(B)$. If $S\subseteq E({\mathcal{I}}(X))$, then $|S|<|E({\mathcal{I}}(X))|=2^n$. Suppose $S$ is not included in $E({\mathcal{I}}(X))$. Then at least one of $S_{\!{\mbox{\tiny $A$}}}$ and $S_{\!{\mbox{\tiny $B$}}}$, say $S_{\!{\mbox{\tiny $A$}}}$, must contain an element that is not an idempotent. Let $k=|S_{\!{\mbox{\tiny $A$}}}|$. By the inductive hypothesis, $|S_{\!{\mbox{\tiny $A$}}}|<2^k$ and $|S_{\!{\mbox{\tiny $B$}}}|\leq2^{n-k}$, and so $|S|=| S_{\!{\mbox{\tiny $A$}}}|\cdot| S_{\!{\mbox{\tiny $B$}}}|<2^k\cdot2^{n-k}=2^n$.
\[leix2\] Let $n=|X|\geq5$. Suppose $S=G\cup T$ is a commutative subsemigroup of ${\mathcal{I}}(X)$ such that $G$ is a nontrivial semiregular subgroup of $\operatorname{Sym}(X)$ and $T$ is a subsemigroup of $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$, where $\{A,B\}$ is a partition of $X$. Then $|S|<{\lambda}_n+1$.
Let $k=|A|$, so $|B|=n-k$. We have $|G|\leq n$ (since $G$ is semiregular) and $|T|\leq|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|\leq{\lambda}_n$ (by Proposition \[pcase1\]). If $k=1$ or $k=n-1$, then $|T|\leq |S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|=n$, and so $|S|\leq n+n=2n<{\lambda}_n+1$ since $n\geq5$ (see the table in Lemma \[lcom4\]).
Suppose $1<k<n-1$. The semigroup $S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$ contains $|A|\cdot|B|=k(n-k)$ nilpotents $[x\,y]$. Let ${\sigma}$ be a nontrivial element of $G$. Then no nilpotent $[x\,y]$ commutes with ${\sigma}$ (by Proposition \[pcen\]), and so such a nilpotent cannot be in $T$. Thus $|T|\leq|S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}|-k(n-k)\leq{\lambda}_n-k(n-k)$. But, since $n\geq5$ and $1<k<n-1$, we have $k(n-k)\geq n$ by elementary algebra, and so $$|S|=|G|+|T|\leq n+{\lambda}_n-k(n-k)\leq n+{\lambda}_n-n={\lambda}_n<{\lambda}_n+1.$$
We can now prove the main theorem of the paper regarding largest commutative subsemigroups of ${\mathcal{I}}(X)$.
\[tgen\] Let $X$ be a set with $n\geq1$ elements and let $m={\left\lfloor\frac{n}{2}\right\rfloor}$. Then:
- If $n<10$, then the maximum cardinality of a commutative subsemigroup of ${\mathcal{I}}(X)$ is $2^n$, and the semilattice $E({\mathcal{I}}(X))$ is the unique commutative subsemigroup of ${\mathcal{I}}(X)$ of order $2^n$.
- Suppose $n\geq10$. Then the maximum cardinality of a commutative subsemigroup of ${\mathcal{I}}(X)$ is $${\lambda}_n+1=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!+1.$$
- If $n$ is even, then there are exactly $\binom{n}{m}$, pairwise isomorphic, commutative subsemigroups of ${\mathcal{I}}(X)$ of order ${\lambda}_n+1$, namely the balanced null monoids $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}\cup\{1\}$.
- If $n$ is odd, then there are exactly $2\binom{n}{m}$, pairwise isomorphic, commutative subsemigroups of ${\mathcal{I}}(X)$ of order ${\lambda}_n+1$, namely the balanced null monoids $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}\cup\{1\}$.
Statement (1) follows immediately from Lemma \[leix1\] and the fact that if $|X|=n$, then $|E({\mathcal{I}}(X))|=2^n$.
To prove (2), suppose $n\geq10$. Each of the balanced null monoids $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}\cup\{1\}$ has order ${\lambda}_n+1$ by Lemma \[lmax\]. If $n$ is even, then $|K|=|L|=m$, and so there are $\binom{n}{m}$ balanced null semigroups $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}$ (since we have $\binom{n}{m}$ choices for $K$ and $L=X-K$ is determined when $K$ has been selected). If $n$ is odd, then the number doubles since we have $\binom{n}{m}$ such semigroups when $|K|=m$ and another $\binom{n}{m}$ when $|K|=n-m$.
Let $S$ be a commutative subsemigroup of ${\mathcal{I}}(X)$ that is different from the balanced null monoids $S_{\!{\mbox{\tiny $K$}}\!,{\mbox{\tiny $L$}}}\cup\{1\}$. Our objective is to prove that $$\label{eq1}
|S|<{\lambda}_n+1=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!+1.$$ Proceeding by induction on $n=|X|$, we suppose that the statement is true for every $X$ with $10\leq|X|<n$. Let $G=S\cap\operatorname{Sym}(X)$ and $T=S-G$.
Suppose $G$ is a semiregular subgroup of $\operatorname{Sym}(X)$ and $T$ is a nilpotent semigroup. If $G$ is trivial, then $T$ is not a balanced null semigroup, and hence $|S| < {\lambda}_n +1$ by Theorem \[tnil\]. So assume that $G \ne \{1\}$. Let $$C=\{c\in X:\mbox{$c\in\operatorname{dom}({\alpha})\cap\operatorname{im}({\beta})$ for some ${\alpha},{\beta}\in T$}\}.$$ If $C=\emptyset$, then $T\subseteq S_{\!{\mbox{\tiny $A$}},{\mbox{\tiny $B$}}}$, where $\{A,B\}$ is a partition of $X$, and so $|S|<{\lambda}_n+1$ by Lemma \[leix2\]. Suppose $C\ne\emptyset$. Then $|G|\leq n$ (since $G$ is semiregular) and $|T|<{\lambda}_n-n$ (by Proposition \[pcase2\]). Thus $|S|=|G|+|T|<n+{\lambda}_n-n={\lambda}_n<{\lambda}_n+1$.
Suppose $G$ is not a semiregular subgroup of $\operatorname{Sym}(X)$ or $T$ is a not a nilpotent semigroup. Then, by Lemmas \[lres1\] and \[lres3\], there is a partition $\{A,B\}$ of $X$ such that $S\cong S_{\!{\mbox{\tiny $A$}}}\times S_{\!{\mbox{\tiny $B$}}}$, where $S_{\!{\mbox{\tiny $A$}}}$ is a commutative subsemigroup of ${\mathcal{I}}(A)$ and $S_{\!{\mbox{\tiny $B$}}}$ is a commutative subsemigroup of ${\mathcal{I}}(B)$. Notice that $1\leq|A|,|B|<|X|=n$. We may assume that $|A|\leq|B|$. Let $k=|A|$. Then $1\leq k<n$ and $|B|=n-k$. We consider three possible cases. [**Case 1.**]{} $k<10$ and $n-k<10$. Then, by Lemma \[leix1\], $|S_{\!{\mbox{\tiny $A$}}}|\leq2^k$ and $|S_{\!{\mbox{\tiny $B$}}}|\leq2^{n-k}$, and so $$|S|=|S_{\!{\mbox{\tiny $A$}}}|\cdot|S_{\!{\mbox{\tiny $B$}}}|\leq2^k\cdot2^{n-k}=2^n<{\lambda}_n+1,$$ where the last inequality is true since $n\geq10$ (see Lemma \[lcom4\] and the table in its proof). [**Case 2.**]{} $k<10$ and $n-k\geq10$. Then, $|S_{\!{\mbox{\tiny $A$}}}|\leq2^k$ (by Lemma \[leix1\]) and $|S_{\!{\mbox{\tiny $B$}}}|\leq{\lambda}_{n-k}+1$ (by Theorem \[tnil\] and the inductive hypothesis). Thus, by (1) of Lemma \[lcom3\], $$|S|=|S_{\!{\mbox{\tiny $A$}}}|\cdot|S_{\!{\mbox{\tiny $B$}}}|\leq2^k({\lambda}_{n-k}+1)<{\lambda}_n+1.$$ [**Case 3.**]{} $k\geq10$ and $n-k\geq10$. Then, by Theorem \[tnil\] and the inductive hypothesis, $|S_{\!{\mbox{\tiny $A$}}}|\leq{\lambda}_k+1$ and $|S_{\!{\mbox{\tiny $B$}}}|\leq{\lambda}_{n-k}+1$. Thus, by (2) of Lemma \[lcom3\], $$|S|=|S_{\!{\mbox{\tiny $A$}}}|\cdot|S_{\!{\mbox{\tiny $B$}}}|\leq({\lambda}_k+1)({\lambda}_{n-k}+1)<{\lambda}_n+1.$$ Hence, in all cases, $|S|<\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!+1$, which concludes the proof of (2).
It follows from Theorem \[tgen\] that every symmetric inverse semigroup ${\mathcal{I}}(X)$ has, up to isomorphism, a unique commutative subsemigroup of maximum order. In comparison, the symmetric group $\operatorname{Sym}(X)$ has, up to isomorphism, a unique abelian subgroup of maximum order if $|X|=3k$ or $|X|=3k+2$, and two abelian subgroups of maximum order if $|X|=3k+1$ [@BuGo89 Theorem 1].
The Clique Number and Diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ {#scdg}
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In this section, we determine the clique number of the commuting graph of ${\mathcal{I}}(X)$ and the diameter of the commuting graph of every nonzero ideal of ${\mathcal{I}}(X)$. The exception is the case of ${\mathcal{G}}({\mathcal{I}}(X))$ when $n=|X|$ is odd and composite, and not a prime power, where we are only able to say that the diameter is either $4$ or $5$.
Let $\Gamma$ be a simple graph, that is, $\Gamma=(V,E)$, where $V$ is a finite non-empty set of vertices and $E\subseteq\{\{u,v\}:u,v\in V, u\ne v\}$ is a set of edges. We will write $u-v$ to mean that $\{u,v\}\in E$. (If ${\mathcal{G}}(S)$ is the commuting graph of a semigroup $S$, then for all vertices $a$ and $b$ of ${\mathcal{G}}(S)$, $a-b$ if and only if $a\ne b$ and $ab=ba$.)
A subset $K$ of $V$ is called a *clique* in $\Gamma$ if $u-v$ for all distinct $u,v\in K$. The *clique number* of $\Gamma$ is the largest integer $r$ such that $\Gamma$ has a clique $K$ with $|K|=r$.
Let $u,w\in V$. A *path* in $\Gamma$ of length $m-1$ ($m\geq1$) from $u$ to $w$ is a sequence of pairwise distinct vertices $u=v_1,v_2,\ldots,v_m=w$ such that $v_i-v_{i+1}$ for every $i\in\{1,\ldots,m-1\}$. The *distance* between vertices $u$ and $w$, denoted $d(u,w)$, is the smallest integer $k\geq0$ such that there is a path of length $k$ from $u$ to $w$. If there is no path from $u$ to $w$, we say that the distance between $u$ and $w$ is infinity, and write $d(u,w)=\infty$. The maximum distance $\max\{d(u,w):u,w\in V\}$ between vertices of $\Gamma$ is called the *diameter* of $\Gamma$. Note that the diameter of $\Gamma$ is finite if and only if $\Gamma$ is connected.
It follows easily from Proposition \[pcen\] that the only central elements of ${\mathcal{I}}(X)$ are the zero and identity transformations. Therefore, the following result is an immediate corollary of Theorem \[tgen\]. (Note that if $|X|=1$, then ${\mathcal{I}}(X)$ is a commutative semigroup.)
\[ccli\] Let $X$ be a set with $n\geq2$ elements and let $m={\left\lfloor\frac{n}{2}\right\rfloor}$. Then:
- If $n<10$, then the clique number of the commuting graph of ${\mathcal{I}}(X)$ is $2^n-2$.
- If $n\geq10$, then the clique number of the commuting graph of ${\mathcal{I}}(X)$ is $${\lambda}_n-1=\sum_{r=0}^m\binom{m}{r}\binom{n-m}{r}r!-1.$$
It is well known (see [@Ho95 Exercises 5.11.2 and 5.11.4]) that ${\mathcal{I}}(X)$ has exactly $n+1$ ideals, $J_0,J_1,\ldots,J_n$, where $$J_r=\{{\alpha}\in{\mathcal{I}}(X):\operatorname{rank}({\alpha})\leq r\}$$ for $0\leq r\leq n$. Each ideal $J_r$ is principal and any ${\alpha}\in{\mathcal{I}}(X)$ of rank $r$ generates $J_r$. The ideal $J_0=\{0\}$ consists of the zero transformation. Our next objective is to find the diameter of the commuting graph of every proper nonzero ideal ${\mathcal{I}}(X)$.
\[ldia1\] Let $n\geq2$. Suppose ${\alpha}\in{\mathcal{I}}(X)-\{0,1\}$ is not an $n$-cycle or a nilpotent of index $n$. Then there exists an idempotent ${\varepsilon}\in{\mathcal{I}}(X)-\{0,1\}$ such that $\operatorname{rank}({\varepsilon})\leq\operatorname{rank}({\alpha})$ and ${\alpha}{\varepsilon}={\varepsilon}{\alpha}$.
Let ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k{\sqcup}\tau_1{\sqcup}\cdots{\sqcup}\tau_m$ be the decomposition of ${\alpha}$ as in Proposition \[pdec\]. Suppose $k,m\geq1$ (that is, ${\alpha}$ contains at least one cycle and at least one chain). Then there is an integer $p\geq1$ such that ${\varepsilon}={\alpha}^p$ is an idempotent different from $0$ and $1$. Clearly, ${\alpha}{\varepsilon}={\varepsilon}{\alpha}$.
Suppose $k=0$. Since ${\alpha}$ is not a nilpotent of index $n$, $\operatorname{span}(\tau_1)\ne X$. Let ${\varepsilon}$ be the idempotent with $\operatorname{dom}({\varepsilon})=\operatorname{span}(\tau_1)$. Then ${\varepsilon}\ne1$ (since $\operatorname{span}(\tau_1)\ne X$), ${\varepsilon}\ne0$ (since $\operatorname{span}(\tau_1)\ne\emptyset$), and ${\alpha}{\varepsilon}={\varepsilon}{\alpha}$ by Proposition \[pcen\]. Suppose $m=0$. Then $k\geq2$ since ${\alpha}$ is not an $n$-cycle. Then ${\alpha}{\varepsilon}={\varepsilon}{\alpha}$ for the idempotent ${\varepsilon}$ with $\operatorname{dom}({\varepsilon})=\operatorname{span}(\rho_1)$.
Note that in all cases, $\operatorname{rank}({\varepsilon})\leq\operatorname{rank}({\alpha})$.
\[ldia2\] Let $n\geq4$. Suppose ${\alpha},{\beta}\in{\mathcal{I}}(X)-\{0,1\}$ such that neither ${\alpha}$ nor ${\beta}$ is an $n$-cycle. Then in the commuting graph ${\mathcal{G}}({\mathcal{I}}(X))$, there is a path from ${\alpha}$ to ${\beta}$ of length at most $4$ such that all vertices in the path have rank at most $\max\{\operatorname{rank}({\alpha}),\operatorname{rank}({\beta})\}$.
Suppose neither ${\alpha}$ nor ${\beta}$ is a nilpotent of index $n$. Then, by Lemma \[ldia1\], there are idempotents ${\varepsilon}_1,{\varepsilon}_2\in{\mathcal{I}}(X)-\{0,1\}$ such that $\operatorname{rank}({\varepsilon}_1)\leq\operatorname{rank}({\alpha})$, $\operatorname{rank}({\varepsilon}_2)\leq\operatorname{rank}({\beta})$, ${\alpha}-{\varepsilon}_1$, and ${\varepsilon}_2-{\beta}$. Since idempotents in ${\mathcal{I}}(X)$ commute, ${\alpha}-{\varepsilon}_1-{\varepsilon}_2-{\beta}$.
Suppose ${\alpha}=[y_1\,y_2\ldots\,y_n]$ is a nilpotent of index $n$ and ${\beta}$ is not a nilpotent of index $n$. Let ${\varepsilon}_1$ be the idempotent with $\operatorname{dom}({\varepsilon}_1)=\{y_1,y_n\}$ (note that $\operatorname{rank}({\varepsilon}_1)\leq\operatorname{rank}({\alpha})$) and ${\varepsilon}_2$ be an idempotent different from $0$ and $1$ such that $\operatorname{rank}({\varepsilon}_2)\leq\operatorname{rank}({\beta})$ and ${\varepsilon}_2-{\beta}$ (such an idempotent exists by Lemma \[ldia1\]). Then ${\alpha}-[y_1\,y_n]-{\varepsilon}_1-{\varepsilon}_2-{\beta}$.
Finally, suppose ${\alpha}=[y_1\,y_2\ldots\,y_n]$ and ${\beta}=[x_1\,x_2\ldots\,x_n]$ are nilpotents of index $n$. If $\{y_1,y_n\}\cap\{x_1,x_n\}=\emptyset$, then $[y_1\,y_n]$ and $[x_1\,x_n]$ commute, and so ${\alpha}-[y_1\,y_n]-[x_1\,x_n]-{\beta}$. Suppose $\{y_1,y_n\}\cap\{x_1,x_n\}\ne\emptyset$. Then, since $n\geq4$, there is $z\in X-\{y_1,y_n,x_1,x_n\}$. Let ${\varepsilon}$ be the idempotent with $\operatorname{dom}({\varepsilon})=\{z\}$. Then, by Proposition \[pcen\], ${\alpha}-[y_1\,y_n]-{\varepsilon}-[x_1\,x_n]-{\beta}$.
\[ldia4\] Let $n\geq2$. Suppose ${\alpha},{\beta}\in{\mathcal{I}}(X)-\{0,1\}$ with ${\alpha}{\beta}={\beta}{\alpha}$. Then
- If ${\alpha}=[x_1\ldots\,x_n]$ is a nilpotent of index $n$, then there is $q\in\{1,\ldots,n-1\}$ such that ${\beta}={\alpha}^q$.
- If ${\alpha}=(x_0\,x_1\ldots\,x_{n-1})$ is an $n$-cycle, then there is $q\in\{1,\ldots,n-1\}$ such that ${\beta}={\alpha}^q$.
Suppose ${\alpha}=[x_1\ldots\,x_n]$. Since ${\beta}\notin\{0,1\}$, it follows by Proposition \[pcen\] that there is $t\in\{1,\ldots,n-1\}$ such that $\operatorname{dom}({\beta})\cap\{x_1,\ldots,x_n\}=\{x_1,\ldots,x_t\}$ and $$x_1{\beta}=x_{n-t+1},\,\,x_2{\beta}=x_{n-t+2},\ldots,\,\,x_t{\beta}=x_n.$$ Thus ${\beta}={\alpha}^q$, where $q=n-t$, and $q\notin\{0,n\}$ (since $1\leq t\leq n-1$). We have proved (1).
Suppose ${\alpha}=(x_0\,x_1\ldots\,x_{n-1})$. Since ${\beta}\ne0$, $\{x_0,x_1,\ldots,x_{n-1}\}\subseteq\operatorname{dom}({\beta})$ by Proposition \[pcen\]. Let $x_q=x_0{\beta}$, where $q\in\{0,1,\ldots,n-1\}$, and note that $q\ne0$ since ${\alpha}\ne1$. Then, by Proposition \[pcen\], $x_i{\beta}=x_{q+i}$ for every $i\in\{0,\ldots,n-1\}$ (where $x_{q+i}=x_{q+i-n}$ if $q+i\geq n$). Thus ${\beta}={\alpha}^q$. We have proved (2).
\[ldia5\] Let $n\geq3$. Then there are nilpotents ${\alpha},{\beta}\in{\mathcal{I}}(X)$ of index $n$ such that $d({\alpha},{\beta})=4$.
Let ${\alpha}=[x_1\,x_2\ldots\,x_k\,y_1\,y_2\ldots\,y_m]$ and ${\beta}=[y_m\ldots\,y_2\,y_1\,x_k\ldots\,x_2\,x_1]$, where $k+m=n$ and $k=\lceil\frac{n}{2}\rceil$. If $n\geq4$, then $d({\alpha},{\beta})\leq4$ by Lemma \[ldia2\]. If $n=3$, then ${\alpha}=[x_1\,x_2\,y_1]-[x_1\,y_1]-{\varepsilon}-[y_1\,x_1]-[y_1\,x_2\,x_1]={\beta}$, where ${\varepsilon}$ is the idempotent with $\operatorname{dom}({\varepsilon})=\{x_2\}$, so $d({\alpha},{\beta})\leq4$.
Note that ${\alpha}$ and ${\beta}$ do not commute, so $d({\alpha},{\beta})\geq2$. Suppose ${\alpha}-{\gamma}-{\delta}-{\beta}$ is a path from ${\alpha}$ to ${\beta}$ of length $3$. By Lemma \[ldia4\], ${\gamma}={\alpha}^p$ and ${\delta}={\beta}^q$ for some $p,q\in\{1,\ldots,n-1\}$. We may assume that $p\geq k$. (If not, then there exists an integer $t$ such that $k\leq pt\leq n-1$, and so ${\alpha}^p$ can be replaced with ${\alpha}^{pt}=({\alpha}^p)^t$ in the path.) Similarly, we may assume that $q\geq k$. Then $${\alpha}^p=[x_1\,y_i]{\sqcup}[x_2\,y_{i+1}]{\sqcup}\cdots{\sqcup}[x_{m-i+1}\,y_m]\mbox{ and }
{\beta}^q=[y_m\,x_j]{\sqcup}[y_{m-1}\,x_{j-1}]{\sqcup}\cdots{\sqcup}[y_{m-j+1}\,x_1],$$ for some $i\in\{1,\ldots,m\}$ and $j\in\{1,\ldots,k\}$ (with $j\in\{1,\ldots,k-1\}$ when $n$ is odd). But then ${\alpha}^p$ and ${\beta}^q$ do not commute (since $x_{m-i+1}({\alpha}^p{\beta}^q)=x_j$ and $x_{m-i+1}\notin\operatorname{dom}({\beta}^q{\alpha}^p)$), which is a contradiction.
We have proved that there is no path from ${\alpha}$ to ${\beta}$ of length $3$. But then there is no path from ${\alpha}$ to ${\beta}$ of length $2$ either since any such path would have the form ${\alpha}-{\alpha}^p-{\beta}$ (and then ${\alpha}-{\alpha}^p-{\beta}^2-{\beta}$ would be a path of length $3$) or ${\alpha}-{\beta}^q-{\beta}$ (and then ${\alpha}-{\alpha}^2-{\beta}^q-{\beta}$ would be a path of length $3$). It follows that $d({\alpha},{\beta})=4$.
\[ldia6\] Let $n\geq3$ and ${\left\lfloor\frac{n-1}{2}\right\rfloor}<r<n-1$. Then there are ${\alpha},{\beta}\in J_r$ such that for every nonzero ${\gamma}\in{\mathcal{I}}(X)$, if ${\alpha}-{\gamma}-{\beta}$, then ${\gamma}=1$.
Consider a nilpotent ${\alpha}=[x\,z_1\ldots\,z_{r-1}\,y]$ of rank $r$ (possible since $r<n-1$). Since $r>{\left\lfloor\frac{n-1}{2}\right\rfloor}$, we have $r>\frac{n-1}2$, and so $2r\geq n$. Therefore, there are pairwise distinct elements $w_1,\ldots,w_{r-1}$ of $X$ such that $\{x,y,x_1,\ldots,x_{r-1},w_1,\ldots,w_{r-1}\}=X$. Let ${\beta}=[y\,w_1\ldots\,w_{r-1}\,x]\in J_r$, and suppose $0\ne{\gamma}\in{\mathcal{I}}(X)$ is such that ${\alpha}-{\gamma}-{\beta}$. We want to prove that ${\gamma}=1$.
Since ${\gamma}\ne0$ and $\operatorname{span}({\alpha})\cup\operatorname{span}({\beta})=X$, we have $\operatorname{dom}({\gamma})\cap\operatorname{span}({\alpha})\ne\emptyset$ or $\operatorname{dom}({\gamma})\cap\operatorname{span}({\beta})\ne\emptyset$. We may assume that $\operatorname{dom}({\gamma})\cap\operatorname{span}({\alpha})\ne\emptyset$. Then, since ${\alpha}{\gamma}={\gamma}{\alpha}$, $x\in\operatorname{dom}({\gamma})$ by Proposition \[pcen\]. Since ${\beta}{\gamma}={\gamma}{\beta}$, $x\in\operatorname{dom}({\gamma})$ and Proposition \[pcen\] imply that $\operatorname{span}({\beta})\subseteq\operatorname{dom}({\gamma})$ and ${\gamma}$ maps ${\beta}$ onto a terminal segment of some chain in ${\beta}$. But ${\beta}$ is a single chain, so ${\gamma}$ must map ${\beta}$ onto ${\beta}$, which is only possible if ${\gamma}$ fixes every element of $\operatorname{span}({\beta})$. We now know that $\operatorname{dom}({\gamma})\cap\operatorname{span}({\beta})\ne\emptyset$. By the foregoing argument, with the roles of ${\alpha}$ and ${\beta}$ reversed, we conclude that ${\gamma}$ must also fix every element of $\operatorname{span}({\alpha})$. Hence ${\gamma}=1$.
We can now determine the diameter of ${\mathcal{G}}(J_r)$ for every $r<n$.
\[tpro\] Let $n=|X|\geq3$ and let $J_r$ be a proper nonzero ideal of ${\mathcal{I}}(X)$. Then:
- The diameter of ${\mathcal{G}}(J_{n-1})$ is $4$.
- If ${\left\lfloor\frac{n-1}{2}\right\rfloor}<r<n-1$, then the diameter of ${\mathcal{G}}(J_r)$ is $3$.
- If $1\leq r\leq{\left\lfloor\frac{n-1}{2}\right\rfloor}$, then the diameter of ${\mathcal{G}}(J_r)$ is $2$.
We first note that for every $r\in\{1,\ldots,n-1\}$, the only central element of $J_r$ is $0$.
To prove (1), observe that $J_{n-1}={\mathcal{I}}(X)-\operatorname{Sym}(X)$. The diameter of ${\mathcal{G}}(J_{n-1})$ is at least $4$ by Lemma \[ldia5\]. If $n\geq4$, then it is at most $4$ by Lemma \[ldia2\].
Let $n=3$ and let ${\alpha},{\beta}\in J_{n-1}-\{0\}$. If ${\alpha}$ or ${\beta}$ is not a nilpotent of index $3$, then $d({\alpha},{\beta})\leq4$ by the proof of Lemma \[ldia2\] (where the assumption $n\geq4$ was only used in the case when both ${\alpha}$ and ${\beta}$ were nilpotents of index $n$).
Let ${\alpha}=[x\,y\,z]$ and ${\beta}$ be distinct nilpotents of index $3$. We want to show that $d({\alpha},{\beta})\leq4$. Since ${\alpha}-[x\,z]$, it suffices to show that $d([x\,z],{\beta})\leq3$. If ${\beta}=[x\,z\,y]$, then $[x\,z]-[x\,y]-[x\,z\,y]$; if ${\beta}=[y\,x\,z]$, then $[x\,z]-[y\,z]-[y\,x\,z]$; if ${\beta}=[y\,z\,x]$, then $[x\,z]-[y\,z]-[y\,x]-[y\,z\,x]$; if ${\beta}=[z\,x\,y]$, then $[x\,z]-[x\,y]-[z\,y]-[z\,x\,y]$; finally, if ${\beta}=[z\,y\,x]$, then $[x\,z]-{\varepsilon}-[z\,x]-[z\,y\,x]$, where ${\varepsilon}$ is the idempotent with $\operatorname{dom}({\varepsilon})=\{y\}$. Thus $d({\alpha},{\beta})\leq4$, which concludes the proof of (1).
To prove (2), suppose ${\left\lfloor\frac{n-1}{2}\right\rfloor}<r<n-1$. Then the diameter of ${\mathcal{G}}(J_r)$ is at least $3$ by Lemma \[ldia6\]. Let ${\alpha},{\beta}\in J_r$. Since $r<n-1$, neither ${\alpha}$ nor ${\beta}$ is an $n$-cycle or a nilpotent of index $n$. Thus, by Lemma \[ldia1\], there are idempotents ${\varepsilon}_1,{\varepsilon}_2\in J_r-\{0\}$ such that ${\alpha}{\varepsilon}_1={\varepsilon}_1{\alpha}$ and ${\beta}{\varepsilon}_2={\varepsilon}_2{\beta}$. Since the idempotents in ${\mathcal{I}}(X)$ commute, we have ${\alpha}-{\varepsilon}_1-{\varepsilon}_2-{\beta}$, so the diameter of ${\mathcal{G}}(J_r)$ is at most $3$.
To prove (3), suppose $1\leq r\leq{\left\lfloor\frac{n-1}{2}\right\rfloor}$. Then the diameter of ${\mathcal{G}}(J_r)$ is at least $2$ since for any distinct $x,y\in X$, the nilpotents $[x\,y]$ and $[y\,x]$ (which are in $J_r$ since $r\geq1$) do not commute. Let ${\alpha},{\beta}\in J_r-\{0\}$. We have $r\leq{\left\lfloor\frac{n-1}{2}\right\rfloor}\leq\frac{n-1}2$, and so $2r\leq n-1<n$. Therefore, $$|\operatorname{im}({\alpha})\cup\operatorname{im}({\beta})|\leq|\operatorname{im}({\alpha})|+|\operatorname{im}({\beta})|\leq r+r=2r<n,$$ and so there is $x\in X$ such that $x\notin\operatorname{im}({\alpha})\cup\operatorname{im}({\beta})$. By the same argument, there is $y\in X$ such that $y\notin\operatorname{dom}({\alpha})\cup\operatorname{dom}({\beta})$. If $x=y$, then ${\alpha}-{\varepsilon}-{\beta}$, where ${\varepsilon}$ is the idempotent with $\operatorname{dom}({\varepsilon})=\{x\}$. If $x\ne y$, then ${\alpha}-[x\,y]-{\beta}$. Thus, the diameter of ${\mathcal{G}}(J_r)$ is at most $2$.
We now want to prove that if $n\geq4$ is even, then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $4$.
\[dali\] [ Let ${\gamma},{\delta}\in{\mathcal{I}}(X)$. We say that ${\gamma}$ and ${\delta}$ are *aligned* if there exists an integer $r\geq2$ and pairwise distinct elements $a_1,\ldots,a_r,c_1,\ldots,c_{r-1},b_1$ of $X$ such that $$\begin{aligned}
{\gamma}&=(a_1\,b_1){\sqcup}(a_2\,c_1){\sqcup}(a_3\,c_2){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-2}){\sqcup}(a_r\,c_{r-1}),\notag\\
{\delta}&=(a_1\,c_1){\sqcup}(a_2\,c_2){\sqcup}(a_3\,c_3){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-1}){\sqcup}(a_r\,b_1).\notag\end{aligned}$$ ]{}
The following lemma follows immediately from Definition \[dali\]
\[lali\] Let ${\gamma},{\delta}\in{\mathcal{I}}(X)$ be aligned. Then, with the notation from [Definition \[dali\]]{}, $${\gamma}-(a_1\,a_2\ldots\,a_r){\sqcup}(b_1\,c_1\ldots\,c_{r-1})-{\delta}.$$
\[lncy\] Let $n=2k=|X|\geq4$ be even. Suppose ${\alpha},{\beta}\in\operatorname{Sym}(X)$ are joins of $k$ cycles of length $2$ with no cycle in common. Then ${\alpha}={\gamma}{\sqcup}{\alpha}'$ and ${\beta}={\delta}{\sqcup}{\beta}'$, where ${\gamma}$ and ${\delta}$ are aligned.
Select any cycle $(a_1\,b_1)$ in ${\alpha}$. Then ${\beta}$ has a cycle $(a_1\,c_1)$ with $c_1\ne b_1$ (since ${\alpha}$ and ${\beta}$ have no cycle in common). Continuing, ${\alpha}$ must have a cycle $(a_2\,c_1)$, and so ${\beta}$ must have either a cycle $(a_2\,b_1)$ or a cycle $(a_2\,c_2)$ with $c_2\ne b_1$. In the latter case, ${\alpha}$ must have a cycle $(a_3\,c_2)$, and so ${\beta}$ must have a cycle $(a_3\,b_1)$ or a cycle $(a_3\,c_3)$ with $c_3\ne b_1$. This process must terminate after at most $k$, say $r$, steps. That is, at step $r$, we will obtain a cycle $(a_{r}\,c_{r-1})$ in ${\alpha}$ and a cycle $(a_r\,b_1)$ in ${\beta}$. Hence $$\begin{aligned}
{\alpha}&=(a_1\,b_1){\sqcup}(a_2\,c_1){\sqcup}(a_3\,c_2){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-2}){\sqcup}(a_r\,c_{r-1}){\sqcup}{\alpha}',\notag\\
{\beta}&=(a_1\,c_1){\sqcup}(a_2\,c_2){\sqcup}(a_3\,c_3){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-1}){\sqcup}(a_r\,b_1){\sqcup}{\beta}',\notag\end{aligned}$$ where ${\alpha}'={\beta}'=0$ if $r=k$. The proof is completed by the observation that ${\gamma}=(a_1\,b_1){\sqcup}(a_2\,c_1){\sqcup}(a_3\,c_2){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-2}){\sqcup}(a_r\,c_{r-1})$ and ${\delta}=(a_1\,c_1){\sqcup}(a_2\,c_2){\sqcup}(a_3\,c_3){\sqcup}\cdots{\sqcup}(a_{r-1}\,c_{r-1}){\sqcup}(a_r\,b_1)$ are aligned.
\[ldia3\] Let $n\geq6$ be composite. Suppose ${\alpha},{\beta}\in{\mathcal{I}}(X)-\{0,1\}$ such that ${\alpha}$ is an $n$-cycle and ${\beta}$ is not an $n$-cycle. Then $d({\alpha},{\beta})\leq4$.
Suppose ${\beta}$ is not a nilpotent of index $n$. Since $n$ is composite, there is a divisor $k$ of $n$ with $1<k<n$. Then ${\alpha}^k\in{\mathcal{I}}(X)-\{0,1\}$ is not an $n$-cycle. Thus, by Lemma \[ldia1\], there are idempotents ${\varepsilon}_1,{\varepsilon}_2\in{\mathcal{I}}(X)-\{0,1\}$ such that ${\alpha}^k-{\varepsilon}_1$ and ${\varepsilon}_2-{\beta}$. Then ${\alpha}-{\alpha}^k-{\varepsilon}_1-{\varepsilon}_2-{\beta}$, and so $d({\alpha},{\beta})\leq4$.
Suppose ${\beta}=[x_1\,x_2\ldots\,x_n]$ is a nilpotent of index $n$. Let $k$ be the largest proper divisor of $n$. Then ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k$, where each $\rho_i$ is a cycle of length $\frac{n}{k}$. Since $n\geq6$, we have $k>2$. Thus, there exists $t\in\{1,\ldots,k\}$ such that $x_1,x_n\notin\operatorname{span}(\rho_t)$. Let ${\varepsilon}$ be the idempotent with $\operatorname{dom}({\varepsilon})=\operatorname{span}(\rho_t)$. Then ${\varepsilon}\ne0,1$ and, by Proposition \[pcen\], ${\alpha}-{\alpha}^k-{\varepsilon}-[x_1\,x_n]-{\beta}$. Hence $d({\alpha},{\beta})\leq4$.
\[tdie\] Let $n=|X|\geq4$ be even. Then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $4$.
Let ${\alpha},{\beta}\in{\mathcal{I}}(X)-\{0,1\}$. We will prove that $d({\alpha},{\beta})\leq4$. If neither ${\alpha}$ nor ${\beta}$ is an $n$-cycle, then $d({\alpha},{\beta})\leq4$ by Lemma \[ldia2\].
Suppose ${\alpha}$ is an $n$-cycle and ${\beta}$ is not an $n$-cycle. If $n\geq6$, then $d({\alpha},{\beta})\leq4$ by Lemma \[ldia3\]. If $n=4$ and ${\beta}$ is not a nilpotent of index $4$, then $d({\alpha},{\beta})\leq4$ again by Lemma \[ldia3\] (where the assumption $n\geq6$ was only used when ${\beta}$ was a nilpotent of index $n$). Let $n=4$, ${\alpha}=(x\,y\,z\,w)$, and ${\beta}=[a\,b\,c\,d]$. Then ${\alpha}^2=(x\,z){\sqcup}(y\,w)$, ${\beta}^3=[a\,d]$, and so it suffices to find a path of length $2$ from $(x\,z){\sqcup}(y\,w)$ to $[a\,d]$. If $\{a,d\}=\{x,z\}$ or $\{a,d\}=\{y,w\}$, then $(x\,z){\sqcup}(y\,w)-{\varepsilon}-[a\,d]$, where ${\varepsilon}$ is the idempotent with $\operatorname{dom}({\varepsilon})=\{a,d\}$. Otherwise, we may assume that $a=x$ and $d=w$, and then $(x\,z){\sqcup}(y\,w)-[x\,y]{\sqcup}[z\,w]-[x\,w]=[a\,d]$. Hence $d({\alpha},{\beta})\leq4$.
Suppose ${\alpha}$ and ${\beta}$ are $n$-cycles. Then for $k=n/2$, ${\alpha}^k$ and ${\beta}^k$ are joins of $k$ cycles of length $2$. Therefore, it suffices to find a path of length $2$ from ${\alpha}^k$ to ${\beta}^k$. If ${\alpha}^k$ and ${\beta}^k$ have a cycle in common, say $(a\,b)$, then ${\alpha}^k-{\varepsilon}-{\beta}^k$, where ${\varepsilon}$ is the idempotent with $\operatorname{dom}({\varepsilon})=\{a,b\}$.
Suppose ${\alpha}^k$ and ${\beta}^k$ have no common cycle. Then ${\alpha}^k={\gamma}{\sqcup}{\alpha}'$ and ${\beta}^k={\delta}{\sqcup}{\beta}'$, where ${\gamma},{\alpha}',{\delta},{\beta}'$ are as in Lemma \[lncy\]. By Lemma \[lali\], there is $\eta\in{\mathcal{I}}(X)$ such that $\operatorname{span}(\eta)=\operatorname{span}({\gamma})=\operatorname{span}({\delta})$ and ${\gamma}-\eta-{\delta}$. It follows that $${\alpha}^k={\gamma}{\sqcup}{\alpha}'-\eta-{\delta}{\sqcup}{\beta}'={\beta}^k.$$
We have proved that $d({\alpha},{\beta})\leq4$ for all ${\alpha},{\beta}\in{\mathcal{I}}(X)$, which shows that the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is at most $4$. Since the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is at least $4$ by Lemma \[ldia5\], the proof is complete.
Suppose $n=2$, say $X=\{x,y\}$. Then the commuting graph ${\mathcal{G}}({\mathcal{I}}(X))$ has one edge, $(x)-(y)$ (recall that in our notation $(x)$ is the idempotent with domain $\{x\}$), and three isolated vertices, $(x\,y)$, $[x\,y]$, and $[y\,x]$. Hence, the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $\infty$.
The following proposition and Theorem \[tpow\] partially solve the problem of finding the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ when $n$ is odd.
\[pdio\] Let $n=|X|\geq3$ be odd. Then:
- If $n$ is prime, then ${\mathcal{G}}({\mathcal{I}}(X))$ is $\infty$.
- If $n$ is composite, then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is either $4$ or $5$.
Suppose $n=p$ is an odd prime. Consider a $p$-cycle ${\alpha}=(x_0\,x_1\ldots x_{p-1})$ and let ${\beta}\in{\mathcal{I}}(X)-\{0,1\}$ with ${\alpha}{\beta}={\beta}{\alpha}$. By Lemma \[ldia4\], ${\beta}={\alpha}^q$ for some $q\in\{1,\ldots,p-1\}$. Thus, since $p$ is prime, ${\beta}$ is also a $p$-cycle. It follows that if ${\gamma}$ is a vertex of ${\mathcal{G}}({\mathcal{I}}(X))$ that is not a $p$-cycle, then there is no path in ${\mathcal{G}}({\mathcal{I}}(X))$ from ${\alpha}$ to ${\gamma}$. Hence ${\mathcal{G}}({\mathcal{I}}(X))$ is not connected, and so the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $\infty$. We have proved (1).
Suppose $n$ is odd and composite (so $n\geq9$). Let ${\alpha},{\beta}\in{\mathcal{I}}(X)-\{0,1\}$. If ${\alpha}$ or ${\beta}$ is not an $n$-cycle, then $d({\alpha},{\beta})\leq4$ by Lemmas \[ldia2\] and \[ldia3\]. Suppose ${\alpha}$ and ${\beta}$ are $n$-cycles. Let $k$ be a proper divisor of $n$ ($1<k<n$). Then ${\alpha}=\rho_1{\sqcup}\cdots{\sqcup}\rho_k$ and ${\beta}={\sigma}_1{\sqcup}\cdots{\sqcup}{\sigma}_k$, where each $\rho_i$ and each ${\sigma}_i$ is a cycle of length $\frac{n}{k}$. Let ${\varepsilon}_1$ and ${\varepsilon}_2$ be the idempotents with $\operatorname{dom}({\varepsilon}_1)=\operatorname{span}(\rho_1)$ and $\operatorname{dom}({\varepsilon}_2)=\operatorname{span}({\sigma}_1)$. Then, ${\alpha}^k,{\beta}^k\ne1$ (since $k<n$), ${\varepsilon}_1,{\varepsilon}_2\ne1$ (since $k>1$), and ${\alpha}-{\alpha}^k-{\varepsilon}_1-{\varepsilon}_2-{\beta}^k-{\beta}$. Hence $d({\alpha},{\beta})\leq5$, and so the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is at most $5$. On the other hand, the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is at least $4$ by Lemma \[ldia5\]. We have proved (2).
We will now prove that when $n=p^k$ is a power of an odd prime $p$, with $k\geq2$, then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $5$.
\[dhag\] [ Let ${\alpha}=\rho_1{\sqcup}\rho_2{\sqcup}\cdots{\sqcup}\rho_k\in\operatorname{Sym}(X)$ and let ${\gamma}\in{\mathcal{I}}(X)$ with ${\alpha}{\gamma}={\gamma}{\alpha}$. We define a partial transformation ${h_{{\gamma}}^{{\alpha}}}$ on the set $A=\{\rho_1,\rho_2,\ldots,\rho_k\}$ of cycles of ${\alpha}$ by: $$\begin{aligned}
\operatorname{dom}({h_{{\gamma}}^{{\alpha}}})&=\{\rho_i\in A:\operatorname{span}(\rho_i)\cap\operatorname{dom}({\gamma})\ne\emptyset\},\notag\\
\rho_i{h_{{\gamma}}^{{\alpha}}}&=\mbox{the unique $\rho_j\in A$ such that $(\operatorname{span}(\rho_i)){\gamma}=\operatorname{span}(\rho_j)$.}\notag\end{aligned}$$ Note that ${h_{{\gamma}}^{{\alpha}}}$ is well defined and injective by Proposition \[pcen\]. ]{}
The case of $n=3^2=9$ is special and we consider it in the following lemma.
\[lne9\] Let $n=|X|=9$. Then there are $9$-cycles ${\alpha}$ and ${\beta}$ in $\operatorname{Sym}(X)$ such that the distance between ${\alpha}$ and ${\beta}$ in ${\mathcal{G}}({\mathcal{I}}(X))$ is $5$.
Let $X=\{1,2,\ldots,9\}$, and consider the following $9$-cycles in $\operatorname{Sym}(X)$: $${\alpha}=(1\,2\,3\,4\,5\,8\,7\,6\,9)\,\mbox{ and }\,{\beta}=(1\,4\,7\,2\,5\,8\,3\,6\,9).$$ We claim that the distance between ${\alpha}$ and ${\beta}$ in ${\mathcal{G}}({\mathcal{I}}(X))$ is $5$. We know that $d({\alpha},{\beta})\leq5$ by Proposition \[pdio\]. Suppose to the contrary that $d({\alpha},{\beta})<5$. Then there are ${\delta},{\gamma},\eta\in{\mathcal{I}}(X)-\{0,1\}$ such that ${\alpha}-{\delta}-{\gamma}-\eta-{\beta}$. Then, by Lemma \[ldia4\], ${\delta}={\alpha}^p$ and $\eta={\beta}^q$ for some $p,q\in\{1,\ldots,8\}$. The exponent $p$ is $3$, $6$, or relatively prime to $9$. In the latter case, there is $t\in\{1,\ldots,8\}$, relatively prime to $9$, such that $pt\equiv1\!\pmod9$. Since ${\gamma}$ commutes with ${\delta}={\alpha}^p$, it also commutes with $({\alpha}^p)^{3t}=({\alpha}^{pt})^3=({\alpha}^1)^3={\alpha}^3$. If $p=6$, then ${\gamma}$ commutes with $({\alpha}^6)^5=({\alpha}^{10})^3=({\alpha}^1)^3={\alpha}^3$. Hence, in either case, ${\gamma}$ commutes with ${\alpha}^3$. By a similar argument, ${\gamma}$ also commutes with ${\beta}^3$, and so $${\alpha}^3=(1\,4\,7){\sqcup}(2\,5\,6){\sqcup}(3\,8\,9)-{\gamma}-(1\,2\,3){\sqcup}(4\,5\,6){\sqcup}(7\,8\,9)={\beta}^3.$$ Since ${\gamma}\ne0$, there is a cycle ${\sigma}$ in ${\beta}^3$ such that $\operatorname{span}({\sigma})\subseteq\operatorname{dom}({\gamma})$ (by Proposition \[pcen\]). Therefore, $1$, $4$, or $7$ is in $\operatorname{dom}({\gamma})$, and so, since ${\gamma}$ commutes with ${\alpha}^3$ and $(1\,4\,7)$ is a cycle in ${\alpha}^3$, we have $(1\,4\,7)\in\operatorname{dom}(h_{\gamma}^{{\alpha}^3})$. There are three possible cases. [**Case 1.**]{} $(1\,4\,7)h_{\gamma}^{{\alpha}^3}=(1\,4\,7)$. Then $1{\gamma}=1$, $4$, or $7$. If $1{\gamma}=1$, then $4{\gamma}=4$ and $7{\gamma}=7$ by Proposition \[pcen\]. But then, since ${\gamma}$ commutes with ${\beta}^3$, ${\gamma}$ must fix every element of every cycle of ${\beta}^3$, that is, ${\gamma}=1$. This is a contradiction. Suppose $1{\gamma}=4$. Then $(1\,2\,3)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$ with $2{\gamma}=5$ and $3{\gamma}=6$. But then $(2\,5\,6)h_{\gamma}^{{\alpha}^3}=(2\,5\,6)$ and $(3\,8\,9)h_{\gamma}^{{\alpha}^3}=(2\,5\,6)$, which is a contradiction since $h_{\gamma}^{{\alpha}^3}$ is injective. If $1{\gamma}=7$, we obtain a contradiction in a similar way. [**Case 2.**]{} $(1\,4\,7)h_{\gamma}^{{\alpha}^3}=(2\,5\,6)$. Then $1{\gamma}=2$, $5$, or $6$. If $1{\gamma}=2$, then $4{\gamma}=5$ and $7{\gamma}=6$, and so $(4\,5\,6)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$ and $(7\,8\,9)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$. This is a contradiction since $h_{\gamma}^{{\beta}^3}$ is injective. If $1{\gamma}=5$, then $4{\gamma}=6$, and so $(1\,2\,3)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$ and $(4\,5\,6)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$, again a contradiction. Finally, if $1{\gamma}=6$, then $7{\gamma}=5$, and so $(1\,2\,3)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$ and $(7\,8\,9)h_{\gamma}^{{\beta}^3}=(4\,5\,6)$, also a contradiction. [**Case 3.**]{} $(1\,4\,7)h_{\gamma}^{{\alpha}^3}=(3\,8\,9)$. In this case, we also obtain a contradiction by the argument similar to the one used in Case 2.
Therefore, the assumption $d({\alpha},{\beta})<5$ leads to a contradiction, and so $d({\alpha},{\beta})\geq5$. Since we already know that $d({\alpha},{\beta})\leq5$, we have $d({\alpha},{\beta})=5$.
\[tpow\] Let $|X|=n=p^k$, where $p$ is an odd prime and $k\ge2$. Then the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $5$.
By Proposition \[pdio\], it suffices to find two $n$-cycles ${\alpha}$ and ${\beta}$ in $\operatorname{Sym}(X)$ such that the distance between ${\alpha}$ and ${\beta}$ in ${\mathcal{G}}({\mathcal{I}}(X))$ is at least $5$. If $n=9$, then such cycles exist by Lemma \[lne9\].
Suppose $n>9$ and let $X=\{1,2,\ldots,n\}$. If ${\alpha},{\beta}\in {\mathcal{I}}(X)$ are $n$-cycles such that ${\alpha}-{\delta}-{\gamma}-\eta-{\beta}$ for some ${\delta},{\gamma},\eta \in {\mathcal{I}}(X)-\{0,1\}$, then, by the argument similar to the one we used in Lemma \[lne9\], we may assume that ${\delta}={\alpha}^q$ and $\eta={\beta}^q$, where $q=p^{k-1}$. Note that then ${\delta}$ and $\eta$ are joins of $q$ cycles, each cycle of length $p$. Consider the following ${\delta},\eta\in\operatorname{Sym}(X)$: $${\delta}=(1 \;\; 2 \;\; \ldots \;\;p) {\sqcup}(p+1\;\; p+2 \;\; \ldots\;\; 2p){\sqcup}\cdots {\sqcup}(n-p+1 \;\; n-p+2\;\; \ldots \;\;n),$$ $$\begin{aligned}
\eta=&~~~\,(1 &&q-1& &2q -2 &&\ldots &&n-3q-p+3 &&n-2q-p+2 &&n-q-p+1) \\
&{\sqcup}(2 &&q && 2q-1 && \ldots &&n-3q-p+4 && n-2q-p+3 && n-q-p+2) \\
&{\sqcup}(3 &&q+1 &&2q &&\ldots && n-3q-p+5 &&n-2q-p+4 &&n-q-p+3) \\
&{\sqcup}(4 &&q+2 &&2q+1 &&\ldots && n-3q-p+6 &&n-2q-p+5 &&n-q-p+4) \\
&~\vdots \\
&{\sqcup}(q-3 &&2q-5 &&3q-6 &&\ldots && n-2q-p-1 &&n-q-p-2 &&n-p-3) \\
&{\sqcup}(q-2 &&2q-4 &&3q-5 &&\ldots && n-2q-p &&n-q-p-1 &&n-p-2) \\
&{\sqcup}(n-p+1 &&2q-3 && 3q-4 &&\ldots && n-2q-p+1 &&n-q-p &&n-p-1) \\
&{\sqcup}(n-p+2 &&n-p+3 &&n-p+4 &&\ldots && n-1 &&n &&n-p) \\\end{aligned}$$ The construction of ${\delta}$ is straightforward. Regarding $\eta$, the last cycle, $$\tau=(n-p+2\,\,\,n-p+3\,\,\,n-p+4\,\,\ldots\,\,\,n-1\,\,\,n\,\,\,n-p),$$ is special. (Its role will become clear in the second part of the proof). If $\tau'=(x_1\,x_2\ldots\,x_p)$ is any other cycle in $\eta$, then $x_{i+1}-x_i=q-1$ for every $i\in\{2,\ldots,p-1\}$. Here and in the following, we assume cycles are always represented by expressions listing the elements in the fixed orders from the definitions of $\delta$ and $\eta$, so that we may speak of the position of an element in a cycle.
Let ${\alpha}$ and ${\beta}$ be $n$-cycles such that ${\alpha}^q={\delta}$ and ${\beta}^q=\eta$. As ${\delta}$ and $\eta$ consist of $q$ disjoint cycles of length $p$, such ${\alpha}$ and ${\beta}$ exist. We claim that $d({\alpha},{\beta})\geq5$. Suppose to the contrary that $d({\alpha},{\beta})<5$. Then, by the foregoing argument, there exists ${\gamma}\in {\mathcal{I}}(X)-\{0,1\}$ with ${\delta}-{\gamma}-\eta$.
Define a binary relation $\sim$ on $X$ by: $x\sim y$ if there exists a cycle $\rho$ in ${\delta}$ or in $\eta$ with $\{x,y\} \subseteq \operatorname{span}(\rho)$. Let $\sim^*$ be the transitive closure of $\sim$. It follows from Proposition \[pcen\] that $\sim$ preserves the following two properties: “${\gamma}$ is defined at $x$” and “${\gamma}$ fixes $x$”. It is then clear that $\sim^*$ preserves these properties as well. We will write $x\sim_{\delta}y$ if $x$ and $y$ are in the same cycle of ${\delta}$, and $x\sim_\eta y$ if $x$ and $y$ are in the same cycle of $\eta$ (so $\sim\,=\,\sim_{\delta}\cup\sim_\eta$).
We claim that $\sim^*\,=X\times X$. Consider the set $A=\{n-q-p+1,n-q-p+2,\ldots,n-p\}$ of the rightmost elements of the cycles in $\eta$. Note that $A$ contains $t=q/p$ multiples of $p$: $$\label{e1tpow}
n-p,\,\,n-2p,\,\ldots,\,\,n-tp.$$ Let $i\in\{1,2,\ldots,t-1\}$. We claim that $n-ip\sim^*n-(i+1)p$. First, we have $n-ip\sim_{\delta}n-ip-p+1$ since $(n-ip-p+1\,\,n-ip-p+2\,\ldots\,\,n-ip)$ is a cycle in ${\delta}$. Next, $n-ip-p+1$ is a rightmost element of a cycle in $\eta$ that is different from $\tau$ (the last cycle). We have already observed that $n-ip-p+1-(q-1)$ is the preceding element in the same cycle. Thus $$n-ip-p+1\sim_\eta n-ip-p+1-(q-1)=n-q-ip-p+2.$$ Further, $n-q-ip-p+2\sim_{\delta}n-q-ip-p+1$, and finally $$n-q-ip-p+1\sim_\eta n-q-ip-p+1+(q-1)=n-ip-p=n-(i+1)p.$$ To summarize, $$n-ip \sim_{\delta}n-ip-p+1\sim_\eta n-q-ip-p+2 \sim_{\delta}n-q-ip-p+1 \sim_\eta n-ip-p= n-(i+1)p.$$ It follows by the transitivity of $\sim^*$ that any two multiples of $p$ from (\[e1tpow\]) are $\sim^*$-related. Let $x,y\in X$. Then there are $z,w\in A$ such that $x\sim_\eta z$ and $y\sim_\eta w$. Now, $z$ must be in some cycle of ${\delta}$ whose rightmost element is a multiple of $p$. Since $z\in A$, that multiple must come from $(\ref{e1tpow})$, that is, $z\sim_{\delta}n-jp$ for some $j\in\{1,2,\ldots,t\}$, where $t=q/p$. Similarly, $w\sim_{\delta}n-lp$ for some $l\in\{1,2,\ldots,t\}$. Hence $$x\sim_\eta z\sim_{\delta}n-jp\sim^*n-lp\sim_{\delta}w\sim_\eta y.$$ Thus $x\sim^* y$, and so $\sim^*\,=X\times X$.
As ${\gamma}\ne 0$, ${\gamma}$ must be defined on some element of $X$. Since $\sim^*$ preserves the statement “${\gamma}$ is defined at $x$” and $\sim^*\,=X\times X$, we have $\operatorname{dom}({\gamma})=X$.
Consider the cycle $\rho=( n-p+1 \;\; n-p+2\;\; \ldots \;\;n)$ in ${\delta}$ and the cycle $$\tau=(n-p+2 \;\; n-p+3 \;\; n-p+4 \;\; \ldots \;\; n-1 \;\; n \;\; n-p)$$ in $\eta$, and note that $\operatorname{span}(\rho)\cap\operatorname{span}(\tau)$ consists of $p-1$ elements. Thus, $\operatorname{span}(\rho h_{\gamma}^{\delta})\cap\operatorname{span}(\tau h_{\gamma}^\eta)$ also consists of $p-1$ elements. However, for all cycles $\rho'$ in ${\delta}$ and $\tau'$ in $\eta$, if $\rho'\ne\rho$, then $\operatorname{span}(\rho')\cap\operatorname{span}(\tau')$ consists of either $1$ or $2$ elements, where $2$ is only possible when $n=p^2$. In the latter case, $p\geq5$ (since $n>9$), and so $p-1>2$. If $n=p^k$ with $k>2$, then $p-1>1$ (even when $p=3$). Hence $\rho h_{\gamma}^{\delta}=\rho$ since otherwise we would have $|\operatorname{span}(\rho h_{\gamma}^{\delta})\cap\operatorname{span}(\tau h_{\gamma}^\eta)|<p-1$. Applying the same argument to $\tau$, we see that $\tau h_{\gamma}^\eta= \tau$.
These two conditions imply that the element $n-p$ that occurs in $\tau$ must be fixed by ${\gamma}$. Since $\sim^*$ preserves the statement “${\gamma}$ fixes $x$” and $\sim^*\,=X\times X$, it follows that ${\gamma}$ fixes every element of $X$. So ${\gamma}=1$, which is a contradiction. We have proved that $d({\alpha},{\beta})\geq5$.
It now follows from Proposition \[pdio\] that the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ is $5$.
The problem of finding the exact value of the diameter of ${\mathcal{G}}({\mathcal{I}}(X))$ when $n$ is odd and divisible by at least two primes remains open. By Lemmas \[ldia2\] and \[ldia3\], $d({\alpha},{\beta})\leq4$ for all ${\alpha},{\beta}\in{\mathcal{I}}(X)$ such that ${\alpha}$ or ${\beta}$ is not an $n$-cycle. So the exact value of the diameter (which is $4$ or $5$) depends on the answer to the following question. [**Question.**]{} Let $n\geq 15$ be odd and divisible by at least two primes. Are there $n$-cycles ${\alpha},{\beta}\in{\mathcal{I}}(X)$ such that $d({\alpha},{\beta})=5$?
We conclude this section with a discussion of the diameter of the commuting graph of the symmetric group $\operatorname{Sym}(X)$. Iranmanesh and Jafarzadeh have proved [@IrJa08 Theorem 3.1] that if $n$ and $n-1$ are not primes, then the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$ is at most $5$. (If $n$ or $n-1$ is a prime, then the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$ is $\infty$.)
Dolzan and Oblak have strengthened this result [@DoOb11 Theorem 4] by showing that if $n$ and $n-1$ are not primes, then the distance between ${\alpha}=(1\,2\,\ldots\,n)$ and ${\beta}=(1\,2\, \ldots\, n-1){\sqcup}(n)$ in ${\mathcal{G}}(\operatorname{Sym}(X))$ is at least $5$ (so the diameter of ${\mathcal{G}}(\operatorname{Sym}(X))$ is exactly $5$). However, their proof contains a gap. They state that if $\rho,{\sigma},\tau\in\operatorname{Sym}(X)$ are such that $\rho-{\sigma}-\tau$ and the length of any cycle in $\rho$ is relatively prime to the length of any cycle in $\tau$, then ${\sigma}$ must fix every point in $X$, and so ${\sigma}=1$. However, this statement is not true, even with the additional assumptions that $\rho$ is the power of an $n$-cycle and $\tau$ is the power of a disjoint join between an $(n-1)$-cycle and a $1$-cycle. Let $X=\{1,2,\ldots,10\}$, and consider $$\rho=(1\,2){\sqcup}(3\,4){\sqcup}(5\,6){\sqcup}(7\,8){\sqcup}(9\,10)\,\,\mbox{ and }\,\,\tau=(1\,3\,5){\sqcup}(2\,4\,6){\sqcup}(7\,8\,9){\sqcup}(10).$$ Then for ${\sigma}=(1\,3\,5){\sqcup}(2\,4\,6){\sqcup}(7){\sqcup}(8){\sqcup}(9){\sqcup}(10)$, we have $\rho-{\sigma}-\tau$ but ${\sigma}\ne1$.
It is possible to fix this gap by taking into account the special form of ${\alpha}$ and ${\beta}$ in the original proof. We do this in the following lemma.
\[ldol\] Let $X=\{1,2,\ldots,n\}$, where neither $n$ nor $n-1$ is a prime. Then, the distance between ${\alpha}=(1\,2\,\ldots\,n)$ and ${\beta}=(1\,2\, \ldots\, n-1){\sqcup}(n)$ in ${\mathcal{G}}(\operatorname{Sym}(X))$ is at least $5$.
Suppose to the contrary that $d({\alpha},{\beta})<5$. Then ${\alpha}-\rho-{\sigma}-\tau-{\beta}$ for some $\rho,{\sigma},\tau\in\operatorname{Sym}(X)-\{1\}$. It easily follows from the proof of Lemma \[ldia4\] that $\rho={\alpha}^m$ and $\tau={\beta}^k$ for some $m\in\{1,\ldots,n-1\}$ and some $k\in\{1,\ldots,n-2\}$. We may assume that $m$ is a proper divisor of $n$. (If $m$ and $n$ are relatively prime, then ${\alpha}^m={\alpha}$, and so we may replace $\rho={\alpha}^m$ in ${\alpha}-\rho-{\sigma}-\tau-{\beta}$ with ${\alpha}^{m'}$, where $m'$ is any proper divisor of $n$. Similarly, if $m=lm'$, where $l$ and $n$ are relatively prime and $m'$ is a proper divisor of $n$, we can replace $\rho={\alpha}^m$ with ${\alpha}^{m'}$.) Similarly, we may assume that $k$ is a proper divisor of $n-1$. Note that $m$ and $k$ are relatively prime. The permutation $\rho={\alpha}^m$ is the join of $m$ cycles, each of length $t=n/m$: $$\label{e1ldol}
\rho={\alpha}^m={\lambda}_1{\sqcup}{\lambda}_2{\sqcup}\cdots{\sqcup}{\lambda}_m.$$ Consider the cyclic group $\mathbb Z_n=\{1,2,\ldots,n\}$ of integers modulo $n$ and the subgroup $\langle m\rangle$ of $\mathbb Z_n$. Then the spans of the cycles in $\rho={\alpha}^m$ are precisely the cosets of the group $\langle m\rangle$. Since $k$ and $m$ are relatively prime, the cosets of $\langle m\rangle$ are $$\langle m\rangle+k,\,\,\langle m\rangle+2k,\,\ldots,\,\,\langle m\rangle+mk.$$ We may order the cycles in (\[e1ldol\]) in such a way that $\operatorname{span}({\lambda}_i)=\langle m\rangle+ik$ for every $i\in\{1,2,\ldots,m\}$.
Since $(n)$ is the only $1$-cycle in $\tau={\beta}^k$, ${\sigma}$ fixes $n$ by Proposition \[pcen\]. Recall that, by Proposition \[pcen\], if ${\sigma}$ fixes some element of a cycle in $\rho$ or in $\tau$, then it fixes all elements of that cycle. Thus ${\sigma}$ fixes all elements of $\operatorname{span}({\lambda}_m)$ (since $\operatorname{span}({\lambda}_m)=\langle m\rangle+mk=\langle m\rangle$ contains $n$). Since $m \le n/2$ and $k \le (n-1)/2$, there is $x\in\operatorname{span}({\lambda}_m)$ such that $x+k\le n-1$ (in standard, non-modular addition). Thus $x$ and $x+k$ are in the same cycle of $\tau$ (since $\tau={\beta}^k$ is a join of $(n)$ and $k$ cycles, each of length $s=(n-1)/k$, and the span of each cycle of length $s$ is closed under addition of $k$ modulo $n-1$). Hence, since ${\sigma}$ fixes $x$, ${\sigma}$ also fixes $x+k$. But $x+k\in\operatorname{span}({\lambda}_1)$ (since $\operatorname{span}({\lambda}_1)=\langle m\rangle+k=(\langle m\rangle+mk)+k$), and so ${\sigma}$ fixes all elements of $\operatorname{span}({\lambda}_1)$.
Applying the foregoing argument $m-2$ more times, to cycles ${\lambda}_1,\ldots,{\lambda}_{m-1}$, will show that ${\sigma}$ fixes all elements of every cycle in $\rho$. Hence ${\sigma}=1$, which is a contradiction. Thus $d({\alpha},{\beta})\geq5$.
Problems {#spro}
========
In the process of proving Theorem \[tgen\], we came across a purely combinatorial conjecture that, if true, could simplify some of the proofs. As this combinatorial problem may be of interest regardless of the commuting graphs, we present it here.
Let $s,t>1$ be natural numbers. Suppose $A$ is an $s\times t$ matrix with entries from some set $S$ such that:
- entries in each row of $A$ are pairwise distinct;
- entries in each column of $A$ are pairwise distinct; and
- there is no $a\in S$ such that $a$ occurs in *every* row of $A$ or $a$ occurs in *every* column of $A$.
For given $s$ and $t$ find the smallest $S$ that satisfies the three conditions above. In particular, is it necessarily true that $A$ contains at least $s+t-1$ distinct entries?
For a graph $G=(V,E)$, denote by $\operatorname{Aut}(G)$ the group of automorphisms of $G$. Recall that $T(X)$ denotes the semigroup of full transformations on $X$. The automorphism groups of the commuting graphs of $T(X)$ and of ${\mathcal{I}}(X)$ are, comparatively to the size of the graphs themselves, very large. We list here their cardinalities for small values of $n=|X|$, which we have obtained using GAP [@Scel92] and GRAPE [@So06]. $$\begin{tabular}{|c|c|c|}\hline
$n$&$|\operatorname{Aut}({\mathcal{G}}({\mathcal{I}}(X)))|$&$|\operatorname{Aut}({\mathcal{G}}(T(X)))|$\\ \hline
$2$&$2^{2}\cdot 3$&$2\cdot3$\\ \hline
$3$&$2^{9}\cdot3$&$2^{5}\cdot 3^{4}$\\ \hline
$4$&$2^{38}\cdot3^5$&$2^{34}\cdot3$\\ \hline
$5$&$2^{231}\cdot 3^{44}\cdot 5^2$&$2^{410}\cdot3^9\cdot5^2$\\\hline
\end{tabular}$$
[Describe the automorphism groups of the commuting graphs of ${\mathcal{I}}(X)$, $T(X)$, and $Sym(X)$. ]{}
The diameter of the commuting graph of $T(X)$ has been determined in [@ArKiKo11 Theorems 2.22].
[Find the clique number of the commuting graph of $T(X)$. ]{}
A related problem is to determine the chromatic number of a given commuting graph.
[Find the chromatic numbers of the commuting graphs of ${\mathcal{I}}(X)$, $T(X)$, and $\operatorname{Sym}(X)$. ]{}
It has been proved in [@ArKiKo11 Theorem 4.1] that for every natural $n$, there exists a semigroup (consisting of idempotents) such that the diameter of its commuting graph is $n$. It has been conjectured that there exists a common upper bound of the diameters of the (connected) commuting graphs of finite groups.
[ Is it true that for every natural $n$, there exists a finite *inverse* semigroup whose commuting graph has diameter $n$? ]{}
The commuting graphs of finite groups have attracted a great deal of attention. There is a parallel concept of the non-commuting graph of a finite group, which has also been the object of intensive study [@AbAkMa06; @Da09; @Ne76; @ZhSh05]. (A *non-commuting graph* of a finite nonabelian group $G$ is a simple graph whose vertices are the non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if $xy\ne yx$.) Once again, the concept carries over to semigroups, but nothing is known about the non-commuting graphs of semigroups.
[ Find the diameters, clique numbers, and chromatic numbers of the non-commuting graphs of $T(X)$ and ${\mathcal{I}}(X)$. Is it true that for every natural $n$, there exists a semigroup whose non-commuting graph has diameter $n$? ]{}
In the present paper and [@ArKiKo11], the commuting graphs of ${\mathcal{I}}(X)$, $T(X)$, and their ideals have been investigated. However, there are many other subsemigroups of ${\mathcal{I}}(X)$ and $T(X)$ that have been intensively studied (see [@vhf; @GaMa09]).
[ Calculate the diameters, clique numbers, and chromatic numbers of commuting and non-commuting graphs of various subsemigroups of ${\mathcal{I}}(X)$ and $T(X)$. ]{}
[**Acknowledgment**]{} The first author was partially supported by FCT through the following projects: PEst-OE/MAT/UI1043/2011, Strategic Project of Centro de Álgebra da Universidade de Lisboa; and PTDC/MAT/101993/2008, Project Computations in groups and semigroups .
The research of the second author leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement no. PCOFUND-GA-2009-246542 and from the Foundation for Science and Technology of Portugal.
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---
abstract: 'The global Torelli theorem for projective ${\mathop{\mathrm {K3}}\nolimits}$ surfaces was first proved by Piatetskii-Shapiro and Shafarevich 35 years ago, opening the way to treat moduli problems for ${\mathop{\mathrm {K3}}\nolimits}$ surfaces. The moduli space of polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of degree $2d$ is a quasi-projective variety of dimension $19$. For general $d$ very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for $d>61$ and for $d=46$, $50$, $54$, $58$, $60$.'
author:
- 'V. Gritsenko, K. Hulek and G.K. Sankaran'
title: The Kodaira dimension of the moduli of K3 surfaces
---
Introduction
============
Moduli spaces of polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces can be identified with the quotient of a classical hermitian domain of type $IV$ and dimension $19$ by an arithmetic group. The general set-up for the problem is the following. Let $L$ be an integral lattice with a quadratic form of signature $(2,n)$ and let $$\label{DL}
{\mathcal D}_L=\{[{\mathbf w}] \in {\mathbb P}(L\otimes {\mathbb C}) \mid
({\mathbf w},{\mathbf w})=0,\ ({\mathbf w},{\overline}{\mathbf w})>0\}^+$$ be the associated $n$-dimensional Hermitian domain (here $+$ denotes one of its two connected components). We denote by ${\mathop{\null\mathrm {O}}\nolimits}(L)^+$ the index $2$ subgroup of the integral orthogonal group ${\mathop{\null\mathrm {O}}\nolimits}(L)$ preserving ${\mathcal D}_L$. We are, in general, interested in the birational type of the $n$-dimensional variety $$\label{FLG}
{\mathcal F}_L(\Gamma)=\Gamma\backslash {\mathcal D}_L$$ where $\Gamma$ is a subgroup of ${\mathop{\null\mathrm {O}}\nolimits}^+(L)$ of finite index. Clearly, the answer will depend strongly on the lattice $L$ and the chosen subgroup $\Gamma$.
A compact complex surface $S$ is a ${\mathop{\mathrm {K3}}\nolimits}$ surface if $S$ is simply connected and there exists a holomorphic $2$-form $\omega_S \in
H(S,\Omega^2)$ without zeros. For example, a smooth quartic in ${\mathbb P}^3({\mathbb C})$ is a ${\mathop{\mathrm {K3}}\nolimits}$ surface and all quartics (modulo projective equivalence) form a (unirational) space of dimension $19$.
The second cohomology group $H^2(S,{\mathbb Z})$ with the intersection pairing is an even unimodular lattice of signature $(3,19)$, more precisely, $$\label{LK3}
H^2(S,{\mathbb Z})\cong L_{{\mathop{\mathrm {K3}}\nolimits}}=3U\oplus 2E_8(-1)$$ where $U$ is the hyperbolic plane and $E_8(-1)$ is the negative definite even lattice associated to the root system $E_8$. The $2$-form $\omega_S$, considered as a point of ${\mathbb P}(L_{{\mathop{\mathrm {K3}}\nolimits}}\otimes
{\mathbb C})$, is the period of $S$. By the Torelli theorem the period of a ${\mathop{\mathrm {K3}}\nolimits}$ surface determines its isomorphism class. The moduli space of all ${\mathop{\mathrm {K3}}\nolimits}$ surfaces is not Hausdorff. Therefore it is better to restrict to moduli spaces of polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces. The moduli of all algebraic ${\mathop{\mathrm {K3}}\nolimits}$ surfaces are parametrised by a countable union of $19$-dimensional irreducible algebraic varieties. To choose a component we have to fix a polarisation. A polarised ${\mathop{\mathrm {K3}}\nolimits}$ surface of degree $2d$ is a pair $(S,H)$ consisting of a ${\mathop{\mathrm {K3}}\nolimits}$ surface $S$ and a primitive pseudo-ample divisor $H$ on $S$ of degree $H^2=2d>0$. If $h$ is the corresponding vector in the lattice $L_{{\mathop{\mathrm {K3}}\nolimits}}$ then its orthogonal complement $$\label{L2d}
h^{\perp}_{L_{{\mathop{\mathrm {K3}}\nolimits}}}\cong L_{2d}=2U\oplus 2E_8(-1)
\oplus \langle -2d \rangle$$ is a lattice of signature $(2,19)$.
The $2$-form $\omega_S$ determines a point of ${\mathcal D}_{L_{2d}}$ modulo the group $${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d}) = \{ g\in {\mathop{\null\mathrm {O}}\nolimits}^+(L_{{\mathop{\mathrm {K3}}\nolimits}}) \mid g(h)=h \}.$$ By the global Torelli theorem ([@P-SS]) and the surjectivity of the period map $$\label{F2d}
{\mathcal F}_{2d}={\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})\setminus {\mathcal D}_{L_{2d}}$$ is the coarse moduli space of polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of degree $2d$. By a result of Baily and Borel [@BB], ${\mathcal F}_{2d}$ is a quasi-projective variety. One of the fundamental problems is to determine its birational type.
For $d=2$, $3$ and $4$ the polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of degree $2d$ are complete intersections in ${\mathbb P}^{d+1}({\mathbb C})$ and the moduli spaces ${\mathcal F}_{2d}$ for such $d$ are classically known. Mukai has extended these results in his papers [@Mu1], [@Mu2] and [@Mu3] to $1\leq d\leq 10$ and $d=17$, $19$, showing that these moduli spaces are also unirational.
In the other direction there are two results of Kondo and of Gritsenko. Kondo [@Ko1] considered the moduli spaces ${\mathcal F}_{2p^2}$ where $p$ is a prime number. (The reason for this choice is that all these spaces are covers of ${\mathcal F}_{2}$.) He proved that these spaces are of general type for $p$ sufficiently large. His result, however, is not effective. Gritsenko [@G] showed a result for level structures: let ${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})(q)$ be the principal congruence subgroup of ${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})$ of level $q$. Then ${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})(q)\setminus {\mathcal D}_{L_{2d}}$ is of general type for any $d$ if $q\ge 3$. In this paper we determine the Kodaira dimension of ${\mathcal F}_{2d}$ without imposing any [*a priori*]{} restriction on $d$.
\[mainthm\] The moduli space ${\mathcal F}_{2d}$ of ${\mathop{\mathrm {K3}}\nolimits}$ surfaces with a polarisation of degree $2d$ is of general type for any $d>61$ and for $d=46$, $50$, $54$, $57$, $58$ and $60$.
If $d\ge 40$ and $d\ne 41$, $44$, $45$ or $47$ then the Kodaira dimension of ${\mathcal F}_{2d}$ is non-negative.
The description of the moduli space ${\mathcal F}_{2d}$ as a quotient of the symmetric space ${\mathcal D}_{L_{2d}}$ by a subgroup of the orthogonal group leads us to study, more generally, quotients of the form ${\mathcal F}_L(\Gamma)=\Gamma\backslash {\mathcal D}_L$. One of the main tools in our proof of the main theorem is the following general result (for a more precise formulation see Theorem \[main\_sings\_theorem\]).
\[sings\_theorem\_intro\] Let $L$ be a lattice of signature $(2,n)$ with $n\ge 9$, and let $\Gamma<{\mathop{\null\mathrm {O}}\nolimits}^+(L)$ be a subgroup of finite index. Then there exists a toroidal compactification ${\overline}{\mathcal F}_L(\Gamma)$ of ${\mathcal F}_L(\Gamma)=\Gamma\backslash{\mathcal D}_L$ such that ${\overline}{\mathcal F}_L(\Gamma)$ has canonical singularities.
We hope that this result will also be important for other applications.
The plan of the paper is as follows. In Section \[orthogonal\] we give the basic definitions that we shall need and explain what the obstructions are to showing that ${\mathcal F}_L(\Gamma)$ is of general type. These obstructions may be called elliptic, cusp and reflective. The elliptic obstructions come from singularities of ${\mathcal F}_L(\Gamma)$ and its compactifications. The cusp obstructions come from infinity, [i.e.]{} from the fact that ${\mathcal F}_L(\Gamma)$ is only quasi-projective. The reflective obstructions come from divisors fixed by $\Gamma$ in its action on the symmetric space ${\mathcal D}_L$.
In Section \[singularities\] we deal with the elliptic obstructions and we show, by an analysis of the toroidal compactifications, that they disappear if $n\ge 9$, and also that there are no fixed divisors at infinity.
In Section \[reflections\] we examine the reflective obstructions by describing the fixed divisors. We do this first for arbitrary $L$ and then in greater detail for $L_{2d}$.
In Section \[specialcusp\] we turn to the cusp obstructions. We describe the structure of the cusps for a lattice $L$ having only cyclic isotropic subgroups in its discriminant group.
In Section \[spinK3\] we study the moduli space ${\mathcal S}{\mathcal F}_{2d}$ of ${\mathop{\mathrm {K3}}\nolimits}$ surfaces with a spin structure. In this case there are few reflective obstructions, and the cusp forms constructed by Jacobi lifting already have the properties we need.
In Section \[Borcherds\] we show how to construct forms with the properties needed for ${\mathcal F}_{2d}$ by pulling back the Borcherds form. This requires us to find a suitable embedding of $L_{2d}$ in $L_{2,26}$, which in turn requires a vector in $E_8$ with square $2d$ that is orthogonal to at most $12$ and at least $2$ roots. We show directly that such a vector exists for large $d$ and use a small amount of computer help to show that it exists for smaller $d$. For some values of $d$ we can find only a vector of square $2d$ orthogonal to $14$ roots. In these cases we can deduce that ${\mathcal F}_{2d}$ has non-negative Kodaira dimension.
[*Acknowledgements:*]{} We have learned much from conversations with many people, but from S. Kondo and N.I. Shepherd-Barron especially. We are grateful for financial support from the Royal Society and the DFG Schwerpunktprogramm SPP 1094 [Globale]{} Methoden in der komplexen [Geometrie]{}, Grant Hu 337/5-3. We are also grateful for the hospitality and good working conditions provided by several places where one or more of us did substantial work on this project: the Max-Planck-Institut f[ü]{}r Mathematik in Bonn; DPMMS in Cambridge and Trinity College, Cambridge; Nagoya University; KIAS in Seoul; Tokyo University; and the Fields Institute in Toronto.
Orthogonal groups and modular forms {#orthogonal}
===================================
Let $L$ be a lattice of signature $(2,n)$, with $n>1$. For any lattice $M$ and field $K$ we write $M_K$ for $M{\otimes}K$. Then ${\mathcal D}_L$ is one of the two connected components of $$\{[{\mathbf w}]\in {\mathbb P}(L_{\mathbb C})\mid ({\mathbf w},{\mathbf w})=0,\ ({\mathbf w},{\overline}{\mathbf w})>0\}.$$ We denote by ${\mathop{\null\mathrm {O}}\nolimits}^+(L)$ the subgroup of ${\mathop{\null\mathrm {O}}\nolimits}(L)$ that preserves ${\mathcal D}_L$. If $\Gamma<{\mathop{\null\mathrm {O}}\nolimits}^+(L)$ is of finite index we denote by ${\mathcal F}_L(\Gamma)$ the quotient $\Gamma\backslash {\mathcal D}_L$, which is a quasi-projective variety by [@BB].
For every non-degenerate integral lattice we denote by $L^\vee={\mathop{\mathrm {Hom}}\nolimits}(L, {\mathbb Z})$ its dual lattice. The finite group $A_L=L^\vee/L$ carries a discriminant quadratic form $q_L$ (if $L$ is even) and a discriminant bilinear form $b_L$, with values in ${\mathbb Q}/2{\mathbb Z}$ and ${\mathbb Q}/{\mathbb Z}$ respectively (see [@Nik2 §1.3]). We define $$\begin{aligned}
{\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}(L) &=& \{g\in {\mathop{\null\mathrm {O}}\nolimits}(L)\mid g|_{A_L}={\mathop{\mathrm {id}}\nolimits}\},\ \text{ and}\\
{\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L) &=& {\widetilde}{\mathop{\null\mathrm {O}}\nolimits}(L)\cap {\mathop{\null\mathrm {O}}\nolimits}^+(L). \end{aligned}$$ The ${\mathop{\mathrm {K3}}\nolimits}$ lattice is $$L_{{\mathop{\mathrm {K3}}\nolimits}}=3U \oplus 2E_8 (-1)$$ where $U$ is the hyperbolic plane and $E_8$ is the (positive definite) $E_8$-lattice. If $h\in L_{{\mathop{\mathrm {K3}}\nolimits}}$ is a primitive vector with $h^2=2d>0$ then its orthogonal complement $h^\perp_{L_{{\mathop{\mathrm {K3}}\nolimits}}}$ is isometric to $$L_{2d}={{\langle{-2d}\rangle}} \oplus 2U \oplus 2E_8 (-1).$$ By [@Nik2 Proposition 1.5.1] $${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}(L_{2d}){\cong}\{ g\in {\mathop{\null\mathrm {O}}\nolimits}(L_{{\mathop{\mathrm {K3}}\nolimits}}) \mid g(h)=h \},$$ and the moduli space ${\mathcal F}_{2d}$ is given by $${\mathcal F}_{2d}={\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d}) \backslash {\mathcal D}_{L_{2d}}.$$
A modular form of weight $k$ and character $\chi\colon \Gamma\to
{\mathbb C}^*$ for a subgroup $\Gamma<{\mathop{\null\mathrm {O}}\nolimits}^+(L)$ is a holomorphic function $F\colon{\mathcal D}_L^\bullet\to {\mathbb C}$ on the affine cone ${\mathcal D}_L^\bullet$ over ${\mathcal D}_L$ such that $$\label{mod-form}
F(tZ)=t^{-k}F(Z)\ \forall\,t\in {\mathbb C}^*,\ \text{ and }\
F(gZ)=\chi(g)F(Z)\ \forall\,g\in \Gamma.$$ A modular form is a cusp form if it vanishes at every cusp. We denote the linear spaces of modular and cusp forms of weight $k$ and character $\chi$ for $\Gamma$ by $M_k(\Gamma,\chi)$ and $S_k(\Gamma,\chi)$ respectively.
\[general\_gt\] Let $L$ be an integral lattice of signature $(2,n)$, $n\ge 9$, and let $\Gamma$ be a subgroup of finite index of ${\mathop{\null\mathrm {O}}\nolimits}^+(L)$. The modular variety ${\mathcal F}_L(\Gamma)$ is of general type if there exists a character $\chi$ of finite order and a non-zero cusp form $F_a\in
S_a(\Gamma,\chi)$ of weight $a<n$ that vanishes along the branch divisor of the projection $\pi\colon {\mathcal D}_L\to {\mathcal F}_L(\Gamma)$.
If $S_n(\Gamma,\det)\neq 0$ then the Kodaira dimension of ${\mathcal F}_L(\Gamma)$ is non-negative.
We let ${\overline}{\mathcal F}_L(\Gamma)$ be a toroidal compactification of ${\mathcal F}_L(\Gamma)$ with canonical singularities and no branch divisors at infinity, which exists by Theorem \[main\_sings\_theorem\]. We take a smooth projective model ${\widehat}{\mathcal F}_L(\Gamma)$ by taking a resolution of singularities of ${\overline}{\mathcal F}_L(\Gamma)$.
Suppose that $F_{nk}\in M_{nk}(\Gamma, \det^k)$. Then, if $dZ$ is a holomorphic volume element on ${\mathcal D}_L$, the differential form $\Omega(F_{nk})=F_{nk}\,(dZ)^k$ is $\Gamma$-invariant and therefore determines a section of the pluricanonical bundle $kK=kK_{{\widehat}{\mathcal F}_L(\Gamma)}$ away from the branch locus of $\pi\colon{\mathcal D}_L\to {\mathcal F}_L(\Gamma)$ and the cusps.
In general $\Omega(F_{nk})$ will not extend to a global section of $kK$. We distinguish three kinds of obstruction to its doing so. There are elliptic obstructions, arising because of singularities given by elliptic fixed points of the action of $\Gamma$; reflective obstructions, arising from the branch divisors in ${\mathcal D}_L$ (divisors fixed pointwise by an element of $\Gamma$ acting locally as a quasi-reflection); and cusp obstructions, arising from divisors at infinity.
In this situation the elliptic obstruction vanishes (and there are no elliptic or reflective obstructions at infinity either) because of the choice of ${\overline}{\mathcal F}_L(\Gamma)$. So $\Omega(F_{nk})$ will extend to a section of $kK$ provided it extends to a general point of each branch divisor and each boundary divisor.
We apply the low-weight cusp form trick, used for example in [@G], [@GH1], [@GS] to show that the cusp obstruction for continuation of the pluricanonical forms on a smooth compactification is small compared with the dimension of $S_{nk}(\Gamma, \det^k)$. Let $N$ be the order of $\chi$ and put $k=2Nl$. Then we consider special elements $F^0_{nk}\in
S_{nk}(\Gamma)$ of the form $$\label{verycusp}
F^0_{nk}=F_a^k F_{(n-a)k}$$ where $F_{(n-a)k}\in M_{(n-a)k}(\Gamma)$ is a modular form of weight $(n-a)k\ge k$. The corresponding differential form $\Omega(F^0_{nk})$ vanishes to order at least $k$ on the boundary of the toroidal compactification $\overline{\mathcal F}_L(\Gamma)$. It follows by the results of [@AMRT] that $\Omega(F^0_{nk})$ extends as a $k$-fold pluricanonical form to the generic point of any boundary divisor of ${\overline}{\mathcal F}_L(\Gamma)$. The reason is that the anticanonical divisor of a toric variety is the sum of the torus-invariant divisors, so $dZ$ has simple poles at all boundary divisors in a toroidal compactification.
Since $F_a$ vanishes at the branch divisors, which are the fixed divisors of reflections by Theorem \[qrefs\_have\_order\_2\], $\Omega(F^0_{nk})$ vanishes there to order $k$, and hence it extends to give a section of $kK$ over ${\widehat}{\mathcal F}_L(\Gamma)$.
Finally, we observe that this gives us an injective map $$M_{(n-a)k}(\Gamma)\hookrightarrow H^0(\widehat{\mathcal F}_L(\Gamma)).$$ But $\dim M_{(n-a)k}(\Gamma)\sim k^n$, as can be seen from [@BB]: a more precise estimate, using the results of [@Mum], can be found in [@GHS1]. Hence it follows that ${\mathcal F}_L(\Gamma)$ is of general type.
Even if we can only find a cusp form of weight $n$ we still get some information, because of the well-known result of Freitag that if $F_n\in S_n(\Gamma, \det)$ then $\Omega(F_n)$ defines an element of $H^0(K_{\widehat{\mathcal F}_L(\Gamma)})$. Therefore the plurigenera do not all vanish: indeed $p_g\ge 1$.
Singularities of locally symmetric varieties {#singularities}
============================================
In this section, we consider the singularities of compactified locally symmetric varieties associated with the orthogonal group of a lattice of signature $(2,n)$. Our main theorem is that for all but small $n$, the compactification may be chosen to have canonical singularities.
\[main\_sings\_theorem\] Let $L$ be a lattice of signature $(2,n)$ with $n\ge 9$, and let $\Gamma<{\mathop{\null\mathrm {O}}\nolimits}^+(L)$ be a subgroup of finite index. Then there exists a toroidal compactification ${\overline}{\mathcal F}_L(\Gamma)$ of ${\mathcal F}_L(\Gamma)=\Gamma\backslash{\mathcal D}_L$ such that ${\overline}{\mathcal F}_L(\Gamma)$ has canonical singularities and there are no branch divisors in the boundary. The branch divisors in ${\mathcal F}_L(\Gamma)$ arise from the fixed divisors of reflections.
Immediate from Corollaries \[can\_sings\_on\_interior\], \[can\_sings\_on\_dim0\_cusps\] and \[can\_sings\_on\_dim1\_cusps\]. The last part is a summary of Theorem \[qrefs\_have\_order\_2\] (an element that fixes a divisor in ${\mathcal D}_L$ has order $2$ on the tangent space) and Corollary \[qrefs\_come\_from\_refs\_in\_L\] (such elements, up to sign, are given by reflections by vectors in $L$).
In fact we prove more than this: for example, ${\mathcal F}_L(\Gamma)$ has canonical singularities if $n\ge 7$ (Corollary \[can\_sings\_on\_interior\]), and our method (which uses ideas from [@Nik1]) gives some information about what non-canonical singularities can occur for small $n$. In order to choose ${\overline}{\mathcal F}_L(\Gamma)$ as in Theorem \[main\_sings\_theorem\] it is enough to take all the fans defining the toroidal compactification to be basic.
The interior {#interior}
------------
For $[{\mathbf w}]\in {\mathcal D}_L$ we define ${\mathbb W}={\mathbb C}.{\mathbf w}$. We put $S=({\mathbb W}\oplus{\overline}{\mathbb W})^\perp\cap L$, noting that $S$ could be $\{0\}$, and take $T=S^\perp\subset L$.
In the case of polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces, $S$ is the primitive part of the Picard lattice and $T$ is the transcendental lattice of the surface corresponding to the period point ${\mathbf w}$.
\[S\_T\_are\_disjoint\] $S_{\mathbb C}\cap T_{\mathbb C}=\{0\}$.
$S_{\mathbb C}$ and $T_{\mathbb C}$ are real ([i.e.]{} preserved by complex conjugation) so it is enough to show that $S_{\mathbb R}\cap T_{\mathbb R}=\{0\}$. If ${\mathbf x}\in T_{\mathbb R}\cap S_{\mathbb R}$ then $({\mathbf x},{\mathbf x})=0$ from the definition of $T$, so it is enough to prove that $S_{\mathbb R}$ is negative definite. The subspace ${\mathbb U}={\mathbb W}\oplus{\overline}{\mathbb W}\subset L_{\mathbb C}$ is also real, so we may write ${\mathbb U}=U_{\mathbb R}{\otimes}{\mathbb C}$, taking $U_{\mathbb R}$ to be the real vector subspace of ${\mathbb U}$ fixed pointwise by complex conjugation. An ${\mathbb R}$-basis for $U_{\mathbb R}$ is given by $\{{\mathbf w}+\bar{\mathbf w}, i({\mathbf w}-\bar{\mathbf w})\}$. But $({\mathbf w}+\bar{\mathbf w},{\mathbf w}+\bar{\mathbf w})>0$ and $(i({\mathbf w}-\bar{\mathbf w}),
i({\mathbf w}-\bar{\mathbf w}))>0$, so $U_{\mathbb R}$ has signature $(2,0)$. Hence $U_{\mathbb R}^\perp$ has signature $(0,n)$, but $S_{\mathbb R}\subset U_{\mathbb R}^\perp$ so $S_{\mathbb R}$ is negative definite.
We are interested first in the singularities that arise at fixed points of the action of $\Gamma$ on ${\mathcal D}_L$. Suppose then that ${\mathbf w}\in L_{\mathbb C}$ and let $G$ be the stabiliser of $[{\mathbf w}]$ in $\Gamma$. Then $G$ acts on ${\mathbb W}$ and we let $G_0$ be the kernel of this action: thus for $g\in G$ we have $g({\mathbf w})=\alpha(g){\mathbf w}$ for some homomorphism $\alpha\colon G \to {\mathbb C}^*$, and $G_0=\ker \alpha$.
\[G\_acts\_on\_T\] $G$ acts on $S$ and on $T$.
$G$ acts on ${\mathbb W}$ and on $L$, hence also on $S=({\mathbb W}\oplus{\overline}{\mathbb W})^\perp\cap L$ and on $T=S^\perp\cap L$.
\[G0\_acts\_trivially\_on\_T\] $G_0$ acts trivially on $T_{\mathbb Q}$.
If ${\mathbf x}\in T_{\mathbb Q}$ and $g\in G_0$ then $$({\mathbf w}, {\mathbf x})=(g({\mathbf w}), g({\mathbf x}))= ({\mathbf w}, g({\mathbf x})).$$ Hence $T_{\mathbb Q}\ni {\mathbf x}-g({\mathbf x})\in L_{\mathbb Q}\cap ({\mathbb W}\oplus{\overline}{\mathbb W}) =S_{\mathbb Q}$, so by Lemma \[S\_T\_are\_disjoint\] we have $g({\mathbf x})={\mathbf x}$.
The quotient $G/G_0$ is a subgroup of ${\mathop{\mathrm {Aut}}\nolimits}{\mathbb W}{\cong}{\mathbb C}^*$ and is thus cyclic of some order, which we call $r_{\mathbf w}$. So by the above, $\mu_{r_{\mathbf w}}{\cong}G/G_0$ acts on $T_{\mathbb Q}$. (By $\mu_r$ we mean the group of $r$th roots of unity in ${\mathbb C}$.)
For any $r\in {\mathbb N}$ there is a unique faithful irreducible representation of $\mu_r$ over ${\mathbb Q}$, which we call ${\mathcal V}_r$. The dimension of ${\mathcal V}_r$ is $\varphi(r)$, where $\varphi$ is the Euler $\varphi$ function and, by convention, $\varphi(1)=\varphi(2)=1$. The eigenvalues of a generator of $\mu_r$ in this representation are precisely the primitive $r$th roots of unity: ${\mathcal V}_1$ is the [$1$-dimensional]{} trivial representation. Note that $-{\mathcal V}_d={\mathcal V}_d$ if $d$ is even and $-{\mathcal V}_d={\mathcal V}_{2d}$ if $d$ is odd.
\[splitting\_of\_T\] As a $G/G_0$-module, $T_{\mathbb Q}$ splits as a direct sum of irreducible representations ${\mathcal V}_{r_{\mathbf w}}$. In particular, $\varphi(r_{\mathbf w})|\dim
T_{\mathbb Q}$.
We must show that no nontrivial element of $G/G_0$ has $1$ as an eigenvalue on $T_{\mathbb C}$. Suppose that $g\in G\setminus G_0$ (so $\alpha(g)\neq 1$) and that $g({\mathbf x}) ={\mathbf x}$ for some ${\mathbf x}\in
T_{\mathbb C}$. Then $$( {\mathbf w},{\mathbf x}) =( g({\mathbf w}), g({\mathbf x})) = \alpha(g)( {\mathbf w}, {\mathbf x}),$$ so $( {\mathbf w},{\mathbf x}) =0$, so ${\mathbf x}\in S_{\mathbb C}\cap T_{\mathbb C}=0$.
\[T\_splits\_for\_g\] If $g\in G$ and $\alpha(g)$ is of order $r$ (so $r|r_{\mathbf w}$), then $T_{\mathbb Q}$ splits as a $g$-module into a direct sum of irreducible representations ${\mathcal V}_r$ of dimension $\varphi(r)$.
Identical to the proof of Lemma \[splitting\_of\_T\].
We are interested in the action of $G$ on the tangent space to ${\mathcal D}_L$. We have a natural isomorphism $$T_{[{\mathbf w}]}{\mathcal D}_L{\cong}{\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},{\mathbb W}^\perp/{\mathbb W})=:V.$$ We choose $g\in G$ of order $m$ and put $\zeta=e^{2\pi i/m}$ for convenience: as $g$ is arbitrary there is no loss of generality. Let $r$ be the order of $\alpha(g)$, as in Corollary \[T\_splits\_for\_g\] (this is called $m$ in [@Nik1] but we want to keep the notation of [@Ko1]). In particular $r|m$. The eigenvalues of $g$ on $V$ are powers of $\zeta$, say $\zeta^{a_1},\ldots,\zeta^{a_n}$, with $0\le
a_i<m$. We define $$\label{RST}
\Sigma(g):=\sum_{i=1}^n a_i/m.$$ Recall that an element of finite order in ${\mathop{\mathrm {GL}}\nolimits}_n({\mathbb C})$ (for any $n$) is called a quasi-reflection if all but one of its eigenvalues are equal to $1$. It is called a reflection if the remaining eigenvalue is equal to $-1$. The branch divisors of ${\mathcal D}_L\to{\mathcal F}_L(\Gamma)$ are precisely the fixed loci of elements of $\Gamma$ acting as quasi-reflections.
\[RST\_for\_g\_nonqref\_and\_phi(r)>4\] Assume that $g\in G$ does not act as a quasi-reflection on $V$ and that $\varphi(r)>4$. Then $\Sigma(g)\ge 1$.
As $\xi$ runs through the $m$th roots of unity, $\xi^{m/r}$ runs through the $r$th roots of unity. We denote by $k_1,\ldots,k_{\varphi(r)}$ the integers such that $0<k_i<r$ and $(k_i,r)=1$, in no preferred order. Without loss of generality, we assume $\alpha(g)=\zeta^{mk_2/r}$ and ${\overline}{\alpha(g)}=\alpha(g)^{-1}=\zeta^{mk_1/r}$, with $k_1\equiv
-k_2 \bmod r$.
One of the ${\mathbb Q}$-irreducible subrepresentations of $g$ on $L_{\mathbb C}$ contains the eigenvector ${\mathbf w}$: we call this ${\mathbb V}_r^{\mathbf w}$ (it is the smallest $g$-invariant complex subspace of $L_{\mathbb C}$ that is defined over ${\mathbb Q}$ and contains ${\mathbf w}$). It is a copy of ${\mathcal V}_r{\otimes}{\mathbb C}$: to distinguish it from other irreducible subrepresentations of the same type we write ${\mathbb V}_r^{\mathbf w}={\mathcal V}_r^{\mathbf w}{\otimes}{\mathbb C}$.
If ${\mathbf v}$ is an eigenvector for $g$ with eigenvalue $\zeta^{mk_i/r}$, $i\neq 1$ (in particular ${\mathbf v}\not\in{\overline}{\mathbb W}$), then ${\mathbf v}\in {\mathbb W}^\perp$ since $({\mathbf v},{\mathbf w})=(g({\mathbf v}),g({\mathbf w}))=\zeta^{mk_i/r}\alpha(g)({\mathbf v}, {\mathbf w})$. Therefore the eigenvalues of $g$ on ${\mathbb V}_r^{\mathbf w}\cap {\mathbb W}^\perp /{\mathbb W}$ include $\zeta^{mk_i/r}$ for $i\ge 3$, so the eigenvalues on ${\mathop{\mathrm {Hom}}\nolimits}({\mathbb W}, {\mathbb V}_r^{\mathbf w}\cap{\mathbb W}^\perp/{\mathbb W})\subset V$ include $\zeta^{mk_1/r}\zeta^{mk_i/r}$ for $i\ge 3$. So, writing $\{a\}$ for the fractional part of $a$, we have $$\begin{aligned}
\label{W-contribution}
\Sigma(g)&\ge&
\sum_{i=3}^{\varphi(r)}\frac{1}{m}\left\{\frac{mk_1}{r}
+\frac{mk_i}{r}\right\}\nonumber\\
&=&
\sum_{i=3}^{\varphi(r)}\left\{\frac{k_1+k_i}{r}\right\}. \end{aligned}$$ Now the proposition follows from the elementary Lemma \[bigphi\] below.
\[bigphi\] Suppose $k_1,\ldots,k_{\varphi(r)}$ are the integers between $0$ and $r$ coprime to $r$, in some order, and that $k_2=r-k_1$. If $\varphi(r)\ge 6$ then $$\sum_{i=3}^{\varphi(r)}\Big\{\frac{k_1}{r}+\frac{k_i}{r}\Big\}\ge 1.$$
If $k_1<k_3<r/2$ then $\left\{\frac{k_1+k_3}{r}\right\}
=\frac{k_1+k_3}{r}$, and $k_4=r-k_3$ so $\left\{\frac{k_1+k_4}{r}\right\} =\frac{k_1+r-k_3}{r}$. Thus $$\left\{\frac{k_1+k_3}{r}\right\}+\left\{\frac{k_1+k_4}{r}\right\}
=\frac{2k_1+r}{r}>1.$$ If $r/2>k_1>r/4$ or $r>k_1>3r/4$ then $(k_1+k_3)+(k_1+k_4)\equiv
2k_1\bmod r$, so $$\left\{\frac{k_1+k_3}{r}\right\}+\left\{\frac{k_1+k_4}{r}\right\}
\equiv\frac{2k_1}{r}\bmod 1.$$ Therefore $\left\{\frac{k_1+k_3}{r}\right\}+\left\{\frac{k_1+k_4}{r}\right\}
>\frac12$, and similarly for $\left\{\frac{k_1+k_5}{r}\right\}+\left\{\frac{k_1+k_6}{r}\right\}$, so the sum is at least $1$.
If $r/2<k_1<3r/4$ then we may take $k_3=1$ and $k_4=r-1$, and then $\left\{\frac{k_1+k_3}{r}\right\}+\left\{\frac{k_1+k_4}{r}\right\}
=1+\frac{2k_1}{r}>1$.
The remaining possibility is that $k_1<r/4$ but $k_1>k_j$ if $k_j<r/2$. But then there is no integer coprime to $r$ between $r/4$ and $3r/4$. As long as $2\lceil r/4\rceil<\lfloor 3r/4\rfloor$, which is true if $r>9$, we may choose a prime $q$ such that $r/4<q<3r/4$, by Bertrand’s Postulate [@HW Theorem 418], and $\gcd(q,r)\neq 1$ so $r=2q$ or $r=3q$. In the first case one of $q\pm 2$ lies in $(r/4,3r/4)$ and is prime to $r$, and in the second case one of $q\pm
1$ or $q\pm 2$ does, unless $r<8$; so this possibility does not occur. The cases $r=7$ and $r=9$, which are not covered by this argument, are readily checked: $2\in (7/4,21/4)$ and $4\in(9/4,27/4)$ are coprime to $r$.
\[RST\_for\_g\_nonqref\_and\_r=1,2\] Assume that $g\in G$ does not act as a quasi-reflection on $V$ and that $r=1$ or $r=2$. Then $\Sigma(g)\ge 1$.
We note first that we may assume $g$ is not of order $2$, because if $g^2$ acts trivially on $V$ but $g$ is not a quasi-reflection then at least two of the eigenvalues of $g$ on $V$ are $-1$, and hence $\sum_{i=1}^n a_i/m\ge 1$. However, $g^2$ does act trivially on $T_{\mathbb C}$, by Corollary \[T\_splits\_for\_g\]. Therefore $g^2$ does not act trivially on $S_{\mathbb C}$. The representation of $g$ on $S_{\mathbb C}$ therefore splits over ${\mathbb Q}$ into a direct sum of irreducible subrepresentations ${\mathcal V}_d$, and at least one such piece has $d>2$. So on the subspace ${\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},{\mathcal V}_d\otimes
{\mathbb C})={\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},({\mathcal V}_d{\otimes}{\mathbb C}\oplus{\mathbb W})/{\mathbb W})\subset V$, the representation of $g$ is $\pm{\mathcal V}_d$ (the sign depending on whether $r=1$ or $r=2$), and choosing two conjugate eigenvalues $\pm\zeta^a$ and $\pm\zeta^{m-a}$ we have $\sum a_i/m\ge 1$.
\[RST\_for\_nonqrefs\_on\_interior\] Assume that $g\in G$ does not act as a quasi-reflection on $V$ and that $n\ge 6$. Then $\Sigma(g)\ge 1$.
In view of Proposition \[RST\_for\_g\_nonqref\_and\_r=1,2\] and Proposition \[RST\_for\_g\_nonqref\_and\_phi(r)>4\], we need only consider $r=3$, $4$, $5$, $6$, $8$, $10$ or $12$. We suppose, as before, that $g$ has order $m$, and we put $k=m/r$.
Consider first a ${\mathbb Q}$-irreducible subrepresentation ${\mathcal V}_d\subset
S_{\mathbb C}$, and the action of $g$ on ${\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},{\mathcal V}_d{\otimes}{\mathbb C})\subset
V$. This is $\zeta^{kc}{\mathcal V}_d$, where $\zeta$ is a primitive $m$th root of unity, and $c$ is some integer with $0<c<r$ and $(c,r)=1$ (the eigenvalue of $g$ on ${\mathbb W}$ is $\zeta^{-kc}$. So the eigenvalues are of the form $\zeta^{b_i/m}$ for $1\le i\le
\varphi(d)$, with $0\le b_i<m$ and the $b_i$ all different mod $m$ but all equivalent mod $l$, where $l=m/d$. Clearly $$\sum_{i=1}^{\varphi(d)}\frac{b_i}{m}\ge \frac{1}{2m}l(\varphi(d)-1)\varphi(d)
=\frac{1}{2d}(\varphi(d)-1) \varphi(d)$$ and it is easy to see that this is $\ge 1$ unless $d\in\{1,\ldots,6,8,10,12,18,30\}$.
By a slightly less crude estimate we can reduce further. For $d>2$ we write $c_{\min}(d)$ for a lower bound for the contribution to the sum $\Sigma(g)$ from ${\mathcal V}_d$ as a subrepresentation of $g$ on $S_{\mathbb C}$, [i.e.]{} $$c_{\min}(d)=\min_{0\le a<d}\sum_{0<b<d,\ (d,b)=1}\left\{\frac{b+a}{d}\right\}.$$ Note that this is a lower bound independently of $r$. For fixed $r$ one has a contribution to $\Sigma(g)$ from ${\mathcal V}_d$ of at most $$\begin{aligned}
\min_{0<c<r}\sum_{0<b<d,\ (d,b)=1}\left\{\frac{bl+kc}{m}\right\}
&=&\min_{0<c<r}\sum_{0<b<d,\ (d,b)=1}\left\{\frac{b}{d}
+\frac{kc}{m}\right\}\\
&\ge& \min_{0<c<r}\sum_{0<b<d,\
(d,b)=1}\left\{\frac{b}{d}+\frac{\lfloor kc/l \rfloor}{d}\right\}\\
&\ge& c_{\min}(d). \end{aligned}$$
It is easy to calculate that $c_{\min}(30)=92/30$ (attained when $a=19$), $c_{\min}(18)=42/18$, $c_{\min}(12)=16/12$, $c_{\min}(10)=12/10$, $c_{\min}(8)=12/8$ and $c_{\min}(5)=6/5$. But $$\label{cmins}
c_{\min}(3)=c_{\min}(6)=1/3, \quad c_{\min}(4)=1/2.$$ Hence we may assume that $r\in\{3,4,5,6,8,12\}$ and $d\in\{1,2,3,4,6\}$ for every subrepresentation ${\mathcal V}{\otimes}{\mathbb C}\subset
S_{\mathbb C}$. The summands of $T_{\mathbb C}$ are all ${\mathcal V}_r{\otimes}{\mathbb C}$. We let $\nu_d$ be the multiplicity of ${\mathcal V}_d$ in $S_{\mathbb C}$ as a $g$-module, and $\lambda$ be the multiplicity of ${\mathcal V}_r$ in $T_{\mathbb C}$. Counting dimensions gives $$\label{dimcount}
\lambda\varphi(r)+\nu_1+\nu_2+2\nu_3+2\nu_4+2\nu_6=n+2.$$ We split into two cases, depending on whether $\varphi(r)=4$ or $\varphi(r)=2$.
[**Case I.**]{} Suppose $\varphi(r)=4$, so $r\in\{5,8,10,12\}$.
If $\lambda>1$ then there will be a ${\mathcal V}_r{\otimes}{\mathbb C}$ not containing ${\mathbb W}$ and this will contribute at least $c_{\min}(r)$ to $\Sigma(g)$, just as if it were contained in $S_{\mathbb C}$ instead of $T_{\mathbb C}$. For $r=5$, $8$, $10$ or $12$ we have $c_{\min}(r)\ge 1$, so we may assume that $\lambda=1$. Moreover in these cases $\varphi(r)=4$, so equation (\[dimcount\]) becomes $$\label{dimcount_phi=4}
\nu_1+\nu_2+2\nu_3+2\nu_4+2\nu_6=n-2.$$ We may assume that $\nu_4\le 1$ and $\nu_3+\nu_6\le 2$, as otherwise those summands contribute at least $1$ to $\Sigma(g)$, by equation (\[cmins\]). The contribution from ${\mathcal V}^{\mathbf w}_r$ was computed in equation (\[W-contribution\]) above: for $\varphi(r)=4$ it is $\frac{k_1+k_3}{r}+\frac{k_1+k_4}{r}$. The contribution from a ${\mathcal V}_1$ (an invariant) is $\frac{k_1}{r}$ and from ${\mathcal V}_2$ (an anti-invariant) it is $\{\frac{k_1}{r}+\frac{1}{2}\}$.
Now we can compute all cases. The contribution from a copy of ${\mathcal V}_d$ is $$\label{d-contribution}
\sum_{(a,d)=1}\left\{\frac{a}{d}+\frac{k_1}{r}\right\}$$ or $\frac{k_1}{r}$ if $d=1$. Half the time ($k_1$ first or third in order of size) the contribution $c^{\mathbf w}$ from ${\mathcal V}^{\mathbf w}_r$ is already at least $1$. In all cases it is at least $\frac{1}{2}$, so we may also assume that $\nu_4=0$. In six of the remaining eight cases we get $\Sigma(g)\ge 1$ unless $L_{\mathbb C}={\mathbb V}_r^{\mathbf w}$ and hence $n=2$: all other possible contributions are greater than $1-c^{\mathbf w}$. The exceptions are $r=5$, $k_1=4$ and $r=10$, $k_1=3$.
For $r=5$, $k_1=4$, contributions from ${\mathcal V}_r^{\mathbf w}$, ${\mathcal V}_1$, ${\mathcal V}_2$, ${\mathcal V}_3$ and ${\mathcal V}_6$ are $\frac{3}{5}$, $\frac{4}{5}$, $\frac{3}{10}$, $\frac{3}{5}$ and $\frac{8}{5}$ respectively. So $\Sigma(g)\ge 1$ unless $\nu_1=\nu_3=\nu_6=0$ and $\nu_2\le 1$, and in particular $n\le
3$.
For $r=10$, $k_1=3$, contributions from ${\mathcal V}_r^{\mathbf w}$, ${\mathcal V}_1$, ${\mathcal V}_2$, ${\mathcal V}_3$ and ${\mathcal V}_6$ are $\frac{3}{5}$, $\frac{3}{10}$, $\frac{8}{10}$, $\frac{6}{10}$ and $\frac{6}{10}$ respectively. So $\Sigma(g)\ge 1$ unless $\nu_2=\nu_3=\nu_6=0$ and $\nu_1\le 1$, and in particular $n\le 3$.
[**Case II.**]{} Suppose $\varphi(r)=2$, so $r\in\{3,4,6\}$.
In this case one summand of $L_{\mathbb C}$ as a $g$-module is the space ${\mathbb W}\oplus{\overline}{\mathbb W}$, which is ${\mathbb V}_r^{\mathbf w}$, a copy of ${\mathcal V}_r{\otimes}{\mathbb C}$. We denote by $\nu_d$ the multiplicity of ${\mathcal V}_d$ in $L_{\mathbb C}/({\mathbb W}\oplus{\overline}{\mathbb W})$ as a $g$-module. Thus $\nu_r$ is the number of copies of ${\mathcal V}_r{\otimes}{\mathbb C}$ in $L_{\mathbb C}$ that are different from ${\mathbb V}_r^{\mathbf w}$. Equation (\[dimcount\]) becomes $$\label{dimcount_phi=2}
\nu_1+\nu_2+2\nu_3+2\nu_4+2\nu_6=n.$$ There are six cases (three values of $r$, and $k_1=1$ or $k_1=r-1$) and we simply compute all contributions in each case using the expression (\[d-contribution\]). For $1$-dimensional summands ($d=1$ or $2$) the lowest contribution is $\frac{1}{6}$ (for $r=3$, $k_1=2$, $d=2$ and for $r=6$, $k_1=1$ and $d=1$). For $2$-dimensional summands the lowest contribution is $\frac{1}{3}$ (for $r=3$, $k_1=2$, $d=3$ and for $r=6$, $k_1=1$, $d=6$). So $\Sigma(g)\ge 1$ unless $n\le 5$.
\[can\_sings\_off\_branch\_divisors\] If $n\ge 6$, then the space ${\mathcal F}_L(\Gamma)$ has canonical singularities away from the branch divisors of ${\mathcal D}_L\to
{\mathcal F}_L(\Gamma)$.
This follows at once from the Reid–Shepherd-Barron–Tai criterion (RST criterion for short) for canonical singularities: see [@Re] or [@T].
[*Remark.*]{} It is easy to classify the types of canonical singularities that can occur for small $n$, by examining the calculations above.
So far we have not considered quasi-reflections. We need to analyse not only quasi-relections themselves but also all elements some power of which acts as a quasi-reflection on $V$: note, however, that Theorem \[RST\_for\_nonqrefs\_on\_interior\] does apply to such elements.
\[qrefs\_have\_order\_2\] Suppose $n>2$. Let $g\in G$ and suppose that $h=g^k$ acts as a quasi-reflection on $V$. Then, as a $g$-module, $L_{\mathbb Q}$ is either ${\mathcal V}_k\oplus\bigoplus_j {\mathcal V}_{2k}$ or ${\mathcal V}_{2k}\oplus\bigoplus_j
{\mathcal V}_{k}$ (that is, one copy of ${\mathcal V}_k$ and some copies of ${\mathcal V}_{2k}$ or vice versa). In particular, $h$ has order $2$.
Suppose that $L_{\mathbb Q}$ decomposes as a $g$-module as ${\mathcal V}_r^{\mathbf w}\oplus\bigoplus_i {\mathcal V}_{d_i}$ for some sequence $d_i\in{\mathbb N}$. The eigenvalues of $h$ on $V$ are all equal to $1$, with exactly one exception. On the other hand, if $\zeta_r$ and $\zeta_{d_i}$ denote primitive $r$th and $d_i$th roots of unity, the eigenvalues of $h$ are certain powers of $\zeta_r$ (on ${\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},{\mathbb V}_r^{\mathbf w}\cap{\mathbb W}^\perp/{\mathbb W})$) and all numbers of the form $\alpha(h)^{-1}\zeta_{d_i}^{ka}$ for $(a,d_i)=1$.
Consider a ${\mathcal V}_d={\mathcal V}_{d_i}$ and put $d'=d/(k,d)$. The eigenvalues of $h$ on ${\mathcal V}_d$ are primitive $d'$th roots of unity: each one occurs with multiplicity exactly $\varphi(d)/\varphi(d')$. However, only two eigenvalues of $h$ may occur in any ${\mathcal V}_d$, and only one (namely $\alpha(h)$) may occur with multiplicity greater than $1$, since if $\xi$ is an eigenvalue of $h$ on ${\mathcal V}_d$, the eigenvalue $\alpha(h)^{-1}\xi$ occurs with the same multiplicity on $V$. Hence $\varphi(d')\le 2$, and if $\varphi(d')=2$ then $\varphi(d)=2$: this last can occur at most once.
Let us consider first the case where for some $d$ we have $\varphi(d)=\varphi(d')=2$. We claim that in this case $n=2$. We must have $d=6$ and $(k,d)=2$, and therefore $\alpha(h)=\omega$, a primitive cube root of unity. There can be no other ${\mathcal V}_d$ summands ([i.e.]{} summands not containing ${\mathbb W}$), because such a ${\mathcal V}_d$ would have $\varphi(d)=1$ and hence give rise to an eigenvalue $\pm\omega^2$ for $h$ on $V$; but the ${\mathcal V}_6$ already gives rise to an eigenvalue for $h$ on $V$ different from $1$. So $L_{\mathbb Q}={\mathcal V}_r^{\mathbf w}\oplus{\mathcal V}_6$. The eigenvalues of $h$ on ${\mathcal V}_r^{\mathbf w}$ are $\omega$ and $\omega^2$, each with multiplicity $\varphi(r)/2$: so $\varphi(r)=2$, otherwise $h$ has the eigenvalue $\omega$ with multiplicity $>1$ on $V$. Hence ${\mathop{\mathrm {rank}}\nolimits}L=4$ and $n=2$.
Since we are assuming that $n\ge 6$, we have $\varphi(d')=1$ for all $d$: that is, the eigenvalues of $h$ on the ${\mathcal V}_d$ part are all $\pm 1$. Put $r'=r/(k,r)$. We claim that $\varphi(r')=1$.
Suppose instead that $\varphi(r')\ge 2$, so $\alpha(h)\neq \pm 1$. Then $\varphi(r)/\varphi(r')\le 2$, since the multiplicity of $\alpha(h)^{-2}\neq 1$ as an eigenvalue of $h$ on $V$ is at least $\varphi(r)/\varphi(r')-1$. But the eigenvalues of $h$ on ${\mathcal V}_r^{\mathbf w}$ are the primitive $r'$th roots of unity. If $\varphi(r')>2$ then these include $\alpha(h)$, $\alpha(h)^{-1}$, $\xi$ and $\xi^{-1}$ for some $\xi$, these being distinct. But then the eigenvalues of $h$ on $V$ include $\alpha(h)^{-1}\xi$ and $\alpha(h)^{-1}\xi^{-1}$, neither of which is equal to $1$. So $\varphi(r')\le 2$
Moreover, if $\varphi(r)/\varphi(r')=2$ then $h$ has the eigenvalue $\alpha(h)^{-2}\neq 1$ on $V$, and any ${\mathcal V}_d$ will give rise to the eigenvalue $\pm\alpha(h)^{-1}\neq 1$; so no such components occur, and $L_{\mathbb Q}={\mathcal V}_r^{\mathbf w}$. Moreover, $\varphi(r)\le 4$ so $n\le 2$.
This shows that if $h$ is a quasi-reflection and $\varphi(r')>1$ then $\varphi(r')=2$; moreover if $n>2$ then $\varphi(r)=\varphi(r')=2$. Hence, if $\varphi(r')>1$, we have $r=6$ and $(r,k)=2$, so again $\alpha(h)=\omega$, a primitive cube root of unity. This time ${\mathbb W}\oplus{\overline}{\mathbb W}={\mathbb V}_r^{\mathbf w}$, so the eigenvalues of $h$ on $V$ all arise from ${\mathcal V}_d$ and since $\varphi(d')=1$ they are equal to $\pm\omega^2\neq 1$. So there is only one of them, that is, $n=1$.
Since we suppose $n>2$, it follows that $\varphi(r')=1$. The theorem follows immediately from this.
\[qrefs\_come\_from\_refs\_in\_L\] The quasi-reflections on $V$, and hence the branch divisors of ${\mathcal D}_L\to{\mathcal D}_F(\Gamma)$, are induced by elements $h\in {\mathop{\null\mathrm {O}}\nolimits}(L)$ such that $\pm h$ is a reflection with respect to a vector in $L$.
The two cases are distinguished by whether $\alpha(h)=\pm 1$. If $\alpha(h)=1$ then the eigenvalues of $h$ on $L_{\mathbb C}$ are $+1$ with multiplicity $1$ and $-1$ with multiplicity $n+1$, so $-h$ is a reflection; if $\alpha(h)=-1$, they are the other way round.
Now suppose that $g\in G$ and that $g^k=h$ is a quasi-reflection, $k>1$. By Theorem \[qrefs\_have\_order\_2\], $h$ has order $2$ so $g$ has order $2k$. We may suppose that the eigenvalues of $g$ on $V$ are $\zeta^{a_1},\ldots,\zeta^{a_n}$, where $\zeta$ is a primitive $2k$th root of unity, $0\le a_i<2k$, $a_n$ is odd and $a_i$ is even for $i<n$.
We need to look at the action of the group ${{\langle{g}\rangle}} /{{\langle{h}\rangle}}$ on $V':=V/{{\langle{h}\rangle}}$. The eigenvalues of $g^l{{\langle{h}\rangle}}$ on $V'$ are $\zeta^{la_1},\ldots,\zeta^{la_{n-1}}, \zeta^{2la_n}$, and we define $$\label{Sigmadash}
\Sigma'(g^l):=\left\{\frac{la_n}{k}\right\}+
\sum_{i=1}^{n-1}\left\{\frac{la_i}{2k}\right\}.$$
\[modified\_RST\] ${\mathcal F}_L(\Gamma)$ has canonical singularities if $\Sigma(g)\ge 1$ for every $g\in\Gamma$ no power of which is a quasi-reflection, and $\Sigma'(g^l)\ge 1$ if $g^k=h$ is a quasi-reflection and $1\le l<k$.
It is easy to see that if $V/{{\langle{g}\rangle}}$ has canonical singularities for every $g\in G$ then $V/G$ has canonical singularities (the converse is false). This follows from the fact that a $G$-invariant form extends to a resolution of $V/G$ if and only if it extends to a resolution of every $V/{{\langle{g}\rangle}}$, which is [@T Proposition 3.1].
If no power of $g$ is a quasi-reflection on $V$ we simply apply the RST criterion. Otherwise, consider $g$ with $g^k=h$ a quasi-reflection as above. By Corollary \[qrefs\_come\_from\_refs\_in\_L\], $V'$ is smooth, and $V/{{\langle{g}\rangle}}{\cong}V'/({{\langle{g}\rangle}}/{{\langle{h}\rangle}})$. So the result follows by applying the RST criterion to the elements $g^l{{\langle{h}\rangle}}$ acting on $V'$.
\[g\^l\_satisfies\_modified\_RST\] If $g^k=h$ is a quasi-reflection and $n\ge 7$ then $\Sigma'(g^l)\ge
1$ for every $1\le l<k$.
In fact we shall show that $\sum_{i=1}^{n-1}\{\frac{la_i}{2k}\}\ge
1$. As in Corollary \[qrefs\_come\_from\_refs\_in\_L\] we have $\alpha(h)=\pm 1$ and this is a primitive $r'$th root of unity; so all the eigenvalues of $h$ on ${\mathcal V}_r^{\mathbf w}$ are equal to $\alpha(h)$. Here, as usual, ${\mathbb W}\oplus{\overline}{\mathbb W}\subset{\mathbb V}_r^{\mathbf w}$ (two copies of ${\mathcal V}_r\otimes{\mathbb C}$ if $r|2$) and we have decomposed $L_{\mathbb C}$ as a $g$-module into ${\mathbb Q}$-irreducible pieces. But exactly one eigenvalue of $h$ on $L_{\mathbb C}$ is $-\alpha(h)=\mp 1$, and this must occur on some summand ${\mathcal V}_d$.
The eigenvalues of $g$ on ${\mathcal V}_d$ are primitive $d$th roots of unity, and in particular they all have the same order. Therefore the eigenvalues of $h$ are either all equal to $1$ (if $\alpha(h)=-1$ and $d|k$) or all equal to $-1$ (if $\alpha(h)=1$ and $d|2k$ but $d$ does not divide $k$). Since the eigenvalue $-\alpha(h)$ on $L_{\mathbb C}$ has multiplicity $1$, it follows that $\varphi(d)=1$, [i.e.]{} $d=1$ or $d=2$.
The eigenvector in $V$ corresponding to $\zeta^{a_n}$ comes from ${\mathcal V}_d$, [i.e.]{} its span is the space ${\mathop{\mathrm {Hom}}\nolimits}({\mathbb W},{\mathcal V}_d{\otimes}{\mathbb C})\subset V$. If we choose a primitive generator $\delta$ of ${\mathcal V}_d\cap L$ we have $\delta^2<0$ since ${\mathcal V}_d\subset U_{\mathbb Q}^\perp$ as in Lemma \[S\_T\_are\_disjoint\], so $L'=\delta^\perp$ is of signature $(2,n-1)$ and ${{\langle{g}\rangle}}/{{\langle{h}\rangle}}$ acts on $L'$ as a subgroup of ${\mathop{\null\mathrm {O}}\nolimits}^+(L')$. But then $\Sigma'(g^l)=\{\frac{la_n}{k}\}+\Sigma(g^l{{\langle{h}\rangle}})$ where $g^l{{\langle{h}\rangle}}\in{\mathop{\null\mathrm {O}}\nolimits}^+(L')$. It is clear that $g^l{{\langle{h}\rangle}}$ cannot be a quasi-reflection on $L'$: if it were, then by Corollary \[qrefs\_come\_from\_refs\_in\_L\] the eigenvalues of $g^l$ on $L'$ are all $\pm 1$, and so is its eigenvalue on ${\mathcal V}_d$, so it has order dividing $2$; so $g^l\in {{\langle{h}\rangle}}$.
Now we apply Theorem \[RST\_for\_nonqrefs\_on\_interior\] to $L'$, using $n-1\ge 6$.
\[can\_sings\_on\_interior\] If $n\ge 7$ then ${\mathcal F}_L(\Gamma)$ has canonical singularities.
Dimension $0$ cusps
-------------------
We now consider the boundary ${\overline}{\mathcal F}_L(\Gamma)\setminus{\mathcal F}_L(\Gamma)$. Boundary components in the Baily-Borel compactification correspond to totally isotropic subspaces $E\subset L_{\mathbb Q}$. Since $L$ has signature $(2,n)$, the dimension of $E$ is $1$ or $2$, corresponding to dimension $0$ and dimension $1$ boundary components respectively. In this section we consider the case $\dim E=1$, that is, isotropic vectors in $L$.
For a cusp $F$ (of any dimension) we denote by $U(F)$ the unipotent radical of the stabiliser subgroup $N(F)\subset \Gamma_{\mathbb R}$ and by $W(F)$ its centre. We let $N(F)_{\mathbb C}$ and $U(F)_{\mathbb C}$ be the complexifications and put $N(F)_{\mathbb Z}=N(F)\cap \Gamma$ and $U(F)_{\mathbb Z}=U(F)\cap \Gamma$.
A toroidal compactification over a $0$-dimensional cusp $F$ coming from a $1$-dimensional isotropic subspace $E$ corresponds to an admissible fan $\Sigma$ in some cone $C(F)\subset U(F)$. We have, as in [@AMRT] $${\mathcal D}_L(F)=U(F)_{\mathbb C}{\mathcal D}_L\subset \check{\mathcal D}_L$$ and in this case $${\mathcal D}_L(F)\cong F\times U(F)_{\mathbb C}=U(F)_{\mathbb C}.$$ Put $M(F)=U(F)_{\mathbb Z}$ and define the torus ${\mathbf T}(F)=U(F)_{\mathbb C}/M(F)$. In general $({\mathcal D}_L/M(F))_\Sigma$ is by definition the interior of the closure of ${\mathcal D}_L/M(F)$ in ${\mathcal D}_L(F)/M(F){\times}_{{\mathbf T}(F)}
X_\Sigma(F)$, [i.e.]{} in $X_\Sigma(F)$ in this case, where $X_\Sigma(F)$ is the torus embedding corresponding to the torus ${\mathbf T}(F)$ and the fan $\Sigma$. We may choose $\Sigma$ so that $X_\Sigma(F)$ is smooth and $G(F):=N(F)_{\mathbb Z}/U(F)_{\mathbb Z}$ acts on $({\mathcal D}_L/M(F))_\Sigma$. The toroidal compactification is locally isomorphic to $X_\Sigma(F)/G(F)$. Thus the problem of determining the singularities is reduced to a question about toric varieties. The result we want will follow from Theorem \[can\_sings\_on\_toric\_quotient\], below. We also need to consider possible fixed divisors in the boundary.
We take a lattice $M$ of dimension $n$ and denote its dual lattice by $N$. A fan $\Sigma$ in $N{\otimes}{\mathbb R}$ determines a toric variety $X_\Sigma$ with torus ${\mathbf T}={\mathop{\mathrm {Hom}}\nolimits}(M,{\mathbb C}^*)=N{\otimes}{\mathbb C}^*$.
\[can\_sings\_on\_toric\_quotient\] Let $X_\Sigma$ be a smooth toric variety and suppose that a finite group $G<{\mathop{\mathrm {Aut}}\nolimits}({\mathbf T})={\mathop{\mathrm {GL}}\nolimits}(M)$ of torus automorphisms acts on $X_\Sigma$. Then $X_\Sigma/G$ has canonical singularities.
It is enough to show that for each $x\in X_\Sigma$ and for each $g\in {\mathop{\mathrm {Stab}}\nolimits}_G(x)$, the quotient $X_\Sigma/{{\langle{g}\rangle}}$ has canonical singularities at $x$.
We consider the subtorus ${\mathbf T}_0={\mathop{\mathrm {Stab}}\nolimits}_{\mathbf T}(x)$ of ${\mathbf T}$, which is given by ${\mathbf T}_0=N_0{\otimes}{\mathbb C}^*$ for some sublattice $N_0\subset
N$, and the quotient torus ${\mathbf T}_1={\mathbf T}/{\mathbf T}_0$. The orbit ${\mathop{\mathrm {orb}}\nolimits}(x)={\mathbf T}.x$ of $x$ is isomorphic to ${\mathbf T}_1$: it corresponds to a cone $\sigma\in\Sigma$ of dimension $$s=\dim\sigma=\dim {\mathbf T}_1 = {\mathop{\mathrm {codim}}\nolimits}{\mathop{\mathrm {orb}}\nolimits}(x),$$ and $N_0$ is the lattice generated by $\sigma\cap N$. More explicitly, ${\mathop{\mathrm {orb}}\nolimits}(x)$ is given locally near $x$ by the equations $\xi_i=0$, where $\xi_i$ are coordinates on ${\mathbf T}_0$. The quotient torus ${\mathbf T}_1$ is naturally isomorphic to $N_1{\otimes}{\mathbb C}^*$, where $N_1=N/N_0$ which is a lattice because $X_\Sigma$ is smooth.
Certainly $x$ determines ${\mathop{\mathrm {orb}}\nolimits}(x)$ and therefore $\sigma$, so $g$ stabilises $\sigma$. If $U_\sigma={\mathop{\mathrm {Hom}}\nolimits}(M\cap \check\sigma,{\mathbb C}^*)$ (semigroup homomorphisms) is the corresponding ${\mathbf T}$-invariant open set, then $U_\sigma$ is $g$-invariant and the tangent spaces to $U_\sigma$ and to $X_\Sigma$ at $x$ are the same: we denote this tangent space by $V$. Choosing a basis for $N_0$ and extending it to a basis for $N$ gives an isomorphism of $U_\sigma$ with ${\mathbb C}^s\times
({\mathbb C}^*)^{n-s}$ (compare [@Od Theorem 1.1.10]). Since $g$ preserves $N_0$ it acts on both factors, by permuting the coordinates and by torus automorphisms respectively. Thus $$V=(N_0{\otimes}{\mathbb C})\oplus{\mathop{\mathrm {Lie}}\nolimits}({\mathbf T}_1)
=(N_0{\otimes}{\mathbb C})\oplus(N_1{\otimes}{\mathbb C})=V_0\oplus V_1$$ as a $g$-module, which is thus defined over ${\mathbb Q}$.
Since $V$ is defined over ${\mathbb Q}$, we may decompose it as a direct sum of ${\mathcal V}_d$s as a $g$-module, with each $d$ dividing $m$, the order of $g$.
Note that if $g$ acts as a quasi-reflection, with eigenvalues $(1,\ldots,1,\zeta)$ then since $g\in{\mathop{\mathrm {GL}}\nolimits}(N)={\mathop{\mathrm {GL}}\nolimits}_n({\mathbb Z})$ we have ${\mathop{\mathrm {tr}}\nolimits}(g)=\zeta+n-1\in{\mathbb Z}$, and therefore $\zeta=-1$ and $g$ is a reflection.
We define $\Sigma(g)$ as we did in equation (\[RST\]) above, and in the event that some power of $g$, say $h=g^k$, acts as a quasi-reflection we define $V'=V/{{\langle{h}\rangle}}$ and $\Sigma'(g^l)$ as we did in equation (\[Sigmadash\]). Now the theorem follows from Proposition \[toric\_RST\] and Proposition \[toric\_modified\_RST\], below.
Note that we only needed to choose $\Sigma$ smooth: no further subdivision is necessary.
A version of Theorem \[can\_sings\_on\_toric\_quotient\] is stated in [@S-B] and proved in [@Sn]. There the variety $X_\Sigma$ is itself allowed to have canonical singularities, but $G$ is assumed to act freely in codimension $1$.
\[toric\_RST\] If $g\in G$ is not the identity, then unless $g$ acts as a reflection, $\Sigma(g)\ge 1$.
If $V$ contains a ${\mathcal V}_d$ with $\varphi(d)>1$ then $g$ has a conjugate pair of eigenvalues and they contribute $1$ to $\Sigma(g)$. The same is true if $V$ contains two copies of ${\mathcal V}_2$. If neither of these is true, then $V={\mathcal V}_2\oplus(n-1){\mathcal V}_1$ and $g$ is a reflection.
\[toric\_modified\_RST\] If $g^k=h$ acts as a reflection, and $g$ has order $m=2k>2$, then $\Sigma'(g^l)\ge 1$ for $1\le l<k$.
Since $m>2$, certainly $V$ contains a ${\mathcal V}_d$ with $\varphi(d)\ge
2$. In such a summand, the eigenvalues of any power of $g$ come in conjugate pairs: in particular, this is true for the eigenvalues of $h$. Therefore the eigenvalues of $h$ on ${\mathcal V}_d$ are equal to $1$ if $\varphi(d)\ge 2$, since the eigenvalue $-1$ occurs with multiplicity $1$. Therefore a pair of conjugate eigenvalues of $g^l$ on ${\mathcal V}_d$ contribute $1$ to $\Sigma'(g^l)$.
\[no\_fixed\_toric\_divisors\] Let $X_\Sigma$ and $g$ be as above. Then there is no divisor in the boundary $X\setminus{\mathbf T}$ that is fixed pointwise by a non-trivial element of ${{\langle{g}\rangle}}$.
Suppose $D$ were such a divisor, fixed pointwise by some element $h\in G$. Then $D$ corresponds to a $1$-parameter subgroup $\lambda\colon{\mathbb C}^*\to {\mathbf T}$. Moreover, $D$ is a toric divisor and is itself a toric variety with dense torus ${\mathbf T}/\lambda({\mathbb C}^*)$. Thus $h\in{\mathop{\mathrm {GL}}\nolimits}(M){\cong}{\mathop{\mathrm {GL}}\nolimits}_n({\mathbb Z})$ acts trivially on ${\mathbf T}/\lambda({\mathbb C}^*)$; but the only such element is $\lambda(t)\mapsto \lambda(t^{-1})$, which does not preserve $D$.
\[can\_sings\_on\_dim0\_cusps\] The toroidal compactification ${\overline}{\mathcal F}_L(\Gamma)$ may be chosen so that on a boundary component over a dimension $0$ cusp, ${\overline}{\mathcal F}_L(\Gamma)$ has canonical singularities, and there are no fixed divisors in the boundary.
Since $\Sigma$ is $G(F)$-invariant, the result follows immediately from Theorem \[can\_sings\_on\_toric\_quotient\] and Lemma \[no\_fixed\_toric\_divisors\]
\[no\_fixed\_divisors\_at\_dim0\] There are no divisors at the boundary over a dimension $0$ cusp $F$ that are fixed by a nontrivial element of $G(F)$.
Note that in this subsection we needed no restriction on $n$.
Dimension $1$ cusps
-------------------
It remains to consider the dimension $1$ cusps. Here we have to be more explicit: we consider a rank $2$ totally isotropic subspace $E_{\mathbb Q}\subset L_{\mathbb Q}$, corresponding to a dimension $1$ boundary component $F$ of ${\mathcal D}_L$. We want to choose standard bases for $L_{\mathbb Q}$ so as to be able to identify $U(F)$, $U(F)_{\mathbb Z}$ and $N(F)_{\mathbb Z}$ explicitly, as is done in [@Sc] for maximal ${\mathop{\mathrm {K3}}\nolimits}$ lattices, where $n=19$. But we shall not be able to choose suitable bases of $L$ itself, as in [@Sc]. The first steps, however, can be done over ${\mathbb Z}$. We define $E=E_{\mathbb Q}\cap L$ and $E^\perp=E^\perp_{\mathbb Q}\cap L$, primitive sublattices of $L$.
\[special\_basis\_for\_L\] There exists a basis ${\mathbf e}_1',\ldots,{\mathbf e}_{n+2}'$ for $L$ over ${\mathbb Z}$ such that ${\mathbf e}_1',{\mathbf e}_2'$ is a basis for $E$ and ${\mathbf e}_1',\ldots,{\mathbf e}_n'$ is a basis for $E^\perp$. Furthermore we can choose ${\mathbf e}_1',\ldots,{\mathbf e}_{n+2}'$ so that $$A=\begin{pmatrix}\delta&0\\ 0&\delta e\end{pmatrix}$$ for some integers $\delta$ and $e$, where $A$ is defined by $$Q':=({\mathbf e}_i',{\mathbf e}_j')=\begin{pmatrix}0&0&A\\ 0&B&C\\
{}^t A&{}^tC&D\end{pmatrix}.$$
We can find a basis with all the properties except for the special form of $A$ by choosing any bases for the primitive sublattices $E$ and $E^\perp$ of $L$. Then the matrix $A$ may be chosen to have the special form given by choosing ${\mathbf e}_1'$, ${\mathbf e}_2'$, ${\mathbf e}_{n+1}'$ and ${\mathbf e}_{n+2}'$ suitably: the numbers $\delta$ and $\delta e$ are the elementary divisors of $A\in{\mathop{\mathrm {Mat}}\nolimits}_{2\times 2}({\mathbb Z})$.
If we are willing to allow two of the basis vectors to be in $L_{\mathbb Q}$ we can achieve much more.
\[good\_Q\_basis\] There is a basis ${\mathbf e}_1,\ldots,{\mathbf e}_{n+2}$ for $L_{\mathbb Q}$ such that ${\mathbf e}_1$ and ${\mathbf e}_2$ form a ${\mathbb Z}$-basis for $E$, and ${\mathbf e}_1,\ldots,{\mathbf e}_n$ form a ${\mathbb Z}$-basis for $E^\perp$, for which $$Q:=({\mathbf e}_i,{\mathbf e}_j)=\begin{pmatrix}
0&0&A\\ 0&B&0\\
A&0&0 \end{pmatrix}$$ with $A$ and $B$ as before.
We start with the basis ${\mathbf e}_1',\ldots,{\mathbf e}_{n+2}'$ from Lemma \[special\_basis\_for\_L\]. Note that $B\in{\mathop{\mathrm {Mat}}\nolimits}_{n-2 \times
n-2}$ has non-zero determinant, because it represents the quadratic form of $L$ on $E_{\mathbb Q}^\perp/E_{\mathbb Q}$. So we put $R=-B^{-1}C\in{\mathop{\mathrm {Mat}}\nolimits}_{n-2\times 2}({\mathbb Q})$ and we take ${\mathbf e}_i$ consisting of the columns of $$N:=\begin{pmatrix}I&0&R'\\ 0&I&R\\ 0&0&I\end{pmatrix},$$ where $R'$ is chosen to satisfy $$D-{}^tCB^{-1}C+{}^tR'A+{}^tAR'=0.$$ Then ${\mathbf e}_i$ is a ${\mathbb Q}$-basis for $L_{\mathbb Q}$ including ${\mathbb Z}$-bases for $E$ and $E^\perp$, as we want, and ${}^tNQ'N=Q$ as required.
\[describe\_NUW\_for\_dim\_1\] The subgroups $N(F)$, $W(F)$ and $U(F)$ are given by $$N(F)=\left\{\begin{pmatrix}U&V&W\\ 0&X&Y\\ 0&0&Z\end{pmatrix}\mid
\begin{matrix}{}^t UAZ=A, {}^tXBX=B, {}^tXBY+{}^tVAZ=0,\\
{}^tYBY+{}^tZAW+{}^tWAZ=0,\ \det U>0\end{matrix}\right\},$$ $$W(F)=\left\{\begin{pmatrix}I&V&W\\ 0&I&Y\\ 0&0&I\end{pmatrix}\mid
BY+{}^tVA=0,\ {}^tYBY+AW+{}^tWA=0\right\},$$ and $$U(F)=\left\{\begin{pmatrix}I&0&\begin{pmatrix}0&ex\\ -x&0\end{pmatrix}\\
0&I&0\\ 0&0&I\end{pmatrix}\mid x\in{\mathbb R}\right\}.$$
This is a straightforward calculation.
As in [@Ko1] we realise ${\mathcal D}_L$ as a Siegel domain and ${\mathcal D}_L(F)=U(F)_{\mathbb C}{\mathcal D}_L(F)$ is identified with ${\mathbb C}\times{\mathbb C}^{n-2}\times {\mathbb H}$. The identification is by choosing homogeneous coordinates $(t_1\colon\ldots\colon t_{n+2})$ on ${\mathbb P}(L_{\mathbb C})$ so that $t_{n+2}=1$ and mapping $t_1\mapsto z\in {\mathbb C}$, $t_{n+1}\mapsto\tau\in{\mathbb H}$ and $t_i\mapsto w_{i-2}\in{\mathbb C}$ for $3\le i
\le n$: the value of $t_2$ is determined by the equation $$\label{siegeldomain}
2\delta et_2=-2\delta z\tau-{}^t\underline{w}B\underline{w}$$ where $\underline{w}\in{\mathbb C}^{n-2}$ is a column vector.
We are interested in the action of $N(F)_{\mathbb Z}=N(F)\cap \Gamma$ on ${\mathcal D}_L(F)$. We denote by ${\underline V}_i$ the $i$th row of the matrix $V$ in Lemma \[describe\_NUW\_for\_dim\_1\].
\[action\_of\_NFZ\] If $g\in N(F)$ is given by $$\begin{pmatrix}U&V&W\\ 0&X&Y\\ 0&0&Z\end{pmatrix},\qquad
Z=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$$ then $g$ acts on ${\mathcal D}_L(F)$ by $$\begin{aligned}
z&{\mapstochar\longrightarrow}& \frac{z}{\det Z}+(c\tau+d)^{-1}\bigg(\frac{c}{2\delta\det Z}
{}^t\underline{w}B\underline{w}
+\underline{V}_1\underline{w}+W_{11}\tau+W_{12}\bigg)\\
\underline{w}&{\mapstochar\longrightarrow}&(c\tau+d)^{-
1}\left(X\underline{w}+Y\begin{pmatrix}\tau\\ 1\end{pmatrix}\right)\\
\tau&{\mapstochar\longrightarrow}& (a\tau+b)/(c\tau+d). \end{aligned}$$
This is also a straightforward calculation. One need only take into account that $$U=\frac{1}{\det Z} \begin{pmatrix}d & -ce\\ b/e & a\end{pmatrix}$$
We must now describe $N(F)_{\mathbb Z}$ and $U(F)_{\mathbb Z}$.
\[Z\_is\_integral\] If $g\in N(F)_{\mathbb Z}$ then $Z\in {\mathop{\mathrm {SL}}\nolimits}_2({\mathbb Z})$, and if $g\in U(F)_{\mathbb Z}$ then $x\in{\mathbb Z}$.
For $Z$, it is enough to show that $Z\in{\mathop{\mathrm {Mat}}\nolimits}_{2\times 2}({\mathbb Z})$, since it acts on ${\mathbb H}$. The condition that $g\in N(F)_{\mathbb Z}$ or $g\in
U(F)_{\mathbb Z}$ is that $N^{-1}gN\in \Gamma$ and in particular ${N'}^{-1}gN\in {\mathop{\mathrm {GL}}\nolimits}_{n+2}({\mathbb Z})$. We calculate this directly: $$N^{-1}gN=\begin{pmatrix}U&V&VB^{-1}C+W-UT+TZ\\ 0&X&Y+XB^{-1}C
-B^{-1}CZ\\ 0&0&Z\end{pmatrix},$$ so $Z$ is integral. In fact, because of ${}^tUAZ=A$ we even have $Z\in
\Gamma_0(e)$.
If $g\in U(F)_{\mathbb C}$ we have in addition $V=0$, $Y=0$, $U=Z=I_2$ and $X=I_{n-2}$, so $VB^{-1}C+W-UT+TZ=W$ and therefore $W$ is integral.
Now we can calculate the action on the tangent space at a point in the boundary. Suppose $g\in G(F)=N(F)_{\mathbb Z}/U(F)_{\mathbb Z}$ has finite order $m>1$. We abuse notation by also using $g$ to denote a corresponding element of $N(F)_{\mathbb Z}$. We choose a coordinate $u=\exp_e(z):=e^{2\pi i
z/e}$ on $U(F)_{\mathbb C}/U(F)_{\mathbb Z}{\cong}{\mathbb C}^*$, where $e$ is as in Lemma \[special\_basis\_for\_L\], because $g\in U(F)_{\mathbb Z}$ acts by $z\mapsto z+ex$. The compactification is given by allowing $u=0$. We suppose that $g$ fixes the point $(0, \underline{w}_0,\tau_0)$. We define $\Sigma(g)$ as we did before, in equation (\[RST\]), as $\sum\{\frac{a_i}{m}\}$ if the eigenvalues are $\zeta^{a_i}$ for $\zeta=e^{2\pi i/m}$.
\[RST\_for\_nonqrefs\_at\_dim1\] If $n\ge 8$ and no power of $g$ acts as a quasi-reflection at $(0,
\underline{w}_0,\tau_0)$ then $\Sigma(g)\ge 1$.
This closely follows [@Ko1 (8.2)]. The action of $g$ on the tangent space is given by $$\begin{pmatrix}\exp_e(t)&0&0\\ *&(c\tau_0+d)^{-1}X&0\\ *&*&(c\tau_0+d)^{-
2}\end{pmatrix}$$ where $t=(c\tau_0+d)^{-1}(c{}^t\underline{w}_0 B\underline{w}_0/2
+\underline{V}_1\underline{w}_0+W_{11}\tau_0+W_{12})/e$, by Lemma \[action\_of\_NFZ\]. Observe that $c\tau_0+d=\xi$ is a (not necessarily primitive) fourth or sixth root of unity, because of the well-known fixed points of ${\mathop{\mathrm {SL}}\nolimits}_2({\mathbb Z})$ on ${\mathbb H}$.
Suppose $X$ is of order $m_X$. We consider the decomposition of the representation $X$, [i.e.]{} of $E_{\mathbb Q}^\perp/E_{\mathbb Q}$ as a $g$-module. It decomposes as a direct sum of ${\mathcal V}_d$. If $\xi\neq\pm 1$ the situation is exactly as in the case $\varphi(r)=2$ at the end of the proof of Theorem \[RST\_for\_nonqrefs\_on\_interior\], except that the right-hand side of equation (\[dimcount\_phi=2\]) is now equal to $n-2$ (that is, ${\mathop{\mathrm {rank}}\nolimits}X$) instead of $n$. Any ${\mathcal V}_d$ contributes at least $c_{\min}(d)$ to $\Sigma(g)$, so we may assume that $\varphi(d)\le 2$; but then the $1$-dimensional summands contribute at least $\frac{1}{6}$ and the $2$-dimensional ones at least $\frac{1}{3}$. Moreover, if $m_X>2$ then $X$ has a pair of conjugate eigenvalues and in the case $\xi=\pm 1$ they contribute $1$ to $\Sigma(g)$.
So we may assume that $m_X=1$ or $m_X=2$, and $\xi=\pm 1$. Since $-1\in\Gamma$ acts trivially on ${\mathcal D}_L$ we may replace $g$ by $-g$ if we prefer, and assume that $\xi=1$. Since $g$ fixes $(0,\underline{w}_0,\tau_0)$ that implies $Z=I$. If also $m_X=1$, so $X=I$, then by Proposition \[action\_of\_NFZ\] we have $$Y\begin{pmatrix}\tau_0\\ 1\end{pmatrix}=\underline{0}$$ and since $\tau_0\not\in{\mathbb Z}$ this implies $Y=0$. But then ${}^tVA=0$ by Lemma \[describe\_NUW\_for\_dim\_1\], so $g\in U(F)_{\mathbb Z}$.
So the remaining possibility is that $Z=I$ and $m_X=2$: thus $U=I$ since ${}^tUAZ=A$, and $c=0$. But then $t$ is a half-integer, because $$\underline{w}_0=X\underline{w}_0+Y\begin{pmatrix}\tau_0\\ 1\end{pmatrix}$$ and the condition $g^2\in U(F)_{\mathbb Z}$ implies that $VX=-V$, that $XY=-Y$ and that $$2W\equiv -VY \bmod \begin{pmatrix}0&e\\ -1&0\end{pmatrix}.$$ So, modulo $e{\mathbb Z}$, we have $$\begin{aligned}
2t&=&2\underline{V}_1\underline{w}_0+2W_{11}\tau_0+2W_{12}\\
&\equiv& 2\underline{V}_1\underline{w}_0-\underline{V}_1Y
\begin{pmatrix}\tau_0\\ 1\end{pmatrix}\\
&\equiv&\underline{V}_1(I+X)\underline{w}_0\\
&\equiv&0. \end{aligned}$$ Thus the eigenvalue $\exp_e(t)$ is $\pm 1$, so in this case all eigenvalues on the tangent space are $\pm 1$ and either $\Sigma(g)\ge
1$ or $g$ acts as a reflection. In particular any quasi-reflections have order $2$.
\[no\_fixed\_divisors\_at\_dim1\] There are no divisors at the boundary over a dimension $1$ cusp $F$ that are fixed by a nontrivial element of $G(F)$.
From the proof of Proposition \[RST\_for\_nonqrefs\_at\_dim1\], any quasi-reflection $g$ has $m_X=2$, and hence fixes a divisor different from $u=0$.
Finally we check the analogue of Proposition \[g\^l\_satisfies\_modified\_RST\]. We define $\Sigma'(g)$ for $g\in G(F)$ exactly as in equation (\[Sigmadash\]).
\[g\^l\_satisfies\_modified\_RST\_at\_dim1\] If $g\in G(F)$ is such that $g^k=h$ is a reflection and $n\ge 9$ then $\Sigma'(g^l)\ge 1$ for every $1\le l<k$.
If the unique eigenvalue of $h$ that is different from $1$ (hence equal to $-1$) is $\exp_e(t)$ then the contribution from $X^l$ to $\Sigma'(g)$ is at least $1$. Otherwise, consider the ${\mathcal V}_d$ (in the decomposition as a $g$-module) in which the exceptional eigenvector ${\mathbf e}_0$ occurs, satisfying $h({\mathbf e}_0)=-{\mathbf e}_0$. We must have $d=1$ or $d=2$, since if $\varphi(d)>1$ the eigenvalue $-1$ for $h$ would occur more than once. But the rest of $X$ ([i.e.]{} the $(n-3)$-dimensional $g$-module $E_{\mathbb Q}^\perp/(E+{\mathbb Q}\,{\mathbf e}_0)$) contributes at least $1$ to $\Sigma(g)$ and hence to $\Sigma'(g)$, as long as $n-3\ge 6$, as was shown in Proposition \[RST\_for\_nonqrefs\_at\_dim1\].
\[can\_sings\_on\_dim1\_cusps\] If $n\ge 9$, the toroidal compactification ${\overline}{\mathcal F}_L(\Gamma)$ may be chosen so that on a boundary component over a dimension $1$ cusp, ${\overline}{\mathcal F}_L(\Gamma)$ has canonical singularities, and there are no fixed divisors in the boundary.
This is immediate from Corollary \[no\_fixed\_divisors\_at\_dim1\], Proposition \[RST\_for\_nonqrefs\_at\_dim1\] and Proposition \[g\^l\_satisfies\_modified\_RST\_at\_dim1\]. In fact there are no choices to be made in this part of the boundary.
Special reflections in ${\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}(L)$ {#reflections}
==============================================================================
Let $L$ be an arbitrary nondegenerate integral lattice, and write $D$ for the exponent of the finite group $A_L=L^\vee/L$. The reflection with respect to the hyperplane defined by a vector $r$ is given by $$\sigma_r\colon l{\mapstochar\longrightarrow}l-\frac{2(l,r)}{(r,r)}r.$$ For any $l\in L$ its [*divisor*]{} ${\mathop{\null\mathrm {div}}\nolimits}(l)$ in $L$ is the positive generator of the ideal $(l,L)$. In other words $l^*=l/{\mathop{\null\mathrm {div}}\nolimits}(l)$ is a primitive element of the dual lattice $L^\vee$. If $r$ is primitive and the reflection $\sigma_r$ fixes $L$, [i.e.]{} $\sigma_r\in
{\mathop{\null\mathrm {O}}\nolimits}(L)$, then we say that $r$ is a reflective vector. In this case $$\label{refldiv}
{\mathop{\null\mathrm {div}}\nolimits}(r)\mid r^2 \mid 2{\mathop{\null\mathrm {div}}\nolimits}(r).$$
\[reflid\] Let $L$ be a nondegenerate even integral lattice. Let $r\in L$ be primitive. Then $\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$ if and only if $r^2=\pm 2$.
For $r^*=r/{\mathop{\null\mathrm {div}}\nolimits}(r)\in L^\vee$ and $\sigma_r\in
{\widetilde}{\mathop{\null\mathrm {O}}\nolimits}(L)$ we get $$\sigma_r(r^*)=-r^*\equiv r^*\mod L.$$ Therefore $2r^*\in L$, ${\mathop{\null\mathrm {div}}\nolimits}(r)=1$ or $2$ (because $r$ is primitive) and $r^2=\pm 2$ or $\pm 4$, because $L$ is even. If $r^2=\pm 2$ then $\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$. If $r^2=\pm 4$, then ${\mathop{\null\mathrm {div}}\nolimits}(r)=2$ by condition (\[refldiv\]). For such $r$ the reflection $\sigma_r$ is in $\widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$ if and only if $$l^\vee-\sigma_r(l^\vee)=\frac{(r,l^\vee)}{2}r=(r^*,l^\vee)r\in L$$ for any $l^\vee\in L^\vee$. Therefore $r^*=r/2\in (L^\vee)^\vee=L$. We obtain a contradiction because $r$ is primitive.
\[reflminusid\] Let $L$ be as in Proposition \[reflid\] and let $r\in L$ be primitive. If $-\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$, [i.e.]{} $\sigma_r|_{A_L}=-{\mathop{\mathrm {id}}\nolimits}$, then
- $r^2=\pm 2D$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=D\equiv 1\mod 2$, or $r^2=\pm D$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=D$ or $D/2$;
- $A_L\cong ({\mathbb Z}/2{\mathbb Z})^m\times ({\mathbb Z}/D{\mathbb Z})$.
In the opposite direction we have
- If $r^2=\pm D$ and either ${\mathop{\null\mathrm {div}}\nolimits}(r)=D$ or ${\mathop{\null\mathrm {div}}\nolimits}(r)=D/2\equiv 1\mod 2$, then $-\sigma_r\in \widetilde
{\mathop{\null\mathrm {O}}\nolimits}(L)$;
- If $r^2=\pm 2D$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=D\equiv 1\mod 2$, then $-\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$.
\(i) $\sigma_r|_{A_L}=-{\mathop{\mathrm {id}}\nolimits}$ is equivalent to the following condition: $$\label{-id}
2l^\vee\equiv \frac{2(r,l^\vee)}{r^2}r\mod L
\qquad\forall\ l^\vee\in L^\vee.$$ It follows that if $r^2=2e$, then $(2L^{\vee})/ L$ is a subgroup of the cyclic group $\langle(r/e)+L\rangle$. Thus $D$ divides $2e$. But by definition of the divisor of the vector $e\mid {\mathop{\null\mathrm {div}}\nolimits}(r)\mid D$, therefore $$e\mid {\mathop{\null\mathrm {div}}\nolimits}(r)\mid 2e\quad\text{and}\quad e\mid D\mid 2e.$$ From this it follows that $(2L^\vee)/L$ is a subgroup of the cyclic group generated by $(r/D)+L$ or $(2r/D)+L$. This implies (ii).
Let us assume that $r^2=\pm 2D$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=D\equiv 0\mod 2$. We have $2l^\vee\equiv \pm \frac{(r,l^\vee)}{D}r\mod L$. If the order of $l^\vee$ in the discriminant group is odd, then $(r,l^\vee)$ is even, since $D$ is even. If the order of $l^\vee$ is even, then $(r,l^\vee)$ is again even, because the order of $2l^\vee$ is $D/2$. Therefore $(r/2,l^\vee)\in {\mathbb Z}$ for all $l^\vee\in L^\vee$. This contradicts the assumption that $r$ is primitive. Thus (i) is proved.
\(iii) Let assume that ${\mathop{\null\mathrm {div}}\nolimits}(r)=D$. In this case $r^*=r/D$ and $
2r^*+L$ is a generator of $(2L^\vee)/L$. According to (ii) we have that for any $l^\vee\in L^\vee$, $2l^\vee= 2xr^*+l'$, where $x\in
{\mathbb Z}$, $l'\in L$. Therefore $$\label{DD}
\frac{(2l^\vee,r)}{r^2}r=
2xr^*\pm \frac{(l',r)}{D}r
\equiv
2xr^*\equiv 2l^\vee\mod L$$ and $-\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$ according to condition (\[-id\]).
Let assume that ${\mathop{\null\mathrm {div}}\nolimits}(r)=D/2\equiv 1 \mod 2$. We have to check condition (\[-id\]) for all elements of order $2$ or $D$ in $A_L$. If ${\mathop{\null\mathrm {ord}}\nolimits}(l^\vee)=2$, then $(2l^\vee,r)\equiv 0\mod D/2$, and also $(l^\vee,r)\equiv 0\mod D/2$, because $D/2$ is odd. It follows that $2(l^\vee,r)/r^2\in {\mathbb Z}$. If $l^\vee$ is an element of order $D$, we have $2l^\vee= 2xr^*+l'$ as above with $r^*=(2r)/D$ and $l'\in L$. Thus $(l',r)$ is even. But $(l',r)$ is also divisible by the odd number $D/2$. Therefore $(l',r)\equiv 0\mod D$ and equation (\[DD\]) is also true.
\(iv) is similar to (iii). $D$ is odd and the group $A_L$ is cyclic with generator $r^*=r/D$. Therefore $l^\vee=xr^*+l'$ for any $l^\vee\in L^\vee$ and $$\frac{(2l^\vee,r)}{r^2}r=\frac{2(xr^*+l',r)}{r^2}r=
2xr^*\pm \frac{2(l',r)}{2D}\equiv 2l^\vee\mod L.$$
\[odddet\] Let $L$ be an even integral lattice and $|A_L|=|\det L|$ be odd. Then
- $\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L)$ if and only if $r^2=\pm
2$;
- $-\sigma_r\in \widetilde{\mathop{\null\mathrm {O}}\nolimits}(L)$ if and only if $r^2=\pm
2D$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=D$.
With ${\mathop{\mathrm {K3}}\nolimits}$ surfaces in mind, we consider in more detail the lattice $L_{2d}=2U\oplus 2E_8(-1)\oplus {{\langle{-2d}\rangle}}$.
\[reflK3\] Let $\sigma_r$ be a reflection in ${\mathop{\null\mathrm {O}}\nolimits}(L_{2d})$ defined by a primitive vector $r\in L_{2d}$. $\sigma_r$ induces $\pm{\mathop{\mathrm {id}}\nolimits}$ on the discriminant form $L_{2d}^\vee/L_{2d}$ if and only if $r^2=\pm 2$ or $r^2=\pm 2d$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=d$ or $2d$.
Any $r\in L_{2d}$ can be written as $r=m+xh$, where $m\in
L_0=2U\oplus2E_8(-1)$ and $h^2=-2d$ ($h$ is primitive).
If $r^2=\pm 2d$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=2d$, then $-\sigma_r\in
{\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}(L_{2d})$ by Proposition \[reflminusid\].
If $r^2=\pm 2d$ and ${\mathop{\null\mathrm {div}}\nolimits}(r)=d$, then $r=dm_0+xh$, where $x^2=1-d(m_0^2/2)$. We see that $$\sigma_r\left(\frac{h}{2d}\right)=\frac{h}{2d}(1-2x^2)-xm_0\equiv
-\frac{h}{2d}\mod L_{2d}.$$
The types of reflections in the full orthogonal group ${\mathop{\null\mathrm {O}}\nolimits}^+(L)$ for $L=L_{2d}^{(0)}=2U\oplus {{\langle{-2d}\rangle}}$ were classified in [@GH2] (for square-free $d$). The result for $L_{2d}=2U\oplus 2E_8(-1)\oplus
{{\langle{-2d}\rangle}}$ is exactly the same, because the unimodular part $2E_8(-1)$ plays no role in the classification.
The reflection $\sigma_r$ is an element of ${\mathop{\null\mathrm {O}}\nolimits}^+(L_{\mathbb R})$ (where $L$ has signature $(2,n)$) if and only if $r^2<0$: see [@GHS1].
The $(-2)$-vectors of $L_{2d}$ form one or two (if $d\equiv 1\mod 4$) orbits with respect to the group $\widetilde {\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d})$. We can also compute the number of $\widetilde{\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d})$-orbits of the $(-2d)$-reflective vectors in Corollary \[reflK3\]. However, in this paper we only need to know the orthogonal complements of $(-2d)$-vectors, which we compute in Proposition \[-2dvect\]. (For the case of $(-2)$-vectors see [@GHS1 §3.6]).
The following lemma, which we use in the proof of Proposition \[-2dvect\], is well-known, but we state it and give a general proof here for the convenience of the reader. Recall that an integral lattice $T$ is called $2$-elementary if $A_T=T^\vee/T\cong
({\mathbb Z}/ 2{\mathbb Z})^m$.
\[2-elementary\] Let $T$ be a primitive sublattice of an unimodular even lattice $M$, and let $S$ be the orthogonal complement of $T$ in $M$. Suppose that there is an involution $\sigma\in {\mathop{\null\mathrm {O}}\nolimits}(M)$ such that $\sigma|_T={\mathop{\mathrm {id}}\nolimits}_T$ and $\sigma|_S=-{\mathop{\mathrm {id}}\nolimits}_S$. Then $T$ and $S$ are $2$-elementary lattices.
Let us consider the inclusions $T\oplus S\subset M\subset T^\vee
\oplus S^\vee$. We have that $(A_S,\ q_{S})\cong (A_T,\ -q_{T})$ because $M$ is unimodular (see [@Nik2]). In particular $[M:T\oplus S]=[T^\vee:T]=[S^\vee:S]$. It follows that $$H=M/(T\oplus S)\cong \phi(M)/S=S^\vee/S=A_S.$$ Here $\phi\colon M\to S^\vee$ is defined by $\phi(m)(s)=(m,s)$ where $s\in S$. The natural projections of the subgroup $H<A_T\oplus A_S$ onto $A_T$ and $A_S$ are injective, therefore the action of $\sigma$ on $A_S$ is completely determined by the action of $\sigma$ on $A_T$. Thus $\sigma$ acts trivially on $A_S$ since it acts trivially on $A_T$. But we assumed that $\sigma(s^\vee)=-s^\vee$ for any $s^\vee\in S^\vee$. It follows that $A_S$ is an abelian $2$-group.
\[-2dvect\] Let $r$ be a primitive vector of $L_{2d}$. If ${\mathop{\null\mathrm {div}}\nolimits}(r)=2d$ then $$r^\perp_{L_{2d}}\cong 2U\oplus 2E_8(-1).$$ If ${\mathop{\null\mathrm {div}}\nolimits}(r)=d$ then either $$r^\perp_{L_{2d}}\cong U\oplus 2E_8(-1)\oplus {{\langle{2}\rangle}} \oplus {{\langle{-2}\rangle}}$$ or $$r^\perp_{L_{2d}}\cong U\oplus 2E_8(-1)\oplus U(2).$$
The lattice $L_{2d}$ is the orthogonal complement of a primitive vector $h$, with $h^2=2d$ in the unimodular ${\mathop{\mathrm {K3}}\nolimits}$ lattice $L_{{\mathop{\mathrm {K3}}\nolimits}}=3U\oplus 2E_8(-1)$. We put $L_r=r^\perp_{L_{2d}}$ and $S_r=(L_r)^\perp_{L_{{\mathop{\mathrm {K3}}\nolimits}}}$.
We note that $L_r$ and $S_r$ have the same determinant: in fact $$\det L_r= \det S_r=4d^2/{\mathop{\null\mathrm {div}}\nolimits}(r)^2=
\begin{cases}
1&\quad\text{if } {\mathop{\null\mathrm {div}}\nolimits}(r)=2d,\\
4&\quad\text{if } {\mathop{\null\mathrm {div}}\nolimits}(r)=d.
\end{cases}$$ To see this, consider a more general situation. Let $N$ be a primitive even nondegenerate sublattice of any even integral lattice $L$ and let $N^\perp$ be its orthogonal complement in $L$. Then we have $$N\oplus N^\perp\subset L\subset L^\vee
\subset N^\vee\oplus (N^\perp)^\vee,$$ where $L/(N\oplus N^\perp)\cong L^\vee/(N^\vee\oplus (N^\perp)^\vee)$. As before we have $\phi\colon L\to N^\vee$, and $\ker(\phi)=N^\perp$. Since $L/(N\oplus N^\perp)\cong \phi(L)/N$ we obtain $$|L/(N\oplus N^\perp)|= |\phi(L)/N|=|\det N|/[N^\vee:\phi(L)],$$ as $|\det N|=[N^\vee:N]$. From the inclusions above $$|\det N|\cdot|\det N^\perp|=(|\det L|)[\phi(M):N]^2=|\det L|\cdot|\det
N|^2/[N^\vee:\phi(L)]^2.$$
In our particular case $L=L_{2d}$, $N={\mathbb Z}r$ and $L_r=N^\perp$. We have $[N^\vee:\phi(L)]={\mathop{\null\mathrm {div}}\nolimits}(r)$, where ${\mathop{\null\mathrm {div}}\nolimits}(r){\mathbb Z}=(r,L)$, and this gives us the formula for the determinant of $L_r$.
If ${\mathop{\null\mathrm {div}}\nolimits}(r)=2d$ then $L_r$ and $S_r$ are are isomorphic to the unique unimodular lattices of signatures $(2,18)$ and $(1,1)$ respectively: that is, $L_r\cong 2U\oplus 2E_8(-1)$ and $S_r\cong U$.
If ${\mathop{\null\mathrm {div}}\nolimits}(r)=d$ then the reflection $\sigma_r$ acts as $-{\mathop{\mathrm {id}}\nolimits}$ on the discriminant group (see Corollary \[reflK3\]). Therefore we can extend $-\sigma_r\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}(L_{2d})$ to an element of ${\mathop{\null\mathrm {O}}\nolimits}(L_{{\mathop{\mathrm {K3}}\nolimits}})$ by putting $(-\sigma_r)|_{{\mathbb Z}h}={\mathop{\mathrm {id}}\nolimits}$. So $\sigma_r$ has an extension $\tilde\sigma_r\in {\mathop{\null\mathrm {O}}\nolimits}(L_{{\mathop{\mathrm {K3}}\nolimits}})$ such that $\tilde\sigma_r|_{L_r}={\mathop{\mathrm {id}}\nolimits}_{L_r}$ and $\tilde\sigma_r|_{S_r}=-{\mathop{\mathrm {id}}\nolimits}_{S_r}$. It follows from Lemma \[2-elementary\] that $L_r$ and $S_r$ are $2$-elementary lattices.
The finite discriminant forms of $2$-elementary lattices were classified by Nikulin in [@Nik3]. The genus of $M$ (and the class of $M$ if $M$ is indefinite) is determined by the signature of $M$, the number of generators $m$ of $A_M$ and the parity $\delta_M$ of the finite quadratic form $q_M\colon A_M\to {\mathbb Q}/2{\mathbb Z}$, which is given by $\delta_M= 0$ if $l^2\in {\mathbb Z}$ for all $l\in M^\vee$ and $\delta_M=1$ otherwise: (see \[Nik3, §3\]). In particular, for an indefinite lattice $S_r$ of rank $2$ and determinant $4$ we have $$S_r\cong
\begin{cases}\ U(2)&\quad\text{if}\quad \delta_{S_r}=0,\\
\ {{\langle{2}\rangle}} \oplus {{\langle{-2}\rangle}}& \quad\text{if}\quad \delta_{S_r}=1.
\end{cases}$$ The class of the indefinite lattice $L_r$ is uniquely defined by its discriminant form. Proposition \[-2dvect\] is proved.
Geometrically the three cases in Proposition \[-2dvect\] correspond to the Néron-Severi group being (generically) $U$, $U(2)$ or ${{\langle{2}\rangle}}\oplus{{\langle{-2}\rangle}}$ respectively. The ${\mathop{\mathrm {K3}}\nolimits}$ surfaces (without polarisation) themselves are, respectively, a double cover of the Hirzebruch surface $F_4$, a double cover of a quadric, and the desingularisation of a double cover of ${\mathbb P}^2$ branched along a nodal sextic.
Special cusp forms. {#specialcusp}
===================
Let $L=2U\oplus L_0$ be an even lattice of signature $(2,n)$ ($n\ge
3$) containing two hyperbolic planes. We write ${\mathcal F}_L={\mathcal F}_L(\widetilde{\mathop{\null\mathrm {O}}\nolimits}^+(L))$ for brevity. A $0$-dimensional cusp of ${\mathcal F}_L$ is defined by a primitive isotropic vector $v$. Any two primitive isotropic vectors of divisor $1$ lie in the same ${\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L)$-orbit, according to the well-known criterion of Eichler (see [@E §10]). We call the corresponding cusp the [*standard $0$-dimensional cusp*]{} of the Baily–Borel compactification ${\mathcal F}_L^{*}$.
Each $1$-dimensional boundary component $F$ of ${\mathcal D}_L$ is isomorphic to the upper half plane ${\mathbb H}$ and in the Baily–Borel compactification this corresponds to adding an (open) curve $\Lambda \backslash {\mathbb H}$, where $\Lambda\subset {\mathop{\mathrm {SL}}\nolimits}_2({\mathbb Q})$ is an arithmetic group which depends on the component $F$. Details of this can be found in [@BB] and [@Sc]. For our purpose we need one general result not contained there.
\[cuspclosure\] Suppose that $L$ is even, and that any isotropic subgroup of the discriminant group $(A_L,q_L)$ is cyclic. Then the closure of every $1$-dimensional cusp in ${\mathcal F}_L^*$ contains the standard $0$-dimensional cusp.
Let $E$ be a primitive totally isotropic rank $2$ sublattice of $L$ and define the lattice $\widetilde E=E_{L^\vee}^{\perp\perp}$ (both orthogonal complements are taken in the dual lattice $L^{\vee}$). We remark that $E\subset \widetilde E$ and that $E=\widetilde E\cap
L$ because $E$ is isotropic and primitive. Thus the finite group $$H_E= E_{L^\vee}^{\perp\perp}/E < A_L$$ is an isotropic subgroup of the discriminant group of $L$. Let us take a basis of $L$ as in Lemma \[special\_basis\_for\_L\]. It is easy to see that $$H_E \cong A^{-1} {\mathbb Z}^2/{\mathbb Z}^2.$$ In the case we are considering, $H_E$ is a cyclic subgroup ($|H_E|^2$ divides $\det L$). Therefore $A={\mathop{\mathrm {diag}}\nolimits}(1,e)$. Thus $E$ contains primitive isotropic vectors with divisors $1$ and $e$, and the first vector defines the standard $0$-dimensional cusp.
[*Remark.*]{} If the discriminant group of $L$ contains a non-cyclic isotropic subgroup then there is a totally isotropic sublattice $E$ of $L$ such that the finite abelian group $H_E$ has elementary divisors $(\delta,\delta e)$ with $\delta>1$. Thus $\det L$ is divisible by $\delta^4e^2$.
Let $L=2U\oplus L_0$ be of signature $(2,n)$ and $u$ be a primitive isotropic vector of divisor $1$. The tube realisation ${\mathcal H}_u$ of the homogeneous domain ${\mathcal D}_L$ at the standard $0$-dimensional cusp is defined by the sublattice $L_1=u^{\perp}/{\mathbb Z}u\cong U\oplus L_0$: $$\label{tube}
{\mathcal H}_u={\mathcal H}(L_1)=\{Z\in L_{1}\otimes {\mathbb C}\ |\
({\mathop{\mathrm {Im}}\nolimits}Z, {\mathop{\mathrm {Im}}\nolimits}Z)>0\}^+,$$ where ${}^+$ denotes a connected component of the domain (see [@G] for details). The modular group ${\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L)$ acting on ${\mathcal H}(L_1)$ contains all translations $Z\to Z+l$ ($l\in L_1$). Therefore the Fourier expansion of a ${\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L)$-modular form $F$ at the standard cusp is $$\label{fourier}
F(Z)=
\sum_{l\in L_1^\vee,\ (l,l)\ge 0}
a(l)\exp(2\pi i (l, Z)).$$
\[cusp\] Let $L$ be an even lattice with two hyperbolic planes such that any isotropic subgroup of the discriminant group of $L$ is cyclic. Let $F$ be a modular form with respect to ${\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L)$. If its Fourier coefficients $a(l)$ at the standard cusp satisfy $a(l)=0$ if $(l,l)=0$, then $F$ is a cusp form.
A standard $1$-dimensional cusp is defined by a primitive totally isotropic sublattice $E_1={{\langle{u, v}\rangle}}$ with ${\mathop{\null\mathrm {div}}\nolimits}(u)={\mathop{\null\mathrm {div}}\nolimits}(v)=1$. We can choose $(u,v)$ in such a way that they generate the maximal totally isotropic sublattice in $U\oplus U$. Let $E$ be an arbitrary primitive totally isotropic sublattice of rank $2$ of $L$ defining a $1$-dimensional cusp of ${\mathcal F}_L$. We can assume that $E={{\langle{u, v'}\rangle}}_{\mathbb Z}$ where $u$ defines the standard $0$-dimensional cusp (see Lemma \[cuspclosure\] above). According to the Witt theorem for the rational hyperbolic quadratic space $L_1\otimes {\mathbb Q}$ there exists $\sigma \in {\mathop{\null\mathrm {O}}\nolimits}(L_1\otimes {\mathbb Q})$ such that $\sigma(v')=v$. We can extend $\sigma$ to an element of ${\mathop{\null\mathrm {O}}\nolimits}^+(L\otimes {\mathbb Q})$ by putting $\sigma(u)=\pm u$. The Siegel operator $\Phi_E$ for the boundary component defined by $E$ has the property $\Phi_E(F\circ \sigma)=\Phi_{\sigma(E)}(F)\circ \sigma$ (see [@BB]). Therefore $$\Phi_E(F)
=\Phi_{\sigma^{-1}E_1}(F)
=\Phi_{E_1}(F\circ \sigma^{-1})\circ \sigma.$$ We can calculate the Fourier expansion of the function under the Siegel operator $\Phi_{E_1}$: $$\begin{aligned}
F\circ \sigma^{-1}&=&
\pm \sum_{l\in L_1^\vee,\,(l,l)>0}
a(l)\exp(2\pi i(l, \sigma^{-1}Z))\nonumber\\
&=&\pm\sum_{l_1\in \sigma L_1^{\vee},\,(l_1,l_1)>0}
a(\sigma^{-1}l_1)\exp(2\pi i(l_1, Z)). \end{aligned}$$ Thus $\Phi_{E}(F)= \Phi_{E_1}(F\circ \sigma^{-1})\circ \sigma\equiv 0$ and $F$ is a cusp form.
In [@G Theorem 3.1] modular forms for $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L)$ are constructed using the arithmetic lifting of a Jacobi form $\phi$. The modular form ${\mathop{\mathrm {Lift}}\nolimits}(\phi)$ is defined by its first Fourier-Jacobi coefficient at a fixed standard $1$-dimensional cusp. In particular, we know the Fourier expansion at the standard $0$-dimensional cusp. Therefore we obtain the following improvement of the result proved in [@G] for square-free $d$.
\[SOF\] Let $L=L_{2d}=2U\oplus 2E_8(-1)\oplus {{\langle{-2d}\rangle}}$. Then the arithmetic lifting ${\mathop{\mathrm {Lift}}\nolimits}(\phi)$ of a Jacobi cusp form $\phi\in
J_{k,1}^{{\mathop{\mathrm {cusp}}\nolimits}}(L_{2d})$ of weight $k$ and index $1$ is a cusp form of weight $k$ for $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ for any $d\ge 1$.
Application: K3 surfaces with a spin structure {#spinK3}
==============================================
Instead of $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})$ and ${\mathcal F}_{2d}$, we may consider the subgroup $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ of $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})$ of index $2$ and the corresponding quotient $${{\mathcal S}{\mathcal F}}_{2d} = \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d}) \backslash {\mathcal D}_{L_{2d}}.$$ If $d>1$ then ${{\mathcal S}{\mathcal F}}_{2d}$ is a double covering of ${{\mathcal F}}_{2d}$. (For $d=1$ the two spaces coincide since $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2})\cong \widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2})/\pm I$.) This double covering has the following geometric interpretation: the domain ${\mathcal D}_{L_{2d}}$ is the parameter space of marked ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of degree $2d$, and dividing out by the group $\widetilde{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})$ identifies all the different markings on a given ${\mathop{\mathrm {K3}}\nolimits}$ surface. Two markings will be identified under the group $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ if and only if they have the same orientation. Hence ${{\mathcal S}{\mathcal F}}_{2d}$ parametrises polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces $(S,h)$ together with an orientation of the lattice $L_h=h^{\perp}$. We shall refer to these as [*oriented*]{} ${\mathop{\mathrm {K3}}\nolimits}$ surfaces. An orientation on a surface $S$ is also sometimes called a [*spin structure*]{} on $S$.
We have seen in Corollary \[reflK3\] that the branch divisor of the map ${\mathcal D}_{L_{2d}} \to {\mathcal F}_{2d}$ is given by the divisors associated to reflections $\sigma_r$ defined by a primitive vector $r$ of length either $r^2=-2$ or $r^2=-2d$. Note that in the first case $\sigma_r$ acts trivially on the discriminant group whereas it acts as $-{\mathop{\mathrm {id}}\nolimits}$ in the second case. Hence $\pm \sigma_r \notin \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ if $r^2=-2$, but $-\sigma_r \in \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ if $r^2=-2d$. It follows that the quotient map ${\mathcal D}_{L_{2d}} \to
{{\mathcal S}{\mathcal F}}_{2d}$ is branched along the $(-2d)$-divisors whereas the double cover $ {{\mathcal S}{\mathcal F}}_{2d} \to {{\mathcal F}}_{2d}$ is branched along the $(-2)$-divisors. In this way the group $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ separates the two types of contributions to our reflective obstructions. The reflective obstructions coming from the $(-2d)$ divisors are less problematic, as we shall see in the next theorem. The $(-2d)$-divisors have a geometric interpretation. The general point on such a divisor is associated to a ${\mathop{\mathrm {K3}}\nolimits}$ surface $S$ whose transcendental lattice $T_S$ has rank $20$ and which admits an involution acting as $-{\mathop{\mathrm {id}}\nolimits}$ on $T_S$. For $d=p^2$ this was shown in ([@Ko1 Prop. 7.4]), and for general $d$ it follows from Corollary \[reflK3\] and the proof of Proposition \[-2dvect\] above.
In [@G] it was proved that the modular variety $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})(q)\backslash {\mathcal D}_{L_{2d}}$, where $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d}(q))$ is the principal congruence subgroup of $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ of level $q\ge 3$, is of general type for any $d\ge 1$. Here we obtain a much stronger result.
\[orientedK3\] The moduli space ${{\mathcal S}{\mathcal F}}_{2d} = \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})
\backslash {\mathcal D}_{L_{2d}}$ of oriented ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of degree $2d$ is of general type if $d\ge 3$.
For $L_{2d}=2U\oplus 2E_8(-1)\oplus {{\langle{-2d}\rangle}}$ the corresponding space of Jacobi cusp forms in $18$ variables is isomorphic (as a linear space) to the space of Jacobi cusp forms of Eichler-Zagier type (see [@G lemma 2.4]) $$J_{k,1}^{{\mathop{\mathrm {cusp}}\nolimits}}(L_{2d})\cong J_{k-8,d}^{{\mathop{\mathrm {cusp}}\nolimits}}(EZ).$$ For $k=17$, this space is non-trivial for any $d\ge 3$. Therefore for any $d\ge 3$ there is a cusp form $F_{17}$ of weight $17$ with respect to $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$.
The ramification divisor of the projection $\pi_{{\mathop{\mathrm {SO}}\nolimits}}\colon
{\mathcal D}_{L_{2d}}\to \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})\backslash {\mathcal D}_{L_{2d}}$ is defined by $(-2d)$-reflections of $L_{2d}$. In Lemma \[oddk\] below we show that the cusp form $F_{17}$ vanishes on the ramification divisors of $\pi_{{\mathop{\mathrm {SO}}\nolimits}}$.
Hence ${\mathcal S}{\mathcal F}_{2d}$ is of general type for $d\ge 3$ by Theorem \[general\_gt\].
\[oddk\] Any modular form $F\in M_{2k+1}(\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d}))$ of odd weight vanishes along the divisors defined by $(-2d)$-reflective vectors.
Let $\sigma_r\in {\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d})$ be a reflection with respect to a $(-2d)$-vector. Then $-\sigma_r\in \widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d})$ (see Corollary \[reflK3\]). For any $z\in {\mathcal D}_{L_{2d}}$ with $(z,r)=0$ and a modular form $F\in M_{2k+1}(\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{2d}))$ we have $$F(z)=F((-\sigma_{r})(z))=F(-z)=(-1)^{2k+1}F(z),$$ so $F(z)\equiv 0$.
We note that ${{\mathcal S}{\mathcal F}}_{2}={{\mathcal F}}_{2}$ is unirational.
The geometric interpretation of the $(-2)$-divisors, which form the ramification of the covering ${{\mathcal S}{\mathcal F}}_{2d} \to {{\mathcal F}}_{2d}$, is that they parametrise those polarised ${\mathop{\mathrm {K3}}\nolimits}$ surfaces whose polarisation is only semi-ample, but not ample. This is due the presence of rational curves on which the polarisation has degree $0$. Thus in the case $d=2$ the map ${{\mathcal S}{\mathcal F}}_{4} \to {{\mathcal F}}_{4}$ is a double cover of the moduli space of quartic surfaces branched along the discriminant divisor of singular quartics. The variety ${{\mathcal F}}_{4}$ is unirational but ${{\mathcal S}{\mathcal F}}_{4}$ is not, since there exists a canonical differential form on it (see [@G]). There is also a cusp form of weight $18$ with respect to $\widetilde{{\mathop{\mathrm {SO}}\nolimits}}^+(L_{4})$ which vanishes on one of the two irreducible components of the ramification divisors for $d=2$. We shall return to this question in a more general context in [@GHS2].
Pull-back of the Borcherds function $\Phi_{12}$. {#Borcherds}
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To construct pluricanonical differential forms on a smooth model of ${\mathcal F}_{2d}$ we shall use the pull-back of the Borcherds automorphic product $\Phi_{12}$.
Let $L_{2,26}=2U\oplus 3E_8(-1)$ be the unimodular lattice of signature $(2,26)$. For later use, we note the following simple lemma.
\[orthcomp2d\] Let $r$ be a primitive reflective vector in $L_{2d}$ with $r^2=-2d$ and let $L_r=r^\perp_{L_{2d}}$ be its orthogonal complement considered as a primitive sublattice of the unimodular lattice $L_{2,26}$. Then $$(L_r)^\perp_{L_{2,26}}\cong E_8(-1), \ E_7(-1)\oplus{{\langle{-2}\rangle}} \ \text{
or }\ D_8(-1).$$
In the proof of Proposition \[-2dvect\] we found $L_r$ and its orthogonal complement $S_r$ in the unimodular lattice $L_{{\mathop{\mathrm {K3}}\nolimits}}=3U+2E_8(-1)$. The discriminant forms of $S_r$ and $K_r=(L_r)^\perp_{L_{2,26}}$ coincide, but $K_r$ is of signature $(0,8)$. The three possible genera of $K_r$ are represented by $E_8(-1)$, $E_7(-1)\oplus \langle -2\rangle$ and $D_8(-1)$. The genera of such lattices contain only one class: one can can prove this well-known fact by analysing sublattices of order $2$ in $E_8$ or simply check it using MAGMA.
The Borcherds function $\Phi_{12}\in M_{12}({\mathop{\null\mathrm {O}}\nolimits}^+(L_{2,26}),\det)$ is the unique modular form of weight $12$ and character $\det$ with respect to ${\mathop{\null\mathrm {O}}\nolimits}^+(L_{2,26})$ (see [@B]).
$\Phi_{12}$ is the denominator function of the fake Monster Lie algebra and it has a lot of remarkable properties. In particular, the zeros of $\Phi_{12}(Z)$ lie on rational quadratic divisors defined by $(-2)$-vectors in $L_{2,26}$, [i.e.]{}, $\Phi_{12}(Z)=0$ if and only if there exists $r\in L_{2,26}$ with $r^2=-2$ such that $(r,Z)=0$ and the multiplicity of the rational quadratic divisor in the divisor of zeros of $\Phi_{12}$ is $1$.
Pulling back this function gives us many interesting automorphic forms (see [@B pp. 200-201], [@GN pp. 257-258]). In the context of the moduli of ${\mathop{\mathrm {K3}}\nolimits}$ surfaces this function was used in [@BKPS] and [@Ko2]. We summarise their results in a suitable form.
Let $l\in E_8(-1)$ satisfy $l^2=-2d$. The choice of $l$ determines an embedding of $L_{2d}$ into $L_{2,26}$ as well as an embedding of the domain ${\mathcal D}_{L_{2d}}$ into ${\mathcal D}_{L_{2,26}}$. We put $R_l=\{r\in
E_8(-1)\mid r^2=-2,\ (r, l)=0\}$, and $N_l=\# R_l$. (It is clear that $N_l$ is even.) Then by [@BKPS] the function $$\label{pb}
\left. F_l=
\frac{\Phi_{12}(Z)}{
\prod_{\{ \pm r\}\in R_l} (Z, r)}
\ \right\vert_{{\mathcal D}_{L_{2d}}}
\in M_{12+\frac{N_l}2}({\widetilde}{\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d}),\, \det)$$ is a non-trivial modular form of weight $12+\frac{N_l}2$ vanishing on all $(-2)$-divisors of ${\mathcal D}_{L_{2d}}$. (As we did in Section \[specialcusp\], we think of a modular form as a function on ${\mathcal D}_L$ rather than ${\mathcal D}_L^\bullet$, by identifying ${\mathcal D}_L$ with a tube domain realisation as in equation (\[tube\]) above.) Moreover it is shown in [@Ko2] that $F_l$ is a cusp form if $d$ is square-free and the weight is odd.
In fact much more is true.
\[cuspform\] The function $F_l$ has the following properties:
- $F_l\in M_{12+\frac{N_l}2}({\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d}),\,
\det)$ and $F_l$ vanishes on all $(-2)$-divisors.
- $F_l$ is a cusp form for any $d$ if $N_l>0$.
- If the weight of $F_l(Z)$ is smaller than $68$ ([i.e.]{}, if $N_l<112$) then $F_l(Z)$ is zero along the branch divisor of the projection $$\pi\colon {\mathcal D}_{L_{2d}}{\longrightarrow}\Gamma_{2d}\backslash {\mathcal D}_{L_{2d}}={\mathcal F}_{2d}.$$
\(i) was proved in [@BKPS], but we repeat some details here for convenience. First, $F_l(Z)$ is holomorphic because of the properties of the divisor of $\Phi_{12}$. Then $F_l(tZ)=t^{-(12+N_l/2)}F_l(Z)$ for any $Z\in {\mathcal D}_{L_{2d}}$. Any $g\in \widetilde {\mathop{\null\mathrm {O}}\nolimits}^+(L_{2d})$ can be extended (by the identity on the orthogonal complement of $L_{2d}$ in $L_{2,26}$) to an element $\tilde g$ of ${\mathop{\null\mathrm {O}}\nolimits}^+(L_{2,26})$. Therefore $F_l(gZ)=\det(g)F_l(Z)$ since $\tilde g(r)=r$ for all roots in $R_l$. This modular form is evidently not identically zero. On the other hand, because it has character $\det$ it vanishes on all divisors of ${\mathcal D}_{L_{2d}}$ which are invariant with respect to $\sigma_r$ with $r^2=-2$, because then $\sigma_r\in {\widetilde}{{\mathop{\null\mathrm {O}}\nolimits}}^+(L_{2d})$.
\(ii) The Fourier expansion of $\Phi_{12}$ at the standard 0-dimensional cusp is defined by the hyperbolic unimodular lattice $L_{1,25}=U\oplus 3E_8(-1)$ (see (\[tube\]) and (\[fourier\])): $$\Phi_{12}(Z) =\sum_{u\in L_{1,25},\, (u,u)=0} \ a(u)\exp(2\pi i
(u,Z)).$$ The weight $12$ is singular, therefore the hyperbolic norm of the index of any non-zero Fourier coefficient is zero.
Let us fix a root $r\in R_l\subset L_{1,25}$ (any root is equivalent to such a root). We denote by $L_r$ the orthogonal complement of $r$ in $L_{1,25}$. We have $Z=Z_r+ zr$, where $Z_r\in {\mathcal H}(L_r)$ and $z\in {\mathbb C}$. We note that $\Phi_{12}(Z_r)\equiv 0$. The function $$\Phi_r(Z_r)=\left.\frac{\Phi_{12}(Z)}{(Z, r)} \,\right\vert_{{\mathcal H}(L_r)}$$ is the first coefficient of the Taylor expansion of the function $\Phi_{12}(Z_r+zr)$ in $z$.
The summation in the Fourier expansion of $\Phi_r(Z_r)$ is taken over the dual lattice $L_r^\vee$. We note that $$L_r \oplus {\mathbb Z}r\subset L_{1,25}\subset L_r^\vee
\oplus {\mathbb Z}(r/2).$$ Let us calculate $$\left.\frac{\partial \Phi_{12}(Z_r+zr)}{\partial z}\right\vert_{z=0}.$$ We get non-zero Fourier coefficient only for indices $u=u_r+m(r/2)$, where $u_r\in L_r^\vee$ and $0\ne m\in {\mathbb Z}$. In this case $(u_r,u_r)=m^2/2>0$. Thus the first derivative has non-zero Fourier coefficient only for indices $u_r$ with positive square. Doing this for every $r$ we see that the Fourier expansion of $F_l$ at the canonical cusp contains only indices with positive hyperbolic norm. Thus $F_l$ is a cusp form.
The components of the branch divisor are divisors $${\mathcal F}_{2d}(r)=\pi(\{Z\in {\mathcal D}_{L_{2d}}\mid (Z,r)=0\})$$ defined by reflective vectors $r\in L_{2d}$, by Corollary \[reflK3\]. For a $(-2)$-vector $r\in L_{2d}$, the form $F_l(Z)$ has a zero along ${\mathcal F}_{2d}(r)$ (see (i)).
Now we can finish the proof using Lemma \[orthcomp2d\]. If $r$ is a $(-2d)$-reflective vector and $L_r=r^\perp_{L_{2d}}$, then the divisor ${\mathcal F}_{2d}(r)$ coincides with the modular projection $\pi({\mathcal D}_{L_r})$ of the homogeneous domain of the lattice $L_r$ of signature $(2,18)$. According to Lemma \[orthcomp2d\], $(L_r)^\perp_{L_{2,26}}$ is a root lattice with $N\ge 112$ roots ($E_8$ has $240$ roots, $E_7$ has $126$ and $D_8$ has $112$). Therefore the Borcherds form $\Phi_{12}$ has a zero of order $N\ge 112>N_l$ along the subdomain ${\mathcal D}_{L_r}$. Thus $F_l$ is zero along the corresponding divisor ${\mathcal F}_{2d}(r)$.
According to Theorem \[cuspform\] and Theorem \[general\_gt\] the main point for us is the following. We want to know for which $2d>0$ there exists a vector $$\label{orth2}
l\in E_8,\ l^2=2d,\ l\ \text{ is orthogonal to at least $2$ and at most $12$
roots.}$$
\[mainineq\] Such a vector $l$ in $E_8$ does exist if one of two inequalities $$\label{mineq}
4N_{E_7}(2d)>28N_{E_6}(2d)+63N_{D_6}(2d)$$ or $$\label{mineqd}
5N_{E_7}(2d)>28N_{E_6}(2d)+63N_{D_6}(2d)+378N_{D_5}(2d)$$ is valid, where $N_L(2d)$ denotes the number of representations of $2d$ by the lattice $L$.
Let us fix a root $a \in E_8$. This choice gives us a realisation of the lattice $E_7$ as a sublattice of $E_8$: $$E_7\cong E_7^{(a)}=a^\perp_{E_8}.$$ We have the following decomposition of the set of roots $R(E_8)$: $$R(E_8)=R(E_7)\sqcup X_{114}\qquad\text{where }
X_{114}=\{c\in R(E_8)\mid c\cdot a \ne 0\}$$ and $|X_{114}|=|R(E_8)|-|R(E_7)|=240-126=114$.
\[A2decomposition\] The roots have the following properties:
- $X_{114}$ is the union of $28$ root systems of type $A_2$ such that $R(A_2^{(i)})\cap R(A_2^{(j)})=\{\pm a\}$ for any $i\ne
j$.
- Let $A_2(a,c)\ne A_2(a,d)$ be two $A_2$-lattices generated by roots $a$, $c$ and $a$, $d$. Then $$A_3(a,c,d)=A_2(a,c)+A_2(a,d)$$ is a lattice of type $A_3$ containing one and only one copy of $A_1$ from $E_7^{(a)}$.
- Let us take three different $A_2(a,c_i)$ ($i=1,2,3$). Then their sum $$S=\sum_{i=1}^3 A_2(a,c_i)$$ is a lattice of type $A_4$ or $D_4$. The first one contains $20$ roots, the second contains $24$ roots. In both cases exactly six roots of $S$ are in $E_7^{(a)}$.
\(i) Recall that $|b\cdot c|\le 2$ for any roots $b$, $c\in R(E_8)$. If $b\cdot c=\pm 2$ then $b=\pm c$. We can assume that $a\cdot c=-1$ (if not we replace $c$ by $-c$). The lattice $A_2(a,c)={\mathbb Z}a+{\mathbb Z}c$ is a lattice of $A_2$-type. Any $A_2$-lattice contains six roots $$R(A_2(a,c))=\{\,\pm a,\ \pm c,\ \pm(a+c)\,\}.$$ $A_2(a,c)$ is generated by any pair of linearly independent roots. Therefore $$A_2(a,c_1)\cap A_2(a,c_2)=\{ \pm a\}$$ if the root lattices are distinct.
\(ii) $c\ne \pm d$ implies that $c\cdot d=0$ or $\pm 1$. Suppose that $c\cdot d=0$. Then the sum of the lattices is of type $A_3$ ($a\cdot
c= a\cdot d=-1$ and $c\cdot d=0$). This lattice contains $12$ roots $$R(A_3(a,c,d))=\pm (a,\ c,\ d,\ a+c, \ a+d; \ a+c+d).$$ The first five roots are elements of $X_{114}$ and $a+c+d\in
E_7^{(a)}$.
If $c\cdot d=1$ then $(a+d)\cdot c=0$ and we come back to the first case. If $c\cdot d=-1$ then $(a+d)\cdot c=-2$, $c=-(a+d)$ and $A_2(a,c)= A_2(a,d)$.
\(iii) As in the proof of $2)$ we can suppose that $c_1\cdot c_2=c_2\cdot c_3=0$ and $c_1\cdot c_3=0$ or $1$.
If $c_1\cdot c_3=1$, then we see that $S$ has a root basis of type $A_4$. .5cm -.5cm
(300,10)(55,10) (120,10) (115,0)[$c_{3}$]{} (120,10)[(1,0)[62]{}]{} (180,10) (170,0)[$-c_{1}$]{} (180,10)[(1,0)[62]{}]{} (240,10) (225,0)[$a+c_{1}$]{} (240,10)[(1,0)[62]{}]{} (300,10) (295,0)[$c_{2}$]{}
$A_4$ has $20$ roots. They are $$\pm(a,\ c_i,\ a+c_i,\ a+c_1+c_2, \ a+c_2+c_3,\ c_1-c_3)
\quad\text{where }i=1,2,3.$$ Only the last three roots belong to $E_7^{(a)}$.
If $c_1\cdot c_3=0$ then the roots $c_1$, $a$, $c_2$, $c_3$ form a basis of $S$. In this case $S$ has type $D_4$ ($a\cdot c_i=-1$ for all $i$ and the other scalar products are zero). This root system contains all roots of $A_4$ except $\pm(c_1-c_3)$ and the roots $$\pm(a+c_1+c_3,\ a+c_1+c_2+c_3,\ 2a+c_1+c_2+c_3).$$ The six roots from $E_7^{(a)}$ are $\pm(a+c_i+c_j)$.
Now we can finish the proof of Theorem \[mainineq\]. Let us assume that every $l\in E_7^{(a)}$ with $l^2=2d>0$ is orthogonal to at least $14$ roots in $E_8$ including $\pm a$. The others are some roots in $E_7^{(a)}$ ($126$ roots), or in $X_{114}\setminus \{\pm a\}$ ($112$ roots). If $l$ is orthogonal to $b\in X_{114}\setminus \{\pm a\} $ then $l$ is orthogonal to the lattice $A_2(a,b)$. Therefore using Lemma \[A2decomposition\] we have $$\label{union}
l\in \bigcup_{i=1}^{28} (A_2^{(i)})^\perp_{E_8} \cup
\bigcup_{j=1}^{63} (A_1^{(j)})^\perp_{E_7}.$$ We recall that $(A_2)^\perp_{E_8}\cong E_6$, $(A_1)^\perp_{E_7}\cong
D_6$ and $(A_1\oplus A_1)^\perp_{E_8}\cong D_6$. Let denote by $n(l)$ the number of components in (\[union\]) containing the vector $l$. We have calculated this vector exactly $n(l)$ times in the sum $$28N_{E_6}(2d)+63N_{D_6}(2d).$$ We shall consider several cases.
(a). Suppose that $l\cdot c\ne 0$ for any $c\in X_{114}\setminus \{\pm
a\}$. Then $l$ is orthogonal to at least $6$ copies of $A_1$ in $E_7^{(a)}$ and $n(l)\ge 6$.
Now we suppose that there exist $c\in X_{114}\setminus \{\pm a\}$ such that $l\cdot c=0$. Then $l$ is orthogonal to $A_2(a,c)$ which is one of the $28$ subsystems of the bouquet $X_{114}$.
(b). If $l$ is orthogonal to only one $A_2^{(i)}$ ($6$ roots) then $l$ is orthogonal to at least $4$ copies of $A_1$ ($8$ roots) in $E_7^{(a)}$. Thus $n(l)\ge 5$.
(c). If $l$ is orthogonal to exactly two $A_2^{(i)}$ and $A_2^{(j)}$ in $X_{114}$ then $l$ is orthogonal to $A_3=A_2^{(i)}+A_2^{(j)}$ having $12$ roots and containing only one $A_1$ from $E_7^{(a)}$. Thus $l$ is orthogonal to another $A_1$ in $E_7^{(a)}$. Therefore $n(l)\ge
4$.
(d). If $l$ is orthogonal to three or more $A_2^{(i)}$ then their sum contains three $A_1\subset E_7^{(a)}$ and $n(l)\ge 6$.
We see that under our assumption $n(l)\ge 4$ for any $l\in E_7^{(a)}$. Therefore we have proved that if every $l\in E_7^{(a)}$ with $l^2=2d$ is orthogonal to at least $14$ roots then $$28N_{E_6}(2d)+63N_{D_6}(2d)\ge 4N_{E_7}(2d).$$ Moreover $n(l)$ can be equal to $4$ only in case (c). In this case $l\in (A_3)^\perp_{E_8}\cong D_5$ and there are $\binom{28}{2}=378$ pairs of $A_2$-subsystems in $X_{114}$. This gives us the second inequality $$28N_{E_6}(2d)+63N_{D_6}(2d)\ge 5N_{E_7}(2d)-378N_{D_5}(2d).$$
The inequalities (\[mineq\]) and (\[mineqd\]) fail only for a finite number of $d$ because their left- and right-hand sides have the asymptotics $O(d^{5/2})$ and $O(d^2)$.
\[P-ex\] A vector $l\in E_8$ satisfying the condition (\[orth2\]) does exist if $d\not \in P_{ex}$, where $$\begin{gathered}
P_{ex}=\{\,1\le m\le 100\ (m\ne 96);\quad 101\le m\le 127\ (m \text{
is odd});
\\
m=110,\ 131, 137,\ 143\,\}. \end{gathered}$$
The Jacobi theta-series of the lattice $E_8$ coincides with the Jacobi-Eisenstein series $E_{4,1}(\tau,z)$ of weight $4$ and index $1$. Let us fix a root $a\in E_8$. We have $$E_{4,1}(\tau,z)=\sum_{l\in E_8} \exp(\pi i\, l^2\tau+2\pi i\, l\cdot a z)=
1+\sum_{m\ge 1} e_{4,1}(m,n) \exp(2\pi m\tau+ nz).$$ $N_{E_7}(2m)=e_{4,1}(m,0)$, since the orthogonal complement of $a$ in $E_8$ is $E_7$.
The Fourier coefficients $e_{4,1}(m,n)$ were calculated in [@EZ]. In particular $$N_{E_7}(2m)=\frac{2^6\pi^3}{15}\frac{L^{Z}_{4m}(3)}{\zeta(3)}\,m^{5/2}$$ where $$L^{Z}_{D}(s)=\sum_{t\ge 1}\frac{\#\{\,x\mod 2t\mid x^2\equiv D\mod
4t\,\}}{t^s}.$$ It is evident that $L^{Z}_{4m}(3)>9/8$ (one has to take only two terms for $t=1$ and $t=2$). Thus $$\label{NE7estimate}
N_{E_7}(2m)>\frac{24\pi^3}{5\zeta(3)}\,m^{5/2}>c(E_7) m^{5/2},$$ where $c(E_7)=123.8$. In fact this estimate is quite good: a computation with PARI shows that $N_{E_7}(314)\approx 124.73\times
(157)^{5/2}$
We can find simple exact formulae for $N_{E_6}(2m)$ and $N_{D_6}(2m)$. Let $\chi_3$ and $\chi_4$ be the unique non-trivial Dirichlet characters modulo $3$ and $4$ respectively. For a Dirichlet character $\chi$ we put $$\sigma_k(m,\chi)=\sum_{d|m} \chi(d)d^k,
\qquad
\tilde\sigma_k(m,\chi)=\sum_{d|m} \chi\left(\frac{m}d\right)d^k.$$
\[NE6andND6\] The number of representations of $2m$ by the quadratic forms $E_6$ and $D_6$ are $$\begin{aligned}
N_{E_6}(2m)&=81\tilde\sigma_2(m,\chi_3)-9\sigma_2(m,\chi_3),\\
N_{D_6}(2m)&=64\tilde\sigma_2(m,\chi_4)-4\sigma_2(m,\chi_4).
\end{aligned}$$
The second identity is well-known. This is the number of representations of $2m$ by six squares. To prove the first identity we consider the theta-series of $E_6$: $$\theta_{E_6}(\tau)=\sum_{l\in E_6}e^{\pi i (l\cdot l)}
\in M_3(\Gamma_0(3),\chi_3)=M_3(\Gamma_1(3)).$$ The dimension of $M_3(\Gamma_1(3))$ is equal to $2$. We can construct a basis with the help of Eisenstein series $G_k^{\alpha}$, where $\alpha=(a,b)\in ({\mathbb Z}/N{\mathbb Z})^2$, $$G_k^{\alpha}(\tau)=\sum_{(n,m)\equiv (a,b)\mod N}\ (n\tau+m)^{-k}.$$ Using the relation $G_k^{\alpha}|_k\gamma=G_k^{\alpha\gamma}$ (where $\gamma\in {\mathop{\mathrm {SL}}\nolimits}_2({\mathbb Z})$) for $k=3$ and $N=3$ we obtain two modular forms in $M_3(\Gamma_1(3))$, namely $G_3^{(0,1)}$ and $G_3^{(1,0)}+G_3^{(1,1)}+G_3^{(1,2)}$. The Fourier expansion of $G_k^{\alpha}$ was found by Hecke (see [@Kob]). Normalising both series we obtain a basis of $M_3(\Gamma_0(3),\chi_3)$ consisting of $$\begin{aligned}
E_3^{(\infty)}(\tau,\chi_3)&=1-9\sum_{m\ge 1}\sigma_2(m,\chi_3)q^m,\\
E_3^{(0)}(\tau,\chi_3)&=\sum_{m\ge 1}\tilde\sigma_2(m,\chi_3)q^m
\qquad\qquad(q=e^{2\pi i \tau}).
\end{aligned}$$ We note that the first series is proportional to $(\eta^3(\tau)/\eta(3\tau))^3$ and it vanishes at the cusp $0$. The second series vanishes at $i\infty$. The lattice $E_6$ has $72$ roots. Therefore $$\label{thE6}
\theta_{E_6}(\tau)=81E_3^{(0)}(\tau,\chi_3)+E_3^{(\infty)}(\tau,\chi_3).$$ This gives us the formula for $N_{E_6}(2m)$. Applying the same method to the theta-series $\theta_{D_6}\in M_3(\Gamma_0(4),\chi_4)$ we obtain that $$\label{thD6}
\theta_{D_6}(\tau)=64E_3^{(0)}(\tau,\chi_4)+E_3^{(\infty)}(\tau,\chi_4),$$ where $$\begin{aligned}
E_3^{(\infty)}(\tau,\chi_4)&=1-4\sum_{m\ge 1}\sigma_2(m,\chi_4)q^m,\\
E_3^{(0)}(\tau,\chi_4)&=\sum_{m\ge 1}\tilde\sigma_2(m,\chi_4)q^m.
\end{aligned}$$
Using these representations we can get an upper bound for $N_{E_6}(2m)$ and $N_{D_6}(2m)$. It is clear that $$\sigma_2(m,\chi_3)
=\chi_3(m)\tilde\sigma_2(m,\chi_3)
\qquad\text{if }\ m\not \equiv 0\mod 3.$$ For any $C\equiv 1\mod 3$ we have the following bound $$\frac{\tilde\sigma_2(m,\chi_3)}{m^2}=\sum_{d|m}\frac{\chi_3(d)}{d^2}<
\sum_{1\le l\le C,\ l\equiv 1\mod 3} l^{-2}+
\bigg(\zeta(2)-\sum_{1\le n\le C+2,}n^{-2}\bigg).$$ Taking $C=19$ we get that for any $m$ not divisible by $3$ $$\label{NE6estimate}
N_{E_6}(2m)=\tilde\sigma_2(m,\chi_3)(81-9\chi_3(m))<c(E_6)m^2,$$ where $c(E_6)=103.69$.
If $m=3^km_1$ then $\sigma_2(m,\chi_3)=\sigma_2(m_1,\chi_3)$, so the last inequality is valid for any $m$. For $D_6$ one can take $C=21$ in a similar sum. As a result we get $$\label{ND6estimate}
N_{D_6}(2m)<c(D_6)m^2,$$ where $c(D_6)=75.13$.
Using the estimates (\[NE7estimate\]), (\[NE6estimate\]) and (\[ND6estimate\]) for $N_L(2m)$, where $L=E_7$, $E_6$ and $D_6$, we obtain that the main inequality (\[mineq\]) of Theorem \[mainineq\] is valid if $$m\ge 238>\left(\frac{28c(E_6)+63c(D_6)}{4c(E_7)}\right)^2.$$ For smaller $m$ we can use another formula for the theta-series of $E_7$ (see [@CS (112)]) $$\label{thetaE7}
\theta_{E_7}(\tau)
=\theta_3(2\tau)^7+7\theta_3(2\tau)^3\theta_2(2\tau)^4,$$ where $$\theta_3(2\tau)=\sum_{n=-\infty}^{\infty} q^{n^2},\qquad
\theta_2(2\tau)=\sum_{n=-\infty}^{\infty} q^{(n+\frac{1}2)^2}.$$ Moreover (see [@CS (87)]) $$\label{thetaDn}
\theta_{D_n}(\tau)
=\frac{1}{2}(\theta_3(\tau)^n+\theta_3(\tau+1)^n).$$ Using (\[thetaE7\]) and (\[thetaDn\]) together with (\[thE6\]) we can compute (using PARI) the first $240$ Fourier coefficients of the function $$5\theta_{E_7}-28\theta_{E_6}-63\theta_{D_6}-378\theta_{D_5}.$$ The indices of the negative coefficients form the set $P_{ex}$ of $d$ for which the inequality (\[mineqd\]) of Theorem \[mainineq\] fails.
Now we are going to analyse the main condition (\[orth2\]) for some $d\in P_{ex}$ from Proposition \[P-ex\]. Moreover we are also looking for vectors with $d\le 61$ orthogonal to exactly $14$ roots. Such vectors produce cusp forms $F_l$ of weight $19$ due to Theorem \[cuspform\].
Let $e_i$ ($1\le i\le 8$) be a euclidean basis of the lattice ${\mathbb Z}^8$ ($(e_i, e_j)=\delta_{ij}$). We consider the Coxeter basis of simple roots in $E_8$ (see [@Bou])
-.5cm
(300,10)(55,10) (100,0) (95,10)[$\alpha_1$]{} (100,0)[(1,0)[42]{}]{} (140,0) (135,10)[$\alpha_3$]{} (140,0)[(1,0)[42]{}]{} (180,0) (175,10)[$\alpha_4$]{} (180,1)[(0,-1)[43]{}]{} (180,-40) (175,-50)[$\alpha_2$]{} (180,0)[(1,0)[42]{}]{} (220,0) (215,10)[$\alpha_5$]{} (220,0)[(1,0)[42]{}]{} (260,0) (255,10)[$\alpha_6$]{} (260,0)[(1,0)[42]{}]{} (300,0) (295,10)[$\alpha_7$]{} (300,0)[(1,0)[42]{}]{} (340,0) (335,10)[$\alpha_8$]{}
where $$\begin{gathered}
\alpha_1=\frac 1{2}(e_1+e_8)-\frac 1{2}(e_2+e_3+e_4+e_5+e_6+e_7),\\
\alpha_2=e_1+e_2,\quad \alpha_k=e_{k-1}-e_{k-2}\ \ (3\le k\le 8)\end{gathered}$$ and $E_8=\langle\alpha_1,\dots \alpha_8\rangle_{\mathbb Z}$.
Let $L_S=\langle \alpha_i\mid i\in S\rangle_{\mathbb Z}\subset E_8$ be a sublattice of $E_8$ generated by some simple roots ($S\subset
\{1,\dots,8\}$). We assume that $\#R (L_S)\le 12$, where $R(L_S)$ is the set of roots of $L_S$. We can find the orthogonal complement of $L_S$ in $E_8$ using fundamental weights $\omega_j$, [i.e.]{} the basis of $E_8$ dual to the basis $\{\alpha_i\}_{i=1}^8$. We have $$L_S^\perp=(L_S)^\perp_{E_8}
=\langle \omega_j\mid j\not\in S\rangle_{\mathbb Z}.$$ Any vector of $L_S^\perp$ is orthogonal to all roots of $L_S$. If $l\in L_S^\perp$ is orthogonal to an additional root $r$ of $E_8$ ($r\not\in R(L_S)$) then we obtain a linear relation on the coordinates of $l$ in the basis $\omega_j$ $(j\not\in S)$. Considering all roots of $E_8$ we can formulate a condition on the coordinates of $l\in L^\perp_S$ to be orthogonal to at most $12$ roots (or to exactly $14$ roots). We shall analyse four different lattices $L_S$.
[**I**]{}. $L_1=4A_1$, $\ \# R(4A_1)=8$ and $L_1^\perp=4A_1$.
We put $$L_1=\langle \alpha_2,\, \alpha_3,\, \alpha_5,\, \alpha_7\rangle_{\mathbb Z}=\langle e_2+e_1,\, e_2-e_1,\, e_4-e_3,\, e_6-e_5\rangle_{\mathbb Z}\cong 4A_1.$$ This root lattice $L_1$ gives us vectors of norm $2d$ for most $d\in
P_{ex}$. $L_1$ is a primitive sublattice of $E_8$. Therefore $L_1^\perp$ is a lattice with the same discriminant form and $L_1^\perp\cong 4A_1$. More exactly, $$L_1^\perp=\langle\,\omega_1,\, \omega_4,\, \omega_6,\, \omega_8\rangle_{\mathbb Z}=\langle\, e_3+e_4,\, e_5+e_6,\, e_7+e_8,\, e_7-e_8\rangle_{\mathbb Z}.$$ This representation follows easily from the formulae for the fundamental weights of $E_8$ (see [@Bou Plat VII]): $$\begin{gathered}
\omega_2=\frac{1}{2}(e_1+\dots+e_7+5e_8),\quad
\omega_3=\frac{1}{2}(-e_1+e_2+\dots+e_7+7e_8),\\
\omega_k=e_{k-1}+\dots+e_7+(9-k)e_8\quad (4\le k\le 8), \quad \omega_1=2e_8. \end{gathered}$$ Any vector $$\label{linL1}
l=m_3(e_3+e_4)+m_5(e_5+e_6)+m_7(e_7+e_8)+m_8(e_7-e_8)\in L_1^\perp$$ is orthogonal to $8$ roots of $L_1$. The root system of $E_8$ contains $112$ integral and $128$ half-integral roots: $$\pm e_i\pm e_j\quad (i<j),\quad
\frac{1}{2}\sum_{i=1}^{8}(-1)^{\nu_i}e_i\quad
\text{with }\ \sum_{i=1}^{8} {\nu_i}\equiv 0\mod 2.$$ If $l$ is orthogonal to a half-integral root $r$ then $$\begin{gathered}
\label{lr1}
2(l\cdot r)=m_7((-1)^{\nu_7}+(-1)^{\nu_8})+m_8((-1)^{\nu_7}-(-1)^{\nu_8})+\\
m_3((-1)^{\nu_3}+(-1)^{\nu_4})+m_5((-1)^{\nu_5}+(-1)^{\nu_6})=0. \end{gathered}$$ We note that only one of $m_7$ or $m_8$ appears. Let us assume that this identity contains three non-zero terms: $m_{7,8}\pm m_3\pm m_5
=0$ (by $m_{7,8}$ we mean $m_7$ or $m_8$). Then $l$ is orthogonal to $4$ additional half-integral roots. There are two choices for $(\nu_1,\nu_2)$ and one can change the sign of the root. A similar result, [i.e.]{} a relation $m_7\pm m_8\pm m_{3,5}=0$ and $4$ additional integral roots, is obtained if $l$ is orthogonal to the integral roots $e_{7,8}\pm e_{3,4}$ or $e_{7,8}\pm e_{5,6}$.
If (\[lr1\]) contains only two non-zero terms then we have a relation of type $m_{7,8}\pm m_{3,5}=0$. In this case $l$ is orthogonal to $8$ additional half-integral roots: there are two choices for $(\nu_3,\nu_4)$ (or $(\nu_5,\nu_6)$), for $(\nu_1,\nu_2)$ and the change of the sign. We can also have $m_{7,8}=0$, and then the number of half-integral roots orthogonal to $l$ is equal to $16$.
If $l$ is orthogonal to an integral root $r\not\in L_1$, which has not been considered above, then we get a relation $m_3=\pm m_5$ or $m_7=\pm m_8$ with $8$ additional roots or $m_{3,5}=0$ with $16$ additional integral roots. For example, if $m_7=m_8$ then $l$ is orthogonal to $\pm(e_8\pm e_{1,2})$; if $m_{3}=0$ then $l$ is orthogonal to $\pm (e_{3,4}\pm e_{1,2})$. Therefore we have proved the following
\[4A1\] $l\in L_1^\perp$ (see (\[linL1\])) is orthogonal to at least $8$ and at most $12$ roots of $E_8$ if and only if
- $m_j\ne 0$ for any $j$ and $m_i\ne m_j$ for any $i\ne j$;
- There is at most one relation of type $m_k=\pm m_i\pm m_j$ for $i<j<k$.
This lemma gives us a set of vectors $l\in L_1^\perp$ with $$l^2=2(m_3^2+m_5^2+m_7^2+m_8^2)=2d\in P_{ex}$$ such that $l$ is orthogonal to $8$ or to $12$ roots of $E_8$. We list these vectors in table [**I-8,12**]{}.
[|c|c||c|c||c|c|]{}\
$d$&$l$&$d$&$l$&$d$&$l$\
$46$&$(1,2,4,5)$&$84$&$(1,3,5,7)$&$110$&$(1,3,6,8)$\
$50$&$(1,2,3,6)$&$85$&$(1,2,4,8)$&$111$&$(1,2,5,9)$\
$54$&$(2,3,4,5)$&$86$&$(3,4,5,6)$&$113$&$(2,3,6,8)$\
$57$&$(1,2,4,6)$&$90$&$(1,2,6,7)$&$117$&$(1,4,6,8)$\
$62$&$(1,3,4,6)$&$91$&$(1,4,5,7)$&$119$&$(2,3,5,9)$\
$63$&$(1,2,3,7)$&$93$&$(2,3,4,8)$&$121$&$(1,2,4,10)$\
$65$&$(2,3,4,6)$&$94$&$(1,2,5,8)$&$123$&$(1,3,7,8)$\
$66$&$(1,2,5,6)$&$95$&$(1,3,6,7)$&$125$&$(3,4,6,8)$\
$70$&$(1,2,4,7)$&$98$&$(2,3,6,7)$&$127$&$(1,3,6,9)$\
$71$&$(1,3,5,6)$&$99$&$(3,4,5,7)$&$131$&$(3,4,5,9)$\
$74$&$(2,3,5,6)$&$102$&$(1,2,4,9)$&$137$&$(2,4,6,9)$\
$78$&$(1,2,3,8)$&$105$&$(1,2,6,8)$&$143$&$(1,5,6,9)$\
$79$&$(1,2,5,7)$&$107$&$(1,3,4,9)$&&\
$81$&$(2,4,5,6)$&$109$&$(2,4,5,8)$&&\
[**II.**]{} $L_2=2A_1\oplus A_2$, $\# R(2A_1\oplus A_2)=10$.
Our second example is the sublattice $$L_2=\langle \alpha_2,\, \alpha_3,\, \alpha_5,\, \alpha_6\rangle_{\mathbb Z}=
\langle e_2+e_1,\, e_2-e_1,\, e_4-e_3,\, e_5-e_4\rangle_{\mathbb Z}\cong 2A_1\oplus A_2.$$ Then using the dual basis $\omega_j$ we obtain that $$\begin{gathered}
\label{linL2}
L_2^\perp =\langle \omega_1, \omega_4,\omega_7,\omega_8\rangle=
\langle e_3+e_4+e_5+e_6, e_6+e_7, e_7-e_8, e_7+e_8\rangle\\
=\left\{l=m_5(e_3+e_4+e_5)+\sum_{i=6}^{8} m_ie_i\mid
m_5+m_6+m_7+m_8\text{ is even}\right\}. \end{gathered}$$ We note that $L_2^\perp$ is not a root lattice.
The vector $l$ is orthogonal to a half-integral root $r$ if $$2(l\cdot r)=m_5((-1)^{\nu_3}+(-1)^{\nu_4}+(-1)^{\nu_5})+m_6(-1)^{\nu_6}+
m_7(-1)^{\nu_7}+m_8(-1)^{\nu_8}\!=0.$$ There are two different cases:
- if $3m_5=\pm m_6\pm m_7\pm m_8$ then there are $4$ half-integral roots orthogonal to $l$, since there are two choices for $(\nu_1,\nu_2)$ and for the sign of $r$;
- if $m_5=\pm m_6\pm m_7\pm m_8$ then there are $12$ half-integral roots orthogonal to $l$, since there are three choices for $(\nu_3,\nu_4, \nu_5)$.
Let us find integral roots of $E_8$ (not in $L_2$) orthogonal to $l$:
- if $m_i=0$ ($i=6$, $7$ or $8$) then there are $8$ roots $\pm(e_{1,2}\pm e_i)$;
- if $m_5=0$ then there are $24$ roots $\pm(e_{1,2}\pm
e_{3,4,5})$;
- if $m_i=\pm m_5$ ($i=6$, $7$ or $8$) then there are $6$ roots $\pm(e_i\mp e_{3,4,5})$;
- if $m_i=\pm m_j$ ($6\le i<j\le 8$) then there are $2$ roots $\pm(e_i\mp e_j)$.
Therefore we obtain
\[2A1\] $l\in L_2^\perp$ (see (\[linL2\])) is orthogonal to exactly $10$ roots of $E_8$ if and only if
- $m_j\ne 0$ for any $j$ and $m_i \ne \pm m_j$ for any $i<j$;
- $km_5\ne \pm m_6\pm m_7\pm m_8$, where $k=1$ or $3$.
Moreover $l\in L_2^\perp$ is orthogonal to exactly $14$ roots of $E_8$ if (i) and (ii) for $k=1$ are valid and there is exactly one relation of type $3m_5=\pm m_6\pm m_7\pm m_8$.
Some $l\in L_2^\perp$ orthogonal to $10$ roots in $E_8$ and having norm $l^2=3m_5^2+m_6^2+m_7^2+m_8^2=2d\in P_{ex}$ are given in table [**II-10**]{}.
[|c|c||c|c||c|c|]{}\
$d$&$l$&$d$&$l$&$d$&$l$\
$58$&$(1;\,2,3,10)$&$75$&$(6;\,1,4,5)$&$89$&$(2;\,6,7,9)$\
$60$&$(3;\,2,5,8)$&$80$&$(3;\,4,6,9)$&$97$&$(4;\,1,8,9)$\
$64$&$(5;\,1,4,6)$&$82$&$(5;\,3,4,8)$&$100$&$(7;\,1,4,6)$\
$67$&$(2;\,4,5,9)$&$83$&$(2;\,1,3,12)$&$101$&$(4;\,1,3,12)$\
$72$&$(3;\,1,4,10)$&$87$&$(6;\,1,4,7)$&$103$&$(8;\,1,2,3)$\
$73$&$(4;\,3,5,8)$&$88$&$(1;\,2,5,12)$&$115$&$(4;\,1,9,10)$\
The vectors from the tables [**I-8,12**]{} and [**II-10**]{} produce cusp forms $F_l(Z)$ of weights $16$, $18$ (table [**I-8,12**]{}) or $17$ (table [**II-10**]{}) for all $d>61$ in the set $P_{ex}$ except $d=68$, $69$, $77$, $92$.
The vectors from $L_2^\perp$ with $l^2=2d$ and $d\le 61$ that are orthogonal to exactly $14$ roots of $E_8$ are given in table [**[II-14]{}**]{}.
[|c|c||c|c||c|c|]{}\
$d$&$l$&$d$&$l$&$d$&$l$\
$40$&$(1;\,2,3,8)$&$48$&$(3;\,1,2,8)$&$55$&$(4;\,1,5,6)$\
$43$&$(2;\,1,3,8)$&$52$&$(1;\,2,4,9)$&$61$&$(2;\,1,3,10)$\
[**III**]{}. $L_3=A_3$, $\# R(A_3)=12$.
The root lattice $A_3$ is maximal. Therefore any sublattice of type $A_3$ in $E_8$ is primitive. Analysing the discriminant form of the orthogonal complement of $A_3$ we obtain that it is isomorphic to $D_5$. We put $$L_3=\langle\,\alpha_2,\,\alpha_4,\,\alpha_3\rangle_{\mathbb Z}=
\langle\,e_2+e_1,\,e_3-e_2,\,e_2-e_1\rangle_{\mathbb Z}\cong A_3.$$ Then $$L_3^\perp=\bigg\{l=\sum_{i=4}^{8}m_ie_i\mid\sum_{i=4}^{8}m_i\equiv
0\mod 2\bigg\}\cong D_5.$$ As above we obtain
\[A3\] $l\in L_3^\perp$ is orthogonal to exactly $12$ roots of $E_8$ if and only if
- $m_j\ne 0$ for any $j$;
- $m_i \ne \pm m_j$ for any $i<j$;
- $\sum_{i=4}^{8} \pm m_i\ne 0$ for any choice of the signs.
Moreover $l\in L_3^\perp$ is orthogonal to exactly $14$ roots of $E_8$ if (i) and (iii) are valid and there is only one relation of type $m_i=\pm m_j$ for $4\le i<j\le 8$.
See table [**III**]{} for several vectors $l\in L_3^\perp$ orthogonal to $N_l$ roots ($N_l=12$ or $14$) in $E_8$ and having norm $l^2=\sum_{i=4}^{8}m_i^2=2d$.
[|c|c|c||c||c|c|]{}\
$d$&$l$&$N_l$&$d$&$l$&$N_l$\
$69$&$(2,3,5,6,8)$&$12$&$53$&$(1,4,4,3,8)$&$14$\
$42$&$(1,3,3,4,7)$&$14$&$54$&$(1,3,3,5,8)$&$14$\
$48$&$(1,1,2,3,9)$&$14$&$56$&$(1,1,5,6,7)$&$14$\
$49$&$(2,2,4,5,7)$&$14$&$59$&$(1,2,2,3,10)$&$14$\
$51$&$(1,6,6,2,5)$&$14$&$63$&$(3,4,4,6,7)$&$14$\
[**IV**]{}. $L_4=A_1\oplus A_2$, $\# R(A_1\oplus A_2)=8$.
For any sublattice $A_1\oplus A_2$ in $E_8$ we see that its orthogonal complement is isomorphic to $A_5$, since $(A_2)^\perp_{E_8}=E_6$ and $(A_1)^\perp_{E_6}=A_5$. We put $L_4=\langle\,\alpha_1,\,\alpha_2,\,\alpha_3\rangle_{\mathbb Z}\cong
A_1\oplus A_2 $. Then $$L_4^\perp=\bigg\{l=\sum_{i=3}^{8} m_ie_i \mid m_8=\sum_{i=3}^{7}m_i \bigg\}.$$ If $l$ is orthogonal to a half-integral root distinct from $\alpha_1$, $\alpha_1+\alpha_3\in L_4$ then we get a relation of the form $$m_{i_1}+\dots+m_{i_k}=0,\quad\text{ where }\quad 3\le i_1<\dots i_k\le 7,\quad
1\le k\le 5.$$ If any relation of this type is valid then $l$ is orthogonal to $4$ additional [*half-integral*]{} roots. Considering the scalar products with [*integral*]{} roots we see that
- if $m_i=0$ ($3\le i\le 8$) then $l$ is orthogonal to $8$ roots $\pm(e_{1,2}\pm e_i)$;
- if $m_i=\pm m_j$ ($3\le i<j\le 8$) then $l$ is orthogonal to $2$ roots $\pm(e_i\mp e_j)$.
We list some cases of these results in table [**IV**]{}.
[|c|c|c||c|c|c|]{}\
$d$&$l$&$N_l$&$d$&$l$&$N_l$\
$68$&$(1,3,4,5,-7;\,8)$&$12$&$92$&$(1,1,2,3,5;\,12)$&$10$\
$77$&$(2,3,4,5,-8;\,6)$&$12$&$40$&$(1,1,2,3,-8;\,-1)$&$14$\
It is possible to formulate a result for this case analogous to Propositions \[4A1\], \[2A1\] and \[A3\], but we do not need it.
An extensive computer search for vectors $l$ orthogonal to at least $2$ and at most $14$ roots for other $d\in P_{ex}$ has not found any.
Now we have everything we need to prove our main theorem, Theorem \[mainthm\]. For $d>61$ and for $d=46$, $50$, $54$, $57$, $58$, $60$ there exists a vector $l$ satisfying condition (\[orth2\]), either by Proposition \[P-ex\] or listed in one of the tables. Hence Theorem \[cuspform\] provides us with a suitable cusp form of low weight. Since the dimension of ${\mathcal F}_{2d}$ is $19$, Theorem \[main\_sings\_theorem\] guarantees the existence of a compactification with only canonical singularities and hence Theorem \[mainthm\] follows by using the low weight cusp form trick, according to Theorem \[general\_gt\].
If $d$ is not as above but $d\ge 40$ and $d\ne 41$, $44$, $45$, $47$ then we have a cusp form of weight $19$ arising from a vector $l$ orthogonal to $14$ roots, listed in one of the tables. This gives rise to a canonical form and hence, by Freitag’s result, the Kodaira dimension of ${\mathcal F}_{2d}$ is non-negative, as stated in Theorem \[general\_gt\].
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S. Mukai, [*Polarized ${\mathop{\mathrm {K3}}\nolimits}$ surfaces of genus $18$ and $20$.*]{} Complex projective geometry (Trieste, 1989/Bergen, 1989), 264–276, London Math. Soc. Lecture Note Ser., [**179**]{}, Cambridge Univ. Press, Cambridge, 1992.
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I. Piatetskii-Shapiro, I. Shafarevich, *A Torelli theorem for algebraic surfaces of type ${\mathop{\mathrm {K3}}\nolimits}$*. Izv. Akad. Nauk SSSR, Ser. Mat. [**35**]{}, 530-572 (1971). English translation in Math. USSR, Izv. [**5**]{} (1971), 547-588 (1972).
M. Reid, [*Canonical $3$-folds.*]{} Journées de Géometrie Algébrique d’Angers 1979, 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.
F. Scattone, [*On the compactification of moduli spaces of algebraic ${\mathop{\mathrm {K3}}\nolimits}$ surfaces.*]{}, Mem. Amer. Math. Soc. [**70**]{}, no. 374, (1987)
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Y. Tai, [*On the Kodaira dimension of the moduli space of abelian varieties*]{}. Invent. Math. [**68**]{} (1982), 425–439.
V.A. Gritsenko\
Université Lille 1\
Laboratoire Paul Painlevé\
F-59655 Villeneuve d’Ascq, Cedex\
France\
[[email protected]]{}
K. Hulek\
Institut für Algebraische Geometrie\
Leibniz Universität Hannover\
D-30060 Hannover\
Germany\
[[email protected]]{}
G.K. Sankaran\
Department of Mathematical Sciences\
University of Bath\
Bath BA2 7AY\
England\
[[email protected]]{}
|
---
abstract: |
The MIT and CXC ACIS teams have explored a number of measures to ameliorate the effects of radiation damage suffered by the ACIS FI CCDs. One of these measures is a novel CCD read-out method called “squeegee mode”. A variety of different implementations of the squeegee mode have now been tested on the I0 CCD.
Our results for the fitted FWHM at Al-K$\alpha$ and Mn-K$\alpha$ clearly demonstrate that all the squeegee modes provide improved performance in terms of reducing CTI and improving spectral resolution. Our analysis of the detection efficiency shows that the so-called squeegee modes “Vanilla” and “Maximum Observing Efficiency” provide the same detection efficiency as the standard clocking, once the decay in the intensity of the radioactive source has been taken into account. The squeegee modes which utilize the slow parallel transfer (“Maximum Spectral Resolution”, “Maximum Angular Resolution”, and “Sub-Array”) show a significantly lower detection efficiency than the standard clocking. The slow parallel transfer squeegee modes exhibit severe grade migration from flight grade 0 to flight grade 64 and a smaller migration into ASCA g7. The latter effect can explain some of the drop in detection efficiency.
There are a few observational penalities to consider in using a squeegee mode. Utilizing any squeegee mode causes a loss of FOV near the aimpoint (4 to 16$''$ strips along the full length of the CCDs), as well as the attendant dead-time increase. Secondly, the cost of the software implementation and its testing will be significant. Lastly, each squeegee mode “flavor” would require lengthy, mode-specific calibration observations. Therefore, since an efficacious, ground-based CTI corrector algorithm is now available (see paper by Plucinsky, Townsley, *et al.* in this proceedings), a scientific judgment will have to be made to determine which, if any, squeegee modes should be developed and calibrated for use by Chandra observers.
author:
- 'Shanil N. Virani and Paul P. Plucinsky'
- 'Catherine E. Grant and Beverly LaMarr'
title: 'Analysis of On-Orbit ACIS Squeegee Mode Data'
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Introduction
============
The ACIS Operations team, combining elements from the CXC and the ACIS MIT$/$IPI team, has been developing and testing the new squeegee modes since the Spring of 2000. In squeegee mode, charge is collected in the top few rows of the CCD and then swept across the imaging array once per readout, thus filling some of the radiation-induced electron traps that cause degraded performance. The development of a novel method of reading out the ACIS CCDs was first developed and tested on ground CCDs that are similar to flight CCDs (see Prigozhin, *et al.* 2000 for a characterization of the radiation damage of ACIS CCDs). This new mode ameliorates some of the effects caused by the radiation damage suffered early in the mission by the FI CCDs. For a more comprehensive discussion on the design and clocking method of squeegee mode, see Bautz and Kissel (2000). A description of the different squeegee modes is included in an internal MIT$/$ACIS Memo by Bautz and Grant (2000). That memo also presents an analysis of the squeegee modes and describes the trade-offs in choosing between the various squeegee modes. In this paper, we present the results of the various squeegee measurements on the I0 CCD.
Test OBSID Date Exp(s) Description
--------- ------- ------------- --------- --------------------------------------------------
Control 62895 20 Feb 2000 9,023.4 I0, 3.2s, standard clocking
($40\mu$s par xfr, no 2x2 sum)
“Control” non-Squeegee
L1 62042 30 May 2000 8,225.5 I0, 3.3s, 16 row sq, 32 row exc win, no 2x2 sum,
$40\mu$s par xfr, 24 flushes, 1010 rev clks
“Vanilla Squeegee”
L2 62019 21 Jun 2000 8,623.6 I0, 1.9s, 16 row sq, 24 row exc window, 2x2 sum,
$320\mu$s par xfr, 24 flushes, 1026+32 rev clks
Maximum Spectral Resolution
L4 62007 01 Jul 2000 8,517.3 I0, 3.3s, 2 row sq, 8 row exc win, no 2x2 sum,
$320\mu$s par xfr, 24 flushes, 1026+32 rev clks
Maximum Angular Resolution
L7 61981 30 Jul 2000 8,551.5 I0, 1.8s, 2 row sq, 8 row exc win, 2x2 sum,
$40\mu$s par xfr, 24 flushes, 1026+32 rev clks
Maximum Observing Efficiency
L9 61949 30 Aug 2000 8,356.9 I0, 1.2s, 2 row sq, 8 row exc win, no 2x2 sum,
$320\mu$s par xfr, 24 flushes, 1026+32 rev clks,
256 row sub-array
: [**List of I0 Squeegee Tests** ]{} []{data-label="datasets"}
This analysis utilized the standard level 0 and level 1 data products produced by the CXC Data System. The datasets included in this analysis are listed in Table 1. The control run (OBSID 62895) was a long charge transfer inefficiency (CTI) measurement. There are five “flavors” of Squeegee discussed in this paper: L1 “Vanilla”, L2 “Maximum Spectral Resolution”, L4 “Maximum Angular Resolution”, L7 “Maximum Observing Efficiency”, and L9 “Sub-Array near the aimpoint”. Table 1 summarizes the important parameters which distinguish one squeegee mode from another: static integration time, number of squeegee rows (2 or 16), number of rows in the exclusion window (8 or 24 or 32), on-chip summing (yes or no), parallel transfer time ($40\mu$s or $320\mu$s), number of frame flushes (always 24 for these squeegees), and number of reverse clocks (1010 or 1026+32). In addition, the table includes the average exposure time for these measurements. With all of these squeegee modes, the exposure time varies from row-to-row. The number listed is the correct exposure time for the middle of the CCD or for the middle of the sub-array for squeegee L9. In the analysis presented in this paper, the row-to-row variation in exposure time has been included by computing the exposure for each of the 32 row elements.
Methodology
===========
Data from each I0 measurement was separated by node and then further reduced to thirty-one, 32-row region files. Spectra files were generated from the CXC level 1 events files. Events were selected in the [ASCA]{} g02346 grade set and were filtered on node and [chipy]{} coordinates. The gain was computed using the so-called “local-gain” method, meaning the fitted peak in each 32 row element is used to compute a conversion from ADUs to eV for that element. The prominent lines of the internal calibration source were then fit using Gaussians; the figure of merit employed was the C-statistic.
Results and Analysis
====================
The results of this analysis are presented in Figures 1 and 2. Figure 1 shows the fitted values of the FWHM in eV for the I0 CCD at 1.5 keV (Al-K$\alpha$) at a focal plane temperature of -110 C, -120 C, and at -120 C using the “Vanilla” squeegee mode. For comparative purposes, the S3 CCD FWHM at -120 C is also overlaid. Figure 2 shows the fitted values of the FWHM in eV for the I0 CCD at 5.9 keV (Mn-K$\alpha$) at a focal plane temperature of -110 C, -120 C, and at -120 C using the “Vanilla” squeegee mode. For comparative purposes, the S3 CCD FWHM at -120 C is also overlaid. What is clear from both plots is that squeegee mode improves the spectral resolution of I0 CCD compared to standard clocking.
Figures 3 and 4 displays the quantities [ASCA]{} [g0234/all grades]{}, [ASCA]{} [g02346/all grades]{}, [ASCA]{} [g7/all grades]{} versus row number, where “all grades” means all events which pass the on-board filtering and make it into telemetry, for the Al-K$\alpha$ and Mn-K$\alpha$ lines. All of these measurements were executed using an upper event amplitude cutoff of 3750 ADUs and a grade filter which rejected flight grades 24, 66, 107, 214, and 255. Figure 5 is a plot of the detection efficiency for each line as a function of row number. We have included a statistical error bar for the count rate plots at the location of the first data element which is representative of the uncertainties in these measurements. In producing Figure 5, we have corrected for the decrease in intensity of the radioactive source as the data span 6 months. See 3.4 for further discussion.
=4.0in =4.0in
=4.0in =4.0in
Al results
----------
=3.0in =3.0in
Figure 3 shows the grade distribution as a function of row number for the squeegee measurements and the control run. The enhanced CTI of the FI CCDs causes events to migrate from [ASCA]{} grades g0234 to higher grades. This can be seen in the control run and the squeegee runs. The effect is quite small at Al-K$\alpha$, never more than 3% for the grade sets listed here. Squeegees L2 and L4 show a very different behavior with regards to grade migration when the individual grades are examined. Analysis of the g0 events as a function of row number show that the percentage of grade g0 events is dropping from $\sim90\%$ in the first resolution element to $\sim20\%$ by the top of the CCD. [ASCA]{} grade 2 is the beneficiary of these counts as it increases from its nominal value of 5% to over 80% of the events at the top of the CCD. [ASCA]{} grade 2 is composed of flight grades 2 (down splits) and 64 (up splits). A further examination of the data reveals that all of the migrating events are going into flight grade 64. The one characteristic which the squeegees L2 and L4 have in common is the slow parallel transfer. Lastly, L2 and L7 both have much higher g0234$/$all grades ratios than the other modes because both L2 and L7 utilize 2x2 summing.
We suggest that this severe grade migration provides a clue to the cause of the lower detection efficiency seen in Figure 5. A common cause of reduced efficiency in a given grade combination is that events are migrating to grades outside of the chosen set. We will defer a full discussion of this until Section 3.4 in which the decay in the intensity of the radioactive source is included.
Mn results
----------
=3.0in =3.0in
The grade distribution versus number row number data are included in Figure 4. The squeegee runs now show greater variability with respect to each other and a larger difference with respect to the control run. Particularly interesting is the grade migration exhibited by squeegee L4. The fraction of g02346 events shows a linear decrease from row 100 to row 400, at which point it flattens out and is much lower than any other measurement. This behavior is also seen in the migration to g7 events. The same effect is seen in L2 but at a reduced level. This is presumably due to the fact that L2 utilizes on-chip summing, which should retain more of the valid X-ray events in the g02346 grade set. The fraction of g02346 and g7 events in the sub-array squeegee L9 lies between L2 and L4. The shorter frame-time of L9 compared to L4 appears to reduce some of this migration indicating that the effect cannot be attributed solely to the short timescale traps.
The grade migration effect seen at Al-K$\alpha$ for squeegees L2 and L4 is more pronounced at Mn-$\rm{K\alpha}$. Analysis of the individual grade “branching ratios” shows an even stronger migration from grades g0, g3, and g4 to g2 than for Al-$\rm{K\alpha}$. Squeegee L4 also shows a steep increase in the g7 events from row 100 to row 500. The flight grade analysis shows that the migrating events are going into flight grade 64. It is interesting to note that the percentage of g2 events is larger in L2 than L4. The difference between these two squeegees is that L2 uses on-chip summing while L4 does not. This suggests that a significant amount of charge is trailing the center of the event by 2-3 or perhaps even more pixels. Lastly, L2 and L7 both have much higher g0234$/$all grades ratios than the other modes because both L2 and L7 utilize 2x2 summing.
Ti results
----------
The results for the Ti-$\rm{K\alpha}$ line are not presented in this paper due to space restrictions. However, the Ti-$\rm{K\alpha}$ data for I0 resemble the Mn-$\rm{K\alpha}$ data. The grade distribution results for Ti are also similar to the Mn results as one would expect given that the performance characteristics of the CCD vary little from 4.5 keV to 5.9 keV.
Correction for Decay of Radioactive Source
------------------------------------------
These squeegee tests were run between 3 and 6 months after the control run. The half-life of the ${\rm Fe^{55}}$ source is 2.7 yr. Therefore, the intensity of the source decays by $\sim6\%$ in three months and by $\sim12\%$ after six months. In Figure 5 we have corrected for this decrease by simply normalizing the detected count rates of the squeegee measurements to the control run. After applying this correction, squeegees L1 and L7 have the same or higher detection efficiency than the control run across the CCD, with the possible exception of L7 at Al-K$\alpha$. The upturn in the detection efficiency for the control run for Al-K$\alpha$ is probably an artifact of the fitting process. It is interesting that the squeegee runs show a decrease of the detection efficiency with row number relative to the control run at Al-K$\alpha$, but squeegees L1 and L7 match or exceed the control run at Mn-K$\alpha$ and Ti-K$\alpha$. L2 appears to be still lower than the control run near the frame-store and drops more rapidly with row number than the control run. L4 still shows the largest drop with row number, but may now be consistent with the control run near the frame-store. The most important conclusion to be drawn from these data is that squeegees L1 and L7 [*have the same detection efficiency as the standard clocking*]{} at Mn-K$\alpha$ and Ti-K$\alpha$.
=3.0in =3.0in
The question still remains as to the lower detection efficiency of L2 and L4. The detection efficiency of L4 is $\sim20\%$ lower than the standard clocking at the top of the CCD. Only about 6% of this difference can be explained by grade migration to [ASCA]{} g7. These measurements were executed with the now-standard on-board rejection of flight grades 24, 66, 107, 214, and 255. Of these, flight grade 66 may be the most interesting because it is the closest to flight grade 64 which is exhibiting the tremendous increase. Perhaps there is also a significant migration to grade 66, which would lead to a lower detection efficiency since these events are never telemetered. It is also possible that flight grade 255 is enhanced by this effect.
The variation with row number for L2 and L4 confirm what our analysis of the S0 data indicated (Virani and Plucinsky 2000). The detection efficiency is changing with row number and the percentage decreases are the same for the squeegee runs which utilize the slow parallel transfer. The severe grade migration effect appears to lower the effective detection efficiency around row 200 for Mn-K$\alpha$ and row 700 for Ti-K$\alpha$. The Al-K$\alpha$ data are not effected until about row 900. With the improved statistical precision of the Ti-K$\alpha$ data, we can start to see the energy dependence of this effect. Clearly, the Mn-K$\alpha$ photons are effected sooner than the Ti-K$\alpha$. This is suggestive of a strong dependence on energy in a rather narrow range from 4.5 to 5.9 keV.
One suggested explanation for the lower detection efficiency is that the analysis is confused by the blending of the K$\alpha$ and K$\beta$ lines as the spectral resolution degrades. We note that this effect should vary in magnitude with row number since the resolution is degrading with row number. However, squeegees L2 and L4 show a discrepancy with respect to the control run in the first 200 rows. Line blending is not an issue in the first 200 rows on I0 since the spectral resolution is still close to pre-launch values and is more than sufficient to resolve the K$\alpha$ and K$\beta$ lines for Mn and Ti. Line blending may be part of the explanation near the top of the CCD as the discrepancy between the squeegees and the control runs increases with row number. Nevertheless, the lower detection efficiency near the frame-store is a puzzling effect which warrants more investigation.
Conclusions
===========
We confirm that all of the tested squeegee modes improve the spectral resolution of the I0 CCD compared to the standard clocking. Our analysis of the detection uniformity indicates that squeegees L1 and L7 have the same detection efficiency as the standard clocking after correcting for the decay in the intensity of the radioactive source, while the squeegees L2, L4, and L9 still exhibit a lower detection efficiency. The discrepancy is as large as 20% for squeegee L4 at the top of the CCD. L2 and L4 also produce a highly spatially-dependent grade distribution. We suggest that the slow parallel transfer of both these modes is the likely explanation for this effect. We suggest that this effect should be investigated further with the hope that a squeegee mode can be developed which optimizes the spectral resolution, the detection efficiency, and the [*uniformity*]{} of the detection efficiency.
Acknowledgments
===============
We thank our ACIS and CXC colleagues, particularly Mark Bautz, Dan Schwartz, and Peter Ford, for many useful ideas over the course of this analysis. SNV and PPP acknowledge support for this research from NASA contract NAS8-39073; CEG and BL acknowledge support for this research from NASA contracts NAS8-37716 and NAS8-38252.
Prigozhin, G., *et al.*, “Characterization of the radiation damage in the Chandra X-ray CCDs”, SPIE Proceedings, vol. 4140, August, 2000, pp. 123-134
Bautz, M. and Kissel, S., “Explanatory Note on Squeegee Mode”, Internal MIT$/$ACIS Memo, 23 May 2000
Bautz, M. and Grant, C., “Choosing an ACIS Squeegee Mode”, Internal MIT$/$ACIS Memo, 15 August 2000
Virani, S. N. and Plucinsky, P. P., “Analysis of ACIS Squeegee Mode Data on Chip S0”, Internal CXC/ACIS Memo, 29 August 2000
|
---
author:
- Dinakar Ramakrishnan
- Song Wang
title: |
A Cuspidality Criterion for the Functorial Product on ${\bf GL
(2) \times GL (3)}$, with a cohomological application
---
Introduction
============
A strong impetus for this paper came, at least for the first author, from a question of Avner Ash, asking whether one can construct non-selfdual, non-monomial cuspidal cohomology classes for suitable congruence subgroups $\Gamma$ of SL$(n, {{\mathbb Z}})$, say for $n=6$. Such a construction, in special examples, has been known for some time for $n=3$ ([@A-G-G], [@vG-T1], [@vG-K-T-V], [@vG-T2]); it is of course not possible for $n=2$. One can without trouble construct non–selfdual, *monomial* classes for any $n = 2 m$ with $m \geq 2$, not just for constant coefficients (see the Appendix, Theorem E). In the Appendix we also construct non-monomial, non-selfdual classes for $n=4$ using the automorphic induction to ${{\mathbb Q}}$ of suitable Hecke character twists of non-CM cusp forms of “weight $2$” over imaginary quadratic fields, but they admit quadratic self-twists and are hence imprimitive. The tack pursued in the main body of this paper, and which is the natural thing to do, is to take a non-selfdual (non-monomial) $n=3$ example $\pi$, and take its functorial product $\boxtimes$ with a cuspidal $\pi'$ on GL$(2)/{{\mathbb Q}}$ associated to a holomorphic newform of weight $4$ for a congruence subgroup of SL$(2, {{\mathbb Z}})$. The resulting (cohomological) $n=6$ example can be shown to be [*non-selfdual*]{} for suitable $\pi'$. (This should be the case for all $\pi'$, but we cannot prove this with current technology – see Remark 4.1.) Given that, the main problem is that it is not easy to show that such an automorphic tensor product $\Pi:= \pi \boxtimes \pi'$, whose modularity was established in the recent deep work of H. Kim and F. Shahidi ([@KSh2000]), is [*cuspidal*]{}. This has led us to prove a precise cuspidality criterion (Theorem A) for this product, not just for those of cohomological type, which hopefully justifies the existence of this paper. The second author earlier proved such a criterion when $\pi$ is a twist of the symmetric square of a cusp form on GL$(2)$ ([@Wa2003]; such forms are essentially selfdual, however, and so do not help towards the problem of constructing non-selfdual classes. One of the reasons we are able to prove the criterion [*in general*]{} is the fact that the associated, degree $20$ exterior cube $L$-function is nicely behaved and analyzable. This helps us rule out, when the forms on GL$(2)$ and GL$(3)$ are non-monomial, the possible decomposition of $\Pi$ into an isobaric sum of two cusp forms on GL$(3)$ (see section 7). This is the heart of the matter.
We will also give a criterion (Theorem B) as to when the base change of $\Pi$ to a solvable Galois extension remains cuspidal. We will derive a stronger result for the cohomological examples (Theorem C), namely that each of them is [*primitive*]{}, i.e., not associated to a cusp form on GL$(m)/K$ for [*any*]{} (possibly non-normal) extension $K/{{\mathbb Q}}$ of degree $d > 1$ with $dm=6$. Furthermore, each of the three main non-selfdual GL$(3)$ examples $\pi$ of [@vG-T1], [@vG-K-T-V] and [@vG-T2] comes equipped, confirming a basic conjecture of Clozel ([@C]), with a certain $3$-dimensional $\ell$-adic representation $\rho_\ell$ whose Frobenius traces $a_p(\rho_\ell)$ agree with the Hecke eigenvalues $a_p(\pi)$ for small $p$. For $\pi'$ on GL$(2)/{{\mathbb Q}}$ defined by a suitable holomorphic newform of weight $4$, with associated Galois representation $\rho'_\ell$, we will show (Theorem D) that the six-dimensional $R_\ell: = \rho_\ell \otimes
\rho'_\ell$, which should correspond to $\Pi$, remains irreducible under restriction to [*any*]{} open subgroup of Gal$(\overline
{{\mathbb Q}}/{{\mathbb Q}})$.
The first author would like to thank Avner Ash for his question and for making comments on a earlier version of this paper, Mahdi Asgari for initially kindling his interest (at Park City, UT) in the problem of establishing a precise [*cuspidality criterion*]{} for the Kim-Shahidi product, and the National Science Foundation for financial support through the grant DMS-0100372. The second author would like to thank James Cogdell and Henry Kim for their interest in his lecture on this work at the Fields Institute Workshop on Automorphic L-functions in May 2003.
The Cuspidality Criterion
=========================
Throughout this paper, by a cusp form on $GL (n)$ (over a global field $F$) we will mean an irreducible, cuspidal automorphic representation $\pi = \pi_\infty \otimes \pi_f$ of $GL_{n}
(\mathbb{A}_{F})$. We will denote its central character by $\omega_\pi$. One says that $\pi$ is [*essentially self-dual*]{} iff its contragredient $\pi^\vee$ is isomorphic to $\pi \otimes
\nu$ for some character $\nu$ of (the idele classes of) $F$; when $n=2$, one always has $\pi^\vee \simeq \pi \otimes
\omega_\pi^{-1}$. Note that $\pi$ is [*unitary*]{} iff $\pi^\vee$ is the [*complex conjugate representation*]{} $\overline \pi$. Given any cusp form $\pi$, we can find a real number $t$ such that $\pi_u: = \pi \otimes \vert . \vert^t$ is unitary.
For any cusp form $\pi'$ on $GL (2)$, put $Ad (\pi') = {\rm
sym}^{2} (\pi') \otimes \omega_{\pi'}^{-1}$ and $A^{4} (\pi') =
{\rm sym}^{4} (\pi') \otimes \omega_{\pi'}^{-2}$. Recall that $\pi'$ is [*dihedral*]{} iff it admits a self-twist by a quadratic character; it is [*tetrahedral*]{}, resp. [*octahedral*]{}, iff $sym^2(\pi')$, resp. $sym^3(\pi')$, is cuspidal and admits a self-twist by a cubic, resp. quadratic, character. (The automorphy of $sym^3(\pi')$ was shown by Kim and Shahidi in [@KSh2000].) We will say that $\pi'$ is of [*solvable polyhedral type*]{} iff it is dihedral, tetrahedral or octahedral.
\[TM:A\] Let $\pi', \pi$ be cusp forms on $GL (2)$, $GL (3)$ respectively over a number field $F$. Then the Kim–Shahidi transfer $\Pi =
\pi \boxtimes \pi'$ on GL$(6)/F$ is cuspidal unless one of the following happens:
[(a)]{.nodecor} $\pi'$ is not dihedral, and $\pi$ is a twist of $Ad (\pi')$;
[(b)]{.nodecor} $\pi'$ is dihedral, $L(s, \pi) = L(s,
\chi)$ for an idele class character $\chi$ of a cubic, non-normal extension $K$ of $F$, and the base change $\pi_K$ is Eisensteinian.
Furthermore, when [(a)]{.nodecor} (resp. [(b)]{.nodecor}) happens, $\Pi$ possesses an isobaric decomposition of type $(2, 4)$ or $(2, 2, 2)$ (resp. of type $(3, 3)$). More precisely, when we are in case [(a)]{.nodecor}, $\Pi$ is of type $(2,2,2)$ if $\pi'$ is tetrahedral, and $(2, 4)$ otherwise.
*Remark:* By [@KSh2000], $\Pi = \pi \boxtimes \pi'$ is automorphic on GL$(6)/F$, and its $L$–function agrees with the Rankin–Selberg $L$–function $L (s, \pi \times \pi')$. Theorem A implies in particular that $\Pi$ is cuspidal if (i) $\pi'$ is not dihedral [*and*]{} (ii) $\pi$ is not a twist of $Ad(\pi')$.
A partial cuspidality criterion was proved earlier by the second author in [@Wa2003]; but he only treated the case when $\pi$ is twist equivalent to the Gelbart–Jacquet symmetric square transfer of some cusp form on $GL (2)$.
\[TM:B\] Let $F$ be a number field and $\pi', \pi$ be cusp forms on $GL (2)/F$, $GL (3)/F$ respectively. Put $\Pi = \pi \boxtimes \pi'$. Assume that $\pi'$ is not of solvable polyhedral type, and $\pi$ not essentially selfdual. Then we have the following:
[(a)]{.nodecor} If $\pi$ does not admit any self twist, $\Pi$ is cuspidal without any self twist. Furthermore, if $\pi$ is not monomial, then $\Pi$ is not induced from any non-normal cubic extension.
[(b)]{.nodecor} If $\pi$ is not of solvable type, i.e., its base change to any solvable Galois extension is cuspidal, $\Pi$ is cuspidal and not of solvable type; in particular, there is no solvable extension $K / F$ of degree $d > 1$ dividing $6$, and a cuspidal automorphic representation $\eta$ of $GL_{6/d} ({{\mathbb A}}_{F})$, such that $L (s, \Pi) = L (s, \eta)$.
*Remark:* If $\pi$ is regular algebraic at infinity, and $F$ is not totally imaginary, then $\pi$ is not monomial (See Lemma \[T:903\]).
We will prove Theorem \[TM:A\] in sections 6 through 8, and Theorem \[TM:B\] in section 9.
Before proceeding with the proofs of these theorems, we will digress and discuss the cohomological application.
Preliminaries on cuspidal cohomology
====================================
The experts can skip this section and go straight to the statement (in section 4) and the proof (in section 5) of Theorems \[TM:C\], \[TM:D\]. Let $$\Gamma \, \subset \, {\rm SL}(n, {{\mathbb Z}}),$$ be a congruence subgroup of $G_n^0: = {\rm SL}(n, {{\mathbb R}})$, which has finite covolume. $\Gamma$ acts by left translation on the symmetric space $X_n^0 : = \, {\rm SL}(n, {{\mathbb R}})/{\rm SO}(n)$. The cohomology of $\Gamma$ is the same as that of the locally symmetric orbifold $\Gamma \backslash X_n^0$. If $H^\ast_{\rm
cont}$ denotes the [*continuous group cohomology*]{}, a version of Shapiro’s lemma gives an isomorphism $$H^\ast(\Gamma, {{\mathbb C}}) \, \simeq \, H^\ast_{\rm cont}(G_n^0, {\mathcal
C}^\infty(\Gamma\backslash G_n^0)).$$ The constant functions are in this space, and the contribution of $H^\ast_{\rm cont}(G_n^0, {{\mathbb C}})$ to $H^\ast(\Gamma, {{\mathbb C}})$ is well understood and plays an important role in Borel’s interpretation of the values of the Riemann zeta function $\zeta(s)$ at negative integers.
We will be interested here in another, more mysterious, piece of $H^\ast(\Gamma, {{\mathbb C}})$, namely its [*cuspidal part*]{}, denoted $H^\ast_{\rm cusp}(\Gamma, {{\mathbb C}})$, which injects into $H^\ast(\Gamma, {{\mathbb C}})$ by a theorem of Borel. Furthermore, one knows by L. Clozel ([@C]) that the cuspidal summand is defined over ${{\mathbb Q}}$, preserved by the Hecke operators. The cuspidal cohomology is represented by cocycles defined by smooth cusp forms in $L^2(\Gamma\backslash G_n^0)$, i.e., one has $$H^\ast_{\rm cusp}(\Gamma, {{\mathbb C}}) \, = \, H^\ast_{\rm cont}(G_n^0,
L^2_{\rm cusp}(\Gamma\backslash G_n^0)^\infty),$$ where $L^2_{\rm cusp}(\Gamma\backslash G_n^0)$ denotes the space of cusp forms, and the superscript $\infty$ signifies taking the subspace of smooth vectors. If $\mathfrak G_{n}$ denotes the [*complexified*]{} Lie algebra of $G_n$, the passage from continuous cohomology to the [*relative Lie algebra cohomology*]{} ([@BoW]) furnishes an isomorphism $$H^\ast_{\rm cusp}(\Gamma, {{\mathbb C}}) \, \simeq \, H^\ast({\mathfrak
G}^0_{n},K: L^2_{\rm cusp}(\Gamma\backslash G_n^0)^\infty).$$ It is a standard fact (see [@BoJ], for example) that the right action of $G_n$ on $L^2_{\rm cusp}(\Gamma\backslash G_n^0)$ is completely reducible, and so we may write $$L^2_{\rm cusp}(\Gamma\backslash G_n^0) \, \simeq \,
\hat{\oplus}_\pi \, m_\pi {\mathcal H}_\pi,$$ where $\pi$ runs over the irreducible unitary representations of $G_n^0$ (up to equivalence), ${\mathcal H}_\pi$ denotes the space of $\pi$, $\hat{\oplus}$ signifies taking the Hilbert direct sum, and $m_\pi$ is the multiplicity. Consequently, $$H^\ast_{\rm cusp}(\Gamma, {{\mathbb C}}) \, \simeq \, {\oplus}_\pi \,
H^\ast({\mathfrak G}_{n,{{\mathbb C}}}^0, K; {\mathcal
H}_\pi^\infty)^{m_\pi}.$$ One knows completely which representations $\pi$ of $G_n^0$ have non-zero $({\mathfrak G}_{n}^0, K)$-cohomology ([@VZ]; see also [@Ku]). An immediate consequence (see [@C], page 114) is the following (with $[x]$ denoting, for any $x \in {{\mathbb R}}$, the integral part of $x$):
\[T:301\] $$H^i_{\rm cusp}(\Gamma, {{\mathbb C}}) \, = \, 0 \quad {\rm unless} \quad d(n)
\leq i \leq d(n)+[(n-1)/2],$$ where $$d(n) = m^2 \quad {\rm if} \quad n=2m \quad {\rm and} \quad d(n) =
m(m+1) \quad {\rm if} \quad n=2m+1.$$
It will be necessary for us to work with the ${{\mathbb Q}}$-group $G_n: =
{\rm GL}(n)$ with center $Z_n$, and also work adelically. Let ${{\mathbb A}}= {{\mathbb R}}\times {{\mathbb A}}_f$ be the adele ring of ${{\mathbb Q}}$, $K_\infty = O(n)$, and $X_n = G_n({{\mathbb R}})/K_n$, whose connected component is $X_n^0$. For any compact open subgroup $K$ of $G_n({{\mathbb A}}_f)$, we have $$S_K: = \, G_n({{\mathbb Q}})Z_n({{\mathbb R}})^0\backslash G_n({{\mathbb A}})/K_\infty K \, \simeq
\, \cup_{j=1}^{r} \Gamma_j\backslash X_n^0, \leqno(3.6)$$ where the $\Gamma_j$ are congruence subgroups of SL$(n, {{\mathbb Z}})$ and $Z_n({{\mathbb R}})^0$ is the Euclidean connected component of $Z_n({{\mathbb R}})$. We need the following, which follows easily from the discussion in section 3.5 of [@C]:
\[T:302\]
1. $$H_{\rm cusp}^\ast(S_K, {{\mathbb C}}) \, \simeq \, \oplus_{\pi \in {\rm
Coh}_K} \, H^\ast(\tilde{\mathfrak G}_{n, \infty}, K_\infty;
\pi_\infty)\otimes \pi_f^K,$$ where $\tilde{\mathfrak G}_{n, \infty}$ consists of matrices in $M_n({{\mathbb C}})$ with purely imaginary trace, and [Coh]{}$_K$ is the set of (equivalence classes) of cuspidal automorphic representations $\pi = \pi_\infty \otimes \pi_f$ of $G_n({{\mathbb A}})$ such that $\pi_f^K \ne 0$, $\pi_\infty$ contributes to the relative Lie algebra cohomology, and $(\omega_\pi)_\infty$ is trivial on $Z({{\mathbb R}})^0$.
2. Suppose $\pi = \pi_\infty \otimes \pi_f$ is a cuspidal automorphic representation of $G_n({{\mathbb A}})$ with $\pi_f^K \ne 0$ such that the restriction $r_\infty$ of the Langlands parameter of $\pi_\infty$ to ${{\mathbb C}}^\ast$ is given by the $n$-tuple $$\begin{aligned}
\{ &(z/\vert z\vert)^{n-1}, ({\overline z}/\vert z\vert)^{n-1},
(z/\vert z\vert)^{n-3}, ({\overline z}/\vert z\vert)^{n-3},
\notag \\
\dots, &(z/\vert z\vert), ({\overline z}/\vert z\vert)\} \otimes
(z\overline z)^{n-1}
\notag\end{aligned}$$ if $n$ is even, and $$\begin{aligned}
\{ &(z/\vert z\vert)^{n-1}, ({\overline z}/\vert z\vert)^{n-1},
(z/\vert z\vert)^{n-3}, ({\overline z}/\vert z\vert)^{n-3},
\notag \\
\dots, &{(z/\vert z\vert)}^{2}, {({\overline z}/\vert z\vert)}^{2},
1\}
\otimes
(z\overline z)^{n-1}
\notag\end{aligned}$$ if $n$ is odd. Then $\pi$ contributes to ${\rm Coh}_K$ in degree $d(n)$.
Given any cohomological $\pi$ as above, the fact that the cuspidal cohomology at any level $K$ has a ${{\mathbb Q}}$-structure ([@C]) preserved by the action of the Hecke algebra ${\mathcal H}_{{\mathbb Q}}(G_f,
K)$ (consisting of ${{\mathbb Q}}$-linear combinations of $K$-double cosets), implies that the $G_f$-module $\pi_f$ is [*rational over a number field*]{} ${{\mathbb Q}}(\pi_f)$. When $n=2$, such a $\pi$ is defined by a holomorphic newform $h$ of weight $2$, and then ${{\mathbb Q}}(\pi_f)$ is none other than the field generated by the Fourier coefficients of $h$.
Non-selfdual, cuspidal classes for $\Gamma \subset {\rm SL}(6, {{\mathbb Z}})$
==============================================================================
The principle of functoriality predicts that given cuspidal automorphic representations $\pi, \pi'$ of $G_n({{\mathbb A}}), G_m({{\mathbb A}})$ respectively, there exists an isobaric automorphic representation $\pi \boxtimes \pi'$ of $G_{nm}({{\mathbb A}})$ such that for every place $v$ of ${{\mathbb Q}}$, one has $$\sigma((\pi \boxtimes \pi')_v) \, \simeq \, \sigma(\pi_v) \otimes
\sigma_v(\pi'_v),$$ where $\sigma$ is the map (up to isomorphism) given by the local Langlands correspondence ([@HaT2000], [@He2000]) from admissible irreducible representations of $G_r({{\mathbb Q}}_v)$ to $r$-dimensional representations of $W'_v$, which is the real Weil group $W_{{\mathbb R}}$ if $v=\infty$ and the extended Weil group $W_{{{\mathbb Q}}_p} \times {\rm SL}(2, {{\mathbb C}})$ if $v$ is defined by a prime number $p$.
This prediction is known to be true for $n=m=2$ ([@Ra2000]), More importantly for the matter at hand, it is also known for $(n,m)=(3,2)$ by a difficult theorem of H. Kim and F. Shahidi ([@KSh2000]).
Put $$T \, = \, T_1 \cup T_2,$$ with $$T_1 = \{53, 61, 79, 89\} \quad {\rm and} \quad T_2 = \{128, 160,
205\}.$$ By the article [@A-G-G] of Ash, Grayson and Green (for $p \in
T_1$), and the works [@vG-T1], [@vG-T2], [@vG-K-T-V] of B. van Geemen, J. Top, et al (for $p \in T_2$), one knows that for every $q \in T$, there is a non-selfdual cusp form $\pi(q)$ on GL$(3)/{{\mathbb Q}}$ of level $q$, contributing to the (cuspidal) cohomology (with constant coefficients) .
\[TM:C\] Let $\pi'$ be a cusp form on GL$(2)/{{\mathbb Q}}$ defined by a non-CM holomorphic newform $g$ of weight $4$, level $N$, trivial character, and field of coefficients $K$. Let $\pi$ denote an arbitrary cusp form on GL$(3)/{{\mathbb Q}}$ contributing to the cuspidal cohomology in degree $2$, and let $\pi(q)$, $q \in T$, be one of the particular forms discussed above. Put $\Pi = \pi \boxtimes
\pi'$ and $\Pi(q) = \pi(q) \boxtimes \pi'$. Then
1. $\Pi$ contributes to the cuspidal cohomology of GL$(6)$.
2. $\Pi(q)$ is not essentially selfdual when $N \leq 23$ and $K = {{\mathbb Q}}$.
3. If $N$ is relatively prime to $q$, then the level of $\Pi(q)$ is $N^3q^2$. Now let $N \leq 23$ and $K={{\mathbb Q}}$. Then $\Pi(q)$ does not admit any self-twist. Moreover, there is no cubic non-normal extension $K/{{\mathbb Q}}$ with a cusp form $\eta$ on GL$(2)/K$ such that $L(s, \Pi(q)) = L(s,
\eta)$, nor is there a sextic extension (normal or not) $E/{{\mathbb Q}}$ with a character $\lambda$ of $E$ such that $L(s, \Pi(q)) = L(s,
\lambda)$.
We note from the [*Modular forms database*]{} of William Stein ([@WSt]) that there exist newforms $g$ of weight $4$ with ${{\mathbb Q}}$-coefficients, for instance for the levels $N = 5, 7, 13, 17,
19, 23$.
*Remark 4.1* Part (b) should be true for any $\pi'$. Suppose that for a given any cusp form $\pi$ on GL$(3)$, cohomological or not, the functorial product $\Pi= \pi \boxtimes
\pi'$ satisfies $\Pi^\vee \simeq \Pi \otimes \nu$ for a character $\nu$. Then at any prime $p$ where $a_p(\pi') \ne 0$, which happens for a set of density $1$, we can of course conclude that $a_p(\pi^\vee) = a_p(\pi)\nu(p)$. But this does not suffice, given the state of knowledge right now concerning the refinement of the strong multiplicity one theorem, to conclude that $\pi^\vee$ is isomorphic to a twist of $\pi$. In the case of the $\pi(q)$, we have information at a small set of primes and we have to make sure that $a_p(\pi') \ne 0$ [*and*]{} $\overline{a_p(\pi)} \ne
a_p(\pi)$ for one of those $p$. The hypothesis that $K = {{\mathbb Q}}$ is made for convenience, however, and the proof will extend to any totally real field.
In [@vG-T1], [@vG-T2], [@vG-K-T-V] one finds in fact an algebraic surface $S(q)$ over ${{\mathbb Q}}$ for each $q \in T_2$, and a $3$-dimensional $\ell$-adic representation $\rho(q)$ (for any prime $\ell$), occurring in $H^2_{\rm et}(S(q)_{\overline {{\mathbb Q}}},
\overline {{\mathbb Q}}_\ell)$, such that $$L_p(s, \rho(q)) \, = \, L_p(s, \pi(q)), \leqno(\ast)$$ for all odd primes $p \leq 173$ not dividing $q$. Here is a conditional result.
\[TM:D\] Let $\pi'$ be a cusp form on GL$(2)/{{\mathbb Q}}$ defined by a non-CM holomorphic newform $g$ of weight $4$, level $N$, trivial character, and field of coefficients ${{\mathbb Q}}$, with corresponding $\overline {{\mathbb Q}}_\ell$-representation $\rho'$. Let $\pi(q), \rho(q), \Pi(q)$ be as above for $q \in
T_2$. Put $R(q) = \rho(q) \otimes \rho'$. Suppose $(\ast)$ holds at all the odd primes $p$ not dividing $q\ell$. Then $R(q)$ remains irreducible when restricted to any open subgroup of Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$.
Proof of Theorems C, D modulo Theorems A, B
===========================================
In this section we will assume the truth of Theorems \[TM:A\] and \[TM:B\].
[*Proof of Theorem \[TM:C\]*]{}: As $\pi'$ is associated to a holomorphic newform of weight $4$, we have $$\sigma(\pi'_\infty)\vert_{{{\mathbb C}}^\ast} \, \simeq \, \left((z/|z|)^3
\oplus (\overline z/|z|)^3\right) \otimes (z\overline z)^3.$$ And since $\pi$ contributes to cohomology, we have (cf. part (ii) of Theorem \[T:301\]) $$\sigma(\pi_\infty)\vert_{{{\mathbb C}}^\ast} \, \simeq \, \left((z/|z|)^2
\oplus 1 \oplus (\overline z/|z|)^2\right) \otimes (z\overline
z)^2.$$ Since $\Pi_\infty$ corresponds to the tensor product $\sigma(\pi_\infty) \otimes \sigma(\pi'_\infty)$, we get part (a) of Theorem \[TM:C\] in view of Theorem \[TM:A\] and part (ii) of Theorem \[T:302\].
Pick any $q$ in $T$ and denote by ${{\mathbb Q}}(\pi(q))$ the field of rationality of the finite part $\pi(q)_f$ of $\pi(q)$. Then it is known by [@A-G-G] that for $q \in T_1$, $${{\mathbb Q}}(\pi(53))={{\mathbb Q}}(\sqrt{-11}), \, {{\mathbb Q}}(\pi(61)) = {{\mathbb Q}}(\sqrt{-3}), \,
{{\mathbb Q}}(\pi(79)) = {{\mathbb Q}}(\sqrt{-15}), \, {{\mathbb Q}}(\pi(89)) = {{\mathbb Q}}(i),$$ while by [@vG-T1], [@vG-K-T-V] and [@vG-T2], $${{\mathbb Q}}(\pi(q)) \, = \, {{\mathbb Q}}(i), \, \, \, \forall \, \, q \in T_2.$$
By hypothesis, $\pi'$ is non-CM, and by part (a), $\Pi(q)$ is cuspidal. Suppose there exists a character $\nu$ such that for some $q \in T$, $$\Pi(q)^\vee \, \simeq \, \Pi(q) \otimes \nu.$$ Comparing central characters, we get $\nu^6 = 1$. We claim that $\nu^2 = 1$. Suppose not. Then there exists an element $\sigma$ of Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$ fixing ${{\mathbb Q}}(\pi(q))$ such that $\nu \ne
\nu^\sigma$. Since $\pi'$ has ${{\mathbb Q}}$-coefficients and $\pi(q)$ has coefficients in ${{\mathbb Q}}(\pi(q))$, we see that $\Pi(q)_{f}$ must be isomorphic to the Galois conjugate $\Pi(q)_{f}^\sigma$, which exists because the cuspidal cohomology group has, by Clozel (see section 3), a ${{\mathbb Q}}$-structure preserved by the Hecke operators. If we put $\mu = \nu/\nu^\sigma \ne 1$, we then see that $\Pi(q)
\simeq \Pi(q) \otimes \mu$. But we will see below that $\Pi(q)$ admits no non-trivial self-twist. This gives the desired contradiction, proving the claim. If $\nu$ is non-trivial, the quadratic extension $F/{{\mathbb Q}}$ it cuts out will need to have discriminant dividing $q^aN^b$ for suitable integers $a, b$. For any prime $p$ which is unramified in $F$, we will have $$\overline a_p(\pi)a_p(\pi') \, = \, \pm a_p(\pi)a_p(\pi').$$ For each $j \leq 3$ and for each $\pi'$ with $N \leq 23$ and $K =
{{\mathbb Q}}$, we can find, using the tables in [@A-G-G], [@vG-T1],[@vG-T2], [@vG-K-T-V] and [@WSt], a prime $p$ such that $a_p(\pi') \ne 0$, $\nu(p) \ne 0$ and $\overline
a_p(\pi) \ne a_p(\pi)$. This proves part (b) of Theorem \[TM:C\].
When $N$ is relatively prime to $q$, the conductor of $\Pi(q)$ must be $N^3q^2$ as can be seen from the way epsilon factors change under twisting (see section 4 of [@Ba-R] for example).
[F]{}rom now on, let $N \leq 23$ and $K = {{\mathbb Q}}$. One knows that as $\pi'$ is holomorphic and not dihedral, the associated Galois representation $\rho'$ remains irreducible when restricted to any open subgroup of Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$. It follows that the base change of $\pi'$ to any solvable Galois extension remains cuspidal. In particular, it is not of solvable polyhedral type. We claim that $\pi(q)$ is not monomial. Indeed, the infinite type of $\pi(q)$ is regular algebraic [@C], and to be monomial there needs to be a cubic, possibly non-normal, extension $K/{{\mathbb Q}}$ which can support an algebraic Hecke character which is [*not*]{} a finite order character times a power of the norm. By [@Weil], for such a character to exist, $K$ must contain a CM field, i.e., a totally imaginary quadratic extension of a totally real field, which forces $K$ to be imaginary. But any cubic extension of ${{\mathbb Q}}$ has a real embedding, and this proves our claim. Note also that as $\pi(q)$ is not essentially self-dual, it is not a twist of the symmetric square of any cusp form, in particular $\pi'$, on GL$(2)/{{\mathbb Q}}$. Now it follows from Theorem \[TM:B\] that $\Pi(q)$ does not admit any self-twist.
Suppose $K$ is a non-normal cubic field together with a cusp form $\eta$ on GL$(2)/K$ such that $L(s, \Pi(q)) = L(s, \eta)$. Let $L$ be the Galois closure of $K$ (with Galois group $S_3$), and let $E$ be the quadratic extension of ${{\mathbb Q}}$ contained in $L$. Then $\Pi(q)_{E}$ will be cuspidal and automorphically induced by the cusp form $\eta_L$ of GL$(3, {{\mathbb A}}_L)$. In other words, $\Pi(q)_{E}$ admits a non-trivial self-twist. To contradict this, it suffices, in view of Theorem \[TM:B\], to show that $\pi(q)_{E}$ admits no self-twist relative to $L/E$, i.e., that $\pi(q)_{E}$ is not automorphically induced by a character $\mu$ of $L$. But as noted above, this forces $L$ to be a totally imaginary number field containing a CM field $L_0$. Then either $L = L_0$ or $L_0 = E$. In the latter case, by [@Weil], $\mu$ will be a finite order character times the pullback by norm of a character $\mu_0$ of $E$, forcing $I_L^E(\mu)$ to be [*not*]{} regular at infinity, and so this case cannot happen. So $L$ itself must be a CM field, with its totally real subfield $F$. Then Gal$(F/{{\mathbb Q}})$ would be cyclic of order $3$ and a quotient of $S_3$, which is impossible. So this case does not arise either. So $\pi(q)_{E}$ doe snot admit any self-twist, and $\Pi(q)$ is not associated to any $\eta$ as above.
Now suppose $L(s, \Pi(q)) = L(s, \lambda)$ for a character $\lambda$ of a sextic field $L$. If $L$ contains a proper subfield $M \ne {{\mathbb Q}}$, then since $m:= [L:M] \leq 3$, one can induce $\lambda$ to $M$ and get an automorphic representation $\beta$ of GL$_m({{\mathbb A}}_{M})$ such that $L(s, \lambda) = L(s, \beta) = L(s,
\Pi(q))$, which is impossible by what we have seen above. So $L$ must not contain any such $M$. But on the other hand, since $\Pi(q)_{\infty}$ is algebraic and regular, we need $L$ to contain, by [@Weil], a CM subfield $L_0$, and hence also its totally real subfield $F$. Either $F = {{\mathbb Q}}$, in which case $L_0$ is imaginary quadratic, or $F \ne {{\mathbb Q}}$. Either way there will be a proper subfield $M$ of degree $\leq 3$, and so the purported equality $L(s, \Pi(q)) = L(s, \lambda)$ cannot happen. We are now done with the proof of Theorem \[TM:C\].
[*Proof of Theorem \[TM:D\]*]{}: By assumption, the $\ell$-adic representation $\rho$ is functorially associated to the cuspidal cohomological form $\pi(q)$ on GL$(3)/{{\mathbb Q}}$ with $q \in
T_2$.
\[T:501\] $\rho$ is irreducible under restriction to any open subgroup.
[*Proof*]{}. Suffices to show that the restriction $\rho_E$ to Gal$(\overline {{\mathbb Q}}/E)$ is irreducible for any finite [*Galois*]{} extension $E/{{\mathbb Q}}$. Pick any such extension and write $G =
$Gal$(E/{{\mathbb Q}})$. Suppose $\rho_E$ is reducible. Then we have [*either*]{}
1. $\rho_E \simeq \tau \oplus \chi$ with $\tau$ irreducible of dimension $2$ and $\chi$ of dimension $1$; [*or*]{}
2. $\rho_E \simeq \chi_1 \oplus \chi_2 \oplus \chi_3$, with each $\chi_j$ one-dimensional.
Let $V$ be the $3$-dimensional $\overline {{\mathbb Q}}_\ell$-vector space on which Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$ acts via $\rho$. Suppose we are in case(i), so that there is a line $L$ in $V$ preserved by Gal$(\overline {{\mathbb Q}}/E)$ and acted upon by $\chi$. Note that $G$ acts on $\{\tau, \chi\}$ and, by the dimension consideration, it must preserve $\{\chi\}$. Hence the line $L$ is preserved by Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$, which contradicts the fact that $\rho$ is irreducible.
So we may assume that we are in case (ii). We claim that $\chi_i
\ne \chi_j$ if $i \ne j$. Indeed, since $\rho$ arises as (the base change to $\overline {{\mathbb Q}}_\ell$ of) a summand of the $\ell$-adic cohomology of a smooth projective variety, it is Hodge-Tate, and so is each $\chi_j$. So each $\chi_j$ is locally algebraic and corresponds to an algebraic Hecke character $\chi_j'$ of $E$. By the identity of the $L$-functions, we will have $L^S(s, \pi) =
\prod_j L^S(s, \chi'_j)$ for a suitable finite set $S$ of places $S$. By the regularity of $\pi$, each $\chi_j'$ must appear with multiplicity one, which proves the claim. Now let $L_j$ denote, for each $j \leq 3$, the (unique) line in $V$ stable under Gal$(\overline {{\mathbb Q}}/E)$ and acted upon by $\chi_i$. And $G$ acts by permutations on the set $\{\chi_1, \chi_2, \chi_3\}$. In other words, there is a representation $r: G \rightarrow S_3$ such that the $G$-action is via $r$. Put $H = {\rm Ker}(r)$, with corresponding intermediate field $M$. Then each $L_j$ is stable under Gal$(\overline {{\mathbb Q}}/M)$, so that $\rho_M \simeq \nu_1 \oplus
\nu_2 \oplus \nu_3$, where each $\nu_j$ is a character of Gal$(\overline {{\mathbb Q}}/M)$. Also, $M/{{\mathbb Q}}$ is Galois with Gal$(M/{{\mathbb Q}})
\subset S_3$. But from the proof of Theorem \[TM:C\] that the base change $\pi_M$ of $\pi$ to any such $M$ is cuspidal. However, if $\nu_j^1$ denotes the algebraic Hecke character of $M$ defined by $\nu_j$, the twisted $L$-function $L^S(s, \pi_M \otimes
{\nu'_j}^{-1})$ will have a pole at $s=1$, leading to a contradiction. We have now proved Lemma 5.1.
Note that Lemma 1 implies in particular that for any finite extension $F/{{\mathbb Q}}$, $\rho_F$ does not admit any self-twist.
\[T:502\] For any finite extension $E/{{\mathbb Q}}$, the restriction $\rho_E$ is not essentially self-dual.
[*Proof*]{}. Again we may assume that $E/{{\mathbb Q}}$ is Galois with group $G$. As before let $V$ denote the space of $\rho$, and suppose that we have an isomorphism $\rho \simeq \rho^\vee \otimes
\nu$, for a character $\nu$. Then there is a line $L$ in $V
\otimes V$ on which Gal$(\overline {{\mathbb Q}}/E)$ acts via $\nu$. By Schur’s lemma (and this is why we have to work over $\overline
{{\mathbb Q}}_\ell$), the trivial representation appears with multiplicity one in $V \otimes V^\vee$. It implies that $\nu$ must appear with multiplicity one in $V \otimes V$. We claim that $V \otimes V$ contains no other character. Indeed, if we have another character $\nu'$, we would have $\rho \simeq \rho \otimes \mu$, where $\mu =
\nu/\nu'$. But as noted above, $\rho_E$ admits no self-twist, and so $\mu = 1$, and the claim is proved. Consequently, the action of $G$ on $V \otimes V$ must preserve $\nu$. In other words, the line $L$ is stable under all of Gal$(\overline {{\mathbb Q}}/{{\mathbb Q}})$, contradicting the fact that $\rho$ is not essentially self-dual. Done
Now consider $R = \rho \otimes \rho'$. We know that both $\rho$ and $\rho'$ remain irreducible upon restriction to any open subgroup and moreover, such a restriction of $\rho$ is [*not*]{} essentially self-dual. It then follows easily that the restriction of $R$ is irreducible.
This finishes the proof of Theorem \[TM:D\].
Proof of Theorem A, Part \#1
============================
By twisting we may assume that $\pi, \pi'$ are unitary, so that $\pi^\vee \simeq \overline \pi$ and ${\pi'}^\vee \simeq
\overline{\pi'}$, with respective central characters $\omega,
\omega'$.
Now we proceed in several steps. Applying Langlands’s classification, ([@La79-1], [@La79-2], [@JS81]), we see that the Kim-Shahidi product $\Pi= \pi \boxtimes \pi'$ must be an isobaric sum of cusp forms whose degrees add up to $6$. Thanks to the Clebsch-Gordon decomposition $${\rm sym}^2(\pi') \boxtimes \pi' \, \simeq \, {\rm sym}^3(\pi')
\boxplus (\pi' \otimes \omega'),$$ $\Pi$ is not cuspidal if $\pi$ is a twist of ${\rm sym}^{2}
(\pi')$.
The list of all the cases when $\Pi$ is not cuspidal is the following:
[**Case I**]{}: [*$\Pi$ has a constituent of degree $1$, i.e., $\Pi = \lambda \boxplus \Pi'$ for some idele class character $\lambda$ and some automorphic representation $\Pi'$ of $GL (5)$.*]{}
[**Case II**]{}: [*$\Pi$ has a constituent of degree $2$, i.e., $\Pi = \tau \boxplus \Pi'$ for some cusp form $\tau$ on $GL
(2)$ and some automorphic representation $\Pi'$ of $GL (4)$.*]{}
[**Case III**]{}: [*$\Pi$ is an isobaric sum of two cusp forms $\sigma_{1}$ and $\sigma_{2}$ on $GL (3)$.* ]{}
We first deal with Cases I and II. We need some preliminaries. First comes the following basic result due to H. Jacquet and J.A. Shalika ([@JS81], [@JS90], and R. Langlands ([@La79-1], [@La79-2]).
\[T:601\]
[(i)]{.nodecor} Let $\Pi$, $\tau$ be isobaric automorphic representations of $GL_{n} (\mathbb{A}_{F})$, $GL_{m}
(\mathbb{A}_{F})$ respectively. Assume that $\tau$ is cuspidal. Then the order of the pole of $L (s, \Pi \otimes \overline{\tau})$ at $s=1$ is the same as the multiplicity of $\tau$ occurring in the isobaric sum decomposition of $\Pi$.
[(ii)]{.nodecor} $L (s, \Pi \times \overline{\Pi})$ has a pole at $s = 1$ of order $m = \sum_{i} m_{i}^{2}$ if $\Pi = \boxplus_{i}
m_{i} \pi_{i}$ is the isobaric decomposition of $\Pi$, with the $\pi_{i}$ being inequivalent cuspidal representations of smaller degree. In particular, $m = 1$ if and only if $\Pi$ is cuspidal.
An $L$–function $L(s)$ is said to be nice if it converges on some right half plane, admits an Euler product of some degree $m$, say, and extends to a meromorphic function of finite order with no pole outside $s = 1$, together with a functional equation related to another $L$–function $L^{\vee} (s)$ given by $$L (s) = W {(d_{F}^{m} N)}^{1/2 - s} L^{\vee} (1 - s),$$ where $W$ is a non-zero scalar.
If $\pi_{1}, \pi_{2}$ are automorphic forms on $GL (m), GL (n)$ respectively, then the Rankin-Selberg $L$-function $L (s, \pi_{1}
\times \pi_{2})$ is known to be nice ([@JPSS], [@MW], [@Sh]). Of course, the product of two nice $L$–functions is nice. Furthermore, we recall the following Tchebotarev-like result for nice $L$–functions ([@JS90]):
\[T:602\] Let $L_{1} (s) = \prod_{v} L_{1, v} (s)$ and $L_{2} (s) = \prod_{v} L_{2, v} (s)$ be two $L$–functions with Euler products, and suppose that they are both of exactly one of the following types:
[(a)]{.nodecor} $L_{i} (s)$ is an Artin $L$–function of some Galois extension;
[(b)]{.nodecor} $L_{i} (s)$ is attached to an isobaric automorphic representation;
[(c)]{.nodecor} $L_{i} (s)$ is a Rankin–Selberg $L$–function of two isobaric automorphic representations.
If $L_{1, v} (s) = L_{2, v} (s)$ for all but finite places $v$ of $F$, then $L_{1} (s) = L_{2} (s)$.
*Proof of Theorem A for Cases I and II.*
Firstly, Case I can never happen. The reason is the following: If $\lambda$ is a constituent of $\Pi = \pi \boxtimes \pi'$, then $L
(s, \Pi \times \bar{\lambda})$ has a pole at $s = 1$ (Lemma \[T:601\]), hence so does $$L (s, \pi' \times \pi \otimes \bar{\lambda}) = L (s, \pi \boxtimes
\pi' \otimes \bar{\lambda}).$$ However, $\pi'$ and $\pi \otimes \bar{\lambda}$ are cuspidal of different degrees, hence $L (s, \pi' \times \pi \otimes
\bar{\lambda})$ is entire, and we get the desired contradiction.
Now we treat Case II, where $\Pi$ has a constituent $\tau$ of degree $2$. We will show that this can happen IF AND ONLY IF $\pi$ is twist equivalent to ${\rm sym}^{2} (\pi')$ in which case $\tau$ is twist equivalent to $\pi'$.
In fact, for each finite $v$ where $\pi$ and $\pi'$ are unramified, $$L_{v} (s, \Pi \otimes \bar{\tau}) = L_{v} (s, \pi
\times (\pi' \boxtimes \bar{\tau})),$$ where $\pi' \boxtimes \bar{\tau}$ is the functorial product of $\pi'$ and $\bar{\tau}$ whose modularity (in GL$(4)$) was established in [@Ra2000]. One may check the following: If $\pi'_{v} = \alpha_{v, 1} \boxplus \alpha_{v, 2}$, $\pi_{v} =
\beta_{v, 1} \boxplus \beta_{v, 2} \boxplus \beta_{v, 3}$, and $\tau'_{v} = \gamma_{v, 1} \boxplus \gamma_{v, 2}$, then both sides of the equality is the same as $\prod_{i, j, k} L (s,
\alpha_{v, i} \beta_{v, j} \bar{\gamma}_{v, k})$ where the product is over all $i, j, k$ such that $1 \leq i, k \leq 2$ and $1 \leq j
\leq 3$.
Hence by Lemma \[T:602\], $$L (s, \Pi \otimes \bar{\tau}) = L (s, \pi \times (\pi' \boxtimes \bar{\tau}))$$
As $\tau$ is a constituent of $\Pi$, the $L$–functions on both sides above have a pole at $s = 1$. As $\pi$ is cuspidal, this means by Lemma \[T:601\], $\bar{\pi}$ is a constituent of $\pi'
\boxtimes \bar{\tau}$. Hence $\pi' \boxtimes \bar{\tau}$ should possess a constituent of degree $1$, namely a character $\mu$.
Thus $L (s, \pi' \times \bar{\tau} \otimes \bar{\mu}) = L (s, \pi'
\boxtimes \bar{\tau} \otimes \bar{\mu})$ has a pole at $s = 1$, implying that $\pi'$ is equivalent to $\tau \otimes \mu$. Hence $$\pi' \boxtimes \bar{\tau} \cong \mu \boxplus Ad (\tau) \otimes
\mu,$$ which means that $\pi \cong Ad (\tau) \otimes \mu \cong Ad
(\pi') \otimes \mu$.
Finally, it is clear that if Case II happens, then $\pi'$ cannot be dihedral. Furthermore, $\Pi$ is Eisensteinian of type $(2, 2,
2)$ if $\pi'$ is tetrahedral, and $(2, 4)$ otherwise. we can see this by observing that $$\pi' \boxtimes Ad (\pi') \cong {\rm sym}^{3} (\pi') \otimes
\omega_{\pi'} \boxplus \pi' \otimes \omega_{\pi'}^{2}$$
Proof of Theorem A, Part \#2
============================
It remains to treat Case III. Here again, $\pi'$ denotes a cusp form on $GL (2)$ and $\pi$ a cusp form on $GL (3)$. Assume that $\Pi = \sigma_{1} \boxplus \sigma_{2}$ where $\sigma_{1}$ and $\sigma_{2}$ are cusp forms on $GL (3)$.
We will divide Case III into two subcases: In this section, we will assume that $\pi'$ is not dihedral. The (sub)case when $\pi'$ is dihedral will be treated in the next section.
The following equality is crucial, and it holds for all cusp forms $\pi'$ on $GL (2)$ and $\pi$ on $GL (3)$:
\[T:701\] $$\begin{aligned}
L (s, &\pi \times \pi'; \Lambda^{3} \otimes \omega_{\pi}^{-1} \chi)
L (s, \pi' \otimes \omega_{\pi'} \chi)
\notag \\ \label{EQ:701}
&= L (s, {\rm sym}^{3} (\pi') \otimes \chi)
L (s, (\pi \boxtimes \pi') \times \tilde{\pi} \otimes \omega_{\pi'} \chi)\end{aligned}$$ where $\omega_{\pi'}$ and $\omega_{\pi}$ are the respective central characters of $\pi'$ and $\pi$.
*Proof of Proposition \[T:701\].*
We claim that both sides of are nice. Indeed, we see that formally, the admissible representation $\Lambda^{3} (\pi
\boxtimes \pi') \otimes \omega_{\pi}^{-1}$ is equivalent to ${\rm
sym}^{3} (\pi') \boxplus (Ad (\pi) \boxtimes \pi' \otimes
\omega_{\pi'})$. So the left hand side is nice. And the right hand side is nice by [@KSh2000], whence the claim. So by Lemma \[T:602\] it suffices to prove this equality given by the Proposition locally at $v$ for almost all $v$. It then suffices to prove the following identity (as admissible representations) for almost all $v$: $$\begin{aligned}
\Lambda^{3} &(\pi'_{v} \boxtimes \pi_{v}) \otimes \omega_{\pi_{v}}^{-1}
\boxplus \pi'_{v} \otimes \omega_{\pi'_{v}}
\notag \\ \label{EQ:702}
&= {\rm sym}^{3} (\pi'_{v}) \boxplus \pi'_{v} \boxtimes
\pi_{v} \boxtimes \tilde{\pi}_{v} \otimes \omega_{\pi'_{v}}\end{aligned}$$
Let $v$ be any place where $\pi'$ and $\pi$ are unramified. Say $\pi'_{v} = \alpha_{v, 1} \boxplus \alpha_{v, 2}$, $\pi_{v} =
\beta_{v, 1} \boxplus \beta_{v, 2} \boxplus \beta_{v, 3}$. Note that $\omega_{\pi'_{v}} = \alpha_{v, 1} \alpha_{v, 2}$ and $\omega_{\pi_{v}} = \beta_{v, 1} \beta_{v, 2} \beta_{v, 3}$. Then it is routine to check that the left and the right hand sides of are equal to the sum of the following terms:
Terms A ($2$ terms): $\alpha_{v, 1}^{3} \boxplus \alpha_{v,
2}^{3}$;
Terms B ($2 \times 4 = 8$ terms): $4$ copies of $(\alpha_{v, 1}
\boxplus \alpha_{v, 2}) \otimes \omega_{\pi'_{v}}$;
Terms C ($12$ terms): $\boxplus_{1 \leq i \leq 2, 1 \leq j \ne
k \leq 3}\, \alpha_{v, i} \omega_{\pi'_{v}} \beta_{v, j}
\beta^{-1}_{v, k}$.
In fact, Terms A, B and C are obtained by expanding the right hand side of . Since $$Ad (\pi_{v}) = 3 \cdot \underline{1} \boxplus (\boxplus_{1 \leq j \ne k \leq 3},
\beta_{v, j} \beta^{-1}_{v, k}),$$ the Terms C and (three of) the Terms B are obtained from $\pi'_{v}
\boxtimes \pi_{v} \boxtimes \tilde{\pi}_{v} \otimes
\omega_{\pi'}$, and the Terms A and (one of) the Terms B arise from ${\rm sym}^{3} (\pi')$.
The left hand side is easy to handle since we have the following:
$$\begin{aligned}
\Lambda^{3} (\pi'_{v} \otimes \pi_{v}) &=
\boxplus_{1 \leq i \leq 2, 1 \leq j, k \leq 3, j \ne k}\,
\alpha_{v, i} \omega_{\pi'_{v}} \beta_{v, j}^{2} \beta_{v, k}
\notag \\
&\boxplus \omega_{\pi_{v}} \alpha_{v, 1}^{3}
\boxplus \omega_{\pi_{v}} \alpha_{v, 2}^{3}
\notag \\
&\boxplus 3 \omega_{\pi_{v}} \alpha_{v, 1} \omega_{\pi'_{v}}
\boxplus 3 \omega_{\pi_{v}} \alpha_{v, 2} \omega_{\pi'_{v}}
\notag\end{aligned}$$
In fact, the $\omega_{\pi_{v}}^{-1}$ twist of the thing above contributes the Terms C, Terms A and (three of) Terms B.
So we have proved , and hence .
Let $\sigma_{1}$ and $\sigma_{2}$ be cusp forms on $GL (3)$.
\[T:702\] Let $\eta_{1}$ and $\eta_{2}$ be the central characters of $\sigma_{1}$ and $\sigma_{2}$ respectively. Then $$\begin{aligned}
L (s, \sigma_{1} \boxplus \sigma_{2}; \Lambda^{3} \otimes \chi')
&= L (s, \eta_{1} \chi') L (s, \eta_{2} \chi')
\notag \\ \label{EQ:703}
&L (s, \sigma_{1} \times \tilde{\sigma}_{2} \otimes \eta_{2} \chi')
L (s, \sigma_{2} \times \tilde{\sigma}_{1} \otimes \eta_{1} \chi')\end{aligned}$$
*Proof. of Lemma \[T:702\].*
This is easy since at each place $v$ where the $\sigma_{i}$ are unramified, $$\Lambda^{3} (\sigma_{1, v} \boxplus \sigma_{2, v})
= \boxplus_{0 \leq i \leq 3}\, \left( \Lambda^{i} (\sigma_{1, v}) \boxtimes
\Lambda^{3 - i} (\sigma_{2, v}) \right),$$
$$\Lambda^{2} (\sigma_{i, v}) \cong \tilde{\sigma}_{i, v} \otimes \eta_{i}$$ and $$\Lambda^{3} (\sigma_{i}) \cong \eta_{i}.$$ Done by applying Lemma \[T:602\].
Before we apply Proposition \[T:701\] and Lemma \[T:702\], let us first investigate a special instance of Case III when $\sigma_{1}$ and $\sigma_{2}$ are both twists of $\pi$:
\[T:703\] If $\pi \boxtimes \pi' = (\pi \otimes \chi_{1}) \boxplus (\pi \boxplus \chi_{2})$ then $${\rm sym}^{3} (\pi') \cong (\pi' \otimes \omega_{\pi'}) \boxplus \chi_{1}^{3}
\boxplus \chi_{2}^{3}$$ Hence if $\pi'$ is not dihedral or tetrahedral, this cannot happen.
*Proof of Lemma \[T:703\].*
Let $v$ be any place where $\pi'_{v}$ and $\pi_{v}$ are unramified. Write $$L_{v} (s, \pi') = {(1 - U_{v}
{(Nv)}^{-s})}^{-1} {(1 - V_{v} {(Nv)}^{-s})}^{-1}$$ and $$L_{v}
(s, \pi) = {(1 - A_{v} {(Nv)}^{-s})}^{-1} {(1 - B_{v}
{(Nv)}^{-s})}^{-1} {(1 - C_{v} {(Nv)}^{-s})}^{-1}.$$
Then $$\begin{aligned}
L_{v} &(s, \pi' \times \pi) =
\notag \\
& {(1 - U_{v} A_{v} {(Nv)}^{-s})}^{-1} {(1 - U_{v} B_{v} {(Nv)}^{-s})}^{-1}
{(1 - U_{v} C_{v} {(Nv)}^{-s})}^{-1}
\notag \\
& {(1 - V_{v} A_{v} {(Nv)}^{-s})}^{-1} {(1 - V_{v} B_{v} {(Nv)}^{-s})}^{-1}
{(1 - V_{v} C_{v} {(Nv)}^{-s})}^{-1}
\notag\end{aligned}$$
Let $X_{v} = \chi_{1} ({\rm Frob}_{v})$ and $Y_{v} = \chi_{2}
({\rm Frob}_{v})$. Then $$\begin{aligned}
L_{v} &(s, \pi \otimes \chi_{1}) L_{v} (s, \pi \otimes \chi_{2}) =
\notag \\
& {(1 - X_{v} A_{v} {(Nv)}^{-s})}^{-1} {(1 - X_{v} B_{v} {(Nv)}^{-s})}^{-1}
{(1 - X_{v} C_{v} {(Nv)}^{-s})}^{-1}
\notag \\
& {(1 - Y_{v} A_{v} {(Nv)}^{-s})}^{-1} {(1 - Y_{v} B_{v} {(Nv)}^{-s})}^{-1}
{(1 - Y_{v} C_{v} {(Nv)}^{-s})}^{-1}
\notag\end{aligned}$$
Consequently we have $$\begin{aligned}
&{{\left\{\,U_{v} A_{v}, U_{v} B_{v}, U_{v} C_{v}, V_{v} A_{v}, V_{v} B_{v}, V_{v} C_{v}\,\right\}}} =
\notag \\ \label{EQ:704}
&{{\left\{\,X_{v} A_{v}, X_{v} B_{v}, X_{v} C_{v}, Y_{v} A_{v}, Y_{v} B_{v}, Y_{v}
C_{v}\,\right\}}}.\end{aligned}$$
In particular, $$(U_{v}^{n} + V_{v}^{n})(A_{v}^{n} + B_{v}^{n} + C_{v}^{n})
=(X_{v}^{n} + Y_{v}^{n})(A_{v}^{n} + B_{v}^{n} + C_{v}^{n})$$ for each positive integer $n$. And besides, taking the products of the elements on each side of and equating, we get $$U_{v}^{3} V_{v}^{3} = X_{v}^{3} Y_{v}^{3}$$
Now we apply the following lemma.
\[T:707\] If $X, Y, U, V, A, B, C$ are nonzero complex numbers such that (for all $n > 0$) $$(U^{n} + V^{n})(A^{n} + B^{n} + C^{n})
= (X^{n} + Y^{n})(A^{n} + B^{n} + C^{n}),$$ and $U^{3} V^{3} = X^{3} Y^{3}$, then ${{\left\{\,U^{3}, V^{3}\,\right\}}} = {{\left\{\,X^{3}, Y^{3}\,\right\}}}$.
*Proof of Lemma \[T:707\].*
If $A^{3} + B^{3} + C^{3} \ne 0$, then $U^{3} + V^{3} = X^{3} +
Y^{3}$. Hence ${{\left\{\,U^{3}, V^{3}\,\right\}}} = {{\left\{\,X^{3}, Y^{3}\,\right\}}}$ as $U^{3}
V^{3} = X^{3} Y^{3}$.
If $A^{3} + B^{3} + C^{3} = 0$, we claim that $A + B + C \ne 0$. Otherwise, $$- 3 A B C = 3 A B (A + B) = {(A + B)}^{3} - (A^{3} + B^{3}) = -C^{3} + C^{3} = 0$$ Thus $a$, $b$ or $c$ is zero. This leads to a contradiction.
In fact we will prove the following statement:
[**Claim:**]{} *If $a, b, c \ne 0$, then $a + b + c$ or $a^{3}
+ b^{3} + c^{3}$ is not zero.*
So we claim also that $A^{9} + B^{9} + C^{9} \ne 0$, and, $A^{2} +
B^{2} + C^{2}$ or $A^{6} + B^{6} + C^{6}$ is not zero.
Hence $$U^{n} + V^{n} = X^{n} + Y^{n}$$ holds for $n = 1$ and $9$, and for one of $2$ or $6$.
If this equality holds for $n = 1$, and $2$, then $$U V = \frac{{(U + V)}^{2} - (U^{} + V^{2})}{2}
= \frac{{(X + Y)}^{2} - (X^{} + Y^{2})}{2} = X Y,$$ implying that ${{\left\{\,U, V\,\right\}}} = {{\left\{\,X, Y\,\right\}}}$, and the lemma will follow.
Now assume that $U^{n} + V^{n} = X^{n} + Y^{n}$ holds for $n = 1$, $6$ or $9$. As we have already assumed that $U^{3} V^{3} = X^{3}
Y^{3}$, $U^{3 n} V^{3 n} = X^{3 n} Y^{3 n}$. So we have $${{\left\{\,U^{n}, V^{n}\,\right\}}} = {{\left\{\,X^{n}, Y^{n}\,\right\}}}$$ for $n = 6$ and $9$.
Without loss of generality, assume that $U^{9} = X^{9}$ and $V^{9}
= Y^{9}$. If $U^{6} = X^{6}$ and $V^{6} = Y^{6}$, then of course we have $U^{3} = X^{3}$ and $V^{3} = Y^{3}$ and the lemma follows. If $U^{6} = Y^{6}$ and $V^{6} = X^{6}$, then $U$, $V$, $X$ and $Y$ have the same norm. However, since $U + V = X + Y$, the pairs $\{U, V\}$ and $\{X, Y\}$ are the same. hence implying the lemma. The reason for this comes from the following statement which is elementary: (Note that even when $U + V = X + Y = 0$, although we cannot directly apply this statement, we still have $U^{3} + V^{3}
= X^{3} + Y^{3} = 0$, so that ${{\left\{\,U^{3}, V^{3}\,\right\}}} = {{\left\{\,X^{3},
Y^{3}\,\right\}}}$.)
[**Statement**]{} *The pair $(z_{1}, z_{2})$ such that $|z_{1}|
= |z_{2}| = R$ and $z_{1} + z_{2} = Z$ is uniquely determined by $R > 0$ and $Z$ with $0 < |Z| < 2 R$.*
So in all cases, ${{\left\{\,U^{3}, V^{3}\,\right\}}} = {{\left\{\,X^{3}, Y^{3}\,\right\}}}$.
*Proof of Lemma \[T:703\]* (contd.)
By the previous lemma, $${{\left\{\,U_{v}^{3}, V_{v}^{3}\,\right\}}}
= {{\left\{\,X_{v}^{3}, Y_{v}^{3}\,\right\}}}$$ at any unramified finite place $v$.
Hence $$\begin{aligned}
L &(s, {\rm sym}^{3} (\pi'_{v}))
\notag \\
&= {(1 - U_{v}^{3} {(Nv)}^{-s})}^{-1}
{(1 - V_{v}^{3} {(Nv)}^{-s})}^{-1}
\notag \\
& {(1 - V_{v}^{2} U_{v} {(Nv)}^{-s})}^{-1}
{(1 - U_{v}^{2} V_{v} {(Nv)}^{-s})}^{-1}
\notag \\
&= {(1 - X_{v}^{3} {(Nv)}^{-s})}^{-1}
{(1 - Y_{v}^{3} {(Nv)}^{-s})}^{-1}
\notag \\
&{(1 - V_{v} \omega_{\pi'_{v}} ({\rm Frob}_{v}) {(Nv)}^{-s})}^{-1}
{(1 - U_{v} \omega_{\pi'_{v}} ({\rm Frob}_{v}) {(Nv)}^{-s})}^{-1}
\notag \\
&= L (s, \chi_{1, v}^{3}) L (s, \chi_{2, v}^{3})
L (s, \pi' \otimes \omega_{\pi'_{v}})
\notag\end{aligned}$$ Here we have used $$U_{v} V_{v} = \omega_{\pi'_{v}} ({\rm Frob}_{v})$$
Hence $${\rm sym}^{3} (\pi'_{v}) \, \simeq \, \chi_{1, v}^{3} \boxplus \chi_{2, v}^{3}
\boxplus
(\pi'_{v} \otimes \omega_{\pi'_{v}}),$$
and by Lemma \[T:602\] we get what we desire, namely, $${\rm sym}^{3} (\pi') \, \simeq \, \chi_{1}^{3} \boxplus \chi_{2}^{3}
\boxplus (\pi' \otimes \omega_{\pi'}).$$
\[T:704\] If $\pi'$ is tetrahedral with ${\rm sym}^{2} (\pi')$ invariant under twisting by a cubic character $\chi$, then $${\rm sym}^{3} (\pi') \, \cong \, (\pi' \otimes \omega_{\pi'}
\chi)
\boxplus (\pi' \otimes \omega_{\pi'} \chi^{-1}).$$
Hence the situation of Lemma \[T:703\] will not happen if $\pi'$ is tetrahedral.
*Proof of Lemma \[T:704\].*
Consider $\pi' \boxtimes {\rm sym}^{2} (\pi') = {\rm sym}^{3}
(\pi') \boxplus (\pi' \otimes \omega_{\pi'})$, which obviously contains $\pi' \otimes \omega_{\pi'}$ as an isobaric constituent .
Since ${\rm sym}^{2} (\pi')$ allows self twists by $\chi$ and $\chi^{-1}$, the isobaric sum above should also contain $\pi'
\otimes \omega_{\pi'} \chi$ and $\pi' \otimes \omega_{\pi'}
\chi^{-1}$. Together with $\pi' \otimes \omega_{\pi'}$, they are pairwise inequivalent, the reason being that if a cusp form on $GL
(2)$ admits a self twist by a character, then such character has to be trivial or quadratic.
Thus, by the uniqueness of the isobaric decomposition, ${\rm
sym}^{3} (\pi')$ should have $\pi' \otimes \omega_{\pi'} \chi$ and $\pi' \otimes \omega_{\pi'} \chi^{-1}$ as its constituents, and there is no room for any other constituent.
*Proof of Case III when $\pi'$ is not dihedral.*
Assume that $\Pi = \pi \boxtimes \pi' = \sigma_{1} \boxplus
\sigma_{2}$ where $\sigma_{1}$ and $\sigma_{2}$ are cusp forms on $GL (3)$ with central characters $\eta_{1}$ and $\eta_{2}$ respectively. Also, assume that $\pi'$ is not dihedral.
*Subcase A: $\pi$ does not allow a self twist by a nontrivial character.*
[F]{}rom (Proposition \[T:701\]) and (Lemma \[T:702\]), we have
$$\begin{aligned}
L (s, {\rm sym}^{3} &(\pi') \otimes \chi)
L (s, (\pi \boxtimes \pi') \times \tilde{\pi} \otimes \omega_{\pi'} \chi)
\notag \\
&= L (s, \pi' \times \pi; \Lambda^{3} \otimes \omega_{\pi}^{-1} \chi)
L (s, \pi' \otimes \omega_{\pi'} \chi)
\notag \\
&= L (s, \eta_{1} \omega_{\pi}^{-1} \chi) L (s, \eta_{2} \omega_{\pi}^{-1} \chi)
L (s, \pi' \otimes \omega_{\pi'} \chi)
\notag \\ \label{EQ:705}
&L (s, \sigma_{1} \times \tilde{\sigma}_{2} \otimes \eta_{2} \omega_{\pi}^{-1} \chi)
L (s, \sigma_{2} \times \tilde{\sigma}_{1} \otimes \eta_{1} \omega_{\pi}^{-1} \chi)\end{aligned}$$
Hence, take $\chi = \omega_{\pi} \eta_{i}^{-1}$, then the right hand side has a pole at $s = 1$ (as the remaining factors do not vanish st $s = 1$). Then the left hand side also has a pole at $s
= 1$.
However, since $\pi'$ is not dihedral, then ${\rm sym}^{3} (\pi')$ is either cuspidal or an isobaric sum of two cusp forms on $GL
(2)$ (Lemma \[T:704\]), so any twisted $L$–function of ${\rm
sym}^{3} (\pi')$ has to be entire. So the only pole at $s = 1$ should come from $L (s, (\pi \boxtimes \pi') \times \tilde{\pi}
\otimes \omega_{\pi'} \omega_{\pi} \eta_{i}^{-1})$.
As $\pi$ is cuspidal, $\Pi = \pi \boxtimes \pi'$ should have both $\sigma'_{i} = \pi \otimes \omega_{\pi'}^{-1} \omega_{\pi}^{-1}
\eta_{i}$ as constituents.
Since $\pi$ is not monomial, it does not allow a self twist. Hence, if $\eta_{1} \ne \eta_{2}$ then $\sigma'_{i}$ are different. Hence $\sigma'_{1}$ and $\sigma'_{2}$ are the only constituents of $\Pi$ which are also twists of $\pi$. Furthermore, if $\eta_{1} = \eta_{2}$, then the order of the pole of both sides of , and hence also of $L (s, (\pi \boxtimes \pi')
\times \tilde{\pi} \otimes \omega_{\pi'} \omega_{\pi}
\eta_{i}^{-1})$, is $2$. Hence $\sigma'_{1} = \sigma_{2}$ should be a constituent of $\Pi$ with multiplicity $2$.
Thus we get an isobaric decomposition of $\Pi$ as a sum of two twists of $\pi$. Thus, from lemma \[T:703\] and Lemma \[T:704\], this cannot happen if $\pi'$ is not dihedral. This completes the treatment of Subcase A.
*Subcase B: $\pi$ admits a self twist by a nontrivial cubic character $\chi$.*
In this subcase, recall that we are assuming $\Pi = \pi \boxtimes
\pi' \cong \sigma_{1} \boxplus \sigma_{2}$, where $\sigma_{1}$ and $\sigma_{2}$ are cusp forms with respective central characters $\eta_{1}$ and $\eta_{2}$.
We claim that $\sigma_{1}$ and $\sigma_{2}$ are also invariant when twisted by $\chi$. Otherwise $\sigma_{i} \otimes \chi$ and $\sigma_{i} \otimes \chi^{-1}$ will be distinct from $\sigma_{i}$, while they should both be constituents of $\Pi \simeq \Pi \otimes
\chi \simeq \Pi$. Hence $\Pi$ has at least degree $3 \times 3 =
9$, which is impossible as it is automorphic on $GL (6)$.
Moreover, let $v$ be any place where $\pi$ and $\pi'$ are unramified. Write (for $i=1,2$) $\pi'_{v} = \alpha_{v} \boxplus
\alpha'_{v}$, $\pi_{v} = \beta_{v} \boxplus \beta_{v} \chi_{v}
\boxplus \beta_{v} \chi_{v}^{-1}$ (this form being implied by $\pi
\cong \pi \otimes \chi$), and $\sigma_{i, v} = \theta_{i, v}
\boxplus \theta_{i, v} \chi_{v} \boxplus \theta_{i, v}
\chi_{v}^{-1}$.
Then we have $$\Pi_{v} = (\alpha_{v} \boxplus \alpha'_{v}) \otimes
\beta_{v}\otimes
(1 \boxplus \chi_{v} \boxplus \chi_{v}^{-1})$$ and $$\sigma_{1, v} \boxplus \sigma_{2, v}
= (\theta_{1, v} \boxplus \theta_{2, v})\otimes
(1 \boxplus \chi_{v} \boxplus \chi_{v}^{-1})$$
Since the sets of all cubes of characters occurring in the previous two isobaric decompositions should be the same, and since $\beta_{v}^{3} = \omega_{\pi_{v}}$ and $\theta_{v}^{3} = \eta_{i,
v}$, we must have $$\begin{aligned}
{{\left\{\,\alpha_{v}^{3} \omega_{\pi_{v}}, {\alpha'_{v}}^{3} \omega_{\pi_{v}}\,\right\}}}
&= {{\left\{\,\alpha_{v}^{3} \beta_{v}^{3}, {\alpha'_{v}}^{3} \beta_{v}^{3}\,\right\}}}
\notag \\
&= {{\left\{\,\theta_{1, v}^{3}, \theta_{2, v}^{3}\,\right\}}}
\notag \\
&= {{\left\{\,\eta_{1, v}, \eta_{2, v}\,\right\}}}
\notag\end{aligned}$$
So $$\begin{aligned}
{\rm sym}^{3} &(\pi'_{v}) \otimes \omega_{\pi_{v}}
\cong \alpha_{v}^{3} \omega_{\pi_{v}} \boxplus {\alpha'_{v}}^{3} \omega_{\pi_{v}}
\boxplus (\alpha_{v} \boxplus \alpha'_{v}) \boxtimes\omega_{\pi'_{v}} \omega_{\pi_{v}}
\notag \\
&\simeq \eta_{1, v} \boxplus \eta_{2, v}
\boxplus \pi'_{v} \otimes \omega_{\pi'_{v}} \omega_{\pi_{v}},
\notag\end{aligned}$$ that is $${\rm sym}^{3} (\pi'_{v}) \cong
\eta_{1, v} \omega_{\pi_{v}}^{-1} \boxplus
\eta_{2, v} \omega_{\pi_{v}}^{-1}
\boxplus \pi'_{v} \otimes \omega_{\pi'_{v}}$$
Thus by Lemma \[T:602\] or the strong multiplicity one theorem, we have $${\rm sym}^{3} (\pi')
\cong \eta_{1} \omega_{\pi}^{-1} \boxplus \eta_{2} \omega_{\pi}^{-1}
\boxplus \pi' \otimes \omega_{\pi'}$$
However, since $\pi'$ is not dihedral, ${\rm sym}^{3} (\pi')$ is by Lemma \[T:704\] either cuspidal or an isobaric sum of two cusp forms on $GL (2)$. This gives a contradiction.
This completes the treatment of Subcase B.
Proof of Theorem A, Part \#3
============================
In this part, we will treat the case when $\pi'$ is dihedral. After that, we will analyze precisely the cuspidality criterion when $\pi$ is an adjoint of a form on GL$(2)$. Again, $F$ denotes a number field.
In fact, we will prove the following:
\[T:801\] Let $\pi', \pi$ be cusp forms on $GL (2)/F, GL (3)/F$ respectively, with $\pi'$ dihedral.
Then $\Pi = \pi \boxtimes \pi'$ is cuspidal unless both the following two conditions hold:
[(a)]{.nodecor} There is a non-normal cubic extension $K'$ of $F$ such that $\pi'_{K'}$ is Eisensteinian; equivalently, $\pi'$ is dihedral of type $D_{6}$.
[(b)]{.nodecor} $\pi$ is monomial and $\pi = I^{F}_{K'} (\chi')$ for some idele class character $\chi'$ of $K$.
If [(a)]{.nodecor} and [(b)]{.nodecor} both hold, then $\Pi$ is an isobaric sum of two cuspidal representations of degree $3$, which are both twist equivalent to $\pi$.
Before we prove this theorem, let us recall that a dihedral Galois representation $\rho'$ of ${{\mathcal G}}_{F}$ is said to be of type $D_{2 n}$ if its projective image is $D_{2 n}$. It is clear that $\rho'$ is not irreducible if and only if $n = 1$ (Note that the projective image of $D_{4 n}$ must be a quotient of $D_{2 n}$ since $D_{4 n}$ has a nontrivial center). If $6 | n$, then $D_{2 n}$ has a unique cyclic subgroup with quotient isomorphic to $D_{6} \cong S_{3}$. Suppose $K'$ is a non-normal cubic extension of $F$, and $\rho'$ restricted to ${{\mathcal G}}_{K'}$ is reducible. Then the projective image of ${{\mathcal G}}_{K'}$ should be a subgroup of that of ${{\mathcal G}}_{F}$ of index $3$, hence is isomorphic to $D_{2 n / 3}$. Thus $n = 3$, and $\rho$ must be dihedral of type $D_{6}$. Similarly, we conclude that if $\pi'$ is dihedral and $\pi_{K'}$ is not cuspidal, then $\pi'$ is of type $D_{6}$.
*Proof of Theorem \[T:801\].*
First assume (a) and (b). Note that $\pi'_{K'} = v_{1} \boxplus v_{2}$, and $\pi = I^{F}_{K'} (\chi')$. Then $$\begin{aligned}
\Pi &= \pi \boxtimes \pi' \cong \pi' \boxtimes I^{F}_{K'} (\chi')
\notag \\
&\cong I^{F}_{K'} (\pi'_{K'} \otimes \chi')
\cong I^{F}_{K'} (v_{1} \chi') \boxplus I^{F}_{K'} (v_{2} \chi')
\notag\end{aligned}$$ Hence $\Pi$ is not cuspidal.
Note that $\pi_{i} = I^{F}_{K'} (v_{i} \chi')$ MUST be cuspidal as from Section 5, $\Pi$ cannot have a character as its constituent.
Next, we prove that if $\Pi$ is not cuspidal, then (a), (b) and the remaining statements of the theorem hold.
*Step 1: $\pi_{K}$ is cyclic cubic monomial.*
Assume that $\pi' = I^{F}_{K} (\tau)$ where $\tau$ is some idele class character of $C_{K}$ with $K$ a quadratic extension of $F$. and also assume that $K / F$ is cut out by $\delta$.
[F]{}rom Section 5, Case I and II cannot happen, so we are in Case III. Say $\Pi = \sigma_{1} \boxplus \sigma_{2}$ where $\sigma_{i}$ are some cusp forms on $GL (3) / F$. As $\pi'$ allows a self twist by $\delta$, so does $\Pi = \pi \boxtimes \pi'$. Thus $\sigma_{1}
\cong \sigma_{2} \otimes \delta$ as the only possible characters that either $\sigma_{i}$ allows (for self-twisting) should be trivial or cubic.
Let $\theta$ be the nontrivial Galois conjugation of $K / F$. Then $\pi'_{K} \cong \tau \boxplus \tau^{\theta}$. Hence the base change $\Pi_{K}= \pi'_{K} \boxtimes \pi_{K}$ is equivalent to $\pi_{K} \otimes \tau$ plus $\pi_{K} \otimes (\tau\circ{\theta})$. As $\Pi = \sigma_{1} \boxplus \sigma_{1} \otimes \delta$ hence $\Pi_{K}$ is equivalent to the isobaric sum of two copies of ${\sigma_{1}}_K$.
Thus $\pi_{K} \otimes \tau \cong \pi_{K} \otimes
(\tau\circ{\theta}) \cong \sigma_{1}$. As $\pi' = I^{F}_{K}
(\tau)$ is cuspidal, $\tau \ne \tau\circ{\theta}$. Hence $\pi_{K}$ is forced to be cyclic monomial.
*Step 2: $\pi$ is non-normal cubic monomial.*
By Step 1, $\mu = \tau^{-1} (\tau\circ{\theta})$ is a cubic character of $C_{K}$. Let $M$ be the cubic field extension of $K$ cut out by $\mu$. As $\mu\circ{\theta} = \mu^{-1}$, $M^{\theta} =
M$, thus $M / F$ is normal, and ${\rm Gal}(M / F) \cong S_{3}$.
Furthermore, $\pi_{K} = I^{K}_{M} (\lambda)$ for some character $\lambda$ of $C_{M}$. And also, $\pi_{M}$ is of the form $\lambda
\boxplus \lambda' \boxplus \lambda''$.
Let $K'$ be a non-normal cubic extension of $F$ contained in $M$. Then $[M : K'] = 2$ and $\pi_{M}$ is a quadratic base change of $\pi_{K'}$.
We claim that $\pi_{K'}$ is Eisensteinian, i.e., not cuspidal. The reason is that, if $\pi_{K'}$ were cuspidal, then its quadratic base change $\pi_{M}$ would be either cuspidal or the isobaric sum of two cusp forms of the same degree. Since $\pi_{K'}$ is a cusp form on $GL (3)$, we see from [@AC] that this is impossible.
So $\pi_{K'}$ must admit a character as an isobaric constituent, which means that $\pi$ is induced from some character of $C_{K'}$.
*Step 3: $\pi'_{K'}$ is not cuspidal, hence $\pi'$ is dihedral of type $D_{6}$.*
Recall that $\pi'_{K} = \tau \boxplus (\tau\circ{\theta}) = \tau
\boxplus \tau \mu$, so that $\pi'_{M} = \tau_{M} \boxplus
\tau_{M}$ as $M / K$ is cut out by $\mu$.
Thus the projective image of $\rho'_{M}$ is trivial, where $\rho'$ is the representation Ind$_K^F(\tau)$ of the Weil group $W_F$, and $\rho'_{M}$ is the restriction of $\rho$ to ${{\mathcal G}}_{M}$. Hence the projective image of $\rho$ must be $D_{6}$.
*Remark:* Even if $\pi'$ is selfdual, $\tau$ may be a character of order $3$ or $6$.
*Step 4: $\sigma_{1}$ and $\sigma_{2}$ are all twist equivalent to $\pi$.*
Observe that $$\begin{aligned}
{(\tau \mu^{-1})}\circ{\theta} &= ({\tau}\circ{\theta}) ({\mu}\circ{\theta})^{-1}
\notag \\
&= \tau \mu \mu = \tau \mu^{-1}.
\notag\end{aligned}$$
So $\tau \mu^{-1}$ is a base change of some character, say $\nu$, of $C_{F}$ to $K / F$.
So $\pi' = I^{F}_{K} (\mu) \otimes \nu$ and $\pi'_{K'} \cong
\nu_{K'} \boxplus \nu_{K'} \delta_{K'}$. By Step 2, we may assume that $\pi = I^{F}_{K'} (\lambda)$ for a character $\lambda$ of a non-normal cubic extension $K'$ of $F$. We get $$\begin{aligned}
\Pi &= \pi \boxtimes \pi' = \pi' \boxtimes I^{F}_{K'} (\lambda)
\notag \\
&= I^{F}_{K'} (\pi'_{K'} \otimes \lambda)
= I^{F}_{K'} (\nu_{K'} \lambda \boxplus \nu_{K'} \lambda \delta_{K'})
\notag \\
&= I^{F}_{K'} (\lambda) \otimes \nu \boxplus I^{F}_{K'} (\lambda)
\otimes \nu \delta
\notag \\
&= (\pi \otimes \nu) \boxplus (\pi \otimes \nu \delta)\end{aligned}$$
Now the proof of Theorem 8.1 is completed.
*Remark.* When $\pi$ is twist equivalent to $Ad (\pi_{0})$, and $\pi'$ is dihedral, we claim that the only way $\Pi = \pi
\boxtimes \pi'$ can be [*not*]{} cuspidal is for $Ad (\pi)$ to be non-normal cubic monomial, implying that $\pi$ is of octahedral type. We get the following theorem which is more precise than the result in [@Wa2003]:
\[T:802\] Let $\pi'$, and $\pi''$ be two cusp form on $GL (2) / F$, and suppose that $\pi''$ is not dihedral. then $\Pi = \pi' \boxtimes Ad (\pi'')$ is cuspidal unless one of the following happens:
[(d)]{.nodecor} $\pi'$ and $\pi''$ are twist equivalent.
[(e)]{.nodecor} $\pi'$ and $\pi''$ are octahedral attached with the same $\tilde{S}_{4}$-extension, and $Ad (\pi')$ and $Ad (\pi'')$ are twist equivalent.
[(f)]{.nodecor} $\pi''$ is octahedral, $\pi' = I^{F}_{K} (\mu) \otimes \nu$ is dihedral, where $\mu$ is the cubic character which is allowed by $Ad (\pi''_{K})$.
The proof of the theorem above uses the following proposition.
\[T:803\] Let $\pi_{1}$ and $\pi_{2}$ are two non-dihedral cusp forms on $GL (2) / F$, and $Ad (\pi_{1})$ and $Ad (\pi_{2})$ are twist equivalent. Then one of the following holds.
[(g)]{.nodecor} $\pi_{1}$ and $\pi_{2}$ are twist equivalent (so that their adjoints are equivalent).
[(h)]{.nodecor} $\pi_{1}$ and $\pi_{2}$ are octahedral attached with the same $\tilde{S}_{4}$-extension, and $Ad (\pi_{1})$ and $Ad (\pi_{2})$ are twist equivalent by a quadratic characters.
*Proof of Proposition \[T:803\].* (cf. [@Ra2000])
It is clear that (g) and (h) imply that $\pi_{1}$ and $\pi_{2}$ are twist equivalent. So it suffices to show the other side.
First assume that $Ad (\pi_{1})$ and $Ad (\pi_{2})$ are equivalent.
Consider $\Pi = \pi_{1} \boxtimes \pi_{2}$. Note that $$\Pi \boxtimes \overline{\Pi} \cong 1 \boxplus Ad (\pi_{1})
\boxplus Ad (\pi_{2}) \boxplus Ad (\pi_{1}) \boxtimes Ad (\pi_{2})$$ admits two copies $1$ as its constituents. Hence $\Pi$ is not cuspidal.
If $\Pi$ contains a character $\nu$, then $\overline{\pi}_{2}
\cong \pi_{1} \otimes \nu$. If $\Pi$ is an isobaric sum of two cusp forms $\sigma_{1}$ and $\sigma_{2}$ on $GL (2)$, then check that $\Lambda^{2} (\pi_{1} \boxtimes \pi_{2})$ is equivalent to $$(Ad (\pi_{1}) \boxplus
Ad (\pi_{2})) \otimes \omega_{\pi_{1}} \omega_{\pi_{2}}$$ which does not contain any character constituent; However, $\Lambda^{2} (\sigma_{1} \boxplus \sigma_{2})$ is equivalent $$\omega_{\sigma_{1}} \boxplus \omega_{\sigma_{1}} \boxplus
(\sigma_{1} \boxtimes \sigma_{2})$$ which contains two $GL (1)$-constituents. Thus we get a contradiction, and (g) holds.
Furthermore, assume that $$Ad (\pi_{2}) \cong Ad (\pi_{1}) \otimes \epsilon$$ where $\epsilon$ is a character. Then $$Ad (\pi_{2}) \cong Ad (\pi_{1}) \otimes \epsilon^{-1}$$ and hence $$Ad (\pi_{1}) \boxtimes Ad (\pi_{1})
\cong Ad (\pi_{2}) \boxtimes Ad (\pi_{2}).$$
However, $$Ad (\pi_{i}) \boxtimes Ad (\pi_{i}) \cong 1
\boxplus Ad (\pi_{i}) \boxplus A^{4} (\pi_{i}).$$ Hence $$Ad (\pi_{1}) \boxplus A^{4} (\pi_{1})
\cong Ad (\pi_{2}) \boxplus A^{4} (\pi_{2}).$$
If $Ad (\pi_{1})$ and $Ad (\pi_{2})$ are equivalent, we get (g). Otherwise, $Ad (\pi_{1})$, which is a nontrivial twist of $Ad (\pi_{2})$, must be contained in $A^{4} (\pi_{2})$.
So $\pi_{1}$ and $\pi_{2}$ are of solvable polyhedral type.
If $\pi_{2}$ is tetrahedral, then $$A^{4} (\pi_{2}) \cong Ad (\pi_{2}) \boxplus \omega \boxplus \omega^{2}$$ where $\omega$ is some cubic character that $Ad (\pi_{2})$ admits as as self-twist. So this cannot happen.
Thus $\pi_{2}$ and $\pi_{1}$ are octahedral, and $$A^{4} (\pi_{2}) \cong I^{F}_{K} (\mu) \boxplus Ad (\pi_{2})
\otimes \epsilon$$ where $K$ is a quadratic extension of $F$ such that $Ad ({\pi_{2}}_{K})$ allows a self twist by $\mu$, and $\epsilon$ is the quadratic character cuts out $K$.
So they must come from the same $\tilde{S}_{4}$-extension, and hence (h) holds.
*Proof of Theorem \[T:802\].*
Set $\pi = Ad (\pi'')$ [F]{}rom what we have seen (including the proof of Theorem \[T:801\]), $\Pi = \pi' \boxtimes Ad (\pi')$ is cuspidal unless (i) $Ad (\pi')$ and $Ad (\pi'')$ are twist equivalent; or (ii) $\pi'$ is dihedral of type $D_{6}$, $\pi = Ad
(\pi'')$ is non-normal cubic monomial, and (a) and (b) of Theorem \[T:801\] hold.
If (ii) holds, then $\pi''$ must be octahedral, and (f) must hold. If (1) holds, then $Ad (\pi')$ and $Ad (\pi'')$ are twist equivalent. Then part (g) of Proposition \[T:803\] implies (d), and part (h) of this proposition implies (e).
The proof of Theorem A is now complete.
Proof of Theorem B
==================
In this part we deduce Theorem \[TM:B\] from Theorem \[TM:A\]. First we need some preliminaries.
Recall that a cusp form $\pi$ on $GL (n)$ over $F$ is essentially selfdual if $\overline{\pi}$ is twist equivalent to $\pi$. Throughout this section, $\pi'$ and $\pi$ denote cusp forms on $GL
(2)$ and $GL (3)$ over $F$. We assume that $\pi'$ is not dihedral, and $\pi$ is not twist equivalent to $Ad (\pi'')$ for any cusp form $\pi''$.
First, from Theorem \[TM:A\], we have the following:
\[T:901\] If $\pi'$ is not of solvable polyhedral type and $\pi$ is not essentially selfdual, then $\Pi = \pi \boxtimes \pi'$ is cuspidal.
\[T:902\] Let $K$ be any solvable extension of $F$. If $\pi$ is not essentially selfdual, and if $\pi_{K}$ does not admit any self twist, then $\pi_{K}$ is not essentially selfdual.
*Proof of Lemma \[T:902\].*
First, assume that $[K : F] = l$ a prime so that $K / F$ is cyclic. Let $\theta \ne 1$ be a Galois conjugation of $K$ over $F$, and $\tau$ be a character cutting out $K / F$.
Assume that $\pi_{K}$ is essentially selfdual. Say $\overline{\pi}_{K} \cong \pi_{K} \otimes \mu$, for a character $\mu$. Applying $\theta$, and being aware of the fact that $\pi_{K}$ is fixed by $\theta$, we get $$\overline{\pi}_{K} \cong \pi_{K} \otimes (\mu\circ{\theta})$$
Since $\pi_{K}$ does not allow a self twist, then $\mu\circ{\theta} = \mu$, hence $\mu$ must be a base change of some character $\alpha$ of $C_{F}$ to $K$.
Hence, $\overline{\pi}$ and $\pi \otimes \alpha$ have the same base change over $K / F$, and thus must be twist equivalent. This shows that $\pi$ is also essentially selfdual.
In general case, let $K_{0} = F
\subset K_{1} \subset \ldots \subset K_{n} = K$ be a tower of cyclic extensions of prime degree. Assume that $\pi_{K_{n}} = \pi_{K}$ is essentially selfdual, then as $\pi_{K_{n}}$ does not allow a self twist, neither does $\pi_{K_{i}}$ for any smaller $i$, thus applying the arguments above inductively, we claim that $\pi_{K_{i}}$ is essentially selfdual. In particular, $\pi$ must be essentially selfdual.
*Proof of Theorem \[TM:B\]*.
First prove (a). $\Pi$ is cuspidal from Corollary \[T:901\]. First, assume only that $\pi$ is not essentially selfdual and does not allow a self twist. Assume that $\Pi$ allows a self twist by some character $\nu$. Without loss of generality, may assume that $\nu$ is of order $2$ or $3$. Let $K / F$ be cut out by $\nu$. Thus $\Pi_{K} = I^{F}_{K} (\eta)$ is Eisensteinian of type $(2, 2, 2)$ or $(3, 3)$.
However, $\pi_{K}$ is cuspidal as $\pi$ does not allow a self twist. From Theorem \[TM:A\] (and the remark at the end of Section 5), $\Pi_{K}$ cannot be of type $(3, 3)$ as $\pi'_{K}$ is not dihedral, type $(2, 2, 2)$ as $\pi'_{K}$ is not tetrahedral (as $\pi'$ and hence $\pi'_{K}$ is not of solvable polyhedral type). Thus $\Pi_{K}$ must be cuspidal, and hence $\Pi$ does not allow a self twist.
Moreover, assume that $\pi$ is not monomial, in particular, $\pi$ is not induced from a non-normal cubic extension. Want to prove that $\Pi$ is not either.
Assume that $\Pi \cong I^{F}_{K'} (\eta)$ where $\eta$ is some cusp form on $GL (2)$ over $K'$ which is non-normal cubic over $F$. Let $M$ be the normal closure of $K'$ over $F$ and $E$ be the unique quadratic subextension of $M$ over $F$. Then $\pi'_{E}$ is still not of solvable polyhedral type. And $\pi_{E}$ is not cyclic monomial as $\pi$ is not monomial. From Lemma \[T:902\], $\pi_{E}$ is not essentially selfdual. Thus the first part of (a) implies that $\Pi_{E}$ does not allow a self twist. Hence $\Pi_{M}$ is still cuspidal. However, $\Pi_{M} \cong {I^{F}_{K'}
(\eta)}_{M} = \eta_{M} \boxplus (\eta_{M}\circ{\theta'}) \boxplus
(\eta_{M}\circ{\theta'^{2}})$ where $\theta'$ is the character cutting out $M / E$. We get a contradiction.
Thus (a) is proved.
Next prove (b). It suffices to prove that $\Pi_{K}$ is cuspidal for any solvable extension $K$ of $F$.
Since $\pi'$ is not of solvable polyhedral type, neither is $\pi'_{K}$. As $\pi$ is not of solvable type, then $\pi_{K}$ must be cuspidal. We claim that $\pi_{K}$ cannot allow a self twist. Otherwise, say $\pi_{K} \cong \pi_{K} \otimes \nu$. $\nu$ must be a cubic character. Let $K_{1} / K$ be cut out by $\nu$, then $\pi_{K_{1}}$ should be Eisensteinian. However, $K_{1} / F$ is contained in some solvable normal extension. Thus $\pi$ is of solvable type. Contradiction.
Hence the claim. From Lemma \[T:902\], $\pi_{K}$ is not essentially selfdual. Corollary \[T:901\] then implies that $\Pi_{K}$ is cuspidal. Thus $\Pi$ is not of solvable type.
Before we finish this section, we would like to prove the following lemma.
\[T:903\] Let $\pi$ be a cusp form on $GL (2 m + 1)$ over $F$. Assume that $\pi$ is regular algebraic at infinity, and $F$ is not totally complex. Then $\pi$ is not monomial.
*Proof of Lemma \[T:903\].*
Assume that $\pi = I^{F}_{K'} (\nu)$ where $K'$ is a field extension of $F$ of degree $2 m + 1$, and $\nu$ is an algebraic character of $C_{K}$. As $F$ is not totally complex, neither is $K'$ as $[K' : F]$ is odd. Thus from Weil ([@Weil]), $\nu$ must be of the form $\nu_{0} {\| \cdot \|}^{k}$ where $\nu_{0}$ is a character of finite order. Thus $I^{F}_{K'} (\nu)$ does not contain any nontrivial algebraic character at infinity, and hence cannot be regular at infinity.
Appendix {#appendix .unnumbered}
========
The object here is to justify the statement made in the Introduction that it is possible to construct, for $n > 2$ even (resp. $n=4$), non-selfdual, [*monomial*]{} (resp. non-monomial, but imprimitive), cuspidal cohomology classes for $\Gamma \subset
{\rm SL}(n, {{\mathbb Z}})$.
\[TM:E\]
1. Fix any integer $m > 1$. Then there exists a cuspidal automorphic representation $\pi$ of GL$(2m, {{\mathbb A}}_{{\mathbb Q}})$, which is not essentially self-dual, contributing to the cuspidal cohomology in degree $m^2$, and admitting a self-twist relative to a character of order $2m$. In fact, this can be done for any, not necessarily constant, coefficient system.
2. There exists a cusp form $\pi$ on GL$(4)/{{\mathbb Q}}$ contributing to the cuspidal cohomology in degree $4$, which is not essentially self-dual. It admits a self-twist relative to a quadratic character, but not relative to any quartic character.
In both cases it will be apparent from the proof below that there are infinitely many such examples, and by the discussion in section 3, they give rise, for arbitrary coefficient systems $V$, to non-selfual, cuspidal cohomology classes in $H^\ast(\Gamma, V)$ for suitable congruence subgroups $\Gamma \subset {\rm SL}(2m,
{{\mathbb Z}})$ ($m \geq 2$). Moreover there are naturally associated Galois representations, which are monomial in case (a), and are imprimitive but non-monomial in case (b).
[*Proof*]{}. (a) For any $m > 1$, fix a finite-dimensional coefficient system $V$. Then a cuspidal automorphic representation $\pi$ of GL$(2m, {{\mathbb A}}_F)$ contributes to the cuspidal cohomology with coefficients in $V$ iff it is algebraic with infinity type (in the [*unitary*]{} normalization): (cf. [@C]) $$\{(z/\vert z\vert)^{k_1}, ({\overline z}/\vert z\vert)^{k_1},
(z/\vert z\vert)^{k_2}, ({\overline z}/\vert z\vert)^{k_2}, \dots,
(z/\vert z\vert)^{k_m}, ({\overline z}/\vert z\vert)^{k_m}\},$$ where $(k_1, k_2, \dots, k_m)$ is an ordered $m$-tuple of integers (determined by $V$) satisfying $$k_1 > k_2 > \dots > k_m.$$ In particular, $\pi_\infty$ is regular. If $V \simeq {{\mathbb C}}$, then as seen in Theorem 3.2, $k_j = 2(m-j)+1$. Now pick any cyclic, totally real extension $F$ of ${{\mathbb Q}}$ of degree $m$, and a totally imaginary quadratic extension $K$ of $F$ which is normal over ${{\mathbb Q}}$. Let $v_1, \dots, v_m$ denote the archimedean places of $F$, and for each $j$ let $w_j$ be a complex place of $K$ above $v_j$. Choose an algebraic Hecke character $\chi$ of of $K$ such that $$\chi_{w_j}(z) \, = \,
\left(\frac{z}{|z|}\right)^{2(m-j)+1}|z\overline z|^{2m-1} \, \,
\forall \, \, j \leq m. \leqno(1)$$ That such a character exists is a consequence of the discussion on page 3 of [@Weil]. To elaborate a bit for the sake of the uninitiated, the necessary and sufficient condition for the existence of $\chi$ as above is that there be a positive integer $M$ such that the following holds for all units $u \in \mathfrak
O_K^\ast$ with components $u_j$ at $w_j$: $$\prod_{j \leq m} \, \left(\frac{u_j}{\overline u_j}\right)^{Mk_j}
\, = \, 1.\leqno(2)$$ But the index of the real units $\mathfrak O_F^\ast$ in $\mathfrak
O_K^\ast$ is finite by the Dirichlet unit theorem, and hence for a suitable $M$, $u_j^M$ is real for all $j$. The desired identity (2) follows.
Next pick a finite order character $\nu$ of $K$ and set $$\Psi \, = \, \chi\nu. \leqno(3)$$ Let $\tau$ be the non-trivial automorphism ([*complex conjugation*]{}) of $K/F$, and $\delta$ the quadratic character of $F$ attached to $K$. Then we may, and we will, choose $\nu$ in such a way that $$\Psi(\Psi\circ\tau) \, \ne \, \mu\circ N_{K/{{\mathbb Q}}}
\leqno(4)$$ for any character $\mu$ of ${{\mathbb Q}}$, which is possible – and this is [*crucial*]{} – because $[K:{{\mathbb Q}}] \geq 4$ and so $F = \{x \in K \,
\vert \, x^\tau = x\} \, \ne \, {{\mathbb Q}}$. Put $$\pi : = \, I_K^{{\mathbb Q}}(\Psi),$$ where $I$ denotes [*automorphic induction*]{} ([@AC]). Note that $\pi$ makes sense because $K/{{\mathbb Q}}$ is solvable and normal, allowing automorphic induction to be defined. By looking at the infinity type (1) we see that $\pi$ is regular and algebraic.
By construction, $\pi_\infty$ contributes to cohomology, and $\pi$ is cuspidal because the infinity type of $\Psi$ precludes it from being $\Psi \circ \sigma$ for any non-trivial automorphism $\sigma$ of $K/{{\mathbb Q}}$. To elaborate, note first that $\eta:=
I_K^F(\Psi)$ is cuspidal and algebraic, corresponding to a Hilbert modular newform on GL$(2)/F$ of the prescribed weight at infinity. Since the automorphic induction is compatible with doing it in stages, $\pi$ is just $I_F^{{\mathbb Q}}(\eta)$, and since $F/{{\mathbb Q}}$ is cyclic, it suffices to check that for any automorphism $\tau$ of $F$, $\eta$ and $\eta\circ \tau$ are not isomorphic, which is clear from the properties of $\Psi$.
It remains to check that $\pi$ is not essentially self-dual, which comes down to checking the same for the $2m$-dimensional representation $\rho$ of $W_{{\mathbb Q}}$ induced by the character $\Psi$ of $W_K$. For this we need, by Mackey, to check that $\Psi^{-1} \ne
(\mu_K)\Psi\circ \sigma$ for any automorphism $\sigma$ of $K$ and any character $\mu$ of ${{\mathbb Q}}$. By our choice of the infinity type, this is clear for any $\sigma \ne \tau$ (and any $\mu$). For $\sigma = \tau$, this is the content of (4). So we are done with the proof of part (a).
\(b) Let $K/{{\mathbb Q}}$ be an imaginary quadratic field and $\beta$ a non-dihedral cusp form of weight $2$ over $K$ with ${{\mathbb Q}}$ coefficients, such that no twist of $\beta$ is a base change from ${{\mathbb Q}}$. Here [*weight 2*]{} signifies the fact that the Langlands parameter of $\beta_\infty$ is given by $$\sigma(\beta_\infty) \, = \, \{z/|z|, \overline z/|z|\} \otimes
(z\overline z)$$ Here are two explicit (known) examples with these properties: First consider the (non-CM) elliptic curve $$E_1: \, y^2+xy \, = \, x^3+(3+\sqrt{-3})x^2/2+(1+\sqrt{-3})x/2$$ over $K_1={{\mathbb Q}}(\sqrt{-3})$. This was shown to be associated to a cusp form $\beta_1$ of weight $2$ and trivial central character on GL$(2)/K_1$ by R. Taylor ([@Ta]) such that $a_P(E_1) =
a_P(\beta_1)$ for a set of primes $P$ of density $1$. (In fact, recent results can be used to show that this equality holds outside a finite set of primes $P$.)
For the second example, set $K_2 = {{\mathbb Q}}(i)$ and $Q$ the prime ideal generated by $8+13i$. Then there is a corresponding cusp form $\beta_2$ of weight $2$, conductor $Q$ and trivial central character, as seen on page 344 of the book [@EGM] by Elstrodt, Grunewald and Mennicke. Its conjugate by the non-trivial automorphism $\theta$ of $K_2$ will have conductor $Q^\theta$ and so no twist of $\beta_2$ can be a base change from ${{\mathbb Q}}$. There is a corresponding elliptic curve $$E_2: \, y^2 +iy \, = \, x^3 +(1+i)x^2 +ix$$ over $K_2$ of conductor $Q$, and one knows for many $P$ that $a_P(E_2) = a_P(\beta_2)$.
Next choose an algebraic Hecke character $\chi$ of $K$ such that $$\chi_{\infty}(z) \, = \, ({z}/{|z|})^2|z\overline z|^{2}.$$ For example, we can choose $\chi$ to be the square of a Hecke character associated to a CM elliptic curve. Now consider, for $j
= 1, 2$, the automorphic induction $$\pi_j : = \, I_K^{{\mathbb Q}}(\beta_j \otimes \chi).$$ The infinity types chosen imply that either $\beta_j \otimes \chi$ is not isomorphic to its transform by the non-trivial automorphism of $K_j$. So $\pi_j$ is a cusp form on GL$(4)/{{\mathbb Q}}$. It is cohomological, as easily seen by its archimedean parameter. That $\pi_j$ is not essentially self-dual is an immediate consequence of the infinity types of $\chi$ and $\beta$. Finally, $\pi_j$ admits a quadratic self-twist, namely by the character of ${{\mathbb Q}}$ associated to $K_j$, but it admits no quartic self-twist as $\beta_j$ is non-dihedral. We are now done.
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Dinakar Ramakrishnan
253-37 Caltech
Pasadena, CA 91125, USA.
[email protected]
Song Wang
Department of Mathematics
Yale University
New Haven, CT 06520, USA.
[email protected]
|
---
abstract: 'We present a set of white-dwarf–main-sequence (WDMS) binaries identified spectroscopically from the Large sky Area Multi-Object fiber Spectroscopic Telescope (LAMOST, also called the Guo Shou Jing Telescope) pilot survey. We develop a color selection criteria based on what is so far the largest and most complete Sloan Digital Sky Survey (SDSS) DR7 WDMS binary catalog and identify 28 WDMS binaries within the LAMOST pilot survey. The primaries in our binary sample are mostly DA white dwarfs except for one DB white dwarf. We derive the stellar atmospheric parameters, masses, and radii for the two components of 10 of our binaries. We also provide cooling ages for the white dwarf primaries as well as the spectral types for the companion stars of these 10 WDMS binaries. These binaries tend to contain hot white dwarfs and early-type companions. Through cross-identification, we note that nine binaries in our sample have been published in the SDSS DR7 WDMS binary catalog. Nineteen spectroscopic WDMS binaries identified by the LAMOST pilot survey are new. Using the 3$\sigma$ radial velocity variation as a criterion, we find two post-common-envelope binary candidates from our WDMS binary sample.'
author:
- 'Juanjuan Ren, Ali Luo, Yinbi Li, Peng Wei, Jingkun Zhao, Yongheng Zhao, Yihan Song, Gang Zhao'
title: 'WHITE-DWARF–MAIN-SEQUENCE BINARIES IDENTIFIED FROM THE LAMOST PILOT SURVEY'
---
Introduction {#sec:intro}
============
White-dwarf–main-sequence (WDMS) binaries consist of a (blue) white dwarf primary and a (red) low-mass main-sequence (MS) companion formed from MS binaries where the primary has a mass $\lesssim 10\ M_{\Sun}$. Most of the WDMS binaries ($\sim 3/4$) are wide binaries [@silvestri2002; @schreiber2010], of which the initial MS binary separation is large enough that the two components will never interact and evolve like single stars [@de1992]. Consequently, the orbital period of these systems will increase because of the mass loss of the primary (the white dwarf precursor). The remaining $\sim 1/4$ of the WDMS binaries are close binaries. When the more massive star of a close binary leaves the MS and evolves into the giant branch or asymptotic giant branch phase, there is a dynamically unstable mass transfer to the MS companion and then the system goes through a common-envelope phase [@iben1993; @zorotovic2010] during which it may continue evolving to an even shorter orbital period through angular momentum loss mechanisms (magnetic braking and gravitational radiation) or undergo a second common envelope. Binary population synthesis models [@willems2004] indicate the bimodal nature of the orbital period distribution of the entire population of WDMS binaries.
During the common-envelope phase of close binaries, the binary separation undergoes a rapid decrease and consequently orbital energy and angular momentum are extracted from the orbit, leading to the ejection of the envelope and exposing a post-common-envelope binary (PCEB). These PCEBs are important objects that may be progenitor candidates of cataclysmic variables [@warner1995] and perhaps Type Ia supernovae [@langer2000]. They can also improve the theory of close binary evolution [@schreiber2003], especially the understanding of the physics of common-envelope evolution [@paczynski1976; @zorotovic2010; @rebassa2012b]. Among the variety of PCEBs (such as sdOB+MS binaries, WDMS binaries, and double degenerates), WDMS binaries are intrinsically the most common ones and the stellar components (white dwarfs and M dwarfs) of WDMS binaries are relatively simple. There are more and more WDMS binaries being found from the Sloan Digital Sky Survey [SDSS; @york2000], making them the most ideal population to help us understand common-envelope evolution. In addition, WDMS binaries, especially eclipsing WDMS binaries, also provide an interesting test of stellar evolution for both the white dwarf primary and the secondary low-mass MS in the binary environment [@nebot2009; @parsons2010; @parsons2011; @parsons2012; @parsons2013; @pyrzas2009; @pyrzas2012].
Until now, a large number of WDMS binaries [@raymond2003; @silvestri2007; @heller2009; @liu2012; @rebassa2012a] have been efficiently identified from the SDSS. @raymond2003 first attempted to study the WDMS pairs using SDSS data from the Early Data Release [@stoughton2002] and Data Release One [@abazajian2003]. They identified 109 WDMS pairs with $g <$ 20th magnitude. @silvestri2006 presented 747 spectroscopically identified WDMS binary systems from the SDSS Fourth Data Release [@adelman2006], and a further 1253 WDMS binary systems [@silvestri2007] from the SDSS Data Release Five [@adelman2007]. @heller2009 identified 857 WDMS binaries from the SDSS Data Release Six [@adelman2008] through a photometric selection method. @liu2012 identified 523 WDMS binaries from the SDSS Data Release Seven [DR7; @abazajian2009] based on their optical and near-infrared color-selection criteria. @rebassa2012a provided a final catalog of 2248 WDMS binaries identified from the SDSS DR7, among which approximately 200 strong PCEB candidates have been found [@rebassa2012a], and they further developed a publicly available interactive online database for these spectroscopic SDSS WDMS binaries.
With these cumulative rich SDSS WDMS binary samples and the identified PCEBs, some important work has been done to test binary population models as well as study the stellar structures, magnetic activity, common-envelope evolution, and other properties. For example, @nebot2011 pointed out that 21%–24% of SDSS WDMS binaries have undergone common-envelope evolution, which is in good agreement with the predictions of binary population models [@willems2004]. @rebassa2011 found that the majority of low-mass ($M\mathrm{_{wd} \lesssim}$ 0.5 $M_{\Sun}$) white dwarfs are formed in close binaries by investigating the white dwarf mass distributions of PCEBs. @zorotovic2012 studied the apparent period variations of eclipsing PCEBs and provided two possible interpretations: second-generation planet formation or variations in the shape of a magnetically active secondary star. @rebassa2013 investigated the magnetic-activity–rotation–age relations for M stars in close WDMS binaries and wide WDMS binaries using what is so far the largest and most homogeneous sample of SDSS WDMS binaries [@rebassa2012a]. They found that M dwarfs in wide WDMS binaries are younger and more active than field M dwarfs and the activity of M dwarfs in close binaries is independent of the spectral type.
As discussed above, several previous studies had developed color selection criteria based on the model colors of binary systems in order to select WDMS binaries from the SDSS [@szkody2002; @raymond2003; @smolcic2004; @eisenstein2006; @silvestri2006; @liu2012]. For example, @raymond2003 developed an initial set of photometric selection criteria ($u - g < 0.45$, $g -r < 0.70$, $r - i > 0.30$, $i - z > 0.40$ to a limiting magnitude of $g <$ 20) to identify binary systems using SDSS photometry and yielded reliable results. Here we present a sample of WDMS binaries also selected with a series of photometric criteria from the spectra of Large sky Area Multi-Object fiber Spectroscopic Telescope [LAMOST, also called the Guo Shou Jing Telescope, GSJT; @cui2012; @zhao2012] pilot survey [@luo2012; @yang2012; @carlin2012; @zhang2012; @chen2012]. We discuss the sample selection and provide stellar parameters of the individual components of some binary systems (e.g., effective temperature, surface gravity, radius, mass and cooling age for the white dwarf; effective temperature, surface gravity, metallicity, spectral type, radius, and mass of the M dwarf).
The structure of this paper is as follows. In Section 2, we describe our sample selection criteria. In Section 3, we present the method of the spectral decomposition of some our WDMS binaries and derive the parameters for the two constituents of the binaries. Section 4 contains the analysis of the spectroscopic parameters of the binaries and the possible PCEB candidates in our sample. Finally, a brief conclusion is provided in Section 5.
Identification of WDMS binaries in the LAMOST pilot survey {#sec:data}
==========================================================
Introduction of the LAMOST Pilot Survey
---------------------------------------
LAMOST (GSJT) is a quasi-meridian reflecting Schmidt telescope [@cui2012] located at the Xinglong Observing Station in the Hebei province of China. A brief description of the hardware and associated software of LAMOST can be found in @cui2012, which is a dedicated review and technical summary of the LAMOST project. The optical system of LAMOST has three major components: a Schmidt correcting mirror Ma (Mirror A), a primary mirror Mb (Mirror B), and a focal surface. The Ma (5.72 m $\times$ 4.40 m) is made up of 24 hexagonal plane sub-mirrors and the Mb (6.67 m $\times$ 6.05 m) has 37 hexagonal spherical sub-mirrors. Both are controlled by active optics. LAMOST has a field of view as large as 20 deg$^2$, a large effective aperture that varies from 3.6 to 4.9 m in diameter (depending on the direction it is pointing), and 4000 fibers installed on the circular focal plane with a diameter of 1.75 m. It has 16 spectrographs and 32 CCD cameras (each spectrograph equipped with two CCD cameras of blue and red channels), so there are 250 fiber spectra in each obtained CCD image. The up-to-date complete lists of LAMOST technical and scientific publications can be found at <http://www.lamost.org/public/publication>
The main aim of LAMOST is the extragalactic spectroscopic survey of galaxies (to study the large-scale structure of the universe) and the stellar spectroscopic survey of the Milky Way [to study the structure and evolution of the Galaxy; @cui2012]. Based on these scientific goals, the LAMOST survey mainly contains two parts: the LAMOST ExtraGAlactic Survey (LEGAS) and the LAMOST Experiment for Galactic Understanding and Exploration (LEGUE) survey of the Milky Way (see @deng2012 for the detailed science plan of LEGUE). It started a scientific spectroscopic survey of more than 10 million objects in 2012 September which will last for about five years. Before that, there was a two-year commissioning survey starting in 2009, and then a pilot spectroscopic survey performed using LAMOST from 2011 October to 2012 June. The LAMOST pilot survey is a test run of the telescope system to check instrumental performance and assess the feasibility of the science goals before the regular spectroscopic survey [@deng2012]. Based on the scientific goals proposed above, some scientists provided sources for LAMOST during the pilot survey. For example, @chen2012 describe the LEGUE disk target selection for the pilot survey. There are about 380 plates observed during the pilot survey, including the sources in the disk, spheroid, and anticenter of the Milky Way; M31 targets; galaxies; and quasars. Figure \[fig:footprint\] shows the footprint of the LAMOST pilot survey [@luo2012].
The LAMOST pilot survey includes nine full-Moon cycles during 2011 October and 2012 June (see @yao2012 for details of the site conditions of LAMOST). Each moon cycle of the pilot survey was divided into three parts, dark nights (five days sequentially before and after the new moon), bright nights (five days sequentially before and after the full moon) and gray nights (the remaining nights in the cycle). In principle, dark nights and bright nights should be used for sources with magnitude 14.5 $< r <$ 19.5 and 11.5 $< r <$ 16.5 respectively, and the corresponding exposure times were 3 $\times$ 30 minutes and 3 $\times$ 10 minutes (depending on the distribution of the brightness of the sources), respectively. The raw CCD data are reduced and analyzed by the LAMOST data reduction system, which mainly includes the two-dimensional (2D) pipeline (for spectra extraction and calibration) and the one-dimensional (1D) pipeline [to classify spectra and measure their redshifts; @luo2012]. What we want to mention is the flux calibration method of LAMOST, which is different from that of the SDSS. Because there is no network of photometric standard stars for LAMOST, @song2012 proposed a relative flux calibration method. They select standard stars (like SDSS F8 subdwarf standards) for each spectrograph field. After observation, if the pre-selected standard stars do not have good spectral quality, they use the Lick spectral index grid method (see @song2012 for details) to select new standard stars. Then the spectral response function for these standard stars is used to calibrate the raw spectra obtained from other fibers of the spectrograph [@song2012].
About one million spectra were obtained during the LAMOST pilot survey (see @luo2012 for the description of the data release of the LAMOST pilot survey). The LAMOST spectra have a low resolving power of $R \sim$ 2000 with wavelength ranging from 3800 $\mathrm{\AA}$ to 9000 $\mathrm{\AA}$. The LAMOST 1D FITS file is named in the form “spec-MMMMM-YYYY$\_$spXX-FFF.fits", where “MMMMM" represents the Modified Julian Date (MJD), “YYYY" is the plan (or plate) identity number, “XX" is the spectrograph identity number, and “FFF" is the fiber identity number. After the reduction of the 1D pipeline, the spectra are classified (flag “class") as either a galaxy, QSO, or star. For the spectra with low quality, we cannot guarantee that the classification given by the 1D pipeline is correct, so we give another flag “final$\_$class” based on the signal-to-noise ratio (S/N). For spectra with S/N $>$ 5, the spectral type of “final$\_$class" is the classification result of the 1D pipeline, while for the spectra with low S/N, the “final$\_$class" uses the visual inspection results. The flags “subclass" and “final$\_$subclass" are the spectral types of stars. The “subclass" is the classification result of the 1D pipeline and the definition of “final$\_$subclass" is the same as “final$\_$class".
WDMS Sample Selection
---------------------
Here we present a compilation of $\sim$ 28 WDMS binary systems from the LAMOST pilot survey. In order to search as many binaries as possible from the LAMOST pilot survey, we make use of the spectra with the “final$\_$class" flag “star" (644,540 stars) and the “final$\_$subclass" belonging to spectral type “M", “K", “WD", or “Binary". After performing these spectral type cuts, we obtained an initial sample set containing 207,596 stars. In order to remove stars that are unlikely WDMS binary candidates from our 207,596 star sample, we plan to reduce the sample by means of the photometric selection method. The LAMOST survey is a multi-object spectroscopic survey that does not conduct an imaging survey like the SDSS. Because of the diversity of the target selection of the pilot survey, not all stars in the LAMOST pilot survey have SDSS magnitudes (by cross matching with the SDSS DR9 photometric catalog we find that about 60% of the 644,540 stars have SDSS magnitudes), so we cannot use the SDSS $ugriz$ magnitude to do color selection and need to develop our own color selection criteria. We convolve the LAMOST spectra with the SDSS $ugriz$ filter response curves to obtain our own LAMOST magnitudes ($u^Lg^Lr^Li^Lz^L$) and colours ($u^L - g^L$, $g^L - r^L$, $r^L - i^L$, $i^L - z^L$).The LAMOST magnitudes we used in this paper are calculated by $$m = -2.5 \log_{10} (F \otimes R),$$ where, $m$ and $R$ represent the magnitude and response of the $ugriz$ filter and $F$ is the flux of the spectrum. For every observing night, we use the stars that have corresponding SDSS fiber magnitudes to fit a linear relationship between the SDSS fiber color and our convolved color to roughly calibrate our convolved color.
Because the SDSS WDMS binary catalog (2248 binaries) has a completeness of nearly $\gtrsim$ 98 $\%$ and represents what is so far the largest and most homogeneous sample set of WDMS binaries [@rebassa2012a], we convolve these 2248 binary spectra with the SDSS $ugriz$ responsing curves (the same as the above procedure of calculating LAMOST convolved color) to help us give the color-cut criteria for the LAMOST sample. Considering that the spectra in the $u^L$ and $z^L$ bands have very small overlap with the photometric $u$ and $z$ band, we neglect the $u^L$ and $z^L$ band photometry and only use $g^L - r^L$ versus $r^L - i^L$ to constrain the color selection criteria. The following are color-cuts we adopted in this paper: $$\begin{array}{lcllcl}
g - r & < & 0.45 + 1.59(r - i), & g -r & > & -0.61 + 0.01(r - i), \\
g - r & > & -1.92 + 1.84(r - i), & g - r & < & 2.02 - 0.38(r - i).
\end{array}$$
After applying these color-cut criteria (the area formed by the black lines in Figure \[fig:colorcut\]) to the 207,596 stars, our sample shrank to 90,079. We then searched for WDMS binary candidates among these 90,079 stars by eye check. The efficiency of the LAMOST blue spectrograph is much lower than the red one [@luo2012], so for the spectra with low S/Ns in the blue band, the Balmer line series are easily superimposed by the noise. But the red band of the spectra have clear M dwarf line features for most of the samples. Based on these considerations, we select the WDMS binary candidates that clearly exhibit an M dwarf in the red band and a resolved white dwarf component with Balmer lines in the blue band of their spectra. In order to identify binaries easily, we decompose the spectra using principal component analysis (PCA) (see @tu2010 for the details of the PCA decomposition). The WDMS binary spectrum is a mixture of a white dwarf spectrum and a M dwarf spectrum, $$D \approx \sum _{i=1} ^{m} a_i w_i + \sum _{j=1} ^{n} a_{m+j} m_j,$$ where, $D$ is the mixed spectrum, $w_i (i=1, 2, \dots, m)$ and $m_j (j=1, 2, \dots, n)$ represent the $m$ white dwarf eigen-spectra, and the $n$ M dwarf eigen-spectra we used to do spectral decomposition, respectively, and $a_i (i=1, 2, \dots, m+n)$ are the coefficients of eigen-spectra. At first, we construct four PCA eigen-spectra ($m=4$) from the white dwarf model spectral library and 12 PCA eigen-spectra ($n=12$) from the M dwarf model spectral library (the details of the white dwarf and M dwarf models can be found in Section 3). Then we determine all the coefficients of the eigen-spectra by the SVD (singular value decomposition) matrix decomposition and orthogonal transformation. After the coefficients of the eigen-spectra have been calculated, the spectrum can be decomposed into two components. By visual investigation of the decomposed spectra of the 90,079 stars decomposed by this PCA method, 33 WDMS binaries from the LAMOST pilot survey are found, of which 3 binaries have been observed more than once by LAMOST (J101616.82+310506.5 has been observed four times, J111035.16+280733.2 and J085900.86+493519.8 have each been observed twice). After eliminating the five duplicated spectra, our final sample includes 28 WDMS binaries (listed in Table 1) with only one DB white-dwarf–M-star binary (J224609.42$+$312912.2). Nine binaries of our WDMS sample have been identified by @rebassa2012a from the SDSS DR7.
Our 28 WDMS binaries show two components in their spectra. In order to ensure the reliability of our binary sample, additional investigations are carried out. For every binary pair in our sample, we check the spectra of its neighboring four fibers to see if there are fiber cross-contaminations brought about by the 2D data reduction pipeline [@luo2012]. For example, the neighboring spectra of the binary spectrum spec-55859-F5902$\_$sp08-169.fits (J221102.56$-$002433.5 ) are spec-55859-F5902$\_$sp08-167.fits, spec-55859-F5902$\_$sp08-168.fits, spec-55859-F5902$\_$sp08-170.fits, and spec-55859-F5902$\_$sp08-171.fits. Their corresponding spectral types are SKY, A0, K7, and F5, respectively. The white dwarf component of this binary has very wide Balmer lines and may not be contaminated by an A0 type neighborhood. Meanwhile, the secondary component shows obvious molecule absorption band features of a M-type star, which also cannot be contaminated by the neighboring K7-type star. So this binary may not suffer from fiber contaminations. For other binaries, if their neighboring spectra have the same spectral types as their binary components, we will use the 2D image for further inspection. We do this check for each binary of our data and find that there are no fiber contaminations in our binary sample.
Table 1 lists the colors of our convolved magnitudes and photometry for our sample cross matched with SDSS DR9 [@ahn2012], 2MASS [@skrutskie2006], and GALEX [@morrissey2007]. The Modified Julian Date (MJD), the plate identity number (PLT), the spectrograph identity number (SPID), and the fiber identity number (FIB) are also provided in Table 1. Figure \[fig:spectra\] shows the 10 spectra of our binary sample on which we do spectral analysis in Section 3 (For the WDMS binaries that have multiple spectra, we only select one spectrum that has relatively good spectral quality to do the spectral analysis). Figure \[fig:coords\] shows the positions of our WDMS binary sample, the SDSS DR7 binaries [@rebassa2012a], and all stars of the LAMOST pilot survey in the Galactic and equatorial coordinates.
Stellar parameters
==================
To investigate the properties of the individual components of our WDMS binary systems, we adopt the template-matching method based on model atmosphere calculations for white dwarfs and M main-sequence stars to separate the two stellar spectra and derive parameters for each system. We only give stellar parameters for the 10 WDMS spectra with relatively high spectral quality (S/N $>$ 5). With the exception of one DB WDMS binary, the remaining 17 binaries not analyzed in this paper are either too noisy or insufficiently flux calibrated for a reasonable spectral analysis. After a good follow-up spectroscopic survey, we will provide stellar parameters for these binaries. The models we used for spectral decomposition and fitting and the parameter estimations for the two constituents are described in the following subsections.
Models
------
The theoretical model grids we used provide a five-dimensional parameter space $(\mathrm{T_{eff}^{WD}}$, $\mathrm{log (g_{WD})}$, $\mathrm{T_{eff}^M}$, $\mathrm{log(g_M)}$, $\mathrm{[Fe/H]_M})$ of white dwarfs and M main-sequence stars. Since most WDMS binaries contain a DA white dwarf, the white dwarf model spectra we used were calculated for pure hydrogen atmospheres (DA white dwarfs) based on the model atmosphere code described by @koester2010, which covers surface gravities of 7$\leq \mathrm{log}(g_\mathrm{WD}) \leq$ 9 with a step size of 0.25 dex and effective temperatures of 6000 K $\leq T_{\mathrm{eff}}^{\mathrm{WD}} \leq$ 90000 K. The step is 500 K, 1000 K, 2000 K, and 5000 K, respectively, when the effective temperatures are in the range of $[$6000 K, 15000 K$]$, $[$15000 K, 30000 K$]$, $[$30000 K, 50000 K$]$, and $[$50000 K, 90000 K$]$. The MARCS models [@gustafsson2008] are used as our MS model grids with a surface gravity range of 3.5 $\leq \mathrm{log}(g_\mathrm{M}) \leq$ 5.5 with step 0.5 dex, the effective temperature range 2500 K $\leq T_{\mathrm{eff}}^{\mathrm{M}} \leq$4000 K with step 100 K, and a metallicity of $\mathrm{[Fe/H]_M} \in \lbrace$ $-$2.0, $-$1.5, $-$1.0, $-$0.75, $-$0.5, $-$0.25, 0.0, 0.25, 0.5, 0.75, 1.0 $\rbrace$.
Methods
-------
We use a template-matching method like @heller2009 to decompose a WDMS binary spectrum into a white dwarf and a MS star simultaneously and derive independent parameters for each component. This method [@heller2009] is able to avoid a mutual dependence of the two scaling factors for WD and M stars since the system of equations can be solved uniquely and also avoids the effects of identifying a local $\chi ^2$ minimum. Our template matching method is not a bona fide weighted $\chi ^2$ minimization technique because we do not divide by $\sigma _i ^2$ (the observational error) in the $\chi ^2$ function of @heller2009 [Equation(3)]. By comparing the fitting method divided by $\sigma _i ^2$ with the method that is not, we note that the template-fitting method that does not divide by $\sigma _i ^2$ gives much better results. The main reason for this is that the observational error given by the LAMOST “.fits" file is the inverse-variance of the flux and may not exactly represent the uncertainty of the flux, so it will distinctly affect the $\chi ^2$ fitting results.
Additionally, the flux calibration of the LAMOST spectra is relative (mentioned in Section 2, or see details in [@song2012]), which means the reddening of the standard stars may affect the flux calibration, especially when the standard stars have very different spacial positions (or reddening) from other stars in the same spectrograph. This may lead to some difference in the continuum between the observational spectra and model spectra. Considering the complexity of reddening, to simplify we empirically incorporate quintic polynomials to Heller’s template-matching method in order to overcome the possible failure of the fit. Therefore, the definition of $\chi^2$ changes to $$\chi ^2 = \sum _{i} ^{n} {\left(F_i - \left(\sum\limits_{j=0}^{5} P_j x_i ^j \right)w_i - \left(\sum\limits_{j=0}^{5} Q_j x_i ^j \right) m_i \right)^2},$$ where $F_i$, $w_i$, and $m_i$ are the observed flux, white dwarf model flux, and M dwarf model flux for each data point in a binary spectrum with a total number of $n$ observed data points. $x_i$ is the wavelength of the spectrum. The quintic polynomials we incorporated are $\sum\limits_{j=0}^{5} P_j x_i ^j$ and $\sum\limits_{j=0} ^{5} Q_j x _i ^j$. Using this revised template-fitting method, we estimate the stellar parameters of the white dwarf component $\lbrace T\mathrm{_{eff}^{WD}}$, $\mathrm{log (}g\mathrm{_{WD})}\rbrace$ and its M-type MS companion $\lbrace T\mathrm{_{eff}^M}$, $\mathrm{log(}g\mathrm{_M)}$, $\mathrm{[Fe/H]_M}\rbrace$ for our binaries. To verify that the quintic polynomials are necessary, we incorporate polynomials of different orders (from zero to five) into our fitting routine. For comparison, we calculate the reduced $\chi ^2$ ($\chi ^2 _{\mathrm{red}}$) like @heller2009. Figure \[fig:poly\] shows the $\chi ^2 _{red}$ distribution for the fitting routine with different order polynomials. It seems that the $\chi^2 _{\mathrm{red}}$ converges when we incorporate five-order polynomials into the fitting routine. An example of a typical WDMS spectrum in our sample and its spectral decomposition are shown in Figure \[fig:example\].
As @heller2009 mentioned, the quality of this spectra-fitting technique is poor in a mathematical context and the standard deviations of the measured parameters are quite weak in terms of physical significance. That means that the unknown systematic errors (due to the incomplete molecular data of the M star model, flux calibration errors, and possible interstellar reddening) of the stellar parameters are larger than the mathematical ones. For a conservative estimate of errors in the measured parameters, we refer to @hugelmeyer2006 and assume an uncertainty of half the model step width for the WDs and MS stars. We assume an uncertainty of 2000 K for WDs with effective temperatures of less than 50,000 K and 100 K for the MS stars. While for $T\mathrm{_{eff} ^{WD}} > 50,000$ K, the uncertainty is given by half of the model step width of 5000 K. Limited by the low resolution of the LAMOST spectra and the step size of our model grid, the accuracy of the surface gravities is given as $\sigma_{\mathrm{log}(g_{\mathrm{WD}})} \approx 0.25$ dex and $\sigma_{\mathrm{log}(g_{\mathrm{M}})} \approx 0.5$ dex. Like @heller2009, we also assume an uncertainty of 0.3 dex for metallicity below $-$1.0 dex. For metallicity over $-$1.0 dex, the corresponding uncertainty is given by half of the model step width of 0.25 dex.
Because the flux-calibration of LAMOST spectra are relative, it is impossible for us to derive the distances of the two components of our WDMS binaries from the best-fitting flux scaling factors that scale the model flux to the observed flux. Once $T\mathrm{_{eff} ^{WD}}$ and $\mathrm{log}(g_{\mathrm{WD}})$ are determined, we estimate the cooling ages, masses and radii for the white dwarfs by interpolating the detailed evolutionary cooling sequences [@wood1995; @fontaine2001]. We use the carbon-core cooling models [@wood1995] with thick hydrogen layers of $q_{H} = M_{H}/M_{*} = 10^{-4}$ for pure hydrogen model atmospheres with effective temperature above 30,000 K. For $T_{eff}$ below 30,000 K, we use cooling models similar to those described in @fontaine2001 but with carbon-oxygen cores and $q_{H} = 10^{-4}$. Additionally, we derive the masses and radii for M-type companions using the empirical effective temperature – spectral type ($T_{\mathrm{eff}}$ – Sp), spectral type – mass (Sp – M) and spectral type –radius (Sp – $R$) relations presented in @rebassa2007. These spectroscopic parameters are listed in Table 2.
Results and Discussion {#sec:results}
======================
Analysis of Stellar Parameters
------------------------------
Our WDMS binary sample is not complete and we only provide stellar parameters for 10 binaries, so the statistical distribution analysis of parameters is not given. The white dwarf effective temperatures are between 21,000 K and 48,000 K and have a mean value 29,900 K. This may suggest that this WDMS binary sample tends to have hot white dwarfs. The mean value of white dwarf surface gravities is around 8.0, which is consistent with the peak value of surface gravity distribution of @rebassa2012a. @yi2013 give an M dwarf catalog from the LAMOST pilot survey, for which the spectral type has a peak around M1 $\sim$ M2. The spectral types of the secondary stars of our WDMS binaries also cluster together around M1.5. All of these imply that our sample favours binaries with hot white dwarfs and early-type companion stars. The white dwarf masses of the 10 binaries for which we provided stellar parameters tend to be higher than the typical peak of WD mass $\sim$ 0.6 $M_\Sun$ [@tremblay2011]. That may be because of the selection effect and the small sample size. Our binary sample is still not large enough for parameter distribution analysis and is waiting to be enlarged by the ongoing LAMOST formal survey.
Radial Velocities
-----------------
For the binary spectra that exhibit the resolved spectral H$\alpha$ $\lambda$ 6564.61 emission line and [Na[I]{}]{} doublet $\lambda\lambda$ 8183.27, and 8194.81 lines, we try to derive radial velocities by fitting the H$\alpha$ emission line with a Gaussian line profile plus a second order polynomial as well as fitting the [Na[I]{}]{} doublet with a double Gaussian profile of fixed separation and a second order polynomial [@rebassa2007]. The total error of the radial velocities is computed by quadratically adding the uncertainty of the LAMOST wavelength calibration [10 km s$^{-1}$; see @luo2012] and the error in the position of the H$\alpha$/[Na[I]{}]{} lines determined from the Gaussian fits. The LAMOST spectra are generally combined from three exposures (which we call ‘subspectra’ following @rebassa2007), with each exposure lasting for either 30 minutes or 15 minutes (see Section 2). We then measure radial velocities for these LAMOST subspectra and the combined spectra. Last, we derive radial velocities for 17 binaries (see Table 3), of which J085900.86+493519.8 and J101616.82+310506.5 have been observed two and four times, respectively, by LAMOST. Figure \[fig:rv\] shows the fitting results of the H$\alpha$ $\lambda$ 6564.61 emission line and the [Na[I]{}]{} doublet $\lambda\lambda$ 8183.27, and 8194.81 lines of two LAMOST spectra.
PCEB Candidates
---------------
From Table 3, 10 binaries have been measured with multiple radial velocities (including the radial velocities measured from LAMOST subspectra and provided by the SDSS WDMS binary catalog [@rebassa2012a]). Following @rebassa2010 and @nebot2011, we consider those systems showing radial velocity variations with 3$\sigma$ significance to be strong PCEB candidates. We used a $\chi^2$ test with respect to the mean radial velocity for the detection of radial velocity variations. If the probability $Q$ that the $\chi^2$ test returns is below 0.0027 (meaning the probability $P(\chi^2)$ of a system showing large radial velocity variation is above 0.9973 where $P(\chi^2)=1-Q$), we can say that we detect 3$\sigma$ radial velocity variation and the corresponding WDMS binary can be considered a strong PCEB candidate. Here we find two PCEB candidates, LAMOST J105421.88$+$512254.1 and LAMOST J122037.01$+$492334.0, using the [Na[I]{}]{} doublet radial velocities. LAMOST J105421.88$+$512254.1 shows 3$\sigma$ radial velocity variation using either [Na[I]{}]{} doublet or H$\alpha$ emission, while for LAMOST J122037.01$+$492334.0, we detect 3$\sigma$ radial velocity variation using the [Na[I]{}]{} doublet. By cross-referencing 195 PCEB candidates provided by @rebassa2012a, we find that LAMOST J105421.88$+$512254.1 has already been identified as a PCEB candidate. Limited by the sample size, we only find one new PCEB candidate: LAMOST J122037.01$+$492334.0.
We estimate the upper limits to orbital periods for these two PCEB candidates in the same way as described in @rebassa2007. Because we do not provide stellar parameters for these two binaries, the white dwarf masses and the secondary star masses are taken from @rebassa2012a. The radial velocity amplitudes of the secondary stars are obtained by using the [Na[I]{}]{} doublet radial velocities (Table 3). Table 4 presents the probability $P(\chi^2)$ of measuring large radial velocity variations for these two binaries and the calculated upper limits to their orbital periods. Both PCEB candidates need intense follow-up spectroscopic observations in order to obtain orbital periods.
Conclusions {#sec:conclusion}
===========
We have presented a catalog of 28 WDMS binaries from the spectroscopic LAMOST pilot survey in this paper. Using the colors of the 2248 binaries from the SDSS WDMS binary catalog, we develop our own color selection criteria for the selection of WDMS binaries from the LAMOST pilot survey based on the LAMOST magnitudes obtained by convolving the LAMOST spectra with the SDSS $ugriz$ filter response curve. This method is efficient for searching for binaries in the spectroscopic survey without having our own optical or infrared photometric data equipment like the LAMOST survey. Using the color selection criteria, we identify 28 WDMS binaries from the LAMOST pilot survey. Nine of these binaries have been published in previous works and 19 of these WDMS binaries are new. For 10 of our binaries, we have used a $\chi^2$ minimization technique to decompose the binary spectra and determine the effective temperatures, surface gravities of the white dwarfs, as well as the effective temperatures, surface gravities, and metallicities for the M-type companions. The cooling ages, masses and radii of white dwarfs are provided by interpolating the white dwarf cooling sequences for the derived effective temperatures and surface gravities. We also derive the spectral types, masses, and radii of the M stars by using empirical spectral type – effective temperature, spectral type – mass, and spectral type – radius relations. In addition, the radial velocities are measured for most of our sample. We also discuss the possible PCEB candidates among the binary systems with multiple spectra in our sample, finally giving two possible PCEB candidates, one of which has already been identified as a PCEB candidate by the SDSS.
The WDMS binary catalog is the first provided by the LAMOST survey, and demonstrates the capability of LAMOST to search for WDMS binaries. With the ongoing formal LAMOST survey, we hope to find many more WDMS binaries and present a more complete catalog of WDMS binaries, which will increase the number of known WDMS binary systems. Enlarging the WDMS binary sample will lead to a deeper understanding of close compact binary star evolution and other follow-up studies.
We thank the anonymous referee for very useful comments and suggestions that greatly improved this paper. We thank Alberto Rebassa-Mansergas for the discussion about the detection of PCEBs. We also thank Detlev Koester for kindly providing the white dwarf atmospheric models and Bengt Edvardsson for useful discussion of the MARCS models. We acknowledge the Web site <http://www.astro.umontreal.ca/~bergeron/CoolingModels> for providing the white dwarf cooling sequences. We thank George Comte for the discussion of M dwarf models and Xiaoyan Chen, Yue Wu, and Fang Zuo for discussion. This study is supported by the National Natural Science Foundation of China under grant Nos. 10973021, 11078019, and 11233004. The Guo Shou Jing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences. The LAMOST Pilot Survey Web site is <http://data.lamost.org/pdr>.
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[lccrrrrrrccccccccccc]{} J014141.83$+$311852.9 & 25.42433 & 31.3147 & 55918 & M31$\_$025N30$\_$M1 & sp15 & 033 & 0.660554 & 0.270636 & 19.393 & 18.429 & 17.53 & 16.446 & 15.826 & 14.522 & 13.816 & 13.619 & 19.521 & 19.581 &\
J025306.36$+$001329.6 & 43.2765 & 0.22491 & 55859 & F5907 & sp14 & 137 & 0.509917 & 0.248582 & 19.359 & 18.916 & 19.308 & 19.818 & 20.025 & 14.173 & 13.546 & 13.272 & 19.601 & 19.429 & e,r\
J052529.17$+$283705.4 & 81.37156 & 28.61819 & 55859 & F5909 & sp04 & 114 & 0.360057 & 0.324585 & ... & ... & ... & ... & ... & 15.354 & 14.596 & 14.368 & ... & ... &\
J052531.26$+$283807.6 & 81.38025 & 28.63545 & 55951 & GAC$\_$080N28$\_$M1 & sp04 & 114 & 0.280857 & 0.361304 & ... & ... & ... & ... & ... & 13.401 & 12.77 & 12.553 & ... & ... &\
J052531.33$+$284549.4 & 81.38058 & 28.76375 & 55859 & F5909 & sp04 & 120 & 0.387878 & 0.332709 & ... & ... & ... & ... & ... & 15.583 & 15.027 & 14.71 & ... & ... &\
J073128.30$+$264353.8 & 112.86793 & 26.73163 & 55921 & GAC$\_$113N28$\_$M1 & sp01 & 044 & 0.510806 & 0.327412 & 18.46 & 17.772 & 17.248 & 16.204 & 15.498 & 14.049 & 13.459 & 13.283 & 19.6 & 18.689 &\
J081934.63$+$024802.7 & 124.8943 & 2.80077 & 55921 & F5592103 & sp08 & 044 & 0.631215 & 0.293536 & 17.721 & 17.255 & 16.561 & 15.686 & 15.074 & 13.708 & 13.076 & 12.897 & ... & ... &\
J081959.21$+$060424.2 & 124.996713 & 6.073391 & 55892 & F9205 & sp06 & 053 & 0.444555 & 0.446157 & 19.665 & 19.433 & 19.456 & 18.811 & 18.154 & 16.58 & 16.357 & 15.847 & 19.188 & 19.421 & r\
J084006.40$+$143025.9 & 130.02669 & 14.50722 & 55977 & F5597707 & sp01 & 145 & 0.304351 & 0.269564 & 18.048 & 18.109 & 17.996 & 17.318 & 16.819 & 15.447 & 14.934 & 14.594 & ... & ... &\
J085900.86$+$493519.8 & 134.75359 & 49.58885 & 56021 & VB3$\_$136N49$\_$V1 & sp03 & 087 & $-$0.034498 & 0.182911 & 16.466 & 15.049 & 13.9 & 13.297 & 12.539 & 11.251 & 10.577 & 10.364 & 16.736 & 16.678 &\
J101616.82$+$310506.5 & 154.07012 & 31.08516 & 55907 & B90705 & sp03 & 065 & 0.634328 & 0.283063 & 17.38 & 16.751 & 16.167 & 15.317 & 14.787 & 13.47 & 12.885 & 12.698 & 17.296 & 17.241 &\
J103514.80$+$390717.7 & 158.8117 & 39.1216 & 55930 & F5593001 & sp04 & 124 & 0.425026 & 0.208033 & 18.423 & 18.178 & 17.919 & 17.301 & 16.859 & 15.565 & 15.034 & 14.849 & 17.768 & 18.07 & s,r\
J105405.25$+$283916.7 & 163.52188 & 28.65466 & 55931 & B5593104 & sp16 & 214 & 0.519861 & 0.284815 & 17.824 & 17.146 & 16.645 & 15.477 & 14.657 & 13.202 & 12.569 & 12.339 & 18.024 & 17.794 & r\
J105421.88$+$512254.1 & 163.59118 & 51.38171 & 55915 & F5591506 & sp11 & 205 & 0.526326 & 0.275433 & 17.258 & 16.778 & 16.134 & 15.271 & 14.729 & 13.455 & 12.813 & 12.575 & 16.46 & 16.817 & s,r\
J111035.16$+$280733.2 & 167.64653 & 28.1259 & 55931 & B5593104 & sp13 & 180 & 0.687018 & 0.232178 & 17.786 & 17.161 & 16.427 & 15.554 & 15.024 & 13.729 & 13.099 & 12.911 & 17.371 & 17.506 &\
J112512.36$+$290545.4 & 171.30153 & 29.09597 & 56018 & B5601803 & sp02 & 195 & 0.228522 & 0.345777 & 18.817 & 18.215 & 17.521 & 16.481 & 15.856 & 14.531 & 13.907 & 13.635 & 18.474 & 18.611 & r\
J114732.25$+$593921.9 & 176.8844 & 59.65611 & 55931 & F5593101 & sp11 & 001 & 0.424084 & 0.243105 & 17.127 & 16.796 & 16.474 & 15.514 & 14.882 & 13.485 & 12.832 & 12.635 & 16.682 & 16.783 &\
J120024.56$+$292310.3 & 180.10235 & 29.3862 & 55961 & B5596106 & sp02 & 085 & 0.432907 & 0.216112 & 20.275 & 17.715 & 16.386 & 15.665 & 15.216 & 14.068 & 13.44 & 13.273 & ... & ... &\
J122037.01$+$492334.0 & 185.1542208 & 49.3927944 & 55923 & F5592306 & sp03 & 077 & 0.63171 & 0.255231 & 18.719 & 18.14 & 17.417 & 16.809 & 16.405 & 15.172 & 14.505 & 14.396 & ... & ... & s,r\
J125555.30$+$560033.0 & 193.98043 & 56.00917 & 55952 & F5595206 & sp04 & 123 & 0.623711 & 0.288381 & 22.695 & 20.013 & 18.701 & 17.558 & 16.898 & 15.682 & 15.063 & 15.027 & ... & ... &\
J131208.11$+$002058.0 & 198.0338083 & 0.3494583 & 55932 & F5593201 & sp16 & 071 & 0.0403807 & 0.232714 & 18.79 & 18.472 & 18.35 & 17.564 & 16.927 & 15.625 & 15.029 & 14.787 & 18.247 & 18.53 & r,c\
J132417.76$+$280755.8 & 201.07402 & 28.13217 & 55975 & B5597505 & sp01 & 195 & 0.477785 & 0.203373 & 17.477 & 16.903 & 16.524 & 15.593 & 14.979 & 13.57 & 12.941 & 12.658 & 17.325 & 17.378 &\
J135635.32$+$084841.7 & 209.1472 & 8.81159 & 56062 & VB3$\_$210N09$\_$V2 & sp10 & 161 & 0.544101 & 0.38414 & 17.524 & 16.798 & 16.057 & 15.021 & 14.403 & 13.042 & 12.416 & 12.169 & 17.387 & 17.417 &\
J150626.53$+$275925.2 & 226.61058 & 27.99036 & 56063 & B5606303 & sp09 & 122 & $-$0.260791 & 0.260247 & 18.258 & 18.177 & 18.053 & 17.5 & 17.043 & 15.852 & 15.253 & 14.698 & 17.347 & 17.602 & r\
J162617.40$+$384029.9 & 246.57251 & 38.67499 & 55999 & B5599908 & sp15 & 040 & 0.253555 & 0.159056 & 17.361 & 16.938 & 16.697 & 15.7 & 15.111 & 12.513 & 11.947 & 11.663 & ... & ... &\
J221102.56$-$002433.5 & 332.76069 & $-$0.40933 & 55859 & F5902 & sp08 & 169 & 0.477924 & 0.309289 & 18.634 & 18.307 & 17.941 & 16.993 & 16.415 & 15.154 & 14.501 & 14.352 & 18.213 & 18.466 &\
J224609.42$+$312912.2 & 341.53925 & 31.48674 & 55863 & B6302 & sp09 & 187 & 0.235289 & 0.387328 & 16.952 & 15.671 & 15.19 & 15.018 & 14.936 & 14.11 & 13.728 & 13.584 & ... & 20.489 &\
J232004.01$+$270623.7 & 350.01673 & 27.10659 & 55878 & B87802$\_$1 & sp15 & 083 & $-$0.305056 & 0.0273775 & 15.852 & 16.072 & 16.378 & 16.168 & 15.811 & 14.609 & 14.074 & 13.783 & 14.747 & 15.203 &
[lccccccccccc]{} J014141.83$+$311852.9 & 21000 & 8.25 & 3700 & 4.0 & 0.00 & 0.78 & 101.6 & 0.011 & 1.0 & 0.464 & 0.480\
J081934.63$+$024802.7 & 32000 & 7.00 & 3600 & 3.5 & $-$0.25 & 0.33 & 5.0 & 0.030 & 1.5 & 0.450 & 0.465\
J084006.40$+$143025.9 & 38000 & 8.50 & 3500 & 4.0 & $-$0.50 & 0.96 & 16.4 & 0.009 & 2.0 & 0.431 & 0.445\
J085900.86$+$493519.8 & 32000 & 8.50 & 3500 & 3.5 & $-$0.75 & 0.95 & 36.4 & 0.009 & 2.0 & 0.431 & 0.445\
J101616.82$+$310506.5 & 25000 & 8.25 & 3600 & 4.5 & $-$0.25 & 0.78 & 50.1 & 0.011 & 1.5 & 0.450 & 0.465\
J111035.16$+$280733.2 & 24000 & 7.75 & 3700 & 4.0 & $-$0.25 & 0.51 & 18.9 & 0.016 & 1.0 & 0.464 & 0.480\
J114732.25$+$593921.9 & 22000 & 8.25 & 3700 & 5.5 & 0.50 & 0.78 & 84.5 & 0.011 & 1.0 & 0.464 & 0.480\
J120024.56$+$292310.3 & 48000 & 7.75 & 3700 & 4.5 & $-$0.25 & 0.58 & 2.3 & 0.017 & 1.0 & 0.464 & 0.480\
J221102.56$-$002433.5 & 23000 & 8.50 & 3600 & 5.0 & 0.25 & 0.94 & 126.2 & 0.009 & 1.5 & 0.450 & 0.465\
J232004.01$+$270623.7 & 34000 & 7.50 & 3300 & 4.0 & 0.00 & 0.45 & 5.0 & 0.020 & 3.5 & 0.350 & 0.359
[lrccccc]{} J014141.83$+$311852.9 & 2455918.0249 & ... & ... & 13.5 & 13.1 & a\
J052531.26$+$283807.6 & 2455951.0953 & 60.7 & 12.6 & 18.6 & 13.7 & a\
J073128.30$+$264353.8 & 2455921.2338 & ... & ... & $-$102.4 & 15.0 & a\
J081934.63$+$024802.7 & 2455921.3048 & 42.9 & 12.7 & 23.7 & 11.7 & a\
& 2455921.2920 & 49.1 & 11.9 & 23.3 & 11.1 & b\
& 2455921.3176 & 42.0 & 12.9 & 21.9 & 12.0 & b\
J084006.40$+$143025.9 & 2455977.1160 & 48.7 & 14.1 & 64.9 & 14.5 & a\
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& 2456021.0742 & ... & ... & 1.9 & 10.4 & b\
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J101616.82$+$310506.5 & 2455907.3713 & $-$16.6 & 12.7 & $-$3.8 & 16.3 & a\
& 2455907.3624 & $-$11.2 & 11.3 & $-$0.3 & 11.0 & b\
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J105405.25$+$283916.7 & 2455931.3041 & 10.3 & 12.1 & ... & ... & a\
& 2455931.2880 & 3.3 & 12.9 & 10.6 & 11.3 & b\
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& 2453826.8103 & 30.0 & 15.0 & 32.4 & 10.4 & c\
& 2453826.7958 & 35.2 & 15.2 & ... & ... & c\
& 2453826.8086 & 35.2 & 15.1 & ... & ... & c\
& 2453826.8231 & 33.3 & 15.2 & ... & ... & c\
& 2454887.2906 & 33.7 & 14.8 & 25.1 & 10.1 & c\
J105421.88$+$512254.1 & 2455915.3897 & 33.0 & 15.4 & 23.0 & 12.9 & a\
& 2455915.4032 & 6.8 & 12.0 & 26.9 & 14.2 & b\
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& 2452669.8635 & 34.8 & 10.5 & ... & ... & c\
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J111035.16$+$280733.2 & 2455931.3041 & ... & ... & $-$13.1 & 14.9 & a\
J112512.36$+$290545.4 & 2456018.1430 & $-$117.2 & 18.3 & ... & ... & a\
& 2453794.9024 & $-$52.0 & 15.3 & $-$41.1 & 10.8 & c\
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J114732.25$+$593921.9 & 2455931.3750 & ... & ... & $-$121.0 & 10.9 & a\
& 2455931.3485 & ... & ... & $-$115.9 & 16.6 & b\
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J135635.32$+$084841.7 & 2456062.1763 & ... & ... & 23.3 & 15.1 & a\
J162617.40$+$384029.9 & 2455999.3847 & $-$25.5 & 11.5 & $-$16.4 & 10.9 & a\
& 2455999.3693 & $-$22.0 & 13.4 & $-$18.7 & 10.9 & b\
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J224609.42$+$312912.2 & 2455863.0592 & $-$18.0 & 10.7 & $-$3.6 & 10.7 & a\
& 2455863.0417 & $-$18.3 & 10.8 & $-$12.6 & 10.9 & b\
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& 2455863.0767 & $-$8.3 & 11.1 & $-$7.8 & 26.2 & b\
J232004.01$+$270623.7 & 2455878.0210 & $-$87.9 & 13.7 & ... & ... & a\
[lrcccccc]{} J105421.88$+$512254.1 & 1.00000 & 0.67 & 0.38 & 2.71\
J122037.01$+$492334.0 & 0.99998 & 0.42 & 0.464 & 2.96\
|
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\[appendixc\] \[subappendixc\]
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**Evidence for “sterile neutrino” dark matter?**
Paolo Gondolo
*Max Planck Institut für Physik, Föhringer Ring 6*
*80805, Munich, Germany*
E-mail: [email protected]
[*Talk presented at the Ringberg Euroconference “New Trends in Neutrino Physics,” Ringberg Castle, Tegernsee, Germany, 24–29 May 1998.*]{}
**Evidence for “sterile neutrino” dark matter?**
Paolo Gondolo
*Max Planck Institut für Physik, Föhringer Ring 6*
*80805, Munich, Germany*
E-mail: [email protected]
Introduction
============
A sophisticated analysis of EGRET data (Dixon et al. 1998) has found evidence for gamma-ray emission from the galactic halo. Filtering the data with a wavelet expansion, Dixon et al. have subtracted an isotropic extra-galactic component and expected contributions from cosmic ray interactions with the interstellar gas and from inverse Compton of ambient photons by cosmic ray electrons, and they have produced a map of the intensity distribution of the residual gamma-ray emission. Besides a few “point” sources, they find an excess in the central region extending somewhat North of the galactic plane, and a weaker emission from regions in the galactic halo. They mention an astrophysical interpretation for this halo emission: inverse Compton by cosmic ray electrons distributed on larger scales than those commonly discussed and with anomalously hard energy spectrum.
I find it intriguing that the angular distribution of the halo emission resembles that expected from pair annihilation of dark matter WIMPs in the galactic halo (Gunn et al. 1978, Turner 1986), and moreover, that the gamma-ray intensity is similar to that expected from annihilations of a thermal relic with present mass density 0.1–0.2 of the critical density (Gondolo 1998a). Namely, the emission at $ b \ge 20^\circ $ is approximately constant at a given angular distance from the galactic center – except for a region around $(b,l) = (60^\circ,45^\circ) $, correlated to the position of the Moon, and a region around $(b,l) = (190^\circ, -30^\circ) $, where there is a local cloud ($b$ and $l$ are the galactic latitude and longitude, respectively). Dixon et al. (1998) argue against the possibility of WIMP annihilations on the base that direct annihilation of neutralinos into photons would give too low a gamma-ray signal. However most of the photons from WIMP annihilations are usually not produced directly but come from the decay of neutral pions generated in the particle cascades following annihilation.
Presently preferred dark matter candidates tend to give a gamma emission which is too low even in the continuum.[^1] In a search for a suitable candidate it is worth examining the impact of various constraints on the properties of the candidate. I will later introduce a working model, so to make the discussion more concrete.
A candidate WIMP can be a thermal relic.
========================================
It is useful to use the WIMP mass $m_\chi$ and its annihilation cross section (times relative velocity at $v=0$) $\sigma v$ as parameters in the discussion. The requirement to approximately match the Dixon et al. maps selects a band in the $\sigma v$–$m_\chi$ plane (band marked “halo $\gamma$’s” in fig. 1). Another band is selected by the requirement that the WIMP is a thermal relic from the early universe, with a relic abundance in the range $ 0.025 < \Omega h^2 < 1$ (band marked “$ 0.025 < \Omega h^2 < 1$” in fig. 1). The intersection of the two bands defines the interesting region.
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\[fig:1\]
The gamma-ray band is obtained as follows. The gamma-ray intensity from WIMP annihilations in the galactic halo from direction with galactic longitude $b$ and galactic latitude $l$ is given by $$\label{phigamma}
\phi_{\gamma}(b,l,\mathord{>}E) = n_{\gamma}(\mathord{>}E) \,
\frac{ \sigma v }{ 4 \pi m_\chi^2 } \, \int \rho_\chi^2 d l .$$ $ \phi_{\gamma}(b,l,\mathord{>}E) $ is in photons/(cm$^2$ s sr), $n_{\gamma}(\mathord{>}E)$ is the number of photons with energy above $E$ generated per WIMP annihilation, $ \sigma v$ is the WIMP annihilation cross section times relative velocity, $m_\chi$ is the WIMP mass, and $ \rho_\chi $ is the WIMP mass density in the halo. The integral in eq. (\[phigamma\]) is along the line of sight in direction $(b,l)$. This integral gives the angular dependence of the gamma-ray flux and depends on details of the dark halo model, which are not well known. For a canonical halo model, $ \rho(r) = \rho_{\rm
loc} ( r_c^2 + R^2 )/( r_c^2 + r^2 ) $, where $ \rho_{\rm loc} $ is the WIMP mass density in the solar neighborhood, $r_c$ is the halo core radius, $R$ is the distance of the Sun from the galactic center, and $r$ is the galactocentric distance. In this case, the integral depends only on the angle $ \psi $ between the direction of observation and the galactic center, $$\label{phiint}
\int \rho_\chi^2 d l = \rho_{\rm loc}^2 R
\frac{x}{2(x-c^2)^{3/2}} \, \left[
\frac{\pi}{2} + \arctan\frac{c}{\sqrt{x-c^2}} +
\frac{ c \sqrt{x-c^2} }{x} \right] .$$ Here $x=1+(r_c/R)^2$ and $c=\cos\psi=\cos b \cos l$. From dynamical studies one finds $\rho_{\rm loc}$ = 0.3–0.5 GeV/cm$^3$, $R$ = 8–8.5 kpc, and $r_c$ = 2–8 kpc. It is interesting to notice that the gamma-ray intensities at $ \psi
= 40^\circ$ and $ \psi = 60^\circ$ are approximately in the ratio 2:1 as on the Dixon et al. maps. Assuming WIMP annihilation into quark-antiquark and lepton-antilepton pairs, fixing $n_{\gamma}(\mathord{>}1{\rm GeV})$ with the Lund Monte-Carlo, varying the halo parameters in the range given above, and matching the observed intensity to eq. (\[phigamma\]) to within 20%, I obtain the required WIMP annihilation cross section as a function of the WIMP mass. This is the band marked “halo $\gamma$’s” in fig. 1.
The second band in fig. 1 comes from the requirement that the WIMP relic density $\Omega h^2$ be in the cosmologically interesting range 0.025–1. ($h$ is the Hubble constant in units of 100km/s/Mpc.) The WIMP relic density is related to the WIMP annihilation cross section through the approximate relation (Kolb & Turner 1990, Gondolo & Gelmini 1991) $$\label{omega}
\sigma v = \frac{ 2.0 \times 10^{-27} \rm{cm^3/s} }{ g g_{\star}^{1/2} x_f
\Omega h^2 } ,$$ where I have assumed that the annihilation cross section is dominated by the $v=0$ term both in the galactic halo and at freeze-out (s-wave annihilation). The freeze-out temperature $x_f m$ can be obtained solving $ x_f^{-1} +
\frac{1}{2} \ln(g_{\star}/x_f) = 80.4 + \ln(g m_\chi \sigma v) $, with $m_\chi$ in GeV and $\sigma v$ in cm$^3$/s. For a Majorana WIMP $g=2$, for a Dirac WIMP $g=4$. $g_{\star}$ is the effective number of relativistic degrees of freedom at freeze-out: $g_{\star} \simeq 81$ before and $g_{\star} \simeq 16$ after the QCD quark-hadron phase transition. Letting $ \Omega h^2 $ vary in the above range gives the relic density band in fig. 1.
The two bands intersect for WIMP masses between 1.2 and 50 GeV. For example, $\Omega = 0.2$ and $H = 60$ km/s/Mpc give the required cross section of $3
\times 10^{-26}$ cm$^3$/s at $m_\chi = 5$ GeV. (This case was presented in Gondolo 1998a.)
Constraints on candidates that couple to the Z boson.
=====================================================
I assume in this section that $\chi\chi$ annihilation and $\chi$–nucleon scattering are dominated by Z boson exchange. In this case, the annihilation cross section reads $$\sigma v = \frac{ G_F^2 } {\pi} \sum_f \beta_f \left[
m_f^2 \left( a_\chi^2 a_f^2 + v_\chi^2 v_f^2 - 2 v_\chi^2 a_f^2 \right) +
2 m_\chi^2 v_\chi^2 \left( a_f^2 + v_f^2 \right) \right] ,$$ where $ \beta_f = (1 - m_f^2/m_\chi^2 ) ^{1/2} $, $a_f=T_f$ and $v_f=T_f-2e_f\sin^2\theta_W$ are the usual axial and vector couplings of fermion $f$ to the Z boson, and $a_\chi$ and $v_\chi$ are the analogous quantities for the $\chi$–Z coupling.
The $\chi$–nucleon scattering cross section $\sigma_{\chi{\scriptscriptstyle\cal N}}$, which is limited by negative direct dark matter searches, is related to the annihilation cross section by crossing symmetry. The experimental bound on $\sigma_{\chi{\scriptscriptstyle\cal N}}$ depends on the WIMP mass and on the spin-dependent or spin-independent character of the interaction (for a recent compilation of limits see Bernabei et al. 1998). For Z boson exchange, the spin-dependent and spin-independent $\chi$–nucleon cross sections read $$\sigma^{\rm SD}_{\chi {\scriptscriptstyle\cal N}} = \frac{ 6 \mu_{\chi{\scriptscriptstyle\cal N}}^2 }{ \pi} G_F^2
a_\chi^2 \left[ \left( \Delta {\rm u} - \Delta {\rm d} \right)^2 + \left(
\Delta {\rm s} \right)^2 \right] ,$$ and $$\sigma^{\rm SI}_{\chi {\scriptscriptstyle\cal N}} = \frac{ \mu_{\chi{\scriptscriptstyle\cal N}}^2 }{ 4 \pi} G_F^2
v_\chi^2 \left[ \left( 1 - 4 \sin^2\theta_W \right)^2 + 1 \right] .$$ Here $\mu_{\chi{\scriptscriptstyle\cal N}} = m_\chi m_{\scriptscriptstyle\cal N}/(m_\chi+m_{\scriptscriptstyle\cal N}) $ is the reduced $\chi$–nucleon mass, and $\Delta {\rm q}$ is the quark q contribution to the spin of the proton. (From neutron and hyperon decay, $\Delta {\rm
u}-\Delta {\rm d} = 1.2573\pm0.0028$, while $\Delta {\rm s}$ is uncertain: $\Delta s=0$ in the naive quark model, $\Delta s=-0.11\pm0.03$ from deep inelastic data, and $\Delta s=-0.15\pm0.09$ from elastic $\nu{\rm p}\to \nu{\rm
p}$ data.)
The coupling of the $\chi$ to the Z boson also gives a contribution to the Z boson decay width, if $m_\chi < m_Z$. Namely, $$\Gamma(Z\to \chi\overline{\chi}) = \frac{ G_F m_Z } { 6 \sqrt{2} \pi }
\left[ a_\chi^2 m_Z^2 \beta_\chi^3 + \beta_\chi v_\chi^2 \left(
m_Z^2+2m_\chi^2 \right) \right] ,$$ with $\beta_\chi = (1 - 4 m_\chi^2/m_Z^2 ) ^{1/2} $. The experimental limit is $\Gamma(Z\to\chi\overline{\chi}) < 5 $ MeV (Barnett et al. 1996).
Once a relation between $a_\chi$ and $v_\chi$ is specified, the experimental bounds on $\sigma_{\chi{\scriptscriptstyle\cal N}}$ and $\Gamma(Z\to\chi\overline{\chi})$ translate into a bound on $\sigma v$. Fig. 2 plots these bounds for two cases: a Dirac particle with $v_\chi=a_\chi$ ($V-A$ interaction), and a Majorana particle with $v_\chi=0$ (axial interaction). For Dirac particles, the interesting region is not fully excluded by these constraints, but for Majorana particles it is. Hence the impact of these constraints is model dependent.
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\[fig:2\]
Constraints from dark matter searches.
======================================
A candidate WIMP has to satisfy constraints from negative dark matter searches. In this section I consider indirect detection through production of rare cosmic rays (antiprotons) and through neutrino production in the Sun and the Earth, and direct detection through elastic scattering off nuclei in a laboratory detector.
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tempboxa = tempboxa > 6in
\[fig:3\]
The most important constraint comes from the measured flux of cosmic ray antiprotons. If the gamma-rays are produced in jets originated by quarks, there is an associated production of antiprotons. The ratio of antiproton and gamma-ray fluxes is independent of the WIMP annihilation cross section and of the local mass density, and the relative number of antiprotons and photons per annihilation is fixed by the physics of jets. The antiproton flux at a ${\rm
\bar{p}}$ kinetic energy $T$ at the top of the atmosphere is $$\phi_{\rm \bar{p}}(T) = \frac{ d N_{\rm \bar{p}}}{ d T} \,
\frac{ \sigma v }{ 4 \pi m_\chi^2 } \, \rho_{\rm
loc}^2 v_{\rm \bar{p}} t_{\rm cont} \mu ,$$ where $d N_{\rm \bar{p}}/ d T$ is the antiproton spectrum per annihilation, $t_{\rm cont}$ is the $\bar{p}$ containment time, which in the diffusion model of Chardonnet et al. (1996) is $t_{\rm cont} \simeq (1+p/3{\rm GeV})^{-0.6} 5
\times 10^{15} $ s, and $ \mu = [ T (T+2m_{\rm\bar{p}}) ] / [ (T+\Delta)
(T+\Delta+2m_{\rm\bar{p}}) ]$ takes into account solar modulation. Fig. 3 shows the bound obtained by taking $\Delta=600$ MeV and imposing $\phi_{\rm
\bar{p}}(150$–$300{\rm MeV}) < 3 \times 10^{-6} $ ${\rm
\bar{p}}$/cm$^2$/s/sr/GeV (Moiseev et al. 1997). For WIMPs heavier than $\sim 10$ GeV, a related constraint on the gamma-ray flux between 300 and 1000 MeV comes from the approximate relation $n_\gamma(\hbox{300--1000MeV}) \simeq
dN_{\bar{\rm p}}(\hbox{150--300MeV})/dT \times 10 {\rm GeV} $. This gives $\phi_\gamma(\hbox{300--1000MeV}) {\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}1.3\times 10^7$ photons/cm$^2$/s/sr if photons originate in quark jets with $m_\chi {\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}10$ GeV. Compared with the observed value of $\sim 8\times 10^7$ photons/cm$^2$/s/sr, it implies too low a gamma-ray flux below 1 GeV. However, these bounds from antiprotons are uncertain because of uncertainties in the antiproton propagation in the galaxy and in the effect of the solar modulation. Moreover, these bounds depend sensitively on the relative branching ratios into the various annihilation channels, and for example can be avoided if the dominant decay channel is leptonic (see next section).
The $\chi$ fermions can also accumulate in the Sun and in the Earth, annihilate therein and produce GeV neutrinos. Accumulation is not efficient for WIMPs lighter than the evaporation mass, which is $\sim 12$ GeV for the Earth and $\sim 3$ GeV for the Sun. The curves marked “Earth” and “Sun” in fig. 3 show approximate constraints obtained from the experimental upper bounds on the flux of through-going muons in the Baksan detector (Suvorova 1997).
Another limit comes from direct searches through the crossing relation between annihilation and scattering cross sections. This relation can be written as $$\mu_{\chi {\scriptscriptstyle\cal N}}^2 \sigma v = f m_\chi^2 c \sigma_{\chi
{\scriptscriptstyle\cal N}} ,$$ where $ c $ is the speed of light and $ f $ accounts for the details of the interactions and of the nuclear structure. The “direct detection” bound in fig. 3 assumes $f=1$, but actually $f$ can range from 0 to infinity. For example, consider a WIMP that couples to quarks and leptons only through exchange of scalar bosons, and consider annihilation through s-channel exchange and scattering through t-channel exchange. For kinematical reasons, if the scalar is CP-even, the annihilation cross section $\sigma v$ at $v=0$ vanishes and the scattering cross section at small velocity is finite, so $f=0$. On the other hand, if the scalar is CP-odd, it is the scattering cross section that vanishes and the annihilation cross section is finite, so $f$ is infinite. Choosing the relative strength of CP-odd and CP-even exchange, the magnitude of the annihilation and scattering cross sections can be tuned. An explicit example is given in the next section.
A candidate WIMP can be a sterile neutrino.
===========================================
The analysis in the previous paragraphs may have been instructive, but the conclusions are so model-dependent that the discussion would be clearer if done in a specific model. Here I show that a particle with the required properties is a sterile right–handed neutrino $\nu_s$ in a model with an extended Higgs sector.
The Higgs sector contains two Higgs doublets $H_1$ and $H_2$ and a Higgs singlet $N$. I assume the following Higgs potential $$\begin{aligned}
\label{eq:V}
V_{\rm Higgs} = &&
\lambda_1 \left( H_1^\dagger H_1 - v_1^2 \right)^2 +
\lambda_2 \left( H_2^\dagger H_2 - v_2^2 \right)^2 +
\lambda_3 \left| N^2 - v_N^2 \right|^2 + \nonumber\\&&
\lambda_4 \left| H_1 H_2 - v_1 v_2 \right|^2 +
\lambda_5 \left| H_1 H_2 - v_1 v_2 + N^2 - v_N^2 \right|^2 +
\lambda_6 \left| H_1^\dagger H_2 \right|^2 .\end{aligned}$$ To fix the notation, $ H_1 H_2 = H_1^0 H_2^0 - H_1^- H_2^+ $. Taking all the $\lambda$’s real and positive guarantees that the absolute minimum of the Higgs potential is at $ \langle H_1^0 \rangle = v_1$, $ \langle
H_2^0 \rangle = v_2 $, and $ \langle N \rangle = v_N $, with the vacuum expectation values of all the other fields vanishing.
Through the Yukawa terms $${\cal L}_{\rm Yukawa} = f_d Q H_1 d_R + f_u Q H_2 u_R + f_e L H_1 e_R + h
\overline{\nu_{sR}^c} \nu_{sR}^{\phantom{c}}\relax N,$$ $H_1$ gives masses to the up–type quarks and the charged leptons, $H_2$ gives masses to the down–type quarks, and $N$ gives a Majorana mass to the right–handed neutrino. There is no mixing of the right–handed neutrino with ordinary neutrinos, otherwise the new neutrino would have decayed in the early universe and would not be in the galactic halo at present.[^2]
There are 2 would-be Goldstone bosons – a charged one $G^\pm$ and a neutral one $G^0$ – and 6 physical Higgs bosons: a charged one $H^\pm$, three neutral “scalars” $S_i$ ($i=1,\dots,3$), and two neutral “pseudoscalars” $P_i$ ($i=1,2$).
The physical charged Higgs field $ H^\pm = H^{\mp *} \sin\beta + H_2^\pm
\cos\beta $ has squared mass $$m_{H^+}^2 = \lambda_6 v^2 .$$ As usual, $v = \sqrt{v_1^2 + v_2^2} $ and $\tan\beta = v_2/v_1$.
The squared mass matrix of the two physical pseudoscalar neutral Higgs fields reads $${\cal M}_P^2 =
\left( \begin{array}{cc}
(\lambda_4 + \lambda_5) v^2 &
2 \lambda_5 v v_N \\
2 \lambda_5 v v_N &
4 (\lambda_3 + \lambda_4) v_N^2
\end{array} \right)$$ in the basis $ \sqrt{2} ( \sin\beta {\rm Im} H_1^0 + \cos\beta {\rm Im} H_2^0 ,
{\rm Im} N )$. The orthogonal combination of fields is the would-be Goldstone boson $G^0$. Let $U^P$ be the unitary matrix that diagonalizes ${\cal M}_P^2$, namely $ P_i = U^P_{i1} A + U^P_{i2} N_I$ where $i=1,2$.
The scalar Higgs boson mass matrix is $${\cal M}_S^2 =
\left( \begin{array}{ccc}
4 \lambda_1 v_1^2 + (\lambda_4 + \lambda_5) v_2^2 &
(\lambda_4 + \lambda_5 ) v_1 v_2 &
2 \lambda_5 v_2 v_N \\
(\lambda_4 + \lambda_5 ) v_1 v_2 &
4 \lambda_2 v_2^2 + (\lambda_4 + \lambda_5) v_1^2 &
2 \lambda_5 v_1 v_N \\
2 \lambda_5 v_2 v_N &
2 \lambda_5 v_1 v_N &
4 (\lambda_3 + \lambda_5) v_N^2
\end{array} \right)$$ in the basis $ \sqrt{2} ( {\rm Re} H_1^0, {\rm Re} H_2^0, {\rm Re} N ) $. Its mass eigenstates $S_i$ ($i=1,\dots,3$) are obtained as $S_i = \sqrt{2} (
U^S_{i1} {\rm Re} H_1^0 + U^S_{i2} {\rm Re} H_2^0 + U^S_{i3} {\rm Re} N )$.
The original Higgs fields can be expressed in terms of the physical fields as $$\begin{aligned}
H_1^0 &=& v_1 + \sqrt{\textstyle{1\over2}}
\left( U^S_{i1} S_i + i U^P_{i1} P_i \sin\beta
\right) , \\
H_2^0 &=& v_2 + \sqrt{\textstyle{1\over2}}
\left( U^S_{i2} S_i + i U^P_{i1} P_i \cos\beta
\right) , \\
N &=& v_N + \sqrt{\textstyle{1\over2}}
\left( U^S_{i3} S_i + i U^P_{i2} P_i \right) .\end{aligned}$$ This gives the interactions of the Higgs bosons with the quarks, the leptons and the right–handed neutrino, $$\begin{aligned}
{\cal L}_{\rm int} &=& \frac{g m_{\nu_s}}{2m_W} \frac{v}{v_N} \left[
U^S_{i3} S_i \overline{\nu}_s \nu_s + i
U^P_{i2} P_i \overline{\nu}_s \gamma_5 \nu_s \right] \\ &+&
\frac{g m_u}{2m_W} \left[
\frac{U^S_{i2}}{\sin\beta} S_i \overline{u} u + i
\cot\beta U^P_{i1} P_i \overline{u} \gamma_5 u \right] \\ &+&
\frac{g m_d}{2m_W} \left[
\frac{U^S_{i1}}{\cos\beta} S_i \overline{d} d + i
\tan\beta U^P_{i1} P_i \overline{d} \gamma_5 d \right]\end{aligned}$$
It is then easy to work out the annihilation cross section, $$\sigma v = \frac{ G_F^2 m_\nu^4} {\pi} \, \frac{v^2}{v_N^2} \,
\left[ \sum_{k=1}^2 \frac{ U^P_{k2} U^P_{k1} }{ m_{P_k}^2 - 4 m_\nu^2 }
\right]^2 \,
\sum_f c_f \beta_f m_f^2 \kappa_f^2 ,$$ and the scattering cross section off nucleons, $$\sigma_{\nu{\scriptscriptstyle\cal N}} = \frac{G_F^2}{\pi} \, \frac{2
m_{\scriptscriptstyle\cal N}^4 m_\nu^4}{\left( m_{\scriptscriptstyle\cal N}
+ m_\nu \right)^2} \, \frac{v^2}{v_N^2} \, \left[ \sum_{j=1}^3
\frac{U^S_{j3}}{m_{S_j}^2} \left( \frac{k_d U^S_{j1}}{\cos\beta} + \frac{ k_u
U^S_{j2}}{\sin\beta} \right) \right]^2 .$$ Here $c_f=3$ for quarks, $c_f=1$ for leptons, $\kappa_f=\cot\beta$ for up–type quarks, and $\kappa_f=\tan\beta$ for down–type quarks and leptons. Moreover, $
k_d = \langle m_d \overline{d} d + m_s \overline{s} s + m_b \overline{b} b
\rangle = 0.21$ and $ k_u = \langle m_u \overline{u} u + m_c \overline{c} c +
m_t \overline{t} t \rangle = 0.15$.
Table 1 lists important quantities for two interesting cases: a 3 GeV neutrino and a 7 GeV neutrino.
tempboxa = tempboxa > 6in
\[tab:1\]
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
“3 GeV” “7 GeV” Experimental/ Observational Values
------------------------------------------------------------------------------- --------------------- ---------------------- ---------------------------------------------------
$m_\nu$ \[GeV\] $3$ $7$
$\phi_\gamma(56^\circ)$ [\[photons($>$1GeV)/cm$^2$/s/sr\]]{} $7.3\times 10^{-7}$ $7.4 \times 10^{-7}$ $\sim 7.6 \times 10^{-7}$
$\Omega$ ($H=60$ km/s/Mpc) $0.11$ $0.12$ $\sim 0.2$
$\phi_{\rm\bar{p}} (200{\rm MeV})$ [\[$\overline{\rm p}$/cm$^2$/s/sr/GeV\]]{} $\sim 0$ $7\times 10^{-6}$ ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}3\times 10^{-6} $
$\sigma_{\chi{\scriptscriptstyle\cal N}}$ \[pb\] $1.2\times 10^{-3}$ $1.3 \times 10^{-4}$ see text
$m_{P_1}, m_{P_2}$ \[GeV\] $5,155$ $4.2,156$
$m_{S_1}, m_{S_2}, m_{S_3}$ \[GeV\] $5,155,220$ $4.2,156,220$
$m_{H^\pm}$ \[GeV\] $110$ $110$ ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$>$}}}53$
$\Gamma(Z\to f\overline{f}P,S)/\Gamma(Z\to f\overline{f})$ $1.3\times $6.3\times 10^{-6}$ ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
10^{-6}$ \raise1pt\hbox{$<$}}}10^{-4}$
“$\sin^2(\beta-\alpha)$” (Zh) $2.3\times 10^{-7}$ $2.3\times 10^{-6}$ ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}10^{-2}$
“$\cos^2(\beta-\alpha)$” (hA) $6 \times 10^{-7} $ $6\times 10^{-6} $ ${\mathrel{\rlap{\lower4pt\hbox{\hskip1pt$\sim$}}
\raise1pt\hbox{$<$}}}0.3$
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
For the “1 GeV” model, the parameters are $h=0.9$, $v/v_N=52$, $\tan\beta=v_2/v_1=40$, and $\lambda_1,\ldots,\lambda_6=g^2$ (the square of the SU(2) coupling constant). The gamma-ray intensity is close to the observed one and the relic density is in the cosmologically interesting range. There is practically no antiproton flux from annihilations because only the $\tau^+\tau^-$ channel is effective. The scattering cross section off nucleons is compatible with the present bound of $2\times 10^{-2}$ pb. Constraints on the Higgs sector are discussed below.
For the “7 GeV” model, the parameters are $h=0.8$, $v/v_N=20$, $\tan\beta=33$, and $\lambda_1,\ldots,\lambda_6=g^2$. The gamma-ray intensity is closed to the observed one, and the relic density is in the cosmologically interesting range. The antiproton flux is slightly higher than the measured one, but, given the big uncertainties in its estimation, it is compatible with it. The scattering cross section is smaller than the present limit of $3\times
10^{-4}$ pb. The Higgs sector is discussed in the following.
Two Higgs bosons ($S_1$ and $P_1$) are light and one has to worry about their production at colliders. But this production is very suppressed because the light Higgs bosons are mostly singlets. First, the LEP bounds on the search of Higgs bosons in two-doublet models are satisfied. Since there is an additional Higgs singlet, $\sin^2(\beta-\alpha)$ in the bound from $e^+ e^- \to Z h$ must be replaced with $| U^S_{11} \cos\beta + U^S_{12} \sin \beta |^2$, and $\cos^2(\beta-\alpha)$ in the bound from $e^+ e^- \to hA$ must be replaced with $ | U^P_{12} U^S_{12} - U^P_{11} U^S_{11} |^2$. In the allowed region, the reinterpreted $\sin^2(\beta-\alpha)$ and $\cos^2(\beta-\alpha)$ are smaller than $5 \times 10^{-5}$, and are not excluded by accelerator searches (Decamp et al. 1992). Secondly, Higgs bremsstrahlung from final state leptons and quarks in Z decays is suppressed by the small mixing between singlet and doublet Higgs bosons. In the present model, $$\label{brems}
\frac{ \Gamma\!(Z\to f\overline{f}X) }{ \Gamma\!(Z\to f\overline{f}) }
= \frac{ \sqrt{2} G_F } { 4 \pi^2} \, g\!\left(\frac{m_X}{m_Z}\right) \,
c_f \, m_f^2 A_{Xf}^2 ,$$ where $A_{P_if} = U^P_{i1} \cot\beta$, $A_{S_if} = U^S_{i1}/\cos\beta$ for up–type quarks, $A_{P_if} = U^P_{i1} \tan\beta$, $A_{S_if} =
U^S_{i2}/\sin\beta$ for down–type quarks and leptons, $c_f = 3$ for quarks and $c_f=1$ for leptons. The function $g(y)$ comes from the phase-space integration, and is $g(y) \cong 1$ at $m_X=12$ GeV. The values obtained in the two examples (see table 1) are lower than the LEP constraint on two leptons + two jets production $ \Gamma\!(Z\to f\overline{f}X) / \Gamma\!(Z\to
f\overline{f}) < 1.5 \times 10^{-4} $ at $m_X = 12 $ GeV (Decamp et al. 1992).
I conclude that these two examples satisfy the experimental and observational constraints considered. A more general scan of the model parameter space is under way (Gondolo 1998b).
Conclusions
===========
I have shown that it may be possible to explain the Dixon et al. gamma-ray emission from the galactic halo as due to halo WIMP annihilations. Not only the intensity and spatial pattern of the halo emission can be matched but also the relic density of the candidate WIMP can be in the cosmologically interesting domain. After a model-independent analysis to learn about the properties of a suitable candidate, I have presented a working model: a sterile neutrino in a model with an extended Higgs sector. Two examples in parameter space indicate the existence of an interesting region in the model parameter space in which present observational and experimental constraints are satisfied and the gamma-ray emission is reproduced.
Acknowledgments
===============
I thank Georg Raffelt and Bernd Kniehl for the invitation to give a talk at this Workshop, Apostolos Pilaftsis and Ara Ioannisian for a discussion on constraints on models with extended Higgs sectors, and Leo Stodolsky for pointing out Higgs bremsstrahlung.
References
==========
Barnett R. M. et al. (Particle Data Group) 1996, [*Phys. Rev.*]{} [**D54**]{}, 1.
Bergström L., Edsjö J., and Ullio P. 1998, astro-ph/9804050 (April 1998).
Bernabei R. et al. 1998, University of Rome preprint ROM2F/98/08 (February 1998).
Chardonnet P., Mignola G., Salati P., and Taillet R. 1996, [ *Phys. Lett.*]{} [**B384**]{}, 161.
Decamp D. et al. 1992, [*Phys. Rep.*]{} [**216**]{}, 253.
Dixon D. D. et al. 1998, in [*Sources and Detection of Dark Matter*]{}, Marina del Rey,
California, February 1998; also astro-ph/9803237 (March 1998).
Gondolo P. and Gelmini G. 1991, [*Nucl. Phys.*]{} [**B360**]{}, 145.
Gondolo P. 1998a, comment to Dixon’s talk in [*Sources and Detection of Dark Matter*]{},
Marina del Rey, California, February 1998 (to appear in the Proceedings).
Gondolo P. 1998b, in preparation.
Gunn J. E. et al. 1978, Ap. J. [**223**]{}, 1015.
Kolb E. W. and Turner M. S. 1990, [*The Early Universe*]{} (Addison-Wesley, Redwood
City).
Moiseev A. et al. (BESS Collab.), 1997, Ap. J. [**474**]{}, 479.
Turner M. 1986, [*Phys. Rev.*]{} [**D34**]{}, 1921.
[^1]: Attempts to increase the flux by clumpiness in the halo (Berström, Edsjö and Ullio 1998) tend to produce either too many antiprotons or too few photons below 1 GeV, in contrast to the Dixon et al. maps (see section 4).
[^2]: F. Vissani has kindly pointed out that a particle that does not mix with ordinary neutrinos should not be called a “neutrino.” I keep this name for lack of a better one.
|
---
abstract: 'This paper constructs cellular resolutions for classes of noncommutative algebras, analogous to those introduced by Bayer–Sturmfels [@BayerSturmfels] in the commutative case. To achieve this we generalise the dimer model construction of noncommutative crepant resolutions of three-dimensional toric algebras by associating a superpotential and a notion of consistency to toric algebras of arbitrary dimension. For abelian skew group algebras and algebraically consistent dimer model algebras, we introduce a cell complex $\Delta$ in a real torus whose cells describe uniformly all maps in the minimal projective bimodule resolution of $A$. We illustrate the general construction of $\Delta$ for an example in dimension four arising from a tilting bundle on a smooth toric Fano threefold to highlight the importance of the incidence function on $\Delta$.'
address: |
Department of Mathematics\
University of Glasgow\
Glasgow\
G12 8QW\
United Kingdom
author:
- Alastair Craw and Alexander Quintero Vélez
title: Cellular resolutions of noncommutative toric algebras from superpotentials
---
Introduction
============
Cellular resolutions were introduced for classes of monomial modules by Bayer–Sturmfels [@BayerSturmfels], generalising the simplicial resolutions for monomial ideals by Bayer–Peeva–Sturmfels [@BPS] and Peeva–Sturmfels [@PeevaSturmfels]. In this paper we develop a noncommutative analogue for certain classes of noncommutative algebra, including skew group algebras for finite abelian subgroups of ${\operatorname{SL}}(n,{\ensuremath{\Bbbk}})$ and superpotential algebras of global dimension three arising from algebraically consistent dimer models. In each case, the minimal bimodule resolution of the algebra is encoded by a collection of cells in a real torus that we call the *toric cell complex*.
We first recall the construction of cellular resolutions of monomial modules over a polynomial ring from Bayer–Sturmfels [@BayerSturmfels]. For a field ${\ensuremath{\Bbbk}}$, set $S:={\ensuremath{\Bbbk}}[x_1,\dots,x_n]$ and consider a monomial $S$-module $M$ with generators $m_1,\dots, m_r \in S$. Let $\Delta$ be a regular cell complex of dimension $n$ with vertex set $\Delta_0=\{1,\dots, r\}$. Label each face $\eta\in \Delta$ by the least common multiple $m_{\eta}$ of the monomials that label its vertices. For any choice of incidence function $\varepsilon$ on $\Delta$, the labelled regular cell complex $\Delta$ defines a complex of free ${\ensuremath{\mathbb{Z}}}^n$-graded $S$-modules $$\begin{aligned}
\label{eqn:BSresolution}
\begin{split}
0 \longrightarrow\bigoplus_{\eta \in \Delta_n}S(-m_{\eta}) & \xlongrightarrow{\partial_{n}} \bigoplus_{\eta^\prime \in \Delta_{n-1}}S(-m_{\eta^\prime}) \xlongrightarrow{\partial_{n-1}} \cdots \\
\cdots & \xlongrightarrow{\partial_2} \bigoplus_{e \in \Delta_1}S(-m_{e}) \xlongrightarrow{\partial_1} \bigoplus_{j \in \Delta_0}S(-m_{j}) \xlongrightarrow{\partial_0} M \longrightarrow 0,
\end{split}
\end{aligned}$$ where $S(-m_{\eta})$ is the free $S$-module with generator $\eta$ in degree $\deg(m_{\eta})$. The maps satisfy $$\partial_{k}(\eta)=\sum_{ {\operatorname{cod}}(\eta',\eta)=1}\varepsilon(\eta,\eta') \frac{m_{\eta}}{m_{\eta'}}\eta'$$ for $\eta \in \Delta_k$, where the sum is taken over all codimension-one faces of $\eta$. Necessary and sufficient conditions for the complex to be acyclic are given [@BayerSturmfels Proposition 1.2], and several classes of examples are presented that satisfy the conditions, in which case the complex is called a *cellular resolution* of $M$.
Before describing our main results we sketch the notion of consistency for toric algebras. Let $\mathscr{E}=(E_0,\dots,E_r)$ denote a collection of reflexive sheaves of rank one on a Gorenstein affine toric variety $X$ of dimension $n$. Our main object of study is the *toric algebra* $A:= {\operatorname{End}}(\bigoplus_{i=0}^r E_i)$ associated to $\mathscr{E}$. Following Craw–Smith [@CrawSmith], we introduce the quiver of sections $Q$ of $\mathscr{E}$, and present $A$ as the quotient of the path algebra of $Q$ by an ideal of relations $J_{\mathscr{E}}$. We use the labelling of arrows in $Q$ to define the superpotential $W$ of $\mathscr{E}$ as a formal sum of cycles in the quiver and, on taking certain higher order derivatives of $W$, we obtain an auxilliary ideal of relations $J_W$ in the path algebra of $Q$. The toric algebra $A$ is *consistent* if the ideals $J_{\mathscr{E}}$ and $J_W$ coincide. Examples include skew group algebras ${\ensuremath{\Bbbk}}[x_1,\dots, x_n]*G$ for finite abelian subgroups $G\subset {\operatorname{SL}}(n,{\ensuremath{\Bbbk}})$ and algebraically consistent dimer model algebras as defined by Broomhead [@Broomhead] in his study of quivers and superpotentials in dimension three.
The notion of consistency is enough to provide a link between $A$ and the toric variety $X$. Indeed, let $\mathcal{M}_{\theta}$ denote the fine moduli space of $\theta$-stable $A$-modules of dimension vector $(1,\dots,1)$ for a generic weight $\theta$, and write $Y_\theta$ for the unique irreducible component of $\mathcal{M}_\theta$ that is birational to $X$. We establish the following result in Theorem \[thm:cohcomp\]:
\[thm:1.1\] For consistent toric algebras $A$, we present an explicit GIT construction of $Y_\theta$ such that the projective birational morphism $Y_\theta\to X$ is obtained by variation of GIT quotient.
Theorem \[thm:1.1\] unifies and extends results by Craw–Maclagan–Thomas [@CMT1] on moduli of McKay quiver representations, and by Mozgovoy [@Mozgovoy] on algebraically consistent dimer models.
We now describe our main result, namely, the construction of the minimal projective bimodule resolution for classes of consistent toric algebras of global dimension $n$ from the toric cell complex $\Delta$ in a real $n$-torus. The key lies in constructing $\Delta$. This is straightforward when $A$ is the skew group algebra for a finite abelian subgroup of ${\operatorname{SL}}(n,{\ensuremath{\Bbbk}})$, in which case $\Delta $ is a regular cell complex. It is considerably more difficult when $A$ is an algebraically consistent dimer model algebra, and in this case the resulting subdivision $\Delta$ of the torus is not even a CW-complex. Nevertheless, $\Delta$ shares several key properties with regular cell complexes which explains our use of the ‘cellular’ terminology (see Remark \[rem:CWcomplex\]). Notably, it admits an incidence function $\varepsilon\colon \Delta\times \Delta\to \{0,\pm 1\}$. In each class of examples as above and for any choice of incidence function $\varepsilon$ on $\Delta$, the toric cell complex $\Delta$ defines a complex of projective $(A,A)$-bimodules $$\begin{aligned}
\label{eqn:NCBSresolution}
\begin{split}
0 \longrightarrow \bigoplus_{\eta \in \Delta_n}A e_{{\operatorname{\mathsf{h}}}(\eta)}\otimes [\eta] \otimes e_{{\operatorname{\mathsf{t}}}(\eta)}A & \xlongrightarrow{d_{n}} \bigoplus_{\eta^\prime \in \Delta_{n-1}}A e_{{\operatorname{\mathsf{h}}}(\eta^\prime)}\otimes [\eta^\prime] \otimes e_{{\operatorname{\mathsf{t}}}(\eta^\prime)} \xlongrightarrow{d_{n-1}}\cdots \\
\cdots & \xlongrightarrow{d_2} \bigoplus_{a \in \Delta_1}A e_{{\operatorname{\mathsf{h}}}(a)}\otimes [a] \otimes e_{{\operatorname{\mathsf{t}}}(a)}A \xlongrightarrow{d_1} A \otimes A \xlongrightarrow{\mu} A \longrightarrow 0,
\end{split}
\end{aligned}$$ where each $e_i\in A$ is a primitive idempotent, where $[\eta]$ are symbols indexed by cells that encode a semigroup grading, and where $\mu\colon A\otimes A \rightarrow A$ is the multiplication map. The maps satisfy $$d_{k}(1 \otimes [\eta] \otimes 1)=\sum_{{\operatorname{cod}}(\eta',\eta)=1}\varepsilon(\eta,\eta') \overleftarrow{\partial}_{\!\eta'}\eta\otimes[\eta'] \otimes \overrightarrow{\partial}_{\!\eta'}\eta.$$ Here, the expressions $\overleftarrow{\partial}_{\!\eta'}\eta$ and $\overrightarrow{\partial}_{\!\eta'}\eta$ are elements of $A$ obtained by right- and left-differentiation of cells (see, for example, Definitions \[def:leftrightderivativesmckay\] and \[def:leftrightderivativesdimer\]). These elements measure the difference between $\eta$ and $\eta^\prime$, and provide the noncommutative analogue of the monomial $m_{\eta}/m_{\eta'}$ from . The following result combines Theorems \[thm:McKayresolution\] and \[thm:dimerresolution\].
\[thm:1.2\] Let $\Delta$ denote the toric cell complex of an abelian skew group algebra or an algebraically consistent dimer model algebra. Then the complex is the minimal projective $(A,A)$-bimodule resolution of $A$.
In each case we refer to as the *cellular resolution* of $A$. For the skew group algebra, we recover the Koszul resolution of $A$ for a suitable choice of $\varepsilon$, and our presentation is reminiscent of that from Tate–Van den Bergh [@TateVandenbergh §3]. For an algebraically consistent dimer model algebra, we exhibit an incidence function $\varepsilon$ for which is the standard resolution associated to a quiver with superpotential in dimension three studied by Ginzburg [@Ginzburg], Mozgovoy-Reineke [@MozgovoyReineke], Davison [@Davison] and Broomhead [@Broomhead].
To conclude, we conjecture that the toric cell complex can be constructed for any consistent toric algebra $A$ whose global dimension $n$ is equal to the dimension of $X$ and, moreover, that the resulting complex is an $(A,A)$-bimodule resolution of $A$. We provide further evidence for this conjecture by examining a representative example arising from a tilting bundle on a smooth toric Fano threefold. More generally, we anticipate a link between the toric cell complex and the coamoeba from Futaki–Ueda [@FutakiUeda] that would describe concretely the mirror Landau-Ginzburg models for smooth toric Fano $n$-folds in the context of Homological Mirror Symmetry.
A direct application of the construction presented here can be made in the study of quiver gauge theories with AdS/CFT gravity duals. As explained by Davey et. al. [@DHMT1], dimer models can be used to describe the gauge theories duals of a class of AdS/CFT backgrounds arising from M$2$-branes placed at a conical Calabi-Yau fourfold. However, the real meaning of dimers in this context is not yet clear. Developing the relationship between the quivers with superpotentials obtained from our construction in dimension four and those arising from the dimer model will hopefully lead to a deeper understanding of this problem.
We now describe the structure of the paper. Section \[sec:toricalgebras\] defines toric algebras and investigates the geometry arising from labelled quivers of sections. The superpotential $W$ and the notion of consistency are presented in Section \[sec:superpotential\], leading to a proof of Theorem \[thm:1.1\]. Section \[sec:McKay\] constructs the toric cell complex $\Delta$ and the resolution in the motivating example of an abelian skew group algebra. We prove in Section \[sec:dimers\] that our superpotential coincides up to sign with the superpotential for an algebraically consistent dimer model algebra, and we use this result to construct $\Delta$ and the resolution in this case. This completes the proof of Theorem \[thm:1.2\]. We present in Section \[sec:conjecture\] the fourfold example which explains why our superpotentials do not involve signs and we conclude with the statement of the main conjecture.
**Conventions** Write ${\ensuremath{\Bbbk}}$ for an algebraically closed field, ${\ensuremath{\Bbbk}}^\times$ for the one-dimensional algebraic torus over ${\ensuremath{\Bbbk}}$, and ${\ensuremath{\mathbb{N}}}$ for the semigroup of nonnegative integers. We do not assume that toric varieties are normal. We often write $p^\pm$ as shorthand for ‘$p^+$ and $p^-$’. Our pictures of cyclic quivers are drawn ‘unwrapped’ to simplify the illustration, and we label each vertex to indicate those vertices that must be identified to reproduce the cyclic quiver from the picture.
**Acknowledgements.** The first author benefited greatly from many conversations with Greg Smith, particular during the MSRI programme in algebraic geometry in 2009. Thanks also to Christian Haase, Akira Ishii, Alastair King, Sergey Mozgovoy, Jan Stienstra, Balázs Szendrői and Michael Wemyss for useful comments and questions. In addition, we thank the anonymous referee for comments. Both authors are supported by EPSRC grant EP/G004048.
Toric algebras from a quiver of sections {#sec:toricalgebras}
========================================
This section introduces the noncommutative toric algebra associated to any collection of rank one reflexive sheaves on a normal affine toric variety $X$. The labelled quivers that encode these algebras also encode the action of an algebraic torus on an auxilliary toric variety, and variation of the resulting GIT quotient produces partial resolutions of $X$. Our toric algebras generalise slightly those from Broomhead [@Broomhead] (compare also the notion of toric $R$-order from Bocklandt [@Bocklandt]).
Toric geometry
--------------
Let $X = {\operatorname{Spec}}R$ be a normal affine toric variety of dimension $n$ with a torus-invariant point. Let $M$ denote the character lattice of the dense torus $T_M:={\operatorname{Spec}}{\ensuremath{\Bbbk}}[M]$ in $X$, and write $N:={\operatorname{Hom}}_{\ensuremath{\mathbb{Z}}}(M,{\ensuremath{\mathbb{Z}}})$ for the dual lattice. There is a strongly convex rational polyhedral cone $\sigma\subset N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ such that $R={\ensuremath{\Bbbk}}[\sigma^\vee\cap M]$. Write $\sigma(1)$ for the set of one-dimensional faces of $\sigma$, set $d:=\vert\sigma(1)\vert$, and let $v_\rho\in N$ denote the primitive lattice point on $\rho\in \sigma(1)$. Each $\rho\in \sigma(1)$ determines an irreducible $T_M$-invariant Weil divisor $D_\rho$ in $X$. These divisors generate the lattice ${\ensuremath{\mathbb{Z}}}^d$ of $T_M$-invariant Weil divisors and the semigroup ${\ensuremath{\mathbb{N}}}^d$ of effective $T_M$-invariant Weil divisors. The map $\deg\colon {\ensuremath{\mathbb{Z}}}^d\to {\operatorname{Cl}}(X)$ sending $D$ to the rank one reflexive sheaf $\mathcal{O}_X(D)$ fits in to the short exact sequence of abelian groups $$\label{eqn:Coxsequence}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}^d @>{\deg}>> {\operatorname{Cl}}(X)@>>> 0,
\end{CD}$$ where the injective map sends $u$ to $\sum_{\rho\in \sigma(1)} \langle u,v_\rho\rangle D_\rho$. The Cox ring of $X$ is the polynomial ring ${\ensuremath{\Bbbk}}[x_\rho :\rho\in \sigma(1)]$ obtained as the semigroup algebra of ${\ensuremath{\mathbb{N}}}^d$, and we have $R\cong {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^d\cap {\operatorname{ker}}(\deg)]$. Since $X$ is normal, every rank one reflexive sheaf on $X$ is of the form $\mathcal{O}_X(D)$ for some Weil divisor class $D\in {\operatorname{Cl}}(X)$, and conversely. For an $R$-module $E$, write $E^\vee:={\operatorname{Hom}}_{R}(E,R)$.
The following result is trivial if the sheaves $E$ and $F$ are invertible.
\[lem:reflexive\] Given rank one reflexive sheaves $E= \mathcal{O}_X(D)$ and $F=\mathcal{O}_X(D^\prime)$, we have that $$\label{eqn:reflexive}
{\operatorname{Hom}}_{\mathcal{O}_X}(E, F) \cong H^0\big(\mathcal{O}_X(D^\prime-D)\big).$$
For any $R$-module $E$ and for any reflexive $R$-module $F$, adjunction gives $${\operatorname{Hom}}_{R}(E,F)\cong {\operatorname{Hom}}_{R}\big(E,{\operatorname{Hom}}_{R}(F^\vee,R)\big) \cong
{\operatorname{Hom}}_{R}(E\otimes F^\vee,R) = (E\otimes F^\vee)^\vee,$$ hence ${\operatorname{Hom}}_{R}(E,F)^{\vee\vee}\cong (E\otimes F^\vee)^{\vee\vee\vee}$. Now, the global sections functor is an equivalence between the category of coherent sheaves on $X$ and the category of finitely generated $R$-modules, and the composition of $H^0(-)$ with the functor $\mathcal{H}om_{\mathscr{O}_X}(-,\mathscr{O}_X)$ is simply ${\operatorname{Hom}}_R(-,R)$. In particular, $${\operatorname{Hom}}_{\mathcal{O}_X}(E,F)^{\vee\vee} \cong H^0\big(X,(E\otimes F^\vee)^{\vee\vee\vee}\big).$$ We now assume that $E= \mathcal{O}_X(D)$ and $F=\mathcal{O}_X(D^\prime)$. Then $F^\vee \cong \mathcal{O}_X(-D^\prime)$ and $(E\otimes F^\vee)^{\vee\vee}\cong \mathcal{O}_X(D-D^\prime)$, see for example Cox–Little–Schenck [@CLS Proposition 8.0.6]. Substitute this into the above and apply $\mathcal{O}_X(D-D^\prime)^\vee = \mathcal{O}_X(D^\prime-D)$ to obtain $${\operatorname{Hom}}_{\mathcal{O}_X}(E, F)^{\vee\vee} \cong H^0\big(\mathcal{O}_X(D^\prime-D)\big).$$ The left hand side is reflexive by Benson [@Benson Lemma 3.4.1(iv)]. This completes the proof.
Quivers of sections
-------------------
Let $Q$ be a finite connected quiver with vertex set $Q_0$, arrow set $Q_1$, and maps ${\operatorname{\mathsf{h}}}, {\operatorname{\mathsf{t}}}\colon Q_1 \to Q_0$ indicating the vertices at the head and tail of each arrow. A nontrivial path in $Q$ is a sequence of arrows $p = a_k \dotsb a_1$ with ${\operatorname{\mathsf{h}}}(a_{j}) = {\operatorname{\mathsf{t}}}(a_{j+1})$ for $1 \leq j < k$. We set ${\operatorname{\mathsf{t}}}(p) = {\operatorname{\mathsf{t}}}(a_{1}), {\operatorname{\mathsf{h}}}(p) = {\operatorname{\mathsf{h}}}(a_k)$ and ${\operatorname{supp}}(p)=\{a_1,\dots, a_k\}$. A cycle is a path $p$ with ${\operatorname{\mathsf{t}}}(p) = {\operatorname{\mathsf{h}}}(p)$. Each $i \in Q_0$ gives a trivial path $e_i$ where ${\operatorname{\mathsf{t}}}(e_i) = {\operatorname{\mathsf{h}}}(e_i) = i$. The path algebra ${\ensuremath{\Bbbk}}Q$ is the ${\ensuremath{\Bbbk}}$-algebra whose underlying ${\ensuremath{\Bbbk}}$-vector space has a basis of paths in $Q$, where the product of basis elements is the basis element defined by concatenation of the paths if possible, or zero otherwise. Let $[{\ensuremath{\Bbbk}}Q,{\ensuremath{\Bbbk}}Q]$ denote the ${\ensuremath{\Bbbk}}$-vector space spanned by all commutators in ${\ensuremath{\Bbbk}}Q$, so ${\ensuremath{\Bbbk}}Q_{\text{cyc}}:={\ensuremath{\Bbbk}}Q/[{\ensuremath{\Bbbk}}Q,{\ensuremath{\Bbbk}}Q]$ has a basis of elements corresponding to cyclic paths in the quiver.
For $r\geq 0$, consider a collection $\mathscr{E}:=(E_0,E_1,\dots,E_r)$ of distinct rank one reflexive sheaves on the affine toric variety $X$. Since $X$ is normal, every such sheaf is of the form $E_i=\mathcal{O}_X(D_i^\prime)$ for some $D_i^\prime\in {\operatorname{Cl}}(X)$. For $0\leq i, j\leq r$, a $T_M$-invariant section $s \in {\operatorname{Hom}}_{\mathcal{O}_X}(E_i,E_j)$ is said to be *irreducible* if it does not factor through some $E_k$ with $k\neq i,j$, that is, the section does not lie in the image of the multiplication map $$H^0\big(X,\mathcal{O}_X(D_j^\prime-D_k^\prime)\big)\otimes_{\ensuremath{\Bbbk}}H^0\big(X,\mathcal{O}_X(D_k^\prime-D_i^\prime)\big)\longrightarrow H^0\big(X,\mathcal{O}_X(D_j^\prime-D_i^\prime)\big)$$ for any $k\neq i,j$, where we use the isomorphism from Lemma \[lem:reflexive\].
The *quiver of sections* of $\mathscr{E}$ is the finite quiver $Q$ in which the vertex set $Q_0 = \{ 0, \dotsc,r \}$ corresponds to the sheaves in $\mathscr{E}$, and where the arrows from $i$ to $j$ correspond to the irreducible sections in ${\operatorname{Hom}}_{\mathcal{O}_X}(E_i,E_j)$.
1. Lemma \[lem:reflexive\] writes ${\operatorname{Hom}}_{\mathcal{O}_X}(E_i,E_j)$ in terms of Weil divisors. To construct $Q$ in practice, write each $E_i\in \mathscr{E}$ as $E_i=\mathcal{O}_X(D_i^\prime)$ for some $D_i^\prime\in {\operatorname{Cl}}(X)$, and compute for every $i,j\in Q_0$ the vertices of the polyhedron ${\operatorname{conv}}({\ensuremath{\mathbb{N}}}^d\cap \deg^{-1}(D^\prime_i-D_j^\prime))$ to obtain the $T_M$-invariant $R$-module generators of ${\operatorname{Hom}}_{\mathcal{O}_X}(E_i,E_j)$. The arrows of $Q$ correspond to the generators of irreducible maps.
2. The quiver $Q$ depends only on differences of effective Weil divisors on $X$. We normalise by choosing $E_0:= \mathcal{O}_X$.
For $a \in Q_1$, write ${\operatorname{div}}(a) := {\operatorname{div}}(s) \in {\ensuremath{\mathbb{N}}}^{d}$ for the divisor of zeroes of the defining section $s \in {\operatorname{Hom}}_{\mathcal{O}_X}(E_i,E_j)$ and, more generally, for any path $p$ in $Q$ we call ${\operatorname{div}}(p) := \sum_{a\in {\operatorname{supp}}(p)} {\operatorname{div}}(a)$ the *label* of $p$. The labelling monomial is $x^{{\operatorname{div}}(p)}:= \prod_{a\in {\operatorname{supp}}(p)} x^{{\operatorname{div}}(a)}\in {\ensuremath{\Bbbk}}[x_\rho : \rho\in \sigma(1)]$.
Consider the two-sided ideal $$J_{\mathscr{E}}:= \big( p^+-p^- \in {\ensuremath{\Bbbk}}Q \mid {\operatorname{\mathsf{h}}}(p^+)={\operatorname{\mathsf{h}}}(p^-),
{\operatorname{\mathsf{t}}}(p^+)={\operatorname{\mathsf{t}}}(p^-), {\operatorname{div}}(p^+) = {\operatorname{div}}(p^-)\big)$$ in the path algebra ${\ensuremath{\Bbbk}}Q$. The quotient $A_{\mathscr{E}}:= {\ensuremath{\Bbbk}}Q/J_{\mathscr{E}}$ is the *toric algebra* of the collection $\mathscr{E}$, and the pair $(Q,J_{\mathscr{E}})$ is the *bound quiver of sections* of the collection $\mathscr{E}$. The phrase ‘bound quiver’ is a synonym for ‘quiver with relations’.
\[lem:algebra\] For $r\geq 0$ and for $\mathscr{E}=(E_0,E_1,\dots,E_r)$, the quiver of sections $Q$ of $\mathscr{E}$ is strongly connected and $A_{\mathscr{E}}\cong {\operatorname{End}}_R\bigl( \bigoplus_{i\in Q_0} E_i \bigr)$. In particular, the centre $Z(A_{\mathscr{E}})$ is isomorphic to $R$.
The top-dimensional cone $\sigma^\vee\subset M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ is obtained by slicing the cone ${\ensuremath{\mathbb{R}}}^d_{\geq 0}$ by ${\operatorname{ker}}(\deg)$, so there exists $u\in \sigma^\vee\cap M$ such that the lattice point $\sum_\rho \langle u,v_\rho\rangle D_\rho$ lies in the interior of ${\ensuremath{\mathbb{R}}}^d_{\geq 0}$. For each $\rho\in \sigma(1)$, set $\mu_\rho:= \langle u,v_\rho\rangle >0$. For $i\in Q_0$, write $E_i=\mathcal{O}_X(D)$ where $D=\sum_{\rho\in \sigma(1)} \lambda_\rho D_\rho$ and choose $k, l\in {\ensuremath{\mathbb{Z}}}$ satisfying $k\mu_\rho\leq \lambda_\rho\leq l\mu_\rho$ for all $\rho\in \sigma(1)$. Then $\sum_{\rho\in \sigma(1)} (l\mu_\rho-\lambda_\rho)D_\rho$ and $\sum_\rho (\lambda_\rho-k\mu_\rho)D_\rho$ are effective divisors, so both ${\operatorname{Hom}}_R(E_0,E_i)$ and ${\operatorname{Hom}}_R(E_i,E_0)$ are nonempty. It follows that $Q$ is strongly connected. The stated isomorphism of ${\ensuremath{\Bbbk}}$-algebras follows as in the proof of [@CrawSmith Proposition 3.3]. To compute the centre, consider the ${\ensuremath{\Bbbk}}$-linear map ${\ensuremath{\Bbbk}}Q_{\mathsf{cyc}}\to R$ determined by sending a cycle $p$ to the section $x^{{\operatorname{div}}(p)}$. This map is surjective by construction of $Q$. Since the centre of $A_{\mathscr{E}}$ is generated by $J_{\mathscr{E}}$-equivalence classes of cycles in $Q$, the isomorphism $Z(A_{\mathscr{E}})\to R$ follows after taking equivalence classes modulo $J_{\mathscr{E}}$.
For any $X={\operatorname{Spec}}(R)$, the quiver of sections $Q$ of the trivial collection $\mathscr{E} = (\mathcal{O}_X)$ has one vertex. If $X={\operatorname{Spec}}({\ensuremath{\Bbbk}})$ then $Q_1=\emptyset$ and $A\cong {\ensuremath{\Bbbk}}$. Otherwise, $Q$ has $m$ loops where the labelling divisors ${\operatorname{div}}(a_1),\dots,{\operatorname{div}}(a_m)$ are the elements in the Hilbert basis of the semigroup $\sigma^\vee\cap M$. The ${\ensuremath{\Bbbk}}$-algebra epimorphism ${\ensuremath{\Bbbk}}Q\to R$ sending $a_i\mapsto x^{{\operatorname{div}}(a_i)}$ for $1\leq i\leq m$ has kernel $J_{\mathscr{E}}$, so the toric algebra $A_{\mathscr{E}}$ is isomorphic to the coordinate ring $R$. In particular, coordinate rings of normal affine toric varieties are toric algebras.
\[exa:F1tilting\] Let $\sigma$ be the cone in ${\ensuremath{\mathbb{R}}}^3$ generated by $v_1=(1,0,1)$, $v_2= (0,1,1)$, $v_3= (-1,1,1)$, $v_4= (0,-1,1)$, so $\sigma$ is the cone over the lattice polygon shown in Figure \[fig:tiltingF1\](a).
For $1\leq \rho\leq 4$, write $D_\rho$ for the Weil divisor in $X={\operatorname{Spec}}{\ensuremath{\Bbbk}}[\sigma^\vee\cap {\ensuremath{\mathbb{Z}}}^3]$ corresponding to the ray of $\sigma$ generated by $v_\rho$. The group ${\operatorname{Cl}}(X)$ is the quotient of the free abelian group generated by $\mathcal{O}_X(D_1)$ and $\mathcal{O}_X(D_4)$, by the subgroup generated by $\mathcal{O}_X(D_1+2D_4)$. The quiver of sections $Q$ of $\mathscr{E} = \big(\mathcal{O}_X, \mathcal{O}_X(D_1),\mathcal{O}_X(D_4), \mathcal{O}_X(D_1+D_4)\big)$ is the cyclic quiver from Figure \[fig:tiltingF1\](b); the quiver is shown in ${\ensuremath{\mathbb{Z}}}^2$, but $\mathcal{O}_X\sim\mathcal{O}_X(D_1+2D_4)$. For $a\in Q_1$ we have $x^{{\operatorname{div}}(a)}\in {\ensuremath{\Bbbk}}[x_1,x_2,x_3,x_4]$, and $$J_{\mathscr{E}}= \left(\begin{array}{c} \! a_6a_3-a_5a_1, \; a_7a_3-a_5a_2, \; a_7a_4a_1-a_6a_4a_2,\; a_3a_9-a_4a_1a_8,\; a_{3}a_{10}-a_4a_2a_8 \! \\
\! a_2a_9-a_{1}a_{10}, \; a_1a_8a_7-a_2a_8a_6, \; a_9a_7-a_{10}a_{6},\; a_8a_6a_4-a_9a_5,\; a_{10}a_{5}-a_8a_7a_4 \!
\end{array}\right).$$ defines the noncommutative toric algebra $A_{\mathscr{E}}={\ensuremath{\Bbbk}}Q/J_{\mathscr{E}}$.
Polyhedral geometry
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The characteristic functions $\chi_{i} \colon Q_0 \to {\ensuremath{\mathbb{Z}}}$ and $\chi_{a} \colon Q_1 \to {\ensuremath{\mathbb{Z}}}$ for $i \in Q_0$ and $a \in Q_1$ form the standard integral bases of the vertex space ${\ensuremath{\mathbb{Z}}}^{Q_0}$ and the arrow space ${\ensuremath{\mathbb{Z}}}^{Q_1}$ respectively. The incidence map ${\operatorname{inc}}\colon {\ensuremath{\mathbb{Z}}}^{Q_1} \to {\ensuremath{\mathbb{Z}}}^{Q_0}$ defined by setting ${\operatorname{inc}}(\chi_{a})=\chi_{{\operatorname{\mathsf{h}}}(a)} - \chi_{{\operatorname{\mathsf{t}}}(a)}$ has image equal to the sublattice ${\operatorname{Wt}}(Q) \subset {\ensuremath{\mathbb{Z}}}^{Q_0}$ of functions $\theta \colon
Q_0 \to {\ensuremath{\mathbb{Z}}}$ satisfying $\sum_{i \in Q_0} \theta_i = 0$. Generalising [@CMT1 Definition 3.2] (compare also [@CrawSmith]), we define $$\pi := ({\operatorname{inc}},{\operatorname{div}})\colon {\ensuremath{\mathbb{Z}}}^{Q_{1}} \to {\operatorname{Wt}}(Q) \oplus {\ensuremath{\mathbb{Z}}}^{d}$$ to be the ${\ensuremath{\mathbb{Z}}}$-linear map sending $\chi_{a}$ to $\bigl( \chi_{{\operatorname{\mathsf{h}}}(a)} - \chi_{{\operatorname{\mathsf{t}}}(a)}, {\operatorname{div}}(a) \bigr)$ for $a\in Q_1$. Let ${\ensuremath{\mathbb{Z}}}(Q)$ and ${\ensuremath{\mathbb{N}}}(Q)$ denote the image under $\pi$ of the lattice ${\ensuremath{\mathbb{Z}}}^{Q_1}$ and the subsemigroup ${\ensuremath{\mathbb{N}}}^{Q_1}$ respectively, and write ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]$ and ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ for the semigroup algebras. Let $\pi_1\colon {\ensuremath{\mathbb{Z}}}(Q)\to {\operatorname{Wt}}(Q)$ and $\pi_2\colon {\ensuremath{\mathbb{Z}}}(Q)\to {\ensuremath{\mathbb{Z}}}^{d}$ denote the first and second projections respectively, and define a group homomorphism $\nu\colon{\operatorname{Wt}}(Q)\to {\operatorname{Cl}}(X)$ by setting $\nu(\chi_i)=E_i$ for all $i\in Q_0$.
\[lem:diagram\] There is a commutative diagram of abelian groups $$\label{eqn:diagram}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}(Q) @>{\pi_1}>> {\operatorname{Wt}}(Q)@>>> 0\\
@. @| @V{\pi_2}VV @V{\nu}VV @. \\
0 @>>> M @>>> {\ensuremath{\mathbb{Z}}}^{d} @>{\deg}>> {\operatorname{Cl}}(X) @>>> 0 \\
\end{CD}$$ where $\pi_2$ identifies the subsemigroup ${\ensuremath{\mathbb{N}}}(Q)\cap {\operatorname{ker}}(\pi_1)$ with $\sigma^\vee\cap M={\ensuremath{\mathbb{N}}}^d\cap {\operatorname{ker}}(\deg)$. In particular, the rank of the lattice ${\ensuremath{\mathbb{Z}}}(Q)$ is $n+r$.
The right-hand square commutes and the bottom row is exact, so it enough to prove that $\pi_2$ yields a ${\ensuremath{\mathbb{Z}}}$-linear isomorphism ${\operatorname{ker}}(\pi_1)\cong M$ which restricts to an isomorphism of semigroups ${\ensuremath{\mathbb{N}}}(Q)\cap {\operatorname{ker}}(\pi_1)\cong{\ensuremath{\mathbb{N}}}^d\cap {\operatorname{ker}}(\deg)$. The proof of [@CMT1 Proposition 4.1] generalises to our setting.
The semigroup ${\ensuremath{\mathbb{N}}}(Q)$ need not be saturated, see Remark \[rem:notsaturated\].
Consider the commutative diagram $$\label{eqn:dualdiagram}
\begin{CD}
0@<<< N @<{\psi^*}<< {\ensuremath{\mathbb{Z}}}(Q)^\vee @<<< {\operatorname{Wt}}(Q)^\vee@<<< 0\\
@. @| @A{\pi_2^*}AA @AAA @. \\
0 @<<< N @<{\iota^*}<< {\ensuremath{\mathbb{Z}}}^{d} @<<< {\operatorname{Cl}}(X)^\vee @<<< 0 \\
\end{CD}$$ dual to . Let $\{\chi_\rho \mid \rho\in \sigma(1)\}$ the standard basis of ${\ensuremath{\mathbb{Z}}}^d$. For each $\rho\in \sigma(1)$, the image of $\chi_\rho$ under the map $\iota^*\colon {\ensuremath{\mathbb{Z}}}^d\to N$ is the primitive generator $v_\rho\in \rho$, so the image of the positive orthant $\{\sum_\rho c_\rho \chi_\rho \mid c_\rho\geq 0\}$ under the linear map $\iota^*\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\colon {\ensuremath{\mathbb{R}}}^d \to N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ is the cone $\sigma$. To establish a similar statement for the top row of , consider the convex polyhedral cone $$C:= \big\{v\in {\ensuremath{\mathbb{Z}}}(Q)^\vee\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\mid \langle u,v\rangle \geq 0 \text{ for all }u\in {\ensuremath{\mathbb{N}}}(Q)\big\}.$$
The image of $C$ under $\psi^*\colon {\ensuremath{\mathbb{Z}}}(Q)^\vee\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\to N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ is the cone $\sigma$.
Lemma \[lem:diagram\] shows that $\pi_2$ identifies the semigroup ${\ensuremath{\mathbb{N}}}(Q)\cap {\operatorname{ker}}(\pi_1)$ with $\sigma^\vee\cap M$, so the ${\ensuremath{\mathbb{R}}}$-linear extension of $\pi_2$ identifies the slice $C\cap {\operatorname{ker}}(\pi_1)$ with the cone $\sigma^\vee$. The result is now immediate from Craw–Maclagan [@CrawMaclagan Corollary 2.10].
The semigroup ${\ensuremath{\mathbb{N}}}(Q)$ is generated by the vectors $\pi(\chi_a)\in {\ensuremath{\mathbb{Z}}}(Q)$ arising from arrows $a\in Q_1$, so $C=\{v\in {\ensuremath{\mathbb{Z}}}(Q)^\vee\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\mid \langle v,\pi(\chi_a)\rangle \geq 0 \text{ for all }a\in Q_1\}$. For any face $\Pi$ of $C$, let ${\operatorname{relint}}(\Pi)$ denote that relative interior of $\Pi$ and define the *support* of $\Pi$ to be $${\operatorname{supp}}(\Pi):= \big\{a\in Q_1 \mid \langle v,\pi(\chi_a)\rangle >0\text{ for all }v\in {\operatorname{relint}}(\Pi)\big\}.$$ To explain the geometric significance of the support, note that the toric variety ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[C^\vee\cap {\ensuremath{\mathbb{Z}}}(Q)]$ is the normalisation of ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ because $C^\vee\cap {\ensuremath{\mathbb{Z}}}(Q)$ is the saturation of ${\ensuremath{\mathbb{N}}}(Q)$. As is standard in toric geometry, a face $\Pi$ of $C$ defines the torus-orbit closure in ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ parametrising points $(w_a) \in{\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]\subseteq \mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ whose coordinates satisfy $w_a= 0$ if and only if $a\in {\operatorname{supp}}(\Pi)$.
\[def:perfectmatching\] A *perfect matching* $\Pi$ of $Q$ is the primitive lattice point on a one-dimensional face of the cone $C$. We also refer to the face itself, or even to the set of arrows ${\operatorname{supp}}(\Pi)$, as the perfect matching. A perfect matching $\Pi$ is *extremal* if $\psi^*(\Pi) = v_\rho$ for some $\rho\in \sigma(1)$.
The terminology ‘perfect matching’ is taken from the special case where the algebra $A$ arises from a dimer model as described in Section \[sec:dimers\].
\[prop:perfmatchlabels\] For $\rho\in \sigma(1)$, the vector $\Pi_\rho:= \pi_2^*(\chi_\rho)$ is an extremal perfect matching of $Q$. In addition, for every $a\in Q_1$ we have $$\label{eqn:supportlabels}
a\in {\operatorname{supp}}(\Pi_\rho)\iff x_\rho \text{ divides }x^{{\operatorname{div}}(a)}.$$
We begin by proving the second statement. An arrow $a$ in $Q$ lies in ${\operatorname{supp}}(\Pi_\rho)$ if and only if $\langle \Pi_\rho,\pi(\chi_a)\rangle >0$. Since $\Pi_\rho:= \pi_2^*(\chi_\rho)$, we have $$\label{eqn:adjoint}
\big\langle \Pi_\rho,\pi(\chi_a)\big\rangle = \big\langle \chi_\rho, \pi_2(\pi(\chi_a))\big \rangle = \big\langle \chi_\rho, {\operatorname{div}}(a)\big\rangle,$$ and this is positive if and only if $x_\rho$ divides $x^{{\operatorname{div}}(a)}$.
For the first statement, note that ${\operatorname{div}}(a)\in {\ensuremath{\mathbb{N}}}^d$ for all $a\in Q_1$ and hence $ \big\langle \Pi_\rho,\pi(\chi_a)\big\rangle\geq 0$ by . It follows that $\Pi_\rho\in C$. Commutativity of diagram shows that $\psi^*(\Pi_\rho)$ is equal to the primitive lattice point $v_\rho$ in $\rho$, so $\Pi_\rho$ is a primitive lattice point in some face of $C$ that we also denote $\Pi_\rho$. To deduce that $\Pi_\rho$ is an extremal perfect matching it remains to show that the face $\Pi_\rho$ has dimension one or, equivalently, that the dual face $F$ in $C^\vee$ has dimension $n+r-1$. The identification of ${\ensuremath{\mathbb{N}}}(Q)\cap {\operatorname{ker}}(\pi_1)$ with $\sigma^\vee\cap M$ from Lemma \[lem:diagram\] and saturatedness of $\sigma^\vee\cap M$ enables us to identify $C^\vee\cap {\operatorname{ker}}(\pi_1)$ with $\sigma^\vee\cap M$. The face $F_\rho$ of $\sigma^\vee$ dual to the cone $\rho$ has dimension $n-1$, and hence [@CrawMaclagan Lemma 2.5] gives $F_\rho=F\cap {\operatorname{ker}}(\pi_1)$. We claim that $F$ intersects ${\operatorname{ker}}(\pi_1)$ transversely, so $F$ has dimension $n-1+r$ as required. To prove the claim, it suffices by Thaddeus [@Thaddeus Lemma 3.3] to show that $F$ is $0$-stable or, equivalently, that the quiver $Q^\prime$ with vertex set $Q_0$ and arrow set $Q_1\setminus{\operatorname{supp}}(\Pi_\rho)$ is strongly connected. In light of , this quiver is obtained from $Q$ by deleting each $a\in Q_1$ for which $x_\rho$ divides $x^{{\operatorname{div}}(a)}$. It follows that $Q^\prime$ is the quiver of sections on the affine toric variety $D_\rho$ defined by the collection $\mathscr{E}^\prime = (E_i\vert_{D_\rho} : i\in Q_0)$. Lemma \[lem:algebra\] implies that $Q^\prime$ is strongly connected.
Together with the multiplicities from , Proposition \[prop:perfmatchlabels\] records the fact that extremal perfect matchings encode the labels in a quiver of sections.
Variation of GIT quotient
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The incidence map of $Q$ determines a ${\operatorname{Wt}}(Q)$-grading of the polynomial ring ${\ensuremath{\Bbbk}}[y_a : a\in Q_1]$ obtained as the semigroup algebra of ${\ensuremath{\mathbb{N}}}^{Q_1}$. The algebraic torus $T:={\operatorname{Hom}}({\operatorname{Wt}}(Q),{\ensuremath{\Bbbk}}^\times)$ of rank $r$ then acts on the affine space $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}:={\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]$, where for $(t_i)\in T$ and $(w_a)\in \mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ we have $$\label{eqn:Taction}(t\cdot w)_{a} = t_{{\operatorname{\mathsf{h}}}(a)}^{\,} w_{a} t_{{\operatorname{\mathsf{t}}}(a)}^{-1}.$$ For any weight $\theta \in {\operatorname{Wt}}(Q)$, let ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]_{\theta}$ denote the $\theta$-graded piece of the coordinate ring of $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$. The GIT quotient $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}{\ensuremath{/\!\!/\!}}_\theta T= {\operatorname{Proj}}(\bigoplus_{j\geq 0} {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]_{j\theta})$ is the categorical quotient of the open subscheme of $\theta$-semistable points in $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ by the action of $T$. We say that a weight $\theta\in {\operatorname{Wt}}(Q)$ is *generic* if every $\theta$-semistable point of $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ is $\theta$-stable, in which case, $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}{\ensuremath{/\!\!/\!}}_\theta T$ is the geometric quotient of the open subscheme of $\theta$-stable points in $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ by $T$.
The map $\pi$ induces a surjective map of semigroup algebras $\pi_*\colon {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]\to {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$ with kernel $$\label{eqn:IQ}
I_{\mathscr{E}} := \big(y^u-y^v \in {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}] \mid u-v\in {\operatorname{ker}}(\pi)\big)$$ that cuts out the affine toric subvariety $\mathbb{V}(I_{\mathscr{E}})$ of $\mathbb{A}_{\ensuremath{\Bbbk}}^{Q_1}$. The incidence map factors through ${\ensuremath{\mathbb{N}}}(Q)$ to define a ${\operatorname{Wt}}(Q)$-grading on ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]$, so the $T$-action on $\mathbb{A}_{\ensuremath{\Bbbk}}^{Q_1}$ restricts to an action on $\mathbb{V}(I_{\mathscr{E}})$. For $\theta \in {\operatorname{Wt}}(Q)$, let ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]_{\theta}$ denote the $\theta$-graded piece and write $$Y_\theta:= \mathbb{V}(I_{\mathscr{E}}){\ensuremath{/\!\!/\!}}_\theta T= {\operatorname{Proj}}\Big(\bigoplus_{j\geq 0} {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]_{j\theta}\Big)$$ for the categorical quotient of the open subset of $\theta$-semistable points in $\mathbb{V}(I_{\mathscr{E}})$.
\[prop:Ytheta\] For any $\theta\in {\operatorname{Wt}}(Q)$, the toric variety $Y_\theta = \mathbb{V}(I_{\mathscr{E}}){\ensuremath{/\!\!/\!}}_\theta T$ admits a projective birational morphism $\tau_\theta\colon Y_\theta \longrightarrow X={\operatorname{Spec}}R$ obtained by variation of GIT quotient.
Lemma \[lem:diagram\] implies that $R={\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^d\cap {\operatorname{ker}}(\deg)]\cong {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)\cap {\operatorname{ker}}(\pi_1)]={\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]^T$, so the variety $X$ is isomorphic to $Y_0={\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}(Q)]^T$. Variation of GIT quotient gives the projective morphism $\tau_\theta\colon Y_\theta \to Y_0$, and it remains to show that $\tau_\theta$ is birational. Each $\theta$-semiinvariant monomial in ${\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]$ is nowhere zero on the dense torus ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]$ of $\mathbb{V}(I_\mathscr{E})$ because the coordinate entries of every such point are all nonzero under the embedding of ${\operatorname{Spec}}({\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)])$ in the dense torus of $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$. It follows that every point of ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]$ is $\theta$-semistable. Since every point of ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]$ is also $0$-semistable, we deduce that the dense torus ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]{\ensuremath{/\!\!/\!}}_\theta T$ of $Y_\theta$ is isomorphic to the dense torus ${\operatorname{Spec}}{\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{Z}}}(Q)]^T$ of $Y_0$.
The morphism $\tau_\theta\colon Y_\theta\to X$ provides a resolution of singularities precisely when $Y_\theta$ is smooth. Note however that $Y_\theta$ need not even be normal, see Remark \[rem:notsaturated\].
Consistency for superpotential algebras {#sec:superpotential}
=======================================
This section introduces the superpotential $W$ and the superpotential algebra $A_W$ of a quiver of sections $Q$ on $X$. This algebra need not be isomorphic to the toric algebra, but when it is we say that the toric algebra is consistent. This implies in particular that the toric variety $Y_\theta$ for generic $\theta$ is the coherent component of a fine moduli space $\mathcal{M}_\theta$ of $\theta$-stable $A_W$-modules.
Superpotential from anticanonical cycles
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Assume from now on that $X$ is Gorenstein, so $(1,\dots,1)\in {\ensuremath{\mathbb{Z}}}^d$ lies in the sublattice $M$ and hence $\sum_{\rho\in \sigma(1)} D_\rho$ is linearly equivalent to zero, giving $\omega_X\cong \mathcal{O}_X$. The primitive lattice point $v_\rho\in N$ on each ray in $\sigma$ lies in an affine hyperplane in $N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$, and $\sigma$ is the cone over the convex polytope $P={\operatorname{conv}}(v_\rho \mid \rho \in \sigma(1))$. For $r\geq 0$, let $\mathscr{E}:=(\mathcal{O}_X,E_1,\dots,E_r)$ be a collection of distinct rank one reflexive sheaves on $X$ with quiver of sections $Q$. The anticanonical divisor $\sum_{\rho\in \sigma(1)} D_\rho$, or equivalently the monomial $\prod_{\rho\in \sigma(1)} x_\rho$ in the Cox ring of $X$, singles out a preferred set of cycles in $Q$ as follows.
\[defn:superpotentialnosigns\] A cycle $p$ in $Q$ is an *anticanonical cycle* if $x^{{\operatorname{div}}(p)}=\prod_{\rho\in \sigma(1)} x_\rho$. Let ${\ensuremath{\mathscr{C}_{\mathsf{ac}}(Q)}}$ denote the set of anticanonical cycles. The *superpotential* of the collection $\mathscr{E}$ is the formal sum of cycles $W := \sum_{p\in {\ensuremath{\mathscr{C}_{\mathsf{ac}}(Q)}}} p\in {\ensuremath{\Bbbk}}Q_{\mathrm{cyc}}$.
Given paths $p,q$ in $Q$, the partial (left) derivative of $q$ with respect to $p$ is $$\partial_qp:=\left\{\begin{array}{cl} r & \text{if }p=rq; \\ 0 & \text{otherwise.}\end{array}\right.$$ Extending by ${\ensuremath{\Bbbk}}$-linearity enables us to take the partial derivative of any element of ${\ensuremath{\Bbbk}}Q$. Define the partial derivative of the superpotential by setting $\partial_qW:= \partial_q(e_{{\operatorname{\mathsf{h}}}(q)}W e_{{\operatorname{\mathsf{h}}}(q)})$ for any path $q$. The expression $\partial_q W$ is simply the sum of all paths $p$ in $Q$ with tail at vertex ${\operatorname{\mathsf{h}}}(q)$, head at vertex ${\operatorname{\mathsf{t}}}(q)$ and labelling monomial $x^{{\operatorname{div}}(p)} = x_1x_2\cdots x_{d}/x^{{\operatorname{div}}(q)}$. For example, $\partial_{e_i}W$ is the sum of all anticanonical cycles that pass through vertex $i\in Q_0$. Consider now the set of paths $$\mathscr{P}:= \left\{ q \text{ in }Q \; \bigg| \begin{array}{c} \partial_q W \text{ is the sum of precisely two paths} \\ \text{ that share neither initial nor final arrow}\end{array} \right\}.$$ The condition that both summands of $\partial_q W$ share neither initial nor final arrow ensures that neither $\partial_{aq} W$ nor $\partial_{qa} W$ is the sum of precisely two paths for $a\in Q_1$.
\[def:Ftermequiv\] The *ideal of superpotential relations* is the two-sided ideal in ${\ensuremath{\Bbbk}}Q$ given by $$J_{W}:= \big( p^+-p^-\in {\ensuremath{\Bbbk}}Q \mid \exists\; q\in \mathscr{P} \text{ such that }\partial_qW = p^++p^-\big).$$ The *superpotential algebra* of $\mathscr{E}$ is $A_{W}:= {\ensuremath{\Bbbk}}Q/J_{W}$. Two paths $p_\pm$ in $Q$ are F*-term equivalent* if there is a finite sequence of paths $p_+=p_0, p_1, \dots, p_{k+1}=p_-$ in $Q$ such that for every $0\leq j\leq k$ we have $p_j- p_{j+1}=q_1(p^+-p^-)q_2$ for paths $q_1, q_2$ in $Q$ and some relation $p^+-p^-\in J_W$.
1. It is sometimes possible to introduce signs in $W$ so that the relevant partial derivatives of $W$ reproduce precisely the generators of $J_W$. Indeed, this is part of the defining data for dimer model algebras, and it is demonstrated for skew group algebras by Bocklandt–Schedler–Wemyss [@BSW]. However, we present in Section \[sec:conjecture\] a relatively simple example in dimension four for which this cannot be done.
2. The F-term equivalence classes of paths form a ${\ensuremath{\Bbbk}}$-vector space basis for $A_W$.
For each generator $p^+-p^-$ of $J_W$, the paths $p^\pm$ share the same head, tail and label so $J_W$ is contained in the ideal $J_{\mathscr{E}}$. If this inclusion is equality then the toric algebra $A_{\mathscr{E}}$ is isomorphic to the superpotential algebra $A_W$. However, this need not be the case as we now illustrate.
\[exa:superpotentialF1tilting\] We consider three collections on the threefold $X$ from Example \[exa:F1tilting\]:
1. For the collection $\mathscr{E}$ from Example \[exa:F1tilting\], the quiver of sections from Figure \[fig:tiltingF1\](b), contains six cycles $p$ with ${\operatorname{div}}(p)=x_1x_2x_3x_4$, giving $$W= a_8a_7a_4a_1 + a_8a_6a_4a_2 + a_9a_5a_2 + a_9a_7a_3 +a_{10}a_6a_3 + a_{10}a_5a_1.$$ It is easy to check that $J_W$ equals the ideal $J_\mathscr{E}$ from Example \[exa:F1tilting\], so $A_W\cong A_\mathscr{E}$.
2. The quiver of sections of $\mathscr{E}^\prime = \big(\mathcal{O}_X, \mathcal{O}_X(D_1),\mathcal{O}_X(D_4)\big)$ and the list of arrows are both shown in Figure \[fig:F1subandsupertilting\](a). We have $W= a_9a_3 + a_7a_4a_1 + a_6a_4a_2$, but in this case $A_W\not\cong A_{\mathscr{E}^\prime}$ because $a_6a_3 - a_5a_1 \in J_{\mathscr{E}^\prime}\setminus J_W$.
3. The quiver of sections of $\mathscr{E}^{\prime\prime} = \big(\mathcal{O}_X, \mathcal{O}_X(D_1),\mathcal{O}_X(D_4), \mathcal{O}_X(D_1+D_4), \mathcal{O}_X(2D_4)\big)$ and the list of arrows are both shown in Figure \[fig:F1subandsupertilting\](b). The superpotential is $$\begin{aligned}
W &= a_{10}a_7a_4a_1 + a_{10}a_6a_4a_2 + a_{11}a_8a_4a_2+ a_{11}a_9a_5a_2 \\
& \quad+ a_{11}a_9a_7a_3 +a_{12}a_8a_4a_1 + a_{12}a_9a_6a_3 + a_{12}a_9a_5a_1,
\end{aligned}$$ and the superpotential algebra $A_W$ is isomorphic to the toric algebra $A_{\mathscr{E}^{\prime\prime}}$ since $$J_W= \left(\begin{array}{c} \! a_5a_1 - a_6a_3,\; a_7a_4a_1 - a_6a_4a_2, \; a_5a_2 - a_7a_3, \; a_{10}a_6 - a_{11}a_8 \; \\ a_{12}a_9a_6 - a_{11}a_9a_7,\: a_{10}a_7 - a_{12}a_8, \: a_{8}a_4 - a_9a_5 \end{array}\right) = J_{\mathscr{E}^{\prime\prime}}.$$
Consistency
-----------
The following notion is adapted from that of algebraic consistency given by Broomhead [@Broomhead] for algebras that arise from superpotentials in a dimer model (see Section \[sec:dimers\]).
\[def:consistent\] A collection $\mathscr{E}$ of rank one reflexive sheaves that encodes a superpotential $W$ is *consistent* if the algebras $A_{\mathscr{E}}$ and $A_W$ are isomorphic. In this case, we say that $A_\mathscr{E}$ is consistent, and write $A$ for brevity if the collection $\mathscr{E}$ is clear from the context.
We begin our study of consistent toric algebras by establishing an important property of the labels on arrows.
\[prop:div(a)\] If $A$ is consistent then $x^{{\operatorname{div}}(a)}$ divides $\prod_{\rho\in \sigma(1)} x_\rho$ for every $a\in Q_1$.
For $a\in Q_1$, Lemma \[lem:algebra\] implies that $e_{{\operatorname{\mathsf{t}}}(a)}Ae_{{\operatorname{\mathsf{t}}}(a)}\cong R$. We consider two cases. Suppose first that there exists $b\in Q_1\setminus\{a\}$ with ${\operatorname{\mathsf{t}}}(b)={\operatorname{\mathsf{t}}}(a)$. Since $Q$ is strongly connected, there exist paths $p, q$ in $Q$ so that the compositions $pa$ and $qb$ are cycles in $Q$ beginning at vertex ${\operatorname{\mathsf{t}}}(a)$. Composing in two ways defines cycles $paqb$ and $qbpa$ with the same head, tail and divisor, so $paqb-qbpa\in J_\mathscr{E}$. Consistency forces $J_\mathscr{E}=J_W$, so $paqb$ and $qbpa$ are F-term equivalent. Since $b\neq a$, there must be a relation $p^+-p^-\in J_W$ with ${\operatorname{\mathsf{t}}}(p^\pm)={\operatorname{\mathsf{t}}}(a)$ for which $a$ lies in the support of one of $p^\pm$. Every such relation is obtained as a partial derivative of $W$, so $x^{{\operatorname{div}}(a)}$ divides $\prod_{\rho\in \sigma(1)} x_\rho$ as required. Suppose otherwise, so there does not exist $b\in Q_1\setminus\{a\}$ with ${\operatorname{\mathsf{t}}}(b)={\operatorname{\mathsf{t}}}(a)$. Then every cycle in $Q$ from ${\operatorname{\mathsf{t}}}(a)$ traverses arrow $a$ and hence for every element $u$ in the Hilbert basis of $\sigma^\vee\cap M$, the corresponding monomial $x^u\in R$ is divisible by $x^{{\operatorname{div}}(a)}$. The monomial $\prod_{\rho\in \sigma(1)} x_\rho$ is a product of such monomials, so $x^{{\operatorname{div}}(a)}$ divides $\prod_{\rho\in \sigma(1)} x_\rho$ as required.
\[cor:allarrowsinW\] If $A$ is consistent then every arrow in $Q$ arises in an anticanonical cycle and hence in a term of the superpotential $W$.
We reinterpret this result by lifting $Q$ to an $M$-periodic quiver in ${\ensuremath{\mathbb{R}}}^d$ using the sequence . The *covering quiver* $\widetilde{Q}$ is the quiver with vertex set $\widetilde{Q}_0=\bigoplus_{i\in Q_0} \deg^{-1}(E_i)$, and with arrow set comprising an arrow $\widetilde{a}$ from each $u\in \deg^{-1}(E_i)$ to $u+{\operatorname{div}}(a)\in \deg^{-1}(E_j)$ for every $a\in Q_1$ from $i$ to $j$. The *label* of $\widetilde{a}$ in $\widetilde{Q}_1$ is the vector ${\operatorname{div}}(\widetilde{a}):={\operatorname{\mathsf{h}}}(\widetilde{a})-{\operatorname{\mathsf{t}}}(\widetilde{a}) ={\operatorname{div}}(a)\in {\ensuremath{\mathbb{N}}}^d$.
\[rem:QinQuotient\] The quiver $Q$ can be recovered from $\widetilde{Q}$ by taking the quotient by the action of $M$. The given embedding of $\widetilde{Q}$ in ${\ensuremath{\mathbb{R}}}^d={\ensuremath{\mathbb{Z}}}^d\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ induces an embedding of $Q$ in ${\ensuremath{\mathbb{R}}}^d/M$.
We now lift the anticanonical cycles from $Q\subset {\ensuremath{\mathbb{R}}}^d/M$ to $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^d$. For each $u\in \widetilde{Q}_0$, let $p$ be an anticanonical cycle in $Q$ that passes through vertex $i:=\deg(u)\in Q_0$. An *anticanonical path from $u$ covering $p$* is any path $\widetilde{p}_u$ in $\widetilde{Q}$ from $u$ to $u +(1,\dots,1)\in \deg^{-1}(E_i)$ whose image in ${\ensuremath{\mathbb{R}}}^d/M$ is the cycle $p$ in $Q$. For $u, u^\prime\in \deg^{-1}(E_i)$, the anticanonical paths from $u$ differ only by translation from the anticanonical paths from $u^\prime$, so we need only study paths from one such vertex. For this, pick a fundamental region in ${\ensuremath{\mathbb{R}}}^d$ for the action of $M$ by choosing a spanning tree in $Q$, and lift to a connected tree in $\widetilde{Q}$. Each $i\in Q_0$ then has a preferred lift $u_i\in \deg^{-1}(E_i)$.
\[def:coveringquiver\] For $i\in Q_0$, let $\widetilde{Q}(i)$ be the quiver in ${\ensuremath{\mathbb{R}}}^d$ with vertex set $$\widetilde{Q}_0(i):=\Big\{v\in \widetilde{Q}_0 \mid\exists\; \text{anticanonical path }\widetilde{p}_{u_i} \text{ that touches }v\Big\}$$ and arrow set $$\widetilde{Q}_1(i):=\Big\{a\in \widetilde{Q}_1\mid\exists\; \text{anticanonical path }\widetilde{p}_{u_i} \text{ that traverses }a\Big\}.$$
\[rem:widetildeQi\]
1. An anticanonical cycle that passes through a vertex more than once gives rise to more than one anticanonical path in a given quiver $\widetilde{Q}(i)$, see Examples \[exa:trivialaction\]-\[exa:conifold\].
2. Since we lift only anticanonical cycles, the vertex set $\widetilde{Q}_0(i)$ is a subset of the set of vertices of the unit hypercube $\mathsf{C}(u_i):=\{u_i+(\lambda_1,\dots,\lambda_d)\in {\ensuremath{\mathbb{R}}}^d \mid 0\leq \lambda_j\leq 1 \text{ for }1\leq j\leq d\}$.
\[exa:trivialaction\] The quiver of sections $Q$ for the trivial collection $\mathscr{E}=(\mathcal{O}_X)$ on $X=\mathbb{A}^n_{\ensuremath{\Bbbk}}$ has one vertex and $n$ loops labelled $x_1, \dots, x_n$. There are $(n-1)!$ anticanonical cycles, and each lifts to $n$ anticanonical paths that emanate from each vertex $u\in {\ensuremath{\mathbb{Z}}}^n$. The support of the quiver $\widetilde{Q}(i)$ in ${\ensuremath{\mathbb{R}}}^n$ is precisely the support of the set of edges of the unit cube $\mathsf{C}(u_0)$.
\[exa:conifold\] For the conifold $X={\operatorname{Spec}}{\ensuremath{\Bbbk}}[x_1,x_2,x_3,x_4]/(x_1x_2-x_3x_4)$, the quiver of sections $Q$ from Figure \[fig:conifold\](a) defines the consistent toric algebra $A$ studied by Szendrői [@Szendroi Figure 1].
The edges of the unit 4-cube $\mathsf{C}(u_0)$ are shown in grey in Figure \[fig:conifold\](b), where the vertices $u_0$ and $u_0+(1,1,1,1)$ are labelled 0 at the bottom and top of the figure respectively. The four anticanonical paths from $u_0$ which cover the pair of anticanonical cycles in $Q$ define the quiver $\widetilde{Q}(0)$ whose vertices and arrows are shown in black in Figure \[fig:conifold\](b). The quiver $\widetilde{Q}(1)$ is similar.
\[cor:arrowsinA\] If $A$ is consistent then the vertex set and arrow set of $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^d$ coincides with the $M$-translates in ${\ensuremath{\mathbb{R}}}^d$ of the vertex set and arrow set of $\bigcup_{i\in Q_0} \widetilde{Q}(i)$.
This is little more than a restatement of Corollary \[cor:allarrowsinW\].
Moduli of quiver representations
--------------------------------
A walk $\gamma$ in $Q$ is an alternating sequence $i_l a_l \dotsb a_1 i_{1}$ of vertices $i_1, \dotsc, i_l$ and arrows $a_1,\dotsc, a_l$ where $a_{k}$ is an arrow between $i_{k}$ and $i_{k+1}$. A walk $\gamma$ is closed if $i_1 = i_l$. If ${\operatorname{\mathsf{t}}}(a_k) = i_{k}$ and ${\operatorname{\mathsf{h}}}(a_k) = i_{k+1}$ then $a_k$ is a *forward* arrow in $\gamma$; otherwise ${\operatorname{\mathsf{t}}}(a_k) = i_{k+1}$, ${\operatorname{\mathsf{h}}}(a_k)
= i_{k}$ and $a_k$ is a *backward* arrow. If $p$ is a path in $Q$ then $p^{-1}$ denotes the walk from ${\operatorname{\mathsf{h}}}(p)$ to ${\operatorname{\mathsf{t}}}(p)$ that traverses backwards each arrow from the support of $p$. For a walk $\gamma$ in $Q$ and for $a\in Q_1$, let ${\operatorname{mult}}_\gamma(a)\in {\ensuremath{\mathbb{Z}}}$ be the number of times $a$ appears as a forward arrow in $\gamma$ minus the number of times it appears as a backwards arrow. Set $v(\gamma):= \sum_{a\in Q_1} {\operatorname{mult}}_\gamma(a) \chi_a\in {\ensuremath{\mathbb{Z}}}^{Q_1}$. Consider the abelian group $$\Lambda = {\ensuremath{\mathbb{Z}}}^{Q_1}/\big(v(p^+) - v(p^-)\in {\ensuremath{\mathbb{Z}}}^{Q_1} \mid\exists\; q\in \mathscr{P} \text{ such that }\partial_qW = p^++p^-\big)$$ and the quotient map ${\operatorname{wt}}\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\to \Lambda$. Define the semigroup $\Lambda_+:= {\operatorname{wt}}({\ensuremath{\mathbb{N}}}^{Q_1})$.
\[lem:Kergens\] If $A$ is consistent then the maps $\pi\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\to {\ensuremath{\mathbb{Z}}}(Q)$ and ${\operatorname{wt}}\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\to\Lambda$ coincide. In particular, $A$ is graded by the semigroup $\Lambda_+= {\ensuremath{\mathbb{N}}}(Q)$.
It suffices to prove that ${\operatorname{ker}}(\pi) = L:=\big(v(p^+) - v(p^-)\in {\ensuremath{\mathbb{Z}}}^{Q_1} : p^+-p^-\in J_W\big)$. For $p^+-p^-\in J_W$, the paths $p^{\pm}$ share the same head, tail and divisor, so $v(p^+) - v(p^-)\in {\operatorname{ker}}(\pi)$. For the opposite inclusion, consider $v\in {\operatorname{ker}}(\pi)$. Since $\pi = ({\operatorname{inc}}, {\operatorname{div}})$ there is a closed walk $\gamma$ in $Q$ with $v(\gamma) \in {\operatorname{ker}}({\operatorname{div}})$. We now use an ‘elongation’ operation to replace $\gamma$ by a more convenient closed walk $\gamma'$ satisfying $v(\gamma')\in {\operatorname{ker}}({\operatorname{div}})$. First, write $\gamma$ as a sequence $\alpha_1^{\,} \alpha_2^{-1} \alpha_3^{\,} \dotsb \alpha_{2\ell -1}^{\,} \alpha_{2 \ell}^{-1}$ that alternates between paths $\alpha_{2i-1}$ ($1\leq i\leq \ell$) comprising forward arrows, and walks $\alpha_{2i}^{-1}$ ($1\leq i\leq \ell$) comprising backward arrows. For $1\leq i\leq \ell$, choose a path $\beta_{2i-1}$ from ${\operatorname{\mathsf{h}}}(\alpha_{2i-1})$ to $0\in Q_0$ and a path $\beta_{2i}$ from $0\in Q_0$ to ${\operatorname{\mathsf{t}}}(\alpha_{2i})$. For $\beta_{0} = \beta_{2 \ell}$, consider $$\beta_{0}^{\,} \alpha_1^{\,}\beta_1^{\,}
\beta_1^{-1}\alpha_2^{-1} \beta_{2}^{-1}\beta_2^{\,} \alpha_3^{\,}
\beta_3^{\,} \beta_3^{-1} \dotsb \alpha_{2 \ell}^{-1}
\beta_{2\ell}^{-1} \, .$$ Each composition $p_{2i+1}:=\beta_{2i}\alpha_{2i+1}\beta_{2i+1}$ is a cycle from $0$, while each $p_{2i}^{-1}:= \beta_{2i-1}^{-1}\alpha_{2i}^{-1}\beta_{2i}^{-1}$ is a closed walk from 0 comprising backwards arrows. Set $\gamma' = p_1p_3\cdots p_{2\ell-1}p_{2\ell}^{-1}p_{2\ell-2}^{-1}\dots p_{2}^{-1}$. Since $v(\gamma')=v(\gamma)= v\in {\operatorname{ker}}(\pi)$, the cycles $\gamma'_+:=p_1p_3\cdots p_{2\ell-1}$ and $\gamma'_-:=p_2p_4\dots p_{2\ell}$ share the same divisor and hence determine the same element in $A_{\mathscr{E}}$. Since $A_{\mathscr{E}}$ is consistent, the cycles $\gamma'_{\pm}$ determine the same element in $A_W$ so they are F-term equivalent. Thus, there exists a finite sequence of cycles $\gamma'_+=\gamma_0, \gamma_1,\dotsb, \gamma_{k+1}=\gamma'_-$ such that for each $0\leq j\leq k$, we have $\gamma_j-\gamma_{j+1} = q_1(p^+-p^-)q_2$ for some relation $p^+-p^-\in J_W$ and paths $q_1, q_2$ in $Q$. Expand $$v = v(\gamma') = v(\gamma'_+) - v(\gamma'_-) = \sum_{0\leq j\leq k} \big(v(\gamma_j) - v(\gamma_{j+1})\big).$$ The first statement follows as $v(\gamma_j) - v(\gamma_{j+1})\in L$ for $0\leq j\leq k$. The second statement follows immediately from the first by setting $\deg(p) = \pi(v(p))\in {\ensuremath{\mathbb{N}}}(Q)$ for any path $p$ in $Q$.
To uncover the geometry encoded in a consistent toric algebra we construct fine moduli spaces of quiver representations. For a path $p$ in $Q$ define $y_p:=\prod_{a\in {\operatorname{supp}}(p)} y_a\in {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]$ and $$I_W:= \big(y_{p^+}-y_{p^-}\in {\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}] \mid p^+-p^-\in J_W\big).$$ This ideal is homogeneous in the ${\operatorname{Wt}}(Q)$-grading, so the subscheme $\mathbb{V}(I_W)\subseteq \mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$ cut out by $I_W$ is invariant under the $T$-action from . For generic $\theta\in {\operatorname{Wt}}(Q)$, the GIT quotient $$\mathcal{M}_\theta:= \mathbb{V}(I_W){\ensuremath{/\!\!/\!}}_\theta T = {\operatorname{Proj}}\Big{(}\bigoplus_{j \geq 0} \big({\ensuremath{\Bbbk}}[{\ensuremath{\mathbb{N}}}^{Q_1}]/I_W)_{j \theta} \Big{)}$$ is the geometric quotient of the open subscheme of $\theta$-stable points of $\mathbb{V}(I_W)$ by the action of $T$. Following King [@King], $\mathcal{M}_\theta$ is the fine moduli space of isomorphism classes of $\theta$-stable representations of $Q$ with dimension vector $(1,1,\dots,1)\in {\ensuremath{\mathbb{N}}}^{Q_0}$ that satisfy the relations $J_W$. If a strictly $\theta$-semistable representation does exist, the resulting categorical quotient $\overline{\mathcal{M}_\theta}:= \mathbb{V}(I_W){\ensuremath{/\!\!/\!}}_\theta T$ is merely the coarse moduli space parametrising S-equivalence classes of $\theta$-semistable representations with dimension vector $(1,1,\dots,1)$ that satisfy $J_W$.
\[thm:cohcomp\] Let $A$ be consistent. For generic $\theta\in {\operatorname{Wt}}(Q)$ there is a commutative diagram $$\begin{CD}
Y_\theta @>>> \mathcal{M}_\theta \\
@V{\tau_\theta}VV @VVV \\
X @>>> \overline{\mathcal{M}_0} \\
\end{CD}$$ where the horizontal maps are closed immersions and the vertical maps are projective morphisms arising from variation of GIT quotient. Moreover, the toric variety $Y_\theta$ is the unique irreducible component of $\mathcal{M}_\theta$ containing the $T$-orbit closures of points of $\mathbb{V}(I_W)\cap ({\ensuremath{\Bbbk}}^\times)^{Q_1}$.
For generic $\theta\in {\operatorname{Wt}}(Q)$, we call $Y_\theta$ the *coherent component* of $\mathcal{M}_\theta$.
To construct the diagram, note that for each $p^+-p^-\in J_W$, the paths $p^{\pm}$ share head, tail and divisor, so $I_W\subseteq I_{\mathscr{E}}$. Therefore $Y_\theta\subseteq \overline{\mathcal{M}_\theta}$ for all $\theta\in {\operatorname{Wt}}(Q)$ which gives the lower horizontal map, and the top map follows since $\overline{\mathcal{M}_\theta} = \mathcal{M}_\theta$ for $\theta$ generic. The vertical maps of the diagram are well known. Lemma \[lem:Kergens\] shows that the vectors $\{v(p^+)-v(p^-)\in {\ensuremath{\mathbb{Z}}}^{Q_1}\}$ generate the lattice ${\operatorname{ker}}(\pi)$. The proof of [@CMT1 Theorem 3.10] applies verbatim to show that $\mathbb{V}(I_{\mathscr{E}})$ is the unique irreducible component of $\mathbb{V}(I_W)$ that does not lie in any coordinate hyperplane of $\mathbb{A}^{Q_1}_{\ensuremath{\Bbbk}}$. The proof of the final statement now follows precisely as in [@CMT1 Theorem 4.3[$(\mathrm{ii})$]{}].
The category of finite dimensional representations satisfying the relations $J_W$ is equivalent to the category ${\operatorname{mod}(\ensuremath{A_W})}$ of finite dimensional left $A_W$-modules. For $V=\bigoplus_{i \in Q_0} {\ensuremath{\Bbbk}}e_i$, this equivalence takes representations of dimension vector $(1,1,\dots,1)\in {\ensuremath{\mathbb{N}}}^{Q_0}$ to $A_W$-modules that are isomorphic as a $V$-module to $V$. For a consistent algebra $A$ and for generic $\theta\in {\operatorname{Wt}}(Q)$, Lemma \[lem:algebra\] then implies that $\mathcal{M}_\theta$ is the fine moduli space of $\theta$-stable ${\operatorname{End}}_R\bigl( \bigoplus_{i \in Q_0} E_i \bigr)$-modules that are isomorphic as a $V$-module to $V$ (see King [@King] for the notion of $\theta$-stability for modules).
Cellular resolution for abelian skew group algebras {#sec:McKay}
===================================================
This section realises the skew group algebra arising from a finite abelian subgroup of ${\operatorname{GL}}(n,{\ensuremath{\Bbbk}})$ as a consistent toric algebra. The geometry encoded by this toric algebra specialises to geometry that arises in the study of the McKay correspondence. The main result introduces the toric cell complex for the skew group algebra and constructs the cellular resolution in this case.
The McKay quiver of sections
----------------------------
Let $G$ be a finite abelian group of ${\operatorname{GL}}(n,\Bbbk)$ containing no quasireflections, where $\Bbbk$ is a field of characteristic not dividing the order of $G$. We may assume that $G$ is contained in the subgroup $(\Bbbk^{\times})^n$ of diagonal matrices with nonzero entries in ${\operatorname{GL}}(n,{\ensuremath{\Bbbk}})$. Setting $\rho_i(g)$ to be the $i$th diagonal element of the matrix $g$ defines $n$ elements $\rho_1,\dots,\rho_n$ of the character group $G^{*}={\operatorname{Hom}}(G,\Bbbk^{\times})$. The *McKay quiver* of $G\subset {\operatorname{GL}}(n,\Bbbk)$ is the quiver $Q$ with vertex set $G^{*}$, and an arrow $a^{\rho}_i$ from $\rho\rho_i$ to $\rho$ for each $\rho \in G^{*}$ and $1 \leq i \leq n$.
The dual action of $G$ on the coordinate ring ${\ensuremath{\Bbbk}}[x_1,\dots,x_n]$ of $\mathbb{A}^n_{\ensuremath{\Bbbk}}$ defines a $G^*$-grading with $\deg(x_i)=\rho_i$, and the $G$-invariant subalgebra $R={\ensuremath{\Bbbk}}[x_1,\dots,x_n]^G$ defines the normal affine toric variety $X={\operatorname{Spec}}R = \mathbb{A}_{\Bbbk}^n/G$. Since $G$ contains no quasireflections, the map assigning to each $\rho \in G^{*}$ the reflexive $R$-module $E_\rho$ spanned over ${\ensuremath{\Bbbk}}$ by semi-invariant polynomials of degree $\rho$ defines an isomorphism ${\operatorname{Cl}}(X)\cong G^*$. The quiver of sections of the collection $$\label{eqn:McKaycollection}
\mathscr{E}=(E_{\rho} \mid \rho \in G^{*})$$ on $X$ coincides with the McKay quiver of $G\subset {\operatorname{GL}}(n,\Bbbk)$, and the labelling monomial of each arrow $a_{i}^{\rho}$ is $x^{{\operatorname{div}}(a_{i}^{\rho})}=x_i$. It follows that the ideal of relations $J_{\mathscr{E}}$ is generated by elements of the form $a_i^{\rho \rho_j}a_j^{\rho}-a_j^{\rho \rho_i}a_i^{\rho}$ with $\rho \in G^{*}$ and $1 \leq i,j \leq n$. Apply [@CMT2 Proposition 2.8] to obtain:
\[lem:skewgroup\] The toric algebra $A_{\mathscr{E}}$ is isomorphic to the skew group algebra $\Bbbk[x_1,\dots,x_n]\ast G$.
Assume now that $G \subset {\operatorname{SL}}(n,\Bbbk)$, so $X$ is Gorenstein. Every anticanonical cycle in $Q$ traverses precisely $n$ arrows, one with each labelling monomial $x_i$ for $1\leq i\leq n$, and without loss of generality we choose the final arrow of any such cycle to have labelling monomial $x_n$ and head at vertex $\rho\in G^*$. Thus, every anticanonical cycle can be written uniquely in the form $$a_{n}^{\rho}a_{\sigma(n-1)}^{\rho\rho_{\sigma(n-1)}}\cdots a_{\sigma(2)}^{\rho\rho_{\sigma(2)}\cdots \rho_{\sigma(n-1)}}a_{\sigma(1)}^{\rho\rho_{\sigma(1)}\cdots \rho_{\sigma(n-1)}}$$ for some $\rho\in G^*$ and some permutation $\sigma$ on $n-1$ letters, so the superpotential for $\mathscr{E}$ is $$\label{eqn:McKaysuperpotential}
W=\sum_{\rho \in G^*}\sum_{\sigma \in \mathfrak{S}_{n-1}} a_{n}^{\rho}a_{\sigma(n-1)}^{\rho\rho_{\sigma(n-1)}}\cdots a_{\sigma(2)}^{\rho\rho_{\sigma(2)}\cdots \rho_{\sigma(n-1)}}a_{\sigma(1)}^{\rho\rho_{\sigma(1)}\cdots \rho_{\sigma(n-1)}} \in {\ensuremath{\Bbbk}}Q_{\mathrm{cyc}},$$ where $\mathfrak{S}_{n-1}$ is the set of permutations on $n-1$ letters. It is straightforward to verify that the ideal of superpotential relations $J_W$ coincides with $J_{\mathscr{E}}$, so we obtain the following result.
The toric algebra $A_{\mathscr{E}}$ is consistent.
\[rem:notsaturated\]
1. The superpotential $\Phi$ for the McKay quiver of $G \subset {\operatorname{SL}}(n,\Bbbk)$ introduced by Bocklandt–Schedler–Wemyss [@BSW §4] counts every anticanonical cycle precisely $n$ times. Thus, ignoring the sign of each term, the superpotential $W$ from equals $\frac{1}{n}\Phi$.
2. Craw–Maclagan–Thomas [@CMT1] introduce the coherent component $Y_{\theta}$ of the fine moduli space $\mathcal{M}_{\theta}$ of $\theta$-stable McKay quiver representations. For $\mathcal{M}_{\theta} = {\ensuremath{G}\operatorname{-Hilb}}$, this recovers Nakamura’s irreducible version $Y_{\theta}={\operatorname{Hilb}^{\ensuremath{G}}}$. Thus, for the subgroup $G\subset {\operatorname{GL}}(6,{\ensuremath{\Bbbk}})$ of order 625 from [@CMT2 Example 5.7], the coherent component ${\operatorname{Hilb}^{\ensuremath{G}}}$ and the variety $\mathbb{V}(I_Q)$ are not normal. In particular, the semigroup ${\ensuremath{\mathbb{N}}}(Q)$ defining $\mathbb{V}(I_Q)$ need not be saturated.
The toric cell complex {#sec:CWcomplex}
----------------------
A *cell* in a topological space is a subspace that is homeomorphic to the closed $k$-dimensional ball $B^k=\{ x \in {\ensuremath{\mathbb{R}}}^k \mid \|x\| \leq 1\}$ for some $k\in {\ensuremath{\mathbb{N}}}$. We use the term $k$-cell when we wish to make explicit the dimension of the cell. A finite *regular cell complex* $\Delta$ is a finite collection of cells in a Hausdorff topological space $\vert\Delta\vert:=\bigcup_{\eta \in \Delta}\eta$ such that we have each of the following: [$(\mathrm{i})$]{} $\emptyset \in \Delta$; [$(\mathrm{ii})$]{} the interiors of the nonempty cells partition $\vert\Delta\vert$; and [$(\mathrm{iii})$]{} the boundary of any cell in $\Delta$ is a union of cells in $\Delta$. Denote by $\Delta_k$ the set of $k$-cells in $\Delta$. The *faces* of a cell $\eta\in \Delta$ are the cells $\eta^\prime$ satisfying $\eta^\prime\subset \eta$, and *facets* of a cell are faces of codimension-one. The prototypical example of a finite regular cell complex is the set of faces of a convex polytope. Note that our cells are the closures of the open cells in the regular cell complexes described in Bruns–Herzog [@BrunsHerzog §6.2].
The most important property of a regular cell complex for this article is the existence of an incidence function $\varepsilon\colon \Delta\times\Delta\rightarrow \{0,\pm 1\}$. To state the definition, recall from [@BrunsHerzog §6.2] that regular cell complexes satisfy the following property: $$\label{eqn:facetsproperty}
\left\{\begin{array}{c} \text{If $\eta\in \Delta_k$ and $\eta^{\prime\prime}\in \Delta_{k-2}$ is a face of $\eta$, there exist precisely two cells} \\
\text{$\eta_1^\prime, \eta_2^\prime\in \Delta_{k-1}$ such that $\eta_j^\prime$ is a face of $\eta$ and $\eta^{\prime\prime}$ is a face of $\eta_j^{\prime}$ for $j=1,2$.}\end{array}\right.$$ An *incidence function* on $\Delta$ is a function $\varepsilon \colon \Delta\times \Delta \longrightarrow \{0,\pm1\}$ such that $\varepsilon(\eta,\eta^\prime)= 0$ unless $\eta^\prime$ is a facet of $\eta$, that $\varepsilon(\eta,\emptyset) = 1$ for all 0-cells $\eta$ and, moreover, that if $\eta\in \Delta_k$ and $\eta^{\prime\prime}\in \Delta_{k-2}$ is a face of $\eta$, then for the cells $\eta_1^\prime, \eta_2^\prime\in \Delta_{k-1}$ from we have $$\label{eqn:signcondition}
\varepsilon(\eta,\eta_1^\prime) \varepsilon(\eta_1^\prime,\eta^{\prime\prime}) +\varepsilon(\eta,\eta_2^\prime) \varepsilon(\eta_2^\prime,\eta^{\prime\prime})= 0.$$ Every regular cell complex $\Delta$ admits an incidence function, and any two such differ by the choice of orientation of each cell, see Bruns–Herzog [@BrunsHerzog Lemma 6.2.1, Theorem 6.2.2].
We now associate a regular cell complex $\Delta$ to the consistent collection $\mathscr{E}$ from on the Gorenstein quotient $X=\mathbb{A}^n_{\ensuremath{\Bbbk}}/G$ for a finite abelian subgroup $G\subset {\operatorname{SL}}(n,{\ensuremath{\Bbbk}})$ of order $r+1$. Let $\{\chi_i \mid 1\leq i\leq n\}$ denote the standard basis of ${\ensuremath{\mathbb{Z}}}^n$. The short exact sequence for $X$ is $$\label{eqn:McKaysequence}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}^n @>{\deg}>> G^*@>>> 0,
\end{CD}$$ where $\deg(\chi_i)=\rho_i$. The covering quiver $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^n$ of the McKay quiver $Q$ has vertex set $\widetilde{Q}_0={\ensuremath{\mathbb{Z}}}^n$, and for each $u\in {\ensuremath{\mathbb{Z}}}^n$ there is an arrow from $u$ to $u+\chi_i$ for $1\leq i\leq n$. Each arrow in $\widetilde{Q}$ is supported on an edge of a unit hypercube $\mathsf{C}(u)\subset {\ensuremath{\mathbb{R}}}^n$. Remark \[rem:QinQuotient\] shows that the image of $\widetilde{Q}$ under the natural projection to the real $n$-torus ${\ensuremath{\mathbb{R}}}^n \rightarrow \mathbb{T}^n:={\ensuremath{\mathbb{R}}}^n/M$ defines an embedding of the McKay quiver $Q$ in $\mathbb{T}^n$, so each arrow of $Q$ with tail at vertex $\rho\in G^*$ is supported on an edge of the image of an $n$-cell $\mathsf{C}(u)$ in $\mathbb{T}^n$ for some $u\in \deg^{-1}(\rho)$. We let $\Delta(\rho)$ denote the set of all cells in $\mathbb{T}^n$ obtained as the image of a face of the hypercube $\mathsf{C}(u)$ for some $u\in \deg^{-1}(\rho)$. The union $\Delta:=\bigcup_{\rho\in G^*} \Delta(\rho)$ is a regular cell complex in $\mathbb{T}^n$ comprising all cells obtained as the projection to $\mathbb{T}^n$ of the faces of the $r+1$ hypercubes $\Delta(\rho)\subset {\ensuremath{\mathbb{R}}}^n$ for $\rho\in G^*$.
\[def:McKaycellcomplex\] The *toric cell complex* for the McKay quiver $Q$ is the finite regular cell complex $\Delta$ in $\mathbb{T}^n$. We also refer to $\Delta$ as the toric cell complex of the subgroup $G \subset {\operatorname{SL}}(n,\Bbbk)$ or, equivalently, of the collection $\mathscr{E}$ from .
\[lem:McKaybijections\] There are canonical bijections between $\Delta_0$ and $Q_0$, between $\Delta_1$ and $Q_1$, and between $\Delta_2$ and the set $\{a_i^{\rho \rho_j}a_j^{\rho}-a_j^{\rho \rho_i}a_i^{\rho} \mid \rho \in G^{*}, 1 \leq i<j \leq n\}$ of minimal generators of $J_\mathscr{E}$.
The bijections for $\Delta_0$ and $\Delta_1$ are described in the construction of $\Delta$ above. As for the final bijection, the closed walk in $Q$ obtained by first traversing the path $a_i^{\rho \rho_j}a_j^{\rho}$ with orientation and then traversing the path $a_j^{\rho \rho_i}a_i^{\rho}$ against orientation lifts to a closed walk in $\widetilde{Q}$ that traverses the boundary of a 2-dimensional face $F$ of the cube $\mathsf{C}(u)$ for each $u\in \deg^{-1}(\rho^\prime)$ with $\rho^\prime = \rho\rho_i\rho_j$. The arrows in the boundary of the resulting 2-cell $\eta\in \Delta$ are precisely the arrows in the relation $a_i^{\rho \rho_j}a_j^{\rho}-a_j^{\rho \rho_i}a_i^{\rho}$. Conversely, every 2-cell arises from a unique relation in this way.
Every cell $\eta\in\Delta$ is the image in $\mathbb{T}^n$ of a face $F\subset\mathsf{C}(u)$ where $u\in {\ensuremath{\mathbb{Z}}}^n$. Write $u_{{\operatorname{\mathsf{h}}}}$ and $u_{{\operatorname{\mathsf{t}}}}$ for the vertices of $F$ that intersect the family of affine hyperplanes $H_\lambda:= \{u\in {\ensuremath{\mathbb{R}}}^n \mid \sum_i u_i=\lambda\}$ at the maximum and minimum value of $\lambda$ respectively. The vertices $u_{{\operatorname{\mathsf{h}}}}$ and $u_{{\operatorname{\mathsf{t}}}}$ depend on the choice of $F$, but their images ${\operatorname{\mathsf{h}}}(\eta)\in \Delta_0$ and ${\operatorname{\mathsf{t}}}(\eta)\in\Delta_0$ in $\mathbb{T}^n$ do not, and we call these the *head* and *tail* vertices of $\eta$. The *divisor* of $\eta$ is the element ${\operatorname{div}}(\eta):=u_{{\operatorname{\mathsf{h}}}}-u_{{\operatorname{\mathsf{t}}}}\in {\ensuremath{\mathbb{N}}}^n$. The following duality property of the toric cell complex is evident from the construction.
\[prop:dualityMcKay\] The map $\tau\colon \Delta\to \Delta$ that assigns to each $\eta\in \Delta_k$ the unique cell $\eta^\prime\in \Delta_{n-k}$ with ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$, ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$ and $x^{{\operatorname{div}}(\eta^\prime)} = \prod_{\rho\in \sigma(1)}x_\rho/x^{{\operatorname{div}}(\eta)}$ is an involution.
We now introduce the notion of right- and left-differentiation of cells with respect to faces. Let $\eta\in \Delta$. For any face $\eta^\prime\subset \eta$ there is a path in $\widetilde{Q}$ from ${\operatorname{\mathsf{h}}}(\eta^\prime)$ to ${\operatorname{\mathsf{h}}}(\eta)$. While this path need not be unique, its image in $Q$ is a well-defined F-equivalence class of paths that we denote $\overleftarrow{\partial}_{\!\eta'}\eta\in A$. Similarly, there is a path in $\widetilde{Q}$ from ${\operatorname{\mathsf{t}}}(\eta)$ to ${\operatorname{\mathsf{t}}}(\eta^\prime)$ that defines an F-equivalence class of paths in $Q$, denoted $\overrightarrow{\partial}_{\!\eta'}\eta\in A$.
\[def:leftrightderivativesmckay\] For $\eta\in \Delta$ and any face $\eta^\prime\subset \eta$, the element $\overleftarrow{\partial}_{\!\eta'}\eta\in A$ is the *left-derivative* of $\eta$ with respect to $\eta'$. Similarly, $\overrightarrow{\partial}_{\!\eta'}\eta\in A$ is the *right-derivative* of $\eta$ with respect to $\eta'$.
For group action of type $\frac{1}{6}(1,2,3)$, Figure \[fig:McKayCW\] illustrates a fundamental region for $\Delta$ in ${\ensuremath{\mathbb{R}}}^3$ (some $k$-cells are repeated for $k<3$). Observe that $\vert\Delta_0\vert = \vert \Delta_3\vert = 6$ and $\vert \Delta_1\vert = \vert \Delta_2\vert = 18$.
(0,-0.3)(9.9,2) (0,0)[A]{} (1.5,0)[B]{}(3,0)[C]{} (4.5,0)[D]{}(6,0)[E]{}(7.5,0)[F]{} (9,0)[G]{} (0,1.5)[A1]{} (1.5,1.5)[B1]{}(3,1.5)[C1]{} (4.5,1.5)[D1]{}(6,1.5)[E1]{}(7.5,1.5)[F1]{} (9,1.5)[G1]{} (1,0.4)[(0,0)[A2]{} (1.5,0)[B2]{}(3,0)[C2]{} (4.5,0)[D2]{}(6,0)[E2]{}(7.5,0)[F2]{} (9,0)[G2]{} (0,1.5)[A3]{} (1.5,1.5)[B3]{}(3,1.5)[C3]{} (4.5,1.5)[D3]{}(6,1.5)[E3]{}(7.5,1.5)[F3]{} (9,1.5)[G3]{}]{}
Let $\eta\in \Delta_3$ denote the 3-cell on the far left of Figure \[fig:McKayCW\] and $\eta^\prime\in \Delta_2$ the facet (drawn horizontally) that contains the 0-cells 0, 1, 2, 3. Then ${\operatorname{\mathsf{h}}}(\eta^\prime)=3$ and ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{h}}}(\eta)=0$. The right derivative is $\overrightarrow{\partial}_{\!\eta'}\eta = e_0\in A$ and the left derivative is the vertical arrow $\overleftarrow{\partial}_{\!\eta'}\eta = a_3^0\in A$.
The cellular resolution
-----------------------
For $0 \leq k \leq n$, consider the ${\ensuremath{\Bbbk}}$-vector space $U_k=\bigoplus_{\eta \in \Delta_k}\Bbbk \cdot [\eta]$, where $[\eta]$ is a formal symbol. Lemma \[lem:Kergens\] shows that the skew group algebra $A$ admits a natural $\Lambda_{+}$-grading, and since each cell $\eta \in \Delta_{k}$ is homogeneous we obtain a $\Lambda_{+}$-grading on $U_{k}$. Identify the semisimple algebra $U_0$ with the subalgebra of $\Bbbk Q$ generated by the trivial paths. Note that $U_k$ is a $(U_0,U_0)$-bimodule, and consider the induced $(A,A)$-bimodule $$P_k=A \otimes_{U_0} U_k \otimes_{U_0} A=\bigoplus_{\eta \in \Delta_k}Ae_{{\operatorname{\mathsf{h}}}(\eta)} \otimes [\eta] \otimes e_{{\operatorname{\mathsf{t}}}(\eta)}A.$$ Note that $P_k$ inherits a $\Lambda_{+}$-grading, called the *total $\Lambda_+$-grading*, in which the degree of a product of homogeneous elements is given by the sum of the degrees in each of the three positions. For $0 \leq k \leq n$, define a morphism $d_k \colon P_k \rightarrow P_{k-1}$ of $\Lambda_+$-graded graded $(A,A)$-bimodules by setting $$d_k(1 \otimes [\eta] \otimes 1)=\sum_{{\operatorname{cod}}(\eta^\prime, \eta) = 1}\varepsilon(\eta,\eta')\; \overleftarrow{\partial}_{\!\eta'}\eta\otimes[\eta'] \otimes \overrightarrow{\partial}_{\!\eta'}\eta,$$ where $\varepsilon$ is an incidence function on $\Delta$. It is convenient to choose $\varepsilon$ to be compatible with the orientation of arrows in $Q$ as follows. Identify each $\eta\in \Delta_1$ with an arrow $a\in Q_1$ according to Lemma \[lem:McKaybijections\], so the 1-cell contains precisely two 0-cells ${\operatorname{\mathsf{h}}}(a), {\operatorname{\mathsf{t}}}(a)\in \Delta_0$. Choosing $\varepsilon(a, {\operatorname{\mathsf{h}}}(a)) = 1$ forces $\varepsilon(a, {\operatorname{\mathsf{t}}}(a)) = -1$ by and hence $$d_1(1 \otimes [a] \otimes 1)=1 \otimes [{\operatorname{\mathsf{h}}}(a)] \otimes a - a \otimes [{\operatorname{\mathsf{t}}}(a)] \otimes 1.$$ Let $\mu \colon P_0=A\otimes_{U_0}A \rightarrow A$ denote the multiplication map.
\[prop:McKaycomplex\] For any choice of incidence function $\varepsilon$, the sequence $$\label{eqn:Koszul}
0 \longrightarrow P_n \xlongrightarrow{d_n} \cdots \xlongrightarrow{d_2} P_1 \xlongrightarrow{d_1} P_0 \xlongrightarrow{\mu} A \longrightarrow 0,$$ is a complex of $\Lambda_+$-graded $(A,A)$-bimodules. Moreover, an alternative incidence function determines a new complex that is naturally isomorphic to that from .
Note that, by our preceding remark, the cokernel of $d_1$ at $P_0$ is just the multiplication map $\mu$. Let us assume $k \geq 2$ and take $\eta \in \Delta_k$. Then $$\begin{aligned}
d_{k-1}(d_k(1 \otimes [\eta] \otimes 1)) &= \sum_{{\operatorname{cod}}(\eta^\prime, \eta) = 1}\varepsilon(\eta,\eta^\prime) \sum_{{\operatorname{cod}}(\eta^{\prime\prime}, \eta^\prime) = 1}\varepsilon(\eta',\eta'') \overleftarrow{\partial}_{\!\eta'}\eta \overleftarrow{\partial}_{\!\eta''}\eta' \otimes[\eta''] \otimes \overrightarrow{\partial}_{\!\eta''}\eta' \overrightarrow{\partial}_{\!\eta'}\eta \\
&= \sum_{{\operatorname{cod}}(\eta^\prime, \eta) = 1}\sum_{{\operatorname{cod}}(\eta^{\prime\prime}, \eta^\prime) = 1}\varepsilon(\eta,\eta')\varepsilon(\eta',\eta'') \; \overleftarrow{\partial}_{\!\eta''}\eta \otimes[\eta''] \otimes \overrightarrow{\partial}_{\!\eta''}\eta.\end{aligned}$$ If $\eta'' \in \Delta_{k-2}$ is a face of $\eta$ then the only contributions in the double sum come from terms involving the facets $\eta'_1, \eta'_2\subset \eta$ from containing $\eta''$. The above sum is therefore $$d_{k-1}(d_k(1 \otimes [\eta] \otimes 1)) = \sum_{{\operatorname{cod}}(\eta^{\prime\prime},\eta) = 2}\Big(\varepsilon(\eta,\eta'_1)\varepsilon(\eta'_1,\eta'')+\varepsilon(\eta,\eta'_2)\varepsilon(\eta'_2,\eta'')\Big) \overleftarrow{\partial}_{\!\eta''}\eta \otimes[\eta''] \otimes \overrightarrow{\partial}_{\!\eta''}\eta,$$ taken as a sum over codimension-two faces of $\eta$. This sum is zero by equation . For the second statement, let $\varepsilon'$ be another incidence function and consider the complex $$0 \longrightarrow P_n \xlongrightarrow{d'_n} \cdots \xlongrightarrow{d'_2} P_1 \xlongrightarrow{d'_1} P_0 \xlongrightarrow{\mu} A \longrightarrow 0$$ determined by $\varepsilon'$. As [@BrunsHerzog Theorem 6.2.2] records, there exists a global sign function $\delta\colon \Delta \rightarrow \{\pm 1 \}$ such that $\varepsilon'(\eta,\eta')=\delta(\eta')\varepsilon(\eta,\eta')\delta(\eta)$ for all $\eta \in \Delta_k$, $\eta' \in \Delta_{k-1}$, $0 \leq k \leq n$. This implies that the bimodule homomorphisms $\phi_k\colon P_k \rightarrow P_k$ given by $\phi_k(1 \otimes [\eta]\otimes 1)=\delta(\eta)\otimes [\eta]\otimes 1$ define a chain map of complexes $\phi\lbdot\!\! \colon (P\lbdot\!\!,d\lbdot\!\!) \rightarrow (P\lbdot\!\!,d_{{^{^{_{_{_{_{\bullet}}}}}}}}'\!\!)$. Since each $\phi_k$ is an isomorphism of $(A,A)$-bimodules, the chain map $\phi\lbdot\!\!$ is an isomorphism of complexes. This completes the proof.
To demonstrate that the complex is a minimal projective $(A,A)$-bimodule resolution of $A$ we choose a suitable incidence function $\varepsilon$ on $\Delta$. For $\eta \in \Delta_k$, let $\eta' \subset \eta$ be a facet. We may write $x^{{\operatorname{div}}(\eta)}=x_{i_1}\cdots x_{i_k}$ and $x^{{\operatorname{div}}(\eta')}=x_{i_1}\cdots \widehat{x_{i_{\nu}}}\cdots x_{i_k}$ for $i_1<\cdots <i_k$, where $ \widehat{x_{i_{\nu}}}$ means that the factor $x_{i_{\nu}}$ is removed. We determine an incidence function on $\Delta$ by setting $$\varepsilon(\eta,\eta')=\left\{
\begin{array}{ll}
(-1)^{\nu} & \text{if ${\operatorname{\mathsf{h}}}(\eta)={\operatorname{\mathsf{h}}}(\eta')$ and $x^{{\operatorname{div}}( \overleftarrow{\partial}_{\!\eta'}\eta)}=x_{i_{\nu}}$;} \\
(-1)^{\nu +1} & \text{if ${\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{t}}}(\eta')$ and $x^{{\operatorname{div}}( \overrightarrow{\partial}_{\!\eta'}\eta)}=x_{i_{\nu}}$.}
\end{array} \right.$$ For a fixed $\eta \in \Delta_k$ with head at $\rho:={\operatorname{\mathsf{h}}}(\eta)$, let $\eta'_1,\dots,\eta'_k$ denote the facets of $\eta$ with head at $\rho$; similarly, let $\overline{\eta}'_1,\dots,\overline{\eta}'_k$ denote the facets of $\eta$ with tail at ${\operatorname{\mathsf{t}}}(\eta)$. For the above choice of $\varepsilon$, the differential $d_k\colon P_k \rightarrow P_{k-1}$ is given by $$\label{eqn:revisedcomplex}
d_k(1\otimes [\eta]\otimes 1)=\sum_{\nu=1}^k(-1)^{\nu} a_{i_{\nu}}^{\rho} \otimes [\eta'_{\nu}]\otimes 1 + \sum_{\nu=1}^k(-1)^{\nu+1} \otimes [\overline{\eta}'_{\nu}]\otimes a_{i_{\nu}}^{\rho \rho_{i_{1}}\cdots \widehat{\rho_{i_{\nu}}}\cdots \rho_{i_{k}}} .$$
\[thm:McKayresolution\] The complex is a minimal projective $(A,A)$-bimodule resolution of $A$; this is the *cellular resolution* of $A$.
Lemma \[lem:skewgroup\] implies that $A$ is Koszul, and Proposition \[prop:McKaycomplex\] ensures that we need only show that the complex with differentials given by coincides with the bimodule Koszul complex. One approach is to write down explicitly the isomorphism between and the bimodule Koszul complex as presented, for example, in Bocklandt–Schedler–Wemyss [@BSW Lemma 6.1]. More directly, by extending the ground category from vector spaces to $(U_0,U_0)$-bimodules, the bimodule Koszul complex constructed by Taylor [@Taylor Equation (4.4)] provides the bimodule Koszul complex of $A$. For the basis $\chi_1,\dots, \chi_n$ of $V:={\ensuremath{\Bbbk}}^n$ and for a cell $\eta \in \Delta_{k}$ with $x^{{\operatorname{div}}(\eta)}=x_{i_1}\cdots x_{i_k}$, the assignment $[\eta] \mapsto \chi_{i_1}\wedge \dots \wedge \chi_{i_k}$ determines an isomorphism from $U_k$ to $\bigwedge^k V$. It follows that the projective $(A,A)$-bimodules $P_k$ are those from the bimodule Koszul complex presented in [@Taylor Equation (4.4)]. In addition, our choice of signs in the differentials from recovers those in Equations (4.1) and (4.4) from [@Taylor]. This completes the proof.
Algebraically consistent dimer models {#sec:dimers}
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This section interprets algebraically consistent dimer models as consistent toric algebras. The main result reproduces the $(A,A)$-bimodule resolution of $A$ from [@Broomhead; @Davison; @MozgovoyReineke] as a cellular resolution, and reconstructs the subdivision of the real two-torus determined by the dimer model in terms of anticanonical cycles in the quiver. The key step associates a label to each arrow in the quiver of the dimer model. The first appearance of this technique in the dimer model literature seems to be Eager [@Eager §6].
On dimer models
---------------
A *dimer model* $\Gamma$ on a torus is a polygonal cell decomposition of the surface of a real two-torus whose vertices and edges form a bipartite graph. Each vertex may be coloured either black or white so that each edge joins a black vertex to a white vertex. The dual cell decomposition of the torus has a vertex dual to every face, an edge dual to every edge, and face dual to every vertex of $\Gamma$. In addition, we orient the edges of this dual decomposition so that a white vertex of the dimer lies on the left as the arrow crosses the dual edge of the dimer. The vertices and edges of this dual decomposition therefore define a quiver $Q=(Q_0, Q_1)$ embedded in the two-torus, with the additional property that the set of faces decomposes as the union $Q_2 = Q_2^+\cup Q_2^-$ of white faces (oriented anticlockwise) and black faces (oriented clockwise).
To each face $F\in Q_2$ we associate the cycle $w_F\in {\ensuremath{\Bbbk}}Q_{\mathrm{cyc}}$ obtained by tracing all arrows around the boundary of $F$. The *superpotential* of the dimer model $\Gamma$ is defined to be $$W_\Gamma := \sum_{F\in Q_2^+} w_F - \sum_{F\in Q_2^-} w_F.$$ For any face $F\in Q_2$ and arrow $a\in {\operatorname{supp}}(w_F)$, choose ${\operatorname{\mathsf{h}}}(a)\in Q_0$ as the starting point of $w_F$ and write $e_{{\operatorname{\mathsf{h}}}(a)}w_Fe_{{\operatorname{\mathsf{h}}}(a)} = aa_{l}\cdots a_1$. The partial derivative of the cycle $w_F$ with respect to $a$ is the path $\partial_a w_F = a_l\cdots a_1$ in $Q$. Extending ${\ensuremath{\Bbbk}}$-linearly gives $\partial_a W_\Gamma\in {\ensuremath{\Bbbk}}Q$ for each $a\in Q_1$, and consider the two-sided ideal $J_\Gamma:= (\partial_a W_\Gamma \mid a\in Q_1)$ in ${\ensuremath{\Bbbk}}Q$. The *superpotential algebra* of $\Gamma$ is $$A_\Gamma:={\ensuremath{\Bbbk}}Q/J_{\Gamma}.$$ A *perfect matching* $\Pi$ of the dimer model is a subset of the edges in $\Gamma$ such that every vertex is the endpoint of precisely one edge. Let ${\operatorname{supp}}(\Pi)$ denote the subset of $Q_1$ dual to the edges in $\Pi$. Since the arrows arising in any given term $\pm w_F$ of $W_\Gamma$ are dual to the set of edges emanating from the corresponding vertex of $\Gamma$, one can rewrite the superpotential in terms of any perfect matching $\Pi$ as $W_\Gamma = \sum_{a\in {\operatorname{supp}}(\Pi)} a\cdot \partial_a W$. Every arrow $a\in Q_1$ occurs in precisely two oppositely oriented faces, so every relation can be written as a path difference $\partial_aW_\Gamma = p_a^+ - p_a^-$, where $p_a^{\pm}$ are paths with tail at ${\operatorname{\mathsf{h}}}(a)$ and head at ${\operatorname{\mathsf{t}}}(a)$. The binomials $\{p_a^+ -p_a^- \in {\ensuremath{\Bbbk}}Q\mid a\in Q_1\}$ are the F*-term relations* of $\Gamma$, and two paths $p_\pm$ in $Q$ are said to be F*-term equivalent* if there is a finite sequence of paths $p_+=p_0, p_1, \dots, p_{k+1}=p_-$ in $Q$ such that for each $0\leq j\leq k$ we have $p_j- p_{j+1}=q_1(p_a^+-p_a^-)q_2$ for some paths $q_1, q_2$ in $Q$ and for some arrow $a\in Q_1$. The F-term equivalence classes of paths form a ${\ensuremath{\Bbbk}}$-vector space basis for $A_\Gamma$.
Several notions of consistency for dimer models have been introduced in the literature, and here we consider that of algebraic consistency due to Broomhead [@Broomhead]. Put simply, a dimer model is *algebraically consistent* if $A_\Gamma$ is isomorphic to an auxilliary algebra constructed from toric data encoded by $\Gamma$. We choose not to reconstruct this toric data here (though see the proof of Proposition \[prop:dimerlabels\]), but we do recall results of Broomhead [@Broomhead] showing that for each algebraically consistent dimer model $\Gamma$ the centre of $A$ is a Gorenstein semigroup algebra $R={\ensuremath{\Bbbk}}[\sigma^\vee\cap M]$ of dimension three and, moreover, that there exists a collection of rank one reflexive $R$-modules $(\mathcal{B}_i \mid i\in Q_0)$ such that $A_\Gamma \cong {\operatorname{End}}_{R}(\bigoplus_{i\in Q_0} \mathcal{B}_i)$. To obtain our preferred normalisation, choose a vertex $0\in Q_0$ and consider instead the $R$-modules $E_i:= \mathcal{B}_i\otimes \mathcal{B}_0^{-1}$ for $i\in Q_0$. Thus, every algebraically consistent dimer model $\Gamma$ defines a collection of rank one reflexive sheaves $$\label{eqn:dimercollection}
\mathscr{E}:=(E_i \mid i\in Q_0)$$ on the Gorenstein toric variety $X:={\operatorname{Spec}}R$ such that $A_\Gamma \cong {\operatorname{End}}_{R}(\bigoplus_{i\in Q_0} E_i)$.
\[lem:dimerqos\] The quiver $Q$ arising from an algebraically consistent dimer model $\Gamma$ is the quiver of sections of the collection $\mathscr{E}$ from . In particular, $A_\Gamma\cong A_\mathscr{E}$.
Let $Q^\prime$ denote the quiver of sections of $\mathscr{E}$, so $Q_0=Q_0^\prime$ by construction. For any $i,j\in Q_0$, the isomorphism $A_\Gamma \cong {\operatorname{End}}_{R}(\bigoplus_{i\in Q_0} E_i)$ implies $e_jA_\Gamma e_i \cong {\operatorname{Hom}}_R(E_i,E_j)$. The set of arrows in $Q$ from $i$ to $j$ provides a basis for the space spanned by irreducible elements of $e_jA_\Gamma e_i$, while the set of arrows in $Q^\prime$ from $i$ to $j$ does likewise for ${\operatorname{Hom}}_R(E_i,E_j)$. This gives $Q_1^\prime = Q_1$ as required. The final statement follows from Lemma \[lem:algebra\].
Labels on arrows in a dimer model
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We may not deduce from Lemma \[lem:dimerqos\] that $\mathscr{E}$ is consistent because the dimer model algebra $A_\Gamma$ is not a priori isomorphic to the superpotential algebra $A_W$ determined by the collection $\mathscr{E}$. To establish the link between $A_\Gamma$ and $A_W$ we investigate the labelling of arrows in $Q$. To begin we present an example that illustrates how our labelling of arrows in $Q$ ties in with the traditional approach to a dimer model $\Gamma$.
\[exa:dimerF1\] Consider the dimer model $\Gamma$ on the real two-torus shown in Figure \[fig:dimerF1\](a) and the quiver $Q$ embedded in the dual cell decomposition from Figure \[fig:dimerF1\](b).
Notice that $Q$ coincides with the quiver from Figure \[fig:tiltingF1\](c), and we list the arrows $a_1,\dots,a_{10}$ in the same way. It is well known that the semigroup algebra $R={\ensuremath{\Bbbk}}[\sigma^\vee\cap M]$ arising from $\Gamma$ is determined by the cone $\sigma$ over the lattice polygon $P$ from Figure \[fig:tiltingF1\](a). The extremal perfect matchings $\Pi_1, \Pi_2, \Pi_3, \Pi_4$ that correspond to the vertices $v_1, v_2, v_3, v_4\in P$ respectively are shown in Figure \[fig:perfectmatchingsF1\].
To compute the labels, note from Figure \[fig:dimerF1\](b) that ${\operatorname{supp}}(\Pi_1) = \{a_1, a_6, a_9\}$. Since $\Pi_1$ is the only extremal perfect matching containing either $a_1$ or $a_6$, Proposition \[prop:perfmatchlabels\] implies that both $x^{{\operatorname{div}}(a_1)}$ and $x^{{\operatorname{div}}(a_6)}$ are pure powers of $x_1$, whereas $a_9\in {\operatorname{supp}}(\Pi_1)\cap{\operatorname{supp}}(\Pi_2)$, so $x_1x_2$ divides $x^{{\operatorname{div}}(a_9)}$. Lemma \[lem:multiplicity\] below shows that the labelling monomial on each arrow in a dimer model is reduced, so $x^{{\operatorname{div}}(a_1)}=x^{{\operatorname{div}}(a_6)}=x_1$ and $x^{{\operatorname{div}}(a_9)}=x_1x_2$. It is now easy to see that the labelling monomials on the arrows of $Q$ are precisely those from Figure \[fig:tiltingF1\](b). The superpotential $$W_\Gamma= -a_8a_7a_4a_1 + a_8a_6a_4a_2 - a_9a_5a_2 + a_9a_7a_3 -a_{10}a_6a_3 + a_{10}a_5a_1$$ coincides, up to the sign of each term, with that from Example \[exa:F1tilting\] and hence $A_\Gamma\cong A_W$.
\[prop:dimerlabels\] Let $Q$ denote the quiver arising from an algebraically consistent dimer model $\Gamma$, and let $\mathscr{E}$ be the collection from . Then each $a\in Q_1$ satisfies $$\label{eqn:label}
x^{{\operatorname{div}}(a)} = \prod_{\{\rho\in \sigma(1) \mid a\in {\operatorname{supp}}(\Pi_\rho)\}} x_\rho,$$ where $\Pi_\rho$ is the unique perfect matching of $\Gamma$ corresponding to the vertex $\rho\in \sigma(1)$.
The quiver of sections $Q$ of $\mathscr{E}$ encodes the commutative diagram from Lemma \[lem:diagram\]. For each $a\in Q_1$, the F-term relation $p_a^+ -p_a^-$ is a binomial contained in the defining ideal of $A_{\Gamma}$. Lemma \[lem:dimerqos\] gives $A_\Gamma\cong A_\mathscr{E}$, so $p_a^+ -p_a^-\in J_\mathscr{E}$ and we deduce that $p_a^{\pm}$ share not only the same head and tail but also the same labelling divisor. This implies that $v(p_a^+) - v(p_a^-)\in {\operatorname{ker}}(\pi)$. The proof of Lemma \[lem:Kergens\] applies verbatim to show that $\pi\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\to {\ensuremath{\mathbb{Z}}}(Q)$ coincides with the map $${\operatorname{wt}}\colon {\ensuremath{\mathbb{Z}}}^{Q_1}\to \Lambda_\Gamma:= {\ensuremath{\mathbb{Z}}}^{Q_1}/\big(v(p_a^+) - v(p_a^-)\in {\ensuremath{\mathbb{Z}}}^{Q_1} \mid \;a\in Q_1\big)$$ from Mozgovoy–Reineke [@MozgovoyReineke §3]. Then ${\ensuremath{\mathbb{N}}}(Q)$ coincides with the semigroup $\Lambda_\Gamma^+:={\operatorname{wt}}({\ensuremath{\mathbb{N}}}^{Q_1})$ that was introduced by Broomhead [@Broomhead Example 5.5] in defining algebraic consistency (compare Mozgovoy [@Mozgovoy Remark 3.8]). It follows that our cone $C$ coincides with the cone dual to $\Lambda_\Gamma^+$ from [@Broomhead], so the perfect matchings from Definition \[def:perfectmatching\] agree with those from [@Broomhead Lemma 2.11]. Each primitive lattice generator $v_\rho\in \rho$ on an extremal ray of the cone $\sigma$ defining $R={\ensuremath{\Bbbk}}[\sigma^\vee\cap M]$ supports only one extremal perfect matching on a dimer model (see [@IshiiUeda2 Proposition 6.5]), so Proposition \[prop:perfmatchlabels\] implies that this perfect matching is $\Pi_\rho=\pi_2^*(\chi_\rho)$ and, moreover, that $a\in {\operatorname{supp}}(\Pi_\rho)$ if and only if $x_\rho$ divides $x^{{\operatorname{div}}(a)}$. If $m_\rho(a)$ denotes the multiplicity of $x_\rho$ in $x^{{\operatorname{div}}(a)}$, we obtain $x^{{\operatorname{div}}(a)} = \prod_{\{\rho\in \sigma(1) \mid a\in {\operatorname{supp}}(\Pi_\rho)\}} x_\rho^{m_\rho(a)}$. Lemma \[lem:multiplicity\] to follow establishes that each $m_\rho(a)$ is either 0 or 1. This completes the proof.
\[lem:multiplicity\] For any algebraically consistent dimer model $\Gamma$ with associated quiver $Q$, and for any arrow $a\in Q_1$, the monomial $x^{{\operatorname{div}}(a)}$ divides $\prod_{\rho\in \sigma(1)} x_\rho$.
In light of , we need only show that $\langle \Pi_\rho, \pi(\chi_a)\rangle \in \{0,1\}$. As Broomhead [@Broomhead §2.3] remarks, we may regard each perfect matching $\Pi$ in $Q$ as a 1-cochain $\pi^*(\Pi)\in ({\ensuremath{\mathbb{Z}}}^{Q_1})^\vee$ with values in $\{0,1\}$, where $\pi^*\colon {\ensuremath{\mathbb{Z}}}(Q)^\vee\to ({\ensuremath{\mathbb{Z}}}^{Q_1})^\vee$ is the natural inclusion. In particular, for any arrow $a\in Q_1$, the dual pairing is simply $\langle \Pi_\rho, \pi(\chi_a)\rangle = \langle \pi^*(\Pi_\rho), \chi_a\rangle \in \{0,1\}$ as required.
If we knew at this stage that $A_\mathscr{E}$ was consistent then Lemma \[lem:Kergens\] could be applied directly in the proof of Proposition \[prop:dimerlabels\] and Proposition \[prop:div(a)\] would make Lemma \[lem:multiplicity\] superfluous. However, we do not establish this fact until Theorem \[thm:unsignedsuperpotentials\] below.
We are now in a position to show that algebraically consistent dimer models define consistent toric algebras. It is convenient to introduce temporarily the unsigned version of the dimer model superpotential, namely, the element $\overline{W}_\Gamma := \sum_{F\in Q_2} w_F$.
\[thm:unsignedsuperpotentials\] For an algebraically consistent dimer model $\Gamma$, the unsigned version of the dimer model superpotential coincides with the superpotential $W$ associated to $\mathscr{E}$, namely $$\overline{W}_\Gamma = \sum_{p\in {\ensuremath{\mathscr{C}_{\mathsf{ac}}(Q)}}} p.$$ In particular, the toric algebra $A_\mathscr{E}$ is consistent.
For any face $F$ in the cell decomposition $\Gamma$, Proposition \[prop:dimerlabels\] implies that $$\label{eqn:divp}
x^{{\operatorname{div}}(w_F)} = \prod_{a\in {\operatorname{supp}}(w_F)} \prod_{\{\rho\in \sigma(1) \mid a\in {\operatorname{supp}}(\Pi_\rho)\}} x_\rho.$$ Since each $\Pi_\rho$ is a perfect matching and since $w_F$ is a term in $\overline{W}_\Gamma$, the set $ {\operatorname{supp}}(w_F)\cap {\operatorname{supp}}(\Pi_\rho)$ consists of precisely one arrow. This gives $x^{{\operatorname{div}}(w_F)} = \prod_{\rho\in \sigma(1)} x_\rho$, so every term of $\overline{W}_\Gamma$ is an anticanonical cycle. Conversely, let $p$ be an anticanonical cycle in $Q$. Fix $\rho\in \sigma(1)$ and write $$\label{eqn:PhiGamma}
W_\Gamma = \sum_{a\in {\operatorname{supp}}(\Pi_\rho)} a \cdot\partial_aW_\Gamma = \sum_{a\in {\operatorname{supp}}(\Pi_\rho)} a (p_a^+-p_a^-).$$ Proposition \[prop:perfmatchlabels\] provides an arrow $a^\prime\in {\operatorname{supp}}(p)\cap {\operatorname{supp}}(\Pi_\rho)$ which gives one summand in , so after choosing a new starting point of $p$ if necessary, we may write $p=a^\prime q$ for some path $q$ in $Q$. Since each term $w_F$ in $\overline{W}_\Gamma$ satisfies $x^{{\operatorname{div}}(w_F)} = \prod_{\rho\in \sigma(1)} x_\rho$, this holds true for the terms $a^\prime p_{a^\prime}^+$ and $a^\prime p_{a^\prime}^-$. But now, each of $q, p_{a^\prime}^+, p_{a^\prime}^-$ is a path in $Q$ from ${\operatorname{\mathsf{h}}}(a^\prime)$ to ${\operatorname{\mathsf{t}}}(a^\prime)$ with labelling monomial $(\prod_{\rho\in \sigma(1)} x_\rho)/x^{{\operatorname{div}}(a^\prime)}$. Since $Q$ is a quiver of sections and since $A_\Gamma$ is algebraically consistent, the ideal $(\partial_a W_\Gamma \mid a\in Q_1)$ must contain each of $\pm(q-p_{a^\prime}^-)$ and $\pm(p_{a^\prime}^+-q)$ in addition to $\pm(p_{a^\prime}^+-p_{a^\prime}^-)$. However, $a^\prime\in Q_1$ is the only arrow from ${\operatorname{\mathsf{t}}}(a^\prime)$ to ${\operatorname{\mathsf{h}}}(a^\prime)$ with label ${\operatorname{div}}(a^\prime)$, and since this arrow appears in precisely two terms of $W_\Gamma$, it follows that $q$ must equal one of $p_a^{\pm}$. This gives $p= w_F$, so the anticanonical cycle $p$ is a term of $\overline{W}_{\Gamma}$ and hence $\overline{W}_\Gamma = \sum_{p\in {\ensuremath{\mathscr{C}_{\mathsf{ac}}(Q)}}} p$.
To show that $A_\mathscr{E}$ is consistent it suffices by Lemma \[lem:dimerqos\] to show that $A_\Gamma\cong A_W$. Since the terms of the superpotentials $W_\Gamma$ and $W$ coincide up to sign we have $\mathscr{P}=Q_1$, so the inclusion $J_W\subseteq J_\mathscr{E} = (\partial_a W_\Gamma \mid a\in Q_1)$ is equality as required.
\[rem:dimerconclusion\] For generic $\theta\in {\operatorname{Wt}}(Q)$, Theorem \[thm:cohcomp\] implies that the crepant resolution $\mathcal{M}_\theta\to X$ from Ishii–Ueda [@IshiiUeda1] coincides with the morphism $\tau_\theta\colon Y_\theta\to Y_0$ obtained by variation of GIT quotient. This strengthens slightly an observation of Mozgovoy [@Mozgovoy Proposition 4.4].
Reconstructing the dimer
------------------------
For an algebraically consistent dimer model $\Gamma$, let $Q$ denote the quiver of sections of the collection $\mathscr{E}$ from . Before introducing the toric cell complex for $Q$, we pause to show how $\Gamma$ can be reconstructed from the covering quiver $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^d$.
List the standard basis of ${\ensuremath{\mathbb{Z}}}^d$ to be compatible with a cyclic order $v_1, \dots, v_d$ on the vertices of the polygon $P\subset N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$, and choose coordinates $N={\ensuremath{\mathbb{Z}}}\langle \mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\rangle$ so that $P$ lies in the affine plane at height one. Write $M= {\ensuremath{\mathbb{Z}}}\langle x,y,z\rangle$ for the dual basis and $B$ for the matrix defining $\iota\colon M\hookrightarrow {\ensuremath{\mathbb{Z}}}^d$ from . Extend scalars on the map $\iota^*$ defined by $B^t$ to obtain $\iota^*_{{\ensuremath{\mathbb{R}}}}\colon {\ensuremath{\mathbb{R}}}^d \to N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. For the standard inner product on ${\ensuremath{\mathbb{R}}}^d$, orthogonal projection $f\colon {\ensuremath{\mathbb{R}}}^d\to M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ is defined by $(B^tB)^{-1}B^t$, so $f$ is simply the composition of $\iota^*_{\ensuremath{\mathbb{R}}}$ with the change of basis $(B^tB)^{-1}$ from $N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ to $M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. For the sublattice $M^\prime ={\ensuremath{\mathbb{Z}}}\langle x,y\rangle\hookrightarrow M$, consider the (not-necessarily-orthogonal) projection $M\to M^\prime$ down the $z$-axis. After extending scalars and composing with $f$, we obtain a map $f^\prime\colon {\ensuremath{\mathbb{R}}}^d\to {\ensuremath{\mathbb{R}}}^2:=M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ whose restriction to ${\ensuremath{\mathbb{Z}}}^d$ fits in to the diagram $$\label{eqn:2torusdiagram}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}^d @>{\deg}>> {\operatorname{Cl}}(X)@>>> 0\\
@. @VVV @VV{f^\prime\vert_{{\ensuremath{\mathbb{Z}}}^d}}V @. @. \\
0 @>>> M^\prime @>>> {\ensuremath{\mathbb{R}}}^2 @>>> \mathbb{T}^2 @>>> 0
\end{CD}$$ where $\mathbb{T}^2:= {\ensuremath{\mathbb{R}}}^2/M^\prime$ is a real two-torus. To study the image $f^\prime(\widetilde{Q})$, lift a spanning tree from $Q$ to ${\ensuremath{\mathbb{R}}}^d$ to obtain a preferred lift $u_i\in \deg^{-1}(E_i)$ for $i\in Q_0$ as in Definition \[def:coveringquiver\]. Set $\mathbf{u}^\prime_i:=f^\prime(u_i)\in {\ensuremath{\mathbb{R}}}^2$ for each $i\in Q_0$, and let $Q^\prime_0\subset {\ensuremath{\mathbb{R}}}^2$ denote the set of all $M^\prime$-translates of such points. Similarly, lift $a\in Q_1$ with tail at $i\in Q_0$ to the unique arrow $\widetilde{a}\in \widetilde{Q}_1$ with tail at $u_i$ and set $\mathbf{v}^\prime_{a}:=f^\prime(\widetilde{a})$. Note that $\mathbf{v}^\prime_{a}$ is the translation of the vector $f^\prime({\operatorname{div}}(a)) = \sum_{\{\rho\in \sigma(1) \mid a\in {\operatorname{supp}}(\Pi_\rho\}} f^\prime(\chi_\rho)$ to the point $\mathbf{u}^\prime_i$. Write $Q^\prime_1\subset {\ensuremath{\mathbb{R}}}^2$ for the set of all $M^\prime$-translates of such vectors.
Let $Q^\prime$ denote the quiver in $\mathbb{T}^2$ defined by the $M^\prime$-periodic quiver in ${\ensuremath{\mathbb{R}}}^2$ with vertex set $Q^\prime_0$ and arrow set $Q^\prime_1$.
Every vertex from $\widetilde{Q}_0$ is an $M$-translate of some $u_i$ and every arrow from $\widetilde{Q}_1$ is an $M$-translate of an arrow $\widetilde{a}$ with tail at some $u_i$, so commutativity of gives $Q^\prime_0 = f^\prime(\widetilde{Q}_0)$ and $Q^\prime_1 = f^\prime(\widetilde{Q}_1)$. Vertices and arrows in $Q^\prime$ may a priori overlap, so it is not obvious that $Q_0^\prime$ and $Q_1^\prime$ form the 0-skeleton and the 1-skeleton of a cell decomposition of $\mathbb{T}^2$. Nevertheless, the following result confirms that this is indeed the case:
\[thm:reconstructingdimer\] Every algebraically consistency dimer model $\Gamma$ is homotopy equivalent to the cell decomposition of $\mathbb{T}^2$ dual to that induced by the subquiver $Q^\prime\subset \mathbb{T}^2$.
The first step is to show that the images $f^\prime(\chi_1), \dots, f^\prime(\chi_d)$ of the standard basis vectors are cyclically ordered in ${\ensuremath{\mathbb{R}}}^2$. For this, let $N\to N^\prime$ denote the map dual to the inclusion $M^\prime \hookrightarrow M$. Explicitly, $N^\prime = N/{\ensuremath{\mathbb{Z}}}\langle \mathbf{e}_z\rangle$ where the vector $\mathbf{e}_z = \sum_{\rho\in \sigma(1)} \iota^*(\chi_\rho)$ is the image of $z$ under the change of basis $B^tB$ from $M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ to $N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. Since $\mathbf{e}_z$ is the sum of the generators of the cone $\sigma$, the vector $\mathbf{e}_z$ lies in the interior of $\sigma$ and hence the cyclic order of the vertices of the slice $P\subset \sigma$ is maintained under the projection to $N^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. It remains to note that the vectors $f^\prime(\chi_1), \dots, f^\prime(\chi_d)$ are obtained from these cyclically ordered vectors in $N^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ by the change of basis from $N^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ to $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$.
We now associate a convex polygon in $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ to every face $F\in Q_2$ in the cell decomposition of $\mathbb{T}^2$ dual to $\Gamma$. Theorem \[thm:unsignedsuperpotentials\] implies that each $F\in Q_2$ determines an anticanonical cycle $p_F$ in $Q$. The lift of $p_F$ is an anticanonical path in $\widetilde{Q}$ whose image under $f^\prime$ is a closed piecewise-linear curve in $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ that traverses arrows in $Q^\prime_1$ arising from the arrows $a\in {\operatorname{supp}}(p_F)$. To see that this curve is the boundary of a convex polygon, consider a pair of complete fans in $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ introduced by Broomhead [@Broomhead §4.4-4.5], namely, the global zig-zag fan $\Xi$ and the local zig-zag fan $\xi(F)$. The two-dimensional cones $\sigma_\rho$ in $\Xi$ are indexed by extremal perfect matchings $\Pi_\rho$, the two-dimensional cones $\sigma_a$ in $\xi(F)$ are indexed by arrows $a\in {\operatorname{supp}}(p_F)$, and $\sigma_\rho\subseteq \sigma_a$ if and only if $a\in {\operatorname{supp}}(\Pi_\rho)$. In particular, equation can be written as $$\label{eqn:zigzaglabels}
x^{{\operatorname{div}}(a)} = \prod_{\{\rho\in \sigma(1) \mid \sigma_a\supseteq \sigma_\rho\}} x_\rho.$$ Since $\Xi$ is the common refinement of the fans $\xi(F)$ for each $F\in Q_2$, [@Broomhead Remark 4.16] implies that the cyclic order of the cones $\sigma_\rho$ in $\Xi$ is the same as the cyclic order of the vertices of the polygon $P$. We deduce from that labels on a given arrow are consecutive and, furthermore, that the boundary of each face $F$ consists of arrows with consecutive labels; these increase around black faces and decrease around white. Every arrow in $Q^\prime$ is an $M^\prime$-translate of a vector $\mathbf{v}_a^\prime = \sum_{\{\rho\in \sigma(1) \mid a\in {\operatorname{supp}}(\Pi_\rho)\}} f^\prime(\chi_\rho)$, and the first step above establishes that the vectors $f^\prime(\chi_1), \dots, f^\prime(\chi_d)$ are cyclically ordered in ${\ensuremath{\mathbb{R}}}^2$, so the set of edges $\{\mathbf{v}^\prime_a \mid a\in{\operatorname{supp}}(p_F)\}$ of the closed piecewise-linear curve is cyclically ordered, clockwise for black faces and anticlockwise for white. It follows that this curve is the boundary of a convex polygon ${\operatorname{\mathcal{P}}}(F)$ in $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$.
It suffices to see that these convex polygons are the 2-cells in a decomposition of $\mathbb{T}^2$ that is homotopy equivalent to that induced by $Q$. For this, fix $i\in Q_0$ and list cyclically all arrows $a_1,b_1,a_2, b_2, \dots a_k, b_k\in Q_1$ with ${\operatorname{\mathsf{t}}}(a_\nu)=i$ and ${\operatorname{\mathsf{h}}}(b_\nu)=i$ for $1\leq \nu\leq k$. Let $F^-_\nu\in Q_2$ denote the unique black face containing $a_\nu, b_\nu$ in its boundary, and similarly, $F^+_\nu\in Q_2$ the white face containing $b_\nu, a_{\nu+1}$, with $a_{k+1}:=a_1$. Since the boundaries of ${\operatorname{\mathcal{P}}}(F^-_\nu)$ and ${\operatorname{\mathcal{P}}}(F^+_\nu)$ are oriented clockwise and anticlockwise respectively, the polygons ${\operatorname{\mathcal{P}}}(F^-_1), {\operatorname{\mathcal{P}}}(F^+_1), \dots, {\operatorname{\mathcal{P}}}(F^-_k), {\operatorname{\mathcal{P}}}(F^+_k)$ glue cyclically around vertex $i$. Note that these polygons do not cycle more than once around $i$, because otherwise the edges dual to the outward-pointing arrows $a_1,\dots, a_k$ cycle more than once around the face dual to $i\in Q_0$ which is absurd. In this way, each vertex $i\in Q_0$ gives rise to a *tile* obtained as union of convex polygons ${\operatorname{\mathcal{T}}}(i):=\bigcup_{1\leq\nu\leq k} ({\operatorname{\mathcal{P}}}(F^-_\nu)\cup {\operatorname{\mathcal{P}}}(F^+_\nu))$. For each incoming arrow $b_\nu$, the tile ${\operatorname{\mathcal{T}}}(i)$ glues to ${\operatorname{\mathcal{T}}}({\operatorname{\mathsf{t}}}(b_\nu))$ along ${\operatorname{\mathcal{P}}}(F_\nu^-)\cup{\operatorname{\mathcal{P}}}(F^+_\nu)$ and, similarly, for each outgoing arrow $a_\nu$, the tile ${\operatorname{\mathcal{T}}}(i)$ glues to ${\operatorname{\mathcal{T}}}({\operatorname{\mathsf{h}}}(a_\nu))$ along ${\operatorname{\mathcal{P}}}(F_\nu^-)\cup{\operatorname{\mathcal{P}}}(F^+_{\nu-1})$. It follows that the convex polygons ${\operatorname{\mathcal{P}}}(F)$ arising from faces $F\in Q_2$ tesselate the plane and, moreover, the 0-skeleton and 1-skeleton coincide with $Q_0^\prime$ and $Q_1^\prime$ respectively. The assignment $F\mapsto {\operatorname{\mathcal{P}}}(F)$ shows that this cell decomposition coincides with that arising from $Q$ up to homotopy.
\[exa:F1tiltingSubdivisionT3\] For the dimer model $\Gamma$ and quiver $Q$ from Example \[exa:dimerF1\], the quivers $\widetilde{Q}(i)$ for $i=0,1$ are drawn in black in Figure \[fig:F1tiltingSubdivisionT3\], each superimposed on a 4-cube $\mathsf{C}(u_i)$ drawn in grey.
To construct the quiver $Q^\prime$ as in Theorem \[thm:reconstructingdimer\], note that $\sigma$ is the cone from Example \[exa:F1tilting\], so the map $\iota\colon M\to {\ensuremath{\mathbb{Z}}}^4$ is defined by the matrix $B$ with columns ${\operatorname{div}}(x)=(1,0,-1,0)$, ${\operatorname{div}}(y)=(0,1,1,-1)$ and ${\operatorname{div}}(z)=(1,1,1,1)$. Orthogonal projection $f^\prime\colon {\ensuremath{\mathbb{R}}}^4\to M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ with respect to the standard inner product on ${\ensuremath{\mathbb{R}}}^4$ is defined by the first two rows of $(B^tB)^{-1}B^t$, namely $$\begin{pmatrix}\frac{5}{9} & \frac{1}{6} & -\frac{4}{9} & -\frac{5}{18} \\ \frac{1}{9} & \frac{1}{3} & \frac{1}{9} & -\frac{5}{9} \end{pmatrix}.$$ A black face $F$ determines the anticanonical path from Figure \[fig:F1tiltingSubdivisionT3\](a) that traverses arrows labelled $x_1, x_2, x_3, x_4$, and this projects via $f^\prime$ to define the anticlockwise boundary of the convex polygon with vertices $\mathbf{u}^\prime_0=(0,0)$, $\mathbf{u}_1^\prime=(\frac{5}{9},\frac{1}{9})$, $\mathbf{u}^\prime_2=(\frac{13}{18},\frac{4}{9})$ and $\mathbf{u}^\prime_3=(\frac{5}{18},\frac{5}{9})$ in the quiver $Q^\prime$ from Figure \[fig:F1SubdivisionT2\](a). A white face determines the adjacent anticanonical path that traverses arrows labelled $x_3, x_2, x_1, x_4$, and this projects to the clockwise boundary of an adjacent polygon in Figure \[fig:F1SubdivisionT2\](a). Each of the six anticanonical paths from Figure \[fig:F1tiltingSubdivisionT3\](a) defines one of the six convex polygons in Figure \[fig:F1SubdivisionT2\](a) with $\mathbf{u}^\prime_0$ as a vertex, and these polygons define ${\operatorname{\mathcal{T}}}(0)$. The paths in $\widetilde{Q}(1)$ from Figure \[fig:F1tiltingSubdivisionT3\](b) define the convex polygons that make up ${\operatorname{\mathcal{T}}}(1)$, and vertices $i=2,3$ are similar. The resulting cell decomposition of $\mathbb{T}^2$ is homotopy equivalent to that from Figure \[fig:dimerF1\](b).
\[rem:notcelldivision\] It is not sufficient for Theorem \[thm:reconstructingdimer\] that $Q$ arises from a consistent toric algebra in dimension three. For example, for the collection $\mathscr{E}^{\prime\prime}$ from Example \[exa:superpotentialF1tilting\][$(\mathrm{iii})$]{}, the vertex labelled 4 in Figure \[fig:F1subandsupertilting\](b) determines $\mathbf{u}_4^\prime=(\frac{4}{9},\frac{8}{9})$ in Figure \[fig:F1SubdivisionT2\](b), and a pair of arrows (drawn as curves) labelled $x_4$ from vertices 1 and 2 cross other arrows at points that are not vertices. In this case, the toric algebra is consistent but it does not arise from an algebraically consistent dimer model ($A_{\mathscr{E}^{\prime\prime}}$ is not Calabi–Yau) and we do not obtain a subdivision of $\mathbb{T}^2$.
The toric cell complex {#the-toric-cell-complex}
----------------------
Let $f\colon {\ensuremath{\mathbb{R}}}^d\to {\ensuremath{\mathbb{R}}}^3:=M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ denote the orthogonal projection onto the subspace spanned by $M\subseteq {\ensuremath{\mathbb{Z}}}^d$ as in the previous subsection. Then by construction the diagram factors through the commutative diagram $$\label{eqn:3torusdiagram}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}^d @>{\deg}>> {\operatorname{Cl}}(X)@>>> 0\\
@. @| @VV{f\vert_{{\ensuremath{\mathbb{Z}}}^d}}V @. @. \\
0 @>>> M @>>> {\ensuremath{\mathbb{R}}}^3 @>>> \mathbb{T}^3 @>>> 0
\end{CD}$$ where $\mathbb{T}^3:= {\ensuremath{\mathbb{R}}}^3/M$ is a real three-torus. For each $i\in Q_0$, our chosen vertex $u_i\in \widetilde{Q}_0$ determines a point $\mathbf{u}_i:=f(u_i)\in {\ensuremath{\mathbb{R}}}^3$. Set $\mathbf{v}_z := f({\operatorname{div}}(z))$, and consider the family of affine planes $$H_\lambda(i):= \left\{\mathbf{u}+\mathbf{u}_i+ \lambda\mathbf{v}_z\in {\ensuremath{\mathbb{R}}}^3=M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\mid \mathbf{u}\in M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}\right\}\quad \text{for }0\leq \lambda\leq 1$$ Note that $H_\lambda(i)$ is the translation by $\lambda\mathbf{v}_z$ of the affine plane through $\mathbf{u}_i$ parallel to $M^\prime\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$. These planes slice the image $f(\mathsf{C}(u_i))$ of the unit box and hence the image $f(\widetilde{Q}(i))$ of the quiver.
\[prop:3cellpolygons\] For $i\in Q_0$, let $n_i$ be the number of arrows $a\in Q_1$ with ${\operatorname{\mathsf{t}}}(a)=i$. Either:
1. $n_i > 2$, when $f(\widetilde{Q}(i))\cap H_\lambda(i)$ is the vertex set of a polygon $P_\lambda(i)$ for all $0< \lambda < 1$, and it is a singleton $P_\lambda(i)$ for $\lambda =0,1$; or
2. $n_i=2$, when there exists $\lambda_{{\operatorname{\mathsf{t}}}}(i)< \lambda_{{\operatorname{\mathsf{h}}}}(i)$ such that $f(\widetilde{Q}(i))\cap H_\lambda(i)$ is the vertex set of
1. a polygon $P_\lambda(i)$ in $H_\lambda(i)$ for $\lambda_{{\operatorname{\mathsf{t}}}}(i)< \lambda < \lambda_{{\operatorname{\mathsf{h}}}}(i);$
2. a line segment $P_{\lambda}(i)$ in $H_\lambda(i)$ for $0<\lambda\leq \lambda_{{\operatorname{\mathsf{t}}}}(i)$ and $\lambda_{{\operatorname{\mathsf{h}}}}(i)\leq \lambda< 1;$
3. a singleton $P_\lambda(i)$ for $\lambda =0,1$.
For $0\leq \lambda\leq 1$, the image under $f$ of any anticanonical path in $\widetilde{Q}(i)$ touches $H_\lambda(i)$ once because the projection of any such path onto the real line spanned by $\mathbf{v}_z$ is bijective onto its image. We proceed by reconstructing the information of the slice $f(\widetilde{Q}(i))\cap H_\lambda(i)$ in ${\ensuremath{\mathbb{R}}}^3$ using our knowledge of the projection to ${\ensuremath{\mathbb{R}}}^2$ from Theorem \[thm:reconstructingdimer\].
Fix $a\in Q_1$ with ${\operatorname{\mathsf{t}}}(a)=i$ and write $q_a^\pm$ for the anticanonical paths from $u_i$ covering $p_a^\pm a$ that begin by traversing the unique lift $\widetilde{a}\in \widetilde{Q}_1(i)$ of $a$. Let $\gamma_a^\pm\colon [0,1]\to {\ensuremath{\mathbb{R}}}^3$ denote the curves with images $f(q_a^\pm)$ in $f(\widetilde{Q}(i))$ such that $\gamma_a^\pm(\lambda)\in H_\lambda(i)$ for $0\leq \lambda \leq 1$. Composing with the projection to ${\ensuremath{\mathbb{R}}}^2$ gives piecewise-linear curves $\overline{\gamma}_a^\pm\colon [0,1]\to {\ensuremath{\mathbb{R}}}^2$ satisfying $\overline{\gamma}_a^\pm(0) = \overline{\gamma}_a^\pm(1) = \mathbf{u}^\prime_i$ that traverse the paths $Q^\prime$ corresponding to $p_a^\pm a$. In the notation of the previous proof, list the arrows $a_1,\dots, a_k\in Q_1$ with tail at vertex $i$. Then as $0\leq \lambda \leq 1$ increases, the set of points $\Omega_\lambda(i):=\{\overline{\gamma}_{a_1}^+(\lambda),\overline{\gamma}_{a_1}^-(\lambda) \dots, \overline{\gamma}_{a_k}^+(\lambda), \overline{\gamma}_{a_1}^-(\lambda)\}$ in ${\ensuremath{\mathbb{R}}}^2$ flow from $\mathbf{u}^\prime_i$ out along $\mathbf{v}^\prime_{a_1}, \dots, \mathbf{v}^\prime_{a_k}$ before splitting and returning to $\mathbf{u}_i^\prime$ along the paths in $Q^\prime$ corresponding to $p_{a_1}^+, p_{a_1}^-, \dots, p_{a_k}^+, p_{a_k}^-$. If $n_i>2$ then $\Omega_\lambda(i)$ comprises three or more points in cyclic order around $\mathbf{u}_i^\prime\in {\ensuremath{\mathbb{R}}}^2$ for $0<\lambda<1$, and hence forms the vertex set of a (nondegenerate) polygon. For $n_i=2$, let $\mathbf{v}^\prime_{a_1}, \mathbf{v}^\prime_{a_2}\in Q^\prime_1$ denote the arrows with tail at $\mathbf{u}_i^\prime$ and $\mathbf{v}^\prime_{b_1}, \mathbf{v}^\prime_{b_2}\in Q^\prime_1$ the arrows with head at $\mathbf{u}_i^\prime$. Set $\lambda_{{\operatorname{\mathsf{t}}}}(i):=\frac{1}{d} \min\{\deg(x^{{\operatorname{div}}(a_1)}),\deg(x^{{\operatorname{div}}(a_2)})\}$ and $\lambda_{{\operatorname{\mathsf{h}}}}(i):=1-\frac{1}{d} \min\{\deg(x^{{\operatorname{div}}(b_1)}),\deg(x^{{\operatorname{div}}(b_2)})\}$. The set $\Omega_\lambda(i)$ comprises two points for $\lambda\leq \lambda_{{\operatorname{\mathsf{t}}}}(i)$ and $\lambda\geq \lambda_{{\operatorname{\mathsf{h}}}}(i)$, but otherwise forms the vertex set of a polygon as above. The result follows since translation by $\mathbf{u}_i+ \lambda\mathbf{v}_z$ canonically identifies $\Omega_\lambda(i)\subset M^\prime \otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ with the slice $f(\widetilde{Q})\cap H_\lambda(i)$ for each $0\leq \lambda\leq 1$.
\[lem:2-celllinesegments\] For $a\in Q_1$, set $i:={\operatorname{\mathsf{t}}}(a)\in Q_0$ and $\lambda_a:= \frac{1}{d}\deg(x^{{\operatorname{div}}(a)})\in {\ensuremath{\mathbb{Q}}}$. Let $q_a^\pm$ be the anticanonical paths in $\widetilde{Q}(i)$ that cover the cycles $ap_a^\pm$ in $Q$. The slice $f(q_a^\pm)\cap H_\lambda(i)$ is the vertex set of a line segment $\ell_{\lambda}(a)$ in $H_\lambda(i)$ for $\lambda_a<\lambda < 1$, and it is a singleton $\ell_\lambda(a)$ for $\lambda=\lambda_a, 1$.
One need only focus attention on a single arrow in the course of the proof above.
Proposition \[prop:3cellpolygons\] and its corollary are introduced to facilitate the following key definition.
\[def:cellcomplex\] We associate to the collection $\mathscr{E}$ on $X$ a set $\Delta$ of closed subsets of $\mathbb{T}^3$. We describe the construction in each dimension separately (compare Remark \[rem:CWcomplex\]):
1. Each $i\in Q_0$ defines a point $\mathbf{u}_i=f(u_i)\in {\ensuremath{\mathbb{R}}}^3$, and we let $\Delta_0$ denote the set of $M$-translates of all such points.
2. Each $a\in Q_1$ with ${\operatorname{\mathsf{t}}}(a)=i$ lifts uniquely to an arrow $\widetilde{a}\in \widetilde{Q}_1$ with tail at $u_i$, and each $a\in Q_1$ with ${\operatorname{\mathsf{h}}}(a)=i$ lifts uniquely to an arrow $\widetilde{a}\in \widetilde{Q}_1$ with head at $u_i$. Let $\Delta_1$ denote the set of $M$-translates of the supports ${\operatorname{supp}}(\widetilde{a})\subset {\ensuremath{\mathbb{R}}}^3$ of all such arrows.
3. For $a\in Q_1$, set $i:={\operatorname{\mathsf{t}}}(a)\in Q_0$ and $\lambda_a:= \frac{1}{d}\deg(x^{{\operatorname{div}}(a)})\in {\ensuremath{\mathbb{Q}}}$. Define $\eta_a:= \bigcup_{\lambda_a\leq \lambda\leq 1}\ell_{\lambda}(a)$ to be the union of the sets from Lemma \[lem:2-celllinesegments\]. Let $\Delta_2$ denote the set of $M$-translates of all such closed subsets $\eta_a$ defined by arrows in $Q$.
4. For $i\in Q_0$, define $\eta_i:= \bigcup_{0\leq \lambda\leq 1} P_{\lambda}(i)$ to be the union of all subsets introduced in Proposition \[prop:3cellpolygons\]. Let $\Delta_3$ denote the set of $M$-translates of such closed subsets $\eta_i$ corresponding to all vertices in $Q$.
Let $\Delta$ denote the set of closed subsets in $\mathbb{T}^3$ determined by these $M$-periodic subsets in ${\ensuremath{\mathbb{R}}}^3$. As before, for $\eta\in \Delta_k$ we write ‘${\operatorname{cod}}(\eta^\prime, \eta)=1$’ as shorthand for those $\eta^\prime\in \Delta_{k-1}$ satisfying $\eta^\prime\subset \eta$.
\[exa:F1tiltingGamma\] Returning to Example \[exa:F1tiltingSubdivisionT3\], the elements of $\Delta$ determined by arrows from $\widetilde{Q}(0)$ as shown in Figure \[fig:F1tiltingSubdivisionT3\](a) are the $M$-translates of the faces of the three-dimensional convex polytope obtained as the convex hull of the vertex set $f(\widetilde{Q}(0))$. We work with the projection, so labels on arrows in this figure should now be $f({\operatorname{div}}(a))\in {\ensuremath{\mathbb{R}}}^3$ rather than ${\operatorname{div}}(a)\in {\ensuremath{\mathbb{Z}}}^4$. After taking into account $M$-periodicity, we see that Figure \[fig:F1tiltingSubdivisionT3\](a) contributes 1 subset to $\Delta_3$, 6 subsets to $\Delta_2$, all 10 subsets to $\Delta_1$ and all 4 subsets to $\Delta_0$. Similarly, Figure \[fig:F1tiltingSubdivisionT3\](b) illustrates $f(\widetilde{Q}(1))$ in ${\ensuremath{\mathbb{R}}}^3$ which contributes 1 subset to $\Delta_3$, 4 to $\Delta_2$, 9 to $\Delta_1$ and all 4 to $\Delta_0$ in $\mathbb{T}^3$. One computes similarly the subsets determined by arrows from $\widetilde{Q}(2)$ and $\widetilde{Q}(3)$ to see that $\Delta$ satisfies $\vert \Delta_0\vert = \vert \Delta_3\vert = 4$ and $\vert \Delta_1\vert = \vert \Delta_2\vert = 10$.
\[rem:CWcomplex\] Example \[exa:F1tiltingGamma\] shows that elements of $\Delta_3$ need not be equidimensional, and subsets from $\Delta_2$ may intersect along their interiors, so $\Delta$ is not a regular cell complex as in Section \[sec:CWcomplex\]. Nevertheless, the closed subsets in $\Delta$ arise from a mild variant of the topological notion of the attaching map of a cell. Indeed, for each subset $\eta\in \Delta_k$ there is a (not necessarily surjective) continuous map $\varphi_\eta \colon B^k\to \eta\subset \mathbb{T}^3$ from the closed $k$-ball such that the boundary satisfies $$\varphi_\eta(S^{k-1}) = \bigcup_{{\operatorname{cod}}(\eta^\prime, \eta)=1} \varphi_\eta(B^k)\cap\eta^{\prime}.$$ Thus, while our closed subsets in $\mathbb{T}^3$ are not actually cells as defined in Section \[sec:CWcomplex\], they are very similar. In addition, Lemma \[lem:dimerincidence\] below shows that $\Delta$ shares key properties with regular cell complexes. We therefore choose to adopt the terminology ‘cells’ and ‘cellular’ from now on.
The *toric cell complex* of the algebraically consistent dimer model $\Gamma$ or, equivalently, of the collection $\mathscr{E}$ from , is the set $\Delta$ comprising closed subsets of $\mathbb{T}^3$ as in Definition \[def:cellcomplex\]. We refer to each element $\eta\in \Delta$ as a *cell* in $\Delta$ (see Remark \[rem:CWcomplex\]).
\[lem:dimerbijection\] There are canonical bijections between $\Delta_0$ and $Q_0$, between $\Delta_1$ and $Q_1$, and between $\Delta_2$ and the set $\{p_a^+ - p_a^- \in {\ensuremath{\Bbbk}}Q\mid a\in Q_1\}$ of minimal generators of the ideal $J_\mathscr{E}$.
This is similar to the proof of Lemma \[lem:McKaybijections\].
For $k\leq 2$ and $\eta\in \Delta_k$, the *head*, *tail* and *label* of $\eta$ are defined to be the head, tail and label respectively of the element of ${\ensuremath{\Bbbk}}Q$ that is associated to $\eta$ by Lemma \[lem:dimerbijection\]. Each $\eta\in \Delta_3$ is constructed from some quiver $\widetilde{Q}(i)$, and we define both the *head* and *tail* of $\eta$ to be the vertex $i\in Q_0$, while the *label* of $\eta$ is $(1,\dots,1)\in {\ensuremath{\mathbb{Z}}}^n$. The notions of right- and left-differentiation of cells with respect to faces are defined precisely as in Section \[sec:McKay\]. Indeed, for $\eta\in \Delta$ and for any face $\eta^\prime\subset \eta$ there is a (not necessarily unique) path in $\widetilde{Q}$ from ${\operatorname{\mathsf{h}}}(\eta^\prime)$ to ${\operatorname{\mathsf{h}}}(\eta)$ that defines (uniquely) an element $\overleftarrow{\partial}_{\!\eta'}\eta\in A$. Similarly, any path in $\widetilde{Q}$ from ${\operatorname{\mathsf{t}}}(\eta)$ to ${\operatorname{\mathsf{t}}}(\eta^\prime)$ defines uniquely an element $\overrightarrow{\partial}_{\!\eta'}\eta\in A$.
\[def:leftrightderivativesdimer\] For $\eta\in \Delta$ and any face $\eta^\prime\subset \eta$, the element $\overleftarrow{\partial}_{\!\eta'}\eta\in A$ is the *left-derivative* of $\eta$ with respect to $\eta'$. Similarly, $\overrightarrow{\partial}_{\!\eta'}\eta\in A$ is the *right-derivative* of $\eta$ with respect to $\eta'$.
Again, the cells of $\Delta$ satisfy a duality property (compare Proposition \[prop:dualityMcKay\]):
The map $\tau\colon \Delta\to \Delta$ that assigns to each $\eta\in \Delta_k$ the unique cell $\eta^\prime\in \Delta_{3-k}$ with ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$, ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$ and $x^{{\operatorname{div}}(\eta^\prime)} = \prod_{\rho\in \sigma(1)}x_\rho/x^{{\operatorname{div}}(\eta)}$ is an involution.
This is evident from the construction.
The cellular resolution
-----------------------
The minimal projective resolution of a dimer model algebra $A$ as an $(A,A)$-bimodule has been studied extensively in the literature [@Broomhead; @Davison; @MozgovoyReineke] under various assumptions on the dimer model. Here we assume algebraic consistency and describe in a uniform way the maps in the resolution using the toric cell complex $\Delta$. As a first step we show that $\Delta$ shares some key properties with regular cell complexes.
\[lem:dimerincidence\] The toric cell complex $\Delta$ of an algebaically consistent dimer model satisfies . In addition, $\Delta$ admits an incidence function, that is, a function $\varepsilon\colon \Delta\times \Delta \to \{0,\pm 1\}$ such that:
1. $\varepsilon(\eta,\eta^\prime)= 0$ unless $\eta^\prime$ is a facet of $\eta$;
2. $\varepsilon(\eta,\emptyset) = 1$ for all 0-cells $\eta$; and
3. if $\eta\in \Delta_k$ and $\eta^{\prime\prime}\in \Delta_{k-2}$ is a face of $\eta$, then for $\eta_1^\prime, \eta_2^\prime\in \Delta_{k-1}$ from we have $$\label{eqn:signconditiondimer}
\varepsilon(\eta,\eta_1^\prime) \varepsilon(\eta_1^\prime,\eta^{\prime\prime}) +\varepsilon(\eta,\eta_2^\prime) \varepsilon(\eta_2^\prime,\eta^{\prime\prime})= 0.$$
Statement is immediate for $k=2$. As for $k=3$, fix a codimension-two face $\eta^{\prime\prime}\in \Delta_1$ of a 3-cell $\eta\in \Delta_3$. Consider three cases: either the tails coincide ${\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{t}}}(\eta^{\prime\prime})$; the heads coincide ${\operatorname{\mathsf{h}}}(\eta)={\operatorname{\mathsf{h}}}(\eta^{\prime\prime})$; or neither heads nor tails coincide. In the first two cases the arrow $a\in Q_1$ whose support is $\eta^{\prime\prime}$ is contained in precisely two anticanonical cycles, say $p_1, p_2$, each of which traverses only arrows supported on 1-cells in $\eta$. In the first case, if $a_j\in {\operatorname{supp}}(p_j)$ denotes the arrow with head at ${\operatorname{\mathsf{h}}}(\eta)$ for $j=1,2$, then the 2-cells $\eta_1^\prime, \eta_2^\prime\subset \eta$ dual to the arrows $a_1, a_2$ are the unique 2-cells in $\eta$ containing $\eta^{\prime\prime}$. In the second case, if $a_j\in {\operatorname{supp}}(p_j)$ denotes the arrow with tail ${\operatorname{\mathsf{t}}}(\eta)$ for $j=1,2$, then the cells $\eta_1^\prime, \eta_2^\prime\subset \eta$ dual to $a_1, a_2$ are the unique 2-cells in $\eta$ containing $\eta^{\prime\prime}$. In the third case, the arrow $a\in Q_1$ whose support is $\eta^{\prime\prime}$ is contained in a unique anticanonical cycle $p$ that traverses only arrows supported on 1-cells in $\eta$. If we let $a_1, a_2\in {\operatorname{supp}}(p)$ denote the unique arrows satisfying ${\operatorname{\mathsf{h}}}(a_1)={\operatorname{\mathsf{h}}}(p)$ and ${\operatorname{\mathsf{t}}}(a_2)={\operatorname{\mathsf{t}}}(p)$, then the 2-cells $\eta_1^\prime, \eta_2^\prime\subset \eta$ dual to the arrows $a_1, a_2$ are the unique 2-cells in $\eta$ containing $\eta^{\prime\prime}$. This establishes .
To construct an incidence function we may assume properties [$(\mathrm{i})$]{} and [$(\mathrm{ii})$]{}. Each cell $\eta\in \Delta_1$ is supported on a unique arrow $a\in Q_1$ and contains precisely two 0-cells ${\operatorname{\mathsf{h}}}(a), {\operatorname{\mathsf{t}}}(a)\in \Delta_0$. Choosing $\varepsilon(\eta, {\operatorname{\mathsf{h}}}(a)) = 1$ forces $\varepsilon(\eta, {\operatorname{\mathsf{t}}}(a)) = -1$ by . Similarly, every 2-cell $\eta\in \Delta_2$ corresponds uniquely to a minimal generator of $J_\mathscr{E}$ and we use the signs from $W_\Gamma$ (see Remark \[rem:signsW\_G\] below) to write this generator as $p_a^+-p_a^- = a_l^+\cdots a_1^+ - a_m^-\cdots a_1^-$ where the boundary of $\eta$ is supported on the arrows $\{a_1^+, \dots ,a_l^+, a_1^-, \dots a_m^-\}$. Identify each 1-cell with the corresponding arrow and choose $\varepsilon(\eta,a_j^+)=1$ for $1\leq j\leq l$, in which case forces $\varepsilon(\eta, a_j^-) = -1$ for $1\le j\leq m$. Finally, each 3-cell $\eta\in \Delta_3$ is such that every facet $\eta^\prime\subset \eta$ satisfies either ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$ or ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$. Choose $\varepsilon(\eta, \eta^\prime) = 1$ if ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$ in which case forces $\varepsilon(\eta, \eta^\prime) = -1$ for ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$.
It remains to show that equation holds for any $\eta\in \Delta_k$ and codimension-two face $\eta^{\prime\prime}\subset \eta$. There are three cases, where either the tails coincide ${\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{t}}}(\eta^{\prime\prime})$, the heads coincide ${\operatorname{\mathsf{h}}}(\eta)={\operatorname{\mathsf{h}}}(\eta^{\prime\prime})$, or neither heads nor tails coincide. For $k=2$, the proof in each case is straightforward because $\eta^{\prime\prime}$ is a 0-cell. For $k=3$, consider the case ${\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{t}}}(\eta^{\prime\prime})$ where $\varepsilon(\eta,\eta_1^\prime)=\varepsilon(\eta,\eta^\prime_2)=-1$. The cell $\eta^{\prime\prime}\in \Delta_1$ is supported on an arrow $a\in Q_1$, and the signs $\varepsilon(\eta^\prime_1,\eta^{\prime\prime}), \varepsilon(\eta^\prime_2,\eta^{\prime\prime})$ differ as required because the pair of anticanonical paths that traverse arrow $a$ have opposite signs in $W_\Gamma$. The case with ${\operatorname{\mathsf{h}}}(\eta)={\operatorname{\mathsf{h}}}(\eta^{\prime\prime})$ is similar. In the final case, the arrow corresponding to $\eta^{\prime\prime}$ lies in a unique anticanonical path and hence $\varepsilon(\eta_1^\prime,\eta^{\prime\prime}) = \varepsilon(\eta_2^\prime,\eta^{\prime\prime})$, but then one of $\eta_1^\prime, \eta_2^\prime$ shares head with $\eta$ while the other shares tail. Thus, the signs $\varepsilon(\eta, \eta_1^{\prime}), \varepsilon(\eta,\eta_2^{\prime})$ differ as required.
Our main result below uses the toric cell complex to provide a simple, uniform description of all terms and differentials in the standard resolution associated to a quiver with superpotential in dimension three as studied by Ginzburg [@Ginzburg], Mozgovoy-Reineke [@MozgovoyReineke], Davison [@Davison] and Broomhead [@Broomhead].
\[thm:dimerresolution\] Let $\Gamma$ be an algebraically consistent dimer model with toric algebra $A$ and let $\varepsilon\colon \Delta\times \Delta \to \{0,\pm 1\}$ be any incidence function on $\Delta$. The minimal bimodule resolution of $A$ is the *cellular resolution* $$0\longrightarrow P_3 \stackrel{d_3}{\longrightarrow} P_2 \stackrel{d_2}{\longrightarrow} P_1 \stackrel{d_1}{\longrightarrow} P_0 \stackrel{\mu}{\longrightarrow} A \longrightarrow 0$$ where for $0\leq k\leq 3$ we have $$P_k:= \bigoplus_{\eta\in \Delta_k} A e_{{\operatorname{\mathsf{h}}}(\eta)} \otimes [\eta] \otimes e_{{\operatorname{\mathsf{t}}}(\eta)} A,$$ where $\mu\colon P_0\to A$ is the multiplication map and where $ d_k\colon P_k\longrightarrow P_{k-1}$ satisfies $$d_k\big(1\otimes[\eta]\otimes 1\big) = \sum_{{\operatorname{cod}}(\eta^\prime,\eta)=1} \varepsilon(\eta,\eta^\prime) \overleftarrow{\partial}_{\!\!\eta^\prime}\eta\otimes [\eta^\prime]\otimes \overrightarrow{\partial}_{\!\!\eta^\prime}\eta.$$
This result can be proved directly by modifying the proofs from [@Broomhead; @Davison; @MozgovoyReineke]. However, we choose instead to consider a particular incidence function that realises precisely the maps from the resolution of Broomhead [@Broomhead]. Then, just as in the proof of Proposition \[prop:McKaycomplex\], choosing any alternative incidence function merely provides an isomorphic resolution of $A$.
Let $\varepsilon\colon \Delta\times \Delta \to \{0,\pm 1\}$ denote the incidence function constructed in the proof of Lemma \[lem:dimerincidence\]. To compute the differentials for this choice of incidence function, note first that $$d_1\big(1\otimes[a]\otimes 1\big) = 1\otimes[{\operatorname{\mathsf{h}}}(a)]\otimes a - a\otimes [{\operatorname{\mathsf{t}}}(a)]\otimes 1.$$ For the 2-cell $\eta\in \Delta$ corresponding to the relation $p_a^+-p_a^- = a_l^+\cdots a_1^+ - a_m^-\cdots a_1^-$, we have $$d_2\big(1\otimes[\eta]\otimes 1\big) = \sum_{j=1}^l a_l^+\cdots a_{j+1}^+\otimes [a_j^+]\otimes a_{j-1}^+\cdots a_1^+ - \sum_{j=1}^m a_m^-\cdots a_{j+1}^-\otimes [a_j^-]\otimes a_{j-1}^-\cdots a_1^- .$$ Finally, consider $\eta\in \Delta_3$ with $i:= {\operatorname{\mathsf{t}}}(\eta)={\operatorname{\mathsf{h}}}(\eta)\in Q_0$. The facets $\eta^\prime\subset \eta$ that satisfy ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$ have left-derivative $\overleftarrow{\partial}_{\!\eta'}\eta = e_i$ and right-derivative $\overrightarrow{\partial}_{\!\eta'}\eta =a$ for an arrow $a\in Q_1$ with ${\operatorname{\mathsf{t}}}(a)=i$. Similarly, the facets $\eta^\prime\subset \eta$ that satisfy ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$ have left-derivative $\overleftarrow{\partial}_{\!\eta'}\eta = a$ for an arrow $a\in Q_1$ with ${\operatorname{\mathsf{h}}}(a)=i$ and right-derivative $\overrightarrow{\partial}_{\!\eta'}\eta =e_i$. In each case relabel the facet as $\eta^\prime_a:=\eta^\prime$ for the corresponding arrow $a\in Q_1$. Then $$\begin{aligned}
d_3\big(1\otimes[\eta]\otimes 1\big) & = & \sum_{{\operatorname{cod}}(\eta_a^\prime,\eta)=1,\; {\operatorname{\mathsf{h}}}(\eta_a^\prime)=i} 1\otimes [\eta^\prime_a]\otimes a - \sum_{{\operatorname{cod}}(\eta_a^\prime,\eta)=1,\; {\operatorname{\mathsf{t}}}(\eta_a^\prime)=i} a\otimes [\eta^\prime_a]\otimes 1\\
& = & \sum_{\{a\in Q_1 \mid {\operatorname{\mathsf{t}}}(a)=i\}} 1\otimes [\eta^\prime_a]\otimes a - \sum_{\{a\in Q_1 \mid {\operatorname{\mathsf{h}}}(a)=i\}} a\otimes [\eta^\prime_a]\otimes 1.\end{aligned}$$ Our differentials are seen to coincide with those from [@Broomhead Theorem 7.3], though note that our convention for composing arrows (where $a^\prime a$ means ‘$a^\prime$ follows $a$’) differs from that in [@Broomhead].
\[rem:signsW\_G\] In the course of the proof we use the signs in the dimer superpotential $W_\Gamma$ to write $p_a^+-p_a^-$, and hence to choose the incidence function $\varepsilon$. However, we chose this incidence function only to reproduce precisely Broomhead’s resolution. Since any incidence function on $\Delta$ suffices for Theorem \[thm:dimerresolution\], knowledge of the signs of $W_\Gamma$ is unnecessary in general.
The cellular resolution conjecture {#sec:conjecture}
==================================
We conclude by formulating a conjecture on the existence of toric cell complexes and cellular resolutions for consistent toric algebras in arbitrary dimension. As a first step we illustrate two key ingredients by presenting an example of a four-dimensional consistent toric algebra. A nice class of such algebras arises from tilting bundles on smooth toric Fano threefolds, and we study here a representative example of this class.
A consistent fourfold example {#sec:conjectureexample}
-----------------------------
Let $X={\operatorname{Spec}}\Bbbk[\sigma^{\vee}\cap M]$ be the Gorenstein toric fourfold determined by the cone $\sigma$ generated by the vectors $v_1=(1,0,0,1)$, $v_2=(0,1,0,1)$, $v_3=(0,0,1,1)$, $v_4=(-1,-1,2,1)$, $v_5=(-1,-1,1,1)$ and $v_6=(0,0,-1,1)$. For $1 \leq \rho \leq 6$, write $D_\rho$ for the toric divisor in $X$ defined by the ray of $\sigma$ generated by $v_\rho$. Then ${\operatorname{Cl}}(X)$ is the quotient of the free abelian group generated by $\mathcal{O}_X(D_1)$, $\mathcal{O}_X(D_5+D_6)$, $\mathcal{O}_X(D_6)$, by the subgroup generated by $\mathcal{O}_X(D_1+D_5+2D_6)$. The singularity $X$ admits several crepant resolutions $\tau \colon Y \rightarrow X$, one of which is given by the total space $\mathrm{tot}(\omega_Z)$ of the canonical bundle of the smooth Fano threefold $Z$ listed as number 11 by Oda [@Oda Figure 2.7]. Consider the collection $$\mathscr{E} = \left(\begin{array}{cc} \mathcal{O}_X, \mathcal{O}_X(D_1), \mathcal{O}_X(2D_1),\mathcal{O}_X(D_6),\mathcal{O}_X(D_5+D_6),\\
\mathcal{O}_X(D_1+D_6), \mathcal{O}_X(D_1+D_5+D_6),\mathcal{O}_X(2D_1+D_6) \end{array}\right)$$ on $X$. This collection is obtained from a tilting bundle[^1] on $Z$ by pulling back each summand via $\mathrm{tot}(\omega_Z)\to Z$ and then pushing forward via the crepant resolution $\mathrm{tot}(\omega_Z)\to X$.
The quiver of sections $Q$ from Figure \[fig:3foldFano11\]
is depicted in ${\ensuremath{\mathbb{Z}}}^3$, but we work in the class group of $X$ and hence take $ \mathcal{O}_X \sim \mathcal{O}_X(D_1+D_5+2D_6)$. If we order the arrows as in Figure \[fig:3foldFano11\](b), then the superpotential is the sum $$\begin{aligned}
W&=a_{22}a_{17}a_{10}a_{5}a_3 + a_{22}a_{16}a_{10}a_{6}a_3 +a_{22}a_{18}a_{15}a_{10}a_6a_1+a_{22}a_{18}a_{15}a_{10}a_5a_2+ a_{22}a_{18}a_{13}a_{8}a_2 \\
& \quad+a_{22}a_{17}a_{10}a_{7}a_{1} + a_{22}a_{17}a_{12}a_{8}a_1 +a_{22}a_{18}a_{14}a_{8}a_1 + a_{22}a_{16}a_{10}a_{7}a_2 +a_{22}a_{16}a_{12}a_{8}a_2 \\
& \quad+ a_{23}a_{21}a_{18}a_{9}a_2 + a_{23}a_{20}a_{15}a_{12}a_4+ a_{23}a_{21}a_{17}a_{12}a_4 + a_{23}a_{21}a_{18}a_{14}a_4 + a_{23}a_{11}a_{7}a_{2} \\
& \quad+ a_{23}a_{11}a_{6}a_{3} + a_{23}a_{20}a_{9}a_{3} + a_{24}a_{11}a_{7}a_{1} +a_{24}a_{11}a_{5}a_{3}+a_{24}a_{21}a_{18}a_{13}a_4 +a_{24}a_{21}a_{16}a_{12}a_4 \\
& \quad + a_{24}a_{19}a_{9}a_{3} + a_{24}a_{21}a_{18}a_{9}a_1 + a_{24}a_{19}a_{15}a_{12}a_4+ a_{25}a_{19}a_{14}a_{4} + a_{25}a_{11}a_{6}a_{1} + a_{25}a_{20}a_{9}a_{1} \\
& \quad + a_{25}a_{11}a_{5}a_{2} + a_{25}a_{19}a_{9}a_{2} + a_{25}a_{20}a_{13}a_{4} + a_{26}a_{21}a_{17}a_{10}a_5 + a_{26}a_{19}a_{14}a_{8} \\
& \quad + a_{26}a_{19}a_{15}a_{10}a_6 + a_{26}a_{21}a_{16}a_{10}a_6 + a_{26}a_{20}a_{13}a_{8} + a_{26}a_{20}a_{15}a_{10}a_5\end{aligned}$$ of all anticanonical cycles in $Q$. By taking partial derivatives, we compute that $$J_W=\left(\begin{array}{rl}
a_6a_1-a_5a_2, a_{25}a_{11} - a_{22}a_{18}a_{15}a_{10}, & a_9a_1 - a_{13}a_4, a_{24}a_{21}a_{18} - a_{25}a_{20} \\
a_{14}a_{4} - a_9a_2, a_{23}a_{21}a_{18} - a_{25}a_{19}, & a_5a_3-a_7a_1, a_{22}a_{17}a_{10} - a_{24}a_{11} \\
a_{9}a_3 - a_{15} a_{12}a_{4}, a_{24}a_{19} - a_{23}a_{20}, & a_{11}a_{6} - a_{20}a_{9}, a_{1}a_{25} - a_3a_{23}\\
a_{11}a_{5} - a_{19}a_{9}, a_2a_{25} - a_3a_{24}, & a_{20}a_{13} - a_{19} a_{14}, a_4a_{25}-a_8a_{26} \\
a_{11}a_{7} - a_{21}a_{18}a_{9}, a_1a_{24} - a_2a_{23}, & a_{19}a_{15} - a_{21}a_{16}, a_{12}a_{4}a_{24} - a_{10}a_{6}a_{26} \\
a_{16}a_{12} - a_{18}a_{13}, a_{8}a_2a_{22} - a_4a_{24}a_{21},& a_7a_2 - a_6a_3, a_{22}a_{16}a_{10}-a_{23}a_{11} \\
a_{17}a_{12} - a_{18}a_{14}, a_8a_1a_{22} - a_4a_{23}a_{21}, & a_{20}a_{15} - a_{21}a_{17}, a_4a_{23}a_{12} - a_{10}a_5a_{26} \\
a_{17}a_{10}a_{5} - a_{16}a_{10}a_{6}, a_3a_{22} - a_{26}a_{21}, & a_{10}a_{7}-a_{12}a_{8}, a_1a_{22}a_{17} - a_2a_{22}a_{16} \\
a_{14}a_{8} - a_{15}a_{10}a_{6}, a_1a_{22}a_{18} - a_{26}a_{19}, & a_{15}a_{10}a_{5} - a_{13}a_{8}, a_2a_{22}a_{18} - a_{26}a_{20}
\end{array}\right).$$ This ideal is equal to $J_{\mathscr{E}}$, so the toric algebra $A:=A_\mathscr{E}$ is consistent.
For each minimal generator $p^+-p^-$ of $J_W$ there exists another minimal generator $q^+-q^-$ with the property that ${\operatorname{\mathsf{t}}}(q^{\pm})={\operatorname{\mathsf{h}}}(p^{\pm})$, ${\operatorname{\mathsf{h}}}(q^{\pm})={\operatorname{\mathsf{t}}}(p^{\pm})$ and $x^{{\operatorname{div}}(q^{\pm})}=\prod_{\rho=1}^6 x_\rho / x^{{\operatorname{div}}(p^{\pm})}$. We list two such pairs on each line in $J_W$ above. This phenomenon is one aspect of the duality property of $\Delta$ described in Proposition \[prop:duality4fold\] below.
The cellular resolution for the fourfold example {#sec:conjectureresolution}
------------------------------------------------
We now sketch the construction of the toric cell complex $\Delta\subset \mathbb{T}^4$ for this example. The analogue of diagram is $$\label{eqn:4torusdiagram}
\begin{CD}
0@>>> M @>>> {\ensuremath{\mathbb{Z}}}^6 @>{\deg}>> {\operatorname{Cl}}(X)@>>> 0\\
@. @| @VV{f\vert_{{\ensuremath{\mathbb{Z}}}^6}}V @. @. \\
0 @>>> M @>>> {\ensuremath{\mathbb{R}}}^4 @>>> \mathbb{T}^4 @>>> 0
\end{CD}$$ where $f\colon {\ensuremath{\mathbb{R}}}^6\to {\ensuremath{\mathbb{R}}}^4:=M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ is orthogonal projection on to the subspace spanned by $M$. Since $A$ is consistent, Corollary \[cor:arrowsinA\] shows that the covering quiver $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^6$ is the union of all $M$-translates of the quivers $\widetilde{Q}(i)$ for $i\in Q_0$. Explicit computation of the image $f(\widetilde{Q}(i))$ for $i\in Q_0$ shows that $\bigcup_{i\in Q_0} f(\widetilde{Q}(i))$ is an embedded quiver in ${\ensuremath{\mathbb{R}}}^4$. Define $\Delta_0$ and $\Delta_1$ to be the union of all $M$-translates of the vertex set and arrow set respectively of this quiver. The resulting subsets of $\mathbb{T}^4$ define the collections $\Delta_0$ of $0$-cells and $\Delta_1$ of 1-cells and, just as in Lemma \[lem:McKaybijections\], there are canonical bijections between $Q_0$ and $\Delta_0$, and between $Q_1$ and $\Delta_1$. Moreover, a lengthy and tedious calculation shows that we may define collections $\Delta_k$ of $M$-periodic subsets in ${\ensuremath{\mathbb{R}}}^4$ for $k=2,3,4$, where the canonical bijection for $\Delta_2$ from Lemma \[lem:McKaybijections\] also holds. The corresponding closed subsets in $\mathbb{T}^4$ define the *toric cell complex* $\Delta\subset \mathbb{T}^4$. As before, each cell $\eta$ in $\Delta$ has a well-defined head ${\operatorname{\mathsf{h}}}(\eta)\in \Delta_0$, tail ${\operatorname{\mathsf{t}}}(\eta)\in \Delta_0$ and label ${\operatorname{div}}(\eta)\in {\ensuremath{\mathbb{N}}}^6$.
In this case, we have $\vert \Delta_0\vert = \vert \Delta_4\vert = 8$, that $\vert \Delta_1\vert = \vert \Delta_3\vert = 26$, and that $\vert \Delta_2\vert = 36$. Explicit computation and inspection shows that $\Delta$ satisfies the following duality property:
\[prop:duality4fold\] The map $\tau\colon \Delta\to \Delta$ that assigns to each $\eta\in \Delta_k$ the unique cell $\eta^\prime\in \Delta_{4-k}$ with ${\operatorname{\mathsf{t}}}(\eta^\prime)={\operatorname{\mathsf{h}}}(\eta)$, ${\operatorname{\mathsf{h}}}(\eta^\prime)={\operatorname{\mathsf{t}}}(\eta)$ and $x^{{\operatorname{div}}(\eta^\prime)} = \prod_{\rho=1}^6x_\rho/x^{{\operatorname{div}}(\eta)}$ is an involution.
Figure \[fig:4foldToricCell\] depicts several cells of $\Delta$. Figure \[fig:4foldToricCell\](a) shows arrow $a_{23}$ and the dual 3-cell, where paths from vertex 0 at the bottom to vertex 0 at the top traverse anticanonical cycles in $Q\subset \mathbb{T}^4$. Figure \[fig:4foldToricCell\](b) shows the 3-cells dual to arrows $a_1$ and $a_{24}$ intersect along a given shaded 2-cell, illustrating that $\Delta$ satisfies property . Note that both 3-cells from Figure \[fig:4foldToricCell\](b) lie in the 4-cell dual to vertex $0$, though we do not draw every edge in the 4-cell for the sake of clarity. Figures \[fig:4foldToricCell\](c) is similar, and shows for example that the 3-cell dual to $a_5$ is not equidimensional.
We verify by an exhaustive examination that $\Delta$ that satisfies property and, in addition, that $\Delta$ admits an incidence function $\varepsilon\colon \Delta\times \Delta\to\{0,\pm 1\}$. As a result, the minimal projective $(A,A)$-bimodule resolution of the consistent toric algebra $A=A_\mathscr{E}$ can be constructed as a cellular resolution. Indeed, it can be shown directly, by adapting the proof in [@Broomhead], that the minimal projective resolution of $A$ as a $(A,A)$-bimodule is $$\label{eqn:4foldresolution}
0 \longrightarrow P_4 \xlongrightarrow{d_4} P_3 \xlongrightarrow{d_3} P_2 \xlongrightarrow{d_2} P_1 \xlongrightarrow{d_1} P_0 \xlongrightarrow{\mu} A \longrightarrow 0,$$ with terms $\displaystyle{P_k=\bigoplus_{\eta \in \Delta_k} Ae_{{\operatorname{\mathsf{h}}}(\eta)}\otimes [\eta] \otimes Ae_{{\operatorname{\mathsf{t}}}(\eta)}}$ and differentials $$d_{k}(1 \otimes [\eta] \otimes 1)=\sum_{{\operatorname{cod}}(\eta',\eta)=1}\varepsilon(\eta,\eta') \overleftarrow{\partial}_{\!\eta'}\eta\otimes[\eta'] \otimes \overrightarrow{\partial}_{\!\eta'}\eta,$$ where $\mu\colon P_0=\bigoplus_{i \in \Delta_0} Ae_{i}\otimes Ae_{i} \rightarrow A$ is the multiplication map.
On signs and syzygies
---------------------
Before stating the main conjecture we make a key observation which explains and justifies our decision to introduce no signs in the superpotentials throughout this paper, namely, that *it is impossible to introduce signs in the superpotential $W$ above so that the generators of $J_W$ can be recovered directly by taking partial derivatives*.
Indeed, consider only terms of $W$ that involve $a_{23}\in Q_1$, namely, those arising in $a_{23}\partial_{a_{23}}W$. Figure \[fig:4foldToricCell\](a) illustrates all seven of the corresponding anticanonical cycles in $Q\subset \mathbb{T}^4$: the 3-cell $\eta_{23}$ dual to arrow $a_{23}$ is drawn as a convex 3-polytope with arrow $a_{23}$ sticking out of the top. The facets of $\eta_{23}$ correspond to those generators of $J_W$ arising from partials of $W$ with respect to paths involving $a_{23}$, e.g., relation $a_{14}{a_4}-a_9a_2$ arises from $\partial_{a_{23}a_{21}a_{18}}W$. We now attempt to introduce signs in $W$ so that the generators of $J_W$ are recovered directly from partial derivatives of $W$. If, say, we fix the sign of $a_{23}a_{21}a_{18}a_{9}a_2$ to be $+1$, then the relation $a_{14}a_{4} - a_9a_2$ forces the sign of $a_{23}a_{21}a_{18}a_{14}a_4$ to be $-1$, but then $a_{17}a_{12} - a_{18}a_{14}$ forces the sign of $a_{23}a_{21}a_{17}a_{12}a_4$ to be $+1$, and so on. By repeating, we hop from one anticanonical cycle to another around the surface of the polytope $\eta_{23}$. There are an odd number of paths, so we obtain sign $-1$ for the original path $a_{23}a_{21}a_{18}a_{9}a_2$ after passing once around $\eta_{23}$. This contradiction shows that introducing signs in $W$ cannot produce the necessary signs in $J_W$.
Comparing the third term $P_3$ in the cellular resolution with the third term in the resolution as described by Butler–King [@ButlerKing (1.1)] shows that the set of 3-cells $\Delta_3$ provides a minimal set of bimodule generators for the space of syzygies $$A\otimes\text{Tor}^A_3(U_0,U_0)\otimes A\cong A\otimes \frac{JI\cap IJ}{I^2+JIJ}\otimes A,$$ where $I:=J_\mathscr{E}$ and $J$ is the ideal in ${\ensuremath{\Bbbk}}Q$ generated by the set of arrows. As Alastair King remarks, the syzygy corresponding to the cell $\eta_{23}\in \Delta_3$ from Figure \[fig:4foldToricCell\](a) provides an equation with signs that includes all seven terms of $W$ involving $a_{23}$. This is indeed the case, but this does not contradict the assertion above. To see this, list all seven relations defined by codimension-one faces of $\eta_{23}$ as $r_1 := a_{14}a_4 - a_9a_2$, $r_2 := a_9a_3 - a_{15}a_{12}a_4$, $r_3 := a_7a_2 - a_6a_3$, $r_4 := a_{11}a_7 - a_{21}a_{18}a_9$, $r_5 := a_{20}a_9-a_{11}a_6$, $r_6 := a_{21}a_{17}-a_{20}a_{15}$ and $r_7 := a_{17}a_{12} - a_{18}a_{14}$. The equation $$\label{eqn:syzygyeqn}
a_{21}a_{18}r_1 + a_{20}r_2+a_{11}r_3 - r_4a_2 - r_5a_3 - r_6a_{12}a_4 = a_{21}r_7a_4\in JIJ$$ shows how to pass between two presentations of the syzygy corresponding to $\eta_{23}\in \Delta_3$: $$s_{23} = [a_{21}a_{18}r_1 + a_{20}r_2+a_{11}r_3] = [r_4a_2 + r_5a_3 + r_6a_{12}a_4]\in \text{Tor}^A_3(U_0,U_0)$$ Multiplying on the left by $a_{23}$ and expanding provides an equation with signs linking the seven terms of $W$ involving $a_{23}$, but each term appears *twice* with opposite signs.
The main conjecture
-------------------
To formulate the cellular resolution conjecture, assume that $A$ is the consistent toric algebra associated to a collection $\mathscr{E}$ on a Gorenstein affine toric variety $X$ of dimension $n$. Let $\widetilde{Q}\subset {\ensuremath{\mathbb{R}}}^d$ denote the covering quiver of the quiver of sections $Q$ of $\mathscr{E}$. Write $f\colon {\ensuremath{\mathbb{R}}}^d\to {\ensuremath{\mathbb{R}}}^n:=M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{R}}}$ for the orthogonal projection. A priori, the vertices and arrows in the image $f(\widetilde{Q})$ may collide and intersect. Nevertheless, in every case considered in this paper, the toric cell complex $\Delta\subset \mathbb{T}^n$ is constructed from an $M$-periodic quiver in ${\ensuremath{\mathbb{R}}}^n$ whose 0-cells and 1-cells are supported in the image $f(\widetilde{Q})\subset {\ensuremath{\mathbb{R}}}^n$, and we suggest that this can always be done if the global dimension of $A$ is equal to $n$. More precisely, we formulate the following conjecture.
If the global dimension of a consistent toric algebra $A$ equals the dimension of $X$, then the toric cell complex $\Delta\subset\mathbb{T}^n$ exists and is constructed as above. Moreover, is the minimal projective $(A,A)$-bimodule resolution of $A$ as in Theorem \[thm:1.2\].
As Remark \[rem:notcelldivision\] shows, consistent toric algebras of global dimension $n$ need not be Calabi–Yau in general. Nevertheless, assuming the conjecture, a sufficient condition for such algebras to be Calabi–Yau can be read off directly from $\Delta$:
Let $A$ be a consistent toric algebra of global dimension $n$ with toric cell complex $\Delta$. If $\Delta$ satisfies the duality property as stated in Proposition \[prop:dualityMcKay\], then $A$ is Calabi–Yau.
The algebras that satisfy the conditions of the corollary, including the example presented in Sections 6.1-6.2, generalise to arbitrary dimension the three-dimensional algebras constructed from algebraically consistent dimer models.
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[^1]: Our chosen Fano $Z$ provides an interesting starting point since the construction of tilting bundles by Bondal [@BondalMFO] does not apply. The tilting bundle here was constructed originally by Greg Smith [@BondalMFO report by Craw].
|
---
abstract: |
We prove that Schrödinger operators with meromorphic potentials $(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+
\frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$ have purely singular continuous spectrum on the set $\{E: L(E)<\delta{(\alpha,\theta)}\}$, where $\delta$ is an explicit function, and $L$ is the Lyapunov exponent. This extends results of [@maryland] for the Maryland model and of [@ayz1] for the almost Mathieu operator, to the general family of meromorphic potentials.
author:
- SVETLANA JITOMIRSKAYA AND FAN YANG
title: Singular continuous spectrum for singular potentials
---
Introduction
============
We study operators of the form: $$\label{analyticmodel}
(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+\frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$$ acting on $l^2(Z)$, where $\alpha \in {{\mathbb R}}\setminus {{\mathbb Q}}$ is the frequency, ${{\mathbb T}}={{\mathbb R}}/{{\mathbb Q}}$, $\theta \in {{\mathbb T}}$ is the phase, $f$ is an analytic function and $g$ is Lipshitz. This class contains all meromorphic potentials and therefore both the almost Mathieu ($f\equiv 1$, $g=\lambda \cos 2\pi\theta$) and Maryland ($f=\sin 2\pi\theta$, $g=\lambda \cos
2\pi\theta$) families as particular cases.
Let $\frac{p_n}{q_n}$ be the continued fraction approximants of $\alpha\in {{\mathbb R}}\setminus {{\mathbb Q}}$.
Assume $g/f$ has $m$ poles, $m\ge 0$. We denote them by $\theta_i,
\;i=1,...,m,$ including multiplicities. We now define index $\delta$ as follows:
$$\label{delta}
\delta( \alpha,\theta) = \limsup_{n\rightarrow \infty} \dfrac{\sum_{i=1}^m \ln\|q_n(\theta-\theta_i) \|_{{{\mathbb R}}/ {{\mathbb Z}}}+ \ln q_{n+1}} {q_n}.$$
where $\|x\|_{{{\mathbb R}}/ {{\mathbb Z}}}=min_{l\in {{\mathbb Z}}}|x-l|$. Let $L(E)$ be the Lyapunov exponent, see (\[L\]). $L$ depends also on $\alpha$ but we suppress it from the notation as we keep $\alpha$ fixed.
Our main result is:
\[sc\]Let $ \delta(\alpha,\theta) $ be as in (\[delta\]). Then
1. $ H_{\alpha, \theta} $ has no eigenvalues on $\{ E:L(E)< \delta(\alpha, \theta)\}$.
2. If $L(E)>0$ for a.e. $E$ (in particular, if $m>0$), then $ H_{\alpha, \theta} $ has purely singular continuous spectrum on $\{ E:L(E)< \delta(\alpha, \theta)\}$.
[**Remark.**]{} Since absence of absolutely continuous spectrum follows from a.e.positivity of the Lyapunov exponents and holds for all unbounded potentials [@simonspencer], part (2) immediately follows from part (1), on which we therefore concentrate.
Recently, there has been an increased interest in obtaining arithmetic conditions (in contrast to a.e. statements) for various quasiperiodic spectral results. In particular, there have been remarkable advances in the theory of the almost Mathieu operator
$$\label{amo}
(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+\lambda \cos 2\pi(\theta+n\alpha)u_n$$
(see e.g. [@lastrev; @jmrev] for the review and background in physics). Define $$\beta=\beta(\alpha)=\limsup_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_n},$$ which describes how Liouvillian $\alpha$ is. We say that $\alpha$ is Diophantine if $\beta(\alpha)=0.$ Note that for almost every phase $\theta$ (only depends on $\alpha$) we have $\delta(\alpha,\theta)=\beta(\alpha).$
It was conjectured in 1994 [@1994] that $\lambda=e^\beta$ is the phase transition point from singular continuous spectrum to pure point spectrum for $\alpha$-Diophantine $\theta$ (and that the transition is at larger $\lambda$ for non-$\alpha$-Diophantine $\theta$). The history of partial results towards this conjecture include [@j; @AJ1]. Recently, Avila, You and Zhou proved [@ayz1]
\[lanaconjecture\] For $\lambda>e^\beta$, the spectrum is pure point with exponentially decaying eigenfunctions for a.e. $\theta,$ and for $1<\lambda<e^\beta$, the spectrum is purely singular continuous for all $\theta$.
[**Remark.**]{} The spectrum is known to be absolutely continuous for all $\alpha,\theta$ for $\lambda<1$ (the final result in [@Avila_2008]).
A fully arithmetic version of the localization statement
\[liu1\] For $\lambda>e^\beta$, the spectrum is pure point with exponential decaying eigenfunctions for $\alpha$-Diophantine $\theta$
was established recently in [@jl1].
Define also $$\label{G.delta}
\gamma=\gamma(\alpha,\theta)=\limsup_{n \to \infty} \dfrac{-\ln \vert\vert 2 \theta+n \alpha \vert\vert_{{{\mathbb R}}/{{\mathbb Z}}} }{\vert n \vert} ~\mbox{.}$$ We say that $\theta$ is $\alpha$-Diophantine if $\gamma(\alpha,\theta)=0.$
It was also conjectured in [@1994; @Jitomirskaya_review_2007] that $\lambda=e^\gamma$ is the phase transition point from singular continuous spectrum to pure point spectrum for Diophantine $\alpha$ (and that the transition is at larger $\lambda$ for non-Diophantine $\alpha$). Partial results towards this conjecture include [@js; @j]. The conjecture was recently fully established in [@jl2]:
\[liu2\] For $\lambda>e^{\gamma(\alpha,\theta)}$, the spectrum is pure point with exponentially decaying eigenfunctions for Diophantine $\alpha,$ and for $\lambda<e^{\gamma(\alpha,\theta)}$, the spectrum is singular continuous for all $\alpha.$
Therefore, for the almost Mathieu operator the precise transition from pure point to singular continuous spectrum is understood for either Diophantine $\alpha$ and all $\theta$ or for all $\alpha$ and $\alpha$-Diophantine $\theta,$ but not yet for all parameters.
Another case with a significant recent arithmetic results is the Maryland model $$\label{maryland}
(H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+\lambda \tan \pi(\theta+n\alpha)u_n$$ It is the prototypical operator of form (\[analyticmodel\]). This model was proposed by Grempel, Fisherman, and Prange [@grempel1982localization] as a linear version of the quantum kicked rotor. It is an exactly solvable example of the family of incommensurate models, thus attracting continuing interest in physics, e.g. [@GKDS2014]. The complete description of spectral transitions for the Maryland model (depending on arithmetic properties of all parameters) was given recently in [@maryland].
Namely, an index $\delta(\alpha,\theta) \in [-\infty,\infty]$ was introduced in [@maryland]: $$\label{marylanddelta}
\delta(\alpha,\theta)=\limsup_{n \rightarrow \infty} \frac{\ln \|q_n(\theta-\frac{1}{2})\|_{{{\mathbb R}}/ {{\mathbb Z}}}+\ln q_{n+1}}{q_n}$$
The main result of [@maryland] regarding the singular continuous part is:
\[jlsingular\][@maryland] $H_{\alpha,\theta}$ has purely singular continuous spectrum on $\{E: L(E)<\delta(\alpha,\theta) \}$.
It is sharp since
\[jlpoint\][@maryland] $H_{\alpha,\theta}$ has pure point spectrum on $\{E: L(E) > \delta(\alpha,\theta) \}$.
Our result therefore is an extension of Theorem \[jlsingular\] (to which Theorem \[sc\] specializes for $f=\cos 2\pi\theta$, $g=\lambda \sin
2\pi\theta$) to the general family of singular potentials. For $f\equiv 1, g=\lambda \cos 2\pi\theta$ we recover the singular continuous part of Theorem \[lanaconjecture\] (note that the proof of [@ayz1] also extends in this case to $f\equiv1$ and a Lipshitz condition on $g$ without many changes). Theorem \[jlpoint\] shows that our result is sharp for the Maryland model. However, Theorems \[liu1\], \[liu2\] show that it is not sharp for the almost Mathieu operator other than for $\alpha$-Diophantine $\theta.$ Based on this, we do not expect sharpness for general Lipshitz or even analytic potentials ($f\equiv
1$), and conjecture that sharpness (that is point spectrum in the complementary regime other than possibly on the transition line) may be a corollary of certain monotonicity.
Preliminaries: cocycle, Lyapunov exponent
=========================================
Assume without loss of generality, $ f(\theta)=(e^{2\pi i\theta}-
e^{2\pi i\theta_1}) \cdots (e^{2\pi i\theta}-
e^{2\pi i\theta_m})$, $m=1,\cdots$. Let $\Theta=\cup_{l=1}^m {\theta_l+{{\mathbb Z}}\alpha+{{\mathbb Z}}}$. From now on we fix $E$ in the spectrum and $\theta\in \Theta^c$ such that $L(E)<\delta(\alpha, \theta)$. We will show $H_{\lambda, \alpha, \theta}$ cannot have an eigenvalue at $E.$
A formal solution of the equation $ H_{\alpha, \theta}u=Eu $ can be reconstructed via the following equation $$\begin{aligned}
\left (\begin{matrix} u_{n+1} \\ u_{n} \end{matrix} \right )= A(\theta+n\alpha)\left (\begin{matrix} u_{n} \\ u_{n-1} \end{matrix} \right )
\end{aligned}$$ where $
A(\theta)=\left (\begin{matrix} E-\frac{g(\theta)}{f(\theta)} & -1 \\ 1 & 0 \end{matrix} \right )
$ is the so-called transfer matrix.
The pair $(\alpha,A)$ is the cocycle corresponding to the operator (\[analyticmodel\]). It can be viewed as a linear skew-product $(x,\omega)\mapsto(x+\alpha,A(x)\cdot \omega)$. Generally, one can define $M_n$ for an invertible cocycle $(\alpha, {{\mathbb M}})$ by $(\alpha,M)^n=(n\alpha,M_n)$, $n \in Z$ so that for $n \geq 0$: $$M_n(x)=M(x+(n-1)\alpha)M(x+(n-2)\alpha) \cdots M(x),$$ and $M_{-n}(x)=M_n(x-n\alpha)$.
The Lyapunov exponent of a cocycle $(\alpha,M)$ is defined by$$L(\alpha,M)=\lim_{n \rightarrow \infty}\frac{1}{n} \int_{{{\mathbb T}}} \ln \|M_n(x)\|\mathrm{d}x.$$
Let $A(x)=\frac{1}{f(x)}D(x)$ where $$\begin{aligned}
D(x)=
\left(
\begin{matrix}
E f(x)- g(x) & -f(x)
\\ f(x) & 0
\end{matrix}
\right)
\end{aligned}$$ be the regular part of $A(x)$. Since $\int_{{{\mathbb T}}} \ln{|f(x)|} \mathrm{d}x=0$, we have $$\label{L}
L(E):=L({\alpha, A} )= L(\alpha, D).$$
\[lana\][@AJ1] Let $\alpha\in {{\mathbb R}}\backslash{{\mathbb Q}}$, $\theta\in{{\mathbb R}}$ and $0\leq j_0 \leq q_{n}-1$ be such that $$\mid \sin \pi(\theta+j_{0}\alpha)\mid = \inf_{0\leq j \leq q_{n}-1} \mid \sin \pi(\theta+j\alpha)\mid ,$$ then for some absolute constant $C>0$, $$-C\ln q_{n} \leq \sum_{j=0,j\neq j_0}^{q_{n}-1} \ln \mid \sin \pi (\theta+j\alpha) \mid+(q_{n}-1)\ln2 \leq C\ln q_n$$
We will also use that the denominators of continued fraction approximants of $\alpha$ satisfy
$$\| k\alpha \|_{{{\mathbb R}}\backslash {{\mathbb Z}}} \geq \| q_n \alpha \|_{{{\mathbb R}}\backslash {{\mathbb Z}}}, 1 \leq k < q_{n+1},$$ and $$\label{contfraction}
\dfrac{1}{2q_{n+1}} \leq \|q_n \alpha\|_{{{\mathbb R}}\backslash {{\mathbb Z}}} \leq \dfrac{1}{q_{n+1}}.$$
A quick corollary of subadditivity and unique ergodicity is the following upper semicontinuity statement:
\[LcontrolA\] (e.g. [@AJ2]) Suppose $(\alpha,A)$ is a continuous cocycle. Then for any $ \varepsilon>0$, there exists $C(\varepsilon)>0$, such that for any $x\in {{\mathbb T}}$ we have $$\|A_n(x) \| \leq C e ^{n(L(A)+\varepsilon)}.$$
\[partf\] Applying this to 1-dimensional continuous cocycles, we get that if $g$ is a continuous function such that $\ln |g| \in L^1({{\mathbb T}})$, then $$|\prod_{l=a}^b g (x+l\alpha) | \leq e^{(b-a+1)(\int \ln|g| \mathrm{d}\theta+\varepsilon)}.$$
Absence of point spectrum
=========================
Let $\varphi$ be a solution to $H_{\alpha, \theta}\varphi =E\varphi$ satisfying $\| \left(
\begin{matrix}
\varphi_0\\
\varphi_{-1}
\end{matrix}
\right)
\|=1$. We have the following restatement of Gordon’s lemma. We state a precise form that will be convenient for us.
\[absenceofpp\] If there exists a constant $c>0$ and a subsequence $q_{n_i}$ of $q_n$ such that the following estimates holds: $$\label{absenceofpp_1}
\|(A_{q_{n_i}}^2(\theta)-A_{2q_{n_i}}(\theta))
\left(
\begin{matrix}
\varphi_0\\
\varphi_{-1}
\end{matrix}
\right)
\|\leq e^{-cq_{n_i}}$$ and $$\label{absenceofpp_2}
\|(A^{-1}_{q_{n_i}}(\theta)
-A^{-1}_{q_{n_i}}(\theta-q_{n_i}\alpha))
\left(
\begin{matrix}
\varphi_0\\
\varphi_{-1}
\end{matrix}
\right)
\|\leq e^{-cq_{n_i}},$$ then we have $$\label{maxineq}
\max\{ \|(\begin{array}{cc}\varphi_{q_{n_i}} \\ \varphi_{q_{n_i}-1}\end{array})\|,
\|(\begin{array}{cc}\varphi_{-q_{n_i}} \\ \varphi_{-q_{n_i}-1}\end{array})\|,
\|(\begin{array}{cc}\varphi_{2q_{n_i}} \\ \varphi_{2q_{n_i}-1}\end{array})\| \} \geq \frac{1}{4}.$$
This is a standard argument, going back to [@Gordon_1976]. The key idea is to use the following two equalities: $$\begin{aligned}
\label{gordon}
\left\lbrace
\begin{matrix}
A_{q_{n_i}}(\theta)-\mathrm{Tr} A_{q_{n_i}}(\theta) \cdot Id + A_{q_{n_i}}^{-1}(\theta)=0\\
A_{q_{n_i}}^2(\theta)-\mathrm{Tr} A_{q_{n_i}}(\theta) \cdot A_{q_{n_i}}(\theta) +Id=0
\end{matrix}
\right.\end{aligned}$$ and separate the cases $|\mathrm{Tr}A_{q_{n_i}} (\theta)| > \frac{1}{2}$,$|\mathrm{Tr}A_{q_{n_i}} (\theta)| < \frac{1}{2}.$
$\hfill{} \Box $
Proof of Theorem \[sc\]
-----------------------
Assume $\varphi$ is a decaying solution of $H_{\alpha, \theta}\varphi = E\varphi$, satisfying $\|(\begin{array}{cc} \varphi_0 \\ \varphi_{-1} \end{array})\|=1$. On one hand, it must be true that for any $\eta>0$, there exists $N$ such that $\|\left(
\begin{matrix}
\varphi_k\\
\varphi_{k-1}
\end{matrix}
\right)
\|\leq \eta$ for $|k|>N$. On the other hand, we will prove the following lemma in section $5$:
\[A\] For any $ \varepsilon >0$ there exists a subsequence $\{q_{n_i}\}$ of $\{q_n\}$ so that we have the following estimates:
$$\label{Ainverse}
\|(A^{-1}_{q_{n_i}}(\theta)
-A^{-1}_{q_{n_i}}(\theta-q_{n_i}\alpha))
\left(
\begin{matrix}
\varphi_0\\
\varphi_{-1}
\end{matrix}
\right)
\| \leq e^{q_{n_i}(L(E)-\delta(\alpha, \theta)+4\varepsilon)},$$
and $$\label{Asquare}
\|(A_{q_{n_i}}^2(\theta)-A_{2q_{n_i}}(\theta))
\left(
\begin{matrix}
\varphi_0\\
\varphi_{-1}
\end{matrix}
\right)
\| \leq e^{q_{n_i}(L(E)-\delta(\alpha, \theta)+4\varepsilon)}.$$
Then combining Lemma \[A\] and Theorem \[absenceofpp\] we get a contradiction, which shows the absence of point spectrum.
key lemmas
==========
Let $| \sin \pi (\theta-\theta_l+j_l \alpha) | =\inf_{0\leq j \leq q_n-1} | \sin \pi (\theta-\theta_l+j\alpha)|$.
\[minimal\]
If $\delta(\alpha,\theta) > 0$, then for any $\varepsilon>0$, there exists a subsequence $q_{n_i}$ of $q_n$ such that the following estimate holds $$\prod_{l=1}^m | \sin \pi (\theta-\theta_l+j_l \alpha) | \geq \frac{e^{q_{n_i}(\delta-\frac{\varepsilon}{2})}}{q_{n_i+1}}.$$
By the definition of $\delta(\alpha, \theta)$, there exists a subsequence $q_{n_i}$ of $q_n $ such that $$\dfrac{\sum_{l=1}^m \ln\|q_{n_i}(\theta-\theta_l) \|+ \ln q_{n_i+1}} {q_{n_i}} >\delta(\alpha,\theta)-\frac{\varepsilon}{4},$$ thus $$\|q_{n_i}(\theta-\theta_1)\| \cdots \|q_{n_i}(\theta-\theta_m)\| > \frac{e^{q_{n_i} (\delta-\frac{\varepsilon}{4})}}{q_{n_i+1}}.$$ In particular, $\|q_{n_i} (\theta -\theta_l)\| > \frac{e^{q_{n_i} (\delta-\frac{\varepsilon}{4})}}{q_{n_i+1}} $ for any $1\leq l \leq m$. Since $$\begin{aligned}
&| \sin \pi (\theta-\theta_l+j_l \alpha)|\\
\geq &2\|(\theta-\theta_l+j_l \alpha)\|\\
\geq &\frac{2\|q_{n_i}(\theta-\theta_l+j_l \alpha)\|}{q_{n_i}} \\
\geq &\frac{2\|q_{n_i}(\theta-\theta_l)\|-\frac{2q_{n_i}}{q_{n_i+1}}}{q_{n_i}} \\
\geq &\frac{\|q_{n_i}(\theta-\theta_l)\|}{q_{n_i}} .
\end{aligned}$$ We have $$\prod_{l=1}^m | \sin \pi (\theta-\theta_l+j_l \alpha) |
\geq \prod_{l=1}^m \frac{\|q_{n_i}(\theta-\theta_l)\|}{q_{n_i}}
> \frac{e^{q_{n_i} (\delta-\frac{\varepsilon}{4})}}{q_{n_i+1}} \cdot \frac{1}{(q_{n_i})^m}
> \frac{e^{q_{n_i}(\delta-\frac{\varepsilon}{2})}}{q_{n_i+1}}$$
$\hfill{} \Box $
\[f\] The following estimate holds
$$\prod_{j=0}^{q_{n_i}-1}|f(\theta+j\alpha )| \geq \dfrac{e^{q_{n_i}(\delta - \varepsilon)}}{q_{n_i+1}}.$$
$$\begin{aligned}
\prod_{j=0}^{q_{n_i}-1}|f(\theta+j\alpha )|
&=2^{m q_{n_i}}\prod_{l=0}^m \prod_{j=0}^{q_{n_i}-1}| \sin \pi (\theta-\theta_l+j\alpha)| \\
&=2^{m q_{n_i}}\left( \prod_{l=0}^m \prod_{j=0,j\neq j_l}^{q_{n_i}-1}| \sin \pi (\theta-\theta_l+j\alpha)| \right) \cdot \left( \prod_{l=0}^m |\sin \pi (\theta-\theta_l+j_l\alpha)| \right).\end{aligned}$$
Combining Lemma $\ref{lana}$ and $\ref{minimal}$, $$\prod_{j=0}^{q_{n_i}-1}|f(\theta+j\alpha )| \geq 2^{mq_{n_i}} e^{m(-C \ln q_{n_i} -(q_{n_i}-1)\ln 2)} \cdot \frac{e^{q_{n_i}(\delta-\frac{\varepsilon}{2})}}{q_{n_i+1}} \geq \frac{e^{q_{n_i}(\delta-\varepsilon)}}{q_{n_i+1}}.$$
$\hfill{} \Box $
proof of lemma \[A\]
====================
We give a detailed proof of (\[Ainverse\]). (\[Asquare\]) could be proved in a similar way.
Let $$A^{-1}(x)=\frac{1}{f(x)} \left( \begin{array}{cc} 0 & f(x) \\ -f(x)& Ef(x)-g(x) \end{array} \right) \triangleq \frac{F(x)}{f(x)}.$$ Consider $$\Psi_{n_i}=\left( A_{q_{n_i}}^{-1} (\theta)-A_{q_{n_i}}^{-1}(\theta- q_{n_i} \alpha)\right) \left( \begin{array}{cc}\varphi_0 \\ \varphi_{-1} \end{array} \right).$$
For simplicity let us introduce some notations: fixing $\theta$, for any function $z(x)$ on ${{\mathbb T}}$ denote $z_j=z(\theta+j\alpha)$; for any matrix function $M(x)$ denote $M^j=M(\theta+j\alpha)$. Then, by telescoping, $$\begin{aligned}
\Psi_{n_i}
=&
\left( \frac{F^0}{f_0} \frac{F^1}{f_1} \cdots \frac{F^{q_{n_i}-1}}{f_{q_{n_i}-1}} - \frac{F^{-q_{n_i}}}{f_{-q_{n_i}}} \cdots \frac{F^{-1}}{f_{-1}} \right)
\left( \begin{array}{cc} \varphi_0 \\ \varphi_{-1} \end{array} \right) \\
=&
\sum_{j=0}^{q_{n_i}-1} \left( \frac{F^0 F^1 \cdots F^{j-1}}{f_0 f_1\cdots f_{j-1}} \right)
\left( \frac{F^j}{f_j} -\frac{F^{-q_{n_i}+j}}{f_{-q_{n_i}+j}}\right)
\left( \frac{F^{-q_{n_i}+j+1} \cdots F^{-1}}{f_{-q_{n_i}+j+1}\cdots f_{-1}} \right) \left( \begin{array}{cc} \varphi_0 \\ \varphi_{-1} \end{array} \right),
\end{aligned}$$ where for $j=0$ the first, and for $j=q_{n_i}-1$ the last, multiple are set to be equal to one.
Thus$$\begin{aligned}
\Psi_{n_i}
=&
\sum_{j=0}^{q_{n_i}-1} \left( \prod_{l=0}^{j-1}\frac{F^l}{f_l} \right)
\left( \frac{F^{j}}{f_{j}} -\frac{F^{-q_{n_i}+j}}{f_{-q_{n_i}+j}}\right)
\left( \begin{array}{cc}\varphi_{-q_{n_i}+j+1} \\ \varphi_{-q_{n_i}+j} \end{array} \right)\\
=&
\sum_{j=0}^{q_{n_i}-1} \left( \prod_{l=0}^{j-1}\frac{F^l}{f_l} \right)
\left( \frac{F^{j}f_{-q_{n_i}+j}-F^{-q_{n_i}+j}f_{-q_{n_i}+j} +F^{-q_{n_i}+j}f_{-q_{n_i}+j} - F^{-q_{n_i}+j}f_{j}}{f_{j}f_{-q_{n_i}+j}}\right)
\left( \begin{array}{cc} \varphi_{-q_{n_i}+j+1} \\ \varphi_{-q_{n_i}+j} \end{array} \right) \\
=&
\sum_{j=0}^{q_{n_i}-1} \left( \prod_{l=0}^{j-1}\frac{F^l}{f_l} \right)
\left( \frac{F^{j}-F^{-q_{n_i}+j}}{f_{j}} \left( \begin{array}{cc} \varphi_{-q_{n_i}+j+1} \\ \varphi_{-q_{n_i}+j} \end{array} \right)
+\frac{f_{-q_{n_i}+j}-f_{j}}{f_{j}}\left( \begin{array}{cc} \varphi_{-q_{n_i}+j} \\ \varphi_{-q_{n_i}+j-1} \end{array} \right) \right).\end{aligned}$$
Since $\phi$ is decaying solution, there exists a constant $C>0$ such that $$\| \left( \begin{array}{cc}\varphi_{k} \\ \varphi_{k-1} \end{array} \right)\| \leq C.$$
Observe that $sup_{\theta} \|F(\theta+q_{n_i}\alpha)-F(\theta)\|<\frac{C}{q_{n_i+1}}$. Now we can get, using Lemma \[LcontrolA\], Remark \[partf\] and Lemma \[f\] in the second inequality $$\begin{aligned}
&\|\left( A_{q_{n_i}}^{-1} (\theta)-A_{q_{n_i}}^{-1}(\theta- q_{n_i} \alpha)\right) \left( \begin{array}{cc}\varphi_0 \\ \varphi_{-1} \end{array} \right)\|\\
\leq &C\sum_{j=0}^{q_{n_i}-1}\frac{\|\prod_{l=0}^{j-1}F^l\|}{q_{n_i+1}|\prod_{l=0}^{j}f_l|}\\
= &C\sum_{j=0}^{q_{n_i}-1}\frac{\|\prod_{l=0}^{j-1}F^l \| |\prod_{l=j+1}^{q_{n_i}-1}f_l |} {q_{n_i+1}|\prod_{l=0}^{q_{n_i}-1}f_l|}\\
\leq &C\frac{q_{n_i} e^{q_{n_i}(L(E)+\varepsilon)} \cdot e^{q_{n_i} \varepsilon}}{e^{q_{n_i} (\delta-\varepsilon)}}\\
\leq & e^{q_{n_i} (L(E)-\delta+4\varepsilon)}.\end{aligned}$$
$\hfill{} \Box$
Acknowledgement {#acknowledgement .unnumbered}
===============
F.Y would like to thank Rui Han for his help, useful discussions and encouragement throughout all the work. We also would like to thank Qi Zhou and Wencai Liu for some suggestions. F.Y was supported by CSC of China (no.201406330007) and the NSFC (no.11571327) and NSF of Shandong Province (grant no.ZR2013AM026). She would like to thank her advisor Daxiong Piao (Professor at Ocean University of China) for supporting her partly. This research was partially supported by NSF DMS-1401204.
[10]{}
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|
---
abstract: 'Time-series of high throughput gene sequencing data intended for gene regulatory network () inference are often short due to the high costs of sampling cell systems. Moreover, experimentalists lack a set of quantitative guidelines that prescribe the minimal number of samples required to infer a reliable model. We study the temporal resolution of data quality of inference in order to ultimately overcome this deficit. The evolution of a Markovian jump process model for the // pathway of proteins and metabolites in the G$_1$ phase of the cell cycle is sampled at a number of different rates. For each time-series we infer a linear regression model of the using the method. The inferred network topology is evaluated in terms of the area under the precision-recall curve (). By plotting the against the number of samples, we show that the trade-off has a, roughly speaking, sigmoid shape. An optimal number of samples corresponds to values on the ridge of the sigmoid.'
author:
- 'Johan Markdahl, Nicolo Colombo, Johan Thunberg, and Jorge Gonçalves [^1][^2]'
bibliography:
- 'cdc2017.bib'
title: |
**Experimental design trade-offs for gene regulatory network inference:\
an *in silico* study of the yeast cell cycle**
---
Introduction
============
Time-series gene expression data provides a series of snapshots of molecular concentrations in gene regulatory networks () [@bar2012studying]. This information is used to infer dynamic models of networks which aid our understanding of how observable phenotypes, diseases, arise from molecular interactions [@kitano2002systems]. As such, time-series data is of importance to fundamental research within systems biology, and potentially also in applications like medical diagnostics, drug development, and therapies [@barabasi2004network]. The advent of high throughput sequencing have made time-series data widely available although it is prohibitively expensive to densely sample gene expression levels. It remains difficult for experimentalists to accurately judge the frequency and distribution of samples needed to infer network structures: for each project, they must navigate the trade-off between oversampling (more samples than necessary, increasing costs with no benefit to inference) and undersampling (too few samples to reliably infer the , potential waste of resources and failure to infer the ) [@bar2004analyzing]. Such costs add up; studies indicate that 85% of research investment in biomedical sciences is wasted, corrsponding to US\$200 billion worldwide in 2010 [@macleod2014biomedical]. This work undertakes an *in silico* study of the impact of the cost number of samples trade-offs on the quality of the output produced by a inference algorithm. Our ultimate goal, to which this paper is a stepping stone, is to formulate guidelines and construct decision support systems to help researches navigate trade-offs such that models of desired quality can be inferred at a minimal cost.
The performance of inference algorithms has been benchmarked against *in silico* and *in vivo* data in a number of comparative studies [@werhli2006comparative; @marbach2010revealing; @marbach2012wisdom; @aderhold2014statistical]. The aforementioned trade-off has received comparatively less attention [@bar2012studying; @sima2009inference; @bar2004analyzing; @sefer2016tradeoffs; @mombaerts2016optimising]. There are of course many works that touch upon it in passing, [@husmeier2003sensitivity], or that pay the price of intentionally oversampling to ensure capturing high-frequency content [@owens2016measuring; @brunton2016discovering]. Early work that take a systematic approach to studying the trade-off are rather abstracts and deal with generalities in broad strokes [@bar2012studying; @sima2009inference; @bar2004analyzing]. For example, [@bar2012studying] states that cyclic processes such as cell cycles and circadean rythms should be sampled uniformly over multiple cycles. In perturbation-response studies, by contrast, most samples should be taken early to capture the transient dynamics.
Only in the past year have results been published to support the common sense notions of navigating the trade-off that are current experimental practices [@sefer2016tradeoffs; @mombaerts2016optimising]. Sefer [@sefer2016tradeoffs] take an in-depth look at the experimental design question of sampling densely versus sampling repeatedly; the former is recommended for the purpose of detecting a spike in the molecule count number of some species. Mombaerts [@mombaerts2016optimising] study the difference between transient and steady-state sampling of the circadian clock in *Arabidopsis thaliana*, finding that the transient contains more information. In a similar vein, this paper establishes that the performance of an inference algorithm that fits a linear model to a pathway in the G$_1$ phase of the cell cycle is comparable to random classifier in the case of 3–6 samples, increases over 7–11 samples, and then flattens out with additional samples giving diminishing returns. Together with [@sefer2016tradeoffs], [@mombaerts2016optimising], this paper represents a first effort to refine previous, rule based experiment trade-off navigation practices [@bar2012studying; @sima2009inference; @bar2004analyzing], into more specific, quantitative guidelines.
Alongside the development of novel inference algorithms, new models have been adopted to generate *in silico* data and represent the dynamics of inferred networks [@milo2002network; @milo2004superfamilies; @tyson2003sniffers; @wilkinson2009stochastic; @karlebach2008modelling]. models exist at different levels of abstraction, from the logical models captured by Boolean networks, over continuous models, systems of ordinary differential equations, to the mesoscopic single molecule models such as chemical reaction networks () whose dynamics are modeled as Markovian jump processes governed by the chemical master equation () [@karlebach2008modelling]. To measure the performance of a inference algorithm, the ground truth in terms of gene expression causal interactions is required. For *in vivo* data, the ground truth is often unavailable and replacing it with a known gold standard poses certain challenges [@sima2009inference; @de2010advantages], making *in silico* studies an attractive alternative [@wilkinson2009stochastic]. In this paper we require *in silico* models to generate output with a wide range of sample rates. We strive to replicate realistic experiment conditions, choosing a detailed *in silico* model of cellular dynamics based on Markovian jump processes to represent key characteristics such as intrinsic noise [@wilkinson2009stochastic; @mcadams1999sa], common network motifs like sparsity [@milo2002network; @milo2004superfamilies], and species with highly different concentrations [@cazzaniga2008modeling].
This paper uses the to model a pathway involved in the G$_1$ phase of the cell cycle [@cazzaniga2008modeling], following the experiment setup of a query driven rather than a global study [@de2010advantages]. A sample is drawn from the probability density function governed by the using a stochastic simulation algorithm (). We then infer a linear autoregressive model to explain the *in silico* data using the method [@tropp2010computational]. provides a basic approach for inference [@marbach2012wisdom], and has the benefit of imposing sparsity on the regression parameters, thereby capturing a characteristic motif. Large regression coefficients suggest the existence of regulatory interactions between species, whereby an interaction topology can be extracted by thresholding the model parameters. The area under the precision-recall curve is used to score the performance of by comparing the inferred topology with that of the simulated by the [@saito2015precision]. We obtain a graph of the trade-off by repeating the inference procedure for data of varying temporal resolution. The main contributions of this paper can be summarized as follows: (i) we establish that the trade-off function which charts performance over number of samples has a sigmoid shape for a pathway in the G$_1$ phase of the cell cycle and the method and (ii) we provide a graph that allows an experimentalist to match a desired quality of inference (for the pathway) with a minimum number of samples.
Research Question and Research Problem {#sec:background}
======================================
Suppose that the experiment budget is somewhat flexible, and that there exist incentives to cut costs. Consider how a biologist conducting a high throughput gene sequencing experiment should navigate the number of samples quality of inference trade-off. Since the cost of undersampling is an incomplete or failed study whereas oversampling amounts to a waste of resources, we express the multiobjective optimization problem, the trade-off, in terms of a hard constraint on the quality of the inferred network: minimize the number of samples required to achieve a certain quality of inference for a given experiment, to optimize marginal costs. For this paper we limit the scope to a particular model of the // pathway in [@cazzaniga2008modeling] and the method applied to inference [@tropp2010computational]. Consider the resolution of gene expressions measurements in cases where additional detail can be purchased at a cost that is higher than that of additional samples, to optimize fixed costs. In particular, we study the cases of including or excluding a phosphoproteomic analysis of , which requires the use of different techniques compared to proteomics and metabolomics [@Larsen2008] (the low molecule count numbers for phosphorylated proteins requires a larger cell culture).
Method {#sec:method}
======
To begin with, *in silico* data is generated from a Markov process model of a pathway in the yeast cell cycle, see Section \[sec:realistic\]. To simulate the model, an efficient solver for the chemical master equation is required as detailed in Section \[sec:chem\]. The model of the pathway is from [@cazzaniga2008modeling], and has been verified against experimental data. The model consists of molecule count numbers for a total of 30 proteins and metabolites and 34 stochastic reactions. It is described in detail in Section \[sec:realistic\]. The output of the simulation is sampled at discrete time-points, whereby a sparse discrete-time state-space model is fitted using the method, see Section \[sec:lasso\]. The translation of the ground truth causal relations from the Markovian jump process model to a discrete-time difference equation based model is done in Section \[sec:causal\]. The evaluation of the model in using precision-recall curves based on the relations established in Section \[sec:causal\] is explained in Section \[sec:ROC\].
The chemical master equation {#sec:chem}
----------------------------
Consider a chemical reaction network () from a mesoscopic, non-deterministic perspective as detailed in [@iglesias2010control]. The system consists of $n$ molecular species $S_1,\ldots,S_n$ contained in a volume $\Omega$. The system is assumed to be well-stirred or spatially homogeneous. Let $\ve{X}(t)=[X_1(t),\ldots,X_n(t)]\mtr\in\N^n$ be a vector whose $i$th element $X_i$ denotes the number of molecules of species $S_i$ at time $t$. The $n$ species interact through $m$ reactions $R_1,\ldots,R_m$ on the form $$\begin{aligned}
R_j:\sum_{l=1}^k n_{j_l} S_{j_l}\smash{\xrightarrow{c_j}}\sum_{l=1}^h m_{j_l} P_{j_l},\label{eq:Rj}\end{aligned}$$ where the left-hand side contain the reactants, the right-hand side the products, and $c_j$ is the stochastic reaction constant. Each reaction $R_i$ defines a transition from some state $\ve[0]{X}\in\N^n$ to $\ve{X}(t)=\ve[0]{X}+\ve[i]{S}$, where $\ve[i]{S}$ is a column of the stoichiometry matrix $\ma{S}=[\ve[1]{S}, \ldots, \ve[m]{S}]$.
To each reaction $R_i$ we associate a function $w_i:\N^n\rightarrow[0,\infty)$ such that $w_i(\ve{X})\diff t$ is the probability that $R_i$ occurs just once in $[t,t+\diff t)$ [@iglesias2010control]. These, so called propensity functions, $w_i$ are given by $c_i$ times the number of distinct molecular reactant combinations for reaction $R_i$ found to be present in $\Omega$ at time $t$ [@Gillespie76]. More specifically, $w_i=c_i$ if $\emptyset\smash{\xrightarrow{c_j}} P$ and $$\begin{aligned}
w_i(X_{j_l},\ldots,X_{j_k})=c_i\prod_{l=1}^k\binom{X_{j_l}}{n_{j_l}},\label{eq:propensity}\end{aligned}$$ if $\sum_{l=1}^k n_{j_l}S_{j_l}\rightarrow P$, where $c_i$ is a stochastic reaction constant, $P$ denotes a sum of chemical products, and $n_{j_l}\in\N$ denote the coefficient of $S_{j_l}$ in $R_i$ as detailed in .
Let $\Prob(\ve{X},t):\N^n\times[0,\infty)\rightarrow[0,1]$ denote the probability that the system is in state $\ve{X}$ at time $t$. The chemical master equation () is a system of coupled differential-difference equations given by $$\begin{aligned}
\tag{\textsc{cme}}
\dot{\Prob}(\ve{X},t)&=\sum_{k=1}^m w_k(\ve{X}-\ve[k]{S}) \Prob(\ve{X}-\ve[k]{S},t)-w_k(\ve{x}) \Prob(\ve{X},t),\label{eq:CME}\end{aligned}$$ one equation for each feasible state $\ve{X}\in\N^n$. Any solution to corresponds to a sample from $\Prob(\ve{x},t)$. Exact closed-form solutions to can only be obtained under rather restrictive assumptions, wherefore most works focus on exact numerical methods, so-called stochastic simulation algorithms (s), approximate numerical methods, the $\tau$-leap algorithm [@gillespie2001approximate; @cao2006efficient], or solving approximations to the such as the chemical Langevin equation [@iglesias2010control].
Gillespie proposes two Monte Carlo s for exact numerical solution of : the first reaction method () [@Gillespie76] and the direct method () [@Gillespie77]. The methods are equivalent since they give the same probability distributions for the first reaction to occur, and the time until its occurrence. The so-called next reaction method () allows for more efficient execution of the first reaction method [@gibson2000efficient]. However, [@gibson2000efficient] underestimated the complexity of the by omitting the cost of managing a priority queue of reaction times [@cao2004efficient]. An optimized version of the () turns out to be more efficient than the [@cao2004efficient]. Additional s have been proposed since then. This paper utilizes the .
The // pathway in ** {#sec:realistic}
--------------------
The // pathway is involed in the regulation of *S. cerevisiae* metabolism and cell cycle progression. A realistic model of 30 proteins and metabolites undergoing 34 reactions is proposed by Cazzaniga [@cazzaniga2008modeling], [@besozzi2012role], see Table \[tab:model\]. See [@williamson2009deterministic] for a deterministic model of the pathway. The pathway is regulated by several control mechanisms, such as the as the feedback cycle ruled by the activity of phosphodiesterase. Feedback and feedforward, directed loops, are common network motifs which pose challanges for many inference algorithms [@marbach2010revealing; @marbach2012wisdom]. The notation in Table \[tab:model\] indicates that two molecules are chemically bound and form a complex. Each complex is treated as a separate variable. For example , , and are four separate variables, three of which appear in reaction one. is however not a variable in this model, as it only appears as part of complexes. The superindex p indicates that a protein is phosphorylated [@Larsen2008]. Note that one effect of the chain of reactions $R_1$–$R_{34}$ in Table \[tab:model\] is to phosphorylate .
Reactants Products
---------- --------------- ----------------- ----------
$R_1$ + 1e0
$R_2$ + 1e0
$R_3$ + 1.5e0
$R_4$ + 1e0
$R_5$ + 1e0
$R_6$ + 1e0
$R_7$ + 1e0
$R_8$ + 1e0
$R_9$ + 3e-2
$R_{10}$ + 7e-1
$R_{11}$ + 1e-3
$R_{12}$ + + 1e[-5]{}
$R_{13}$ + + + 1e-3
$R_{14}$ + 1e-5
$R_{15}$ + ([2]{}) 1e-5
$R_{16}$ + ([2]{}) ([3]{}) 1e-5
$R_{17}$ + ([3]{}) ([4]{}) 1e-5
$R_{18}$ ([4]{}) + ([3]{}) 1e-1
$R_{19}$ ([3]{}) + ([2]{}) 1e-1
$R_{20}$ ([2]{}) + 1e-1
$R_{21}$ + 1e-1
$R_{22}$ ([4]{}) [2]{} + [2]{}() 1e0
$R_{23}$ + [2]{} 1e0
$R_{24}$ [2]{} + [2]{} 1e0
$R_{25}$ + + 1e-6
$R_{26}$ + 1e-1
$R_{27}$ + 1e-1
$R_{28}$ + 7.5e0
$R_{29}$ + + 1e-4
$R_{30}$ + 1e-4
$R_{31}$ + 1e0
$R_{32}$ + 1.7e0
$R_{33}$ + + 1e1
$R_{34}$ + + 1e-2
: Stochastic model of the <span style="font-variant:small-caps;">r</span>as/c<span style="font-variant:small-caps;">amp</span>/<span style="font-variant:small-caps;">pka</span> pathway [@cazzaniga2008modeling]. Each row of the table represents a reaction on the form of . \[tab:model\]
Cazzaniga use the $\tau$-leap algorithm of Gillespie [@gillespie2001approximate; @cao2006efficient] to solve the model in Table \[tab:model\] approximately. The stochastic reaction constants in Table \[tab:model\] have been tuned relatively to each other, but not absolutely wherefore the time-scale of the simulations is given in an unspecified unit [@cazzaniga2008modeling]. We prefer to use a known time-scale since the minimum sample time is bounded below for *in vivo* experiments. Experimental results establish that initially rises to a maximum and then decreases to steady-state with a settling time of 3-5 minutes [@rolland2000glucose]. By repeating that experiment *in silico*, [@cazzaniga2008modeling] establish that 3–5 minutes correspond to 1000 units of simulation time. The *in vivo* experiment included 15 samples from the evolution of over 7 minutes [@rolland2000glucose]. <span style="font-variant:small-caps;">lcsb</span> experimentalists confirm that we can sample *in vivo* systems at most twice per minute due to technological limitations, corresponding to at most 6–10 samples per 1000 units of simulation time. The initial molecule copy numbers from [@cazzaniga2008modeling] are given in Table \[tab:copy\]. The numbers reflect realistic assumptions regarding the contents of a single cell of based on calculations and experimental data. However, in high throughput gene sequencing experiments, a large number of cells are sampled from a culture and destroyed in the process [@alberts1997molecular]. The molecule counts in each sample correspond to a sum of around 50 000 to 100 000 cells. Since any two cells can be in different stages of the cell cycle, their molecule counts may not agree aside from the approximately 10% difference that is due to intrinsic stochastic variation [@alon2006introduction]. This problem is addressed by synchronizing the cell cycles to evolve in phase, for which a number of techniques are available [@futcher1999cell]. Under the assumption of *in vivo* data being from a synchronized processes, it is thus justified to study a single cell *in silico*.
--------- ----- ----- ----- ------- ------- ----- ------- ----- ------- ------- -------
Species
Number 2e2 3e2 2e2 1.4e3 2.5e3 4e3 6.5e3 2e5 1.5e6 5.0e6 2.4e7
--------- ----- ----- ----- ------- ------- ----- ------- ----- ------- ------- -------
Network inference method {#sec:lasso}
------------------------
inference problems involve many species but few samples and is thus underdetermined [@de2010advantages]. A well established network motif, sparsity, that each species interact with only a few other species, is imposed to reduce the number of solutions [@alon2006introduction]. Sparsity also protects the inferred model against overfitting without having to deal with the combinatorial explosion that other methods for model selection such as those based on the Akaike or Bayesian information criteria face. A basic problem in compressive sampling, to find the sparsest solution to a linear system of equations in terms of the number of nonzero entries, is -hard [@natarajan1995sparse] and difficult to approximate [@amaldi1998approximability] wherefore the use of convex relaxations and other heuristic methods is commonplace [@tropp2010computational]. A dynamical system is usually not the object of study in compressive sampling [@candes2008introduction], although techniques from that field can be used for inference. To adopt a convex relaxation of the sparse approximation technique to time-series we use the idea of minimizing an error.
To explain the discrete data $\ve{X}(t)\in\N^n$ for all $t\in[0,\infty)$, we adopt a discrete-time system model, $$\begin{aligned}
\vh[k+1]{X}&=\ve{f}(\Delta t_k,\vh[k]{X})+\ve[k]{\varepsilon},\end{aligned}$$ where $\vh[k]{X}\in\R^n$ models $\ve{X}(t_k)$, $\Delta t_k=t_{k+1}-t_k$, and $\ve[k]{\varepsilon}$ is white noise. For the sake of simplicity we take $\ve{f}:\R^{n}\rightarrow\R^n$ to be a linear function, $$\begin{aligned}
\vh[k+1]{X}&=\ma{A}(\Delta t_k)\vh[k]{X}+\ve[k]{\varepsilon}.\label{eq:linear}\end{aligned}$$ Since the propensity functions of the are nonlinear, the model will not capture all the species interdependencies and we cannot expect a zero error in the limit of infinite samples. However, rather than adding a large dictionary of terms that are linear in parameters but nonlinear in the explanatory variables we prefer to adopt a minimal model. The limit would anyhow not be approached in practice due to the low temporal resolution of data, and there is merit to using linear models since certain nonlinear models are prone to overfitting [@aderhold2014statistical]. Since the // pathway is part of a cell cycle, we take the advice of [@bar2012studying] and adopt a uniform sample rate, $\Delta t_k=\Delta t\in(0,\infty)$ in . This requires some post-processing of the data.
The output of the consists of the molecule count numbers and time instances for each reaction during a timespan $[0,T]$. To create discrete-time samples $(\ve{X}(t_k))_{k=0}^{N-1}$ with $t_0=0$, $t_k=T$, $t_{i+1}-t_i=\Delta t$, for all $i=0,\ldots,N-1$ we use the function `interp1` that interpolates linearly based on the data obtained from the and rounds each sample to the nearest point in $\N^n$. The output from the contains a number of time-points on the order of $10^8$ whereas $T$ is on the order of $10^3$, so any error due to the interpolation and rounding is negligible. Since the molecule count numbers vary greatly in order of magnitude, see Table \[tab:copy\], we introduce new variables by scaling each time series $(X_i(t_k))_{k=0}^{N-1}$ by a constant equal to one over $\max_{k}X_i(t_k)$ to facilitate the optimization [@wright1999numerical]. For future reference, we let the rescaling be given by a diagonal matrix $\ma{D}\in\R^{n\times n}$. Assume that the output of the previous steps is given by $(\ve[k]{Y})_{k=0}^{N-1}$, where $\ma[k]{Y}=\ve{H}(\ma{D}\ve{X}(t_k))$, and that we are interested in modeling the evolution of $\ve[k]{Z}=\ve{G}(\ve[k]{Y})$, where both $\ve{H}:\R^n\rightarrow\R^q$ and $\ve{G}:\R^q\rightarrow\R^p$ are linear ‘permutation’ maps that may exclude some elements. The maps are given the following interpretation: $\ma{H}$ selects the species that correspond to actual measurements, while the matrix $\ma{G}$ selects the species whose interdependencies we wish to infer. This allows us to remove species whose dynamics are faster than we can realistically sample, which behave as a constant with added white noise in steady state. Such species are detected by their time-series having a constant mean and approximately zero autocorrelation. In theory, a distinction is made between the cases of full state measurements for which good theoretical results exists and the case of hidden nodes which is more difficult [@gonccalves2008necessary]. For *in vivo* experiments, the case of hidden nodes is prevalent. Indeed, the real // pathway is influenced by species which are not represented in Table \[tab:model\] [@cazzaniga2008modeling; @besozzi2012role].
Let $\|\cdot\|_1:\R^{n\times n}\rightarrow[0,\infty)$ denote the entry-wise matrix $1$-norm given by $\|\ma{A}\|_1=\sum_{i,j}|\ma[ij]{A}|$, while $\|\cdot\|_2:\R^n\rightarrow[0,\infty)$ denote the Euclidean vector norm. The least absolute shrinkage and selection operator () is an algorithm for solving sparse linear systems of equations and a key tool in compressive sensing. Using the model to create an error to be minimized, the model is fitted to the data $(\ma{Z}(t_k))_{k=0}^{N-1}$ by solving in the Lagrangian form $$\begin{aligned}
\tag{\textsc{lasso}}
\min_{\ma{B}\in\R^{p\times p }} \frac{1}{N}\sum_{k=0}^{N-1}\|\ma[k+1]{Z}-\Delta t\ma{B}\ma[k]{Z}\|^2_2+\lambda\|\ma{B}\|_1,\label{eq:lasso}\end{aligned}$$ where the regularization parameter $\lambda\in[0,\infty)$ affects, roughly speaking, the trade-off between the goodness of fit and the sparsity of the regression parameters $\ma{B}\in\R^{p\times p}$. The matrix $\ma{B}$ is a submatrix of $\ma{A}$ in , up to a change of basis. The $\frac1N$ and $\Delta t$ parameters are included to reduce the sensitivity of $\ma{B}$ to changes in the sample rate.
Consider that $M$ replicates of an experiment has yielded $M$ datasets $\mathcal{I}_i$, $i=1,\ldots M$, to be used for identification. For each $\mathcal{I}_i$, we infer a set of models $\ma{B}(\mathcal{I}_i,\lambda)$ using the method for a range $[0,b]$ of values of $\lambda$. To determine the best value of the regularization parameter $\lambda$, we compare the ability of the models $\ma{B}(\mathcal{I}_i,\lambda)$ to predict the time-evolution of a validation data set $\mathcal{V}_{j(i)}$, $j(i)\in\{1,\ldots,K\}$, where $j(i)$ is selected at random. The validation data $\mathcal{V}_{j(i)}$ is the output of an experiment where the model organism is subjected to somewhat different conditions than for $\mathcal{I}_i$. For each set $\mathcal{I}_i$, we select the model that satisfies $$\begin{aligned}
\lambda=\argmin_{\mu\in[0,b]}\sum_{k=0}^{N-1}\|\ma[k+1]{Z}(\mathcal{V}_{j(i)})-\Delta t\ma{B}(\mathcal{I}_i,\mu)\ma[k]{Z}(\mathcal{V}_{j(i)})\|^2_2,\end{aligned}$$ where $\ma[k]{Z}(\mathcal{V}_{j(i)})$ is data from $\mathcal{V}_{j(i)}$. In an *in vivo* setting, this approach corresponds to the common practice of a replicate experiment used to validate the original. Experiments that involve synchronization, in particular, should be repeated at least twice using different methods of synchronization since the process may induce artifacts in the cells [@futcher1999cell].
Modelling causal relations {#sec:causal}
--------------------------
We wish to study causal relations in the . From the output of the *in silico* experiment, all we know are changes in the molecule count numbers. A manipulation and invariance view of causality is hence appropriate: if, roughly speaking, after changing one gene we measure a change in the molecule count number of a protein, the gene is a direct or indirect cause of that change [@illari2014causality]. This idea is epitomized by the gene knock-out experiment, the procedure of deactivating one or more genes at a time. However, such experiment designs suffer from a combinatorial explosion as we increase the number of genes to be manipulated, and does not account for redundancies in gene functionality [@illari2014causality]. As such, it is desirable to be able to reliably infer regulatory interactions from time-series data of cell cycles rather than gene knockout experiments.
The causal relations underlying the reactions in Table \[tab:model\] can be visualized using a hypergraph $\mathcal{H}$ where each reaction corresponds to a hyperedge, see Fig. \[fig:causal\]. Note in particular that the graph is rather sparse, as is consistent with the assumption of Section \[sec:lasso\]. To translate the ground truth into the modeling framework that we have adopted, equation , corresponds to converting the directed hypergraph in Fig. \[fig:causal\] into a directed graph with self-loops, $$\begin{aligned}
\mathcal{D}=(\mathcal{V},\mathcal{F}),\label{eq:D}\end{aligned}$$ where $\mathcal{V}=\{1,\ldots,30\}$ represents all the species in Table \[tab:model\] and $\mathcal{F}=\cup_{i=1}^3\mathcal{A}_i$, where $$\begin{aligned}
\mathcal{A}_1&=\{(i,j)\in\mathcal{V}\times\mathcal{V}\,|\,n_i S_i+\ldots\smash{\xrightarrow{c_k}} n_jP_j+\ldots,i\neq j\},\\
\mathcal{A}_2&=\left\{(i,j)\in\mathcal{V}\times\mathcal{V}\,|\,n_i S_i+n_jS_j\ldots\smash{\xrightarrow{c_k}} \sum_{l\neq i}n_lP_l\right\},\\
\mathcal{A}_3&=\{(i,i)\in\mathcal{V}\times\mathcal{V}\}.\end{aligned}$$ Each arc in $\mathcal{A}_1$ represents a reactant and a product, each arc in $\mathcal{A}_2$ two reactants of which at least one is consumed during the reaction, and each self-loop in $\mathcal{A}_3$ represent the fact that species which do not react persist existing. Note that one difference between the causality represented by $\mathcal{H}$ and $\mathcal{D}$: all species on the left-hand side of a reaction must be present for it to occur, but that requirement cannot be captured by a system of the form . This would require to include terms that are bilinear in the explanatory variables.
We adopt the following approach to approximately infer the topology. Given estimated values of the regression parameters $\ma{B}$, we assign a topology $\mathcal{G}(r)=(\mathcal{V},\mathcal{E}(r))$, where $\mathcal{U}=\{u_1,\ldots,u_q\}$ corresponds to the set of measured species, $\mathcal{V}=\{v_1,\ldots,v_p\}\subseteq\mathcal{U}$ is the set of species whose dynamics we wish to infer, $\mathcal{E}(r)=\{(i,j)\in\mathcal{U}\times\mathcal{U}\,|\,|\ma[ij]{B}|\geq r\}$ are the causal relations, and $r\in[0,\max_{i,j}|B_{ij}|]$ is a threshold. By varying the threshold different causal models are obtained. The matrix $\ma{B}$ relate to $(\ve{X}(t_k))_{k=0}^{N-1}$ via the rescaling matrix $\ma{D}$ which is required for the optimization solver to converge. We could remove this dependence but it is our experience that the validation procedure gives a better result if we rescale $\mathcal{V}_{j(i)}$ (see Section \[sec:lasso\]) rather than $\ma{B}$.
![\[fig:causal\]Directed hypergraph $\mathcal{H}$ of the causal relations expressed by reactions $R_1$–$R_{34}$ in Table \[tab:model\]. The hyperedges go from the reactants (no arrow) to the products (arrow). Hyperedges with arrows at both ends indicate that a reaction $R_i$ is reversed by another reaction $R_j$, for some $i,j\in\{1,\ldots,34\}$.](Images/hypergraph_linear.eps){width="53.00000%"}
Performance measure {#sec:ROC}
-------------------
To evaluate the performance of the network inference algorithm we focus on the relation of the inferred network topology to that of the ground truth $\mathcal{D}$ given by . We use a criteria known as the area under the precision-recall curve (). Given an inferred representation of causal relations $G(r)$ and the ground truth $\mathcal{D}$, we can calculate the ratio of true positives to all estimated positives (precision, $|\{e\in\mathcal{E}(r) \cap\mathcal{F}\}|/|\mathcal{E}(r)|$) and that of true positives to all positives (recall, $|\{e\in\mathcal{E}(r)\cap\mathcal{F}\}|/|\mathcal{F}|$). These are coordinates in -space, the unit square $[0,1]^2$ with precision on the ordinate and recall on the abscissa. By varying $r\in[0,\infty)$ we obtain a right to left curve from the point $(1,|\mathcal{F}|/|\mathcal{V}|^2)$ to some point in set $\{(0,s)\,|\,s\in[0,1]\}$. The area under this curve is the . By plotting the against the number of samples, we establish how the quality of inference depends on the temporal resolution of data, the trade-off function.
Let us make these notions more precise. A partition $\mathcal{P}=(t_k)_{k=0}^{N-1}$ of a time interval $[0,T]$ is a sequence of real numbers such that $t_0=0< t_1<\ldots<t_{N-1}=T$ [@abbott2001understanding]. Consider a number of partitions $\mathcal{P}_1,\ldots,\mathcal{P}_l$ of $[0,T]$ and the data corresponding to each partition $\mathcal{I}_i=(\ve{X}(t_k))_{t_k\in \mathcal{P}_j}$. The trade-off function is the discrete graph of the obtained from inferring a model $\ma{B}(\mathcal{I}_j)$ which can be thresholded into a network $\mathcal{G}(r)$ over the sampling frequency $|\mathcal{P}_j|/T$. In this paper $T$ is constant, wherefore we plot the against the number of samples $|\mathcal{P}_j|$. Although we define the trade-off function without specifying all details, it is clear that it depends on the inference method, in our case .
Aside from the trade-off function that each experiment yields, we can consider a sample median trade-off function as the median over multiple experiments, and a true median trade-off function. The true trade-off function depends on the method used for inference. It is however clear that its value for zero samples is zero, and it seems likely that it converges to a constant in the limit of infinite samples although performance may deteriorate due to numerical reasons. If we know that to be the case, we can always prune samples and thereby reduce the sample rate to some practical value. As such, we expect the trade-off function to increase from 0 to some value in $[0,1]$ as $|\mathcal{P}_j|\rightarrow\infty$, or at least to increase in the case of sufficiently many samples.
Although the is popular, it should be noted that there are other goodness of fit indices, curves [@fawcett2006introduction], or three-way s [@mossman1999three] and their respective integrals. We prefer the since it is known to give a more realistic measure of performance than the when the distribution of positive and negative instances is heavily skewed [@saito2015precision]. This is the case for inference due to the sparseness of the network. Random performance for the is given by the number of true instances divided by the total number of instances, $|\mathcal{F}|/|\mathcal{V}|^2$. An issue that benchmark and comparative studies face is that different methods are to some extent complimentary, and their ranking depends on the type of network considered [@marbach2010revealing; @marbach2012wisdom]. In this paper, we are interested in studying the performance of an algorithm relative to the quality of its input, relative to itself. Fortunately, this relative performance should be less sensitive to the choice of inference algorithm, goodness of fit index, type of model, and type of network than is the benchmark of one algorithm or comparative studies that benchmark multiple algorithms.
Results {#sec:results}
=======
We simulated 40 cells using the , each run encompassing $10^8$ reactions, resulting in datasets whose time span include $[0,3000]$. We keep the first 1500 time units, which correspond to 4.5–7.5 minutes [@cazzaniga2008modeling]. Realistically, this implies that we may sample 9–15 times at most (see Section \[sec:realistic\]). The output of the simulation in the case of 15 samples is given in Fig. \[fig:15\]. The intrinsic noise does not influence the overall shape of the trajectories, rather it is most pronounced in the species with low molecule count numbers such as and . Fig \[fig:val\] depicts a second set of 3 cells that is used as validation data (see Section \[sec:lasso\]). The validation data is simulated from the glucose starved cell condition obtained by setting the initial value of the metabolite to $1.5\cdot10^6$ instead of $5\cdot10^6$ [@cazzaniga2008modeling].
The species in the model evolve over different time intervals, wherefore some are dormant or have already reached steady-state while others go through a transient state. This is typical of the cell cycle, where different genes are expressed during different phases. While the dense data $\smash{(X_i(\tau_k))_{k=0}^{10^8-1}}$ from the is not white noise on $[100,1500]$, the autocorrelation dissipate with time wherefore the sampled data $(X_i(t_k))_{k=1}^{N-1}$ on a time partition of length $N$ may be white noise. Species that are either white noise (, , , , , , , , , <span style="font-variant:small-caps;">r</span>), constant or practically constant after rescaling (, , , ), on $(t_k)_{k=1}^{N-1}$ are removed from the inference and evaluation process, compare with the 15 point time-series in Fig. \[fig:15\]–\[fig:val\]. It is possible to build a model of given sufficently many samples from the interval $[0,100]$, but that would not be consistent with our assumption of slow sampling, at most two samples per minute.
![Twenty five draws from the solution to for the reactions given by Table \[tab:model\]–\[tab:copy\] sampled 15 times uniformly over \[0,1500\]. \[fig:15\]](Images/25_experiments.eps){width="48.00000%"}
Fig. \[fig:to\] displays the trade-off function for the cases of 3–25 samples. The performance of a random classifier over this data yields an of approximately 0.2. For the cases of 3–6 samples, we note that performs on par with the random classifier. The performance in case of 7–15 samples is better than average with at least 95% certainty (pointwise for each number of samples). Note that there is a trend of increasing performance with increasing samples. Cases of comparatively good or poor performance, like that of 7 and 14 samples respectively can partly be explained by variation in the data. Although not displayed in Fig. 4, more than 25 samples give diminishing returns with respect to the . By identifying the true trade-off function with the sample medians, we could imagine that the shape of the trade-off function is approximately captured by a continuous sigmoid curve.
Consider the inclusion or exclusion of a phosphoproteomic study, whether the species , , and are measured or not. Fig. \[fig:to\] is based on *in silico* experiments that include phosphoproteomics. The regression parameters $\ma{B}$ of the best performing model with an of $0.41$ is displayed in Fig. \[fig:B\]. Note that neither could not be explained using the other data (last row have no true positives), nor is it helpful in explaining the other variables (last column is zero). The protein contributes a true positive ( in its column) but it is mostly white noise followed by a short and noisy evolution. While the trajectory of is discernable in Fig. \[fig:15\], care must be taken as it becomes less so when the number of samples are reduced. However, is well explained with all positives identified on its row, and also manages to explain the evolution of , with two out of four true positives in its column. To have a true positive on the diagonal may not seem impressive, but it is valuable since it indicates that the model makes sense, that it has some explanatory power aside from mere data fitting.
About 80% of microarray time series in 2006 were short with lengths of 3–8 time points [@ernst2006stem]. For a study of the // pathway in where inference is done using the method, such time-series would not suffice to infer the topology of the underlying network. It may still be possible to predict how the organism would react to changes in its environment, such as the difference between normal and low glucose levels as represented by the trajectories in Fig. \[fig:15\] and Fig. \[fig:val\] respectively. However, that model would not give us clues about the regulatory interactions inside the cell. In theory, it would be possible for an experimentalist that desires such an understanding to consult Fig. \[fig:to\] and read off the minimum number of samples required to achieve a certain value of the . In practice, the generality of our results need to be increased before it can become a useful tool in the laboratory.
Discussion {#sec:discussion}
==========
This paper studies the trade-off between quality of inferred gene regulatory network models versus the temporal resolution of data in the case of full and partial state measurements corresponding to an experiment setup that either includes or excludes phosphoproteomics. The goodness of fit is characterized using the area under the curve of the precision-recall curve (). In theory, experimentalists who desires a particular value may consult our graph of the trade-off function to see how many samples are needed to achieve that quality of inference. They can also determine if an increase in the number of samples, or the inclusion of phosphoproteomics, is worthwhile compared to their additional marginal and fixed experimental costs respectively. In practice, it is however clear that additional studies are needed before such a tool becomes mature enough to be of actual use in the laboratory. This paper should be considered as a proof-of-concept study. As such, its purpose is to establish a framework, showcasing how a study of the aforementioned trade-off can be conducted from simulation of data to the evaluation of an inference algorithm.
![Heat map of $\ma{B}$ with the ground truth as black dots.\[fig:B\]](Images/three_validation.eps){width="48.00000%"}
![Heat map of $\ma{B}$ with the ground truth as black dots.\[fig:B\]](Images/trade-off40.eps){width="50.00000%"}
![Heat map of $\ma{B}$ with the ground truth as black dots.\[fig:B\]](Images/ppp.eps){width="50.00000%"}
[^1]: J. Markdahl, J. Thunberg, and J. Gonçalves are with the Luxembourg Centre for Systems Biomedicine (<span style="font-variant:small-caps;">lcsb</span>), University of Luxembourg. Corresponding author: [[email protected]]{}
[^2]: N. Colombo is with the Department of Statistical Science, University College London (<span style="font-variant:small-caps;">ucl</span>).
|
---
abstract: 'We present for the first time a determination of the energy dependence of the isoscalar $\pi\pi$ elastic scattering phase-shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum we obtain the $S$-wave phase-shift up to the $K\overline{K}$ threshold. Calculations are performed at two values of the $u,d$ quark mass corresponding to $m_\pi = 236, 391$ MeV and the resulting amplitudes are described in terms of a $\sigma$ meson which evolves from a bound-state below $\pi\pi$ threshold at the heavier quark mass, to a broad resonance at the lighter quark mass.'
author:
- 'Raul A. Briceño'
- 'Jozef J. Dudek'
- 'Robert G. Edwards'
- 'David J. Wilson'
bibliography:
- 'shortbib.bib'
title: ' Isoscalar $\pi\pi$ scattering and the $\sigma$ meson resonance from QCD '
---
*Introduction:* Meson-meson scattering has long served as a tool to investigate the fundamental theory of strong interactions, quantum chromodynamics (QCD). The isoscalar channel, where all flavor quantum numbers are equal to zero, is dominated at low energies by $\pi\pi$ scattering, but despite experimental data on elastic $\pi\pi$ scattering being in place for many decades [@Protopopescu:1973sh; @*Hyams:1973zf; @*Grayer:1974cr; @*Estabrooks:1974vu], the existence of the lowest lying resonance with spin zero, the $f_0(500)/\sigma$, has only recently been demonstrated with certainty [@Caprini:2005zr; @GarciaMartin:2011jx]. This is astonishing given the crucial role played by such a state in our understanding of, for example, spontaneous chiral symmetry breaking [@GellMann:1960np], and long-range contributions to the nuclear force [@Johnson:1955zz]. The difficulty comes from the especially short lifetime of the $\sigma$ which causes it to lack the simple narrow “bump” signature associated with longer-lived resonances. It is the use of dispersive analysis techniques [@Roy:1971tc], which build in constraints from the causality and crossing symmetry of scattering amplitudes, when applied to the experimental data, which have led to an unambiguous signal for a $\sigma$ resonance. These techniques have ensured that the location of the corresponding pole singularity, located far into the complex energy plane, can now be stated with a high level of precision [@Pelaez:2015qba; @Agashe:2014kda].
In principle it should be possible within QCD to calculate the scalar, isoscalar $\pi\pi$ scattering amplitude and the contribution of the $\sigma$ resonance to it, but the non-perturbative nature of the theory at low energies leaves us with limited calculational tools. The most powerful current approach is *lattice QCD*, in which the quark and gluon fields are discretized on a space-time grid of finite size, allowing numerical computation by averaging over large numbers of possible field configurations generated by Monte-Carlo. In particular, from the time-dependence of correlation functions calculated in this way, we can extract a discrete spectrum of states whose dependence on the volume of the lattice can be related to meson-meson scattering amplitudes [@Luscher:1990ux; @*Rummukainen:1995vs; @*Kim:2005gf; @*Fu:2011xz; @*Leskovec:2012gb; @He:2005ey; @*Hansen:2012tf; @*Briceno:2012yi; @*Guo:2012hv].
Calculations of the scalar, isoscalar channel have long been considered to be among the most challenging applications of lattice QCD. In order to be successful here it is necessary to evaluate all quark propagation diagrams contributing to the correlation functions, to reliably extract a large number of states in the spectrum, and to determine and interpret the energy dependence of the scattering amplitude in the elastic region. To date no calculation has overcome all these challenges [@Alford:2000mm; @*Prelovsek:2010kg; @*Fu:2013ffa; @*Wakayama:2014gpa; @*Howarth:2015caa; @*Bai:2015nea].
In this Letter we show that by combining a number of novel techniques whose application we have pioneered over the past few years, we can meet all these challenges and provide the first determinations of the scalar, isoscalar scattering amplitude within QCD. By utilizing *distillation* [@Peardon:2009gh], we are able to evaluate with good statistical precision all required quark propagation diagrams, including those which feature quark-antiquark annihilation. By diagonalizing matrices of correlation functions [@Michael:1985ne; @*Dudek:2007wv; @*Blossier:2009kd] using a large basis of composite QCD operators with relevant quantum numbers [@Dudek:2009qf; @Dudek:2010wm; @Dudek:2010ew; @Dudek:2011tt; @Thomas:2011rh; @Dudek:2012gj; @Dudek:2012xn] we are able to make robust determinations of spectra, and by considering multiple lattice volumes and moving frames, we are able to map out the energy dependence of the $\pi\pi$ scattering amplitude over the entire elastic region.
We perform our calculations with two different values of the degenerate $u,d$ quark mass, corresponding to pions of mass 236 MeV and 391 MeV. We find that for the lighter mass, the scattering amplitude is compatible with featuring a $\sigma$ appearing as a broad resonance, which closely resembles the experimental situation. As the quark mass is increased we find that the $\sigma$ evolves into a stable bound-state lying below $\pi\pi$ threshold.
*Correlation functions and the finite-volume spectrum:*
The discrete spectrum of hadronic eigenstates of QCD in a finite volume is extracted from two-point correlation functions, ${ C_{ab }(t,t';\vec{P})=\langle 0|\mathcal{O}^{}_a(t,\vec{P})\mathcal{O}^\dag_b(t',\vec{P})|0\rangle }$, with spatial momentum $\vec{P}=\frac{2\pi}{L}\big[n_x,n_y,n_z\big],$ where $n_i\in \mathbb Z$ in an $L \times L \times L$ box. We use a large basis of interpolating fields, $\mathcal{O}_a$, from two classes. The first are “single-meson”-like operators [@Dudek:2009qf; @Dudek:2010wm; @Thomas:2011rh] which resemble a $q\bar{q}$ construction of definite momentum, $(\bar\psi\mathbf{\Gamma}\psi)_{\vec{P}}$, where $\mathbf{\Gamma}$ are operators acting in spin, color and position space [@Peardon:2009gh]. Both $u\bar{u} + d\bar{d}$ and $s\bar{s}$ flavor constructions are included [@Dudek:2011tt; @Dudek:2013yja]. The second class of operators are those resembling a pair of pions, “$\pi\pi$”, with definite relative and total momentum, ${ \sum_{\hat{p}_1, \hat{p}_2} w_{\vec{p}_1, \vec{p}_2; \vec{P}} \, (\bar\psi\mathbf{\Gamma_1}\psi)_{\vec{p}_1} (\bar\psi\mathbf{\Gamma_2}\psi)_{\vec{p}_2} }$ [@Dudek:2012gj], projected into isospin=0. Each isovector pion-like operator is constructed as the particular linear superposition, in a large basis of single-meson operators, that maximally overlaps with the pseudoscalar ground state [@Thomas:2011rh; @Dudek:2012gj].
We compute matrices of correlation functions $C_{ab}(t,t';\vec{P})$ using multiple single-meson operators along with several relative momentum constructions, ${\vec{p}_1+\vec{p}_2=\vec{P}}$, of the $\pi\pi$-like operators[^1]. This kind of operator basis has been used successfully in the determination of scattering amplitudes in the $\pi\pi$ $I=1$ channel [@Dudek:2012xn; @Wilson:2015dqa] and the coupled-channel $(\pi K, \eta K)$ [@Dudek:2014qha; @*Wilson:2014cna] and $(\pi \eta, K\overline{K})$ [@Dudek:2016cru] cases.
After integration over the quark fields appearing in the path-integral representation of $C_{ab}(t,t';\vec{P})$, we find that a variety of topologies of quark propagation diagrams appear, shown schematically in Fig. \[fig:wicks\]. Correlators with $\pi\pi$-like operators at $t$ and $t'$, for instance, require both *connected* pieces $(a),(b)$ and partially $(c)$ and completely $(d)$ *disconnected* pieces which feature quark propagation from a time $t$ to the same time $t$. Computation of these propagation objects has historically been a major challenge for lattice QCD. Within the distillation approach we utilize, determining these objects becomes manageable, and by obtaining them for all timeslices, $t$, good signals can be garnered by averaging correlation functions for fixed time separations over the whole temporal extent of the lattice. The factorization of operator construction, inherent in distillation, allows for the reuse of these propagation objects and those used here have been previously computed and used in other projects that featured quark annihilation [@Dudek:2012xn; @Dudek:2013yja; @Wilson:2015dqa; @Dudek:2014qha; @Wilson:2014cna; @Dudek:2016cru; @Briceno:2015dca; @Briceno:2016kkp].
In Fig. \[fig:corr\] we show the contributions of the various diagrams to an example correlation function having an operator $\pi_{[000]} \pi_{[110]}$ at both $t'=0$ and $t$, where we observe that all diagrams are evaluated with good statistical precision. In general delicate cancellations between different contributing diagrams can be present in isoscalar correlation functions, and our approach is seen to be capable of accurately capturing these.
We computed correlation matrices for total momentum $\vec{P} = [000], [100], [110], [111]$ and $[200]$, extracting multiple states in the spectrum of each using variational analysis of the type described in Ref. [@Dudek:2010wm]. Details of the dynamical lattices, which include degenerate light $u,d$ quarks and a heavier $s$ quark, and which have spatial lattice spacing $a_s \sim 0.12 \, \mathrm{fm}$, can be found in Refs. [@Lin:2008pr; @Wilson:2015dqa]. For the 391 MeV pion case we computed with three lattice volumes, $16^3$, $20^3$ and $24^3$, while for the 236 MeV pion case we used a single larger $32^3$ volume. The extracted spectra are shown in Figure \[fig:spec\]. In this first study we will restrict our attention to energies below the $K\overline{K}$ threshold from which we can determine the $\pi\pi$ elastic scattering phase-shift.
*Scattering amplitudes and the $\sigma$-pole:* Under the well-justified approximation of neglecting kinematically suppressed higher partial waves, the $L\times L \times L$ finite-volume spectrum is related to the $S$-wave $\pi\pi$ elastic scattering phase-shift by $$\cot \delta_0(E_\mathsf{cm}) + \cot \phi(P, L) = 0 \label{luscher}$$ where $\phi(P,L)$ is a known function which differs according to $\vec{P}$ [@Luscher:1990ux; @*Rummukainen:1995vs; @*Kim:2005gf; @*Fu:2011xz; @*Leskovec:2012gb]. This provides a one-to-one mapping between the discrete finite-volume energies determined in lattice QCD and the infinite-volume scattering phase-shift evaluated at those energies.
In Figure \[fig:amps\] we present the phase-shifts determined from the spectra shown in Figure \[fig:spec\]. A simple-minded approach to parameterizing the energy dependence of these scattering amplitudes neglects the explicit contribution of any left-hand cut[^2], leaving significant freedom in choice of functional form. We find that we can obtain good descriptions of the lattice spectra for many unitarity-preserving choices of parameterization – Figure \[fig:amps\] shows one illustrative example, which uses a single-channel $K$-matrix featuring a pole plus a constant, and a Chew-Mandelstam phase-space (see Ref. [@Wilson:2014cna] and references therein), the corresponding description of the finite volume spectrum being shown in orange in Figure \[fig:spec\]. In previous studies [@Dudek:2012xn; @Wilson:2015dqa; @Dudek:2014qha; @Wilson:2014cna; @Dudek:2016cru] of amplitudes featuring narrow resonances, we observed very little variation in the pole position of the resonance with parameterization choice variation.
In the 391 MeV pion case, we find that all parameterizations which successfully describe the finite volume spectra have a pole on the real energy axis below $\pi\pi$ threshold on the physical Riemann sheet, which we interpret as the $\sigma$ appearing as a bound state of mass $758(4)$ MeV. Considering the amplitude determined with 236 MeV pions, we observe in Figure \[fig:amps\] a qualitative change of behavior in the phase-shift curve to a form which does not resemble either that expected for a bound-state or that of a narrow elastic resonance. We find that all successful descriptions of the spectrum have a pole on the second Riemann sheet with a large imaginary part, which we interpret as the $\sigma$ appearing as a broad resonance. Because the amplitude, determined from the finite-volume spectrum, is only constrained on the real energy axis, which is far from the pole position, there is a significant variation in the precise determination of the location of the pole under reasonable variations of the parameterization form[^3]. This is the same phenomenon that is observed when experimental $\pi\pi$ phase-shift data are used to fix parameters in amplitude models that do not build in dispersive constraints [@Pelaez:2015qba].
Figure \[fig:poles\] shows the complex energy plane illustrating the extracted pole position, $s_0 = (E_\sigma - \tfrac{i}{2} \Gamma_\sigma)^2$, for a range of parameterization choices. We also show the coupling, $|g_{\sigma \pi\pi}|$, extracted from the residue of the $t$-matrix at the pole, $g_{\sigma \pi\pi}^2 = \lim_{s\to s_0} (s_0 -s) \, t(s)$.
*Summary and outlook:* In this Letter we have, for the first time, determined the low-lying spectra of the scalar-isoscalar channel of QCD in a box, including all required quark propagation diagrams. From the finite-volume spectra we have extracted the $\pi\pi$ elastic scattering amplitude which shows qualitatively different behavior at the two pion masses considered, 236 MeV and 391 MeV, with the heavier mass featuring a $\sigma$ appearing as a stable bound-state.
The amplitude parameterizations we explored to describe the finite-volume spectrum determined with 236 MeV pions all feature a $\sigma$ appearing as a broad resonance, but the pole position is not precisely determined, showing variation with parameterization choice. We believe that this comes about because our parameterizations, while maintaining elastic unitarity, do not necessarily respect the analytical constraints placed on them by causality and crossing symmetry. In the future we plan to adapt dispersive approaches so that they are applicable to describing the lattice data, and we expect this will allow us to pin down the $\sigma$ pole position with precision directly from QCD.
With constrained amplitude forms in hand, it will become appropriate to perform calculations with lighter $u,d$ quarks, such that we move closer to the physical pion mass, in order to make direct comparison with the experimental situation. It will also be useful to examine pion masses between the 236 MeV and 391 MeV considered here to determine how the transition we have observed from bound-state to resonance is manifested – a suggestion from unitarized chiral perturbation theory [@Hanhart:2008mx; @*Nebreda:2010wv] has the coupling $g_{\sigma \pi \pi}$, which one might conclude from Figure \[fig:poles\] is approximately independent of quark mass, having a divergent behavior somewhere near $m_\pi \sim 300$ MeV.
Our calculational techniques allow us to determine finite-volume spectra above the $K\overline{K}$ threshold, and by considering such energies within a coupled-channel analysis, we expect to be able to study any $f_0(980)$-like resonance that may appear. Such a state is anticipated as an isospin partner of the $a_0$ resonance which we observed near the $K\overline{K}$ threshold in a recent 391 MeV pion mass calculation [@Dudek:2016cru]. A comprehensive study of the light scalar meson nonet ($\sigma, \kappa, a_0, f_0$) within first-principles QCD will then be possible. The finite-volume approach can also be extended to study the coupling of these states to external currents [@Briceno:2014uqa; @Briceno:2015csa; @Briceno:2015dca; @Briceno:2016kkp; @Briceno:2015tza; @Bernard:2012bi; @Agadjanov:2014kha; @Lellouch:2000pv] – by examining the current virtuality dependence of the form-factors evaluated at the resonance pole, we expect to be able to infer details of the constituent structure of the scalar mesons.
*Acknowledgments:* We thank our colleagues within the Hadron Spectrum Collaboration, and in particular, thank Bálint Joó for help. We also thank Kate Clark for use of the [QUDA]{} codes. RAB would like to thank I. Danilkin for useful discussions in the preparation of the manuscript. The software codes [Chroma]{} [@Edwards:2004sx], [QUDA]{} [@Clark:2009wm; @Babich:2010mu], [QUDA-MG]{} [@Clark:SC2016], [QPhiX]{} [@ISC13Phi], and [QOPQDP]{} [@Osborn:2010mb; @Babich:2010qb] were used for the computation of the quark propagators. The contractions were performed on clusters at Jefferson Lab under the USQCD Initiative and the LQCD ARRA project. This research was supported in part under an ALCC award, and used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This research is also part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources. Gauge configurations were generated using resources awarded from the U.S. Department of Energy INCITE program at Oak Ridge National Lab, and also resources awarded at NERSC. RAB, RGE and JJD acknowledge support from U.S. Department of Energy contract DE-AC05-06OR23177, under which Jefferson Science Associates, LLC, manages and operates Jefferson Lab. JJD acknowledges support from the U.S. Department of Energy Early Career award contract DE-SC0006765. DJW acknowledges support from the Isaac Newton Trust/University of Cambridge Early Career Support Scheme \[RG74916\].
[^1]: We also include several “$K\overline{K}$”-like operators, of analogous construction to the “$\pi\pi$” operators, although they are not vital in the determination of the spectrum below $K\overline{K}$-threshold.
[^2]: which is present due to crossing symmetry, but which is distant from the physical scattering region for heavy pions.
[^3]: see Refs. [@Wilson:2014cna; @Dudek:2016cru] for the kinds of variation we consider.
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abstract: 'In a previous article, an algorithm for discovering therapeutic targets in Boolean networks modeling disease mechanisms was introduced. In the present article, the updates made on this algorithm, named kali, are described. These updates are: i) the possibility to work on asynchronous Boolean networks, ii) a smarter search for therapeutic targets, and iii) the possibility to use multivalued logic. kali assumes that the attractors of a dynamical system correspond to the phenotypes of the modeled biological system. Given a logical model of a pathophysiology, either Boolean or multivalued, kali searches for which biological components should be therapeutically disturbed in order to reduce the reachability of the attractors associated with pathological phenotypes, thus reducing the likeliness of pathological phenotypes. kali is illustrated on a simple example network and shows that it can find therapeutic targets able to reduce the likeliness of pathological phenotypes. However, like any computational tool, kali can predict but can not replace human expertise: it is an aid for coping with the complexity of biological systems.'
author:
- Arnaud Poret
bibliography:
- 'kali\_updates.bib'
date: 10 November 2016
title: 'Therapeutic target discovery using Boolean network attractors: updates from kali'
---
Copyright 2016 Arnaud Poret
This article is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, visit <https://creativecommons.org/licenses/by-nc-sa/4.0/>
`[email protected]`\
IRCCyN – École Centrale de Nantes
Introduction
============
In a previous article, an algorithm for *in silico* therapeutic target discovery was presented in its first version [@poret2014silico]. In the present article, updates made on this algorithm, named kali, are described. The complete background was introduced in the previous article whose some important concepts are recalled in page .
kali still belongs to the logic-based modeling formalism [@le2015quantitative; @wynn2012logic; @morris2010logic], mainly Boolean networks [@albert2014boolean; @wang2012boolean], and keeps its original goal: searching for therapeutic interventions aimed at healing the supplied pathologically disturbed biological network. Such a network is intended to model the mechanisms of the studied disease and is on what kali operates. In this work, therapeutic interventions are combinations of targets, these combinations being named bullets. Targets are components of the considered network, such as enzymes or transcription factors, and can be subjected to inhibitions or activations. This is what bullets specify: which targets and which action to apply on them.
The pivotal assumption on which kali is based is that the attractors of a dynamical system, such as a Boolean network, are associated with the phenotypes of the modeled biological system, such as a signaling pathway. In other words: attractors model phenotypes [@jaeger2014bioattractors]. This assumption was successfully used in several works [@cho2016attractor; @gan2016analysis; @davila2015modeling; @crespo2013detecting; @fumia2013boolean; @cheng2013biomolecular; @creixell2012navigating] and makes sense since the steady states of a dynamical system, the attractors, should correspond to the steady states of the modeled biological system, the phenotypes.
In the mean time, various works using logical modeling with applications in therapeutic innovation was published. For example, Melody Morris and colleagues used fuzzy logic through their constrained fuzzy logic formalism [@morris2016systematic]. This formalism aims at giving more quantitative abilities to logical models. They applied this approach on a molecular interaction network dedicated to hepatocellular carcinoma microenvironment. They successfully predicted the impact of several combinations of kinase inhibitors on key transcription factors. An other example is the work of Hyunho Chu and colleagues [@chu2015precritical]. They built a molecular interaction network involved in colorectal tumorigenesis and studied its dynamics, particularly its attractors and their basins, with stochastic Boolean modeling. They highlighted what they named the flickering, that is the displacement of the system from one basin to an other one due to stochastic noise. They suggest that flickering is involved in pushing the system from a physiological state to a pathological one during colorectal tumorigenesis.
Concerning kali, three updates were done: i) adding the possibility to work on asynchronous Boolean networks, ii) making the selection of therapeutic bullets smarter, and iii) adding the possibility to use multivalued logic.
Handling asynchronous updating
------------------------------
To compute the behavior of a discrete dynamical system such as a Boolean network, its variables have to be iteratively updated. These iterative updates, also named time steps, can be made synchronously or not [@garg2008synchronous]. If all the variables are simultaneously updated at each iteration then the network is synchronous, otherwise it is asynchronous.
\[async\] Compared to asynchronous updating, synchronous updating is easier to compute. However, when a biological network is simulated synchronously, it is assumed that all its components evolve simultaneously, which can be not acceptable depending on what is modeled. The asynchronous updating is frequently built so that only one randomly selected variable is updated at each iteration. This allows to capture two important features: i) biological entities do not necessarily evolve simultaneously, and ii) noise due to randomness can affect when a biological interaction takes place [@szekely2014stochastic; @buiatti2013randomness; @ullah2010stochastic]. This is particularly true at the molecular scale, such as with signaling pathways, where macromolecular crowding and Brownian motion can impact the firing of biochemical reactions [@rivas2016macromolecular].
Therefore, the choice between a synchronous or an asynchronous updating may depend on the model, the computational resources and the acceptability of the assumptions implied by synchrony. Knowing that the luxury is to have the choice, kali can now use synchronous and asynchronous updating.
Managing basin sizes for therapeutic purposes
---------------------------------------------
Until now, kali requires therapeutic bullets to remove all the attractors associated with pathological phenotypes, named pathological attractors. This criterion for selecting therapeutic bullets is somewhat drastic. A smoother criterion should enable to consider more targeting strategies, and then more possibilities for counteracting diseases. However, it could also unravel less effective therapeutic bullets, but being too demanding potentially leads to no results and loss of nonetheless interesting findings.
The therapeutic potential of bullets could be assessed by estimating their ability at reducing the size of the pathological basins, namely the basins of the pathological attractors. This is a more permissive criterion since therapeutic bullets no longer have to necessarily remove the pathological attractors. Reducing the size of a pathological basin renders the corresponding pathological attractor less reachable, and then the associated pathological phenotype less likely. This new criterion includes the previous one since removing an attractor means reducing its basin to the empty set. Consequently, therapeutic bullets obtainable with the previous criterion are still obtainable.
Extending to multivalued logic
------------------------------
One of the main limitations of Boolean models is that variables can take only two values, which can be quite simplistic. Depending on what is modeled, such as activity level of enzymes or abundance of gene products, considering more than two levels can be more appropriate. Without leaving the logic-based modeling formalism, one solution is to extend Boolean logic to multivalued logic [@rescher1968many]. With multivalued logic, a finite number $h$ of values in $[0;1]$ is used, allowing variables to model more than two levels. For example, the level $0.5$ can be introduced in order to model partial activations of enzymes, or moderate concentrations of gene products.
Methods
=======
Additional definitions
----------------------
In addition to the concepts defined in the previous article [@poret2014silico], and briefly recalled in page , here are some supplementary definitions:
- **physiological state space**: the state space $S_{physio}$ of the physiological variant
- **pathological state space**: the state space $S_{patho}$ of the pathological variant
- **testing state space**: the state space $S_{test}$ of the pathological variant under the influence of a bullet
- **physiological basin**: the basin $B_{physio,i}$ of a physiological attractor $a_{physio,i}$
- **pathological basin**: the basin $B_{patho,i}$ of a pathological attractor $a_{patho,i}$
- **$n$-bullet**: a bullet made of $n$ targets
Handling asynchronous updating
------------------------------
To implement asynchronous updating, the corresponding algorithms of BoolNet were adapted. BoolNet is an R[^1] package for generation, reconstruction and analysis of Boolean networks [@mussel2010boolnet]. Asynchronous updating is implemented so that only one randomly selected variable is updated at each iteration. This random selection is made according to a uniform distribution and implies that the network is no longer deterministic. To do so, given a Boolean network, BoolNet uses the three following functions:
- **AsynchronousAttractorSearch**: this function computes the attractor set of the supplied Boolean network by using the two following functions
- **ForwardSet**: this function computes the forward reachable set of a state and considers it as a candidate attractor
- **ValidateAttractor**: this function checks if a forward reachable set is a terminal strongly connected component (terminal SCC), that is an attractor
The forward reachable set $Fwd_{\boldsymbol{x}} \subset S$ of a state $\boldsymbol{x} \in S$ is the set made of the states reachable from $\boldsymbol{x}$, including $\boldsymbol{x}$ itself. A terminal SCC is a set $tSCC \subset S$ made of the forward reachable sets of its states: $\forall \boldsymbol{x} \in tSCC$, $Fwd_{\boldsymbol{x}} \subset tSCC$. As a consequence, when a terminal SCC is reached, the system can not escape it: this is an attractor in the sense of asynchronous Boolean networks [@saadatpour2010attractor]. Asynchronous Boolean networks with random updating are not deterministic: their attractors are no longer deterministic sequences of states, namely cycles, but terminal SCCs. To find an attractor, a long random walk is performed in order to reach an attractor with high probability. This candidate attractor is then validated, or not, by checking if it is a terminal SCC.
Managing basin sizes for therapeutic purposes
---------------------------------------------
To implement the new criterion for selecting therapeutic bullets, kali considers a bullet as therapeutic if it increases $card\ \bigcup B_{physio,i}$ in $S_{test}$ without creating *de novo* attractors. Knowing that the attractors are either physiological or pathological, increasing $card\ \bigcup B_{physio,i}$ is equivalent to decreasing $card\ \bigcup B_{patho,i}$. The goal is to increase the physiological part of the pathological state space, or equivalently to decrease its pathological part. Consequently, a pathologically disturbed biological network receiving such a therapeutic bullet tends to, but not necessarily reaches, an overall physiological behavior. However, as with the previous criterion, it does not ensure that all the physiological attractors are preserved. *A fortiori*, it does not ensure that their basin remains unchanged. This means that a therapeutic bullet can also alter the reachability of the physiological attractors. Nevertheless, as with the previous criterion, this is a matter of choice between a therapeutic bullet or not.
\[threshold\] The therapeutic potential of a bullet is expressed by its gain. It is displayed as follow: $x\% \to y\%$ where $card\ \bigcup B_{physio,i}=x\%$ of $card\ S_{patho}$ and $y\%$ of $card\ S_{test}$. Therefore, in order to increase the physiological part of the pathological state space, a therapeutic bullet has to make $y \geq x$. Note that $y=x$ is allowed. In this particular case, it is conceivable that several pathological basins changed in size without changing the size of their union. In other words, the composition of the pathological part changed while its size did not. This can be therapeutic if, for example, the basin of a weakly pathological attractor increased at the expense of the basin of a heavily pathological attractor.
The increase of the physiological part can be subjected to a threshold $\delta$: $y \geq x$ becomes $y-x \geq \delta$. As $x$ and $y$, $\delta$ is expressed in $\%$ of $card\ S_{patho \slash test}$. This threshold is introduced to allow the stringency of kali to be tuned. By the way, using this threshold also decreases the probability to obtain misassessed therapeutic bullets due to roundoff errors, or sampling errors when the state space is too big to be wholly computed.
A therapeutic bullet as defined by the previous criterion, namely which removes all the pathological attractors, makes *de facto* $card\ \bigcup B_{physio,i}=100\%$ of $card\ S_{test}$. As already mentioned, the previous criterion is included in this new one: therapeutic bullets obtainable with the former are also obtainable with the latter.
Extending to multivalued logic
------------------------------
Multivalued logic requires suitable operators to be introduced. One solution is to use a mathematical formulation of the Boolean operators which also works with multivalued logic, just as the Zadeh operators. These operators are a generalization of the Boolean ones proposed for fuzzy logic by its pioneer Lotfi Zadeh [@zadeh1965fuzzy]. Their mathematical formulation is:
[r C l]{} x y&=&min(x,y)\
x y&=&max(x,y)\
x&=&1-x
With a $h$-valued logic, $card\ S=h^{n}$, which raises more computational difficulties than with Boolean logic. The same applies to the testable bullets since there are $h^{r}$ possible modality arrangements and then $(n! \cdot h^{r})/(r! \cdot (n-r)!)$ possible bullets, where $r$ is the number of targets per bullet and $n$ the number of nodes in the network. To illustrate how kali works with multivalued logic without overloading it, a $3$-valued logic is used where $\lbrace 0,0.5,1 \rbrace$ is the domain of value: $x_{i} \in \lbrace 0,0.5,1 \rbrace$. $0$ and $1$ have the same meaning as in Boolean logic while $0.5$ is an intermediate truth degree which can be seen as an intermediate level of activity or abundance depending on what is modeled. By the way, $S=\lbrace 0,0.5,1 \rbrace^{n}$ and $moda_{i} \in \lbrace 0,0.5,1 \rbrace$.
Example network
---------------
To illustrate what results from the updates made on kali, a fictive network is used. This fictive network is specifically designed for illustration and is not intended to model a real biological phenomenon. This example network is depicted in page . Among the three updates made on kali, only the asynchronous updating and the management of basin sizes are illustrated. Multivalued logic is illustrated in page .
{width="67.00000%"}
Below are the Boolean equations encoding the example network:
[r C l]{} do&=&do\
factory&=&factory\
energy&=&(energy task) factory\
locker&=&energy\
releaser&=&do\
sequester&=&releaser\
activator&=&do locker\
effector&=&activator sequester\
task&=&effector
The do instruction and the factory are the two inputs: they are equal to themselves. The equation of energy tells that energy is present if the factory is active, even if the task is running: the factory has a sufficient capacity. However, if the factory is not active, energy disappears as soon as the task is initiated. Concerning the activator and the effector, their equations tell that their respective inhibitor takes precedence: whatever is the state of the other nodes, if the inhibitor is active then the target is not.
The physiological variant $\boldsymbol{f}_{physio}$ is the network as described above. The pathological variant $\boldsymbol{f}_{patho}$ is the network plus a constitutive inactivation of the locker: the execution of the task no longer considers if sufficient energy is available. Consequently, $f_{locker}$ becomes $locker=0$ in $\boldsymbol{f}_{patho}$.
Implementation, code availability and license
---------------------------------------------
kali, together with the example network, is implemented in Go[^2], tested with Go version go1.7.3 linux/amd64 under Arch Linux[^3]. It is freely available on GitHub[^4] at <https://github.com/arnaudporet/kali>, licensed under the GNU General Public License[^5]. The core of kali in pseudocode can be found in page .
Results
=======
Attractor sets
--------------
The example network is computed asynchronously over the whole state space using Boolean logic. The resulting attractors can be studied according to four variables: the do instruction, the factory, the locker and the task. Note that in the initial conditions, energy can be present without a running factory. In this case, if the do instruction is sent then the energy is consumed by the task but not remade by the factory. With the physiological variant, the locker is expected to stop the task. However, with the pathological variant, an abnormal behavior is expected since the locker is constitutively inactivated. Below are the computed attractors:
- $A_{physio}$:
attractor basin ($\%$ of $card\ S_{physio}$) $do$ $factory$ $energy$ $locker$ $task$
--------------- ------------------------------------ ------ ----------- ---------- ---------- --------
$a_{physio1}$ $17.8\%$ $0$ $0$ $0$ $1$ $0$
$a_{physio2}$ $7.2\%$ $0$ $0$ $1$ $0$ $0$
$a_{physio3}$ $25\%$ $0$ $1$ $1$ $0$ $0$
$a_{physio4}$ $25\%$ $1$ $0$ $0$ $1$ $0$
$a_{physio5}$ $25\%$ $1$ $1$ $1$ $0$ $1$
- $A_{patho}$:
attractor basin ($\%$ of $card\ S_{patho}$) $do$ $factory$ $energy$ $locker$ $task$
--------------- ----------------------------------- ------ ----------- ---------- ---------- --------
$a_{patho1}$ $18.4\%$ $0$ $0$ $0$ $0$ $0$
$a_{physio2}$ $6.6\%$ $0$ $0$ $1$ $0$ $0$
$a_{physio3}$ $25\%$ $0$ $1$ $1$ $0$ $0$
$a_{patho2}$ $25\%$ $1$ $0$ $0$ $0$ $1$
$a_{physio5}$ $25\%$ $1$ $1$ $1$ $0$ $1$
With the physiological variant, the behavior is as expected: the task runs only if the do instruction is sent and only if the factory is able to remade the consumed energy. With the pathological variant, two pathological phenotypes represented by $a_{patho1}$ and $a_{patho2}$ appear. $a_{patho1}$ is pathological because the locker is inactive while there is no available energy. However, it is weakly pathological since the do instruction is not sent: an operational locker is not mandatory since there is no task to stop. In contrast, $a_{patho2}$ is heavily pathological since an operational locker is required to stop the task in absence of energy supply. In the fictive cell bearing this example network, $a_{patho2}$ could drain all the energy content and then bring the cell to thermodynamical death. Moreover, $a_{patho2}$ should not be neglected since its basin occupies $25\%$ of the pathological state space.
Therapeutic bullets
-------------------
Bullets are assessed for their therapeutic potential on the pathological variant $\boldsymbol{f}_{patho}$ according to the new criterion. The goal is to decrease the size of the pathological basins $B_{patho,i}$. All the bullets made of one to two targets are computed with a threshold of $5\%$. This choice of the threshold value is quite arbitrary. It tells that if the physiological part $\bigcup B_{physio,i}$ of the pathological state space $S_{patho}$ occupies $x\%$ of it, then to be therapeutic a bullet has to bring this value above $(x+5)\%$. Therefore, the increases below this threshold are considered not significant by kali. Even the choice to use a threshold could be arbitrary, as discussed in the section page .
Knowing that $card\ \bigcup B_{physio,i}=56.6\%$ of $card\ S_{patho}$, with a threshold of $5\%$ the $1,2$-bullets have to make $card\ \bigcup B_{physio,i} \geq (56.6+5)\%=61.6\%$ of $card\ S_{test}$ to be considered therapeutic. Below are the returned therapeutic bullets:
- $1$-therapeutic bullets:
$B_{physio1}$ $B_{physio2}$ $B_{physio3}$ $B_{physio4}$ $B_{physio5}$ $B_{patho1}$ $B_{patho2}$
-------------- ---------- --------------- --------------- --------------- --------------- --------------- -------------- -------------- ---------- -------
$do[0]$ $56.6\%$ $\to$ $64.4\%$ $0\%$ $14.4\%$ $50\%$ $0\%$ $0\%$ $35.5\%$ $0\%$
$factory[1]$ $56.6\%$ $\to$ $100\%$ $0\%$ $0\%$ $50\%$ $0\%$ $50\%$ $0\%$ $0\%$
- $2$-therapeutic bullets:
$B_{physio1}$ $B_{physio2}$ $B_{physio3}$ $B_{physio4}$ $B_{physio5}$ $B_{patho1}$ $B_{patho2}$
-------------- ---------------- --------------- --------------- --------------- --------------- --------------- -------------- -------------- --------- ---------- -------
$do[0]$ $factory[1]$ $56.6\%$ $\to$ $100\%$ $0\%$ $0\%$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[1]$ $factory[1]$ $56.6\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $100\%$ $0\%$ $0\%$
$do[0]$ $energy[1]$ $56.6\%$ $\to$ $100\%$ $0\%$ $50\%$ $50\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[0]$ $locker[0]$ $56.6\%$ $\to$ $64.1\%$ $0\%$ $14.1\%$ $50\%$ $0\%$ $0\%$ $35.9\%$ $0\%$
$do[0]$ $releaser[0]$ $56.6\%$ $\to$ $62.9\%$ $0\%$ $12.9\%$ $50\%$ $0\%$ $0\%$ $37.1\%$ $0\%$
$do[0]$ $sequester[1]$ $56.6\%$ $\to$ $62.5\%$ $0\%$ $12.5\%$ $50\%$ $0\%$ $0\%$ $37.5\%$ $0\%$
$do[0]$ $activator[0]$ $56.6\%$ $\to$ $64.8\%$ $0\%$ $14.8\%$ $50\%$ $0\%$ $0\%$ $35.2\%$ $0\%$
$do[0]$ $effector[0]$ $56.6\%$ $\to$ $67.8\%$ $0\%$ $17.8\%$ $50\%$ $0\%$ $0\%$ $32.2\%$ $0\%$
$do[0]$ $task[0]$ $56.6\%$ $\to$ $73.2\%$ $0\%$ $23.2\%$ $50\%$ $0\%$ $0\%$ $26.8\%$ $0\%$
$factory[1]$ $energy[1]$ $56.6\%$ $\to$ $100\%$ $0\%$ $0\%$ $50\%$ $0\%$ $50\%$ $0\%$ $0\%$
$factory[1]$ $locker[0]$ $56.6\%$ $\to$ $100\%$ $0\%$ $0\%$ $50\%$ $0\%$ $50\%$ $0\%$ $0\%$
where $x[y]$ means that the variable $x$ has to be set to the value $y$. For example, the bullet $do[0]\ factory[1]$ tells that the do instruction has to be abolished while the factory has to be maintained active.
All the returned therapeutic bullets not removing all the pathological attractors exhibit the ability to suppress the basin of $a_{patho2}$ while increasing the one of $a_{patho1}$. Certainly, removing all the pathological attractors should be better, but knowing the $a_{patho2}$ is more pathological than $a_{patho1}$, such therapeutic bullets can nevertheless be interesting. With the previous criterion, namely removing all the pathological attractors, these therapeutic bullets are not obtainable, thus highlighting fewer therapeutic strategies.
Some of the found therapeutic bullets enable physiological attractors required by the pathological variant to react properly to the do instruction. For example, the bullet $factory[1]$ enables $a_{physio3}$ and $a_{physio5}$ which correspond respectively to “no do, no task” and “do the task, energy supply”. However, the remaining of the therapeutic bullets, such as $do[0]\ releaser[0]$ or $do[1]\ factory[1]$, either disable or force the do instruction, thus either suppressing or forcing the task. A network which can not do the task or, at the opposite, which permanently does it, may not be therapeutically interesting even if energy is supplied.
None of the found therapeutic bullets suggest to reverse the constitutive inactivation of the locker by making it constitutively active. This highlight that applying the opposite action of the pathological disturbance is not necessarily a solution, which can appear counterintuitive. This might be due to the fact that biological entities subjected to pathological disturbances belong to complex networks, resulting in behaviors which can not be predicted by mind [@koutsogiannouli2013complexity; @koch2012modular]. This is where computational tools, and their growing computing capabilities, can help owing to their integrative power [@jessica2016multi; @walpole2013multiscale; @fisher2010executable; @voit2008steps; @fischer2008mathematical].
Also, none of the found therapeutic bullets allow the recovery of all the physiological attractors: there are no golden bullets. In a general manner, the components of a biological network should be able to take several states, such as enzymes which should be active when suitable. Consequently, healing a pathologically disturbed biological network by maintaining some of its components in a particular state should not allow the recovery of a complete and healthy behavior. This is a limitation of the method implemented in kali. This limitation is common in biomedicine while not necessarily being an issue. For example, statins are well known lipid-lowering drugs widely used in cardiovascular diseases with proven benefits [@mihos2014cardiovascular; @schooling2014statins]. They inhibit an enzyme, the HMG-CoA reductase, and they do it constantly, just as the targets are modulated in the therapeutic bullets returned by kali. The HMG-CoA reductase is component of a complex metabolic network and maintaining it in an inhibited state should not allow this network to run properly, maybe causing some adverse effects. Nevertheless, such as with all drugs, this is a matter of benefit-risk ratio. All of this is to say that there are no perfect method for counteracting diseases and that computational tools, such as kali, can help scientists but can certainly not replace their expertise. Human expertise is mandatory to assess the concrete meaning and usability of the results, and ultimately to take decisions.
Conclusion
==========
kali can now work on asynchronous Boolean networks, in addition to synchronous ones. This is probably the most important update which had to be done. Indeed, the asynchronous updating is frequently used by biomodelers since it is supposed to be more realistic, as discussed in the section page . Therefore, a computational tool aimed at working on models built by the scientific community, such as kali, has to handle this updating scheme. It should be noted that there are more than one asynchronous updating scheme. The one implemented in kali is the most popular and is named the general asynchronous updating: one randomly selected variable is updated at each iteration. However, other asynchronous updating methods exist. For example, with the random order updating, all the variables are updated at each iteration but in a randomly built order. Implementing various asynchronous updating schemes in kali may be a required future improvement.
kali now uses a new criterion for selecting therapeutic bullets which brings a wider range of targeting strategies intended to push pathological behaviors toward physiological ones. This new criterion is based on a more permissive assumption stating that reducing the reachability of pathological attractors is therapeutic. For an *in silico* tool, such as kali, being a little bit more permissive may be important since theoretical findings have to outlive the bottleneck separating prediction to reality. With a too strict assessment of therapeutic bullets, the risk of highlighting too few candidate targets, or to miss some interesting ones, could be hight. Moreover, predicted does not necessarily mean true: an *in silico* prediction of apparently poor interest can reveal itself of great interest, and *vice versa*.
This new criterion also brings a finer assessment of therapeutic bullets since all the percentages between $card\ \bigcup B_{physio,i}$ in $S_{patho}$ and $100\%$ in $S_{test}$ are considered. With the previous criterion, the only therapeutic potential was $card\ \bigcup B_{physio,i}=100\%$ in $S_{test}$, thus reducing the assessment to therapeutic or not. However, things are not necessarily so dichotomous but rather nuanced, so the assessment of therapeutic bullets should be nuanced too.
kali can now compute with multivalued logic. Allowing variables to take an arbitrary finite number of values should enable to more accurately model biological processes and produce more fine-tuned therapeutic bullets. However, this accuracy and fine-tuning are at the cost of an increased computational requirement. Indeed, the cardinality of the state space depends on the size of the model and the used logic. Therefore, the size of the model and the used logic should be balanced: the smaller the model is, the more variables should be finely valued. For example, for an accurate therapeutic investigation, the model should only contain the essential and specific pieces of the studied pathophysiology, modeled by a finely valued logic. On the other hand, for a broad therapeutic investigation, a more exhaustive model can be used but modeled by a coarse-grained logic. Finally, it should be noted that the ultimate multivalued logic is the infinitely valued one, which is fuzzy logic [@zadeh1988fuzzy]. With fuzzy logic, the whole $[0;1]$ is used to valuate variables, which might bring the best accuracy for the qualitative modeling formalism [@poret2014enhancing; @morris2011training; @aldridge2009fuzzy].
Two additional improvements are envisioned for kali. The first one is to allow *de novo* attractors to appear in $A_{test}$. For example, it is conceivable that a bullet greatly decreases the pathological basins while creating a new attractor not belonging to $A_{physio}$ nor to $A_{patho}$. Such a *de novo* attractor might be defined as not physiological, and then pathological. However, if it is weakly pathological and induced by a bullet which greatly decreases the basin of other, and heavier, pathological attractors, such a case should be allowed to be investigated.
The second improvement is to allow partial matching when checking if an attractor is associated with a physiological phenotype by comparing it to the physiological attractors. Currently, an attractor which does not match a physiological attractor is considered pathologic. However, it is conceivable that some variables not exhibiting a physiological behavior in an attractor do not pathologically impact the associated phenotype. To allow such a case to be considered, some variables within attractors should be allowed to not be matched when assessing the associated phenotype. This suggests the concept of decisive variables. Decisive variables would be variables whose the behavior in the attractors is sufficient to biologically interpret the associated phenotype. Therefore, kali could allow non-decisive variables to not be matched. Ultimately, this could allow the modeler to specify himself what a physiological attractor is without computing a physiological variant.
Appendices
==========
Appendix 1: recall of previous concepts {#brief}
---------------------------------------
### Biological networks
A network is a digraph $G=(V,E)$ where $V=\lbrace v_{1},\ldots,v_{n}\rbrace$ is the set containing the nodes and $E=\lbrace (v_{i,1},v_{j,1}),\ldots,(v_{i,m},v_{j,m})\rbrace \subset V^{2}$ the set containing the edges. In practice, nodes represent entities and edges represent binary relations $R \subset V^{2}$ involving them: $v_{i}\ R\ v_{j}$ [@zhu2007getting]. It indicates that the node $v_{i}$ exerts an influence on the node $v_{j}$. For example, in gene regulatory networks [@emmert2014gene], $v_{i}$ can be a transcription factor while $v_{j}$ a gene product. The edges are frequently signed so that they indicate if $v_{i}$ exerts a positive or a negative influence on $v_{j}$, such as inhibitions or activations.
### Boolean networks
A Boolean network is a network where nodes are Boolean variables $x_{i}$ and edges $(x_{i},x_{j})$ the $is\ input\ of$ relation: $x_{i}\ is\ input\ of\ x_{j}$. Each $x_{i}$ has $b_{i} \in [\![0,n]\!]$ inputs. Depending on the updating scheme, at each iteration $k \in [\![k_{0},k_{end}]\!]$, one or more $x_{i}$ are updated using their associated Boolean function $f_{i}$ and their inputs, as in the following pseudocode representing a synchronous updating:
**for** $k \gets k_{0},\ldots,k_{end}$\
$x_{1} \gets f_{1}(x_{1,1},\ldots,x_{1,b_{1}})$\
$\vdots$\
$x_{n} \gets f_{n}(x_{n,1},\ldots,x_{n,b_{n}})$\
**end for**\
which can be written in a more concise form:
**for** $k \gets k_{0},\ldots,k_{end}$\
$\boldsymbol{x} \gets \boldsymbol{f}(\boldsymbol{x})$\
**end for**\
where $\boldsymbol{f}=(f_{1},\ldots,f_{n})$ is the Boolean transition function and $\boldsymbol{x}=(x_{1},\ldots,x_{n})$ the state vector. The value of $\boldsymbol{x}$ belongs to the state space $S=\lbrace 0,1\rbrace^{n}$ which is the set containing the possible states.
The set $A=\lbrace a_{1},\ldots,a_{p}\rbrace$ containing the attractors is the attractor set. An attractor $a_{i}$ is a collection of states $(\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{q})$ such that once the system reaches a state $\boldsymbol{x}_{j} \in a_{i}$, it can subsequently visit the states of $a_{i}$ and no other ones: the system can not escape. The set $B_{i} \subset S$ containing the $\boldsymbol{x} \in S$ from which $a_{i}$ can be reached is its basin of attraction, or simply basin.
### Definitions
- **physiological phenotype**: A phenotype which does not impair the life quantity /quality of the organism which exhibits it.
- **pathological phenotype**: A phenotype which impairs the life quantity /quality of the organism which exhibits it.
- **variant (of a biological network)**: Given a biological network, a variant is one of its versions, namely the network plus eventually some modifications.
- **physiological variant**: A variant which produces only physiological phenotypes. This is the biological network as it should be, namely the one of healthy organisms.
- **pathological variant**: A variant which produces at least one pathological phenotype, or which fails to produce at least one physiological phenotype. This is a dysfunctional version of the biological network, namely a version found in ill organisms.
- **physiological attractor set**: The attractor set $A_{physio}$ of the physiological variant.
- **pathological attractor set**: The attractor set $A_{patho}$ of the pathological variant.
- **physiological Boolean transition function**: The Boolean transition function $\boldsymbol{f}_{physio}$ of the physiological variant.
- **pathological Boolean transition function**: The Boolean transition function $\boldsymbol{f}_{patho}$ of the pathological variant.
- **physiological attractor**: An attractor $a_{i}$ such that $a_{i} \in A_{physio}$. Note that it does not exclude the possibility that $a_{i} \in A_{patho}$ in addition to $a_{i} \in A_{physio}$.
- **pathological attractor**: An attractor $a_{i}$ such that $a_{i} \notin A_{physio}$.
- **modality**: The perturbation $moda_{i} \in \lbrace 0,1 \rbrace$ applied on a node $v_{j}$ of the network, either activating ($moda_{i}=1$) or inactivating ($moda_{i}=0$). At each iteration, $moda_{i}$ overwrites $f_{j}(\boldsymbol{x})$ making $x_{j}=moda_{i}$.
- **target**: A node $targ_{i}$ of the network on which a $moda_{i}$ is applied.
- **bullet**: A couple $(c_{targ},c_{moda})$ where $c_{targ}=(targ_{1},\ldots,targ_{r})$ is a combination without repetition of targets and $c_{moda}=(moda_{1},\ldots,moda_{r})$ an arrangement with repetition of modalities. $moda_{i}$ is intended to be applied on $targ_{i}$.
Appendix 2: multivalued case {#multivalued}
----------------------------
Below is the multivalued version of the example network:
[r C l]{} do&=&do\
factory&=&factory\
energy&=&max(min(energy,1-task),factory)\
locker&=&1-energy\
releaser&=&do\
sequester&=&1-releaser\
activator&=&min(do,1-locker)\
effector&=&min(activator,1-sequester)\
task&=&effector
where the Boolean operators are replaced by the Zadeh ones. To take advantage of multivalued logic, $f_{locker}$ becomes $locker=min(1-energy,0.5)$ in $\boldsymbol{f}_{patho}$. This equation tells that the locker is actionable when required, namely when there is no energy, but that it is unable at being fully operational due to some pathological defects: the maximal value of $f_{locker}$ in $\boldsymbol{f}_{patho}$ is $0.5$. As mentioned in the article, $0.5$ can be interpreted as an incomplete activation, or an incomplete inhibition depending on what is modeled. Consequently, the activator is at most partly inhibited by the locker when no energy is available, allowing the task to be nevertheless performed. However, the task itself will be moderately performed.
### Attractor sets
Below are the computed attractors:
- $A_{physio}$:
attractor basin ($\%$ of $card\ S_{physio}$) $do$ $factory$ $energy$ $locker$ $task$
---------------- ------------------------------------ ------- ----------- ---------- ---------- --------
$a_{physio1}$ $6.1\%$ $0$ $0$ $0$ $1$ $0$
$a_{physio2}$ $4.5\%$ $0$ $0$ $0.5$ $0.5$ $0$
$a_{physio3}$ $2.5\%$ $0$ $0$ $1$ $0$ $0$
$a_{physio4}$ $9.7\%$ $0$ $0.5$ $0.5$ $0.5$ $0$
$a_{physio5}$ $1.8\%$ $0$ $0.5$ $1$ $0$ $0$
$a_{physio6}$ $10.8\%$ $0$ $1$ $1$ $0$ $0$
$a_{physio7}$ $6.5\%$ $0.5$ $0$ $0$ $1$ $0$
$a_{physio8}$ $4.8\%$ $0.5$ $0$ $0.5$ $0.5$ $0.5$
$a_{physio9}$ $10.3\%$ $0.5$ $0.5$ $0.5$ $0.5$ $0.5$
$a_{physio10}$ $10.6\%$ $0.5$ $1$ $1$ $0$ $0.5$
$a_{physio11}$ $7.3\%$ $1$ $0$ $0$ $1$ $0$
$a_{physio12}$ $3.2\%$ $1$ $0$ $0.5$ $0.5$ $0.5$
$a_{physio13}$ $10.3\%$ $1$ $0.5$ $0.5$ $0.5$ $0.5$
$a_{physio14}$ $11.6\%$ $1$ $1$ $1$ $0$ $1$
- $A_{patho}$:
attractor basin ($\%$ of $card\ S_{patho}$) $do$ $factory$ $energy$ $locker$ $task$
---------------- ----------------------------------- ------- ----------- ---------- ---------- --------
$a_{patho1}$ $6.2\%$ $0$ $0$ $0$ $0.5$ $0$
$a_{physio2}$ $4.7\%$ $0$ $0$ $0.5$ $0.5$ $0$
$a_{physio3}$ $2.2\%$ $0$ $0$ $1$ $0$ $0$
$a_{physio4}$ $9.7\%$ $0$ $0.5$ $0.5$ $0.5$ $0$
$a_{physio5}$ $1.8\%$ $0$ $0.5$ $1$ $0$ $0$
$a_{physio6}$ $10.8\%$ $0$ $1$ $1$ $0$ $0$
$a_{patho2}$ $5.5\%$ $0.5$ $0$ $0$ $0.5$ $0.5$
$a_{physio8}$ $5.8\%$ $0.5$ $0$ $0.5$ $0.5$ $0.5$
$a_{physio9}$ $10.3\%$ $0.5$ $0.5$ $0.5$ $0.5$ $0.5$
$a_{physio10}$ $10.6\%$ $0.5$ $1$ $1$ $0$ $0.5$
$a_{patho3}$ $7.3\%$ $1$ $0$ $0$ $0.5$ $0.5$
$a_{physio12}$ $3.2\%$ $1$ $0$ $0.5$ $0.5$ $0.5$
$a_{physio13}$ $10.3\%$ $1$ $0.5$ $0.5$ $0.5$ $0.5$
$a_{physio14}$ $11.6\%$ $1$ $1$ $1$ $0$ $1$
$a_{physio1}$, $a_{physio3}$, $a_{physio6}$, $a_{physio11}$ and $a_{physio14}$ are the physiological attractors found in the Boolean case, with a different numbering due to additional attractors coming from multivalued logic. Indeed, given that $\lbrace 0,1 \rbrace \subset \lbrace 0,0.5,1 \rbrace$ and that the Zadeh operators also work with Boolean logic, the Boolean results are still obtainable. The same does not apply to the pathological attractors because $f_{locker}$ in $\boldsymbol{f}_{patho}$ differs between the Boolean and multivalued case.
For example, $a_{physio13}$ indicates that the do instruction is sent while energy is partly supplied. Consequently, the locker is partly activated resulting in a partial inhibition of the activator. The task is thus moderately performed despite full do instruction. Concerning the pathological attractors, as an example, $a_{patho3}$ indicates that the do instruction is sent in total absence of energy supply. Consequently, the locker should be fully activated to prevent the task to be performed. However, due to some pathological defects, it is at most partly activated. Therefore, the task is performed in total absence of energy. However, since the locker is partly operational, the task is not performed at its maximum rate.
Among the pathological attractors, $a_{patho1}$ can be considered weakly pathological. In $a_{patho1}$, the locker should be fully activated since there is no energy. However, there is no do instruction and therefore no task to stop. On the other hand, $a_{patho2}$ and $a_{patho3}$ are more pathological since the task is performed while no energy is available.
### Therapeutic bullets
Below are the returned therapeutic bullets:
- $1$-therapeutic bullets:
---------------- -------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- --------------- --------------- -------------- ----------
$B_{physio1}$ $B_{physio2}$ $B_{physio3}$ $B_{physio4}$ $B_{physio5}$ $B_{physio6}$ $B_{physio7}$ $B_{physio8}$ $B_{physio9}$
$B_{physio10}$ $B_{physio11}$ $B_{physio12}$ $B_{physio13}$ $B_{physio14}$ $B_{patho1}$ $B_{patho2}$ $B_{patho3}$
$factory[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $29.3\%$ $6.1\%$ $0\%$ $0\%$ $0\%$ $32.2\%$
$0\%$ $0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$ $0\%$
$factory[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $35.4\%$ $0\%$ $0\%$ $0\%$
$32.2\%$ $0\%$ $0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$
---------------- -------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- --------------- --------------- -------------- ----------
- $2$-therapeutic bullets:
---------------- ---------------- --------------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- --------------- -------------- -------------- ----------
$B_{physio1}$ $B_{physio2}$ $B_{physio3}$ $B_{physio4}$ $B_{physio5}$ $B_{physio6}$ $B_{physio7}$ $B_{physio8}$ $B_{physio9}$
$B_{physio10}$ $B_{physio11}$ $B_{physio12}$ $B_{physio13}$ $B_{physio14}$ $B_{patho1}$ $B_{patho2}$ $B_{patho3}$
$do[0]$ $factory[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $84\%$ $16\%$ $0\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[0]$ $factory[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $100\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[0.5]$ $factory[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $100\%$
$0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[0.5]$ $factory[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[1]$ $factory[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[1]$ $factory[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $0\%$ $100\%$ $0\%$ $0\%$ $0\%$
$do[0]$ $energy[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $34.9\%$ $0\%$ $32.1\%$ $33\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$do[0]$ $task[0]$ $81\%$ $\to$ $89\%$ $0\%$ $11.6\%$ $12.3\%$ $21.3\%$ $10.8\%$ $33\%$ $0\%$ $0\%$ $0\%$
$0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $11\%$ $0\%$ $0\%$
$do[0.5]$ $task[0.5]$ $81\%$ $\to$ $89.4\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $24.3\%$ $32.1\%$
$33\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $10.6\%$ $0\%$
$factory[0]$ $energy[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $35.4\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $32.2\%$ $0\%$
$0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$
$factory[0.5]$ $energy[0.5]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $35.4\%$ $0\%$ $0\%$ $0\%$ $0\%$ $32.2\%$
$0\%$ $0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$ $0\%$
$factory[1]$ $energy[1]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $35.4\%$ $0\%$ $0\%$ $0\%$
$32.2\%$ $0\%$ $0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$
$factory[1]$ $locker[0]$ $81\%$ $\to$ $100\%$ $0\%$ $0\%$ $0\%$ $0\%$ $0\%$ $35.4\%$ $0\%$ $0\%$ $0\%$
$32.2\%$ $0\%$ $0\%$ $0\%$ $32.4\%$ $0\%$ $0\%$ $0\%$
---------------- ---------------- --------------- --------------- --------------- ---------------- ---------------- ---------------- ---------------- ---------------- --------------- -------------- -------------- ----------
For example, the therapeutic bullet $factory[1]\ locker[0]$ may be interesting. It suppresses all the pathological attractors while maintaining three physiological attractors allowing the network to properly respond to the three possible levels of the do instruction. Moreover, the basins of these three physiological attractors, namely $a_{physio6}$, $a_{physio10}$ and $a_{physio14}$, equally span the state space, making them equally reachable. On the other hand, the therapeutic bullet $do[0.5]\ factory[0.5]$ seems to be less interesting. While this bullet also suppresses all the pathological attractors, it enables only one physiological attractor. In this physiological attractor, namely $a_{physio9}$, all the variables are at their intermediate level. Consequently, the network can not fulfill its switching function.
Appendix 3: core of kali {#pseudocode}
------------------------
Below is the core of kali in pseudocode derived from its Go[^6] sources, freely available on GitHub[^7] at <https://github.com/arnaudporet/kali> under the GNU General Public License[^8].
### Defined types
**structure** $Attractor$\
**field** $Name$\
**field** $Basin$\
**field** $States$\
**end structure**\
**structure** $Bullet$\
**field** $Targ$\
**field** $Moda$\
**field** $Gain$\
**field** $Cover$\
**end structure**
### Parameters
$\boldsymbol{nodes}$\
$\Omega$\
$sync$\
$whole$\
$max_{S}$\
$max_{k}$\
$n_{targ}$\
$max_{targ}$\
$max_{moda}$\
$\delta$
### Functions
**function** $DoTheJob(\boldsymbol{f}_{physio},\boldsymbol{f}_{patho},n_{targ},max_{targ},max_{moda},max_{S},max_{k},\delta,\\ \indent sync,\boldsymbol{nodes},\Omega,whole)$\
\
$n \gets Size(\boldsymbol{nodes})$\
**select** $whole$\
**case** $0$\
$S \gets GenArrangs(\Omega,n,max_{S})$\
**case** $1$\
$S \gets GenSpace(\Omega,n)$\
**end select**\
$A_{physio} \gets ComputeAttractorSet(\boldsymbol{f}_{physio},S,\varnothing,max_{k},0,sync,\emptyset)$\
$A_{patho} \gets ComputeAttractorSet(\boldsymbol{f}_{patho},S,\varnothing,max_{k},1,sync,A_{physio})$\
$A_{versus} \gets GetVersus(A_{patho})$\
$C_{targ} \gets GenCombis(\lbrace 1,\ldots,n\rbrace,n_{targ},max_{targ})$\
$C_{moda} \gets GenArrangs(\Omega,n_{targ},max_{moda})$\
**if** $A_{versus} \neq \emptyset$\
$B_{therap} \gets ComputeTherapeuticBullets(\boldsymbol{f}_{patho},S,C_{targ},C_{moda},max_{k},\\ \indent \indent \delta,sync,A_{physio},A_{patho},A_{versus})$\
**end if**\
**return** $S,A_{physio},A_{patho},A_{versus},C_{targ},C_{moda},B_{therap}$\
**end function**\
$Size(container)$ returns the number of items in $container$.\
$GenSpace(\Omega,n)$ returns the state space of the vectors made of $n$ values from $\Omega$.\
$GenArrangs(\Omega,n,max_{arrang})$ returns $max_{arrang}$ arrangements with repetition made of $n$ elements from $\Omega$. If $max_{arrang}$ exceeds its maximal possible value then it is automatically decreased to its maximal possible value.\
$GenCombis(\Omega,n,max_{combi})$ returns $max_{combi}$ combinations without repetition made of $n$ elements from $\Omega$. If $max_{combi}$ exceeds its maximal possible value then it is automatically decreased to its maximal possible value.\
$A_{physio}$ is computed without bullet ($b \gets \varnothing$), without reference set ($A_{ref} \gets \emptyset$) and with physiological setting ($setting \gets 0$).\
$A_{patho}$ is computed without bullet ($b \gets \varnothing$), with reference set ($A_{ref} \gets A_{physio}$) and with pathological setting ($setting \gets 1$).\
$A_{versus}$ is not a true attractor set but the set containing the pathological attractors: $A_{versus} \subset A_{patho}$. $A_{patho}$ can contains physiological attractors if the pathological variant exhibits some of them. However, $A_{versus}$ only contains the pathological attractors.\
Therapeutic bullets are computed only if there are pathological basins to shrink, namely only if $A_{versus} \neq \emptyset$.\
**function** $\boldsymbol{f}_{physio}(\boldsymbol{x})$\
\
$\boldsymbol{y}[1] \gets \boldsymbol{f}_{physio}[1](\boldsymbol{x})$\
$\vdots$\
$\boldsymbol{y}[n] \gets \boldsymbol{f}_{physio}[n](\boldsymbol{x})$\
**return** $\boldsymbol{y}$\
**end function**\
**function** $\boldsymbol{f}_{patho}(\boldsymbol{x})$\
\
$\boldsymbol{y}[1] \gets \boldsymbol{f}_{patho}[1](\boldsymbol{x})$\
$\vdots$\
$\boldsymbol{y}[n] \gets \boldsymbol{f}_{patho}[n](\boldsymbol{x})$\
**return** $\boldsymbol{y}$\
**end function**\
**function** $ComputeAttractor(\boldsymbol{f},\boldsymbol{x}_{0},b,max_{k},sync)$\
\
**select** $sync$\
**case** $1$\
$a.States \gets ReachCycle(\boldsymbol{f},\boldsymbol{x}_{0},b)$\
**case** $0$\
**for**\
$a.States \gets GoForward(\boldsymbol{f},Walk(\boldsymbol{f},\boldsymbol{x}_{0},b,max_{k}),b)$\
**if** $IsTerminal(a,\boldsymbol{f},b)$\
**break**\
**end if**\
**end for**\
**end select**\
**return** $a$\
**end function**\
**function** $ComputeAttractorSet(\boldsymbol{f},S,b,max_{k},setting,sync,A_{ref})$\
\
$A \gets \lbrace \rbrace$\
**select** $setting$\
**case** $0$\
$name \gets a_{physio}$\
**case** $1$\
$name \gets a_{patho}$\
**end select**\
**for** $i \gets 1,\ldots,Size(S)$\
$a \gets ComputeAttractor(\boldsymbol{f},S[i],b,max_{k},sync)$\
**if** $\exists i_{A}: A[i_{A}]=a$\
$A[i_{A}].Basin \gets A[i_{A}].Basin+1$\
**else**\
$a.Basin \gets 1$\
$A \gets A \cup \lbrace a\rbrace$\
**end if**\
**end for**\
**for** $i \gets 1,\ldots,Size(A)$\
$A[i].Basin \gets 100 \cdot A[i].Basin/Size(S)$\
**end for**\
**return** $SetNames(A,name,A_{ref})$\
**end function**\
**function** $ComputeTherapeuticBullets(\boldsymbol{f}_{patho},S,C_{targ},C_{moda},max_{k},\delta,sync,\\ \indent A_{physio},A_{patho},A_{versus})$\
\
$B_{therap} \gets \lbrace \rbrace$\
$b.Gain[1] \gets Sum(GetCover(A_{physio},A_{patho}))$\
**for** $i_{1} \gets 1,\ldots,Size(C_{targ})$\
**for** $i_{2} \gets 1,\ldots,Size(C_{Moda})$\
$b.Targ \gets C_{targ}[i_{1}]$\
$b.Moda \gets C_{Moda}[i_{2}]$\
$A_{test} \gets ComputeAttractorSet(\boldsymbol{f}_{patho},S,b,max_{k},1,sync,A_{physio})$\
$b.Gain[2] \gets Sum(GetCover(A_{physio},A_{test}))$\
**if** $IsTherapeutic(b,A_{test},A_{versus},\delta)$\
$b.Cover \gets GetCover(A_{physio} \cup A_{versus},A_{test})$\
$B_{therap} \gets B_{therap} \cup \lbrace b\rbrace$\
**end if**\
**end for**\
**end for**\
**return** $B_{therap}$\
**end function**\
$Sum(container)$ returns the sum of the items of $container$.\
$b.Gain$ is a couple $(gain1,gain2)$ where:
- $gain1$ is $card\ \bigcup B_{physio,i}$ in $S_{patho}$
- $gain2$ is $card\ \bigcup B_{physio,i}$ in $S_{test}$
expressed in $\%$ of $card\ S_{patho}$ and $\%$ of $card\ S_{test}$ respectively.\
$b.Cover$ stores the size of the physiological and pathological basins in the testing state space.\
**function** $GetCover(A_{1},A_{2})$\
\
$\boldsymbol{cover} \gets ()$\
**for** $i \gets 1,\ldots,Size(A_{1})$\
**if** $\exists i_{2}: A_{2}[i_{2}]=A_{1}[i]$\
$\boldsymbol{cover} \gets Append(\boldsymbol{cover},A_{2}[i_{2}].Basin)$\
**else**\
$\boldsymbol{cover} \gets Append(\boldsymbol{cover},0)$\
**end if**\
**end for**\
**return** $\boldsymbol{cover}$\
**end function**\
$Append(container,item)$ returns $container$ with $item$ added to it.\
**function** $GetVersus(A_{patho})$\
\
$A_{versus} \gets \lbrace \rbrace$\
**for** $i \gets 1,\ldots,Size(A_{patho})$\
**if** $IsSubString(A_{patho}[i].Name,patho)$\
$A_{versus} \gets A_{versus} \cup \lbrace A_{patho}[i]\rbrace$\
**end if**\
**end for**\
**return** $A_{versus}$\
**end function**\
$IsSubString(s_{1},s_{2})$ checks if $s_{2}$ is a substring of $s_{1}$.\
**function** $GoForward(\boldsymbol{f},\boldsymbol{x}_{0},b)$\
\
$fwd \gets \lbrace \boldsymbol{x}_{0}\rbrace$\
$\boldsymbol{stack} \gets (\boldsymbol{x}_{0})$\
**for**\
$\boldsymbol{x} \gets \boldsymbol{stack}[Size(\boldsymbol{stack})]$\
$\boldsymbol{stack} \gets \boldsymbol{stack}[1,\ldots,Size(\boldsymbol{stack})-1]$\
$\boldsymbol{y} \gets \boldsymbol{f}(\boldsymbol{x})$\
**for** $i \gets 1,\ldots,Size(\boldsymbol{y})$\
$\boldsymbol{z} \gets \boldsymbol{x}$\
$\boldsymbol{z}[i] \gets \boldsymbol{y}[i]$\
$\boldsymbol{z} \gets Shoot(\boldsymbol{z},b)$\
**if** $\boldsymbol{z} \notin fwd$\
$fwd \gets fwd \cup \lbrace \boldsymbol{z}\rbrace$\
$\boldsymbol{stack} \gets Append(\boldsymbol{stack},\boldsymbol{z})$\
**end if**\
**end for**\
**if** $Size(\boldsymbol{stack})=0$\
**break**\
**end if**\
**end for**\
**return** $fwd$\
**end function**\
**function** $IsTerminal(a,\boldsymbol{f},b)$\
\
**for** $i \gets 1,\ldots,Size(a.States)$\
**if** $GoForward(\boldsymbol{f},a.States[i],b) \neq a.States$\
**return** $false$\
**end if**\
**end for**\
**return** $true$\
**end function**\
**function** $IsTherapeutic(b,A_{test},A_{versus},\delta)$\
\
**if** $b.Gain[2]-b.Gain[1] \geq \delta$\
**for** $i \gets 1,\ldots,Size(A_{test})$\
**if** $IsSubString(A_{test}[i].Name,patho) \land A_{test}[i] \notin A_{versus}$\
**return** $false$\
**end if**\
**end for**\
**return** $true$\
**else**\
**return** $false$\
**end if**\
**end function**\
**function** $ReachCycle(\boldsymbol{f},\boldsymbol{x}_{0},b)$\
\
$\boldsymbol{cycle} \gets (\boldsymbol{x}_{0})$\
$\boldsymbol{x} \gets \boldsymbol{x}_{0}$\
**for**\
$\boldsymbol{x} \gets Shoot(\boldsymbol{f}(\boldsymbol{x}),b)$\
**if** $\exists i: \boldsymbol{cycle}[i]=\boldsymbol{x}$\
$\boldsymbol{cycle} \gets \boldsymbol{cycle}[i,\ldots,Size(\boldsymbol{cycle})]$\
**break**\
**else**\
$\boldsymbol{cycle} \gets Append(\boldsymbol{cycle},\boldsymbol{x})$\
**end if**\
**end for**\
**return** $\boldsymbol{cycle}$\
**end function**\
**function** $SetNames(A,name,A_{ref})$\
\
$y \gets A$\
$k \gets 1$\
**for** $i \gets 1,\ldots,Size(A)$\
**if** $\exists i_{ref}: A_{ref}[i_{ref}]=A[i]$\
$y[i].Name \gets A_{ref}[i_{ref}].Name$\
**else**\
$y[i].Name \gets CatStrings(name,ToString(k))$\
$k \gets k+1$\
**end if**\
**end for**\
**return** $y$\
**end function**\
$CatStrings(s_{1},s_2)$ returns the concatenation of $s_{1}$ and $s_{2}$.\
$ToString(item)$ returns the string corresponding to $item$.\
This function names the attractors of $A$ according to a reference set $A_{ref}$. If an attractor of $A$ also belongs to $A_{ref}$ then its name in $A_{ref}$ is used, otherwise the default name, numbered with $k$, is used.\
**function** $Shoot(\boldsymbol{x},b)$\
\
$\boldsymbol{y} \gets \boldsymbol{x}$\
**for** $i \gets 1,\ldots,Size(b.Targ)$\
$\boldsymbol{y}[b.Targ[i]] \gets b.Moda[i]$\
**end for**\
**return** $\boldsymbol{y}$\
**end function**\
**function** $Walk(\boldsymbol{f},\boldsymbol{x}_{0},b,max_{k})$\
\
$\boldsymbol{x} \gets \boldsymbol{x}_{0}$\
**for** $k \gets 1,\ldots,max_{k}$\
$\boldsymbol{y} \gets \boldsymbol{f}(\boldsymbol{x})$\
$i \gets RandInt(1,Size(\boldsymbol{x}))$\
$\boldsymbol{x}[i] \gets \boldsymbol{y}[i]$\
$\boldsymbol{x} \gets Shoot(\boldsymbol{x},b)$\
**end for**\
**return** $\boldsymbol{x}$\
**end function**\
$RandInt(a,b)$ returns a randomly selected integer between $a$ and $b$ according to a uniform distribution.\
[^1]: <https://www.r-project.org/>
[^2]: <https://golang.org>
[^3]: <https://www.archlinux.org>
[^4]: <https://github.com>
[^5]: <https://www.gnu.org/licenses/gpl.html>
[^6]: <https://golang.org>
[^7]: <https://github.com>
[^8]: <https://www.gnu.org/licenses/gpl.html>
|
---
abstract: 'We introduce and study a construction of higher derived brackets generated by a (not necessarily inner) derivation of a Lie superalgebra. Higher derived brackets generated by an element of a Lie superalgebra were introduced in our earlier work. Examples of higher derived brackets naturally appear in geometry and mathematical physics. From a totally different viewpoint, we show that higher derived brackets arise when one wants to turn the inclusion map of a subalgebra of a differential Lie superalgebra, with a given complementary subalgebra, into a fibration. (For a non-Abelian complementary subalgebra, this leads to a generalization of $L_{\infty}$-algebras with dropped or weakened (anti)symmetry of the brackets.)'
---
\[firstpage\] $ $
[Higher Derived Brackets for Arbitrary Derivations]{}
[by Theodore Th. Voronov]{}
Introduction
============
Higher derived brackets were introduced by the author in [@tv:higherder], motivated by physical and differential-geometric examples. The starting point in the construction was an element $\D$ in a Lie superalgebra $L$ provided with a direct sum decomposition $L=K\oplus V$ into two subalgebras, where $V$ is Abelian. Then a sequence of symmetric brackets on $V$ is ‘derived’ from $\D$ in the same way as the partial derivatives of a function: $$\{\underbrace{{{\xi}},\ldots,{{\xi}}}_{n}\}_{\D}=P(-\operatorname{ad}{{\xi}})^n\D,$$ for coinciding even arguments, $P$ denoting the projector on $V$. (In particular examples this analogy becomes exact.) It was proved that the Jacobiators for the higher derived brackets of an odd $\D$ are equal to the higher derived brackets of $\D^2$. In particular, this leads to $L_{\infty}$-algebras and algebras related with them.
In this paper we introduce and study higher derived brackets generated by an arbitrary derivation $D\co L\to L$, which does not have to be inner. See formula . (The case of non-inner derivations was touched on in the final version of [@tv:higherder] without proofs.) As in [@tv:higherder], we make use of the decomposition $L=K\oplus V$. The subalgebra $V$ is still assumed to be Abelian, though at the end of the paper we briefly discuss how this condition can be relaxed.
Our first main result is Theorem \[thmjac\], which we state and prove in Section \[secjac\]. It says that the Jacobiators for the higher derived brackets generated by an odd derivation $D$ are equal to the brackets generated by $D^2$. So it is an analog of a similar statement for $\D$. However, the presently available proof of Theorem \[thmjac\] is technically much harder. Notice also that strictly speaking, the theorem about brackets generated by $\D$ is not a corollary of the theorem for $D$ because of the possible presence of a ‘background’ in the former.
Secondly, we establish a relation between higher derived brackets and homotopical algebra. This is done in Section \[sechomot\]. The main result is Theorem \[thmcocyl\]. The question of whether the higher derived brackets of $\D$ defined in our paper [@tv:higherder] can be interpreted in the framework of homotopical algebra was asked by the anonymous referee of [@tv:higherder]. In fact, he suggested linking them with the notion of a ‘left cone’ (i.e., a cocone, or a homotopy fiber in topologists’ language). This question happened to be very fruitful. The proper framework for it is when the brackets are generated by an arbitrary derivation $D$. In Section \[sechomot\] we show that such a homotopical-algebraic interpretation is indeed possible. We show that the higher derived brackets of $D$ appear as part of the brackets in $\Pi L\oplus V$ that naturally arise from the condition that the canonical differential in $\Pi L\oplus V$ (viewed as a cone or a cocylinder) respects an algebra structure extended from $L$, and we prove that the latter brackets make $\Pi L\oplus V$ an $L_{\infty}$-algebra if $D^2=0$. Thus we arrive at an alternative approach to higher derived brackets.
Behind Theorem \[thmjac\] one can recognize a more general algebraic statement. If one considers the Lie superalgebra $\operatorname{Der}L$ of derivations of $L$ and the Lie superalgebra $\operatorname{Vect}V$ of vector fields on $V$, both w.r.t. the commutator, then it is possible to see that the construction of higher derived brackets gives a homomorphism $\operatorname{Der}L\to \operatorname{Vect}V$. By identifying vector fields with multilinear operations on $V$ specified by their Taylor expansion at zero, we arrive at the statement that $V$ becomes a ‘generalized’ $L_{\infty}$-algebra ‘over’ the Lie superalgebra $\operatorname{Der}L$ (that is, there is a family of brackets parametrized by elements of $\operatorname{Der}L$ obeying ‘Jacobi type’ relations following the relations in $\operatorname{Der}L$). This is a new algebraic notion. We discuss it briefly. As mentioned, we also briefly discuss the possibility of dropping the condition that $V$ be Abelian. By doing so, we arrive at higher derived brackets that are not necessarily symmetric. This leads to another generalization of $L_{\infty}$-algebras, which we hope to analyze further elsewhere.
*Terminology and notation.* We use the ‘super’ language and conventions; in particular, a vector space always means a ${{\mathbb Z_{2}}}$-graded vector space, and we freely identify it with the corresponding vector supermanifold; multilinearity, symmetry, antisymmetry, derivations, etc., are always understood in the ${{\mathbb Z_{2}}}$-graded sense. $\Pi$ stands for the parity reversion functor, and the parity of homogeneous objects is denoted by a tilde, i.e. ${{\tilde a}}=0$ or ${{\tilde a}}=1$ if $a\in A_0$ or $a\in A_1$ respectively, for $a$ in a ${{\mathbb Z_{2}}}$-graded module $A$.
*Acknowledgements.* I wish to thank the anonymous referee of paper [@tv:higherder] for remarks that motivated the “homotopical” part of this work, and Hovhannes Khudaverdian for numerous fruitful discussions. I am very grateful to Peter Symonds as well as to the referee of the present paper for their help in improving the style of exposition.
Construction of Higher Derived Brackets {#secjac}
=======================================
The algebraic setup is as follows. We are given a Lie superalgebra $L$ and a decomposition of $L$ into a sum of two subalgebras: $$L=K\oplus V.$$ Let $P\co L\to V$ be the projector on $V$ parallel to $K$, i.e., $V={{\mathop{\mathrm{Im}}}}P$, $K=\operatorname{Ker}P$.
Consider an arbitrary derivation $D\co L\to L$, either even or odd.
\[def.brack\] The [*$k$-th (higher) derived bracket of $D$*]{} is a multilinear operation $$\underbrace{V\times \ldots \times V}_{\text{$k$ times}}\to V$$ given by the formula $$\label{eqdefderbrack}
\{a_1,\ldots,a_k\}_D:=P[\ldots[[Da_1,a_2],a_3],\ldots,a_k]$$ where $a_i\in V$. Here $k=1,2,3,\ldots \ $.
The derived brackets have the same parity as the parity of the derivation $D$.
In this paper we construct higher derived bracket for an arbitrary derivation $D$. Higher derived brackets were first defined in [@tv:higherder] in the case when $D$ is an inner derivation, $D=\operatorname{ad}\D$ for some $\D\in L$.
The binary derived bracket when $P={{\mathrm{id}}}$, i.e, $L=V$, $K=0$, was introduced by Yvette Kosmann-Schwarzbach [@yvette:derived] following an idea of Koszul, and independently by the author [@tv:lectures93etc] (unpublished). In [@yvette:derived] a slightly more general setting was considered, $L$ being a Loday (Leibniz) algebra. Derived brackets have numerous applications, see [@yvette:derived2] for a survey.
Suppose that $V$ is an Abelian subalgebra. Then the derived brackets are symmetric (in the ${{\mathbb Z_{2}}}$-graded sense).
**From now on we assume that $V$ is Abelian.**
(Later we shall discuss whether this requirement can be relaxed.)
A symmetric multilinear operation is defined by its value on coinciding even arguments (to be more precise, this is true if extending scalars to include as many ‘odd constants’ as necessary, is allowed). For the higher derived brackets, if ${{\xi}}$ is even, we have $$\label{eqxxx}
\{\underbrace{{{\xi}},\ldots,{{\xi}}}_{n}\}_D=(-1)^{n-1}P\left(\operatorname{ad}{{\xi}}\right)^{n-1} D{{\xi}}$$ for any $n=1,2,\ldots \ $, regardless of the parity of $D$.
We want to investigate if in addition to symmetry the derived brackets can satisfy other identities such as the Jacobi identity. Consider the binary bracket. Notice first that it is symmetric, not antisymmetric, compared to the bracket on a Lie algebra. To turn symmetry into antisymmetry we have to reverse parity and consider $\Pi V$. The bracket induced on $\Pi V$ will be even (as a Lie bracket should be) if the bracket in $V$ is odd. Therefore, to ask about (analogs of) the Jacobi identity for the higher derived brackets makes sense when the derivation $D$ is odd.
\[sec\_const.ex1\] Consider a vector space $V$. Let $L=\operatorname{Der}S(V^*)$. Elements of $L$ can be viewed as polynomial vector fields on $V$. Let $\xi^i$ be the linear coordinates on $V$ corresponding to a basis $(e_i)$ in $V$. We can consider vectors in $V$ as vector fields with constant coefficients, so $V\subset L$ will be an Abelian subalgebra. Then $L=K\oplus V$, where the subalgebra $K$ consists of all vector fields vanishing at the origin. The projector $P$ maps every vector field to its value at the origin (constant vector field). Consider an arbitrary vector field $$X=X^j({{\xi}})\,{{\frac{\partial {}}{\partial {{{\xi}}^j}}}},$$ even or odd, and consider the derived brackets on $V$ generated by the derivation $\operatorname{ad}X$, $$\{u_1,\ldots,u_k\}_{X}:=\{u_1,\ldots,u_k\}_{\operatorname{ad}X}=\left[\ldots\left[X,u_1\right],\ldots,u_k\right](0)\,.$$ One can see that $$\{e_{i_1},\ldots,e_{i_k}\}_{X}=(\pm)\, X^j_{i_1 \ldots i_k}\,e_j$$ where $$X^j_{i_1 \ldots i_k}=\frac{\partial^k X^j}{\partial{{\xi}}^{i_1}\ldots
\partial{{\xi}}^{i_k}}\,(0)\,.$$ In particular, consider a quadratic vector field: $$Q=\frac{1}{2}\,\xi^i\xi^j Q_{ji}^k{{\frac{\partial {}}{\partial {{{\xi}}^k}}}}\,.$$ Then the $k$-th derived bracket of $Q$ is zero unless $k=2$, and the binary derived bracket is given by $$\label{eqbrackqbin}
\{e_i,e_j\}_Q=(\pm)\,Q_{ij}^ke_k\,.$$ Suppose that $Q$ is odd. By a direct check one can see that *the Jacobi identity for the bracket is equivalent to the condition $Q^2=0$.* (More precisely, the usual graded Jacobi identity is valid for the antisymmetric bracket in $\Pi V$.) $Q$ then can be identified with the Chevalley–Eilenberg differential in the standard cochain complex of the resulting Lie (super)algebra.
Odd vector fields with square zero are known as *homological*. Example \[sec\_const.ex1\] shows that the structure of a Lie superalgebra on a vector space corresponds to a quadratic homological vector field. If we drop the condition that the homological vector field be quadratic, we obtain ‘$L_{\infty}$-algebras’ or ‘strong homotopy Lie algebras’ where the Jacobi identity for a binary bracket holds up to homotopy, which is a ternary bracket, and in its turn satisfies an analog of the Jacobi identity up to a homotopy, and so on. Describing algebraic structures by derivations of square zero is a very general principle dating back to Nijenhuis in 1950’s and used in recent works of Kontsevich (his “formality theorem” implying the existence of deformation quantization of Poisson manifolds, see [@kontsevich:quant]).
Let $V$ be a vector space endowed with a sequence of $k$-linear odd symmetric operations denoted by braces. Here $k=0,1,2,3,\ldots \ $.
\[def.jacobiator\] The *$n$-th Jacobiator* is the following expression: $$J^n(a_1,\ldots,a_n)=\sum_{k+l=n}\,\sum_{\text{$(k,l)$-shuffles}}
(-1)^{\e}
\{\{a_{\s(1)},\ldots,a_{\s(k)}\},a_{\s(k+1)},\ldots,a_{\s(k+l)}\}$$ where the sign $(-1)^{\e}$ is given by the usual sign rule for permutations of homogeneous elements of $V$.
Let us recall the definition of $L_{\infty}$-algebras (due to Lada and Stasheff [@lada:stasheff]).
An *$L_{\infty}$-algebra*, or *strong(ly) homotopy Lie algebra*, is a vector space $V$ endowed with a sequence of $k$-linear odd symmetric operations, $k=0,1,2,3,\ldots \ $, such that all the Jacobiators vanish.
We gave the definition in the form most convenient for our purposes. If one wishes to directly include the case of ordinary Lie algebras, the term $L_{\infty}$-algebra should be applied to the structure on the opposite space, i.e., $\Pi V$, where the corresponding operations are antisymmetric and are even for an even number of arguments and odd otherwise. Also, it is often assumed that the $0$-bracket is zero. The $0$-bracket is sometimes referred to as the ‘background’.
\[prop.linfandhom\]There is a one-to-one correspondence between $L_{\infty}$-algebra structures on $V$ and formal homological vector fields on $V$: $$Q=Q^k(\xi)\,{{\frac{\partial {}}{\partial {{{\xi}}^k}}}}=\left(Q^k_0+{{\xi}}^iQ_i^k+\frac{1}{2}\,\xi^i\xi^j
Q_{ji}^k+\frac{1}{3!}\,\xi^i\xi^j{{\xi}}^l Q_{lji}^k+\ldots
\right){{\frac{\partial {}}{\partial {{{\xi}}^k}}}}\,.$$
This proposition is a well-known fact. What we can give here is an explicit invariant expression for the correspondence: *the brackets in an $L_{\infty}$-algebra corresponding to a homological field $Q$ are the higher derived brackets of $Q$*, $$\label{equuQ}
\{u_1,\ldots,u_k\}_{Q}=\left[\ldots\left[Q,u_1\right],\ldots,u_k\right](0)$$ (this generalizes Example \[sec\_const.ex1\]).
Let us return to our abstract setup. Consider the higher derived brackets of an *odd* derivation $D$. We get a sequence of odd symmetric multilinear operations on $V$. By definition the $0$-ary operation is zero. What about the Jacobiators?
\[thmjac\] Suppose that $D$ preserves the subalgebra $K=\operatorname{Ker}P$. Then the $n$-th Jacobiator of the derived brackets of $D$ equals the $n$-th derived bracket of $D^2$: $$\label{eqjacobimain}
J^n_D(a_1,\ldots,a_n)=\{a_1,\ldots,a_n\}_{D^2}\,,$$ for all $n=1,2,3,\ldots \ $.
Let us make two comments before giving the proof.
Firstly, notice that in our setup the condition that $L=K\oplus V$ is the sum of subalgebras where $V$ is Abelian can be expressed by the identities $$\label{eqvabel}
[Pa, Pb]=0$$ and $$\label{eqdistrib}
P[a,b]=P[Pa,b]+P[a,Pb],$$ for all $a,b$ (a “distributivity law” for $P$). Notice that is also equivalent to vanishing of the Nijenhuis bracket of $P$ with itself.
Secondly, the condition in the theorem that $D(K)\subset K$ can be written as the identity $$\label{eqpdp}
PDP=PD\,.$$ Condition already appears if we check the Jacobiator of order one: $$J^1_D(a)=\{\{a\}_D\}_D=PDPDa=PDDa=\{a\}_{D^2},$$ if $PDP=PD$. (Notice that in general $D$ does not have to preserve $V$. Indeed, if $D$ preserves $V$, then all the derived brackets of $D$ starting from the binary bracket, will vanish.)
To simplify the notation, let us omit temporarily the subscript $D$ from the brackets and Jacobiators. Since the Jacobiators are multilinear symmetric functions, it is sufficient to consider them for coinciding even arguments. Denote $J^n({{\xi}}, \ldots, {{\xi}})$ where ${{\xi}}$ is even by $J^n({{\xi}})$. From Definition \[def.jacobiator\] and equation we clearly obtain $$J^n({{\xi}})= \sum_{l=0}^{n-1}C_n^l
\{\{\underbrace{{{\xi}},\ldots,{{\xi}}}_{n-l}\},\underbrace{{{\xi}},\ldots,{{\xi}}}_{l}\},$$ where $C_n^l=\frac{n!}{l!(n-l)!}$ is the binomial coefficient, in our case appearing as the number of $(n-l,l)$-shuffles. It follows that $$\begin{gathered}
J^n({{\xi}})=
\sum_{l=0}^{n-1}C_n^lP[\ldots[D\{\underbrace{{{\xi}},\ldots,{{\xi}}}_{n-l}\},
\underbrace{{{\xi}}],\ldots,{{\xi}}]}_{l}=\\
\sum_{l=0}^{n-1}C_n^lP[\ldots[DP(-1)^{n-l-1}
(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}},\underbrace{{{\xi}}],\ldots,{{\xi}}]}_{l}=\\
\sum_{l=0}^{n-1}C_n^l (-1)^{n-l-1} (-1)^l P(\operatorname{ad}{{\xi}})^l
DP(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}=\\
\sum_{l=0}^{n-1}C_n^l (-1)^{n-1} P(\operatorname{ad}{{\xi}})^l DP(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\,.\end{gathered}$$ Consider $(\operatorname{ad}{{\xi}})^l DP$. We want to move $D$ to the left. Since $$\operatorname{ad}{{\xi}}\cdot D-D\cdot \operatorname{ad}{{\xi}}=-\operatorname{ad}(D{{\xi}})\,,$$ as one can easily check, it follows that for any $l{\geqslant}1$ $$\begin{gathered}
(\operatorname{ad}{{\xi}})^l DP=(\operatorname{ad}{{\xi}})^{l-1}(\operatorname{ad}{{\xi}}\cdot D-D\cdot \operatorname{ad}{{\xi}}+D\cdot
\operatorname{ad}{{\xi}})P=\\
(\operatorname{ad}{{\xi}})^{l-1}(-\operatorname{ad}(D{{\xi}})+D\cdot
\operatorname{ad}{{\xi}})P=-(\operatorname{ad}{{\xi}})^{l-1}\operatorname{ad}(D{{\xi}})P=
-\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}}, P(.)
\right]\end{gathered}$$ where we used $\operatorname{ad}{{\xi}}\cdot P=0$. Substituting this into the formula above we obtain $$J^n({{\xi}})=\sum_{l=1}^{n-1} C_n^l(-1)^nP\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]
+(-1)^{n-1}PDP(\operatorname{ad}{{\xi}})^{n-1}D{{\xi}}$$ or $$(-1)^n J^n({{\xi}})=\sum_{l=1}^{n-1} C_n^l P\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]
-PD(\operatorname{ad}{{\xi}})^{n-1}D{{\xi}}$$ (where we also used ). We can re-arrange the first sum by adding it to itself in the reverse order and dividing by two: $$\begin{gathered}
\sum_{l=1}^{n-1} C_n^l P\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]=\\
\frac{1}{2}\,\left(\sum_{l=1}^{n-1} C_n^l P\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]+
\sum_{l=1}^{n-1} C_n^l P\left[(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}}\right]\right)=\\
\frac{1}{2}\,\sum_{l=1}^{n-1} C_n^l \Bigl(P\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]+ P\left[(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}},
P(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}}\right]\Bigr)\,.\end{gathered}$$ Noticing that $\left[(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}},P(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}}\right]=
\left[P(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]$, because ${{\xi}}$ is even and $D{{\xi}}$ is odd, and using the distributivity relation , we find the following expression for the Jacobiator: $$(-1)^n J^n({{\xi}})=
\frac{1}{2}\,\sum_{l=1}^{n-1} C_n^l P\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]-PD(\operatorname{ad}{{\xi}})^{n-1}D{{\xi}}$$ or $$\begin{gathered}
\label{eqjacob1}
(-1)^n J^n({{\xi}})=
\frac{1}{2}\,P\sum_{l=1}^{n-1} C_n^l \left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},
(\operatorname{ad}{{\xi}})^{n-l-1}D{{\xi}}\right]-\\
P\left[D,(\operatorname{ad}{{\xi}})^{n-1}\right]D{{\xi}}- P(\operatorname{ad}{{\xi}})^{n-1}D^2{{\xi}}\,.\end{gathered}$$ We shall now analyze the term $\left[D,(\operatorname{ad}{{\xi}})^{n-1}\right]D{{\xi}}$. Using the formula for the commutator $[A,B^N]$ for arbitrary operators $A$, $B$, we get $$\begin{gathered}
\left[D,(\operatorname{ad}{{\xi}})^{n-1}\right]D{{\xi}}=\sum_{i+j=n-2}(\operatorname{ad}{{\xi}})^i[D,\operatorname{ad}{{\xi}}](\operatorname{ad}{{\xi}})^jD{{\xi}}= \\
\sum_{i+j=n-2}(\operatorname{ad}{{\xi}})^i\operatorname{ad}(D{{\xi}})(\operatorname{ad}{{\xi}})^jD{{\xi}}= \sum_{i+j=n-2}(\operatorname{ad}{{\xi}})^i\left[D{{\xi}}, (\operatorname{ad}{{\xi}})^jD{{\xi}}\right]=\\
\sum_{i+j=n-2}\,\,\sum_{r+s=i}
C_i^r\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}}, (\operatorname{ad}{{\xi}})^{s+j}D{{\xi}}\right]=\sum_{i=0}^{n-2}\sum_{r=0}^i
C_i^r\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}}, (\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]=\\
\sum_{r=0}^{n-2}\sum_{i=r}^{n-2}
C_i^r\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}}, (\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]\,.\end{gathered}$$ Since in the internal sum the commutators do not depend on the index of summation $i$, they can be taken out of the sum. It is possible to apply a well-known identity for sums of binomial coefficients (see, e.g. [@bron-sem:1981 p. 153]): $$\sum_{i=r}^{m} C_i^r=C_r^r+C_{r+1}^r+\ldots+C_{m}^r=
C_r^0+C_{r+1}^1+\ldots+C_{m}^{m-r}=C_{m+1}^{m-r},$$ where in our case $m=n-2$. Hence $$\sum_{i=r}^{n-2} C_i^r=C_{n-1}^{n-2-r},$$ and we arrive at the equality $$\label{eqkomdiad1}
\left[D,(\operatorname{ad}{{\xi}})^{n-1}\right]D{{\xi}}=
\sum_{r=0}^{n-2}C_{n-1}^{n-2-r}\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}},
(\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]\,.$$ Notice that since $D{{\xi}}$ is odd and the bracket is symmetric, the left-hand side contains similar terms, with $r$ and $r'$, where $r=n-2-r'$. Hence, this sum can be re-arranged by writing it twice in opposite orders and dividing by two: $$\begin{gathered}
\left[D,(\operatorname{ad}{{\xi}})^{n-1}\right]D{{\xi}}= \\
\frac{1}{2}\left(\sum_{r=0}^{n-2}C_{n-1}^{n-2-r}\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}},
(\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]+
\sum_{r=0}^{n-2}C_{n-1}^{r}\left[(\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}},
(\operatorname{ad}{{\xi}})^{r}D{{\xi}}\right]\right)=\\
\frac{1}{2}\sum_{r=0}^{n-2}\left(C_{n-1}^{r+1}+C_{n-1}^{r}\right)
\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}},(\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]=\\
\frac{1}{2}\sum_{r=0}^{n-2} C_{n}^{r+1}
\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}},(\operatorname{ad}{{\xi}})^{n-2-r}D{{\xi}}\right]=
\frac{1}{2}\sum_{l=1}^{n-1} C_{n}^{l}
\left[(\operatorname{ad}{{\xi}})^{l-1}D{{\xi}},(\operatorname{ad}{{\xi}})^{n-1-l}D{{\xi}}\right]\,,\end{gathered}$$ which coincides with the first term in the formula for the Jacobiator . Substituting into , we see that the first two terms cancel, and we finally obtain $$(-1)^n J^n({{\xi}})=-P(\operatorname{ad}{{\xi}})^{n-1}D^2{{\xi}}$$ or $$J^n_D({{\xi}})=(-1)^{n-1}
P(\operatorname{ad}{{\xi}})^{n-1}D^2{{\xi}}=\{\underbrace{{{\xi}},\ldots,{{\xi}}}_{n}\}_{D^2}$$ for an arbitrary even ${{\xi}}$. This implies identity for all elements of $V$.
We say that the derivation $D$ is of *order* $r$ with respect to the subalgebra $V$ if for all $a_1,\ldots, a_{r+1}\in V$ $$\left[\ldots\left[Da_1,a_2\right],\ldots,a_{r+1}\right]=0.$$ Here $r=0,1,2,\ldots\,$
If $D$ is of order $r$, all the derived $k$-brackets of $D$ vanish for $k{\geqslant}r+1$.
For an odd derivation $D$, if the order of $D^2$ is $r$, then the higher derived brackets of $D$ satisfy all the Jacobi identities with ${\geqslant}r+1$ arguments,
If the order of $D^2$ is zero, i.e., $D^2(V)=0$, in particular if $D^2=0$, then the higher derived brackets of an odd derivation $D$ define an $L_{\infty}$-algebra.
Proposition \[prop.linfandhom\] shows that all $L_{\infty}$-algebras are obtained in this way.
Examples
========
All examples of higher derived brackets naturally arising in applications are for the case when $D=\operatorname{ad}\D$ is an inner derivation given by some element $\D$. This is the situation where higher derived brackets were first introduced in [@tv:higherder]. An analog of Theorem \[thmjac\] was proved there for brackets generated by $\D$. (That proof is simpler than the above proof of Theorem \[thmjac\] for general $D$.) Let us clarify the relation between the higher derived brackets of an element $\D\in L$ as introduced in [@tv:higherder] and the higher derived brackets of a derivation $D\co L\to L$ as in Definition \[def.brack\].
Any element $\D\in L$, of course, gives an inner derivation $D=\operatorname{ad}\D\co L\to L$, and the higher derived brackets of the derivation $D=\operatorname{ad}\D$ $$\{a_1,\ldots,a_k\}_{D}=P\left[\ldots\left[(\operatorname{ad}\D)a_1,a_2\right],\ldots,a_k\right],$$ coincide with the brackets defined in [@tv:higherder], $$\{a_1,\ldots,a_k\}_{\D}=P\left[\ldots\left[\left[\D,a_1\right],a_2\right],
\ldots,a_k\right],$$ where $k=1,2,3,\ldots \ $. However, for $\D$ there is a natural notion of a $0$-bracket (no arguments, a distinguished element), $$\{{\varnothing}\}_{\D}=P\D\,,$$ which is not defined for arbitrary derivations $D$. The Jacobiators for the higher derived brackets of $\D$ include this $0$-bracket and start with the $0$-th Jacobiator $\{\{{\varnothing}\}_{\D}\}_{\D}$. At the same time, the $0$-ary bracket is assumed to be zero in all the Jacobiators for a general $D$ and it does not appear in our Theorem \[thmjac\]. There is no obvious way of incorporating the $0$-th bracket into the picture for a general derivation $D$. If $P\D\neq 0$, that means that $\D\notin K$, hence there is no guarantee that $\operatorname{ad}\D(K)\subset K$, which is a condition of Theorem \[thmjac\]. The calculation of $J^1_D(a)$ above shows that some sort of condition is needed (and at least a weaker condition $PD^2P=PDPDP$ is necessary). Therefore, Theorem 1 of [@tv:higherder], to which Theorem \[thmjac\] is an analog, does not follow from Theorem \[thmjac\], in general.
We can summarize by saying that the theory developed in [@tv:higherder] is a particular case of the theory developed here if $P(\D)=0$, i.e., $\D\in K$. Then, in particular, $(\operatorname{ad}\D)(K)\subset K$ and Theorem \[thmjac\] applies.
We shall leave open the question of how the theory for non-inner derivations can be modified to incorporate an $0$-ary bracket.
With having this in mind, there are some examples of higher derived brackets, all coming from inner derivations. They are given for illustrative purposes only. More details can be found in [@tv:higherder]. See also [@tv:graded], [@tv:laplace2].
The setup of Example \[sec\_const.ex1\]. $L=\operatorname{Vect}V$, where $V$ is a vector space, $P\co X\mapsto X(0)$ is a projection onto the Abelian subalgebra of vector fields with constant coefficients. For an odd vector field $Q$ such that $Q(0)=0$ we get the higher derived brackets on $V$, $k=1,2,\ldots, \
$, $$\{u_1,\ldots,u_k\}_Q=\left[\ldots\left[Q,u_1\right],\ldots,u_k\right](0).$$ They define an $L_{\infty}$-algebra with e zero background (‘strict’ in the terminology of [@tv:higherder]) if $Q^2=0$, and this is a canonical description of all (strict) $L_{\infty}$-algebra structures on the space $V$.
$L=\operatorname{End}A$ for a commutative associative algebra with unit $A$ and $V=A$. The projector $P$ maps an operator $\D$ to $\D
1\in A\subset \operatorname{End}A$. The higher derived brackets of $\operatorname{ad}\D$ for an operator $\D$ such that $\D1=0$ are the ‘Koszul operations’ (see [@koszul:crochet85]) $$\{a_1,\ldots,a_k\}_{\D}=\left[\ldots\left[\D,a_1\right],\ldots,a_k\right]1,$$ $k=1,2,3,\ldots, ...\ $. For a differential operator of order $n$ the brackets with more than $n$ arguments vanish and the top bracket is the symbol of $\D$. An odd operator $\D$ satisfying $\D^2=0$ provides an example of a ‘homotopy Batalin–Vilkovisky algebra’.
$L={C^{\infty}}(T^*M)$, $V={C^{\infty}}(M)$, $P$ is the restriction on $M$, and $i^*\co{C^{\infty}}(T^*M)\to{C^{\infty}}(M)$, where $i\co M\to T^*M$. For a Hamiltonian $S\in{C^{\infty}}(T^*M)$ such that $i^*S=0$, on functions on $M$ there are derived brackets $$\{f_1,\ldots,f_k\}_{S}=i^*\left(\ldots\left(S,f_1\right),\ldots,f_k\right),$$ $k=1,2,3,\ldots\ $, where in the right-hand side stand the canonical Poisson brackets on $T^*M$. If $S$ is odd (for a nontrivial example $M$ should be a supermanifold) and satisfies $(S,S)=0$, we get ‘higher Schouten brackets’ on ${C^{\infty}}(M)$ giving an example of a ‘homotopy Schouten algebra’.
Similarly, let $L={C^{\infty}}(\Pi T^*M)$, $V={C^{\infty}}(M)$ and let $P$ be the restriction on $M$. For a multivector field $\psi\in{C^{\infty}}(\Pi
T^*M)$ such that $i^*\psi =0$, on functions on $M$ there are derived brackets $$\{f_1,\ldots,f_k\}_{\psi}=i^* {{[\![}}\ldots {{[\![}}\psi,f_1
{{]\!]}},\ldots,f_k {{]\!]}},$$ $k=1,2,3,\ldots \, $, where on the right-hand side we have the canonical Schouten brackets on $\Pi T^*M$. Since the canonical Schouten brackets are odd, for an even $\psi$ the derived brackets have alternating parity (even for an even number of arguments, odd for odd). If ${{[\![}}\psi,\psi{{]\!]}}=0$, these ‘higher Poisson brackets’ on functions on $M$ give an example of a ‘homotopy Poisson algebra’.
Other examples of higher derived brackets which we shall not consider here, are ‘homotopy Lie algebroids’, which are an analog of $L_{\infty}$-algebras in the world of algebroids, and the non-linear analogs in the world of graded manifolds [@tv:graded] (see also [@roytenberg:graded]). We hope to be able to say more about such examples elsewhere.
It is a good question whether a genuinely non-inner derivation can naturally occur in examples of higher derived brackets coming from differential geometry or physics.
Relation with Homotopy Theory {#sechomot}
=============================
Now we shall show how our construction of the (higher) derived brackets arises naturally if one wishes to consider the homotopy theory of Lie superalgebras.
Let us re-formulate the setup in a way convenient for this purpose. We have a Lie superalgebra $L$ with a decomposition $L=K\oplus V$, where $K$ and $V$ are subalgebras. Consider an odd derivation $D$ such that $D(K)\subset K$, and from the start assume that $D$ is of square zero. Hence we have an inclusion of differential Lie superalgebras $$i\co K\to L$$ and a given complement for the image of $i$, which is called $V$. ($V$ is *not*, in general, a differential subalgebra.)
There is an idea, familiar to topologists, that every map can be made into a fibration by appropriately replacing a space by a homotopy equivalent one. More precisely, if we have a category where a “weak equivalence”, “fibration” and “cofibration” make sense (i.e., a Quillen model category [@quillen:hoalgebra67]), the following diagram is called a *cocylinder diagram*: $$\begin{diagram}[small]
X & & \rTo^f & & Y \\
& \rdTo_j & & \ruTo_p & \\
& & Z & & \\
\end{diagram}$$ if $j$ is a cofibration and weak equivalence, and $p$ is a fibration. Then $Z$ is also denoted by $\operatorname{Cocyl}f$. (To refresh the intuition, recall that for topological spaces that are cell complexes, cofibrations are just inclusions of subspaces, fibrations are ‘Serre fibrations’, i.e., maps satisfying the covering homotopy property, and weak equivalences are maps inducing isomorphisms of all homotopy groups. In this case, a cocylinder for any map $f\co
X\to Y$ may be obtained as a subspace in $X\times Y^I$ consisting of all pairs $(x,{{\gamma}})$ where ${{\gamma}}\co I\to Y$ is a path such that ${{\gamma}}(0)=f(x)$.)
Can we do this (in an algebraic context) for the inclusion $K\to L$?
To begin with let us temporarily forget about the algebra structure. Consider just an arbitrary inclusion of complexes $$i\co K\to L$$ such that there is a given complementary subspace $V$ (not a subcomplex!) and $L=K\oplus V$. In the context of this paper, a *complex* is simply a vector space with an odd operator of square zero.
Let $P$ be the projector onto $V$ parallel to $K$. The space $V$ becomes a complex with the differential $PD$. Introduce into $L\oplus
\Pi V$ an operator $d$ as follows: $$\label{eqdincocyl}
d(x,\Pi a):=\bigl(Dx, -\Pi P(x+Da)\bigr), $$ for $x\in L$, $a\in V$. It is then straightforward to show that $d^2=0$. Consider the maps $j\co K\to L\oplus \Pi V$ and $p\co
L\oplus \Pi V\to L$, where $j\co x\mapsto (x,0)$, $p\co (x,\Pi
a)\mapsto x$.
\[lemcocyl\] The following diagram $$\begin{diagram}[small]
K & & \rTo^i & & L \\
& \rdTo_j & & \ruTo_p & \\
& & L\oplus \Pi V & & \\
\end{diagram}$$ is a cocylinder diagram in the category of complexes, i.e., the maps $j$ and $p$ are chain maps, $i=p\circ j$, the map $j\co K\to
L\oplus \Pi V$ is a monomorphism [(‘cofibration’)]{.nodecor} and a quasi-isomorphism [(‘weak homotopy equivalence’)]{.nodecor}, and the map $p\co L\oplus \Pi V\to L$ is an epimorphism [(‘fibration’)]{.nodecor}.
This is immediate. A quasi-inverse for $j$ is the map $$q\co (x,\Pi a)\mapsto (1-P)(x+Da).$$
As is well known, for maps of complexes there are canonical constructions of cylinders and cocylinders; they are modelled on the (co)chain complexes corresponding to the canonical topological cylinders and cocylinders. For a particular chain map it might be more convenient to consider a ‘smaller’ cylinder or cocylinder than the one featured by the standard construction. This is exactly what happens in our case. The standard cocylinder construction applied to the inclusion $i\co K\to L$ would not yield the complex $L\oplus
\Pi V$ as in Lemma \[lemcocyl\], instead it would give a bigger complex $K\oplus L\oplus \Pi L=K\oplus K\oplus V\oplus \Pi K\oplus
\Pi V$ that is homotopy equivalent to $L\oplus \Pi V$. One should also note that the complex $L\oplus \Pi V$ essentially coincides with the standard (co)cone of the projection $L\to V$. See the Appendix.
It follows from Lemma \[lemcocyl\] that the space $\Pi V=\operatorname{Ker}p$, taken with the differential $-\Pi PD$, is a homotopy fiber of the inclusion of complexes $i\co K\to L=K\oplus V$.
Now we want to ‘turn the algebra structure on’. To this end, since we have been working with $V$ rather than $\Pi V$, let us first apply a parity shift to the cocylinder diagram above. Then we have the cocylinder diagram $$\begin{diagram}[small]
\Pi K & & \rTo^i & & \Pi L \\
& \rdTo_j & & \ruTo_p & \\
& & \Pi L\oplus V & & \\
\end{diagram}$$ for the inclusion of complexes $\Pi K\to \Pi L$. In particular, the differential in $\Pi L\oplus V$ is $$\label{eqdifincoc2}
d\co (\Pi x,a)\mapsto (-\Pi Dx, P(x+Da))$$ (which is the differential in the standard cone, see the Appendix, of the projection of complexes $(L,D)$ onto $(V, PD)$).
Let us restore our framework. Assume as above that $L$ is a Lie superalgebra with $D$ being a derivation, and that $V$ is an Abelian subalgebra. The Lie bracket in $L$ induces an odd bracket in $\Pi
L$: $$\label{eqbrackpixpiy}
\{\Pi x, \Pi y\}=\Pi[x,y](-1)^{\tilde x}.$$ Is it possible to extend this to a bracket on the whole of $\Pi
L\oplus V$?
There exist an odd binary bracket on $\Pi L\oplus V$ extending that on $\Pi L$ such that the operator acts as a derivation. It is given by the formulae $$\begin{aligned}
\{\Pi x, a\}&=P[x,a], \label{eqbrackpixa}\\
\{a, b\}&= P[Da,b]\label{eqbrackab}\end{aligned}$$ for arbitrary $x\in L$ and $a,b\in V$.
As a starting point we use formula for the bracket on $\Pi L$, where $\Pi x$ and $\Pi y$ are considered as elements of $\Pi L\oplus V$. Apply $d$ given by to $\{\Pi x,\Pi y\}$ and require that the Leibniz rule be satisfied: $$\label{eqleibpixpiy}
d\{\Pi x,\Pi y\}=-\{d\Pi x,\Pi y\}-(-1)^{{{\tilde x}}+1}\{\Pi x,d\Pi y\}$$ for all $x,y\in L$ (notice that the parity in ‘sits’ at the opening bracket, hence the signs). Expanding $d$ by , so that $d\Pi x=-\Pi Dx+Px$, and treating the brackets between elements of $\Pi L$ and $V$ as unknown, we find that the failure of for $x=y$ and ${{\tilde x}}=1$ is the difference $\{Px,\Pi x\}-P[Px,x]$. Replacing $Px$ by an arbitrary element of $V$, we arrive at the above definition . Now assume and require the Leibniz rule for this new bracket: $$\label{eqleibpixa}
d\{\Pi x,b\}=-\{d\Pi x,b\}+(-1)^{{{\tilde x}}}\{\Pi x,db\}$$ for all $x\in L$, $b\in V$. Here $d\Pi x=-\Pi Dx+Px$, $da=PDa$, and we treat the bracket in $V$ as unknown. The failure of equals $\{Px,b\}+(-1)^{{{\tilde x}}}P\left[Px,Db\right]$. Denoting $Px=a\in V$, we arrive at the formula $\{a,b\}=-(-1)^{{{\tilde a}}}P[a,Db]$ or, equivalently, $$\{a,b\}=P[Da,b]$$ as the necessary and sufficient condition of . This is our derived bracket for $k=2$. The Leibniz rule for $\{a,b\}$ is now satisfied automatically and does not bring any new relations.
Defining the bracket by formula is a sufficient condition for . A more detailed analysis shows that is also necessary at least when $x\in K$. Hence the condition that the operator acts as a derivation defines the bracket in an essentially unique way.
One can see that a binary bracket defined in this way on $\Pi
L\oplus V$ will not satisfy the Jacobi identity exactly, thus giving rise to a ternary bracket, and so on. Define the higher brackets on $\Pi L\oplus V$ as follows: $$\begin{aligned}
\{\Pi x, a_1,\ldots,a_n\}&=P[\ldots[x,a_1],\ldots,a_n], \label{eqbrackxaaa}\\
\{a_1,\ldots,a_n\}&=P[\ldots[Da_1,a_2],\ldots,a_n],\label{eqbrackaaa}\end{aligned}$$ where $n{\geqslant}1$. As an unary bracket take the differential , and set the $0$-ary bracket to zero. All the other brackets except those obtainable by symmetry, are defined to be zero. Formulae , directly extend , to many arguments, and formula is the familiar higher derived bracket on $V$ for all $k$.
\[thmcocyl\] The set of brackets , and , together with , define on the space $\Pi L\oplus V$ the structure of an $L_{\infty}$-algebra.
We shall prove that all the brackets – satisfy all the generalized Jacobi identities. Consider the Jacobiator $J^n$ in $\Pi
L\oplus V$ with $n$ arbitrary arguments. Without loss of generality we can assume that each of the arguments is either in $\Pi L$ or $V$. We claim that there can be no non-trivial Jacobiators with more than $3$ arguments in $\Pi L$. Indeed, $J^n$ is a sum of terms of the form $$\bigl\{\{\underbrace{\_,\_,\_}_{k}\},\underbrace{\_,\_,\_,\_}_{l}\bigr\}$$ where $k+l=n$ and $k{\geqslant}1$. If there occur $4$ elements of $\Pi L$ or more, then among those $k$ or $l$ arguments there must be at least $2$ in $\Pi L$, and it should be exactly $k=2$ and $l=2$, since there are no brackets involving $3$ arguments in $\Pi L$. Then the internal bracket also takes values in $\Pi L$, hence we get $3$ arguments in $\Pi L$ for the external bracket, so it must vanish. Consider the Jacobiators that contain exactly $3$ arguments from $\Pi L$. By a similar analysis one can see that the only potential non-vanishing Jacobiator is for $n=3$, which is exactly the Jacobiator in $\Pi L$ and it vanishes since $L$ is a Lie superalgebra. This leaves the Jacobiators with exactly $1$ or $2$ arguments in $\Pi L$. (The Jacobiators with all arguments in $V$ vanish by Theorem \[thmjac\] applied to $D$ such that $D^2=0$.) They are as follows: $$\begin{gathered}
\label{eqjacone1}
J^{p+1}\left(\Pi x,a_1,\ldots,a_p\right)=\\
\sum_{k=1}^p\sum_{\text{$(k,p-k)$-shuffles}}
(-1)^{{{\tilde x}}+1}(-1)^{\e(\s;a_1,\ldots,a_p)}\,
\left\{\Pi
x,\{a_{\s(1)},\ldots,a_{\s(k)}\},a_{\s(k+1)},\ldots,a_{\s(p)}\right\}+\\
\sum_{k=0}^p\sum_{\text{$(k,p-k)$-shuffles}}
(-1)^{\e(\s;a_1,\ldots,a_p)}\,
\left\{\{\Pi x,a_{\s(1)},\ldots,a_{\s(k)}\},a_{\s(k+1)},\ldots,a_{\s(p)}\right\}\end{gathered}$$ and $$\begin{gathered}
\label{eqjactwo1}
J^{p+2}(\Pi x,\Pi y,a_1,\ldots,a_p)=
\left\{\{\Pi x,\Pi y\},a_1,\ldots,a_p\right\}+\\
\sum_{k=0}^p\sum_{\text{$(k,p-k)$-shuffles}}\!\!\!(-1)^{\e(\s;a_1,\ldots,a_p)}
\Biggl(
(-1)^{{{\tilde x}}+1}\left\{\Pi x,\{\Pi y,a_{\s_(1)},\ldots,a_{\s(k)}\},
a_{\s(k+1)},\ldots,a_{\s(p)}\right\} \\
+(-1)^{({{\tilde y}}+1)({{\tilde x}}+{{\tilde a}}_{\s(1)}+\ldots+{{\tilde a}}_{\s(k)})}
\left\{\Pi y, \{\Pi x,a_{\s_(1)},\ldots,a_{\s(k)}\},
a_{\s(k+1)},\ldots,a_{\s(p)}\right\}
\Biggr).\end{gathered}$$ Here $x,y\in L$, $a_i\in V$. By $(-1)^{\e(\s;a_1,\ldots,a_p)}$ we denoted the sign arising from the action of a permutation $\s$ on the product of $p$ commuting homogeneous variables of parities ${{\tilde a}}_1,\ldots,{{\tilde a}}_p$. The equalities $J^{p+1}=0$ and $J^{p+2}=0$ can be informally perceived, respectively, as expressing the fact that taking a bracket with $\Pi x$ acts, in a sense, as a derivation, and that taking a bracket with $\{\Pi x, \Pi y\}$ acts, in a sense, as the commutator of brackets with $\Pi x$ and with $\Pi y$. (All this in a generalized sense, involving partitions and shuffles). Hence these equalities are intuitively plausible. Let us prove them. For this sake consider $x=y$ and $a_i={{\xi}}$ for all $i$, where ${{\tilde x}}=1$, $\tilde{{\xi}}=0$. Then and reduce to $$\begin{gathered}
\label{eqjacone2}
J^{p+1}(\Pi x,{{\xi}}):=J^{p+1}(\Pi x,\underbrace{{{\xi}},\ldots,{{\xi}}}_{p})=\\
\sum_{k=1}^pC_p^k\bigl\{\Pi x, \{\underbrace{{{\xi}},\ldots,{{\xi}}}_k\},
\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\}+
\sum_{k=0}^pC_p^k\bigl\{\{\Pi x, \underbrace{{{\xi}},\ldots,{{\xi}}}_k\},
\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\}\end{gathered}$$ and $$\begin{gathered}
\label{eqjactwo2}
J^{p+2}(\Pi x, {{\xi}}):=J^{p+2}(\Pi x,\Pi x, \underbrace{{{\xi}},\ldots,{{\xi}}}_{p})=\\
\bigl\{\{\Pi x,\Pi x\},\underbrace{{{\xi}},\ldots,{{\xi}}}_p\bigr\}+
2 \sum_{k=0}^p C_p^k
\bigl\{\Pi x,\{\Pi x,\underbrace{{{\xi}},\ldots,{{\xi}}}_k\},
\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\},\end{gathered}$$ respectively. Here $C_p^k$ denotes the binomial coefficient. Substituting the definitions of the brackets in , we get after a simplification $$\begin{gathered}
J^{p+1}(\Pi x,{{\xi}})=\{-\Pi Dx+Px,\underbrace{{{\xi}},\ldots,{{\xi}}}_{p}\,\}+\\
\sum_{k=1}^pC_p^k \Bigl(\bigl\{\Pi x, P[\ldots[D{{\xi}},
\underbrace{{{\xi}}],\ldots,{{\xi}}]}_{k-1},\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\}
+
\bigl\{P[\ldots[[x,\underbrace{{{\xi}}],{{\xi}}],\ldots,{{\xi}}]}_{k},
\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\}\Bigr)=\\
(-1)^{p+1}P(\operatorname{ad}{{\xi}})^pDx +(-1)^pP(\operatorname{ad}{{\xi}})^pDPx+\\
\sum_{k=1}^pC_p^k \Bigl(
P(-1)^{p-1}(\operatorname{ad}{{\xi}})^{p-k}\left[x,P(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}\right]
+P(-1)^{p}(\operatorname{ad}{{\xi}})^{p-k}DP(\operatorname{ad}{{\xi}})^kx \Bigr)\end{gathered}$$ or $$\begin{gathered}
(-1)^pJ^{p+1}(\Pi x,{{\xi}})=
-P(\operatorname{ad}{{\xi}})^pDx + P(\operatorname{ad}{{\xi}})^pDPx+\\
\sum_{k=1}^pC_p^k \Bigl(-
P(\operatorname{ad}{{\xi}})^{p-k}\left[x,P(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}\right] +P
(\operatorname{ad}{{\xi}})^{p-k}DP(\operatorname{ad}{{\xi}})^kx \Bigr)\,.\end{gathered}$$ Using the identity $(\operatorname{ad}{{\xi}})^kDP=-[(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}},P(\,.\,)]$, for $k{\geqslant}1$ (see the proof of Theorem \[thmjac\]), we can re-write this as $$\begin{gathered}
\label{eqjacone3}
(-1)^pJ^{p+1}(\Pi x,{{\xi}})=P\Biggl(
- (\operatorname{ad}{{\xi}})^pDx -[(\operatorname{ad}{{\xi}})^{p-1}D{{\xi}},Px]+\\
\sum_{k=1}^{p-1}C_p^k \Bigl(- \left[(\operatorname{ad}{{\xi}})^{p-k}
x,P(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}\right] -
\left[(\operatorname{ad}{{\xi}})^{p-k-1}D{{\xi}},P(\operatorname{ad}{{\xi}})^kx\right] \Bigr)\\
-\left[x,P(\operatorname{ad}{{\xi}})^{p-1}D{{\xi}}\right]+DP(\operatorname{ad}{{\xi}})^px\Biggr)=\\
- P(\operatorname{ad}{{\xi}})^pDx +PD(\operatorname{ad}{{\xi}})^px-P[(\operatorname{ad}{{\xi}})^{p-1}D{{\xi}},x]-
\sum_{k=1}^{p-1}C_p^k P\left[(\operatorname{ad}{{\xi}})^{p-k}
x,(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}\right]=\\
P\left[D,(\operatorname{ad}{{\xi}})^p\right]x-
\sum_{k=1}^{p}C_p^k P\left[(\operatorname{ad}{{\xi}})^{p-k} x,(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}\right]
$$ where we used identities and . Now, by arguing in the same way as we did when deducing the expression for the commutator of $D$ and $(\operatorname{ad}{{\xi}})^{N}$ acting on $D{{\xi}}$ in the proof of Theorem \[thmjac\], we can deduce the equality $$\begin{gathered}
\left[D,(\operatorname{ad}{{\xi}})^{p}\right]x=
\sum_{r=0}^{p-1}C_{p}^{p-1-r}\left[(\operatorname{ad}{{\xi}})^{r}D{{\xi}},
(\operatorname{ad}{{\xi}})^{p-1-r}x\right]=\\
\sum_{k=1}^{p}C_{p}^{k}\left[(\operatorname{ad}{{\xi}})^{k-1}D{{\xi}}, (\operatorname{ad}{{\xi}})^{p-k}x\right].\end{gathered}$$ Notice that since $x$ is odd, ${{\xi}}$ is even, and $D$ is odd, in the Lie bracket above both arguments are odd, so the order is irrelevant. We immediately see that the two terms in the last line of cancel, and thus for all $x$ and ${{\xi}}$ $$J^{p+1}(\Pi x,{{\xi}})=0,$$ as desired. Now consider $J^{p+2}(\Pi x,{{\xi}})$. Substituting the definitions of the brackets into , we get $$\begin{gathered}
J^{p+2}(\Pi x,x)=
-\bigl\{ \Pi [x,x],\underbrace{{{\xi}},\ldots,{{\xi}}}_p\bigr\}+
2 \sum_{k=0}^p C_p^k
\bigl\{\Pi x,P(-\operatorname{ad}{{\xi}})^{k}x,
\underbrace{{{\xi}},\ldots,{{\xi}}}_{p-k}\bigr\}=\\
-(-1)^pP(\operatorname{ad}{{\xi}})^p[x,x]+
2(-1)^p\sum_{k=0}^pC_p^kP(\operatorname{ad}{{\xi}})^{p-k}\left[x,P(\operatorname{ad}{{\xi}})^kx\right],\end{gathered}$$ or $$\begin{gathered}
(-1)^{p+1} J^{p+2}(\Pi x,x)=
P(\operatorname{ad}{{\xi}})^p[x,x]-
2 \sum_{k=0}^pC_p^kP(\operatorname{ad}{{\xi}})^{p-k}\left[x,P(\operatorname{ad}{{\xi}})^kx\right]=\\
P(\operatorname{ad}{{\xi}})^p[x,x]-
2 \sum_{k=0}^pC_p^kP\left[(\operatorname{ad}{{\xi}})^{p-k}x,P(\operatorname{ad}{{\xi}})^kx\right]=\\
P(\operatorname{ad}{{\xi}})^p[x,x]-P\sum_{k=0}^pC_p^k\left[(\operatorname{ad}{{\xi}})^{p-k}x,(\operatorname{ad}{{\xi}})^kx\right]=\\
P(\operatorname{ad}{{\xi}})^p[x,x]-P(\operatorname{ad}{{\xi}})^p[x,x]=0,\end{gathered}$$ where we used the commutativity of $V$ and identity . Thus for all $x$ and ${{\xi}}$ $$J^{p+2}(\Pi x,{{\xi}})=0,$$ as desired. This completes the proof of the theorem.
A remarkable fact about the formulae for the brackets in $\Pi
L\oplus V$ is that they arise naturally if one wants to extend the bracket in $\Pi L$ keeping the differential a derivation. Of course, the crucial and much harder thing is to prove that they indeed give the structure of an $L_{\infty}$-algebra as stated by Theorem \[thmcocyl\]. The subspace $V$ is a subalgebra (even an ideal) with respect to this structure, and the induced brackets are exactly the higher derived brackets.
\[corcone\] The complex $\Pi L\oplus V$, with operations defined as above, is a cocylinder for $i\co \Pi K\to \Pi L$ in the category of $L_{\infty}$-algebras. $V$ with the higher derived brackets of $D$ is a homotopy fiber (or a cocone), in this category, for the inclusion of differential Lie superalgebras.
The idea of relating the higher derived brackets of $\D$ with homotopical algebra was proposed by the referee of the first version of [@tv:higherder]. He conjectured, for the ${{\mathbb Z}}$-graded case, an interpretation of these brackets in terms of a ‘homotopy left cone’ (cocone, in our terminology) and suggested a formula of type for the extended brackets. In this section we showed that the conjecture about a homotopical-algebraic interpretation of higher derived brackets is correct, in the natural setup where the brackets are generated by an arbitrary odd derivation $D$. Corollary \[corcone\] gives the precise statement.
The considerations of this section give an alternative and quite unexpected, viewpoint of higher derived brackets. For a given derivation $D$, which is assumed to be a differential, the construction of the complex $\Pi L\oplus V$, viewed as a cone (for $L\to V$) or a cocylinder (for $\Pi K\to \Pi L$) is canonical. The higher derived brackets of $D$ appear as an answer to the question of how to extend the algebra structure to $\Pi L\oplus V$ from $L$.
Notice also that although homological or homotopical algebra requires $D^2=0$ from the start, we never directly used this identity in the proof of Theorem \[thmcocyl\], except where we referred to Theorem \[thmjac\] in the particular case when $D^2=0$; hence it seems reasonable that the homotopical-algebraic picture can be rephrased in a way allowing to incorporate a possibly non-zero $D^2$.
Generalizations and Discussion
==============================
Let us return to Theorem \[thmjac\] and see what information can be extracted from it if one does not immediately set $D^2$ equal to zero. To be able to make a precise statement, notice that our construction of higher derived brackets allows extension of scalars, in the following sense.
Consider an arbitrary commutative superalgebra ${{\Lambda}}$ with unit (a good example is the Grassmann algebra with $N$ generators, ${{\Lambda}}={{\Lambda}}_N$) and the tensor product $L\otimes {{\Lambda}}$. It is a Lie superalgebra over ${{\Lambda}}$, and we have $\operatorname{Der}_{{{\Lambda}}}(L\otimes
{{\Lambda}})=(\operatorname{Der}L)\otimes {{\Lambda}}$. Thus the higher derived brackets can be constructed from $D\in \operatorname{Der}_{{{\Lambda}}}(L\otimes {{\Lambda}})$, i.e., a derivation with coefficients in ${{\Lambda}}$. They will be operations on $V\otimes
{{\Lambda}}$. (In particular, brackets generated by $D\in \operatorname{Der}L$ can be considered on $V\otimes {{\Lambda}}$ for any ${{\Lambda}}$ and this explains why it is sufficient to check the Jacobiators only on even arguments.) Clearly, Theorem \[thmjac\] remains valid. Now, the map which assigns to a derivation $D$ all its higher derived brackets is a linear operation in the sense that it commutes with sums and with multiplication by scalars. Now we shall make use of the following obvious algebraic statement: *if a linear map of Lie superalgebras maps the squares of odd elements to squares, for all extensions of scalars by various ${{\Lambda}}$, then it is a Lie algebra homomorphism.* (Indeed, by polarization, it maps all brackets of odd elements to the brackets; then by using suitable odd constants, even elements can be turned into odd, and after that the constants can be eliminated.)
An arbitrary sequence of multilinear symmetric operations on $V$ can be encoded in a (formal) vector field $X$, which serves as their generating function, so that the operations are obtained as the higher derived brackets of $X$: $$\{u_1,\ldots,u_k\}_X=\left[\ldots\left[X,u_1\right],\ldots,u_k\right](0)$$ where $u_i\in V$, $X\in\operatorname{Vect}V$, as in . If we restrict ourselves to formal vector fields, this correspondence will be one-to-one. The sequence of the Jacobiators of the brackets derived from $X$ has the vector field $X^2$ as the generating function (this is a very special case of Theorem \[thmjac\], but can be seen directly).
Consider now an arbitrary derivation $D\co L\to L$. Denote the vector field on $V$ corresponding to the higher derived brackets of $D$, by $Q_D$. Theorem \[thmjac\] then can be re-formulated as the equality $$\label{eqqdd}
(Q_D)^2=Q_{D^2}$$ for all odd $D$. Having in mind the above remarks, we see that Theorem \[thmjac\] is equivalent to the following.
The correspondence $D\mapsto Q_D$ is a homomorphism of Lie superalgebras $\operatorname{Der}L\to \operatorname{Vect}V$, i.e., $$\label{eqqd1d2}
[Q_{D_1}, Q_{D_2}]=Q_{[D_1,D_2]}$$ for all $D_1, D_2\in \operatorname{Der}L$.
(It is an interesting question whether there is a more direct way of constructing a vector field on $V$ from the following data: the homological field specifying the Lie bracket in $L$ and a derivation $D$.)
Let $\mathfrak g$ be a Lie superalgebra and $V$ a vector space. We call the space $V$ a *generalized $L_{\infty}$-algebra over $\mathfrak g$* (or: a *$\mathfrak g$-parametric $L_{\infty}$-algebra*) if there is given a homomorphism $\mathfrak
g\to \operatorname{Vect}V$. We can visualize this as (sequences of) brackets in $V$ parametrized by elements of $\mathfrak g$. Relations between elements of $\mathfrak g$ give rise to ‘generalized Jacobi identities’ in $V$ between the corresponding brackets.
If $\mathfrak g$ has dimension $0|1$, with a single odd basis element $Q$ satisfying $Q^2=0$, then we get a usual $L_{\infty}$-algebra structure.
If $\mathfrak g$ has dimension $1|1$, with a basis $H,Q$ with $H$ even, $Q$ odd, satisfying $Q^2=H$, then a generalized $L_{\infty}$-algebra over $\mathfrak g$ is the same as an arbitrary sequence of odd symmetric brackets that a priori are not subject to any relations. (In fact, there are some relations that are always satisfied, they are the ‘mixed’ Jacobi identities for odd brackets and their Jacobiators, corresponding to the identity $[H,Q]=0$.)
Apart from these two opposite extremes there should be other interesting examples.
Another attractive direction is to study the higher derived brackets where $V$ in the decomposition $L=K\oplus V$ is not assumed Abelian. Notice that this is exactly the case in the original definition of a (binary) derived bracket: given a Lie superalgebra $L$ and an odd derivation $D\co L\to L$, then for arbitrary $a,b\in
L$ $$\label{eqderived}
[a,b]_D:=[Da,b]$$ (we use a sign convention convenient for the comparison with ). This is a particular case of for $k=2$ if $L=V$ and $K=0$. It is known that the derived bracket is not, in general, symmetric: $$\label{eqdersym}
[a,b]_D-(-1)^{{{\tilde a}}{{\tilde b}}}[b,a]_D=D[a,b]$$ (in typical applications it is possible to restrict to an Abelian subalgebra, thus restoring symmetry and making it into a different special case of for $k=2$ with a ‘hidden’ $P$).
In the context of $L=K\oplus V$ where $V$ is not necessarily Abelian, the $k$-th derived brackets defined by satisfy the identity $$\begin{gathered}
\label{eqddersym}
\{a_1,\ldots,a_i,a_{i+1},\ldots,a_k\}_D-(-1)^{{{\tilde a}}_i{{\tilde a}}_{i+1}}
\{a_1,\ldots,a_{i+1},a_i,\ldots,a_k\}_D=\\
\{a_1,\ldots,[a_i,a_{i+1}],\ldots,a_k\}_D\end{gathered}$$ for the transposition of two adjacent arguments, for all $a_1,\ldots,a_k\in V$ and all $i=1,\ldots,k-1$. Here on the right-hand side we have the $(k-1)$-th derived bracket with the Lie bracket of the arguments $a_i$ and $a_{i+1}$ inserted at the $i$-th position.
The proof is not hard and we omit it. Formula generalizes .
It is known that the classical derived bracket, though not symmetric, satisfies the Jacobi identity, defining an odd Loday algebra if $D^2=0$. What about analogs for higher derived brackets? What is the precise list of relations in an algebraic structure defined by the higher derived brackets if $V$ is non-Abelian? (It includes an even Lie bracket as well as a sequence of odd brackets and may be called an ‘$L_{\infty}$-algebra on a Lie algebra background’.) It may be possible to make use of a homotopic-algebraic approach such as in Section \[sechomot\]. We hope to consider these questions elsewhere.
Appendix. Standard cylinders and cocylinders
============================================
Here we collect, for reference purposes, the formulae for the standard constructions of cylinders and cocylinders of chain maps (compare, e.g., [@sgelman]). They all originate in topological constructions of the cylinder $X\times I$ and cocylinder $X^I$.
A *complex* is a (${{\mathbb Z_{2}}}$-graded) vector space equipped with an odd operator $d$ such that $d^2=0$. A *map* or a ‘chain map’ is an even linear map commuting with $d$.
Let $f\co X\to Y$ be a map of complexes.
The standard *cylinder* diagram for $f\co X\to Y$ is the commutative diagram $$\begin{diagram}[small]
X & & \rTo^f & & Y \\
& \rdTo_j & & \ruTo_p & \\
& & \operatorname{Cyl}f & & \\
\end{diagram}$$ where $$\operatorname{Cyl}f=X\oplus \Pi X \oplus Y$$ with the differential given by $$d(x_1,x_2,\Pi y)=(dx_1-x_2, \Pi (-dx_2), dy+f(x_2)).$$ The maps $j$ and $p$ are given by the formulae $$\begin{aligned}
j(x)&=(x,0,0)\\
p(x_1,\Pi x_2,y)&=f(x_1)+y,\end{aligned}$$ and $p$ is a quasi-isomorphism with a quasi-inverse map $i\co Y\to
\operatorname{Cyl}f$, $i(y)=(0,0,y)$. The *cone* of $f$ is the cofiber of $j$, i.e., $\operatorname{Cyl}f/j(X)$. Hence $$\operatorname{Con}f=\Pi X\oplus Y$$ with the differential $$d(\Pi x,y)=(\Pi (-dx), dy+f(x)).$$
In a similar way, the standard *cocylinder* diagram for $f\co
X\to Y$ is the commutative diagram $$\begin{diagram}[small]
X & & \rTo^f & & Y \\
& \rdTo_j & & \ruTo_p & \\
& & \operatorname{Cocyl}f & & \\
\end{diagram}$$ where $$\operatorname{Cocyl}f=X\oplus Y \oplus \Pi Y$$ with the differential given by $$d(x,y_1,\Pi y_2)=(dx, dy_1, \Pi (f(x)-y_1-dy_2)).$$ The maps $j$ and $p$ are given by the formulae $$\begin{aligned}
j(x)&=(x,f(x),0)\\
p(x,y_1,\Pi y_2)&=y_1,\end{aligned}$$ and $j$ is a quasi-isomorphism with a quasi-inverse map $q\co \operatorname{Cocyl}f\to X$, $q(x,y_1,\Pi y_2)=x$. The *cocone* of $f$ is the fiber (kernel) of $p$. Hence $$\operatorname{Cocon}f=X\oplus \Pi Y$$ with the differential $$d(x,\Pi y)=(dx, \Pi (f(x)-dy)).$$
It follows that $\Pi \operatorname{Con}f=\operatorname{Con}f^{\Pi}=\operatorname{Cocon}(-f)$; i.e., up to a sign, the cone and cocone of a chain map $f$ are related by the parity shift functor. In the main text, the complex $L\oplus \Pi V$ appearing there as a cocylinder of the inclusion of complexes $i\co
K\to L$, can be alternatively viewed as the canonical $\operatorname{Cocon}(-P)$ or as $\Pi \operatorname{Con}P$ where the projector $P$ is treated as a map $L\to
V$, so $V$ with the differential $PD$ is considered as a quotient complex of $L$ (rather than a subspace of $L$).
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Dr Theodore Voronov\
School of Mathematics\
University of Manchester\
Sackville Street\
Manchester M60 1QD\
United Kingdom\
`[email protected]`
\[lastpage\]
|
---
abstract: '[In this work, we pursue further consequences of a general formalism for non-covariant gauges developed in an earlier work (hep-th/0205042). We carry out further analysis of the additional restrictions on renormalizations noted in that work. We use the example of the axial gauge $ A_{3}=0 $. We find that if multiplicative renormalization together with ghost-decoupling is to hold, the “prescription-term” (that defines a prescription) cannot be chosen arbitrarily but has to satisfy certain non-trivial conditions (over and above those implied by the validity of power counting) arising from the WT identitites associated with the residual gauge invariance. We also give a restricted class of solutions to these conditions. ]{}'
author:
- '[Satish D. Joglekar]{}[^1]'
title: 'Additional considerations in the definition and renormalization of non-covariant gauges'
---
Department of Physics, I.I.T. Kanpur, Kanpur 208016 [\[]{}INDIA[\]]{}
The Yang-Mills theory in gauges other than the Lorentz gauges have been a subject of wide research [@bns; @l; @1]. These gauges have been used in a variety of Standard Model calculations and in formal arguments in gauge theories [@bns; @l] (as well as in string theories). As compared to the covariant gauges, these gauges have, however, not been fully developed [@dev]. Recently, an approach that gives the *definition* of non-covariant gauges in a Lagrangian path-integral formulation, which moreover is compatible with the Lorentz gauges by construction, has been given [@jm00] and exploited [@jm99] in various contexts such as those related to the axial, planar and the Coulomb gauges. A general path-integral framework, suggested by these results, that attempts to treat all these gauges formally but *rigorously* and hopefully *completely* ( i.e. including the treatment of all their problems) was presented recently [@j02]. Several new observations regarding these gauges were recently made from such a framework by simple and direct considerations [@j02]. This work presents further results regarding the nature of the renormalization in axial gauges based on this formulation and the results in [@j02].
It was suggested in [@j02] that many of the ways of defining non-covariant gauges including the one based on ref. [@jm00] can be formulated as a special case of the path-integral[^2] $$\label{ii}
W[J,K,\overline{K};\xi ,\overline{\xi }]=\int D\phi \; exp\{iS_{eff}[A,c,\overline{c},\psi ]+\varepsilon O[\phi ]+source-terms\}$$ obtained by including *an* $ \epsilon $-term[^3]. We recognize that in order that (\[ii\]) is mathematically well-defined, this $ \epsilon $-term must, in particular, break the residual gauge invariance[^4] completely. In addition, to keep the discussion general enough and to cover many of the ways suggested for dealing with these gauges, we do not necessarily limit $ \epsilon $ to have dimension two in the following, nor do we restrict $ O $ to have local nature[^5].
We note that the various prescriptions, say the Leibbrandt-Mandelstam (L-M) prescription for the light-cone gauges and the CPV for axial gauges etc, can be understood[^6] as special cases of (\[ii\]) [\[]{}with rather complicated nonlocal $ O $[\]]{} and thus the following discussion should include these as special cases (For more details, see ref. [@prep]). Generally, the axial poles are treated by giving a way of interpreting the poles. They amount to replacing the naive propagator (with $ \lambda \rightarrow 0 $)$$\label{nai}
\frac{-i}{k^{2}}\left[ g_{\mu \nu }-\frac{k_{\mu }\eta _{\nu} +k_{\nu }\eta _{\mu }}{\eta .k}+\frac{(\eta ^{2}+\lambda k^{2})k_{\mu }k_{\nu }}{(\eta .k)^{2}}\right]$$ [\[]{}that is obtained from the action by inverting the quadratic form in it for $ k^{2}\neq 0;\; \eta .k\neq 0 $[\]]{} by a modified propagator valid for all $ k $. The latter, in turn, can be obtained by inverting the quadratic form in a modified action in $ A $, that formally differs from the original action by quadratic $ O(\varepsilon A^{2}) $ terms [@prep]; where $ \epsilon $ is a parameter appearing in the pole prescription.
In this work, we wish to elaborate on one of the essential new observation made in [@j02] and to bring out further the power of that observation and to show that it leads to new conclusions. This observation pertains to the fact that a careful treatment of the renormalization of gauge theories, formulated by the path-integral in (\[ii\]), ought also to take an account the import of the extra relations that follow from the presence of the residual gauge invariance as formulated by the IRGT[^7] WT-identities in [@j02]. These were formulated in [@j02] using a generalized version of infinitesimal residual gauge invariance in the BRS space[^8]. In this work, we wish to draw attention to several observations using these. We will show, in particular, that a prescription (such as those one considers commonly [@bns; @l]), given by any *fixed*[^9] *$ \epsilon O $* term, may lead to the IRGT WT-identities that are not compatible with the expected form of renormalization together with ghost decoupling. We shall also show that, if renormalization (in its expected form) with a given $ \varepsilon O $ is possible[^10] at all, we may generally *need renormalization of the prescription term* and this possibility, moreover, is not always necessarily consistent with the IRGT WT-identities. **Later in this work, we shall formulate the conditions on $ O $ under which the latter interpretation becomes possible. As we shall later see, this observation does not look surprising when seen in the light of the present framework, where as suggested in [@j02], we may be required to deal with the *entire action $ S_{eff}+\varepsilon \int d^{4}xO $, including the symmetry breaking term $ \varepsilon O $* while discussing renormalization*.* An obvious question at this point would be why one needs to care about the renormalization of the $ \epsilon $-term at all, if we are going to take the limit $ \epsilon \rightarrow 0 $ in the answer. This is suggested by the role of the $ \epsilon $-term and the observations made in [@j02] regarding it. In particular, we wish to draw attention to the fact that the limit $ \epsilon \rightarrow 0 $ in (\[ii\]) is highly nontrivial as putting $ \epsilon =0 $ in it leads to an ill-defined path-integral leading to very many unacceptable consequences [@j02]. We will elaborate on it further at a later stage.
We shall illustrate this point with the help of the axial gauge $ A^{\alpha }_{3}=0 $. Consider the following set of defining properties and/or assumptions:
1. $ A_{3}=0 $: spatial axial gauge
2. Multiplicative renormalization of the type:$$\label{mulren}
A_{3}=\widetilde{Z}^{1/2}A^{R}_{3};\; \; \; \; A_{\mu }=Z^{1/2}_{3}A^{R}_{\mu };\; \; \; \mu =0,1,2\; \; \; g=Z_{1}g^{R}$$ leading to renormalized Green’s functions that are finite and well-defined in all momentum domains.
3. Ghost decoupling: so that we may assume that the ghost action can be taken as $$\label{gh}
S_{gh}\equiv \int d^{4}x\{-\overline{c}^{\alpha }\partial _{3}c^{\alpha }+i\varepsilon \overline{c}^{\alpha }c^{\alpha }\}$$
4. Path-integral formulation of axial gauges with a prescription for the gauge propagator poles implemented by a *fixed* quadratic term of the form[^11] $$\label{pres}
-i\varepsilon \int d^{4}xO[A]=-i\varepsilon \int d^{4}x\int d^{4}yA_{\mu }^{\alpha }(x)a^{\mu \nu }(x,y)A_{\nu }^{\alpha }(y)$$ where $ O[A] $ is (generally) a nonlocal operator.
In the following, we shall first show that the above set is not necessarily compatible *unless certain additional restrictions, (spelt out later) are satisfied by $ O[A] $.*
We shall implement the $ A_{3}=0 $ gauge by the use of the Nakanishi-Lautrup “b”- field. This method has also been used in the early literature on axial gauges by Kummer [@k75]. We write the generating functional of Green’s functions of the gauge field as $$\label{i}
W[J]=\int DADbDcD\overline{c}\exp \left\{ iS_{eff}+\varepsilon \int d^{4}xO[A]+i\int d^{4}x\; JA\right\}$$ where$$S_{eff}=S_{0}-\int d^{4}x\; b^{\alpha }A_{3}^{\alpha }+S_{gh}$$ We note that $$\label{gf}
\int Db\exp \left\{ -i\int d^{4}xb^{\alpha }A_{3}^{\alpha }\right\} \sim \prod _{\alpha ,x}\delta (A_{3}^{\alpha }(x))$$ has been used in dropping the $ A_{3} $-dependence in the ghost-action (\[gh\]). We may do the same in $ O[A] $ and assume that it has no $ A_{3} $-dependence.
We now consider the following infinitesimal transformations based on the residual gauge-invariance of the action (without the $ \epsilon $-term) (following [@j02], we call them the IRGT) with $ \theta ^{\alpha }=\theta ^{\alpha }(x_{0},x_{1},x_{2}) $$$\begin{aligned}
A_{\mu }^{\alpha }(x)\rightarrow A_{\mu }^{\alpha }(x)+\partial _{\mu }\theta ^{\alpha }-gf^{\alpha \beta \gamma }A_{\mu }^{\beta }(x)\theta ^{\gamma }(x) & & \nonumber \\
X^{\alpha }(x)\rightarrow X^{\alpha }(x)-\; gf^{\alpha \beta \gamma }X^{\beta }(x)\theta ^{\gamma }(x) & & \nonumber \\
X\equiv A_{3},b,c,\overline{c,}\, \partial _{3}c & \label{irgt} \end{aligned}$$ We note that under this IRGT, $ S_{eff} $ and $ \overline{c}c $ are invariant. We, now, carry out the IRGT in $ W $ of (\[i\]) and equate the change to zero. We thus obtain,$$\label{rwt1}
<<\int d^{4}x\left\{ J_{\mu }^{\alpha }(x)D^{\alpha \gamma }_{\mu }-i\varepsilon \Delta O^{\gamma }[A]\right\} \theta ^{\gamma }(x)>>\; =\; 0$$ where we have expressed the change in $ O $ under (\[irgt\]) as$$\label{do}
\int d^{4}x\; O\rightarrow \int d^{4}x\; O+\int d^{4}x\; \Delta O^{\gamma }\theta ^{\gamma }(x)$$ and we have defined, for any X,$$\label{db}
<<X>>\equiv \int DADbDcD\overline{c}X\exp \left\{ iS_{eff}+\varepsilon \int d^{4}xO[A]+i\int d^{4}x\; JA\right\}$$ In view of the fact that $ \theta ^{\gamma }=\theta ^{\gamma }(x_{0},x_{1},x_{2}) $ can be varied arbitrarily, we find that (\[rwt1\]) leads us to,$$\begin{aligned}
0\; =\; <<\int dx_{3}\left\{ D^{\alpha \gamma }_{\mu }J^{\gamma \mu }(x)+i\varepsilon \Delta O^{\alpha }[A]\right\} >> & & \nonumber \\
=\; <<\int dx_{3}\left\{ \sum _{\mu \neq 3}[\partial ^{\mu }J_{\mu }^{\alpha }(x)+gf^{\alpha \beta \gamma }J_{\mu }^{\beta }(x)A^{\gamma \mu }(x)]+i\varepsilon \Delta O^{\alpha }[A]\right\} >> & & \nonumber \\
=\; \int dx_{3}\left\{ \sum _{\mu \neq 3}\left[ \partial ^{\mu }J_{\mu }^{\alpha }(x)W[J]-igf^{\alpha \beta \gamma }J_{\mu }^{\beta }(x)\frac{\delta W[J]}{\delta J_{\mu }^{\gamma }(x)}\right] +i\varepsilon <<\Delta O^{\alpha }[A]>>\right\} & & \label{rwt} \end{aligned}$$ In the above, we have dropped the term $ \sim A_{3} $ using the $ \delta - $function in (\[gf\]). We remark that, as emphasized in [@j02], the last term can have a finite limit as $ \epsilon \rightarrow 0 $ (even in tree approximation) and its presence cannot just be ignored.
The above identity is over and above the *usual formal* BRST-WT identity (in which no account of the $ \epsilon $-term is taken) and as pointed out in [@j02], the renormalization has to be compatible (or made compatible) with it. We now discuss, in the light of (\[rwt\]), various possibilities regarding the pole prescription treatment . Before proceeding, we shall note that
1. If $ O[A] $ is a local quadratic term $ \sim A_{\mu }^{\alpha }(x)A^{\alpha \mu }(x) $ then $ \Delta O^{\alpha }[A]\sim \sum _{\mu \neq 3}\partial ^{\mu }A_{\mu }^{\alpha } $ is linear in A. We further note that under the assumption of the multiplicative renormalization, $ Z_{3}^{-1/2}\Delta O^{\alpha }[A] $ is a finite operator.
2. If $ \int d^{4}xO[A] $ is a non-local quadratic term $ \int d^{4}x\int d^{4}y\; A_{\mu }^{\alpha }(x)a^{\mu \nu }(x,y)A_{\nu }^{\alpha }(y) $ then$$\Delta O^{\alpha }[A]=2\int d^{4}y\left\{ -\partial _{x}^{\mu }a^{\mu \nu }(x,y)A_{\nu }^{\alpha }(y)+gf^{\alpha \beta \gamma }A_{\mu }^{\beta }(x)a^{\mu \nu }(x,y)A_{\nu }^{\gamma }(y)\right\}$$ and has two terms: One is linear in A and the other is quadratic in A and is moreover a composite operator. We express this, in obvious notations, as $ \Delta O[A]\equiv \Delta _{1}O+\Delta _{2}O $.
**A SPECTATOR PRESCRIPTION TERM**
It is usually assumed [@bns; @l]that the prescription for treating the axial gauge propagator is unaffected by renormalization and so is “$ \epsilon $”. Thus, in this case, we are effectively assuming that the term $ \epsilon O[A] $ is unaffected during the renormalization process. We shall call this case the “ spectator prescription term”.
In case one above of a local quadratic $ O[A] $, the renormalizations of each of the three terms in (\[rwt\]) has been assumed to be multiplicative with scales: Z$ _{3}^{-1/2} $; Z$ _{1} $ and Z$ _{3}^{1/2} $. These would be compatible only if $ Z_{1}=1=Z_{3} $. This would, of course, contradict a non-trivial value for $ \beta $-function which is (expected to be) gauge-independent and hence must be the same as the Lorentz gauges.
The discussion for the case 2 above, is a special case of the discussion given below for the “ renormalized prescription term” and we shall see that it is required that $ O[A] $ must satisfy certain constraints. More comments are made later.
**RENORMALIZED PRESCRIPTION TERM**
We shall now explore, however, another (and a more general) possibility in which the (\[rwt\]) is made consistent with renormalization. We shall not insist on keeping the $ \epsilon $-term fixed in form, but allow it to be modified under the renormalization process. Thus we are allowing for a “renormalization of prescription”. We shall now explore the restrictions on $ O $, under which this is possible. We assume that renormalization replaces the $ \varepsilon O[A] $ term by[^12] say $ \varepsilon \{O[A]+\widetilde{O[A]\}} $ (where $ \widetilde{O[A]} $ depends on the regularization parameter). We need not any further treat $ \epsilon $ as a parameter that can be rescaled, as the definition of $ \widetilde{O[A]} $ can absorb effects of such a scaling. The (\[rwt\]) then is replaced by the renormalized version of the (\[rwt\]), viz.$$\label{renrwt}
\int dx_{3}\left\{ \sum _{\mu \neq 3}\left[ \partial ^{\mu }J_{\mu }^{\alpha }(x)W[J;\varepsilon ]-igf^{\alpha \beta \gamma }J_{\mu }^{\beta }(x)\frac{\delta W[J;\varepsilon ]}{\delta J_{\mu }^{\gamma }(x)}\right] +i\varepsilon <<\Delta O^{\alpha }[A]+\Delta \widetilde{O}^{\alpha }[A]>>\right\}$$ Further analysis of (\[renrwt\]) will have to be carried out under a restricted but “reasonable” set of assumptions spelt out later in various places. First of all, we shall assume that $ O[A] $ is of net dimension two. We shall write, in obvious notations, $ \Delta \widetilde{O[A]}\equiv \Delta _{1}\widetilde{O[A]}+\Delta _{2}\widetilde{O[A]} $; where the two pieces are respectively linear and quadratic in A[^13]. We multiply the identity by $ Z^{1/2}_{3} $ and express the equation in terms of the renormalized quantities[^14]:$$\begin{aligned}
\int dx_{3}\sum _{\mu \neq 3}\left[ \partial ^{\mu }J_{\mu }^{R\alpha }(x)W^{R}[J^{R};\varepsilon ]-iZ_{1}Z^{1/2}_{3}g^{R}f^{\alpha \beta \gamma }J_{\mu }^{R\beta }(x)\frac{\delta W^{R}[J^{R};\varepsilon ]}{\delta J_{\mu }^{R\gamma }(x)}\right] & & \nonumber \\
=-i\varepsilon Z_{3}\int dx_{3}<<\Delta _{1}O^{\alpha }[A^{R}]+\Delta _{1}\widetilde{O}^{\alpha }[A^{R}]>> & & \nonumber \label{rwt11} \\
-i\varepsilon Z^{1/2}_{3}\int dx_{3}<<\Delta _{2}O^{\alpha }[A]+\Delta _{2}\widetilde{O}^{\alpha }[A]>> & \label{rwt11} \end{aligned}$$ Let us now discuss the above equation in the 1-loop approximation. We express $ Z_{3}=1+z_{3} $ etc. and look at the divergent part of (\[rwt11\]). We find,$$\begin{aligned}
& i(z_{1}+\frac{1}{2}z_{3})g^{R}f^{\alpha \beta \gamma }J_{\mu }^{R\beta }(x)\frac{\delta W^{R}[J^{R};\varepsilon ]}{\delta J_{\mu }^{R\gamma }(x)} & \nonumber \\
= & i\varepsilon \int dx_{3}<<z_{3}\Delta _{1}O^{\alpha }[A^{R}]+\Delta _{1}\widetilde{O}^{\alpha }[A^{R}]>> & \nonumber \\
& +i\varepsilon \int dx_{3}<<\Delta _{2}O^{\alpha }[A]>>^{div}_{i}A^{R}_{i}++i\varepsilon \frac{z_{3}}{2}\int dx_{3}\Delta _{2}O^{\alpha }[A^{R}]W^{R}[J^{R}] & \nonumber \\
& +i\varepsilon \int dx_{3}<<\Delta _{2}O^{\alpha }[A]>>^{div}_{mn}A^{R}_{m}A^{R}_{n}+i\varepsilon \int dx_{3}<<\Delta \widetilde{_{2}O}^{\alpha }[A]>> & \label{rwt12} \end{aligned}$$ where we have expressed (in obvious notations) the linear and the quadratic terms in $ <<\Delta _{2}O[A]>>^{div} $ in one loop approximation[^15]. We note that the usual BRST WT-identities, which hold when one stays away from external momenta satisfying $ k.\eta =0 $, imply that, should the multiplicative renormalization as postulated be possible, we have $$\label{z1z3}
z_{1}+\frac{1}{2}z_{3}=0$$ We now compare the $ O[A] $ and $ O[A^{2}] $ terms on both sides :It leads us to two constraints:$$\label{con1}
0=\int dx_{3}<<z_{3}\Delta _{1}O^{\alpha R}[A]+\Delta _{1}\widetilde{O}^{\alpha R}[A]>>+\int dx_{3}<<\Delta _{2}O^{\alpha }[A]>>^{div}_{i}A^{R}_{i}$$ $$\label{con2}
0=\int dx_{3}\left[ \frac{1}{2}z_{3}\Delta _{2}O^{\alpha }[A^{R}]W[0]+<<\Delta _{2}O^{\alpha }[A]>>^{div}_{mn}A^{R}_{m}A^{R}_{n}+<<\Delta _{2}\widetilde{O}^{\alpha }[A^{R}]>>\right]$$ These constraints determine the unknowns $ \Delta _{1}\widetilde{O}^{\alpha }[A^{R}] $ and $ \Delta _{2}\widetilde{O}^{\alpha }[A^{R}] $. In addition, there is the requirement that these can be written as the IRGT variation of some $ \int d^{4}x\widetilde{O}[A] $. Moreover, this term $ \varepsilon \int d^{4}x\widetilde{O}[A] $ when added to the action should make, say, the inverse propagator $ \Gamma _{\mu \nu }(k,\varepsilon ) $ in 1-loop finite. If there is a solution to these conditions, then only one can interpret this as the “renormalization of prescription”.
To summarize, up-to 1-loop order, the IRGT WT-identity can be made consistent with renormalization in the assumed form by “ renormalization of prescription” if :
[\[]{}1[\]]{} There exists an $ \int d^{4}x\widetilde{O}[A] $, such that its IRGT variation $ \Delta \widetilde{O} $ can be expressed as $ \Delta \widetilde{O[A]}\equiv \Delta _{1}\widetilde{O[A]}+\Delta _{2}\widetilde{O[A]} $ ; where $ \Delta _{1}\widetilde{O}^{\alpha }[A^{R}] $ and $ \Delta _{2}\widetilde{O}^{\alpha }[A^{R}] $ satisfy the constraints (\[con1\]) and (\[con2\]) ;
[\[]{}2[\]]{} The counterterm $ \varepsilon \int d^{4}x\widetilde{O}[A] $ makes $ \Gamma _{\mu \nu }(k,\varepsilon ) $ finite;
[\[]{}3[\]]{} The usual power counting holds to this order for the renormalization of *nonlocal* operator $ \Delta _{2}O^{\alpha }[A] $.
(These spell out the sufficient conditions).
Finally, we note that the case 2 of a “spectator prescription term” is a special case of the above discussion with $ \Delta \widetilde{O} $ deleted. Thus, in this case, it is necessary that (\[con1\]) and (\[con2\]) hold with the terms $ \widetilde{\Delta _{1}O} $ and $ \widetilde{\Delta _{2}O} $ deleted.
We add some conclusions that follow from an analysis of the above conditions. The analysis of these conditions shows that:
1. Let us suppose that the (arbitrary) function $ a_{\mu \nu }(x-y)=a_{\nu \mu }(y-x) $ in (\[pres\]) be such that the power counting in momentum space *in terms of the external momenta* holds for the one-loop diagrams contributing to $ <<\Delta _{2}O^{\alpha }[A]>>_{i} $ and $ <<\Delta _{2}O^{\alpha }[A]>>_{mn} $ in the sense that the divergence in the first is a monomial in $ p $ of degree 1; and that in latter a monomial of degree zero. [\[]{}Note: $ \int d^{4}x\Delta _{2}O^{\alpha }[A]\equiv 0 $[\]]{}.
2. Then, in momentum space, $ <<\Delta _{2}O^{\alpha }[A]>>^{div}_{i} $ is of the form $ p_{\mu }\Delta ^{\mu \nu } $ (with $ \Delta ^{\mu \nu } $ a *constant* matrix); and the divergence *from the one-loop diagram* contributing to $ <<\Delta _{2}O^{\alpha }[A]>>^{div}_{mn} $ vanishes on account of the fact that $ \int d^{4}x\Delta _{2}O^{\alpha }[A]\equiv 0 $ for any $ a_{\mu \nu } $.
3. In such a case, a solution to the conditions (\[con1\]) and (\[con2\]) exists provided $ \Delta ^{\mu \nu } $is symmentric for ($ \mu $,$ \nu $= 0,1,2) and is given by,$$\label{sol}
\int d^{4}x\widetilde{O[A]}=-z_{3}\int d^{4}xO[A]+\frac{1}{2}\int d^{4}x\Delta '^{\mu \nu }A^{\alpha }_{\mu }A^{\alpha \mu }$$ where $ \Delta '_{\mu \nu }=\Delta _{\mu \nu } $ for all $ (\mu ,\nu ) $ except that $ \Delta '_{3i}=\Delta _{i3};i=0,1,2 $ .
4. If we further assume that the divergence in $ \Gamma _{\mu \nu }(p,\varepsilon ) $ proportional to $ \epsilon $ is , by the assumed validity of power counting in terms of external momenta, also a constant, and therefore independent of momentum $ p $, then the condition [\[]{}2[\]]{} above is also satisfied by this solution.
The above solution (\[sol\]) has been given under certain conditions sufficient for its existence. The main restriction on $ O $ seems to come from (1) the requirement of power-counting as enumerated above; and (2) the symmetry requirement on $ \Delta ^{\mu \nu } $ mentioned above in 2.
We shall note further that while the present analysis has arrived at its results using a specific form of path-integral definition of non-covariant gauges of (\[i\]), we expect an equivalent set of conclusions should follow from any other way of defining these gauges. This formalism has enabled us to see the existence of and to arrive at these conclusions in a easy and direct manner. No such analysis seems to have been carried out in the context of attempts at defining the non-covariant gauges [@bns; @l] in other ways.
**A QUALITATIVE EXPLANATION**
We shall now explain the results qualitatively. Consider the inverse propagator $ \Gamma _{\mu \nu } $ for the gauge-field in one loop approximation. There is a contribution to the $ \epsilon $-dependent terms to this order. For momenta $ k $, such that $ k.\eta \neq 0 $, the $ \epsilon $ terms as a whole are negligible (as $ \epsilon \rightarrow 0 $). In this sector, the usual multiplicative renormalization does the job of making the inverse propagator finite, if $ \epsilon $-terms are ignored. Nonetheless, in the *3-dimensional* subspace $ k.\eta =0 $, the quantity $ k^{\mu }\Gamma _{\mu \nu }k^{\nu } $ obtained by taking the longitudinal projection of $ \Gamma $ has only $ \epsilon $-terms remaining **(and the inverse of which tends to infinity as $ \epsilon \rightarrow 0 $*). These also receive divergences ; which need not generally be removed by the field-renormalization.* (Recall that there was no such subspace in the case of Lorentz gauge that needs to be worried about). One may be required to perform an extra renormalization on the $ \epsilon $-term (This may have to be checked in each case).
At this point, one may ask the justifiable question, as to whether the renormalization of the $ \epsilon $-term should matter at all, since we mean to take the $ \epsilon $ to zero in the end!. Earlier, we have already made some comments based on [@j02]. In addition, we recall that there are several examples [@bns; @l] where the change of prescription has altered (1) the nature and the presence of divergences (2) value of gauge-invariant quantities[^16]. This makes us strongly suspect that this sector in momentum space is important enough.
Now, $ S_{eff} $ is invariant under IRGT. Any prescription breaks the residual gauge invariance in a particular manner. It is not obvious that the physical quantities so calculated using it will be gauge-independent. This is controlled by the behavior of the path-integral under infinitesimal residual gauge transformations as formulated by IRGT WT-identities[^17]. Under IRGT, the path-integral changes solely due to the “symmetry breaking” term $ \epsilon O $ in addition to the source term. The form of divergence in the variation in the source term is restricted by the *assumptions* we made in the beginning. This restriction then becomes imposed on the divergences that can arise from the variation of the $ \epsilon $-term via IRGT WT-identity (and such terms can have non-vanishing contributions as $ \epsilon \rightarrow 0 $ [@j02]). These are additional restrictions on $ O $, and it not *a priori* obvious that they will be obeyed.
**SUMMARY AND CONCLUSIONS**
We shall now summarize our conclusions. We considered the formalism for non-covariant gauges presented in [@j02], where the “prescription” is imposed via an $ \varepsilon \int d^{4}xO[A] $ term added to the action. We found this formalism lead us in an easy manner to an additional consideration that is required in the definition and renormalization of these gauges. We illustrated this for the $ A_{3}=0 $ gauge. This fact, which was brought out in [@j02], has been further elaborated and analyzed here. We see that the usual expectations of multiplicative renormalization together with ghost decoupling are not automatically compatible with every prescription term $ \varepsilon \int d^{4}xO[A] $; there are additional constraints that have to be satisfied further by it (which are implied by the IRGT WT-identities). We also pointed out the need to have to deal with renormalization of $ \epsilon $-terms carefully. These considerations do not seem to have been taken into account so far in attempts to define noncovariant gauges.
**ACKNOWLEDGMENT**
I would like to acknowledge support from Department of Science and Technology, India via grant for the project No. DST/PHY/19990170.
[10]{} A. Bassetto, G. Nardelli, and R. Soldati, Yang-Mills Theories in Algebraic Non-covariant Gauges (World Scientific, Singapore, 1991) and references therein. G. Leibbrandt, Non-covariant Gauges (World Scientific, Singapore, 1994) and references therein. See also, Physical and non-standard gauges (Springer Verlag 1990)P.Gaigg, W.Kummer and M.Schweda (Editors) See references 5-9 for some recent works; also see the references therein. See e.g. references in [\[]{}1,2[\]]{} as well as ref. [\[]{} 8,9[\]]{} and those therein. S. D. Joglekar, and A. Misra, Int. J. Mod. Phys.A 15, 1453 (2000); Erratum *ibid*A15, 3899(2000); S. D. Joglekar, and A. Misra, J. Math. Phys. 41, 1755-1767 (2000). S. D. Joglekar, and A. Misra, Mod. Phys. Lett. A14, 2083 (1999); ***ibid*** A15, 541-546 (2000); Int. J. Mod. Phys.A 16, 3731 (2001); S. D. Joglekar, Mod. Phys. Lett. A15, 245-252 (2000); Int. J. Mod. Phys.A 16, 5043 (2001); S. D. Joglekar, and B. P. Mandal, Int. J. Mod. Phys.A 17, 1279 (2002). S.D.Joglekar “Some Observations on Non-covariant Gauges and the $ \epsilon $-term”- hep-th/0205045 ;Mod.Phys. Lett.A 17, 2581 (2002) L. Baulieu, and D. Zwanziger, Nucl. Phys. B548, 527-562 (1999) . G. Leibbrandt, Nucl. Phys. Proc. Suppl. 90, 19 (2000); G. Heinrich and G. Leibbrandt, Nucl. Phys B575,359 (2000) W. Kummer, Nucl. Phys. B 100, 106 (1976) S.D.Joglekar (in preparation). A. Andrasi and J.C.Taylor, Nucl.Phys. **B310**,222 (1988)
[^1]: email address: *[email protected]*
[^2]: In the following, we use $ \phi $ to generically denote all fields.
[^3]: We may often require an $ \epsilon $-term of the form $ \varepsilon \int d^{4}xO[A,c,\overline{c};\varepsilon ] $; i.e. with an $ \epsilon $-dependent $ O $.
[^4]: A definition of the generalized residual gauge-invariance in the BRS-space has been given in [@j02].
[^5]: We do not however imply that *any* such $ \epsilon $-term will necessarily be appropriate to define a gauge theory *compatible with* the Lorentz gauges. Existence (and construction) of an **$ \epsilon $-term which will serve *this purpose* is already known however. See e.g. [@jm00] and 5$ ^{th} $ of ref. [@jm99].
[^6]: We however note some of the complications in the interpretation of *double* poles in CPV. See e.g. references [@bns; @l].
[^7]: IRGT stands for the abbreviation of “infinitesimal residual gauge transformations” as formulated in [@j02].
[^8]: These, in particular, deal with the Green’s functions with an external momentum in certain non-trivial domains (such as $ \eta .k=0 $ for axial gauges) and their form is generally dependent on the specific “prescription term”. The content of these *is not* covered by the *usual* BRST WT-identities. As shown in [@j02], however, the rigorous BRST WT-identity arising from (\[i\]), that takes into account the $ \epsilon $-term carefully, does cover IRGT WT-identities.
[^9]: As argued later, the usual ways of giving prescription for poles corresponds to the addition of a *fixed* term $ \varepsilon O $ in the action.
[^10]: The renormalization scheme with a particular $ \epsilon O $ could, for example, be obstructed by a lack of validity of usual power counting [@bns; @l]
[^11]: We do not include ghosts in O since we have assumed ghost decoupling in 3 above.
[^12]: With the assumption of ghost-decoupling, $ O[A] $ cannot mix with a $ \overline{c}^{\alpha }c^{\alpha } $ like operators involving ghosts.
[^13]: This amounts to the *assumption* that the usual power counting works for the prescription at hand.
[^14]: We are going to assume that the renormalized Green’s functions are finite functions of $ \epsilon $ for $ \epsilon $ in some interval (0, $ \varepsilon _{0} $). We require this especially since in axial gauges, it has been found that there can be finite contributions to diagrams from $ \varepsilon \bullet \frac{1}{\varepsilon } $type terms (See e.g. Ref. [@at89]). In any case, the $ \epsilon \rightarrow 0 $ limit is to be taken only at the end of the calculation.
[^15]: We are again *making an assumption* that the naive power-counting will work here also. Moreover, note that in evaluating $ <<\Delta _{2}O^{\alpha }[A]>>^{div}_{mn} $ , we need to pay attention to the fact that there are *unrenormalized* coupling and fields in $ \Delta _{2}O^{\alpha }[A] $ that do contribute to the divergence.
[^16]: Here, we recall that two different prescriptions $ \epsilon O $ and $ \epsilon O' $ *may not be related by a residual gauge transformation,* and hence they need not lead to identical physical results. Moreover, neither of these need coincide with the Lorentz gauge result for analogous reasons.
[^17]: As mentioned earlier, these have been shown to be contained in the BRST-identities for the *net* action including the $ \epsilon $-term in [@j02].
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abstract: 'By interpreting the well-known, qualitative criteria for the existence of quantum error correction (QEC) codes by Knill and Laflamme from a quantitative perspective, we propose a figure of merit for assessing a QEC scheme based on the average fidelity between codewords. This enables us to quantify the performance of a protocol as a whole, including errors beyond the correctable set. Various examples are calculated for photonic qubit codes dealing with the experimentally relevant case of photon loss, demonstrating the advantages of the new measure. In the context of continuous-variable QEC, our codeword-overlap measure can be used to reproduce, in a different way with no need for calculating entanglement measures, a previous result concerning the impossibility of improving transmission of Gaussian states through Gaussian channels via Gaussian operations alone.'
author:
- Ricardo Wickert
- 'Peter van Loock$^{1,2,}$'
bibliography:
- 'losbenches.bib'
title: |
Assessing quantum error correction:\
fidelity and entanglement measures with application to photonic codes
---
Introduction
============
Operating a device or scheme in the microscopic domain places formidable demands on the purity and stability of the diverse systems involved, augmented by fundamental limitations of Quantum Mechanics [@NoCloning]. In this context, ingenious mechanisms to avoid the undesirable effects of decoherence play a fundamental role in many areas of Quantum Information Processing (QIP), most notably, enabling large-scale, fault-tolerant quantum computation [@ShorFT] and the communication of quantum bits across large distances [@QREncoding]. Such mechanisms typically rely on two different solutions, namely, those provided by Quantum Error Correction (QEC) [@gottesmanphd] and teleportation [@BennettTeleport] combined with Entanglement Distillation or Entanglement Purification Protocols (EPPs) [@duerbriegel].
As these techniques mature over a multitude of implementations, with many advanced protocols already finding applications in real-world conditions, the necessity to define suitable measures to compare and rank different schemes becomes of uttermost importance. In particular, the need arises to find quantities which can be easily computed for the widening range of experimentally accessible states currently used in QIP. Two main measures, in different variants, currently share widespread acceptance: the entanglement degree is traditionally seen as the golden measure when defining the ’quantumness’ of a channel, and is the benchmarking criterion of choice when evaluating EPPs and quantum memory devices [@Nathan]. Conversely, in the realm of QEC, fidelity measures, in particular, the worst-case fidelity [@KLCriteria] or entanglement fidelity [@schumacher], have historically been employed. The connection between these approaches has been investigated in different regimes [@Bennett; @duerbriegel], but it is not always a straightforward one: Naively, one could expect codes which improve such fidelities to lead to higher safeguarded entanglement, which is indeed true in certain cases; nevertheless, there are examples [@bjork; @DebbieApprox; @EberlyNew] where a code will not safeguard any entanglement, up to and including causing entanglement sudden death (ESD) [@eberly], while still improving input-output or entanglement fidelities under certain conditions. In this contribution, we propose a new measure based on the average fidelity between codewords of a given alphabet, emphasizing that it satisfies certain desirable features. We present initially a motivating argument starting from well-known criteria for the existence of QEC codes, and proceed with examples comparing the entanglement (as measured, in the case of qubits, by the concurrence [@Wooters]) to our average codeword fidelity for a variety of photonic schemes, with particular attention given to codes designed to protect a logical qubit against amplitude damping [@Yamamoto]. In the spirit of [@scottprobfail], we go beyond the analysis traditionally restricted to the correctable error set and establish a comparison method for general schemes. However, here we impose no limitations to a code’s distance or a priori assumptions on the operating regime [@Aschauer] (correction or detection modes [@ashik]). Finally, in the context of continuous-variable QEC with the logical states being infinite-dimensional, we employ our new measure to re-obtain a known result [@CerfNoGo] concerning the impossibility of performing QEC for Gaussian signals and channels when restricted to a Gaussian toolbox.
Error Correction Conditions
===========================
Consider a completely positive and trace-preserving map $\mathcal{E}$ corresponding to a - possibly noisy - transmission/evolution channel. Its action on an arbitrary state $\varrho$ is given by $$\begin{aligned}
\mathcal{E}(\varrho) = \sum_{k} A_{k}^{} \, \varrho \, A_{k}^{\dagger} \; ,
\label{KrausEq}\end{aligned}$$ dubbed the Kraus operator-sum representation [@KrausBook]. Based on this channel decomposition, the Knill-Laflamme (K-L) criteria [@KLCriteria; @BKK] establish that an error correction map $\mathcal{R}$ satisfying $\mathcal{R}\circ\mathcal{E}=\mathcal{I}d$ exists provided it holds that $$\begin{aligned}
\langle \chi_i | A^{\dagger}_{k}
A_{l} | \chi_j \rangle
= \delta(i-j) \lambda_{k,l} \; ,
\label{KLConds}\end{aligned}$$ where $|\chi_i\rangle$ are the codewords from a given input alphabet, and the $\lambda_{k,l}$ define how different errors skew the codespace; in a simplified interpretation, it requires that different codewords remain orthogonal after the action of the channel, and that different errors must effect the same deformation across the input alphabet. Exact satisfiability of the above can only be achieved in certain scenarios [@MintertNew]; at the same time, allowing for small deviations in the orthogonality and deformability requirements - or, alternatively, for $\mathcal{R} \circ \mathcal{E}$ to be close, but not necessarily equal, to the identity [@BenyApprox2; @barnumknill] - enables one to obtain more efficient codes [@DebbieApprox; @BenyApprox]. The discrepancies may be employed to bound the entanglement fidelity obtained with the use of the code [@leungstudent].
Before proceeding, a few notes are in order. First, in achieving the codewords, an encoding step is implicitly assumed, and, for notation purposes, it is condensed together with the transmission map into a single channel [^1]; in the case of no encoding/direct transmission, one can simply regard the encoding as the identity map, $\mathcal{I}d$. Furthermore, we note $\delta(i-j)$ corresponds to the more general case of a continuous alphabet; in the case of qubits (or other discrete-variable codewords), this should correspond to a Kroenecker delta, $\delta_{ij}$.
Motivated by the idea of employing a primarily qualitative criteria as a potentially *quantitative* measure [@QuantitativeWitness], we will explore taking into account violations from the exact satisfiability of Eq. (\[KLConds\]) (instead of neglecting such violations up to a certain order in the channel parameters [@DebbieApprox]). Deviations from Eq. (\[KLConds\]) can be broken down in two qualitatively different types: first, violations of $\delta(i-j)$, which lead different codewords to overlap, and thus reduce the distinguishability between the input alphabet; second, departures from $\lambda_{k,l}$, which in the non-violated case is strictly independent of the codewords, but in the violated case may affect different codewords in unequal manner (and hence may deform a superposition of codewords). Obviously this latter deformation can only take place when the prior effect is also present, but the converse is not necessary [^2]. We’ll dub codes (i.e., encoding together with transmission channels) in which the entire alphabet is affected uniformly as “non-deformable", whereas those with codeword-dependent skewness will be called “deformable".
Non-deformable codes will not preclude the distinguishability to decrease, neither prevent different pairs from suffering varying deformations: the label only guarantees, for any two orthogonal states, the overlap between the original and the resulting states to be the same. In the case of experimentally relevant amplitude damping channels, direct transmission of the $|0\rangle$ and $|1\rangle$ states (taken here as the occupation numbers of a bosonic mode like in so-called single-rail encoding) yields a deformable code: While the vacuum state is unaffected, the single-excitation $|1\rangle$ is taken to $(1-\gamma)|1\rangle\langle1| + \gamma|0\rangle\langle0|$. That the conjugate basis would be uniformly affected does not alter the classification of the code; in fact, for every deformable code there exists at least one superposition in which the deformations are equal. However, considering the dual-rail encoding, $|0\rangle_{L}=|0\rangle|1\rangle$ and $|1\rangle_{L}=|1\rangle|0\rangle$ - and assuming, of course, equal dampening in both rails (modes) - one finds again a non-deformable encoding, as *any* superposition of the codewords will result in an equal, global reduction of the length of the logical state vector. With the above considerations in mind, one could conceive employing the amount by which the criteria in Eq. (\[KLConds\]) have been violated as a measure to rank different channels. We aim at a measure which, instead of classifying codes by the number and kind of errors it is capable of handling, should deliver information precisely about the operators the scheme is not capable of correcting, quantifying just how much it causes the input alphabet to skew and overlap. To this purpose, we construct the *codeword overlap* of a map by considering the output fidelities obtained from a pair of orthogonal input states, and then averaging this quantity over the entire input alphabet: $$\begin{aligned}
\label{fidoverlap}
F^{CW} = \int dQ \operatorname{Tr} \left[\sqrt{\sqrt{\rho_Q} \rho_{\tilde{Q}} \sqrt{\rho_Q}}\right] \; .\end{aligned}$$ Here, $\rho_Q$ and $\rho_{\tilde{Q}}$ correspond to the outputs originating from a pair of orthogonal input states $|Q\rangle$ and $|\tilde{Q}\rangle$, with $|Q\rangle \perp |\tilde{Q}\rangle$. The above loosely corresponds to an integration of Eq. (\[KLConds\]), with two important distinctions: first, since the performance of a code should be basis-independent, we cover the surface of the Bloch sphere, taking each pair of diametrically opposed states as possible codewords; second, instead of acting with the Kraus operators individually, we consider the full channel’s effect on the codewords, which enables it to be employed even in channels were a decomposition in form of Eq. (\[KrausEq\]) is not known. When considering encodings for spaces larger than qubits (*e.g.*, the qudit codes proposed in [@Yamamoto]), one should per – doneform the integration considering opposing states in the surface of the corresponding Bloch hypersphere [@BlochHypersphere].[^3] This, however, precludes a straightforward generalization for continuous-variable (CV) encodings; nevertheless, one may consider only a truncated set, i.e., a qudit encoding where $d \rightarrow \infty$.
Photon-loss qubit codes
=======================
Here we consider the ubiquitous binary logical basis, that is, with the information encoded in orthogonal states $|0\rangle_L$ and $|1\rangle_L$. The transmission of such states via optical fibers corresponds generally to a lossy process, with the fiber absorbing or “losing" photons along its length. This corresponds to a Gaussian channel of particular relevance, dubbed an *amplitude damping* channel. It is characterized by a loss parameter, $\gamma$, and described, in the Kraus representation of Eq. (\[KrausEq\]), by an infinite sum ($k=0...\infty$) of operators of the form $$\begin{aligned}
\label{amplitudedampingkrausoperators}
A_k = \sum_{n=k}^\infty \sqrt{\binom{n}{k}} \sqrt{(1-\gamma)^{n-k} \gamma^k} |n-k\rangle \langle n| \; .\end{aligned}$$
Different encodings have been designed to protect qubits against the errors caused by these operators, and are presented below to illustrate the usage of the new measure. The first code considered is the aforementioned dual-rail encoding, which provides *detection* capabilities only: $$\begin{aligned}
|0\rangle_L = |0 1\rangle \; \mbox{,} \quad |1\rangle_L = |1 0 \rangle \; .\end{aligned}$$ Whenever both modes are found to be in the vacuum state, the information has to be transmitted anew (error detection mode) or, after the decoding step, the output is replaced by the mixed qubit $\rho = \frac{1}{2} |0\rangle\langle0| + \frac{1}{2} |1\rangle\langle1|$ (error correction mode, which will be considered here).
The ubiquitous three-qubit repetition code, $$\begin{aligned}
|0\rangle_L = |0 0 0\rangle \; \mbox{,} \quad |1\rangle_L = |1 1 1 \rangle \; ,\end{aligned}$$ is capable of correcting any single bit-flip error, and can also be employed against amplitude damping errors, assuming photon loss to be a particular case of bit-flip, with a highly asymmetrical behaviour in which only one of the logical states is affected.
Codes can also be constructed by exploring higher occupations of the bosonic modes. For instance, in [@Yamamoto], the authors develop the following encoding, capable of correcting the loss of up to one quanta to the environment: $$\begin{aligned}
|0\rangle_L = \frac{|40\rangle + |04\rangle}{\sqrt{2}} \; \mbox{,} \quad |1\rangle_L = |22 \rangle \; .\end{aligned}$$
Allowing for codes which do not exactly satisfy Eq. (\[KLConds\]) enables one to achieve more economical encodings. One such was proposed by Leung *et al*. [@DebbieApprox], $$\begin{aligned}
|0\rangle_L = \frac{|0000\rangle + |1111\rangle}{\sqrt{2}} \; \mbox{,} \quad
|1\rangle_L = \frac{|0011\rangle + |1100\rangle}{\sqrt{2}} \; ,\end{aligned}$$ which, assuming the damping constant $\gamma$ be kept low, is capable of correcting the loss of one excitation to $O(\gamma^2)$.
Finally, the usage of infinite-dimensional carriers is one that offers numerous advantages in the realm of QIP [@LloydBraunstein; @RalphMilburn]. One may thus consider the encoding of qubits on coherent states, which are innately non-orthogonal: $$\begin{aligned}
|0\rangle_L = |-\alpha\rangle \; \mbox{,} \quad |1\rangle_L = |\; \alpha\rangle \; .\end{aligned}$$ Here, the encoding can be seen as a channel which irreversibly reduces the distinguishability; in this case, a state is no longer orthogonal to the state found on the diametrically opposing point of the Bloch sphere, with the exception of those lying in the equator. Error correction codes have also been developed in this regime, such as the protocol in [@GVR], which effects, up to a normalization, $$\begin{aligned}
|-\alpha\rangle \rightarrow \left( |-\alpha\rangle + |\alpha\rangle \right)^{\otimes N} \; \mbox{,} \;
|\; \alpha\rangle \rightarrow \left( |-\alpha\rangle - |\alpha\rangle \right)^{\otimes N} \; ,
\label{coherentencoding}\end{aligned}$$ capable of correcting $\lfloor \frac{N-1}{2} \rfloor$ amplitude-damping induced phase-flip errors. We note, however, that not only the transmissivity of the channel contributes to this scheme’s performance: also the size of the superposition, which regulates the amount of overlap in the input alphabet, affects the transmission characteristics (see *e.g.* [@OurPaper; @OurFuturePaper]).
We proceed by calculating the average codeword overlap for each of the above encodings and comparing it to a different figure of merit, namely, the amount of entanglement preserved after employing the protocol at hand to transmit one half of a two-qubit maximally entangled state (further details are provided in Appendix A). The measures are computed as a function of the channel loss parameter $\gamma$, in the case of the discrete-variable encodings, or for a fixed channel transmissivity, as a function of the coherent-state superposition size $|\alpha|$.
![(Color online) Codeword overlap (top) and concurrence (bottom), as a function of the damping parameter $\gamma$, for different codes: dual-rail, Eq. (5) (blue), three-qubit repetition, Eq. (6) (red), bosonic, Eq. (7) (green, dashed), and four-qubit, Eq. (8) (black, dotted) [@OurFuturePaper].[]{data-label="fig:worstcase"}](overXeta37 "fig:")\
![(Color online) Codeword overlap (top) and concurrence (bottom), as a function of the damping parameter $\gamma$, for different codes: dual-rail, Eq. (5) (blue), three-qubit repetition, Eq. (6) (red), bosonic, Eq. (7) (green, dashed), and four-qubit, Eq. (8) (black, dotted) [@OurFuturePaper].[]{data-label="fig:worstcase"}](concXeta37 "fig:")\
The results are presented in Figs. 1 and 2. In the former, we observe that the overlaps are zero when the channel approaches perfect transmission: as expected, the orthogonal input codewords remain orthogonal; equally, the entanglement remains maximum. As the losses increase, the overlaps grow; at the same time, the entanglement diminishes. Alone, this fact is highly unsurprising; however, we remark that also the *ordering* established through one measure is reflected on the other. That is to say, the code which offers the best performance, in terms of codeword fidelities, in a given regime, also returns the highest safeguarded entanglement; as the parameters vary, the relative ordering changes as well, and this is observed in both figures of merit. However, the exact point in which a change of ordering occurs is not strictly always the same: between certain encodings, the channel parameters in which a crossing occurs may differ by small amounts. This is particularly noticeable when examining the three-qubit repetition codes, which is highly deformable and suggests that variations may be related to the codeword-dependent skewness.
All of the above suggests a strong link between the overlapping properties of the output alphabet, and the amount of entanglement capable of being safeguarded by means of a given scheme. Nevertheless, we note that no a priori reason exists to suggest that the crossing points should be precisely the same: after all, we are dealing with measures of apparently different character. A further observation is in order: while certain encodings lead to entanglement sudden death [@eberly; @EberlyNew; @bjork], the equivalent catastrophic breakdown in terms of overlaps (average codeword fidelity equalling unity) does not happen in any encoding until the channel becomes fully lossy. This would suggest that the new measure is capable of portraying certain characteristics which would be otherwise lost in an analysis solely based on the entanglement.
\
 \[fig:full066\]
In Fig. 2, we observe the behavior of different coherent-state encodings as a function of the superposition size $|\alpha|$. Again, a good agreement is found between the ordering obtained through concurrence and codeword overlaps. Notable is the difference in the case of direct transmission: while it is possible to obtain a maximally entangled state in the limit of $\alpha \rightarrow 0$, the alphabet as a whole becomes indistinguishable and thus impractical for encoding purposes - a feature reflected by the codeword overlap measure.
Gaussian Error Correction No-Go
===============================
Optical modes of the electromagnetic field, whose quadratures satisfy the canonical commutation relations, provide a natural testbed for a wide range of quantum information concepts [@PeterReview]. Here, logical information is encoded by means of a truly *continuous* logical alphabet; for instance, the codewords defined by the position eigenstates $\{ |x\rangle_L \}$. In this scenario, the so-called Gaussian operations - defined as those that map Gaussian states into Gaussian states - are of great relevance due to the ease in which they can be implemented with current experimental resources.
Nevertheless, the capabilities of the Gaussian set are not without restrictions. Given the fragility of the quantum resources in face of ubiquitous decoherence mechanisms, a significant limitation is that such operations are incapable of distilling higher entanglement from less entangled Gaussian states [@Giedke; @Eisert; @Fiurasek], or, in close relation, unable to protect Gaussian states from the widespread class of Gaussian errors [@CerfNoGo].
Interestingly, Gaussian transformations alone suffice to suppress and correct non-Gaussian errors acting on arbitrary states, but in particular, the Gaussian transformations also suffice when the non-Gaussian errors act upon Gaussian input states [@BraunsteinQEC; @Aoki; @niset2; @PeterNote]; equally worth noting is the fact that a Gaussian error channel, when acting upon specific non-Gaussian input states, can exhibit stochastic, non-Gaussian behaviour [@CMM], however, in this case Gaussian encoding and decoding procedures alone appear to be incapable of correcting such errors, and non-Gaussian operations should be accounted for [@GVR].
In this section, we’ll employ the codeword overlap measure in order to explore what kind of statements it allows us to make in the context of continuous-variable QEC, in particular, in relation to the aforementioned No-Go theorem for all-Gaussian QEC [@CerfNoGo]. Most importantly, such an approach enables us to obtain fundamental insights in a rather distinct way, without the need for calculating an entanglement measure (such as the logarithmic negativity employed in Ref. [@CerfNoGo]). We note that the quantity as developed in Eq. (\[fidoverlap\]) is based on averaging the fidelity between the channel-output states which originate from two orthogonal input states. However, here, in order to include the possibility of non-orthogonal input states as it is typically the case for a Gaussian alphabet, we’ll consider two general states: if the output overlap between two arbitrary states cannot be reduced through QEC, then, in particular, it also holds that the overlap between two orthogonal states (after their channel transmission) will not decrease, and integrating over all possible pairs of states will furthermore not result in a lower quantity, thus establishing the desired result.
The fidelity between two states $\rho_1$ and $\rho_2$ is given, in the characteristic function formalism, by $$\begin{aligned}
F = \frac{1}{\pi^N} \int d^Nx \; \chi_{\rho_1}(x) \; \chi_{\rho_2}(-x) \quad .\end{aligned}$$ For Gaussian states with zero mean, the characteristic function is given by $\chi_{\rho_{i}}=e^{-\frac{1}{2}x^T\sigma_{i}x}$, and we can re-write the above expression as $$\begin{aligned}
F = \frac{1}{\pi^N} \int d^Nx \; e^{-\frac{1}{2} x^T \sigma_1 x} \; e^{-\frac{1}{2} (-x)^T \sigma_2 (-x)} \quad ,\end{aligned}$$ which evaluates to [@Scutaru] $$\begin{aligned}
F = \frac{2}{\sqrt{\Delta + \delta} - \sqrt{\delta}} \quad ,
\label{Forig}\end{aligned}$$ with $\Delta = \det\left(\sigma_1 + \sigma_2\right)$ and $\delta = \left(\det \sigma_1 - 1\right) \left(\det \sigma_2 - 1\right)$. Now, the action of a Gaussian channel on the level of the covariance matrices is given by $\gamma \rightarrow M \gamma M^{T} + N$ (see Appendix B for details). Taking this transformation in acount, the resulting fidelity is $$\begin{aligned}
F^{\prime} = \frac{2}{\sqrt{\Delta^{\prime} + \delta^{\prime}} - \sqrt{\delta^{\prime}}} \quad ,
\label{Fprime}\end{aligned}$$ where $\Delta^{\prime} = \det\left(M\sigma_1M^{T} + M\sigma_2M^{T} + 2N\right)$, and $\delta^{\prime} = \left(\det (M \sigma_1 M^{T} + N) - 1\right) \left(\det (M \sigma_2 M^{T} + N)- 1\right)$.
Now, to establish a relationship between $F$ and $F^{\prime}$, we must evaluate the resulting effect from $M$ and $N$ in the above (for detailed arguments see Appendix C). We are faced with three relevant cases: (i) $|\det M| = 1$, (ii) $|\det M| > 1$ and (iii) $|\det M| > 1$. The first case is trivially evaluated: when $\det N=0$, this corresponds to a Gaussian unitary, which, as expected, yields exactly the original fidelity, *i.e.*, $F^{\prime}=F$. For $\det N > 0$, representing the addition of classical (thermal) noise, one observes the difference between the square roots in Eq. (\[Fprime\]) to diminish, and thus the fidelity to increase. The second case also finds a straightforward solution based on the same argument. The third case, however, contemplates channels which do not necessarily induce a spreading of the Gaussian state, and requires a more careful analysis. In this case, the action of the map results in a “contraction" towards a common state. This is clearly exemplified by the prototypical amplitude damping channel, using two displaced thermal states as inputs: as the loss parameter $\gamma$ increases, the states are gradually attracted towards the vacuum state; however, the increased purity (and correspondingly reduced values of $\det \sigma_i$) plays no role in diminishing the overlaps.
We note furthermore that displacements (*i.e.*, shifts in the first-order moments) bear no effect in the above, simply rescaling $F$ and $F^{\prime}$ by a fixed amount.
One can then conclude that, except when $\mathcal{E}$ corresponds to a symplectic operation, either the additional noise will cause a spreading of the Gaussian state, or the states will be contracted towards a common state. In any case, the fidelity to any other state subject to the same action is bound to increase. When the decoding operation is equally symplectic, one can at best re-obtain the original state (through $\mathcal{R} = \mathcal{E}^{-1}$), assuming, of course, that the encoding operation already had the best choice of unitary operations.
In the above discussion, eventually we considered input states of a Gaussian nature, thus reproducing the known No-Go result for all-Gaussian QEC [@CerfNoGo] in terms of our fidelity-based codeword criterion and independently of entanglement measures. The treatment of Gaussian codeword pairs above is analogous to our earlier treatment for the qubit codes, and the argumentation follows through depending on the assumption of a Gaussian input alphabet (see Appendix C).
Conclusions and Outlook
=======================
We have defined a quantitative measure based indirectly on the amount by which the well-known quantum error correction criteria by Knill and Laflamme are violated, proposing the use of averaged codeword fidelities as a figure of merit to evaluate different quantum protocols. By quantifying how distinguishable originally orthogonal inputs emerge from an error channel, the measure finds a natural interpretation as a translation, into the quantum regime, of the classical coding theory notion of *confusability* of an alphabet [@Shannon; @*Lovasz].
While initially harder to compute, by requiring the evaluation of the fidelity between two generally mixed states, the codeword overlap measure does not require the optimization (minimization) necessary to obtain the worst-case fidelity. At the same time, it accurately depicts the alphabet’s distinguishability behaviour which fails to be portrayed by means of the entanglement fidelity or conventional entanglement measures. We emphasize that properly describing this distinguishability, both in an ideal and in a noisy or lossy quantum channel, becomes particularly important when the ideal (quantum) codewords are already non-orthogonal, as it is often the case when the quantum information carriers are continuous-variable oscillator states.
The employment of the codeword overlap reflects *qualitatively* the behaviour found when quantifying the performance by means of other figures of merit, in particular here the concurrence. In other words, whenever the variation of a certain channel parameter causes one scheme to improve (reduce) the average codeword fidelity in comparison to another scheme, one also observes that the first scheme will result in an improved figure in the safeguarded entanglement. This matching was also found to be *quantitatively* exact for certain choices of codes, however, the precise reasons for such behaviour are hitherto unknown.
We have also employed the new measure to reaffirm the impossibility of improving the transmission of Gaussian states, subject to Gaussian noisy channels, through Gaussian operations alone. In this case, the No-Go result was found independently of an entanglement measure, solely based upon the overlap of Gaussian codeword pairs.
Finally, since the measure is, in principle, accessible and computable in those instances where entanglement is found hard to quantify (*i.e.*, infinite-dimensional non-Gaussian states), we expect these results to shed light in other aspects of (photonic) error correction in particular and quantum information in general.
$\quad$\
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank Carlo Cafaro for comments to the manuscript and acknowledge discussions with Dominik Hörndlein in the early stages of this project. Portions of this work were carried out while R.W. was visiting the Institute for Quantum Computing in Waterloo, Canada. He is grateful for the hospitality and inspiring discussions with the Optical Quantum Communication Theory group. Financial assistance from the “Collaborative Training in Quantum Information Processing" program is gratefully acknowledged. This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG) via its Emmy Noether Program, and by the Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung - BMBF) by means of the HIPERCOM project.
Calculating the concurrence and average codeword fidelity
=========================================================
In obtaining the codeword overlap curves depicted in Figs. (1) and (2), the following procedure is used:
For each encoding, an input qubit $|Q\rangle$ is prepared in the state $\cos \frac{w}{2}|{0}\rangle_L + e^{i \theta}\sin \frac{w}{2}|{1}\rangle_L$; concomitantly, a second state is prepared, $|\tilde{Q}\rangle = \sin \frac{w}{2}|{0}\rangle_L - e^{i \theta} \cos \frac{w}{2}|{1}\rangle_L$, ensuring that $\langle \tilde{Q} | Q \rangle = 0$. Each of those states is then subject to the transmission channel characterized by the operators in Eq. (\[amplitudedampingkrausoperators\]). Followed by each code’s respective decoding procedure, the following outputs are found:
Direct transmission results in $$\begin{aligned}
\label{amplitudedampingrho1}
&\rho_{Q,direct} = \nonumber \\
&\frac{1}{2}
\left(
\begin{array}{cc}
1+ \gamma + \cos w - \gamma \cos w & e^{-i \theta} \sqrt{1-\gamma} \sin w \\
e^{i \theta} \sqrt{1-\gamma} \sin w & (\gamma - 1) (\cos w -1)
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
\label{amplitudedampingrho2}
&\rho_{\tilde{Q},direct} = \nonumber \\
&\frac{1}{2}
\left(
\begin{array}{cc}
1+ \gamma + \cos w + (1- \gamma) \cos w & -e^{-i \theta} \sqrt{1-\gamma} \sin w \\
-e^{i \theta} \sqrt{1-\gamma} \sin w & -(\gamma-1)(\cos w + 1)
\end{array}
\right) \; .\end{aligned}$$ $$\begin{aligned}
&\mbox{To the dual-rail encoding corresponds} \nonumber \\
\label{dr-output1}
& \rho_{Q,dual-rail} = \nonumber \\
& \frac{1}{2}
\left(
\begin{array}{cc}
(1+(-1+\gamma ) \cos w) & e^{-i \theta } (-1+\gamma ) \sin w \\
e^{i \theta } (-1+\gamma ) \sin w & (1+\cos w-\gamma \cos w) \\
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
\label{dr-output2}
& \rho_{\tilde{Q},dual-rail} = \nonumber \\
&\frac{1}{2}
\left(
\begin{array}{cc}
(1+\cos w-\gamma \cos w) & - e^{-i \theta } (-1+\gamma ) \sin w \\
- e^{i \theta } (-1+\gamma ) \sin w & (1+(-1+\gamma ) \cos w) \\
\end{array}
\right) \; .\end{aligned}$$
The 3-qubit code produces $$\begin{aligned}
\label{amplitudedampingrho3}
\rho_{Q,3-qubit} &=
\frac{1}{2}
\left(
\begin{array}{cc}
1+(3-2 p) p^2+(p-1)^2 (1+2 p) \cos w & e^{-i \theta } (1-p)^{3/2} \sin w \\
e^{i \theta } (1-p)^{3/2} \sin w & (p-1)^2 (2+4 p) \sin^2 \frac{w}{2}
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
\label{amplitudedampingrho4}
\rho_{\tilde{Q},3-qubit} &=
\frac{1}{2}
\left(
\begin{array}{cc}
1+(3-2 p) p^2-(p-1)^2 (1+2 p) \cos w & -e^{-i \theta } (1-p)^{3/2} \sin w \\
-e^{i \theta } (1-p)^{3/2} \sin w & (p-1)^2 (1+2 p) (1+\cos w)
\end{array}
\right) \; .\end{aligned}$$
For the bosonic encoding, one finds $$\begin{aligned}
\label{bosonicrho1}
\rho_{Q,bosonic} &=
\frac{1}{2}
\left(
\begin{array}{cc}
1-(\gamma -1)^3 (1+3 \gamma ) \cos w & - e^{-i \theta } (\gamma -1)^3 (1+3 \gamma ) \sin w \\
- e^{i \theta } (\gamma -1)^3 (1+3 \gamma ) \sin w & 1+(\gamma -1)^3 (1+3 \gamma ) \cos w \\
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
\label{bosonicrho2}
\rho_{\tilde{Q},bosonic} &=
\frac{1}{2}
\left(
\begin{array}{cc}
1+(\gamma -1)^3 (1+3 \gamma ) \cos w & e^{-i \theta } (\gamma -1)^3 (1+3 \gamma ) \sin w \\
e^{i \theta } (\gamma -1)^3 (1+3 \gamma ) \sin w & 1-(\gamma -1)^3 (1+3 \gamma ) \cos w \\
\end{array}
\right) \; .\end{aligned}$$
Finally, for the “approximate" encoding, the outputs are $$\begin{aligned}
\label{bjorkrho1}
&\rho_{Q,approximate} = \nonumber \\
&\left(s
\begin{array}{cc}
\frac{1}{2} \left( 1+\gamma ^2 (2 \gamma -1)+(\gamma-1 )^2 (1+2 \gamma ) \cos w \right) & \frac{1}{4} e^{i \theta } \left(\gamma ^2-\gamma ^3+e^{-2 i \theta } \left(2+\gamma ^2 (3 \gamma -5)\right)\right) \sin w \\
\frac{1}{4} e^{-i \theta } \left(\gamma ^2-\gamma ^3+e^{2 i \theta } \left(2+\gamma ^2 (3 \gamma -5)\right)\right) \sin w &
\frac{1}{2} \left( 1+\gamma ^2-2 \gamma ^3-(\gamma -1 )^2 (1+2 \gamma ) \cos w \right) \\
\end{array}
\right)\end{aligned}$$ and $$\begin{aligned}
\label{bjorkrho2}
&\rho_{\tilde{Q},approximate} = \nonumber \\
&\left(
\begin{array}{cc}
\frac{1}{2} \left(1+\gamma ^2 (-1+2 \gamma )-(-1+\gamma )^2 (1+2 \gamma ) \cos w\right) & \frac{1}{4} e^{-i \theta } (-1+\gamma ) \left(2+\gamma
\left(2+\left(-3+e^{2 i \theta }\right) \gamma \right)\right) \sin w \\
\frac{1}{4} e^{-i \theta } \left((-1+\gamma ) \gamma ^2+e^{2 i \theta } \left(-2+(5-3 \gamma ) \gamma ^2\right)\right) \sin w & \frac{1}{2}
\left(1+\gamma ^2-2 \gamma ^3+(-1+\gamma )^2 (1+2 \gamma ) \cos w\right) \\
\end{array}
\right) .\end{aligned}$$
With the above, Eq. (\[fidoverlap\]) is then computed by means of a numeric integration procedure over $w$ and $\theta$. At each value of $\gamma$, approximately 1000 states are employed to produce the average, although for less deformable encodings a smaller sample already resulted in an adequate agreement with the asymptotic behaviour.
For evaluating the entanglement-safeguarding capabilities of a scheme, the maximally entangled state $\frac{1}{\sqrt{2}}\left( |0\rangle |0\rangle_L + |1\rangle |1\rangle_L \right)$ is employed, transmitting the encoded mode through the lossy channel, afterwards followed by the decoding procedures. We note that the decoding operations reduce the encoded state back to a qubit subspace, therefore allowing the entanglement to be computed on a $2\times2$ Hilbert space.
For the entanglement analysis developed in the text, we consider Wooters’ concurrence, which is obtained through $$\label{conceq}
C = \max\{0,\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4} \} \; .$$ Here, $\lambda_i$ are the eigenvalues, in decreasing order, of $\rho \tilde{\rho}$, where $\tilde{\rho} = ( \sigma_{y,1} \otimes \sigma_{y,2} ) \rho^{*} ( \sigma_{y,1} \otimes \sigma_{y,2} )$, and $\sigma_{y,i}$ is the Pauli $Y$ operator in the $i$-th mode.
In the case of coherent-state encodings evaluated in Fig. (2), the input state for both direct transmission and the different encodings is $$|Q\rangle = \frac{1}{\sqrt{N(\alpha)}} ( \sqrt{w} |-\alpha\rangle + e^{i \theta} \sqrt{1-w} |\alpha\rangle ) \quad ,$$ with $0 \geq w \geq 1$, $0 \geq \theta \geq \pi$, and the normalization constant $N(\alpha) = 1 + 2 \cos \theta \sqrt{w(1-w)} e^{-2|\alpha|^2}(ab^*+a^*b)$, required due to the non-orthogonal nature of the coherent-state alphabet. When considering the error correction protocol [@GVR], the deterministic operation of the scheme is considered; in other words, the transformation in Eq. (\[coherentencoding\]) is only obtained for large superposition sizes, with smaller $|\alpha|$ incurring an erroneous component (see ref. [@OurFuturePaper] for details). This induces in a double-tradeoff between the superposition size and the figure of merit being considered: while on the one hand, larger “cat states" are desired - not only due to their inherenty larger distinguishability, but also for the operation of the non-Gaussian Hadamard gates found in the protocol - on the other hand, this effects an increased channel-induced phase-flip probability.
For computing the concurrence in the later case, two-mode maximally entangled states are employed, $$\label{maxentstate}
|\Phi^-\rangle = \frac{1}{\sqrt{2-2e^{-4|\alpha|^2}}} (|\alpha,\alpha\rangle - |-\alpha,-\alpha\rangle) \quad .$$ Again, the first mode is kept while the second is sent through the error-correcting scheme. Finally, for the above calculations, it helps to express the density matrices in terms of an orthogonal basis $\{ |u_{\alpha}\rangle,|v_{\alpha}\rangle \}$, such that $$\begin{aligned}
|\alpha\rangle = \mu_\alpha |u_{\alpha}\rangle& + \nu_\alpha |v_{\alpha}\rangle \\
|-\alpha\rangle = \mu_\alpha |u_{\alpha}\rangle& - \nu_\alpha |v_{\alpha}\rangle \nonumber \\
\mbox{with } \mu_\alpha = \left(\frac{1+e^{-2|\alpha|^2}}{2}\right)^{\frac{1}{2}}& \; \textrm{and} \; \nu_\alpha = \left(\frac{1-e^{-2|\alpha|^2}}{2}\right)^{\frac{1}{2}} \; .\nonumber\end{aligned}$$
Gaussian Formalism
==================
Gaussian States
---------------
Here we briefly review the Gaussian formalism, adopting the convention from [@Giedke].
Given the Weyl operators $$\begin{aligned}
W(x) = e^{-ix^TR},\end{aligned}$$ where $x \in \mathbb{R}^{2n}$ and $R=(X_1,P_1,..,X_n,P_n)^T$, with the commutator relations $[X_j,P_k]=i\delta_{jk}$, a Gaussian state $\rho$ is defined as having a Gaussian characteristic function $\chi_{\rho}(x) = \mbox{tr}[{\rho}W(x)]$. An equivalent definition can be given in terms of the state’s Wigner function, which can be obtained from the characteristic function by a Fourier transform, and is also Gaussian-shaped. One defines the first and second order moments $\textbf{d}$ and $\gamma$, respectively the displacement vector and the covariance matrix (CM), by $d_i = \langle x_i \rangle$ and $\gamma_{ij} = \langle x_i x_j + x_j x_i\rangle - 2 d_i d_j$. In most cases, $\textbf{d}$ can be set to zero without loss of generality. The CM satisfies $\gamma = \gamma^T \geq i J_n$, where we define the symplectic matrix $$\begin{aligned}
J_n = \bigoplus_{k=1}^{n} J_1 \, , \quad \quad J_1 = \left(
\begin{array}{cr}
0 & -1 \\
1 & 0 \end{array} \right) .\end{aligned}$$ The displacement vector and covariance matrix completely determine the state $\rho$, whose density operator can be written as $$\begin{aligned}
\rho = \pi^{-n} \int_{\mathbb{R}^{2n}} dx \, e^{ - \frac{1}{4}x^T \gamma x + i d^{T}x} W(x) \; .
\label{opform}\end{aligned}$$
Of special relevance is the maximally entangled state (MES), corresponding to an infinitely squeezed two-mode squeezed state (TMSS), with CM $$\begin{aligned}
\lim_{r \rightarrow \infty}\gamma(r) = \left(
\begin{array}{cc}
A_r & C_r \\
C_r & A_r \end{array} \right) ,
\label{TMSS}\end{aligned}$$ where $A_r = \cosh r \mathbb{1}$ and $C_r = \sinh r \Lambda$ are both $2n \times 2n$ matrices, and $$\begin{aligned}
\Lambda = \mbox{diag} \left(1, -1, 1, -1, ... , 1, -1\right) .\end{aligned}$$
Gaussian Operations
-------------------
A Gaussian channel is defined as a map $ \mathcal{E} $ taking Gaussian states into Gaussian states, cf. $\rho^{\prime}=\mathcal{E}(\rho)$. Following the Choi-Jamiolkowski isomorphism between completely positive maps and positive operators [@choi], to every Gaussian map $\mathcal{E}$ there corresponds an operator $\hat{E}$, $$\begin{aligned}
\hat{E}_{12} = \lim_{r\rightarrow\infty} \left(\mathcal{E} \otimes \mathbb{1} \right) \left( |\phi \rangle_{12} \langle \phi | \right) \: ,
\label{choieq}\end{aligned}$$ this equation allowing us to re-interpret the transmission of an arbitrary state $\rho$ through $\mathcal{E}$ as a teleportation using $\hat{E}_{12}$ as the entangled resource state, *i.e.*, $$\begin{aligned}
\mathcal{E}(\rho) \propto \mbox{tr}_2[\hat{E}_{12}^{T_2}\rho_2] = \mbox{tr}_{23} (\hat{E}_{12}\rho_{3} |\phi\rangle_{23} \langle \phi | ) .
\label{eqdotraco}\end{aligned}$$ One should note that, since $\mathcal{E}$ maps Gaussian states into Gaussian states, and $|\phi\rangle$ in Eq. (\[choieq\]) can be taken as the limit of a Gaussian state, $\hat{E}$ must itself correspond to a Gaussian operator which, similarly to (\[opform\]), can also be written as $$\begin{aligned}
\hat{E} = \int_{\mathbb{R}^{2n}} dx \, e^{ - \frac{1}{4}x^T \Gamma x + i D^{T}x - C} W(x) ,
\label{opform2}\end{aligned}$$ with appropriately-defined CM $\Gamma$, displacement vector $D$ and a normalization constant $C$. Now, by employing Eq. (\[eqdotraco\]) and replacing $\hat{E}_{12}$ with the operator in Eq. (\[opform2\]), one can obtain the action of $\mathcal{E}$ on a general state, ie, obtain $\gamma^{\prime}$ and $d^{\prime}$ in $\mathcal{E}: \rho_{\gamma,d} \rightarrow \rho_{\gamma^{\prime},d^{\prime}}$ from $\Gamma$ and $D$ [^4].
We now consider the set of operations which can be implemented by augmenting our system with additional (Gaussian) ancillary states, performing Gaussian unitary operations over the whole combined system and discarding (tracing over) the ancillas, thus obtaining the class of Gaussian completely positive trace-preserving (CPTP) maps. The action of such a map on a state with CM $\gamma$ is given by $$\begin{aligned}
\gamma \rightarrow M \gamma M^{T} + N \: ,
\label{MgammaM}\end{aligned}$$ with $M$ real and $N \geq 0$ real and symmetric. We must also have $$\begin{aligned}
\det N \geq (\det M -1 )^2 ,
\label{relNM}\end{aligned}$$ lest the complete-positivity requirement be violated. The Gaussian operator corresponding to this operation has the CM $$\begin{aligned}
\Gamma = \lim_{r \rightarrow \infty} \left(
\begin{array}{cc}
M^{T} A_r M + N & M^{T} C_r \\
C_r M & A_r \end{array} \right) .
\label{MAM}\end{aligned}$$
The solutions to Eq. (\[relNM\]) with $N=0$ (adding no extra noise) and $\det M = 1$ (preserving the sum of areas) are dubbed symplectic transformations. In the quantum-optical context, these are exemplified by unitaries such as squeezers, phase shifters, or lossless beam-splitters. One can easily verify those to be the channels with minimal entanglement degradation [@CerfNoGo], or equally, those preserving the newly-developed codeword overlap measure.
Finally, we note that the identity map, $\mathcal{I}$, is obtained in Eq. (\[MgammaM\]) by taking $M = \mathbb{1}$ and $N=0$ (being thus a symplectic operation), and reduces the expression (\[MAM\]) to the CM of the maximally entangled ($r \rightarrow \infty$) TMSS, Eq. (\[TMSS\]).
Fidelity between Gaussian states
================================
The effect of a Gaussian channel on the level of the covariance matrices is given by $\gamma \rightarrow M \gamma M^{T} + N$. In order to facilitate the treatment of such expression when evaluating fidelities, we note that the above Gaussian channel can be parametrized by means of a singular value decomposition. This gives rise to an equivalent channel characterised by the matrices $$\begin{aligned}
\label{symplecticdecomp}
M' &= S V M U \; \quad \; \mbox{ and} \\
N' &= S V N V^T S \nonumber\end{aligned}$$ Since $U$, $V$ and $S$ are all symplectic, the overlaps between the original and transformed channel remain unaltered. Now, $\det M' = \det M$ (and equally for $N'$ and $N$); furthermore, without loss of generality we can choose $M$ as proportional to the identity, i.e., $M \propto \eta \mathbb{1}$. Doing so greatly simplifies the expression for the fidelity after the Gaussian operations, since one can then basically employ the original determinants, up to a scaling factor.
Now, to establish a relationship between $F$ and $F^{\prime}$, we must evaluate the resulting effect from $M$ and $N$ in the above. We are faced with three relevant cases: (i) $|\det M| = 1$, (ii) $|\det M| > 1$ and (iii) $|\det M| > 1$.
The first case, $|\det M| = 1$, with $\det N=0$ corresponds to a Gaussian unitary and is trivially evaluated, yielding, as expected, exactly the original fidelity
For $\det N > 0$, representing the addition of classical (thermal) noise, and equivalently for the second case, $|\det M| > 1$, with help of the parametrization in Eq. (\[symplecticdecomp\]), one trivially observes the difference between the square roots in Eq. (\[Fprime\]) to diminish, and thus the fidelity to increase. The later holds for quantum-limited maps [@Solomon] where $N$ bounds Eq. (\[relNM\]); and evidently for those cases where a second, (classical) noise channel follows.
The third and final case, however, involves a more elaborate analysis. Using the expression for the symplectic transformation Eq. (\[symplecticdecomp\]) in Eq. (\[Fprime\]), expressing the relevant quantities in terms of the parameter $\eta$ governing $M$ and $N$, and employing the physicality constraints for Gaussian channels (see Eq. (\[relNM\]) in Appendix B), one finds, after long but otherwise straightforward calculations, that $$\begin{aligned}
\label{thebigresult}
F - F^{\prime} > 0 \;\end{aligned}$$ has an empty solution set; and thus, the fidelity is indeed bound to increase.
Finally, in order to fully appreciate that the treatment with codeword pairs of Gaussian states developed in section IV is analogous to our earlier treatment for the qubit codes in section III, and also that the argument follows through depending on the assumption of a Gaussian input alphabet, notice the following. A pair of continuous-variable codewords may be written as $|\psi_1\rangle = \int \, dx \, \psi_1(x)\,|x\rangle_L$ and $|\psi_2\rangle = \int \, dx \, \psi_2(x)\,|x\rangle_L$, similar to our $|Q\rangle$ and $|\tilde Q\rangle$ states for qubits, but with the general Bloch-sphere parameters (App. A) this time replaced by two general wave functions $\psi_1(x)$ and $\psi_2(x)$; the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are also no longer restricted to be orthogonal. Now assuming that the logical states $|x\rangle_L$ that span the continuous codespace are the result of an arbitrary, unitary Gaussian encoding operation acting upon a one-mode basis state $|x\rangle$ together with an arbitrary Gaussian multi-mode ancilla state $|{\rm Gaussian}\rangle$, $|x\rangle_L = \hat U_{\rm Gaussian} \left( |x\rangle\otimes |{\rm Gaussian}\rangle \right)$, we obtain for the two input states, $$\begin{aligned}
|\psi_i\rangle &=& \int \, dx \, \psi_i(x)\,|x\rangle_L \\
&=& \int \, dx \, \psi_i(x)\,\hat U_{\rm Gaussian} \left( |x\rangle\otimes |{\rm Gaussian}\rangle \right)
\nonumber\\
&=& \hat U_{\rm Gaussian} \,\left( \int \, dx \, \psi_i(x) |x\rangle\otimes |{\rm Gaussian}\rangle \right)\,
\nonumber\end{aligned}$$ with $i=1,2$. Under the assumption of two Gaussian wave functions $\psi_1(x)$ and $\psi_2(x)$, it is guaranteed that the resulting state above is again a Gaussian state. However, for two arbitrary wave functions $\psi_1(x)$ and $\psi_2(x)$, the resulting states can be non-Gaussian, and the averaging for obtaining our overlap measure would have to include non-Gaussian codewords as well (corresponding to the more general scenario of arbitrary signal states subject to Gaussian encoding/decoding operations and Gaussian error channels). In this latter case, the above analysis for the Gaussian overlaps would no longer suffice. Therefore, we must assume that $\psi_1(x)$ and $\psi_2(x)$ represent [*any*]{} pair of [*Gaussian*]{} wave functions, and the above arguments follow through. Note that this conclusion is robust against a change of the basis of the original signal state, $|x\rangle\rightarrow |n\rangle$ (with $|n\rangle$ being the photon number basis), in which case $$\begin{aligned}
|\psi_i\rangle &=& \sum_n c_{i,n}\,|n\rangle_L \\
&=&
\sum_n c_{i,n}\,\hat U_{\rm Gaussian}\left( |n\rangle\otimes |{\rm Gaussian}\rangle \right)\nonumber\\
&=&
\hat U_{\rm Gaussian}\left( \sum_n c_{i,n}\, |n\rangle\otimes |{\rm Gaussian}\rangle \right)\,,
\nonumber\end{aligned}$$ and where $\hat U_{\rm Gaussian}\left( |n\rangle\otimes |{\rm Gaussian}\rangle \right)$ is generally a non-Gaussian state, but $\hat U_{\rm Gaussian}\left( \sum_n c_{i,n}\, |n\rangle\otimes |{\rm Gaussian}\rangle \right)$ is a Gaussian state, provided that $|\psi_i\rangle = \sum_n c_{i,n}\,|n\rangle$ is a Gaussian state too.
[^1]: Explicitly, this means that the initial signal Hilbert space is extended, typically by adding a sufficient set of ancilla states and applying a global encoding unitary on the signal-ancillae system. The total system is then subject to a global channel transmission, which is usually composed of the original channel acting individually and independently on the signal and ancilla subsystems. In the optical setting, enlarging the Hilbert space may mean either adding extra modes or allowing for higher photon numbers with the same number of modes. In the latter case, there are no additional channels that must act on auxiliary modes.
[^2]: More precisely, when taking each channel Kraus operator separately like in Eq. (\[KLConds\]), of course each Kraus effect can affect different codewords differently, even though their overlap remains unaffected. An example is the amplitude damping channel acting on a single-rail qubit basis, for which we have $\langle 0 | A^{\dagger}_{0}
A_{0} | 0 \rangle = \lambda_{0,0} = 1$, $\langle 1 | A^{\dagger}_{0}A_{0} | 1 \rangle = \lambda_{0,0} = 1 - \gamma$, but also $\langle 0 | A^{\dagger}_{0}A_{0} | 1 \rangle = 0$ (see later, Eq. (\[amplitudedampingkrausoperators\])). In other words, the length of vector $| 1 \rangle$ is reduced for any non-zero $\gamma$ and that of vector $| 0 \rangle$ remains unity, while they would perfectly maintain their orthogonality. However, note that by taking into account a second Kraus effect, which together with the first Kraus effect forms a trace-preserving qubit channel, the deformation is indeed accompanied by a reduction of the codeword distinguishability: $\langle 0 | A^{\dagger}_{0}A_{1} | 1 \rangle = \sqrt{\gamma}$. Our codeword overlap measure, as defined in Eq. (\[fidoverlap\]) for the whole trace-preserving channel, therefore captures both deformation and non-orthogonality at the same time (see below).
[^3]: The form above is of course not exactly convenient when considering such encodings of higher dimensions. Already in the case of a qutrit ($d=3$) this is easily seen: an arbitrary logical state has the general form $a{\ensuremath{|0\rangle}}_L + b{\ensuremath{|1\rangle}}_L + c{\ensuremath{|2\rangle}}_L$, and a possible codeword is ${\ensuremath{|0\rangle}}_L$, which has infinitely many states orthogonal to it (for instance, any state of the form $b{\ensuremath{|1\rangle}}_L + c{\ensuremath{|2\rangle}}_L$ with $b^2+c^2=1$). Thus, in order to calculate the measure, for every ${\ensuremath{|Q\rangle}}$ state, one has to consider the average fidelity to *each* of the possible ${\ensuremath{|\tilde{Q}\rangle}}$ orthogonal states.
[^4]: The interested reader should follow sections II and III in ref. [@Giedke] for a complete discussion of this procedure.
|
---
abstract: 'We complete Satake’s classification of Kuga fiber varieties by showing that if a representation $\rho$ of a hermitian algebraic group satisfies Satake’s necessary conditions, then some multiple of $\rho$ defines a Kuga fiber variety.'
address: 'Department of Mathematics, Mail Stop 561, East Carolina University, Greenville, NC 27858, USA'
author:
- Salman Abdulali
title: Classification of Kuga fiber varieties
---
Introduction
============
Kuga fiber varieties [@Kuga; @Kuga2018] are families of abelian varieties ${\mathcal A}\to {\mathcal V}$, where ${\mathcal V}= \Gamma \backslash G({\mathbb R})^0 / K$ is an arithmetic variety, and ${\mathcal A}$ is the pullback of the universal family of abelian varieties over a Siegel modular variety. Here, $G$ is a semisimple algebraic group over ${\mathbb Q}$ such that $G({\mathbb R})$ is of hermitian type, $K$ is a maximal compact subgroup of $G({\mathbb R})^0$, and $\Gamma$ is an arithmetic subgroup of $G({\mathbb Q})$. A Kuga fiber variety is constructed from a symplectic representation $\rho \colon G \to Sp(2n, {\mathbb Q})$ which is equivariant with a holomorphic map $\tau \colon X \to {\mathfrak S}_n$, where $X = G({\mathbb R})^0 / K$ is the symmetric domain belonging to $G$, and ${\mathfrak S}_n$ is the Siegel space of degree $n$. Kuga assumed that ${\mathcal V}$ is compact; we do not make this assumption.
Kuga’s original motivation was to prove the Ramanujan conjecture, a goal achieved by Deligne [@Deligne; @DeligneWeil]. Kuga fiber varieties, which include Shimura’s <span style="font-variant:small-caps;">pel</span>-families [@Shimura], have played a central role in the arithmetic theory of automorphic forms [@KugaShimura; @Ohta1; @Ohta2]. These varieties are also key to the study of algebraic cycles on abelian varieties and abelian schemes [@Abdulali1994c; @Abdulali2002; @Gordon1; @Gordon2; @HallKuga; @Kuga1982; @Mumford1969; @Tjiok]; indeed, the concept of the Hodge group (or Mumford-Tate group) of an abelian variety arose in the context of Kuga fiber varieties [@Mumford1966]. Another area in which Kuga fiber varieties play a key role is in the study of $K3$-surfaces, via the Kuga-Satake construction of abelian varieties associated to $K3$-surfaces [@KugaSatake], as in Deligne’s proof of the Weil conjectures for these surfaces [@DeligneK3].
We consider the following problem in this paper: Given an arithmetic variety ${\mathcal V}$, classify all Kuga fiber varieties over it. Equivalently, given the group $G$, find all representations of it into a symplectic group which are equivariant with holomorphic maps of the corresponding symmetric domains. From another point of view, this problem is equivalent to the classification (up to isogeny) of the semisimple parts of the Hodge groups of abelian varieties, together with their action on the first cohomology of the abelian variety. This problem was raised by Kuga [@Kuga] in the 1960’s, and partially answered by Satake [@Satake1964; @Satake1965a; @Satake1965b; @SatakeBoulder; @Satake1966; @Satake1967; @Satakebook; @Satake1997]. Addington [@Addington] completed Satake’s classification for ${\mathbb Q}$-simple groups of type II (orthogonal groups) and type III (symplectic groups). This was partially extended to non-simple groups of type III by Abdulali [@Abdulali1; @Abdulali1988].
Deligne [@Deligne1979] and Milne [@Milne2013] considered this problem from a somewhat different point of view. Their results are similar to those of Satake. Milne states the theorem for all suitable representations of a ${\mathbb Q}$-simple group, though he proves it only in the situations where Satake proved it, and he does not deal fully with non-simple groups. It is important to deal with non-simple groups because the semisimple part of the Hodge group of a simple abelian variety need not be simple. We give an example of such an abelian variety in Section \[example\]. Further examples may be found in Satake [@SatakeBoulder]\*[Remark 2, p. 356]{}, Kuga [@Kuga1984]\*[§5]{}, and Abdulali .
Green, Griffiths, and Kerr [@GreenGriffithsKerr2012; @GreenGriffithsKerr2013] and Patrikis [@Patrikis] have considered the more general problem of classifying the Hodge groups of Hodge structures of higher weight. They completely classify the reductive groups which are Mumford-Tate groups of polarizable Hodge structures of arbitrary weight; however the representations of the groups on the Hodge structures have not been classified.
The key to our classification is a reduction to the *rigid* case, which is much easier. In the proof of our main theorem (Theorem \[maintheorem\]) we reduce the general case to the rigid case, which is proved in Theorem \[rigid\]. The inspiration for this strategy comes from the construction of families of families of abelian varieties by Kuga and Ihara [@KugaIhara], and the related concept of “sharing” in Kuga [@Kuga1984]\*[§5, p. 277]{}.
All representations are finite-dimensional and algebraic. For a finite field extension $E$ of a field $F$, we let $\operatorname{Res}_{E/F}$ be the restriction of scalars functor, from schemes over $E$ to schemes over $F$. For an algebraic or topological group $G$, we denote by $G^0$ the connected component of the identity.
Kuga fiber varieties {#kuga}
====================
In this section we give an overview of the construction of Kuga fiber varieties. Our primary purpose is to fix the notations and terminology; for details we refer to Satake [@Satakebook].
Groups of hermitian type
------------------------
Let $G$ be a group of hermitian type. This means that $G$ is a semisimple real Lie group, and $X := G^0/K$ is a bounded symmetric domain for a maximal compact subgroup $K$ of $G^0$. Denote by $1$ the identity element of $G$. Let ${\mathfrak g}:= \operatorname{Lie}G$ be the Lie algebra of $G$, ${\mathfrak k}:= \operatorname{Lie}K$, and let ${\mathfrak g}= {\mathfrak k}\oplus {\mathfrak p}$ be the corresponding Cartan decomposition. Differentiating the natural map $\nu \colon G^0 \to X$ induces an isomorphism of ${\mathfrak p}$ with $T_o(X)$, the tangent space of $X$ at the base point $o = \nu(1)$, and there exists a unique $H_0 \in Z({\mathfrak k})$, called the *$H$-element* at $o$, such that $\operatorname{ad}H_0 | {\mathfrak p}$ is the complex structure on $T_o (X)$.
Equivariant holomorphic maps
----------------------------
Let $G_1$ and $G_2$ be groups of hermitian type, with symmetric spaces $X_1$ and $X_2$ respectively. Let $H_0$ and $H'_0$ be $H$-elements at base points $o_1 \in X_1$ and $o_2 \in X_2$ respectively. Let $\rho \colon G_1 \to G_2$ be a homomorphism of Lie groups. We say that $\rho$ satisfies the *$H_1$-condition* relative to the $H$-elements $H_0$ and $H'_0$ if $$\label{h1}
[d \rho (H_0) - H'_0, d \rho (g)] = 0 \qquad \text{for all }g \in {\mathfrak g}.$$ The stronger condition $$\label{h2}
d \rho (H_0) = H'_0$$ is called the *$H_2$-condition*. If either of these is satisfied, then there exists a unique holomorphic map $\tau \colon X_1 \to X_2$ such that $\tau (o_1) = o_2$, and the pair $(\rho, \tau)$ is equivariant in the sense that $$\tau (g \cdot x) = \rho(g) \cdot \tau (x) \qquad \text{for all } g \in G^0, \ x \in X.$$ In fact, Clozel [@Clozel] has shown that if $G_2$ has no exceptional factors, then the $H_1$-condition is equivalent to the existence of an equivariant holomorphic map.
The Siegel space
----------------
Let $E$ be a nondegenerate alternating form on a finite-dimensional real vector space $V$. The symplectic group $Sp(V, E)$ is a Lie group of hermitian type; the associated symmetric domain is the *Siegel space* $$\begin{gathered}
{\mathfrak S}(V, E) = \{\, J \in GL(V) \mid J^2 = -I \text{ and } \\
E(x,Jy) \text{ is symmetric, positive definite}\, \}.\end{gathered}$$ $Sp(V, E)$ acts on ${\mathfrak S}(V, E)$ by conjugation. The $H$-element at $J \in {\mathfrak S}(V, E)$ is $J/2$.
\[h2lemma\] Let $G$ be a group of hermitian type with symmetric domain $X$, and let $E$ be a nondegenerate alternating form on a finite-dimensional real vector space $V$. Let $\rho \colon G \to Sp(V, E)$ satisfy the $H_2$-condition with respect to $H$-elements $H_0$ and $H'_0 = J_0/2$ at base points $o \in X$ and $J_0 \in {\mathfrak S}(V, E)$, respectively. Then $J_0 \in \rho(G)$.
Since $J_0$ is a complex structure on $V$, there exists a basis of $V$ with respect to which $J_0 = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$, where $2n = \dim V$. Then, using the $H_2$-condition, we calculate that $J_0 = \exp (\frac{\pi}{2} J_0) = \exp(\pi H'_0) = \exp(d\rho (\pi H_0)) = \rho (\exp(\pi H_0))$.
The fiber varieties
-------------------
We shall say that an algebraic group $G$ over a subfield of ${\mathbb R}$ is of hermitian type if the Lie group $G({\mathbb R})$ is of hermitian type. Now let $G$ be a connected, semisimple algebraic group of hermitian type over ${\mathbb Q}$. Assume that $G$ has no nontrivial, connected, normal subgroup $H$ such that $H({\mathbb R})$ is compact.
Let $E$ be a nondegenerate alternating form on a finite-dimensional rational vector space $V$. The symplectic group $Sp(V, E)$ is then a ${\mathbb Q}$-algebraic group of hermitian type. We write ${\mathfrak S}(V, E)$ for ${\mathfrak S}(V_{{\mathbb R}}, E_{{\mathbb R}})$. Let $\rho \colon G \to Sp(V, E)$ be a representation defined over ${\mathbb Q}$, which satisfies the $H_1$ condition with respect to the $H$-elements $H_0$ and $H'_0 = J/2$. Let $\tau \colon X \to {\mathfrak S}(V, E)$ be the corresponding equivariant holomorphic map. Let $\Gamma$ be a torsion-free arithmetic subgroup of $G({\mathbb Q})$, and $L$ a lattice in $V$ such that $\rho(\Gamma)L = L$. Then the natural map $${\mathcal A}= (\Gamma \ltimes_\rho L) \backslash (X \times V_{{\mathbb R}})
\longrightarrow {\mathcal V}:= \Gamma \backslash X$$ is a morphism of smooth quasiprojective algebraic varieties (Borel [@Borel1972]\*[Theorem 3.10, p. 559]{} and Deligne [@900]\*[p. 74]{}), so that ${\mathcal A}$ is a fiber variety over ${\mathcal V}$ called a *Kuga fiber variety*. The fiber ${\mathcal A}_P$ over any point $P \in {\mathcal V}$ is an abelian variety isomorphic to the torus $V_{{\mathbb R}}/L$ with the complex structure $\tau (x)$, where $x$ is a point in $X$ lying over $P$.
We say that a representation $\rho \colon G \to GL(V)$ defines a Kuga fiber variety if $\rho (G)$ is contained in a symplectic group $Sp(V, E)$, and $\rho$ satisfies the $H_1$-condition with respect to some $H$-elements.
Satake’s Classification
=======================
Necessary Conditions {#necessary}
--------------------
In a series of papers [@Satake1964; @Satake1965a; @Satake1965b; @SatakeBoulder; @Satake1966; @Satake1967; @Satakebook; @Satake1997] Satake classified the $H_1$-representations of a given hermitian group into a symplectic group. We summarize his results below. Let $G$ be a connected, semisimple, linear algebraic group over ${\mathbb Q}$. Assume that $G({\mathbb R})^0$ is of hermitian type, and has no nontrivial, connected, normal ${\mathbb Q}$-subgroup $H$ with $H({\mathbb R})$ compact. After replacing $G$ by a finite covering, if necessary, we may write $${\mathfrak g}_{{\mathbb R}} = \bigoplus_{j=0}^s {\mathfrak g}_j, \qquad G({\mathbb R}) = G_0 \times G_1 \times \dots \times G_s,$$ where $G_0$ is compact, each $G_j$ is a noncompact absolutely simple Lie group for $j > 0$, and, each ${\mathfrak g}_j = \operatorname{Lie}(G_j)$. Suppose $\rho \colon G \to Sp(V, E)$ is a symplectic representation satisfying the $H_1$-condition. Then,
1. For $j = 1, \dots s$, we have that $G_j$ is one of the following:
1. Type I: $SU(p, q)$ with $p \geq q \geq 1$;
2. Type II: $SU^-(n, {\mathbb H})$ with $n \geq 5$ (this is the group that Helgason [@Helgason]\*[p. 445]{} calls $SO^{\star}(2n)$);
3. Type III: $Sp(2n, {\mathbb R})$ with $n \geq 1$;
4. Type IV: $Spin(p, 2)$ with $p \geq 1, p \neq 2$.
2. \[stability\] Let $\rho'$ be a nontrivial ${\mathbb C}$-irreducible subrepresentation of $\rho_{{\mathbb C}}$. Then, for some index $j$ ($1 \leq j \leq s$), we have that $\rho'$ is equivalent to $\rho_0 \otimes \rho_j$, where $\rho_0$ is a representation of $G_{0,{\mathbb C}}$, and, $\rho_j$ is a representation of $G_{j,{\mathbb C}}$. We call this the *stability* condition.
3. Fix an index $j$ with $1 \leq j \leq s$. Let $\rho_j$ be an irreducible subrepresentation of $V_{{\mathbb R}}$ considered as a $G_j$-module. Then $\rho_j$ is either trivial or given by one of the following:
1. \[supq\] If $G_j = SU(p,q)$ with $p \geq q \geq 2$, then $\rho_{j,{\mathbb C}}$ is the direct sum of the standard representation of $G_{j,{\mathbb C}} = SL_{p+q}({\mathbb C})$ and its contragredient; it satisfies the $H_2$-condition if and only if $p=q$.
2. If $G_j = SU(p, 1)$, then $\rho_j$ is one of the following:
1. \[sup1gen\] $\bigwedge^k \oplus \bigwedge^{p+1-k}$, for some $k$ with $1 \leq k < \frac{p+1}{2}$;
2. $\bigwedge^k$ with $k = \frac{p+1}{2}$, and $p \equiv 1 \pmod{4}$;
3. the direct sum of two copies of $\bigwedge^k$ with $k = \frac{p+1}{2}$, and $p \equiv 3 \pmod{4}$.
The $H_2$-condition is satisfied if and only if $k = \frac{p+1}{2}$.
3. If $G_j = SU^-(n, {\mathbb H})$ with $n \geq 5$, then $\rho_{j,{\mathbb C}}$ is the direct sum of two copies of the standard representation. The $H_2$-condition is satisfied in this case.
4. If $G_j = Sp(2n, {\mathbb R})$, then $\rho_j$ is the standard representation, and satisfies the $H_2$-condition.
5. If $G_j = Spin(p,2)$ with $p \geq 1$ and $p$ odd, then
1. $\rho_j$ is the spin representation if $p \equiv 1, 3 \pmod{8}$;
2. $\rho_{j,{\mathbb C}}$ is the direct sum of two copies of the spin representation if $p \equiv 5, 7 \pmod{8}$.
In both cases, $\rho_j$ satisfies the $H_2$-condition.
6. If $G_j = Spin(p,2)$ with $p \geq 4$, and $p$ even, then $\rho_j$ is
1. one of the two spin representations if $p \equiv 2 \pmod{8}$;
2. the direct sum of two copies of a spin representation if $p \equiv 6 \pmod{8}$;
3. the direct sum of the two spin representations if $p \equiv 0 \pmod{4}$.
In each case, $\rho_j$ satisfies the $H_2$-condition.
We note that the above conditions imply that $\rho$ is self-dual.
A sufficient condition
----------------------
Satake showed that the necessary conditions listed in §\[necessary\] are sufficient if we make an additional assumption.
\[sataketype\] Let $G$ be a ${\mathbb Q}$-simple hermitian group, and write $G({\mathbb R}) = \prod_{\alpha \in S} G_{\alpha}$ where each $G_{\alpha}$ is an absolutely simple real algebraic group. Let $\rho$ be a representation of $G$ satisfying the conditions of §. Assume further that each irreducible subrepresentation of $\rho_{{\mathbb C}}$ is nontrivial on $G_{\alpha}$ for exactly one $\alpha$. Then some multiple of $\rho$ defines a Kuga fiber variety.
More on type I {#nonh2}
--------------
We now take a closer look at the $H_1$-representation $\rho \colon SU(p,q) \to Sp(V, E)$ given by item (\[supq\]) of the list in §\[necessary\]. We recall the matrix representation of this given by Satake. Let $J_0 \in {\mathfrak S}(V, E)$ be the base point. The eigenvalues of $J_0$ on $V_{{\mathbb C}}$ are $i$ and $-i$, and we take a basis of $V_{{\mathbb C}}$ with respect to which the matrix of $J_0$ is $\begin{pmatrix} iI_m & 0 \\ 0 & -iI_m \end{pmatrix}$. The Lie algebra of $Sp(V, E)$ with respect to this basis is given by $${\mathfrak s}{\mathfrak p}(V, E) =
\left\{ \begin{pmatrix} A & B \\ C & -{}^tA \end{pmatrix} \, \middle| \, B, C \text{ symmetric} \right\},$$ and the $H$-element is $H'_0 = \begin{pmatrix} \frac{i}{2}I_m & 0 \\ 0 & -\frac{i}{2}I_m \end{pmatrix}$.
With respect to a suitable basis the Lie algebra of $SU(p,q)$ is given by $${\mathfrak s}{\mathfrak u}(p,q) = \left\{
\begin{pmatrix} X_1 & X_{12} \\ {}^t \overline{X}_{12} & X_2 \end{pmatrix} \in {\mathfrak s}{\mathfrak l}_{p+q}({\mathbb C}) \, \middle| \,
\begin{matrix} X_1 \in M_p({\mathbb C}), X_2 \in M_q({\mathbb C}) \\ {}^t\overline{X}_j = -X_j (j=1,2) \end{matrix}
\right\},$$ and an $H$-element is given by $$H_0 = \begin{pmatrix} \frac{qi}{p+q} I_p & 0 \\ 0 & -\frac{pi}{p+q} I_q \end{pmatrix}.$$ Then, $d\rho \colon {\mathfrak s}{\mathfrak u}(p,q) \to {\mathfrak s}{\mathfrak p}_{2p+2q}$ is given by $$\begin{pmatrix} X_1 & X_{12} \\ {}^t \overline{X}_{12} & X_2 \end{pmatrix} \mapsto
\begin{pmatrix} \overline{X}_2 & 0 & 0 & {}^tX_{12} \\
0 & X_1 & X_{12} & 0 \\
0 & {}^t\overline{X}_{12} & X_2 &0 \\
\overline{X}_{12} & 0 & 0 & \overline{X}_1 \end{pmatrix}.$$ We extend $d\rho$ to ${\mathfrak u}(p,q)$, and denote by $$\bar{\rho} \colon U(p,q) \to Sp(2p+2q, {\mathbb R})$$ the corresponding map of Lie groups which extends $\rho$. Let $$\bar{H}_0^{p,q} = \begin{pmatrix} \frac{i}{2}I_p & 0 \\ 0 & -\frac{i}{2}I_q \end{pmatrix}.$$ Then $d\bar{\rho} (\bar{H}_0) = H'_0$. It follows, as in the proof of Lemma \[h2lemma\], that $J_0 \in \bar{\rho} (U(p,q))$.
Consider, next, the $H_1$-representation $\rho \colon SU(p, 1) \to Sp(2m, {\mathbb R})$ given in item (\[sup1gen\]) of the list in §\[necessary\]. We extend it to a representation $\bar{\rho} \colon U(p, 1) \to Sp(2m, {\mathbb R})$. Let $$\tilde{H}_0 = \begin{pmatrix} \frac{i}{2k}I_p & 0 \\ 0 & \frac{1-2k}{2k}i \end{pmatrix}.$$ Then $$\bigwedge^k (\tilde{H}_0) = \begin{pmatrix} \frac{i}{2}I_p' & 0 \\ 0 & -\frac{i}{2}I_q' \end{pmatrix}
= \bar{H}_0^{p',q'},$$ where $p' = \binom{p}{k}$ and $q' = \binom{p}{k-1}$. From this we see that $d\bar{\rho} (\tilde{H}_0) = H'_0$, the $H$-element of $Sp(2m, {\mathbb R})$. It follows, as in the proof of Lemma \[h2lemma\], that $J_0 \in \bar{\rho} (U(p,1))$.
The rigid case
==============
Statement of the theorem
------------------------
Let $G$ be an algebraic group over ${\mathbb Q}$ of hermitian type, and $\rho$ a representation of $G$ satisfying Satake’s conditions in §\[necessary\]. By the stability condition (\[stability\]) of §\[necessary\], every irreducible subrepresentation of $\rho_{{\mathbb C}}$ is nontrivial on at most one noncompact factor of $G({\mathbb R})$. We say that $\rho$ is *rigid*, if every irreducible subrepresentation of $\rho_{{\mathbb C}}$ is nontrivial on exactly one noncompact factor of $G({\mathbb R})$. We begin by classifying the rigid representations which define Kuga fiber varieties.
\[rigid\] Let $G$ be a semisimple connected algebraic group over ${\mathbb Q}$ such that $G({\mathbb R})^0$ is of hermitian type and has no compact factors defined over ${\mathbb Q}$. Let $\rho$ be a representation of $G$ satisfying Satake’s conditions in §. If $\rho$ is rigid, then some multiple of $\rho$ defines a Kuga fiber variety.
The rest of this section is devoted to the proof of this theorem.
Beginning of the proof
----------------------
Without loss of generality we assume that $G$ is simply connected, and $\rho$ is nontrivial and a multiple of a ${\mathbb Q}$-irreducible representation (see Satake [@Satakebook]\*[p. 189]{}).
Write $G = \prod_{j=1}^t G_j$, where each $G_j$ is a simple group of hermitian type. Then there are totally real number fields $F_j$, and absolutely simple groups ${\widetilde G}_j$ over $F_j$, such that $G_j = \operatorname{Res}_{F_j/{\mathbb Q}} {\widetilde G}_j$ for $1 \leq j \leq t$. Let $F$ be the smallest Galois extension of ${\mathbb Q}$ containing all the $F_j$, and ${\mathcal G}= \operatorname{Gal}(F/{\mathbb Q})$. Let $S_j$ be the set of embeddings of $F_j$ into ${\mathbb R}$, and let $S$ be the disjoint union of the $S_j$’s. For $\alpha \in S$, we let $j(\alpha)$ be the unique index such that $\alpha \in S_{j(\alpha)}$. We note that $F$ is a totally real field, ${\mathcal G}$ acts on $S$, and the orbits of this action are the sets $S_j$. For $\alpha \in S$, we let $G_{\alpha} = \widetilde{G}_{j(\alpha)} \otimes_{F_{j(\alpha)}, \alpha} F$. Then $G_F = \prod_{\alpha \in S} G_{\alpha}$.
Let $S_0 = \{ \alpha \in S \mid G_{\alpha, {\mathbb R}} \text{ is not compact} \}$. An $H$-element of $G_{{\mathbb R}}$ is given by $H_0 = \sum_{\alpha \in S_0} H_{0, \alpha}$, where $H_{0, \alpha}$ is an $H$-element of $G_{\alpha, {\mathbb R}}$.
Let $\rho_0 \colon G_{{\mathbb C}} \to GL(V_0)$ be a ${\mathbb C}$-irreducible subrepresentation of $\rho_{{\mathbb C}}$. Let $$M = \{ \alpha \in S \mid \rho_0 \text{ is nontrivial on } G_{\alpha,{\mathbb C}} \},$$ and let $\alpha_0$ be the unique element of $M$ such that $G_{\alpha_0, {\mathbb R}}$ is not compact. Write $\rho_0 = \otimes_{\alpha \in M} \rho'_{\alpha}$, where $\rho'_{\alpha}$ is an irreducible representation of $G_{\alpha, {\mathbb C}}$. Either $\rho'_{\alpha}$, or the sum of two copies of $\rho'_{\alpha}$, or, the direct sum of $\rho'_{\alpha}$ and its complex conjugate is defined over ${\mathbb R}$. Let $\rho_{\alpha}$ be the real representation such that $\rho'_{\alpha}$ equals $\rho_{\alpha, {\mathbb C}}$, the direct sum of two copies of $\rho_{\alpha, {\mathbb C}}$, or the direct sum of $\rho_{\alpha, {\mathbb C}}$ and its complex conjugate, respectively.
For each $\alpha \in M$, let $\hat{\rho}_{\alpha} = \sum_{\sigma \in {\mathcal G}} \rho_{\alpha}^{\sigma}$. Then $\hat{\rho}_{\alpha}$ is a representation of $G_{j(\alpha)}$ satisfying the hypotheses of Theorem \[sataketype\], so some multiple of it defines a Kuga fiber variety. By Satake’s construction (see [@Satakebook]\*[§IV.6, Theorems 6.1, 6.2, 6.3]{}), this representation is defined over $F_{j(\alpha)}$ in the sense that it is the restriction from $F_{j(\alpha)}$ to ${\mathbb Q}$ of a symplectic representation $$\tilde{\rho}_{\alpha} \colon {\widetilde G}_{j(\alpha)} \to Sp({\widetilde V}_{\alpha}, {\widetilde E}_{\alpha}).$$ Here, ${\widetilde E}_{\alpha}$ is an $F_{j(\alpha)}$-bilinear alternating form on ${\widetilde V}_{\alpha}$, and $E_{\alpha} = \operatorname{Tr}_{{F_{j(\alpha)}}/{\mathbb Q}} {\widetilde E}_{\alpha}$ is a $G_{j(\alpha)}$-invariant ${\mathbb Q}$-bilinear alternating form on $V_{\alpha} = \operatorname{Res}_{{F_{j(\alpha)}}/{\mathbb Q}}{\widetilde V}_{\alpha}$. Let $V_{\alpha} = \operatorname{Res}_{F_{j(\alpha)}/{{\mathbb Q}}} {\widetilde V}_{\alpha}$. Then $V_{\alpha} \otimes F_{j(\alpha)} = \oplus_{\sigma \in {\mathcal G}} {\widetilde V}_{\alpha}^{\sigma}$ (up to multiplicity), and ${\widetilde V}_{\alpha}^{\sigma}$ is the representation space of $\rho_{\alpha}^{\sigma}$. For each $\alpha \in S_{j(\alpha)}$, there is a complex structure $J_{\alpha}$ on ${\widetilde V}_{\alpha,{\mathbb R}}$ such that ${\widetilde E}_{\alpha} (x, J_{\alpha}y)$ is a symmetric, positive definite form. Moreover, $\rho_{\alpha_0} \colon G_{\alpha_0, {\mathbb R}} \to Sp({\widetilde V}_{\alpha_0,{\mathbb R}}, {\widetilde E}_{\alpha_0})$ satisfies the $H_1$-condition with respect to the $H$-elements $H_{0, \alpha_0}$ and $\frac{1}{2} J_{\alpha_0}$.
Construction of a symmetric form
--------------------------------
We claim that for each $\alpha \in M$ there exists a $G_{\alpha}$-invariant, $F_{j(\alpha)}$-bilinear positive definite symmetric form $\gamma_{\alpha}$ on ${\widetilde V}_{\alpha}$. The space of symmetric positive definite $G_{\alpha, {\mathbb R}}$-invariant forms on ${\widetilde V}_{\alpha,{\mathbb R}}$ is an open subset of the space of symmetric forms; it is nonempty because ${\widetilde E}_{\alpha} (x, J_{\alpha}y)$ is such a form. Therefore it contains an $F_{j(\alpha)}$-rational point $\gamma_{\alpha}$. This proves the claim.
We next claim that when $\alpha = \alpha_0$ we have $$\label{J_invariance}
\gamma_{\alpha_0} (J_{\alpha_0}x, J_{\alpha_0}y) = \gamma_{\alpha_0} (x,y).$$ If $\rho_{\alpha_0}$ satisfies the $H_2$-condition, then Lemma \[h2lemma\] shows that $J_{\alpha_0} $ belongs to the image of $G_{\alpha_0}({\mathbb R})$ under $\rho_{\alpha_0}$, so (\[J\_invariance\]) is a consequence of $\gamma_{\alpha_0}$ being $G_{\alpha_0}$-invariant. Finally, we consider the situation when the $H_2$-condition is not satisfied. Then we are in either case (\[supq\]) or case (\[sup1gen\]) of §\[necessary\]. In both cases, $G_{\alpha}$ is a special unitary group $SU(W, h)$. We can extend $\hat{\rho}_{\alpha}$ to a representation of the full unitary group $U(W, h)$. We have seen in §\[nonh2\] that $J_{\alpha_0}$ belongs to the image of $U(W,h)$, so now we can argue as before to prove (\[J\_invariance\]) in this situation.
Observe that (\[J\_invariance\]) is equivalent to $$\label{J_alternate}
\gamma_{\alpha_0} (x, J_{\alpha_0}y) = - \gamma_{\alpha_0} (y,J_{\alpha_0}x),$$ since $J_{\alpha_0}$ is a complex structure.
Construction of an alternating form
-----------------------------------
Let ${\widetilde V}= \otimes_{\alpha \in M} ({\widetilde V}_{\alpha} \otimes_{F_{j(\alpha)}} F)$. Define an $F$-bilinear alternating form ${\widetilde E}$ on ${\widetilde V}$ by $${\widetilde E}(\otimes_{\alpha} x_{\alpha}, \otimes_{\alpha} y_{\alpha}) =
\sum_{\alpha \in M} \Bigg( {\widetilde E}_{\alpha} (x_{\alpha}, y_{\alpha}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha}} \gamma_{\beta} (x_{\beta}, y_{\beta}) \Bigg).$$ Then ${\widetilde E}$ is ${\widetilde G}$-invariant, where ${\widetilde G}= \prod_{j=1}^t {\widetilde G}_j$. Next, we define a ${\mathbb Q}$-bilinear alternating form $E$ on $V = \operatorname{Res}_{F/{\mathbb Q}} {\widetilde V}$ by $$E(x,y) = \operatorname{Tr}_{F/{\mathbb Q}} {\widetilde E}(x,y).$$ Then $E$ is $G$-invariant.
Construction of a complex structure
-----------------------------------
We next construct a complex structure ${\widetilde J}$ on ${\widetilde V}_{{\mathbb R}} = \otimes_{\alpha \in M} ({\widetilde V}_{\alpha} \otimes_{F_{j(\alpha)}} {\mathbb R})$. We have seen that we have a complex structure $J_{\alpha_0}$ on ${\widetilde V}_{\alpha_0, {\mathbb R}}$ such that ${\widetilde E}_{\alpha_0} (x, J_{\alpha_0} y)$ is symmetric and positive definite. Define ${\widetilde J}$ on ${\widetilde V}_{{\mathbb R}}$ by ${\widetilde J}(\otimes x_{\alpha}) = \otimes I_{\alpha} x_{\alpha}$, where $I_{\alpha_0} = J_{\alpha_0}$, and $I_{\alpha}$ is the identity for $\alpha \neq \alpha_0$. Then ${\widetilde J}$ is a complex structure.
For $x = \otimes x_{\alpha}, y = \otimes y_{\alpha} \in {\widetilde V}_{{\mathbb R}}$, we have $$\begin{aligned}
{\widetilde E}(x, {\widetilde J}y) & = \sum_{\alpha \in M} \Bigg( {\widetilde E}_{\alpha} (x_{\alpha}, I_{\alpha} y_{\alpha}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha}} \gamma_{\beta} (x_{\beta}, I_{\beta} y_{\beta}) \Bigg) \\
& = {\widetilde E}_{\alpha_0} (x_{\alpha_0}, J_{\alpha_0} y_{\alpha_0}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha_0}} \gamma_{\beta} (x_{\beta}, y_{\beta}) \\
& \qquad + \sum_{\substack{\alpha \in M \\ \alpha \neq \alpha_0}} \Bigg( {\widetilde E}_{\alpha} (x_{\alpha}, y_{\alpha}) \gamma_{\alpha_0}(x_{\alpha_0}, J_{\alpha_0} y_{\alpha_0}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha \\ \beta \neq \alpha_0}} \gamma_{\beta} (x_{\beta}, y_{\beta}) \Bigg)\\
& = {\widetilde E}_{\alpha_0} (y_{\alpha_0}, J_{\alpha_0} x_{\alpha_0}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha_0}} \gamma_{\beta} (y_{\beta}, x_{\beta}) \\
& \qquad + \sum_{\substack{\alpha \in M \\ \alpha \neq \alpha_0}} \Bigg( {\widetilde E}_{\alpha} (y_{\alpha}, x_{\alpha}) \gamma_{\alpha_0}(y_{\alpha_0}, J_{\alpha_0} x_{\alpha_0}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha \\ \beta \neq \alpha_0}} \gamma_{\beta} (y_{\beta}, x_{\beta}) \Bigg)\\
& = \sum_{\alpha \in M} \Bigg( {\widetilde E}_{\alpha} (y_{\alpha}, I_{\alpha} x_{\alpha}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha}} \gamma_{\beta} (y_{\beta}, I_{\beta} x_{\beta}) \Bigg) \\
& = {\widetilde E}(y, {\widetilde J}x),\end{aligned}$$ because ${\widetilde E}_{\alpha_0} (x, J_{\alpha_0} y)$ and $\gamma_{\alpha}$ are symmetric, $I_{\alpha}$ is the identity for $\alpha \neq \alpha_0$, ${\widetilde E}_{\alpha_0} (x_{\alpha_0}, J_{\alpha_0} y_{\alpha_0})$ and $\gamma_{\alpha_0}(x_{\alpha_0}, J_{\alpha_0} y_{\alpha_0})$ are alternating, and using (\[J\_alternate\]). Thus ${\widetilde E}(x, {\widetilde J}y)$ is a symmetric form on ${\widetilde V}_{{\mathbb R}}$. It follows that $E (x, {\widetilde J}y)$ is symmetric.
If $x=y$ we have $$\begin{aligned}
{\widetilde E}(x, {\widetilde J}x) &=
\sum_{\alpha \in M} \Bigg( {\widetilde E}_{\alpha} (x_{\alpha}, I_{\alpha} x_{\alpha}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha}} \gamma_{\beta} (x_{\beta}, I_{\beta} x_{\beta}) \Bigg) \\
&= {\widetilde E}_{\alpha_0} (x_{\alpha_0}, J_{\alpha_0} x_{\alpha_0}) \prod_{\substack{\beta \in M \\ \beta \neq \alpha_0}} \gamma_{\beta} (x_{\beta}, x_{\beta}),\end{aligned}$$ so ${\widetilde E}(x, {\widetilde J}y)$ is positive definite.
Our next task is to define a complex structure $J$ on $V_{{\mathbb R}}$, where $V = \operatorname{Res}_{F/{\mathbb Q}} {\widetilde V}$. Now, $V_{{\mathbb R}} = \bigoplus_{\sigma \in {\mathcal G}} {\widetilde V}_{{\mathbb R}}^{\sigma}$, so it is sufficient to define a complex structure ${\widetilde J}^{\sigma}$ on ${\widetilde V}_{{\mathbb R}}^{\sigma}$ for each $\sigma \in {\mathcal G}$. When $\sigma$ is the identity, we have already defined ${\widetilde J}$ on ${\widetilde V}_{{\mathbb R}}$. In the same manner we can define ${\widetilde J}^{\sigma}$ for each $\sigma \in {\mathcal G}$, such that ${\widetilde E}^{\sigma} (x, {\widetilde J}^{\sigma} y)$ is symmetric and positive definite on ${\widetilde V}_{{\mathbb R}}^{\sigma}$. Then $J = \sum_{\sigma \in {\mathcal G}} {\widetilde J}^{\sigma}$ is a complex structure on $V_{{\mathbb R}}$.
Conclusion of the proof
-----------------------
Since each ${\widetilde E}^{\sigma} (x, {\widetilde J}^{\sigma} y)$ is symmetric, so is $E(x, Jy)$. It remains to show that $E(x, Jy)$ is positive definite. For each $\sigma \in {\mathcal G}$, let $\alpha(\sigma)$ be the unique element of $M^{\sigma} \cap S_0$. Then, we have $$\begin{aligned}
E(x, Jx) &= \sum_{\sigma \in {\mathcal G}} {\widetilde E}^{\sigma} (x, {\widetilde J}^{\sigma}x) \\
&= \sum_{\sigma \in {\mathcal G}} \Bigg( {\widetilde E}_{\alpha(\sigma)} (x_{\alpha(\sigma)}, {\widetilde J}^{\alpha(\sigma)}x_{\alpha(\sigma)})
\prod_{\substack{\beta \in M^{\sigma} \\ \beta \neq \alpha(\sigma)}} \gamma_{\beta} (x_{\beta}, x_{\beta}) \Bigg) \\
&= \sum_{\sigma \in {\mathcal G}} Q^{\sigma} (x),\end{aligned}$$ where $$Q^{\sigma} (x) = {\widetilde E}_{\alpha(\sigma)} (x_{\alpha(\sigma)}, {\widetilde J}^{\alpha(\sigma)}x_{\alpha(\sigma)})
\prod_{\substack{\beta \in M^{\sigma} \\ \beta \neq \alpha(\sigma)}} \gamma_{\beta} (x_{\beta}, x_{\beta})$$ is a symmetric form on $V_{{\mathbb R}}$. For $\sigma$ equal to the identity, we know that $Q(x) = {\widetilde E}(x,Jx)$ is positive definite. Therefore there exists a positive integer $N$ such that $$Q'(x) = NQ(x) + \sum_{\substack{\sigma \in {\mathcal G}\\ \sigma \neq \operatorname{id}}} Q^{\sigma}(x)$$ is positive definite (see Addington [@Addington]\*[Lemma 4.9, p. 80]{}). For each $j$ ($1 \leq j \leq t$), let $c_j \in F_j$ be such that $\alpha(c_j) > N$ if $\alpha \in S_0$, and $0 < \alpha(c_j) < 1$ if $\alpha \notin S_0$. Replace each ${\widetilde E}_{\alpha}$ by $c_{j(\alpha)}{\widetilde E}_{\alpha}$. Then $E(x, Jy)$ is positive definite.
An $H$-element for $Sp(V, E)$ is given by $\frac{1}{2} J$. Since $H_0 = \sum_{\alpha \in S_0} H_{0, \alpha}$ is an $H$-element of $G$, and $\rho_{\alpha}$ satisfies the $H_1$-condition with respect to $H_{0, \alpha}$ and $\frac{1}{2} J_{\alpha}$ whenever $\alpha \in S_0$, it follows from our construction that $\rho$ satisfies the $H_1$-condition, and therefore defines a Kuga fiber variety.
The general case
================
We next derive the general case from the rigid case.
\[maintheorem\] Let $\rho$ be a representation of $G$ satisfying Satake’s conditions in §. Then some multiple of $\rho$ defines a Kuga fiber variety.
We keep the notations used in the proof of Theorem \[rigid\]. Without loss of generality we assume that $\rho$ is a primary representation, i.e., a multiple of an irreducible representation. An irreducible subrepresentation $\rho_0$ of $\rho_{{\mathbb C}}$ is said to be *rigid* if it is nontrivial on some noncompact factor of $G({\mathbb R})$. We define the *index of rigidity* of $\rho$ to be the cardinality of the set $\{ \sigma \in {\mathcal G}\mid \mu^{\sigma} \text{ is rigid} \}$, where $\mu$ is an irreducible subrepresentation of $\rho_{{\mathbb C}}$. It depends only on $\rho$, and not on the choice of $\mu$.
Suppose $\rho$ is not rigid. Then there exists a subrepresentation $\mu$ of $\rho_{{\mathbb C}}$ such that $\mu$ is trivial on all noncompact factors of $G_{{\mathbb R}}$. Let ${\mathcal G}_1 = \{ \sigma \in {\mathcal G}\mid \mu^{\sigma} = \mu \}$. Then $\rho_{{\mathbb C}}$ is equivalent to a multiple of $\sum_{\sigma \in {\mathcal G}/{\mathcal G}_1} \mu^{\sigma}$.
Let $\alpha_0 \in S_j$ be such that $\mu$ is nontrivial on $G_{\alpha_0}$. Extend $\alpha_0$ to an embedding of $F$ into ${\mathbb R}$, and denote it again by $\alpha_0$. Let $B$ be a quaternion algebra over $F$ which splits at $\alpha_0$ and ramifies at all other infinite places. Let $SL_1(B)$ be the group of norm $1$ units of $B$, and $H = \operatorname{Res}_{F/{\mathbb Q}} SL_1(B)$. Then $H({\mathbb R}) = \prod_{\alpha \in \overline{S}} H_{\alpha}$, where $\overline{S}$ is the set of embeddings of $F$ into ${\mathbb R}$, and $H_{\alpha} = H \otimes_{F, \alpha} {\mathbb R}$.
Define a representation of $G \times H$ by $$\rho_1 = \sum_{\sigma \in {\mathcal G}} \mu^{\sigma} \otimes p_{\alpha_0}^{\sigma},$$ where $p_{\alpha_0} \colon H({\mathbb R}) \to H_{\alpha_0} = SL_2({\mathbb R})$ is the projection. Since $\rho_1$ is invariant under any automorphism of ${\mathbb C}$, some multiple $n \rho_1$ of $\rho_1$ is defined over ${\mathbb Q}$. Now $p_{\alpha_0}^{\sigma} = p_{\alpha_0^{\sigma}}$. If $\sigma$ is not the identity then $\alpha_0^{\sigma} \neq \alpha_0$. Since $H_{\alpha}$ is compact for $\alpha \neq \alpha_0$, we see that $n \rho_1$ satisfies the stability condition. We verify that $n \rho_1$ satisfies all of Satake’s conditions.
Now, $\mu^{\sigma} \otimes p_{\alpha}^{\sigma}$ is rigid whenever $\mu^{\sigma}$ is rigid, and it is also rigid when $\sigma$ is the identity. Hence the index of rigidity of $n \rho_1$ is greater than the index of rigidity of $\rho$. Continuing this process, if necessary, we will eventually get a representation $\tilde{\rho}$ whose index of rigidity is the cardinality of ${\mathcal G}$, i.e., one which is rigid. Then Theorem \[rigid\] implies that some multiple of $\tilde{\rho}$ defines a Kuga fiber variety. Since the restriction of $\tilde{\rho}$ to $G$ is a multiple of $\rho$, this completes the proof.
An example {#example}
==========
Let $F = {\mathbb Q}(\sqrt{3})$. Let $\alpha_1, \alpha_2$ be the embeddings of $F$ into ${\mathbb R}$. Let $B$ be a quaternion algebra over $F$ which splits at $\alpha_1$ and ramifies at $\alpha_2$. Let ${\widetilde G}_1$ be the group of norm $1$ units in $B$, and, $G_1 = \operatorname{Res}_{F/{\mathbb Q}} {\widetilde G}_1$. Let ${\widetilde G}_2$ be a group over $F$ such that ${\widetilde G}_2 \otimes_{F, \alpha_1} {\mathbb R}= SU(5,1)$ and ${\widetilde G}_2 \otimes_{F, \alpha_2} {\mathbb R}= SU(6,0)$. Let $G_2 = \operatorname{Res}_{F/{\mathbb Q}} {\widetilde G}_2$. Let $G = G_1 \times G_2$. Then $$G({\mathbb R}) = SL_2({\mathbb R}) \times SU(2) \times SU(5, 1) \times SU(6, 0).$$
We shall classify all representations of $G$ which define Kuga fiber varieties. Let $$\begin{aligned}
& p_1 \colon G({\mathbb R}) \to SL_2({\mathbb R}),\\
& p_2 \colon G({\mathbb R}) \to SU(2),\\
& p_3 \colon G({\mathbb R}) \to SU(5, 1),\\
& p_4 \colon G({\mathbb R}) \to SU(6, 0),\end{aligned}$$ be the projections. Then, the representations of $G$ defining Kuga fiber varieties are equivalent over ${\mathbb R}$ to linear combinations of the following:
1. $p_1 \oplus p_2$,
2. $p_1 \otimes p_2$,
3. $\bigwedge^k p_3 \oplus \bigwedge^k p_4$,
4. $\left( \bigwedge^j p_3 \otimes \bigwedge^k p_4 \right) \oplus \left( \bigwedge^k p_3 \otimes \bigwedge^j p_4 \right)$,
5. $\left( p_1 \otimes \bigwedge^k p_4 \right) \oplus \left( p_2 \otimes \bigwedge^k p_3 \right)$.
Of these, the first four are direct sums of representations of either $G_1$ alone, or $G_2$ alone. The last one is a representation of the product group in an essential manner. A general fiber of the corresponding Kuga fiber variety is a simple abelian variety; the semisimple part of its Hodge group is isogenous to $G = G_1 \times G_2$.
|
---
abstract: 'We find general solutions to the generating-function equation $\sum c_q^{(X)} z^q = F(z)^X$, where $X$ is a complex number and $F(z)$ is a convergent power series with $F(0) \neq 0$. We then use these results to derive finite expressions containing only integers or simple fractions for partition functions and for Euler, Bernoulli, and Stirling numbers.'
author:
- Jerome Malenfant
title: 'Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers'
---
\[1\][Lemma]{} \[1\][Lemma]{}
Introduction and main result
============================
Generating functions are often used as a compact way to define special number sequences and functions as the coefficients in a power-series expansion of more elementary functions. One example of this is the partition function $p(n)$, (sequence A000041 in OEIS [@integer]), the number of partitions of $n$ into positive integers. It has the generating function [@Partition] $$\begin{aligned}
\sum_{q=0}^{\infty} p(q) z^q = \prod_{k=1}^{\infty} \frac{1}{1-z^k} = 1 +z + 2z^2 +3z^3 +5z^4 + \cdots
\end{aligned}$$ Some other examples of generating functions are those for the Bernoulli and the Euler numbers:
$$\begin{aligned}
\sum_{q=0}^{\infty} \frac{ B_q}{q!} ~z^q &=& \frac{z}{e^z-1} ,\\
\sum _{q=0}^{\infty} \frac{E_{2q}}{(2q)!}~ z^{2q}&=& \frac{1}{\cosh z}
\end{aligned}$$
The solutions of generating-function equations can often be found from recursion relations. Some of them can also be calculated directly by using various expressions, such as Laplace’s determinental formula for the Bernoulli numbers [@Korn], which we write in the form: $$\begin{aligned}
B_n = n! \left| \begin{array}{ccccc}
1 & 0 & \cdots &~~ 0 & ~~1 \\
\frac{1}{2!} & 1 & ~ & ~~ 0 & ~~ 0 \\
\vdots & ~ & \ddots & ~ & ~~\vdots \\
\frac{1}{n!} & \frac{1}{(n-1)!} &~ & ~~1 & ~~ 0 \\
\frac{1}{ (n+1)!}&\frac{1}{n!} & \cdots & ~~ \frac{1}{2!} & ~~0
\end{array} \right|
\end{aligned}$$ Vella [@Vella] has derived expressions for $B_n$ and for $E_n$ as sums over the partitions (or, alternatively, over the compositions) of $n$ using a method based on the Faa di Bruno formula for the higher derivatives of composite functions. Concerning the partition function, Rademacher [@Partition; @Rade] derived an exact formula, an improvement over the Hardy-Ramanujan asymptotic formula, which however involves an infinite sum of rather complicated, non-integer terms. More recently, Bruinier and Ono [@Ono] have derived an explicit formula for the partition function as a finite sum of algebraic numbers that requires finding a sufficiently precise approximation to an auxiliary function.
In the following theorem, we present methods for solving generating-function equations for the case where the right-hand side is expressible in terms of a convergent, non-zero-near-the-origin power series. We will then apply this theorem to solve for the partition function and other objects which have the appropriate generating functions. In the following, the notations $$\begin{aligned}
\left( \begin{array}{c} n \\ q_1, \ldots , q_K \end{array} \right)
\equiv \frac{ n!}{q_1! \cdots q_K!} ~ \delta_{n,q_1+ \cdots + q_K}
\end{aligned}$$ and $$\begin{aligned}
\left( \begin{array} {c} X \\ N \end{array} \right) \equiv \frac{X(X-1) \cdots (X-N+1)}{N!}
\end{aligned}$$ will denote, respectively, multinomial coefficients and generalized binomial coefficients.
Our main result is:
Let $F(z)$ be a holomorphic function in a neighborhood of the origin with the series expansion $ \sum _{q=0}^{\infty} a_qz^q$, with $a_0 \neq 0$, and let $X$ be a complex number. Then the coefficients $c_p^{(X)}$ in the generating-function equation $$\begin{aligned}
\sum_{p=0}^{\infty} c^{(X)}_pz^p = F(z)^X
\end{aligned}$$ are given by the two equivalent expressions: $$\begin{aligned}
({\rm I}) ~~~~ c_p^{(X)} &=& \sum_{ 0 \leq k_1, \ldots, k_p \leq p}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{p, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p},
\end{aligned}$$ where $ K \equiv k_1 + \cdots + k_p$, and $$\begin{aligned}
({\rm II}) ~~~~ c_p^{(X)} &=& a_0^{(p+1)X} \left|
\left( \begin{array}{cccc}
a_0 & ~ & ~ & ~ \\
a_1& a_0 &~ \mbox{{\rm \Huge 0}} & ~ \\
\vdots & ~& \ddots ~~&~ \\
a_{p} & a_{p-1} & \cdots & a_0 \end{array}
\right)^{\mbox{ -X}}+ \left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
- 1/a_0^X \end{array} \right) \right|.
\end{aligned}$$
The proof of (II) requires some additional machinery and will be postponed to Section II.\
$Proof~ of ~(\rm {I})$: Since $( a_0 + a_1z + \cdots ) ^X- ( a_0 + a_1z + \cdots + a_pz^p)^X \sim z^{p+1}$, the coefficient of the $z^p$ term is the same in the two sums, so in determining the $c_p^{(X)}$ coefficient we need deal only with the finite sum. Using the generalized binomial theorem, we write $$\begin{aligned}
\left( \sum_{q=0}^{p} a_qz^q \right) ^X
&=& \sum_{N=0}^{\infty} \left( \begin{array} {c} X \\ N \end{array} \right) a_0^{X-N} \left( \sum_{q=1}^{p}
a_qz^q \right) ^N \nonumber\\
&=& \sum_{N=0}^{\infty} \left( \begin{array} {c} X \\ N \end{array} \right) a_0^{X-N}
\sum_{ 0 \leq k_1, \ldots, k_p \leq N} \left( \begin{array}{c} N \\ k_1, \ldots, k_p \end{array} \right)
a_1^{k_1} \cdots a_p^{k_p}~z^{\sum mk_m} .
\end{aligned}$$ In the second line we’ve applied a multinomial expansion to $(\sum_{q=1}^p a_qz^q)^N$. We now write $z^{\sum mk_m} = \sum_{s=0}^{\infty} z^s \delta_{s, \sum mk_m}$ and interchange the order of the sums: $$\begin{aligned}
\left( \sum_{q=0}^{p} a_qz^q \right) ^X
& =& \sum_{s=0}^{\infty} z^s \sum_{ 0 \leq k_1, \ldots, k_p \leq s} \sum_{N=0}^{\infty}
\left( \begin{array} {c} X \\ N \end{array} \right)
\left( \begin{array}{c} N \\ k_1, \ldots, k_p \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-N} a_1^{k_1} \cdots a_p^{k_p}
\end{aligned}$$ After the interchange, the $k_q$’s are fixed in the sum over $N$ and only the $N = k_1+\cdots + k_p~( \equiv K)$ term contributes. We are interested in the coefficient of $z^p$ in this sum, which is thus $$\begin{aligned}
c_p^{(X)} &=& \sum_{ 0 \leq k_1, \ldots, k_p \leq p}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{p, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p}.
\end{aligned}$$ As stated above, this is also the coefficient of $z^p$ in the infinite sum. QED.
The Kroneker delta in eq.(6) restricts the sums over the $k$’s to a sum over the partitions of $p$. The multinomial coefficient in this formula counts the number of unique ways the parts can be ordered. As a sum instead over compositions, eq.(6) becomes $$\begin{aligned}
c_p^{(X)} &=& \sum_{K=0}^p \left( \begin{array} {c} X \\ K \end{array} \right)
\sum_{ 1 \leq q_1, \ldots, q_K \leq p}
\delta_{p, q_1+ \cdots +q_K} ~a_0^{X-K} a_{q_1} \cdots a_{q_K} ;
\end{aligned}$$ As a corollary, we have the multinomial identity:
For fixed $p$, $$\begin{aligned}
\sum_{K=0}^p (-1)^K \sum_{1 \leq q_1, \ldots , q_K \leq p} \left( \begin{array}{c} p \\
q_1, \ldots , q_K \end{array} \right) =(-1)^p
\end{aligned}$$
$Proof$: We set $X=-1$ and $a_q =1/q!$, so that $(\sum a_qz^q)^X = e^{-z}$ and so $ c_p^{(-1)} = (-1)^p/p!$. Then, from eq, (7), $$\begin{aligned}
\frac{(-1)^p}{p!} &=& \sum_{K=0}^p (-1)^K \sum_{ 1 \leq q_1, \ldots, q_K \leq p}
\frac{ \delta_{p, q_1+ \cdots +q_K}}{q_1! \cdots q_K!} .
\end{aligned}$$ Multiplying both sides by $p!$ gives the result.\
QED
We now apply Part (I) of Theorem 1 to eq.(1) and use Euler’s pentagonal theorem: $$\begin{aligned}
\prod_{k=1}^{\infty} (1-z^k) = 1-z -z^2 + z^5 + z^7 -z^{12} -\cdots .
\end{aligned}$$ where the exponents $0,1,2,5,7,12, \ldots $ are generalized pentagonal numbers (sequence A001318): $q_m = (3m^2-m)/2,~ m = 0,\pm 1, \pm 2, \cdots$ [@Partition]. $p(n)$ is then equal to a sum over the pentagonal partitions of $n$: $$\begin{aligned}
p(n) &=& \sum_{ 0 \leq k_1, k_2, k_5,\ldots k_{q_{M}} \leq n} (-1)^{\sum k_{q_{2m}} }
\left( \begin{array}{c} K \\ k_1, k_2, k_5, \ldots ,k_{q_M} \end{array} \right)
\delta_{n, \sum k_{q_{m}} q_m}
\end{aligned}$$ where $q_M$ is the largest GPN $\leq n$. Eq.(10) thus expresses $p(n)$ as a finite sum of integers. The number of terms in the sum is the number of partitions of $n$ into generalized pentagonal numbers, (sequence A095699). For example, 9 has 10 pentagonal partitions, ($9 = 7+2 = 7+1+1 =\cdots = 1 + \cdots +1$), and $p(9)$ is $$\begin{aligned}
p(9) &=&- \left( \begin{array}{c} 2 \\ 0,1,0,1 \end{array} \right) - \left( \begin{array}{c} 3 \\ 2,0,0,1 \end{array} \right)
- \left( \begin{array}{c} 3 \\ 0,2,1,0 \end{array} \right)
- \left( \begin{array}{c} 4 \\ 2,1,1,0 \end{array} \right) - \left( \begin{array}{c} 5 \\ 4,0,1,0 \end{array} \right)\\
&& + \left( \begin{array}{c} 5 \\ 1,4,0,0 \end{array} \right)
+ \left( \begin{array}{c} 6 \\ 3,3,0,0 \end{array} \right) +
\left( \begin{array}{c} 7 \\ 5,2,0,0 \end{array} \right)
+ \left( \begin{array}{c} 8 \\ 7,1,0,0 \end{array} \right) +
\left( \begin{array}{c} 9 \\ 9,0,0,0 \end{array} \right) .
\end{aligned}$$
From the as-yet-to-be-proven Part II of the theorem, $p(n)$ is also expressible as a $(k+1) \times (k+1)$ determinant, for any integer $k \geq n$; $$\begin{aligned}
p(n) &=& \left| \begin{array} {ccccccc}
~~1 & ~~0 & ~~0 & ~~0& ~~\cdots & ~~0 &~~0 \\
-1 & ~~1 & ~~0 & ~~0 &~&~& ~~\vdots \\
-1 & -1 & ~~1 &~~0 &~& ~ & ~~0 \\
~~0 & -1 & -1 & ~~1&~& ~& ~~1 \\
~~\vdots & ~ & ~ &~&\ddots & ~& ~~0 \\
d_{k-1}& d_{k-2} & d_{k-3} &d_{k-4}&~& ~~1 &~~ \vdots\\
d_k & d_{k-1} &d_{k-2} &d_{k-3}& \cdots & -1 & ~~0
\end{array} \right|
\begin{array}{c}~ \\ ~ \\ ~\\ ~ \\
\left. \begin{array}{c} ~ \\ ~\\ ~ \end{array} \right\} \\
~ \end{array}
\begin{array}{c} ~\\~\\~ \\~\\ n\\ ~ \end{array}
\end{aligned}$$ where, for $0 \leq q \leq k,$ $$\begin{aligned}
d_q = ({\rm sequence~A010815}) = \left\{ \begin{array} {l} (-1)^m ~{\rm if~} q=q_m, ~m= 0, \pm 1, \pm 2, \ldots , \\
~~~ 0 ~~{\rm otherwise}, \end{array} \right. ,
\end{aligned}$$ This reduces to the form stated in the theorem by successive expansions by minors along the top row. Form (11) will prove more useful in some of the following discussions, but it can be reduced (by expansions in minors) to the $n \times n$ determinant, $$\begin{aligned}
\begin{array}{c} {\rm GPN's} \\ \hline 0\\1\\2\\~\\~\\5\\~\\7 \\~\\~\\~\\~\\12\\ \vdots \\~\\~\\~
\end{array} ~~~~
p(n) = \left| ~~\begin{array}{cccccccccccccccc}
~1& -1 & ~\\
~1& ~1& -1 & ~\\
~0& ~1& ~1& -1 & ~\\
~0& ~0& ~1& ~1& -1 & ~\\
-1 & ~0& ~0& ~1&~1 & -1 & ~\\
~0& -1 & ~0& ~0& ~ 1&~1& -1 & ~\\
-1& ~0& -1 & ~0& ~0& ~ 1&~1& -1 & ~\\
~0& -1& ~0& -1& ~0 &~ 0& ~1&~1& -1 & ~\\
~0& ~0& -1& ~0& -1 & ~0& ~0& ~ 1&~1& -1 & ~\\
~0& ~0& ~0& -1 & ~ 0& -1 & ~0& ~0& ~ 1&~1& -1 & ~\\
~ 0& ~0&~ 0& ~0& -1 & ~0& -1 & ~0& ~0& ~1&~1& -1 & ~\\
~1 & ~0&~ 0& ~0& ~ 0& -1 & ~0& -1 & ~0& ~0& ~ 1&~1& -1&~ \\
~0& ~1&~ 0& ~0&~ 0& ~0& -1 & ~ 0& -1 &~ 0&~ 0& ~ 1&~1& ~\\
~ \vdots & ~ & ~&~&~&~&~&~&~&~&~&~&~&~&\ddots &
\end{array} ~~\right| _{(n \times n)}
\end{aligned}$$
Matrix formalism
================
Matrices which are constant along all diagonals are Toeplitz matrices. We will be concerned in this section with lower-triangular Toeplitz (LTT) matrices. A $nondegenerate$ LTT matrix is one with non-zero diagonal elements; if the diagonal elements are equal to 1, it is then a $unit$ LTT matrix.
Infinite-dimensional LTT matrices have the form $$\begin{aligned}
A = \left( \begin{array}{ccccc}
a_0 & ~&~&~&~ \\
a_1 & a_0 & ~& ~~\mbox{ \Huge 0} &~\\
a_2 & a_1 & a_0 &~&~ \\
\vdots & ~ & ~& ~& \ddots
\end{array} \right) .
\end{aligned}$$ The determinant of a $k$-dimensional lower-triangular matrix is the product of it’s diagonal elements; if the matrix is also Toeplitz, then its determinant is $a_0^k$.
We define the infinite-dimensional lower shift matrix $J$ as $$\begin{aligned}
J \equiv \left( \begin{array}{ccccc}
~0 ~& ~&~&~&~ \\
~1 ~& ~0~ & ~& \mbox{\Huge 0} &~\\
~0 ~& ~1~ &~ 0 &~&~ \\
\vdots & ~ & ~& ~& \ddots
\end{array} \right).
\end{aligned}$$ The $p$th power of $J$ has elements $(J^p)_{ij} = \delta_{p,i-j} $; these matrices obey the relations $J^pJ^q = J^qJ^p = J^{p+q}$. Any infinite-dimensional LTT matrix can be expanded out in non-negative powers of $J$: $$\begin{aligned}
\left( \begin{array}{ccccc}
a_0 & ~&~&~&~ \\
a_1 & a_0 & ~& ~~\mbox{ \Huge 0} &~\\
a_2 & a_1 & a_0 &~&~ \\
\vdots & ~ & ~& ~& \ddots
\end{array} \right) = \sum_{q=0}^{\infty} a_q J^q.
\end{aligned}$$ The product of two infinite-dimensional LTT matrices is $$\begin{aligned}
AB = \sum_{q=0}^{\infty} a_q J^q~ \sum_{s=0}^{\infty} b_s J^s
= \sum_{p=0}^{\infty} J^p \sum _{q=0}^p a_q b_{p-q} = B A
\end{aligned}$$ A similar expression can be written for finite-dimensional LTT matrices, with the $J$’s replaced by finite lower shift matrices. LTT matrices, whether finite- or infinite-dimensional, thus commute with one another.
Let $A$ be an infinite-dimensional nondegenerate LTT matrix: $$\begin{aligned}
A = \sum_{q=0}^{\infty} a_qJ^q , ~~ a_0 \neq 0.
\end{aligned}$$ Then the inverse of $A$ is an LTT matrix with coefficients $$\begin{aligned}
b_p &=& \frac{1}{a_0^{k+1}} \left| \begin{array}{cccccc}
a_0 & 0 & \cdots & ~& 0 &~ \\
a_1& a_0 & ~ & ~&~ & ~\\
\vdots & ~& \ddots &~& ~ & ~ \\
a_{k-1} &~&~&~& a_0& ~ \\
a_{k} & a_{k-1} & \cdots &~& a_1&~ \end{array}
\begin{array}{c}
~0~\\
\vdots\\
~1~ \\
\vdots\\
~0~ \end{array} \right| \begin{array}{c}
~\\
~\\
~ \\
\left. \begin{array}{c} ~ \\ ~ \end{array} \right\} \\
~ \end{array}
\begin{array}{c}
~\\
~\\
~ \\
p\\
~ \end{array}
\end{aligned}$$ for any integer $k \geq p$.
$Proof$: We have $$\begin{aligned}
\sum_{q=0}^{\infty} a_q J^q~ \sum_{s=0}^{\infty} b_s J^s
= \sum_{p=0}^{\infty} J^p \sum _{q=0}^p a_q b_{p-q} .
\end{aligned}$$ But $$\begin{aligned}
\sum_{q=0}^p a_{q}b_{p-q} = \frac{1}{a_0^{k+1}} \left| \begin{array}{cccccc}
a_0 & 0 & \cdots & ~& 0 &~ \\
a_1& a_0 & ~ & ~&~ & ~\\
\vdots & ~& \ddots &~& ~ & ~ \\
a_{k-1} &~&~&~& a_0& ~ \\
a_{k} & a_{k-1} & \cdots &~& a_1&~ \end{array}
\begin{array}{c}
~0~\\
\vdots\\
~a_0~ \\
\vdots\\
~a_p~ \end{array} \right| = \left\{ \begin{array} {l} 1 ~~~~{\rm if~} p =0,\\ 0 ~~~~{\rm if~} p>0, \end{array} \right.
\end{aligned}$$ and therefore $\sum_{s=0}^{\infty} b_s J^s = A^{-1}$. QED
It is clear that this proof still holds if the $J$’s are replaced by finite-dimensional lower shift matrices and the sums over $q$ and $s$ are finite. Thus, the coefficients of the inverse of a finite $k$-dimensional (nondegenerate) LTT matrix are given by the same formula, and we have the result:
The inverse of a $k$-dimensional LTT matrix is equal to the $k$-dimensional truncation of the inverse of the corresponding infinite-dimensional LTT matrix.
I.e., $$\begin{aligned}
\left( \begin{array}{ccccc}
b_0 & ~&~&~&~ \\
b_1 & b_0 & ~& ~~ \mbox{{\rm \Huge 0}} &~\\
b_2 & b_1 & b_0 \\
\vdots & ~& ~& \ddots & ~\\
\end{array} \right) = \left( \begin{array}{ccccc}
a_0 & ~&~&~&~ \\
a_1 & a_0 & ~& ~~\mbox{{\rm \Huge 0}} &~\\
a_2 & a_1 & a_0 \\
\vdots & ~ & ~& \ddots & ~\\
\end{array} \right) ^{-\mbox {1}}
\end{aligned}$$ iff $$\begin{aligned}
\left( \begin{array}{ccccc}
b_0 & ~&~&~&~ \\
b_1 & b_0 & ~& ~~ \mbox{{\rm \Huge 0}} &~\\
\vdots & ~& \ddots &~ & ~\\
b_{k-1} & b_{k-2} & \cdots & ~& b_0
\end{array} \right) = \left( \begin{array}{ccccc}
a_0 & ~&~&~&~ \\
a_1 & a_0 & ~& ~~\mbox{{\rm \Huge 0}} &~\\
\vdots & ~ & \ddots &~ & ~\\
a_{k-1} & a_{k-2} & \cdots & ~& a_0
\end{array} \right) ^{-\mbox {1}}
\end{aligned}$$ for all $k$.
If $A= \sum_{q=0}^{\infty} a_pJ^p$ is an infinite-dimensional nondegenerate LTT matrix and $X$ is a complex number, then $A$ raised to the power $X$ is $$\begin{aligned}
A^X = \sum_{s=0}^{\infty} J^s \sum_{ 0 \leq k_1, \ldots, k_s \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_s \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_s^{k_s}
\end{aligned}$$
$Proof$: As before, we first consider the finite-sum case. Let $z$ be some nonzero complex number. We then define ${\bf Z }$ as the infinite row vector $$\begin{aligned}
{\bf Z } = (1, ~z,~z^2,~z^3, \ldots ).
\end{aligned}$$ Multiplying [**Z** ]{} by $J$ on the right, we have ${\bf Z} J = z {\bf Z}$. Therefore, ${\bf Z} J^q = z^q {\bf Z}$ and, by linearity, $ {\bf Z} (\sum c_q J^q) =( \sum c_q z^q) {\bf Z}$. Then, if $F(z)$ is a holomorphic, nonzero function of $z$ in a neighborhood $U$ of the origin, $ {\bf Z} F(J) = F(z) {\bf Z}$ for all $ z \in U$, where $F(J)$ is the LTT matrix obtained by replacing $z$ by $J$ in the Taylor series expansion of $F(z)$. Since $ (\sum_{q=0}^{p} a_pz^p)^X ,~ a_0 \neq 0$ is such a function, we can write, $$\begin{aligned}
{\bf Z} \left( \sum_{q=0}^p a_qJ^q \right)^X &=& \left( \sum_{q=0}^{p} a_qz^q \right)^X {\bf Z} \\
&=& \sum_{s=0}^{\infty} z^s \sum_{ 0 \leq k_1, \ldots, k_p \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p} ~ {\bf Z} \nonumber \\
&=& {\bf Z} ~ \sum_{s=0}^{\infty} J^s \sum_{ 0 \leq k_1, \ldots, k_p \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p} .\nonumber
\end{aligned}$$ $z$ is otherwise arbitrary, so we have $$\begin{aligned}
\left( \sum_{q=0}^p a_qJ^q \right)^X &=& \sum_{s=0}^{\infty} J^s \sum_{ 0 \leq k_1, \ldots, k_p \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right)
\left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p} .
\end{aligned}$$ For fixed $s < p$, $k_{s+1} = \cdots =k_p =0$ in the sums over the $k$’s as a result of the restriction $ s=k_1+ \cdots + pk_p$: $$\begin{aligned}
\left( \sum_{q=0}^p a_qJ^q \right)^X &=& \sum_{s=0}^{p-1} J^s \sum_{ 0 \leq k_1, \ldots, k_s \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right) \left( \begin{array}{c} K \\ k_1, \ldots, k_s \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_s^{k_s} \nonumber \\
&+& \sum_{s=p}^{\infty} J^s \sum_{ 0 \leq k_1, \ldots, k_p \leq s}
\left( \begin{array} {c} X \\ K \end{array} \right) \left( \begin{array}{c} K \\ k_1, \ldots, k_p \end{array} \right)
\delta_{s, \sum mk_m} ~a_0^{X-K} a_1^{k_1} \cdots a_p^{k_p} .
\end{aligned}$$ The equality in the lemma is then demonstrated by letting $p \rightarrow \infty$ in this equation. QED\
\
We now have the necessary machinery to prove the second part of Theorem 1.\
\
$Proof~ of ~Thm~1, {\rm (II)}$: From the proof of Lemma 3, the coefficients in the expansion of $ \left(\sum a_p J^p \right)^X$ in powers of $J$ are the same as in the expansion of $ \left(\sum a_p z^p \right)^X$ in powers of $z$; i.e., $$\begin{aligned}
\sum_{q=0}^{\infty} c_q z^q = \left( \sum_{q=0}^{\infty} a_q z^q \right)^X ~~
{\rm iff}~~ \sum_{q=0}^{\infty} c_q J^q = \left( \sum_{q=0}^{\infty} a_q J^q \right)^X
\end{aligned}$$ We can therefore use Lemma 1, written in the form $$\begin{aligned}
b_p &=& \frac{1}{a_0^{p+1}} \left| \left( \begin{array}{cccccc}
a_0 & ~ & ~ & ~& ~ &~ \\
a_1& a_0 & ~ & \mbox{{\rm \Huge 0}} &~ & ~\\
\vdots & ~& \ddots &~& ~ & ~ \\
a_{p-1} &~&~&~& a_0& ~ \\
a_{p} & a_{p-1} & \cdots &~& a_1& ~a_0 \end{array}
\right)
+ \left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0 ~\\
~\vdots ~\\
0 \\
-a_0 \end{array} \right) \right| ,
\end{aligned}$$ (taking $k=p$), with the replacement $ A= \sum a_p J^p \rightarrow A= \left(\sum a_p J^p \right)^{-X}$ (and thus $a_0 \rightarrow 1/a_0^X$), to solve for $c_p^{(X)}$ as a determinant in the form (II) in the theorem. QED.
In a straightforward fashion, eq. (23) generalizes to $$\begin{aligned}
\sum_{p=0}^{\infty} c_p z^p &=& \left( \sum_{p=0}^{\infty} a_p^{(1)}z^p \right) ^{X_1}
\cdots \left( \sum_{p=0}^{\infty} a_p^{(n)}z^p \right) ^{X_n} \nonumber \\
{ \rm iff}~~\sum_{p=0}^{\infty} c_p J^p &=& \left( \sum_{p=0}^{\infty} a_p^{(1)}J^p \right) ^{X_1}
\cdots \left( \sum_{p=0}^{\infty} a_p^{(n)}J^p \right) ^{X_n} .
\end{aligned}$$ The $c_p$ coefficients are then: $$\begin{aligned}
c_p &=& \left( (a_0^{(1)})^{X_1} \cdots (a_0^{(n)})^{X_n} \right)^{p+1} \left|
\left( \begin{array}{cccc}
a_0^{(1)} & ~ & ~ & ~ \\
a_1^{(1)}& a_0^{(1)} &~ \mbox{{\rm \Huge 0}} & ~ \\
\vdots & ~& \ddots ~~&~ \\
a_{p}^{(1)} & a_{p-1}^{(1)} & \cdots & a_0^{(1)} \end{array}
\right)^{-X_1}\cdots
\left( \begin{array}{cccc}
a_0^{(n)} & ~ & ~ & ~ \\
a_1^{(n)}& a_0^{(n)} &~ \mbox{{\rm \Huge 0}} & ~ \\
\vdots & ~& \ddots ~~&~ \\
a_{p}^{(n)} & a_{p-1}^{(n)}& \cdots & a_0^{(n)} \end{array}
\right)^{-X_n}\right. \nonumber\\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\left.
+ \left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
- (a_0^{(1)})^{-X_1}\cdots (a_0^{(n)})^{-X_n} \end{array} \right) \right|.
\end{aligned}$$ Gradshteyn and Ryzhik [@Grad] give an equivalent expression for these coefficients for the case $n=2,~X_1=1, ~X_2 =-1$.
Returning to the partition function: MacMahon’s recurrence relation [@Partition], $$\begin{aligned}
p(n) -p(n-1) -p(n-2) + p(n-5)+p(n-7) -p(n-12) - p(n-15) + \cdots =0,
\end{aligned}$$ follows directly from expression (11); setting $k=n$ in that equation, the sum on the right in (27) is equal to the determinant $$\begin{aligned}
\left| \begin{array} {ccccccc}
~~1 & ~~0 & ~~0 & ~~0& ~~\cdots & ~~0 &~~1 \\
-1 & ~~1 & ~~0 & ~~0 &~&~& -1 \\
-1 & -1 & ~~1 &~~0 &~& ~ & -1 \\
~~0 & -1 & -1 & ~~1&~& ~& ~~0 \\
~~\vdots & ~ & ~ &~&\ddots & ~& ~~\vdots \\
~ & ~ & ~ &~&~& ~~1 &~~d_{n-1}\\
d_n & d_{n-1} &d_{n-2} &d_{n-3}& \cdots & -1 & ~~d_n
\end{array} \right| ,
\end{aligned}$$ which is zero since the first and last columns are equal.
Expressed in terms of the $J$ matrices, expression (12) for $p(n)$ is $$\begin{aligned}
p(n) &=& \det \left[ -J^T + \sum_{m>0}^{q_m < n+1} (-1)^{m+1} \left( ~J^{(m-1)(3m+2)/2} + J^{(m+1)(3m-2)/2}~ \right) \right]_{n \times n}\end{aligned}$$ where $J^T$ is the transpose of $J$ and the notation $[ ~~]_{n \times n}$ means the $n \times n$ truncation of an infinite-dimensional matrix. And, by relation (23), we have the compact matrix equivalent of $p(n)$’s generating-function equation: $$\begin{aligned}
\left( \begin{array} {ccccc}
p(0) & 0 & 0 & \cdots &~~0 \\
p(1) & p(0) & 0 &~& ~~0 \\
p(2) & p(1) & p(0) &~& ~~0 \\
~~\vdots & ~ & ~ &\ddots & ~~\vdots \\
p(k) & p(k-1) & p(k-2) & \cdots & ~~p(0)
\end{array} \right) &=& \left( \begin{array} {ccccc}
~~1 & ~~0 & ~~0 & ~~\cdots &~~0 \\
-1 & ~~1 & ~~0 &~& ~~0 \\
-1 & -1 & ~~1 &~& ~~0 \\
~~\vdots & ~ &~~ &\ddots & ~~\vdots \\
d_k & d_{k-1} &d_{k-2} & \cdots & ~~1
\end{array} \right)^{-1} .
\end{aligned}$$
The generating function for the number of partitions in which no part occurs more than $D$ times is [@Partition] $$\begin{aligned}
\prod_{k=1}^{\infty} \frac { 1-x^{(D+1)k} } {1-x^k} = \frac { 1-x^{D+1} - x^{2(D+1)} + x^{5(D+1)} + x^{7(D+1)} - \cdots }
{1-x-x^2 + x^5 +x^7 - \cdots }.\end{aligned}$$ We have then from (26), with $X_1 =-X_2 =1$, $$\begin{aligned}
p_D(n) = \left| \begin{array} {ccccc}
~~1 & ~~0 & ~~\cdots & ~~0 &~~ t_0 \\
-1 & ~~1 &~&~~0& ~~ t_1 \\
~~\vdots & ~ &\ddots & ~& ~~\vdots \\
d_{n-1} & d_{n-2} & ~ & ~~1 &~~ t_{n-1} \\
d_n & d_{n-1} & \cdots & -1 & ~~ t_{n}
\end{array} \right| ,
\end{aligned}$$ where $$\begin{aligned}
t_q = \left\{ \begin{array} {l} (-1)^m ~{\rm if~} q=(D+1)q_m, ~m= 0, \pm 1, \pm 2, \ldots
~~~~~~~\\~~~ 0 ~~{\rm otherwise}. \end{array} \right.
\end{aligned}$$ By expanding this determinant by minors along the last column, and using the expression (11), we get the relation $$\begin{aligned}
p_D(n) = p(n) + \sum_{m} (-1)^m p(n- (D+1)q_m) ,
\end{aligned}$$ which is a generalization of (27), the $D=0$ case. If we now take $D=1$, $p_1(n) \equiv q(n)$ = the number of partitions of $n$ into distinct integers (A000009). This is also, from a result due to Euler [@PartitionQ], the number of partitions of $n$ into $odd$ integers, and so we have for the odd partition function $$\begin{aligned}
q(n) = p(n) - p(n-2)-p(n-4) + p(n-10) + \cdots +(-1)^m p(n-2p_m) + \cdots .
\end{aligned}$$
Application to Bernoulli, Euler and Stirling numbers
====================================================
Vella’s expression for the $n$th Bernoulli number, (eq.(a) in his Theorem 11) is, in my notation, $$\begin{aligned}
B_n &=& \sum_{K=0}^n \frac{ (-1)^K }{1+K} \sum_{1 \leq q_1, \ldots , q_K \leq n} \left( \begin{array}{c} n \\
q_1, \ldots , q_K \end{array} \right) .
\end{aligned}$$ Applying Part I of Theorem 1 to the generating function (2a) for Bernoulli numbers, we have
$$\begin{aligned}
B_n &=& n!\sum_{0 \leq k_1, \cdots ,k_n \leq n }
\left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \delta_{n,\sum_{m=1}^n mk_m }
\left( \frac {-1~}{2!} \right)^{k_1} \left( \frac{-1~}{3!} \right)^{k_2} \cdots \left( \frac {-1~}{(n+1)!} \right)^{k_n} .
\end{aligned}$$
As a sum over compositions, this expression becomes $$\begin{aligned}
B_n &=& \sum_{K=0}^n (-1)^K \sum_{1 \leq q_1, \ldots , q_K \leq n}
\frac{1}{(q_1+1) \cdots (q_K+1)} \ \left( \begin{array}{c} n \\
q_1, \ldots , q_K \end{array} \right),
\end{aligned}$$
to be compared to Vella’s result above.
Another expression for the (even-numbered) Bernoulli numbers is obtained from the generating function $$\begin{aligned}
\sum _{q=0}^{\infty} \frac{(2-2^{2q})B_{2q}}{(2q)!}~ z^{2q} &=& \frac{z}{\sinh z}.
\end{aligned}$$ From this we have
$$\begin{aligned}
B_{2p} &=& \frac{(2p)!}{2-2^{2p}}\sum_{0 \leq k_1, \ldots, k_p \leq p}
\left( \begin{array}{c} K \\ k_1, \ldots , k_p \end{array} \right)
\delta_{p,\sum_{m=1}^p mk_m } \left( \frac{-1~}{3!} \right)^{k_1} \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times \left( \frac{-1~}{5!} \right)^{k_2}
\cdots \left( \frac{-1~}{(2p+1)!} \right)^{k_p} , \\
&=& \frac{1}{2-2^{2p}} \sum_{K=0}^p (-1)^K \sum_{1 \leq q_1, \cdots , q_K \leq p}
\frac{1}{(2q_1+1) \cdots (2q_K+1)} \left( \begin{array}{c} 2p \\
2q_1, \ldots , 2q_K \end{array} \right),
\end{aligned}$$
while from the generating function for the Euler numbers (A000364), eq.(2b), we obtain the two expressions,
$$\begin{aligned}
E_{2p}&=& (2p)! \sum_{0 \leq k_1, \ldots, k_p \leq p}~ \left( \begin{array}{c} K \\ k_1, \ldots , k_p \end{array} \right)
\delta_{p,\sum_{m=1}^p mk_m } \left( \frac{-1~}{2!} \right)^{k_1} \left( \frac{-1~}{4!} \right)^{k_2}
\cdots \left( \frac{-1~}{(2p)!} \right)^{k_p} , \\
&=& \sum_{K=0}^p (-1)^K \sum_{1 \leq q_1, \ldots , q_K \leq p} \left( \begin{array}{c} 2p \\
2q_1, \ldots , 2q_K \end{array} \right).
\end{aligned}$$
$B_{2p}$ and $E_{2p}$ can thus both be expressed as sums over the even compositions of $2p$. Since there are no even compositions of odd numbers, these expressions can be extended to include all of the odd-numbered numbers except for $B_1$. We then have, more generally, $$\begin{aligned}
B_{n} &=& \frac{1}{2-2^{n}} \sum_{K=0}^{ \lfloor n/2 \rfloor } (-1)^K \sum_{1 \leq q_1, \ldots , q_K \leq \lfloor n/2 \rfloor }
\frac{1}{(2q_1+1) \cdots (2q_K+1)} \left( \begin{array}{c} n \\
2q_1, \ldots , 2q_K \end{array} \right),\\ && ( n \neq 1) ,\nonumber
\end{aligned}$$ $$\begin{aligned}
E_{n} &=& \sum_{K=0}^{ \lfloor n/2 \rfloor } (-1)^K \sum_{1 \leq q_1, \ldots , q_K \leq \lfloor n/2 \rfloor }
\left( \begin{array}{c} n \\ 2q_1, \ldots , 2q_K \end{array} \right)
\end{aligned}$$ This expression for $E_{n}$, in a different notation, was previously derived by Vella, (eq. (c) in Thm. 11 [@Vella]). $E_n$ can however also be expressed as a sum over the $odd$ partitions/compositions of $n-1$:
For n $>$1, $$\begin{aligned}
E_{n} = \sum_{N=1}^{\lfloor n/2 \rfloor } (-1)^N \sum_{1 \leq q_1, \ldots ,q_{2N-1} \leq \lfloor n/2 \rfloor }
\left( \begin{array}{c} n-1\\ 2q_1-1, \ldots , 2q_{2N-1}-1 \end{array} \right)
\end{aligned}$$
\
$Proof.$ The sum is over all compositions of $n-1$ that contain an odd number of odd parts, which is an empty set if $n-1$ is even. Therefore $E_n =0$ for odd $n$, and we only have to prove this equation for even $n$. The equality is trivially true for $n=2$. We will show that both $E_{2p}$ and $$\begin{aligned}
a_{2p} \equiv \sum_{N=1}^p (-1)^N \sum_{1 \leq q_1, \ldots ,q_{2N-1} \leq p} \left( \begin{array}{c} 2p-1\\
2q_1-1, \ldots , 2q_{2N-1}-1 \end{array} \right)
\end{aligned}$$ satisfy the same recursion relation $$\begin{aligned}
a_{2p} = - 1 -\sum_{q=1}^{p-1} \left( \begin{array}{c} 2p-1 \\ 2q \end{array} \right) 2^{2q-1} a_{2p-2q}.
\end{aligned}$$ Then the equality would, by induction, be valid for all $p$.
To prove this for $E_{2p}$, consider the expression $$\begin{aligned}
\frac{d}{dz} ~\frac{1}{\cosh z} + 2 \sinh z + \cosh 2z ~ \frac{d}{dz} ~\frac{1}{\cosh z} .
\end{aligned}$$ This is equal to zero as a result of the identity $\cosh 2z = 2 \cosh ^2z-1$. Expanding out $\sinh z$ and $1/\cosh z$ and performing the differentiation in the 1st and 3rd terms, we have, for each power of $z$, $$\begin{aligned}
\frac{ E_{2p}}{(2p-1)!} + \frac{ 2}{(2p-1)!} + \sum_{q=0}^{p-1} \frac{2^{2q} E_{2p-2q}}{(2q)!(2p-2q-1)!} =0,
\end{aligned}$$ which can be rearranged into the form of the recursion relation (43) above.
On the other hand we have $$\begin{aligned}
a_{2p} &=&- 1 + \sum_{N=2}^p (-1)^N
\sum_{q_1=1}^{p-1} \sum_{q_2=1}^{p-1} \sum_{k_1=1}^{p-1} \cdots \sum_{k_{2N-3}=1}^{p-1}
\left( \begin{array}{c} 2p-1 \\ 2q_1-1, 2q_2-1, 2k_1-1, \ldots , 2k_{2N-3}-1 \end{array} \right), ~\nonumber \\
\end{aligned}$$ where we’ve separated off the $N=1$ term and made the substitutions $q_3 \rightarrow k_1, \ldots,
q_{2N-1} \rightarrow k_{2N-3}$. We have then, with $N = L+1$, $$\begin{aligned}
a_{2p} &=& - 1+\sum_{q_1=1}^{p-1}\sum_{q_2=1}^{p-1} \frac{1} {(2q_1-1)!(2q_2-1)!}~ \frac{ (2p-1)! }
{(2p-2q_1-2q_2+1)!} \nonumber\\
&& ~~~~~~~~~ \times \sum_{L=1}^{p-q_1-q_2 +1} (-1)^{L+1} \sum_{k_1=1}^{p-q_1-q_2+1}
\cdots \sum_{k_{2L-1}=1}^{p-q_1-q_2+1}
\left( \begin{array}{c} 2p-2q_1-2q_2+1 \\2k_1-1, \ldots , 2k_{2L-1}-1 \end{array} \right).
\end{aligned}$$ Setting $q =q_1+q_2-1$, this is $$\begin{aligned}
a_{2p} &=& - 1 -\sum_{q=1}^{p-1} ~\sum_{q_1=1}^{q} \left( \begin{array}{c} 2q \\ 2q_1-1 \end{array} \right) ~
\frac{(2p-1)!}{(2q)!(2p-2q-1)! } \nonumber \\
&& ~~~~~~~~~~~~~ \times \sum_{L=1}^{p-q} (-1)^{L} \sum_{k_1=1}^{p-q}
\cdots \sum_{k_{2L-1}=1}^{p-q} \left( \begin{array}{c} 2p-2q-1 \\2k_1-1, \ldots , 2k_{2L-1}-1 \end{array}
\right).
\end{aligned}$$ The sum over $q_1$ equals $2^{2q-1}$, evaluated by expanding $(1+1)^{2q} -(1-1)^{2q}$ binomially. The remaining sum over $L$ and $k_1, \ldots, k_{2L-1}$ is $a_{2p-2q}$, and we again get relation (43). QED
The sum-over-partitions form follows in a straightforward fashion: $$\begin{aligned}
E_{2p} &=& (-1)^{p-1} (2p-1)! \sum_{0 \leq k_1, \ldots, k_p \leq 2p-1}
\left( \begin{array} {c} K \\ k_1, \ldots , k_p \end{array} \right)
\delta_{2p-1,\sum (2m-1)k_m } \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \left( \frac{-1~}{1!} \right)^{k_1} \left( \frac{1}{3!} \right)^{k_2}
\cdots \left( \frac{(-1)^p}{(2p-1)!} \right)^{k_p} .
\end{aligned}$$ The sign of each term in this sum is $(-1)^{p-1+ \sum mk_m}$, while the sign in the sum-over-compositions form is $(-1)^N$. To check that the signs agree, note that $2p-1 = \sum (2m-1)k_m = 2 \sum mk_m -K$. But $K$ is the number of factorials in the denominator of each term, which in the sum-over-compositions form is $2N-1$. So $ \sum mk_m = N+p-1$ and $(-1)^{p-1+ \sum mk_m} = (-1)^N$.
As an example of the even and the odd expansions for $E_{n}$, we have: $$\begin{aligned}
E_{10} &=&10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!}
- \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \nonumber \\
&=&~9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!}
+\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) = -50,521
\end{aligned}$$
In a similar fashion, Theorem 1 can be applied to the Euler and Bernoulli polynomials, using their generating functions:
$$\begin{aligned}
\sum_{q=0}^{\infty} \frac {E_q(x)}{q!}~ z^q &=& \frac{2e^{xz}}{e^z+1}
= 2\left\{ \sum _{k=0}^{\infty} z^k~ \frac{(1-x)^k+(-x)^k}{k!} \right\}^{-1},\\
\sum_{q=0}^{\infty} \frac{ B_q(x)}{q!} ~z^q &=& \frac{ze^{xz}}{e^z-1}
= \left\{ \sum_{k=0}^{\infty} z^k ~\frac{( 1-x)^{k+1} - (-x)^{k+1}}{(k+1)!} \right\}^{-1}.
\end{aligned}$$
The results are straightforward and we omit writing out the explicit expressions.
Bell polynomials [@Bell] are defined as $$\begin{aligned}
B_{n,k}(x_1, \ldots , x_{n-k+1}) &=& \sum_{0\leq k_0, \ldots, k_{n-k} \leq n} \delta_{k,k_0+\cdots +k_{n-k}}
\delta _{n,\sum_m mk_m} \nonumber \\
&& ~~~~~~~\times \frac{n!}{k_0! \cdots k_p!}~\left( \frac{x_1}{1!} \right) ^{k_0} \cdots
\left( \frac{x_{n-k+1}}{(n-k+1)!} \right)^{k_{n-k}} .
\end{aligned}$$ Stirling numbers of the 2nd kind are equal to the values of these polynomials at $x_1=x_2= \cdots =x_k=1$: $$\begin{aligned}
S(n,n-p) &=& \left. B_{n,n-p}(x_1,x_2, \ldots , x_{p+1}) \right|_{x_1= \cdots =x_{p+1} =1} \nonumber \\
&=& \frac{n!}{ (n-p)!} \sum_{0\leq k_1, \ldots, k_p \leq p} \left( \begin{array}{c} n-p\\K \end{array} \right)
\left( \begin{array} {c} K \\ k_1, \ldots , k_p \end{array} \right)
\delta _{p,\sum mk_m} \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\times \left( \frac{1}{2!} \right)^{k_1} \left( \frac{1}{3!} \right)^{k_2}
\cdots \left( \frac{1}{(p+1)!} \right)^{k_p} ,
\end{aligned}$$ where we’ve set $k=n-p$ and in the last line we’ve summed over $k_0$. A similar expression for Stirling numbers of the 1st kind can be found using their relation to the $n$-th order Bernoulli numbers [@Korn], $$\begin{aligned}
s(n,n-p)= \left( \begin{array}{c}
n-1\\
p \end{array} \right) B_{p}^{(n)},
\end{aligned}$$ which have the generating function $$\begin{aligned}
\sum_{q=0}^{\infty} \frac{B_q^{(k)}}{q!} ~ z^q = \frac{z^k}{(e^z-1)^k} .
\end{aligned}$$ Using Theorem 1, we get for $s(n,n-p)$: $$\begin{aligned}
s(n,n-p) &=& \frac{(n-1)!}{(n-p-1)!}\sum_{0 \leq k_1, \ldots , k_{p} \leq p}
\left( \begin{array}{c} n+K-1\\K \end{array} \right)
\left( \begin{array} {c} K \\ k_1, \ldots , k_p \end{array} \right) \delta_{p,\sum mk_m} \nonumber \\
&& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \left( \frac{-1~}{2!} \right)^{k_1}
\left( \frac{-1~}{3!} \right)^{k_2} \cdots \left( \frac{-1}{(p+1)!} \right)^{k_p} .
\end{aligned}$$
Laplace’s formula (3) for the Bernoulli numbers corresponds to expression (36a) and follows from equation (2a) and Part (II) of Theorem 1. Other matrix representations we can derive from this theorem are:
$$\begin{aligned}
B_{2p} &=& - \frac{(2p)!}{2^{2p}-2} \left| \begin{array}{cccccc}
1 & 0 & 0 & ~ & \cdots & ~ 1 \\
\frac{1}{3!} & 1 & 0 &~& ~ & ~0 \\
\frac{1}{5!} & \frac{1}{3!} & 1 & ~ & ~& ~\vdots \\
\vdots & ~ & ~ & \ddots &~& ~ \\
\frac{1}{(2p+1)!}&\frac{1}{(2p-1)!} &\frac{1}{(2p-3)!} & \cdots & \frac{1}{3!} ~& 0
\end{array} \right| ; \\
E_{2p} &=& (2p)! ~~\left| \begin{array}{cccccc}
1 & 0 & 0 & ~ & \cdots & ~ 1 \\
\frac{1}{2!} & 1 & 0 &~& ~ & ~0 \\
\frac{1}{4!} & \frac{1}{2!} & 1 & ~ & ~& ~\vdots \\
\vdots & ~ & ~ & \ddots &~& ~ \\
\frac{1}{(2p)!}&\frac{1}{(2p-2)!} &\frac{1}{(2p-4)!} & \cdots & \frac{1}{2!} ~& 0
\end{array} \right| ; \\
S(n,n-p) &=& \frac{n!}{(n-p)!} ~\left| ~I + \left( \begin{array}{cccc}
1 & ~ & ~ & ~ \\
\frac{1}{2!}& 1 &~~~~ \mbox{{\rm \Huge 0}} & ~ \\
\vdots & ~& ~~\ddots &~ \\
\frac{1}{(p+1)!} & \frac{1}{p!} & ~~\cdots & ~~~~~1 \end{array}
\right)^{\mbox{\it{n-p}}} \times
\left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
- 1 \end{array} \right) ~ \right|;\\
s(n,n-p) &=& \frac{(n-1)!}{(n-p-1)!}
\left|~ \left( \begin{array}{cccc}
1 & ~ & ~ & ~ \\
\frac{1}{2!}& 1 &~~~~ \mbox{{\rm \Huge 0}} & ~ \\
\vdots & ~& ~~\ddots &~ \\
\frac{1}{(p+1)!} & \frac{1}{p!} & ~~\cdots & ~~~~~1 \end{array}
\right)^{\mbox{\it{n}}} +
\left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
- 1 \end{array} \right) ~ \right|.
\end{aligned}$$
Note that the size of the determinant for $B_{2p}$ in (56a) is $(p+1)\times(p+1)$, (as is the one for $E_{2p}$ in (56b)), compared with $(2p+1)\times(2p+1)$ using Laplace’s formula.
Conclusion
==========
We have derived compact, closed-form expressions for partition functions and for Bernoulli, Euler and Stirling numbers that contain only “simple” numbers and that require either finite summations or finding the determinants or inverses of matrices. In particular, the partition function $p(n)$ is given directly by a sum of integers, the number of terms in the sum being the value at $n$ of the pentagonal partition function.
I wish to thank Yonko Millev for invaluable assistance.
$p(5k+4), ~p(7k+5)$, and $p(25k+24)$ determinants
==================================================
For $n= 5k+4, 7k+5$, or $25k+24$, the dimension of the matrix in eq.(12) can be reduced to $k+1$ by using one of the identities below by Ramanujan [@Partition; @Ram]:
$$\begin{aligned}
\sum_{k=0}^{\infty} p(5k+4)q^k &=& 5~ \frac{(q^5)^5_{\infty}}{(q)^6_{\infty} },\\
\sum_{k=0}^{\infty} p(7k+5)q^k &=& 7~ \frac{(q^7)^3_{\infty}}{(q)^4_{\infty} } + 49q~ \frac{(q^7)^7_{\infty}}{(q)^8_{\infty} },\\
\sum_{k=0}^{\infty} p(25k+24)q^k &=& 5^2 \cdot 63 \frac{(q^5)^6_{\infty}}{(q)^7_{\infty} }
+ 5^5 \cdot 52q~ \frac{(q^5)^{12}_{\infty}}{(q)^{13}_{\infty} } +5^7 \cdot 63q^2~ \frac{(q^5)^{18}_{\infty}}{(q)^{19}_{\infty} }
\nonumber \\
&&~~~~ + 5^{10} \cdot 6q^3~ \frac{(q^5)^{24}_{\infty}}{(q)^{25}_{\infty} }
+ 5^{12} \cdot q^4~ \frac{(q^5)^{30}_{\infty}}{(q)^{31}_{\infty} } ,
\end{aligned}$$
where $(q)_{\infty} \equiv \prod_{m=1}^{\infty} (1-q^m)$. The first identity, combined with eq.(26), gives us
$$\begin{aligned}
\frac{ p(5k+4)} {5} &=& \left| \begin{array} {c} ~\\~\\~\\~\\~ \end{array} \left( \sum_{m \geq 0} (-1)^m(2m+1)J^{m(m+1)/2}
\right)^2 \right. \nonumber \\
&& + \left. \left( I+ \sum_{m=\pm1,\pm2, \ldots }(-1)^{m} J^{5m(3m-1)/2} \right)^5 \times \left( \left.
\begin{array}{cccc}
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
~0~ \\
- 1 \end{array} \right) \right|_{ (k+1) \times (k+1)} \nonumber \\
&=& \left| \begin{array}{cccccccccc}
~1& ~ & ~&~&~&~&~&~&~&~1\\
-6& ~1& ~ & ~&~&~&~&~&~&~0\\
~9& -6& ~1& ~ & ~&~&~&~&~&~0\\
~10& ~9& -6& ~1 & ~ & ~&~&~&~&~0\\
-30 & ~10 & ~9& -6&~1 & ~ &~&~& ~&~0\\
~0& -30 & ~10& ~9& -6&~1& ~ &~&~& -5\\
~11& ~0& -30 & ~10& ~9& -6&~1 & ~ &~&~0\\
~42& ~11& ~0& -30 & ~10& ~9& -6&~ & ~ &~0\\
~ \vdots & ~&~&~&~&~&~&~~\ddots &~& ~\vdots
\end{array} \right|_{(k+1) \times (k+1) },
\end{aligned}$$
while the second one gives $$\begin{aligned}
\frac{ p(7k+5)} {7} &=& \left| \begin{array} {c} ~\\~\\~\\~\\~ \end{array} \left(I + \sum_{m=\pm1,\pm2, \ldots } (-1)^{m}
J^{m(3m-1)/2} \right)^8 \right. \nonumber \\
&& + \left\{ \left( \sum_{m \geq 0} (-1)^m(2m+1)J^{7m(m+1)/2} \right) \left(I + \sum_{m=\pm1,\pm2, \ldots }
(-1)^{m} J^{m(3m-1)/2} \right)^4 \right. \nonumber \\ && + \left. \left. 7J \left(I + \sum_{m=\pm1,\pm2, \ldots } (-1)^{m}
J^{7m(3m-1)/2} \right)^7 \right\}
\times \left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
~0~ \\
- 1 \end{array} \right) \right|_{ (k+1) \times (k+1)} \nonumber \\
&=& \left| \begin{array}{cccccccccc}
~1& ~ & ~&~&~&~&~&~&~&~1\\
-8& ~1& ~ & ~&~&~&~&~&~&~3\\
~20& -8& ~1& ~ & ~&~&~&~&~&~2\\
~0& ~20& -8& ~1 & ~ & ~&~&~&~&~8\\
-70 & ~0 & ~20& -8&~1 & ~ &~&~& ~&-5\\
~64& -70 & ~0& ~20& -8&~1& ~ &~&~& -4\\
~56& ~64& -70 & ~0& ~20& -8&~1 & ~ &~&-10\\
~ 0&~56& ~64& -70 & ~0& ~20& -8&~ & ~ &~5\\
~\vdots & ~&~&~&~&~&~&~~\ddots &~& ~ \vdots
\end{array} \right|_{(k+1) \times (k+1)}
\end{aligned}$$
(where we’ve used the Jacobi identity $(1-z-z^2+z^5+z^7 - \cdots)^3= 1-3z+5z^3-7z^6+9z^{10} + \cdots $). The matrices above thus consist of an LTT “base” part and a “tower” part. The coefficients of the base matrix for $p(5k+4)$ correspond to sequence A000729, the coefficients in the expansion of $\left( \prod_k (1-x^k)\right)^6$, while the coefficients of powers of $J^5$ in the tower part is sequence A000728. Likewise, sequence A000731 gives the base matrix for $p(7k+5)$, while the tower part involves a combination of sequences A000727, A000730, and A010816.
From the 3rd identity, the coefficients for the base matrix for $p(25k+24)$ are given by the expansion $$\begin{aligned}
(q)^{31}_{\infty} &=& 1 -31q+434q^2-3565q^3
+18445q^4 - 57505q^5 +70091q^6 + 227447 q^7 + \cdots ,
\end{aligned}$$ (sequence A010836); the tower part is from the expansion $$\begin{aligned}
&& 63 ~ (q)^{24}_{\infty} (q^5)^{6}_{\infty} + 5^3 \cdot 52q ~(q)^{18}_{\infty} (q^5)^{12}_{\infty}
+5^5 \cdot 63q^2 ~(q)^{12}_{\infty} (q^5)^{18}_{\infty} \nonumber \\ && ~~~~+ 5^{8} \cdot 6q^3 ~(q)^{6}_{\infty}
(q^5)^{24}_{\infty} + 5^{10} \cdot q^4~ (q^5)^{30}_{\infty} \\
&& ~~~~~= 63 + 4988q + 95751q^2 +766014q^3 + 3323665q^4 + 8359848q^5 \nonumber \\
&& ~~~~~~+ 10896075 q^6 -6659766 q^7 + \cdots ,\nonumber
\end{aligned}$$ and so we have, to this order, $$\begin{aligned}
\frac{p(25k+24)}{25} = \left| \begin{array}{cccccccccc}
~1& ~ & ~&~&~&~&~&~&~&~63\\
-31& ~1& ~ & ~&~&~&~&~&~&~4988\\
~434& -31& ~1& ~ & ~&~&~&~&~&~95751\\
-3565& ~434& -31& ~1 & ~ & ~&~&~&~&~766014\\
~18445 & -3565 & ~434& -31&~1 & ~ &~&~& ~&~3323665\\
-57505& ~18445 & -3565& ~434& -31&~1& ~ &~& ~&~8359848\\
~70091& -57505& ~18445 & -3565& ~434& -31&~1 & ~ &~&~10896075\\
~ 227447 & ~70091& -57505& ~18445 & -3565& ~434& -31& ~ &~& -6659766~\\
~\vdots & ~&~&~&~&~& ~&~~\ddots &~& ~\vdots
\end{array} \right|_{ (k+1) \times (k+1)}
\end{aligned}$$ We list below some sample calculations of partition functions using these determinants: $$\begin{aligned}
p(24) &=&5 \cdot \left| \begin{array}{ccccc} ~1 &~&~&~&~1 \\ -6 & ~1& ~&~&~0\\ ~9 & -6& ~1& ~&~0\\~10&~9&-6&~1&~0\\
-30&~10&~9& -6&~0 \end{array} \right| = 1575; \\ ~\\
p(40) &=& 7 \cdot \left| \begin{array}{cccccc} ~1 &~&~&~&~&~1 \\ -8 & ~1& ~&~&~&~3\\ ~20 & -8& ~1& ~&~&~2\\
~0&~20&-8&~1&~& ~8\\-70&~0&~20& -8&~1&-5\\ ~64&-70&~0&~20& -8&-4
\end{array} \right| = 37338; \\ ~\\
p(199) &=& 25 \cdot \left| \begin{array}{cccccccc}
~1& ~ & ~&~&~&~&~&63\\
-31& ~1& ~ & ~&~&~&~&4988\\
~434& -31& ~1& ~ & ~&~&~&95751\\
-3565& ~434& -31& ~1 & ~ & ~&~&766014\\
~18445 & -3565 & ~434& -31&~1 & ~& ~&3323665\\
-57505& ~18445 & -3565& ~434& -31&~1& ~&8359848\\
~70091& -57505& ~18445 & -3565& ~434& -31& 1 &~10896075\\
~227447 & ~70091& -57505& ~18445 & -3565& ~434& -31 &-6659766\\
\end{array} \right| \\~\\ &=& 3646072432125.
\end{aligned}$$
$p(5k+a)$ and $p(25k+a)$ determinants
======================================
One can ask if it’s possible to fill in some of the gaps in equations (A2a), (A2b) and (A5) and to get expressions for $p(5k+a)$, etc., for other values of $a$. In the following we will consider the problem of generalizing eqs. (A2a) and (A5). Ramanujan [@Ram] derived the relation $$\begin{aligned}
\frac{ (q^5)_{\infty}}{(q^{1/5})_{\infty}}
&=& \frac{(J_1^4+3qJ_2) +q^{1/5}(J_1^3+2qJ_2^2) +q^{2/5} (2J_1^2+qJ_2^3)
+q^{3/5}(3J_1+qJ_2^4) +5q^{4/5}}{J_1^5 -11q + q^2J_2^5}
\end{aligned}$$ (his eq.(20.5)), where he defined the functions $J_1(q)$ and $J_2(q)$ by the equation $$\begin{aligned}
\frac{ (q^{1/5})_{\infty}} {(q^5)_{\infty}}= J_1-q^{1/5}+q^{2/5}J_2.
\end{aligned}$$ $J_1$ and $J_2$ are series expansions in $q$ with integer coefficients and exponents. Ramanujan then proved the identities $$\begin{aligned}
J_1^5 -11q + q^2J_2^5
&=& \frac {(q)^6_{\infty} } { (q^5)^6_{\infty}}; ~~J_1J_2 =-1.
\end{aligned}$$ ($J_1(q)$ and $J_2(q)$ are given by sequences A003823 and A007325, respectively.) From the first identity and from (B1) we have $$\begin{aligned}
\sum_{n=0}^{\infty} p(n) q^{n/5} &=& \left[ (J_1^4+3qJ_2) +q^{1/5}(J_1^3+2qJ_2^2) +q^{2/5} (2J_1^2+qJ_2^3)\right.
\nonumber \\
&& ~~~~~~~~~~ \left. +q^{3/5}(3J_1+qJ_2^4) +5q^{4/5}\right] ~\frac{ (q^5)^5_{\infty}}{(q)^6_{\infty}},
\end{aligned}$$ and so $$\begin{aligned}
\sum_{k=0}^{\infty} p(5k) q^{k} &=& (J_1^4+3qJ_2)~\frac{ (q^5)^5_{\infty}}{(q)^6_{\infty}},\\
\sum_{k=0}^{\infty} p(5k+1) q^{k} &=& (J_1^3+2qJ_2^2) ~\frac{ (q^5)^5_{\infty}}{(q)^6_{\infty}},\\
\sum_{k=0}^{\infty} p(5k+2) q^{k} &=& (2J_1^2+qJ_2^3) ~\frac{ (q^5)^5_{\infty}}{(q)^6_{\infty}},\\
\sum_{k=0}^{\infty} p(5k+3) q^{k} &=& (3J_1+qJ_2^4) ~\frac{ (q^5)^5_{\infty}}{(q)^6_{\infty}},
\end{aligned}$$ in addition to (A1a). Let $$\begin{aligned}
(q^{1/5})_{\infty} = G_1 -q^{1/5}(q^5)_{\infty} + q^{2/5} G_2 ; ~~ J_{1,2} = \frac{ G_{1,2}}{(q^5)_{\infty}}.
\end{aligned}$$ Then, for $a=0,1,2,3,4$, $$\begin{aligned}
\sum_{k=0}^{\infty} p(5k+a) q^{k} &=&~\frac{ 1}{(q)^6_{\infty}} \left[ ~F_{a+1}(q^5)^{a+1}_{\infty} G_1^{4-a}
+F_{4-a} q(q^5)^{4-a}_{\infty} G_2^{a+1} ~\right]
\end{aligned}$$ where $F_n$ is the $n$-th Fibonacci number, with $F_0 =0$. We then have, making the replacement $q \rightarrow J$, $$\begin{aligned}
p(5k+a) &=& \left| ~(J)^6_{\infty} + \left[~F_{a+1}(J^5)^{a+1}_{\infty} G_1^{4-a}(J)
+F_{4-a} J(J^5)^{4-a}_{\infty} G_2^{a+1}(J)~ \right]
\times \left( \left.
\begin{array}{cccc}
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} & ~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0~\\
\vdots\\
~0~ \\
- 1 \end{array} \right) \right|_{ (k+1)\times(k+1)} \nonumber \\
&=& \left| \begin{array}{cccccccccc}
~1& ~ & ~&~&~&~&~&~&~&X_0\\
-6& ~1& ~ & ~&~&~&~&~&~&X_1\\
~9& -6& ~1& ~ & ~&~&~&~&~&X_2\\
~10& ~9& -6& ~1 & ~ & ~&~&~&~&X_3\\
-30 & ~10 & ~9& -6&~1 & ~ &~&~& ~&X_4\\
~0& -30 & ~10& ~9& -6&~1& ~ &~&~& X_5\\
~11& ~0& -30 & ~10& ~9& -6&~1 & ~ &~&X_6\\
~42& ~11& ~0& -30 & ~10& ~9& -6&~ & ~ &X_7\\
~ \vdots & ~&~&~&~&~&~&~~\ddots &~& ~\vdots
\end{array} \right|_{(k+1) \times (k+1)},
\end{aligned}$$ where the elements $X_n(=X_n^{(a)})$ in the tower matrix are determined for each value of $a$ by the expansion $$\begin{aligned}
F_{a+1}(J^5)^{a+1}_{\infty} G_1^{4-a}(J) +F_{4-a} J(J^5)^{4-a}_{\infty} G_2^{a+1}(J)= X_0I +X_1J +X_2J^2 + \cdots
\end{aligned}$$
The $G$’s can be expressed in terms of the Ramanujan theta function: $$\begin{aligned}
G_1(q) = \frac{f(-q^2,-q^3)^2}{ f(-q,-q^2)} ;~~
G_2(q) = -\frac{f(-q,-q^4)^2}{ f(-q,-q^2)} ;~~
f(a,b) = \sum_{n= -\infty}^{\infty} a^{n(n+1)/2}b^{n(n-1)/2}.
\end{aligned}$$ However, it is more convenient to write them as the series expansions $$\begin{aligned}
G_1(q) &=& -1 + \sum_{k=0}^{\infty} q^{k(30k-1)} \left[ 1 + q^{2k} +q^{12k+1} -q^{20k+3} -q^{30k+7}
-q^{32k+8} -q^{42k+14} + q^{50k+20} \right]\\
&=& 1 +q-q^3 -q^7 -q^8-q^{14} + q^{20} + q^{29} + q^{31} + q^{42} - q^{52} + \cdots, \\
G_2(q) &=& \sum_{k=0}^{\infty} q^{k(30k+7)} \left[ -1 + q^{6k+1} -q^{10k+2} +q^{16k+4} +q^{30k+11}
-q^{36k+15} +q^{40k+18} - q^{46k+23} \right] \\
&=& -1 +q-q^2 +q^4+q^{11} -q^{15} + q^{18} - q^{23} - q^{37}+q^{44} -q^{49} +q^{57} + \cdots ,
\end{aligned}$$ which follow directly from (B5). ($G_1(q) $ corresponds to sequence A113681, and $-G_2(q)$ to sequence A116915.) In column-vector form, the first terms in the expansions from eq. (B8) are: $$\begin{aligned}
{\bf X}^{(0)} = \left( \begin{array}{c} ~~1 \\ ~~1 \\ ~~9 \\ -3 \\ -11 \\ -10 \\ ~~10 \\ -10 \\ \vdots~ \end{array} \right);
~~{\bf X}^{(1)} = \left( \begin{array}{c} ~~1 \\ ~~5 \\ -1 \\ ~~4 \\ -10 \\ -7 \\-5 \\ ~~2 \\ \vdots ~\end{array} \right);
~~{\bf X}^{(2)} = \left( \begin{array}{c} ~~2 \\ ~~3 \\ ~~5 \\ -10 \\ ~~3 \\ -9 \\-11 \\ -8 \\ \vdots ~\end{array} \right);
~~{\bf X}^{(3)} = \left( \begin{array}{c} ~~3 \\ ~~4 \\ -4 \\ ~~7 \\ -16 \\ ~~3 \\ -17 \\ -13 \\ \vdots ~\end{array} \right);\end{aligned}$$ with the coefficients for ${\bf X}^{(4)}$ being given in (A2a).
We can generalize eq.(A5) to an expression for $p(25k +a)$ for $a=4,9,14$ and 19 following Ramanujan’s derivation of the identity (A1c)[@Ram]. We make the replacement $q \rightarrow q^{1/5}$ in eq. (A1a) and get $$\begin{aligned}
\sum_{k=0}^{\infty} p(5k+4)q^{k/5} = 5~ \frac{(q)^5_{\infty}}{(q^{1/5})^6_{\infty} }
=5 ~ \frac{(q)^5_{\infty}}{ (q^5)_{\infty}^6 } ~\frac{1} { (J_1-q^{1/5}+q^{2/5}J_2)^6} .
\end{aligned}$$ To simplify the notation, we define $ x \equiv q/J_1^5=-qJ_2^5 $. Then $$\begin{aligned}
J_1-q^{1/5}+q^{2/5}J_2= J_1 (1-x^{1/5}-x^{2/5}) = J_1 (1+x^{1/5}/\phi )(1-\phi~x^{1/5})
\end{aligned}$$ where $\phi = ( \sqrt{5} +1)/2$, the golden ratio. We have that $$\begin{aligned}
\frac{1}{1+ ax^{1/5} } = \frac{1- ax^{1/5} + a^2x^{2/5} - a^3x^{3/5} +a^4x^{4/5}} {1+ a^5x} .
\end{aligned}$$ Then $$\begin{aligned}
\frac{1} { 1-x^{1/5}-x^{2/5}}&=& \frac{1} { (1+x^{1/5}/\phi)(1-\phi x^{1/5})} \nonumber \\ &=&
\frac{1-x^{1/5}/\phi + x^{2/5}/\phi^2- x^{3/5}/\phi^3 +x^{4/5}\/\phi^4} {1+x/\phi^5}
~ \frac{1+\phi x^{1/5} + \phi^2x^{2/5}+\phi^3x^{3/5} +\phi^4x^{4/5}} {1-\phi^5x} \nonumber \\
&=& \frac{ 1-3x + (1+2x) x^{1/5}+ (2-x)x^{2/5}+ (3+x) x^{3/5}+5x^{4/5} } { 1-11x-x^2 }
\end{aligned}$$ The denominator in (B13) is $$\begin{aligned}
1-11x-x^2 = \frac{J_1^5 -11q +q^2J_2^5 } {J_1^5} = \frac{1}{J_1^5} \frac {(q)^{6}_{\infty} } { (q^5)^6_{\infty}}
\end{aligned}$$ The numerator in (B13) is to be raised to the sixth power in eq.(B10). We define the functions $H_n(x)$ to be series expansions in $x$ with integer coefficients and exponents such that $$\begin{aligned}
\left[ 1-3x + (1+2x) x^{1/5}+ (2-x)x^{2/5}+ (3+x) x^{3/5}+5x^{4/5} \right]^6 &=& H_1(x) +H_2(x)x^{1/5}
+ H_3(x)x^{2/5} \nonumber \\
&& ~~~ + H_4(x) x^{3/5} + H_5(x) x^{4/5}
\end{aligned}$$ Expanding the left side of this equation and collecting terms, we get $$\begin{aligned}
H_1(x) &=& -98 x^9 +9939 x^8 -107712 x^7 + 167031 x^6 - 27918 x^5 + 127011 x^4 + 160552 x^3 +
32784 x^2 + 858 x+1\\
H_2(x) &=& ~~27 x^9 -4806 x^8 + 78758 x^7 - 171984 x^6 +98667 x^5 +78986 x^4 + 176592 x^3+52644 x^2
+ 2138 x +6\\
H_3(x) &=& \; -6 x^9 +2138 x^8 -52644 x^7 + 176592 x^6 - 78986 x^5 + 98667 x^4 + 171984 x^3 +
78758 x^2 + 4806 x+27 \\
H_4(x) &=& ~~~~ x^9 - 858 x^8 + 32784 x^7 - 160552 x^6 + 127011 x^5 + 27918 x^4 + 167031 x^3 +
107712 x^2 + 9939 x+98\\
H_5(x) &=& ~~~~~~~~~~~~\, 315 x^8 -18640 x^7 + 139305 x^6 -127020 x^5 + 106425 x^4 + 127020 x^3 +
139305 x^2 + 18640 x+315
\end{aligned}$$ Then, changing back to the variable $q$, eq.(B10) is $$\begin{aligned}
\sum_{k=0}^{\infty} p(5k+4)q^{k/5} &=& 5~ \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }~\left( J_1^{24} H_1(q)
+ J_1^{23}H_2(q)q^{1/5}+ J_1^{22} H_3(q)q^{2/5} +J_1^{21}H_4(q) q^{3/5}+ J_1^{20} H_5(q) q^{4/5}\right)
\end{aligned}$$ and we have $$\begin{aligned}
\sum_{k=0}^{\infty} p(25k+4)q^{k} &=& 5~ \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }
\left[ ~ J_1^{24} + 858~ qJ_1^{19}+ 32784~ q^2J_1^{14} +160552~ q^3J_1^{9} + 127011~ q^4J_1^4 \right. \\
&& \left.~~~~~~~~~~~~ + 27918~ q^5J_2+ 167031 ~q^6J_2^{6} +107712~q^7J_2^{11}+9939~ q^8J_2^{16}
+98~ q^9J_2^{21} \right] \\~ \\
\sum_{k=0}^{\infty} p(25k+9)q^{k} &=& 5~ \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }
\left[ ~ 6J_1^{23} + 2138~ qJ_1^{18}+ 52644~ q^2J_1^{13} +176592~ q^3J_1^{8} + 78986~ q^4J_1^3 \right. \\
&& \left.~~~~~~~~~~~~ + 98667~ q^5J_2^2+ 171984 ~q^6J_2^{7} +78758~q^7J_2^{12}
+4806~ q^8J_2^{17}+ 27~q^9J_2^{22} \right] \\ ~\\
\sum_{k=0}^{\infty} p(25k+14)q^{k} &=& 5~ \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }
\left[ ~ 27J_1^{22} + 4806~ qJ_1^{17}+ 78758~ q^2J_1^{12} +171984~ q^3J_1^{7} + 98667~ q^4J_1^2 \right. \\
&& \left.~~~~~~~~~~~~ + 78986~ q^5J_2^3+ 176592 ~q^6J_2^{8} +52644~q^7J_2^{13}
+2138~ q^8J_2^{18}+6~ q^9J_2^{23} \right] \\ ~\\
\sum_{k=0}^{\infty} p(25k+19)q^{k} &=& 5~ \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }
\left[~ 98J_1^{21}+ 9939 ~qJ_1^{16} + 107712 ~ q^2J_1^{11} + 167031~ q^3J_1^{6} + 27918 ~q^4 J_1 \right. \\
&&\left. ~~~~~~~~~~~~ + 127011~ q^5J_2^{4} +160552~ q^6J_2^{9}
+ 32784 ~q^7J_2^{14}+858 ~q^8J_2^{19} + q^9J_2^{24} \right] \\ ~ \\
\sum_{k=0}^{\infty} p(25k+24)q^{k} &=& 5^2 \frac{(q^5)^{30}_{\infty}}{ (q)^{31}_{\infty} }
\left[ ~ 63 J_1^{20} +3728 ~qJ_1^{15}+27861~ q^2J_1^{10}+25404~ q^3J_1^{5} + 21285~ q^4\right. \\
&& ~~~~~~~~~~~~~ \left. +25404 ~q^5J_2^{5}+ 27861~ q^6J_2^{10}
+3728~ q^7J_2^{15} + 63 ~q^8 J_2^{20}\right]
\end{aligned}$$ The expression on the right in the last equation above reduces to Ramanujan’s result in (A1c) upon the substitutions $$\begin{aligned}
J_1^5 +q^2J_2^5 &=& X +11~q, \\
J_1^{10} +q^4J_2^{10} &=& X^2 +22~q X +123~q^2,\\
J_1^{15} +q^6J_2^{15} &=& X^3 +33~qX^2 +366~q^2 X +1364~q^3,\\
J_1^{20} +q^6J_2^{20} &=& X^4 +44~q X^3 + 730~q^2X^2 +5412~q^3 X +15127~q^4,
\end{aligned}$$ where $$\begin{aligned}
X &\equiv& \frac {(q)^{6}_{\infty} } { (q^5)^6_{\infty}} .
\end{aligned}$$ As before, the $Z_n$ coefficients in the $p(25k+a)$ determinant $$\begin{aligned}
p(25k+a) = 5 \cdot \left| \begin{array}{cccccccccc}
~1& ~ & ~&~&~&~&~&~&~&Z_0\\
-31& ~1& ~ & ~&~&~&~&~&~& Z_1\\
~434& -31& ~1& ~ & ~&~&~&~&~& Z_2\\
-3565& ~434& -31& ~1 & ~ & ~&~&~&~& Z_3\\
~18445 & -3565 & ~434& -31&~1 & ~ &~&~& ~& Z_4\\
-57505& ~18445 & -3565& ~434& -31&~1& ~ &~& ~& Z_5\\
~70091& -57505& ~18445 & -3565& ~434& -31&~1 & ~ &~& Z_6\\
~ 227447 & ~70091& -57505& ~18445 & -3565& ~434& -31& ~ &~& Z_7 \\
~\vdots & ~&~&~&~&~& ~&~~\ddots &~& ~\vdots
\end{array} \right|_{ (k+1) \times (k+1)}
\end{aligned}$$ are obtained by an expansion in powers of $q$ of the numerators on the RHS ’s of these generating-function equations. We have, for $a=4,9,14, 19$, $$\begin{aligned}
{\bf Z}^{(4)} = \left( \begin{array}{c} 1 \\ 882 \\ 49362 \\ 768246 \\ 5380497 \\ 20802996 \\ 47413915 \\ 46923084 \\
\vdots~ \end{array} \right);
~~{\bf Z}^{(9)} = \left( \begin{array}{c} 6 \\ 2276 \\ 92646 \\ 1198566 \\ 7354172 \\ 25710039 \\ 51224670 \\ 39450895 \\
\vdots ~\end{array} \right);
~~{\bf Z}^{(14)} = \left( \begin{array}{c} 27 \\ 5400 \\ 166697 \\ 1811682 \\ 9871992 \\ 30828786 \\ 55015749 \\
20079168 \\ \vdots ~\end{array} \right);
~~{\bf Z}^{(19)} = \left( \begin{array}{c} 98 \\ 11997 \\ 287316 \\ 2672825 \\12906450 \\ 36553962 \\ 54917174
\\ 2443563\\ \vdots ~\end{array} \right).\end{aligned}$$
$\sum p(n)x^{n}$ determinants
==============================
Expression (11) for $p(n)$ can be used to express finite sums of the form $\sum p(n)x^{n}$ as determinants. We have $$\begin{aligned}
\sum_{n=0}^k p(n) x^{n} = x^{k} \sum_{n=0}^k x^{n-k}p(n)
= x^{k} \left| \begin{array} {cccccccc}
~~1 & ~ & ~ & ~ &~&~ &~&~~1 \\
-1 & ~~1 &~&~&~&~&~& ~~1/x\\
-1 & -1 & ~~1 & ~ &~&~&~& ~~1/x^2\\
~0 & -1 &-1&~~ \ddots &~&~&~& ~~1/x^3\\
~\vdots & ~ &~ &~~\ddots & ~&~&~& ~~\vdots \\
d_{k-1}& d_{k-2} &~&~&~&~& ~~1 &~~1/x^{k-1} \\
d_k & d_{k-1} & \cdots &~&~&~& -1 & ~~1/x^k
\end{array} \right|_{(k+1)\times (k+1)}
\end{aligned}$$ The tower part of the matrix can be expressed as $$\begin{aligned}
(I- J/x)^{-1} \left( \left. \begin{array}{cccc}
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & ~ & ~&~ \\
~ & \mbox{{\rm \Huge 0}} &~ &~ \\
~ & ~ & ~ &~\\
~ & ~ & ~&~ \\
~ & ~ & ~&~ \end{array} \right|
\begin{array}{c}
~1~\\
~0 ~\\
~0~\\
~\vdots ~\\
~0~\\
0 \\
-1 \end{array} \right)_{(k+1)\times (k+1)} . \nonumber
\end{aligned}$$ We now multiply the determinant in eq. (C1) by the determinant of $(I-J/x)$, (which is equal to 1), and get $$\begin{aligned}
\sum_{n=0}^k p(n) x^{n}&=& x^k \left| \begin{array} {cccccccc}
~~1 & ~ & ~ & ~ &~&~ &~&~~1 \\
-1-1/x & ~~1 &~&~&~&~&~& ~~0\\
-1+1/x & -1-1/x & ~~1 & ~ &~&~&~& ~~0\\
~1/x & -1+1/x &-1-1/x &~~ \ddots &~&~&~& ~~0\\
~\vdots & ~ &~ &~~\ddots & ~&~&~& ~~\vdots \\
d_{k-1}-d_{k-2}/x & d_{k-2}-d_{k-3}/x &~&~&~&~& ~~1 &~~0 \\
d_{k}-d_{k-1}/x & d_{k-1} -d_{k-2}/x & \cdots &~&~&~& -1-1/x & ~~0
\end{array} \right|_{(k+1)\times (k+1)} \nonumber \\ \nonumber ~\\
&=& \left| \begin{array} {cccccccc}
x+1 & -x & ~ & ~ &~&~ &~&~ \\
x-1 & x+1 &-x &~&~&~&~& ~\\
-1 & x-1 & x+1 & ~~~~\ddots &~&~&~& ~\\
0 & -1 & x-1 &~~~~ \ddots &~&~&~& ~\\
\vdots & ~ &~ &~~~~\ddots & ~&~&~& ~ \\
d_{k-2}-xd_{k-1} & d_{k-3}-xd_{k-2} &~&\cdots &~&~& x+1 &-x \\
d_{k-1} -xd_{k} & d_{k-2} -xd_{k-1} & ~& \cdots &~&~& x-1 & x+1
\end{array} \right|_{k\times k} ,
\end{aligned}$$ where in the last line we’ve expanded the $(k+1)$-dimensional determinant by minors along the final column and then taken the factor $(-1)^k x^k$ inside the resulting $k$-dimensional determinant, multiplying each of the columns by $-x$. The final result can be written in a more compact notation as $$\begin{aligned}
\sum_{n=0}^{k} p(n) x^n = \det \left[ -xJ^T +I +\sum_{m > 0}^{q_m<k+1} (-1)^m \left[ J^{m(3m-1)/2} +J^{m(3m+1)/2}
-x J^{(m-1)(3m+2)/2} -x J^{(m+1)(3m-2)/2} \right] \right] _{k \times k} .
\end{aligned}$$
[99]{} The On-Line Encyclopedia of Integer Sequences. http://oeis.org. Weisstein, Eric W. “Partition Function P.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html. Korn and Korn, [*Mathematical Handbook for Scientists and Engineers*]{}, Section 21.5, (in Russian), (McGraw-Hill, 1968). Vella, David C., “Explicit Formulas for Bernoulli and Euler Numbers”, [*Integers*]{} [**8**]{}, 2008. Rademacher, H. “On the Partition Function .” Proc. London Math. Soc. 43, 241-254, 1937. Bruinier and Ono, “Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms“. http://arxiv.org/abs/1104.1182v1/ Wikipedia, “Bell polynomials”. http://en.wikipedia.org/wiki/Bell polynomials. Gradshteyn and Ryzhik ([*Table of Integrals, Series, and Products*]{}, p. 14, (Academic,New York 1980). Weisstein, Eric W. “Partition Function Q.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionQ.html. Berndt and Ono, ”Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary". http://www.math.wisc.edu/ ono/reprints/044.pdf. Weisstein, Eric W. “Ramanujan Theta Functions.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/PartitionFunctionP.html.
\
2000 Mathematics Subject Classification: Primary: 05A17; Secondary: 11Y35, 11B68.\
Keywords: partition function, Bernoulli numbers, Euler numbers, Sterling numbers.\
(Concerned with sequences A000009, A000041, A000364, A000728, A000729, A000730, A000731, A001318, A010815, A010816, A010836, A095699, A113681, and A116915)
|
---
abstract: 'We study the shadows of the fully non-linear, asymptotically flat Einstein-dilaton-Gauss-Bonnet (EdGB) black holes (BHs), for both static and rotating solutions. We find that, in all cases, these shadows are *smaller* than for *comparable* Kerr BHs, $i.e.$ with the same total mass and angular momentum [under similar observation conditions]{}. In order to compare both cases we provide quantitative shadow parameters, observing in particular that the differences in the shadows mean radii are never larger than the percent level. Therefore, generically, EdGB BHs $cannot$ be excluded by (near future) shadow observations alone. On the theoretical side, we find no clear signature of some exotic features of EdGB BHs on the corresponding shadows, such as the regions of negative (Komar, say) energy density outside the horizon. We speculate that this is due to the fact that the Komar energy interior to the light rings (or more precisely, the surfaces of constant radial coordinate that intersect the light rings in the equatorial plane) is always smaller than the ADM mass, and consequently the corresponding shadows are smaller than those of comparable Kerr BHs. The analysis herein provides a clear example that it is the light ring impact parameter, rather than its “size", that determines a BH shadow.'
author:
- |
[Pedro V. P. Cunha]{}$^{1,2}$, $^{1}$, $^{3}$,\
[Jutta Kunz]{}$^{3}$ and [Eugen Radu]{}$^{1}$\
\
$^{1}$[Departamento de Física da Universidade de Aveiro and ]{}\
[Centre for Research and Development in Mathematics and Applications (CIDMA), ]{}\
[Campus de Santiago, 3810-183 Aveiro, Portugal]{}\
`[email protected]; [email protected]; [email protected]`\
\
$^{2}$[CENTRA, Departamento de Física, Instituto Superior Técnico]{}\
[Universidade de Lisboa, Avenida Rovisco Pais 1, 1049, Lisboa, Portugal]{}\
\
$^{3}$[Institute of Physics, University of Oldenburg, Oldenburg, 26111, Germany]{}\
`[email protected]; [email protected]`
title:
---
Introduction
============
Ultraviolet theoretical inconsistencies of Einstein’s General Relativity, such as its non-renormalizability [@'tHooft:1974bx; @Deser:1974cz; @Deser:1974xq] and the existence of singularities, have since long motivated the suggestion that higher curvature corrections should be taken into account, in an improved theory of gravity (see $e.g.$ [@Stelle:1976gc]). Inclusion of a finite set of such higher curvature corrections, however, generically leads to runaway modes (Ostrogradsky instabilities [@Ostrogradsky:1850fid]) in the classical theory and a breakdown of unitarity due to ghosts, in the quantum theory. These undesirable properties can be simply diagnosed, at the level of the classic field equations, by the presence of third order time (and consequently also space, by covariance) derivatives. A natural way around this problem is to require a self-consistent model, obtained as a truncation of the higher curvature expansion, to yield a set of field equations without such higher order derivatives.
Lovelock [@Lovelock:1971yv] first established, for vacuum gravity, what are the allowed curvature combinations so that the field equations have no higher than second order time derivatives. It turns out that, in a Lagrangian, these combinations are simply the Euler densities, particular scalar polynomial combinations of the curvature tensors of order $n$. Since the $n^{th}$ Euler density is a topological invariant in spacetime dimension $D=2n$ and yields a non-dynamical contribution to the action in dimensions $D\leqslant 2n$, an immediate corollary is that, in $D=4$ vacuum gravity, the most general Lovelock theory is a combination of the 0$^{th}$ and 1$^{st}$ Euler density, or in other words, General Relativity with a cosmological constant. The 2$^{nd}$ Euler density, known as the Gauss-Bonnet (GB) combination, is a topological invariant in $D=4$ and does not contribute to the dynamical equations of motion if included in the action.
There is, however, a simple and natural way to make the GB combination dynamical in a $D=4$ theory: couple it to a dynamical scalar field. This is actually a model that emerges naturally in string theory [@Zwiebach:1985uq] (see also [@Kanti:1995vq] for a discussion on this point), where the scalar field is the dilaton, and can be considered as a simple effective model to investigate the consequences of higher curvature corrections in $D=4$ gravity. The corresponding model takes the name of Einstein-dilaton-Gauss-Bonnet (EdGB) theory and is described by the action in section \[section\_model\] below.
Black holes (BHs) in EdGB theory were first shown to exist, in spherical symmetry, by Kanti et al. [@Kanti:1995vq], wherein they were obtained numerically. These solutions, which moreover are perturbatively stable along their main branch [@Kanti:1997br], are asymptotically flat, regular on and outside an event horizon, and describe a horizon surrounded by a non-trivial dilaton profile. They circumvent some well-known no (real) scalar hair theorems, namely those by Bekenstein [@Bekenstein:1972ny; @Bekenstein:1995un] (see [@Herdeiro:2015waa] for a recent review), due to the non minimal coupling of the dilaton to the geometry and the fact that if one associates some *effective matter* with the GB term, then this represents *exotic matter*, violating the typical energy conditions. One manifestation of this *effective exotic matter* is that the BH solutions have regions of negative energy density outside the horizon. Another manifestation is that there is a minimal mass for BHs, determined by the GB coupling. We remark that the scalar hair of this BHs has no-independent conserved charge, thus being called *secondary*. See, $e.g.$ [@Torii:1996yi; @Alexeev:1996vs; @Melis:2005ji; @Chen:2006ge; @Chen:2008hk] for further discussions of these spherically symmetric solutions and some charged generalizations.[^1]
Rotating BHs in EdGB theory were found, fully non-linearly in [@Kleihaus:2011tg; @Kleihaus:2015aje] (see also [@Pani:2009wy; @Pani:2011gy; @Ayzenberg:2014aka; @Maselli:2015tta] for perturbative studies). A minimal mass depending on the GB coupling still exists for these rotating solutions and, as a novel physical feature, some (small) violations of the Kerr bound in terms of ADM quantities are observed. Again, regions with negative energy density exist outside the horizon.
In this paper, we shall investigate how the dGB term impacts on one particular observable feature of a BH: its shadow [@Falcke:1999pj]. BH shadows can be roughly described as the silhouette produced by the BH when placed in front of a bright background. They are determined by the BH absorption cross section for light at high frequencies. Over the last few years there has been a renewed theoretical interest in this old concept, first discussed for the Kerr BH by Bardeen [@Bardeen:1973tla], due to observational attempts to measure the BH shadow of the supermassive BHs in our galactic center as well as that in the centre of M87 [@Lu:2014zja]. In particular, in [@Cunha:2015yba; @Vincent:2016sjq; @Cunha:2016bjh], the shadows of a type of hairy BHs that connect continuously to Kerr, within General Relativity and with matter obeying all energy conditions, called Kerr BHs with scalar hair [@Herdeiro:2014goa; @Herdeiro:2014ima; @Herdeiro:2015gia], have been studied. It has been pointed out that, generically, these shadows are smaller than those of a comparable Kerr BH, $i.e.$ a vacuum rotating BH with the same total mass and angular momentum. A possible interpretation of this qualitative behaviour is the following: the total mass (and angular momentum) of the hairy BHs is now partly stored in the scalar field outside the horizon; in particular the existence of some energy outside the region of unstable spherical photon orbits, also referred to as photon region (see section \[subsection\_Ligh-rings\]) [@Grenzebach:2014fha], implies that less energy exists inside this region and hence the light rings should be smaller (within an appropriate measure) as compared to their vacuum counterparts and consequently so should be the shadows.
The above interpretation raises an interesting question in relation to the BHs in EdGB theory. Since these have negative energy densities outside the horizon, how do these regions of *effective exotic matter* impact on their shadows? In particular could there be a negative energy contribution outside the photon region that is sufficiently large to increase the shadow size with respect to a vacuum counterpart? We remark that for other non-vacuum solutions with physical matter, $i.e.$ obeying all energy conditions, the size of the shadow typically decreases with respect to the size of a comparable vacuum Kerr BH – see $e.g.$ [@Takahashi:2005hy] for electrically charged BHs. However, larger shadows have also been observed, $e.g.$, in extended Chern-Simons gravity [@Amarilla:2010zq] or brane world BHs [@Amarilla:2011fx] which possess *effective exotic matter*, similarly to EdGB. Nevertheless, we shall see that for EdGB the shadows are always smaller with respect to the vacuum case, with the maximal deviation being of the order of only a few percent. For some work on BH shadows in different models see [@Cunha:2015yba; @Grenzebach:2014fha; @Amarilla:2010zq; @Amarilla:2011fx; @Tretyakova:2016ale; @Abdolrahimi:2015rua; @Abdolrahimi:2015kma; @Shipley:2016omi; @Abdujabbarov:2016hnw; @Amir:2016cen; @Johannsen:2015qca; @Amarilla:2013sj], and in particular [@Younsi:2016azx] for perturbative EdGB BHs.
This paper is organized as follows. In Section \[section\_model\] we describe the EdGB model and present its field equations. An overview of the known BH solutions in this model, both static and stationary, is also provided there, together with the corresponding domain of existence and limiting cases. Then, in Section \[section\_shadows\] we present the shadows for a representative sample of solutions and interpret the patterns obtained. We close with a discussion in Section \[section\_discussion\].
The model and solutions {#section_model}
========================
The field equations and general results
-----------------------------------------
We consider the Einstein-dilaton-Gauss-Bonnet (EdGB) model, described by the following action[^2] $$\begin{aligned}
S=\frac{1}{16 \pi }\int d^4x \sqrt{-g} \left[R - \frac{1}{2}
(\partial_\mu \phi)^2
+ \alpha e^{-\gamma \phi} R^2_{\rm GB} \right],
\label{act}\end{aligned}$$ where $\phi$ is the dilaton field, $\alpha $ is a parameter with units (length)$^2$ and $R^2_{\rm GB} = R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma}
- 4 R_{\mu\nu} R^{\mu\nu} + R^2$ is the GB combination. Also, $\gamma$ is an input parameter of the theory[^3], with most of the studies assuming $\gamma=1$. [Both $\gamma$ and $\phi$ are dimensionless.]{}
Varying the action (\[act\]) with respect to $g_{\mu\nu}$, we obtain[^4] the Einstein equations: $$\begin{aligned}
\label{EGB-eq1}
G_{\mu\nu}= T_{\mu\nu}^{\rm (eff)} \ ,
\end{aligned}$$ where $G_{\mu\nu}$ is the standard Einstein tensor and the effective energy-momentum tensor reads $$\begin{aligned}
\label{Teff}
T_{\mu\nu}^{\rm (eff)}\equiv \frac12\biggl[\nabla_{\mu}\phi\nabla_{\nu} \phi -\frac12 g_{\mu\nu}(\nabla\phi)^2\biggr]
-\alpha e^{-\gamma\phi}T_{\mu\nu}^{\rm (GBd)} \ ,
\end{aligned}$$ where the full expression for $T_{\mu\nu}^{\rm (GBd)}$ can be found in [@Kleihaus:2015aje]. Varying the action (\[act\]) with respect to the dilaton field, on the other hand, yields the scalar equation of motion, which reads: $$\begin{aligned}
\label{dil-eq}
\Box \phi =\alpha \gamma e^{-\gamma\phi} R^2_{\rm GB} \ .\end{aligned}$$
The EdGB model possesses BH and wormhole [@Kanti:2011jz] solutions, but no particle-like solitonic configurations are known (for a review, see the recent work [@Blazquez-Salcedo:2016yka]), although the coupling to matter leads, $e.g.$, to neutron stars [@Pani:2011xm; @Kleihaus:2016dui]. Note that in contrast to the GR case, all EdGB solutions (with $\alpha\neq 0$) have been obtained numerically.
In terms of the spherical-type coordinates $r,~\theta$ and $\varphi$, all known EdGB solutions possess at least two Killing vectors $\xi=\partial/\partial t$ and $ \zeta=\partial/\partial \varphi$ (where $t$ is the time coordinate). Then a generic metric ansatz can be written as $$\begin{aligned}
\label{metric}
ds^2=g_{rr}dr^2+g_{\theta\theta}d\theta^2+g_{\varphi\varphi}d\varphi^2+2 g_{\varphi t} d\varphi dt+g_{tt}dt^2~,\end{aligned}$$ where $g_{\mu \nu}$ and the scalar $\phi$ are functions of $(r,\theta)$. Moreover, we can set $\phi(\infty)=0$ without any loss of generality [(any other choice would correspond to a rescaling of the radial coordinate in [@Kleihaus:2015aje])]{}. The ADM (Arnowitt-Deser-Misner) mass $M$ and angular momentum $J$ are read off, as usual, from the asymptotic expansion $$\begin{aligned}
\label{asym}
g_{tt} =-1+\frac{2M}{r}+\dots,~~g_{\varphi t}=-\frac{2J}{r}\sin^2\theta+\dots~.~~\end{aligned}$$
One can also define a global dilaton measure $D$ from the asymptotic expansion of the scalar field, $\phi=-D/r+\dots$ which however is not an independent quantity, since the dilaton field does not qualify as primary hair [@Kanti:1995vq], [@Kleihaus:2015aje].
The static EdGB black holes
---------------------------
Consider for the moment the static, spherically symmetric solutions ($J=0$). Close to the event horizon, these solutions possess an approximate expression as a power series in $r-r_H$, where $r_H$ is the radial coordinate of the horizon. [ In particular, in Schwarzschild coordinates one finds $\phi(r)=\phi_H+\phi_1(r-r_H)+\dots$, where $\phi_1$ satisfies a quadratic equation (see $e.g.$ [@Kanti:1995vq], [@Torii:1996yi], [@Alexeev:1996vs], [@Kleihaus:2015aje]). Since the scalar field is real,]{} the discriminant of the quadratic equation is required to be positive, yielding the condition: $$\begin{aligned}
\label{cond1}
1-96\alpha^2\gamma^2\frac{e^{-2\gamma \phi_H}}{A_H^2/(16\pi^2)} \geq 0,
\end{aligned}$$ where $A_H$ is the event horizon area. Eventually, this condition will be violated after some *limiting solution* is reached, beyond which solutions cease to exist in the parameter space. For a given $\gamma$, all solutions can be obtained continuously in the parameter space. When appropriately scaled they form a line, starting from the smooth GR limit ($\phi\to 0$ as $\alpha \to 0$), and ending at the limiting solution. The existence of the latter places a lower bound on the BH horizon radius. It actually also implies the existence of a lower bound on the BH mass. In particular, as discussed in [@Kanti:1995vq; @Pani:2009wy], the static EdGB solutions with $\gamma=1$ are limited to the parameter range $0\leq\alpha/M^2\lesssim 0.1728$. A rather similar behaviour holds[^5] for $\gamma\neq 1$.
Solutions no longer exist if the ratio $\alpha/M^2$ is larger than a critical value, which decreases with increasing $\gamma$. The configuration at this maximal value is dubbed the *critical solution*, which needs not to coincide with the limiting solution. In particular, for large enough $\gamma$, the solution line can be extended backwards in $\alpha/M^2$, into a “secondary branch”, after the critical configuration is reached [@Guo:2008hf]; this secondary branch eventually terminates at the limiting solution. Some of these features can be seen in an $(\alpha,D)$-diagram of solutions with different $\gamma$, as shown in Figure \[DM\] (left). In particular, notice how for sufficiently large $\gamma$ values it is possible to have two different values of $D/M$ for the same $\alpha/M^2$, which indicates the presence of two branches. According to arguments from catastrophy theory, the stability should change at the critical solution, so that the solutions along the secondary branch will be unstable [@Torii:1996yi].
![ (Left) Domain of existence of static EdGB BHs in a $D/M$ vs. $\alpha/M^2$ diagram with several values of $\gamma$. [The points $a$ and $b$ depict the limiting and critical solutions respectively for $\gamma=10$.]{} (Right) Domain of existence of spinning solutions with $\gamma=1$. The set of considered (spinning) solutions in Fig. \[fig-r\] and Fig. \[fig-sig\] are shown here as highlighted points. []{data-label="DM"}](./spherical1.eps "fig:"){width="9.cm"}![ (Left) Domain of existence of static EdGB BHs in a $D/M$ vs. $\alpha/M^2$ diagram with several values of $\gamma$. [The points $a$ and $b$ depict the limiting and critical solutions respectively for $\gamma=10$.]{} (Right) Domain of existence of spinning solutions with $\gamma=1$. The set of considered (spinning) solutions in Fig. \[fig-r\] and Fig. \[fig-sig\] are shown here as highlighted points. []{data-label="DM"}](./DM.eps "fig:"){width="9.cm"} (-409.5,47) (-423,68.5) (-409.5,45)[ [$b$]{}]{} (-423,68.5)[ [$a$]{}]{}
The spinning EdGB black holes
-----------------------------
[Spherically symmetric BHs typically possess spinning generalizations]{}. However, so far only the $\gamma=1$ case has been explored in the literature. These BHs were first obtained at the fully non-linear level in [@Kleihaus:2011tg] (see also [@Pani:2009wy; @Pani:2011gy; @Ayzenberg:2014aka; @Maselli:2015tta] for perturbative results). Similar to the GR case, these BHs possess a $\mathbb{Z}_2$ symmetry along the equatorial plane ($\theta=\pi/2$) and are obtained by solving the field equations and subject to appropriate boundary conditions that are detailed in [@Kleihaus:2015aje].
The domain of existence of EdGB BHs is bounded by four sets of solutions: $i)$ the set of static ($i.e.$ spherically symmetric) EdGB BHs with $J=0$; $ii)$ the set of extremal ($i.e.$, zero temperature) EdGB BHs; $iii)$ the set of critical solutions; and $iv)$ the set of GR solutions – the Kerr/Schwarzschild BHs with $\alpha=0$. In Fig. \[fig-r\] and Fig. \[fig-sig\] the boundary line displayed includes the sets $ii)$ and $iii)$.
The general critical solutions are the rotating generalization of the static case, while the extremal set does not appear to be regular on the horizon.[^6] Moreover, the mass of the EdGB rotating BHs is always bounded from below, whereas the angular momentum can (slightly) exceed the Kerr bound, which is given by [$J \leqslant M^2$]{}. Further details on these aspects together with various plots of the domain of existence are found in [@Kleihaus:2015aje]. Here we give the domain of existence in $(\alpha,D)$-variables \[Figure 1 (right)\] and in $(\alpha,J)$-variables (Figure \[fig-r\]).
Shadows {#section_shadows}
=======
Light rings {#subsection_Ligh-rings}
-----------
As it is well described in the literature, the Kerr spacetime supports unstable photon orbits with a fixed [Boyer-Lindquist]{} radial coordinate, $i.e.$, the photon region [@Bardeen:1973tla]. A subset of the latter is restricted to the equatorial plane $(\theta=\pi/2)$, and comprises two independent circular photon orbits with opposite rotation senses, dubbed here as *light rings*. Such orbits are not unique to the Kerr spacetime and have an intrinsic relation to the BH shadow. In particular, unstable light rings embody a threshold of stability between equatorial null geodesics that scatter to infinity and ones that plunge into the BH. Consequently, light rings account for the shadow edge in observations restricted to the equatorial plane’s line of sight (provided both exist). Following [@Cunha:2016bjh], the light ring positions can be obtained by analysing the following condition in the equatorial plane: $$\partial_rh_\pm=0,\quad \textrm{with}\quad h_\pm=\frac{-g_{t\varphi}\pm\sqrt{g^2_{t\varphi}-g_{tt}g_{\varphi\varphi}}}{g_{tt}}.$$ Recalling the Kerr case, each sign $\pm$ leads to one of the two light rings. Curiously, although the EdGB BHs discussed in this paper are fully non-linear solutions (rather than perturbations of Kerr), the light ring qualitative structure still appears to be the same as in Kerr. However, notice that for other families of solutions this is not always the case. For instance, multiple light rings can appear for BHs with scalar hair, some of which are stable [@Cunha:2016bjh].\
Characterizing the shadow
-------------------------
Assuming that a suitable light source is present to provide contrast, a BH casts a black region in an observer’s sky, commonly called the BH shadow. Although some characteristics are observer dependent [@Vincent:2016sjq], the size and shape of the shadow are essentially a manifestation of the spacetime properties close to the BH, depending for instance on the light ring characteristics. Consequently, instructive physics can be inferred from such observations.\
Consider the dummy shadow in Fig. \[fig-shadow\], represented in the image plane of the observer.
![Representation of a BH shadow in the $(x,y)$ image plane of the observer. []{data-label="fig-shadow"}](./shadow.eps "fig:"){width="6cm"} (-60,113.3)[$P$]{} (-76.7,93.3)[$r'$]{} (-120,160)[$y$]{} (-16.7,60)[$x$]{} (-93.3,55)[$C$]{} (-106.7,70)[$O$]{} (-43,70)[$x_2$]{} (-160,70)[$x_1$]{}
A Cartesian parametrization $(x,y)$ is used, where the $x$-axis is defined to be parallel to the azimuthal Killing vector $\zeta= \partial/\partial\varphi$ at the observer’s position. The origin $(0,0)$ of this coordinate system, defined as point $O$ in Fig. \[fig-shadow\], corresponds to the direction pointing towards the center of the BH $-\partial/\partial r$ (from the reader into the paper).
The point $C$ in the figure, taken to be the center of the shadow, is such that its abscissa is given by $x_C=(x_{\textrm{max}}+x_{\textrm{min}})/2$, where $x_{\textrm{max}}$ and $x_{\textrm{min}}$ are respectively the maximum and minimum abscissae of the shadow’s edge. If the observer is in the equatorial plane ($\theta=\pi/2$), which will be assumed throughout the paper, then the shadow inherits along the $x$-axis the spacetime reflection symmetry, giving $y_C=0$. Since the points $C$ and $O$ need not to coincide, a specific feature of a shadow is the [displacement]{} $x_C$ between the shadow and the center of the image plane $O$.
A generic point $P$ on the shadow’s edge is at a distance $r'$ from $C$, which is defined as $r'\equiv \sqrt{{y_P}^2 + {(x_P-x_C)}^2}$. Given the line element $ds^2=dx^2 + dy^2$, the perimeter $\mathcal{P}$ of the shadow, its [average radius]{} $\bar{r}$ and the [deviation from sphericity]{} $\sigma_r$ are defined by: $$\oint ds \equiv \mathcal{P},\qquad \bar{r}\equiv \frac{1}{\mathcal{P}}\oint r'\,ds,\qquad \sigma_r=\sqrt{\frac{1}{\mathcal{P}}\oint {\left(1-\frac{r'}{\bar{r}}\right)}^2\,ds}.$$ All these parameters are expressed in units of the ADM mass $M$.\
In some cases, it is possible to compare the shadow parameters of a given EdGB solution with the ones from a Kerr BH with the same ADM mass $M$ and angular momentum $J$. Hence, let us also define the relative deviations to the Kerr case[^7]: $$\delta_r =\frac{\bar{r}-\bar{r}_\textrm{kerr}}{\bar{r}_\textrm{kerr}},\qquad\delta_\sigma =\frac{\sigma_r-\sigma_\textrm{kerr}}{\sigma_\textrm{kerr}},\qquad \delta_{x_C}=\frac{ x_C -{x_C}_\textrm{kerr} }{{x_C}_\textrm{kerr}}.$$
Rotating EdGB BHs
-----------------
Due to the existence of a hidden constant of motion - the Carter constant - the edge of the Kerr shadow can be obtained in a closed analytical form [@Bardeen:1973tla; @Grenzebach:2014fha; @Cunha:2016bpi]. However, EdGB BHs are not expected to have such a property, since they all appear to be of Petrov type I [@Kleihaus:2015aje]. This is consistent with the perturbative results in [@Ayzenberg:2014aka]. As a consequence, in general the shadow of the latter has to be obtained numerically through the standard *backwards ray-tracing* framework [@Johannsen:2015qca; @Psaltis:2010ww]. In order to generate a virtual image of the shadow, this method requires propagating null geodesics “backwards in time”, where a high frequency approximation is assumed, starting from the observer’s position and determining the source of each light ray. Different points in the image plane correspond to different directions in the observer’s sky, and hence to different initial conditions of the geodesic equations. The shadow is precisely the set of all those initial conditions which induce geodesics with endpoints on the event horizon, when propagated backwards in time. Since the event horizon is not a source of any light (classically), the shadow actually embodies a lack of radiation[^8].\
The geodesic propagation method described above is necessary to compute most of the shadow edge. However, the points $x_1$ and $x_2$ in Fig. \[fig-shadow\], where the edge intersects the $x$-axis, can be computed using a highly precise local method. In particular, for an observer in the equatorial plane, light rings are the orbits responsible for these intersection points. The impact parameter $\eta=L/E$ will play here a crucial role, where $E$ and $L$ are respectively the photon’s energy and axial angular momentum with respect to a static observer at infinity. Moreover, these quantities are constants of geodesic motion, connected to the Killing vectors of the spacetime $\xi=\partial/\partial t$ and $\zeta=\partial/\partial \varphi$. The function $h_\pm$ will now be helpful again, as the value of $\eta$ in a given light ring orbit is provided simply by $\eta=h_\pm$, computed at that position [@Cunha:2016bjh].
The precise relation between the image coordinate $x$ and the impact parameter $\eta$ depends on the choice for the observer’s frame, but also on how $x$ is constructed in terms of observation angles. Following [@Cunha:2016bjh; @Cunha:2016bpi], the $x$ coordinate is defined to be directly proportional to an observation angle $\beta$ along that axis: $x=-\tilde{R}\,\beta,$ where the perimetral radius $\tilde{R}\equiv \sqrt{g_{\varphi\varphi}}$ is computed at the observer’s position. By computing the projection of the photon’s 4-momentum onto a ZAMO frame [@Cunha:2016bjh; @Cunha:2016bpi], the relation $\sin\beta = \eta/(A_0+\eta\,B_0)$ can be derived (if $y=0$), where the following quantities are computed at the position of the observer: $A_0=g_{\varphi\varphi}/\sqrt{D},\quad B_0=g_{t\varphi}/\sqrt{D}$, with $D\equiv g_{t\varphi}^2-g_{tt}g_{\varphi\varphi}$. This leads to the relation (with $y=0$): $$x=-\tilde{R}\arcsin\left(\frac{\eta}{A_0+\eta B_0}\right).$$ For the sake of the argument, consider also a very far away observer ($r\to \infty$). In these conditions we obtain the very simple relation $x=-\eta$. By computing $\eta_1$ and $\eta_2$ for each of the two light rings, we can obtain the shadow radius $\bar{r}_x$ on the $x$-axis simply with $\bar{r}_x=|x_1-x_2|/2$, where each $x$ is evaluated from the respective $\eta$. Notice that this is a local method, in the sense that it does not require the evolution of a geodesic throughout the spacetime. Hence, obtaining a very precise $\bar{r}_x$ value only depends on knowing $\eta$ at the light rings with sufficiently high accuracy. Furthermore, by comparing this $\bar{r}_x$ value with the one obtained with ray-tracing, we can estimate that the precision of the latter is around $\sim 0.08\%$.\
The data of the EdGB shadows, computed with ray-tracing, is represented in Fig. \[fig-r\] and Fig. \[fig-sig\], where a dilaton coupling $\gamma=1$ is assumed. The observer is always placed in the equatorial plane, at a radial coordinate such that $\tilde{R}=\sqrt{g_{\varphi\varphi}}=15M$.\
![Representation of $(\bar{r}-4.68M)$ (left) and $\delta_r$ (right) for EdGB solutions with $\gamma=1$, in a $\alpha/M^2$ vs. $J/M^2$ diagram. Each circle [radius]{} is proportional to the quantity represented, [with some values also included for reference]{}. All the values of $\delta_r$ are negative, with the maximum deviation to Kerr being around $\simeq -1.5\%$. []{data-label="fig-r"}](./radius.eps "fig:"){width="9.8cm"}![Representation of $(\bar{r}-4.68M)$ (left) and $\delta_r$ (right) for EdGB solutions with $\gamma=1$, in a $\alpha/M^2$ vs. $J/M^2$ diagram. Each circle [radius]{} is proportional to the quantity represented, [with some values also included for reference]{}. All the values of $\delta_r$ are negative, with the maximum deviation to Kerr being around $\simeq -1.5\%$. []{data-label="fig-r"}](./variation.eps "fig:"){width="9.8cm"} (-425,260)[[$\bar{r}-4.68M$]{}]{} (-130,260)[[$\delta_r$]{}]{} (-500,65)[$\simeq$ 0.24]{} (-503,207)[$\simeq$ 0.1]{} (-58,168)[$\simeq-0.9$]{} (-58,46)[$\simeq-1.5$]{} (-205,205)[$\simeq-0.08$]{}
![(Left) Representation of $|\delta_\sigma|$ for EdGB solutions with $\gamma=1$, in a $\alpha/M^2$ vs. $J/M^2$ diagram. Each circle [radius]{} is proportional to the quantity represented, [with some values also included for reference]{}. All the values of $\delta_\sigma$ are negative. (Right) Depiction of the shadow edge of a EdGB BH with $\gamma=1$ and $(\alpha/M^2,J/M^2)\simeq(0.172,0.41)$, yielding $\bar{r}\simeq 4.85$, $\sigma=0.3$, $x_C=0.84$; the radial deviation $\delta_r$ with respect to the comparable Kerr case is $\simeq -1.35\%$. [The observer is at a perimetral radius $15M$.]{} []{data-label="fig-sig"}](./sigma.eps "fig:"){width="9.8cm"}![(Left) Representation of $|\delta_\sigma|$ for EdGB solutions with $\gamma=1$, in a $\alpha/M^2$ vs. $J/M^2$ diagram. Each circle [radius]{} is proportional to the quantity represented, [with some values also included for reference]{}. All the values of $\delta_\sigma$ are negative. (Right) Depiction of the shadow edge of a EdGB BH with $\gamma=1$ and $(\alpha/M^2,J/M^2)\simeq(0.172,0.41)$, yielding $\bar{r}\simeq 4.85$, $\sigma=0.3$, $x_C=0.84$; the radial deviation $\delta_r$ with respect to the comparable Kerr case is $\simeq -1.35\%$. [The observer is at a perimetral radius $15M$.]{} []{data-label="fig-sig"}](./shadow-demo.eps "fig:"){width="9.8cm"} (-410,260)[[$|\delta_\sigma|$]{}]{} (-355,217)[$\sim 7$]{} (-430,165)[$\sim 1$]{}
In the left of Fig. \[fig-r\], the size of each circle represents the value of the shadow radius $\bar{r}$ for several EdGB solutions. In order to make the differences across the solution space more apparent, the circle [radius]{} is proportional to $\bar{r}-4.68M$. In other words, a vanishing circle (in this plot only) represents $\bar{r}=4.68M$. With this depiction, it is clear that - as a rule of thumb - increasing either $J$ or $\alpha$ decreases the shadow size. However, from an observational[^9] point of view, it is much more relevant to compare the shadow prediction of an EdGB model with the one of a comparable[^10] Kerr BH with the same $M$ and $J$. In particular, on the right of Fig. \[fig-r\] the relative differences of the shadow size $\delta_r$ with respect to Kerr is represented in a circle plot. All deviations are negative, with the largest ones (in absolute) around $\simeq -1.5\%$. As (another) rule of thumb, increasing $\alpha/M^2$ appears to lead to larger radial deviations from Kerr. In particular, the spherically symmetric EdGB line ($J=0$) includes some of the largest $|\delta_r|$ values. As a side note, the data represented by the smallest circles in the right of Fig. \[fig-r\] correspond to deviations around $\sim 0.08\%$, which is about the estimated numerical accuracy.\
For completeness, the deviations[^11] of $\sigma_r$ with respect to Kerr are represented in the left of Fig. \[fig-sig\]. Curiously, all values of $\delta_\sigma$ are negative, which means that EdGB shadows are more “circular” than the corresponding Kerr case. Hence, the GB term appears to soften the spin deformations that exist on the Kerr shadows. Moreover, notice how the largest $|\delta_\sigma|$ values can be found close to the critical boundary in solution space. Additionally, the deviations $\delta_{x_C}$ can be both positive and negative, although a plot for this quantity is not shown.
In order to display an illustrative shadow case, in the right of Fig. \[fig-sig\] we have the representation of a EdGB shadow edge in the image plane, together with the comparable Kerr one. Although the difference between the curves is barely visible, amounting to a variation of only $\simeq-1.35\%$ in the shadow size, the case here depicted has one of the largest values of $|\delta_r|$ for $\gamma=1$. Such an example reinforces the idea that shadow observations are very unlikely to constrain EdGB BH models in the near future.
Static EdGB BHs
---------------
Until this point we discussed only the shadows of EdGB solutions for dilatonic coupling $\gamma=1$. Repeating the above analysis for other values of $\gamma$ would be rather cumbersome. Nevertheless, as discussed in the previous subsection, some of the largest $\bar{r}$ deviations occur within the static case. Therefore this can be considered as an incentive to explore other values of $\gamma$, while restricting ourselves to $J=0$. This will provide some insight on the effect of the $\gamma$ parameter without much more effort.\
For the static case ($J=0$) the shadow is a circle due to the spherical symmetry of the spacetime. Using this property, we have $\bar{r}=\bar{r}_x$, which allows us to use the high precision method described before, thus obtaining the shadow edge without having to resort to any ray-tracing. Notice that in this case $\sigma_r$ and $x_C$ are both zero due to the spherical symmetry.\
![Representation of $\delta_r$ for static EdGB BHs, computed with respect to the Schwarzschild case. Data for different $\gamma$ values is displayed as a function of $\alpha/M^2$. All deviations are negative. [The displayed lines only interpolate the numerical data, with colors red, green, blue, pink and light blue respectively for $\gamma=\{0.5,1,2,5,10\}$. The observer’s perimetral radius was set at $15M$.]{} []{data-label="fig-gamma"}](./gamma.eps){width="11cm"}
The radial deviations $\delta_r$ of static EdGB shadows with respect to those of a comparable Schwarzschild BH are represented in Fig. \[fig-gamma\], for different $\gamma$ values. The data suggests a scenario where for a fixed value of $\alpha/M^2$ the deviations on the stable branches are larger if we increase $\gamma$; however, after entering the domain of the secondary (unstable) branches, $\gamma$ has to decrease in order to yield larger deviations. Furthermore, for a given $\gamma$, the maximum deviation always appears to occur at the limiting solution, with this maximal deviation being larger for smaller $\gamma$ values. For instance, $\gamma=0.5$ can lead to shadows $\simeq 2\%$ smaller than for Schwarzschild, whereas for $\gamma=1$ all deviations are below $1.5\%$.
Discussion {#section_discussion}
==========
The shadow of a EdGB BH is always smaller than the comparable Kerr one. However, the deviations observed are always smaller (in modulus) than a few percent ($\sim 1\%$). Since such differences are below the expected resolution of planned observations [($\sim 6\%$ as anticipated in ]{}[@Johannsen:2016vqy]), it is unlikely that in the near future any shadow measurement can exclude or restrict EdGB models. [Nevertheless, the present study was not exhaustive; it leaves, for instance, studies for different inclinations and distances as future work.]{}\
Since EdGB theory possesses unusual features such as *effective exotic matter*, it might come as a surprise that there are no significant effects at the level of the shadow. However, this *effective exotic matter* is concentrated close to the horizon, such that there is no negative energy contribution outside the photon region that could significantly affect the shadow’s size. At the same time any near-horizon odd effects are concealed from a remote observer by the shadow.\
It may come as another surprise, that the light ring size[^12] of EdGB BHs can, for instance, change by as much as $\simeq 4\%$, when considering the static case with $\gamma=0.5$, and this effect will increase with further decreasing $\gamma$. The natural question is then: why are the deviations in the shadow size not larger? For the sake of the argument consider the static case, where it becomes clear that the critical ingredient for the shadow radius is the impact parameter $\eta$, and not the light ring size. Naturally, there is a strong correlation between both concepts, but at the end of the day what matters is the value of the impact parameter. We would like to point out that this observation is often not clear enough in the literature: a large variation of the light ring size does not have to lead to equally large variations of the shadow radius.
Acknowledgments {#acknowledgments .unnumbered}
===============
P.C. is supported by Grant No. PD/BD/114071/2015 under the FCT-IDPASC Portugal Ph.D. program and by the Calouste Gulbenkian Foundation under the Stimulus for Research Program 2015. C.H. and E. R. acknowledge funding from the FCT-IF programme. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690904 and by the CIDMA project UID/MAT/04106/2013. B.K. and J.K. gratefully acknowledge discussions with Arne Grenzebach, Claus Lämmerzahl and Volker Perlick, as well as support by the DFG Research Training Group 1620 “Models of Gravity” and by the grant FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-606096).
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[^1]: BH solutions of a closely related Horndeski model can be found in [@Sotiriou:2013qea; @Sotiriou:2014pfa].
[^2]: In this work we shall use geometric units $c = G = 1$.
[^3]: Since the system possesses the symmetry $\gamma \to-\gamma$, $\phi\to -\phi$, it is enough to consider strictly positive values of $\gamma$. Furthermore, in order to have a non-trivial coupling to the dilaton field, $\gamma\neq 0$.
[^4]: We follow the conventions in Ref. [@Guo:2008hf].
[^5]: Note that solutions seem to exist for any nonzero value of $\gamma$.
[^6]: Perturbing the extremal vacuum solution in $\alpha$, the scalar field/metric develops singularities at the poles in first/second order in $\alpha$.
[^7]: An analytical expression for the Kerr shadow, as seen by an observer with zero angular momentum (ZAMO), can be found in [@Cunha:2016bpi].
[^8]: We are implicitly assuming that there is no glowing matter in front of the BH.
[^9]: [For a given BH under observation, the quantities $M$, $J$ and $\tilde{R}$ are all assumed to be known.]{}
[^10]: [The shadows are comparable if $M$, $J$ and the observation distance $\tilde{R}=\sqrt{g_{\varphi\varphi}}$ are the same.]{}
[^11]: [Additional measures of EdGB shadow shapes are possible, but they resemble closely Kerr ones.]{}
[^12]: The perimetral radius $\sqrt{g_{\varphi\varphi}}$ in $M$ units can be used as an invariant measure for the light ring size.
|
---
abstract: 'This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings, respectively, in the context of (locally) essential domains.'
address:
- 'Dipartimento di Matematica, Università degli Studi “Roma Tre”, 00146 Roma, Italy'
- 'Department of Mathematics, P.O. Box 5046, KFUPM, Dhahran 31261, Saudi Arabia'
author:
- 'M. Fontana'
- 'S. Kabbaj'
title: Essential Domains and Two Conjectures in Dimension Theory
---
[^1]
[^2]
Introduction
============
Let $R$ be an integral domain with quotient field $K$ and let $\Int(R):=\{f \in K[X]: f(R) \subseteq R\}$ be the ring of integer-valued polynomials over $R$. Considerable work, part of it summarized in Cahen-Chabert’s book [@CC2], has been concerned with various aspects of integer-valued polynomial rings. A central question concerning $\Int(R)$ is to describe its prime spectrum and, hence, to evaluate its Krull dimension. Several authors tackled this problem and many satisfactory results were obtained for various classes of rings such as Dedekind domains [@Ch1; @Ch2], Noetherian domains [@Ch2], valuation and pseudo-valuation domains [@CH], and pseudo-valuation domains of type $n$ [@T]. A well-known feature is that $\dim(R[X])-1
\leq \dim(\Int(R))$ for any integral domain $R$ [@Ca1]. However, the problem of improving the upper bound $\dim(\Int(R))\leq \dim_{v}(R[X])=\dim_{v}(R)+1$ [@FIKT1], where $\dim_{v}(R)$ denotes the valuative dimension of $R$, is still elusively open in general. It is due, in part, to the fact that the fiber in $\Int(R)$ of a maximal ideal of $R$ may have any dimension [@Ca1 Example 4.3] (this stands for the main difference between polynomial rings and integer-valued polynomial rings). Noteworthy is that all examples conceived in the literature for testing $\dim(\Int(R))$ satisfy the inequality $\dim(\Int(R)) \leq \dim(R[X])$. In [@FIKT1; @FIKT2], we undertook an extensive study, under two different approaches, in order to grasp this phenomenon. We got then further evidence for the validity of the conjecture:
($\mathcal{C}_{1}$).
The current situation can be described as follows; ($\mathcal{C}_{1}$) turned out to be true in three large -presumably different- classes of commutative rings; namely: (a) Krull-type domains, e.g., unique factorization domains (UFDs) or Krull domains [@Gr2; @FIKT2]; (b) pseudo-valuation domains of type $n$ [@FIKT2]; and (c) Jaffard domains [@FIKT1; @Ca1].
A finite-dimensional domain $R$ is said to be Jaffard if $\dim(R[X_{1}, ..., X_{n}]) = n + \dim(R)$ for all $n \geq 1$; equivalently, if $\dim(R) = \dim_{v}(R)$ [@ABDFK; @BK; @DFK; @G; @J]. The class of Jaffard domains contains most of the well-known classes of finite-dimensional rings involved in dimension theory of commutative rings such as Noetherian domains [@K], Prüfer domains [@G], universally catenarian domains [@BDF], stably strong S-domains [@Kab; @MM]. However, the question of establishing or denying a possible connection to the family of Krull-like domains is still unsolved. In this vein, Bouvier’s conjecture (initially, announced during a 1985 graduate course at the University of Lyon I) sustains that:
($\mathcal{C}_{2}$).
As the Krull property is stable under adjunction of indeterminates, the problem merely deflates to the existence of a Krull domain $R$ with $1+\dim(R)\lneqq \dim(R[X])$. It is notable that the rare non-Noetherian finite-dimensional UFDs or Krull domains existing in the literature do defeat ($\mathcal{C}_{2}$), since all are Jaffard [@AM; @Da1; @Da2; @DFK; @Fu; @G2]. So do the examples of non-Prüfer finite-dimensional Prüfer $v$-multiplication domains (PVMDs) [@Gr1; @HMM; @MZ; @Z]; as a matter of fact, these, mainly, arise as polynomial rings over Prüfer domains or as pullbacks, and both settings either yield Jaffard domains or turn out to be inconclusive (in terms of allowing the construction of counterexamples) [@ABDFK; @FG]. In order to find the missing link, one has then to dig beyond the context of PVMDs.
Essential domains happen to offer such a suitable context for ($\mathcal{C}_{2}$) as well as a common environment for both conjectures ($\mathcal{C}_{1}$) and ($\mathcal{C}_{2}$), though these have developed in two dissimilar milieus. An integral domain $R$ is said to be [*essential*]{} if $R$ is an intersection of valuation rings that are localizations of $R$ [@Gr3]. As this notion does not carry up to localizations, $R$ is said to be [*locally essential*]{} if $R_{p}$ is essential for each $p\in
\Spec(R)$. Notice that the locally essential domains correspond to the $P$-domains in the sense of Mott and Zafrullah [@MZ]. PVMDs and almost Krull domains [@G p. 538] are perhaps the most important examples of locally essential domains. Recall that Heinzer constructed in [@H2] an example of an essential domain that is not locally essential. Also, it is worth noticing that Heinzer-Ohm’s example [@HO] of an essential domain which is not a PVMD is, in fact, locally essential (cf. [@MZ Example 2.1]). Finally recall that a Krull-type domain is a PVMD in which no non-zero element belongs to an infinite number of maximal $t$-ideals [@Gr1]. We have thus the following implications within the family of Krull-like domains:
-------------- ------------------- --------------
UFD
$\downarrow$
Krull
$\swarrow$ $\searrow$
Krull-type Almost Krull
$\downarrow$
PVMD $\swarrow$
$\searrow$
Locally Essential
$\downarrow$
Essential
-------------- ------------------- --------------
The purpose of this note is twofold. First, we state a result that widens the domain of validity of ($\mathcal{C}_{1}$) to the class of locally essential domains. It is well-known that ($\mathcal{C}_{1}$) holds for Jaffard domains too [@FIKT1; @Ca1]. So one may enlarge the scope of study of ($\mathcal{C}_{2}$) -discussed above- and legitimately raise the following problem:
($\mathcal{C'}_2$)
Clearly, an affirmative answer to ($\mathcal{C'}_2$) will definitely defeat ($\mathcal{C}_2$); while a negative answer will partially resolve ($\mathcal{C}_2$) for the class of (locally) essential domains. Our second aim is to show that the rare constructions of non-trivial (locally) essential domains (i.e., non-PVMD) existing in the literature yield Jaffard domains, putting therefore ($\mathcal{C'}_2$) under the status of open problem. Consequently, a settlement of ($\mathcal{C}_2$) seems -at present- out of reach.
Result and example
==================
In the first part of this section, we establish the following result.
For any locally essential domain $R$, $\dim(\Int(R))=\dim(R[X])$.
Assume that $R$ is finite-dimensional and $R\not=K$, where $K$ denotes the quotient field of $R$. Let $R=\bigcap_{p\in\Delta}R_{p}$ be a locally essential domain, where $\Delta\subseteq\Spec(R)$. Set:
$\begin{array}{ll}
\Delta_{1}:= &\{p\in\Delta: R_{p}\ \textup{is a DVR}\}\\
\Delta_{2}:= &\{p\in\Delta: R_{p}\ \textup{is a valuation domain but
not a DVR}\}.
\end{array}$\
We wish to show first that $\dim(\Int(R))\leq\dim(R[X])$. Let $M$ be a maximal ideal of $\Int(R)$ such that $\dim(\Int(R))=\htt(M)$ and let $\mathcal{M}:= M\cap R$. Without loss of generality, we may assume that $\mathcal{M}$ is maximal in $R$ with a finite residue field. We always have $R_{\mathcal{M}}[X]\subseteq (\Int(R))_{\mathcal{M}}\subseteq
\Int(R_{\mathcal{M}})$ [@CC1 Corollaires (4), p. 303]. If $\mathcal{M}\in\Delta_{1}$, then $R_{\mathcal{M}}[X]$ is a two-dimensional Jaffard domain [@S2 Theorem 4] and [@ABDFK Proposition 1.2]. So the inclusion $R_{\mathcal{M}}[X]\subseteq(\Int(R))_{\mathcal{M}}$ yields $\dim((\Int(R))_{\mathcal{M}})\leq \dim_{v}(R_{\mathcal{M}}[X])=\dim(R_{\mathcal{M}}[X])$. Thus, $\dim((\Int(R))_{\mathcal{M}})=\dim(R_{\mathcal{M}}[X])=2$. If $\mathcal{M}\in\Delta_{2}$, then $\Int(R_{\mathcal{M}})=R_{\mathcal{M}}[X]=
(\Int(R))_{\mathcal{M}}$ [@CC1 Exemples (5), p. 302]. If $\mathcal{M}\notin\Delta$, then $R_{\mathcal{M}}=\bigcap_{p\in\Delta, p\subsetneqq\mathcal{M}}R_{p}$ since $R$ is a locally essential domain. So that by [@CC1 Corollaires (3), p. 303] $\Int(R_{\mathcal{M}})=\bigcap_{p\in\Delta, p\subsetneqq\mathcal{M}}\Int(R_{p})
=\bigcap_{p\in\Delta, p\subsetneqq\mathcal{M}}R_{p}[X]=R_{\mathcal{M}}[X]=
(\Int(R))_{\mathcal{M}}$. In all cases, we have $\dim(\Int(R))=
\dim((\Int(R))_{\mathcal{M}})=\dim(R_{\mathcal{M}}[X])\leq\dim(R[X])$, as desired.
We now establish the inverse inequality $\dim(R[X])\leq\dim(\Int(R))$. Let $M$ be a maximal ideal of $R[X]$ such that $\dim(R[X])=\htt(M)$, and $\mathcal{M}:= M\cap R$. Necessarily, $\mathcal{M}$ is maximal in $R$. Further, we may assume that $\mathcal{M}$ has a finite residue field. If $\mathcal{M}\in\Delta_{2}$ or $\mathcal{M}\notin\Delta$, similar arguments as above lead to $R_{\mathcal{M}}[X]=(\Int(R))_{\mathcal{M}}$ and hence to the conclusion. Next, suppose $\mathcal{M}\in\Delta_{1}$. Then, $\Int(R_{\mathcal{M}})$ is a two-dimensional (Prüfer) domain [@Ch3]. Let $(0)\subsetneqq P_{1}\subsetneqq P_{2}$ be a maximal chain of prime ideals in $\Spec(\Int(R_{\mathcal{M}}))$. Clearly, it contracts to $(0)\subsetneqq \mathcal{M}R_{\mathcal{M}}$ in $\Spec(R_{\mathcal{M}})$. Further, by [@Ca1 Corollaire 5.4], $P_{1}$ contracts to $(0)$. Therefore, by [@Ch2 Proposition 1.3], $P_{1}=fK[X]\cap\Int(R_{\mathcal{M}})$, for some irreducible polynomial $f\in K[X]$. This yields in $\Spec(R_{\mathcal{M}}[X])$ the maximal chain: $$(0)\subsetneqq P_{1}\cap R_{\mathcal{M}}[X]=fK[X]\cap
R_{\mathcal{M}}[X]\subsetneqq P_{2}\cap R_{\mathcal{M}}[X],$$ which induces in $\Spec(\Int(R))_{\mathcal{M}})$ the following maximal chain: $$(0)\subsetneqq P_{1}\cap \Int(R))_{\mathcal{M}}=
fK[X]\cap \Int(R))_{\mathcal{M}}\subsetneqq P_{2}\cap \Int(R))_{\mathcal{M}}.$$ Consequently, in all cases, we obtain $\dim(R[X])=\dim(R_{\mathcal{M}}[X])=\dim((\Int(R))_{\mathcal{M}})$ $\leq\dim(\Int(R))$, to complete the proof of the theorem.
>From [@HO Proposition 1.8] and [@G Exercise 11, p. 539] we obtain the following.
Let $R$ be a PVMD or an almost Krull domain. Then $\dim(\Int(R))=\dim(R[X])$.
In the second part of this section, we test the problem ($\mathcal{C'}_{2}$) -set up and discussed in the introduction- for the class of non-PVMD (locally) essential domains. These occur exclusively in Heinzer-Ohm’s example [@HO] and Heinzer’s example [@H2] both mentioned above. The first of which is a 2-dimensional Jaffard domain [@MZ Example 2.1]. Heinzer’s example [@H2], too, is a 2-dimensional Jaffard domain by [@DFK Theorem 2.3]. Our next example shows that an enlargement of the scope of this construction -still- generates a large family of essential domains with nonessential localizations of arbitrary dimensions $\geq 2$ -but unfortunately- that are Jaffard domains.
\[3.1\] For any integer $r\geq2$, there exists an $r$-dimensional essential Jaffard domain $D$ that is not locally essential.
Notice first that the case $r=2$ corresponds to Heinzer’s example mentioned above. In order to increase the dimension, we modify Heinzer’s original setting by considering Kronecker function rings via the $b$-operation. For the sake of completeness, we give below the details of this construction.
Let $R$ be an integral domain, $K$ its quotient field, $n$ a positive integer (or $n=\infty$), and $X,X_{1}, ..., X_{n}$ indeterminates over $K$. The $b$-operation on $R$ is the a.b. star operation defined by $I^{b}:=\bigcap\{IW: W\ \textup{is a
valuation overring of}\ R\}$, for every fractional ideal $I$ of $R$. Throughout, we shall use $\Kr_{K(X)}(R, b)$ to denote the Kronecker function ring of $R$ defined in $K(X)$ with respect to the $b$-operation on $R$; and $R(X_{1}, ..., X_{n})$ to denote the Nagata ring associated to the polynomial ring $R[X_{1}, ..., X_{n}]$, obtained by localizing the latter with respect to the multiplicative system consisting of polynomials whose coefficients generate $R$.
Let $r$ be an integer $\geq 2$. Let $k_{0}$ be a field and $\{X_{n}: n\geq 1\}$, $Y$, $\{Z_{1}, ..., Z_{r-1}\}$ be indeterminates over $k_{0}$. Let $n$ be a positive integer. Set: $$\begin{array}{lll}
k_{n}:=k_{0}(X_{1}, ..., X_{n}) &; & k:=\bigcup_{n\geq 1}k_{n}\\
F_{n}:=k_{n}(Z_{1}, ..., Z_{r-1}) &; & F:=\bigcup_{n\geq 1}F_{n}=k(Z_{1}, ..., Z_{r-1})\\
K_{n}:=F_{n}(Y) &; & K:=\bigcup_{n\geq 1}K_{n}=F(Y)\\
M_{n}:=YF_{n}[Y]_{(Y)} &; & M:=\bigcup_{n\geq 1}M_{n}=YF[Y]_{(Y)} \\
A_{n}:= k_{n}+M_{n} &; & A:=\bigcup_{n\geq 1}A_{n}=k+M\\
V_{n}:=F_{n}[Y]_{(Y)} &; & V:=\bigcup_{n\geq
1}V_{n}=F[Y]_{(Y)}.
\end{array}$$
Note that, for each $n\geq 1$, $V$ and $V_{n}$ (resp., $A$ and $A_{n}$) are one-dimensional discrete valuation domains (resp., pseudo-valuation domains) and $\dim_{v}(A)=\dim_{v}(A_{n})$ $=r$. For each $n\geq 1$, set $X'_{n}:=\frac{1+YX_{n}}{Y}$. Clearly, $K_{n}=K_{n-1}(X_{n})=K_{n-1}(X'_{n})$. Next, we define inductively two sequences of integral domains $\big(B_{n}\big)_{n\geq 2}$ and $\big(D_{n}\big)_{n\geq 1}$ as follows: $$\begin{array}{lll}
&;& D_{1}:= A_{1}\\
B_{2}:=\Kr_{K_{1}(X'_{2})}(D_{1},b) &;& D_{2}:=B_{2}\cap A_{2}\\
B_{n}:=\Kr_{K_{n-1}(X'_{n})}(D_{n-1},b) &;& D_{n}:=B_{n}\cap
A_{n},\textup{ for } n\geq 3.
\end{array}$$
For $ n\geq 2$, let ${\mathcal M}_{n} := M_{n} \cap D_{n}\ (
\subset D_{n} = B_{n}\cap A_{n} \subseteq A_{n}$), where $M_{n}$ is the maximal ideal of $A_{n}$.
\[3.4\] (1) $B_{n}$ is an $r$-dimensional Bezout domain.\
(2) $B_{n}\cap K_{n-1}=D_{n-1}\subseteq A_{n-1}=A_{n}\cap K_{n-1}$.\
(3) $D_{n}\cap K_{n-1}=D_{n-1}$.\
(4) $D_{n}[X'_{n}]=B_{n}$ and $(D_{n})_{{\mathcal
M}_{n}}=D_{n}[\frac{1}{YX'_{n}}]=A_{n}$, with $\frac{1}{X'_{n}}$ and $YX'_{n}\in D_{n}$.\
(5) ${\mathcal M}_{n}$ is a height-one maximal ideal of $D_{n}$ with ${\mathcal M}_{n}\cap K_{n-1}={\mathcal M}_{n-1}$.\
(6) For each $q\in \Spec(D_{n})$ with $q\not={\mathcal M}_{n}$ there exists a unique prime ideal $Q\in \Spec(B_{n})$ contracting to $q$ in $D_{n}$ and $(D_{n})_{q}=(B_{n})_{Q}$.\
(7) $B_{n}=\bigcap\{(D_{n})_{q}: q\in \Spec(D_{n})\ \textup{and}\ q\not={\mathcal M}_{n}\}$.\
(8) For each $q'\in \Spec(D_{n-1})$ with $q'\not={\mathcal
M}_{n-1}$ there exists a unique prime ideal $q\ (\not={\mathcal M}_{n})\in \Spec(D_{n})$ contracting to $q'$ in $D_{n-1}$ such that $(D_{n})_{q}=(D_{n-1})_{q'}(X'_{n})$.
Similar arguments as in [@H2] lead to (1)-(7).\
(8) By (7), $B_{n-1}\subseteq(D_{n-1})_{q'}$, and hence $(D_{n-1})_{q'}$ is a valuation domain in $K_{n-1}$ of dimension $\leq r$ containing $D_{n-1}$. Since $B_{n}$ is the Kronecker function ring of $D_{n-1}$ defined in $K_{n-1}(X'_{n})$ by all valuation overrings of $D_{n-1}$, then $(D_{n-1})_{q'}$ has a unique extension in $K_{n-1}(X'_{n})$, which is a valuation overring of $B_{n}$, that is, $(D_{n-1})_{q'}(X'_{n})$. By (7), the center $q$ of $(D_{n-1})_{q'}(X'_{n})$ in $D_{n}$ is the unique prime ideal of $D_{n}$ lying over $q'$ and $(D_{n})_{q}=(D_{n-1})_{q'}(X'_{n})$.
Set $D:=\bigcup_{n\geq 1}D_{n}$ and ${\mathcal M}:=\bigcup_{n\geq
2}{\mathcal M}_{n}$. It is obvious that $D \subseteq
A=\bigcup_{n\geq 1}A_{n}$.
\[3.5\] $D_{\mathcal M}=A$ and ${\mathcal M}$ is a height-one maximal ideal in $D$.
It is an easy consequence of Claim \[3.4\](4), since ${\mathcal M}_{n} ={\mathcal M}\cap D_{n}$, for each $n$.
Let $q\in \Spec(D)$ with $q\not={\mathcal M}$. Then, for some $m\geq 2$, we have in $D_{m}$, $q_{m}:=q\cap D_{m}\not={\mathcal
M}_{m} ={\mathcal M}\cap D_{m}$. So, by Claim \[3.4\](6), $B_{m}\subseteq(D_{m})_{q_{m}}$, hence $(D_{m})_{q_{m}}$ is a valuation overring of $B_{m}$ of dimension $\leq r$, whence, by Claim \[3.4\](8), $D_{q}=\bigcup_{n\geq
1}(D_{n})_{q_{n}}=(D_{m})_{q_{m}}(X'_{m+1}, ...)$ is still a valuation domain of dimension $\leq r$.
\[3.6\] $D=\bigcap\{D_{q}: q\in \Spec(D)\ \textup{and}\ q\not={\mathcal
M}\}$.
Similar to [@H2].
>From Claims \[3.5\] and \[3.6\] we obtain:
$D$ is an essential domain with a nonessential localization and $\dim(D)=\dim_{v}(D)=r$.
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[^1]: The first author was partially supported by a research grant MIUR 2001/2002 (Cofin 2000-MM01192794). The second author was supported by the Arab Fund for Economic and Social Development
[^2]: This work was done while both authors were visiting Harvard University
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title: '[^1]'
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[^1]: Submitted to the editors .
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abstract: 'We evaluate the matter-antimatter asymmetry produced by emission of fermionic carriers from vortons which are assumed to be destabilized at the electroweak phase transition. The velocity of contraction of the vorton, calculated through the decrease of its magnetic energy, originates a chemical potential which allows a baryogenesis of the order of the observed value. This asymmetry is not diluted by reheating if the collapse of vortons is distributed along an interval of $\sim 10^{-9}$sec.'
author:
- |
Luis Masperi [^1] [^2]\
Centro Atómico Bariloche and Instituto Balseiro,\
Comisión Nacional de Energía Atómica and Universidad Nacional de Cuyo,\
8400 San Carlos de Bariloche, Argentina
- |
Milva Orsaria [^3]\
Laboratorio TANDAR,\
Comisión Nacional de Energía Atómica, Av. del Libertador 8250,\
1429 Buenos Aires, Argentina
title: BARYOGENESIS THROUGH GRADUAL COLLAPSE OF VORTONS
---
Introduction
============
The matter-antimatter asymmetry in the universe is one of the well established facts of cosmology. There are many possible mechanisms to generate this baryonic density due to phenomena which presumably occurred in the first fraction of second after the big-bang but all of them suffer some criticism. They include also methods involving cosmic strings or other topological defects produced in some of the phase transitions produced in the universe.
In this work we present a baryogenesis model based on possibly very abundant closed cosmic strings called vortons stabilized by superconducting currents, which might lose this stability at the electroweak phase transition. The distinctive feature of our mechanism is that we follow the process of contraction of vortons and that we assume that they do not destabilize all at the same time.
In Section 2 we give a survey of the baryogenesis methods which have some connection with our proposal. In Section 3 we briefly describe vortons and their relationship with Grand Unified Theories (GUT). Section 4 reminds the instantaneous decay of vortons and the reheating which causes the dilution of matter-antimatter asymmetry. In Section 5 we present our scenario of gradual decay of vortons indicating how the reheating problem may be solved. Section 6 contains the details of the calculation of the contraction velocity which, for the case of charged carriers, is based on the decrease of the associate magnetic energy. In Section 7 we evaluate by tunneling the probability of emission of carriers which gives way to the asymmetry produced by each vorton. Section 8 shows how the variation of magnetic field due to contraction produces a chemical potential which allows our baryogenesis to be of the order of the expected one. Section 9 contains some conclusions.
Methods to generate matter - antimatter asymmetry
=================================================
From the nucleosynthesis of light elements there is a costraint for the baryonic density which, related to entropy density to give an invariant value, is [@Kolb]
$$\frac{n_B}s=10^{-11}-10^{-10}\text{ . } \label{e1}$$
To explain this asymmetry, if one starts from a symmetric universe, three conditions are required [@A.D.Sakharov] : i ) non-conservation of baryonic number, ii ) violation of C and CP to distinguish particle from antiparticle, iii ) period of non-equilibrium to allow different number of particles and antiparticles.
The method of baryogenesis closest to experimental verification is that which corresponds to the electroweak phase transition [@A.D.Dolgov] provided it is a first-order one. Expanding bubbles of the broken-symmetry phase would produce in its wall a chemical potential due to the variation of a CP violating phase $\theta $ compared to the external symmetric medium, where the active sphalerons would generate the baryonic density. The latter would not be erased because the bubble expansion would include it in the broken-symmetry phase where sphaleron processes are very slow. Due to the rate of sphaleron transitions in the high-temperature phase one would obtain
$$\frac{n_B}s\simeq \frac{\alpha _w^4}{g^{*}}\text{ }\Delta \theta \text{ , }
\label{e2}$$
where the weak coupling is such that $\alpha _w\simeq 10^{-3}$ and the number of zero-mass modes at the electroweak temperature $T\sim 100GeV$ is $%
g^{*}\simeq 100.$ Therefore the observed asymmetry is reproduced if $\Delta
\theta \sim 10^{-2}.$ However this mechanism with Standard Model ingredients is not possible because the phase transition turns out to be of second order for the experimental bound on the Higgs mass and the CP violation is not enough.
A solution which would include not too high-energy elements beyond the Standard Model is afforded by the Minimal Supersymmetric Standard Model. But this model would be severely constrained because to give an enough first-order phase transition the Higgs boson should be light $m_H<100GeV$as well as the stop $m_{\widetilde{t}}\leq 200GeV$, and to allow a large enough variation of the CP violation parameter without entering in conflict with the neutron electric dipole moment the lower generations of squarks should be very heavy[@M.Carena], though this last condition might be relaxed[@A.Riotto].
On the other extreme of the energy range, a possibility of baryogenesis would be given by the decay of GUT Higgs and gauge bosons which should be produced out of equilibrium requiring $T\sim 10^{16}GeV$. The generated baryonic density would be
$$\frac{n_B}s\simeq \frac \varepsilon {g^{*}} \label{e3}$$
where, with the asymmetry produced by one of these superheavy particles $%
\varepsilon \sim 10^{-8},$ there would be agreement with the expected value.
A problem is here that at these extremely high temperatures magnetic monopoles would have been produced with the consecuent overclosure of the universe density, as well as very heavy cosmic strings which might originate undesirable inhomogeneities. It is anyhow difficult to explain such high T from the reheating at the end of inflation, unless the non-linear quantum effects of preheating give way to an explosive heavy particle production out of equilibrium[@E.W.Kolb].
Cosmic strings and vortons
==========================
Cosmic strings are topological defects which appear in a phase transition when an abelian symmetry additional to the standard model is broken. To avoid the monopole problem we may assume that the universe reached a temperature for which the GUT symmetry G was already broken
$$G\rightarrow SU\left( 3\right) _C\times SU\left( 2\right) _L\times U\left(
1\right) _Y\times \widetilde{U}\left( 1\right) \text{ .} \label{e4}$$
If at a slightly lower temperature, let us say $10^{15}GeV$, also the symmetry $\widetilde{U}\left( 1\right) $ is broken the cosmic strings will be produced[@T.W.B.Kibble]. They will become superconducting[@E.Witten] depending on the group G and the details of the Higgs mechanism for the breaking of $\widetilde{U}\left( 1\right) $. A superconducting current will appear for those fermionic carriers which acquire mass due to the coupling with a Higgs field which winds the string and originates zero modes inside it. The superconducting current classically stabilizes closed loops through a number $N$ related to the angular momentum due to the carriers inside them.
It is not necessary that the carriers are charged[@R.L.Davis]. In fact if $G=SO\left( 10\right) $ the only particle which acquires mass at the $%
\widetilde{U}\left( 1\right) $ phase transition is $\nu _R$ which may have a zero mode inside the string. On the other hand if $G=E_6$ several fermions acquire mass at the $\widetilde{U}\left( 1\right) $ breaking and some of them, which may give superconducting currents, are charged and with baryonic number.
For normal cosmic strings it is interesting[@P.Bhattacharjee] that if the emission of superheavy bosons at present time is normalized to explain the flux of ultra-high energy cosmic rays, their decay in the past may give the expected baryogenesis provided that the asymmetry per particle is six orders of magnitude higher than that necessary in Eq.(3).
The stabilized superconducting closed loops are called vortons[@Shellard]. Their number density, mass, length and quantum decay probability depend on the coincidence or not of the scales of string formation and appearance of superconductivity in them[@R.Brandenberger]. If both scales coincide at $%
m_x$ the vorton density is
$$n_v\simeq \left( \frac{m_x}{m_{pl}}\right) ^{\frac 32}T\text{ }^{\text{3}}%
\text{ , } \label{e5}$$
its energy $E_v\simeq N$ $m_x$, radius $R\simeq $ $N$ $m_x^{-1}$ and $N\sim
10$ if $m_x\sim 10^{15}GeV$. If the superconductivity scale is smaller than the formation one, the density is smaller and vortons are more stable for quantum decay.
The density Eq.(5) overcloses the universe in a way similar to that of monopoles, if there is not a collapse of vortons for some reason. If this is produced at high energy when the carriers are $\nu _R$, a lepton asymmetry appears which may be converted into baryon asymmetry by sphaleron processes to give the expected value with adequately large CP violation parameter[@R.Jeannerot]. Alternatively, if superconductivity appears at much lower temperature, i.e. at the supersymmetric scale $\sim 1TeV$, there are models predicting that vortons which subsequently decay below the electroweak temperature may release baryonic charge in agreement with the expected one[@Riotto], again assuming an adequate CP violation factor.
It must be noted that if the scales of formation and superconductivity coincide and the vorton density is decreased for some process to be constrained by the critical density of universe, the quantum decay probability might be enough to explain the high energy cosmic rays[@L.Masperi].
Instantaneous decay of vortons at the electroweak transition
============================================================
Trying to include as few ingredients as possible, we will adopt the point of view that vortons have obtained the superconducting property at the same scale of formation, and that they lose their stability at the well established electroweak transition. This may occur if, due to the new Higgs mechanism at this scale, the zero modes acquire a small mass[@S.C.Davis]. It is not required that the transition is of first order.
If vortons disappear instantaneously, since they contain roughly N heavy bosons the produced baryonic density is
$$\frac{n_B}s=\left( \frac{m_x}{m_{pl}}\right) ^{\frac 32}\frac{N\varepsilon }{%
g^{*}}\text{ , } \label{e6}$$
which will be very small if the asymmetry due to each particle X is of the same order of that of GUT bosons assumed in Eq.(3).
Furthermore, since vortons behave as non-relativistic matter, its density which is very small at formation becomes equivalent to that of radiation at $%
T\sim 10^8GeV$ and dominates on it by 6 orders of magnitude at the electroweak scale. Therefore if at this temperature vortons transform instantaneously into light particles, i.e.radiation, there will be a reheating according to
$$\rho _v\left( T_{EW}\right) =N\text{ }m_x\left( \frac{m_x}{m_{pl}}\right)
^{\frac 32}T_{EW}^{\text{3}}=\rho _{_R}\left( T_{reh}\right) =g^{*}T_{reh}^4%
\text{ ,} \label{e7}$$
which gives $T_{reh}\simeq 10^{\frac 72}GeV$.
This instantaneous reheating would produce an increase of the entropy density of $\left. \left( \frac{Treh}{T_{EW}}\right) ^3\simeq 10^{\frac
92}\right. $ times with the corresponding dilution [@W.B.Perkins] of the baryogenesis of Eq.(6) in the same factor.
According to this scenario the universe would be initially dominated by radiation, then from $T\sim 10^8GeV$ to $T_{EW}$ by vortons and after the reheating to $T_{reh}$ again by radiation till $t\sim 10^{11}\sec .$ when finally non-relativistic matter takes over.
Gradual collapse of vortons
===========================
The alternative that we wish to present corresponds to the plausible situation that vortons are not destabilized all at the same time when reaching the electroweak temperature. Due to the Higgs mechanism that will be working in this phase transition, we expect a probability that a vorton loses its zero modes and starts its collapse. Without attempting to calculate this probability for destabilization, we remark that the temperature will remain constant at $T_{EW}\sim 100GeV$ if vortons decay during an interval such that the universe expands its scale from $a_1$ to $%
a_2$ when all is transformed to radiation
$$a_{1}^{3}\text{ }N\text{ }m_{x}\left( \frac{m_{x}}{m_{pl}}\right) ^{\frac{3}{%
2}}T_{EW}^{3}=a_{2}^{3}\text{ }g^{*}T_{EW}^{4}\text{ .} \label{e8}$$
The space scale would therefore increase in two orders of magnitude and, using $\frac{a_1}{a_2}=\left( \frac{t_1}{t_2}\right) ^{\frac 23}$, if the process starts at $t_1\sim 10^{-12}\sec $ it would be completed at $t_2\sim
10^{-9}\sec $. The advantage is now that the total increase of entropy, which is similar to that of instantaneous destabilization, is distributed in a larger volume. Baryogenesis would not be diluted at the beginning of the interval but only at the end with a factor $\left( \frac{a_2}{a_1}\right) ^3$so that the average dilution would be $\sim \frac 12.$ It is reasonable to think that the collapse of vortons keeps the temperature constant because as soon as there is a tendence to reheating the symmetry is restored and the destabilization of vortons stops.
Furthermore, we will follow the contraction of each vorton obtaining baryogenesis not by the presumably small asymmetry in the decay of bosons X, but from the emission of charged baryonic carriers during the collapse. The resulting asymmetry per vorton may turn out to be larger due to the chemical potential which will appear in the wall of the vorton because of the non-equilibrium process of contraction, resulting in a different emission of fermions and antifermions.
Velocity of contraction during vorton decay
===========================================
The evaluation of the velocity of contraction of the vorton after its destabilization at the electroweak temperature is crucial for determining the non-equilibrium process.
One possibility of calculation, which may be applied to the case of neutral carriers, is to consider that stabilization is abruptly lost at $T_{EW}$ so that the string contracts due to a constant tension $\mu \sim m_x^2$. If the string mass were constant and the initial radius is $R\sim \frac N{m_x}$, the relation between the velocity and each radius r would be
$$\text{v}^2\simeq 2\text{ }\frac{R-r}R\text{ .} \label{e9}$$
Considering that the vorton mass decreases linearly with its radius in the rest frame, including the Lorentz factor and being at the initial stage of the contraction when the iterative approximation may be used, the velocity turns out to be $$\begin{aligned}
\text{v} &=&\frac 1{\sqrt{2}}\arctan \frac{\sqrt{2}N}{m_xt}\left[ 1-\left(
\frac{m_xt}N\right) ^2\right] ^{\frac 12}+2\arcsin \left( \frac{m_xt}N\right)
\label{e10} \\
&&-\frac 3{2\sqrt{2}}\ln \left( \left| \frac{m_xt+\sqrt{2}N\left[ 1-\left(
\frac{m_xt}N\right) ^2\right] ^{\frac 12}}{m_xt-\sqrt{2}N\left[ 1-\left(
\frac{m_xt}N\right) ^2\right] ^{\frac 12}}\right| \right) -\frac \pi {2\sqrt{%
2}}\text{ .} \nonumber\end{aligned}$$
To have the relation between velocity and radius which replaces Eq.(9), v of Eq.(10) should be integrated on time. It is clear that v will vary from 0 to 1 when r = 0 so that the time of collapse will be $t_{c}\geq \frac{N}{m_{x}}$ .
For charged carriers, in which we are more interested, the velocity of contraction may be calculated in an easier way. We consider the decay of a vorton as a succession of transitions between superconducting states of numbers N, N-1,...keeping the value of the current I. Looking at a classical average, one will see a loop with increasing contraction velocity with the corresponding relativistic factor in its mass which will be compensated by the decrease of the magnetic energy that is defined in the broken-symmetry phase.
The balance for the vorton when its radius is r and the associated magnetic field is B compared with the initial one B$_i,$ will be
$$\frac{r}{R}\text{ }N\text{ }m_{x}\left( \frac{1}{\sqrt{1-\text{v}^{\text{2}}}%
}-1\right) =\frac{1}{2}\int d\mathbf{\rho }\left( B_{i}^{2}-B^{2}\right)
\text{ }. \label{e11}$$
Since in general we will expect
$$\frac{1}{2}\int d\mathbf{\rho }\text{ }B^{2}=k\text{ }I^{2}\text{ }r
\label{e12}$$
and being $I=\frac{N}{2\pi R}$ , we will have
$$1-\text{v}^2=\frac 1{\left( 1+\frac{k_i\text{ }R-k_f\text{ }r}{4\pi ^2\text{
}r}\right) ^2}\text{ ,} \label{e13}$$
where the coefficient k$_{f}$ at the end of the collapse may be different from the initial one k$_{i}$.
At the beginning, when $R-r$ is small and $k_i=k_f$
$$\text{v}^2\simeq \frac{k_i}{2\pi ^2}\text{ }\frac{R-r}R\text{ ,} \label{e14}$$
which is analogous to the previous estimation due to constant tension. For $%
r\rightarrow 0$ the velocity Eq.(13) will tend to 1.
An important ingredient for our evaluation of baryogenesis will be the calculation of the coefficients k.
In the first part of the contraction the vorton will be certainly well represented by a loop of radius r that, lying in the x-y plane, will give a magnetic potential
$$A_\varphi \left( \rho ,\theta \right) =I\text{ }r\text{ }\int_0^{2\pi
}d\varphi ^{\prime }\frac{\cos \varphi ^{\prime }}{\left( \rho ^2+r^2-2r\rho
\sin \theta \cos \varphi ^{\prime }\right) ^{\frac 12}}\text{ .} \label{e15}$$
For large distances $\rho \gg r$ Eq.(15) gives the dipole approximation for the magnetic field
$$B_\rho =\frac{2m\cos \theta }{\rho ^3}\text{ , }B_\theta =\frac{m\sin \theta
}{\rho ^3}\text{ ,} \label{e16}$$
with $m=\pi r^2I$ .
For distances much smaller than the radius $\rho \ll r$
$$B_\rho =\frac{2\pi I}r\text{ }\cos \theta \text{ , }B_\theta =-\text{ }\frac{%
2\pi I}r\text{ }\sin \theta \text{ ,} \label{e17}$$
whereas the exact expressions from Eq.(15) are
$$\begin{aligned}
B_{\rho } &=&\cot \theta \text{ }I\text{ }\frac{r}{\rho }\text{ }%
\int_{0}^{2\pi }d\varphi ^{\prime }\text{ }\frac{\cos \varphi ^{\prime }}{%
\left( \rho ^{2}+r^{2}-2r\rho \sin \theta \cos \varphi ^{\prime }\right) ^{%
\frac{1}{2}}} \\
&&+I\text{ }r^{2}\cos \theta \text{ }\int_{0}^{2\pi }d\varphi ^{\prime }%
\text{ }\frac{\cos ^{2}\varphi ^{\prime }}{\left( \rho ^{2}+r^{2}-2r\rho
\sin \theta \cos \varphi ^{\prime }\right) ^{\frac{3}{2}}}\end{aligned}$$
$$\begin{aligned}
B_{\theta } &=&-I\text{ }\frac{r}{\rho }\text{ }\int_{0}^{2\pi }d\varphi
^{\prime }\text{ }\frac{\cos \varphi ^{\prime }}{\left( \rho
^{2}+r^{2}-2r\rho \sin \theta \cos \varphi ^{\prime }\right) ^{\frac{1}{2}}}
\label{e18} \\
&&+I\text{ }r\text{ }\rho \text{ }\int_{0}^{2\pi }d\varphi ^{\prime }\text{ }%
\frac{\cos \varphi ^{\prime }}{\left( \rho ^{2}+r^{2}-2r\rho \sin \theta
\cos \varphi ^{\prime }\right) ^{\frac{3}{2}}} \nonumber \\
&&-I\text{ }r^{2}\sin \theta \text{ }\int_{0}^{2\pi }d\varphi ^{\prime }%
\text{ }\frac{\cos ^{2}\varphi ^{\prime }}{\left( \rho ^{2}+r^{2}-2r\rho
\sin \theta \cos \varphi ^{\prime }\right) ^{\frac{3}{2}}}\text{ .}
\nonumber\end{aligned}$$
For small $\sin \theta $, Eq.(18) may be approximated by
$$B_\rho =\frac{2\pi r^2I}{\left( \rho ^2+r^2\right) ^{\frac 32}}\text{ }\cos
\theta \text{ , }B_\theta =\frac{\pi r^2I}{\left( \rho ^2+r^2\right) ^{\frac
32}}\text{ }\frac{\rho ^2-2r^2}{\rho ^2+r^2}\text{ }\sin \theta \text{ .}
\label{e19}$$
We may evaluate the magnetic energy, except for the x-y plane, taking the dipole approximation Eq.(16) for $\rho >3$ $r$, the small $\rho $ approximation Eq.(17) for $\rho <\frac r3$ , and the intermediate expression Eq.(19) for $\frac r3<\rho <3$ $r$ since it matches well with the other ones at these values. In this way one obtains a contribution to k in Eq.(12) of $%
3.5\pi ^3$ , but it is still necessary to add the contribution near the plane x-y.
For this last part, the contribution will come essentially from the region close to the loop. For $\theta =\frac \pi 2$ , $B_\theta $ of Eq.(18) will have a logaritmic divergence for $\rho =r$ coming from $\varphi ^{\prime
}\sim 0$, which is regularized considering the width of the loop and that inside it $B=0$ to be a superconducting medium.
With the approximation of keeping a region $\alpha $ of integration of $%
\varphi ^{\prime }$ near to zero and comparing the finite contribution to $%
B_\theta $ with that coming from the exact evaluation of Eq. (18) for $\rho
=r$, its turns out that $\alpha \simeq 0.77$ . Taking now for $\rho =r+\eta $ the approximation of keeping up to terms $\varphi ^{\prime \text{ }2}$ in Eq.(18), for $\frac{\left| \eta \right| }r<<1$, one obtains
$$B_\theta \simeq \frac 1{\left( \alpha ^2r^2+\eta ^2\right) ^{\frac 12}}\text{
}\frac{2\alpha Ir}\eta +\left[ \ln \frac{\left| \eta \right| }r-\ln \left(
\alpha +\sqrt{\alpha ^2+\frac{\eta ^2}{r^2}}\right) \right] \frac Ir\text{ .
} \label{e20}$$
A reasonable estimation of the magnetic energy near the string, considering that it must correspond to the stages of contraction starting from $R\sim
\frac N{m_X\text{ }}$ and being $\eta \sim \frac 1{m_X}$ , is to take the above approximation for $B_\theta $ in an external region of size $\eta $ around it. It turns out that this contribution to the coefficient $k$ will be $\sim 2\pi ^3.$
Therefore the total contribution of magnetic energy when the decaying vorton may still be considered as a thick loop corresponds to
$$k_i\simeq 5.5\pi ^3\text{ . } \label{e21}$$
With this value, the expression Eq.(14) for the velocity of contraction using the magnetic energy is of the same order of that given by Eq.(9) for the initial stage with constant string tension.
But it is not always correct to consider the contracting vorton as a loop. At the final stage when the radius is of the order of the width it may be better to represent it as a sphere with currents running inside it around the z-axis. Approximating each disc at angle $\theta ^{\prime }$ by an effective loop of radius $\eta \sin \theta ^{\prime }$, the magnetic potential will be
$$A_\varphi \left( \rho ,\theta \right) =I\eta \int_0^\pi d\theta ^{\prime
}\sin \theta ^{\prime }\int_0^{2\pi }d\varphi ^{\prime }\frac{\cos \varphi
^{\prime }}{\left[ \rho ^2+\eta ^2-2\rho \eta \left( \sin \theta \sin \theta
^{\prime }\cos \varphi ^{\prime }+\cos \theta \cos \theta ^{\prime }\right)
\right] ^{\frac 12}}\text{ .} \label{e22}$$
For large distances $\rho >>\eta $ , Eq.(22) gives again the dipole limit
$$A_\varphi \left( \rho ,\theta \right) =I\frac{\pi \eta ^2}{\rho ^2}\text{ }%
\frac \pi 2\text{ }\sin \theta \text{ . } \label{e23}$$
On the other hand, near the sphere $\rho =\eta +\delta $ the contribution to the magnetic field will come mainly from a region $\alpha $ in the integration over $\varphi ^{\prime }$ for small values such that only terms $%
\varphi ^{\prime }$ $^2$are kept and a region $\beta $ in the integration over $\theta ^{\prime }$ such that $\theta ^{\prime }\simeq \theta $. The result is
$$\begin{aligned}
B_\rho \left( \delta ,\theta \right) &=&2\beta \cos \theta \text{ }I\eta
\text{ }\left[ \frac 1{2c\left( \eta +\delta \right) }\text{ }\left( 1+\frac{%
\delta ^2}{2c^2}\right) \ln \left( \frac{\alpha c}\delta +\sqrt{1+\frac{%
\alpha ^2c^2}{\delta ^2}}\right) \right. \\
&&\left. -\frac 1{\eta +\delta }\text{ }\frac{\alpha \delta }{4c^2}\sqrt{1+%
\frac{\alpha ^2c^2}{\delta ^2}}+\frac 12\text{ }\frac{\alpha \eta }{\delta
c^2}\text{ }\frac{\sin ^2\theta }{\sqrt{1+\frac{\alpha ^2c^2}{\delta ^2}}}%
\right] \text{ ,}\end{aligned}$$
$$\begin{aligned}
B_\theta \left( \delta ,\theta \right) &=&2\beta \sin \theta \text{ }I\eta
\left\{ -\left[ \frac 1{2c\left( \eta +\delta \right) }\left( 1+\frac{\delta
^2}{2c^2}\right) +\frac \delta {2c^3}\right] \ln \left( \frac{\alpha c}%
\delta +\sqrt{1+\frac{\alpha ^2c^2}{\delta ^2}}\right) \right. \label{e24}
\\
&&\left. +\frac 1{\eta +\delta }\text{ }\frac{\alpha \delta }{4c^2}\text{ }%
\sqrt{1+\frac{\alpha ^2c^2}{\delta ^2}}-\frac 12\text{ }\frac{\alpha \eta }{%
\delta c^2}\text{ }\frac{\sin ^2\theta }{\sqrt{1+\frac{\alpha ^2c^2}{\delta
^2}}}\text{ }+\frac \alpha {\sqrt{1+\frac{\alpha ^2c^2}{\delta ^2}}}\left(
\frac 1{\delta ^2}+\frac 1{2c^2}\right) \right\} \text{ ,} \nonumber\end{aligned}$$
where $c=\sqrt{\eta ^2+\eta \delta }\sin \theta $ .
The limit of Eq.(24) for small values of $\sin \theta $ is
$$B_\rho \rightarrow 2\beta \alpha \cos \theta \frac{I\eta }{\delta \rho }%
\text{ },\text{ }B_\theta \rightarrow 2\beta \alpha \sin \theta \frac{I\eta
^2}{\rho \delta ^2}\text{ , } \label{e25}$$
whereas for $\theta \sim \frac \pi 2$ and $\alpha \sim 0.77$ , $B_\rho \sim
0 $ and a numerical evaluation of Eq.(24) gives
$$B_\theta \left( \delta ,\theta =\frac \pi 2\right) \simeq 0.35\text{ }\beta
\text{ }\frac I\delta \text{ .} \label{e26}$$
We now take two regions for evaluating the magnetic energy : that for large $%
\rho $ where the field corresponds to the dipole approximation and that close to the sphere, since inside it the field is zero for a superconducting medium. The two regions match reasonably well for $\delta =\eta $ and the lower limit for the integration is $\rho =\frac 32$ $\eta $ because $\eta $ was the average radius of the disc in the x-y plane.
Therefore the coefficient for the magnetic energy when the vorton is approximated by a sphere turns out to be, with $\beta \sim \alpha $,
$$k_f\simeq 35\text{ .} \label{e27}$$
Probability of emission of carriers
===================================
We will calculate the matter-antimatter asymmetry per vorton through the emission of fermions and antifermions by quantum tunneling. This corresponds to the transition e.g. from a state of vorton with number $N$ to another with number $N-1$ plus a fermion of mass $m_x$ with conservation of angular momentum.
It must be stressed that this channel is not the dominant one for the contraction of the string since the corresponding partial lifetime is much longer than the actual time of collapse. But it turns out to be the most effective one for baryogenesis since, due to the chemical potential produced by the non-equilibrium process of contraction, the probability for emission of baryons will be substancially different from that of antibaryons. In comparison, other channels which eliminate pieces of string due to the destabilization produced at the electroweak transition will give through the decay of heavy bosons a rather small amount of matter-antimatter asymmetry as discussed in Section 4.
We evaluate the tunneling process semiclassically. The height of the barrier will be of the order of $m_x$ since it corresponds to the increase of energy when the massless carrier inside the string is put outside it with the same momentum. Additionally, the width of the barrier is the displacement of the carrier such that, always conserving angular momentum and taking into account the one - step contraction of the string, the energy of the configuration is equal to the initial one. This displacement turns out to be of the order of the radius r of the emitting string [@L.Masperi].
Therefore, the emission probability in the string rest frame will be
$$\Gamma _0\simeq m_x^2\text{ }re^{-m_xr}\text{ .} \label{e28}$$
Considering that the probability in laboratory frame requires the relativistic factor for the dilatation of time
$$\Gamma =\Gamma _0\sqrt{1-\text{v}^2}\text{ ,} \label{e29}$$
and that the difference between emission of particle and antiparticle is given by its multiplication times
$$-\frac \mu T=\text{v }\Delta \text{ ,} \label{e30}$$
where $\mu $ is the chemical potential and $\Delta $ will depend on a specific contribution, the asymmetry due to a vorton during all the time of its collapse will be
$$\varepsilon _v=\Delta \text{ }m_x^2\int_\eta ^Rdr\text{ }r\text{ }\sqrt{1-%
\text{v}^2}\text{ }e^{-m_xr}\text{ . } \label{e31}$$
Defining $y=m_x$ $r$ , and being $R\simeq \frac N{m_x}$ and $\eta \simeq
\frac 1{m_x}$, from Eq.(13) one has
$$\varepsilon _v=\Delta \int_1^Ndy\frac{y^2\text{ }e^{-y}}{\frac{k_i}{4\pi ^2}%
N+\left( 1-\frac{k_f}{4\pi ^2}\right) y}\text{ . } \label{e32}$$
Due to the fact that $k_i$ corresponds always to the loop approximation of vorton but $k_f$ may correspond either to loop or to sphere approximations, the integral of Eq.(32) must be splitted into two parts
$$\frac{\varepsilon _v}\Delta =\int_{N_s}^Ndy\text{ }\frac{y^2\text{ }e^{-y}}{%
\widetilde{k}N+\left( 1-\widetilde{k}\right) y}+\int_1^{N_s}dy\text{ }\frac{%
y^2\text{ }e^{-y}}{\widetilde{k}N+\left( 1-\widetilde{k}_f\right) y}=I_1+I_2%
\text{ , } \label{e33}$$
where we have called $\widetilde{k}\simeq 5$ and $\widetilde{k}_f\simeq 1$ according to Eqs. (21) and (27).
These integrals can be done exactly but it is instructive also to calculate them expanding the denominators in powers of $\frac{\left( \widetilde{k}%
-1\right) y}{\widetilde{k}N}<1$ and $\frac{\left( \widetilde{k}_f-1\right) y%
}{\widetilde{k}N}<1$ giving
$$I_1=\frac 1{\widetilde{k}N}\text{ }\sum_{n=0}^\infty \left( \frac{\widetilde{%
k}-1}{\widetilde{k}N}\right) ^n\left. \left\{ \left( n+2\right) !-\left[
y^{n+2}+\left( n+2\right) \text{ }y^{n+1}+...\left( n+2\right) !\right]
e^{-y}\right\} \right| _{N_s}^N \label{e34}$$
and for $I_2$ a similar expression where the coefficient in the numerator is $\widetilde{k}_f-1$ and the limits 1 and $N_s$.
For order $n=0$ there is no influence of the difference between $\widetilde{k%
}$ and $\widetilde{k}_f$ and of the value of $N_s$.
$$\frac{\varepsilon _v^{(0)}}\Delta =\frac 1{\widetilde{k}N}\left[ \frac
5e-\left( N^2+2N+2\right) \text{ }e^{-N}\right] \simeq 0.03\text{ . }
\label{e35}$$
The contribution of $n=1$ adds, taking $N_s=2$,
$$\begin{aligned}
\frac{\varepsilon _v^{(1)}}\Delta &=&\frac 1{\left( \widetilde{k}N\right)
^2}\left[ \left( \widetilde{k}_f-1\right) \frac{16}e+\left( \widetilde{k}-%
\widetilde{k}_f\right) \left( N_s^3+3N_s^2+6N_s+6\right) e^{-N_s}\right.
\label{e36} \\
&&\left. -\left( \widetilde{k}-1\right) \left( N^3+3N^2+6N+6\right)
e^{-N}\right] \nonumber \\
&\simeq &0.005\text{ .} \nonumber\end{aligned}$$
Therefore we may expect $\frac{\varepsilon _v^{}}\Delta $ close to $0.1$ .
In fact the exact evaluation of the asymmetry per vorton gives
$$\begin{aligned}
I_1 &=&\left[ \frac{\left( \gamma y+\widetilde{k}N\right) ^2}{2\gamma ^3}-%
\frac{2\widetilde{k}N}{\gamma ^3}\left( \gamma y+\widetilde{k}N\right) +%
\frac{\left( \widetilde{k}N\right) ^2}{\gamma ^3}\ln \left( \gamma y+%
\widetilde{k}N\right) \right] e^{-y} \label{e37} \\
&&-\frac{e^{-y}}{2\gamma ^3}\left\{ \left[ \gamma \left( y+1\right) +%
\widetilde{k}N\right] ^2-4\left( \widetilde{k}N\right) ^2+\gamma ^2\right\}
-e^{-y}\frac{\left( \widetilde{k}N\right) ^2}{\gamma ^3}\ln \left( \gamma y+%
\widetilde{k}N\right) \nonumber \\
&&+\left. \frac{\left( \widetilde{k}N\right) ^2}{\gamma ^3}e^{\frac{%
\widetilde{k}N}\gamma }\ln \left( \gamma y+\widetilde{k}N\right) +\frac{%
\left( \widetilde{k}N\right) ^2}{\gamma ^3}e^{\frac{\widetilde{k}N}\gamma
}\sum_{n=1}^\infty \frac{\left( -1\right) ^n}{nn!}\left( y+\frac{\widetilde{k%
}N}\gamma \right) ^n\right| _{N_s}^N\text{ , } \nonumber\end{aligned}$$
where $\gamma =1-\widetilde{k}$, and a similar expression for $I_2$ with $%
\gamma =1-\widetilde{k_f}$ and the limits 1 and $N_s$.
The numerical computation with the above values of $\widetilde{k}$ and$%
\widetilde{\text{ }k_f}$ and $N_s=2$ gives $I_1=0.0487$ and $I_2=0.0078$ so that
$$\frac{\varepsilon _v}\Delta =0.0565\text{ .} \label{e38}$$
All what we still need to calculate is the chemical potential to have the numerical value of $\Delta $.
It must be added that we assume that inside the string one has the high-temperature phase in thermal equilibrium so that there matter-antimatter symmetry is kept.
Chemical potential
==================
In the outer part of the string, the non-equilibrium process of contraction will produce a chemical potential which should be otherwise zero for a non-conserved charge as the baryonic one.
In our case the chemical potential may have two sources. One of them is traditional as it appears in the expanding bubbles of electroweak baryogenesis. In the Hamiltonian the baryonic density appears multiplied by $%
-\frac{d\theta }{dt}$ where $\theta $ is a CP violating phase which is nonzero outside the string. During the contraction of the latter, points which are crossed by its external wall pass $\theta $ from $0$ to a finite value $\Delta \theta >0$ so that
$$\mu =-\frac{d\theta }{dt}=\frac{\Delta \theta }\eta \text{ v}<0\text{ ,}
\label{e39}$$
and therefore emission of matter is favoured on antimatter. Since the emission probability must be multiplied by $-\frac \mu {T\text{ }}$ and $%
\eta \sim \frac 1T$ in the high-temperature phase, in our expression of $%
\varepsilon _v$ Eq.(31) $\Delta =\Delta \theta $ which, as said in Section 2 should be $\sim 0.01$ to be in agreement with the bound of the electric dipole moment of neutron. This contribution might be too small to give the expected baryogenesis with our mechanism.
But our collapsing superconducting loop has another source of chemical potential due to the magnetic field that it generates. Outside the external wall of the string, which is where the emission occurs, these will be a potential multiplying the fermionic density with charge q in the Hamiltonian
$$\mu =q\int_0^\varphi \frac \partial {\partial t}A_\varphi \text{ }r\text{ }%
d\varphi ^{\prime }\text{ , } \label{e40}$$
corresponding to the electric field generated by time variation of $%
A_\varphi $ due to the contraction of the loop.
It is interesting to note that this contribution to chemical potential can be also thought as the difference of a phase if one thinks that in the wall of the string the magnetic potential $A_\varphi $ will produce a change of phase of the fermionic field which can be compensated by the transformation
$$\Psi \left( \varphi \right) \longrightarrow e^{iq\int_0^\varphi A_\varphi
\text{ }dl\text{ }}\text{ }\Psi \left( \varphi \right) \text{ . }
\label{e41}$$
But in so doing the kinetic term of the Dirac energy will acquire a contribution of the tipe of $\mu $ Eq.(40) times the fermionic density due to the time variation of $A_\varphi $ .
To evaluate the contribution of Eq.(40) one must calculate $A_\varphi $ of Eq.(15) for $\rho =r+\eta $ and $\theta =\frac \pi 2$ and derivate it at fixed $\rho $ with respect to time due to the variation of r. The most important contribution to $A_\varphi $ is
$$A_\varphi \left( \rho =r+\eta ,\theta =\frac \pi 2\right) \simeq -2\text{ }I%
\text{ }\ln \left( \frac{\rho -r}r\right) \text{ ,} \label{e42}$$
so that
$$\left. \frac{\partial A_\varphi }{\partial t}\right| _\rho \simeq -2\frac
I\eta \text{ v , v}=-\frac{dr}{dt}\text{ .} \label{e43}$$
Because of the definition of Eq.(40), it will correspond to take an average of the potential between $0$ and $2\pi $, i.e.
$$\left\langle \text{ }\mu \right\rangle =-q2\pi \frac{Ir}\eta \text{ v .}
\label{e44}$$
Considering again the factor $-\frac \mu T$ which multiplies the emission probability, we have
$$\Delta =q2\pi Ir\text{ .} \label{e45}$$
This coefficient will vary during the contraction. At the beginning $r\sim
R\simeq \frac N{m_x}$ and being the electric charge of the carrier $q\sim
0.1 $ and $I=\frac{m_x}{2\pi }$ it turns out that $\Delta \simeq 0.1$ , which is larger than the previous contribution to $\mu $ .
Therefore we obtain that $\varepsilon _v\sim 0.01$ and our asymmetry due to gradual collapse of vortons will be
$$\frac{n_B}s\simeq \left( \frac{m_x}{m_{pl}}\right) ^{\frac 32}\frac{%
\varepsilon _v}{g*}\sim 10^{-10}\text{ .} \label{e46}$$
Considering that $\Delta $ may decrease towards the end of the collapse in one order of magnitude and accepting a moderate dilution effect due to gradual transformation of vortons into radiation as discussed in Section 5, $%
\frac{n_B}s$ might not go below $10^{-11}$ which is the lower limit of the acceptable baryogenesis.
Conclusions
===========
We have found that the emission of fermions from vortons destabilized at the electroweak transition during their collapse may supply the matter-antimatter asymmetry required by nucleosynthesis. This avoids on the one hand the necessity of very high temperature of reheating after inflation to produce superheavy bosons of GUT, and on the other the requirement of first order for the electroweak transition needed for the production of bubbles.
It is clear that our evaluation gives only a possible order of magnitude for the baryogenesis with this mechanism. A more precise result would require a calculation of the emission probability beyond the semiclassical approach, as well as a detailed analysis of the disappearance of zero modes in vortons to estimate the time interval for their destabilization.
It is interesting to note that if a particular Grand Unification model loses not all its zero-modes at the electroweak temperature and a part of present dark matter is due to vortons, their emission might explain the observed high-energy cosmic rays, so that this phenomenon would be linked to that of baryogenesis.
**Acknowledgments**
This research was partially supported by CONICET PICT 0358. L.M. wishes to thank the hospitality of the Physics Dept. of Università di Napoli and Università della Calabria at Cosenza, where parts of this work were performed.
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[^1]: Present address: Centro Latinoamericano de Física, Av. Wenceslau Bráz 71 Fundos, 22290 - 140 Río de Janeiro, Brazil.
[^2]: E - mail : [email protected]
[^3]: E - mail : [email protected]
|
---
abstract: 'We present recent results on noise-induced transitions in a nonlinear oscillator with randomly modulated frequency. The presence of stochastic perturbations drastically alters the dynamical behaviour of the oscillator : noise can wash out a global attractor but can also have a constructive role by stabilizing an unstable fixed point. The random oscillator displays a rich phenomenology but remains elementary enough to allow for exact calculations : this system is thus a useful paradigm for the study of noise-induced bifurcations and is an ideal testing ground for various mathematical techniques. We show that the phase is determined by the sign of the Lyapunov exponent (which can be calculated non-perturbatively for white noise), and we derive the full phase diagram of the system. We also investigate the effect of time-correlations of the noise on the phase diagram and show that a smooth random perturbation is less efficient than white noise. We study the critical behaviour near the transition and explain why noise-induced transitions often exhibit intermittency and multiscaling : these effects do not depend on the amplitude of the noise but rather on its power spectrum. By increasing or filtering out the low frequencies of the noise, intermittency and multiscaling can be enhanced or eliminated.'
author:
- 'Sébastien Auma\^ itre$^1$, Kirone Mallick$^2$, François Pétrélis$^3$'
title: 'Noise-induced bifurcations, Multiscaling and On-Off intermittency'
---
Introduction
============
Most patterns observed in nature are created by instabilities that occur in an uncontrolled noisy environment : convection in the atmospheric layers and in the mantle are subject to inhomogeneous and fluctuating heat flux; sand dunes are formed under winds with fluctuating directions and strengths. These fluctuations usually affect the control parameters driving the instabilities, such as the Rayleigh number which is proportional to the imposed temperature gradient in natural convection. These fluctuations act multiplicatively on the unstable modes. In the same spirit, the evolution of global quantities, averaged under small turbulent scales, can be represented by a nonlinear equation with fluctuating global transport coefficients that reflect the complexity at small scales. For instance, it has been shown that the temporal evolution of the total heat flux in rotating convection can be described by a non–linear equation with a multiplicative noise [@Neufeld]. The dynamo instability that describes the growth of the magnetic field of the earth and the stars because of the motion of conducting fluids in their cores, is usually analyzed in similar terms : the magnetic field is expected to grow at large scale, forced by a turbulent flow. Here again, the parameters controlling the growth rate of the field are fluctuating [@Sweet].
Since the theoretical predictions of Stratonovich [@Strato], and the experimental works of Kawaboto, Kabashima and Tsuchiya [@Kawakubo], it is well known that the phase diagram of a system can be drastically modified by the presence of noise [@graham; @lefever]. Because of stochastic fluctuations of the control parameter, the critical value of the threshold may change and noise can delay or favor a phase transition [@vankampen; @gardiner; @risken; @anishchenko]. When a physical system subject to noise undergoes bifurcations into states that have no deterministic counterparts, the stochastic phases generated by randomness have specific characteristics (such as their scaling behavior and the associated critical exponents) allowing to define new universality classes [@munoz1].
A straightforward approach to study the effect of noise on a bifurcation diagram would consist in analyzing the nonlinear Langevin equation that governs the system. However, the interplay between noise and nonlinearity results in subtle effects that make nonlinear stochastic differential equations hard to handle [@vankampen; @wax; @arnold]. One of the simplest systems that can be used as a paradigm for the study of noise-induced phase transitions is the random frequency oscillator (for a recent and detailed monograph devoted to this subject see [@gitterman]). A deterministic oscillator with damping evolves towards the equilibrium state of minimal energy that represents the unique asymptotic state of the system. However, if the frequency of the oscillator is a time-dependent variable, the behavior may change : due to continuous energy injection into the system through the frequency variations, the system may sustain non-zero oscillations even in the long time limit if the amplitude of the noise is large enough.
In this work, we present recent results on noise-induced transitions in a nonlinear oscillator with randomly modulated frequency. The ouline of this article is as follows.
In section \[sec:LyapBif\], we show that an explicit calculation of the Lyapunov exponent allows to determine exactly the phase diagram of the noisy Duffing oscillator [@philkir1; @philkir2]. In the case of a single-well oscillator, a large enough noise destabilizes the origin (which is a fixed point for the deterministic dynamics). The double-well oscillator presents a richer structure : the unstable fixed point becomes a global attractor of the stochastic dynamics when the amplitude of the noise lies in a well-defined interval : this is a typical example of a reentrant transition (which was observed earlier in the more elaborate setting of spatially extended systems [@vdbtorral]). These properties are only qualitatively modified when the noise has temporal correlations.
In section \[sec:scaling\], we discuss the scaling exponents associated with the order parameter in the vicinity of the transition. For a deterministic transition, it is well-known that the different types of bifurcations can be classified by some characteristic exponents. We show, in the stochastic case, that these exponents can be modified by the noise : the dynamical variable has a non-trivial scaling behaviour and its time series exhibits on-off intermittency. Following [@seb1; @seb2], we explain that multiscaling and intermittency are intimately linked with each other and that their existence is due to the presence of low frequency components in the noise spectrum. By using a random process with a spectral structure richer than that of white or Ornstein-Uhlenbeck noise, we show that the deterministic exponents can be recovered by a suitable low-band flitering of the noise. This fact implies that the concept of critical exponents can be ambiguous for noise-induced bifurcations and is certainly not as useful as it is for deterministic phase transitions.
Section \[sec:Poincare\] presents a discussion on perturbative perturbative expansion of the nonlinear Langevin equation. Whereas most studies rely on Fokker-Planck type evolution equations for the Probability Distribution Function (PDF) in which the noise is integrated out by mapping a stochastic ordinary differential equation into a deterministic partial differential equation in the phase space of the system, it is also possible to perform a direct perturbative expansion of the nonlinear Langevin equation by using the classical Poincaré-Lindstedt method [@luecke1; @luecke2]. This technique has the advantage as compared to the more traditional approaches of being applicable to a noise with an arbitrary spectrum (whereas Fokker-Planck equations are only valid for white-noise). However, due to the presence of low-frequencies in the noise spectrum, the perturbative expansion breaks down and diverges. This divergence is at the origin of the multiscaling behaviour and of the anomalous scaling exponents. The last section is devoted to concluding remarks.
Stochastic Bifurcations of a random parametric oscillator {#sec:LyapBif}
=========================================================
In this section, we present the phase diagram of a nonlinear oscillator with a randomly modulated frequency. The case when the frequency is a periodic function of time is a classical problem known as the Mathieu oscillator [@drazin; @nayfeh; @kevorkian]. The phase diagram of the (damped) Mathieu oscillator is obtained by calculating the Floquet exponents (defined as the characteristic growth rates of the amplitude of the system). This phase diagram presents an alternance of stable regions in which the system evolves towards its equilibrium state and of unstable regions in which, because of parametric resonance, the amplitude of the oscillations grows without bound. A nonlinear term is needed to saturate these oscillations. When the frequency of the pendulum is a random process, the role of the Floquet exponents is taken over by the Lyapunov exponents. The system undergoes a bifurcation when the largest Lyapunov exponent, defined as the growth rate of the logarithm of the energy, changes its sign. Thus, the Lyapunov exponent vanishes on the critical surface that separates the phases in the parameter space. This criterion involving the sign of the Lyapunov exponent resolves the ambiguities that were found in studies of the stability of higher moments [@bourret; @bourretFrisch; @lindenberg3] and has a firm mathematical basis [@arnold]. In recent works [@philkir1; @philkir2], we have obtained the exact phase diagram of the random oscillator when the frequency is a Gaussian white noise. In the last part of this section, we discuss the effect of non-vanishing time-correlation in the noise [@kirPeyneau].
Instability induced by noise: the single-well oscillator
--------------------------------------------------------
The equation for the amplitude $x$ of a random Duffing oscillator with a fluctuating frequency is $$\ddot x + \gamma \dot x + (\omega^2 + \xi(t))\, x + \lambda x^3 = 0 \, ,
\label{randomOsc1}$$ where $ \gamma $ is the (positive) friction coefficient, $\omega$ the mean frequency and$\lambda$ the coefficient of the cubic non-linear term. The random fluctuations of the frequency are represented by a Gaussian white noise $\xi(t)$ of zero mean-value and of amplitude ${\mathcal D}$ $$\begin{aligned}
\langle \xi(t) \rangle &=& 0 \, ,\nonumber \\
\langle \xi(t) \xi (t') \rangle &=& {\mathcal D} \, \delta(t - t') \, .
\label{statxi}\end{aligned}$$ In this work, all stochastic differential equations are interpreted according to Stratonovich calculus. By rewriting time and amplitude in dimensionless units, $ t := \omega t$ and $ x:= \lambda^{1/2} \omega^{-1} x,$ Eq. (\[randomOsc1\]) becomes $$\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} +
\alpha \, \frac{\mathrm{d} x}{\mathrm{d} t} + x + x^3 = x \, \Xi(t) ,
\label{dissipDuff}$$ where $\Xi(t)$ is a delta-correlated Gaussian variable with vanishing mean value and with correlations given by $$\begin{aligned}
\langle \Xi(t) \Xi (t') \rangle &=& \Delta \, \delta(t - t') \, .
\label{defXi}\end{aligned}$$ The parameters $$\alpha = \gamma/\omega \,\,\,\,\, \hbox{ and }
\,\,\,\,\, \Delta = {\mathcal D}/\omega^3 \label{parametre}$$ correspond to dimensionless dissipation rate and to noise strength, respectively.
The equation (\[randomOsc1\]) has the origin of the phase space, $ \dot x = x = 0$, as a [*fixed point*]{}. When the system is deterministic, [*i.e.*]{}, when ${\mathcal D} = 0$, the origin is a global attractor for any $\gamma > 0$ : for any initial condition we know that $x(t) \to 0$ and $\dot x(t) \to 0$ when $t \to \infty$.
However, in presence of noise, the asymptotic behaviour of the system becomes more complex. In Fig. \[fig:traj\], we present, for three values of the parameters, a trajectory in the phase plane $(x, v = \dot x)$ characteristic of the oscillator’s behaviour, using the numerical one step collocation method advocated in [@mannella]. Initial conditions are chosen far from the origin, with an amplitude of order $1$. Noisy oscillations are observed for small values of the damping parameter $\alpha$ \[See Fig. \[fig:traj\].a\]. For larger values of $\alpha$ the origin becomes a global attractor for the dynamics \[See Fig. \[fig:traj\].b\]. Increasing the noise amplitude $\Delta$ at constant $\alpha$ makes the origin unstable again \[See Fig. \[fig:traj\].c\].
![Phase plane plots of a typical trajectory of the noisy nonlinear oscillator. Eq. (\[dissipDuff\]) is integrated numerically with a time step $\delta t = 5 \ 10^{-4}$. (a) $\alpha = 0.5$, $\Delta = 20$, $ t \le 100$; (b) $\alpha = 2.0 $, $\Delta = 20$, $t \le 100$; (c) $\alpha = 2.0$, $\Delta = 60$, $ t \le 100$. Early-time relaxation towards noisy oscillations has been omitted for clarity in cases (a) and (c). The trajectory spirals down towards the origin in case (b).[]{data-label="fig:traj"}](fig1a.eps){height="6.0cm"}
![Phase plane plots of a typical trajectory of the noisy nonlinear oscillator. Eq. (\[dissipDuff\]) is integrated numerically with a time step $\delta t = 5 \ 10^{-4}$. (a) $\alpha = 0.5$, $\Delta = 20$, $ t \le 100$; (b) $\alpha = 2.0 $, $\Delta = 20$, $t \le 100$; (c) $\alpha = 2.0$, $\Delta = 60$, $ t \le 100$. Early-time relaxation towards noisy oscillations has been omitted for clarity in cases (a) and (c). The trajectory spirals down towards the origin in case (b).[]{data-label="fig:traj"}](fig1b.eps "fig:"){height="6.0cm"} ![Phase plane plots of a typical trajectory of the noisy nonlinear oscillator. Eq. (\[dissipDuff\]) is integrated numerically with a time step $\delta t = 5 \ 10^{-4}$. (a) $\alpha = 0.5$, $\Delta = 20$, $ t \le 100$; (b) $\alpha = 2.0 $, $\Delta = 20$, $t \le 100$; (c) $\alpha = 2.0$, $\Delta = 60$, $ t \le 100$. Early-time relaxation towards noisy oscillations has been omitted for clarity in cases (a) and (c). The trajectory spirals down towards the origin in case (b).[]{data-label="fig:traj"}](fig1c.eps "fig:"){height="6.0cm"}
In other words, for the stochastic oscillator, the following property is true : For a given $\gamma$ there is a [*critical noise amplitude*]{} $\mathcal{D}_c(\gamma)$ such that $$\begin{aligned}
&\mathcal{D} < \mathcal{D}_c(\gamma)& \,\,\, \,\,\, \dot x, x \to 0
\,\,\, \hbox{ when } t \to \infty \nonumber \\
&\mathcal{D} > \mathcal{D}_c(\gamma)& \,\,\, \,\,\,
\dot x, x \to \hbox{ Oscillating Stationary State} \nonumber\end{aligned}$$ The main problem is to calculate the value of $\mathcal{D}_c$ as a function of $\gamma$ and the other parameters of the problem.
For random dynamical systems, it was recognized early on that various ‘naive’ stability criteria [@bourret; @bourretFrisch], obtained by linearizing the dynamical equation around the origin, lead to ambiguous results. This feature is in contrast with the deterministic case for which the bifurcation threshold is obtained without ambiguity by studying the eigenstates of the linearized equations [@manneville]. [*For the random oscillator, trying to determine the critical noise amplitude by studying the stability of the moments of the amplitude of the linearized equations leads to inconsistent results*]{}. More precisely, consider the random harmonic oscillator, obtained by linearizing equation (\[randomOsc1\]) near the origin : $$\ddot x + \gamma \dot x + (\omega^2 + \xi(t))\, x = 0 \, .
\label{randomOsc2}$$ The second moment $ \langle x^{2} \rangle$ of the amplitude of the linearized equation converges to 0 in the long time limit if $$\frac{ \mathcal{D}}{\omega^3} < 2 \frac{\gamma}{\omega} \, .$$ The fourth moment $ \langle x^{4} \rangle$ converges to 0 if $$\frac{ \mathcal{D}}{\omega^3} <
\frac{2\gamma}{3\omega} \, \frac{3\gamma^2 + 4 \omega^2}
{3\gamma^2 + 2 \omega^2}\, .$$ More generally, the moment $ \langle x^{2p} \rangle$ is stable if $$\frac{ \mathcal{D}}{\omega^3}
< \frac{2}{2p -1} \frac{\gamma}{\omega}
{\mathcal F}_p \left( \frac{\gamma}{\omega} \right) \, .$$ In fact, it was conjectured in [@bourretFrisch] and proved in [@lindenberg3] that, in a linear oscillator with arbitrarily small parametric noise, all moments beyond a certain order grow exponentially in the long time limit. In figure \[fig:moments\], we plot the stability diagram for the moments of order 2,4 and 6. We observe that the stability range becomes narrower when the order of the moment becomes higher. This means that any criterion based on finite mean displacement, momentum, or energy of the linearized dynamical equation is not adequate to show the stability of the initial problem and the bifurcation threshold of a nonlinear random dynamical system cannot be determined simply from the moments of the linearized system. In practice, the phase diagram of the non-linear random oscillator was usually determined in a perturbative manner by using weak noise expansions [@luecke1; @luecke2; @landa; @drolet; @landaMc].
![Stability diagram of the moments of the single-well oscillator. The full curves from top to bottom represent the stability lines of $\langle x^2 \rangle$, $\langle x^4 \rangle$ and $\langle x^6 \rangle$. The dashed curve is the line on which the Lyapunov exponent vanishes.[]{data-label="fig:moments"}](moment.eps){height="7.0cm"}
The reason for this failure is that in the linearized problem the statistics of the variable $x$ is dominated by very large and very rare events that induce the divergence of moments of high order. These rare events are suppressed by the nonlinearity and the bifurcation threshold becomes well defined. We face the following paradoxes : (i) the stability region of the origin $\dot x = x =0$ for the non-linear equation has a well defined threshold for any value of the noise amplitude : below this threshold the origin is a global attractor of the dynamics. On the contrary the linearized equation does not have a clear-cut bifurcation threshold. (ii) Close to the origin the non-linear term is irrelevant but the nonlinearity has to be taken into account to suppress the rare events that spoil the statistical behaviour of the system.
The problem is to find the correct stability criterion for the non-linear equation that can be formulated as in the deterministic case on the linear equation and that would allow explicit calculations.
Such a criterion does exist and is based on the Lyapunov exponent that measures the growth rate of the random harmonic oscillator’s energy $$\Lambda = \lim_{t \to \infty} \frac{1}{2t} \langle \log E \rangle
\,\,\,\, \hbox{ with } \,\,\, E = \frac{\dot x^2}{2} + \omega^2
\frac{x^2}{2} \, ,
\label{eq:defLyap}$$ where $x(t)$ is a solution of equation (\[randomOsc2\]). It has been shown (see [@philkir1] and references therein) that when the Lyapunov exponent is negative the Fokker-Planck equation has a unique stationary solution which is the Dirac delta function at the origin of the phase space. This means that the origin is a stable global absorbing state. However, when the Lyapunov exponent is positive and extended, stationary probability distribution function exists and describes an oscillatory asymptotic state of the nonlinear random oscillator. These features are reminiscent of Anderson localization and there exists indeed a mapping between the random oscillator and a 1d localization model [@tessieri; @jmluck].
The equation of the transition line that determines the stability of the origin for the nonlinear oscillator with a random phase is therefore given by $$\boxed{ \,\,\,\, \,\,\,\,
\Lambda(\gamma, \omega, \mathcal{D}) = 0 \, . \,\,\,\, \,\,\,\, }
\label{eq:stabline}$$ The following exact closed formula for the Lyapunov exponent of the system can be derived when the random frequency modulation is a Gaussian white noise [@hansel; @imkeller; @philkir1] : $$\begin{aligned}
\Lambda &=& \frac{\gamma}{2} \left( \frac{ \int_0^\infty \sqrt{u} \,\,
\rm{e}^{-\psi(u)} \rm{d}u }{ \int_0^\infty \frac{\rm{d}u}{ \sqrt{u}}
\,\, \rm{e}^{-\psi(u)} } -1 \right) \label{eq:Lyap} \, \\
\hbox{ with }
\psi(u) &=& \frac{2\gamma^3}{ \mathcal{D} }
\left\{ \left(\frac{\omega^2}{\gamma^2} - \frac{1}{4} \right) u
+ \frac{u^3}{12} \right\} \, . \end{aligned}$$
![Phase diagram of the single-well oscillator with multiplicative noise. The critical curve separates an absorbing and an oscillatory phase in the $(\alpha,\Delta)$ plane. The dashed line represents the small noise approximation (see text).[]{data-label="fig:diagphase1"}](digrphase.eps){height="6.0cm"}
Inserting this formula in the stability criterion (\[eq:stabline\]) and using dimensionless variables we obtain the equation of the transition line in the ($\alpha$, $\Delta$) plane. For a given value of $\Delta$, the critical value of the damping $\alpha = \alpha_c(\Delta)$ for which the Lyapunov exponent vanishes is given by $$\alpha_c = {\displaystyle { \frac {\int_0^{+\infty}
\mathrm{d}u \; {\sqrt u} \;
\exp\left\{-\frac{2}{\Delta} \left( ( 1 - \frac{\alpha_c^2}{4} ) u
+ \frac{u^3}{12} \right) \right\} }
{\int_0^{+\infty} \frac{\mathrm{d}u}{\sqrt u}
\exp\left\{-\frac{2}{\Delta} \left( ( 1 - \frac{\alpha_c^2}{4} ) u
+ \frac{u^3}{12} \right) \right\} } }. }
\label{alphac}$$ The critical curve $\alpha = \alpha_c(\Delta)$ is represented in Fig. \[fig:diagphase1\]. It separates two regions in parameter space: for $ \alpha < \alpha_c$ (resp. $\alpha > \alpha_c$) the Lyapunov exponent is positive (resp. negative), the stationary PDF is an extended function (resp. a delta distribution) of the energy and the origin is unstable (resp. stable). For small values of the noise amplitude, it can be shown that the exact formula (\[alphac\]) reduces at first order to the linear relation $\alpha_c = \frac{\Delta }{4},$ in agreement with previous perturbative calculations. For large values of the noise, we obtain $ \alpha_c \simeq 0.656 \Delta^{1/3} \, .$ We emphasize that the phase diagram drawn in Fig. \[fig:diagphase1\] is exact for all values of the noise amplitude $\Delta$ and the damping parameter $\alpha$.
Stabilisation by noise: the double-well oscillator
--------------------------------------------------
In a classical calculation, Kapitza (1951) showed that the unstable upright position of an inverted pendulum is stabilised if its suspension axis undergoes sinusoidal vibrations of high enough frequency. Analytical derivations of the stability limit are based on perturbative approaches, [*i.e.*]{}, in the limit of small forcing amplitudes [@landau; @nayfeh; @barma].
When the sinusoidal vibrations of the suspension axis are replaced by a white noise, an [*exact*]{}, non-perturbative, stability analysis can be performed [@philkir2]. In particular, for the stochastic inverted pendulum, we have shown that the unstable fixed point can be stabilized by noise and have discovered the existence of a noise-induced reentrant transition.
The general equation for the Duffing oscillator subject to multiplicative noise is given by : $$\ddot x + \gamma \dot x -
(\mu + \xi(t))\, x + x^3 = 0 \, .
\label{eq:doublewell}$$ When $\mu < 0$, the non-linear potential has a single-well. This case is the same as the one discussed in the previous section : the origin is deterministically a global attractor that can become unstable in presence of noise. When $\mu > 0$, the oscillator is subject to a double-well potential and the origin is deterministically unstable.
![The double-well oscillator with parametric noise[]{data-label="fig:2well"}](dwell.eps){height="5.0cm"}
In presence of noise, the correct criterion for stability analysis is again based on the sign of the Lyapunov exponent. In order to take into account both possible signs of $\mu$, it is useful to define the following set of dimensionless parameters : $$\alpha = \frac{\mu}{\gamma^2}
\,\, \,\, \hbox{ and } \Delta =
\frac{ {\mathcal D} }{ \gamma^3 } \, .
\label{defalpha}$$
An exact calculation of the Lyapunov exponent [@philkir2] then allows to draw the phase diagram represented in figure (\[fig:diagphase2\]).
![Phase diagram of the double-well oscillator subject to multiplicative Gaussian white noise. The solid line is the locus where $\Lambda(\alpha,\Delta) = 0$. The bifurcation line $\alpha = 0$ of the noiseless dynamical system is drawn for comparison (dotted line). For $\Delta \le \Delta^* \simeq 3.55$ (resp. $\Delta \ge \Delta^*$), the origin is stabilised (resp. destabilised) by the stochastic forcing in the range $0 \le \alpha \le \alpha_c(\Delta)$ (resp. $\alpha_c(\Delta) \le \alpha \le 0$).[]{data-label="fig:diagphase2"}](diagr2.eps){height="6.5cm"}
We observe that when $\alpha < 0.21\ldots$, there exits a range of noise amplitudes for which the origin is [*stabilized by noise*]{} : for $\Delta$ such that $\Delta_1(\alpha) < \Delta < \Delta_2(\alpha)$ the origin becomes an attractive fixed point of the stochastic dynamics. The values $\Delta_1(\alpha)$ and $\Delta_2(\alpha)$ that determine the stability interval are known analyticaly. We emphasize that these functions can not be calculated perturbatively by using a small noise expansion when $\alpha$ is finite. For $\alpha > 0.21\ldots$, the stability interval does not exist anymore : the origin is always unstable and the non-equilibrium stationary state exhibits an oscillatory behaviour.
![Stability range for the double-well oscillator : when $\alpha < 0.21$, the unstable equilibrium point is stabilized by noise for noise amplitudes between $\Delta_1$ and $\Delta_2$.[]{data-label="fig:Stab"}](figStab.eps){height="2.2cm"}
The effect of colored noise
---------------------------
We now discuss the phase diagram of an oscillator whose frequency is a random process with finite time memory. More precisely, we consider the case of an Ornstein-Uhlenbeck noise $x(t)$ of correlation time $\tau$ : $$\begin{aligned}
\langle \xi(t) \xi(t') \rangle &=& \frac{{\mathcal D}_1}{2 \tau}
\exp\left( - |t - t'|/\tau \right) \, \, .
\label{statxiOU}\end{aligned}$$ When $\tau \to 0$, the process $\xi(t)$ becomes identical to the white noise. The Lyapunov exponent of the system $\Lambda(\omega, \gamma, {\mathcal D}, \tau)$ becomes now a function of $\tau$ also.
From a physical point of view, the influence of a finite correlation time on the shape of the critical curve is an interesting open question: due to the finite correlation time of the noise, the random oscillator is a non-Markovian random process and there exists no closed Fokker-Planck equation that describes the dynamics of the Probability Distribution Function (P.D.F.) in the phase space. This non-Markovian feature hinders an exact solution in contrast with the white noise case where a closed formula for the Lyapunov exponent was found. For the single-well stochastic oscillator, we have calculated [@kirPeyneau] this Lyapunov exponent by using different approximations and have derived the phase diagram. The main role of the correlation time of the noise, as can be seen in Figure \[figpeyn\], is to enhance the stability region. When $\tau$ grows, amplitude of the noise required to destabilize the origin becomes bigger. This effect can be seen quantitavely in the following exact asymptotic expansions (see also [@crauel]) : $$\begin{aligned}
\hbox{ For small values of the noise amplitude~:} \,\,\,
\alpha_c &\simeq& \frac{\Delta }{4 (1 + 4 \omega^2\tau^2)} \, , \\
\hbox{ For large values of the noise amplitude~:} \,\,\,
\alpha_c &\simeq& C \Big( \frac{\Delta}{\omega\tau} \Big)^{1/4} \, , \end{aligned}$$ where the constant $C$ is of order 1. Comparing with the white noise result we observe that the behaviour of the critical curve for large noise is modified in presence of time correlations; the asymprtotic exponent is 1/3 for white noise and 1/4 for colored noise even if the correlation time $\tau$ is small. The presence of a non-vanishing correlation time thus modifies the scaling characteristics of the system. This change of scaling at large noise is related to the differentiability properties of the random potential as shown in [@delyon] : the white noise is continuous but nowhere differentiable whereas the Ornstein-Uhlenbeck process has a first derivative but no second derivative.
For the double-well oscillator subject to Ornstein-Uhlenbeck noise, the results of numerical computations of the phase diagram are shown in figure \[fig:diagphase3\]. Here also we observe that the noise-induced[*stabilization*]{} of the deterministically unstable fixed point becomes less efficient in presence of time correlations. In the weak noise limit, this curve agrees with the prediction of [@luecke1]:$\alpha_c(\Delta) \sim \Delta/(2 (1 + \tau))$. For a noise amplitude of order 1, the bifurcation line is qualitatively similar to that obtained with white noise. However, depending on the value of $\Delta$, the value of the bifurcation point $\alpha_c(\Delta)$ is not necessarily a monotonic function of $\tau$. A precise understanding of this non-monotonous behaviour is lacking and the analytical theory of the stability of the double-well oscillator with time-correlated noise still remains to be done. Although the obtention of exact results for the Ornstein-Uhlenbeck noise seems unlikely, various recent works [@vandenbroeck] indicate that the Poisson process provides a useful model for the study of time correlation effects in stochastic systems. The advantage of the Poisson process is that it is amenable to exact analytic calculations.
![Phase diagram of the double-well oscillator subject to Ornstein-Uhlenbeck noise with correlation time $\tau$. The bifurcation lines for $\tau = 0.1$ and $1.0$ are obtained from numerical calculations of the Lyapunov exponent. For comparison, we draw the (analytic) white noise line. In this figure the dimensionless parameters are given by $ \alpha = \frac{\omega^2}{\gamma^2} $ and $ \Delta = \frac{ {\mathcal D} }{ \gamma^3 }\,.$ []{data-label="fig:diagphase3"}](diagr3.eps){height="6.5cm"}
We conclude this section by emphasizing that the results obtained here for the oscillator with multiplicative noise could be used for other stochastic systems. For example, it has been shown by Schimansky-Geier et al. [@schimansky] that the random Duffing oscillator with additive noise undergoes a phase transition that does not manifest itself in the stationary P.D.F. (which is simply given by the Gibbs-Boltzmann formula). This subtle phase transition, which affects the properties of the random attractor of motion in phase space, can be formulated mathematically as a bifurcation in an associated linear oscillator subject to a multiplicative noise with a a finite correlation time. If we approximate this noise by an effective Ornstein-Uhlenbeck process, the system becomes identical to the one studied here.
Scaling behaviour near the bifurcation threshold {#sec:scaling}
================================================
For deterministic Hopf bifurcation, the amplitude of the order parameter $x$ exhibits a normal scaling behaviour in the vicinity of the transition line; if $\epsilon$ denotes the distance from threshold of the control parameter, we have in the deterministic case : $$\begin{aligned}
\langle x^{2n} \rangle \sim \epsilon^n \, .
\label{Hopfnormal}\end{aligned}$$ We now consider the case of the stochastic oscillator. In figure \[fig:mult\], we plot the behaviour of the amplitude moments in the vicinity of the reentrant transition in a double-well oscillator. The damping rate $\alpha$ is fixed at a given value and the noise strength $\Delta$ is chosen such that $0 < \Delta -\Delta_c(\alpha) \ll 1$, where $\Delta_c(\alpha)$ is the critical value. In this case, the parameters of the system are tuned just above the bifurcation threshold and the Lyapunov exponent is slightly positive: $0 < \Lambda \ll 1$. We observe that even-order moments of the amplitude scale linearly with the distance to threshold in the vicinity of the bifurcation line (note that odd-order moments are equal to zero by symmetry) : $$\begin{aligned}
\boxed{ \,\,\,\, \,\,\,\, \langle x^{2n} \rangle \sim \epsilon \, .
\,\,\,\, \,\,\,\, }
\label{Hopfstoch}\end{aligned}$$ A similar scaling behaviour is obtained by simulating the single-well oscillator or by replacing white noise by an Ornstein-Uhlenbeck process. Such a behaviour was also observed in random maps [@pikovsky] and in stochastic fields [@munoz2]. Such a strong multiscaling seems to be a typical feature of stochastic bifurcations and has been noticed a long ago [@graham] in first order stochastic differential equations. The fact that the stochastic oscillator exhibits a similar behaviour can be understood ‘physically’ by noticing that the second order time derivative becomes irrelevant in the long-time limit. The mathematical explanation for this behaviour is also elementary [@philkir1] : near the bifurcation threshold, when the Lyapunov exponent $\Lambda \to 0$, the stationary Probability Distribution Function $P_{{\rm stat}}$ exhibits a power law divergence at the origin $(\dot x = x = 0)$, and the region near the origin dominates the statistics : it can be shown that near $E=0$, the stationary energy distribution scales as $ P_{{\rm stat}}(E) \sim E^{c\Lambda -1} $ where $c$ is a positive constant. From this formula, one readily deduces that all moments of the energy (and therefore all moments of the amplitude of the oscillator) are of the order $\Lambda$ which itself grows linearly with $\epsilon$, the distance from threshold.
Another remarkable feature of the order parameter near threshold is that the time series $ t \mapsto x(t)$ exhibits on-off intermittency. Again this intermittent behaviour, first discovered in coupled dynamical systems [@yamada] and in a system of reaction-diffusion equations [@pikovskyonoff], is believed to be generic (see e.g., [@spiegel]) when an unstable system is coupled to a system that evolves in an unpredictable manner (multiplicative noise).
Multiscaling and intermittency lead however to the following very puzzling problems :
The multiscaling behaviour given in equation (\[Hopfstoch\]) is very clearly seen in computer experiments but does not seem to be observed in ‘real’ experiments. Similarly, It is surprising that, despite the genericity of the on-off intermittency mechanism (which is well established mathematically) this effect has scarcely been reported in experiments. One might expect that any careful experimental investigation of an instability should reveal on-off intermittency when the system is close to the onset of instability, and is hence sensitive to unavoidable experimental noise in the control parameters.
In well known works [@luecke1; @luecke2], it was predicted, using perturbation theory, that the scaling of the stochastic bifurcation should be [*the same*]{} as that of the deterministic Hopf bifurcation, equation (\[Hopfnormal\]). This problem is analyzed in section \[sec:Poincare\].
![Behaviour of the moments near the reentrant bifurcation threshold. The noisy double-well oscillator subject to parametric white noise is simulated with a fixed parameter $\alpha = 0.2$. The critical value of the noise amplitude corresponding to the reentrant transition is found to be $\Delta_c \simeq 0.627$ : for a noise amplitude $\Delta$ belonging to the interval (0.627–1.70) the origin is stabilized by the noise. Even-order moments $\langle x^2 \rangle$, $\langle x^4 \rangle$ and $\langle x^6 \rangle$ are plotted versus the distance to threshold $\epsilon = (\Delta_c - \Delta)/\Delta_c$ (symbols). Dashed lines respecting a linear behaviour $\langle x^{2n} \rangle \propto \epsilon$ are drawn to guide the eye. Inset: the mean square position (measured in the stationary regime) is non-zero for $\Delta < 0.627$ and $\Delta > 1.70$, i.e., the stationary state is extended.[]{data-label="fig:mult"}](fig.souscritique.eps){height="6.5cm"}
Problem 1 is solved in [@seb1; @seb2]. In these works, it is shown that although the noise strength controls the transition between the absorbing and the oscillatory states, the amplitude of the noise [*is not the only relevant*]{} parameter responsible for multiscaling and intermittency. Rather, these effects are due to the zero-frequency mode of the noise; in other words these effects depend on the Power Spectrum Density (PSD) of the noise and not on its overall amplitude. In order to identify which part of the PSD of the random forcing really affects the dynamics, one needs a random perturbation with a spectral density more complex than that of white noise or of Ornstein-Uhlenbeck noise. A useful type of noise [@schimanskyharmonic] is the [*harmonic*]{} noise $\zeta(t)$ whose autocorrelation function is given by $$\begin{aligned}
\langle \zeta(t)\,\zeta(t+\tau) \rangle_s =
A \exp(-2\pi \eta |\tau|) \left(\cos(2\pi \Omega \tau) +
\frac{\eta}{\Omega} \sin(2\pi \Omega |\tau|)\right) \, , \,\,\,
\,\,\, \label{selfcorrbr}\end{aligned}$$ where $A$ is the noise amplitude; the corresponding PSD is $$C(\nu)=\frac{A \eta (\Omega^2+\eta^2)}
{\pi[(\Omega^2+\eta^2-\nu^2)^2+2\nu^2\eta^2]}\,.
\label{spectrecol}$$ The value of the PDS at zero frequency is therefore given by $$S = A \eta/\left[\pi(\eta^2+\Omega^2)\right]\, .$$ For harmonic noise, the amplitude $A$ and the zero frequency amplitude $S$ can be tuned [*independently*]{}.
In figure \[fig:Intermittence\], we plot the time series of a Duffing oscillator with a random frequency perturbation given by a [*harmonic*]{} noise $\zeta(t)$. We observe that the zero frequency amplitude $S$ is the pertinent parameter controlling the intermittent regime : on-off intermittency disappears when the value of $S$ is lowered. For the simple first-order model, $$\dot{x}=(a+\zeta(t)) x-x^3\,,
\label{eqbase}$$ it is possible to determine analytically the transition line between intermittent behaviour and non-intermittent behaviour in the $(S,a)$ plane, using the cumulant expansion introduced by Van Kampen [@vankampen]. Intermittency occurs when $$0<\frac{a}{S}<1 \,.
\label{eqcriterium}$$ This criterion is checked numerically in figure \[fig:Dgphaseb\].
Finally, it is possible to prove that multiscaling is a consequence of the intermittent behaviour of the order parameter $x(t)$ [@seb2]. Therefore as the zero frequency noise amplitude is reduced, both on-off intermittency and multiscaling are suppressed and normal scaling (\[Hopfnormal\]) is recovered.
This analysis solves Problem 1 by explaining why many experimental investigations on the effect of a multiplicative noise on an instability do not display on-off intermittency. If the noise is high-pass filtered, as often required for experimental reasons, then the regime of intermittent behavior disappears. This is the case for instance in [@Francois2]: a ferrofluidic layer undergoes the Rosensweig instability and peaks appear at the surface. The layer is then subject to a multiplicative noise through random vertical shaking. Close to the deterministic onset, the unstable mode submitted to a colored noise does not display intermittency. Another experimental obstacle to observe intermittency is the presence of additive noise that destroys the symmetry of the system.
![Temporal traces of the amplitude of a Duffing oscillator subject to harmonic noise of constant amplitude $A=0.05$. As the value $S$ of the zero frequency amplitude of the noise is decreased from top to bottom, intermittent behaviour is suppressed.[]{data-label="fig:Intermittence"}](Xal005def.eps){height="7.0cm"}
![Intermittency phase diagram for the solution of (\[eqbase\]) with harmonic noise. ($\triangle$) : intermittent behavior, ($\circ$) : non-intermittent behavior. The full line is the transition curve predicted by (\[eqcriterium\]).[]{data-label="fig:Dgphaseb"}](Dgphaseb.eps){height="7.0cm"}
Remarks on the perturbative analysis of a noisy Hopf bifurcation {#sec:Poincare}
================================================================
Problem 2 questions the relevance of perturbative expansions for studying noise induced bifurcations. Consider, for example, the parametrically driven damped anharmonic oscillator that naturally appears in the study of many instabilities [@fauve]. Such a system is described by the following equation : $$m \ddot{x} + m \gamma \dot{x} =
\left(\epsilon + \Delta \xi(t) \right) x - x^3 \, ,
\label{eq:Lucke1}$$ where $\epsilon$ is the control parameter and the modulation $\xi(t)$ is of arbitrary dynamics and statistics: it can be a periodic function or a random noise. For small driving amplitudes $\Delta$, Lücke and Schank [@luecke1] have performed a Poincaré-Lindstedt expansion called the Poincaré-Lindstedt expansion [@drazin; @kevorkian; @kleinert] (see [@khrustalev] for applications to field-theory). They have obtained an expression for the threshold $\epsilon_c(\Delta)$ (at first order in $\Delta$). Their result has been verified both numerically and experimentally and is also in agreement with the exact result obtained for the Gaussian white noise (in this case a closed formula is available for $\epsilon_c(\Delta)$ for arbitrary values of $\Delta$). Another result obtained in [@luecke1; @luecke2] is the scaling of the moments near the threshold, $$\langle x^{2n} \rangle = s_n
\left( \epsilon - \epsilon_c(\Delta) \right)^{n}
+ {\mathcal O}\left( ( \epsilon - \epsilon_c)^{n + 1} \right) \, ,
\label{eq:Luckemoment}$$ where the constant $s_n$ depends on $\xi(t)$ and on $\Delta$. The moments have a [*normal scaling*]{} behaviour : $ \langle x^{2n} \rangle$ scales as $\langle x^2 \rangle^n$. The bifurcation scaling exponent is equal to 1/2 and is the same as that of a deterministic Hopf bifurcation. However, this expression does not agree with the results for random iterated maps, for the random parametric oscillator and with recent studies on On-Off intermittency [@pikovsky; @philkir1; @seb1]. These works predict that the variable $x$ is intermittent and that the moments of $x$ exhibit [*anomalous scaling*]{}, $$\langle x^{2n} \rangle \simeq \kappa_n (\epsilon - \epsilon_c)
\,\,\, \hbox{ for all } \,\,\, n >0 \, ,
\label{eq:anomalous}$$ [*i.e.*]{}, all the moments grow linearly with the distance from threshold. This multiscaling behaviour, confirmed by numerical simulations for a Gaussian white noise, was derived using effective Fokker-Planck equations. The origin of the contradiction between equations (\[eq:Luckemoment\]) and (\[eq:anomalous\]) lies in the divergences that appear in the Poincaré-Linsdtedt expansion. This fact, identified in [@luecke2], implies that the results of [@luecke1] are valid only for noises that do not have low frequencies.
A simple model
--------------
The importance of low frequencies in the noise spectrum can be seen analytically in a a model technically simpler than equation (\[eq:Lucke1\]). To simplify our discussion we work on the first order stochastic equation : $$\dot{x} =
\left( \epsilon + \Delta \xi(t) \right) x - x^{3} \, ,
\label{eq:Hopf}$$ that we already encountered in the previous section. Here, the noise $\xi(t)$ is a Gaussian stationary random process with zero mean value and with a correlation function given by $$\langle \xi(t) \xi(t') \rangle = {\mathcal D}(|t - t'|) \, .
\label{eq:corr}$$ The power spectrum of the noise, which is non-negative thanks to the Wiener-Khinchin theorem [@vankampen], is the Fourier transform of the correlation function $$\hat{\mathcal D}(\omega) = \int_{-\infty}^{+\infty} {\rm d}t \exp(i\omega t)
\langle \xi(t) \xi(0) \rangle =
\int_{-\infty}^{+\infty} {\rm d}t \exp(i\omega t)
{\mathcal D}(|t|) \, .
\label{eq:PSD}$$ Applying elementary dimensional analysis to equation (\[eq:Hopf\]), we obtain the following scaling relations: $$x \sim t^{{1}/{(2p)}} \, , \,\,\,\,\,\,\,\,
\xi \sim t^{-1/2} \, , \,\,\,\,\,\,\,\,
\epsilon \sim \Delta^2 \sim t^{-1 } \, .
\label{eq:dimensions}$$ The dimension of the noise $\xi$ is so chosen as to render the power spectrum $\hat{\mathcal D}(\omega)$ a dimensionless function.
We first remark that the presence of noise does not modify the bifurcation threshold; indeed the Lyapunov exponent of the system is given by $\Lambda = \epsilon $ and therefore the bifurcation always occurs at $\epsilon_c(\Delta) = 0$.
When $\xi(t)$ is a Gaussian white noise, the stationary solution of the Fokker-Planck equation corresponding to equation (\[eq:Hopf\]) is given by $$P_{{\rm stat}}(x) = \frac{ 2}
{\Gamma( \alpha) (\Delta^2)^{\alpha} }
x^{2\alpha -1}
\exp\left(-\frac{x^{2}}{\Delta^2} \right) \,\,\, \,\,\,
{\rm with } \,\,\, \alpha = \frac{\epsilon}{\Delta^2} \, ,
\label{eq:PDFblanc}$$ where $\Gamma$ represents the Euler Gamma-function. The bifurcation threshold is given by $\epsilon = 0$; for $\epsilon < 0 $, the solution (\[eq:PDFblanc\]) is not normalizable : the stationary PDF is the Dirac distribution $\delta(x)$ localized at the absorbing fixed point $x =0$. For $\epsilon > 0 $, the solution (\[eq:PDFblanc\]) is normalizable and the moments of $x$ are given by $ \langle x^{2n} \rangle = \frac{ \Gamma( \alpha + n) }
{\Gamma( \alpha) } (\Delta^2)^{n} \, . $ In the vicinity of the threshold, $\epsilon$ is small and we recover multiscaling : the moments scale linearly with $\epsilon$, [*i.e.*]{}, $$\langle x^{2n} \rangle \simeq \epsilon \, (\Delta^2)^{n -1}
\Gamma(n) \, .
\label{eq:scalmomentblanc}$$
Exact solution for arbitrary noise
----------------------------------
For an arbitrary noise $\xi(t)$, it is still possible to obtain an exact solution of equation (\[eq:Hopf\]) for all times. First, we shall derive some remarkable identities satisfied by the exact solution.
We define the differential operator ${\mathcal L}$ as follows, $${\mathcal L} = \frac{ {\rm d}}{ {\rm d}t} - \Delta \xi(t) \, .
\label{eq:defL}$$ Let us call $y_1(t)$ the solution of the adjoint equation ${\mathcal L}^ \dagger y_1 = 0 $, which is given by $$y_1(t) = \exp\left( - \Delta \int_0^t \xi(u){\rm d}u\right) \, .
\label{eq:defy1}$$ Multiplying equation (\[eq:Hopf\]) by the function $y_1$ and taking average values, we obtain $$\langle y_1 {\mathcal L} x \rangle =
\langle \epsilon y_1 x - y_1 x^3 \rangle \, .$$ Integrating the left hand side of this equation by parts and taking into account the fact that $y_1$ is in the kernel of the adjoint operator ${\mathcal L}^ \dagger$, we derive the following relation $$\epsilon = \frac{ \langle y_1 x^3 \rangle }
{ \langle y_1 x \rangle } +
\frac{ {\rm d}}{ {\rm d}t} \ln \langle y_1 x \rangle \, .
\label{eq:solvability}$$ Similarly, we have $$\epsilon = \langle x^2 \rangle +
\frac{ {\rm d}}{ {\rm d}t} \langle \ln (y_1 x) \rangle \, .
\label{eq:solvability2}$$
Dividing both sides of equation (\[eq:Hopf\]) by $x^3$ and using the auxiliary variable $U = 1/x^2$, we observe that equation (\[eq:Hopf\]) becomes a [*linear*]{} first order stochastic in $U$ : $$\frac{1}{2} \dot U = 1 - ( \epsilon + \Delta \xi) U \, .
\label{eq:varU}$$ This equation can be solved exactly for all time by using the method of variation of constants. Introducing the initial value $x(0) = \lambda$ that has the dimensions $$\lambda \sim t^{{1}/{2}} \, ,
\label{eq:dimlambda}$$ we obtain the following explicit formula for $x$ : $$x(t) = \frac{ \lambda \exp\left( \epsilon t + \Delta B_t \right)}
{ \sqrt{ 1 + 2 \lambda^2 \int_0^t
\exp\left(2 \epsilon u + 2\Delta B_u \right) \rm{d}u \, } \,\,\,\, } \, .
\label{eq:formulex(t)}$$ We have defined here the auxiliary random variable $B_t$ : $$B_t = \int_0^t \xi(u){\rm d}u \, .
\label{eq:defB}$$ Because $\xi$ is taken to be a Gaussian random process, $B_t$ is also Gaussian.
Importance of low frequencies in the noise spectrum
---------------------------------------------------
In order to determine the scaling on the moments of $x$, we must evaluate expressions of the type $\langle \exp \Delta B_t \rangle$. Because $B_t$ is a Gaussian random variable, this quantity is given in terms of the variance of $B_t$ : $$\begin{aligned}
\langle B_t^2 \rangle =
\int_0^t\int_0^t \langle \xi(u) \xi(v) \rangle{\rm d}u {\rm d}v
= \int_0^t\int_0^t {\mathcal D}(|u-v|) \rangle{\rm d}u {\rm d}v
= \int_{-\infty}^{+\infty}
\frac{ 1 - \cos\omega t}{\omega^2} \,\,
\frac{ \hat{\mathcal D}(\omega) {\rm d}\omega}{\pi}\, .
\label{eq:varB}\end{aligned}$$ The last integral is well defined at $\omega = 0$ (the time $t$ introduces an effective low frequency cut-off for $\omega \sim 1/t$). The behaviour of $\langle B_t^2 \rangle$ for $t \to \infty$ depends on the behaviour of $\hat{\mathcal D}(\omega)$ at $\omega \to 0$. The following two cases must be distinguished :
\(i) The spectrum of the noise vanishes at low frequencies, [*i.e.*]{}, ${\mathcal D}(0) = 0$. Because $\hat{\mathcal D}(\omega)$ is an even function of $\omega$, we suppose that $\hat{\mathcal D}(\omega) \sim \omega^2 $ for $\omega \sim 0$ (we disgard non-analytic behaviour of the power spectrum at the origin. Such non-analyticity would correspond to long tails in the correlation function of the noise).
\(ii) The power spectrum of the noise is finite at $\omega = 0$, [*i.e.*]{}, ${\mathcal D}(0) > 0$.
In case (i), the long time limit of equation (\[eq:varB\]) is readily derived and we obtain (by using the Riemann-Lebesgue lemma) $$\langle B_t^2 \rangle \rightarrow \int_{-\infty}^{+\infty}
\frac{ \hat{\mathcal D}(\omega) {\rm d}\omega}{ \pi \, \omega^2} \,\,\,\,\,
\hbox{ when } \,\,\,\,\, t \to \infty \,.$$ The variance of $B_t$ has a [*finite*]{} limit at large times.
In case (ii), the integral on the right hand side of equation (\[eq:varB\]) diverges when $ t \to \infty$ and its leading behaviour is $$\langle B_t^2 \rangle = \frac{ t }{\pi } \int_{-\infty}^{+\infty}
\frac{ 1 - \cos u }{ u^2} \,\,
\hat{\mathcal D}\left(\frac{u}{t}\right) {\rm d}u
\rightarrow \hat{\mathcal D}(0) t \, .$$ The variance of $B_t$ grows linearly with time in the long time limit.
Behaviour of the moments
------------------------
We have seen that the behaviour of the variance of $B_t$ in the long time limit crucially depends on the on the behaviour of the noise spectrum at low frequencies. When ${\mathcal D}(0) > 0$ the noise $\xi(t)$ acts dominantly as a white noise and the multiscaling behaviour (\[eq:scalmomentblanc\]) is recovered when $t \to \infty$.
When ${\mathcal D}(0) = 0$, the random variable $B_t$ has a finite variance even when $t \to \infty$. Using equation (\[eq:formulex(t)\]) and keeping only the dominant terms in the long time limit, we obtain $$x(t) \simeq
\frac{ \exp\left( \epsilon t + \Delta B_t \right)}
{ \sqrt{ 2 \int_0^t
\exp\left(2 \epsilon u + 2\Delta B_u \right) \rm{d}u \, } \,\,\,\, } \, .
\label{eq:formule2x(t)}$$ Therefore, we have $$x^{2n}(t) \simeq
\Big( \,\,\,
2 \int_0^t \exp\left(-2 \epsilon (t-u) + 2\Delta (B_u- B_t) \right)
\rm{d}u \,\,\, \Big)^{-n}
\label{eq:xpower2n}$$ The fluctuations of the random variable being of order one, the integral on the r.h.s. of this equation is dominated by the contribution of the linear term $ -2\epsilon (t-u)$. The contribution is maximal in the region $u \simeq t$. Therefore, we have $$\langle x^{2n} \rangle \simeq C_n \epsilon^n \, ,
\label{eq:scalmom}$$ where the constant $C_n$ depends a priori on the statistical properties of the noise. This equation predicts that subject to a noise without low frequencies, the amplitude $x(t)$ exhibits a normal scaling behaviour identical to that of the deterministic case.
Conclusion
==========
The stochastic oscillator is an ideal model to study the effect of random perturbations on a nonlinear dynamical system. Various effects of noise can be demonstrated on this simple model : noise can shift bifurcation thresholds, can create new phases by destabilizing (resp. stabilizing) stable (resp. unstable) fixed points, can induce reentrant behaviour. The relevant parameter that determines the long time behaviour of the system is the Lyapunov exponent of the underlying linearized dynamics. Exact results can be derived for white noise and for dichotomous Poisson noise. When the system is coupled to the smooth Ornstein-Uhlenbeck process, analytical treatments have to rely on various approximations, however qualitative aspects on time-correlations are well understood.
The effect of noise on the scaling behaviour is subtle : intermittency and multiscaling can appear if the relative weight of the zero-frequency mode of noise is large enough. If low frequencies are filtered out, the noise-induced bifurcation becomes qualitatively similar to the deterministic transition. These features are difficult to extract from a perturbative analysis of the Langevin equation : they appear as divergences in the perturbative expansion which must be resummed in order to get correct results. Indeed, multiscaling of the order parameter can not be revealed from a finite order truncation of the expansion.
Despite the formal simplicity of the problem, exact results have been derived only recently and many questions still remain to be addressed. For example, little is known about the relevance of the noise spectrum on barrier crossing problems and on stochastic resonance. Analytical results for higher dimensional systems and for stochastic fields are rare : such results could be relevant for the study of the trapping of quasi one-dimensional Bose-Einstein condensates in random potentials [@aspect].
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abstract: 'We report a THz reflectarray metasurface which uses graphene as active element to achieve beam steering, shaping and broadband phase modulation. This is based on the creation of a voltage controlled reconfigurable phase hologram, which can impart different reflection angles and phases to an incident beam, thus replacing bulky and fragile rotating mirrors used for terahertz imaging. This can also find applications in other regions of the electromagnetic spectrum, paving the way to versatile optical devices including light radars, adaptive optics, electro-optical modulators and screens.'
author:
- 'M. Tamagnone $^{1,2,*,\dagger}$, S. Capdevila$^{1,\dagger}$, A. Lombardo$^{3}$, J. Wu$^{3}$, A. Centeno$^{4}$, A. Zurutuza$^{4}$, A. M. Ionescu$^{5}$, A. C. Ferrari$^{3}$, J. R. Mosig$^{1}$'
title: Graphene Reflectarray Metasurface for Terahertz Beam Steering and Phase Modulation
---
{width="180mm"}
{width="180mm"}
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Metasurfaces are planar devices based on a periodic or quasi-periodic bi-dimensional array of cells (typically dielectric or metallic elements placed on a layered substrate) capable of manipulating impinging light to obtain various functionalities, such as focusing[@khorasaninejad2016metalenses; @headland2016dielectric], beam steering and shaping[@HumPerruisseau-Carrier2014; @DebogovicBartolicPerruisseau-Carrier2014; @CarrascoTamagnoneMosigEtAl2015], unilateral propagation[@TamagnoneFallahiMosigEtAl2014], polarization control[@mueller2017metasurface; @NiuWithayachumnankulUpadhyayEtAl2014], frequency filtering[@HasaniTamagnoneCascanteEtAl2016; @mittra1988techniques], non-linear phenomena[@lee2014giant], and dynamic modulation[@sun2016optical; @TamagnoneFallahiMosigEtAl2014; @LiYu2013; @Sensale-RodriguezYanRafiqueEtAl2012; @kim2016vanadium; @sherrott2017experimental; @miao2015widely; @YaoShankarKatsEtAl2014]. When optically tuneable materials are embedded in the cells, metasurfaces can be designed to dynamically steer a beam in different directions. This can be achieved at microwave frequencies in reflectarray (RA) metasurfaces using, e.g., micro-electrical-mechanical-systems (MEMS)[@HumPerruisseau-Carrier2014; @DebogovicBartolicPerruisseau-Carrier2014], or voltage controlled capacitors[@HumPerruisseau-Carrier2014; @RodrigoJofrePerruisseau-Carrier2013]. Extending beam steering to THz, infrared and visible frequencies is challenging due to the scarcity of compact tuneable elements operating at shorter wavelengths.
This technological issue can be solved by using single layer graphene (SLG), which is an ideal material for photonics and optoelectronics due to its rich physics and gate-tuneable optical properties[@BonaccorsoSunHasanEtAl2010]. Compared to bulk materials, the possibility of inducing high carrier densities in SLG is the key to achieve optical tuneability both for optical intraband processes [@Sensale-RodriguezYanRafiqueEtAl2012; @TamagnoneMoldovanPoumirolEtAl2016b] and for interband processes[@LiuYinUlin-AvilaEtAl2011] (which are relevant for optoelectronic modulators and photodetectors[@KoppensMuellerAvourisEtAl2014]). Furthermore, high mobility of SLG allows mid infrared plasmon-polaritons, also tuneable by gating[@rodrigo2015mid; @YanLowZhuEtAl2013].
SLG is ideally suited to modulate terahertz waves because of its high mobility and easy integration on Si technology. The mobility is linked to the massless nature of carriers in SLG and allows for a larger conductivity tuneability range than Si, for a given carrier density interval [@Sensale-RodriguezYanRafiqueEtAl2012]. Unlike radio-frequency MEMS[@HumPerruisseau-Carrier2014; @DebogovicBartolicPerruisseau-Carrier2014], SLG does not require packaging [@Sagade2015], its switching speed is several order of magnitude faster[@LiuYinUlin-AvilaEtAl2011] and it does not suffer from reliability issues[@HumPerruisseau-Carrier2014; @DebogovicBartolicPerruisseau-Carrier2014]. Tuneable capacitors, instead, are dominated by resistive losses above the microwave range[@HumPerruisseau-Carrier2014]. Thus, SLG is an ideal choice for THz modulation[@sherrott2017experimental; @miao2015widely; @Sensale-RodriguezYanRafiqueEtAl2012].
We report a THz reflectarray metasurface exploiting SLG as active element to achieve beam steering, shaping and broadband phase modulation. Our device achieves dynamical beam steering thanks to an array of cells (including metal and gated SLG) built on a reflective substrate, see Fig.\[fig:Figure1\]a. This consists of a dielectric spacer layer (20 $\mu$m float zone Si, with dielectric constant 11.7 at 1 THz[@headland2016dielectric]), on a metallic reflective film (140 nm Ag on 60 nm Al)[@TamagnoneMoldovanPoumirolEtAl2016b; @HasaniTamagnoneCascanteEtAl2016]. An additional AlO$_2$ layer is used to gate the SLG, allowing for the dynamical tuning of its conductivity and hence of its optical behavior at THz frequencies. High resistivity Si (for our sample the resistivity is 10 k$\Omega\cdot$cm) is transparent at THz frequencies[@HasaniTamagnoneCascanteEtAl2016], but allows injected carriers to charge the gate capacitance and tune SLG via field effect[@Sensale-RodriguezYanRafiqueEtAl2012]. The unit cell in Figs.\[fig:Figure1\]b,c is inspired by bow tie antennas[@Balanis2005], with two trapezoidal Au elements that concentrate the impinging electromagnetic energy in a 3 $\mu$m narrow gap where SLG is placed.
Fig.\[fig:SupplFigure1\] in Methods M1 illustrates the RA substrate fabrication process flow. Our device requires a substrate comprising a reflective conductive ground plane and a dielectric spacer with thickness in the order of a quarter wavelength (in the material itself). We achieve this by using high resistivity Si as the dielectric spacer. An anodic bonding process is used to bond a metallic reflective layer (Ag + Al) to a supporting glass substrate. Fig.\[fig:SupplFigure2\] in Methods M1 summarizes the fabrication of the RA starting from the substrate chips.
The beam steering RA can work as intended only if the cell can modulate its reflection coefficient between two values ($\Gamma_\mathrm{ON}$, $\Gamma_\mathrm{OFF}$) with a phase modulation $\pi$ (see Methods \[met:radpattheo\]). The amplitude of the reflection coefficient should be maximized and remain constant in the two states. This is equivalent to creating a metasurface where each cell has a tuneable surface impedance, since the reflection coefficient $\Gamma$ and the surface impedance $Z_S$ are related by[@Balanis2005; @TamagnoneMoldovanPoumirolEtAl2016b]:
$$\label{seqn:surfaceimped}
\Gamma=\frac{Z_S-\eta}{Z_S+\eta}$$
where $\eta = \sqrt{\mu_0\varepsilon_0^{-1}} \simeq 377\:\Omega$ is the free space impedance, $\mu_0$ is the vacuum magnetic permeability and $\varepsilon_0$ is the vacuum dielectric permittivity. The cell can then be designed to obtain suitable values of $Z_\mathrm{S}$ starting from the SLG’s sheet impedance $Z_\mathrm{g} = \sigma^{-1}$, where $\sigma$ is SLG’s conductivity. This can be changed via electric field gating between a maximum ($\sigma_\mathrm{ON}$) and minimum ($\sigma_\mathrm{OFF}$). It is possible to control the reflection phase in a binary way (two opposite values of the phase) if the metasurface is designed to have complete absorption ($\Gamma=0$) for $Z_\mathrm{S}=\sqrt{Z_\mathrm{S,ON}\cdot Z_\mathrm{S,OFF}}$. Because $\Gamma=0$ implies $Z_\mathrm{S}=\eta$, the approximate design condition becomes $\sqrt{Z_\mathrm{S,ON}\cdot Z_\mathrm{S,OFF}}=\eta$. The geometric average is used here, to ensure that $\Gamma_\mathrm{ON}=-\Gamma_\mathrm{OFF}$, providing binary phase modulation with the same amplitude in two states. The metasurface design must therefore achieve $Z_\mathrm{S}=\eta$ when $Z_\mathrm{g}=\sqrt{Z_\mathrm{g,ON}\cdot Z_\mathrm{g,OFF}}$. For our samples we measure $Z_\mathrm{g,ON}=800 \;\Omega$, $Z_\mathrm{g,OFF}=4000 \;\Omega$. Therefore $\sqrt{Z_\mathrm{g,ON}\cdot Z_\mathrm{g,OFF}}= 1789 \;\Omega = 4.75 \eta$. This implies that the cell must be designed to scale down SLG’s impedance to a factor 4.75 to be at the optimal working point.
To achieve this, we first chose a Salisbury screen configuration[@Salisbury]. This consists of a dielectric spacer on a reflective metallic layer[@TamagnoneMoldovanPoumirolEtAl2016b]. The spacer is a Si layer having thickness $t$:
$$\label{seqn:salisbury}
t=\frac{\lambda_0}{4n}=\frac{c}{4nf_0}$$
where $f_0 = 1$ THz is the design frequency, $\lambda_0$ is the corresponding free space wavelength, $c$ is the speed of light, and $n=\sqrt{11.7}$ is the Si refractive index. The purpose of this structure is to cancel the contribution of the reflective layer to the free space impedance, obtaining $Z_\mathrm{S}\simeq 0$ in absence of other structures on top of the spacer, as discussed in Ref.. From Eq.\[seqn:salisbury\] we get $t = 21.9$ $\mu$m. For our experiments we use $t = 20$ $\mu$m due to limitations in the available silicon on insulator (SOI) wafers.
The metallic structure in Fig.\[fig:SupplFigure4\] is chosen to concentrate the impinging field on a SLG rectangular load over a broad-band, hence the choice of the bow-tie antenna element. SLG is prolonged on one side, to contact an additional Au bias line used to improve the connectivity of the column, so that the applied voltage is uniform even in case of cracks in one or more of the SLG loads. Voltage is applied to both ends of the column.
The cells ($20\times100$ $\mu$m$^2$) are smaller than half of the wavelength (300 $\mu$m at 1 THz). Each reflects the incident waves with a reflection coefficient that can be modulated applying different voltages to SLG. Numerical simulations and measurements of the reflection coefficient are in Fig.\[fig:Figure1\]d-f. These measurements are performed by gating all the cells with the same voltage and then measuring the overall reflection coefficient $\Gamma$ of the surface, Fig.\[fig:Figure1\]d. $\Gamma$ is a complex number describing both the amplitude and the phase of the reflected wave, with the phase delay normalized with respect to a reference mirror (Au deposited on the same chip directly on Al$_2$O$_3$). A THz fiber-coupled time domain system is used to measure $\Gamma$ (see Methods \[met:measurement\]), focusing the incident beam on a small area of the array to avoid probing areas outside it. Note that, because of the subwavelength nature of the array, only one reflected beam exists, without diffraction.
The unit cell geometry is optimized so that, at the target design frequency of 1.05 THz, different $\sigma$ cause the reflection coefficient to vary from one value to its opposite, passing close to the total absorption condition ($\Gamma=0$). In this way, by switching the cell between these two states (ON and OFF) a local phase modulation of $\pi$ can be achieved. This is similar to the concept proposed in Refs. and demonstrated experimentally at microwave frequencies in Refs.. The slight shift measured reflection coefficient with respect to the simulations visible in the figures is due to fabrication tolerances.
Beam steering can then be achieved by switching the cell state in the array so that a dynamical and reconfigurable phase hologram is created, obtaining a fully reconfigurable RA. We focus on beam steering and shaping in one plane. This allows for a simplification of the control network, whereby all cells belonging to the same column are connected together, and each column can be controlled by an individual voltage.
The far electric field radiation pattern obtained illuminating the array with a plane wave having electric field amplitude $E_0$ and angle of incidence ${\theta_\mathrm{i}}$ (in our case fixed to 45$^\circ$) can be estimated based on the interference of discrete radiators[@Balanis2005]:
$$\begin{aligned}
\label{eqn:mainformula}
E(\theta,r)=E_0\,g(r)\,{f_\mathrm{C}}(\theta)\,{f_\mathrm{A}}(\theta) = E_0\,g(r)\,{f_\mathrm{C}}(\theta)
\nonumber\\ \,\sum_{n=1}^{N} \Gamma_n\,e^{\,jnk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}\end{aligned}$$
where $\theta$ is the deflection angle, ${f_\mathrm{C}}(\theta)$ is the radiation pattern of a single isolated column, ${f_\mathrm{A}}(\theta)$ is called *array factor* [@Balanis2005], $\Gamma_n$ is the reflection coefficient of the n-th cell, N is the total number of cells in the array, $k_0=2\pi/\lambda$ is the wavenumber, L is the cell width, r is the distance from the RA and $g(r)=r^{-1}exp(-jkr)$. The ${f_\mathrm{C}}(\theta)$ factor is negligible here, as it does not show sharp variations in $\theta$ due to the sub-wavelength size of the unit cell. The summation (hence ${f_\mathrm{A}}(\theta)$) is maximized when its elements are in phase. If a linear phase profile is created setting the $\Gamma_n$ elements such that $\Gamma_n=e^{jn\phi}$ then the maximum (hence the reflected beam direction) is obtained for $\theta=\arcsin\left(\sin{\theta_\mathrm{i}}-\frac{\phi}{k_0 L}\right)$, which can be changed dynamically tuning the phase profile. It is possible to show (see Methods M3 for a full mathematical derivation) that this principle still holds if the reflection phase is quantized to just two values (0 and $\pi$) for all the elements, thus reducing the gradient to a periodic set of segments with phase 0 alternated with segments of phase $\pi$. The periodicity $P$ of the pattern expressed in terms of number of cells is then given by $P=2\pi/\phi$ and by the beam steering law:
$$\label{eqn:steer}
\theta=\arcsin\left(\sin{\theta_\mathrm{i}}-\frac{\lambda}{PL}\right)$$
where $\lambda$ is the wavelength.
If $P$ is an even integer the pattern consists of a repetitions of a supercell of $P$ cells (with $P/2$ cells set to phase 0 and $P/2$ to $\pi$). Fig.\[fig:Figure2\]a illustrates the case $P=4$. However, it is possible to generalize the pattern to odd and even fractional values of $P$ using a pattern generation technique described in Methods M4, thus achieving continuous beam steering.
Beam-steering with integer $P$ (from 4 to 8) is shown in Fig.\[fig:Figure2\]b, for the voltage patterns in Fig.\[fig:Figure2\]c. The angular steering range reaches 25$^\circ$. The beam is well-formed with the exception of small side lobes which appear for odd $P$, due to the technique used to emulate odd and fractional $P$ values. Fig.\[fig:Figure2\]d plots the beam steering as a function of frequency, compared with the prediction of Eq.\[eqn:steer\], while Figs.\[fig:Figure2\]e,f demonstrate the continuous beam-steering achieved with fractional $P$.
Our device can also reconfigure the beam shape. This is achieved by smoothly changing the local P from one extreme to the other of the array using a chirped pattern, as shown in Fig.\[fig:Figure2\]k, thus having slightly different deflection angles across the array, emulating a parabolic profile. The device operates as a parabolic mirror with tunable curvature, which we use here to generate a wider beam, Fig.\[fig:Figure2\]g. The same principle can be used to achieve tuneable focusing (limited here to one dimension).
Besides focusing and widening the beam, more complex operations can be performed. E.g., Fig.\[fig:Figure2\]h plots the generation of a double beam by filling two halves of the array with patterns having different $P$ (changing abruptly in the middle of the array, as shown in the dual-beam pattern in Fig.\[fig:Figure2\]k). Another important application is the ability to manipulate an impinging THz pulse at the time domain level. This is possible because the incident pulse reaches at different times each RA element. Therefore, the voltage pattern selected on the array is transferred to the time response of the system (within some limits due to the spectral response of each cell and to the total size of the illuminated portion of the array). Figs.\[fig:Figure2\]i,j show that a periodic pattern with $P=6$ generates a sinusoid (of finite time duration due to the finite size of the array). The chirped pattern used for beam broadening gives a chirped sinusoid in the time response. Similar transformations can be achieved with more complex patterns.
We now consider the effect of shifting a periodic pattern (with $P = 2, 4, 6$) of a finite number of cells, and we verify that the corresponding time domain sinusoid is similarly de-phased. This is equivalent to the phase shift associated to a lateral translation of an optical grating[@lee2013displacement], but the movement here is emulated by the reconfigurable control patterns. The experiment is illustrated in Fig.\[fig:Figure3\]a and the measurements, better represented in the frequency domain, are plotted in Figs.\[fig:Figure3\]b-d. These are the complex reflection coefficients for each of the aforementioned cases, and for each possible shift of the pattern. E.g., the $P=4$ patterns can be shifted in 4 ways, with shifts of 0, 1, 2, 3 cells, while shifting of 4 cells is identical to 0 and so on. Each cell shift corresponds to a phase delay of $2\pi/P$ regardless of the beam frequency. This scheme, here referred to as *geometrical PSK* (phase shift keying[@blahut1987principles]), provides a way to perform a precise phase modulation on a wide band (60GHz at 1THz).
In summary, we reported a reconfigurable RA metasurface for terahertz waves using SLG. Beam steering, shaping and modulation were achieved. Our results demonstrate that graphene can be embedded in metasurfaces providing an unprecedented control and modulation capabilities for THz beams, with applications for adaptive optics, sensing and telecommunications. Our approach can be extended to mid infrared, and to two dimensional beam steering, by using individual cell control.
Acknowledgements {#acknowledgements .unnumbered}
================
We dedicate this work to the memory of Prof. Julien Perruisseau-Carrier. We thank Giancarlo Corradini, Cyrille Hibert, Julien Dorsaz, Joffrey Pernollet, Zdenek Benes, and the rest of EPFL CMi staff for the useful discussions. We acknowledge funding from the EU Graphene Flagship, the Swiss National Science Foundation (SNSF) grants 133583 and 168545, the Hasler Foundation (Project 11149), ERC Grant Hetero2D, EPSRC grant nos. EP/509 K01711X/1, EP/K017144/1, EP/N010345/1, EP/M507799/5101 and EP/ L016087/1.
Methods {#methods .unnumbered}
=======
Fabrication process flow {#met:fab}
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The starting point is a Si on insulator (SOI) wafer (produced by Ultrasil Corp.) having a device layer with the required characteristics for our dielectric spacer with 20 $\mu$m thickness and high resistivity $\sim$ 1 k$\Omega\cdot$cm, Fig.\[fig:SupplFigure1\]a. Ag is deposited (e-beam evaporation) to create the reflective layer, followed by an Al layer (vacuum is not broken between the two depositions, Fig.\[fig:SupplFigure1\]b). The Al coated face of the SOI wafer is then bonded via anodic bonding (Fig.\[fig:SupplFigure1\]c) to a borosilicate glass wafer (Borofloat 33, very similar to Pyrex in composition[@borofloat]), acting as a support for the device layer, which is too fragile to be handled alone given its thickness. Bonding is performed at atmospheric pressure with a Suss Microtec SB6 tool immediately after evaporation. Glass wafers are also cleaned in a hot piranha bath immediately before bonding to remove organic impurities. A second borosilicate wafer is used below the one to be bonded, as a sacrificial substrate to collect the excess Na ions, thus preventing contamination. To prevent accidental bonding of the two borosilicate wafers, the sacrificial substrate is thinned using wafer grinding, and the non-polished surface is placed in contact with the borosilicate substrate to be bonded.
The aim of the following steps is to eliminate the SOI handle and box layer, to expose the device layer. This is done by first grinding the Si handle wafer down to 100 $\mu$m (Fig.\[fig:SupplFigure1\]d). This is a mechanically aggressive process, therefore further thinning could damage the substrate or cause the failure of the bonding. The remaining Si is dry-etched using a fluorine-based chemistry, with a process having 200:1 selectivity with respect to SiO$_2$, Fig.\[fig:SupplFigure1\]e. This ensures that the box layer survives the process, preserving the device layer as well. The box is then dissolved in HF 49% (Fig.\[fig:SupplFigure1\]f), selected over buffered oxide etch (BHF) because it etches faster SiO$_2$ [@williams1996etch] and, unlike BHF, does not attack Al [@williams1996etch].
The gate oxide (200 nm Al$_2$O$_3$) is prepared using atomic layer deposition (ALD) on all the wafer, as shown in Fig.\[fig:SupplFigure1\]g. Afterwards, dual layer photo-litography (LOR + AZ1512) is performed, followed by evaporation of 100 nm Au with an adhesion layer of 5 nm Cr and liftoff (Fig.\[fig:SupplFigure1\]h,i). During this step, the reference mirrors (one for each chip), bonding pads, and dicing markers are defined on the full wafer (Fig.\[fig:SupplFigure3\]a). Oxygen plasma de-scum is performed prior to the evaporation to ensure maximum adhesion, important for the subsequent wire-bonding step. The wafer is then diced (Fig.\[fig:SupplFigure1\]j) using an automatic dicing saw (Disco DAD-321). During the dicing process, the wafer is protected by a photo-resist layer, then stripped in remover on each chip, lifting the dicing residues.
SLG is grown on Cu foil (99.8% purity) by chemical vapor deposition (CVD) on a tube furnace as for Ref.. The Cu foil is annealed in H$_2$ (flow 20 sccm) at 1000 $^\circ$C for 30 min. After annealing, CH$_4$ (flow 5 sccm) is introduced for 30 min while keeping the temperature at 1000 $^\circ$C, leading to the growth of SLG. This is then transferred onto the Al$_2$O$_3$/Si/Ag/Pyrex by wet transfer (Figure \[fig:SupplFigure2\]a,b)[@bae], where polymethyl methracrylate (PMMA) is used as a sacrificial layer to support SLG during Cu etching in ammonium persulfate[@bae]. After transfer, PMMA is dissolved in acetone. Raman spectroscopy is used to monitor the sample quality throughout the process by using a Renishaw inVia spectrometer equipped with 100X objective and a 2400 groves/mm grating at 514.5 nm. Representative Raman spectra of SLG placed onto the Al$_2$O$_3$/Si/Ag/Pyrex substrate are shown in Fig.\[fig:SupplFigure9\]. The spectrum of graphene on Cu shows not significant D peak, indicating a negligible defect density [@NN2013]. After transfer, the position of G peak is 1590 cm$^{-1}$ and its full width at half maximum is 17 cm$^{-1}$, the position of 2D peak is 2692 cm$^{-1}$, while the ratio of the 2D to G peaks intensities, I(2D)/I(G), is $\sim$1.84 and the ratio of their areas, A(2D)/A(G), is $\sim$4, indicating a Fermi level $\sim$0.2-0.4eV and a charge carrier concentration $\sim$10$^{12}$ cm$^{-2}$ [@Das]. The D peak is present in the spectrum of the transferred SLG, suggesting that some defects have been introduced during the process. From I(D)/I(G) $\sim$0.13 and given the Fermi level, we can estimate a defect density $\sim$7x10$^{10}$ cm$^{-2}$ [@Bruna2014; @cancado].
SLG is then e-beam patterned using PMMA resist followed by oxygen plasma and stripping in acetone at 45 $^\circ$C, Fig.\[fig:SupplFigure2\]c,d). Apart from patterning SLG in the RA, during this process all the bonding and traces are also exposed to the oxygen plasma to ensure that no SLG remains on them, to avoid short circuits. Subsequently, a new e-beam lithography (MMA + PMMA) is performed to define the metallic antennas via evaporation and liftoff in acetone, Figs. \[fig:SupplFigure2\]e,f, \[fig:SupplFigure3\]a,b. Finally the chip is glued to the PCB substrate and all the columns are connected via wire-bonding to the PCB traces, Fig.\[fig:SupplFigure2\]g). The ground plane is contacted laterally with Ag paint.
Measurement setup and post processing {#met:measurement}
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The RA is characterized using a commercial fiber coupled THz time domain setup (by Menlo Systems, model TERA K15 mark II). The THz emitter is mounted at 45 degrees of incidence with respect to the RA, while the detector is placed on a motorized rotation stage. The system is first aligned in transmission to maximize the THz signal intensity, and subsequently in reflection, using a reference mirror mounted on the sample holder. This is mounted on a translation stage (motorized XY linear stages plus manual Z stage), to automatically alternate between sample and reference mirror. All measurements are normalized with respect to the reference mirror, created on the same chip of the RA during the optical lithography process on the full wafer.
Two different measurement modes are used, Figure \[fig:SupplFigure6\]:
- **Collimated** The beam is collimated and impinges on a large area of the sample ($\sim$1 cm$^2$). The reduced spread of the angular components of the beam (when decomposed into a superposition of plane waves) ensures precise measurement of angles and radiation patterns, but part of the beam interacts with the area around the RA.
- **Focused**: The beam is focused by an additional pair of lenses so that it impinges completely inside the array. However, this requires larger spread of the angular components, hence this mode is not accurate for angles and radiation pattern measurement. Instead, it is used to measure the reflection coefficient and the efficiency of the array.
The measurements in Fig.\[fig:Figure1\] are performed in the *focused* mode, while those in Figs.\[fig:Figure2\],\[fig:Figure3\] use the *collimated* mode. The latter is to be preferred to characterize the geometric phase shift keying (G-PSK, Fig.\[fig:Figure3\]), since precise phase modulation relies on the interaction between the beam and all of the columns of the RA, which can be illuminated completely only in the collimated mode.
The drawback of the collimated mode is the interaction of the beam with areas outside the RA metasurface. This can be addressed with the following post-processing method. For radiation pattern characterization, two measurements are performed, one with the chosen control sequence, and the other with the opposite (logical NOT) control sequence. In this way, the steered beam will have opposite phase in the two cases (see Method \[met:radpattheo\]) and the radiation pattern can be extracted by subtracting (frequency by frequency and angle by angle) these two measurements. Any contribution from the area outside the array is canceled by the subtraction. For the G-PSK case, the same is accomplished by subtracting from all the signals the average in the complex plane (frequency by frequency). An overall phase factor $e^{-j(\omega\tau+\phi)}$ is removed from all the traces. A unique value of the delay $\tau$ is used for all the symbols in each G-PSK measurement. This is done to remove the free-space phase delay of the measured beam, due to slight differences in the paths when measuring the array and the reference mirror. Similarly, the removed phase factor $\phi$ is unique for all the traces, and it is used to align the symbols to the real and imaginary axes of the complex plane.
Radiation pattern theory {#met:radpattheo}
------------------------
The array geometry does not depend on the $y$ direction since the array has a periodicity smaller than half wavelength in that direction, independently of the control pattern. We assume that the incident wave is propagating in the $xz$ plane (i.e. $k_y=0$). In the low cell-to-cell coupling approximation[@HumPerruisseau-Carrier2014], the electric field of an antenna array in the $x$ direction (assuming equidistant elements separated by $L$) in far field conditions is given by[@HumPerruisseau-Carrier2014; @Balanis2005]:
$$\begin{aligned}
\label{seqn:mainformula}
E(\theta,r) = g(r)\sum_{n=1}^{N} w_n{f_\mathrm{C}}(\theta)\,e^{\,jnk_0L\sin\theta} = g(r){f_\mathrm{C}}(\theta)\nonumber\\ \sum_{n=1}^{N}w_n\,e^{\,jnk_0L\sin\theta}\end{aligned}$$
where ${f_\mathrm{C}}$ is the single cell radiation pattern, $k_0$ is the wavenumber, $w_n$ is the amplitude associated to the n-th element. For a RA we can write $w_n$ as the product of the incident field at the element position times a reflection coefficient:
$$\label{seqn:wn}
w_n=\Gamma_n\, {E_\mathrm{i}}(x=nL,z=0)$$
where we assume for simplicity and without loss of generality that the RA is in the $z=0$ plane. The electric field ${E_\mathrm{i}}$ of an incident wave (with incident angle ${\theta_\mathrm{i}}$ with respect to the normal) is given by:
$$\label{seqn:einc}
{E_\mathrm{i}}(x,z)=E_0\,e^{-jk_0(x\sin{\theta_\mathrm{i}}+z\cos{\theta_\mathrm{i}})}$$
Combining Eqs.\[seqn:einc\],\[seqn:wn\],\[seqn:mainformula\] we get:
$$\begin{aligned}
\label{seqn:mainformula2}
E(\theta) = E_0 \, g(r) {f_\mathrm{C}}(\theta) \sum_{n=1}^{N}\Gamma_n\,e^{\,jnk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}=\nonumber\\ E_0 \, g(r) {f_\mathrm{C}}(\theta){f_\mathrm{A}}(\theta)\end{aligned}$$
where we define the *array factor* ${f_\mathrm{A}}(\theta)$ as:
$$\label{seqn:af}
{f_\mathrm{A}}(\theta) = \sum_{n=1}^{N}\Gamma_n\,e^{\,jnk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}$$
We notice that, if all the reflection coefficients are phase-modulated of $\pi$ (thus reversing their sign), the total phase of the scattered field will also be out of phase of $\pi$, which is used to suppress the background in our measurements. If all the cells have the same $\Gamma$, then:
$$\label{seqn:uniformarray}
E(\theta) = E_0\, g(r)\, {f_\mathrm{C}}(\theta)\,\Gamma\sum_{n=1}^{N}\,e^{\,jnk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}$$
A maximum in the reflection is obtained for $\theta={\theta_\mathrm{i}}$ (which is the direction of the specular reflection) where all the contributions of the summation add in phase. More generally, a maximum is obtained if:
$$\label{seqn:refl}
k_0 L(\sin\theta-\sin{\theta_\mathrm{i}})=2\pi l \qquad l\in \mathbb{Z}$$
for $\lambda=2\pi/k_0$:
$$\label{seqn:refl2}
\sin\theta-\sin{\theta_\mathrm{i}}= l \frac{\lambda}{L} \qquad l\in \mathbb{Z}$$
Because $\lambda/L=3$ in our case, only the specular reflection $l=0$ satisfies this condition, for any ${\theta_\mathrm{i}}$.
Super-cells can be implemented by creating periodic patterns of reflection coefficients, fulfilling the condition $\Gamma_n=\Gamma_{n+P}$, where $P$ is a positive integer number of cells in the super-cell. If a super-cell with periodicity $P$ is implemented, then the array factor can be rewritten, by splitting the summation in two levels: an external sum over all the super-cells, and an internal one over the cells in a super-cell:
$$\begin{aligned}
\label{seqn:supercelldecomp}
{f_\mathrm{A}}(\theta) =
\sum_{n=0}^{N/P}\sum_{m=1}^{P}\Gamma_m\,e^{\,j(nP+m)k_0L(\sin\theta-\sin{\theta_\mathrm{i}})}=\nonumber\\
\sum_{m=1}^{P}\Gamma_m\,e^{\,jmk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}\sum_{n=0}^{N/P}\,e^{\,jnPk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}\end{aligned}$$
We can then define the *supercell factor* ${f_\mathrm{SC}}$:
$$\label{seqn:supercell}
{f_\mathrm{SC}}(\theta) =\sum_{m=1}^{P}\Gamma_m\,e^{\,jmk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}$$
and the *superarray factor* ${f_\mathrm{SA}}$:
$$\label{seqn:superarray}
{f_\mathrm{SA}}(\theta) =\sum_{n=0}^{N/P}\,e^{\,jnPk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}$$
and use them to decompose the array factor as ${f_\mathrm{A}}(\theta)={f_\mathrm{SC}}(\theta) {f_\mathrm{SA}}(\theta)$. The final expression for the electric far field is then:
$$\begin{aligned}
\label{seqn:mainformula3}
E(\theta) \;=\; E_0 \, g(r) f(\theta) \;=\; E_0 \, g(r) {f_\mathrm{C}}(\theta){f_\mathrm{A}}(\theta) \;=\nonumber\\ \; E_0 \, g(r) {f_\mathrm{C}}(\theta) {f_\mathrm{SC}}(\theta) {f_\mathrm{SA}}(\theta)\end{aligned}$$
The angular part of the radiation pattern $f(\theta)$ is decomposed in three factors (ordered here from the most directive to the least):
- ${f_\mathrm{SA}}$, associated to the super-array, identifies a set of possible directions where light can be scattered, and behaves as a diffraction grating.
- ${f_\mathrm{SC}}$, associated to the super-cell, gives different weights to the possible diffraction orders accordingly to the phase of the cells in the supercell.
- ${f_\mathrm{C}}$, associated to the cell, slowly varying with $\theta$ in the subwavelength case and does not affect significantly the final pattern.
The directions of diffracted beams launched by the array is then given by ${f_\mathrm{SA}}(\theta)$, and for each beam the following must be satisfied:
$$\label{seqn:grating}
\sin\theta-\sin{\theta_\mathrm{i}}= l \frac{\lambda}{PL} \qquad l\in \mathbb{Z}$$
Evaluating ${f_\mathrm{SC}}$ for each of the diffracted beams:
$$\begin{aligned}
\label{seqn:fscl}
{f_\mathrm{SC}}(l)=\sum_{m=1}^{P}\Gamma_m\,e^{\,jmk_0L(\sin\theta-\sin{\theta_\mathrm{i}})}=\nonumber\\ \sum_{m=1}^{P}\Gamma_m\,e^{\,j 2 \pi m l /P} \qquad l\in \mathbb{Z}\end{aligned}$$
{width="180mm"}
In our case ${\theta_\mathrm{i}}=45^\circ$ and we are operating at 1 THz. Also, let us consider the simplest case, with $N$ even integer and with a supercell formed by $N/2$ cells with reflection phase $0^\circ$ followed by $N/2$ cells with reflection phase $180^\circ$. $l=-1$ represents the steered beam of interest. The specular reflection ($l=0$) is suppressed because ${f_\mathrm{SC}}$ vanishes for $\theta={\theta_\mathrm{i}}$. This is due to the fact that in our super-cell half of the cells have a reflection coefficient $\Gamma$ and the remaining ones $-\Gamma$, so the total sum is zero.
Then evaluating the summation for the considered supercell:
$$\begin{aligned}
\label{seqn:evenN}
{f_\mathrm{SC}}(l)=\sum_{m=1}^{P}\Gamma_m\,e^{\,j 2 \pi m l /P}=\nonumber\\
\Gamma \left(\sum_{m=1}^{P/2}\,e^{\,j 2 \pi m l /P}-\sum_{m=P/2+1}^{P}\,e^{\,j 2 \pi m l /P}\right)\end{aligned}$$
The same cancelation holds for any even value of $l$. Any beam for odd values of $l$ different from 1 and -1 is also strongly attenuated. For all our choices of period ($P$ between 4 and 8) the $l=1$ beam does not exist, since no real $\theta$ solves Eq.\[seqn:grating\] for the chosen beam wavelength and incident angle. Imperfections in the array may still cause smaller side lobes, i.e. unwanted beams for $l\neq-1$. This is especially true for odd and fractional values of $P$, where the analysis of Eq.\[seqn:evenN\] does not apply rigorously (though it still describes qualitatively the situation).
The geometric PSK operation can be understood inspecting ${f_\mathrm{SC}}$ for $l=-1$, and noting that a translation of any supercell pattern described as $\Gamma_m \leftarrow \Gamma_{(m+1) \mathrm{mod} P}$ creates a phase shift $2\pi/P$ (for an infinite array approximation):
$$\label{seqn:QPSK1}
{f_\mathrm{SC}}(l=-1)=
\sum_{m=1}^{P}\Gamma_m\,e^{\,-j 2 \pi m /P}$$
$$\begin{aligned}
\label{seqn:QPSK2}
\sum_{m=1}^{P}\Gamma_{(m+1)\mathrm{mod} P}\,e^{\,-j 2 \pi m /P}=
\sum_{r=0}^{P-1}\Gamma_{r}\,e^{\,-j 2 \pi(r-1)/P}=\nonumber\\
e^{j2\pi/P}\sum_{r=1}^{P}\Gamma_{r}\,e^{\,-j 2 \pi r/P}\end{aligned}$$
The software Ansys HFSS is used to compute and optimize the reflection coefficient of the cells. In particular, the effect of the metallic bias lines on the cell impedance is reduced thanks to the optimization. Simulations are performed using the Floquet boundary conditions (equivalent to Bloch periodic boundary conditions, see for example[@TamagnoneFallahiMosigEtAl2014]), with an incident angle $45^\circ$. The optimized bias line width is 2 $\mu$m, so that its inductance per unit length is sufficient to reduce its effect on the structure.
Control unit and list of control sequences {#met:control}
------------------------------------------
The RA is glued on a support PCB substrate and each column is wirebonded to a PCB trace. Both ends of each column are connected to a high voltage transistor stage, and all the stages are controlled by two Arduino units. Fig.\[fig:SupplFigure5\]a is an Arduino unit connected to a PCB driver with 40 transistor stages. A similar unit controls the remaining 40 columns. Fig.\[fig:SupplFigure5\]b shows a circuit schematic of the high voltage stage, while Fig.\[fig:SupplFigure5\]c illustrates the support PCB substrate. Fig.\[fig:SupplFigure5\]d has two PCB drivers connected to the support PCB. The high voltage stage can switch on and off the voltage of each column, reaching values close to the supply voltage in one case and close to ground in the other. The needed positive and negative gate voltages are achieved by applying an intermediate voltage on the retroreflector of the RA, so that the voltage difference is negative in one state and positive in the other one. The total voltage is set via a high voltage DC generator, and the retroreflector voltage is controlled by a potentiometer.
Tables \[tab:SupplTable1\], \[tab:SupplTable2\], \[tab:SupplTable3\], \[tab:SupplTable4\] show the control sequences used in each of our experiments. These are programmed into the Arduino module by the control computer. The latter can control the Arduino modules, the rotary stage, the XY stage, the high voltage generator and the terahertz setup, so that the measurements are completely automatized.
ON and OFF states (or equivalently 1 and 0) correspond to the following voltages between SLG and retroreflector: $V_\mathrm{ON}=26V$, $V_\mathrm{OFF}=-44 V$. The sequences are generated by discretizing sinusoids and chirped sinusoids with different periods, which leads to quasi-periodic signals for odd and fractional P values.
[|l|l|]{}
Sequence name & Sequence\
$\phantom{\lnot}$ All OFF
------------------------------------------------------------------------
& 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000\
$\phantom{\lnot}$ All ON & 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111\
$\phantom{\lnot}$ Period 4
------------------------------------------------------------------------
& 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001\
$\lnot$ Period 4 & 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110\
$\phantom{\lnot}$ Period 5 & 11001100 01110011 00011100 11000111 00110001 11001100 01110011 00011100 11000111 00110001\
$\lnot$ Period 5 & 00110011 10001100 11100011 00111000 11001110 00110011 10001100 11100011 00111000 11001110\
$\phantom{\lnot}$ Period 6 & 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100\
$\lnot$ Period 6 & 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011\
$\phantom{\lnot}$ Period 7 & 10001110 00011110 00111000 01111000 11100001 11100011 10000111 10001110 00011110 00111000\
$\lnot$ Period 7 & 01110001 11100001 11000111 10000111 00011110 00011100 01111000 01110001 11100001 11000111\
$\phantom{\lnot}$ Period 8 & 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001\
$\lnot$ Period 8 & 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110\
[|l|l|]{}
Sequence name & Sequence\
$\phantom{\lnot}$ All OFF
------------------------------------------------------------------------
& 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000\
$\phantom{\lnot}$ All ON & 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111\
$\phantom{\lnot}$ Period 5.5
------------------------------------------------------------------------
& 01110001 11001110 00111001 11000111 00111000 11100011 00011100 01100011 10001100 01110001\
$\lnot$ Period 5.5 & 10001110 00110001 11000110 00111000 11000111 00011100 11100011 10011100 01110011 10001110\
$\phantom{\lnot}$ Period 5.75 & 11100111 00011100 01110001 11001110 00111000 11100011 10001100 01110001 11000111 00011000\
$\lnot$ Period 5.75 & 00011000 11100011 10001110 00110001 11000111 00011100 01110011 10001110 00111000 11100111\
$\phantom{\lnot}$ Period 6 & 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110\
$\lnot$ Period 6 & 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001\
$\phantom{\lnot}$ Period 6.25 & 00111100 01110001 11000111 00011110 00111000 11100011 10000111 00011100 01110001 11000011\
$\lnot$ Period 6.25 & 11000011 10001110 00111000 11100001 11000111 00011100 01111000 11100011 10001110 00111100\
$\phantom{\lnot}$ Period 6.5 & 01110001 11100011 10001111 00011100 01111000 11100001 11000111 00001110 00111000 01110001\
$\lnot$ Period 6.5 & 10001110 00011100 01110000 11100011 10000111 00011110 00111000 11110001 11000111 10001110\
[|l|l|]{}
Sequence name & Sequence\
$\phantom{\lnot}$ All OFF
------------------------------------------------------------------------
& 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000\
$\phantom{\lnot}$ All ON & 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111\
$\phantom{\lnot}$ Period 6
------------------------------------------------------------------------
& 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110\
$\lnot$ Period 6 & 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001\
$\phantom{\lnot}$ Chirp 1 & 01110011 10001100 01110001 10001110 00111000 11100011 10001110 00011100 01111000 11100001\
$\lnot$ Chirp 1 & 10001100 01110011 10001110 01110001 11000111 00011100 01110001 11100011 10000111 00011110\
$\phantom{\lnot}$ Chirp 2 & 00111001 11001110 01110001 11001110 00111000 11100011 10001111 00011100 00111000 01111000\
$\lnot$ Chirp 2 & 11000110 00110001 10001110 00110001 11000111 00011100 01110000 11100011 11000111 10000111\
$\phantom{\lnot}$ Dual beam & 11001100 11001100 11001100 11001100 11001100 11110000 11110000 11110000 11110000 11110000\
$\lnot$ Dual beam & 00110011 00110011 00110011 00110011 00110011 00001111 00001111 00001111 00001111 00001111\
[|l|l|]{}
Sequence name & Sequence\
All OFF
------------------------------------------------------------------------
& 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000\
All ON & 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111\
4-GPSK Symbol 0
------------------------------------------------------------------------
& 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001 10011001\
4-GPSK Symbol 1 & 00110011 00110011 00110011 00110011 00110011 00110011 00110011 00110011 00110011 00110011\
4-GPSK Symbol 2 & 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110 01100110\
4-GPSK Symbol 3 & 11001100 11001100 11001100 11001100 11001100 11001100 11001100 11001100 11001100 11001100\
6-GPSK Symbol 0
------------------------------------------------------------------------
& 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100\
6-GPSK Symbol 1 & 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000\
6-GPSK Symbol 2 & 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001\
6-GPSK Symbol 3 & 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011\
6-GPSK Symbol 4 & 11000111 00011100 01110001 11000111 00011100 01110001 11000111 00011100 01110001 11000111\
6-GPSK Symbol 5 & 10001110 00111000 11100011 10001110 00111000 11100011 10001110 00111000 11100011 10001110\
8-GPSK Symbol 0
------------------------------------------------------------------------
& 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001 11100001\
8-GPSK Symbol 1 & 11000011 11000011 11000011 11000011 11000011 11000011 11000011 11000011 11000011 11000011\
8-GPSK Symbol 2 & 10000111 10000111 10000111 10000111 10000111 10000111 10000111 10000111 10000111 10000111\
8-GPSK Symbol 3 & 00001111 00001111 00001111 00001111 00001111 00001111 00001111 00001111 00001111 00001111\
8-GPSK Symbol 4 & 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110 00011110\
8-GPSK Symbol 5 & 00111100 00111100 00111100 00111100 00111100 00111100 00111100 00111100 00111100 00111100\
8-GPSK Symbol 6 & 01111000 01111000 01111000 01111000 01111000 01111000 01111000 01111000 01111000 01111000\
8-GPSK Symbol 7 & 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000 11110000\
Device efficiency {#met:efficiency}
-----------------
Our RA is based on the modulation of the reflected signal from each cell due to the variation of $\sigma$ with voltage. Ref. demonstrated that specific upper bounds to the efficiency of such two-state modulators exist, and depend uniquely on the conductivity $\sigma_\mathrm{ON}$ and $\sigma_\mathrm{OFF}$ of graphene in the two states. If the corresponding cell reflection coefficients in the two states are $\Gamma_\mathrm{ON}$ and $\Gamma_\mathrm{OFF}$ then this bound is given by the following inequality[@TamagnoneFallahiMosigEtAl2014]:
$$\begin{aligned}
\label{seqn:bound1}
\gamma_\mathrm{mod}\triangleq \frac{|\Gamma_\mathrm{ON}-\Gamma_\mathrm{OFF}|^2}{(1-|\Gamma_\mathrm{ON}|^2)(1-|\Gamma_\mathrm{OFF}|^2)} \nonumber\\ \leq
\gamma_\mathrm{R}\triangleq \frac{|\sigma_\mathrm{ON}-\sigma_\mathrm{OFF}|^2}{4\;\mathrm{Re}(\sigma_\mathrm{ON})\;\mathrm{Re}(\sigma_\mathrm{OFF})}\end{aligned}$$
$\sigma$ at THz frequencies can be estimated analytically with the Drude model, as [@TamagnoneFallahiMosigEtAl2014]:
$$\label{seqn:Drude}
\sigma_\mathrm{ON,OFF}=(R_\mathrm{ON,OFF}\;(1+j\omega\tau))^{-1}$$
In our samples we have $R_\mathrm{ON}=800\;\Omega$, $R_\mathrm{OFF}=4000\;\Omega$, $\tau=45$ fs. From Eq. \[seqn:bound1\] we get:
$$\label{seqn:bounddrude}
\gamma_\mathrm{R} = \frac{(R_\mathrm{ON}-R_\mathrm{OFF})^2(1+\omega^2\tau^2)}{4\;R_\mathrm{ON}\;R_\mathrm{OFF}}\simeq 0.856$$
For the RA, ideally, the reflection coefficients must have opposite phases and same absolute values. In practice this happens only approximately, and the actual deflected signal is proportional to the difference $D\triangleq\Gamma_\mathrm{ON}-\Gamma_\mathrm{OFF}$, while the sum $S\triangleq\Gamma_\mathrm{ON}+\Gamma_\mathrm{OFF}$ is responsible for an unwanted specular reflection component. Expressing the bound in $S$ and $D$ we get from Eq. \[seqn:bound1\]:
$$\begin{aligned}
\label{seqn:boundSD}
\gamma_\mathrm{mod}\;=\;\frac{16|D|^2}{(4-|S|^2-|D|^2)^2-4(\mathrm{Re}(SD^*))^2}\;\nonumber\\ \ge\;\frac{16|D|^2}{(4-|D|^2)^2}\end{aligned}$$
and:
$$\label{seqn:boundD}
\frac{16|D|^2}{(4-|D|^2)^2}\;\le\gamma_\mathrm{R}$$
Eq. \[seqn:boundD\] can now be inverted to find the maximum achievable $|D|$ with our SLG parameters. We get:
$$\label{seqn:boundinv}
D\;\le\;|D|_\mathrm{max}\triangleq\sqrt{\frac{4(\gamma_\mathrm{R}+2-2\sqrt{\gamma_\mathrm{R}+1})}{\gamma_\mathrm{R}}}\;\simeq\;0.7832$$
From the measurements in Fig.\[fig:Figure1\], $|D|\simeq 0.5$ at peak efficiency, below the computed $|D|_\mathrm{max}$. This is likely due to losses in the metal, therefore there is room for improvement in the cell design. The final power efficiency of the deflected beam is given by the average differential reflection coefficient ($|D|/2\simeq0.25$, i.e. $\sim-$12 dB, while the efficiency of the optimal device would be $\sim-8$ dB. $|D|_\mathrm{max}$ is a strict bound that cannot be exceeded[@TamagnoneFallahiMosigEtAl2014], however its value can be increased using SLG having higher mobility or with gate oxide with better breakdown voltage and hence larger SLG tunability. The efficiency can be further increased by reducing the temperature to increase mobility and reduce the effect of thermal carriers.
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|
---
abstract: |
A review on the application of Melnikov's method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative, oscillator systems by weak harmonic excitations is presented, including diverse applications such as chaotic escape from a potential well, chaotic solitons in Frenkel-Kontorova chains, and chaotic charged particles in the field of an electrostatic wave packet.
**Keywords: chaos control; Melnikov method; homoclinic chaos; heteroclinic chaos; harmonic perturbation**
author:
- |
<span style="font-variant:small-caps;">Ricardo Chacon</span>\
*Departamento de Electrónica e Ingeniería Electromecánica,*\
* Escuela de Ingenierías Industriales, Universidad de Extremadura,*\
*Apartado Postal 382, E-06071 Badajoz, Spain*
title: 'Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations'
---
Introduction
============
During the past 15 years or so, diverse techniques of non-feedback chaos control have been proposed (Chen & Dong 1998) that may be roughly classified into three types: (i) The parametric excitation of an experimentally adjustable parameter; (ii) Entrainment to the target dynamics; and (iii) The application of a coordinate-independent (or dipole) external periodic excitation. It is shown below that techniques (i) and (iii) may be unified in a general setting for the class of dissipative systems considered in this present review. There exists numerical, experimental, and theoretical evidence that the period of the most effective chaos-controlling excitations usually is a rational fraction of a certain period associated with the uncontrolled system, although the effectiveness of incommensurate excitations has also been demonstrated in some particular cases (Chacón & Martínez 2002). Indeed, resonances between the chaos-controlling excitation and (i) a periodic chaos-inducing excitation, (ii) an unstable periodic orbit embedded in the chaotic attractor, (iii) a natural period in a period doubling route to chaos, or (iv) a period associated with some peak in the power spectrum, have been considered in diverse successful chaos-controlling excitations. This is not really surprising since these types of resonances are closely related to each other. For instance, when a damped, harmonically forced oscillator exhibits a steady chaotic state, the power spectrum corresponding to a given state variable typically presents its main peaks at frequencies which are rational fractions of the chaos-inducing frequency for certain ranges of the chaos-inducing amplitude. The extensive literature concerning experimental, theoretical, and numerical studies of non-feedback methods is frankly unapproachable because of its volume in a review of the present type. Therefore, only pioneering key work (from the author's viewpoint) is mentioned in the following. The effectiveness of periodic parametric excitations in suppressing chaos was shown by Alekseev & Loskutov (1987). Hübler & Lüscher (1989) discussed how a nonlinear oscillator can be driven toward a given target dynamics by means of resonant excitations. Braiman & Goldhirsch (1991) provided numerical evidence to show the possibility of inhibiting chaos by an additional periodic coordinate-independent excitation. Salerno (1991) showed, by the analysis of a phase-locked map, the possibility of suppressing chaos in long biharmonically driven Josephson junctions. Chacón & Díaz Bejarano (1993) discussed a new way to reduce or suppress steady chaotic states, by only altering the geometrical shape of weak periodic perturbations. Kivshar *et al*. (1994) showed analytically and numerically that the suppression of chaos may be effectively achieved by applying a high-frequency parametric force to a chaotic dynamical system. Experimental control of chaos by weak periodic excitations has been demonstrated in many diverse systems, including magnetoelastic systems (Ditto *et al.* 1990), ferromagnetic systems (Azevedo & Rezende 1991), electronic systems (Hunt 1991), laser systems (Roy *et al.* 1992; Meucci *et al*. 1994; Chizhevsky & Corbalán 1996; Uchida *et al.* 1998), chemical reactions (Petrov *et al.* 1993; Alonso *et al.* 2003), neurological systems (Schiff *et al.* 1994), and plasma systems (Ding *et al.* 1994).
This paper summarizes some main results concerning the application of Melnikov's method (Melnikov 1963; Arnold 1964; Guckenheimer & Holmes 1983; Wiggings 1990) to the problem of control of chaos in low-dimensional, non-autonomous and dissipative, oscillator systems by small-amplitude harmonic perturbations. Specifically, the class of systems considered is described by the differential equation$$\overset{..}{x}+\frac{dU(x)}{dx}=-d(x,\overset{.}{x})+p_{c}(x,\overset{.}{x})F_{c}(t)+p_{s}\left( x,\overset{.}{x}\right) F_{s}(t), \tag{1.1}$$where $U(x)$ is a nonlinear potential, $-d(x,\overset{.}{x})$ is a generic dissipative force which may include constant forces and time-delay terms, $p_{c}(x,\overset{.}{x})F_{c}(t)$ is a chaos-inducing excitation, and $p_{s}\left( x,\overset{.}{x}\right) F_{s}(t)$ is an as yet undetermined suitable chaos-controlling excitation, with $F_{c}(t)$, $F_{s}(t)$ being harmonic functions of initial phases $0,\Theta $, and frequencies $\omega
,\Omega $, respectively. It is worth mentioning that Melnikov's method imposes on (1.1) some additional limitations: the excitation, time-delay, and dissipation terms are weak perturbations of the underlying conservative system $\overset{..}{x}+dU(x)/dx=0$ which has a separatrix. The original work of Melnikov (1963) was generalized by Arnold (1964) to a particular instance of a time-periodic Hamiltonian perturbation of a two-degree-of-freedom integrable Hamiltonian system. Fifteen years later, Holmes (1979) was the first to apply Melnikov's method (to a damped forced two-well Duffing oscillator) in the west. From then on the method began to be popular. Chow *et al.* (1980) rediscovered Melnikov's results using alternative methods and emphasized that homoclinic and subharmonic bifurcations are closely related. Through the 1980s a great variety of extensions and generalizations of Melnikov's approach were developed (Greenspan 1981; Holmes & Marsden 1982; Lerman & Umanski 1984; Greundler 1985; Salam 1987; Schecter 1987; Wiggings 1987). The interested reader is referred to the books by Lichtenberg & Lieberman (1983), Guckenheimer & Holmes (1983), Wiggings (1988), and Arrowsmith & Place (1990) for more details and references. The work of Cai *et al.* (2002) provides the simplest extension of Melnikov's method to include perturbational time-delay terms.
The application of Melnikov's method to controlling chaos in low-dimensional systems by weak periodic perturbations began in about 1990. Indeed, Lima & Pettini (1990) provided a heuristic argument to extend the idea that parametric perturbations can modify the stability of hyperbolic or elliptic fixed points, in the phase space of linear systems, to the case of nonlinear systems, and hence that parametric perturbations could also provide a means to reduce or suppress chaos in nonlinear systems. They used for the first time the Melnikov method to analytically demonstrate this conjecture in the case of a damped driven two-well Duffing oscillator subjected to a chaos-suppressing parametric excitation. However, their insufficient analysis of the corresponding Melnikov function led them into gross errors in their final results and conclusions. Specifically, they failed both theoretically to demonstrate the sensitivity of the suppression scenario to the initial phase of the chaos-suppressing excitation and to find it numerically. They also failed theoretically to predict the suppression of chaos in the case of subharmonic resonances (between the chaos-inducing and chaos-suppressing excitations) higher than the main one. Although a part of their erroneous analysis of the Melnikov function originated from a mistake in its calculation (Cuadros & Chacón 1993; Lima & Pettini 1993), its main weakness was in not providing a correct necessary and *sufficient* condition for the Melnikov function to always have the same sign (i.e., for the frustration of homoclinic bifurcations). For the two-well Duffing oscillator that they considered, such a correct necessary and sufficient condition was first deduced for the general case of subharmonic resonances by Chacón (1995*a*), where the extremely important role of the initial phase (of the chaos-suppressing excitation) on the suppression scenario was demonstrated theoretically. Cicogna & Fronzoni (1990) studied the suppression of chaos in the Josephson-junction model$$\overset{..}{\phi }+\left[ 1+\xi \cos \left( \Omega t+\theta \right) \right]
\sin \phi =-\delta \overset{.}{\phi }+\gamma \cos \left( \omega t\right) ,
\tag{1.2}$$where the parametric excitation $\xi \cos \left( \Omega t+\theta \right)
\sin \phi $ is the chaos-suppressing excitation, for the single case of the main resonance $\Omega =\omega $ by using Melnikov's method. Their insufficient analysis of the Melnikov function (in particular, that of the role played by the initial phase $\theta $) led them also into gross mistakes in their final conclusions. On the contrary, it was demonstrated by Chacón (1995*b*) that the effect of the above parametric excitation in (1.2) for the general case of subharmonic resonances ($\Omega =p\omega $, $p$ an integer) is to suppress the chaotic behavior when a *suitable* initial phase is used and only for *certain* ranges of its amplitude. It was also shown (Chacón 1995*b*) for the first time that such suitable initial phases are compatible with the surviving natural symmetry under the parametric excitation. It was also conjectured (Chacón 1998) that such maximum survival of the symmetries of solutions from a broad and relevant class of systems, subjected both to primary chaos-inducing and chaos-suppressing excitations, corresponds to the optimal choice of the suppressory parameters; specifically, to particular values of the initial phase differences between the two types of excitations for which the amplitude range of the suppressory excitation is maximum. Rajasekar (1993) applied Melnikov's method to study the suppression of chaos in the Duffing-van der Pol oscillator$$\overset{..}{x}-\alpha ^{2}x+\beta x^{3}=-p\left( 1-x^{2}\right) \overset{.}{x}+f\cos \left( \omega t\right) +\eta \cos \left( \Omega t+\Omega \phi
\right) , \tag{1.3}$$where the additional forcing $\eta \cos \left( \Omega t+\Omega \phi \right) $ is the chaos-suppressing excitation, for the single case of the main resonance $\Omega =\omega $. He pointed out the relevant role of the initial phase (of the chaos-suppressing excitation) on the suppression scenario for the first time, and he also deduced the analytical expression of the boundaries of the regions in the $\eta -\phi $ phase plane where homoclinic chaos is inhibited. A generalization of Rajasekar's approach concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos is discussed in the work of Chacón (2002). There, general analytical expressions are derived from the analysis of generic Melnikov functions providing the boundaries of the regions as well as the enclosed area in the amplitude-initial phase plane of the chaos-suppressing excitation where homoclinic/heteroclinic chaos is inhibited. Also, a criterion based on the aforementioned area was deduced and shown to be useful in choosing the most suitable of the possible chaos-suppressing excitations. Cicogna & Fronzoni (1993) analyzed the Melnikov function associated with the family of systems $$\overset{..}{x}=f(x)-\delta \overset{.}{x}+\gamma \cos \left( \omega
t\right) +\varepsilon g\left( x\right) \cos \left( \Omega t+\theta \right) ,
\tag{1.4}$$where $\varepsilon g\left( x\right) \cos \left( \Omega t+\theta \right) $ is the chaos-suppressing excitation, for the single case of the main resonance $\Omega =\omega $. They deduced both the suitable suppressory values of the initial phase $\theta $ and the associated chaotic threshold function $\left( \gamma /\delta \right) _{th}$ when the chaos-suppressing excitation acts on the system. General results (Chacón 1999) concerning suppression of homoclinic/heteroclinic chaos were derived on the basis of Melnikov's for the family (1.1) for the general case of subharmonic resonance ($\Omega =p\omega $, $p$ an integer). There, a generic analytical expression was deduced for the maximum width of the intervals of the initial phase $\Theta $ for which homoclinic/heteroclinic bifurcations can be frustrated. It was also demonstrated that $\left\{ 0,\pi /2,\pi ,3\pi /2\right\} $ are, in general, the only optimal values of such initial phase, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation. The work of Chacón (2001*a*) presents general results concerning *enhancement* or maintenance of chaos for the family (1.1), where the connection with the results on chaos suppression was discussed in a general setting. It was also demonstrated that, in general, a second harmonic excitation can reliably play an enhancer or inhibitor role by solely adjusting its initial phase. The work of Chacón (2001*b*) provides a preliminary Melnikov-method-based approach concerning suppression of chaos by a chaos-suppressing excitation which satisfies an *ultrasubharmonic* resonance condition with the chaos-inducing excitation. This approach was further applied to the problem of the inhibition of chaotic escape from a potential well by *incommensurate* escape-suppressing excitations (Chacón & Martínez 2002).
Basic theoretical approach
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To illustrate the theoretical approach with a paradigmatic example, consider a single Josephson junction subjected to a nonlinear dissipative term and driven by two harmonic excitations (Chacón *et al.* 2001)$$\overset{..}{x}+\sin x=-\alpha (1+\gamma \cos x)\overset{.}{x}+F\sin \left(
\omega t\right) +\beta F\sin \left( \Omega t+\Theta \right) , \tag{2.1}$$where $x$ and time are dimensionless variables, and $\overset{.}{x}$ is proportional to the difference of potential between the two superconductors. It is also assumed that the terms of dissipation and excitation are regarded as weak perturbations and $\beta F\sin \left( \Omega t+\Psi \right) $ is the chaos-suppressing excitation. The nonlinear dissipative term appears in the study of a single Josephson junction when the conditions are such that the interference effects between the pair and quasiparticle currents should be taken into account (Barone & Paterno 1982). The application of the Melnikov method to (2.1) yields the Melnikov function$$M^{\pm }\left( t_{0}\right) =-C\pm A\sin \left( \omega t_{0}\right) \pm
B\sin \left( \Omega t_{0}+\Theta \right) , \tag{2.2}$$with$$\begin{aligned}
C &\equiv &8\alpha \left( 1+\gamma /3\right) , \notag \\
A &\equiv &2\pi F\func{sech}\left( \frac{\pi \omega }{2}\right) , \notag \\
B &\equiv &2\pi \beta \func{sech}\left( \frac{\pi \Omega }{2}\right) .
\TCItag{2.3}\end{aligned}$$Turning to the general case (1.1), let us assume that such a family of systems satisfies the requirements of the Melnikov method. Then, the application of the method to (1.1) provides the generic Melnikov function$$M_{h,h^{\prime }}^{\pm }\left( \tau _{0}\right) =D\pm Ahar\left( \omega \tau
_{0}\right) +Bhar^{\prime }\left( \Omega \tau _{0}+\Psi _{h,h^{\prime
}}^{\pm }\right) , \tag{2.4}$$where $\limfunc{har}\left( \tau \right) $ means indistinctly $\sin \left(
\tau \right) $ or $\cos \left( \tau \right) $, and $A$ is a non-negative function, while $D,B$ can be non-negative or negative functions, depending upon the respective parameters for each specific system. In particular, $D$ contains the effect of the damping, time-delay terms, and constant forces. In the absence of time-delay terms and constant drivings, $D<0$, while one has the three cases $D\gtrless 0$ when a constant driving and a time-delay term act on the system. Also, $A$ and $B$ contain the effect of the chaos-inducing and chaos-controlling excitations, respectively. Note that changing the sign of $B$ is equivalent to having a fixed shift of the initial phase: $B\rightarrow -B\Longleftrightarrow \Psi _{h,h^{\prime
}}^{\pm }\rightarrow $ $\Psi _{h,h^{\prime }}^{\pm }\pm \pi $, where the two signs before $\pi $ apply to each of the sign superscripts of $\Psi $. Therefore, $B$ will be considered (for example) as a positive function in the following. As phase and initial time $\tau _{0}$ are not fixed, one may study the simple zeros of $M_{h,h^{\prime }}^{\pm }\left( \tau _{0}\right) $ by choosing quite freely the trigonometric functions in (2.4). Therefore, consider, for instant, the Josephson junction given by (2.1). It is worth noting that the Melnikov functions $M_{h,h^{\prime }}^{\pm }\left( \tau
_{0}\right) $, $M^{\pm }\left( t_{0}\right) $ (cf. (2.4) and (2.2), respectively) are connected by *linear* relationships which are known for each specific system (1.1):$$\begin{aligned}
\tau _{0} &=&\tau _{0}\left( t_{0},\omega \right) , \TCItag{2.5} \\
\Psi _{h,h^{\prime }}^{\pm } &=&\Psi _{h,h^{\prime }}^{\pm }\left( \Theta ,\frac{\Omega }{\omega }\right) . \notag\end{aligned}$$Therefore, the control theorems associated with any Melnikov function $M_{h,h^{\prime }}^{\pm }\left( \tau _{0}\right) $ can be straightforwardly obtained from those associated with $M^{\pm }\left( t_{0}\right) $ (Chacón 1999).
Suppression of chaos
--------------------
As is well-known, the Melnikov method provides estimates in parameter space for the appearance of homoclinic (and heteroclinic) bifurcations, and hence for transient chaos. This means that in most of cases only *necessary* conditions for steady chaos (strange chaotic attractor) are obtained from the method. Therefore, one may always get sufficient conditions for the inhibition of even transient chaos (frustration of homoclinic/heteroclinic bifurcation) and, *a fortiori*, for the inhibition of the steady chaos that ultimately arises from such a homoclinic/heteroclinic bifurcation. This is the principal foundation of the utility of Melnikov method in predicting the suppression of (steady) chaos when a homoclinic/heteroclinic bifurcation occurs prior to its emergence. For the Josephson junction (2.1) one has the following theorem (Chacón *et al.* 2001):
Let $\Omega =p\omega $, $p$ an integer, such that, for $M^{+}\left(
t_{0}\right) $ $\left( M^{-}\left( t_{0}\right) \right) $, $p=\frac{4m-1-2\Theta /\pi }{4n+1}$ $\left( p=\frac{4m+1-2\Theta /\pi }{4n-1}\right) $ is satisfied for some integers $m$ and $n$. Then $M^{\pm }\left(
t_{0}\right) $ always has the same sign, specifically $M^{\pm }\left(
t_{0}\right) <0$, if and only if the following condition is satisfied:$$\begin{aligned}
\beta _{\min } &<&\beta \leqslant \beta _{\max }, \notag \\
\beta _{\min } &\equiv &\left( 1-\frac{C}{A}\right) R, \notag \\
\beta _{\max } &\equiv &\frac{R}{p^{2}}, \notag \\
R &\equiv &\frac{\cosh \left( \pi \Omega /2\right) }{\cosh \left( \pi \omega
/2\right) }. \TCItag{2.6}\end{aligned}$$
Now, the following remarks are in order.
First, one can test the suppression theorem theoretically by considering the limiting Hamiltonian case $\left( \alpha =0\right) $. Notice that, in the absence of dissipation, (2.6) must be rewritten as $\beta _{\min }\leqslant
\beta \leqslant \beta _{\max }$, $\beta _{\min }\equiv R$, $\beta _{\max
}\equiv R/p^{2}$, since $\beta _{\min }$ cannot now be zero. Thus, one obtains (Chacón *et al.* 2001) $\Omega =\omega $, $\Theta =\pi $, and $\beta =1$ as a necessary and sufficient condition for suppressing stochasticity. (This result can be trivially obtained, to *first perturbative order*, from (2.2), (2.3) with $\alpha =0$, i.e., having $M^{\pm }\left( t_{0}\right) =0$ for all $t_{0}$.
Second, the lower threshold for the chaos-suppressing amplitude, $\beta
_{\min }$, takes into account the *strength* of the initial chaotic state through the factor $\left( 1-C/A\right) $, since one usually finds that the corresponding maximal Lyapunov exponent $\lambda ^{+}$ increases as the ratio $C/A$ decreases over a certain range of parameters. Therefore, for fixed-chaos inducing and chaos-suppressing frequencies (and hence $R$ fixed), one would expect that $\beta _{\min }$ will increase as $\lambda
^{+} $ is increased. Note that the corresponding upper threshold, $\beta
_{\max }$, does not verify this important property, which is because $\beta
_{\max }$ arises from a *necessary* condition for the necessary condition yielding $\beta _{\min }$ to be also a sufficient condition. This means that $\beta _{\max }$ slightly underestimates the upper threshold for the chaos-suppressing amplitude, as is numerically and experimentally observed in different instances. It is worth noting that this remark holds for any Melnikov function (2.4).
Third, the asymmetry between the upper and lower homoclinic orbits (cf. (2.2), (2.3)) gives rise to two distinct sets of *optimal* initial phases that are suitable for suppressing chaos. The optimal suppressory values of $\Theta $ (hereafter denoted as $\Theta _{opt}$) are those values allowing the widest amplitude ranges for the chaos-suppressing excitation (the use of the adjective is justified below in the discussion of the suitable intervals of initial phase difference for taming chaos). Indeed, Theorem 1 requires having $\Theta =\Theta _{opt}\equiv \pi ,\pi /2,0,3\pi /2$ $\left( \pi ,3\pi /2,0,\pi /2\right) $ for $p=4n-3,4n-2,4n-1,4n$ $\left(
n=1,2,...\right) $, respectively, in order to inhibit chaos when one considers orbits initiated near the upper (lower) homoclinic orbit. These distinct values are those compatible with the *surviving natural symmetry* under the additional forcing. Indeed, the dissipative, harmonically driven Josephson junction ($\beta =0$) is invariant under the transformation $$\begin{aligned}
x &\rightarrow &-x, \notag \\
t &\rightarrow &t+\frac{\left( 2n+1\right) \pi }{\omega }, \TCItag{2.7}\end{aligned}$$where $n$ is an integer, i.e., if $\left[ x(t),\overset{.}{x}(t)\right] $ is a solution of (2.1) with $\beta =0$, then so is $\left[ -x(t+\left(
2n+1\right) \pi /\omega ),-\overset{.}{x}(t+\left( 2n+1\right) \pi /\omega )\right] $. This pair of solutions may be essentially the same in the sense that they may differ by an integer number of complete cycles, i.e., $$x(t)=-x\left[ t+\left( 2n+1\right) \pi /\omega \right] +2\pi l, \tag{2.8}$$with $l$ an integer, and they are termed symmetric. Otherwise, the time-shifting and sign reversal procedure yields a different solution, and the two solutions are termed broken-symmetric. When $\beta >0$ and $\Theta $ is arbitrary the aforementioned natural symmetry is generally broken. The reason for that breaking is $$\sin \left( \Omega t+\Theta \right) \neq \sin \left[ \Omega t+\Theta +\left(
2n+1\right) \pi \Omega /\omega \right] , \tag{2.9}$$for arbitrary $\omega ,\Omega ,$ and $\Theta $. Assuming a resonance condition $\Omega =p\omega $, the survival of the above symmetry implies$$\sin \left( p\omega t+\Theta \right) =\left( -1\right) ^{p+1}\sin \left(
p\omega t+\Theta \right) . \tag{2.10}$$Obviously, this is only the case for $p$ an *odd* integer. For $p$ an even integer, one has the new transformation \[$x\rightarrow -x$, $t\rightarrow t+\left( 2n+1\right) \pi /\omega $, $\Theta \rightarrow \Theta
\pm \pi $\]. In other words, if $\left[ x(t),\overset{.}{x}(t)\right] $ is a solution for a value $\Theta $, then so is $\left[ -x(t+\left( 2n+1\right)
\pi /\omega ),-\overset{.}{x}(t+\left( 2n+1\right) \pi /\omega )\right] $ for $\Theta \pm \pi $. Thus, this explains the origin of the differences between the corresponding (at the same resonance order) allowed $\Theta
_{opt}$ values for the two homoclinic orbits. Similar results have been found for the damped, driven pendulum mounted on a vertically oscillating point of suspension (Chacón 1995*b*). Therefore, this maximum symmetry principle appears to be the common background in the mechanism of regularization by the application of resonant excitations.
Fourth, the width of the allowed interval $\left] \beta _{\min },\beta
_{\max }\right] $ for regularization is $$\Delta \beta \equiv \beta _{\max }-\beta _{\min }=\left[ \frac{C}{A}-\frac{p^{2}-1}{p^{2}}\right] R, \tag{2.11}$$with $R$ given by (2.6). Since $R$ is a positive function, there always exists a *maximum resonance order* $p_{\max }$ for suppression of homoclinic (and heteroclinic) chaos, for each fixed initial chaotic state (i.e., $C/A$ fixed), which is $$p_{\max }\equiv \left[ \left( 1-\frac{C}{A}\right) ^{-1/2}\right] ,
\tag{2.12}$$where the brackets indicate integer part. From (2.12), one sees that $p_{\max }$ increases as the ratio $C/A$ is increased, which would associated with the decrease of the corresponding maximal Lyapunov exponent over a certain range of parameters. For a given set of parameters satisfying the above theorem's hypothesis, as the resonance order $p$ is increased the allowed interval $\left] \beta _{\min },\beta _{\max }\right] $ shrinks quickly for *low* frequencies. This means that initial chaotic states cannot necessarily be regularized to periodic attractors of arbitrary *long* period, since numerical experiments indicate that the regularized responses are typically a period-1 attractor for $p=1$ and a period-2 attractor for $p=2$. On the other hand, the asymptotic behavior $\Delta \beta \left( \omega \rightarrow
\infty \right) =\infty $ (the remaining parameters being held constant) means that chaotic motion is not possible in this limit, as expected.
Fifth, to establish the suppression theorem corresponding to any Melnikov function (2.4), it is enough to transform $M_{h,h^{\prime }}^{\pm }\left(
\tau _{0}\right) $ into the form given by (2.2). Therefore, taking into account (2.5) and the aforementioned $\Theta _{opt}$ values, one finds that in general there exist at most four suitable optimal values for the suppressory initial phase difference between the two (commensurate: $\Omega
=p\omega $) excitations: $0,\pi /2,\pi ,3\pi /2$.
Sixth, It has been stated above that the suitable values of the initial phase difference (between the two excitations involved) given by Theorem 1 are optimal, in the sense that they allow the widest amplitude ranges for the chaos-suppressing excitation. One therefore could expect reliable control of the dynamics over certain suitable phase difference intervals, which would be centered on such optimal values, although this would imply a reduction of the respective amplitude ranges. It has been deduced (Chacón 1999; Chacón *et al.* 2001) that there always exists a *maximum-range* interval $$\left[ \Theta _{opt}-\Delta \Theta _{\max },\Theta _{opt}+\Delta \Theta
_{\max }\right] , \tag{2.13}$$of permitted initial phase differences for homoclinic/heteroclinic chaos inhibition, where$$\Delta \Theta _{\max }\equiv \arcsin \left( \frac{C}{A}\right) . \tag{2.14}$$For each value of $\Theta $ belonging to this interval there exists a reduced interval (with regard to the limiting case where the only suitable values of $\Theta $ are $\Theta _{opt}$) of amplitudes of the chaos-suppressing excitation which is$$\begin{aligned}
\beta _{\min }\left( \Theta =\Theta _{opt}\pm \Delta \Theta \right) &<&\beta
\leqslant \beta _{\max }\left( \Theta =\Theta _{opt}\pm \Delta \Theta
\right) , \notag \\
\beta _{\min }\left( \Theta =\Theta _{opt}\pm \Delta \Theta \right) &\equiv
&\left( 1-\frac{C}{A}\right) R\sec \left( \Delta \Theta \right) , \notag \\
\beta _{\max }\left( \Theta =\Theta _{opt}\pm \Delta \Theta \right) &\equiv &\frac{R\cos \left( \Delta \Theta \right) }{p^{2}}, \TCItag{2.15}\end{aligned}$$where $R$ is given by (2.6) and $0\leqslant \Delta \Theta \leqslant \Delta
\Theta _{\max }$. Thus, the width of the range for the chaos-suppressing amplitude is $$\Delta \beta \left( \Theta =\Theta _{opt}\pm \Delta \Theta \right) =\left\{
\frac{\cos \left( \Delta \Theta \right) }{p^{2}}-\frac{1-C/A}{\cos \left(
\Delta \Theta \right) }\right\} R, \tag{2.16}$$i.e., for fixed $C,A$ and $\Delta \Theta $ there always exists a *maximum resonance order* $p_{\max }$ for homoclinic chaos suppression which is$$p_{\max }\left( \Theta =\Theta _{opt}\pm \Delta \Theta \right) =\left[ \frac{\cos \left( \Delta \Theta \right) }{\sqrt{1-\sin \left( \Delta \Theta _{\max
}\right) }}\right] , \tag{2.17}$$where the brackets indicate integer part. Also, one can put $$\Delta \Theta _{\max }\simeq \frac{C}{A}+O\left[ \left( \frac{C}{A}\right)
^{3}\right] . \tag{2.18}$$Thus, one can use a *linear* approximation for $\Delta \Theta _{\max
} $ suitable for chaotic motions arising away from the limiting case of tangency between the stable and unstable manifolds $\left( C/A\ll 1\right) $. It is worth mentioning that the last inequality is usually associated with the observation of steady chaos (strange chaotic attractor).
Enhancement of chaos
--------------------
It has been mentioned above that the mechanism for suppressing homoclinic (and heteroclinic) chaos is the frustration of a homoclinic/heteroclinic bifurcation, which prevents the appearance of horseshoes in the dynamics. Chacón (2001*a*) showed that the enhancement of the initial chaos is achieved by moving the system from the homoclinic tangency condition *even more* than in the initial situation with no second periodic excitation. Thus, enhancement of chaos can mean increasing the duration of a chaotic transient, passing from transient to steady chaos, or increasing the maximal Lyapunov exponent. Consider again that the family of systems modeled by (1.1) satisfies the requirements of the Melnikov method. Similarly to the preceding discussion of the suppression of chaos, one can assume any particular form of the Melnikov function (2.4) to discuss the enhancement of chaos. Therefore, consider, for instance, the following nonlinearly damped, biharmonically driven, two-well Duffing oscillator:$$\overset{..}{x}-x+\beta x^{3}=-\delta \overset{.}{x}\left\vert \overset{.}{x}\right\vert ^{n-1}+F\cos \left( \omega t\right) -\eta \beta x^{3}\cos \left(
\Omega t+\Theta \right) , \tag{2.19}$$where $\eta ,\Omega ,$ and $\Theta $ are the normalized amplitude factor, frequency, and initial phase, respectively, of the chaos-controlling parametric excitation $\left( 0<\eta \ll 1\right) $, and $\beta ,\delta
,n,F, $ and $\omega $ are the normalized parameters of the potential coefficient, damping coefficient, damping exponent, chaos-inducing amplitude, and chaos-inducing frequency, respectively $\left( 0<\delta ,F\ll
1,\beta >0,n=1,2,...\right) $. The application of the Melnikov method to (2.19) yields the Melnikov function$$M^{\pm }(t_{0})=D\pm A\sin \left( \omega t_{0}\right) -C\sin \left( \Omega
t+\Theta \right) , \tag{2.20}$$with$$\begin{aligned}
D &\equiv &-\delta \left( \frac{2}{\beta }\right) ^{\left( n+1\right)
/2}B\left( \frac{n+2}{2},\frac{n+1}{2}\right) , \notag \\
A &\equiv &\left( \frac{2}{\beta }\right) ^{1/2}\pi \omega F\func{sech}\left( \frac{\pi \omega }{2}\right) , \notag \\
C &\equiv &\frac{\pi \eta }{6\beta }\left( \Omega ^{4}+4\Omega ^{2}\right)
\func{csch}\left( \frac{\pi \Omega }{2}\right) , \TCItag{2.21}\end{aligned}$$where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying integrable two-well Duffing oscillator ($\delta
=F=\eta =0$), and where $B\left( m,n\right) $ is the Euler beta function. It has been demonstrated (Chacón 2001*a*) that, in general, a second harmonic excitation can reliably play an enhancer or inhibitor role *solely* from adjusting its initial phase. The Melnikov function $M^{+}(t_{0}) $ will be used here to illustrate the approach to the enhancement of chaos. Indeed, consider that, in the absence of any second parametric excitation $\left( C=0\right) $, the associated Melnikov function $M_{0}^{+}(t_{0})=-\left\vert D\right\vert +A\sin \left( \omega t_{0}\right)
$ changes sign at some $t_{0}$, i.e., $\left\vert D\right\vert \leqslant A$. If one now lets the second excitation act on the system such that $C\leqslant A-\left\vert D\right\vert $, this relationship represents a sufficient condition for $M^{+}(t_{0})$ to change sign at some $t_{0}$. Thus, a necessary condition for $M^{+}(t_{0})$ to always have the same sign $\left( M^{+}(t_{0})<0\right) $ is $C>A-\left\vert D\right\vert \equiv
C_{\min }$. It was above mentioned (Chacón 1999) that a sufficient condition for $C>C_{\min }$ to also be a sufficient condition for inhibiting chaos is $\Omega =p\omega $ (subharmonic resonance condition), $C\leqslant
C_{\max }\equiv A/p^{2}$, $p$ an integer, and that $M_{0}^{+}(t_{0})$ and $-C_{\min ,\max }\sin \left( \Omega t_{0}+\Theta \right) $ are *in opposition*. This condition yields the optimal suppressory values $\Theta
_{opt}^{\sup }\equiv \Theta _{opt}$. It was demonstrated (Chacón 2001*a*) that imposing $M_{0}^{+}(t_{0})$ to be *in phase* with $-C_{\min ,\max }\sin \left( \Omega t_{0}+\Theta \right) $ is a *sufficient* condition for $M^{+}(t_{0})$ change sign at some $t_{0}$. This condition provides the optimal enhancer values of the initial phase, $\Theta
_{opt}^{enh}$, in the sense that $M^{+}(t_{0})$ presents its highest maximum at $\Theta _{opt}^{enh}$, i.e., one obtains the maximum gap from the homoclinic tangency condition. Now, the following remarks are in order.
First, for a given homoclinic orbit forming (part of) a separatrix, one has in general (i.e., for any Melnikov function (2.4)) that $$\left\vert \Theta _{opt}^{\sup }-\Theta _{opt}^{enh}\right\vert =\pi ,
\tag{2.22}$$for each resonance order.
Second, for $C=C_{\min }$ there always exists a *maximum-range* interval$$\left[ \Theta _{opt}^{enh}-\Delta \Theta _{\max }^{enh}\left( C=C_{\min
}\right) ,\Theta _{opt}^{enh}+\Delta \Theta _{\max }^{enh}\left( C=C_{\min
}\right) \right] \tag{2.23}$$of permitted initial phases for enhancement of chaos in the sense that, for values of $\Theta $ belonging to that interval, the maxima of $M^{+}(t_{0})$ are higher than those of $M_{0}^{+}(t_{0})$. Similarly, for $C=C_{\max }$ there always exists a *different* maximum-range interval $$\left[ \Theta _{opt}^{enh}-\Delta \Theta _{\max }^{enh}\left( C=C_{\max
}\right) ,\Theta _{opt}^{enh}+\Delta \Theta _{\max }^{enh}\left( C=C_{\max
}\right) \right] \tag{2.24}$$of allowed initial phases for enhancement of chaos, and also$$\Delta \Theta _{\max }^{enh}\left( C=C_{\max }\right) \geqslant \Delta
\Theta _{\max }^{enh}\left( C=C_{\min }\right) , \tag{2.25}$$which is a consequence of the dissipation. It must be emphasized that the definition of $\Theta _{opt}^{enh}$ is general; i.e., it refers to any resonance and any Melnikov function (2.4).
Third, for general separatrices, i.e., those formed by several homoclinic and (or) heteroclinic loops, the above scenario of control of chaos holds for *each* homoclinic (heteroclinic) orbit. However, it is common to find that the different homoclinic (heteroclinic) orbits of a given separatrix yield *distinct* $\Theta _{opt}^{enh}$ values. This is a consequence of the survival of the symmetries existing in the absence of the second excitation. Thus, the actual scenario is usually more complicated. For instance, let $\Theta _{opt,r}^{\sup }$, $\Theta _{opt,l}^{\sup }$ be the optimal values associated with the right and left homoclinic orbits, respectively, of a typical separatrix with a figure-of-eight loop, as in the two-well Duffing oscillator (2.19). One then obtains that the best chance for enhancing chaos should now be at $\Theta _{opt}^{enh}\sim \left( \Theta
_{opt,r}^{enh}-\Theta _{opt,l}^{enh}\right) /2\,\func{mod}\left( 2\pi
\right) .$ See Chacón (2001*a*) for more details.
Further developments
--------------------
The case of subharmonic resonance between the chaos-inducing and chaos-controlling frequencies has been briefly discussed above. However, a number of theoretical (Salerno 1991; Salerno & Samuelsen 1994), numerical (Braiman & Goldhirsch 1991), and experimental (Uchida *et al.* 1998) studies show that chaos can be reliably controlled by other non-subharmonic resonances. The work of Chacón (2001*b*) presents a Melnikov-method-based approach concerning reduction of homoclinic and heteroclinic instabilities for the family of systems (1.1) where the harmonic excitations verify an ultrasubharmonic resonance condition: $\Omega
/\omega =p/q$, $q>1\left( p\neq q\right) $, $p,q$ positive integers and $\Omega \left( \omega \right) $ the chaos-suppressing (inducing) frequency. Such general results can be used to approach the case of *incommensurate* chaos-suppressing excitations by means of a series of ever better rational approximations, which are the successive convergents of the infinite continued fraction associated with the irrational ratio $\Omega
/\omega $. This procedure has been much employed in characterizing strange non-chaotic attractors in quasiperiodically forced systems as well as in studying phase-locking phenomena in both Hamiltonian and dissipative systems. To illustrate the method one intentionally chooses the golden section $\Omega /\omega =\Phi \equiv \left( \sqrt{5}-1\right) /2$, since it is the irrational number which is the worst approximated by rational numbers (in the sense of the size of the denominator). As is well-known, the golden section can be approximated by the sequence of rational numbers $\left(
\Omega /\omega \right) _{i}=F_{i-1}/F_{i}$ where $F_{i}=1,1,2,3,5,8,...$, are the Fibonacci numbers such that $\lim_{i\rightarrow \infty }\left(
\Omega /\omega \right) _{i}=\Phi $. For each $\left( \Omega /\omega \right)
_{i}$ one replaces each quasiperiodically excited system $$\overset{..}{x}+\frac{dU(x)}{dx}=-d(x,\overset{.}{x})+p_{c}(x,\overset{.}{x})har(\omega t)+p_{s}\left( x,\overset{.}{x}\right) har^{\prime }\left( \Phi
\omega t+\Psi _{h,h^{\prime }}\right) \tag{2.26}$$by the respective periodically excited system$$\overset{..}{x}+\frac{dU(x)}{dx}=-d(x,\overset{.}{x})+p_{c}(x,\overset{.}{x})har(\omega t)+p_{s}\left( x,\overset{.}{x}\right) har^{\prime }\left( \frac{F_{i-1}}{F_{i}}\omega t+\Psi _{h,h^{\prime }}\right) \tag{2.27}$$giving a sequence of periodically excited systems whose associated frequencies satisfy an ultrasubharmonic resonance condition. The work of Chacón & Martínez (2002) applied this approach to the problem of the reduction of chaotic escape from a potential well using the simple model$$\overset{..}{x}=x-\beta x^{2}-\delta \overset{.}{x}+\gamma \sin \left(
\omega t\right) -\beta \eta x^{2}\sin \left( \Omega t+\Theta \right) ,
\tag{2.28}$$where $\beta \eta x^{2}\sin \left( \Omega t+\Theta \right) $ is the escape-suppressing excitation. They found that, for irrational escape-suppressing frequencies, the effective escape-reducing initial phases are found to lie close to the *accumulation* points of the set of suitable initial phases that are associated with the complete series of convergents up to the convergent giving the chosen rational approximation.
A Melnikov-method-based approach (Chacón 2002) was presented concerning the *relative effectiveness* of harmonic excitations in suppressing homoclinic (and heteroclinic) chaos of the family (1.1) for the main resonance between the chaos-inducing and chaos-suppressing excitations. A criterion based on the area in the suppressory amplitude/initial phase parameter plane, where suppression of homoclinic chaos is guaranteed, was deduced and shown to be useful in choosing the most suitable of the possible chaos-suppressing excitations. Additionally, the choice of the most suitable chaos-suppressing excitation was shown to exhibit *sensitivity* to the particular initial chaotic state.
The work of Chacón *et al.* (2003) presents general findings concerning control of chaos for the family $$\overset{..}{x}+\frac{dU(x)}{dx}=-d\left( x,\overset{.}{x}\right)
+\sum_{i=1}^{N}h_{ch,i}\left( x,\overset{.}{x}\right)
F_{ch,i}(t)+\sum_{j=1}^{M}h_{co,j}\left( x,\overset{.}{x}\right) F_{co,j}(t),
\tag{2.29}$$where $U(x)$ is a general potential, $-d\left( x,\overset{.}{x}\right) $ represents a generic dissipative force, $\sum_{i=1}^{N}h_{ch,i}\left( x,\overset{.}{x}\right) F_{ch,i}(t)$ is a general multiple chaos-inducing excitation, and $\sum_{j=1}^{M}h_{co,j}\left( x,\overset{.}{x}\right)
F_{co,j}(t)$ is an as yet undetermined suitable multiple chaos-controlling excitation, with $F_{ch,i}(t),\,F_{co,j}(t)$ being harmonic functions of common frequency $\omega $ and initial phases $0$ ($i=1,...,N$), $\varphi
_{j}$ ($j=1,...,M$). The effectiveness of this approach in suppressing spatio-temporal chaos of chains of identical chaotic coupled oscillators was demonstrated through the example of coupled Duffing oscillators, where coherent oscillations were achieved under *localized* control.
The work of Chacón *et al.* (2002) studied the robustness of the suppression of bidirectional chaotic escape of a harmonically driven oscillator from a quartic potential well by the application of weak parametric excitations. It was numerically shown that Melnikov-method-based theoretical predictions also work for harmonic escape-inducing excitations in the presence of external noise, and for chaotic-escape-inducing excitations having a sharp Fourier component with a sufficiently high power.
The method proposed in the work of Lenci & Rega (2003) consists of choosing the shape of external and/or parametric periodic excitations, which permits one avoid, in an optimal manner, a homoclinic bifurcation. They numerically investigated the effectiveness of the control method with respect to the basin erosion and escape phenomena of a perturbed Helmholtz oscillator.
Some applications
=================
Taming chaotic escape from a potential well
-------------------------------------------
The work of Chacón *et al.* (1996, 1997, 2001) and Balibrea *et al.* (1998) applies the above Melnikov-method-based approach to the problem of chaotic escape from a potential well. This is a general and ubiquitous phenomenon in science. Indeed, one finds it in very distinct contexts: the capsizing of a boat subjected to trains of regular waves (Thompson 1989), the stochastic escape of a trapped ion induced by a resonant laser field (Chacón & Cirac, 1995), and the escape of stars from a stellar system (Contopoulos *et al*. 1993) are some important examples. Remarkably, such complex escape phenomena can often be well described by a low-dimensional system of differential equations. The case considered by Chacón and coworkers is that where escape is induced by an external periodic excitation added to the model system, so that, before escape, chaotic transients of unpredictable duration (due to the fractal character of the basin boundary) are usually observed for orbits starting from chaotic generic phase space regions (such as those surrounding separatrices), in both dissipative and Hamiltonian systems. In particular, Chacón *et al.* (1996) studied the simplest model for a universal chaotic escape situation:$$\overset{..}{x}-x+\beta x^{2}=-\delta \overset{.}{x}+\gamma \sin \left(
\omega t\right) +\binom{-\beta \eta x^{2}\sin \left( \Omega t+\Theta \right)
}{\beta \eta x\sin \left( \Omega t+\Theta \right) }, \tag{3.1}$$where $\beta \eta x^{2}\sin \left( \Omega t+\Theta \right) $ and $\beta \eta
x\sin \left( \Omega t+\Theta \right) $ are the (independently considered) escape-suppressing parametric excitations. It was demonstrated that the parametric excitation of the linear (quadratic) term suppress chaotic escape more efficiently than that of the quadratic (linear) term for small (large) driving periods of the primary chaos-inducing excitation. Chacón *et al*. (1997) studied the inhibition of chaotic escape of a driven oscillator from the cubic potential well that typically models a metastable system close to a fold:$$\overset{..}{x}+x-\beta x^{2}=-\binom{\delta _{1}\overset{.}{x}}{\left(
\delta _{2}x^{2}+\delta _{3}x^{4}\right) \overset{.}{x}}+\gamma \cos \left(
\omega t\right) -\eta x\cos \left( \Omega t+\Theta \right) , \tag{3.2}$$ where $\delta _{1}\overset{.}{x}$ and $\left( \delta _{2}x^{2}+\delta
_{3}x^{4}\right) \overset{.}{x}$ are the (independently considered) linear and nonlinear damping terms, respectively. They demonstrated that the effectiveness of a parametric excitation at suppressing chaotic escape from such a cubic potential well diminishes as the system approaches a period-1 *parametric resonance*, and that, for linear damping, the parametric excitation inhibits chaotic escape more efficiently than for nonlinear damping. The role of a nonlinear damping term, proportional to the *n*th power of the velocity, on the escape-inhibition scenario is considered in the work of Chacón *et al.* (2001):$$\overset{..}{x}+x-x^{2}=-\delta \overset{.}{x}\left\vert \overset{.}{x}\right\vert ^{n-1}+\gamma \cos \left( \omega t\right) +\eta x^{2}\cos \left(
\Omega t+\Theta \right) , \tag{3.3}$$where $\eta x^{2}\cos \left( \Omega t+\Theta \right) $ is the escape-suppressing parametric excitation. In this case, the effectiveness of the parametric excitation of the quadratic potential well at inhibiting chaotic escape diminishes as the system approaches either a period-1 or a period-2 parametric resonance. Also, the effectiveness of the parametric excitation in the presence of the nonlinear dissipative force is less than for a linear dissipative force.
Taming chaotic solitons in Frenkel-Kontorova chains
---------------------------------------------------
Control of chaos in spatially extended systems is one of the most important and challenging problems in the field of nonlinear dynamics. Instances of possible applications include the stabilization of superconducting Josephson-junction arrays (Barone & Paterno 1982), periodic patterns in optical turbulence, and semiconductor laser arrays (Schöll 2001), to cite just a few. Martínez & Chacón (2004) presented a Melnikov-method-based general theoretical approach to control chaotic *solitons* in damped, noisy and driven Frenkel-Kontorova chains. Specifically, they studied the model$$\begin{aligned}
\overset{..}{x}_{j}+\frac{K}{2\pi }\sin \left( 2\pi x_{j}\right)
&=&x_{j+1}-2x_{j}+x_{j-1}-\alpha \overset{.}{x}_{j}+F\cos \left( \omega
t\right) \notag \\
&&+\beta F\cos \left( \Omega t+\varphi \right) +\xi \left( t\right) ,
\TCItag{3.4}\end{aligned}$$where $\beta F\cos \left( \Omega t+\varphi \right) $ is the chaos-suppressing excitation, and $\xi \left( t\right) $ is a bounded noise term. They obtained an effective equation of motion governing the dynamics of the soliton center of mass for which they deduced Melnikov-method-based predictions concerning the regions in the control parameter space where homoclinic bifurcations are frustrated. Numerical simulations indicated that such theoretical predictions can be reliably applied to the original Frenkel-Kontorova chains, even for the case of *localized* application of the soliton-taming excitations. It is worth mentioning that the same effectiveness of such a localized control in suppressing spatio-temporal chaos of chains of identical chaotic coupled oscillators was demonstrated through the example of coupled Duffing oscillators (Chacón *et al.* 2003).
Taming chaotic charged particles in the field of an electrostatic wave packet
-----------------------------------------------------------------------------
The interaction of charged particles with an electrostatic wave packet is a basic and challenging problem appearing in diverse fundamental fields such as astrophysics, plasma physics, and condensed matter physics. While the Hamiltonian approach to this problem is suitable in many physical contexts, the consideration of dissipative forces seems appropriate in diverse phenomena such as the stochastic heating in the dynamics of charged particles interacting with plasma oscillations. In any case, stochastic (chaotic) dynamics already appears (can appear) when the wave packet solely consists of two electrostatic plane waves. Such a non-regular behavior of the charged particles may yield undesirable effects on a number of technological devices such as the destruction of magnetic surfaces in tokamaks. Thus, apart from its general intrinsic interest, the problem of regularization of the dissipative dynamics of charged particles in an electrostatic wave packet by a small-amplitude uncorrelated wave (which is added to the initial wave packet) is especially relevant in plasma physics. Chacón (2004) considered the simplest model equation to examine this problem:$$\begin{aligned}
\overset{..}{x}+\delta \overset{.}{x} &=&-\frac{e}{m}\left[ E_{0}\sin \left(
k_{0}x-\omega _{0}t\right) +E_{c}\sin \left( k_{c}x-\omega _{c}t\right) \right] \notag \\
&&-\frac{e}{m}E_{s}\sin \left( k_{s}x-\omega _{s}t\right) , \TCItag{3.5}\end{aligned}$$where $E_{c}\sin \left( k_{c}x-\omega _{c}t\right) $ and $E_{s}\sin \left(
k_{s}x-\omega _{s}t\right) $ are the chaos-inducing and chaos-suppressing waves, respectively. In a reference frame moving along the main wave $E_{0}\sin \left( k_{0}x-\omega _{0}t\right) ,$ (3.5) transforms into a perturbed pendulum equation which is capable of being studied by means of Melnikov's method. Two suppressory mechanisms were identified: One mechanism requires chaos-inducing and chaos-suppressing waves to have both commensurate wavelengths and commensurate relative phase velocities, while the other allows chaos to be tamed when these quantities are incommensurate.
Conclusions and open problems
=============================
The present review summarizes some of the main results and applications of a preliminary theoretical approach to control chaos in dissipative, non-autonomous dynamical systems, capable of being studied by Melnikov's method, by means of periodic excitations. Diverse extensions and applications of the current theory remain to be developed. Among them:
\(i) To obtain the boundaries of the regularization regions in the control parameter space for the case of a general resonance (not just the main) between the involved excitations.
\(ii) To extend the theoretical approach to (some family of) multidimensional systems capable of being studied by (some generalized version of) Melnikov's method.
\(iii) To develop a multiharmonic control theory beyond the main resonance case.
\(iv) To extend the theoretical approach for the case of periodic excitations to the case of random excitations.
\(v) To obtain analytical approximations of the regularized responses for the deterministic case of a general resonance between the chaos-inducing and chaos-suppressing excitations.
\(vi) To extend the current theory described for harmonic excitations to the case of general periodic excitations (both chaos-inducing and chaos-controlling). In particular, the *waveform effect* should be taken into account in the control scenario.
\(vii) To extend the current theory to the case where the chaos-controlling excitation is a parametric excitation of the amplitude of the chaos-inducing excitation, as well as to the case where it is a parametric excitation of the frequency of the chaos-inducing excitation.
\(viii) To apply the current theory to ratchet systems to improve the directed energy transport.
\(ix) To apply the current theory to control chaotic population oscillations between two coupled Bose-Einstein condensates with time-dependent asymmetric potential and damping.
The author acknowledges financial support from Spanish MCyT and European Regional Development Fund (FEDER) program through BFM2002-00010 project.
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abstract: 'A method is described for using resonant x-ray scattering to separately quantify the charge (valence) modulation and the strain wave associated with a charge density wave. The essence of the method is a separation of the atomic form factor into a “raw" amplitude, $f_R(\omega)$, and a valence-dependent amplitude, $f_D(\omega)$, which in many cases may be determined independently from absorption measurements. The advantage of this separation is that the strain wave follows the average quantity $|f_R(\omega) + <\!\! v \!\!> f_D(\omega)|^2$ whereas the charge modulation follows only $|f_D(\omega)|^2$. This allows the two distinct modulations to be quantified separately. A scheme for characterizing a given CDW as Peierls-like or Wigner-like follows naturally. The method is illustrated for an idealized model of a one-dimensional chain.'
author:
- 'P. Abbamonte'
bibliography:
- 'oChainPRB-v2.bib'
title: 'Charge Modulations vs. Strain Waves in Resonant X-Ray Scattering'
---
Charge density waves (CDWs) are pervasive in condensed matter, arising in di- and trichalcogenides, conducting polymers, transition metal oxides etc. However they may form by a variety of mechanisms. The best understood is the Peierls mechanism in which a gap at the Fermi level opens, and a charge (valence) modulation forms, through the presence of a strain wave. The basis for the Peierls mechanism is the electron-phonon interaction which drives pairing between an electron and a hole with equal but opposite wave vector, ${\bf k}_F$, resulting in a charge modulation with total momentum $2{\bf k}_F$. A CDW may also be driven by electron interactions, however, an idealized example being the Wigner crystal (WC) ground state of a dilute electron gas, which is driven by direct Coulomb[@wigner]. Because of the close analogy between charge density waves and superconductivity [@gruner], and the occurance of both in many systems such as the dichalcogenides and the copper-oxides in which superconductivity is not clearly understood, it would be extremely useful to distinguish between Peierls-like and Wigner-like (i.e. interaction-driven) mechanisms in practice[@WCxtal].
The main distinction between a Peierls CDW and a WC is the presence of a lattice distortion in the former case that is not required in the latter. However in any real material even a WC must have at least some incidental lattice distortion if only because of electrostatics. Therefore, in practice, distinguishing between these two phenomena is a quantitative rather than a qualitative matter.
X-ray diffraction is sensitive to both charge modulations and strain waves. Scattering from both can be enhanced, through different resonant mechanisms, by tuning the x-ray photon energy to an atomic core transition, i.e. so-called resonant x-ray scattering (RXS). Here a general method is proposed for using RXS to distinguish between scattering from a charge modulation and scattering from a strain wave, even when the two coexist. This method allows the charge and strain amplitudes to be quantified independently. A system classifying a given charge density wave as either Peierls or Wigner in origin naturally arises from this separation. The method is demonstrated for the idealized case of a one-dimensional chain.
Separating the atomic form factor
=================================
In quantum electrodynamics, the cross section for elastic scattering of photons from a material is[@sakurai]
$$M_{\bf q} = \frac{e^2}{2mc^2} \, \rho_{\bf q} + \frac{e^2}{\hbar m^2c^2} \sum_n \frac{<0|\hat{{\bf p}} \cdot {\bf A}|n><n|\hat{{\bf p}} \cdot {\bf A}|0>}{\omega-\omega_n+i\gamma}$$
where $\rho_{\bf q}$ is a Fourier component of the total electron density, [**q**]{} is the momentum transfer, $\hat{{\bf p}}$ is the momentum operator, $\hat{{\bf A}}$ is the photon field operator (vector potential), and $\omega$ is the photon energy. The first of these two terms is equivalent to classical Thomson scattering, does not depend on $\omega$, and gives rise to the “normal" dispersion of optical constants in matter, i.e. in SI units $\chi_{\bf q} = - r_e \lambda^2 \rho_{\bf q}/4 \pi^2$, where $r_e$ is the classical electron radius and $\lambda=2\pi c / \omega$ [@henke]. The second, resonant term corresponds to scattering via virtual transitions between core and valence states, is highly dependent on $\omega$, and is the origin of anomalous dispersion[@james]. This term, sometimes referred to as the Kramers-Heisenberg formula, has been shown to provide direct sensitivity to valence band charge and spin ordering in condensed matter [@gibbs; @hannon; @kao; @castleton; @wilkins; @dhesi; @thomas; @schuessler; @usScience; @usSCO; @usLBCO], provided $\omega$ coincides with an edge threshold.
How can one distinguish between a charge moduation and a strain wave? One might be tempted to just compare the relative sizes of nonresonant and resonant scattering on the assumption that a lattice distortion, which translates all the electrons in an atom including those in the core, contributes mainly to the former and valence band effects mainly to the latter. Indeed, the Kramers-Heisenburg formula alone is normally used to analyze near-edge scattering data [@castleton; @wilkins; @thomas; @schuessler]. However one must realize two things. First, the valence modulation itself should scatter in the Thomson channel as well since it corresponds to a real, albeit small, charge modulation. Second, it was realized long ago[@parratt] that the resonant term also contributes to scattering far from the edges via virtual, off-shell processes. The optical contants of Cu metal, for example, exhibit no region of ‘normal’ dispersion from the infrared up to 100 keV[@parratt]. So the resonant term is omnipresent, a trait that is commonly exploited in crystallographic phasing methods[@hendrickson]. More to the point, both the valence and lattice modulations of a CDW scatter in both channels, and a more appropriate division of the cross section than Eq. 1 is needed.
Here it is proposed to divide the cross section instead into a ‘raw’ and ‘valence-dependent’ component. More specifically, assuming the form factor for an atom, $f$, is a function of both $\omega$ and the valence state, $v$, it is proposed to make the separation
$$f(\omega,v)=f_R(\omega) + v f_D(\omega).$$
Here $f_R$ is the ‘raw’ part of the atomic scattering factor and $f_D$ describes how it changes with valence[@arbitrary]. Like the full $f$, $f_R$ is dimensionless (i.e. ‘electrons’) and converges to $Z^*$ as $\omega \rightarrow \infty$[@henke]. $f_R$ contains both resonant and nonresonant processes but only those, such as edge jumps, that are independent of the detailed electronic structure of the atom.
$f_D$ has units of electrons/valence and can be thought of the scattering power of the valence modulation. Like $f_R$ this quantity contains both resonant and nonresonant processes, however the nonresonant contribution will be extremely small. One can expect $f_D$ to be large only for $\omega$ near the threshold of an absorption edge, where the intermediate states involve transitions directly into the valence band.
There are two advantages to the division in Eq. 2. First, as will be illustrated in the next section, scattering from a strain wave will follow only $|f(\omega,<\!\!v\!\!>)|^2$, where $<\!\!v\!\!>$ is the [*average*]{} atomic valence of the material, while scattering from the valence modulation follows $|f_D(\omega)|^2$. By measuring the photon energy dependence of a CDW reflection, provided $f_D$ and $f_R$ are known, one can determine separately the charge and strain amplitudes as well as their relative phase.
Second, and most importantly, in many cases $f_R$ and $f_D$ can be independently determined from doping-dependent x-ray absorption (XAS) data. More specifically, $f$ is related to the absorption coefficient [@henke] by
$$Im[n(\omega,<\!\!v\!\!>)] =-\frac{r_e N \lambda^2}{2 \pi V_c} Im \left [f_R(\omega) + <\!\!v\!\!> f_D(\omega) \right ]$$
where $n$ is the complex index of refraction, $N$ is the number of atoms in a unit cell, and $V_c$ is the unit cell volume. So, by measuring the absorption coefficient on two or more samples of different $<\!\!v\!\!>$, $Im[f_R(\omega)]$ and $Im[f_D(\omega)]$ can be determined by solving a system of equations at each value of $\omega$. The real parts can then by determined by Kramers-Kronig transform. Without having to appeal to a specific model, then, the charge and strain amplitudes can be determined. XAS acts as a reference against which resonant scattering measurements can be calibrated.
Use of this procedure rests on several assumptions. First, one must assume that the scattering processes are local and that an atomic form factor, $f$, is definable. This will be valid as long as $v_g \tau \ll \lambda/2$, where $v_g$ is the group velocity of the core electron-hole pair, $\tau$ is its radiative lifetime, and $\lambda$ is the x-ray wavelength. For measurements near edges with sharp white lines, such as the $L$ edges of the transition metals, one can expect this condition to be well satisfied. Next one must assume that changes in XAS spectra due to changes in $<\!\! v \!\!>$, determined by comparing samples with different chemical composition, indeed arise from valence effects and not extrinsic phenomena such as changing crystal structures etc. Next, one must assume that changes in $f$ are small i.e. are linear in $v$, which should be true as long as the CDW amplitude is not too large. It is [*not* ]{} necessary that the sample have a rigid or noninteracting band structure. In determining $f_R$ and $f_D$ from XAS one implicitly makes the assumption that the electronic structure of every point in the CDW corresponds to the [*average*]{} electronic structure in a sample with that valence. Finally, one assumes that x-ray absorption measurements are a good measure of the forward scattering amplitude, i.e. that there are not hidden loss processes such as photoelectron production. Violation of this last point has already been observed [@usLBCO] but is not a significant effect.
One-dimensional chain
=====================
To see how $f_R$ and $f_D$ enter a resonant scattering experiment consider the idealized case of a one-dimensional, monatomic chain, as shown in Fig. 1. Imposed upon this chain are a longitudinal strain wave, described by a set of displacements $u_n$, and a valence modulation, described by a set of valence states, $v_n$. For simplicity it is assumed that the modulations are sinusoidal, i.e.
$$u_n = u_0 \cos(k n a)$$
$$v_n = \, <\!\! v \!\!> + \, v_0 \cos(k n a + \phi)$$
where $u_n$ implies that the $n$th atom resides at position $r_n = na + u_n$, where $a$ is the average lattice parameter. $u_0$ and $v_0$ are the aplitudes of the strain and charge waves, respectively, $<\!\! v \!\!>$ is the average atomic valence, and $k$ is the wave vector of the CDW. The quantity $\phi$ allows for the fact that, while the two modulations have the same wave vector, there may be a phase difference between them.
The integrated intensity of a Bragg reflection is proportional to the square of the scattering amplitude, $\rho_q$, which is given by
$$\rho_q(\omega)=\frac{1}{N} \sum_{n=1}^N f(\omega,v_n) \; e^{i q u_n}$$
where $f(\omega,v_n)$ is as defined by eq. (2) and $N = 2\pi/ka$ is the number of sites in the supercell, i.e. the dimensionless CDW period [@comm]. $q$ is the momentum transfer along the chain and is restricted to discrete values $q=2\pi l/Na$, where $l$ is an integer. If one multiplies out the terms one arrives at four distinct contributions to the scattering amplitude.
$$\rho_q(\omega)=\rho_q^B(\omega) + \rho_q^v(\omega) + \rho_q^u(\omega) + \rho_q^{uv}(\omega)$$
The first term,
$$\rho_q^B(\omega) = \frac{1}{N} \; f(\omega,<\!\! v \!\!>) \; \sum_{n=1}^N e^{i q n a},$$
is the “Bragg" term and corresponds to resonant x-ray scattering off the average lattice. This quantity is independent of $u_0$ and $v_0$ and is nonzero whenever $q = 2\pi m/a$, where $m$ is an integer, i.e. at the Bragg points of the undistorted chain. This term always has exactly the value $f(\omega,<\!\! v \!\!>)$, which demonstrates that regular Bragg scattering has the energy dependence of $|f(\omega,<\!\! v \!\!>)|^2 = |f_R(\omega)+<\!\! v \!\!> f_D(\omega)|^2$, i.e. simply tracks the average scattering factor of the atomic lattice. $\rho^B_k$ is highly resonant but nonzero for all values of $\omega$.
The next term
$$\rho_q^v(\omega) = \frac{1}{N} \, v_0 \, f_D(\omega) \, \sum_{n=1}^N \cos(k n a + \phi) e^{i q n a }$$
is the “valence" term and corresponds to resonant x-ray scattering off the valence modulation. This quantity is nonzero only for $q=\pm k$ and has the value $\rho_k^v(\omega) = v_0 f_D(\omega) \exp(i\phi)/2$. This demonstrates that resonant scattering from the valence modulation tracks only $|f_D(\omega)|^2$ and is proportional to $v_0^2$. Since $f_D$ is significant only near an edge, one can expect to see scattering from a valence modulation only over a narrow energy range near threshold. This is the experimental signature of a valence modulation in resonant x-ray scattering.
The third term is the “strain" term and corresponds to scattering off the lattice distortion. Provided the size of the distortion is small, i.e. $\exp(iqr_n) = \exp(iqna) (1 + iqu_n)$, the strain term has the value
$$\rho_q^u(\omega) = \frac{1}{N} \, i \, q \, u_0 \, f(\omega,<\!\! v \!\!>) \, \sum_{n=1}^N \cos(k n a) e^{i q n a}.$$
This quantity is also nonzero only if $q=\pm k$ and reduces to $\rho_k^u(\omega) = i q u_0 f(\omega,<\!\! v \!\!>)/2$. Scattering from the strain wave is proportional to $(k u_0)^2$, as expected, and like the Bragg term follows $|f(\omega,<\!\! v \!\!>)|^2$ so is visible at all values of $\omega$. Evidently all structural scattering is alike in its adherence to the average $f(\omega,<\!\! v \!\!>)$. Unlike the Bragg term, however, the strain term occurs at the same $q$ as the valence scattering and and the two may coherently interfere. We will see that, if this interference is visible, it provides a means to determine the phase shift, $\phi$.
The final term is somewhat unexpected and apparently has not been addressed before. It is a mixed term corresponding to coherent displacement of the valence modulation. Written out, it has the form
$$\rho_q^{uv}(\omega) = \frac{1}{2N} i q v_0 u_0 f_D(\omega) \sum_{n=1}^N [ \cos(2 k n a + \phi) + cos(\phi) ] e^{i q r_n}$$
This quantity is nonzero only if $q= \pm 2k$ and, while it arises from both strain and valence scattering, it tracks only $f_D$. $\rho^{uv}$ is not a multiple-scattering effect; it is a sign that if both charge and strain modulations are present the total modulation is anharmonic. Observation of this term, i.e. by tuning the x-ray energy near threshold and scanning around $2k$, would be a strong validation of our approach. However this term is extremely small.
A generic edge
==============
The various scattering processes are best illustrated with a specific model of $f_R$ and $f_D$ near an absorption edge for a single atom. As was argued earlier, $f_R$ describes processes that are material- and valence- independent, such as the edge jump, and $f_D$ describes only those processes that depend on the atomic valence, $v$. As an illustrative model of $f_R$, we consider a generic edge jump, i.e. a dielectric susceptibility whose imaginary part has the form
$$Im[ \, \chi(\omega) \, ] = \frac{J}{\omega} \; \theta(\omega-\omega_e).$$
where $\omega_e$ is the edge energy and $J$ is the size of the jump. $\chi$ must satisfy the Kramers-Kronig relations so its real part is given by the integral
$$Re[ \, \chi(\omega) \, ] = \frac{2 J}{\pi} \int_{\omega_e}^\infty \frac{d\omega'}{(\omega')^2 - \omega^2} = -\frac{J}{\pi\omega}\log \left | \frac{\omega_e-\omega}{\omega_e+\omega} \right |.$$
This can be considered a generic analytic model for an idealized edge. The atomic scattering factor is related to this susceptibility by[@batterman]
$$f_R(\omega) = - \frac{V_c \, \omega^2}{r_e \, c^2} \; \chi(\omega)$$
where $\chi$ is in cgs units. $V_c$ here is the volume per atom. Bear in mind that $f$ in general is a tensor, but for simplicity we will treat it as a scalar here. It is important to point out that this quantity actually diverges as $\omega \rightarrow \infty$ but near the edge it is well behaved.
Eq. 14 is for an isolated edge. However in practice there are other transitions that are far away but still contribute to $f$. To account for these we add a constant “background" scattering factor, $f_0$, i.e.
$$f_R(\omega) \rightarrow f_R(\omega) + f_0$$
Using the notation $f(\omega) = f^1(\omega)+if^2(\omega)$, writing it out explicitly, $f_R$ has the form
$$f^1_R(\omega) = -\Delta \, \frac{\omega}{\pi \omega_e}\log \left | \frac{\omega_e-\omega}{\omega_e+\omega} \right | + f_0^1$$
$$f^2_R(\omega) = \Delta \, \frac{\omega}{\omega_e} \; \theta(\omega-\omega_e) + f_0^2$$
where $\Delta=V_c J / r_e c^2$.
The valence-dependent form factor, $f_D$, could take on many forms. In high temperature superconductors, for example, it actually exhibits a sign change[@usLBCO]. For present, illustrative purposes, we will simply take it to be a Lorentzian, i.e.
$$f_D(\omega) = \frac{A}{\omega-\omega_0-i\gamma}$$
In principle $f_D$ should have an energy-independent component representing Thomson scattering from the valence modulation, but in situations of interest this is small.
For numerics, we will take the parameter values $f_1^0 = 9.50$, $f_2^0 = 0.246$, $\Delta = 4.114$, $A=2$ eV, and $\gamma=0.2$ eV, $\omega_e=534$ eV, and $\omega_0=528.6$ eV. While it is intended that this illustration be general these parameters are quite a good model of the oxygen K edge. $f_R$ and $f_D$ for these parameter values are plotted in Fig. 2.
For the CDW itself we use the parameters $v_0 = 0.1$ electron, $u_0 = 0.1 a$, $k=2\pi/4a$, and $<\!\! v \!\!> = 0.12$. The scattered intensity at $q=k$ is given by square of the total scattering amplitude, $|\rho^u_k(\omega)+\rho^v_k(\omega)|^2$. The strain and charge scattering amplitudes can coherently interfere, so must be added before squaring. The resulting quantity depends explicitly on the phase, $\phi$.
The quantity $|\rho^u_k(\omega)+\rho^v_k(\omega)|^2$ is plotted against energy in Fig. 3. For this plot $\phi$ is taken to be zero. Spectra are shown for a pure charge wave, a pure strain wave, and a composite wave. Notice that scattering from the charge-only wave is nonzero only in the region in which $f_D$, displayed in Fig. 2, is nonzero. The strain-only wave, in contrast, is visible at (almost) all energies. This affirms ones intuition that scattering from structural distortions should be visible at all energies, but the valence modulation only near the edge. If both modulations are present the lineshape is not simply the sum of the two because of the non-trivial dependence on the phase factor, $\phi$.
To illustrate this phase dependence the line shape is plotted in Fig. 4 for vaious values of $\phi$. Notice that not only the intensity of the various features but in fact the entire spectral lineshape depends sensitively on $\phi$. Therefore, if both strain and charge scattering are simultaneously visible, and $f_R$ and $f_D$ are determined independently from XAS, it should be possible to objectively determine the phase from a one-parameter fit to this shape.
peierls vs. wigner cdws
=======================
Distinguishing between a Peierls CDW and a more exotic CDW driven by many-body interactions, such as a stripe phase or a Wigner crystal, is a quantitative rather than a qualitative matter. This is because in any real material even a CDW driven purely by electron-electron interactions, because of electrostatics, must still be accompanied by a lattice distortion, though it may be small. Identifying a given CDW as either Peierls-like or Wigner-like requires a quantitative comparison between the charge and lattice amplitudes. Once $u_0$ and $v_0$ have been determined by the procedure just outlined, then, a quantity of great interest is the ratio
$$W = \frac{v_0}{u_0},$$
This quantity has units of length$^{-1}$ and describes the degree to which the CDW is Wigner-like, i.e. diven by many-body interactions rather than a Peierls distortion. For example, a perfect Wigner crystal with no lattice distortion at all would have $W=\infty$. A typical Peierls CDW on the other hand, such as that in NbSe$_3$, has a lattice distortion of approximately $u_0 \sim 0.01 \AA$ and a charge amplitude of $v_0 \sim 0.1$, giving $W=10 \AA^{-1}$. We propose that a CDW with $W$ less than about 20 should be considered a Peierls CDW. If $W > 100$ the CDW probably arises at least partly from many-body effects and should be considered “exotic". It is likely that CDWs can exist over the entire continuum of values of $W$. It would be particularly enlightening to determine the $W$ values for several CDW materials, such as the copper-oxides and the dichalcogenides, that also exhibit superconductivity.
summary
=======
A descripton of resonant x-ray scattering was introduced in which the atomic scattering factor is divided into raw- and valence-dependent amplitudes, $f_R$ and $f_D$. The advantage of this division is that resonant x-ray scattering from the strain wave component of a CDW tracks the average form factor $|f_R(\omega) + <\!\! v \!\!> f_D(\omega)|^2$ whereas the charge (valence) scattering tracks only $|f_D(\omega)|^2$. In many cases $f_R$ and $f_D$ can be independently determined from x-ray absorption measurements on materials with different average valence, combined with Kramers-Kronig analysis. This provides a means to separately quantify the charge and strain components of a CDW. In this framework one can define a quantity “$W$" which provides a quantitative means to characterize a given CDW as Peierls-like, Wigner-like, or anywhere on the continuum between.
The author thanks M. V. Klein for a critical reading of the manuscript. This work was funded by the Materials Sciences and Engineering Division, Office of Basic Energy Sciences, U.S. Department of Energy under grant No. DE-FG02-06ER46285.
|
---
abstract: 'U-Net and its variants have been demonstrated to work sufficiently well in biological cell tracking and segmentation. However, these methods still suffer in the presence of complex processes such as collision of cells, mitosis and apoptosis. In this paper, we augment U-Net with Siamese matching-based tracking and propose to track individual nuclei over time. By modelling the behavioural pattern of the cells, we achieve improved segmentation and tracking performances through a re-segmentation procedure. Our preliminary investigations on the Fluo-N2DH-SIM+ and Fluo-N2DH-GOWT1 datasets demonstrate that absolute improvements of up to 3.8 % and 3.4% can be obtained in segmentation and tracking accuracy, respectively.'
author:
- |
Deepak K. Gupta$^*$\
University of Amsterdam\
`[email protected]`\
Nathan de Bruijn[^1]\
University of Amsterdam\
`[email protected]`\
Andreas Panteli\
University of Amsterdam\
`[email protected]`\
Efstratios Gavves\
University of Amsterdam\
`[email protected]`\
bibliography:
- 'egbib\_temp.bib'
title: 'Tracking-Assisted Segmentation of Biological Cells'
---
Introduction
============
Small sized biomedical data, such as images of biological cells or lymphocytes, are often difficult to acquire and even harder to visualise due to the restricting scale of particles, the low resolution scans and the viscosity of moving cells [@saltz2018spatial; @swiderska2019learning]. As illustrated in Figure \[fig:sample\], the features of the nuclei in each frame are not always clearly visible and their position is very volatile. Being able to detect individual cells and track their trajectory through time will help automate treatment observation and disease spread detection [@coudray2018classification].
![Schematic representation of cell states in two different frames (modified from an image of HL60 cells from the Fluo-N2DH-SIM+ dataset). One cell fades out (cell death) and two collide, in a later frame.[]{data-label="fig:sample"}](images/data_sample.png)
@unet introduced the U-Net architecture, which has demonstrated state-of-the-art performance on many biomedical image segmentation tasks [@unet; @falk2019u]. Since then, several cell tracking approaches have utilised the success of U-Net to boost their performances [@li2018h]. However, due to the constant change of the position, shape and status of the cells in the data, most such approaches fail to accurately detect cells that combine, split or die (leave the image) [@christ2016automatic].
In this paper, we propose to improve the segmentation of biological cells by augmenting with Siamese matching. The initial U-Net segmentation results are combined with cell detection from a Siamese matching-based tracker [@tao2016siamese] to improve the cell recognition in subsequent frames. Next, a dedicated detection mechanism for cell collisions, mitosis (splitting of cells) and apoptosis (cell death) is introduced which attempts to model the movement behaviour of the nuclei. Based on the corrections, we re-segment the cells using random walker algorithm. [@grady2006random].
Approach
========
![An illustration of Mitosis detection through cell tracking[]{data-label="fig:tracking"}](images/tracking.pdf){width="\linewidth"}
We propose to use Siamese matching approach to detect the cell collisions as well as mitosis. Our model combines the U-Net segmentation with an object tracking mechanism in order to improve upon the initial predictions. The various steps are described below.
**Initial Segmentation.** The initial segmentation is done using the U-Net implementation of [@unet], and the results are used as the baseline segmentation for the chosen sequence. During training, data augmentation consisting of random flips and shifts is employed, and as a final step, a nearest neighbour interpolation algorithm is used convert the U-Net results of $512 \times 512$ pixels to the desired resolution. After this process is finished, cells are detected and defined as being connected regions of positively labelled pixels in the segmentation map.
**Tracking.** To predict the new location of a cell in a consecutive frame, we use the Siamese tracker of [@siamfc] and use a pre-trained model, without any further training on cellular footage. SiamFC is adapted for grayscale images and a search space of $150\times 150$ pixels is used. Tracking is done in the forward as well as the backward directions. During tracking, given that our segmentation and location predictions are independent, we refine the tracking performance and detect the occurrence of mitosis and collision events.
Let $I_t$ denote the $t^{\text{th}}$ frame in a sequence of length $T$, and $\mathcal{S}_t = \{ C_t^1,...,C_t^K\}$ be the set of detected cells in this frame. These are used to initialize the tracker at step $t$. For cell $C^i_t$, we refer to the predicted locations by the tracker in $I_{t+1}$ and $I_{t-1}$ as forward $(F_t^i)$ and backward $(B^i_t)$ predictions, respectively. Collision and mitosis are then detected as follows.
**Collision detection.** Collision refers to scenarios where two cells share a fraction of their boundary, and this can often be mistaken as a single cell during segmentation. When processing a new frame $I_t$, where $t > 1$, we start off by performing collision detection in which a cell $C_{t}^i$ is considered to be a lump of multiple individual cells if the centroids of two or more cells in $\mathcal{S}_{t-1}$ lie within the tracked region $B_{t}^i$. If this is the case, $C_{t}^i$ is re-segmented according to a procedure based on Random Walker algorithm. More details on this are described later in Re-segmentation section. This collision detection procedure continues until each cell in $I_t$ matches at most one cell in $I_{t-1}$ or until the re-segmentation procedure fails and does not yield improvement anymore.
**Mitosis detection.** We then continue by matching cells in $\mathcal{S}_{t-1}$ to the detected cells in $I_{t}$. This matching happens in a manner similar to the collision detection, namely a cell $C_{t-1}^i$ is matched to a cell $C_{t}^i$ if the centroid of $C_{t}^i$ is inside the region $F_{t-1}^i$. Different from collision detection, however, $C_{t-1}^i$ is also matched to $C_{t}^i$ if the centroid of the region $F_{t-1}^i$ lies within the boundaries of the cell $C_{t}^i$. This matching procedure yields a set of matches for each cell $C_{t-1}^i$, which we denote as $M_{t-1}^i$ and its size as $|M_{t-1}^i|$. The state of =cell $C_{t-1}^i$ is then determined according to:
$$\begin{aligned}
\label{continuation}
C_{t-1}^i-\text{state} &=
\begin{cases}
\text{Apoptosis}, & |M_{t-1}^i| = 0\\
M_{t-1,1}^i, & |M_{t-1}^i| = 1\\
\text{Mitosis}, & \text{otherwise}
\end{cases}\end{aligned}$$
In case of mitosis, the cell splits, thus the tracking of $C_{t-1}^i$ ends and the cells in $|M_{t-1}^i|$ are initialised with two new trackers which have $C_{t-1}^i$ as their parent. After continuations have been determined for all cells in $\mathcal{S}_{t-1}$, cells in $\mathcal{S}_{t}$ that are not linked to any cell in $I_{t-1}$ are interpreted as newly detected cells which start their life in $I_{t}$ without link to a parent cell. An illustration is shown in Figure \[fig:tracking\].
**Re-segmentation.** In case of a detected collision of two or more cells into a cell $C_{t}^i$, we re-segment $C_{t}^i$ in such a manner that the new number of segments matches the number of colliding cells. This is achieved using the random walker segmentation algorithm as described in [@grady2006random]. To prevent over-segmentation of the cell $C_{t}^i$, which adversely affects segmentation accuracy, we use the relative position of the centroids of the cells in $I_{t-1}$ as the seeds for the segmentation algorithm. An illustration of re-segmented cells is shown in Figure \[fig:segmentation\].
Results
=======
The results of this work are compared against the vanilla U-Net, [@unet], as the baseline approach. The accuracy scores, as described in [@CTC], are used as the decisive metric for model performance. The segmentation and tracking performance of the two methods is listed in Table \[tab:seg\_results\]. As can be seen, the method proposed outperforms the baseline approach on both datasets in both segmentation and tracking with a maximum increase in performance of 3.8% and 3.4% on the segmentation and tracking of the Fluo-N2DH-SIM+ 02 collection, respectively.
The increased performance of the method proposed by this work indicates that the mitosis, apoptosis and cell fusion interpretation of the collision detection mechanism indeed help the tracker network improve the segmentation performance. Furthermore, this finding highlights the importance of modelling the movement behaviour of the cells to better capture the pattern nature of cells in consecutive frames.
-------------------- ----------- ---------------- ------------------------- ----------------
**U-Net** **Our method** **U-Net + SiamTracker** **Our method**
Fluo-N2DH-SIM+ 01 0.919 **0.924** 0.986 **0.992**
Fluo-N2DH-SIM+ 02 0.800 **0.838** 0.922 **0.956**
Fluo-N2DH-GOWT1 01 0.739 **0.746** 0.973 **0.978**
Fluo-N2DH-GOWT1 02 0.827 **0.837** 0.869 **0.875**
-------------------- ----------- ---------------- ------------------------- ----------------
: Accuracy results for segmentation and tracking of cells. Our method includes the modules for detecting collisions, mitosis and apoptosis on top of U-Net and Siamese matching.[]{data-label="tab:seg_results"}
Conclusions
===========
Deep learning trackers such as U-Net work well on biological cell tracking problems. However, they still suffer in the presence of events such as cell collisions, mitosis and apoptosis. In this work, we proposed to combine Siamese matching [@tao2016siamese; @siamfc] with U-Net [@unet] to accurately identify these difficult scenarios. Siamese matching-based detection efficiently provides a more precise understanding and modelling of individual cells, thereby providing better grasp of latent information about them. Based on the proposed approach, we obtained absolute improvements of up to 3.8 % and 3.4% for segmentation and tracking accuracy, respectively. Our future work includes rigorously testing the proposed methodology on a more diverse set of cell tracking data.
[^1]: Equal contribution.
|
---
abstract: 'Magnetoresistance in the spin-density wave (SDW) state of [[(TMTSF)$_2$PF$_6$]{}]{} is known to exhibit a rich variety of the angular dependencies when a magnetic field [${{\mathbf{B}}}$]{} is rotated in the [${{{{\mathbf{b''}}}\text{--}{{\mathbf{c^*}}}}}$]{}, [${{{{\mathbf{a}}}\text{--}{{\mathbf{b''}}}}}$]{} and [${{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}}$]{}planes. In the presence of a magnetic field the [quasiparticle]{} spectrum in the SDW with imperfect nesting is quantized. In such a case the minimum [quasiparticle energy]{} depends both on the magnetic field strength $|B|$ and the angle $\theta$ between the field and the crystal direction [${{\mathbf{a}}}$]{}, [${{\mathbf{b''}}}$]{} or [${{\mathbf{c^*}}}$]{}. This approach describes rather satisfactory the magnetoresistance above [[${T^\star}\approx 4\un{K}$]{}]{}.'
author:
- 'B. Korin-Hamzić[^1]'
- 'M. Basletić'
- 'K. Maki'
title: 'Magnetoresistance in the SDW state of [[(TMTSF)$_2$PF$_6$]{}]{} above [[${T^\star}\approx 4\un{K}$]{}]{}; Novel effect due to the Landau quantization'
---
Introduction
============
Since the discovery of the superconductivity in [[(TMTSF)$_2$PF$_6$]{}]{} in 1979 [@JeromeJP80] the Bechgaard salts or the highly anisotropic organic conductors [[(TMTSF)$_2$X]{}]{} (where TMTSF is tetramethyltetraselenafulvalene and X is anion PF$_6$, AsF$_6$, ClO$_4$ …) are well known because of a variety of physical phenomena related to their low dimensionality. Their rich phase diagram exhibits various low temperature phases under pressure and/or in magnetic field among which the spin density wave (SDW), field induced SDW (FISDW) with quantum Hall effect, and unconventional superconductivity are the most intriguing [@IshiguroBook98; @LeePRL02]. The quasi-one-dimensionality (1D) is a consequence of the crystal structure, where the TMTSF molecules are stacked in columns in the [${{\mathbf{a}}}$]{} direction (along which the highest conductivity occurs), and the resulting anisotropy in conductivity is commonly taken to be $\sigma_a : \sigma_b : \sigma_c \approx t_a^2 : t_b^2 : t_c^2\approx 10^5:10^3:1$, where $t_i$ are the tight binding transfer integrals. These materials are open-orbit metals with the Fermi surface (FS) consisting of a pair of weakly modulated sheets in the [${\mathbf{b}}$]{} and [${\mathbf{c}}$]{} directions.
The SDW ground state of the organic conductors [[(TMTSF)$_2$X]{}]{} has been the subject of considerable experimental and theoretical investigations in the last twenty years [@IshiguroBook98]. It is caused by the almost perfect nesting of the two sheets of the FS, due to the instability of the (1D) electron gas with strong electron-electron interactions. The opening of the energy gap ($\Delta$) in the conduction band leads to the semiconducting transport properties. The external pressure increases the transverse coupling and above about 8 kbars the SDW ground state is suppressed in favor of superconductivity.
One of the most intriguing features of the [[(TMTSF)$_2$X]{}]{} is their anomalous behaviour in a magnetic field. There are several angular effects of magnetoresistance known in the literature, like the variety of resonance-like features, known as the geometrical resonance or the Lebed resonance (or the magic angle effect)[@LebedJETP86], the Danner-Chaikin oscillation [@DannerPRL95] and the third angular effect [@OsadaPRL96], and all of them are mostly connected with the metallic phase of [[(TMTSF)$_2$X]{}]{}. Only few measurements of the magnetoresistance anisotropy, as far as we know, have been performed in the SDW ground state at ambient pressure [@UlmetJPL85; @BasleticEPL93; @KorinHamzicEPL98; @KorinHamzicJP99].
[[(TMTSF)$_2$PF$_6$]{}]{} is one of the most investigated [[(TMTSF)$_2$X]{}]{} compound. It is metallic down to ${T_{\text{SDW}}}\approx 12\un{K}$ where the metal-semiconductor transition into SDW ground state occurs and below which the resistance displays an activated behavior. It is known that SDW in [[(TMTSF)$_2$PF$_6$]{}]{} undergoes another transition at ${T^\star}\approx
{T_{\text{SDW}}}/3$ (at $3.5-4\un{K}$ at ambient pressure) [@LebedJETP86; @DannerPRL95], but the nature of the possible subphases remains controversial. We have recently shown [@KorinHamzicEPL98; @KorinHamzicJP99; @BasleticPRB02] that the temperature dependent magnetoresistance anisotropy changes abruptly below 4 K, indicating a possible phase transition at ${T^\star}\approx 4\un{K}$.
In this paper we shall concentrate ourselves to the magnetoresistance (MR) behavior above ${T^\star}$, and we propose that it can be understood in terms of the Landau quantization of the [quasiparticle]{} spectrum in a magnetic field $B$, where the imperfect nesting plays the crucial role. The experimental MR data of [[(TMTSF)$_2$PF$_6$]{}]{} at $T=4.2\un{K}$ will be compared with our new theoretical model. The discussion of the magnetoresistance below ${T^\star}$, in terms of unconventional SDW (or USDW), will be presented elsewhere [@BasleticPRB02; @KorinHamzicIJMPB01].
Experiment
==========
The measurements were done in magnetic fields up to 5 T with directions of the current along the different crystal axis, and for different orientations of the applied magnetic field. A rotating sample holder enabled the sample rotation around a chosen axis over a range of $190{^\circ}$. The experimental MR data are for [${{\mathbf{c^*}}}$]{} ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}), [${{\mathbf{b'}}}$]{} ([${{{\mathbf{j}}\|{\mathbf{b'}}}}$]{}) and [${{\mathbf{a}}}$]{} ([${{{\mathbf{j}}\|{\mathbf{a}}}}$]{}) axis and for different orientations of magnetic field. The single crystals used come all from the same batch. Their [${{\mathbf{a}}}$]{} direction is the highest conductivity direction, the [${{\mathbf{b'}}}$]{} direction (with intermediate conductivity) is perpendicular to [${{\mathbf{a}}}$]{} in the [${{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}}$]{} plane, and [${{\mathbf{c^*}}}$]{} direction (with the lowest conductivity) is perpendicular to the [${{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}}$]{}plane (and ${{{\mathbf{a}}}\text{--}{\mathbf{b}}}$). The room temperature conductivity values are: $\sigma_a =
500{\un{(\Omega\,\text{cm})^{-1}}}$, $\sigma_b = 20{\un{(\Omega\,\text{cm})^{-1}}}$ and $\sigma_c = 1/35{\un{(\Omega\,\text{cm})^{-1}}}$.
The MR (defined as ${\Delta\rho/\rho_0}= [\rho(B)-\rho(0)]/\rho(0)$) was measured in various four probe arrangements on samples cut from the long crystals. In the case of $\rho_a$ ([${{{\mathbf{j}}\|{\mathbf{a}}}}$]{}), $\rho_b$ ([${{{\mathbf{j}}\|{\mathbf{b'}}}}$]{}) and $\rho_c$ ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}), two pairs of the contacts were placed on the opposite [${{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}}$]{}, [${{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}}$]{} and [${{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}}$]{} surfaces, respectively.
The four configurations that will be analyzed in this work are shown on Fig 1, , and :\
[*i)*]{} [Fig. \[fig:config\]]{} shows the case when the current direction is along the [${{\mathbf{b'}}}$]{} axis and the magnetic field is rotated in the [${{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}}$]{} plane ([${{{\mathbf{j}}\|{\mathbf{b'}}}}$]{}, [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{}) perpendicular to the current direction. $\theta$ is the angle between [${{\mathbf{B}}}$]{} and the [${{\mathbf{a}}}$]{} axis, [[*i.e.*]{}]{} $\theta=0$ for [${{{\mathbf{B}}\|{\mathbf{a}}}}$]{} and $\theta=90{^\circ}$ for [${{{\mathbf{B}}\|{\mathbf{c^\star}}}}$]{}.\
[*ii)*]{} [Fig. \[fig:config\]]{} shows the case when the current direction is along the [${{\mathbf{c^*}}}$]{} axis and the magnetic field is rotated in the [${{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}}$]{} plane ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}, [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{}). $\theta$ is the angle between [${{\mathbf{B}}}$]{}and the [${{\mathbf{b'}}}$]{} axis, [[*i.e.*]{}]{} $\theta=0$ for [${{{\mathbf{B}}\|{\mathbf{b'}}}}$]{} and $\theta=90{^\circ}$ for [${{{\mathbf{B}}\|{\mathbf{c^\star}}}}$]{}.\
[*iii)*]{} [Fig. \[fig:config\]]{} shows the case when the current direction is along the [${{\mathbf{c^*}}}$]{} axis and the magnetic field is rotated in the [${{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}}$]{} plane ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}, [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}})}}}}$]{}) perpendicular to the current direction. $\theta$ is the angle between [${{\mathbf{B}}}$]{} and the [${{\mathbf{b'}}}$]{} axis, [[*i.e.*]{}]{} $\theta=0$ for [${{{\mathbf{B}}\|{\mathbf{b'}}}}$]{} and $\theta=90{^\circ}$ for [${{{\mathbf{B}}\|{\mathbf{a}}}}$]{}.\
[*iv)*]{} [Fig. \[fig:config\]]{} shows the case when the current direction is along the [${{\mathbf{a}}}$]{} axis and the magnetic field is rotated in the [${{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}}$]{} plane ([${{{\mathbf{j}}\|{\mathbf{a}}}}$]{}, [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{}) perpendicular to the current direction. $\theta$ is the angle between [${{\mathbf{B}}}$]{} and the [${{\mathbf{c^*}}}$]{} axis, [[*i.e.*]{}]{} $\theta=0$ for [${{{\mathbf{B}}\|{\mathbf{c^\star}}}}$]{} and $\theta=90{^\circ}$ for [${{{\mathbf{B}}\|{\mathbf{b'}}}}$]{}.
Model, Results and Discussion
=============================
If we limit ourselves to SDW above [[${T^\star}\approx 4\un{K}$]{}]{}, it is well established that the [quasiparticle energy]{} is given by [@KorinHamzicEPL98; @HuangPRB90]: $$\label{eq:QPEnergy}
E({{\mathbf{k}}}) = \sqrt{\eta^2 + \Delta^2} - {\varepsilon_0}\cos 2bk_y\,,$$ where $\eta = \sqrt{ v_a^2 \left(k_x-k_F\right)^2 + v_c^2 k_z^2 }$ is the [quasiparticle energy]{} in the normal state ($v_a$ and $v_c$ are Fermi velocities in [${{\mathbf{a}}}$]{} and [${{\mathbf{c^*}}}$]{} direction), $\Delta\approx 34\un{K}$ is the order parameter for SDW and ${\varepsilon_0}\approx 13\un{K}$ is the parameter characterizing the imperfect nesting [@KorinHamzicEPL98]. In a presence of a magnetic field $B$ the [quasiparticle]{} orbit is quantized. This can be readily seen from the [quasiparticle energy]{} landscape as shown in [Fig. \[fig:landscape\]]{}.
In [Fig. \[fig:landscape\]]{}a we show the [quasiparticle energy]{} for SDW with perfect nesting. Quasiparticle energy has the minima at $k_x=\pm k_F$, which is independent of $k_y$ and consequently, the [quasiparticle]{} orbit is open. On the other hand, for SDW with imperfect nesting, there are minima for [quasiparticle energy]{} at $k_x=\pm k_F$ and $k_y =
\pm \pi/2b$ (see [Fig. \[fig:landscape\]]{}b). Therefore, in the presence of a magnetic field, [quasiparticle]{} will circle around these minima, [[*i.e.*]{}]{} closed orbits appear and they will be quantized.
We expand the [quasiparticle energy]{} in [Eq. (\[eq:QPEnergy\])]{} for small $(k_x-k_F)^2$ and $k_y^2$. In the presence of a magnetic field within different planes ([Fig. \[fig:config\]]{}) the minimum [quasiparticle energy]{} ([[*i.e.*]{}]{} the energy gap) associated with the lowest Landau level is given by: $$\begin{aligned}
\label{eq:EnergyGapAbove}
E(B,\theta) & = & \Delta - {\varepsilon_0}+ \sqrt{\frac{{\varepsilon_0}}{\Delta}}\,v_a b e B{\sqrt{{\sin^2\theta+\gamma_{2}\cos^2\theta}}}\,,
\text{\ \ for}{{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}\\
& = & \Delta - {\varepsilon_0}+ \frac{v_a v_c b e}{2 \Delta} B{\sqrt{{\sin^2\theta+\gamma_{3}\cos^2\theta}}}\,,
\text{\ \ for}{{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}})}}}}\\
& = & \Delta - {\varepsilon_0}+ \sqrt{\frac{{\varepsilon_0}}{\Delta}}\,v_a b e B|\sin\theta|\,,
\text{\ \ for }{{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}})}}}}\,,\end{aligned}$$ where $\gamma_2 = \gamma_3^{-1} = \left( v_c/2b \right)^2 / {\varepsilon_0}\Delta$. In general, the [quasiparticle energy]{} gap increases linearly with $|B|$ and depends on angle $\theta$ (as defined in [Fig. \[fig:config\]]{}). For $B=0$ the resistance is given as $\rho_i \propto \exp \left[ \beta E(0,0) \right]$. For $B \neq 0$ and for $\omega_c\tau > 1$ ($\omega_c$ – cyclotron frequency; $\tau$ – scattering rate; we also assume that the [quasiparticle]{} scattering rate is independent of [[${\mathbf{k}}$]{}]{}) we may write down the magnetoresistance as: $$\begin{aligned}
\label{eq:MRzzBbc}
\rho_{zz}(T,B) & = &
\exp \left\{
\beta(\Delta-{\varepsilon_0}) \left[ 1+A_2 B {\sqrt{{\sin^2\theta+\gamma_{2}\cos^2\theta}}} \right]
\right\} \nonumber \\
& & \times \left( 1+C_2 B {\sqrt{{\sin^2\theta+\gamma_{2}\cos^2\theta}}} \right)\,, \text{\ \ for}{{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}\end{aligned}$$ $$\begin{aligned}
\label{eq:MRzzBab}
\rho_{zz}(T,B) & = &
\exp \left\{
\beta(\Delta-{\varepsilon_0}) \left[ 1+A_3 B {\sqrt{{\sin^2\theta+\gamma_{3}\cos^2\theta}}} \right]
\right\} \nonumber \\
& & \times \left( 1+C_3 B {\sqrt{{\sin^2\theta+\gamma_{3}\cos^2\theta}}} \right)\,, \text{\ \ for}{{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}})}}}}\,,\end{aligned}$$ with: $$\label{eq:AAzz}
A_2 = \sqrt{ \frac{{\varepsilon_0}}{\Delta} } \, \frac{v_a b e} {\Delta-{\varepsilon_0}}\,,\quad
A_3 = \frac{v_a v_c e}{2\Delta} \, \frac{1}{\Delta-{\varepsilon_0}} \,.$$
We compare now our experimental data with the above equations. Figs. \[fig:MRzzBbc\] and \[fig:MRzzThbc\] show magnetic field dependence of MR ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}, [${{{\mathbf{B}}\|{\mathbf{c^\star}}}}$]{} and [${{{\mathbf{B}}\|{\mathbf{b'}}}}$]{}) and the angular dependence of MR ([${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{}, [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{}) at 4.2 K, respectively. The solid lines are fits to [Eq. (\[eq:MRzzBbc\])]{}. The analogous results for [${{{\mathbf{j}}\|{\mathbf{c^\star}}}}$]{} but different magnetic field rotation ([${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{b'}}}}})}}}}$]{}) is given on Figs. \[fig:MRzzBab\] and \[fig:MRzzThab\]. The solid lines are fits to [Eq. (\[eq:MRzzBab\])]{}.
On the other hand, for [${{{\mathbf{j}}\|{\mathbf{b'}}}}$]{} and [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{} we obtain: $$\label{eq:MRyy}
\frac{\Delta\rho_{yy}(T,B)}{\rho_{yy}(T,0)} =
\exp \left\{ \beta(\Delta-{\varepsilon_0}) A_1 B |\sin\theta| \right\}
\left( 1+ C_1 B |\sin\theta| \right)\,,$$ where $A_1 = \sqrt{ {\varepsilon_0}/\Delta } \, v_a b e / (\Delta-{\varepsilon_0})$. We give the excess magnetoresistance rather than the MR itself, as in this configuration the [quasiparticle energy]{} gap for $B=0$ is only 8 K [@KorinHamzicEPL98], instead of all other cases where $\Delta-{\varepsilon_0}= 21\un{K}$. The experimental results for [${{{\mathbf{j}}\|{\mathbf{b'}}}}$]{} and [${{\mathbf{B}}}$]{} in [${{{{\mathbf{a}}}\text{--}{{\mathbf{c^*}}}}}$]{} plane are shown on Figs. \[fig:MRyyBac\] and \[fig:MRyyThac\]. The solid lines are fits to [Eq. (\[eq:MRyy\])]{}.
It is evident that the present model (Eq. (\[eq:MRzzBbc\]–\[eq:MRyy\])) describes both the $B$ and $\theta$ dependence of MR rather well (fits on Figs. \[fig:MRzzBbc\]–\[fig:MRyyThac\] are from good to excellent). The fitting procedure yielded $\Delta-{\varepsilon_0}=21\un{K}$, $A_2 = 0.014\un{T^{-1}}$, $\gamma_2 = 0.85$, $C_2 = 0.38\un{T^{-1}}$, $A_3 = 0.00905\un{T^{-1}}$, $\gamma_3 = 3.1$, $C_3 = 0$, $A_1 = 0.048\un{T^{-1}}$, $C_1 = 2.14\un{T^{-1}}$.
First, from $A_2$ and $\gamma_2$ we can extract the [${{\mathbf{a}}}$]{} axis coherence length $\xi_a = v_a/\Delta =
120$ Å and $v_c/v_a = 7.33\times 10^{-2}$. The ratio $v_c/v_a$ appears to be somewhat larger than the one previously determined ($v_c/v_a = 1.7\times 10^{-2}$ [@IshiguroBook98]) but the difference is minor. Furthermore, $A_3$ gives $\xi_a = 100$ Å, where we used our $v_c/v_a$ value. We can, therefore, conclude that $\xi_a$ is consistent in these two configurations. On the other hand, $\gamma_3$ gives $\xi_a = 5.4$ Å (the value that appears to be too small) and $A_1$ gives $\xi_a = 410$ Å (which is somewhat larger, but the difference of a factor 3.5 may be acceptable).
Finally, we comment the result for the highest conductivity direction [${{\mathbf{a}}}$]{}, [[*i.e.*]{}]{} $\rho_{xx}(T,B)$ for [${{\mathbf{B}}}$]{} in the [${{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}}$]{} plane, perpendicular to the current direction. The experimental results for the magnetic field dependence of MR ([${{{\mathbf{j}}\|{\mathbf{a}}}}$]{}, [${{{\mathbf{B}}\|{\mathbf{c^\star}}}}$]{} at 4.2 K and 2.2 K) and the angular dependence of MR at 4.2 K and 2.2 K are shown on Figs. \[fig:MRxxBbc\] and \[fig:MRxxThbc\]. We notice that the angular dependencies of MR for $T=4.2\un{K}$ and $2.2\un{K}$ look very similar (see [Fig. \[fig:MRxxThbc\]]{}). Namely, the fitted $\rho_{xx}(T,B)$ for both $T=4.2\un{K}$ and $2.2\un{K}$ (Figs. \[fig:MRxxBbc\] and \[fig:MRxxThbc\], solid lines), are well accounted for by: $$\label{eq:MRxx}
\rho_{xx}(B,\theta) \propto 1 + C_1 B|\cos\theta|\,,$$ where $\theta$ is the angle $\measuredangle({{\mathbf{B}}},{{\mathbf{c^*}}})$. Here $C_1 = 0.55\un{T^{-1}}$ and $1.6\un{T^{-1}}$ for $T=4.2\un{K}$ and $2.2\un{K}$, respectively.
The linear $B$ dependence of $\rho_{xx}$ for [${{{\mathbf{{{\mathbf{B}}}}}\|{\mathbf{({{{{\mathbf{b'}}}\text{--}{{\mathbf{c^*}}}}})}}}}$]{} in SDW state of [[(TMTSF)$_2$PF$_6$]{}]{} ([Fig. \[fig:MRxxBbc\]]{}) is well known [@IshiguroBook98], though it is not understood. Therefore, we may conclude that $\rho_{xx}(T,B)$ is not sensitive to the [quasiparticle]{} spectrum we are considering here.
Conclusion
==========
In summary, we have derived the expression of the magnetoresistance based on the Landau quantization of the [quasiparticle]{} orbit in SDW with imperfect nesting. At the qualitative level these expressions give excellent description of experimental magnetoresistance results. From fitting procedure, we can deduce the physical content like $\xi_a$ and $v_c/v_a$, which both look very reasonable. The origin of a few cases where $\xi_a$ has not a consistent value remains to be clarified. In any case, we may conclude that the Landau quantization of the [quasiparticle energy]{} describes most of salient features of the angular dependence of the magnetoresistance in SDW in [[(TMTSF)$_2$PF$_6$]{}]{} above ${T^\star}\approx 4\un{K}$.
A parallel study of the magnetoresistance for $T<{T^\star}$ based on possible USDW in addition to the already existing SDW, will be reported elsewhere [@KorinHamzicIJMPB01; @BasleticPRB02].
This experimental work was performed on samples prepared by K. Bechgaard. We acknowledge useful discussion with A. Hamzić and S. Tomić.
[99]{}
, (Springer) 1998, 2nd ed.
;
submitted to Phys. Rev. B
submitted to Int. J. Mod. Phys. B
[^1]: E-mail:
|
---
abstract: 'A recent CMS analysis has reported the observation of an excess in the invariant mass distribution of the opposite-sign same-flavour lepton pair, which can be interpreted as a kinematic edge due to new physics. Using collider simulation tools, we recast relevant LHC search results reported by ATLAS and CMS collaborations in order to determine constraints on supersymmetric models that could produce the observed features. In particular, we focus on models involving cascade decays of light-flavour squarks and sbottoms. We find no favourable supersymmetry scenario within our exploration that could explain the origin of the excess when other LHC constraints are taken into account.'
---
KCL-PH-TH/2015-07\
LCTS/2015-03\
UT-15-06\
[**A closer look at a hint of SUSY at the 8 TeV LHC** ]{}
.5in
[ Philipp Grothaus$^{(a,}$[^1]$^)$, Seng Pei Liew$^{(b,}$[^2]$^)$ and Kazuki Sakurai$^{(a,}$[^3]$^)$ ]{}
.3in
[*$^a$Department of Physics, King’s College London, London WC2R 2LS, UK\
$^b$Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\
*]{}
Introduction
============
The search for supersymmetry (SUSY) as an extension of the Standard Model (SM) is one major target of the Large Hadron Collider (LHC) physics program. However, we have thus far found no definitive evidence of SUSY based on the first run of the LHC despite dedicated searches on many fronts. Even so, there is an analysis presented by the CMS collaboration that could be showing the first signs of SUSY [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa]. In the analysis, two leptons, jets and missing energy are looked for in the final states. It is found that there is an excess ($130^{+48}_{-49}$ events in the “central" region) in the invariant mass distribution of the opposite-sign same-flavour (OSSF) lepton pair, corresponding to a significance of 2.6 $\sigma$.
The excess of the signal is fitted kinematically as a triangular-shape edge at $m_{\ell \ell}= 78.7 \pm 1.4~\GEV$. Such a kinematic edge is a characteristic signal of SUSY, where a SUSY particle undergoes a two-stage two-body decay. The kinematic edge formed by a pair of leptons can be interpreted as the cascade decay of a neutralino: $\ntwo \to \sl^{\pm}\ell^{\mp}\to \ell^{\pm}\ell^{\mp}\none$ (on-shell slepton decay) [@Allanach:2000kt]. It is also possible to interpret the edge as a three-body decay signal of a neutralino, $\ntwo \to \ell^{\pm}\ell^{\mp}\none$, where the lepton pair is produced via an off-shell $Z$ (off-shell $Z$ decay). The shape of the edge would be more rounded as compared to the two-stage two-body decay, but as shown in the original CMS analysis, the three-body decay still provides a good fit. The direct production of $\ntwo$ is too small to reproduce the dilepton excess, however its production can be boosted if coloured sparticles subsequently decay into $\ntwo$. The explanation of the dilepton excess in terms of coloured sparticles is consistent with the CMS analysis, since events with jet multiplicity are selected and counted.
In this work we perform a detailed study on the possibility of explaining the dilepton excess with several SUSY models taking into account a comprehensive list of LHC constraints from a number of ATLAS and CMS direct SUSY searches. In order to accurately estimate the LHC constraints and simulate many analyses systematically, we use the automated simulation tool [Atom]{} [@atom]. We take a bottom-up approach by considering simplified SUSY models with minimal content of particles at low energy to reproduce the excess optimally. As will be discussed in the following sections, light-flavour squarks and sbottoms are potential candidates that satisfy these criteria. Some of these models have already been studied in earlier works [@1409.3532; @1410.4998]. [^4] Here, we will confront our simplified models with various direct SUSY search constraints such that their viability is tested in great detail. We will show that the light-flavour squarks and sbottom models we consider in this paper are strongly constrained when providing a large enough contribution to the dilepton excess.
Our paper is organised as follows. In the next section, we describe the selection criteria of the CMS dilepton analysis. In section \[sc:susy\], we consider SUSY models that can possibly reproduce the required features of the dilepton edge. In section \[sc:sim\], we describe the procedure of our simulation and analysis. We discuss our results and their interpretations in Section \[sc:re\]. Conclusions are drawn in Section \[sc:con\].
CMS dilepton analysis
=====================
CMS reported an excess of events in the dilepton plus missing energy channel [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa] in the 8 TeV, 19.4 fb$^{-1}$ data. The analysis requires an OSSF lepton pair with $p_T > 20$ GeV. It also requires $\ge 2$ jets with $p_T > 40$ GeV and $\met > 150$ GeV or $\ge 3$ jets with $p_T > 40$ GeV and $\met > 100$ GeV. The excess is observed in the central region where both leptons satisfy $|\eta_{\rm lep}| < 1.4$. It exhibits an $edge$ in the dilepton invariant mass distribution around $m_{ \ell \ell} = 78$ GeV. The counting experiment in the $m_{ \ell \ell} \in [20,70]$ GeV region shows an excess of $\sim 130$ events over the Standard Model expectation, which corresponds to a standard deviation of 2.6 $\sigma$.
SUSY interpretations of the dilepton edge {#sc:susy}
=========================================
In this paper we consider simplified SUSY models that capture the essence needed for explaining the observed dilepton excess. Generalizations of SUSY models given here are straightforward.
It is known that the OSSF dilepton pair in the decay of the second lightest neutralino $\ntwo$ via on-shell slepton and off-shell $Z$ exhibit an edge-like shape at m\_[edge]{} = m\_ &:& *\^ \^\^\^, \[eq:edge1\]\
m\_[edge]{} = m\_ - m\_&:& \^\^, \[eq:edge2\] respectively, in the $m_{\ell \ell}$ distribution. In order to obtain a large enough production cross section to fit the excess and to have $\ge 2$ high $p_T$ jets required in the event selection, we consider production of coloured SUSY particles, which may subsequently decay into $\ntwo$. In this paper we consider two scenarios: light-flavour squark and sbottom production.*
Squark scenarios
----------------
In the squark scenario, we consider the production of pairs of light-flavour squarks. This scenario assumes the first two generations of squarks (both left and right-handed) to be mass degenerate and within the reach of the LHC, whilst the third generation squarks and gluinos are decoupled. We also assume that the second lightest neutralino is mostly Wino-like or an admixture of Wino and Higgsinos and the lightest neutralino is mostly Bino-like. In this setup the lighter chargino, $\cone$, is naturally introduced as a $SU(2)_L$ partner of the $\ntwo$ and their masses have to be close to each other. Since the right-handed squarks do not couple to the Wino and only very weakly couple to the Higgsinos, they decay predominantly into a quark and the $\none$, whereas the left-handed squarks can decay to either $\cone$, $\ntwo$ or $\none$ depending on the Wino-Higgsino mixing in the $\cone$ and $\ntwo$.
We consider two models according to the $\cone$ and $\ntwo$ decay modes. The first model is the [*on-shell slepton*]{} model, where we assume the right-handed selectron and smuon in the low energy spectrum so that the $\ntwo$ decays predominantly into an OSSF lepton pair and the $\none$ via the on-shell slepton. We decouple the left-handed slepton doublets, ($\tilde \nu_L, \tilde \ell_L$), to maximise the signal rate, otherwise the $\ntwo$ could also decay into a pair of neutrinos and the $\none$ via the on-shell $\tilde \nu_L$.[^5]
Any models that lead to multi-lepton final states are severely constrained by the multi-lepton plus missing energy searches [@CMS-PAS-SUS-13-002]. To avoid these constraints a large branching ratio of the $\tilde q_L \to q \none$ mode is necessary in this model. We assume
BR( q\_L q + / / ) = 10 / 5 / 85 %. This can be achieved if $\cone$ and $\ntwo$ have large Higgsino components because the squarks couple to the Higgsinos with small Yukawa couplings. We will see in section \[sc:re\_sq\] that our conclusion is not sensitive to variations of the branching ratios.
In the squark with on-shell slepton model we then have the following cascade decays q\_L q q \^*\^q \^\^&:& 5%,\
q\_L q q *\^q \^&:& 10%,\
q\_L q &:& 85%,\
q\_R q &:& 100%. If one of the pair produced squarks undergoes the first decay chain, the final state may contain an OSSF dilepton plus two energetic jets, and such events can contribute to the CMS dilepton excess.**
![Decay chains of squark scenarios ([*left*]{}: on-shell slepton model, [*right*]{}: off-shell $Z$ model).[]{data-label="fig:squark_model"}](figures/squark_model.pdf){width="12cm"}
The second model we consider in this paper is the [*off-shell $Z$*]{} model, where the $\ntwo$ decays via the off-shell $Z$ into an OSSF dilepton pair and the $\none$. Unlike in the on-shell slepton model we here need a large branching ratio of $\tilde q_L \to q \ntwo$ such that the small leptonic branching ratio of the off-shell $Z$ into electrons and muons (about 6 %) is compensated. We assume BR( q\_L q + / / ) = 66 / 33 / 1 %, which can be realised by assuming $\cone$ and $\ntwo$ are Wino-like. In the squark with off-shell $Z$ model we have the following decay chains q\_L q q f |f ([via]{} Z\^\*) &:& 33%,\
q\_L q q f |f\^ ([via]{} W\^\*) &:& 66%,\
q\_L q &:& 1%,\
q\_R q &:& 100%. The signal events can be obtained if one of the pair produced squarks undergoes the first decay chain and the $\ntwo$ decays via the $Z^*$ into the dilepton pair and the $\none$. A schematic picture of the squark scenarios is shown in Fig. \[fig:squark\_model\].
Sbottom scenarios {#sec:sbottom_scenario}
-----------------
Another way of interpreting the CMS dilepton excess is to assume that the observed dileptons in the excessive events come from cascade decays of bottom squarks. Unlike in the squark scenario, the decay mode to charginos, $\tilde b_1 \to t \cone$, is kinematically forbidden if $m_{\tilde b_1} < m_t + m_{\cone}$. We consider this case because the decay mode to charginos is more constrained due to emergence of top quarks. Similarly to the squark scenario we consider [*on-shell slepton*]{} and [*off-shell $Z$*]{} models according to the $\ntwo$ decay mode.
In the on-shell slepton scenario the $\ntwo$ may decay either via a right-handed charged slepton or a left-handed charged slepton and sneutrino. We will treat these two cases separately in our analysis.
If the mediating slepton is right-handed, $\ntwo$ predominantly decays into two charged leptons and $\none$, and the events tend to have more than two leptons in the final state. Such models are severely constrained by the multi-lepton plus missing energy searches as we have previously discussed. To avoid these constraints, we assume 70% of sbottoms decay into a bottom quark and a $\none$ and the rest of sbottoms decay into a bottom quark and a $\ntwo$. This situation can be achieved if $\ntwo$ is Wino-like and $\tilde b_1$ has a large component of $\tilde b_R$. We have the following decay chains in the sbottom with on-shell slepton model. b\_1 b b \^*\^b \^\^&:& 30%,\
b\_1 b &:& 70%.*
In the case where the mediating slepton is left-handed, sneutrinos are introduced as SU(2) partners of charged sleptons. We assume sneutrinos and charged sleptons are mass degenerate and $\ntwo$ decays democratically into charged sleptons and sneutrinos. b\_1 b b \^*\^b \^\^&:& 25%,\
b\_1 b b b &:& 25%,\
b\_1 b &:& 50%. The schematic picture of these cases is shown in Fig. \[fig:sbottom\_slep\].*
![Decay chains of on-shell slepton mediated sbottom scenarios ([*left*]{}: left-handed slepton model, [*right*]{}: right-handed slepton model).[]{data-label="fig:sbottom_slep"}](figures/sbottom_slepton.pdf){width="12.5cm"}
Let us discuss the off-shell $Z$ model for the sbottom scenario. Analogous to the squark with off-shell $Z$ model, we need sbottoms to have a sizeable decay branching ratio to $\ntwo$ in order to have large enough dilepton event rates. One way to realise this situation is to have a Higgsino-like $\ntwo$, a mostly right-handed $\tilde b_1$ and to assume a large sbottom-bottom-Higgsino coupling due to a large ${\rm tan} \beta$. It is shown in [@1410.4998] that for ${\rm tan} \beta=50$, $m_{\tilde b_1} \simeq 330~\GEV$ and a Higgsino mass parameter $\mu\simeq 290$ around 44% of sbottoms decay to the roughly mass-degenerate $\ntwo$ and $\tilde \chi_3^0$. This model point predicts about 1-$\sigma$ less events than the central fit without being excluded. In order to explore in more detail the parameter region that could possibly contribute to the excess, we expand the study of this scenario by varying the parameters $M_1$, $\mu$ and $m_{\tilde b_1}$, while fixing ${\rm tan} \beta=50$. The mass spectrum and particle decay branching ratio of this simplified model are calculated using [SPheno]{} [@Porod2003; @Porod2011].
Alternatively one can obtain a large branching ratio of sbottom decaying to $\ntwo$ by assuming $\tilde b_1$ is left-handed and $\ntwo$ is Wino-like. Due to $SU(2)$ gauge invariance a left-handed top squark, $\tilde t_1$, is necessarily included in the low energy spectrum. For simplicity, we assume $m_{\tilde b_1}=m_{\tilde t_1}$. We consider the following decay chains for the left-handed sbottom with off-shell $Z$ model. b\_1 q q f |f ([via]{} Z\^\*) &:& 100%,\
t\_1 q q f |f\^ ([via]{} W\^\*) &:& 100%. A schematic picture of the off-shell $Z$ sbottom scenarios is shown in Fig. \[fig:sbottom\_Z\].
![Decay chains of off-shell $Z$ mediated sbottom scenarios ([*left*]{}: right-handed sbottom model, [*right*]{}: left-handed sbottom model).[]{data-label="fig:sbottom_Z"}](figures/sbottom_Z.pdf){width="10.5cm"}
The squark and sbottom scenarios are the priorities of this work, but let us also touch on the possibilities of explaining the dilepton excess with the remaining coloured sparticles in SUSY, namely gluino and stop. Gluinos can decay into $\ntwo$ via an intermediate squark, not much different from the squark or sbottom scenario other than a larger jet multiplicity. For stop, its decay into a top quark would lead to an extra lepton that plays no role at explaining the dilepton excess. It is not clear how gluino or stop could explain the dilepton excess without inducing additional jet or leptonic constraints, and hence we are not going to study these scenarios further in this work.
The simulation setup {#sc:sim}
====================
In this section we describe our procedure to calculate the contribution to the CMS dilepton excess and the constraints from other ATLAS and CMS SUSY searches.
The production cross section, $\sigma_{\rm prod}$, for light-flavour squarks is calculated using [Prospino 2]{} [@Beenakker1996] with the gluino mass set to 3.5 TeV. For the sbottom cross section we use results from the LHC SUSY Cross Section Working Group based on [@Kramer2012]. We create SLHA files of our simplified models for the event generation and pass them to [Pythia 6.4]{} [@Sjostrand2006] to generate a total number of $10 \cdot \sigma_{\rm prod } \cdot \mathcal{L}$, with maximal $5 \cdot 10^5$, events, where $\mathcal{L} = 19.4$ fb$^{-1}$ is the integrated luminosity at the CMS dilepton analysis. We then run [Atom]{} [@atom] on the generated HepMC event files to estimate the efficiencies, $\epsilon$, of the signal regions defined in all the ATLAS and CMS analyses that will be used in this work. The application examples and validation of [Atom]{} is found in [@Papucci:2011wy; @Papucci:2014rja; @Kim:2014eva]. We have implemented the CMS dilepton analysis in [Atom]{} and validated it using the cut-flow tables given by the CMS collaboration based on the $\tilde b_1 \to b \ntwo \to b \ell^+ \ell^- \none$ simplified model. The comparison in the number of expected signal events calculated by [Atom]{} and CMS is shown in Appendix \[ap:validation\]. We also cross-checked some of the analyses with another simulation tool [CheckMATE]{} [@1312.2591].
From the obtained cross section and efficiency, the SUSY contribution to the CMS dilepton excess is calculated as $N_{\ell \ell} = \sigma_{\rm eff} \cdot {\cal L}$, where the effective cross section, $\sigma_{\rm eff}$, is defined as the cross section after the event selection: $\sigma_{\rm eff} = \epsilon \cdot \sigma_{\rm prod}$. For the other ATLAS and CMS analyses the 95% CL upper limit on $\sigma_{\rm eff}$, $\sigma_{\rm UL}$, is reported for each signal region by the collaborations. We define a useful measure for exclusion by $R = \sigma_{\rm eff}/\sigma_{\rm UL}$. If $R > 1$ is found for one of the signal regions, the model is likely to be excluded, although one needs to combine all the signal regions statistically to draw a definite conclusion. However, we do not attempt to combine these signal regions because there are non-trivial correlations among them which originate from the uncertainties on $e.g.$ the jet energy scale, the lepton efficiency and luminosity, and it is not possible for us to combine the signal regions correctly. Instead, in the next section we will look at the exclusion measure $R$ individually to understand which signal regions are sensitive to the model points.
channel search for arXiv or CONF-ID refs
------------------------------------- -------------------------------------------------------------- --------------------- ---------------------------------------------
[$2{\rm -}6 j + 0 \ell + \met$]{} $\tilde{q},\tilde{g}$ ATLAS-CONF-2013-047 [@ATLAS-CONF-2013-047]
1405.7875 [@1405.7875]
$2 b + 0 \ell + \met$ $\tilde{t},\tilde{b}$ 1308.2631 [@1308.2631]
[$4j + 1 \ell +\met$]{} $\tilde{t}$ ATLAS-CONF-2013-037 [@ATLAS-CONF-2013-037]
$\geq 2 j + \geq 1 \ell + \met$ $\tilde{q},\tilde{g}~(1~{\rm or}~2 \ell)$ ATLAS-CONF-2013-062 [@ATLAS-CONF-2013-062]
[$2j + 2 \ell + \met$]{} dilepton edge CMS-PAS-SUS-12-019 [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa]
$2j + \ell^{\pm} \ell^{\pm} + \met$ $\tilde{q},\tilde{g},\tilde{t},\tilde{b}~({\rm SS\ lepton})$ ATLAS-CONF-2013-007 [@ATLAS-CONF-2013-007]
$2j + 2 \ell + \met$ $\tilde{t} (2 \ell)$ ATLAS-CONF-2013-048 [@ATLAS-CONF-2013-048]
1403.4853 [@1403.4853]
[$2,3 \ell + \met$]{} $\tilde{\chi}^{\pm},\tilde{\chi}^0,\tilde{ \ell }$ 1404.2500 [@1404.2500]
1405.7570 [@1405.7570]
$ 3 \ell + \met$ $\tilde{\chi}^{\pm},\tilde{\chi}^0$ 1402.7029 [@1402.7029]
$\geq 3 \ell + \met$ $\tilde{\chi}^{\pm},\tilde{\chi}^0$ CMS-PAS-SUS-13-002 [@CMS-PAS-SUS-13-002]
: LHC searches used in this paper to test the viability of the simplified models.[]{data-label="tab:search"}
In Table \[tab:search\] we list the ATLAS and CMS analyses we consider in this work. We include the multi-jet [@ATLAS-CONF-2013-047; @1405.7875] and di-$b$ jet [@1308.2631] analyses, jets plus single [@ATLAS-CONF-2013-037] or two lepton [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa; @ATLAS-CONF-2013-007] (including same-sign (SS) dilepton [@ATLAS-CONF-2013-007]) analyses [@ATLAS-CONF-2013-062] and multi-lepton analyses [@1404.2500; @1405.7570; @CMS-PAS-SUS-13-002; @1402.7029]. In the next section we investigate whether the SUSY models can fit the CMS dilepton excess taking the constraints from these analyses into account.
Results {#sc:re}
=======
Squark scenarios {#sc:re_sq}
----------------
![Signal rate and $R$-values for the squark models. The left panel presents the intermediate slepton, the right panel the off-shell $Z$ scenario. []{data-label="fig:squark_results"}](figures/squark_slepton.pdf "fig:"){width="7.9cm"} ![Signal rate and $R$-values for the squark models. The left panel presents the intermediate slepton, the right panel the off-shell $Z$ scenario. []{data-label="fig:squark_results"}](figures/squark_z.pdf "fig:"){width="7.9cm"}
In Fig. \[fig:squark\_results\] we show the results of our numerical calculation for the squark scenario. In the plots the black curves represent the SUSY contribution, $N_{\ell \ell}$, normalised by the best fit value 130. The green bands correspond to the 1$\sigma$ region of the fit. In the same plots we show also the exclusion measure, $R$, for a few signal regions that are particularly sensitive to the models. The region where any $R$ is greater than 1 is strongly disfavoured.
In the left panel of Fig. \[fig:squark\_results\] we show $N_{\ell \ell}/130$ and $R$ as functions of $m_{\tilde q}$ for the squark with on-shell slepton model. We fix $m_{\ntwo} = 495$ GeV and $m_{\none} = 416$ GeV. For these masses, there are no constraints from the chargino-neutralino direct searches. The right-handed slepton mass is fixed at 450 GeV such that $m_{\rm edge}$ in Eq. (\[eq:edge1\]) is 78 GeV, which is the optimal value for the CMS dilepton excess.
As can be seen, this model can fit the excess only in the region where $m_{\tilde q} \lsim 650$ GeV. However, this region is strongly disfavoured by the L110 signal region (shown in the blue curve) in the ATLAS stop search [@1403.4853]. This signal region requires the same final state ($2 j + 2 \ell + \met$) as the CMS dilepton analysis, in particular an OS lepton pair with $p_T > 25$ GeV and at least two jets with $p_T > 20$ GeV. The condition $m_{T2} > 110$ GeV is also imposed, which is very effective to reduce the $t \bar t$ and $WW + {\rm jets}$ backgrounds. One can see that the sensitivity of this signal region decreases as the $m_{\tilde q}$ increases due to the reduction of the production cross section. Nevertheless, the signal rate in the dilepton excess also decreases in the same way since these analyses employ similar event selection. Consequently there is no region in the plot where the SUSY events can fit the dilepton excess avoiding the exclusion from the other searches. This conclusion is robust against our assumption on the branching ratios, $Br(\tilde q_L \to q + \cone/\ntwo/\none) = 10/5/85$%, since the L110 signal region constrains the same channel as in the CMS dilepton analysis.
One can also see that in the $m_{\tilde q} > 680$ GeV region, the 2jm signal region in the ATLAS multi-jet search [@1405.7875] becomes sensitive and rules out the model points. This signal region is characterised by the requirement of at least two jets with $p_T > 130$ and 60 GeV and a moderately large effective mass, $m_{\rm eff} \equiv \sum_i |p_{Ti}^{j40}| + \met > 1200$ GeV, where $p_{Ti}^{j40}$ is the $i$-th high $p_T$ jet with $p_T > 40$ GeV. The events with an electron or muon with $> 10$ GeV are rejected in this analysis. The 2jm signal region targets the $\tilde q \tilde q \to q \none q \none$ topology, which is indeed the dominant event topology in this model since $Br(\tilde q_R \to q \none) = 100$% and $Br(\tilde q_L \to q \none) = 85$%.[^6] Due to the harsh cut on the $m_{\rm eff}$, the 2jm signal region is sensitive to the models with large mass gaps between $\tilde q$ and $\none$. This is the reason why the sensitivity increases as $m_{\tilde q}$ increases until the point ($m_{\tilde q} \simeq 850$ GeV) at which a rapid degradation of the squark production cross section finally turns the sensitivity down.
In the right panel of Fig. \[fig:squark\_results\], we plot the $N_{\ell \ell}/130$ and $R$ as functions of $m_{\tilde q}$ for the squark with off-shell $Z$ model, where we fix $m_{\ntwo} = 478$ GeV and $m_{\none} = 400$ GeV so that $m_{\rm edge}$ in Eq. (\[eq:edge2\]) is 78 GeV. One can see that the SUSY contribution is too small to account for the dilepton excess, whilst this region is severely constrained by the 2jm and 4jl signal regions in the ATLAS multi-jet search [@1405.7875]. Compared to the on-shell slepton model, the rate of an OSSF lepton from a squark cascade decay is small: $Br(\tilde q_L \to q \ntwo) \cdot Br(\ntwo \to Z^* \none) \cdot Br( Z^* \to \ell^+ \ell^-) \simeq 0.33 \cdot 1 \cdot 0.06 \simeq 2 \,\%$, though we took a maximal value $33\,\%$ for $Br(\tilde q_L \to q \ntwo)$ assuming $\ntwo$ and $\cone$ to be Wino-like. Instead, $\ntwo$ and $\cone$ have large branching ratio to hadronic modes via $Z^*$ and $W^*$ which makes the off-shell $Z$ model more prone to be excluded by the ATLAS multi-jet search [@1405.7875] compared to the on-shell slepton model due to the lepton veto cut in the analysis. The 2jm signal region constrains mostly $\tilde q_R \tilde q_R \to q \none q \none$ topology and the sensitivity peaks around $m_{\tilde q} \simeq 900$ GeV with $m_{\none} = 400$ GeV, similarly to the on-shell slepton model. On the other hand, the 4jm signal region requires at least 4 jets ($p_T > 130, 60, 60, 60$ GeV) and looks at the jets not only from the squark decay, $\tilde q \to j \tilde \chi$ ($\tilde \chi = \none, \ntwo$ or $\cone$), but also from hadronic $\cone$ and $\ntwo$ decays and initial state radiation. Due to the milder cut $m_{\rm eff} > 1000$ GeV, the sensitivity peaks at a much lower squark mass.
We conclude that for the squark models it is very difficult to fit the observed CMS dilepton excess if the ATLAS stop search [@1403.4853] and the ATLAS multi-jet search [@1405.7875] are both considered.
Sbottom scenarios {#sec:res_sbottom}
-----------------
![Signal rate and $R$-values for the on-shell left-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Lslepton"}](figures/LH-slepton_diff-50.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell left-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Lslepton"}](figures/LH-slepton_diff-90.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell left-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Lslepton"}](figures/LH-slepton_diff-130.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell left-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Lslepton"}](figures/LH-slepton_diff-170.pdf "fig:"){width="7.5cm"}
In this section we present the results for the sbottom scenarios, starting with the on-shell left-handed slepton model. In Fig. \[fig:sbottom\_Lslepton\] we show $N_{\ell \ell}/130$ and $R$ of the most constraining signal regions as functions of $m_{\tilde b_1}$. As discussed previously, we assume $m_{\ntwo} < m_{\tilde b_1} - m_t$ to avoid tops in the decay chains that would lead to more stringent constraints. Within this condition we examine four different mass gaps: $\Delta m \equiv m_{\tilde{b}_1} - m_{\ntwo} =$ 50, 90, 130 and 170 GeV. The left-handed slepton mass is fixed at $m_{\tilde{\ell}_L} = m_{\ntwo} - 40$ GeV and $m_{\none}$ is set for each combination of $m_{\ntwo}$ and $m_{\tilde{\ell_L}}$ such that $m_{\rm edge}$ in Eq. (\[eq:edge1\]) is 78 GeV. The intermediate slepton can either be a sneutrino or a charged slepton and the branching ratio of $\ntwo$ into these two states is assumed to be equal. Therefore, only half of the produced $\ntwo$ decay into an OSSF dilepton and a $\none$.
In Fig. \[fig:sbottom\_Lslepton\] we see that a good fit can be obtained for sbottom masses between 420 and 520 GeV, depending on $\Delta m$. However, these model points are strongly disfavoured by the L100 and L110 signal regions of the ATLAS stop search [@1403.4853]. The event selection in the L100 signal region is very similar to the L110 signal region which we briefly described in the previous subsection. The difference is that in the L100 signal region the lepton $p_T$ requirement is raised to $(p_T^{\ell 1}, p_T^{\ell 2}) > (100, 50)$ GeV and $m_{T2} > 100$ GeV is imposed. As the lepton $p_T$ requirement is raised with respect to L110, L100 is especially sensitive to larger mass gaps.
![Signal rate and $R$-values for the on-shell right-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Rslepton"}](figures/RH-slepton_diff-50.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell right-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Rslepton"}](figures/RH-slepton_diff-90.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell right-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Rslepton"}](figures/RH-slepton_diff-130.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the on-shell right-handed slepton mediated sbottom models.[]{data-label="fig:sbottom_Rslepton"}](figures/RH-slepton_diff-170.pdf "fig:"){width="7.5cm"}
In Fig. \[fig:sbottom\_Rslepton\] we show the contribution to the excess and the constraints from other searches in the on-shell right-handed slepton model for the four different $\Delta m$, similarly to Fig. \[fig:sbottom\_Lslepton\]. In this scenario $\ntwo$ decays into an OSSF dilepton and a $\none$ with the branching ratio of 100%. As can be seen, the results are similar to the left-handed slepton case and the region where the model gives a good fit is strongly disfavoured by the L110 and L100 signal regions of the ATLAS stop search. The similarity of the results amongst the left- and right-handed slepton scenarios can be understood because L110 and L100 constrain the same final state ($2j + 2 \ell + \met$) as that is targeted in the CMS dilepton analysis and the kinematics of the dilepton events are similar between these scenarios. We conclude that it is difficult to attribute the CMS dilepton excess to the sbottom with on-shell slepton models if the constraint from the ATLAS stop search [@1403.4853] is taken into account.
We now turn to the sbottom with off-shell $Z$ models. The first model we investigate is the right-handed sbottom model where $\ntwo$ and $\nthree$ are mostly Higgsino-like and $\none$ is mostly Bino-like. In this model the masses of three lightest neutralinos are calculated from the parameters, $\mu, M_1$ and $\tan\beta$, fixing $M_2$ at 3.5 TeV. Since we assume $\mu > M_1$, we have $m_{\nthree} \sim m_{\ntwo} \sim \mu$ and $m_{\none} \sim M_1$ and both $\ntwo$ and $\nthree$ can contribute to the excess through their decays into an off-shell $Z$ boson and a $\none$. The decay rate of the sbottom into the Higgsino states is dictated by the sbottom-bottom-Higgsino coupling which is proportional to $\tan\beta$. In order to have a large signal rate, we take $\tan\beta = 50$ in our numerical scan. We again examine four different mass gaps $\Delta m =$ 50, 90, 130, and 170 GeV between the sbottom and $\ntwo$. To this end we vary $\mu$ such that $\ntwo$ takes the desired mass set by $\Delta m$. $M_1$ is chosen such that $m_{\ntwo} - m_{\none} = 70$ GeV. The mass of the lightest sbottom is calculated from given parameters fixing the left-handed third generation squarks mass, $m_{\tilde Q_3}$, at 1.5 TeV. A table with paramter values for each model point can be found in the Appendix \[ap:pmssm\].
![Signal rate and $R$-values for the off-shell $Z$-mediated right-handed sbottom-higgsino models. []{data-label="fig:r_sbottom_results"}](figures/RHSB-higgsino_dm_50.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated right-handed sbottom-higgsino models. []{data-label="fig:r_sbottom_results"}](figures/RHSB-higgsino_dm_90.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated right-handed sbottom-higgsino models. []{data-label="fig:r_sbottom_results"}](figures/RHSB-higgsino_dm_130.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated right-handed sbottom-higgsino models. []{data-label="fig:r_sbottom_results"}](figures/RHSB-higgsino_dm_170.pdf "fig:"){width="7.5cm"}
In Fig. \[fig:r\_sbottom\_results\] we show our results again in terms of $N_{\ell \ell}/130$ and $R$. First we note that the strong constraint from L100 and L110 observed in the on-shell slepton models is relaxed. To understand this we compare the distributions of $m_{T2}$, a kinematical variable used both in the L100 and L110 signal regions, between the on-shell right-handed slepton (blue) and off-shell $Z$ models (red) in Fig. \[fig:mT2\_histo\] at similar mass spectra. We take $(m_{\tilde b_1}, m_{\ntwo}) = (400, 230)$ GeV and fix $m_{\none}$ such that $m_{\rm edge} \simeq 78$ GeV for both models. For the on-shell slepton model we take $m_{\tilde \ell} = 190$ GeV. In Fig. \[fig:mT2\_histo\] we see that the off-shell $Z$ model tends to give smaller $m_{T2}$ compared to the on-shell slepton model. The solid (dashed) vertical black line represents the event selection cut on the $m_{T2}$ variable employed in the L100 (L110) signal region. As can be seen, the off-shell $Z$ model is less sensitive to the the L100 and L110 signal regions than the on-shell slepton model.
What can also be seen from Fig. \[fig:r\_sbottom\_results\] is that for $\Delta m > 90$ GeV the SRA mCT150 signal region in the ATLAS di-bottom analysis [@1308.2631] is constraining and most of the preferred region of the dilepton excess is indeed disfavoured by this signal region. This signal region looks for two energetic $b$-jets with $p_T > 130$ and $50$ GeV in events with $m_{\rm CT} > 150$ GeV[^7] and $\met > 150$. Events containing an electron ($p_T > 7$ GeV) or a muon ($p_T > 6$ GeV) are rejected in this analysis. This signal region is more constraining for larger $\Delta m$ because the event selection requires two energetic $b$-jets.
![Histogram of $m_{T2}$-distribution for sbottom production for both the off-shell $Z$ and on-shell RH-slepton mediated case. The sbottom mass is 400 GeV and $m_{\ntwo} = 230$ GeV. For the off-shell slepton mediated case we have $m_{\tilde{\mu}_R}=190$ GeV and $m_{\none}=151$ GeV. The black vertical line indicates the cut for the limiting signal region SRA mCT150.[]{data-label="fig:mT2_histo"}](figures/mT2_histo.pdf){width="9cm"}
For $\Delta m = 50$ and 90 GeV we find the regions where the observed excess can be explained at 1-$\sigma$ level without $R > 1$ from other searches. This result is consistent with the findings reported in [@1410.4998]. However these regions are already in tension with other searches. In particular the ATLAS stop search [@1403.4853] and the ATLAS di-bottom search [@1308.2631] give $R \lsim 1$ in these regions.
![Signal rate and $R$-values for the off-shell $Z$-mediated left-handed sbottom model. Pure sbottom production is indicated by dashed lines and combined sbottom and stop production by solid lines. []{data-label="fig:sbottom_Zresults"}](figures/LHSB_diff-50.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated left-handed sbottom model. Pure sbottom production is indicated by dashed lines and combined sbottom and stop production by solid lines. []{data-label="fig:sbottom_Zresults"}](figures/LHSB_diff-90.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated left-handed sbottom model. Pure sbottom production is indicated by dashed lines and combined sbottom and stop production by solid lines. []{data-label="fig:sbottom_Zresults"}](figures/LHSB_diff-130.pdf "fig:"){width="7.5cm"} ![Signal rate and $R$-values for the off-shell $Z$-mediated left-handed sbottom model. Pure sbottom production is indicated by dashed lines and combined sbottom and stop production by solid lines. []{data-label="fig:sbottom_Zresults"}](figures/LHSB_diff-170.pdf "fig:"){width="7.5cm"}
In Fig. \[fig:sbottom\_Zresults\] we show $N_{\ell \ell}/130$ and $R$ as functions of $m_{\tilde b_1}$ in the left-handed sbottom model where $\ntwo$ ($\cone$) is assumed to be Wino-like and decays predominantly to an off-shell $Z$ ($W$) and a Bino-like $\none$. We again show the results for four different mass gaps and fix $m_{\none} = m_{\ntwo} - 70$ GeV to fit the central value of the counting experiment. As we have mentioned in section \[sc:susy\], we assume the presence of the top squark, $\tilde t_1$, with $m_{\tilde t_1} = m_{\tilde b_1}$. The solid curves in Fig. \[fig:sbottom\_Zresults\] represent the results with both $\tilde b_1 \tilde b^*_1$ and $\tilde t_1 \tilde t^*_1$ production processes. To see the impact of the tilde $\tilde t_1 \tilde t^*_1$ production on the result, we also plot the contribution to $N_{\ell \ell}/130$ and $R$ from $\tilde b_1 \tilde b^*_1$ by dashed curves.
One can see from Fig. \[fig:sbottom\_Zresults\] that for $\Delta m = 50$ and 90 GeV the model is strongly constrained by the SL5j signal region in the ATLAS jets plus 1-2 lepton analysis [@ATLAS-CONF-2013-062]. This signal region requires a soft single electron (muon) with $p_T \in [10, 25]$ $([6, 25])$ GeV and veto additional electron (muon) with $p_T > 10$ (6) GeV. It also requires $\geq 5$ jets with $p_T > (180, 25, 25, 25, 25)$ GeV. The SL5j signal region is more sensitive to the $\tilde t_1 \tilde t^*_1$ topology where one of the stops decays hadronically $\tilde t_1 \to b \cone \to b W^* \none \to b q q' \none$ and the other decays leptonically $\tilde t_1 \to b \cone \to b W^* \none \to b \ell \nu \none$, because event selection requires a single lepton. We also note that the SL5j signal region becomes less sensitive for larger $\Delta m$ because the leptons from the stop cascade decay chain are boosted in this case and do not pass the low $p_T$ requirement ($< 25$ GeV) efficiently. However, for larger $\Delta m$ the SRA mCT150 signal region becomes constraining. In particular the preferred region of the dilepton excess is disfavoured by this signal region at $\Delta m = 170$ GeV.
As a result we find a good fit to the dilepton excess at $\Delta m = 130$ GeV and $m_{\tilde b_1} \in [350, 400]$ GeV, although this region is already in tension with the SL5j signal region in the ATLAS jets plus 1-2 lepton analysis. In addition, let us note that there is an additional constrain on the $\none-{\tilde t_1}$ mass plane from CMS single-lepton analysis [@1308.1586], which is not included in our analysis. This analysis does not use the cut-and-count method but rather uses a BDT multivariate method, which prevents us from implementing this analysis. While recasting this analysis is out of the scope of this work, it is worthwhile to deduce its constraint on our models. Specifically, the exclusion contour on the $\none-{\tilde t_1}$ mass plane with chargino mass fixed at $m_{\cone}=0.25~m_{\tilde t_1} + 0.75~m_{\none}$ in the CMS analysis is most relevant to the allowed parameter space in our study ($m_{\cone}\simeq 0.3~m_{\tilde t_1} + 0.7~m_{\none}$). At $m_{\tilde t_1} \simeq 380$ GeV the CMS analysis excludes $m_{\none} \lsim 200$ GeV, whilts $m_{\none} = 180$ GeV at $m_{\tilde t_1} = 380$ GeV in the bottom left plot ($\Delta m = 130$ GeV) in Fig \[fig:sbottom\_Zresults\].
![Variation of $\tilde{b}_1$ branching ratio into $\ntwo$ for off-shell $Z$ mediated left-handed sbottom scenario with no stop production. The color indicates $N_{\ell \ell}/130$ and the black curves are lines of constant $R_{\rm max}$. Only large wino branching ratios can provide a good fit to the excess. []{data-label="fig:sbottom_2d"}](figures/LHSB_brBino.pdf){width="12cm"}
It is interesting to note that the difference between the black solid and black dashed curves are small, whereas the difference is large amongst the blue solid and blue dashed curves in the bottom left plot ($\Delta m = 130$ GeV) in Fig \[fig:sbottom\_Zresults\]. This means that the $\tilde b_1 \tilde b_1^*$ production gives the main contribution to the dilepton excess, while the model is disfavoured mainly by the additional $\tilde t_1 \tilde t_1^*$ production. Before concluding our study we show in Fig. \[fig:sbottom\_2d\] the contribution to the dilepton excess and the constraint from other searches in the $m_{\tilde b_1}$ versus $BR(\tilde b_1 \to b \ntwo)$ plane concerning only the $\tilde b_1 \tilde b_1^*$ production. In this study we assume $BR(\tilde b_1 \to b \none) = 1 - BR(\tilde b_1 \to b \ntwo)$. The region is divided into 3 colours, red, green and blue, which correspond to under, good and over fit of the dilepton excess, respectively. The $R$ value of the most constraining signal region is shown in the black contours. One can see that a good fit is found for $m_{\tilde b_1} \in [340, 380]$ GeV and $BR(\tilde b \to b \ntwo) \gsim 0.8 $ without having $R > 1$ from other searches. Within our exploration we did not find the models where the sbottom is mostly right-handed and $BR(\tilde b \to b \ntwo) \gsim 0.8$. However, this result indicates that models that have a large cross section of the topology equivalent to $\tilde b_1 \to b \ntwo \to b Z^* \none$ can in principle explain the CMS dilepton excess avoiding constraints from other ATLAS and CMS direct SUSY searches.
Conclusions {#sc:con}
===========
One straightforward supersymmetric interpretation of the observed dilepton excess by CMS [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa] is the cascade decays of light-flavour and bottom squarks. In this paper, we studied and tested the viability of promising SUSY models by deriving constraints on these from various direct SUSY searches using the automated simulation tool [Atom]{}.
In order to obtain a contribution to the dilepton excess from SUSY events, we considered the decay of the second lightest neutralino, $\ntwo$, via either an off-shell $Z$ or an intermediate on-shell slepton. The $\ntwo$ itself arises from a light-flavour squark or sbottom decay. We investigated in total six possible simplified models, see figures \[fig:squark\_model\], \[fig:sbottom\_slep\] and \[fig:sbottom\_Z\].
We found that all of these models are already in strong tension with the experimental data once we demand a good fit to the dilepton excess. In particular strong limits arise from an earlier neglected ATLAS stop search [@1403.4853] with identical final state topology. This analysis alone rules out the interpretation of the excess in terms of an intermediate (left- or right-handed) on-shell slepton for both light squark and sbottom production, see left panel of Fig. \[fig:squark\_results\], Fig. \[fig:sbottom\_Lslepton\] and Fig. \[fig:sbottom\_Rslepton\] respectively. We showed that if multijet plus missing energy searches are taken into account, the off-shell $Z$ scenario with squark production is strongly disfavoured and noted that it is not able to give a sizeable contribution to the dilepton signal region, as can be seen in the right panel of Fig. \[fig:squark\_results\].
We confirmed the result reported in [@1410.4998] and showed that the right-handed sbottom model with Higgsino-like $\ntwo$ and $\nthree$ decaying predominantly into an off-shell $Z$ can explain the dilepton excess at 1-$\sigma$ level, although the model is already in tension with the ATLAS di-bottom search and the ATLAS stop search, as can be seen in Fig. \[fig:r\_sbottom\_results\]. This tension can be ameliorated if the left-handed sbottom model with Wino-like $\ntwo$ is considered. However, the left-handed stop is necessarily introduced in this model and that creates another tension with the ATLAS jets plus 1-2 lepton analysis as be seen in Fig. \[fig:sbottom\_Zresults\].
We also showed in Fig. \[fig:sbottom\_2d\] that in a simplified model that only has sbottom production the dilepton excess can be explained avoiding constraints from other searches in the region where $m_{\tilde b_1} - m_{\ntwo} \sim 130$ GeV, $m_{\tilde{b}} \sim 360$ GeV and $BR(\tilde b_1 \to b \ntwo) \gsim 0.8$, although we did not find a corresponding model point in the context of the MSSM within our exploration. This results may indicate that a more non-trivial SUSY scenario should be considered to explain the CMS dilepton excess.
[**Note added:**]{} Shortly after this paper was submitted to arXiv, CMS updated their result [@Khachatryan:2015lwa] and reported most of the excessive events are observed associated with at least one $b$-jet. This new information further disfavours the squark scenario, which does not change our conclusion. Shortly after the CMS update, ATLAS released their new analysis of the jets plus SFOS dilepton channel [@Aad:2015wqa]. They explicitly looked at the signal region employed in the CMS dilepton analysis and did not find any significant excess. This casts a doubt that observed dilepton excess is merely due to the statistical fluctuation or background mismodeling. The next run of the LHC will provide a definitive answer to this question.
Validation {#ap:validation}
==========
Here we show the validation results of our implementation of CMS-PAS-SUS-12-019/1502.06031 [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa] and the ATLAS stop search with two lepton final state [@1403.4853].
The benchmark point considered in the CMS analysis has a sbottom of mass 400 GeV decaying via $\tilde{b} \rightarrow\ntwo b$ with 100%. The second lightest neutralino then undergoes an off-shell $Z$ decay with SM branching ratios. We show the good agreement between the CMS results and our implemented analysis in [Atom]{} in table \[tbl:val\_cms\]. There, we give the event numbers in the central and forward signal regions as quoted by the CMS collaboration and their ratio to our results obtained with [Atom]{}.
Additionally, we provide validation results for the stop search because of the strong constraints that we derive from this analysis. The ATLAS benchmark scenario consists of a stop decaying to $\cone + b$ with 100% probability followed by a decay of $\cone$ via a $W$ into $\none$ and Standard Model particles. We show our validation in table \[tbl:val\_atlas\]. In this table we present event numbers for the same-flavour (SF) and different-flavour (DF) case as given by ATLAS and their ratio to our results in the column [Atom]{}/Exp.
$( m_{\tilde{b}},m_{\ntwo} ) = ( 400,150 )~\GEV$ Central [Atom]{}/Exp Forward [Atom]{}/Exp
-------------------------------------------------- ----------------- -------------- ---------------- --------------
$N_{\rm jets}~\geq 2$(no $\met$ requirement) $242.7\pm 2.8$ 1.04 $34.2\pm 1.1$ 0.77
$N_{\rm jets}~\geq 3$(no $\met$ requirement) $186.2\pm 2.5$ 1.09 $25.6\pm 0.9$ 0.76
$\met > 100~\GEV$(no $N_{\rm jets}$ requirement) $152.5\pm 2.1$ 1.03 $19.8\pm 0.8$ 0.98
$\met > 150~\GEV$(no $N_{\rm jets}$ requirement) $85.0\pm 1.5$ 0.93 $10.4\pm 0.5$ 0.87
Signal region $132.4 \pm 2.0$ 1.031 $17.0 \pm 0.7$ 0.937
: Validation table for our implementation of the CMS-PAS-SUS-12-019/1502.06031 analysis [@CMS-PAS-SUS-12-019; @Khachatryan:2015lwa] in [Atom]{}. \[tbl:val\_cms\]
$( m_{\tilde{t}},m_{\cone},m_{\none} ) = ( 400, 390, 195 )~\GEV$ SF [Atom]{}/Exp DF [Atom]{}/Exp
------------------------------------------------------------------ -------- -------------- -------- --------------
$\Delta \phi > 1$ 1834.9 1.09 2390.1 1.06
$\Delta \phi_{b}$ 1402.8 1.07 1800.5 1.07
$m_{T2} > 90~\GEV$ 396.5 1.02 500.0 1.09
$m_{T2} > 120~\GEV$ 211.8 1.01 284.4 1.1
$m_{T2} > 100~\GEV$, $p_{T,{\rm jet}} > 100~\GEV$ 21.7 1.4 35.0 0.99
$m_{T2} > 110~\GEV$, $p_{T,{\rm jet}} > 20~\GEV$ 86.0 0.95 116.1 0.89
: Validation table for our implementation of the ATLAS stop search with two leptons [@1403.4853] in [Atom]{}. \[tbl:val\_atlas\]
Parameter Values for pMSSM scan {#ap:pmssm}
===============================
In table \[tbl:pmssm\] we give additinal pMSSM input parameters as well as the sum of the branching ratio of $\tilde{b}_1$ to $\ntwo$, $\nthree$. These points were used to scan the right-handed sbottom-Higgsino model. Calculation of the physical SUSY masses and branching ratios was done using SPheno [@Porod2003; @Porod2011].
$\Delta m (\tilde{b}, \ntwo)=$50 $\Delta m (\tilde{b}, \ntwo)=$90 $\Delta m (\tilde{b}, \ntwo)=$130 $\Delta m (\tilde{b}, \ntwo)=$170
--------------------- ---------------------------------- ---------------------------------- ----------------------------------- -----------------------------------
$m_{\tilde{b}}=$280 (229,157,0.48) (183,157,0.63) (140,157,0.69) (96,157,0.63)
$m_{\tilde{b}}=$320 (270,215,0.48) (228,215,0.63) (183,215,0.69) (140,215,0.67)
$m_{\tilde{b}}=$360 (307,263,0.46) (269,263,0.63) (228,263,0.69) (183,263,0.7)
$m_{\tilde{b}}=$400 (345,310,0.46) (307,310,0.63) (269,310,0.7) (228,310,0.71)
: Additional information about the right-handed sbottom-higgsino model. We give values for ($\mu$, $m_{\tilde{b}_R}$,$\sum_{i=2,3} BR(\tilde{b} \rightarrow b \, N_i)$) for each model point. All masses and mass differences are given in GeV. See text for more details. \[tbl:pmssm\]
Acknowledgment {#acknowledgment .unnumbered}
==============
We are grateful to J. S. Kim, K. Rolbiecki, and J. Tattersall for collaborations during the early stages of this work. P.G. is supported by an ERC grant. S.P.L. is supported by JSPS Research Fellowships for Young Scientists and the Program for Leading Graduate Schools, MEXT, Japan. The work of K.S. was supported in part by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research Council via the Advanced Investigator Grant 267352.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: See [@Allanach:2015ria] for a non-SUSY interpretation of the observed excess.
[^5]: In our setup the $\ntwo$ decays predominantly into muon pairs through the Higgsino component of the $\ntwo$.
[^6]: We note that in [@1405.7875] ATLAS does not exclude the region where $m_{\none} = 416$ GeV in the squark-neutralino simplified model. We, on the other hand, exclude this neutralino mass for a certain range of the squark mass (see Fig. \[fig:squark\_results\] (left)). This is because our squark production cross section is larger than the ATLAS’s value since we set the gluino mass at 3.5 TeV in which the contribution from the gluino exchange diagram is still sizeable.
[^7]: $m_{\rm CT} \equiv \sqrt{ (E_T^{b_1} + E_T^{b_2})^2 - ({\bf p}_T^{b_1} - {\bf p}_T^{b_2})^2}$, where $E_T$ and ${\bf p}_T$ are the transverse energy and the transverse momentum vector, respectively.
|
---
abstract: 'We propose a topic-guided variational autoencoder (TGVAE) model for text generation. Distinct from existing variational autoencoder (VAE) based approaches, which assume a simple Gaussian prior for the latent code, our model specifies the prior as a Gaussian mixture model (GMM) parametrized by a neural topic module. Each mixture component corresponds to a latent topic, which provides guidance to generate sentences under the topic. The neural topic module and the VAE-based neural sequence module in our model are learned jointly. In particular, a sequence of invertible Householder transformations is applied to endow the approximate posterior of the latent code with high flexibility during model inference. Experimental results show that our TGVAE outperforms alternative approaches on both unconditional and conditional text generation, which can generate semantically-meaningful sentences with various topics.'
author:
- |
Wenlin Wang$^{1}$, Zhe Gan$^{2}$, Hongteng Xu$^{1,3}$, Ruiyi Zhang$^{1}$, Guoyin Wang$^{1}$,\
**Dinghan Shen**$^{1}$, **Changyou Chen**$^{4}$, **Lawrence Carin**$^{1}$\
$^{1}$Duke University, $^{2}$Microsoft Dynamics 365 AI Research,\
$^{3}$Infinia ML, Inc, $^{4}$University at Buffalo\
[[email protected] ]{}\
bibliography:
- 'textVAE.bib'
title: 'Topic-Guided Variational Autoencoders for Text Generation'
---
Introduction
============
Text generation plays an important role in various natural language processing (NLP) applications, such as machine translation [@cho2014learning; @sutskever2014sequence], dialogue generation [@li2017adversarial], and text summarization [@nallapati2016abstractive; @rush2015neural]. As a competitive solution to this task, the variational autoencoder (VAE) [@kingma2013auto; @rezende2014stochastic] has been widely used in text-generation systems [@bowman2015generating; @hu2017toward; @serban2017hierarchical]. In particular, VAE defines a generative model that propagates latent codes drawn from a simple prior through a decoder to manifest data samples. The generative model is further augmented with an inference network, that feeds observed data samples through an encoder to yield a distribution on the corresponding latent codes.
Compared with other potential methods, $e.g.$, those based on generative adversarial networks (GANs) [@yu2017seqgan; @guo2017long; @zhang2017adversarial; @zhang2018sequence; @chen2018adversarial], VAE is of particular interest when one desires not only text generation, but also the capacity to infer meaningful latent codes from text. Ideally, semantically-meaningful latent codes can provide high-level guidance while generating sentences. For example, when generating text, the vocabulary could potentially be narrowed down if the input latent code corresponds to a certain topic (*e.g.*, the word “military” is unlikely to appear in a sports-related document).
However, in practice this desirable property is not fully achieved by existing VAE-based text generative models, because of the following two challenges. First, the sentences in documents may associate with different semantic information (*e.g.*, topic, sentiment, etc.) while the latent codes of existing VAE-based text generative models often employ a simple Gaussian prior, which cannot indicate the semantic structure among sentences and may reduce the generative power of the decoder. Although some variants of VAE try to impose some structure on the latent codes [@jiang2016variational; @dilokthanakul2016deep], they are often designed with pre-defined parameter settings without incorporating semantic meanings into the latent codes, which may lead to over-regularization [@dilokthanakul2016deep].
The second issue associated with VAE-based text generation is “posterior collapse,” first identified in @bowman2015generating. With a strong auto-regressive decoder network (*e.g.*, LSTM), the model tends to ignore the information from the latent code and merely depends on previous generated tokens for prediction. Several strategies are proposed to mitigate this problem, including making the decoder network less auto-regressive ($i.e.$, using less conditional information while generating each word) [@yang2017improved; @shen2017deconvolutional], or bridging the amortization gap (between the log-likelihood and the ELBO) using semi-amortized inference networks [@kim2018semi]. However, these methods mitigate the issue by weakening the conditional dependency on the decoder, which may fail to generate high-quality continuous sentences.
To overcome the two problems mentioned above, we propose a topic-guided variational autoencoder (TGVAE) model, permitting text generation with designated topic guidance. As illustrated in Figure \[fig: illustration\], TGVAE specifies a Gaussian mixture model (GMM) as the prior of the latent code, where each mixture component corresponds to a topic. The GMM is learnable based on a neural topic model — the mean and diagonal covariance of each mixture component is parameterized by the corresponding topic. Accordingly, the degree to which each component of the GMM is used to generate the latent code and the corresponding sentence is tied to the usage of the topics. In the inference phase, we initialize the latent code from a GMM generated via the encoder, and apply the invertiable Householder transformation [@bischof1994orthogonal; @sun1995basis] to derive the latent code with high flexibility and low complexity.
As shown in Figure \[fig: illustration\_summarization\], besides unconditional text generation, the proposed model can be extended for conditional text generation, *i.e.,* abstractive text summarization [@nallapati2016abstractive] with an attention module. By injecting the topics learned by our model (semantic information), we are able to make better use of the source document and improve a sequence-to-sequence summarization model [@sutskever2014sequence].
We highlight the contributions of our model as follows: (*i*) A new Topic-Guided VAE (TGVAE) model is proposed for text generation with designated topic guidance. (*ii*) For the model inference, Householder flow is introduced to transform a relatively simple mixture distribution into an arbitrarily flexible approximate posterior, achieving powerful approximate posterior inference. (*iii*) Experiments for both unconditional and conditional text generation demonstrate the effectiveness of the proposed approach.
Model
=====
The proposed TGVAE, as illustrated in Figure \[fig: illustration\], consists of two modules: a neural topic model (NTM) and a neural sequence model (NSM). The NTM aims to capture long-range semantic meaning across the document, while the NSM is designed to generate a sentence with designated topic guidance.
Neural Topic Model
------------------
Let $\dv\in \mathbb{Z}_+^D$ denote the bag-of-words representation of a document, with $\mathbb{Z}_+$ denoting non-negative integers. $D$ is the vocabulary size, and each element of $\dv$ reflects a count of the number of times the corresponding word occurs in the document. Let $a_n$ represent the topic assignment for word $w_n$. Following @miao2017discovering, a Gaussian random vector is passed through a softmax function to parameterize the multinomial document topic distributions. Specifically, the generative process of the NTM is $$\begin{aligned}
\thetav \sim \mathcal{N}(0, \Imat)&, \quad \tv = g(\thetav) \,, \\
\quad a_n \sim \text{Discrete}(\tv)&, \quad w_n \sim \text{Discrete}(\betav_{a_n})\,, \nonumber\end{aligned}$$ where $\mathcal{N}(0, \Imat)$ is an isotropic Gaussian distribution, $g(\cdot)$ is a transformation function that maps sample $\thetav$ to the topic embedding $\tv$, defined here as $g(\thetav) = \mbox{softmax}(\hat{\Wmat}\thetav + \hat{\bv})$, where $\hat{\Wmat}$ and $\hat{\bv}$ are trainable parameters; $\betav_{a_n}$ represents the distribution over words for topic $a_n$; $n\in [1,N_d]$, and $N_d$ is the number of words in the document. The marginal likelihood for document $\dv$ is:
The second equation in (\[eqn:eq1\]) holds because we can marginalize out the sampled topic words $a_n$ by
where $\betav = \{\betav_i\}_{i=1}^{T}$ are trainable parameters of the decoder; $T$ is the number of topics and each $\betav_i\in \R^D$ is a topic distribution over words (all elements of $\betav_i$ are nonnegative, and sum to one).
Neural Sequence Model
---------------------
Our neural sequence model for text generation is built upon the VAE proposed in @bowman2015generating. Specifically, a continuous latent code $\zv$ is first generated from some prior distribution $p(\zv)$, based on which the text sequence $\yv$ is then generated from a conditional distribution $p(\yv|\zv)$ parameterized by a neural network (often called the decoder). Since the model incorporates a latent variable $\zv$ that modulates the entire generation of the sentence, it should be able to capture the high-level source of variation in the data.
#### Topic-Guided Gaussian Mixture Prior
The aforementioned intuition is hard to be captured by a standard VAE, simply imposing a Gaussian prior on top of $\zv$, since the semantic information associated with a document intrinsically contains different subgroups (such as topics, sentiment, etc.). In our model, we consider incorporating the topic information into latent variables. Our model assumes each $\zv$ is drawn from a *topic-dependent* GMM, that is, $$\begin{aligned}
p(\zv | \betav, \tv) &= \sideset{}{_{i=1}^T}\sum t_i \mathcal{N}( \muv(\betav_i), \sigmav^2(\betav_i)) ~\nonumber \\
\muv(\betav_i) &= f_{\mu}(\betav_i) ~\nonumber \\
\sigmav^2(\betav_i) &= \text{diag}(\exp{(f_\sigma(\betav_i))}) \,,
\label{eq:meancov}\end{aligned}$$ where $t_i$ is the usage of topic $i$ in a document and $\betav_i$ is the $i$-th topic distribution over words. Both of them are inherited from the NTM discussed above. Both $f_\mu(\cdot)$ and $f_\sigma(\cdot)$ are implemented as feedforward neural networks, with trainable parameters $\Wmat_\mu$ and $\Wmat_\sigma$, respectively. Compared with a normal GMM prior that sets each mixture component to be $\mathcal{N}(0,\Imat)$, the proposed topic guided GMM prior provides semantic meaning for each mixture component, and hence makes the model more interpretable and controllable for text generation.
#### Decoder
The likelihood of a word sequence $\yv = \{y_m\}_{m=1}^M$ conditioned on the latent code $\zv$ is defined as: $$\begin{aligned}
p(\yv|\zv) &= p(y_1|\zv)\sideset{}{_{m=2}^{M}}\prod p(y_m | y_{1:m-1}, \zv) ~\nonumber \\
&=p(y_1|\zv)\sideset{}{_{m=2}^{M}}\prod p(y_m | \hv_m) \,,\end{aligned}$$ where the conditional probability of each word $y_m$ given all the previous words $y_{1:m-1}$ and the latent code $\zv$ is defined through the hidden state $\hv_m$: $\hv_m = f(\hv_{m-1}, y_{m-1}, \zv)$, where the function $f(\cdot)$ is implemented as a Gated Recurrent Unit (GRU) cell [@cho2014learning] in our experiments.
Inference
=========
The proposed model (see Figure \[fig: illustration\]) takes the bag-of-words as input and embeds a document into a topic vector. The topic vector is then used to reconstruct the bag-of-words input, and the learned topic distribution over words is used to model a topic-dependent prior to generate a sentence in the VAE setup. Specifically, the joint marginal likelihood can be written as: $$\begin{aligned}
p(\yv, \dv | \betav) = \int_\thetav &\int_{\zv} p(\thetav) p(\dv | \betav, \thetav) \nonumber \\
&\cdot p(\zv | \betav, \thetav) p(\yv|\zv) \,d\thetav d\zv \,. ~\label{jointmarginal}\end{aligned}$$ Since direct optimization of (\[jointmarginal\]) is intractable, auto-encoding variational Bayes is employed [@kingma2013auto]. Denote $q(\thetav | \dv)$ and $q(\zv | \yv)$ as the variational distributions for $\thetav$ and $\zv$, respectively. The variational objective function, also called the evidence lower bound (ELBO), is constructed as
By assuming $$\begin{aligned}
q(\thetav|\dv) = \mathcal{N} (\thetav|g_{\mu}(\dv),\text{diag}(\exp{(g_\sigma(\dv))})), \nonumber\end{aligned}$$ where both $g_\mu(\cdot)$ and $g_\sigma(\cdot)$ are implemented as feed-forward neural networks, the re-parameterization trick [@kingma2013auto] can be applied directly to build an unbiased and low-variance gradient estimator for the $\mathcal{L}_t$ term in (\[eqn:elbo\]). Below, we discuss in detail how to approximate the $\mathcal{L}_s$ term in (\[eqn:elbo\]) and infer an arbitrarily complex posterior for $\zv$. Note that $\zv$ is henceforth represented as $\zv_K$ in preparation for the introduction of Householder flows.
Householder Flow for Approximate Posterior
------------------------------------------
Householder flow [@zhang2017learning; @tomczak2016improving] is a volume-preserving normalizing flow [@rezende2015variational], capable of constructing an arbitrarily complex posterior $q_K(\zv_K|\yv)$ from an initial random variable $\zv_0$ with distribution $q_0$, by composing a sequence of invertible mappings, *i.e.*, $\zv_K = f_K \circ \cdots \circ f_2 \circ f_1 (\zv_0)$. By repeatedly applying the chain rule and using the property of Jacobians of invertible functions, $q_K(\zv_K|\yv)$ is expressed as:
where $| \det \frac{\partial f_k}{\partial \zv_{k-1}}|$ is the absolute value of the Jacobian determinant. Therefore, the $\mathcal{L}_s$ term in (\[eqn:elbo\]) may be rewritten as
Here $q_0(\zv_0|\yv)$ is also specified as a GMM, *i.e.*, $q_0 (\zv_0 | \yv) = \sum_{i=1}^T \pi_i(\yv) \mathcal{N}(\muv_i(\yv), \sigmav^2_i(\yv))$. As illustrated in Figure \[fig: illustration\], $\yv$ is first represented as a hidden vector $\hv$, by encoding the text sequence with an RNN. Based on this, the mixture probabilities $\piv$, the means and diagonal covariances of all the mixture components are all produced by an encoder network, which is a linear layer with the input $\hv$. In (\[eqn:normalizing\_flow\_objective\]), the first term can be considered as the reconstruction error, while the remaining two terms act as regularizers, the tractability of which is important for the whole framework.
#### KL Divergence between two GMMs
Since both the prior $p(\zv_K|\betav,\thetav)$ and the initial density $q_0(\zv_0|\yv)$ for the posterior are GMMs, the calculation of the third term in (\[eqn:normalizing\_flow\_objective\]) requires the KL divergence between two GMMs. Though no closed-form solutions exist, the KL divergence has an explicit upper bound [@dilokthanakul2016deep], shown in Proposition 1.
**Proposition 1.** *For any two mixture densities* $p=\sum_{i=1}^{n} \pi_i g_i$ *and* $\hat{p}=\sum_{i=1}^{n} \hat{\pi}_i \hat{g}_i$, *their KL divergence is upper-bounded by*
*where equality holds if and only if* $\frac{\pi_ig_i}{\sum_{i=1}^{n}\pi_ig_i} = \frac{\hat{\pi}\hat{g_i}}{\sum_{i=1}^n\hat{\pi}\hat{g_i}}$.
*Proof.* With the log-sum inequality
Since the KL divergence between two Gaussian distributions has a closed-form expression, the upper bound of the KL divergence between two GMMs can be readily calculated. Accordingly, the third term in (\[eqn:normalizing\_flow\_objective\]) is upper bounded as
where the expectation $\mathbb{E}_{q(\thetav | \dv)}[\cdot]$ can be approximated by a sample from $q(\thetav | \dv)$.
#### Householder Flow
Householder flow [@tomczak2016improving] is a series of Householder transformations, defined as follows. For a given vector $\zv_{k-1}$, the reflection hyperplane can be defined by a Householder vector $\vv_t$ that is orthogonal to the hyperplane. The reflection of this point about the hyperplane is $$\begin{aligned}
\zv_k = \left(\mathbf{I} - 2\frac{\vv_k\vv_k^T}{||\vv_k||^2}\right)\zv_{k-1} = \Hmat_k \zv_{k-1} \,,
~\label{hf}\end{aligned}$$ where $\Hmat_k = \mathbf{I} - 2\frac{\vv_k\vv_k^T}{||\vv_k||^2}$ is called the *Householder matrix*. An important property of the *Householder matrix* is that the absolute value of the Jacobian determinant is equal to 1, therefore $\sum_{k=1}^K \log \Big| \det \frac{\partial f_k}{\partial \zv_{k-1}}\Big| = \sum_{k=1}^K \log |\det \Hmat_k| = 0$, significantly simplifying the computation of the lower bound in (\[eqn:normalizing\_flow\_objective\]). For $k=1,\ldots,K$, the vector $\vv_k$ is produced by a linear layer with the input $\vv_{k-1}$, where $\vv_0=\hv$ is the last hidden vector of the encoder RNN that encodes the sentence $\yv$.
Finally, by combining (\[eqn:elbo\]), (\[eqn:normalizing\_flow\_objective\]) and (\[eqn:gmm\_kl\]), the ELBO can be rewritten as $$\begin{aligned}
\mathcal{L} \geq \mathcal{L}_t + \mathbb{E}_{q_0(\zv_0|\yv)} [\log p(\yv|\zv_K)] - \mathcal{U}_{KL} \,.\end{aligned}$$
Extension to text summarization
-------------------------------
When extending our model to text summarization, we are interested in modeling $p(\yv,\dv|\xv)$, where $(\xv, \yv)$ denotes the document-summary pair, and $\dv$ denotes the bag-of-words of the input document. The marginal likelihood can be written as $p(\yv, \dv |\xv) = \int_\thetav \int_{\zv} p(\thetav) p(\dv | \betav, \thetav) p(\zv | \betav, \thetav) p(\yv|\xv,\zv) \,d\thetav d\zv$. Assume the approximate posterior of $\zv$ is only dependent on $\xv$, *i.e.*, $q(\zv|\xv)$ is proposed as the variational distribution for $\zv$. The ELBO is then constructed as $$\begin{aligned}
\mathcal{L} = &\mathcal{L}_t
+ \mathbb{E}_{q(\zv | \xv)}\left[ \log p(\yv|\xv, \zv) \right] \nonumber \\
&- \mathbb{E}_{q(\thetav | \dv)}\left[\text{KL}\left( q(\zv | \xv) || p(\zv|\betav,\thetav)\right)\right] \,,\end{aligned}$$ where $\mathcal{L}_t$ is the same as used in (\[eqn:elbo\]). The main difference when compared with unconditional text generation lies in the usage of $p(\yv|\xv, \zv)$ and $q(\zv | \xv)$, illustrated in Figure \[fig: illustration\_summarization\]. The generation of $\yv$ given $\xv$ is not only dependent on a standard Seq2Seq model with attention [@nallapati2016abstractive], but also affected by $\zv$ (*i.e.*, $\zv_K$), which provides the high-level topic guidance.
Diversity Regularizer for NTM
-----------------------------
Redundancy in inferred topics is a common issue existing in general topic models. In order to address this, it is straightforward to regularize the row-wise distance between paired topics to diversify the topics. Following @xie2015diversifying [@miao2017discovering], we apply a topic diversity regularization while carrying out the inference.
Specifically, the distance between a pair of topics is measured by their cosine distance $a(\betav_i, \betav_j) = \arccos \left( \frac{|\betav_i\cdot \betav_j|}{\|\betav_i\|_2 \|\betav_j\|_2}\right)$. The mean angle of all pairs of $T$ topics is $\phi = \frac{1}{T^2}\sum_i\sum_j a(\betav_i, \betav_j)$, and the variance is $\nu = \frac{1}{T^2}\sum_i\sum_j(a(\betav_i, \betav_j) - \phi)^2$. Finally, the topic-diversity regularization is defined as $R=\phi - \nu$.
Related Work
============
The VAE was proposed by @kingma2013auto, and since then, it has been applied successfully in a variety of applications [@gregor2015draw; @kingma2014semi; @chen2017continuous; @wang2018zero; @shen2018nash]. Focusing on text generation, the methods in @miao2017discovering [@miao2016neural; @srivastava2017autoencoding] represent texts as bag-of-words, and @bowman2015generating proposed the usage of an RNN as the encoder and decoder, and found some negative results. In order to improve the performance, different convolutional designs [@semeniuta2017hybrid; @shen2017deconvolutional; @yang2017improved] have been proposed. A VAE variant was further developed in @hu2017toward to control the sentiment and tense of generated sentences. Additionally, the VAE has also been considered for conditional text generation tasks, including machine translation [@zhang2016variational], image captioning [@pu2016variational], dialogue generation [@serban2017hierarchical; @shen2017conditional; @zhao2017learning] and text summarization [@li2017deep; @miao2016language]. In particular, distinct from the above works, we propose the usage of a topic-dependent prior to explicitly incorporate topic guidance into the text-generation framework.
The idea of using learned topics to improve NLP tasks has been explored previously, including methods combining topic and neural language models [@ahn2016neural; @dieng2016topicrnn; @lau2017topically; @mikolov2012context; @wang2017topic], as well as leveraging topic and word embeddings [@liu2015topical; @xu2018distilled]. Distinct from them, we propose the use of topics to guide the prior of a VAE, rather than only the language model (*i.e.*, the decoder in a VAE setup). This provides more flexibility in text modeling and also the ability to infer the posterior on latent codes, which could be useful for visualization and downstream tasks.
Neural abstractive summarization was pioneered in @rush2015neural, and it was followed and extended by @chopra2016abstractive. Currently the RNN-based encoder-decoder framework with attention [@nallapati2016abstractive; @see2017get] remains popular in this area. Attention models typically work as a keyword detector, which is similar to topic modeling in spirit. This fact motivated us to extend our topic-guided VAE model to text summarization.
Experiments
===========
We evaluate our TGVAE on text generation and text summarization, and interpret its improvements both quantitatively and qualitatively.
Text Generation
---------------
**Dataset** We conduct experiments on three publicly available corpora: APNEWS, IMDB and BNC.[^1] APNEWS[^2] is a collection of Associated Press news articles from 2009 to 2016. <span style="font-variant:small-caps;">IMDB</span> is a set of movie reviews collected by @maas2011learning, and <span style="font-variant:small-caps;">BNC</span> [@BNCConsortium2007] is the written portion of the British National Corpus, which contains excerpts from journals, books, letters, essays, memoranda, news and other types of text. For the three corpora, we tokenize the words and sentences, lowercase all word tokens, and filter out word tokens that occur less than 10 times. For the topic model, we remove stop words in the documents and exclude the top $0.1\%$ most frequent words and also words that appear less than 100 documents. A summary statistics is provided in Table \[Table:datasetsStatistics\].
**Evaluation** We first compare the perplexity of our neural sequence model with a variety of baselines. Further, we evaluate BLEU scores on the generated sentences, noted as *test*-BLEU and *self*-BLEU. *test*-BLEU (higher is better) evaluates the quality of generated sentences using a group of real test-set sentences as the reference, and *self*-BLEU (lower is better) mainly measures the diversity of generated samples [@zhu2018texygen].
**Setup** For the neural topic model (NTM), we consider a 2-layer feed-forward neural network to model $q(\thetav | \dv)$, with 256 hidden units in each layer; ReLU is used as the activation function. The hyper-parameter $\lambda$ for the neural topic model diversity regularizer is fixed to $0.1$ across all the experiments. All the sentences in the paragraph are used to obtain the bag-of-words presentation $\dv$. The maximum number of words in a paragraph is set to 300. For the neural sequence model (NSM), we use bidirectional-GRU as the encoder and a standard GRU as the decoder. The hidden state of our GRU is fixed to $600$ across all the three corpora. For the input sequence, we fix the sequence length to 30. In order to avoid overfitting, dropout with a rate of $0.4$ is used in each GRU layer.
**Baseline** We test the proposed method with different numbers of topics (components in GMM) and different numbers of Householder flows ($i.e.$, $K$), and compare it with six baselines: (*i*) a standard language model (LM); (*ii*) a standard variational RNN auto-encoder (VAE); (*iii*) a Gaussian prior-based VAE with Householder Flow (VAE+HF); (*iv*) a standard LSTM language model with LDA as additional feature (LDA+LSTM); (*v*) Topic-RNN [@dieng2016topicrnn], a joint learning framework which learns a topic model and a language model simultaneously; (*vi*) TDLM [@lau2017topically], a joint learning framework which learns a convolutional based topic model and a language model simultaneously.
**Results** The results in Table \[perplexity\] show that the models trained with a VAE and its Householder extension does not outperform a well-optimized language model, and the KL term tends to be annealed with the increase of $K$. In comparison, our TGVAE achieves a lower perplexity upper bound, with a relative larger $\mathcal{U}_{KL}$. We attribute the improvements to our topic guided GMM model design, which provides additional topical clustering information in the latent space; the Householder flow also boosts the posterior inference for our TGVAE. We also observe consistent improvements with the number of topics, which demonstrates the efficiency of our TGVAE.
To verify the generative power of our TGVAE, we generate samples from our *topic-dependent* prior and compare various methods on the BLEU scores in Table \[testBLEU\]. With the increase of topic numbers, our TGVAE yields consistently better *self*-BLEU and a boost over *test*-BLEU relative to standard VAE models. We also show a group of sampled sentences drawn from a portion of topics in Table \[Table:generateSentences\]. Our TGVAE is able to generate diverse sentences under topic guidance. When generating sentences under a mixture of topics, we draw multiple samples from the GMM and take $\zv$ as the averaged sample.
Though this paper focuses on generating coherent topic-specific sentences rather than the learned topics themselves, we also evaluate the topic coherence [@lau2017topically] to show the rationality of our joint learning framework. We compute topic coherence using normalized PMI (NPMI). In practice, we average topic coherence over the top 5/10/15/20 topic words. To aggregate topic coherence score, we further average the coherence scores over topics. Results are summarized in Table \[Table:coherence\].
Text Summarization
------------------
**Dataset** We further test our model for text summarization on two popular datasets. First, we follow the same setup as in @rush2015neural and consider the <span style="font-variant:small-caps;">Gigawords</span> corpus[^3], which consists of $3.8$M training pair samples, $190$K validation samples and 1,951 test samples for evaluation. An input-summary pair consists of the first sentence and the headline of the source articles. We also evaluate various models on the <span style="font-variant:small-caps;">DUC-2004</span> test set[^4], which has $500$ news articles. Different from <span style="font-variant:small-caps;">Gigawords</span>, each article in <span style="font-variant:small-caps;">DUC-2004</span> is paired with four expert-generated reference summaries. The length of each summary is limited to $75$ bytes.
**Evaluation** We evaluate the performance of our model with the ROUGE score [@lin2004rouge], which counts the number of overlapping content between the generated summaries and the reference summaries, *e.g.*, overlapped n-grams. Following practice, we use F-measures of ROUGE-1 (RF-1), ROUGE-2 (RF-2) and ROUGE-L (RF-L) for <span style="font-variant:small-caps;">Gigawords</span> and Recall measures of ROUGE-1 (RR-1), ROUGE-2 (RR-2) and ROUGE-L (RR-L) for <span style="font-variant:small-caps;">DUC-2004</span>.
**Setup** We have a similar data tokenization as we have in text generation. Additionally, for the vocabulary, we count the frequency of words in both the source article the target summary, and maintain the top 30,000 tokens as the source article and target summary vocabulary. For the NTM, we further remove top $0.3\%$ words and infrequent words to get a topic model vocabulary in size of $8000$. For the NTM, we follow the same setup as our text generation. In the NSM, we keep using bidirectional-GRU as the encoder and a standard GRU as the decoder. The hidden state is fixed to $400$. An attention mechanism [@bahdanau2014neural] is applied in our sequence-to-sequence model.
**Baseline** We compare our method with the following alternatives: (*i*) a standard sequence-to-sequence model with attention [@bahdanau2014neural] (Seq2Seq); (*ii*) a model similar to our TGVAE, but without the usage of the topic-dependent prior and Householder flow (Var-Seq2Seq); and (*iii*) a model similar to our TGVAE, but without the usage of the topic dependent prior (Var-Seq2Seq-HF).
**Results** The results in Table \[Table:Rouge\] show that our TGVAE achieves better performance than a variety of strong baseline methods on both <span style="font-variant:small-caps;">Gigawords</span> and <span style="font-variant:small-caps;">DUC-2004</span>, demonstrating the practical value of our model. It is worthwhile to note that recently several much more complex CNN/RNN architectures have been proposed for abstract text summarization, such as SEASS [@zhou2017selective], ConvS2S [@gehring2017convolutional], and Reinforced-ConvS2S [@wang2018reinforced]. In this work, we focus on a relatively simple RNN architecture for fair comparison. In such a way, we are able to conclude that the improvements on the results are mainly from our topic-guided text generation strategy. As can be seen, though the Var-Seq2Seq model achieves comparable performance with the standard Seq2Seq model, the usage of Householder flow for more flexible posterior inference boosts the performance. Additionally, by combining the proposed topic-dependent prior and Householder flow, we yield further performance improvements, demonstrating the importance of topic guidance for text summarization.
To demonstrate the readability and diversity of the generated summaries, we present typical examples in Table \[tab:sample\]. The words in blue are the topic words that appear in the source article but do not exist in the reference, while the words in red are neither in the reference nor in the source article. When the topic information is provided, our model is able to generate semantically-meaningful words which may not even exist in the reference summaries and the source articles. Additionally, with our topic-guided model, we can always generate a summary with meaningful initial words. These phenomena imply that our model supplies more insightful semantic information to improve the quality of generated summaries.
Finally, to demonstrate that our TGVAE learns interpretable topic-dependent GMM priors, we draw multiple samples from each mixture component and visualize them with t-SNE [@maaten2008visualizing]. As can be seen from Figure \[tsne\], we have learned a group of separable *topic-dependent* components. Each component is clustered and also maintains semantic meaning in the latent space, *e.g.*, the clusters corresponding to the topic “stock” and “finance” are close to each other, while the clusters for “finance” and “disease” are far away from each other. Additionally, to understand the topic model we have learned, we provide the top 5 words for 10 randomly chosen topics on each dataset (the boldface word is the topic name summarized by us), as shown in Table \[Table:topicwords\].
Conclusion
==========
A novel text generator is developed, combining a VAE-based neural sequence model with a neural topic model. The model is an extension of conditional VAEs in the framework of unsupervised learning, in which the topics are extracted from the data with clustering structure rather than predefined labels. An effective inference method based on Householder flow is designed to encourage the complexity and the diversity of the learned topics. Experimental results are encouraging, across multiple NLP tasks.
[^1]: These three datasets can be downloaded from https://github.com/jhlau/topically-driven-language-model.
[^2]: https://www.ap.org/en-gb/
[^3]: https://catalog.ldc.upenn.edu/ldc2012t21
[^4]: http://duc.nist.gov/duc2004
|
---
abstract: 'Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $v\in V$ such that ${\rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.'
author:
- Benjamin Martin
date: 'October 8, 2015'
title: Generic stabilisers for actions of reductive groups
---
Introduction {#sec:intro}
============
Let $G$ be a reductive linear algebraic group over an algebraically closed field $k$ and let $V$ be a quasi-projective $G$-variety. For convenience, we assume throughout the paper that $G$ permutes the irreducible components of $V$ transitively (the extension of our results to the general case is straightforward). An important question in geometric invariant theory is the following: what can we say about generic stabilisers for the $G$-action? For instance, given $v\in V$, what does the stabiliser $G_v$ tell us about the stabilisers $G_w$ for $w$ near $v$? Define $V_0$ to be the set of points $v\in V$ such that the stabiliser $G_v$ has minimal dimension. The basic theory tells us that $V_0$ is open (Lemma \[lem:stabdimcty\]). Here is a deeper result [@BR Prop. 8.6]: if $V$ is affine and there exists an étale slice through $v$ for the $G$-action then there exists an open neighbourhood $U$ of $v$ such that for all $w\in U$, $G_w$ is conjugate to a subgroup of $G_v$. In particular, if ${\rm dim}(G_v)$ is minimal in this case then $G_w^0$ is conjugate to $G_v^0$ for all $w\in U$. The existence of an étale slice requires, among other conditions, that $V$ be affine and the orbit $G\cdot v$ be closed and separable. If $V$ is affine and $k$ has characteristic 0 then every $v\in V$ such that $G\cdot v$ is closed admits an étale slice, but if $k$ has positive characteristic then it can happen that there are no étale slices at all, since, for example, orbits need not be separable.
In this paper we prove some results about properties of generic stabilisers. Most previous work in this area has dealt with affine varieties and/or fields of characteristic zero only. Our results hold for quasi-projective varieties and in arbitrary characteristic, although in some cases we get stronger results in characteristic zero. We need no assumptions on the existence of or properties of closed orbits, and we allow $G$ to be non-connected.
Let $V_{\rm red}= \{v\in V_0\mid\mbox{$G_v$ is reductive}\}$. It is possible for $V_{\rm red}$ to be empty (see Example \[exmp:unipotent\]). Our first main result implies that if $V_{\rm red}$ is nonempty then generic stabilisers are reductive.
\[thm:main\] $V_{\rm red}$ is an open subvariety of $V$.
A key ingredient in the proof is the Projective Extension Theorem (see Lemma \[lem:projextn\]).
We mention two related results. First, it follows from [@rich72acta Cor. 9.1.2] that if $G$ is a complex linear algebraic group—not necessarily reductive—and $V$ is a smooth algebraic transformation space for $G$ then $V_{\rm red}$ is open. Second, V. Popov proved the following [@pop] (cf. [@LunaVust]). Let $G$ be a connected linear algebraic group—not necessarily reductive, and in arbitrary characteristic—such that $G$ has no nontrivial characters, and let $V$ be an irreducible normal algebraic variety on which $G$ acts such that the divisor class group ${\rm Cl}(V)$ has no elements of infinite order. Then generic orbits $G$-orbits on $V$ are closed if generic $G$-orbits on $V$ are affine, and the converse also holds if $V$ is affine.
Richardson proved that if $G$ is reductive and $V$ is an affine $G$-variety then an orbit $G\cdot v$ is affine if and only if the stabiliser $G_v$ is reductive [@Rich77 Thm. A]. Suppose $V$ is affine and there exists a closed orbit $G\cdot v$ of maximum dimension; then the union of the closed orbits of maximal dimension is open in $V$ [@New Prop. 3.8]. It follows from Richardson’s result that there is an open dense set of points $v\in V$ such that $G_v$ is reductive. Theorem \[thm:main\] extends this to the case when generic orbits are not closed, without the affineness assumption.
Richardson’s result discussed above gives the following immediate corollary to Theorem \[thm:main\] (note that $G_v$ has minimal dimension if and only if the orbit $G\cdot v$ has maximum dimension).
\[cor:affine\] Suppose $V$ is affine. Then the set $v\in V$ such that ${\rm dim}(G\cdot v)$ is maximal and $G\cdot v$ is affine is open.
We give an application of Theorem \[thm:main\]. Nisnevi[č]{} [@nisnevic] proved the following result when ${\rm char}(k)= 0$ and $t= 1$[^1]; he also proved that the subset $A$ is nonempty in this special case.
\[thm:subgpint\] Let $M,H_1,\ldots, H_t$ be subgroups of a reductive group $G$ such that $M$ is reductive. Let $$A= \{(g_1,\ldots, g_t)\in G^t\mid \mbox{$M\cap g_1H_1g_1^{-1}\cap \cdots \cap g_tH_tg_t^{-1}$ is reductive and has minimal dimension}\}.$$ Then $A$ is open.
We do not know in general whether $A$ can be empty in positive characteristic, not even when $t= 1$ and $H_1= M$.
If generic stabilisers are reductive, it is reasonable to try to pin down which reductive subgroups of $G$ actually appear as stabilisers. We say that a subgroup $H$ of $G$ is a [*principal stabiliser*]{} for the $G$-variety $V$ if there is a nonempty open subset $O$ of $V$ such that $G_v$ is conjugate to $H$ for all $v\in O$. We then say that $V$ has a [*principal orbit type*]{}. Under our assumptions on $G$ and $V$, a principal stabiliser is unique up to conjugacy if it exists. Richardson proved that if ${\rm char}(k)= 0$ and $V$ is smooth and affine then a principal stabiliser exists [@rich72 Prop. 5.3].
It turns out that in positive characteristic, the condition of conjugacy of the stabilisers is too strong: Example \[exmp:notprinc\] below shows that even if generic stabilisers are connected and reductive, a principal stabiliser need not exist. To obtain a result, we need to weaken the notion of principal stabiliser. Let $M\leq G$ and let $P$ be a minimal R-parabolic subgroup of $G$ containing $M$ (see Section \[sec:prelims\] for the definition of R-parabolic subgroups), let $L$ be an R-Levi subgroup of $P$ and let $\pi_L\colon P{\rightarrow}L$ be the canonical projection. It can be shown that up to $G$-conjugacy, $\pi_L(M)$ does not depend on the choice of $P$ and $L$ (cf. [@GIT Prop. 5.14(i)]). We define ${\mathcal D}(M)$ to be the conjugacy class $G\cdot \pi_L(M)$, and we call this the [*$G$-completely reducible degeneration*]{} of $M$ (see Section \[sec:Gcr\] for the definition of $G$-complete reducibility). Our second main result says that the ${\mathcal D}(G_v)$ are equal for generic $v$.
\[thm:genericstab\] There exist a $G$-completely reducible subgroup $H$ of $G$ and a nonempty open subset $O$ of $V$ such that ${\mathcal D}(G_v)= G\cdot H$ for all $v\in O$.
If $G$ is connected and every stabiliser is unipotent then ${\mathcal D}(G_v)= 1$ for all $v\in V$, so we don’t learn much about the structure of the stabilisers. Under some extra hypotheses, however, we can deduce the existence of a principal stabiliser.
\[cor:Gcrprinc\] Suppose there is a nonempty open subset $O$ of $V$ such that $G_v$ is $G$-completely reducible for all $v\in O$. Then the subgroup $H$ from Theorem \[thm:genericstab\] is a principal stabiliser for $V$.
\[cor:char0princ\] Suppose ${\rm char}(k)= 0$ and $V_{\rm red}$ is nonempty. Then the subgroup $H$ from Theorem \[thm:genericstab\] is a principal stabiliser for $V$.
If we restrict ourselves to the identity components of stabilisers then we get slightly stronger results.
\[thm:genericstab\_weak\_conn\] Suppose $V_{\rm red}$ is nonempty. There exists a connected $G$-completely reducible subgroup $H$ of $G$ such that ${\mathcal D}(G_v^0)= G\cdot H$ for all $v\in V_{\rm red}$.
In fact, we prove a version of Theorem \[thm:genericstab\_weak\_conn\] which applies even when $V_{\rm red}$ is empty (see Theorem \[thm:genericstab\_conn\]).
We briefly explain our approach to the proof of Theorems \[thm:genericstab\] and \[thm:genericstab\_weak\_conn\]. We may regard the subgroups $G_v$ as a family of subgroups of $G$ parametrised by $V$. There is no obvious way to endow a set of subgroups of $G$ with a geometric structure, so instead we follow the approach of R.W. Richardson [@rich2], [@rich] and consider the set of tuples that generate these subgroups.
Let $N\in {\mathbb N}$. Define $C= C_N= \{(v,g_1,\ldots, g_N)\mid v\in V, g_1,\ldots, g_N\in G_v\}$. We call $C$ the [*stabiliser variety*]{} of $V$.
Our results follow from a study of the geometry of $C$, using the theory of character varieties and the theory of $G$-complete reducibility. A major technical problem is that $C$ can be reducible even when $G$ is connected and $V$ is irreducible, so the projection into $V$ of a nonempty open subset of $C$ need not be dense (see Remarks \[rem:genericGirstab\] and \[rem:notgeneric\], for example). The situation is better if we consider only the identity components of stabilisers: we can work with a canonically defined subvariety $\widetilde{C}$ of $C$ with nicer properties (see Lemma \[lem:cpts\]).
The paper is laid out as follows. Section \[sec:prelims\] contains preliminary material. In Section \[sec:mainthm\] we prove Theorems \[thm:main\] and \[thm:subgpint\]. Section \[sec:Gcr\] reviews $G$-complete reducibility and Section \[sec:preceq\] introduces a technical tool needed in Section \[sec:genericstab\], where we prove Theorem \[thm:genericstab\] and Corollaries \[cor:Gcrprinc\] and \[cor:char0princ\]. We study the irreducible components of $C$ in Section \[sec:stabcpts\] and prove Theorem \[thm:genericstab\_weak\_conn\]. The final section contains some examples.
Preliminaries {#sec:prelims}
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Throughout the paper, $N$ denotes a positive integer, $G$ is a reductive linear algebraic group—possibly non-connected—over an algebraically closed field $k$ and $V$ is a quasi-projective $G$-variety over $k$. The stabiliser variety $C_N$ depends on the choice of $N$, but to ease notation we suppress the subscript and write just $C$. All subgroups of $G$ are assumed to be closed. If $H$ is a linear algebraic group then we write $\kappa(H)$ for the number of connected components of $H$, $R_u(H)$ for the unipotent radical of $H$ and $\alpha_H$ for the canonical projection $H{\rightarrow}H/R_u(H)$. The irreducible components of $H^N$ are the subsets of the form $H_1\times\cdots \times H_N$, where each $H_i$ is a connected component of $H$. If $X'$ is a subset of a variety $X$ then we denote the closure of $X'$ in $X$ by $\overline{X'}$. Below we will use the following results on fibres of morphisms (cf. [@Bo AG.10.1 Thm.]): if $f\colon X{\rightarrow}Y$ is a dominant morphism of irreducible quasi-projective varieties then for all $y\in Y$, every irreducible component of $f^{-1}(y)$ has dimension at least ${\rm dim}(X)- {\rm dim}(Y)$, and there is a nonempty open subset $U$ of $Y$ such that if $y\in U$ then equality holds. More generally, if $Z$ is a closed irreducible subset of $Y$ and $W$ is an irreducible component of $f^{-1}(Z)$ that dominates $Z$ then ${\rm dim}(W)\geq {\rm dim}(Z)+ {\rm dim}(X)- {\rm dim}(Y)$.
The next result is [@New Lem. 3.7].
\[lem:stabdimcty\] Let a linear algebraic group $H$ act on a quasi-projective variety $W$. For any $t\in {{\mathbb N}}\cup \{0\}$, the set $\{w\in W\mid {\rm dim}(H_w)\geq t\}$ is closed.
Our assumption that $G$ permutes the irreducible components of $V$ transitively implies that these components all have the same dimension, which we denote by $n$, and also that nonempty open $G$-stable subsets of $V$ are dense. In particular, the open subset $V_0$ is dense; we denote the dimension of $G_v$ for any $v\in V_0$ by $r$.
The group $G$ acts on $G^N$ by simultaneous conjugation: $g\cdot (g_1,\ldots, g_N)= (gg_1g^{-1},\ldots, gg_Ng^{-1})$. We define $\phi\colon C{\rightarrow}G^N$ and $\eta\colon C{\rightarrow}V$ to be the canonical projections. We allow $G$ to act on $C$ in the obvious way, so that $\phi$ and $\eta$ are $G$-equivariant.
We recall an approach to parabolic subgroups and Levi subgroups using cocharacters [@spr2 Sec. 8.4], [@BMR Lem. 2.4, Sec. 6]. We denote by $Y(G)$ the set of cocharacters of $G$. The subgroup $P_\lambda:= \{g\in G\mid \lim_{a{\rightarrow}0} \lambda(a)g\lambda(a)^{-1}\ \mbox{exists}\}$ is called an [*R-parabolic subgroup*]{} of $G$, and the subset $L_\lambda:= C_G(\lambda(k^*))$ is called an [*R-Levi subgroup*]{} of $P_\lambda$. An R-parabolic subgroup $P$ is parabolic in the sense that $G/P$ is complete, and $P^0$ is a parabolic subgroup of $G^0$. If $G$ is connected then an R-parabolic (resp. R-Levi) subgroup is a parabolic (resp. Levi) subgroup, and every parabolic subgroup $P$ and every Levi subgroup $L$ of $P$ arise in this way. The normaliser $N_G(P)$ of a parabolic subgroup $P$ of $G^0$ is an R-parabolic subgroup. The subset $\{g\in G\mid \lim_{a{\rightarrow}0} \lambda(a)g\lambda(a)^{-1}= 1\}$ is the unipotent radical $R_u(P_\lambda)$, and this coincides with $R_u(P_\lambda^0)$. We denote the canonical projection from $P_\lambda$ to $L_\lambda$ by $c_\lambda$. There are only finitely many conjugacy classes of R-parabolic subgroups [@Mar Prop. 5.2(e)].
We finish with some results that are well known; we give proofs here as we could not find any in the literature. These results are not needed in the proofs of Theorems \[thm:main\] and \[thm:subgpint\].
\[lem:bddfib\] Let $\psi\colon X{\rightarrow}Y$ be a morphism of quasi-projective varieties over $k$. There exists $d\in {\mathbb N}$ such that any fibre of $\psi$ has at most $d$ irreducible components.
By noetherian induction on closed subsets of $X$ and $Y$, we are free to pass to open affine subvarieties of $X$ and $Y$ whenever this is convenient. So assume that $X$, $Y$ are affine and let $R$, $S$ be the co-ordinate rings of $X$, $Y$ respectively. Suppose first that $X$ and $Y$ are irreducible and that $\psi$ is finite and dominant. By a simple induction argument, we can assume that $R=S[f]$ for some $f\in R$. Let $m(t)=t^d+ a_{d-1}t^{d-1}+\cdots +a_0$ be the minimal polynomial of $f$ with respect to the quotient field of $S$. Passing to open subvarieties, we can assume that the $a_i$ are defined on $Y$. Let $y\in Y$. If $x\in X$ with $\psi(x)=y$ then we have $f(x)^d+ a_{d-1}(y)f(x)^{d-1}+\cdots +a_0(y)=0$; it follows that there can be at most $d$ such values of $x$. Thus the fibres of $\psi$ have cardinality at most $d$.
Now consider the general case. Passing to open subvarieties, we can assume that $X$, $Y$ are irreducible and affine and that $\psi$ is dominant. We can write $R=S[f_1,\ldots,f_t]$ for some $t$ and some $f_1,\ldots,f_t\in R$. After reordering the $f_i$ if necessary, there exists $s$ with $0\leq s\leq t$ such that $f_1,\ldots,f_s$ are algebraically independent over $S$ and $f_{s+1},\ldots,f_t$ are algebraic over $S[f_1,\ldots,f_s]$. The inclusion $S\subseteq S[f_1,\ldots,f_s]\subseteq R$ corresponds to a factorisation of $\psi$ as $\psi=X\stackrel{\psi'}{{\rightarrow}} Y'\stackrel{g}{{\rightarrow}} Y$, where $Y'$ is the affine variety with co-ordinate ring $S[f_1,\ldots,f_s]$. Then ${\rm dim}(X)= {\rm dim}(Y')$ and $\psi'$ is dominant. By passing to open affine subvarieties, we can assume that $\psi'$ is finite and $Y'$ is normal. By the special case above, the cardinality of the fibres of $\psi'$ is bounded by some $d$.
Suppose that for some $y\in Y$, the fibre $F:=\psi^{-1}(y)$ has $d+1$ distinct irreducible components, say $F_1,\ldots,F_{d+1}$. The fibre $F':=g^{-1}(y)$ is clearly isomorphic to $k^s$ and we have $F=(\psi')^{-1}(F')$. Since $\psi'$ is finite and $Y'$ is normal, every irreducible component of $F$ has dimension $s$ and is mapped surjectively to $F'$ [@Hum 4.2 Prop. (b)]. But this means that for generic $y'\in F'$, $(\psi')^{-1}(y')$ has at least $d+1$ elements, a contradiction. We deduce that $F$ has at most $d$ irreducible components, as required.
Applying Lemma \[lem:bddfib\] to the map $\eta\colon C{\rightarrow}V$, we see there is a uniform bound on $\kappa(G_v)$ as $v$ ranges over the elements of $V$, since the number of irreducible components of $G_v^N$ is $\kappa(G_v)^N$. We denote the least such bound by $\Theta$.
\[lem:refined\_baire\] Let $\Omega/k$ be a proper extension of algebraically closed fields. Let $t\in {\mathbb N}$ and let $X$ be an $\Omega$-defined constructible subset of $\Omega^t$. Let $\{X_i\mid i\in I\}$ be a family of $k$-defined constructible subsets of $\Omega^t$ such that $X\subseteq \bigcup_{i\in I} X_i$. Then there exists $i\in I$ such that $X\cap X_i$ has nonempty interior in $X$. Moreover, there exists a finite subset $F$ of $I$ such that $X\subseteq \bigcup_{i\in F} X_i$.
Clearly we can reduce to the case when $X$ and each of the $X_i$ is irreducible and locally closed in $\Omega^t$. The second assertion follows from the first by Noetherian induction on closed subsets of $X$, so it is enough to prove the first assertion. Let $m= {\rm dim}(X)$. It suffices to show that ${\rm dim}(X\cap X_i)= m$ for some $i\in I$. We use induction on $m$. The result is trivial if $m= 0$. Choose polynomials $f_1,\ldots, f_m\in k[T_1,\ldots, T_t]$ such that the restrictions of the $f_i$ to $X$ form a subset of the co-ordinate ring $\Omega[X]$ that is algebraically independent over $\Omega$. Define $f\colon \Omega^t{\rightarrow}\Omega^m$ by $f(x)= (f_1(x),\ldots, f_m(x))$; note that $f$ is $k$-defined. Any proper closed subset of $X$ is a union of irreducible components of dimension less than $m$. By induction, we are therefore free to replace $X$ with any nonempty open subset of $X$, so we can assume that $f|_{X}$ gives a finite map from $X$ onto an open subset of $\Omega^m$. Then $f(X)\subseteq \bigcup_{i\in I} f(X_i)$ and each $f(X_i)$ is $k$-constructible. It is enough to prove that $f(X)\cap f(X_i)$ has nonempty interior in $f(X)$. Hence we can assume without loss that $t= m$ and $X$ is an open subset of $\Omega^m$.
Let $\pi\colon \Omega^m{\rightarrow}\Omega$ be the projection onto the first co-ordinate. Since $X$ is an open and dense subset of $\Omega^m$, $\pi(X)$ is a dense constructible subset of $\Omega$, so $\Omega\backslash \pi(X)$ is finite. Hence there exists $y\in \pi(X)$ such that $y\not\in k$. Let $\widetilde{X}= X\cap \pi^{-1}(y)$. Then $\widetilde{X}$ is an $\Omega$-defined locally closed subset of $\Omega^m$, $\widetilde{X}$ is irreducible of dimension $m- 1$ and $\widetilde{X}\subseteq \bigcup_{i\in I} X_i$. By induction, there exists $j\in I$ such that $\widetilde{X}\cap X_j$ has an irreducible component of dimension $m-1$. Hence $\pi^{-1}(y)\cap X_j$ has an irreducible component of dimension at least $m-1$. Note that we retain our assumption that the $X_i$ are irreducible. Now $X_j$ cannot be contained in $\pi^{-1}(y)$ because $\pi^{-1}(y)$ has no $k$-points, so $\pi^{-1}(y)\cap X_j$ is a proper closed subset of $X_j$. Hence ${\rm dim}(X_j)= m$, as required.
\[cor:qcmpct\] Let $\Omega$ be an uncountable algebraically closed field. Let $t\in {\mathbb N}$ and let $X$ be an $\Omega$-defined constructible subset of $\Omega^t$. Let $\{X_i\mid i\in I\}$ be a countable family of $\Omega$-constructible subsets of $X$ such that $X\subseteq \bigcup_{i\in I} X_i$. Then there exists $i\in I$ such that $X_i$ has nonempty interior in $X$. Moreover, there exists a finite subset $F$ of $I$ such that $X\subseteq \bigcup_{i\in F} X_i$.
Each of the $X_i$ is defined over a subfield of $\Omega$ that is finitely generated over the algebraic closure of the prime field, so there exists a countable subfield $k$ of $\Omega$ such that each of the $X_i$ is defined over $k$. Since $k$ is countable and $\Omega$ is not, $\Omega/k$ is a proper field extension. Now apply Lemma \[lem:refined\_baire\].
\[cor:qcmpctirred\] If $X$ is irreducible and the $X_i$ are closed in Corollary \[cor:qcmpct\] then there exists $i\in I$ such that $X\subseteq X_i$.
This is immediate from Corollary \[cor:qcmpct\].
Proof of Theorem \[thm:main\] {#sec:mainthm}
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We now prove our first main result.
\[lem:projextn\] Let $X$ be a quasi-projective variety, let $Y$ be a projective variety and let $Z$ be a closed subvariety of $X\times Y$. Then the projection of $Z$ onto $X$ is a closed variety.
Choose a covering of $X$ by open affine subvarieties $X_1,\ldots, X_m$. A subset $S$ of $X$ (resp. $X\times Y$) is closed if and only if its intersection with $X_i$ (resp. $X_i\cap Y$) is closed for all $i$, so we can assume that $X$ is affine. The result now follows from the Projective Extension Theorem [@clo Ch. 8, Sec. 5, Thm. 6].
\[lem:closedcrit\] Let $P$ be an R-parabolic subgroup of $G$ and let $W$ be a closed $P$-stable subset of $V$. Then $G\cdot W$ is closed in $V$.
Set $D= \{(v,g)\in V\times G\mid g^{-1}\cdot v\in W\}$, a closed subvariety of $V\times G$. We let $P$ act on $V\times G$ by $h\cdot (v,g)= (v,gh^{-1})$; then $D$ is $P$-stable as $W$ is. Let $\pi_P\colon G{\rightarrow}G/P$ be the canonical projection and define $\theta\colon V\times G{\rightarrow}V\times G/P$ by $\theta(v,g)= (v,\pi_P(g))$. Since $\pi_P$ is smooth, $\pi_P$ is flat, so $(\theta,V\times G/P)$ is a geometric quotient by [@bongartz Lem. 5.9(a)]. Then $\theta$ takes closed $P$-stable subvarieties of $V\times G$ to closed subvarieties of $V\times G/P$, so $\theta(D)$ is a closed subvariety of $V\times G/P$. Note that the projection of $\theta(D)$ onto $V$ is $G\cdot W$. Lemma \[lem:projextn\] implies that $G\cdot W$ is closed in $V$, so we are done.
\[rem:Pclosed\] We record one corollary (cf. [@sikora Prop. 27]). Recall that $G$ acts on $G^N$ by simultaneous conjugation. Let $P$ be an R-parabolic subgroup of $G$. Then $G\cdot P^N$ is closed in $G^N$. This follows immediately from Lemma \[lem:closedcrit\], taking $V= G^N$ and $W= P^N$.
\[prop:redcrit\] Let $P$ be an R-parabolic subgroup of $G$ with unipotent radical $U$. Set $V_P= \{v\in V_0\mid {\rm dim}(P_v)= r\}= \{v\in V_0\mid G_v^0\leq P\}$ and $V_{P,t}= \{v\in V_P\mid {\rm dim}(U_v)\geq t\}$. Then $G\cdot V_{P,t}$ is closed in $V_0$ for each $t$.
This follows from Lemma \[lem:closedcrit\] (applied to $V_0$), as each $V_{P,t}$ is $P$-stable and closed in $V_0$ (Lemma \[lem:stabdimcty\]).
We show that $G_v$ is non-reductive if and only if $v\in \bigcup_P G\cdot V_{P,1}$, where the union is over a set of representatives of the conjugacy classes of R-parabolic subgroups of $G$. Since there are only finitely many R-parabolic subgroups up to conjugacy and each subset $G\cdot V_{P,1}$ is closed in $V_0$ (Proposition \[prop:redcrit\]), this suffices to prove the theorem.
If $v\in G\cdot V_{P,1}$—say, $g\cdot v\in V_{P,1}$—then $G_v^0\leq g^{-1}Pg$ and $G_v^0$ contains a positive-dimensional subgroup $M$ of $g^{-1}Ug= R_u(g^{-1}Pg)$, so $G_v^0$ is not reductive: for $G_v^0$ normalises the connected unipotent subgroup of $g^{-1}Ug$ generated by the $G_v^0$-conjugates of $M$. Hence $G_v$ is not reductive, either. Conversely, if $v\in V_0$ and $G_v$ has nontrivial unipotent radical $H$ then we can pick a minimal R-parabolic subgroup $P$ of $G$ containing $G_v$; then $H\leq R_u(P)$ (see the paragraph following Lemma \[lem:countable\]), so $v\in G\cdot V_{P,1}$. The result now follows.
\[rem:Vmin\] More generally, set $V(t)= \{v\in V_0\mid {\rm dim}(R_u(G_v))\geq t\}$. A similar argument to the one above shows that $V(t)= \bigcup_P G\cdot V_{P,t}$, where the union is over a set of representatives of the conjugacy classes of R-parabolic subgroups of $G$, so $V(t)$ is closed. In particular, define $V_{\rm min}= \{v\in V_0\mid {\rm dim}(R_u(G_v))\ \mbox{is minimal}\}$; then $V_{\rm min}$ is a nonempty open subset of $V_0$. Note that $V_{\rm min}= V_{\rm red}$ if $V_{\rm red}$ is nonempty.
We finish the section with the proof of Theorem \[thm:subgpint\]. Each coset space $G/H_i $ is quasi-projective, and the reductive group $M$ acts on $G/H_i$ by left multiplication. Let $V= G/H_1\times\cdots \times G/H_t$, with $M$ acting on $V$ by the product action. For any $(g_1,\ldots, g_t)\in G^t$, the stabiliser $M_{(g_1H_1,\ldots, g_tH_t)}$ is equal to $M\cap g_1H_1g_1^{-1}\cap \cdots \cap g_tH_tg_t^{-1}$. Hence the set $A$ equals the preimage of $V_{\rm red}$ under the map from $G^t$ to $V$ that sends $(g_1,\ldots, g_t)$ to $(g_1H_1,\ldots, g_tH_t)$. But $V_{\rm red}$ is open by Theorem \[thm:main\], so $A$ is open. This completes the proof.
In the set-up in the proof of Theorem \[thm:subgpint\], we do not know whether the subgroups $M\cap g_1H_1g_1^{-1}\cap \cdots \cap g_tH_tg_t^{-1}$ are all conjugate for generic $(g_1,\ldots, g_t)$. This is the case, however, if these subgroups are $G$-completely reducible for generic $(g_1,\ldots, g_t)$ (cf. Example \[exmp:guralnick\]).
$G$-complete reducibility and orbits of tuples {#sec:Gcr}
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Let $H$ be a subgroup of $G$. We say that $H$ is [*$G$-completely reducible*]{} ($G$-cr) if whenever $H$ is contained in an R-parabolic subgroup $P$ of $G$, there is an R-Levi subgroup $L$ of $P$ such that $H$ is contained in $L$. This notion is due to Serre [@serre2]; see [@serre1] and [@serre1.5] for more details. In particular, we say that $H$ is [*$G$-irreducible*]{} ($G$-ir) if $H$ is not contained in any proper R-parabolic subgroup of $G$ at all; then $H$ is $G$-cr. A $G$-cr subgroup of $G$ is reductive (cf. [@BMR Sec. 2.5 and Thm. 3.1]), and the converse holds in characteristic 0. A linearly reductive subgroup is $G$-cr, while a nontrivial unipotent subgroup of $G^0$ is never $G$-cr. A normal subgroup of a $G$-cr subgroup is $G$-cr [@BMR Thm. 3.10]. We denote by ${\mathcal C}(G)_{\rm cr}$ the set of conjugacy classes of $G$-cr subgroups of $G$.
\[lem:countable\] ${\mathcal C}(G)_{\rm cr}$ is countable.
Let $F$ be the algebraic closure of the prime field. Then $G$ has an $F$-structure, by [@Mar Prop. 3.2]. By [@Mar Thm. 10.3] and [@BMR Thm. 3.1], any $G$-cr subgroup of $G$ is $G$-conjugate to an $F$-defined subgroup. But $G(F)$ has only countably many $G(F)$-conjugacy classes of $G(F)$-cr subgroups since $F$ is countable. The result follows.
Let $H$ be a subgroup of $G$. Let $P=P_\lambda$ be minimal amongst the R-parabolic subgroups of $G$ that contain $H$. Then $c_\lambda(H)$ is an $L_\lambda$-ir subgroup of $L_\lambda$ (see the proof of [@GIT Prop. 5.14(i)]), so $c_\lambda(H)$ is $G$-cr. As observed in Section \[sec:intro\], $c_\lambda(H)$ does not depend on the choice of $\lambda$ up to conjugacy, and we set ${\mathcal D}(H)= G\cdot c_\lambda(H)$. We have ${\mathcal D}(H)= G\cdot H$ if and only if $c_\lambda(H)$ is conjugate to $H$ if and only if $H$ is $G$-cr [@GIT Prop. 5.14(i)]. For any $\mu\in Y(G)$ such that $H\leq P_\mu$, if $H$ is $G$-cr then $c_\mu(H)$ is conjugate to $H$, and if $c_\mu(H)$ is $G$-ir then $L_\mu= G$, so $H= c_\mu(H)$ is $G$-ir. Since $c_\lambda(H)$ is reductive, $R_u(H)\leq R_u(P_\lambda)$ and $H$ is reductive if and only if $H\cap R_u(P_\lambda)$ is finite if and only if ${\rm dim}(H)= {\rm dim}(c_\lambda(H))$. Moreover, ${\rm dim}(C_G(H))\leq {\rm dim}(C_G(c_\lambda(H)))$, with equality if and only if $H$ is $G$-cr [@GIT Thm. 5.8(ii)], and ${\rm dim}(c_\lambda(H))= {\rm dim}(H)- {\rm dim}(R_u(H))$. If $M\leq H$ and $\alpha_H(M)= H/R_u(H)$ then ${\mathcal D}(M)= {\mathcal D}(H)$.
If ${\rm char}(k)= 0$ then $H$ has a Levi subgroup $M$ by [@hochschild VIII, Thm. 4.3]: that is, $H$ has a reductive subgroup $M$ such that $H\cong M\ltimes R_u(H)$. Then $c_\lambda(H)= c_\lambda(M)$ is conjugate to $M$, since $M$ is $G$-cr, so ${\mathcal D}(H)= G\cdot M$.
The paper [@BMR] laid out an approach to the theory of $G$-complete reducibility using geometric invariant theory; we briefly review this now. As described in Section \[sec:intro\], the idea is to study subgroups of $G$ indirectly by looking instead at generating tuples for subgroups. Given $s\in {\mathbb N}$ and ${\mathbf g}= (g_1,\ldots, g_s)\in G^s$, we denote by ${\mathcal G}({\mathbf g})$ or ${\mathcal G}( g_1,\ldots, g_s)$ the closed subgroup generated by $g_1,\ldots, g_s$. If $H$ is of the form ${\mathcal G}(g_1,\ldots, g_s)$ for some $g_1,\ldots, g_s\in G$ then we say that $H$ is [*topologically finitely generated*]{}, and we call ${{\mathbf g}}$ a [*generating $s$-tuple*]{} or [*generating tuple*]{} for $H$. The structure of the set of generating $s$-tuples is complicated; for instance, if $H= k^*$ and $k$ is solid (Definition \[defn:solid\]) then both $\{{{\mathbf h}}\in H^s\mid {\mathcal G}({{\mathbf h}})= H\}$ and $\{{{\mathbf h}}\in H^s\mid {\mathcal G}({{\mathbf h}})\neq H\}$ are dense in $H^s$, even when $s= 1$.
Recall that $G$ acts on $G^N$ by simultaneous conjugation. We call the quotient space $G^N/G$ a [*character variety*]{} and we denote the canonical projection from $G^N$ to $G^N/G$ by $\pi_G$. If $\lambda\in Y(G)$ then we abuse notation and denote the map $c_\lambda\times \cdots\times c_\lambda\colon P_\lambda^N{\rightarrow}L_\lambda^N$ by $c_\lambda$. We have $\pi_G({\mathbf g})= \pi_G(c_\lambda({\mathbf g}))$ and ${\mathcal G}(c_\lambda({{\mathbf g}}))= c_\lambda({\mathcal G}({{\mathbf g}}))$ for all ${\mathbf g}\in P_\lambda^N$. If ${{\mathbf g}}\in P_\lambda^N$ and ${{\mathbf g'}}\in P_{\lambda'}^N$ such that $G\cdot c_\lambda({{\mathbf g}})$ and $G\cdot c_{\lambda'}({{\mathbf g'}})$ are closed then $\pi_G({{\mathbf g}})= \pi_G({{\mathbf g'}})$ if and only if $c_\lambda({{\mathbf g}})$ and $c_{\lambda'}({{\mathbf g'}})$ are conjugate (see [@New Cor. 3.5.2]). In particular, if $G\cdot {{\mathbf g'}}$ is closed then we can take $\lambda'= 0$, so $\pi_G({{\mathbf g}})= \pi_G({{\mathbf g'}})$ if and only if $c_\lambda({{\mathbf g}})$ is conjugate to ${{\mathbf g'}}$.
We need a condition on the field to ensure that reductive groups are topologically finitely generated.
\[defn:solid\] An algebraically closed field is [*solid*]{} if either it has characteristic 0 or it has characteristic $p> 0$ and is transcendental over ${{\mathbb F}}_p$.
The next result allows us to understand subgroups of $G$ by studying generating tuples; several of the results stated above for subgroups have equivalent formulations given for tuples below.
\[prop:topfg\] [@Mar Lem. 9.2]. Suppose $k$ is solid. Let $H$ be a reductive algebraic group and suppose that $N\geq \kappa(H)+ 1$. Then there exists ${\mathbf h}\in H^N$ such that ${\mathcal G}({\mathbf h})= H$.
Proposition \[prop:topfg\] fails if $k= {\overline}{{{\mathbb F}}_p}$: for then any topologically finitely generated subgroup of $G$ is finite. This is the reason for some of the technical complexity in what follows. We can, however, formulate the results of this section for arbitrary $k$: for example, by using the notion of a “generic tuple” [@GIT Defn. 5.4]. Even when $k$ is solid, non-reductive subgroups need not be topologically finitely generated (for example, a topologically finitely generated subgroup of a unipotent group in positive characteristic is finite). This is why we need to work with $H/R_u(H)$ rather than just $H$ in Definition \[defn:preceq\].
The next result is [@BMR Cor. 3.7].
Let ${\mathbf g}\in G^N$. Then the orbit $G\cdot {\mathbf g}$ is closed if and only if ${\mathcal G}({\mathbf g})$ is $G$-cr.
Let $H$ be a reductive subgroup of $G$. The inclusion of $H^N$ in $G^N$ gives rise to a morphism $\Psi^G_H\colon H^N/H{\rightarrow}G^N/G$, given by $\Psi^G_H(\pi_H({\mathbf h}))= \pi_G({\mathbf h})$ for ${\mathbf h}\in H^N$. The next result is [@Mar Thm. 1.1].
\[thm:finiteness\] The morphism $\Psi^G_H$ is finite. In particular, $\Psi^G_H(H^N/H)$ is closed in $G^N/G$.
\[rem:genericGir\] (i) The set $(G^N)_{\rm ir}:= \{{\mathbf g}\in G^N\mid \mbox{${\mathcal G}({\mathbf g})$ is $G$-ir}\}$ is open; this was proved in [@Mar Cor. 8.4] but it also follows from Remark \[rem:Pclosed\].
\(ii) Suppose $V$ is irreducible, $N\geq 2$ and there exists $v\in V_0$ such that $G_v^0$ is $G$-ir. Then $\phi^{-1}((G^N)_{\rm ir})$ is a nonempty open $G$-stable subset of $C$ by (i), and it follows from arguments in Section \[sec:stabcpts\] that $\eta(\phi^{-1}((G^N)_{\rm ir}))$ is a dense subset of $V$ (cf. Remark \[rem:genericGirstab\]). This means that generic stabilisers are “large” in the sense of not being contained in any proper R-parabolic subgroup of $G$. On the other hand, we can interpret Lemma \[lem:stabdimcty\] as saying that generic stabilisers are “small”. This special case illustrates the tension between largeness and smallness, from which several of our results spring.
The partial order $\preceq$ {#sec:preceq}
===========================
In this section we introduce a technical tool which we need for the proof of Theorem \[thm:genericstab\]. For simplicity, we assume throughout the section that $k$ is solid; see Remark \[rem:nonsolid\] for a discussion of arbitrary $k$.
\[defn:preceq\] Let $H,M$ be subgroups of $G$. We define $G\cdot H\preceq G\cdot M$ if there exist $s\in {\mathbb N}$, ${{\mathbf h}}\in H^s$ and ${{\mathbf m}}\in M^s$ such that $\alpha_H({\mathcal G}({{\mathbf h}}))= H/R_u(H)$ and $\pi_G({{\mathbf m}})= \pi_G({{\mathbf h}})$. (It is clear that this does not depend on the choice of subgroup in the conjugacy classes $G\cdot H$ and $G\cdot M$.) We define $G\cdot H\prec G\cdot M$ if $G\cdot H\preceq G\cdot M$ and $G\cdot H\neq G\cdot M$.
\[lem:preccrit\] Let $H,M\leq G$. Then $G\cdot H\preceq G\cdot M$ if and only if ${\mathcal D}(H)\preceq {\mathcal D}(M)$.
Pick $\lambda,\mu\in Y(G)$ such that $H\leq P_\lambda$, $c_\lambda(H)$ is $G$-cr, $M\leq P_\mu$ and $c_\mu(M)$ is $G$-cr. Since ${\mathcal D}(H)= G\cdot c_\lambda(H)$ and ${\mathcal D}(M)= G\cdot c_\mu(M)$, it is enough to show that $G\cdot H\preceq G\cdot M$ if and only if $G\cdot c_\lambda(H)\preceq G\cdot c_\mu(M)$.
So suppose $G\cdot H\preceq G\cdot M$. There exist $s\in {\mathbb N}$, ${{\mathbf m}}= (m_1,\ldots, m_s)\in M^s$ and ${{\mathbf h}}= (h_1,\ldots, h_s)\in H^s$ such that $\alpha_H({\mathcal G}({{\mathbf h}}))= H/R_u(H)$ and $\pi_G({{\mathbf m}})= \pi_G({{\mathbf h}})$. Then $c_\mu({{\mathbf m}})\in c_\mu(M)^s$ and $\pi_G(c_\mu({{\mathbf m}}))= \pi_G(c_\lambda({{\mathbf h}}))$. Now $c_\lambda(H)$ is reductive, so $c_\lambda(R_u(H))= 1$. It follows that ${\mathcal G}(c_\lambda({{\mathbf h}}))= c_\lambda({\mathcal G}({{\mathbf h}}))= c_\lambda(H)$. This shows that $G\cdot c_\lambda(H)\preceq G\cdot c_\mu(M)$.
Conversely, suppose $G\cdot c_\lambda(H)\preceq G\cdot c_\mu(M)$. There exist $s\in {\mathbb N}$, ${{\mathbf y}}= (y_1,\ldots, y_s)\in c_\mu(M)^s$ and ${{\mathbf x}}= (x_1,\ldots, x_s)\in c_\lambda(H)^s$ such that ${\mathcal G}({{\mathbf x}})= c_\lambda(H)$ and $\pi_G({{\mathbf y}})= \pi_G({{\mathbf x}})$. The maps $c_\lambda\colon H^s{\rightarrow}c_\lambda(H)^s$ and $c_\mu\colon M^s{\rightarrow}c_\mu(M)^s$ are surjective, so there exist ${{\mathbf h}}= (h_1,\ldots, h_s)\in H^s$ and ${{\mathbf m}}= (m_1,\ldots, m_s)\in M^s$ such that $c_\lambda({{\mathbf h}})= {{\mathbf x}}$ and $c_\mu({{\mathbf m}})= {{\mathbf y}}$.
As $c_\lambda(H)$ is reductive, $R_u(H)\leq R_u(P_\lambda)$. As $(R_u(P_\lambda)\cap H)^0$ is a connected normal unipotent subgroup of $H$, we must have $(R_u(P_\lambda)\cap H)^0\leq R_u(H)$, and it follows that $(R_u(P_\lambda)\cap H)^0= R_u(H)$. Choose $h_{s+1},\ldots, h_{s+t}\in R_u(P_\lambda)\cap H$ such that the $\alpha_H(h_i)$ for $s+ 1\leq i\leq s+ t$ generate the finite group $(R_u(P_\lambda)\cap H)/R_u(H)$. Set ${{\mathbf h}}'= (h_1,\ldots, h_s,h_{s+1},\ldots, h_{s+t})\in H^{s+t}$, ${{\mathbf x}}'= (x_1,\ldots, x_s,1,\ldots, 1)\in c_\lambda(H)^{s+t}$, ${{\mathbf m}}'= (m_1,\ldots, m_s,1,\ldots, 1)\in M^{s+t}$ and ${{\mathbf y}}'= (y_1,\ldots, y_s,1,\ldots, 1)\in c_\mu(M)^{s+t}$. Then $c_\lambda({{\mathbf h}}')= {{\mathbf x}}'$ and $c_\mu({{\mathbf m}}')= {{\mathbf y}}'$; moreover, $\alpha_H({\mathcal G}({{\mathbf h}}'))= H/R_u(H)$ by construction.
To finish, it is enough to show that $\pi_G({{\mathbf x}}')= \pi_G({{\mathbf y}}')$. As $\pi_G({{\mathbf x}})= \pi_G({{\mathbf y}})$ and ${\mathcal G}({{\mathbf x}})= c_\lambda(H)$ is $G$-cr, there exists $\nu\in Y(G)$ such that ${\mathcal G}({{\mathbf y}})\leq P_\nu$ and $c_\nu({{\mathbf y}})$ is conjugate to ${{\mathbf x}}$. It is then immediate that ${\mathcal G}({{\mathbf y}}')\leq P_\nu$ and $c_\nu({{\mathbf y}}')$ is conjugate to ${{\mathbf x}}'$. Hence $\pi_G({{\mathbf x}}')= \pi_G({{\mathbf y}}')$, as required.
\[lem:quotient\] Let $H,M\leq G$. Suppose that $H$ is $G$-cr. Then $G\cdot H\preceq {\mathcal D}(M)$ if and only if $G\cdot H\preceq G\cdot M$ if and only if there exist $\lambda\in Y(G)$ and $M_1\leq P_\lambda\cap M$ such that $c_\lambda(M_1)$ is conjugate to $H$.
The first equivalence follows from Lemma \[lem:preccrit\]. We prove the second equivalence. As $H$ is $G$-cr, $H$ is reductive. Suppose $G\cdot H\preceq G\cdot M$. There exist $s\in {\mathbb N}$, ${{\mathbf h}}\in H^s$ and ${{\mathbf m}}\in M^s$ such that ${\mathcal G}({{\mathbf h}})= H$ and $\pi_G({{\mathbf m}})= \pi_G({{\mathbf h}})$. Set $M_1= {\mathcal G}({{\mathbf m}})$. As $H= {\mathcal G}({{\mathbf h}})$ is $G$-cr, there exist $\lambda\in Y(G)$ and $g\in G$ such that $M_1\leq P_\lambda$ and $c_\lambda({{\mathbf m}})= g\cdot{{\mathbf h}}$. Then $c_\lambda(M_1)= c_\lambda({\mathcal G}({{\mathbf m}}))= {\mathcal G}(c_\lambda({{\mathbf m}}))= {\mathcal G}(g\cdot {{\mathbf h}})= g{\mathcal G}({{\mathbf h}})g^{-1}= gHg^{-1}$, as required.
Conversely, suppose there exist $\lambda\in Y(G)$ and $M_1\leq P_\lambda\cap M$ such that $c_\lambda(M_1)$ is conjugate to $H$. Pick $s\geq \kappa(H)+ 1$. By Proposition \[prop:topfg\], there exists ${{\mathbf h}}\in H^s$ such that ${\mathcal G}({{\mathbf h}})= H$. We can pick ${{\mathbf m}}\in M_1^s$ such that $c_\lambda({{\mathbf m}})$ is conjugate to ${{\mathbf h}}$. Then $\pi_G({{\mathbf m}})= \pi_G(c_\lambda({{\mathbf m}}))= \pi_G({{\mathbf h}})$, so $G\cdot H\preceq G\cdot M$, and we are done.
\[lem:transitive\] Let $H,M,K\leq G$. If $G\cdot H\preceq G\cdot M$ and $G\cdot M\preceq G\cdot K$ then $G\cdot H\preceq G\cdot K$.
Suppose $G\cdot H\preceq G\cdot M$ and $G\cdot M\preceq G\cdot K$. By Lemma \[lem:preccrit\], we can assume $H,M$ and $K$ are $G$-cr. By Lemma \[lem:quotient\], there exist $\lambda\in Y(G)$ and $K_1\leq P_\lambda\cap K$ such that $c_\lambda(K_1)$ is conjugate to $M$. Replacing $(K,\lambda)$ with a conjugate of $(K,\lambda)$ if necessary, we can assume that $c_\lambda(K_1)= M$. Pick $s\in {\mathbb N}$, ${{\mathbf h}}\in H^s$ and ${{\mathbf m}}\in M^s$ such that ${\mathcal G}({{\mathbf h}})= H$ and $\pi_G({{\mathbf h}})= \pi_G({{\mathbf m}})$. There exists ${{\mathbf k}}\in K_1^s$ such that $c_\lambda({{\mathbf k}})= {{\mathbf m}}$. Then $\pi_G({{\mathbf k}})= \pi_G(c_\lambda({{\mathbf k}}))= \pi_G({{\mathbf m}})= \pi_G({{\mathbf h}})$, so $G\cdot H\preceq G\cdot K$.
If $H$ and $M$ are subgroups of $G$ and $H$ is conjugate to a subgroup of $M$ then $G\cdot H\preceq G\cdot M$ (and so ${\mathcal D}(H)\preceq {\mathcal D}(M)$ by Lemma \[lem:preccrit\]); in particular, $G\cdot H\preceq G\cdot H$. For without loss we can assume that $H\leq M$, and if we take $s\geq \kappa(H/R_u(H))+ 1$ then by Proposition \[prop:topfg\] we can choose ${{\mathbf m}}= {{\mathbf h}}\in H^s$ such that $\alpha_H({{\mathbf h}})$ generates the reductive group $H/R_u(H)$. The following example shows that the converse is false, even when $H$ and $M$ are $G$-cr.
\[exmp:not\_subgp\] Let ${\rm char}(k)= 2$, let $G= {\rm SL}_8(k)$ and let $M$ be ${\rm PGL}_3(k)$ embedded in $G$ via the adjoint representation on ${\rm Lie}(M)\cong k^8$. Since ${\rm Lie}(M)$ is a simple $M$-module, $M$ is $G$-cr (in fact, $G$-ir). It follows from elementary representation-theoretic arguments that $M$ contains exactly two subgroups of type $A_1$ up to $M$-conjugacy: the derived group $H_1$ of a Levi subgroup of a rank 1 parabolic subgroup of $M$, and the image $H_2$ of ${\rm SL}_2(k)$ under the map ${\rm SL}_2(k){\rightarrow}{\rm SL}_3(k){\rightarrow}M$, where the first arrow is the adjoint representation of ${\rm SL}_2(k)$ and the second is the canonical projection. It is easily checked that $H_1$ is $M$-cr but $H_2$ is not: in fact, there exists $\lambda\in Y(M)$ such that $c_\lambda(H_2)= H_1$.
Now $H_1$ is not $G$-cr because ${\rm Lie}(H_1)$ is an $H_1$-stable submodule of ${\rm Lie}(M)$ and $H_1$ does not act completely reducibly on ${\rm Lie}(H_1)$. Choose $\mu\in Y(G)$ such that $H_1\leq P_\mu$ and $H:= c_\mu(H_1)$ is $G$-cr. We have $G\cdot H_1\preceq G\cdot M$ as $H_1\leq M$, so $G\cdot H\preceq G\cdot M$ by Lemma \[lem:preccrit\]. We claim that $H$ is not $G$-conjugate to a subgroup of $M$. First, $H$ is not $G$-conjugate to $H_1$ because $H$ is $G$-cr but $H_1$ is not. If $H$ is $G$-conjugate to $H_2$ then $H_2$ is $G$-cr, so $H_1= c_\lambda(H_2)$ is $G$-conjugate to $H_2$; but then $H$ is $G$-conjugate to $H_1$, a contradiction. This proves the claim.
We do, however, have the following result.
\[lem:antisymm\] Let $H,M\leq G$. If $G\cdot H\preceq G\cdot M$ and $G\cdot M\preceq G\cdot H$ then ${\mathcal D}(H)= {\mathcal D}(M)$. In particular, if $H$ and $M$ are $G$-cr then $G\cdot H= G\cdot M$.
By Lemma \[lem:preccrit\], we can assume $H$ and $M$ are $G$-cr; in particular, $H$ and $M$ are reductive. By Lemma \[lem:quotient\], there exist $\lambda\in Y(G)$ and $M_1\leq P_\lambda\cap M$ such that $c_\lambda(M_1)$ is conjugate to $H$. Replacing $(M,\lambda)$ with a conjugate of $(M,\lambda)$ if necessary, we can assume that $c_\lambda(M_1)= H$. We have $${\rm dim}(H)= {\rm dim}(c_\lambda(M_1))\leq {\rm dim}(M_1)\leq {\rm dim}(M).$$ By symmetry, ${\rm dim}(M)\leq {\rm dim}(H)$, so $${\rm dim}(H)= {\rm dim}(c_\lambda(M_1))= {\rm dim}(M_1)= {\rm dim}(M).$$ It now follows that $$\kappa(H)= \kappa(c_\lambda(M_1))\leq \kappa(M_1)\leq \kappa(M).$$ By symmetry, $\kappa(M)\leq \kappa(H)$, so $$\kappa(H)= \kappa(c_\lambda(M_1))= \kappa(M_1)= \kappa(M).$$ This implies that $M_1= M$ since $M_1\leq M$, so $H= c_\lambda(M)$. But $M$ is $G$-cr, so $M$ is conjugate to $H$. This completes the proof.
The next result follows immediately from Lemmas \[lem:transitive\] and \[lem:antisymm\].
\[cor:po\] The relation $\preceq$ is a partial order on ${\mathcal C}(G)_{\rm cr}$.
\[rem:dcc\] The proof of Lemma \[lem:antisymm\] shows that if $H$ and $M$ are $G$-cr subgroups of $G$ and $G\cdot H\prec G\cdot M$ then either ${\rm dim}(H)< {\rm dim}(M)$, or ${\rm dim}(H)= {\rm dim}(M)$ and $\kappa(H)< \kappa(M)$. It follows that ${\mathcal C}(G)_{\rm cr}$ satisfies the descending chain condition with respect to $\preceq$.
Given a reductive subgroup $H$ of $G$, set $S(H)= \{{{\mathbf g}}\in G^N\mid \pi_G({{\mathbf g}})\in \Psi^G_H(H^N/H)\}$. Theorem \[thm:finiteness\] implies that $S(H)$ is closed.
\[lem:Psiimage\] Let ${{\mathbf g}}\in G^N$ and let $H\leq G$ be reductive. Then ${{\mathbf g}}\in S(H)$ if and only if $G\cdot {\mathcal G}({{\mathbf g}})\preceq G\cdot H$ if and only if ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq {\mathcal D}(H)$.
We prove the first equivalence. If ${{\mathbf g}}\in S(H)$ then there exists ${{\mathbf h}}\in H^N$ such that $\pi_G({{\mathbf h}})= \pi_G({{\mathbf g}})$, so $G\cdot {\mathcal G}({{\mathbf g}})\preceq G\cdot H$ as ${{\mathbf g}}$ generates ${\mathcal G}({{\mathbf g}})$. Conversely, suppose $G\cdot {\mathcal G}({{\mathbf g}})\preceq G\cdot H$. Set $M= {\mathcal G}({{\mathbf g}})$. Then ${\mathcal D}(M)\preceq {\mathcal D}(H)$ by Lemma \[lem:preccrit\]. Choose $\mu\in Y(G)$ such that $H\leq P_\mu$ and $c_\mu(H)$ is $G$-cr. Choose $\nu\in Y(G)$ such that $M\leq P_\nu$ and $c_\nu(M)$ is $G$-cr. Then ${\mathcal D}(H)= G\cdot c_\mu(H)$ and ${\mathcal D}(M)= G\cdot c_\nu(M)$. By Lemma \[lem:quotient\], there exist $K\leq c_\mu(H)$ and $\lambda\in Y(G)$ such that $G\cdot c_\lambda(K)= G\cdot c_\nu(M)$. There exists ${{\mathbf k}}\in K^N$ such that $G\cdot c_\lambda({{\mathbf k}})= G\cdot c_\nu({{\mathbf g}})$. There exists ${{\mathbf h}}\in H^N$ such that $c_\mu({{\mathbf h}})= {{\mathbf k}}$. We have $\pi_G({{\mathbf h}})= \pi_G(c_\mu({{\mathbf h}}))= \pi_G({{\mathbf k}})= \pi_G(c_\lambda({{\mathbf k}}))= \pi_G(c_\nu({{\mathbf g}}))= \pi_G({{\mathbf g}})$, so ${{\mathbf g}}\in S(H)$, as required.
The second equivalence follows from Lemma \[lem:preccrit\].
To prove our results in Section \[sec:genericstab\], we need to investigate the behaviour of the relation $\preceq$ under field extensions. We assume for the rest of the section that $N\geq \Theta+ 1$. Fix a $G$-cr subgroup $H$ of $G$ such that $N\geq \kappa(H)+ 1$.
Define $B_H= \{v\in V\mid {\mathcal D}(G_v)= G\cdot H\}$.
Let $v\in V$. For all ${{\mathbf g}}\in G_v^N$, we have ${\mathcal G}({{\mathbf g}})\leq G_v$, and so ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq {\mathcal D}(G_v)$. Moreover, since $N\geq \Theta+ 1\geq \kappa(G_v)+1\geq \kappa(G_v/R_u(G_v))+1$, there exists ${{\mathbf g'}}\in G^N$ such that $\alpha_{G_v}({\mathcal G}({{\mathbf g'}}))= G_v/R_u(G_v)$ by Proposition \[prop:topfg\], so ${\mathcal D}({\mathcal G}({{\mathbf g'}}))= {\mathcal D}(G_v)$. Lemma \[lem:transitive\] now implies that ${\mathcal D}(G_v)\preceq G\cdot H$ if and only if ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq {\mathcal D}(H)$ for all ${{\mathbf g}}\in G_v^N$ if and only if $\pi_G({{\mathbf g}})\in S(H)$ for all ${{\mathbf g}}\in G_v^N$, where the last equivalence follows from Lemma \[lem:Psiimage\]. This is the case if and only if the following formula holds: $$\label{eqn:leq}
(\forall {{\mathbf g}}\in G_v^N) \ (\exists {{\mathbf h}}\in H^N) \ \pi_G({{\mathbf h}})= \pi_G({{\mathbf g}}).$$ Conversely, $G\cdot H\preceq {\mathcal D}(G_v)$ if and only if there exist $M_1\leq G_v$ and $\lambda\in Y(G)$ such that $M_1\leq P_\lambda$ and $c_\lambda(M_1)$ is conjugate to $H$ (Lemma \[lem:quotient\]). This is the case if and only if the following formula holds: $$\label{eqn:geq}
(\exists {{\mathbf g}}\in G_v^N) \ (\exists g\in G) \ \pi_G({{\mathbf g}})= g\cdot {{\mathbf h_0}},$$ where ${{\mathbf h_0}}$ is a fixed element of $H^N$ such that ${\mathcal G}({{\mathbf h}}_0)= H$. For, given $ {{\mathbf g}}\in G_v^N$ and $g\in G$ such that $\pi_G({{\mathbf g}})= g\cdot {{\mathbf h_0}}$, we set $M_1= {\mathcal G}({{\mathbf g}})$; conversely, given $M_1\leq G_v$ and $\lambda\in Y(G)$ such that $M_1\leq P_\lambda$ and $g\in G$ such that $c_\lambda(M_1)= gHg^{-1}$, we choose ${{\mathbf g}}\in M_1^N$ such that $c_\lambda({{\mathbf g}})= g\cdot {{\mathbf h_0}}$.
We summarise the above argument as follows.
\[lem:constructible\] Let $H$ be a $G$-cr subgroup of $G$ such that $N\geq \kappa(H)+ 1$. Then $B_H\subseteq V$ is the set of solutions to the formulas Eqn. (\[eqn:leq\]) and Eqn. (\[eqn:geq\]). In particular, $B_H$ is constructible.
\[rem:nonsolid\] It can be shown that Lemma \[lem:constructible\] holds for arbitrary $k$, where we take ${{\mathbf h}}$ to be a generic tuple for $H$ in the sense of [@GIT Defn. 5.4]. To do this, one replaces generating tuples with generic tuples in the definition of $\preceq$ and makes the obvious modifications to the arguments of this section.
Proof of Theorem \[thm:genericstab\] {#sec:genericstab}
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We assume throughout the section that $N\geq \Theta+ 1$.
We will show that there is a $G$-cr subgroup $H$ of $G$ such that $N\geq \kappa(H)+ 1$ and $B_H$ has nonempty interior. By Lemma \[lem:constructible\] and Remark \[rem:nonsolid\], it is enough to prove this after extending the ground field to an uncountable algebraically closed field $\Omega$ (recall from the proof of Lemma \[lem:countable\] that any $G(\Omega)$-cr subgroup of $G(\Omega)$ is $G(\Omega)$-conjugate to a $k$-defined $G$-cr subgroup). Thus we can assume without loss that $k$ is uncountable (and hence solid).
Let $D_1,\ldots, D_t$ be the irreducible components of $C$ such that ${\overline}{\eta(G\cdot D_j)}= V$ for $1\leq j\leq t$—it follows from Lemma \[lem:cpts\](b) below that there is at least one such component—and let $D_1',\ldots, D_{t'}'$ be the other irreducible components of $C$. Let $V'= V\backslash \bigcup_{j=1}^{t'} {\overline}{\eta(G\cdot D_{j}')}$. For $1\leq j\leq t$, set $E_j= \{(v,{{\mathbf g}})\in D_j\mid \alpha_{G_v}({\mathcal G}({{\mathbf g}}))= G_v/R_u(G_v)\}$; note that $E_j$ is neither closed nor open in general, and if $(v,{{\mathbf g}})\in E_j$ then ${\mathcal D}({\mathcal G}({{\mathbf g}}))= {\mathcal D}(G_v)$. For any $v\in V'$, $N\geq \Theta+ 1\geq \kappa(G_v)+1\geq \kappa(G_v/R_u(G_v))+1$, so by Proposition \[prop:topfg\] there exists ${{\mathbf g}}\in G^N$ such that $\alpha_{G_v}({\mathcal G}({{\mathbf g}}))= G_v/R_u(G_v)$. Then $(v,{{\mathbf g}})\in D_j$ for some $1\leq j\leq t$, so $(v,{{\mathbf g}})\in E_j$. Hence $\bigcup_{1\leq j\leq t} \eta(G\cdot E_j)\supseteq V'$. As $G$ permutes the irreducible components of $V$ transitively, $\eta(G\cdot E_m)$ is dense in $V$ for some $1\leq m\leq t$.
Choose $G$-cr subgroups $H_i$ such that ${\mathcal H}:= \{H_i\mid i\in I\}$ is a set of representatives for the conjugacy classes in ${\mathcal C}(G)_{\rm cr}$; by Lemma \[lem:countable\], $I$ is countable. Let $\Lambda= \{H_i\mid G\cdot D_m\subseteq \phi^{-1}(S(H_i))\}$. Then $G\in \Lambda$, so $\Lambda$ is nonempty. By Remark \[rem:dcc\], we can pick $H\in \Lambda$ such that $H$ is minimal with respect to $\preceq$. We claim that $G\cdot D_j\subseteq \phi^{-1}(S(H))$ for all $1\leq j\leq t$. To prove this, let $(v,{{\mathbf g}})\in D_j$ such that $v\in \eta(E_m)$. There exists ${{\mathbf g}}'\in G_v^N$ such that $(v,{{\mathbf g}}')\in E_m$. Then $(v,{{\mathbf g}}')\in \phi^{-1}(S(H))$, so ${{\mathbf g}}'\in S(H)$. Now ${\mathcal G}({{\mathbf g}})\leq G_v$, so ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq {\mathcal D}(G_v)= {\mathcal D}({\mathcal G}({{\mathbf g}}'))\preceq G\cdot H$ by Lemma \[lem:Psiimage\]. Hence $(v,{{\mathbf g}})\in \phi^{-1}(S(H))$ by Lemma \[lem:Psiimage\]. As $S(H)$ is $G$-stable, it now follows that if $(v,{{\mathbf g}})\in D_j$ and $v\in \eta(G\cdot E_m)$ then $(v,{{\mathbf g}})\in \phi^{-1}(S(H))$. But $\eta^{-1}(\eta(G\cdot E_m))\cap D_j$ is dense in $D_j$ as $\eta(G\cdot E_m)$ is dense in $V$, so $D_j\subseteq \phi^{-1}(S(H))$. As $S(H)$ is $G$-stable, $G\cdot D_j\subseteq \phi^{-1}(S(H))$, as claimed. It follows from Lemma \[lem:Psiimage\] that ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq G\cdot H$ for all $1\leq j\leq t$ and all $(v,{{\mathbf g}})\in G\cdot D_j$. In particular, for any $v\in V'$, there exist $j$ and ${{\mathbf g'}}\in G^N$ such that $(v,{{\mathbf g'}})\in E_j$, so ${\mathcal D}(G_v)= {\mathcal D}({{\mathbf g'}})\preceq G\cdot H$.
To finish, we show that $B_H$ has nonempty interior in $V$. Suppose otherwise. As $B_H$ is constructible (Lemma \[lem:constructible\]), ${\overline}{B_H}$ is a proper closed subset of $V$, so $V\backslash B_H$ is a $G$-stable subset with nonempty interior. Now $\eta(\phi^{-1}(S(H))$ is dense in $V$ as it contains $\eta(G\cdot D_m)$. Hence there is a nonempty open $G$-stable subset $O$ of $\eta(\phi^{-1}(S(H)))\cap V'$ such that $B_H\cap O$ is empty. Let $v\in O$ and let ${{\mathbf g}}\in G^N$ such that $(v,{{\mathbf g}})\in D_m$. Then ${\mathcal D}({\mathcal G}({{\mathbf g}}))\preceq {\mathcal D}(G_v)\preceq G\cdot H$; but $v\not\in B_H$, so ${\mathcal D}(G_v)\neq G\cdot H$, and it follows from Corollary \[cor:po\] that ${\mathcal D}({\mathcal G}({{\mathbf g}}))\prec G\cdot H$. Hence ${\mathcal D}({\mathcal G}({{\mathbf g}}))= G\cdot H_i$ for some $i\in I$ such that $G\cdot H_i\prec G\cdot H$. Lemma \[lem:Psiimage\] now implies that $\eta^{-1}(O)\cap D_m\subseteq \bigcup_{i\in I'} \phi^{-1}(S(H_i))$, where $I':= \{i\in I\mid G\cdot H_i\prec G\cdot H\}$. By Corollary \[cor:qcmpctirred\], there exists $i\in I'$ such that $\eta^{-1}(O)\cap D_m\subseteq \phi^{-1}(S(H_i))$. Since $\eta^{-1}(O)\cap D_m$ is a nonempty open subset of $D_m$ and $\phi^{-1}(S(H_i))$ is closed and $G$-stable, $G\cdot D_m\subseteq \phi^{-1}(S(H_i))$. But $G\cdot H_i\prec G\cdot H$, which contradicts the minimality of $H$. We conclude that $B_H$ has nonempty interior in $V$ after all. Finally, since $G\cdot H= {\mathcal D}(G_v)$ for some $v\in V$, we have $\kappa(H)\leq \kappa(G_v)\leq \Theta$, so $N\geq \kappa(H)+ 1$. This completes the proof.
We can assume $O$ is $G$-stable. By Theorem \[thm:genericstab\], there is a nonempty open $G$-stable subset $O'$ of $V$ and a $G$-cr subgroup $H$ of $G$ such that ${\mathcal D}(G_v)= G\cdot H$ for all $v\in O'$. Now $O\cap O'$ is a nonempty open $G$-stable subset of $V$, and for all $v\in O\cap O'$, ${\mathcal D}(G_v)= G\cdot H$. Since $G_v$ is $G$-cr for $v\in O\cap O'$, $G_v$ is conjugate to $H$. It follows that $V$ has a principal stabiliser.
In particular, the hypotheses of Corollary \[cor:Gcrprinc\] are satisfied if ${\rm char}(k)= 0$ and $V_{\rm red}$ is nonempty, since then $V_{\rm red}$ is open by Theorem \[thm:main\] and for all $v\in V_{\rm red}$, $G_v$—being reductive—is $G$-cr. This proves Corollary \[cor:char0princ\].
\[rem:Leviconj\] Here is a generalisation of Corollary \[cor:char0princ\]. If ${\rm char}(k)= 0$ and $O$ is as in Theorem \[thm:genericstab\] then $G\cdot M_v= {\mathcal D}(G_v)= G\cdot H$ for all $v\in O$, where $M_v$ is any Levi subgroup of $G_v$.
Irreducible components of the stabiliser variety {#sec:stabcpts}
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In this section we study the irreducible components of the stabiliser variety $C$. We use the information we obtain to prove results analogous to those in Section \[sec:genericstab\], but for the subgroups $G_v^0$ rather than the subgroups $G_v$. We assume throughout the section that $N\geq 3$.
\[lem:cpts\] (a) Let $D$ be an irreducible component of $C$ such that $\eta(G\cdot D)$ is dense in $V$. Then ${\rm dim}(D)= n+ Nr$ and for all $v\in V_0$, the fibre $\left(\eta|_D\right)^{-1}(v)$ either is empty or has dimension $Nr$ and is isomorphic (via $\phi$) to a union of irreducible components of $G_v^N$.\
(b) There is a unique closed subset $\widetilde{C}$ of $C$ such that $\widetilde{C}$ contains $V\times \{{\mathbf 1}\}$, $\widetilde{C}$ is a union of irreducible components of $C$ and $G$ permutes these irreducible components transitively. The variety $\widetilde{C}$ is the closure of the set $\{(v,{\mathbf g})\mid v\in V_0, {\mathbf g}\in (G_v^0)^N\}$, and each irreducible component of $\widetilde{C}$ has dimension $n+ Nr$.
Clearly it is enough to prove the result when $G$ is connected and $V$ is irreducible, so we assume this.
\(a) Define $f\colon V\times G^N {\rightarrow}V\times V^N$ by $$f(v,{\mathbf g})= (v,g_1\cdot v,\ldots, g_N\cdot v).$$ Let $Y$ be the closure of the image of $f$. Let $\Delta$ be the diagonal in $V\times V^N$; then $C= f^{-1}(\Delta)$. The variety $Y$ is irreducible because $G$ and $V$ are irreducible. Let $v\in V$ and let ${\mathbf g}\in G^N$. Then $f^{-1}(v,g_1\cdot v,\ldots, g_N\cdot v)= \{v\}\times g_1G_v,\ldots, g_NG_v$. Hence irreducible components of generic fibres of $f$ over $Y$ have dimension $Nr$. It follows that ${\rm dim}(Y)= {\rm dim}(V\times G^N)- Nr = n+ N{\rm dim}(G)- Nr= n+ N({\rm dim}(G)- r)$. Since $\eta(D)$ is dense in $V$, $f(D)$ is dense in $\Delta$. Hence ${\rm dim}(D)\geq {\rm dim}(\Delta)+ Nr= n+ Nr$.
If $v\in \eta(D)\cap V_0$ and $Z$ is an irreducible component of $\left(\eta|_D\right)^{-1}(v)$ then ${\rm dim}(Z)\geq {\rm dim}(D)- {\rm dim}(V)\geq n+ Nr- n= Nr$. But $\phi(\eta^{-1}(v))$ is a subset of $G_v^N$ and the irreducible components of $G_v^N$ all have dimension ${\rm dim}(G_v^N)= Nr$. This forces $Z$ to be isomorphic (via $\phi$) to an irreducible component of $G_v^N$. Hence irreducible components of generic fibres of $\eta|_D$ have dimension $Nr$, which implies that ${\rm dim}(D)= n+ Nr$. Part (a) now follows.\
(b) Since $V\times \{{\mathbf 1}\}$ is irreducible, there is some irreducible component $\widetilde{C}$ of $C$ such that $\widetilde{C}$ contains $V\times \{{\mathbf 1}\}$. For any $v\in V_0$, let $Z$ be an irreducible component of the fibre $\left(\eta|_{\widetilde{C}}\right)^{-1}(v)$ such that $(v,{{\mathbf 1}})\in Z$. By part (a), ${\rm dim}(Z)= Nr$, so $Z$ is isomorphic via $\phi$ to an irreducible component of $G_v^N$. But the only component of $G_v^N$ that contains ${\mathbf 1}$ is $(G_v^0)^N$, so $\{v\}\times (G_v^0)^N\subseteq Z$. Hence $\widetilde{C}$ contains the closure of $\{(v,{\mathbf g})\mid v\in V_0, {\mathbf g}\in (G_v^0)^N\}$—call this closure $C'$.
Let $A_1,\ldots, A_m$ be the irreducible components of $C'$ such that ${\overline}{\eta(A_j)}= V$ (there is at least one, since $\eta(C')= V$). Let $s_i= {\rm dim}(A_i)$ for $1\leq i\leq m$ and let $\eta_i\colon A_i{\rightarrow}V$ be the restriction of $\eta$. There is a nonempty open subset $U$ of $V$ such that for all $v\in U$, $\eta^{-1}(v)\subseteq A_1\cup\cdots \cup A_m$ and every irreducible component of $\eta_i^{-1}(v)$ has dimension $s_i- n$. Since $\{v\}\times (G_v^0)^N\subseteq C'$ for all $v\in V_0$, if $v\in U\cap V_0$ then $\eta_j^{-1}(v)$ must contain $\{v\}\times (G_v^0)^N$ for some $1\leq j\leq m$, which forces $s_j\geq n+ Nr$. But ${\rm dim}(\widetilde{C})= n+ Nr$ by part (a), so $A_j$ must be the whole of $\widetilde{C}$, so $C'= \widetilde{C}$. This completes the proof.
\[rem:pathologies\] The dimension inequality in Lemma \[lem:cpts\](a) can fail if $\eta(G\cdot D)$ is not dense in $V$ (Example \[exmp:unipotent\]). Moreover, $\widetilde{C}$ need not contain the whole of $\{(v,{\mathbf g})\mid v\in V, {\mathbf g}\in (G_v^0)^N\}$: see Examples \[exmp:cosets\](a) and \[exmp:unipotent\].
If $G$ is connected and $V$ is irreducible then $\widetilde{C}$ is irreducible and $G$-stable. More generally, any irreducible component of $C$ is $G$-stable in this case.
We call $\widetilde{C}$ the [*connected-stabiliser variety*]{} of $V$.
\[cor:finstab\] If $r= 0$ then $\widetilde{C}= V\times \{{{\mathbf 1}}\}$.
The irreducible components of $V\times \{{{\mathbf 1}}\}$ are isomorphic via $\eta$ to the irreducible components of $V$, so they are permuted transitively by $G$ and each has dimension $n$. It follows from the dimension formula in Lemma \[lem:cpts\](a) that these irreducible components are irreducible components of $C$. The result now follows from Lemma \[lem:cpts\](b).
We denote by $\widetilde{\phi}\colon \widetilde{C}{\rightarrow}G^N$ and $\widetilde{\eta}\colon \widetilde{C}{\rightarrow}V$ the restrictions to $\widetilde{C}$ of $\phi$ and $\eta$, respectively, and if $v\in V$ then we denote $\widetilde{\phi}(\widetilde{\eta}^{-1}(v))$ by $F_v$. If $v\in V_0$ then $(G_v^0)^N\subseteq F_v$; we do not know whether equality holds for all $v\in V_0$, or even for generic $v\in V_0$.
We now give a counterpart to Theorem \[thm:genericstab\]. In the connected case, we obtain slightly more information: we can describe ${\mathcal D}(G_v^0)$ for all $v\in V_{\rm min}$ (recall the definition of $V_{\rm min}$ from Remark \[rem:Vmin\]).
\[thm:genericstab\_conn\] There exists a connected $G$-completely reducible subgroup $H$ of $G$ such that:
- for all $v\in V_{\rm min}$, ${\mathcal D}(G_v^0)= G\cdot H$.
- $\widetilde{C}\subseteq \widetilde{\phi}^{-1}(S(H))$.
In particular, if $V_{\rm red}$ is nonempty then ${\mathcal D}(G_v^0)= G\cdot H$ for all $v\in V_{\rm red}$.
By Theorem \[thm:genericstab\], there exist a $G$-cr subgroup $H'$ of $G$ and a $G$-stable open subset $O$ of $V$ such that ${\mathcal D}(G_v)= G\cdot H'$ for all $v\in O$. Set $H= (H')^0$; then $H$ is $G$-cr as $H\unlhd H'$. Let $t$ be the minimal dimension of ${\rm dim}(R_u(G_v))$ for $v\in V_0$. The $G$-stable open sets $O$ and $V_{\rm min}$ have nonempty intersection, so there exists $v\in V_{\rm min}\cap O$ such that ${\mathcal D}(G_v)= G\cdot H'$ and ${\rm dim}(R_u(G_v))= t$. This yields ${\rm dim}(H')= {\rm dim}(G_v)- {\rm dim}(R_u(G_v))= r- t$.
By the proof of Theorem \[thm:genericstab\], $\widetilde{C}\subseteq \widetilde{\phi}^{-1}(S(H'))$. Let $v\in V_{\rm min}$ and choose $\lambda\in Y(G)$ such that ${\mathcal D}(G_v)= c_\lambda(G_v)$. Then $c_\lambda(G_v)$ is $G$-cr and $c_\lambda(G_v^0)= c_\lambda(G_v)^0$ is a normal subgroup of $c_\lambda(G_v)$, so $c_\lambda(G_v^0)$ is $G$-cr. It follows that ${\mathcal D}(G_v^0)= G\cdot c_\lambda(G_v^0)$. We want to prove that $c_\lambda(G_v^0)$ is conjugate to $H$: that is, we want to prove that $$\label{eqn:conj_conn}
(\exists m\in G) \ [(\forall g\in G_v^0) \ c_\lambda(g)\in mHm^{-1} \wedge (\forall h\in H) \ (\exists g\in G_v^0) \ c_\lambda(g)= mhm^{-1}].$$ Since (\[eqn:conj\_conn\]) is a first-order formula, this is a constructible condition. Hence it is enough to prove that it holds after extending $k$ to any larger algebraically closed field. So without loss of generality we assume $k$ is solid.
By Proposition \[prop:topfg\], we can choose ${{\mathbf g'}}\in (G_v^0)^N$ such that $\alpha_{G_v^0}({\mathcal G}({{\mathbf g'}}))= G_v^0/R_u(G_v^0)$. There exists ${{\mathbf h}}\in (H')^N$ such that $\pi_G({{\mathbf h}})= \pi_G({{\mathbf g'}})$. Let $K= {\mathcal G}({{\mathbf h}})$. Now $c_\lambda({\mathcal G}({{\mathbf g'}}))= c_\lambda(G_v^0)$ is $G$-cr, so there exists $\mu\in Y(G)$ such that $c_\mu({{\mathbf h}})$ is conjugate to $c_\lambda({{\mathbf g'}})$. Then $c_\lambda({\mathcal G}({{\mathbf g'}}))$ is conjugate to $c_\mu({\mathcal G}({{\mathbf h}}))$. But ${\rm dim}(c_\lambda({\mathcal G}({{\mathbf g'}})))= {\rm dim}(c_\lambda(G_v^0))= {\rm dim}(H)\geq {\rm dim}(K)\geq {\rm dim}(c_\mu(K))= {\rm dim}(c_\mu({\mathcal G}({{\mathbf h}})))$, which forces ${\rm dim}(K)$ to equal ${\rm dim}(H)$. Hence $K\supseteq H$. Now $c_\mu(K)$ is conjugate to $c_\lambda(G_v^0)$, which is connected, so $c_\mu(K)= c_\mu(H)$. But $c_\mu(H)$ is conjugate to $H$ since $H$ is $G$-cr, so we deduce that $c_\lambda(G_v^0)$ is conjugate to $H$. Hence ${\mathcal D}(G_v^0)= G\cdot H$. This proves part (a). Moreover, if ${{\mathbf g}}\in (G_v^0)^N$ then $c_\lambda({\mathcal G}({{\mathbf g}}))= {\mathcal G}(c_\lambda({{\mathbf g}}))$ is conjugate to a subgroup of $H$, so there exists ${{\mathbf h}}\in H^N$ such that $c_\lambda({{\mathbf g}})$ is conjugate to ${{\mathbf h}}$; hence $(v,{{\mathbf g}})\in \widetilde{\phi}^{-1}(S(H))$. As $\{(v,{{\mathbf g}})\mid v\in V_{\rm min}, {{\mathbf g}}\in (G_v^0)^N\}$ is dense in $\widetilde{C}$ by Lemma \[lem:cpts\](b) and Remark \[rem:Vmin\], $\widetilde{C}\subseteq \widetilde{\phi}^{-1}(S(H))$. This proves part (b).
The next result is the counterpart to Corollaries \[cor:Gcrprinc\] and \[cor:char0princ\]. We omit the proof, which is similar.
\[cor:Gcrprinc\_conn\] Suppose there is a nonempty open subset $O$ of $V$ such that $G_v^0$ is $G$-cr for all $v\in O$ (in particular, this condition holds if ${\rm char}(k)= 0$ and $V_{\rm red}$ is nonempty)). Let $H$ be the connected $G$-cr subgroup from Theorem \[thm:genericstab\_conn\]. Then $G_v^0$ is conjugate to $H$ for all $v\in V_{\rm red}$.
\[rem:genericGirstab\] Suppose there exists $v\in V_0$ such that $G_v^0$ is $G$-ir. Then $G_v$ is $G$-cr, so $v\in V_{\rm red}$, so $V_{\rm red}$ is nonempty. We have ${\mathcal D}(G_v^0)= G\cdot H$ by Theorem \[thm:genericstab\_conn\](a). As $G_v^0$ is $G$-cr, $G\cdot G_v^0= G\cdot H$. It follows that $H$ is $G$-ir and $G_w^0$ is conjugate to $H$ for all $w\in V_{\rm red}$; in particular, $G_w^0$ is $G$-ir for all $w\in V_{\rm red}$. The analogous result for the full stabiliser $G_v$ is false (cf. Remarks \[rem:notgeneric\] and \[rem:genericGir\](ii), and Examples \[exmp:cosets\](c) and \[exmp:unipotent\]). However, if $O$ is as in Theorem \[thm:genericstab\] and there exists $v\in O$ such that $G_v$ is $G$-ir then an argument like the one above shows that $V$ has a $G$-ir principal stabiliser.
Theorem \[thm:genericstab\_conn\] gives rise to the following counterpart to Remark \[rem:Leviconj\] for $G_v^0$; the proof is similar.
Suppose ${\rm char}(k)= 0$. Then $H$ is conjugate to a Levi subgroup of $G_v^0$ for all $v\in V_{\rm min}$.
We give a criterion to ensure that the fibres of $\widetilde{\eta}$ are irreducible. Define $\widetilde{C}_{\rm min}= \widetilde{\eta}^{-1}(V_{\rm min})$.
Suppose ${\rm char}(k)= 0$ and $N\geq \Theta+ 1$. Then $\widetilde{C}_{\rm min}= \{(v,{{\mathbf g}})\mid v\in V_{\rm min}, {{\mathbf g}}\in (G_v^0)^N\}$.
Let $H$ be as in Theorem \[thm:genericstab\_conn\]. Let $v\in V_{\rm min}$, and suppose $F_v$ properly contains $(G_v^0)^N$. Then $F_v$ contains an irreducible component $D\neq (G_v^0)^N$ of $G_v^N$ by Lemma \[lem:cpts\](a). Set $K= G_v$, set $M= K/R_u(K)$ and let $\alpha_K\colon K{\rightarrow}M$ be the canonical projection. Let $K_1$ be the subgroup of $K$ generated by $K^0$ together with the components of each of the tuples in $D$; then $K_1$ properly contains $K^0$. As $R_u(G_v)$ is connected, $M_1:= \alpha_K(K_1)$ properly contains $M^0$. In particular, $M_1$ is reductive. By [@Mar Lem. 9.2], there exists ${{\mathbf g}}\in D$ such that $\alpha_K({{\mathbf g}})$ generates $M_1$. Hence $G\cdot K_1\preceq G\cdot {\mathcal G}({{\mathbf g}})$. Now ${{\mathbf g}}\in \widetilde{C}$, so $G\cdot {\mathcal G}({{\mathbf g}})\preceq G\cdot H$ (Theorem \[thm:genericstab\_conn\](b)). It follows from Lemma \[lem:transitive\] that $G\cdot K_1\preceq G\cdot H$.
We have $G\cdot H= {\mathcal D}(K^0)$ by choice of $v$ and Theorem \[thm:genericstab\_conn\], so $G\cdot H\preceq G\cdot K^0$ by Lemma \[lem:preccrit\]. Now $G\cdot K^0\preceq G\cdot K_1$ as $K^0\leq K_1$, so $G\cdot H\preceq G\cdot K_1$ by Lemma \[lem:transitive\]. It follows from Lemmas \[lem:preccrit\] and \[lem:antisymm\] that $G\cdot H= {\mathcal D}(K_1)$. Now ${\mathcal D}(K_1)= G\cdot M_1$ as $M_1$ is reductive and ${\rm char}(k)= 0$, so $G\cdot H= G\cdot M_1$. But this is impossible as $H$ is connected and $M_1$ is not. We conclude that $F_v= (G_v^0)^N$ after all. The result now follows.
We have seen that we obtain stronger results if we know that generic stabilisers (or their identity components) are $G$-cr. Reductive subgroups are always $G$-cr in characteristic 0, but things are more complicated in positive characteristic. Our next result shows that if this $G$-complete reducibility condition fails for connected stabilisers then it fails badly: we prove that if there exists $v\in V_0$ such that $G_v^0$ is reductive but not $G$-cr then generic elements of $V$ have the same property.
\[prop:genericnoncr\_conn\] Let $H$ be as in Theorem \[thm:genericstab\_conn\]. Let $$\widetilde{B}_H'= \{v\in V_{\rm min}\mid \mbox{$G_v^0$ is not $G$-cr}\}.$$ If $\widetilde{B}_H'$ is nonempty then $\widetilde{B}_H'$ has nonempty interior.
Note that ${\mathcal D}(G_v^0)= G\cdot H$ for all $v\in \widetilde{B}_H'$, by Theorem \[thm:genericstab\_conn\]. The argument below shows that $\widetilde{B}_H'= V_{\rm min}\cap \widetilde{\eta}(\widetilde{\phi}^{-1}(U))$, where $U$ is the open set defined below, so $\widetilde{B}_H'$ is constructible. It follows as in the proof of Theorem \[thm:genericstab\_conn\] that we can extend the ground field $k$; hence we can assume $k$ is solid.
Suppose $\widetilde{B}_H'$ is nonempty. Let $v\in \widetilde{B}_H'$. We can choose ${{\mathbf g}}\in (G_v^0)^N$ such that $\alpha_{G_v^0}({\mathcal G}({{\mathbf g}}))= G_v^0/R_u(G_v^0)$ (Proposition \[prop:topfg\]) and such that $1\neq g_N\in R_u(G_v^0)$ if $G_v^0$ is non-reductive. This ensures that ${\mathcal G}({{\mathbf g}})$ is not $G$-cr. As $H$ is $G$-cr and ${\mathcal D}(G_v^0)= G\cdot H$, there exists $\lambda\in Y(G)$ such that $G_v^0\leq P_\lambda$ and $c_\lambda(G_v^0)$ is conjugate to $H$. Then $c_\lambda({\mathcal G}({{\mathbf g}}))= c_\lambda(G_v^0)$ is conjugate to $H$, so ${\rm dim}(G_{{{\mathbf g}}})= {\rm dim}(C_G({\mathcal G}({{\mathbf g}})))< {\rm dim}(C_G(H))$, since ${\mathcal G}({{\mathbf g}})$ is not conjugate to $H$ (as ${\mathcal G}({{\mathbf g}})$ is not $G$-cr). Consider $G^N$ regarded as a $G$-variety. Let $U$ be the set of all ${{\mathbf m}}\in G^N$ such that ${\rm dim}(G_{{\mathbf m}})< {\rm dim}(C_G(H))$; then $U$ is an open neighborhood $U$ of ${{\mathbf g}}$, by Lemma \[lem:stabdimcty\].
Let $E= \{(w,{{\mathbf g}})\in \widetilde{C}\cap \widetilde{\phi}^{-1}(U)\cap \widetilde{\eta}^{-1}(V_{\rm min})\mid {{\mathbf g}}\in (G_w^0)^N\}$. By Lemma \[lem:cpts\](b), $E$ is dense in $\widetilde{C}$, so $\widetilde{\eta}(E)$ is dense in $V$. To complete the proof, it is enough to show that $\widetilde{\eta}(E)\subseteq \widetilde{B}_H'$: for then $\widetilde{B}_H'$, being constructible and dense, has nonempty interior. So let $w\in \widetilde{\eta}(E)$. Pick ${{\mathbf m}}$ such that $(w,{{\mathbf m}})\in E$. Then ${{\mathbf m}}\in (G_w^0)^N\cap U$, so ${\rm dim}(C_G({{\mathbf m}}))< {\rm dim}(C_G(H))$, so ${\rm dim}(C_G(G_w^0))< {\rm dim}(C_G(H))$ also. It follows by running the argument above for $G_v^0$ in reverse that $G_w^0$ is not $G$-cr. Hence $w\in \widetilde{B}_H'$, as required.
\[rem:notgeneric\] A similar argument establishes the following. Let $H$ be as in Theorem \[thm:genericstab\]. If there exists $(v,{{\mathbf g}})\in C$ such that $v\in V_0$, ${\mathcal D}(G_v)= G\cdot H$ and $G_v$ is not $G$-cr then there is an open neighbourhood $U$ of $(v,{{\mathbf g}})\in C$ such that for all $(w,{{\mathbf g'}})\in U$, $G_w$ is not $G$-cr. But this does not yield an analogue of Proposition \[prop:genericnoncr\_conn\] for $G_v$ (see Example \[exmp:cosets\](b))—the problem is that $\eta(U)$ need not be dense in $V$.
Examples {#sec:ex}
========
In this section we present some examples that show the limits of our results and illustrate some of the phenomena that can occur. We assume $N\geq \Theta+ 1$.
\[exmp:cosets\] We consider a special case of the set-up from the proof of Theorem \[thm:subgpint\]. Let $G= {\rm PGL}_2(k)$, let $M\leq G$ and let $V$ be the quasi-projective variety $G/M$ with $G$ acting by left multiplication. We assume that $M\cap gMg^{-1}= 1$ for generic $g\in G$ (this will hold in all the cases we consider). Then $G_w= 1$ for generic $w\in V$, so the subset $H$ from Theorem \[thm:genericstab\] is $1$, and $\widetilde{C}= V\times \{{{\mathbf 1}}\}$ by Corollary \[cor:finstab\]. In particular, $F_w= \{{{\mathbf 1}}\}$ for all $w\in V$. Let $v= M\in G/M$; then $G_v= M$.\
(a) Let $M$ be a maximal torus of $G$. Then $F_v$ is properly contained in $M^N$, so we see that $F_v$ need not contain all of $(G_v^0)^N$ when $v\not\in V_0$ (cf. Remark \[rem:pathologies\]). The subset $B_H$ is dense but not closed in $V$, as ${\mathcal D}(G_v)= G\cdot M$.\
(b) Let $M= \langle x\rangle$, where $x\in G$ is a nontrivial unipotent element. Then $V= V_0= V_{\rm red}$ and $G_w$ is unipotent for all $w\in V$, so ${\mathcal D}(G_v)= \{1\}$ for all $w\in V$ (where $1$ denotes the trivial subgroup). Now $G_w= 1$ is $G$-cr for generic $w\in V$ but $G_v$ is not $G$-cr. Hence the set $\{w\in V_{\rm red}\mid {\mathcal D}(G_w)= G\cdot H\ \mbox{and $G_v$ is not $G$-cr}\}$ is nonempty but not dense in $V$ (cf. Remark \[rem:notgeneric\]). The irreducible components of $C$ apart from $\widetilde{C}$ do not dominate $V$.\
(c) Let $M= {\rm PGL}_2(q)$, where $q$ is a power of the characteristic $p$. We have $V= V_0= V_{\rm red}$. Now $M$ is $G$-ir, so the set $\{w\in V_{\rm red}\mid G_w\ \mbox{is $G$-ir}\}$ is nonempty but not dense in $V$. Moreover, the set $O$ from Theorem \[thm:genericstab\] does not contain the whole of $V_{\rm red}$.
\[exmp:unipotent\] Suppose $G$ is connected and not a torus. Let $m\in {{\mathbb N}}$ and let $V$ be the variety of $m$-tuples of unipotent elements of $G$, with $G$ acting on $V$ by simultaneous conjugation. We claim that $\{(1,\ldots, 1)\}\times G^N$ is an irreducible component of $C$. For let $D$ be an irreducible component of $C$ such that $\{(1,\ldots, 1)\}\times G^N\subseteq D$. Consider the element $(1,\ldots, 1,{{\mathbf g}})\in D$, where the components of ${{\mathbf g}}\in G^N$ are all regular semisimple elements of $G$. There is an open neighborhood $O$ of ${{\mathbf g}}$ in $G^N$ consisting of tuples of regular semisimple elements. If $(v_1,\ldots, v_m,{{\mathbf g'}})\in \phi^{-1}(O)$ then each component of ${{\mathbf g'}}$ is a regular semisimple element of $g$ centralising the unipotent elements $v_1,\ldots, v_m$ of $G$. But this forces $v_1,\ldots, v_m$ to be 1. It follows that $D= \{(1,\ldots, 1)\}\times G^N$, as claimed. Hence $\eta(D)= \{(1,\ldots, 1)\}$ and $\eta(D)\cap V_0$ is empty (note also that if $m$ is large enough then the dimension inequality from Lemma \[lem:cpts\](a) is violated). We see that the set $\{w\in V\mid G_w^0\ \mbox{is $G$-ir}\}$ is nonempty but not dense in $V$.
It is not hard to show that $F_{(1,\ldots, 1)}\subseteq \{g\cdot {{\mathbf g}}\mid {{\mathbf g}}\in U^N\}$, where $U$ is a maximal unipotent subgroup of $G$; in particular, we see as in Example \[exmp:cosets\](a) that $F_v$ need not contain all of $(G_v^0)^N$ when $v\not\in V_0$. Moreover, since the centraliser of a nontrivial unipotent subgroup of a connected group can never be reductive, the only reductive stabiliser is $G_{(1,\ldots, 1)}$, so $V_{\rm red}$ is empty.
\[exmp:notprinc\] Let $X$ be an affine variety and let $M$ be a reductive linear algebraic group. Suppose we are given a morphism $f\colon X\times M{\rightarrow}X\times G$ of the form $f(x,m)= (x,f_x(m))$, and suppose further that each $f_x\colon M{\rightarrow}G$ is a homomorphism of algebraic groups. Set $K_x= {\rm im}(f_x)$. Define actions of $G$ and $M$ on $X\times G$ by $g\cdot (x,g')= (x,gg')$ and $m\cdot (x,g')= (x,g'f_x(m)^{-1})$. These actions commute with each other, so we get an action of $G$ on the quotient space $V:= (X\times G)/M$.
Now suppose moreover that ${\rm dim}(K_x)$ is independent of $x$. Then the $M$-orbits on $X\times G$ all have the same dimension, so they are all closed. This means the canonical projection $\varphi$ from $X\times G$ to $V$ is a geometric quotient, so its fibres are precisely the $M$-orbits [@New Cor. 3.5.3]. A straightforward calculation shows that for any $(x,g)\in X\times G$, the stabiliser $G_{\varphi(x,g)}$ is precisely $gK_xg^{-1}$. It follows that if $X$ is infinite and the subgroups $K_x$ are pairwise non-conjugate as $x$ runs over the elements of a dense subset of $X$ then $V$ has no principal stabiliser.
Here is a simple example. Let $G= {\rm SL}_2(k)$, let $X= k$ and let $M= C_p\times C_p= \langle \gamma_1,\gamma_2\mid \gamma_1^p= \gamma_2^p= [\gamma_1,\gamma_2]= 1\rangle$. Define $f\colon X\times M{\rightarrow}X\times G$ by $f(x,m)= (x,f_x(m))$, where $f_x(\gamma_1^{m_1}\gamma_2^{m_2}):=
\left(
\begin{array}{cc}
1 & m_1x+ m_2x^2 \\
0 & 1
\end{array}
\right)
$. It is easily checked that $f$ has the desired properties, so $V:= (X\times M)/G$ has no principal stabiliser. Note also that generic stabilisers are nontrivial finite unipotent groups, but the element $v= \varphi(0,1)$ has trivial stabiliser.
Here is an example where the stabilisers are connected. Daniel Lond [@lond Sec. 6.5] produced a family, parametrised by $X:= k$, of homomorphisms from $M:= {\rm SL}_2(k)$ to $G:= B_4$ in characteristic 2 with pairwise non-conjugate images. Using this one can construct a morphism $f\colon X\times M{\rightarrow}X\times G$ with the desired properties, giving rise to a $G$-variety $V:= (X\times M)/G$ having no principal stabiliser and with all stabilisers connected and reductive. Results of David Stewart give rise to a similar construction for $G= F_4$ in characteristic 2 [@stewart Sec. 5.4.3].
\[exmp:guralnick\] We now give an example where there is a point with trivial stabiliser but generic stabilisers are finite and linearly reductive, using another special case of the set-up from the proof of Theorem \[thm:subgpint\]. We describe a recipe for producing such examples, given in [@BGS Cor. 3.10]. Take a simple algebraic group $G$ of rank $s$ in characteristic not 2 and set $M= C_G(\tau)$, where $\tau$ is an involution that inverts a maximal torus of $G$. Then the affine variety $G/M$, with $M$ acting by left multiplication, has precisely one orbit that consists of points with trivial stabiliser. Let $V= G/M\times G/M$ with the product action of $G$. Then generic stabilisers of points in $V$ are 2-groups of order $2^s$, but $V$ contains points with trivial stabiliser. Thus $V= V_0= V_{\rm red}$ and $\widetilde{C}= V\times \{{{\mathbf 1}}\}$. Since 2-groups are linearly reductive—and hence $G$-cr—in characteristic not 2, the $G$-cr subgroup $H$ from Theorem \[thm:genericstab\] must be a 2-group of order $2^s$, and moreover, $H$ is a principal stabiliser for $V$ by Corollary \[cor:Gcrprinc\]. The set $\{v\in V_{\rm red}\mid {\mathcal D}(G_v)= G\cdot H\}$ does not contain the whole of $V_{\rm red}$ (cf. Theorem \[thm:genericstab\_conn\]).
We claim that there is at least one irreducible component $D$ of $C$ such that ${\overline}{\eta(D)}= V$ but $\eta(D)\neq V$. Let $D_1,\ldots, D_t$ be the irreducible components of $V$ apart from $C$. Then $\bigcup_{i= 1}^t {\overline}{\eta(D_i)}= V$, so $\eta(D_j)$ is dense in $V$ for some $1\leq j\leq t$. There are only finitely many conjugacy classes of nontrivial elements of $G$ of order dividing $2^s$, and each such conjugacy class is closed because in characteristic not 2, elements of order a power of 2 are semisimple. Hence there are regular functions $f_1, \ldots, f_m\colon G{\rightarrow}k$ for some $m$ such that for all $g\in G$, $g$ is a nontrivial element of order dividing $2^s$ if and only if $f_1(g)= \cdots = f_m(g)= 0$. For $1\leq l\leq N$, let $Z_l$ be the closed subset $\{(v,g_1,\ldots, g_N)\in C\mid f_1(g_l)=\cdots = f_m(g_l)= 0\}$ of $C$ and let $Z= Z_1\cup\cdots \cup Z_N$. If $(v,{{\mathbf g}})\in D_j\backslash (D_j\cap \widetilde{C})$ then ${{\mathbf g}}\neq 1$, so some component of ${{\mathbf g}}$ is a nontrivial element of $G$ of order dividing $2^s$, so $(v,{{\mathbf g}})\in Z$. Hence the open dense subset $D_j\backslash (D_j\cap \widetilde{C})$ of $D_j$ is contained in $Z$, and it follows that $D_j\subseteq Z$. This implies that if $v\in V$ and $G_v= 1$ then $v\not\in \eta(D_j)$.
: The author acknowledges the financial support of Marsden Grant UOA1021. He is grateful to Robert Guralnick for introducing him to the reductivity problem for generic stabilisers and for helpful discussions. He also thanks V. Popov for drawing his attention to references [@rich72acta], [@pop] and [@LunaVust], and the referee for helpful comments.
[00]{} T.C. Burness, R.M. Guralnick, J. Saxl, *On base sizes for algebraic groups*, to appear in J. Eur. Math. Soc. [arXiv:1310.1569 \[math.GR\]]{}.
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[^1]: Wallach also has a proof in this case [@wallach].
|
---
abstract: 'The anomalous magnetic moments of the electron and the muon are interesting observables, since they can be measured with great precision and their values can be computed with excellent accuracy within the Standard Model (SM). The current experimental measurement of this quantities show a deviation of a few standard deviations with respect to the SM prediction, which may be a hint of new physics. The fact that the electron and the muon masses differ by two orders of magnitude and the deviations have opposite signs makes it difficult to find a common origin of these anomalies. In this work we introduce a complex singlet scalar charged under a Peccei–Quinn-like (PQ) global symmetry together with the electron transforming chirally under the same symmetry. In this realization, the CP-odd scalar couples to electron only, while the CP-even part can couple to muons and electrons simultaneously. In addition, the CP-odd scalar can naturally be much lighter than the CP-even scalar, as a pseudo-Goldstone boson of the PQ-like symmetry, leading to an explanation of the suppression of the electron anomalous magnetic moment with respect to the SM prediction due to the CP-odd Higgs effect dominance, as well as an enhancement of the muon one induced by the CP-even component.'
author:
- Jia Liu
- 'Carlos E.M. Wagner'
- 'Xiao-Ping Wang'
bibliography:
- 'referencelist.bib'
title: A light complex scalar for the electron and muon anomalous magnetic moments
---
Introduction
============
The Standard Model (SM) provides a precise theoretical framework for the description of all known interactions in nature. The SM description of the interaction of quarks and leptons with electroweak gauge bosons has been probed at the per-mille level, being hence sensitive to quantum corrections to the tree-level results [@Tanabashi:2018oca]. No significant deviations from the SM predictions have been found.
Since Schwinger’s first computation of the electron anomalous magnetic moment of the electron, it was realized that its measurement can provide an accurate test of Quantum Electrodynamics (QED), and subsequently of the SM, describing the interactions of fundamental particles in nature. The QED contribution [@Schwinger:1948iu; @Sommerfield:1957zz; @Petermann:1957hs; @PhysRevLett.47.1573; @Kinoshita:1990wp; @Laporta:1996mq; @Degrassi:1998es; @Kinoshita:2004wi; @Kinoshita:2005sm; @Passera:2006gc; @Kataev:2006yh; @Passera:2006gc; @Aoyama:2007mn; @Aoyama:2012wj; @Aoyama:2012wk; @Laporta:2017okg; @Aoyama:2017uqe; @Volkov:2017xaq; @Volkov:2018jhy] to the anomalous magnetic moment of the electron and the muon is today known up to 5-loop order [@Tanabashi:2018oca; @Mohr:2000ie; @Czarnecki:1998nd].
The QED contribution, although dominant, is not the only one affecting the anomalous magnetic moments. The hadronic contributions [@Jegerlehner:1985gq; @Lynn:1985sq; @Swartz:1994qz; @Martin:1994we; @Eidelman:1995ny; @Krause:1996rf; @Davier:1998si; @Jegerlehner:1999hg; @Jegerlehner:2003qp; @Melnikov:2003xd; @deTroconiz:2004yzs; @Bijnens:2007pz; @Davier:2007ua] become quite relevant and can be accurately computed from dispersion relations describing the electron-positron collisions with hadrons in the final states. Moreover, the weak interaction effects [@Czarnecki:1995wq; @Czarnecki:1995sz; @Czarnecki:1996if; @Czarnecki:2002nt; @Heinemeyer:2004yq; @Gribouk:2005ee], although suppressed by powers of the weak gauge boson masses, become also relevant at the level of accuracy provided by today’s computations. Finally, there is a component of the hadronic contribution, the so-called light-by-light contribution [@Bijnens:1995xf; @Hayakawa:1997rq; @Knecht:2001qf; @Knecht:2001qg; @RamseyMusolf:2002cy; @Melnikov:2003xd; @deTroconiz:2004yzs; @Prades:2009tw; @Kataev:2012kn; @Kurz:2015bia; @Colangelo:2017qdm], which cannot be obtained experimentally and hence has to be estimated by theoretical methods.
Quite importance for these determinations is an accurate measurement of the fine structure constant. The authors of Ref. [@Parker191] use the recoil frequency of Cesium-133 atoms in a matter-wave interferometer to determine the mass of the Cs atom, and obtain the most accurate value of the fine structure constant to date. By combining it with theory [@Aoyama:2014sxa; @Mohr:2015ccw], they obtain the electron magnetic dipole moment to be $$\begin{aligned}
\Delta a_e \equiv a_e^{\rm exp} - a_e^{\rm SM} = (-88 \pm 36)\times 10^{-14},
\label{eq:g-2-e}\end{aligned}$$ which implies the deviation has a negative sign and presents a $2.4~\sigma$ discrepancy [@Parker191; @Jegerlehner:2018zrj; @Davoudiasl:2018fbb] between the SM prediction and experimental measurements [@PhysRevLett.100.120801; @Hanneke:2010au]. On the other hand, the muon magnetic dipole moment has $3.7~\sigma$ discrepancy with a positive sign, opposite to the $a_e$ deviation [@Blum:2018mom; @Bennett:2006fi], $$\begin{aligned}
\Delta a_\mu \equiv a_\mu^{\rm exp} - a_\mu^{\rm SM} = (2.74 \pm 0.73)\times 10^{-9}.
\label{eq:g-2-mu}\end{aligned}$$
The $a_\mu$ deviation is of the same order of the weak corrections and hence can be naturally explained by physics at the weak scale. As it was first stressed in Ref. [@Giudice:2012ms], assuming similar corrections to $a_e$, due to the dependence on the square of lepton mass, they become of the order of $\Delta a_e \simeq 0.7 \times 10^{-13}$. Therefore, they cannot lead to an explanation of the $a_e$ anomaly. Moreover, if the interactions affecting electron and muon sector would be the same, one would expect deviations of the same sign and not of opposite signs as observed experimentally, Eqs. (\[eq:g-2-e\]) and (\[eq:g-2-mu\]).
To simultaneously explain the two anomalies, the interactions should violate lepton flavor universality in a delicate way, to contribute negatively for electrons while positively for muons. Recently, the authors of Ref. [@Davoudiasl:2018fbb] have provided a solution with one CP-even real scalar coupled to both $e$ and $\mu$ with different couplings. To achieve negative contribution to $g-2$ of electron, they further require that this scalar contribute to $a_e$ via a 2-loop Barr-Zee diagram with the sign of the coupling specifically chosen to lead to the require effect. Another recent work [@Abu-Ajamieh:2018ciu], also discusses both scalar and pseudo-scalar with 2-loop Barr-Zee, Light-By-Light and Vacuum Polarization diagrams. In an independent work, the authors of Ref. [@Crivellin:2018qmi] have, instead, added both $SU(2)_{L}$ doublet and singlet vector-like heavy leptons, which couple to the SM leptons via Yukawa interaction. The origin of different sign to $\Delta a_{e/\mu}$ comes from the sign of the off-diagonal Yukawa coupling between heavy lepton and SM lepton.
In this work, we shall assume that the reason for the discrepancy in sign of the deviations of $a_e$ and $a_\mu$ with respect to the SM has to do, in part, with a difference in mass of the bosons interacting with these particles at the loop level. Moreover, we shall assume these bosons to proceed from a singlet complex scalar, with electrons coupling to the CP-odd and CP-even components in a similar way, but with the CP-odd effects becoming dominant due to the small mass of the corresponding scalar. On the other hand, we shall assume that the muons interact mainly with the CP-even component. We shall achieve these properties by imposing an appropriate PQ-like symmetry, under which both the complex scalar and the electron are charged. The CP-odd component may be hence naturally light, since it could be a pseudo-Goldstone boson of the PQ-like symmetry. The explanation of the deviation of $a_\mu$, on the other hand, is similar to the one proposed in several works in the literature [@Kinoshita:1990aj; @Zhou:2001ew; @Barger:2010aj; @TuckerSmith:2010ra; @Chen:2015vqy; @Liu:2016qwd; @Batell:2016ove; @Marciano:2016yhf; @Wang:2016ggf].
This article is organized as follows. In section \[sec:g-2\], we describe the scalar and pseudo-scalar corrections to the anomalous magnetic moments of the electron and muon. In section \[sec:EFTModel\], we present an effective field theory description of our model, describing the interactions of the leptons with the complex scalar after imposing the PQ-like symmetry. In section \[sec:UVModel\], we present an ultraviolet (UV) completion of the effective theory. In section \[sec:discussionElecMuon\], we discuss the phenomenology constraints on the UV complete model. We reserve section \[sec:conclusions\] for our conclusions.
g-2 anomalies for electron and muon {#sec:g-2}
===================================
In our approach, the new physics only comes from the scalar sector, where a singlet light complex scalar $\phi$ solves both $\Delta a_{e/\mu}$. We use the fact that the contributions to $g-2$ of scalars with scalar and pseudo-scalar coupling to leptons are of opposite sign. The pseudo-scalar $\phi_I$ from $\phi$ contributes only to $\Delta a_e$ because of a global PQ-like symmetry and the CP symmetry, while the CP-even scalar $\phi_R$ is responsible for the contributions to $\Delta a_\mu$. Therefore, the relative sign between $\Delta a_e$ and $\Delta a_\mu$ has its origin from the CP properties of scalars.
In the following we begin with a generic Yukawa coupling of a scalar to electron or muon. To be specific, a scalar with both scalar and pseudo-scalar couplings to leptons, $ S \bar{\ell} \left(g_R + i g_I \gamma_5 \right) \ell $, it can contribute to the anomalous magnetic dipole moment as [@PhysRevD.5.2396; @Leveille:1977rc] $$\begin{aligned}
\Delta a_\ell = \frac{1}{8 \pi^2} \int^1_0 dx
\frac{(1-x)^2 \left((1+x)g_R^2 - (1-x) g_I^2 \right)}{(1-x)^2+x \left( m_S/m_\ell \right)^2} .\end{aligned}$$
However, if a real scalar has both non-zero scalar and pseudo-scalar couplings, $g_R$ and $g_I$, respectively, the CP is violated and lepton electric dipole moment will be generated. To avoid this constraint, we require CP conservation that each scalar has either scalar or pseudo-scalar couplings. In particular, we assume the presence of a pseudo-scalar $\phi_I$ that couples to electron and a CP-even scalar which couples to muon as $$\begin{aligned}
\mathcal{L}_{\rm int} = i g_{\phi_I}^e \phi_I \bar{e} \gamma_5 e + g^\mu_{\phi_R} \phi_R \bar{\mu} \mu .\end{aligned}$$
We show the parameter space for $\Delta a_{e/\mu}$ in Eq. (\[eq:g-2-e\]) and Eq. (\[eq:g-2-mu\]) in Fig. \[fig:independentcoupling\] and the relevant constraints for the couplings are added in the plot. For the coupling to electrons, using electron beam, the beam dump experiments E137 [@Bjorken:1988as], E141 [@Riordan:1987aw], and Orsay [@Davier:1989wz] may produce scalars via Bremsstrahlung-like process. The scalar would travel macroscopic distances and decay back to electron pairs. The lack of observation of such events results in the orange shaded exclusion region [@Batell:2016ove; @Liu:2016qwd] in Fig. \[fig:independentcoupling\] (a). The JLab experiment HPS [@Battaglieri:2014hga] projection for scalars [@Batell:2016ove] is plotted as a region bounded by the dot-dashed dark cyan line as well.
The BaBar collaboration searches for dark photons through the process $e^+ e^- \to \gamma A'$ [@Lees:2014xha], where $A' \to \ell^+ \ell^-$ decays democratically. Ref. [@Knapen:2017xzo] recasts the results and give constraints for scalars via $e^+ e^- \to \gamma S$, which is shown in green shaded region in Fig. \[fig:independentcoupling\] (a). In the BaBar study, $A'\to \mu^+ \mu^-$ channel is more sensitive than $e^+ e^-$. The constraint for scalar from [@Knapen:2017xzo] applies for ${\rm BR}(S \to \mu^+\mu^-) \gg {\rm BR}(S \to e^+ e^-)$, which is the case for coupling proportional to lepton mass. If the scalar decays to $e^+e^-$ dominantly, the limit will be weaker by an order one factor. The process $e^+ e^- \to \gamma S$ at Belle II [@Abe:2010gxa; @Kou:2018nap] has also been studied to obtain the projected sensitivity [@Batell:2016ove], which is plotted as dot-dashed green line in Fig. \[fig:independentcoupling\] (a). In the lower mass region, the KLOE collaboration provides the constraints for a similar process [@Anastasi:2015qla], and these constraints have been re-interpreted into bounds on the scalar couplings in Ref. [@Alves:2017avw].
For the coupling to muon, the BaBar collaboration searches the dark photon with muonic coupling via the $e^+ e^- \to \mu^+ \mu^- A'$ process [@TheBABAR:2016rlg], with $A' \to \mu^+ \mu^-$. It has been re-casted by the authors of Ref. [@Batell:2016ove; @Batell:2017kty] for a scalar with muonic coupling and we plotted the excluded region in Fig. \[fig:independentcoupling\] (b) by the shaded green area. The future projection for Belle-II [@Batell:2017kty; @Kou:2018nap] is also shown, bounded by the dot-dashed green line.
At the LHC Run-I, the ATLAS collaboration has searched for exotic Z decays, $Z \to 4 \mu$ [@Aad:2014wra] with both 7 TeV and 8 TeV data. It has been interpreted as a constraint on $Z \to \mu^+ \mu^- S$ by Ref. [@Batell:2017kty], which is shown in Fig. \[fig:independentcoupling\] (b) as a shaded brown region. Ref. [@Batell:2017kty] has also projected this limit for high luminosity LHC (HL-LHC) and we show it as a region bounded by the dot-dashed brown line. Recently, the CMS collaboration has studied the exotic Z decay process $Z \to Z' \mu^+ \mu^-$ at 13 TeV with integrated luminosity $77.3~{\rm fb}^{-1}$ [@Sirunyan:2018nnz], which constrained the production cross-section and exotic Z decay BR($Z \to Z' \mu^+ \mu^-$) as a function of the $Z'$ mass. We recast this constraint for a scalar which couples to muon and plotted as shaded red region in Fig. \[fig:independentcoupling\] (b). Since the ATLAS search for exotic Z decay $Z \to 4 \mu$ [@Aad:2014wra] does not require a dilepton resonance from the four muon, its HL-LHC projection is weaker than the CMS 13 TeV limit with $77.3~{\rm fb}^{-1}$ [@Sirunyan:2018nnz].
For beam dump experiments, whether $\phi_R$ is long-lived is crucial. If $\phi_R$ couples to muons only, it can only decay to diphoton when $m_{\phi_R} < 2 m_\mu$ which could be long-lived. The beam dump constraints could apply in this case due to its small coupling to photons [@Batell:2017kty]. However, in our model, $\phi_R$ will also couple to electrons with the same coupling strength as $\phi_I$. Therefore, the beam dump constraints do not apply for $\phi_R$ under the assumption that it is heavier than $\phi_I$.
We only plotted the relevant limits for the EFT model in Fig. \[fig:independentcoupling\]. For readers who are interested in more detailed future sensitivity projections and new proposals from beam dump, collider searches and cosmology constraints for light scalar coupled to leptons, they can be found in Refs. [@Batell:2016ove; @Knapen:2017xzo; @Batell:2017kty] and references therein.
----- -----
(a) (b)
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EFT model with a light complex scalar {#sec:EFTModel}
======================================
In this section, we demonstrate at the effective field theory (EFT) level that a complex scalar $\phi$, accompanied with some symmetry assumption can simultaneously solve the $\Delta a_{e}$ and $\Delta a_\mu$ anomalies. The gauge charge of $\phi$ and the global $U(1)_{\rm PQ}^{e}$ charges are presented in Table. \[tab:gaugecharge\].
filed $SU(2)_L$ $U(1)_Y$ $U(1)_{\rm PQ}^{e}$
-------- ----------- --------------- ---------------------
$H$ 2 $\frac{1}{2}$ 0
$\phi$ 1 0 -2
$L_e$ 2 $\frac{1}{2}$ 1
$e_R$ 1 -1 -1
: All particles with $ SU(2)_L \times U(1)_Y \times U(1)_{\rm PQ}^{e}$ charge specified, where $U(1)_{\rm PQ}^{e}$ is a global Peccei-Quinn-like symmetry. $H$ and $L_e$ ($e_R$) are SM Higgs and left-handed (right-handed) electron, while $\phi$ is the new light singlet complex scalar. []{data-label="tab:gaugecharge"}
Given the particle content and charge in Table. \[tab:gaugecharge\], we can write down the effective theory Lagrangian as $$\begin{aligned}
\mathcal{L}_{\rm EFT} = \frac{\phi^*}{\Lambda_e} \bar{L}_e H e_R + y_\mu \bar{L}_\mu H \mu_R
+ \frac{\phi^* \phi}{\Lambda_\mu^2} \bar{L}_\mu H \mu_R + H.c. ,
\label{eq:EFT-Lag}\end{aligned}$$ where $\Lambda_{e,\mu}$ are interaction scales, $H$ is the SM Higgs, $L_{e,\mu}$ are SM left-handed doublets for leptons and $e_R, ~ \mu_R$ are the right-handed SM leptons. In principle, the tau leptons could also appear in the last two terms in Eq. (\[eq:EFT-Lag\]), thus flavor violation coupling can be generated. We postpone the discussion of this issue to section \[sec:discussionElecMuon\]. Both the SM Higgs and the new scalar $\phi $ can get vacuum expectation values (vevs), $$\begin{aligned}
H = \frac{1}{\sqrt{2}} \left(v + h + i G^0 \right), \quad
\phi = \frac{1}{\sqrt{2}} \left(v_\phi + \phi_R + i \phi_I \right) .\end{aligned}$$
For the electron, its mass can only come from the first term which is a dimension 5 operator, while the muon mass can come from the second and third term. It is straight forward to obtain the following relations $$\begin{aligned}
& m_e = \frac{v v_\phi}{2 \Lambda_e}, \quad m_{\mu} = \frac{y_\mu v}{\sqrt{2}}
+ \frac{v v_\phi^2}{2\sqrt{2} \Lambda_{ \mu}^2} ,
\label{eq:masses} \\
&g_{\phi_{R}}^{e, {\rm EFT}} = - g_{\phi_{I}}^{e, {\rm EFT}} = \frac{v}{2 \Lambda_e}=\frac{m_e}{v_\phi} ,
\quad g_{\phi_{R}}^{\mu, {\rm EFT}} = \frac{v_\phi v}{\sqrt{2} \Lambda_\mu^2} .
\label{eq:gIEFT}\end{aligned}$$ We find that the CP-odd $\phi_I$ and CP-even scalars $\phi_R$ couples to electron with the same strength. For the electron anomalous magnetic dipole, the contributions from the two scalars have opposite signs. To obtain negative $\Delta a_{e}$, the $\phi_I$ contribution has to be larger than the $\phi_R$ one, which can be satisfied by requiring $m_{\phi_I} \ll m_{\phi_R}$. We emphasize that such requirement is natural to achieve, because if $U(1)_{\rm PQ}^{e}$ is spontaneously broken, the Goldstone $\phi_I$ is massless. However, we have to downgrade the continuous global symmetry to a discrete one, for example, adding a soft breaking term, e.g. $\mu_4^2 \phi_I^2$ term to give mass to $\phi_I$. It can also get mass from hidden confinement scale [@Marques-Tavares:2018cwm]. The mass of $\phi_R$ is not dictated by symmetry breaking, thus can be larger.
In the EFT model, we have 6 free parameters, $\Lambda_e$, $\Lambda_\mu$, $y_\mu$, $v_\phi$, $m_{\phi_I}$ and $m_{\phi_R}$. With the electron and muon masses, we can eliminate $\Lambda_e$ and $y_\mu$. To fit the anomalous magnetic moment $\Delta a_{e}$, we further eliminate $v_\phi$. From the electron sector, only $m_{\phi_I}$ is a free parameter, though is limited to a small range $10-100$ MeV from the constraints in Fig. \[fig:independentcoupling\] (a). We choose $m_{\phi_I} \sim 15$ MeV as our benchmark, which also implies $g_{\phi_I}^e \sim 10^{-4}$. Let us stress that for $\Delta a_{e}$, the 1-loop [@PhysRevLett.65.21] correction is suppressed by the electron mass, and hence the 2-loop Barr-Zee diagram could be dominant if $\phi_{I}$ couples to other heavy charged fermions [@Giudice:2012ms; @Davoudiasl:2018fbb]. In our case, however, the $\phi_I$ only couples to the electron due to the PQ charge assignment and thus the 2-loop contribution is much smaller than the 1-loop one [@TuckerSmith:2010ra].
The $\Delta a_{\mu}$ defines a band in $\Lambda_\mu$ and $m_{\phi_R}$ region as well. As a result, after applying two lepton mass and $\Delta a_{e / \mu}$ requirements, we are left with 2 degree of freedom (d.o.f.) as $m_{\phi_{I}}$ and $m_{\phi_{R}}$. We list a benchmark point with $m_{\phi_R} \sim 15$ MeV and $m_{\phi_R} \sim 0.15$ GeV as an example in Table. \[tab:benchmark1\]. In Fig. \[fig:EFT-fit\], we show the fits for $\Delta a_{e/\mu}$ anomalies with the parameters $v_{\phi}$, $m_{\phi_I}$, $m_{\phi_R}$, and $\Lambda_{ \mu}$.
$v_{\phi}$ (GeV) $m_{\phi_I}$ (MeV) $\Lambda_e$ (GeV) $\Lambda_{ \mu}$ (GeV) $m_{\phi_R}$ (GeV )
------------------ -------------------- -------------------- ------------------------ ---------------------
$4.7 $ 15 $1.12 \times 10^6$ $1080$ 0.15
: The benchmark for EFT model. The parameter $y_\mu$ is determined by muon mass which is not listed here. The EFT model has 2 d.o.f., $m_{\phi_{I}}$ and $m_{\phi_{R}}$, after applying all the constraints and signal requirements. The change of $m_{\phi_R}$ only affects $\Lambda_{ \mu}$, while $v_\phi$ and $\Lambda_e$ are already fixed by the electron mass and $\Delta a_e$. $m_{\phi_I}$ is limited to a small range $10-100$ MeV by relevant constraints. This benchmark is labeled as a black star in Fig. \[fig:independentcoupling\]. []{data-label="tab:benchmark1"}
In the EFT model, we further consider the possibility that the muon mass comes from the dimension 6 operator, e.g. when $y_\mu =0$. In this case, $\Lambda_{ \mu} = 135$ GeV is enforced by the muon mass. It implies that $g_{\phi_{R}}^{\mu, {\rm EFT}} \approx 0.045$ and the $\phi_R$ mass is around $26 - 50$ GeV. In this case, there is no free parameter left in the EFT model. This possibility is constrained by the recent analysis of the CMS 13 TeV data with $77.3~{\rm fb}^{-1}$ [@Sirunyan:2018nnz] shown in Fig. \[fig:independentcoupling\] (b), that restrict $\phi_R$ masses smaller than 38.5 GeV is excluded. Although masses of the order of 40 GeV would be allowed, leading to values of $a_\mu$ which deviate by less than 1 $\sigma$ from the experimental value, one more issue with this region of parameters is that $\Lambda_{ \mu}$ is around 135 GeV, which implies new physics should be much lighter than in the original benchmark. We leave the exploration of this parameter space for future work.
UV complete model with a light complex scalar {#sec:UVModel}
=============================================
In this section, we show the UV completion of the EFT Lagrangian in Eq. (\[eq:EFT-Lag\]). The particle content of the UV model is listed in Table \[tab:Higgsdoublet\]. It contains three Higgs doublet $\Phi_{1,2,3}$, where $\Phi_2$ will become the SM-like Higgs. A SM singlet complex scalar $\phi$ transforms under an approximate $U(1)$ PQ-like symmetry, while $\Phi_1$, $L_e$ and $e_R$ also transform under it. The symmetry has to be softly broken to allow a massive $\phi_I$. $\Phi_{2,3}$ have no global charge assigned.
filed $SU(2)_L$ $U(1)_Y$ $U(1)_{\rm PQ}^{e}$
---------- ----------- --------------- ---------------------
$\Phi_1$ 2 $\frac{1}{2}$ 2
$\Phi_2$ 2 $\frac{1}{2}$ 0
$\Phi_3$ 2 $\frac{1}{2}$ 0
$\phi$ 1 0 -2
$L_e$ 2 $\frac{1}{2}$ 1
$e_R$ 1 $-1$ -1
: The particles under $ SU(2)_L \times U(1)_Y \times U(1)_{\rm PQ}^{e}$ , where $U(1)_{\rm PQ}^{e}$ is a global Peccei-Quinn-like symmetry. The Higgs doublet $\Phi_1$ and $\Phi_3$ are supposed to be heavy degrees of freedom, which are integrated out in the effective theory. The mixing between the scalars are assumed to be small and $\Phi_2$ will be the SM-like Higgs. []{data-label="tab:Higgsdoublet"}
----- ----- -----
(a) (b) (c)
----- ----- -----
The electron sector
-------------------
We need the Higgs doublet $\Phi_1$ charged under $U(1)_{\rm PQ}^{e}$ to generate the dimension 5 operator in the EFT Lagrangian which is responsible for the electron mass. The relevant Feynman diagram is shown in Fig. \[fig:EFTgeneration\] (a), where the heavy $\Phi_1$ is integrated out. The relevant UV Lagrangian for the electron sector is given by, $$\begin{aligned}
\mathcal{L}_{\rm UV}^e & = V(\Phi_1, \Phi_2)_{\rm 2HDM}^{U(1)} +\left( y_e \bar{L}_e \Phi_1 e_R + H.c.\right)
\nonumber \\
&+ V(\phi) + \frac{1}{2} \mu_4^2 \phi_I^2 \nonumber \\
& + (\phi^* \phi) \left( \lambda_5 \Phi^\dagger_1 \Phi_1
+ \lambda_6 \Phi^\dagger_2 \Phi_2 \right)
+ \mu_8 \left(\Phi^\dagger_1 \Phi_2 \phi^* + H.c. \right) .
\label{eq:LagPhi1}\end{aligned}$$ After getting a vev, the neutral component in each of the Higgs doublets is $$\begin{aligned}
\Phi_j^0 = \frac{1}{\sqrt{2}} \left(v_j + h_j + i a_j \right),\end{aligned}$$ where we assume $v_3 \ll v_1 \ll v_2$. For further simplicity, we assume the alignment limit that $\Phi_2 \approx H$ and the mixing angles between $\Phi_i$ and $\phi$ are small. We neglect $\Phi_3$ at this moment, since it is not necessary for generating the EFT operators in the electron sector.
In Eq. (\[eq:LagPhi1\]), the coefficients are all real, as required by CP conservation. In the first line, the scalar potential $V(\Phi_1, \Phi_2)_{\rm 2HDM}^{U(1)}$ [@Branco:2011iw] is the usual two Higgs doublet model (2HDM) potential subject to the global $U(1)$ charge. The Yukawa coupling for the electron is mediated by $\Phi_1$ only. In the second line, the singlet scalar potential $ V(\phi)$ contains the quadratic $\phi^* \phi$ and quartic $(\phi^* \phi)^2$ terms satisfying the global $U(1)_{\rm PQ}^{e}$ symmetry. However, we explicitly add the $\mu_4^2 \phi_I^2$ term to break $U(1)_{\rm PQ}^{e}$ softly, since otherwise $\phi_I$ will be a massless pseudo-Goldstone boson. In the third line [^1], the $\mu_8$ term is special because it contributes to the splitting of the mass for CP-odd scalars with respect to the CP-even ones. Regarding the CP-odd sector, the mass eigenstates are a heavy massive $A^0$, a Goldstone boson $G^0$ eaten by Z gauge boson and a remaining pseudo-Goldstone $\phi_I^{'}$ for the global $U(1)_{\rm PQ}^{e}$. In the small mixing setup, the mass eigenstates $A^0$, $G^0$ and $\phi_I^{'}$ are mostly $a_1$, $a_2$ and $\phi_I$, respectively.
Following [@Liu:2017xmc], the mass for $A^0$ and $\phi_I^{'}$ and their mixing between different states are given by $$\begin{aligned}
& m_{A^0}^2 = - \mu_8 v_2 \frac{v_1^2 + v_\phi^2}{\sqrt{2} v_1 v_\phi} ,
\quad m_{\phi_I^{'}}^2 = \mu_4^2 \frac{v_\phi^2}{v_1^2 + v_\phi^2} ,
\label{eq:CPoddmass}\\
& a_1 = \frac{v_\phi}{\sqrt{v_1^2 + v_\phi^2}} A^0 + \frac{v_1}{v} G^0
- \frac{v_1}{\sqrt{v_1^2 + v_\phi^2}} \phi_I^{'} +\mathcal{O}\left( \frac{\mu_4^2}{\mu_8 v_2}\right)
\label{eq:CPoddmixing1},\\
& \phi_I = \frac{v_1}{\sqrt{v_1^2 + v_\phi^2}} A^0 + 0 \times G^0
+ \frac{v_\phi}{\sqrt{v_1^2 + v_\phi^2}} \phi_I^{'} +\mathcal{O}\left( \frac{\mu_4^2}{\mu_8 v_2}\right),
\label{eq:massmixing}\end{aligned}$$ where $v \equiv \sqrt{v_1^2 + v_2^2}$ and we have taken only the leading term under assumption $v_2 \gg v_{\phi}, v_1$. If we further impose $v_\phi \gg v_1$, then our assumption that scalar mixing is small can be satisfied. From the mixing in the UV model, we can calculate the coupling $g_{\phi_I}^e$ that $$\begin{aligned}
g_{\phi_I}^{e, {\rm UV}} = - \frac{y_e}{\sqrt{2}} \frac{v_1}{\sqrt{v_1^2 + v_\phi^2}} =
- \frac{m_e}{\sqrt{v_1^2 + v_\phi^2}} .
\label{eq:gphiIeUV}\end{aligned}$$ After integrating out $\Phi_1$, one can also obtain the interaction scale $\Lambda_e$ that $$\begin{aligned}
\frac{1}{\Lambda_e} = y_e \frac{\mu_8}{ m_{A_0}^2} . \label{eq:Lambdae}\end{aligned}$$ In Eq. (\[eq:Lambdae\]), due to CP conservation, the integrated particle should be the CP-odd component in $\Phi_1$, thus the denominator is the mass of $A_0$ squared. Applying Eq. (\[eq:CPoddmass\]) and Eq. (\[eq:Lambdae\]), one can check that $g_{\phi_I}^{e, {\rm EFT}} $ in Eq. (\[eq:gIEFT\]) agrees with $g_{\phi_I}^{e, {\rm UV}} $. One can also see that the mass of $A^0$ can be easily as large as 1 TeV if $\mu_8$ is electroweak scale and $v_\phi/v_1$ is large.
The muon sector
---------------
In this section, we describe the UV model which can generate the dimension 6 operator in $\mathcal{L}_{\rm EFT}$, which is responsible for the $\phi_R$ coupling to muons. A third Higgs doublet $\Phi_3$ is essential and it has to carry the same quantum charge as SM-like Higgs $\Phi_2$. The relevant Lagrangian is $$\begin{aligned}
\mathcal{L}_{\rm UV}^{\mu} & = V(\Phi_2, \Phi_3)_{\rm 2HDM}
+ (y_{\mu} \bar{L}_\mu \Phi_2 \mu_R + y_{\mu 3} \bar{L}_\mu \Phi_3 \mu_R + H.c.) \nonumber \\
& + V(\phi) + (\phi^* \phi) \left( \lambda_6 \Phi^\dagger_2 \Phi_2+ \lambda_8 \Phi^\dagger_3 \Phi_3 \right) \nonumber \\
& + \lambda_9 (\phi^* \phi) \left( \Phi^\dagger_2 \Phi_3 + H.c. \right)
+ \mu_9 \left(\Phi^\dagger_1 \Phi_3 \phi^* + H.c. \right) ,
\label{eq:LagPhi3}\end{aligned}$$ where the coefficients are real.
The first line in Eq. (\[eq:LagPhi3\]) contains a general 2HDM scalar potential $V(\Phi_2, \Phi_3)_{\rm 2HDM}$. The last two terms in that line are the Yukawa couplings for the muon. We will again assume hierarchical vevs, $v_3 \ll v_1 \ll v_\phi \ll v_2$, so that the muon mass predominantly comes from $\Phi_2$ and $y_{\mu 3}$ is free from the muon mass constraint. The second line contains the scalar potential for $\phi$ and the quartic coupling between $\phi$ and $\Phi_{2,3}$. Since $v_3\sim 0$, if we require $\lambda_6 \ll 1$, the quartic term in the second line does not induce a large mixing between the different scalars[^2]. Since $\Phi_2$ and $\Phi_3$ have the same quantum numbers, the potential $V(\Phi_2,\Phi_3)_{\rm 2HDM}$ may include a quadratic term $m_{23}^2 \Phi_3^\dagger \Phi_2 + H.c.$, while the third line contains the term proportional to $\lambda_9$ which may also lead to a similar term when $\phi$ acquires a vev. These two terms contribute to the dimension 4 and 6 operators responsible for the muon mass and the coupling of $\phi_R$ to the muons in the effective field theory described by $\mathcal{L}_{\rm EFT}$, Eq. (\[eq:EFT-Lag\]). Finally, the term proportional to the trilinear mass parameter $\mu_9$, in combination with the $\mu_8$-induced interactions, can also contribute to the $\phi_R$ coupling to muons. Although all these contributions may coexist, we shall treat them in a separate way for simplicity of presentation.
### Generating the operators from quartic scalar interactions
The term proportional to the $\lambda_9$ coupling in Eq. (\[eq:LagPhi3\]) can generate the Feynman diagram depicted in Fig. \[fig:EFTgeneration\] (b), which can lead, after integrating out $\Phi_3$, to the coupling of $\phi_R$ to muons in the EFT, Eq. (\[eq:EFT-Lag\]). This coupling is given by $$\begin{aligned}
g_{\phi_R}^{\mu, {\rm EFT}} = \frac{v_\phi v}{\sqrt{2} \Lambda_{ \mu}^2} =
y_{\mu 3} \lambda_9 \frac{v_\phi v_2}{\sqrt{2} m_{h_3}^2} , \label{eq:gEFTmuphiR4point}\end{aligned}$$ where $m_{h_3}^2$ is the CP-even scalar mass from $\Phi_3$. The interaction scale $\Lambda_{ \mu}$ is related to the heavy Higgs parameters by the relation $$\begin{aligned}
\frac{1}{\Lambda_{ \mu}^2} = \frac{y_{\mu 3} \lambda_9}{m_{h_3}^2}
\label{eq:mh3case1} .\end{aligned}$$
Given the fact that the $\lambda_{9}$ term gives the off-diagonal mass terms between $\phi_R$ and $h_3$, we can calculate the mass matrix and obtain the mixing angle, $$\begin{aligned}
& M^2_{\phi_R h_3} =
\left(\begin{array}{cc}
m_{\phi_R}^2 & \lambda_9 v_\phi v_2
\\
\lambda_9 v_\phi v_2 & m_{h_3}^2
\end{array}\right) , \\
& \sin \theta_{\phi_R h_3} \approx \lambda_9 \frac{v_\phi v_2 }{m_{h_3}^2} . \label{eq:quarticMIX}\end{aligned}$$ Assuming that $h_3$ and $\phi_R$ only have a small mixing between themselves ($\sin \theta_{\phi_R h_3} \ll 1$) and negligible mixing with other fields, the coupling between $\phi_R$ and the muon from the UV model is $$\begin{aligned}
g_{\phi_R}^{\mu, {\rm UV}} = \frac{y_{\mu 3}}{\sqrt{2}} \sin \theta_{\phi_R h_3} .\end{aligned}$$ One can easily check that it agrees with $g_{\phi_R}^{\mu, {\rm EFT}}$ in Eq. (\[eq:gIEFT\]) and Eq. (\[eq:gEFTmuphiR4point\]).
In the UV model, the $\lambda_9$ and $\lambda_8$ terms in $\mathcal{L}_{\rm UV}^{\mu}$ contain only $\phi^* \phi$, thus $\phi_I$ couples to muon only in quadrature and can not contribute to $\Delta a_{\mu}$. Given that $\Lambda_{ \mu}$ needs to be about $1080$ GeV (see Table \[tab:benchmark1\]), the scalar boson $h_3$ can be easily heavier than $\mathcal{O}(1)$ TeV, as can be seen from Eq. (\[eq:mh3case1\]).
### Generating the operators from triplet scalar interaction {#sec:MuCouplingfromTriplet}
We can generate the CP-even scalar $\phi_R$ coupling to muon via Fig.\[fig:EFTgeneration\] (c), after integrating out the heavy $h_1$ and $h_3$ scalar bosons. As emphasized above, it requires the simultaneous action of the two triple scalar couplings $\mu_9 \Phi^\dagger_1
\Phi_3 \phi^*$ and $\mu_8 \Phi^\dagger_1 \Phi_2 \phi^*$. According to [@Liu:2017xmc], under the assumption $v_1 \ll v_\phi \ll v_2$, $m_{h_1}^2$ is the same order as $m_{A_0}^2$ in Eq. (\[eq:CPoddmass\]), what is also confirmed in the full UV model calculation presented in Appendix \[sec:CPoddUV\]. The EFT coupling between $\phi_R$ and muon can be computed as $$\begin{aligned}
g_{\phi_R}^{\mu, {\rm EFT}} =
\frac{y_{\mu 3} v_\phi v_2 \mu_8 \mu_9}{\sqrt{2} m_{h_1}^2 m_{h_3}^2} \approx
y_{\mu 3} \frac{v_1 \mu_9}{m_{h_3}^2}, \label{eq:gEFTmuphiR}\end{aligned}$$ where $m_{h_{1, 3}}^2$ are the CP-even scalar mass from $\Phi_{1,3}$. The interaction scale $\Lambda_{ \mu}$ in this case is $$\begin{aligned}
\frac{1}{\Lambda_{ \mu}^2} = \frac{y_{\mu 3} \mu_8 \mu_9}{m_{h_1}^2 m_{h_3}^2} .\end{aligned}$$
In the UV model, the $\phi_R$ coupling to muon again comes from mixing with $h_3$. We calculate the mass matrix and obtain the mixing angle via $\mu_9 \Phi^\dagger_1
\Phi_3 \phi^*$ term, $$\begin{aligned}
& M^2_{\phi_R h_3} =
\left(\begin{array}{cc}
m_{\phi_R}^2 & \mu_9 v_1
\\
\mu_9 v_1 & m_{h_3}^2
\end{array}\right) , \\
& \sin \theta_{\phi_R h_3} \approx \frac{\mu_9 v_1 }{m_{h_3}^2}, \label{eq:MixphiRh3}\end{aligned}$$ where again we find that $g_{\phi_R}^{\mu, {\rm UV}} \equiv y_{\mu 3} \sin \theta_{\phi_R h_3} $ agrees with $g_{\phi_R}^{\mu, {\rm EFT}}$ again. In the above discussion, we did not include off-diagonal terms with $h_{1,2}$. The $\phi_R$-$h_3$ mixing, may be modified through the mixing with them. We did the full calculation in the 3HDM plus a singlet complex scalar in Appendix \[sec:CPoddUV\]. The result contains more terms than Eq. (\[eq:MixphiRh3\]), but one can tune down some parameters to converge to this result, while keeping the $\Phi_{1,3}$ scalars heavy. Such tuning is also in agreement of the initial assumption that the mixing between different scalars is small, see Appendix \[sec:CPoddUV\].
From the benchmark point, we can see that a coupling $g_{\phi_R}^{\mu, {\rm EFT}} \sim 0.7 \times 10^{-3}$ can fit the $\Delta a_\mu$ anomaly. One can infer the mass square $m_{h_3}^2 \simeq 10^3 y_{\mu 3} v_1
\mu_9$ from Eq. (\[eq:gEFTmuphiR\]). With a large $\mu_9 \simeq \mathcal{O}({\rm few})$ TeV and $v_1 \sim 1$ GeV, $h_3$ mass can be larger than $\mathcal{O}(1)$ TeV.
Given the fact that the $\phi_I^{'}$ mass is much smaller than $\phi_R$, any small mixing between $\phi_I^{'}$ and the CP-odd components of $\Phi_2$ and $\Phi_3$, would induce a coupling of $\phi_I^{'}$ to muons, that could make the contribution from $\phi_I^{'}$ to $\Delta a_\mu$ larger than the one of $\phi_R$. However, in the full calculation within the 3HDM plus singlet scalar potential, presented in Appendix \[sec:CPoddUV\], the mass eigenstate $\phi_I^{'}$ only mixes with $a_1$ in the leading order of $v_1/v_\phi$. In fact, the absence of mixing with $a_{2,3}$ can be simply understood from the pseudo-Goldstone nature of this particle. Thus, the components of $\phi_I^{'}$ are approximately described by Eq. (\[eq:CPoddmixing1\]), and $\phi_I^{'}$ only couples to electrons, as occurs in the EFT model.
Phenomenology constraints {#sec:discussionElecMuon}
=========================
There are several important phenomenological constraints to address, once moving from EFT model to the UV model.
Heavy scalars and anomalous magnetic moments
--------------------------------------------
One relevant constraint is the contribution of the heavy scalars to the anomalous magnetic moments. Although these scalars are integrated out in the EFT, they may contribute in a relevant way. At large mass, the CP-even scalar contribution to the lepton g-2 is approximately given by $\Delta a_{\ell}^{\rm even} \approx g_{S \ell}^2/(8 \pi^2) m_{\ell}^2/m_{S}^2 ( \log(m_{S}^2/m_{\ell}^2) -7/6)$, while CP-odd scalar contributes as $\Delta a_{\ell}^{\rm odd} \approx g_{S \ell}^2/(8 \pi^2)
m_{\ell}^2/m_{S}^2 (- \log(m_{S}^2/m_{\ell}^2) +11/6 )$ [@Giudice:2012ms]. Neglecting the mild dependence on log terms, the anomalous magnetic moments are hence proportional to $g_{\ell}^2 / m_S^2$.
In the UV model, the light scalar couples to leptons via mixing with the heavy ones for the pseudo-scalar case, where the mixing angles are related to vevs due to pseudo-Goldstone nature. Therefore, for light scalar contribution dominating over the heavy scalar one, the relation $\sin^2\theta > m_{\rm light}^2 / m_{\rm heavy}^2$ must be satisfied, where $\sin\theta$ is the mixing angle, while $m_{\rm light}$ and $m_{\rm heavy}$ are the light and heavy scalar masses. The mixing angles do not significantly depend on the mass of the light scalars, $m_{\rm light}$, thus one can always tune down light scalar mass to meet the requirement. It is easy to find that the $a_1$ contributions to the $e$ anomalous magnetic moments is sub-dominant than the $\phi_I$ ones due to the small values of the lightest pseudo-scalar mass, while satisfying the benchmark requirements.
However, for CP-even scalar $h_3$ mixing, the mixing angle $\sin\theta$ is proportional to $m_{h_3}^{-2}$. Therefore, $\sin^2\theta \propto m_{h_3}^{-4}$ and the $\sin^2\theta > m_{\rm light}^2 / m_{\rm heavy}^2$ condition actually provides an upper bound on the $h_3$ mass. If we choose the benchmark presented in Table \[tab:benchmark1\], with $v_\phi = 4.7$ GeV, $m_{\phi_R} =0.15$ GeV, and $g_{\phi_R}^{\mu, \text{EFT}} = 0.7 \times 10^{-3}$, for the cases in which the effective low energy couplings are induced by quartic (triplet) scalar interactions, the $h_3$ contribution would be smaller than $\phi_R$ provided that $$m_{h_3} < \lambda_9 \times 7.7~{\rm TeV}~ \; (m_{h_3} < 6.7 \times \mu_9 v_1 \text{GeV}^{-1}).
\label{eq:upperbmh3}$$ To satisfy $g_{\phi_R}^{\mu, \text{EFT}} = 0.7 \times 10^{-3}$, for quartic (triplet) scalar interactions, one should further demand that $m_{h_3} = 1~ \text{TeV} \times \sqrt{\lambda_9 y_{\mu3}}$ ($m_{h_3} = 37.8 \sqrt{y_{\mu3} \mu_9 v_1}$), as can be seen from Eqs. (\[eq:gEFTmuphiR4point\]) and (\[eq:gEFTmuphiR\]). These requirements can be achieved easily, with $\lambda_9 \sim 1$ and $y_{\mu 3} \sim 1$ for quartic case, while $\mu_9 \sim 1$ TeV, $v_1 \sim 1$ GeV and $y_{\mu 3} \sim 1$ for triplet case. It is worth mentioning that $m_{h_3}$ is about 1 TeV in both cases. Therefore, we conclude that in the UV model, under the hierarchical vevs and heavy $\Phi_{1,3}$ assumptions, the heavy scalars do not contribute to the anomalous magnetic moment in a relevant way.
Moreover, we comment that the way of generating dimension 6 operators in the EFT model is not restricted to those ones depicted in Fig. \[fig:EFTgeneration\] (b) and (c). Since $\Phi_{2,3}$ have the same quantum number, the scalar potential $V(\Phi_2, \Phi_3)_{\rm 2HDM} $ interactions are only weakly constrained and could induce large $h_2$-$h_3$ mixing. As discussed in Appendix \[sec:CPoddUV\], and emphasized before, in the presence of large $\lambda_6$ or $\mu_8$ terms, these mixing effects can lead to large contributions to the dimension 6 operator in the EFT model. Let us stress, however, that large $\lambda_6$ and $\mu_8$ terms can also induce large mixing between $h_2$-$\phi_R$. Such possibility beyond the scope of the EFT model and is in tension with our initial assumption that mixing between different scalars are all small.
Scalar interactions and relevant phenomenology
----------------------------------------------
The next constraint is the decay channels modified by scalar interactions. In the EFT model, $\phi_I$ decays to $e^+ e^-$, while $\phi_R$ decays to $e^+ e^-$ with same coupling as $\phi_I$. $\phi_R$ can also decay to $\mu^+ \mu^-$ if kinematics allowed. With the scalar potential from UV model, e.g. 3HDM plus singlet scalar in Appendix \[sec:CPoddUV\], there are a few phenomenologically relevant decay channels, $\phi_R^{'} \to
\phi_I^{'} \phi_I^{'}$, $H_2^0 \to \phi_I^{'} \phi_I^{'}$ and $H_2^0 \to \phi_R^{'} \phi_R^{'}$, where $\phi_{I,R}^{'}$ and $H_2^0$ are mass eigenstates of CP-even (CP-odd) light scalars and SM Higgs. According to the mixing matrix for both CP-even (odd) scalars in Appendix \[sec:CPoddUV\], the triple scalar couplings between the mass eigenstates $\phi_{I,R}^{'}$ and $H_2^0$ can be calculated, $$\begin{aligned}
\mathcal {L}_{\rm tri}= \left(\lambda_\phi -\lambda_6\left(\frac{\lambda_6}{4\lambda_2} + \frac{\mu_8 v_1}{\sqrt{2}\lambda_2 v_2 v_\phi}\right)\right)v_\phi \phi^{'2}_I\phi^{'}_R+ \left(\frac{\lambda_6}{2} +\frac{\mu_8 v_1}{\sqrt{2} v_2 v_\phi}\right)v_2 \left(\phi^{'2}_I H_2^0 + \phi^{'2}_R H_2^0\right) .\end{aligned}$$
First, from our benchmark, we have $v_\phi = 4.7$ GeV, $m_{\phi_I^{'}} \approx 15 $ MeV and $v_1 \ll v_\phi$. The CP-even scalar $\phi_R^{'}$ has a coupling which is about $10^{-4} (10^{-3})$ to electrons (muons) respectively, while its coupling to pairs of $\phi_I^{'}$ is $2\lambda_{\phi} v_\phi$. Thus, $\phi_R^{'}$ will dominantly decay into $\phi_{I}^{'}$ pairs. Assuming $m_{\phi_R^{'}} \sim \sqrt{\lambda_{\phi}} v_\phi$, the branching ratio of its decay into $e^- e^+$ ($\mu^+ \mu^-$) will be about $\sim 10^{-8}$ ($10^{-6}$) respectively. Then, the previous constraints on $\phi_R^{'} $ shown in Fig. \[fig:independentcoupling\] (b), which are based on the assumption $BR(\phi_R^{'} \to \ell^+ \ell^-)\sim 1$, should be revised. At low energy electron colliders, the relevant search channels are $e^+ e^- \to \gamma \phi'_R$ and $e^+ e^- \to \phi_R^{'*} \to \phi'_I \phi'_I$, governed by the electron coupling and $e^+ e^- \to \mu^+ \mu^- \phi'_R$ governed by the muon coupling. Since $\phi'_R \to \phi'_I \phi'_I \to 4e$, there are multiple leptons in the final state. Although the BaBar experiment has searched for new physics in similar channels, for instance $e^+ e^- \to h' A'$, $h' \to A' A'$ and $A' \to \ell^+ \ell^-$ [@Lees:2012ra] and $e^+ e^- \to W' W' \to 2(\ell^+ \ell^-)$ in exclusive mode [@Aubert:2009af], it has not explored the invariant mass regions consistent with $m_{\phi_I^{'}}$. However, BaBar has the capability of lowering the invariant mass threshold, as has been shown in the 2014 search for dark photons via $\gamma A'$ channel [@Lees:2014xha], where the BaBar collaboration extended the di-electron resonance channel to $m_{e^+ e^-} > 0.015$ GeV, and fits for $m_{A'} > 0.02$ GeV. We believe it would be important to reanalyze their searches by imposing similar bounds on the dielectron invariant mass. Moreover, since $\phi_I^{'}$ and $ \phi_R^{'}$ are pretty light, they will be very boosted at high energy colliders and form lepton jets [@ArkaniHamed:2008qp; @Baumgart:2009tn; @Bai:2009it; @Katz:2009qq]. The proper lifetime of $\phi_I^{'}$ in the benchmark is about $c\tau \approx 10^{-3} {\rm cm}$, thus it will appear as a prompt lepton jet in a low energy lepton collider, but displaced lepton jet at the LHC. The displacement could help the search at the LHC, to separate the signal from the SM background, for example photon conversions. However, the invariant mass of the di-electron or even four lepton events coming from $\phi_R^{'}$ might be too low for the LHC experiments to detect them.
Second, we discuss the exotic SM Higgs decay channels $H_2^0 \to \phi_I^{'} \phi_I^{'} \to 2(e^+e^-)$ and $H_2^0 \to \phi_R^{'} \phi_R^{'} \to 4(e^+e^-)$. It is clear that if $\lambda_6$ is of $\mathcal{O}(1)$, then the SM-like Higgs will dominantly decay to those light scalars thus one needs $\lambda_6 \ll 1$. The ratio $\mu_8 v_1/(\sqrt{2} v_2 v_\phi)$ should also be small. To obtain a $H_2^0 \to \phi_I^{'} \phi_I^{'} , \phi_R^{'} \phi_R^{'}$ branching ratio smaller than $1\%$, the coefficient $\lambda_6$ or $\mu_8 v_1/(\sqrt{2} v_2 v_\phi)$ should be $\lesssim 10^{-3}$, thus $\mu_8 v_1 \lesssim 1.7 ~{\rm GeV}^{2}$. If we tune down both $\mu_8$ and $v_1$, the $A_1^0 / H_1^0$ masses, of about $\sqrt{v_\phi v_2 (\mu_8/v_1)}$ (see Appendix \[sec:CPoddUV\]), can still remain as heavy as $\sim 300$ GeV, with $\mu_8 \sim 10$ GeV and $v_1 \sim 0.1$ GeV. Interestingly, in the electron sector, we have $m_e = y_e v_1/\sqrt{2}$, which suggests $v_1 \gtrsim m_e$ and one can further decrease $v_1$ to make $A_1^0 / H_1^0$ heavier. Furthermore, according to Eq. (\[eq:gphiIeUV\]), the coupling $g_{\phi_I}^e$ is not affected by a small $v_1$. One should note that, as we mentioned before, in the case $g_{\phi_R}^\mu$ is generated from triplet scalar interactions, we have from Eq. (\[eq:gEFTmuphiR\]) that for the benchmark presented in Table \[tab:benchmark1\], $m_{h_3} = 37.8 \sqrt{y_{\mu 3} \mu_9 v_1} $. Hence, if we take $v_1 = 0.1$ GeV while keeping $\mu_9 \sim 1$ TeV and $y_{\mu 3} \sim 1$, the mass $m_{h_3} $ goes down to $\sim 380$ GeV and will become smaller for smaller values of $v_1$. However, for the case $g_{\phi_R}^\mu$ is generated from quartic scalar interaction, Eqs. (\[eq:gEFTmuphiR4point\]), the mass $m_{h_3}$ does not have a strong dependence on $v_1$ and hence could remain heavy even for very small values of $v_1$.
Charged lepton flavor violation
-------------------------------
In this section, we discuss the possible flavor changing neutral current (FCNC) constraint. Since the muon and the tau leptons have the same quantum number, in the EFT Lagrangian, Eq. (\[eq:EFT-Lag\]), the muon leptons can be substituted with tau leptons. Moreover, in the UV model, the two Higgs doublets $\Phi_{2,3}$ have the same quantum charge and hence admit the same couplings. After the charged lepton mass matrix diagonalization, a possible misalignment between the lepton mass and Yukawa couplings can induce off-diagonal Yukawa couplings to muons and taus, see also a recent review [@Lindner:2016bgg] on $\Delta a_\mu$ and lepton flavor violation. To avoid the appearance of FCNC, one can assume minimal flavor violation (MFV) [@Chivukula:1987py; @DAmbrosio:2002vsn] to align the couplings of $\Phi_{3}$ with the $\Phi_2$ ones. In the case of MFV, $\Phi_3$ will also couple to muon and tau lepton with diagonal couplings weighted by the lepton masses. Heavy Higgs bosons, which couple only to leptons and gauge bosons are difficult to test at hadron colliders. Under the MFV assumption, however, the light scalar $\phi_R^{'}$ couples in a relevant way to $\tau$ leptons and is constrained to have a mass between $30 - 200$ MeV in order to be consistent with precision electroweak constraints associated with loop corrections to $Z \to \tau^+ \tau^-$ [@Abu-Ajamieh:2018ciu].
While MFV can solve the FCNC constraint for heavy scalars, the constraints on the light scalar couplings remain severe. This is represented by the LFV decay $\tau \to \mu \phi'_R \to \mu + 2(e^+ e^-)$. The total width of $\tau$ is very small, $2.27 \times 10^{-12} $ GeV and the current limit on the three lepton LFV decay is ${\rm BR}(\tau \to \mu e^+e^-) < 1.8 \times 10^{-8}$ [@Tanabashi:2018oca]. This limit is easy to satisfy because $ {\rm BR}(\tau \to \mu e^+e^-) = {\rm BR}(\tau \to \mu \phi'_R) \times {\rm BR}(\phi'_R \to e^+e^-) $ for our benchmark point and ${\rm BR}(\phi'_R \to e^+e^-)\sim 10^{-8}$ as discussed above. However, since $\phi_R'$ can decay into pairs of $\phi_I'$, there is a potential flavor violation in the channel $\tau \to \mu + 2(e^+ e^-)$. We did not find limits on this channel at the PDG [@Tanabashi:2018oca], but if the limits were of the same order as the one on ${\rm BR}(\tau \to \mu e^+e^-) $, it will imply $y_{\tau \mu }^{\phi_R} \lesssim 10^{-10}$. Since $y_{\tau \mu }^{\phi_R} = \sin \theta_{\phi_R h_3} y_{\tau \mu }^{h_3}$, and the mixing angle is about $10^{-3}$, one should restrict the LFV coupling $y_{\tau \mu }^{h_3} $ down to $10^{-7}$. Therefore, the alignment of the lepton Yukawa couplings must be enforced by a symmetry. The most natural candidate would be an extra global $U(1)_\mu \times U(1)_\tau$ symmetry, which is vector-like when applied to fermions unlike the chiral $U(1)_{\rm PQ}^e$. These symmetries forbid the off-diagonal terms between charged lepton species, and then the charged lepton mass matrix is diagonal and LFV is not present in the charged lepton sector.
Others constraints and discussion
---------------------------------
Besides the FCNC issue, the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix for the lepton sector needs to be generated. Given the global $U(1)_{\rm PQ}^e \times U(1)_\mu \times U(1)_\tau$ symmetry, the Yukawa matrices of the SM charged and neutral leptons are diagonal. However, assuming a see-saw mechanism, one can generate the PMNS matrix from mixing in the heavy sterile neutrino sector [@Pascoli:2006ci; @King:2013eh; @Akhmedov:2013hec], by assuming that the mass terms of the sterile neutrino $m^N_{ij} N^c_i N_j$ softly break the global symmetry (see, for instance, the review, Ref. [@Xing:2015fdg], for the case of $U(1)_{\mu - \tau}$).
Finally, we briefly mention that a $\phi'_I$ mass around 15 MeV, as required to satisfy $\Delta a_e$ and the other relevant phenomenological constraints, is accidentally within the mass region necessary to explain the so-called anomaly, observed by the Atomki collaboration [@Krasznahorkay:2015iga]. Addressing this anomaly would imply a coupling of the singlet scalar to quarks, something that is beyond the scope of our work. Let us stress, however, that the authors of Ref. [@Ellwanger:2016wfe; @Alves:2017avw] concluded that this possibility is subject to relevant constraints from low energy meson experiments that can only be avoided by assuming specific coupling structures in the quark sector.
Conclusions {#sec:conclusions}
===========
We have presented a scenario with a light complex scalar which can simultaneously accommodate the anomalies in the electron and muon anomalous magnetic moments. The interesting feature is that the same complex scalar induces positive contributions to $a_\mu$ and negative contributions to $a_e$. This is achieved by assuming that the CP-even component is much heavier than the CP-odd component and having the CP-odd scalar scalar coupled only to electrons, while the CP-even couples to both the electron and muon fields. This scenario may be realized in a natural way by introducing an approximate PQ-like symmetry and assuming that the CP-odd scalar is a pseudo-Goldstone boson associated to its spontaneous breakdown. The EFT model can then be written down directly and cope with the anomalies, while evading all the existing constraints.
We also analyzed how to generate such EFT model from a Standard Model extension containing multiple Higgs doublets. While the additional heavy Higgs doublet masses may be as large as 1 TeV, flavor changing neutral currents may be avoided by assuming a global symmetry in the lepton sector, broken softly in the neutrino sector. Furthermore, the heavy scalars contribution to the anomalous magnetic moments is much smaller than the one of the light scalars due to the small masses of the CP-odd and even component of the complex scalar compared to the ones of the heavy Higgs bosons. For the light complex scalar, its CP-odd and even components could be potentially reached by future B-factories and the HL-LHC. Looking for multiple prompt lepton jets in low energy electron collider and displaced lepton jets from exotic SM Higgs decay at LHC is also a promising way to find those light scalars.
Acknowledgments {#acknowledgments .unnumbered}
===============
Work at University of Chicago is supported in part by U.S. Department of Energy grant number DE-FG02-13ER41958. Work at ANL is supported in part by the U.S. Department of Energy under Contract No. DE-AC02-06CH11357. The work of CW was partially performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. We would like to thank Zhen Liu, Ian Low, Joshua T. Ruderman, Emmanuel Stamou, Lian-Tao Wang, and Neal Weiner for useful discussions and comments. JL acknowledges support by Oehme Fellowship.
The CP-even and CP-odd scalars in full UV model {#sec:CPoddUV}
===============================================
We consider the full UV model with three Higgs doublet $\Phi_{1,2,3}$ and one singlet complex scalar $\phi$, where $\Phi_1$ and $\phi$ carries global $U(1)_{PQ}^{e}$ charge. The general scalar potential is $$\begin{aligned}
V=&\mu_1^2 \Phi_1^\dag \Phi_1 + \mu_2^2 \Phi_2^\dag \Phi_2 + \mu_3^2 \Phi_3^\dag \Phi_3 + \mu_\phi^2 \phi^* \phi
- m_{23}^2 \left( \Phi_2^\dag \Phi_3 + \Phi_3^\dag \Phi_2\right) + \frac{1}{2}\mu_4^2 \phi_I^2 \nonumber \\
& + \lambda_1 \left(\Phi_1^\dag \Phi_1\right)^2 + \lambda_2 \left(\Phi_2^\dag \Phi_2 \right)^2 + \lambda_{3} \left( \Phi_3^\dag \Phi_3\right)^2+ \lambda_\phi \left(\phi^* \phi \right)^2 + \lambda_4 \left( \Phi_1^\dag \Phi_1\right)\left( \Phi_2^\dag \Phi_2\right) \nonumber \\
&+ \lambda_5 \left( \Phi_1^\dag \Phi_1\right)\left( \phi^*\phi \right) + \lambda_6 \left( \Phi_2^\dag \Phi_2\right) \left( \phi^* \phi\right) + \lambda_7 \left(\Phi_2^\dag \Phi_1 \right) \left(\Phi_1^\dag \Phi_2\right) + \lambda_8 \left( \Phi_3^\dag \Phi_3 \right) \left( \phi^* \phi\right) \nonumber \\
&+ \lambda_9 \left( \Phi_2^\dag \Phi_3 + \Phi_3^\dag \Phi_2\right) \left( \phi^* \phi\right) +\mu_8 \left( \Phi_1^\dag \Phi_2 \phi^*+ H.c.\right) + \mu_9 \left( \Phi_1^\dag \Phi_3 \phi^* + H.c.\right) \nonumber \\
& + \lambda_{23}^a \left(\Phi_2^\dag \Phi_2\right)\left( \Phi_3^\dag \Phi_3\right)
+ \lambda_{23}^b \left(\Phi_2^\dag \Phi_3\right)\left( \Phi_3^\dag \Phi_2\right) \nonumber \\
&+ \lambda_{23}^c \left[\left(\Phi_2^\dag \Phi_3\right)^2 + \left( \Phi_3^\dag \Phi_2\right) ^2 \right] + \left(\lambda_{23}^d \Phi_2^\dag \Phi_2 + \lambda_{23}^e \Phi_3^\dag \Phi_3\right)\left(\Phi_2^\dag \Phi_3 +\Phi_3^\dag \Phi_2\right) + \ldots\end{aligned}$$ where we only written the scalar potential contributions, Eq. (\[eq:LagPhi1\]) and Eq. (\[eq:LagPhi3\]), which are relevant to the computation of $\Delta a_{e,\mu}$. The “$\ldots$" denotes the irrelevant terms like $(\Phi_1^\dagger \Phi_1)(\Phi_3^\dagger \Phi_3)$ etc, which are neglected to avoid a too cumbersome computation.
Minimizing the scalar potential, one obtains the following relations $$\begin{aligned}
\mu_1^2 =&-\left[ \lambda_1 v_1^2 + \frac{\left(\lambda_4 +\lambda_7\right)}{2} v_2^2 +\frac{ \lambda_5}{2} v_\phi^2 +\frac{v_\phi}{\sqrt{2}v_1} \left(\mu_8 v_2+\mu_9 v_3\right)\right] , \nonumber \\
\mu_2^2=&-\left[ \lambda_2 v_2^2 + \frac{\lambda_{23}^a + \lambda_{23}^b + 2\lambda_{23}^c}{2}v_3^2 +\frac{\left(\lambda_4 +\lambda_7\right) v_1^2}{2} + \frac{\lambda_6 v_\phi^2}{2} \right. \nonumber \\
&\left. + \frac{v_3}{2v_2}\left(3\lambda_{23}^d v_2^2 +\lambda_{23}^e v_3^2 + \lambda_9 v_\phi^2 - 2 m_{23}^2 \right) + \frac{\mu_8 v_1v_\phi}{\sqrt{2}v_2}\right] ,\nonumber \\
\mu_3^2 =&-\left[ \lambda_3 v_3^2 + \frac{\lambda_{23}^a + \lambda_{23}^b + 2\lambda_{23}^c}{2}v_2^2 + \frac{v_2}{2v_3} \left(\lambda_{23}^d v_2^2 + 3\lambda_{23}^e v_3^2 - 2m_{23}^2\right)+\frac{v^2_\phi}{2v_3}\left(\lambda_8 v_3+\lambda_9 v_2 \right) + \frac{\mu_9 v_1 v_\phi}{\sqrt{2} v_3}\right] ,\nonumber \\
\mu_\phi^2 =& -\left[ \lambda_\phi v_\phi^2 +\frac{\lambda_5 v_1^2 +\lambda_6 v_2^2 +\lambda_8 v_3^2 + 2\lambda_9 v_2v_3 }{2} + \frac{v_1}{\sqrt{2} v_\phi }\left(\mu_8 v_2 + \mu_9 v_3\right) \right] .\end{aligned}$$ We can diagonalize the mass matrix of CP-even or CP-odd scalars and obtain the mass in the leading order under the assumption $v_3 \ll v_1 \ll v_\phi \ll v_2$ and $v_2 \sim \mu_{8,9} \sim m_{23}$. The results for the CP-odd scalars are given by the eigenvalues $$\begin{aligned}
m_{A^0_1}^2 & \approx -\frac{ v_\Phi}{\sqrt{2} v_1} \left(\mu_8 v_2 + \mu_9 v_3\right) , \\
m_{A^0_3}^2 & \approx \frac{ v_2\left(2 m_{23}^2 - \lambda_{23}^d v_2^2-\lambda_9 v_\phi^2\right) - \sqrt{2} \mu_9 v_1 v_\phi}{2 v_3}
- 2 \lambda_{23}^c v_2^2 ,
\\
m_{\phi'_I}^2& \approx \mu_4^2 ,\end{aligned}$$ where $A^0_2$ is the massless Goldstone associated with the breakdown of the electroweak symmetry. $A_{1,2,3}^0$ and $\phi_{I}^{'}$ are the mass eigenstates, while $a_{1,2,3}^0$ and $\phi_{I}$ are flavor states. If the results contain not only the leading terms, we always put the leading term on the left and the sub-leading term on the right. The $4\times 4$ mixing matrix for CP-odd scalars in the leading order is given by, $$\begin{aligned}
\left(\begin{array}{c} a_1 \\ a_2 \\ a_3 \\ \phi_I\end{array}\right) \approx
\left(\begin{array}{cccc}
1 & \frac{v_1}{v} & U_{13}^A & \frac{- v_1}{v_\phi} \\
- \frac{v_1}{v} & 1& -\frac{v_3}{v_2} & \frac{v_1^2}{v_2 v_\phi}\\
- U_{13}^A & \frac{v_3}{v} & 1
& \mathcal{O}\left(\frac{v_{1,3, \phi}^3}{v_2^3} \right) \\
\frac{v_1}{v_\phi} & \mathcal{O}\left(\frac{v_{1,3, \phi}^3}{v_2^3} \right) &
U_{43}^A & 1
\end{array}\right)
\left(\begin{array}{c} A^0_1 \\ G^0 \\ A^0_3 \\ \phi'_I\end{array}\right) ,\end{aligned}$$ where $v=\sqrt{v_1^2 + v_2^2 + v_3^2}$, $v_1\ll v_\phi$, and $\mathcal{O}\left({v_{1,3, \phi}^3}/{v_2^3} \right)$ means at least three orders in small parameter expansion. $$\begin{aligned}
U_{13}^A &=\frac{\sqrt{2}\mu_9 v_\phi v_1}{2 m_{23}^2 v_1 - \lambda_{23}^d v_2^2v_1-v_\phi\left( \lambda_9v_1v_\phi -\sqrt{2}\mu_8 v_3\right)} \frac{v_3}{v_2} \simeq \frac{\sqrt{2}\mu_9 v_\phi}{2 m_{23}^2 - \lambda_{23}^d v_2^2} \frac{v_3}{v_2} ,\nonumber \\
U_{43}^A& =\frac{\sqrt{2} \mu_9 v_1^2 }{2 m_{23}^2 v_1 - \lambda_{23}^d v_2^2v_1 -\lambda_9 v_\phi^2 v_1+\sqrt{2}\mu_8 v_\phi v_3} \frac{v_3}{v_2}\simeq \frac{\sqrt{2} \mu_9 v_1 }{2 m_{23}^2 - \lambda_{23}^d v_2 } \frac{v_3}{v_2} \simeq \frac{v_1}{v_\phi}U_{13}^A.\end{aligned}$$
The calculation for CP-even scalars are similar, with $H_{1,2,3}^0$ and $\phi_{R}^{'}$ being the mass eigenstates, while $h_{1,2,3}^0$ and $\phi_{R}$ being the flavor states. The eigenvalues for CP-even scalars are, $$\begin{aligned}
m_{H^0_1}^2 \approx &- \frac{\left(v_\phi ^2 + v_1^2\right)}{\sqrt{2} v_1v_\phi}\left(\mu_8 v_2+\mu_9 v_3 \right) -\frac{\mu_8 v_\phi v_1}{\sqrt{2} v_2} \simeq - \frac{v_\phi}{\sqrt{2} v_1}\left(\mu_8 v_2+\mu_9 v_3 \right), \\
m_{H^0_2}^2 \approx &2 \lambda_2 v_2^2
+ \frac{2 \lambda_{23}^d v_2 v_3 \left(3 \lambda_{23}^d v_2^2 -4 m_{23}^2 \right)}{\lambda_{23} v_2^2 - 2 m_{23}^2} , \\
m_{H^0_3}^2 \approx & \frac{\left(2 m_{23}^2 - \lambda_{23}^d v_2^2-\lambda_9 v_\phi^2\right)v_2}{2 v_3} -\frac{\mu_9 v_1 v_\phi}{\sqrt{2} v_3}+\frac{v_3 \left(2m_{23}^2-3\lambda_{23}^d v_2^2\right)^2}{2v_2\left(m_{23}^2 - \lambda_{23}^d v_2^2\right)}
+\frac{3\lambda_{23}^e v_2v_3}{2} \nonumber \\
\approx & \frac{2 m_{23}^2 v_2 - \lambda_{23}^d v_2^3}{2 v_3} + \mathcal{O}\left(\frac{v_{1,3, \phi}}{v_2} \right), \\
m_{\phi'_R}^2 \approx & \left(2 \lambda_\phi -\frac{\lambda_6^2}{2 \lambda_2} \right) v_\phi^2
- \frac{\sqrt{2} \lambda_6 \mu_8 v_\phi v_1 }{\lambda_2 v_2} - \frac{\mu_8^2 v_1^2}{\lambda_2 v_2^2}.\end{aligned}$$ We see that under the hierarchical vevs assumption, $m_{A_1^0} \approx m_{H_1^0}$ and $m_{A_2^0} \approx m_{H_2^0}$, while $m_{\phi_R^{'}} > m_{\phi_I^{'}} $. The mixing matrix for CP-even scalars is given by, $$\begin{aligned}
\left(\begin{array}{c} h_1 \\ h_2 \\ h_3 \\ \phi_R \end{array}\right) \approx
\left(\begin{array}{cccc}
1 & \frac{(2 \lambda_2 + \lambda_6) v_1}{2 \lambda_2 v_2}
& \frac{\sqrt{2} \mu_9 v_\phi}{2 m_{23}^2 - \lambda_{23}^d v_2^2} \frac{v_3}{v_2} & \frac{v_1}{v_\phi} \\
- \frac{v_1}{v_2} & 1 & \frac{2 m_{23}^2 - 3 \lambda_{23}^d v_2^2}{\lambda_{23}^d v_2^2 - 2 m_{23}^2} \frac{v_3}{v_2}
&- \frac{\sqrt{2}\mu_8 v_1+\lambda_6 v_\phi v_2}{2 \lambda_2 v_2^2} \\
\frac{\sqrt{2}\mu_9 v_\phi }{ \lambda_{23}^d v_2^2 - 2 m_{23}^2 } \frac{v_3}{v_2}
& \frac{ 3 \lambda_{23}^d v_2^2 - 2 m_{23}^2 }{\lambda_{23}^d v_2^2 - 2 m_{23}^2} \frac{v_3}{v_2}& 1
& U_{34} \\
- \frac{v_1}{v_\phi} & \frac{\sqrt{2}\mu_8 v_1 +\lambda_6 v_\phi v_2}{2\lambda_2 v_2^2}
& \frac{\sqrt{2} \mu_9 v_1 + 2\lambda_9 v_\phi v_2 }{2 m_{23}^2 -\lambda_{23}^d v_2^2} \frac{v_3}{v_2} & 1
\end{array}\right)
\left(\begin{array}{c} H^0_1 \\ H^0_2 \\ H^0_3 \\ \phi'_R \end{array}\right) , \end{aligned}$$ where we have $$\begin{aligned}
U_{34} \approx \frac{1}{m_{H_3}^2} \left( -\sqrt{2} \mu_9 v_1 -\lambda_9 v_2 v_\phi
-\frac{m_{23}^2 \mu_8 v_1}{\sqrt{2} \lambda_2 v_2^2} - \frac{\lambda_6 m_{23}^2 v_\phi}{2 \lambda_2 v_2}
+ \frac{3 \lambda_{23}^d \mu_8 v_1}{2 \sqrt{2} \lambda_2}
+ \frac{3 \lambda_{23}^d \lambda_6 v_2 v_\phi}{4 \lambda_2}
\right) \label{eq:U34} .\end{aligned}$$ We see clearly that the above $\mu_9 v_1$ ($\lambda_9 v_2 v_\phi$) in $U_{34}$ terms match with $\sin\theta_{\phi_R h_3}$ in Eq. (\[eq:MixphiRh3\]) and Eq. (\[eq:quarticMIX\]) from the $2\times 2$ mass matrix calculation. The last four terms with $\lambda_2 v_2^2$ in the denominator show additional contributions to the mixing, whose effects can be tuned down by further assuming $\lambda_6, ~\lambda_{23}^d \ll 1$ and $m_{23} < v_2$, while still keeping the scalars $\Phi_{1,3}$ heavy.
[^1]: It is termed as leptonic Higgs portal in [@Batell:2016ove], where a real singlet scalar example is demonstrated.
[^2]: As it is discussed in Appendix \[sec:CPoddUV\], the presence of large $\lambda_6$ or $\mu_8$ term combined with large $h_2$-$h_3$ mixing from $V(\Phi_2, \Phi_3)_{\rm 2HDM} $ can lead to relevant contributions to the dimension 6 operator at low energy.
|
---
abstract: 'In this paper we study Ulrich ideals and modules of Cohen-Macaulay local rings from various points of view. Developing the general theory, we deal with those of numerical semigroup rings, rings of finite CM-representation type and rational local rings of dimension two.'
address:
- 'S. Goto and K. Ozeki: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki 214-8571, Japan'
- 'R. Takahashi: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan/Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, USA'
- 'K.-i. Watanabe and K. Yoshida: Department of Mathematics, College of Humanities and Sciences, Nihon University, 3-25-40 Sakurajosui, Setagaya-Ku, Tokyo 156-8550, Japan'
author:
- 'Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-ichi Watanabe, Ken-ichi Yoshida'
title: Ulrich ideals and modules
---
[^1] [^2] [^3]
Introduction
============
The purpose of this paper is to report the study of Ulrich ideals and modules with a generalized form. We shall explore their structure and establish, for given Cohen–Macaulay local rings, the ubiquity of these kinds of ideals and modules.
Ulrich modules with respect to maximal ideals in our sense, that is MGMCM (aximally enerated aximal ohen–acaulay) modules were introduced by [@U; @BHU] and have been closely explored in connection to the representation theory of rings. Our motivation has started, with a rather different view-point, from the naive question of why the theory of MGMCM modules works so well. We actually had an occasion [@G] to make a heavy use of it and wanted to know the reason.
To state the main results, let us begin with the definition of Ulrich ideals and modules. Throughout this paper, let $A$ be a Cohen–Macaulay local ring with maximal ideal ${\mathfrak{m}}$ and $d = \dim A \geq 0$. Let $I$ be an ${\mathfrak{m}}$–primary ideal of $A$ and let $$\operatorname{gr}_I(A) = \bigoplus_{n \ge 0}I^n/I^{n+1}$$ be the associated graded ring of $I$. For simplicity, we assume that $I$ contains a parameter ideal $Q = (a_1, a_2, \ldots, a_d)$ of $A$ as a reduction. Notice that this condition is automatically satisfied, if the residue class field $A/{\mathfrak{m}}$ of $A$ is infinite, or if $A$ is analytically irreducible and $\dim A = 1$. Let ${\mathrm{a}}(\operatorname{gr}_I(A))$ denote the $a$–invariant of $\operatorname{gr}_I(A)$ ([@GW Definition 3.1.4]).
\[1.1\] We say that $I$ is an Ulrich ideal of $A$, if the following conditions are satisfied.
1. $\operatorname{gr}_I(A)$ is a Cohen–Macaulay ring with ${\mathrm{a}}(\operatorname{gr}_I(A)) \le 1-d$.
2. $I/I^2$ is a free $A/I$–module.
Condition (1) of Definition \[1.1\] is equivalent to saying that $I^2 = QI$ ([@GW Remark 3.1.6]). Hence every parameter ideal is an Ulrich ideal. When $I = {\mathfrak{m}}$, Condition (2) is naturally satisfied and Condition (1) is equivalent to saying that the Cohen–Macaulay local ring $A$ possesses maximal embedding dimension in the sense of J. Sally [@S1], namely the equality $${\mathrm{v}}(A) = {\mathrm{e}}(A) + d -1$$ holds true, where ${\mathrm{v}}(A)$ (resp. ${\mathrm{e}}(A)$) denotes the embedding dimension of $A$, that is the minimal number $\mu_A({\mathfrak{m}})$ of generators of the maximal ideal ${\mathfrak{m}}$ (resp. the multiplicity ${\mathrm{e}}_{\mathfrak{m}}^0(A)$ of $A$ with respect to ${\mathfrak{m}}$).
\[Umodule\] Let $M$ be a finitely generated $A$-module. Then we say that $M$ is an Ulrich $A$–module with respect to $I$, if the following conditions are satisfied.
- $M$ is a maximal Cohen-Macaulay $A$-module, that is $\operatorname{depth}_AM = d$. (Hence the zero module is not maximal Cohen-Macaulay in our sense).
- $\e_I^0(M)=\ell_A(M/IM)$.
- $M/IM$ is $A/I$-free.
Here $\e_I^0(M)$ denotes the multiplicity of $M$ with respect to $I$ and $\ell_A(M/IM)$ denotes the length of the $A$–module $M/IM$.
When $M$ is a maximal Cohen–Macaulay $A$–module, we have $${\mathrm{e}}_I^0(M) = {\mathrm{e}}_Q^0(M) = \ell_A(M/QM) \ge \ell_A(M/IM),$$ so that Condition (2) of Definition \[Umodule\] is equivalent to saying that $IM = QM$. Therefore, if $I = {\mathfrak{m}}$, Condition (2) is equivalent to saying that ${\mathrm{e}}_{\mathfrak{m}}^0(M) = \mu_A(M)$, that is $M$ is maximally generated in the sense of [@BHU]. Similarly to the case of Ulrich ideals, every Ulrich $A$–module with respect to a parameter ideal is free.
Our purpose is to explore the structure of Ulrich ideals and modules in the above sense and investigate how many Ulrich ideals and modules exist over a given Cohen–Macaulay local ring $A$.
This paper consists of nine sections. In Sections 2 and 3 we will summarize basic properties of Ulrich ideals and modules. Typical examples we keep in mind shall be given. Higher syzygy modules of Ulrich ideals are Ulrich modules (Theorem \[3.2\]), which we will prove in Section 3. Several results of this paper are proven by induction on $d = \dim A$. In Section 3 we shall closely explain the induction technique, which is due to and dates back to W. V. Vasconcelos [@V].
The converse of Theorem \[3.2\] is also true. We will show in Section 4 that our ideal $I$ is Ulrich, once the higher syzygy modules of $I$ are Ulrich $A$–modules with respect to $I$ (Theorem \[4.1\]). We will discuss in Section 5 the problem of when the canonical dual of Ulrich $A$–modules are again Ulrich.
It seems natural and interesting to ask if how many Ulrich ideals which are not parameter ideals are contained in a given Cohen–Macaulay local ring. The research about this question is still in progress and we have no definitive answer. In Section 5 we shall study the case where $A$ is a numerical semigroup ring over a field $k$, that is $$A = k[[t^{a_1}, t^{a_2}, \ldots, t^{a_\ell}]] \ \ \subseteq k[[t]],$$ where $0 < a_1, a_2, \ldots, a_\ell \in \Bbb Z$ $(\ell > 0)$ with $\operatorname{GCD}(a_1, a_2, \ldots, a_\ell) = 1$ and $k[[t]]$ denotes the formal power series ring. We also restrict our attention to the set ${\mathcal X}^g_A$ of Ulrich ideals $I$ of $A$ which are generated by monomials in $t$ but not parameter ideals, namely $\mu_A(I) > 1$. Then ${\mathcal X}^g_A$ is a finite set (Theorem \[6.1\]). We will show in Section 6 the following structure theorem of those Ulrich ideals also, when $A$ is a Gorenstein ring, that is the case where the semigroup $$H = \langle a_1, a_2, \ldots, a_\ell \rangle = \left\{\sum_{i = 1}^\ell c_i a_i\,\Bigg|\,0 \le c_i \in \Bbb Z \right\}$$ generated by the integers $a_i's$ is symmetric.
\[6.4\] Suppose that $A$ is a Gorenstein ring and let $I$ be an ideal of $A$. Then the following conditions are equivalent.
1. $I \in {\mathcal X}_A^g$.
2. There exist elements $a, b \in H$ such that
1. $a < b$, $I = (t^a, t^b)$,
2. $c \not\in H$, $2c \in H$,
3. the numerical semigroup $H_1 = H + \left<c \right>$ is symmetric, and
4. $a = \min \{h \in H \mid c + h \in H\}$,
where $c = b-a$.
As a consequence, we will show that for given integers $1 < a < b$ with $\operatorname{GCD}(a,b)= 1$, the numerical semigroup ring $A = k[[t^a,t^b]]$ contains at least one Ulrich ideal generated by monomials in $t$ which is not a parameter ideal if and only if $a$ or $b$ is even (Theorem \[H=(a,b)\]).
Section 7 is devoted to the analysis of minimal free resolutions of Ulrich ideals. Let $I$ be an Ulrich ideal of $A$ and put $n = \mu_A(I)$. Let $$\Bbb F_{\bullet} : \cdots \to F_i \overset{\partial_i}{\to} F_{i-1} \to \cdots \to F_1 \overset{\partial_1}{\to} F_0 = A \overset{\varepsilon}{\to} A/I \to 0$$ be a minimal free resolution of the $A$–module $A/I$ and put $\beta_i = \operatorname{rank}_AF_i$. We then have the following (Theorem \[7.1\]).
\[betti\] The following assertions hold true.
- $A/I \otimes_A \partial_i=0$ for all $i \geq 1$.
- $$\beta_i= \left\{
\begin{array}{ll}
(n-d)^{i-d}{\cdot}(n-d+1)^d & (d \le i),\\
\binom{d}{i}+(n-d){\cdot}\beta_{i-1} & (1 \leq i \leq d),\\
1 & (i=0).
\end{array}
\right.$$
Hence $\beta_i=\binom{d}{i}+(n-d){\cdot}\beta_{i-1}$ for all $i \geq 1$.
What Theorem \[betti\] says is that, thanks to the exact sequence $$0 \to Q \to I \to (A/I)^{n-d} \to 0,$$ a minimal free resolution of the Ulrich ideal $I$ is isomorphic to the resolution induced from the direct sum of $n-d$ copies of $\Bbb F_{\bullet}$ and the minimal free resolution of $Q = (a_1, a_2, \ldots, a_d)$, that is the truncation $${\Bbb L}_{\bullet} : 0 \to K_d \to K_{d-1} \to \cdots \to K_1 \to Q \to 0$$ of the Koszul complex ${\Bbb K}_{\bullet}(a_1, a_2, \ldots, a_d; A)$ generated by the $A$–regular sequence $a_1, a_2, \ldots, a_d$. As consequences, we get that $\Bbb F_\bullet$ is eventually periodic, if $A$ is a Gorenstein ring and that for Ulrich ideals $I$ and $J$ which are not parameter ideals, one has $I = J$, once $$\operatorname{Syz}_A^i(A/I) \cong \operatorname{Syz}_A^i(A/J)$$ for some $i \ge 0$, where $\operatorname{Syz}_A^i(A/I)$ and $\operatorname{Syz}_A^i(A/J)$ denote the $i$-th syzygy modules of $A/I$ and $A/J$ in their minimal free resolutions, respectively. The latter result eventually yields the following, which we shall prove in Section 7. Recall here that a Cohen-Macaulay local ring is said to be of finite CM–representation type if there exist only finitely many nonisomorphic indecomposable maximal Cohen–Macaulay modules.
If $A$ is a Cohen–Macaulay local ring of finite CM–representation type, then $A$ contains only finitely many Ulrich ideals which are not parameters.
When $A$ is of finite CM–representation type and of small dimension, we are able to determine all the Ulrich ideals of $A$, which we will perform in Sections 8 and 9.
Unless otherwise specified, throughout this paper, let $(A,{\mathfrak{m}})$ be a Cohen–Macaulay local ring of dimension $d \ge 0$ and let $I$ be an ${\mathfrak{m}}$–primary ideal of $A$ which contains a parameter ideal $Q = (a_1, a_2, \ldots, a_d)$ as a reduction. We put $n =\mu_A(I)$, the number of elements in a minimal system of generators of $I$.
Ulrich ideals
=============
The purpose of this section is to summarize basic properties of Ulrich ideals. To begin with, let us recall the definition.
\[2.1\] We say that $I$ is an Ulrich ideal of $A$, if $I^2=QI$ and the $A/I$–module $I/I^2$ is free.
We note the following.
\[2.2\] Let $R$ be a Cohen-Macaulay local ring with maximal ideal ${\mathfrak{n}}$ and dimension $d \ge 0$. Let $F=R^r$ with $r > 0$ and let $A=R \ltimes F$ be the idealization of $F$ over $R$. Then $A$ is a Cohen–Macaulay local ring with maximal ideal ${\mathfrak{m}}= {\mathfrak{n}}\times F$ and $\dim A = d$. Let $\q$ be an arbitrary parameter ideal of $R$ and put $Q=\q A$. Then the ideal $I=\q \times F$ of $A$ contains the parameter ideal $Q$ as a reduction. We actually have $I^2 = QI$ and $I/I^2$ is $A/I$–free, so that $I$ is an Ulrich ideal of $A$ with $n = \mu_A(I) = r + d > d$. Hence the local ring $A$ contains infinitely many Ulrich ideals which are not parameters.
It is routine to check that $I^2 = QI$, while $I/I^2 = \q/\q^2 \times F/\q F$ is a free module over $A/I = R/\q$.
\[2.3\] Suppose that $I^2 =QI$. Then$:$
1. $\e_{I}^0(A) \le (\mu(I)-d+1)\ell_A(A/I)$ holds true.
2. The following conditions for $I$ are equivalent$:$
1. Equlaity holds in $(1)$.
2. $I$ is an Ulrich ideal.
3. $I/Q$ is a free $A/I$-module.
\(1) Since $Q$ is generated by an $A$-sequence, $\ell_A(Q/I^2) = \ell_A(Q/QI)=d \cdot \ell_A(A/I)$. Hence $\ell_A(I/I^2) = \ell_A(A/Q) + \ell_A(Q/QI)-\ell_A(A/I)
=\e_{I}^0(A)+ (d-1) \cdot \ell_A(A/I)$.
On the other hand, since there exists a natural surjection $(A/I)^{\mu_A(I)} \to I/I^2$, we have $\ell_A(I/I^2) \le \mu_A(I)\cdot \ell_A(A/I)$. Thus we obtain the required inequality.
\(2) $(a)\Leftrightarrow (b):$ Equality holds true if and only if the surjection as above $(A/I)^{\mu_A(I)}\to I/I^2$ is an isomorphism, that is, $I/I^2$ is a free $A/I$-module. This is equivalent to saying that $I$ is an Ulrich ideal.
$(b)\Leftrightarrow (c):$ We look at the canonical exact sequence $$0 \to Q/I^2 \overset{\varphi}{\to} I/I^2 \to I/Q \to 0$$ of $A/I$–modules. Then, since $I^2 = QI$, $Q/I^2 = A/I \otimes_{A/Q}Q/Q^2$ is $A/I$–free, whence $I/I^2$ is $A/I$–free if $I/Q$ is $A/I$–free. Conversely, if $I/I^2$ is $A/I$–free, then the $A/I$-module $I/Q$ has projective dimension at most $1$. As $A/I$ is Artinian, $I/Q$ is $A/I$–free.
We need the following result in Section 3.
\[2.4\] Suppose that the residue class field $A/{\mathfrak{m}}$ of $A$ is infinite. Then the following conditions are equivalent.
1. $I$ is an Ulrich ideal of $A$.
2. For every minimal reduction $\q$ of $I$, $I^2 \subseteq \q$ and the $A/I$–module $I/\q$ is free.
Thanks to Lemma \[2.3\], we have only to show $(2) \Rightarrow (1)$. We may assume $n > d > 0$. Let us choose elements $x_1, x_2, \ldots, x_n$ of $I$ so that $I = (x_1, x_2, \ldots, x_n)$ and the ideal $(x_{i_1}, x_{i_2}, \ldots, x_{i_d})$ is a reduction of $I$ for every choice of integers $1 \le i_1 < i_2 < \ldots < i_d \le n$.
We will firstly show that $I/I^2$ is $A/I$–free. Let $c_1, c_2, \ldots, c_n \in A$ and assume that $\sum_{i=1}^nc_ix_i\in I^2\subseteq\q$. Let $1 \le i \le n$ and choose a subset $\Lambda \subseteq \{1, 2, \ldots, n \}$ so that $\sharp\Lambda=d$ and $i \not\in \Lambda$. We put $\q = (x_j \mid j \in \Lambda)$. Then, because $\mu_A(I/\q) = n -d$ and $I/\q = (\overline{x_j} \mid j \not\in \Lambda)$, $\{\overline{x_j}\}_{j \not\in \Lambda}$ form an $A/I$–free basis of $I/\q$, where $\overline{x_j}$ denotes the image of $x_j$ in $I/\q$. Therefore, $c_j \in I$ for all $j \not\in \Lambda$, because $\sum_{j \not\in \Lambda}c_j\overline{x_j}= 0$ in $I/\q$; in particular, $c_i \in I$. Hence $I/I^2$ is $A/I$–free.
Let $\q = (x_1, x_2, \ldots, x_d)$ and let $y \in I^2 \subseteq \q$. We write $y = \sum_{i=1}^dc_ix_i$ with $c_i \in A$. Then, since $\sum_{i=1}^dc_i\overline{x_i} = 0$ in $I/I^2$, we get $c_i \in I$ for all $1 \le i \le d$, because the images $\overline{x_i}$ of $x_i$ ($1 \le i \le n)$ in $I/I^2$ form an $A/I$–free basis of $I/I^2$. Thus $y \in \q I$, so that $I^2 = \q I$ and $I$ is an Ulrich ideal of $A$.
\[2.5\] Even though $I^2 \subseteq Q$ and $I/Q$ is $A/I$–free for some minimal reduction $Q$, the ideal $I$ is not necessarily an Ulrich ideal, as the following example shows. Let $A = k[[t^4,t^5,t^6]] \subseteq k[[t]]$, where $k[[t]]$ denotes the formal power series ring over a field $k$. Let $ I = (t^4, t^5)$ and $Q=(t^4)$. Then $I^4 = Q I^3$ but $I^3 \ne Q I^2$, while $I^2 \subseteq Q$ and $I/Q \cong A/I$.
For each Cohen–Macaulay $A$–module $M$ of dimension $s$, we put $${\mathrm{r}}_A(M) = \ell_A(\operatorname{Ext}_A^s(A/{\mathfrak{m}},M))$$ and call it the Cohen–Macaulay type of $M$. Suppose that $A$ is a Gorenstein ring. Then $I$ is said to be *good*, if $I^2 = QI$ and $Q : I = I$ ([@GIW]). With this notation Ulrich ideals of a Gorenstein ring are characterized in the following way.
\[2.6\] Assume that $I$ is not a parameter ideal and put $n = \mu_A(I) > d$.
1. If $I$ is an Ulrich ideal of $A$, then $(\mathrm{i})$ $Q : I = I$ and $(\mathrm{ii})$ $(n-d){\cdot}{\mathrm{r}}(A/I) \le {\mathrm{r}}(A)$, whence $n \le {\mathrm{r}}(A) + d$.
2. Suppose that $A$ is a Gorenstein ring. Then the following conditions are equivalent.
1. $I$ is an Ulrich ideal of $A$.
2. $I$ is a good ideal of $A$ and $\mu_A(I) = d+ 1$.
3. $I$ is a good ideal of $A$ and $A/I$ is a Gorenstein ring.
Since $I/Q \cong (A/I)^{n-d}$ and $n > d$, assertion (i) is clear. This isomorphism also shows $(n-d){\cdot}{\mathrm{r}}(A/I) = {\mathrm{r}}_A(I/Q) \le {\mathrm{r}}(A/Q) = {\mathrm{r}}(A),$ which is assertion (ii).
We now suppose that $A$ is a Gorenstein ring. If $I$ is an Ulrich ideal of $A$, then by assertion (i) $I$ is a good ideal of $A$ and $(n-d){\cdot}{\mathrm{r}}(A/I)=1$, so that $n = d+1$ and $A/I$ is a Gorenstein ring. Conversely, assume that $I$ is a good ideal of $A$. If $n = d+1$, then since $I/Q$ is a cyclic $A$–module with $Q:I = I$, we readily get $I/Q \cong A/I$, whence $I$ is an Ulrich ideal of $A$ by Lemma \[2.3\]. If $A/I$ is a Gorenstein ring, then $I/Q$ is a faithful $A/I$–module. Since in general a finitely generated faithful module over an Artinian Gorenstein local ring is free, $I/Q$ is $A/I$-free.
We close this section with the following examples.
\[2.7\] Let $k$ be a field.
1. Let $A = k[[t^4, t^6, t^{4\ell - 1}]]$ ($ \ell \ge 2$). Then $I = (t^4, t^6)$ is an Ulrich ideal of $A$ containing $Q = (t^4)$ as a reduction.
2. Let $q, d \in \Bbb Z$ such that $d \ge 1$ and $q \ge 2$. Let $R = k[[X_1, \ldots, X_d, X_{d+1}]]$ be the formal power series ring and let $$A = R/(X_1^2 + \cdots + X_d^2 + X_{d+1}^q).$$ Let $x_i$ be the image of $X_i$ in $A$ and put $I = (x_1, \ldots, x_d, x_{d+1}^\ell)$ where $\ell = \lceil \frac{q}{2} \rceil$. Then $I$ is an Ulrich ideal of $A$ with $\mu_A(I) = d + 1$.
3. Let $K/k~(K \ne k)$ be a finite extension of fields and assume that there are no proper intermediate fields between $K$ and $k$. Let $V=K[[t]]$ be the formal power series ring over $K$ and put $A = k[[Kt]]~\subseteq V$. Then the ring $A$ contains a unique Ulrich ideal, that is ${\mathfrak{m}}= tV$, except parameter ideals.
\(1) Let us identify $A = k[[X, Y, Z]]/(X^3-Y^2, Z^2 - X^{2\ell -2}Y)$, where $k[[X,Y,Z]]$ denotes the formal power series ring. We then have $I^2 = QI + (t^{12}) = QI$, while $Q : I = I$. In fact, since $A/Q = k[[Y,Z]]/(Y^2, Z^2)$ and $(0) : y = (y)$ in $k[[Y,Z]]/(Y^2, Z^2)$ where $y$ denotes the image of $Y$ in $k[[Y,Z]]/(Y^2, Z^2)$, the equality $Q : I = I$ follows.
\(2) Let $Q = (x_1, x_2, \ldots, x_d)$ (resp. $Q = (x_2, \ldots, x_d, x_{d+1}^\ell)$) if $q = 2\ell$ (resp. $q = 2\ell + 1$). It is standard to check that $I$ is good. The assertion follows from Corollary \[2.6\].
\(3) $A$ is a Noetherian local ring of dimension one, because $V = K[[t]]$ is the normalization of $A$ which is a module–finite extension of $A$ with $A: V = tV ={\mathfrak{m}}$. Hence there are no proper intermediate subrings between $V$ and $A$. Let $I$ be an Ulrich ideal of $A$ and assume that $I$ is not a parameter ideal. Choose $a \in I$ so that $Q= (a)$ is a reduction of $I$. Let $\frac{I}{a} = \{\frac{x}{a} \mid x \in I\}$. Then, because $\frac{I}{a} = A[\frac{I}{a}]$ and $\frac{I}{a} \ne A$, we get $I = aV$. Hence $I = A:V = tV = {\mathfrak{m}}$, because $I = Q : I$.
Let $A$ be the ring of Example \[2.7\] (3) and put $I = t^n{\mathfrak{m}}$ with $0 < n \in \Bbb Z$. Then $I \cong {\mathfrak{m}}$ as an $A$–module but $I$ is not Ulrich, since $I \ne {\mathfrak{m}}$. This simple fact shows that for given ${\mathfrak{m}}$–primary ideals $I$ and $J$ of $A$, $I$ is not necessarily an Ulrich ideal of $A$, even though $I \cong J$ as an $A$–module and $J$ is an Ulrich ideal of $A$.
Ulrich modules
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In this section we shall explain the basic technique of induction which dates back to [@V]. Firstly, let us recall the definition of an Ulrich module.
Let $M$ be a finitely generated $A$-module. Then we say that $M$ is an Ulrich $A$–module with respect to $I$, if the following conditions are satisfied.
- $M$ is a maximal Cohen–Macaulay $A$-module.
- $IM = QM$.
- $M/IM$ is $A/I$-free.
If $d = 1$ and $I$ is an Ulrich ideal of $A$, then $I$ is an Ulrich $A$–module with respect to $I$. More generally, higher syzygy modules of Ulrich ideals are Ulrich $A$–modules, as the following theorem shows. For each $i \ge 0$, let $\operatorname{Syz}_A^i(A/I)$ denote the $i$–th syzygy module of $A/I$ in a minimal free resolution.
\[3.2\] Let $I$ be an Ulrich ideal of $A$ and suppose that $I$ is not a parameter ideal of $A$. Then for all $i\ge d$, $\operatorname{Syz}_A^i(A/I)$ is an Ulrich $A$–module with respect to $I$.
Theorem \[3.2\] is proven by induction on $d$. Let $I$ be an arbitrary ${\mathfrak{m}}$–primary ideal of a Cohen–Macaulay local ring $A$ of dimension $d \ge 0$ and assume that $I$ contains a parameter ideal $Q = (a_1, a_2, \ldots, a_d)$ as a reduction. We now suppose that $d > 0$ and put $a = a_1$. Let $$\overline{A} = A/(a), \ \ \overline{I} = I/(a), \ \ \text{and}\ \ \overline{Q} = Q/(a).$$ We then have the following.
\[3.3\] If $I$ is an Ulrich ideal of $A$, then $\overline{I}$ is an Ulrich ideal of $\overline{A}$.
We have $\ol{I}^2 = \ol{Q}{\cdot}\ol{I}$, since $I^2 = QI$, while $\ol{I}/\ol{Q} = I/Q$ and $\ol{A}/\ol{I} = A/I$. Hence by Lemma \[2.3\] $\ol{I}$ is an Ulrich ideal of $\ol{A}$, because $\ol{I}/\ol{Q}$ is $\ol{A}/\ol{I}$–free.
\[3.4\] Suppose that $I/I^2$ is $A/I$-free. Then $$\Syz_A^i(A/I)/a \Syz_A^i(A/I) \cong \Syz_{\overline{A}}^{i-1}(A/I) \oplus \Syz_{\overline{A}}^{i}(A/I)$$ for all $i \geq 1$.
We look at the minimal free resolution $$\Bbb F_{\bullet} : \cdots \to F_i \overset{\partial_i}{\to} F_{i-1} \to \cdots \to F_1 \overset{\partial_1}{\to} F_0 = A \overset{\varepsilon}{\to} A/I \to 0$$ of $A/I$. Then, since $a$ is $A$–regular and $a{\cdot}(A/I) = (0)$, we get exact sequences $$(3.5)\ \ \ \cdots \to F_i/aF_i \to F_{i-1}/aF_{i-1} \to \cdots \to F_1/aF_1 \to I/aI \to 0$$ and $$0 \to A/I \overset{\varphi}{\to} I/aI \to \ol{A} \to A/I \to 0$$ of $\ol{A}$–modules, where $\varphi (1) = \ol{a}$, the image of $a$ in $I/aI$. We claim that $\varphi$ is a split monomorphism. Namely
\[3.6\] $I/aI \cong A/I \oplus \ol{I}.$
Let $n = \mu_A(I)$ and write $I=(x_1, x_2, \ldots, x_n)$ with $x_1 = a$. Then $I/aI=A \ol{x_1}+\sum_{i=2}^nA\ol{x_i}$, where $\ol{x_i}$ denotes the image of $x_i$ in $I/aI$. To see that $A \ol{x_1}\cap \sum_{i=2}^nA\ol{x_i} = (0)$, let $c_1, c_2, \ldots, c_n \in A$ and assume that $c_1\ol{x_1} + \sum_{i=2}^nc_i\ol{x_i} = 0$. Then, since $c_1a + \sum_{i=2}^nc_ix_i \in aI$, we have $$(c_1 - y)a + \sum_{i=2}^nc_ix_i = 0$$ for some $y \in I$. Now remember that $I/I^2 \cong (A/I)^{n}$. Hence the images of $x_i$ ($1 \le i \le n$) in $I/I^2$ form an $A/I$–free basis of $I/I^2$, which shows $c_1 - y \in I$. Thus $c_1 \in I$, so that $c_1\ol{x_1}= 0=\sum_{i=2}^nc_i\ol{x_i}$ in $I/aI$. Hence $\varphi$ is a split monomorphism and $I/aI \cong A/I \oplus \ol{I}$.
By (3.5) and Claim we get $$\Syz_A^i(A/I)/a \Syz_A^i(A/I) \cong \Syz_{\overline{A}}^{i-1}(A/I) \oplus \Syz_{\overline{A}}^{i}(A/I)$$ for $i \ge 2$. See Claim for the isomorphism in the case where $i = 1$.
We are now in a position to prove Theorem \[3.2\].
Suppose that $I$ is an Ulrich ideal of $A$ with $\mu_A(I) > d$. If $d = 0$, then $I^2 = (0)$ and $I$ is $A/I$–free, so that $\Syz_A^i(A/I) \cong (A/I)^{n^i}$ for all $i \ge 0$, which are certainly Ulrich $A$–modules with respect to $I$. Let $d > 0$ and assume that our assertion holds true for $d-1$. Let $\ol{A}=A/(a)$ and $\ol{I} = I/(a)$, where $a = a_1$. Then $\ol{I}$ is an Ulrich ideal of $\ol{A}$ by Lemma \[3.3\] and for all $i \ge d$ $$\Syz_A^i(A/I)/a \Syz_A^i(A/I) \cong \Syz_{\overline{A}}^{i-1}(A/I) \oplus \Syz_{\overline{A}}^{i}(A/I)$$ by Lemma \[3.4\]. Because the right hand side of the above isomorphism is an Ulrich $\ol{A}$–module with respect to $\ol{I}$, it readily follows that $\Syz_A^i(A/I)$ is an Ulrich $A$–module with respect to $I$.
Vasconcelos [@V] proved that if a given ideal $I$ in a Noetherian local ring $A$ has finite projective dimension and if $I/I^2$ is $A/I$–free, then $I$ is generated by an $A$–regular sequence. In his argument the key observation is Claim in Proof of Lemma \[3.4\], which shows every Ulrich ideal of finite projective dimension is a parameter ideal. Hence, inside regular local rings, Ulrich ideals are exactly parameter ideals.
Relation between Ulrich ideals and Ulrich modules
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Let $(A,{\mathfrak{m}})$ be a Cohen–Macaulay local ring of dimension $d \ge 0$ and let $I$ be an ${\mathfrak{m}}$–primary ideal of $A$ which contains a parameter ideal $Q = (a_1, a_2, \ldots, a_d)$ as a reduction. With this notation the converse of Theorem \[3.2\] is also true and we have the following.
\[4.1\] The following conditions are equivalent.
- $I$ is an Ulrich ideal of $A$ with $\mu_A(I) > d$.
- For all $i \ge d$, $\Syz_A^i(A/I)$ is an Ulrich $A$-module with respect to $I$.
- There exists an Ulrich $A$–module $M$ with respect to $I$ whose first syzygy module $\Syz_A^1(M)$ is an Ulrich $A$–module with respect to $I$.
When $d>0$, then we can add the following condition.
- $\mu_A(I) > d$, $I/I^2$ is $A/I$-free, and $\Syz_A^i(A/I)$ is an Ulrich $A$-module with respect to $I$ for some $i \geq d$.
The proof of Theorem \[4.1\] is based on the following.
\[4.2\] Suppose that $ 0 \to X \to F \to Y \to 0 $ is an exact sequence of finitely generated $A$–modules such that $F$ is free, $X \subseteq {\mathfrak{m}}F$, and $Y$ is an Ulrich $A$–module with respect to $I$. Then the following conditions are equivalent.
- $X= \Syz_A^1(Y)$ is an Ulrich $A$–module with respect to $I$.
- $I$ is an Ulrich ideal of $A$ with $\mu_A(I) > d$.
Since $Y$ is a maximal Cohen–Macaulay $A$–module, $X$ is also a maximal Cohen–Macaulay $A$–module if $X \ne (0)$. Look at the exact sequence $$0 \to X/QX \to F/QF \to Y/QY \to 0$$ and we get $$X/QX \cong (I/Q)^r$$ where $r = \operatorname{rank}_AF > 0$, because $Y/QY \cong (A/I)^r$ and $X \subseteq {\mathfrak{m}}F$. Remember that this holds true for any parameter ideal $Q$ of $A$ which is contained in $I$ as a reduction.
$(2) \Rightarrow (1)$ Since $I^2 \subseteq Q$, we get $I{\cdot}(X/QX) = (0)$, so that $IX = QX$. Because $I/Q \cong (A/I)^{n-d}$ ($n = \mu_A(I) > d$) by Lemma \[2.3\], $X \ne (0)$ and $X/IX = X/QX$ is a free $A/I$–module. Hence $X$ is an Ulrich $A$-module with respect to $I$.
$(1) \Rightarrow (2)$ Enlarging the residue class field of $A$ if necessary, we may assume that the field $A/{\mathfrak{m}}$ is infinite. Since $X/IX \cong (I/Q)^r$ and $X/IX$ is $A/I$–free, we have $I^2 \subseteq Q$ and $I/Q$ is $A/I$–free. Thus $I$ is an Ulrich ideal of $A$ by Proposition \[2.4\]. Notice that $I \ne Q$, because $(I/Q)^r \cong X/IX \ne (0)$. Hence $\mu_A(I) > d$.
Let us prove Theorem \[4.1\].
$(1) \Rightarrow (2)$ See Theorem \[3.2\].
$(2) \Rightarrow (3)$ This is obvious.
$(2) \Rightarrow (4)$ and $(3) \Rightarrow (1)$ See Lemma \[4.2\].
Suppose that $d > 0$ and we will prove $(4) \Rightarrow (1)$. Let $a = a_1$ and put $\ol{A} = A/(a)$, $\ol{I} = I/(a)$, and $\ol{Q} = Q/(a)$. Then $$\Syz_A^i(A/I)/a\Syz_A^i(A/I) \cong \Syz_{\ol{A}}^{i-1}(A/I) \oplus \Syz_{\ol{A}}^{i}(A/I)$$ by Lemma \[3.4\] and $\Syz_{\ol{A}}^{i-1}(A/I) \oplus \Syz_{\ol{A}}^{i}(A/I)$ is an Ulrich $\ol{A}$–module with respect to $\ol{I}$, since $\Syz_A^i(A/I)$ is an Ulrich $A$–module with respect to $I$. Therefore $\Syz_{\ol{A}}^{i-1}(A/I) \ne (0)$. If $\Syz_{\ol{A}}^{i}(A/I) = (0)$, then $\ol{I}$ has finite projective dimension, so that $\ol{I} = \ol{Q}$ (see Theorem \[3.2\] or [@V]), which is impossible because $I \ne Q$. Hence $\Syz_{\ol{A}}^{i}(A/I) \ne (0)$. Consequently, $\ol{I}$ is an Ulrich ideal of $\ol{A}$, thanks to the implication $(3) \Rightarrow (1)$, and hence $I^2 \subseteq Q$ and $I/Q$ is $A/I$–free, because $\ol{I}^2 = \ol{Q}{\cdot}\ol{I}$ and $\ol{I}/\ol{Q}$ is $\ol{A}/\ol{I}$–free. It is now easy to check that $I^2 = QI$. In fact, write $I = (x_1, x_2, \ldots, x_n)$ ($n = \mu_A(I)$) with $x_i = a_i $ for $1 \le i \le d$. Then, for each $y \in I^2$, writing $y = \sum_{i=1}^dc_ix_i$ with $c_i \in A$, we see that $\sum_{i=1}^dc_i\ol{x_i} = 0$ in $I/I
^2$ where $\ol{x_i}$ denotes the image of $x_i$ in $I/I^2$. Consequently, $c_i \in I$ for all $1 \le i \le d$, because $\{\ol{x_i}\}_{1 \le i \le n}$ forms an $A/I$–free basis of $I/I^2$. Hence $y \in QI$, so that $I^2 = QI$, which shows $I$ is an Ulrich ideal of $A$.
([@BHU Example (2.6)]) Suppose $d = 0$ and let $\ell = \min \{\ell \in \Bbb Z \mid {\mathfrak{m}}^{\ell} = (0) \}$. Then ${\mathfrak{m}}^{\ell - 1}$ is an Ulrich $A$–module with respect to ${\mathfrak{m}}$, but if $\ell > 2$, ${\mathfrak{m}}$ itself is not an Ulrich ideal of $A$, since ${\mathfrak{m}}^2 \ne (0)$. Therefore an ${\mathfrak{m}}$–primary ideal $I$ is not necessarily an Ulrich ideal of $A$, even though there exists an Ulrich $A$–module with respect to $I$. More precisely, let $A = k[[t]]/(t^3)$ where $k[[t]]$ is the formal power series ring over a field $k$. We look at the exact sequence $$0 \to {\mathfrak{m}}^2 \to A \overset{t}{\to} A \to A/{\mathfrak{m}}\to 0.$$ Then ${\mathfrak{m}}^2 = \Syz_A^2(A/{\mathfrak{m}})$ is an Ulrich $A$–module with respect to ${\mathfrak{m}}$, but ${\mathfrak{m}}$ is not an Ulrich ideal of $A$. This shows the implication $(4) \Rightarrow (1)$ in Theorem \[4.1\] does not hold true in general, unless $ d = \dim A > 0$.
Duality
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Let ${\mathrm{K}}_A$ be the canonical module of $A$ ([@HKun]). In this section we study the question of when the dual $M^\vee = \Hom_A(M, {\mathrm{K}}_A)$ of an Ulrich $A$–module $M$ is Ulrich. Our answer is the following.
\[5.1\] Let $M$ be an Ulrich $A$–module with respect to $I$. Then the following conditions are equivalent.
1. $M^{\vee} = \Hom_A(M, {\mathrm{K}}_A)$ is an Ulrich $A$–module with respect to $I$.
2. $A/I$ is a Gorenstein ring.
Notice that $M^{\vee}$ is a maximal Cohen–Macaulay $A$–module ([@HKun Satz 6.1]). Since $IM = QM$ and $M^{\vee}/QM^\vee \cong \Hom_{A/Q}(M/QM, {\mathrm{K}}_{A/Q})$ ([@HKun Korollar 6.3]), we get $IM^\vee = Q M^\vee$, while $$\Hom_{A/Q}(M/IM, {\mathrm{K}}_{A/Q}) \cong \Hom_{A/Q}(A/I, {\mathrm{K}}_{A/Q})^m \cong ({\mathrm{K}}_{A/I})^m$$ by [@HKun Korollar 5.14], because $M/IM \cong (A/I)^m$ where $m = \mu_A(M) > 0$. Hence $$M^\vee/IM^\vee = M^\vee/ Q M^\vee \cong ({\mathrm{K}}_{A/I})^m.$$ Therefore $M^\vee$ is an Ulrich $A$–module with respect to $I$ if and only if ${\mathrm{K}}_{A/I}$ is a free $A/I$–module, that is $A/I$ is a Gorenstein ring.
As an immediate consequence of Corollary \[2.6\] and Theorem \[5.1\], we get the following, where $M^* =\Hom_A(M,A)$ for each $A$–module $M$.
\[5.2\] Suppose that $A$ is a Gorenstein ring and let $I$ be an Ulrich ideal of $A$. Let $M$ be a maximal Cohen–Macaulay $A$–module. Then the following conditions are equivalent.
1. $M^*$ is an Ulrich $A$–module with respect to $I$.
2. $M$ is an Ulrich $A$–module with respect to $I$.
Suppose that $A$ is a Gorenstein ring and let $M$ be a maximal Cohen–Macaulay $A$–module with a minimal free resolution $$\cdots \to F_i \to \cdots \to F_2 \overset{\partial_2}{\to} F_1 \overset{\partial_1}{\to} F_0 \to M \to 0.$$ Let $\Syz_A^1(M) = \operatorname{Im}\partial_1$ and put $\operatorname{Tr}M = \operatorname{Coker}\partial_1^*$, the Auslander transpose of $M$. Then we get the presentation $$0 \to M^* \to F_0^* \overset{\partial_1^*}{\to} F_1^* \to \operatorname{Tr}M \to 0$$ of $\operatorname{Tr}M$, so that $[\operatorname{Tr}M]^* = \Syz_A^2(M)$. Because the dual sequence $$0 \to M^* \to F_0^* \overset{\partial_1^*}{\to} F_1^* \overset{\partial_2^*}{\to} F_2^* \to \cdots \to F_i^* \to \cdots$$ is exact, $\operatorname{Tr}M$ is a maximal Cohen–Macaulay $A$–module, if $\operatorname{Tr}M \ne (0)$, that is the case where $M$ is not free. Notice that $$M^* = \Syz_A^2(\operatorname{Tr}M),$$ if $M$ contains no direct summand isomorphic to $A$.
With this notation, when $A$ is a Gorenstein ring, we can modify Lemma \[4.2\] in the following way.
\[5.3\] Suppose that $A$ is a Gorenstein ring and let $I$ be an Ulrich ideal of $A$ which is not a parameter ideal. Let $M$ be a maximal Cohen–Macaulay $A$–module and assume that $M$ contains no direct summand isomorphic to $A$. Then the following conditions are equivalent.
1. $M$ is an Ulrich $A$–module with respect to $I$.
2. $M^*$ is an Ulrich $A$–module with respect to $I$.
3. $\Syz_A^1(M)$ is an Ulrich $A$–module with respect to $I$.
4. $\operatorname{Tr}M$ is an Ulrich $A$–module with respect to $I$.
\(1) $\Leftrightarrow$ (2) See Corollary \[5.2\].
\(1) $\Rightarrow$ (3) See Theorems \[3.2\] and \[4.1\].
\(3) $\Rightarrow (1)$ Let $X = \Syz_A^1(M)$ and look at the presentation $0 \to X \to F_0 \to M \to 0$ of $M$ such that $F_0$ is a finitely generated free $A$–module and $X \subseteq {\mathfrak{m}}F_0$. Take the $A$–dual and we get the exact sequence $0 \to M^* \to F_0^* \to X^* \to 0.$ Then by Corollary \[5.2\], $X^*$ is an Ulrich $A$–module with respect to $I$. Therefore $\Syz_A^1(X^*)$ is by Lemma \[4.2\] an Ulrich $A$–module with respect to $I$, if $\Syz_A^1(X^*) \ne (0)$. On the other hand, because $$M^* \cong \Syz_A^1(X^*) \oplus A^r$$ for some $r \ge 0$ and because the reflexive $A$–module $M$ contains no direct summand isomorphic to $A$, we have $M \cong [\Syz_A^1(X^*)]^*$. Hence by Corollary \[5.2\], $M$ is an Ulrich $A$–module with respect to $I$.
\(1) $\Rightarrow (4)$ Because $M$ is an Ulrich $A$–module with respect to $I$, $[\operatorname{Tr}M]^* = \Syz_A^2(M)$ is by Theorem \[3.2\] an Ulrich $A$–module with respect to $I$. Hence $\operatorname{Tr}M$ is by Corollary \[5.2\] an Ulrich $A$–module with respect to $I$.
\(4) $\Rightarrow (2)$ This follows from Theorem \[3.2\], since $M^* = \Syz_A^2(\operatorname{Tr}M)$.
Ulrich ideals of numerical semigroup rings
==========================================
It seems interesting to ask, in a given Cohen–Macaulay local ring $A$, how many Ulrich ideals are contained, except parameter ideals. If $A$ is regular, we have nothing ([@V]), but in general cases the research is still in progress and we have no definitive answer. Here let us note a few results in a rather special case, that is the case where $A$ is a numerical semigroup ring over a field.
Let $k$ be a field. Let $a_1, a_2, \ldots, a_\ell > 0 $ ($\ell \ge 1$) be integers with $\operatorname{GCD}(a_1, a_2, \ldots, a_\ell) = 1$. We put $$H = \langle a_1, a_2, \ldots, a_\ell \rangle = \left\{\sum_{i=1}^\ell c_ia_i\,\Bigg|\,0 \le c_i \in \Bbb Z\right\}$$ which is the numerical semigroup generated by $a_i's$. Let $$A = k[[t^{a_1}, t^{a_2},\ldots, t^{a_\ell}]] \ \ \subseteq \ \ k[[t]],$$ where $V = k[[t]]$ is the formal power series ring over $k$. Then the numerical semigroup ring $A$ of $H$ is a one-dimensional complete local integral domain with $V$ the normalization. Let ${\mathcal X}_A^g$ denote the set of Ulrich ideals of $A$ which are not parameter ideals of $A$ but generated by monomials in $t$.
We then have the following.
\[6.1\] The set ${\mathcal X}_A^g$ is finite.
Let $I \in {\mathcal X}_A^g$ and put $a = \min \{h \in H \mid t^h \in I \}$. Then $Q = (t^a)$ is a reduction of $I$, since $t^aV = IV$. Therefore $I^2 = t^aI$. As $I/Q\subseteq IV/Q=t^aV/t^aA\cong V/A$, we have $\ell_A(I/Q) \le \ell_A(V/A)= \sharp (\Bbb N \setminus H),$ which yields ${\mathfrak{m}}^q {\cdot} (I/Q) = (0)$ where $q = \sharp (\Bbb N \setminus H)$. Therefore ${\mathfrak{m}}^q \subseteq I$, since $I/Q \cong (A/I)^{n-1}$ by Lemma \[2.3\] where $n = \mu_A(I) > 1$. Thus the set ${\mathcal X}_A^g$ is finite, because the set $\{h \in H \mid t^h \not\in {\mathfrak{m}}^q \}$ is finite.
Let us examine the following example.
\[6.2\] ${\mathcal X}_{k[[t^3, t^5, t^7]]}^g = \{{\mathfrak{m}}\}$.
We put $A = k[[t^3, t^5, t^7]]$. As ${\mathfrak{m}}^2 = t^3{\mathfrak{m}}$, we get ${\mathfrak{m}}\in {\mathcal X}_{A}^g$. Let $I \in
{\mathcal X}_{A}^g$. Then $1 < \mu_A(I) \le 3 = {\mathrm{e}}_{\mathfrak{m}}^0(A)$ ([@S2]). Suppose that $\mu_A(I) = 2$ and write $I = (t^a, t^b)$ with $a, b \in H$, $a < b$. Then $Q = (t^a)$ is a reduction of $I$ and $I/Q = (\ol{t^b}) \cong A/I$, where $\ol{t^b}$ denotes the image of $t^b$ in $I/Q$. Hence $I =(t^h \mid h \in H, h + (b-a) \in H),$ as $(t^a) : t^b = I$. Therefore, since $H \ni c$ for all $c \ge 5$, we get $t^3, t^5, t^7 \in I$ if $b - a \ge 2$, so that $I ~= {\mathfrak{m}}$. This is impossible, because $\mu_A({\mathfrak{m}}) = 3$. If $b -a =1$, then $I =(t^5, t^6, t^7)$, which is also impossible. Hence $\mu_A(I) \ne 2$.
Let $I = (t^a, t^b, t^c)$ with $a,b,c \in H$ such that $a < b < c$. We put $Q = (t^a)$. Then $I/Q = (\ol{t^b}, \ol{t^c}) \cong (A/I)^2.$ Hence $(t^a) : t^c = I$, so that $I = (t^h \mid h \in H, h + (c-a) \in H)$. Because $c-a \ge 2$, we see $t^3, t^5, t^7 \in I$, whence $I = {\mathfrak{m}}$ as is claimed.
When $A$ is a Gorenstein ring, that is the case where the semigroup $H$ is symmetric, we have the following characterization of Ulrich ideals generated by monomials.
\[6.4\] Suppose that $A=k[[t^{a_1},t^{a_2}, \ldots, t^{a_\ell}]]$ is a Gorenstein ring and let $I$ be an ideal of $A$. Then the following conditions are equivalent.
1. $I \in {\mathcal X}_A^g$.
2. $I=(t^a, t^b)$ ($a,b\in H, a<b$) and if we put $c=b-a$, the following conditions hold.
1. $c \not\in H$, $2c \in H$,
2. the numerical semigroup $H_1 = H + \left< c \right>$ is symmetric, and
3. $a = \min \{h \in H \mid h + c \in H\}$.
$(1) \Rightarrow (2)$ We have $\mu_A(I) = 2$ (Corollary \[2.6\]). Let us write $I = (t^a, t^b)$ ($a, b \in H$, $a < b$) and put $Q = (t^a)$. Then $I^2 = QI$. Therefore $t^{2b} \in (t^{2a}, t^{a+b})$, whence $t^{2b} \in (t^{2a})$, because $t^b \not\in Q = (t^a)$. Thus $b-a \not\in H$ but $2(b-a) \in H$. We put $c = b-a$ and let $$B = k[[t^{a_1}, t^{a_2} , \ldots, t^{a_{\ell}}, t^{c}]]$$ be the semigroup ring of $H_1 = H + \left<c \right>$. Then, since $2c \in H$, we see $B = A + At^c$, so that $t^aB = t^a A + t^b A = I$. Because $I/Q = t^aB/t^a A \cong B/A$ and $I/Q \cong A/I$, we have $I = A : B$. Hence $B$ is a Gorenstein ring, because ${\mathrm{K}}_B \cong A:B = I$ by [@HKun Satz 5.22] and $I = t^aB$. Thus $H_1 = H + \left<c\right>$ is symmetric. Assertion (iii) is now clear, since $I = Q : I = (t^h \mid h \in S)$ and $IV = t^aV$ where $S=\{h \in H \mid h + c \in H\}$.
$(2) \Rightarrow (1)$ We put $Q = (t^a)$. Then $I^2 = QI$ and $I \ne Q$ by (i) and (ii). We must show $I = Q : I$. Let $B = A[t^c]$. Then, since $t^{2c} \in A$, we get $B = A + At^c$, so that $t^aB = I$. Hence $A:B = Q : I$. Let $J = A : B$. We then have $J = fB$, because $A : B \cong {\mathrm{K}}_B$ and $B = k[[t^{a_1}, t^{a_2} , \ldots, t^{a_{\ell}}, t^{c}]]$ is a Gorenstein ring by (iii). Hence $$\frac{I}{t^a} = B = \frac{J}{f}.$$ On the other hand, because $J=Q:I = (t^h \mid h \in S)$ where $S = \{h \in H \mid h + c \in H\}$, by (iv) $Q$ is a reduction of $I$ (notice that $t^a V = IV$), whence $J^2 = t^a J$ (remember that $J^2 = fJ$). Consequently $$\frac{J}{t^a} = \frac{J}{f} = \frac{I}{t^a},$$ whence $I = J = Q : I$. Thus $I/Q = (\ol{t^b}) \cong A/I$ where $\ol{t^b}$ is the image of $t^b$ in $I/Q$. Hence $I \in {\mathcal X}_A^g$ as claimed.
\[6.5\] Let $a \ge 5$ be an integer. Then ${\mathcal X}_{k[[t^a, t^{a+1}, \ldots, t^{2a-2}]]}^g = \emptyset$.
We put $H = \left<a, a+1, \ldots, 2a - 2 \right>$. Then $H$ is symmetric. Let $c \in \Bbb Z$. Assume that $c \not\in H$ but $2c \in H$ and put $H_1 = H + \left< c \right>$. Then $H_1 \setminus H = \{c, 2a-1\}$ and it is routine to check that $H_1$ is never symmetric, whence ${\mathcal X}_{k[[t^a, t^{a+1}, \ldots, t^{2a-2}]]}^g = \emptyset$ by Theorem \[6.4\].
Using the characterization of Ulrich ideals of Theorem \[6.4\], we can determine all the Ulrich ideals of semigroups rings when $H$ is generated by $2$ elements. For that purpose, we recall the following result of [@W].
\[Wat\][@W Proposition 3] Let $H = \left<a, b, c \right>$ be a symmetric numerical semigroup generated minimally by $3$ integers. Then changing the order of $a,b,c$ if necessary, we can write $b = b'd, c= c'd$ where $d>1, \GCD(a,d)=1$ and $a\in \left< b', c' \right>$.
Next we determine the structure of $H_1= H + \left< c \right>$ in Theorem \[6.4\] when $H=\left< a, b \right>$
\[6main\] Let $H=\left< a, b \right>$ and $H_1= H + \left< c \right>$ be symmetric numerical semigroups, where $a,b >1$ are relatively coprime integers and $c$ is a positive integer satisfying $c\not\in H$ and $2c\in H$. Then after changing the order of $a,b$ if necessary, one of the following cases occur.
1. $H=\left< 2, 2\ell +1 \right>$ and $c= 2m+1$ with $0\le m< \ell$,
2. $a= 2c/d$, where $d= \GCD(b,c)$ is odd and $d \ge 1$.
3. $a=2d$, where $d= \GCD(a,c)>1, c/d$ is odd and $1\le c/d < b$.
If $H=\left< 2, 2\ell +1 \right>$, then obviously the case (1) occurs. Henceforth we assume $2\not\in H$ and $c\ne 1$.
If $H_1= H + \left< c \right>$ is generated by $2$ elements and $H_1=
\left< b, c\right>$, we may assume
1. $a = mb + nc$ and
2. $2c = pa + qb$
for some non-negative integers $m,n,p,q$. From $\mathrm{(i)}$ and $\mathrm{(ii)}$ we get $$2a = 2mb + 2nc = npa + (2m + nq)b.$$ Hence we must have $0\le np\le 2$ and if $np=1$, $a\in \left< b\right>$, a contradiction. If $np=0$, since $a,b$ are relatively coprime, we must have $b=2$, contradicting our hypothesis $2\not\in H$. If $np=2$, then $m=q=0$ and $a= nc$ and since $np=2$ and $c \not\in H$, we must have $a=2c$.
Now, we assume that $H_1$ is minimally generated by $3$ elements and $a,b>2$. Then by Lemma \[Wat\], we may assume $$b = b'd, c= c'd,$$ where $d>1, \GCD(a,d)=\GCD(b',c')=1$ and $a\in \left< b', c' \right>$. Then we have
1. $a = mb' + nc'$ and
2. $2c = pa + qb$.
We can put $p = p'd$ and $2c' = p'a + qb'$. Note that $n\ne 0$ since $a,b$ are relatively coprime. We have the following equality. $$2a = 2mb' + 2nc' = np'a + (2m + nq) b'.$$ Again, we must have $0\le np'\le 2$ and if $np'=1$, $a\in \left< b'\right>$, a contradiction. If $np'=2$, then $m=q=0$ and we must have $a=2c'$, $(a,b,c) = (2c', b'd, c'd)$, with $d$ odd and $d>1$.
If $p'=0$, then we have $2c = qb$, or $2c' = qb'$ and we must have $b'=2$. Now, let us interchange $a$ and $b$. Then $a=2d$ and $c=c'd$. Since $H_1$ is symmetric, $b>c'$ by Lemma \[Wat\]. This is our case (3).
\[H=(a,b)\] Suppose that $A=k[[t^a,t^b]]$ with $\GCD(a,b)=1$ and $I=(t^{\alpha}, t^{\beta})\in {\mathcal X}_A^g$ with $c= \beta -\alpha>0$. Then either $a$ or $b$ is even and we can determine $c, \alpha$ as follows.
1. If $H= \left< 2, 2\ell +1 \right>$, then $\alpha = 2q$ with $1\le q\le \ell$ and $\beta= 2\ell+1$. In this case, $\sharp {\mathcal X}_A^g=\ell$.
2. If $a$ is even and $a>2$, then we can take $c= ac'/2$, where $c'$ is an odd integer with $1\le c' < b$. In this case, $\alpha=\min \{h \in H \mid h + c \in H\}$ and $\sharp {\mathcal X}_A^g= (b-1)/2$.
We have shown in Lemma \[6main\] that $a$ or $b$ is even. If $H= \left< 2, 2\ell +1 \right>$, it is easy to see that $c$ is odd and $1\le c < 2\ell +1$.
If $a$ is even and $a>2$, then by Theorem \[6.4\] and Lemma \[6main\], the number $c $ is determined as one of the following cases.
1. $c= ad/2$, where $d$ is a proper divisor of $b$ (including $d=1$) and
2. $c= ac' /2$ where $c'$ is an odd integer with $\GCD(a,c')=1, c'< b$ and $\alpha=\min \{h \in H \mid h + c \in H\}$.
But we can easily see that the case (i) is included in the case (ii).
\[6.3\] The following assertions hold true.
1. ${\mathcal X}_{k[[t^3, t^5]]}^g = \emptyset$.
2. If $A=k[[t^8,t^{15}]]$, then $c= \beta -\alpha$ is one of the integers $4,12,20,28,36,44$ and $52$. Hence ${\mathcal X}_{A}^g = \{ ( t^{8i}, t^{60})\}_{1\le i\le7}$.
3. ${\mathcal X}_{k[[t^4, t^6, t^{4\ell -1}]]}^g = \{(t^4, t^6), (t^{4\ell - 4}, t^{4 \ell - 1}),(t^{4(\ell - q) - 6}, t^{4\ell - 1}), (t^{4(\ell - q) - 8}, t^{4\ell - 1})\}_{0 \le q \le \ell - 3},$ where $\ell \ge 2$. (For the proof, see §8).
Structure of minimal free resolutions of Ulrich ideals
======================================================
Let $(A, {\mathfrak{m}})$ be a Cohen-Macaulay local ring of dimension $d \ge 0$ and let $I$ be an ${\mathfrak{m}}$–primary ideal of $A$ which contains a parameter ideal $Q=(a_1,a_2,\cdots,a_d)$ as a reduction. The purpose of this section is to explore the structure of minimal free resolutions of Ulrich ideals.
Throughout this section, we assume that $I$ is an Ulrich ideal of $A$. Let $${\Bbb F}_{\bullet}: \cdots \to F_i \overset{{\partial}_i}{\to} F_{i-1} \to \cdots \to F_1 \overset{{\partial}_1}{\to} F_0=A \overset{\varepsilon}{\to} A/I \to 0$$ be a minimal free resolution of the $A$–module $A/I$. We put $\beta_i=\rank_AF_i = \beta_i^A(A/I)$, the $i$-th Betti number of $A/I$, and $n = \mu_A(I) = \beta_1 \ge d$.
We begin with the following.
\[7.1\] The following assertions hold true.
- $A/I \otimes_A \partial_i=0$ for all $i \geq 1$.
- $$\beta_i= \left\{
\begin{array}{ll}
(n-d)^{i-d}{\cdot}(n-d+1)^d & (d \le i),\\
\binom{d}{i}+(n-d){\cdot}\beta_{i-1} & (1 \leq i \leq d),\\
1 & (i=0).
\end{array}
\right.$$ Hence $\beta_i=\binom{d}{i}+(n-d){\cdot}\beta_{i-1}$ for all $i \geq 1$.
We proceed by induction on $d$. If $d = 0$, then $\Syz_A^i(A/I) \cong (A/I)^{n^i}$ ($ i \ge 0$) and the assertions are clear. Assume that $d >0$ and that our assertions hold true for $d-1$. Let $a=a_1$ and put $\ol{A}=A/(a)$, $\ol{I}=I/(a)$. Then by Lemma \[3.4\] $$\Syz_A^i(A/I)/a \Syz_A^i(A/I) \cong \Syz_{\ol{A}}^{i-1}(A/I) \oplus \Syz_{\ol{A}}^{i}(A/I)$$ for each $i \ge 1$. Hence we get $$\beta_i = \ol{\beta}_{i-1} + \ol{\beta}_i$$ for $i \ge 1$, where $\ol{\beta}_i = \beta_i^{\ol{A}}(A/I)$. We put $\ol{n}=n-1$ and $\ol{d}=d-1$. Then, thanks to the hypothesis of induction on $d$, we get for each $i \ge d$ that $$\begin{aligned}
\beta_i&=&\ol{\beta}_{i-1}+\ol{\beta_i}\\
&=& (\ol{n}-\ol{d})^{i-1-\ol{d}}{\cdot}(\ol{n}-\ol{d}+1)^{\ol{d}}+(\ol{n}-\ol{d})^{i-\ol{d}}{\cdot}(\ol{n}-\ol{d}+1)^{\ol{d}}\\
&=& (n-d)^{i-d}{\cdot}(n-d+1)^{d-1}+(n-d)^{i-d+1}{\cdot}(n-d+1)^{d-1}\\
&=& (n-d)^{i-d}{\cdot}(n-d+1)^{d-1}{\cdot}\left[1+(n-d)\right]\\
&=& (n-d)^{i-d}{\cdot}(n-d+1)^{d}.\end{aligned}$$
Let $1 \leq i \leq d$. If $i = 1$, then $\beta_i=\beta_1=n=\binom{d}{1}+(n-d){\cdot}\beta_0=\binom{d}{i}+(n-d){\cdot}\beta_{i-1}$. If $2 \leq i \leq d-1$, then $$\begin{aligned}
\beta_i&=&\ol{\beta}_{i-1}+\ol{\beta}_i\\
&=& \textstyle\left[\binom{\ol{d}}{i-1}+(\ol{n}-\ol{d}){\cdot}\ol{\beta}_{i-2}\right]+\left[\binom{\ol{d}}{i}+(\ol{n}-\ol{d}){\cdot}\ol{\beta}_{i-1}\right]\\
&=& \textstyle\left[\binom{d-1}{i-1}+(n-d){\cdot}\ol{\beta}_{i-2}\right]+\left[\binom{d-1}{i}+(n-d){\cdot}\ol{\beta}_{i-1}\right]\\
&=& \textstyle\binom{d}{i}+(n-d){\cdot}\left[\ol{\beta}_{i-2}+\ol{\beta}_{i-1}\right] \\
&=& \textstyle\binom{d}{i}+(n-d){\cdot}\beta_{i-1}.\end{aligned}$$
Suppose that $i=d \geq 2$. We then have $$\begin{aligned}
\beta_i=\beta_d &=& \ol{\beta}_{d-1} + \ol{\beta}_d\\
&=& \textstyle\binom{\ol{d}}{\ol{d}} + (\ol{n}- \ol{d}){\cdot}\ol{\beta}_{d-2}+ \ol{\beta}_d\\
&=& \textstyle\binom{d}{d} + (n -d){\cdot}\ol{\beta}_{d-2} + \ol{\beta}_d,\end{aligned}$$ while $$\begin{aligned}
\textstyle\binom{d}{d} + (n-d){\cdot}\beta_{d-1}
&=& \textstyle\binom{d}{d} + (n-d){\cdot}\left[\ol{\beta}_{d-2} + \ol{\beta}_{d-1}\right]\\
&=& \textstyle\binom{d}{d} + (n-d){\cdot}\ol{\beta}_{d-2} + (n-d){\cdot}\ol{\beta}_{d-1}\\
&=& \textstyle\binom{d}{d} + (n-d){\cdot}\ol{\beta}_{d-2} + (\ol{n} - \ol{d}){\cdot}\ol{\beta}_{d-1}\\
&=& \textstyle\binom{d}{d} + (n-d){\cdot}\ol{\beta}_{d-2} + \ol{\beta}_d.\end{aligned}$$ Hence $$\beta_i= \left\{
\begin{array}{ll}
(n-d)^{i-d}{\cdot}(n-d+1)^d & (d \le i),\\
\binom{d}{i}+(n-d){\cdot}\beta_{i-1} & (1 \leq i \leq d),\\
1 & (i=0),
\end{array}
\right.$$ so that $\beta_i=\binom{d}{i}+(n-d){\cdot}\beta_{i-1}$ for all $i \geq 1$.
Because $$\cdots \to F_i/aF_i \to F_{i-1}/aF_{i-1} \to \cdots \to F_1/aF_1 \to I/aI \to 0$$ is a minimal free resolution of the $\ol{A}$–module $I/aI$ and because $$I/aI \cong A/I \oplus \ol{I}$$ by Claim in the proof of Lemma \[3.4\], we see $A/I \otimes_A\partial_i=\ol A/\ol I\otimes_{\ol A}\ol\partial_i= 0$ for $i > 1$ from the induction hypothesis, where $\ol\partial_i:=\ol A\otimes_A\partial_i$. As $A/I\otimes_A\partial_1 = 0$ obviously, this proves Theorem \[7.1\].
Suppose that $d > 0$ and we look at the exact sequence $$(\sharp) \ \ \ 0 \to Q \overset{\iota}{\to} I \to I/Q \to 0$$ of $A$–modules, where $\iota : Q \to I$ is the embedding. Remember now that a minimal free resolution of $Q$ is given by the truncation $$\Bbb L_{\bullet}: 0 \to K_d \to \cdots \to K_1 \to Q \to 0$$ of the Koszul complex ${\Bbb K}_{\bullet} = \Bbb K_\bullet (a_1, a_2, \ldots, a_d; A)$ generated by the $A$–regular sequence $a_1, a_2, \ldots, a_d$ and a minimal free resolution of $I/Q = (A/I)^{n-d}$ is given by the direct sum $\Bbb G_{\bullet}$ of $n - d$ copies of $\Bbb F_{\bullet}$. Then by the horseshoe lemma, a free resolution of $I$ is induced from $\Bbb L_{\bullet}$ and $\Bbb G_{\bullet}$ via exact sequence $(\sharp)$ above. With this notation, what Theorem \[7.1\] says is the following.
\[7.2\] In the exact sequence $0 \to Q \overset{\iota}{\to} I \to (A/I)^{n-d} \to 0$, the free resolution of $I$ induced from $\Bbb L_{\bullet}$ and $\Bbb G_{\bullet}$ is a minimal free resolution.
For example, suppose that $A$ is a Gorenstein ring with $\dim A = 0$ and assume that $I \ne (0)$. Then $I = (x)$ for some $x \in A$ (Lemma \[2.6\]). Because $(0) : I = I$, a minimal free resolution of $A/I$ is given by $$\Bbb F_{\bullet} \cdots \to A \overset{x}{\to} A \overset{x}{\to} A \to A/I \to 0.$$
We similarly have the following.
\[7.3\] Suppose that $A$ is a Gorenstein ring with $\dim A = 1$. Let $I$ be an ${\mathfrak{m}}$–primary ideal of $A$ containing $Q = (a)$ as a reduction. Assume that $I$ is an Ulrich ideal of $A$ which is not a parameter ideal. Then $\mu_A(I) = 2$. We write $I = (a,x)$ ($x \in A$). Then $x^2 = ay$ for some $y \in I$, because $I^2 = aI$. With this notation, a minimal free resolution of $A/I$ is given by
$$\Bbb F_{\bullet} : \cdots \to A^2 {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\begin{pmatrix}
-x&-y\\
a & x
\end{pmatrix}}}}
A^2 {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\begin{pmatrix}
-x&-y\\
a & x
\end{pmatrix}}}}A^2 {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\begin{pmatrix}
a&x
\end{pmatrix}}}} A \overset{\varepsilon}{\to} A/I \to 0.$$
It is standard to check that $\Bbb F_{\bullet}$ is a complex of $A$–modules. To show that $\Bbb F_{\bullet}$ is exact, let $f,g \in A$ and assume that $af + xg = 0$. Then, since $g \in Q : I =I$, we may write $g = ag_1 + xg_2$ with $g_i \in A$. Then, because $af + xg = af + a(xg_1 + yg_2) = 0$, we get $f = -(xg_1+yg_2)$, so that $$\begin{pmatrix}
f\\
g
\end{pmatrix} = \begin{pmatrix} -(xg_1 + yg_2)\\
ag_1 + xg_2
\end{pmatrix}
= \begin{pmatrix}
-x&-y\\
a & x
\end{pmatrix}
\begin{pmatrix}
g_1 \\
g_2
\end{pmatrix}.$$ Therefore, if $f,g \in A$ such that $$\begin{pmatrix}
-x&-y\\
a&x
\end{pmatrix}\begin{pmatrix}f\\
g
\end{pmatrix}= \binom{0}{0},$$ we then have $$\begin{pmatrix}
f\\
g
\end{pmatrix} = \begin{pmatrix}
-x&-y\\
a & x
\end{pmatrix}
\begin{pmatrix}
g_1\\
g_2
\end{pmatrix}$$ for some $g_i \in A$, because $af + xg = 0$. Hence $\Bbb F_\bullet$ is a minimal free resolution of $A/I$.
As we have seen in Example \[7.3\], minimal free resolutions of Ulrich ideals of a Gorenstein local ring are eventually periodic. Namely we have the following.
\[7.4\] The following assertions hold true.
- $\Syz_A^{i+1}(A/I) \cong [\Syz_A^{i}(A/I)]^{n-d}$ for all $i \geq d$.
- Suppose that $A$ is a Gorenstein ring. Then one can choose a minimal free resolution ${\Bbb F}_{\bullet}$ of $A/I$ of the form $$\cdots \to F_d \overset{{\partial}_{d+1}}{\to} F_{d} \overset{\partial_{d+1}}{\to} F_d \overset{{\partial}_{d}}{\to} F_{d-1} \to \cdots \to F_1 \overset{\partial_1}{\to} F_0 = A \overset{\varepsilon}{\to} A/I \to 0,$$ that is $F_{d+i}=F_d$ and $\partial_{d+i+1}=\partial_{d+1}$ for all $i \geq 1$.
\(1) This is clear, because $$\Syz_A^{i+1}(A/I) = \Coker~\partial_{i+2} \cong \left[\Coker~\partial_{i+1}\right]^{n-d} = \left[\Syz_A^i(A/I)\right]^{n-d}$$ for all $i \ge d$ (see Corollary \[7.2\]).
\(2) By Example \[7.3\] we may assume $n > d \ge 2$; hence $n = d+1$. By Corollary \[7.2\] there exist isomorphisms $\alpha : F_{d+2} \tilde{\to} F_{d+1}$ and $\beta : F_{d+1} \tilde{\to} F_d$ which make the following diagram $$\begin{array}{ccccccc}
F_{d+2} & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_{d+2}}}} & F_{d+1} & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_{d+1}}}} & F_{d} & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_d}}} & F_{d-1}\\
{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\alpha\,$}}$ }} & &{\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\beta\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }} & & {\Big\downarrow
\llap{$\vcenter{\hbox{$\scriptstyle\,$}}$ }}\\
F_{d+1} & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_{d+1}}}} & F_d & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_d}}} & F_{d-1} & {\smash{\mathop{\hbox to 1cm{\rightarrowfill}}\limits^{\partial_{d-1}}}} & F_{d-2}
\end{array}$$ commutative. Then a simple diagram chase will show that the sequence $$\cdots \to F_{d+1} \overset{\beta^{-1}\partial_{d+1}}{\to} F_{d+1} \overset{\beta^{-1}\partial_{d+1}}{\to} F_{d+1} \overset{\partial_d \beta}{\to} F_{d-1} \overset{\partial_{d-1}}{\to} F_{d-2}$$ is exact.
Assertion (1) of Corollary \[7.4\] shows that Ulrich modules with respect to $I$ obtained by syzygies $\Syz_A^i(A/I)$ ($i \ge d$) are essentially of one kind. To see this phenomenon more precisely, let ${\mathrm{I}}_1(\partial_i)$ ($i \ge 1$) denote the ideal of $A$ generated by the entries of the matrix $\partial_i : F_i \to F_{i-1}$. We then have the following.
\[7.5\] Suppose that $\mu_A(I) > d$. Then ${\mathrm{I}}_1(\partial_i)=I$ for all $i \geq 1$.
The assertion is obvious, if $d = 0$ (remember that $\Syz_A^i(A/I) \cong (A/I)^{n^i}$ for all $i \ge 0$). Therefore induction on $d$ easily shows that ${\mathrm{I}}_1(\partial_i) + Q = I$ for all $i \ge 1$ (use Lemma \[3.4\]).
Suppose now that $d > 0$. Then we get ${\mathrm{I}}_1(\partial_i) \supseteq Q$ for all $1 \le i \le d$, because by Corollary \[7.2\] the truncation $$\Bbb L_{\bullet} : 0 \to K_d \to K_{d-1} \to \cdots \to K_1 \to Q \to 0$$ of the Koszul complex $\Bbb K_{\bullet} = \Bbb K_{\bullet}(a_1, a_2, \ldots, a_d;A)$ is a subcomplex of the truncation $${\Bbb M}_{\bullet} \cdots \to F_{d+1} \to F_d \to \cdots \to F_1 \to I \to 0$$ of the minimal free resolution $\Bbb F_{\bullet}$ of $A/I$ and $K_i$ is a direct summand of $F_i$ for each $1 \le i \le d$. Hence ${\mathrm{I}}_1(\partial_i) = I$ if $1 \le i \le d$. On the other hand, Corollary \[7.2\] shows also that ${\mathrm{I}}_1(\partial_{i+1}) = {\mathrm{I}}_1(\partial_i)$ for $i \ge d+1$. Consequently, to see that ${\mathrm{I}}_1(\partial_i) = I$ for all $i \ge d+1$, it suffices to show ${\mathrm{I}}_1(\partial_{d+1}) \supseteq Q$ only, which is obviously true, because by Corollary \[7.2\] the matrix $\partial_{d+1}$ has the form $$\partial_{d+1} = \begin{pmatrix}
*\\
\partial_d^{\oplus n-d}
\end{pmatrix}\ \ \ (n -d > 0)$$ with ${\mathrm{I}}_1(\partial_d) = I$. This completes the proof of Theorem \[7.5\].
The following result is a direct consequence of Corollary \[7.4\] and Theorem \[7.5\].
\[7.6\] Let $(A,{\mathfrak{m}})$ be a Cohen–Macaulay local ring of dimension $d \ge 0$. Let $I$ and $J$ be ${\mathfrak{m}}$–primary ideals of $A$ containing some parameter ideals of $A$ as reductions. Suppose that both $I$ and $J$ are Ulrich ideals of $A$ with $\mu_A(I) > d$ and $\mu_A(J) > d$. If for some $i \ge0$ $$\Syz_A^i(A/I) \cong \Syz_A^i(A/J),$$ then $I = J$.
For a given Cohen–Macaulay local ring $A$ let ${\mathcal X}_A$ denote the set of Ulrich ideals $I$ of $A$ which contains parameter ideals as reductions but $\mu_A(I) > d$. Then as a consequence of Corollary \[7.6\], we get the following. Remember that $A$ is said to be of finite CM-representation type, if there exist only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay $A$–modules.
\[7.7\] If $A$ is of finite CM–representation type, then the set ${\mathcal{X}}_A$ is finite.
Let $I \in {\mathcal{X}}_A$. Then $\mu_A(I) \le {\mathrm{r}}(A) + d$ (Corollary \[2.6\]). Let $${\mathcal{S}}= \{\left[\Syz_A^d(A/I) \right] \mid I \in {\mathcal{X}}_A \},$$ where $\left[\Syz_A^d(A/I) \right]$ denotes the isomorphism class of the maximal Cohen–Macaulay $A$–module $\Syz_A^d(A/I)$. Then because $$\beta_d^{A}(A/I) = (\mu_A(I) - d + 1)^d \le \left[{\mathrm{r}}(A) +1\right]^d,$$ the minimal number $\mu_A(\Syz_A^d(A/I))$ of generators for $\Syz_A^d(A/I)$ has an upper bound which is independent of the choice of $I \in {\mathcal{X}}_A$. Hence the set ${\mathcal{S}}$ is finite, because $A$ is of finite CM-representation type. Thus the set ${\mathcal{X}}_A$ is also finite, because ${\mathcal{X}}_A$ is a subset of ${\mathcal{S}}$ by Corollary \[7.6\].
Let us explore the following example in order to illustrate Theorem \[7.7\].
\[7.8\] Let $A = k[[X, Y, Z]]/(Z^2 - XY)$ where $k[[X,Y,Z]]$ is the formal power series ring over a field $k$. Then ${\mathcal{X}}_A = \{{\mathfrak{m}}\}$.
Let $x, y$, and $z$ be the images of $X, Y$, and $Z$ in $A$, respectively. Then ${\mathfrak{m}}^2 = (x,y){\mathfrak{m}}$, so that ${\mathfrak{m}}\in {\mathcal{X}}_A$. Let $I \in {\mathcal{X}}_A$ and put $X = \Syz_A^2(A/I)$. Then $\mu_A(I) = 3$ (Corollary \[2.6\]), $\rank_AX = 2$, and $\mu_A(X)=4$ (Theorem \[7.1\]). Therefore, because $A$ and ${\mathfrak{p}}= (x,z)$ are the indecomposable maximal Cohen–Macaulay $A$–modules (up to isomorphism), we get $$X \cong {\mathfrak{p}}\oplus {\mathfrak{p}}\cong \Syz_A^2(A/{\mathfrak{m}}),$$ so that $I = {\mathfrak{m}}$ by Corollary \[7.6\]. Thus ${\mathcal{X}}_A = \{{\mathfrak{m}}\}$ as claimed.
Ulrich ideals of one-dimensional Gorenstein local rings of finite CM–representation type
========================================================================================
In the preceding section we observed that every Cohen-Macaulay local ring of finite CM-representation type admits only finitely many nonparameter Ulrich ideals (Theorem \[7.7\]). In this section, we consider giving complete classification of those ideals, and do it for Gorenstein local rings of dimension one under some mild assumptions. To achieve our purpose, we use techniques from the representation theory of maximal Cohen-Macaulay modules. Let us begin with recalling several definitions and basic facts stated in Yoshino’s book [@Y].
[@Y (2.8) and (3.11)]\[8.5\] Let $A$ be a $d$-dimensional Cohen-Macaulay complete local ring. Suppose that $A$ is an [*isolated singularity*]{}, that is, the local ring $A_{\mathfrak{p}}$ is regular for every nonmaximal prime ideal ${\mathfrak{p}}$ of $A$. Let $M$ be a nonfree indecomposable maximal Cohen-Macaulay $A$-module. Then we define the [*Auslander-Reiten translation*]{} of $M$ by: $$\tau M=\Hom_A(\Syz_A^d(\operatorname{Tr}M),{\mathrm{K}}_A).$$ Here ${\mathrm{K}}_A$ denotes the canonical module of $A$.
\[8.6\] With the notation of Definition \[8.5\], assume that $A$ is Gorenstein with $d=1$. Then one has an isomorphism $\tau M\cong\Syz_A^1(M)$.
Since $M$ is nonfree and indecomposable, there exists an exact sequence $$\cdots \xrightarrow{\partial_2} F_1 \xrightarrow{\partial_1} F_0 \xrightarrow{\partial_0} F_{-1} \xrightarrow{\partial_{-1}} \cdots$$ of finitely generated free $A$-modules whose $A$-dual is also exact such that $\Im\,\partial_i\subseteq{\mathfrak{m}}F_{i-1}$ for all integers $i$ and $\Im\,\partial_0=M$. We see from this exact sequence that $\tau M=(\Syz_A^1(\operatorname{Tr}M))^\ast\cong(\Im(\partial_1^\ast))^\ast\cong\Im\,\partial_1=\Syz_A^1(M)$.
Let $A$ be a Cohen-Macaulay local ring. The [*Auslander-Reiten quiver*]{} $\Gamma_A$ of $A$ is a graph consisting of vertices, arrows and dotted lines. The vertices are the isomorphism classes of indecomposable maximal Cohen-Macaulay $A$-modules. For nonfree indecomposable maximal Cohen-Macaulay modules $M$ and $N$, the vertex $[M]$ is connected by a dotted line with the vertex $[N]$ if and only if $M\cong\tau N$ and $N\cong\tau M$. We refer to [@Y (5.2)] for details. For a $1$-dimensional hypersurface, the Auslander-Reiten quiver finds all the pairs of maximal Cohen-Macaulay modules one of which is the first syzygy of the other:
\[8.7\] Let $A$ be a local hypersurface of dimension one. Let $M,N$ be nonfree indecomposable maximal Cohen-Macaulay $A$-modules. Then the following are equivalent.
1. $M\cong\Syz_A^1(N)$.
2. $N\cong\Syz_A^1(M)$.
3. In $\Gamma_A$ the vertices $[M],[N]$ are connected by a dotted line.
Since $A$ is a hypersurface and $M,N$ are nonfree indecomposable, we have $M\cong\Syz_A^2(M)$ and $N\cong\Syz_A^2(N)$ (cf. [@Y (7.2)]). By Lemma \[8.6\], we obtain the equivalence.
Throughout the rest of this section, let $A$ be a $1$-dimensional Gorenstein local ring. We denote by ${\mathcal{C}}_A$ the set of nonisomorphic maximal Cohen-Macaulay $A$-modules $M$ without nonzero free summand such that $\Syz_A^1(M)\cong M$ and $\mu_A(M)=2$. The following statement relates the notion of Ulrich ideals with the representation theory of maximal Cohen-Macaulay modules.
\[8.8\] Let $A$ be a $1$-dimensional Gorenstein local ring. Then one has the inclusion ${\mathcal{X}}_A\subseteq{\mathcal{C}}_A$.
If an ideal $I$ of $A$ has a nonzero free summand, then we can write $I=(x)\oplus J$ for some nonzerodivisor $x$ and ideal $J$ of $A$. Since $xJ\subseteq(x)\cap J=(0)$, we have $J=(0)$, and $I=(x)$. Thus every ideal that is an element of $\X_A$ does not have a nonzero free summand. The assertion now follows from Corollaries \[2.6\] and \[7.4\].
Let $A$ be a $1$-dimensional Gorenstein complete equicharacteristic local ring with algebraically closed residue field $k$ of characteristic $0$. Suppose that $A$ has finite CM-representation type. Then $A$ is a [*simple singularity*]{}, namely, one has a ring isomorphism $$A\cong k[[x,y]]/(f),$$ where $f$ is one of the following:
1. $x^2+y^{n+1} \quad(n\ge1)$,
2. $x^2y+y^{n-1} \quad (n\ge4)$,
3. $x^3+y^4$,
4. $x^3+xy^3$,
5. $x^3+y^5$.
For the details, see [@Y (8.5), (8.10) and (8.15)]. In this case, we can make a complete list of the nonparameter Ulrich ideals.
\[8.9\] With the above notation, the set $\X_A$ is equal to:
1. $\begin{cases}
\{(x,y),(x,y^2),\dots,(x,y^{\frac{n}{2}})\} & \text{if $n$ is even},\\
\{(x,y),(x,y^2),\dots,(x,y^{\frac{n-1}{2}}),(x,y^{\frac{n+1}{2}})\} & \text{if $n$ is odd}.
\end{cases}$
2. $\begin{cases}
\{(x^2,y),(x+\sqrt{-1}y^\frac{n-2}{2},y^\frac{n}{2}),(x-\sqrt{-1}y^\frac{n-2}{2},y^\frac{n}{2})\} & \text{if $n$ is even},\\
\{(x^2,y),(x,y^\frac{n-1}{2})\} & \text{if $n$ is odd}.
\end{cases}$
3. $\{(x,y^2)\}$.
4. $\{(x,y^3)\}$.
5. $\emptyset$.
Thanks to Proposition \[8.8\], the set $\X_A$ is contained in ${\mathcal{C}}_A$, so it is essential to calculate ${\mathcal{C}}_A$. It is possible by looking at the Auslander-Reiten quiver $\Gamma_A$ of $A$, which is described in [@Y]. More precisely, by virtue of Proposition \[8.7\], all elements of ${\mathcal{C}}_A$ are direct sums of modules corresponding to vertices of $\Gamma_A$ connected by dotted lines. Once we get the description of ${\mathcal{C}}_A$, we can find elements of ${\mathcal{C}}_A$ belonging to $\X_A$, by making use of Corollary \[2.6\].
\(1) The case $({\mathrm{A}}_n)$ with $n$ even:
It follows from [@Y (5.11) and (5.12)] that $${\mathcal{C}}_A=\{(x,y),(x,y^2),\dots,(x,y^\frac{n}{2})\}.$$ Applying Corollary \[2.6\] to $I=(x,y^i)$ and $Q=(y^i)$ for $1\le i\le\frac{n}{2}$, we see that $(x,y^i)$ is an Ulrich ideal. Hence ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x,y),(x,y^2),\dots,(x,y^\frac{n}{2})\}$.
\(2) The case $({\mathrm{A}}_n)$ with $n$ odd:
We use the same notation as in [@Y (9.9)]. It is seen by [@Y Figure (9.9.1)] that $${\mathcal{C}}_A\subseteq\{M_1,M_2,\dots,M_\frac{n-1}{2},N_+\oplus N_-\}$$ holds. For $1\le j\le n+1$, the sequence $$\begin{CD}
A^2 @>{\left(\begin{smallmatrix}
x & y^j \\
y^{n+1-j} & -x
\end{smallmatrix}\right)}>> A^2 @>{\text{nat}}>> (x,y^j) @>>> 0
\end{CD}$$ is exact, which shows that $M_j$ is isomorphic to the ideal $(x,y^j)$ of $A$. Since $N_+\oplus N_-\cong M_\frac{n+1}{2}$, we have ${\mathcal{C}}_A=\{(x,y),(x,y^2),\dots,(x,y^{\frac{n-1}{2}}),(x,y^{\frac{n+1}{2}})\}$. Applying Corollary \[2.6\] to $I=(x,y^j)$ and $Q=(y^j)$ yields that $(x,y^j)$ is an Ulrich ideal for $1\le j\le\frac{n+1}{2}$. Therefore ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x,y),(x,y^2),\dots,(x,y^{\frac{n-1}{2}}),(x,y^{\frac{n+1}{2}})\}$.
\(3) The case $({\mathrm{D}}_n)$ with $n$ odd:
We adopt the same notation as in [@Y (9.11)], except that we use $A'$ instead of $A$ there. By [@Y Figure (9.11.5)] we have the inclusion relation $${\mathcal{C}}_A\subseteq\{A'\oplus B,X_1\oplus Y_1,M_1\oplus N_1,X_2\oplus Y_2,\dots,M_\frac{n-3}{2}\oplus N_\frac{n-3}{2},X_\frac{n-1}{2}\}.$$ Taking into account the minimal number of generators, we observe that ${\mathcal{C}}_A=\{A'\oplus B,X_\frac{n-1}{2}\}$. Since $(0):y=(x^2+y^{n-2})$, $(0):(x^2+y^{n-2})=(y)$ and $(x^2+y^{n-2})\cap (y)=(0)$, we have $$A'\oplus B=A/(y)\oplus A/(x^2+y^{n-2})\cong (x^2+y^{n-2})\oplus (y)=(x^2+y^{n-2},y)=(x^2,y).$$ As $X_\frac{n-1}{2}\cong Y_\frac{n-1}{2}\cong (x,y^\frac{n-1}{2})$, we get ${\mathcal{C}}_A=\{(x^2,y),(x,y^\frac{n-1}{2})\}$. Put $I=(x^2,y)\supseteq Q=(x^2-y)$. Then $QI=(x^4+y^{n-1},y^2(1+y^{n-3}))=(x^4,y^2)=I^2$, since $1+y^{n-3}\in A$ is a unit as $n\ge4$. We see that $A/Q$ is Artinian, whence $Q$ is a parameter ideal of $A$. It is straightforward that $Q:I=I$ holds, and Corollary \[2.6\] shows that $(x^2,y)$ is an Ulrich ideal. Also, using Corollary \[2.6\] for $I:=(x,y^\frac{n-1}{2})\supseteq Q:=(x)$, we observe that $(x,y^\frac{n-1}{2})$ is an Ulrich ideal. Thus, we obtain ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x^2,y),(x,y^\frac{n-1}{2})\}$.
\(4) The case $(D_n)$ with $n$ even:
We adopt the same notation as in [@Y (9.12)], except that we use $A'$ instead of $A$ there. It follows from [@Y Figure (9.12.1)] that $${\mathcal{C}}_A\subseteq\{ A'\oplus B,X_1\oplus Y_1,M_1\oplus N_1,X_2\oplus Y_2,\dots,X_\frac{n-2}{2}\oplus Y_\frac{n-2}{2},C_+\oplus D_+,C_-\oplus D_-\}.$$ Restricting to the modules generated by at most two elements, we have ${\mathcal{C}}_A=\{ A\oplus B,C_+\oplus D_+,C_-\oplus D_-\}$. Similarly to (3), we get isomorphisms $A'\oplus B \cong(x^2,y)$ and $C_\pm\oplus D_\pm\cong(y^\frac{n}{2},x\mp\sqrt{-1}y^\frac{n-2}{2})$. Hence ${\mathcal{C}}_A=\{(x^2,y),(y^\frac{n}{2},x-\sqrt{-1}y^\frac{n-2}{2}),(y^\frac{n}{2},x+\sqrt{-1}y^\frac{n-2}{2})\}$. We have $(x^2,y)\in{\mathcal{X}}_A$ similarly to (3).
Let us consider the ideal $I=(y^\frac{n}{2},x-\sqrt{-1}y^\frac{n-2}{2})$. Set $Q=((x-\sqrt{-1}y^\frac{n-2}{2})+y(x+\sqrt{-1}y^\frac{n-2}{2}))$. To check that $I,Q$ satisfy the assumptions of Corollary \[2.6\], we apply the change of variables $x\mapsto\sqrt{-1}x,\ y\mapsto y$ and put $n=2m+2$ with $m\ge1$. We may assume: $$A=k[[x,y]]/(x^2y-y^{2m+1}),\ I=(y^{m+1},x-y^m),\ Q=((x-y^m)+y(x+y^m)).$$ Note that $xy^{m+1}-y^{2m+1}=\frac{1}{2}(x^2+y^{2m})y-y^{2m+1}=0$ in the residue ring $A/(x-y^m)^2$. Hence $I^2=(y^{2m+2},(x-y^m)^2)$ and $QI=((x-y^m)y^{m+1}+y^{m+2}(x+y^m),(x-y^m)^2)=(2y^{2m+2},(x-y^m)^2)$, from which $I^2=QI$ follows. Clearly, $I$ contains $Q$. We have $$\begin{aligned}
A/Q & = k[[x,y]]/(x^2y-y^{2m+1},(1+y)x-(1-y)y^m) \\
& = k[[x,y]]/(((1+y)x)^2y-(1+y)^2y^{2m+1},(1+y)x-(1-y)y^m) \\
& = k[[x,y]]/(((1-y)y^m)^2y-(1+y)^2y^{2m+1},x-(1+y)^{-1}(1-y)y^m) \\
& = k[[x,y]]/(-4y^{2m+2},x-(1+y)^{-1}(1-y)y^m) \cong k[[y]]/(y^{2m+2}).\end{aligned}$$ This especially says that $Q$ is a parameter ideal, and the isomorphism corresponds $I/Q=y(y^m,x)A/Q$ to $y^{m+1}k[[y]]/(y^{2m+2})$. Hence $(Q:_AI)/Q=(0:_{A/Q}I/Q)=I/Q$, and therefore $Q:I=I$. Now we can apply Corollary \[2.6\], and see that $I$ is an Ulrich ideal.
The change of variables $x\mapsto-x,\ y\mapsto y$ shows that $(y^\frac{n}{2},x+\sqrt{-1}y^\frac{n-2}{2})$ is an Ulrich ideal. Thus ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x^2,y),(y^\frac{n}{2},x-\sqrt{-1}y^\frac{n-2}{2}),(y^\frac{n}{2},x+\sqrt{-1}y^\frac{n-2}{2})\}$.
\(5) The case $({\mathrm{E}}_6)$:
We adopt the same notation as in [@Y (9.13)], except that we use $A'$ instead of $A$ there. By [@Y Figure (9.13.1)] we have $${\mathcal{C}}_A\subseteq\{M_2,X,A'\oplus B,M_1\oplus N_1\}.$$ We observe that $\mu_A(A'\oplus B)=6$, $\mu_A(M_1\oplus N_1)=\mu_A(X)=4$ and $M_2\cong (x^2,y^2)\cong(x^3,xy^2)=(y^4,xy^2)\cong(x,y^2)$. Hence ${\mathcal{C}}_A=\{ M_2\}=\{ (x,y^2)\}$. Applying Corollary \[2.6\] to $I=(x,y^2)$ and $Q=(x)$, we get ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x,y^2)\}$.
\(6) The case $({\mathrm{E}}_7)$:
We adopt the same notation as in [@Y (9.14)], except that we use $A'$ instead of $A$ there. According to [@Y Figure (9.14.1)], $${\mathcal{C}}_A\subseteq\{ A'\oplus B,C\oplus D,M_1\oplus N_1,M_2\oplus N_2,X_1\oplus Y_1,X_2\oplus Y_2,X_3\oplus Y_3\}$$ holds. We see that $\mu_A(C\oplus D)=\mu_A(M_1\oplus N_1)=\mu_A(M_2\oplus N_2)=4$, $\mu_A(X_1\oplus Y_1)=\mu_A(X_2\oplus Y_2)=6$, $\mu_A(X_3\oplus Y_3)=8$ and $A'\oplus B\cong(x,y^3)$. Hence ${\mathcal{C}}_A=\{(x,y^3)\}$. Using Corollary \[2.6\] for $I=(x,y^3)$ and $Q=(x-y^3)$, we get $I\in\X_A$. Therefore ${\mathcal{X}}_A={\mathcal{C}}_A=\{(x,y^3)\}$.
\(7) The case $({\mathrm{E}}_8)$:
We use the same notation as in [@Y (9.15)]. By [@Y Figure (9.15.1)] we have $${\mathcal{C}}_A\subseteq\{ A_1\oplus B_1,A_2\oplus B_2,C_1\oplus D_1,C_2\oplus D_2,M_1\oplus N_1,M_2\oplus N_2,X_1\oplus Y_1,X_2\oplus Y_2\}.$$ We have $\mu_A(M_i\oplus N_i)=4$, $\mu_A(A_i\oplus B_i)=6$ and $\mu_A(C_i\oplus D_i)=8$ for $i=1,2$, and have $\mu_A(X_1\oplus Y_1)=12$ and $\mu_A(X_2\oplus Y_2)=10$. Consequently, we get ${\mathcal{X}}_A={\mathcal{C}}_A=\emptyset$.
The proof of Theorem \[8.9\] yields the following result.
\[8.10\] Let $A$ be a $1$-dimensional complete equicharacteristic Gorenstein local ring with algebraically closed residue field of characteristic $0$. If $A$ has finite CM-representation type, then one has ${\mathcal{X}}_A={\mathcal{C}}_A$.
Without the assumption that $A$ has finite CM-representation type, the equality in Corollary \[8.10\] does not necessarily hold true even if $A$ is a $1$-dimensional complete intersection (cf. Remark \[2.5\]).
Ulrich ideals and modules of two-dimensional rational singularities
===================================================================
In this section, we always assume that $A$ is a two-dimensional excellent rational singularity with unique maximal ideal $\m$ containing an algebraically closed field $k$ of characteristic $0$, unless otherwise specified. (Many results in this section hold true if $k$ is an algebraically closed field of positive characteristic. For simplicity, we assume that $k$ has characteristic $0$.) In particular, $A$ is an excellent normal local domain and there exists a resolution of singularities $X$ of $\Spec A$ with $\H^1(X,\mathcal{O}_X)=0$; see [@Li1; @Li2]. The main purpose of this section is to consider the following problems$:$
1. Classify all Ulrich ideals $I$ of $A$.
2. Let $I$ be an $\m$-primary ideal of $A$. Classify all Ulrich $A$-modules with respect to $I$.
3. Let $I$ be an $\m$-primary ideal of $A$. Suppose that there exists an Ulrich $A$-module with respect to $I$. Then determine the structure of $I$.
As an answer to the first problem, we give a complete list of Ulrich ideals of any two-dimensional rational double point in terms of (anti-nef) cycles. Moreover, we show that any Ulrich ideal $I$ is *good*, that is, $I$ is integrally closed and is represented by the minimal resolution of singularities (Notice that this definition is different from that of Corollary \[2.6\]. But Lemma \[GIW-good\] justifies it. See also [@GIW] or [@WY]). Furthermore, we give a criterion for good ideals $I$ to be Ulrich. As an application, we show that any Ulrich ideal of a two-dimensinoal rational singularity is a Gorenstein ideal.
As for the second problem, we give a complete answer in the case of rational double points using the McKay correspondence. As for the third problem, we show that $I$ is an Ulrich ideal if and only if there exists an Ulrich $A$-module with respect to $I$ in the case of rational double points. Notice that these problems remain still open for non-Gorenstein cases.
In what follows, let $A$ be a two-dimensional excellent rational singularity as above. Let $\varphi \colon X \to \Spec A$ be a resolution of singularities with $E=\varphi^{-1}({\mathfrak{m}}) = \bigcup_{i=1}^r E_i$. In the set $\mathcal{C}$ of cycles supported on $E$, we define a partial order $\le$ as follows: for $Z$, $Z' \in \mathcal{C}$, $Z \le Z'$ if every coefficient of $E_i$ in $Z'-Z$ is nonnegative. A cycle $Z=\sum_{i=1}^r a_i E_i$ is called *effective* if $a_i$ is a non-negative integer for every $i$.
Let $X \to \Spec A$ be a resolution of singularities of a rational singularity $A$ with exceptional divisors $E=\bigcup_{i=1}^r E_i$. Then since the intersection matrix $[E_i E_j]_{1 \le i,j \le r}$ is negative definite, there exists the unique $\mathbb{Q}$-divisor $\K_X$, the *canonical divisor*, such that the arithmetic genus $$p_a(E_i):=\dfrac{E_i^2+\K_X E_i}{2} + 1
$$ is equal to zero. If $E_i^2=\K_X E_i=-1$, then $E_i \cong \mathbb{P}^1$ is called a *$(-1)$-curve*. We say that $X$ is a *minimal resolution* if $X$ contains no $(-1)$-curve. Such a resolution is essentially unique.
Let $I$ be an $\m$-primary ideal of $A$. Then $I$ is said to be *represented* on $X$ if the sheaf $\mathcal{O}_X$ is invertible, that is, there exists an anti-nef cycle $Z$ with supported in $E$ such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ and $I=\H^0(X,\mathcal{O}_X(-Z))$. The product of two integrally closed ideals of $A$ is also integrally closed.
Now we recall the notion of good ideals of rational singularities Let $I$ be an $\m$-primary ideal of $A$. Then $I$ is called *good* if $I$ is represented on the minimal resolution of singularities; see also Lemma \[GIW-good\]. There is a one-to-one correspondence between the set of integrally closed $\m$-primary ideals of $A$ that are represented on $X$ and the set of anti-nef cycles $Z=\sum_{i=1}^{r} a_iE_i$ on $X$.
The following fact is well-known.
\[good-known\] Let $A$ be a two-dimensional $($not necessarily Gorenstein$)$ rational singularity, and $\varphi \colon X \to \Spec A$ denotes the minimal resolution of singularities. Then$:$
1. The minimum element $($say, $Z_0$$)$ among all non-zero anti-nef cycles on $X$ exists. This cycle $Z_0$ is called the *fundamental cycle* on $X$ which corresponds to the maximal ideal $\m$. In particular, $\m = \H^0(X,\mathcal{O}_X(-Z_0))$ is a good ideal.
2. If $I=\H^0(X,\mathcal{O}_X(-Z)$ and $J = \H^0(X,\mathcal{O}_X(-Z'))$ are good ideals of $A$, then $IJ=\H^0(X,\mathcal{O}_X(-(Z+Z'))$ is also a good ideal.
3. If $I=\H^0(X,\mathcal{O}_X(-Z))$, then $\e_I^0(A)=-Z^2$.
The colength $\ell_A(A/I)$ can also be determined by the anti-nef cycle $Z$; see Kato’s Riemann-Roch formula (see Lemma \[RR\]).
Ulrich ideals of rational double points
---------------------------------------
The first main goal of this subsection is to give a complete classfication of Ulrich ideals of any two-dimensional Gorenstein rational singularity (rational double point) $A$ and determine all of the Ulrich $A$-modules with respect to those ideals. The main tools are the McKay correspondence and Kato’s Riemann-Roch formula.
First we recall the definition of rational double points.
\[RDP-def\] Let $A$ be a two-dimensional complete Noetherian local ring with unique maximal ideal ${\mathfrak{m}}$ containing an algebraically closed field $k$. Then $A$ is said to be a *rational double point* if it is isomorphic to the hypersurface $k[[x,y,z]]/(f)$, where $f$ is one of the following polynomials$:$ $$\begin{array}{cll}
(A_n) & z^2 + x^2 + y^{n+1} & (n \ge 1), \\
(D_n) & z^2 + x^2y + y^{n-1} & (n \ge 4), \\
(E_6) & z^2+x^3+y^4, & \\
(E_7) & z^2+x^3+xy^3, & \\
(E_8) & z^2+x^3+y^5. &
\end{array}$$
Note that $A$ is a two-dimensional Gorenstein rational singularity (of characteristic $0$) if and only if the $\m$-adic completion $\widehat{A}$ is a rational double point in the above sense.
The following result plays a key role in this subsection.
\[GIW-good\] Suppose that $A$ is a two-dimensional rational double point. Let $I$ be an $\m$-primary ideal of $A$. Then the following conditions are equivalent$:$
1. $I$ is a good ideal.
2. $I$ is integrally closed and represented on the minimal resolution of singularities $\varphi \colon X \to \Spec A$, that is, there exists an anti-nef cycle $Z$ on $X$ such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ is invertible and $I=\H^0(X,\mathcal{O}_X(-Z))$.
Note that any Ulrich ideal of a Gorenstein local ring is good by Corollary \[2.6\]. We first characterize Ulrich ideals of $A$ in terms of anti-nef cycles using this lemma.
\[Uideal-cond\] Let $A$ be a two-dimensional rational double point, and let $I$ be an $\m$-primary ideal of $A$. Then the following conditions are equivalent$:$
1. $I$ is an Ulrich ideal of $A$ with $\mu_A(I) > 2$.
2. $I$ is integrally closed, and there exists an anti-nef cycle $Z$ on the minimal resolution of singularities $\varphi \colon X \to \Spec A$ such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$, $I=\H^0(X,\mathcal{O}_X(-Z))$ and $-ZZ_0=2$, where $Z_0$ denotes the fundamental cycle on $X$.
We may assume that $I$ is good. In particular, we can take an anti-nef cycle $Z$ on $X$ such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ and $I=\H^0(X,\mathcal{O}_X(-Z))$. Then it is enough to show that $\mu(I)=3$ if and only if $-ZZ_0=2$. Indeed, since $I$ and $I{\mathfrak{m}}$ are good by [@GIW Corollary 7.9], we have $$\begin{aligned}
\mu(I) & = & \ell_A(I/{\mathfrak{m}}I) = \ell_A(A/{\mathfrak{m}}I) - \ell_A(A/I)
= \frac{\e_{{\mathfrak{m}}I}^0(A)}{2} - \frac{\e_I^0(A)}{2} \\
&=& \frac{-(Z+Z_0)^2}{2} - \frac{-Z^2}{2}
= \frac{-Z_0^2}{2} -ZZ_0 = 1 - ZZ_0,\end{aligned}$$ where the last equality follows from $-Z_0^2 = \e_\m^0(A) = 2$.
\[Udieal-rem\] Note that $-ZZ_0=2$, that is, $-(Z-Z_0)Z_0=0$ means that every coefficient of $E_i$ in $Z - Z_0$ is zero for any $E_i$ with $E_i Z_0 < 0$.
Next, we characterize Ulrich $A$-modules with respect to a good ideal $I$. In order to achieve this, we recall the McKay correspondence, and Kato’s Riemann-Roch formula.
Let $A$ be a two-dimensional rational double point. Then $A$ can be regarded as an invariant subring $B^G$, where $B=k[[s,t]]$ is a formal power series ring over $k$, and $G$ is a finite subgroup of $\mathrm{SL}(2,k)$.
\[McKay\] Let $A=B^G$ as above. Then$:$
1. The ring $A$ is of finite CM–representation type. Let $\{M_i\}_{i=0}^r$ be the set of isomorphism classes of indecomposable maximal Cohen-Macaulay $A$-modules, where $M_0=A$. Then $B \cong \bigoplus_{i=0}^r M_i^{n_i}$, where $n_i = \rank_A M_i$.
2. The fundamental cycle is given by $Z_0=\sum_{i=1}^r n_i E_i$ so that if we choose indices suitably, then $c(\mathcal{M}_i)E_j = \delta_{ij}$ for $1 \le i,j \le r$, where $c(*)$ denotes the Chern class and $\mathcal{M}_i = f^{*}(M_i)/\text{torsion}$.
\[RR\] Let $A$ be as above, and let $I=\H^0(X,\mathcal{O}_X(-Z))$ be a good ideal. Then for any maximal Cohen-Macaulay $A$-module $M = \bigoplus_{i=0}^r M_i^{k_i}$, we have $$\ell_A(M/IM) = \rank_A M \cdot \ell_A(A/I) + \sum_{i=1}^r k_i \cdot
c(\mathcal{M}_i)Z.$$
Let $M$ be a maximal Cohen-Macaulay $A$-module. Suppose that $M'$ is a direct summand of $M$. If $M$ is an Ulrich $A$-module with respect to $I$, then so is $M'$. Therefore it suffices to characterize the $M_i$’s that are Ulrich $A$-modules in order to classify Ulrich $A$-modules with respect to $I$.
\[UlrichMod-RDP\] Let $A$ be a two-dimensional rational double point. Let $\{M_i\}_{i=0}^r$ be the set of indecomposable maximal Cohen-Macaulay $A$-modules such that $c(\mathcal{M}_i)E_j = \delta_{ij}$ for all $1 \le i,j \le r$. Assume that $I$ is a good ideal such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ and $I=\H^0(X,\mathcal{O}_X(-Z))$ and $Z=\sum_{j=1}^r a_j E_j$. Then the following conditions are equivalent$:$
1. $M_i$ is an Ulrich $A$-module with respect to $I$.
2. $a_i = n_i \cdot \ell_A(A/I)$.
By Kato’s Riemann-Roch formula (Lemma \[RR\]), we have $$\ell_A(M_i/IM_i)
= n_i \cdot \ell_A(A/I) + c(\mathcal{M}_i) Z
= n_i \cdot \ell_A(A/I) + a_i.$$ On the other hand, since $I$ is good, we have $$\e_I^0(M_i) = \e^0_I(A) \cdot \rank_A M_i = 2 \cdot \ell_A(A/I) \cdot n_i.$$ $(1) \Longrightarrow (2):$ If $M_i$ is an Ulrich $A$-module with respect to $I$, then $\ell_A(M_i/IM_i) = \e_I^0(M_i)$. Hence $a_i=n_i \cdot \ell_A(A/I)$.
$(2) \Longrightarrow (1):$ By the argument as above, we have $$\begin{aligned}
\ell_A(M_i/IM_i)
& = & \e_I^0(M_i) \\
& = & 2n_i \cdot \ell_A(A/I) \\
& = & \e_\m^0(A) \cdot n_i \cdot \ell_A(A/I) \\
& = & \e_{\mathfrak{m}}^0(M_i) \cdot \ell_A(A/I). \end{aligned}$$ Furthermore, since $M_i$ is an Ulrich $A$-module with respect to ${\mathfrak{m}}$, we get $$\ell_A(M_i/IM_i) = \mu_A(M_i) \cdot \ell_A(A/I).$$ This implies that $M_i/IM_i$ is a free $A/I$-module. Thus $M_i$ is an Ulrich $A$-module with respect to $I$.
\[Ineq-RDP\] Under the same notation as in Proposition \[UlrichMod-RDP\], we have $$a_i \le n_i \cdot \ell_A(A/I)$$ for every $i$ since $\e_I^0(M_i) \ge \ell_A(M_i/IM_i)$ holds true.
The following theorem is the main result in this section. Let $I$ be a good ideal of $A$ and let $Z$ be an anti-nef cycle on $X$ such that $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ and $I=\H^0(X,\mathcal{O}_X(-Z))$. Then we call $Z$ an *Ulrich cycle* if $I$ is an Ulrich ideal.
Now let us illustrate the main theorem by the following example. Let $Z=2E_1+2E_2+3E_3+4E_4+3E_5+2E_6$ be an Ulrich cycle of a rational double point $A=k[[x,y,z]]/(x^3+y^4+z^2)$, and put $I=\H^0(X,\mathcal{O}_X(-Z))$. Then since $Z$ is an anti-nef cycle on the minimal resolution $X \to \Spec A$ with supported on $E=\bigcup_{i=1}^6 E_i$, $Z$ can be described as the following$:$
(400,35)(-20,0) (-10,10)[$Z=$]{} (25,18)[[$2$]{}]{} (25,0)[[$E_2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$3$]{}]{} (55,0)[[$E_3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$4$]{}]{} (83,0)[[$E_4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$3$]{}]{} (115,0)[[$E_5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$2$]{}]{} (145,0)[[$E_6$]{}]{} (150,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (95,26)[[$E_1$]{}]{} (90,34)
Furthermore, the following picture means that the corresponding Ulrich $A$-module $M$ with respect to $I$ is given by $M \cong M_2^{a} \oplus M_6^{b}$ for some integers $a,b \ge 0$.
(400,35)(-20,0) (-10,10)[$Z=$]{} (25,18)[[$2$]{}]{} (25,0)[[$E_2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$3$]{}]{} (55,0)[[$E_3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$4$]{}]{} (83,0)[[$E_4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$3$]{}]{} (115,0)[[$E_5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$2$]{}]{} (145,0)[[$E_6$]{}]{} (150,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (95,26)[[$E_1$]{}]{} (90,34)
We are now ready to state the main theorem in this section.
\[Main-RDP\] Let $A$ is a two-dimensional rational double point. Let $\varphi \colon X \to \Spec A$ be the minimal resolution of singularities with $E=\varphi^{-1}({\mathfrak{m}}) = \bigcup_{i=1}^r E_i$, the exceptional divisor on $X$. Then all Ulrich cycles $Z_k$ of $A$ and all indecomposable Ulrich $A$-modules with respect to $I_k=\H^0(X,\mathcal{O}_X(-Z_k))$ are given by the following$:$
$\bullet$ $(A_n)$ $x^2 + y^{n+1}+z^2$
When $n=2m$, the complete list of all Ulrich cycles is given by the following$:$
(400,20)(-20,0) (-10,10)[$Z_k=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[15]{}]{} (84,10)[$\cdots$]{} (100,12)[(1,0)[15]{}]{} (115,18)[[$k$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (140,18)[[$k+1$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (170,18)[[$k+1$]{}]{} (180,12) (185,12)[(1,0)[15]{}]{} (204,10)[$\cdots$]{} (220,12)[(1,0)[15]{}]{} (230,18)[[$k+1$]{}]{} (240,12) (245,12)[(1,0)[20]{}]{} (260,18)[[$k+1$]{}]{} (270,12) (275,12)[(1,0)[20]{}]{} (295,18)[[$k$]{}]{} (300,12) (305,12)[(1,0)[15]{}]{} (324,10)[$\cdots$]{} (340,12)[(1,0)[15]{}]{} (355,18)[[$2$]{}]{} (360,12) (365,12)[(1,0)[20]{}]{} (385,18)[[$1$]{}]{} (390,12) (148,6)[$\underbrace{\phantom{aaaaaaaaaaaaaaaaaaaa}}$]{} (198,-10)[[$n-2k$]{}]{}
for $k=0,1,\cdots,m-1(=\frac{n}{2}-1)$. Then $\ell_A(A/I_k)=k+1$ for each $k=0,1,\ldots,m-1$.
When $n=2m+1$, the complete list of all Ulrich cycles is given by the following$:$
(400,20)(-20,0) (-10,10)[$Z_k=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[15]{}]{} (84,10)[$\cdots$]{} (100,12)[(1,0)[15]{}]{} (115,18)[[$k$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (140,18)[[$k+1$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (170,18)[[$k+1$]{}]{} (180,12) (185,12)[(1,0)[15]{}]{} (204,10)[$\cdots$]{} (220,12)[(1,0)[15]{}]{} (230,18)[[$k+1$]{}]{} (240,12) (245,12)[(1,0)[20]{}]{} (260,18)[[$k+1$]{}]{} (270,12) (275,12)[(1,0)[20]{}]{} (295,18)[[$k$]{}]{} (300,12) (305,12)[(1,0)[15]{}]{} (324,10)[$\cdots$]{} (340,12)[(1,0)[15]{}]{} (355,18)[[$2$]{}]{} (360,12) (365,12)[(1,0)[20]{}]{} (385,18)[[$1$]{}]{} (390,12) (148,6)[$\underbrace{\phantom{aaaaaaaaaaaaaaaaaaaa}}$]{} (198,-10)[[$n-2k$]{}]{}
for $k=0,1,\cdots,m(=\frac{n-1}{2})$. Then $\ell_A(A/I_k)=k+1$ for each $k=0,1,\ldots,m$.
$\bullet$ $(D_n)$ $x^2y + y^{n-1}+z^2$ $(n \ge 4)$
When $n=2m$, the complete list of all Ulrich cycles is given by the following$:$
(400,30)(-20,0) (-10,10)[$Z_k=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (85,18)[[$3$]{}]{} (90,12) (95,12)[(1,0)[15]{}]{} (114,10)[$\cdots$]{} (130,12)[(1,0)[15]{}]{} (140,18)[[$2k+2$]{}]{} (150,12) (155,12)[(1,0)[15]{}]{} (174,10)[$\cdots$]{} (190,12)[(1,0)[15]{}]{} (195,18)[[$2k+2$]{}]{} (210,12) (215,14)[(1,1)[11]{}]{} (215,10)[(1,-1)[11]{}]{} (230,26) (230,-2) (226,32)[[$k+1$]{}]{} (226,4)[[$k+1$]{}]{} (145,6)[$\underbrace{\phantom{aaaaaaaaaaa}}$]{} (158,-10)[[$n-2k-3$]{}]{}
for $k=0,1,\ldots,m-2(=\frac{n-4}{2})$.
(400,30)(-20,0) (-10,10)[$Z_{m-1}=$]{} (35,18)[[$1$]{}]{} (40,12) (45,12)[(1,0)[20]{}]{} (65,18)[[$2$]{}]{} (70,12) (75,12)[(1,0)[20]{}]{} (95,18)[[$3$]{}]{} (100,12) (105,12)[(1,0)[15]{}]{} (124,10)[$\cdots$]{} (140,12)[(1,0)[15]{}]{} (145,18)[[$2m-3$]{}]{} (160,12) (165,12)[(1,0)[25]{}]{} (177,18)[[$2m-2$]{}]{} (195,12) (200,14)[(1,1)[11]{}]{} (200,10)[(1,-1)[11]{}]{} (215,26) (215,-2) (211,32)[[$m-1$]{}]{} (211,4)[[$m$]{}]{}
(400,30)(-20,0) (-10,10)[$Z_{m}=$]{} (35,18)[[$1$]{}]{} (40,12) (45,12)[(1,0)[20]{}]{} (65,18)[[$2$]{}]{} (70,12) (75,12)[(1,0)[20]{}]{} (95,18)[[$3$]{}]{} (100,12) (105,12)[(1,0)[15]{}]{} (124,10)[$\cdots$]{} (140,12)[(1,0)[15]{}]{} (145,18)[[$2m-3$]{}]{} (160,12) (165,12)[(1,0)[25]{}]{} (177,18)[[$2m-2$]{}]{} (195,12) (200,14)[(1,1)[11]{}]{} (200,10)[(1,-1)[11]{}]{} (215,26) (215,-2) (211,32)[[$m$]{}]{} (211,4)[[$m-1$]{}]{}
(400,30)(-20,0) (-10,10)[$Z_{m+1}=$]{} (35,18)[[$2$]{}]{} (40,12) (45,12)[(1,0)[20]{}]{} (65,18)[[$2$]{}]{} (70,12) (75,12)[(1,0)[20]{}]{} (95,18)[[$2$]{}]{} (100,12) (105,12)[(1,0)[15]{}]{} (124,10)[$\cdots$]{} (140,12)[(1,0)[15]{}]{} (155,18)[[$2$]{}]{} (160,12) (165,12)[(1,0)[25]{}]{} (187,18)[[$2$]{}]{} (195,12) (200,14)[(1,1)[11]{}]{} (200,10)[(1,-1)[11]{}]{} (215,26) (215,-2) (211,32)[[$1$]{}]{} (211,4)[[$1$]{}]{}
Then $\ell_A(A/I_k)=k+1$ for each $k=0,1,\ldots,m-2$, $\ell_A(A/I_{m-1})=\ell_A(A/I_{m})=m$ and $\ell_A(A/I_{m+1})=2$, where $m=\frac{n}{2}$.
When $n=2m+1$, the complete list of all Ulrich cycles is given by the following$:$
(400,30)(-20,0) (-10,10)[$Z_k=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (85,18)[[$3$]{}]{} (90,12) (95,12)[(1,0)[15]{}]{} (114,10)[$\cdots$]{} (130,12)[(1,0)[15]{}]{} (140,18)[[$2k+2$]{}]{} (150,12) (155,12)[(1,0)[15]{}]{} (174,10)[$\cdots$]{} (190,12)[(1,0)[15]{}]{} (195,18)[[$2k+2$]{}]{} (210,12) (215,14)[(1,1)[11]{}]{} (215,10)[(1,-1)[11]{}]{} (230,26) (230,-2) (226,32)[[$k+1$]{}]{} (226,4)[[$k+1$]{}]{} (145,6)[$\underbrace{\phantom{aaaaaaaaaaa}}$]{} (158,-10)[[$n-2k-3$]{}]{}
for $k=0,1,\ldots,m(=\frac{n-1}{2})$.
(400,30)(-20,0) (-14,10)[$Z_{m-1}=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (85,18)[[$3$]{}]{} (90,12) (95,12)[(1,0)[15]{}]{} (114,10)[$\cdots$]{} (130,12)[(1,0)[15]{}]{} (150,12) (155,12)[(1,0)[15]{}]{} (160,18)[[$2m-2$]{}]{} (175,12) (180,12)[(1,0)[25]{}]{} (194,18)[[$2m-1$]{}]{} (210,12) (215,14)[(1,1)[11]{}]{} (215,10)[(1,-1)[11]{}]{} (230,26) (230,-2) (226,32)[[$m$]{}]{} (226,4)[[$m$]{}]{}
(400,30)(-20,0) (-10,10)[$Z_{m}=$]{} (35,18)[[$2$]{}]{} (40,12) (45,12)[(1,0)[20]{}]{} (65,18)[[$2$]{}]{} (70,12) (75,12)[(1,0)[20]{}]{} (95,18)[[$2$]{}]{} (100,12) (105,12)[(1,0)[15]{}]{} (124,10)[$\cdots$]{} (140,12)[(1,0)[15]{}]{} (155,18)[[$2$]{}]{} (160,12) (165,12)[(1,0)[25]{}]{} (187,18)[[$2$]{}]{} (195,12) (200,14)[(1,1)[11]{}]{} (200,10)[(1,-1)[11]{}]{} (215,26) (215,-2) (211,32)[[$1$]{}]{} (211,4)[[$1$]{}]{}
Then $\ell_A(A/I_k)=k+1$ for each $k=0,1,\ldots,m-2$, $\ell_A(A/I_{m-1})=m$ and $\ell_A(A/I_{m})=2$.
$\bullet$ $(E_6)$ $x^3 + y^4+z^2$
All Ulrich cycles of $A$ are the following $Z_0$ and $Z_1$:
(200,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$3$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$2$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$1$]{}]{} (150,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (90,34)
(200,35)(-20,0) (-10,10)[$Z_1=$]{} (25,18)[[$2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$3$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$2$]{}]{} (150,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (90,34)
Then $\ell_A(A/I_{k})=k+1$ for each $k=0,1$.
$\bullet$ $(E_7)$ $x^3 + xy^3+z^2$
All Ulrich cycles of $A$ are the following $Z_0$, $Z_1$ and $Z_2$:
(300,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$3$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$2$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$1$]{}]{} (180,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (90,34)
(200,35)(-20,0) (-10,10)[$Z_1=$]{} (25,18)[[$2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$4$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$6$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$4$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$2$]{}]{} (180,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$3$]{}]{} (90,34)
(200,35)(-40,0) (-10,10)[$Z_2=$]{} (25,18)[[$2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$4$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$6$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$4$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$3$]{}]{} (180,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$3$]{}]{} (90,34)
Then $\ell_A(A/I_{k})=k+1$ for each $k=0,1,2$.
$\bullet$ $(E_8)$ $x^3 + y^5+z^2$
All Ulrich cycles of $A$ are the following $Z_0$ and $Z_1$:
(400,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$4$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$6$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$4$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$3$]{}]{} (180,12) (185,12)[(1,0)[20]{}]{} (205,18)[[$2$]{}]{} (210,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$3$]{}]{} (90,34)
(400,35)(-20,0) (-10,10)[$Z_1=$]{} (25,18)[[$4$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$7$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (80,18)[[$10$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$8$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$6$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$4$]{}]{} (180,12) (185,12)[(1,0)[20]{}]{} (205,18)[[$2$]{}]{} (210,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$5$]{}]{} (90,34)
Then $\ell_A(A/I_{k})=k+1$ for each $k=0,1$.
In the previous section, we give a complete list of the nonparameter Ulrich ideals for one-dimensional simple singularities. We can also do it for two-dimensional simple singularities (rational double points). Compare the result below with Theorem \[8.9\].
\[RDP-Uideal\] With the same notation as in Theorem $\ref{Main-RDP}$, the set ${\mathcal{X}}_A$ is equal to$:$
----------------- ------------------------------------------------------------ ---------------------
$(A_n)$ $\{(x,y,z),(x,y^2,z),\ldots,(x,y^m,z)\}$ if $n=2m$ is even;
$\{(x,y,z),(x,y^2,z),\ldots,(x,y^{m+1},z)\}$ if $n=2m+1$ is odd.
\[2mm\] $(D_n)$ $\{(x,y,z),(x,y^2,z),\ldots,(x,y^{m-1},z),$
$(x+\sqrt{-1}y^{m-1},y^{m},z),(x-\sqrt{-1}y^{m-1},y^m,z)$,
$(x^2,y,z)\}$ if $n=2m$ is even;
$\{(x,y,z),(x,y^2,z),\ldots,(x,y^{m},z),(x^2,y,z)\}$ if $n=2m+1$ is odd.
\[2mm\] $(E_6)$ $\{(x,y,z),(x,y^2,z) \}$.
\[2mm\] $(E_7)$ $\{(x,y,z),(x,y^2,z),(x,y^3,z)\}$.
\[2mm\] $(E_8)$ $\{(x,y,z),(x,y^2,z) \}$.
----------------- ------------------------------------------------------------ ---------------------
One can easily see that any ideal $I$ as above can be written as the form $I=Q+(z)$, where $Q$ is a parameter ideal of $A$ and $I^2 =QI$, $\ell_A(A/Q)= 2 \cdot \ell_A(A/I)$ and $\mu(I)=3$. Hence those ideals $I$ are Ulrich.
On the other hand, Theorem \[Main-RDP\] implies that $\sharp{\mathcal{X}}_A=m$ (resp. $m+1$, $m+2$, $m+1$, $2$, $3$ ,$2$) if $A$ is a rational double point of type $(A_{2m})$ (resp. $(A_{2m+1})$, $(D_{2m})$, $(D_{2m+1})$, $(E_6)$, $(E_7)$, $(E_8)$). Hence the set as above coincides with ${\mathcal{X}}_A$, respectively.
In what follows, we prove Theorem \[Main-RDP\]. The following lemma is elementary, but it plays the central role in the proof of the theorem.
\[Fund-linear\] Consider the following linear inequalities $$\left\{
\begin{array}{rcl}
x_1 & = & 1, \\
x_2 & \le & 2x_1, \\
x_1+x_3 & \le & 2x_2, \\
x_2+x_4 & \le & 2x_3, \\
& \vdots & \\
x_{m-2} + x_m & \le & 2x_{m-1}.
\end{array}
\right.$$ Then all positive integer solutions are given by $$(x_1,\ldots,x_m) = (1,2,\ldots,k,k+1,\ldots,k+1) \quad
\text{for $k=0,1,\ldots,m-1$}.$$
We prove this by induction on $m$. When $m=1$, it is clear. When $m=2$, $1 \le x_2 \le 2x_1 =2$ implies that $x_2=1$ or $2$. Hence the required solutions are $(x_1,x_2)=(1,1),(1,2)$.
Now suppose that $m \ge 2$ and the assertion holds true for $m-1$. Assume that all $x_i$ are positive integers. Then the first inequality $1 \le x_2 \le 2x_1 =2$ implies that $x_2=1$ or $2$. Suppose $x_2=1$. Then $x_3 \le 2x_2 - x_1 = 1$. Hence $x_3=1$. Similarly we obtain that $x_4=\cdots = x_m=1$.
Next suppose that $x_2=2$. Then we may assume that $x_i \ge 2$ for every $i \ge 2$ by the argument as above. Indeed, if $x_i=1$ for some $2 \le i \le m$, then $x_{i-1}=1$. By repeating this process, we obtain that $x_2=1$. If we put $x_i'=x_{i+1}-1$ for every $i=1,\ldots,m-1$, then we get the following linear inequalities: $$\left\{
\begin{array}{rcl}
x_1' & = & 1, \\
x_2' & \le & 2x_1', \\
x_1'+x_3' & \le & 2x_2', \\
x_2'+x_4' & \le & 2x_3', \\
& \vdots & \\
x_{m-3}' + x_{m-1}' & \le & 2x_{m-2}'.
\end{array}
\right.$$ Since $x_i'$ are positive integers, we have $(x_1',\ldots,x_{m-1}') = (1,2,\ldots,k-1,k,\ldots,k)$ for some $k$ with $1 \le k \le m-1$ by the induction hypothesis. This meams that $(x_1,\ldots,x_m) = (1,2,\ldots,k,k+1,\ldots,k+1)$ for some $k = 1,\ldots,m$, as required.
Let $\varphi \colon X \to \Spec A$ be the minimal resolution of singularities. Since any Ulrich ideal $I$ is good, we can find a suitable anti-nef cycle $Z$ on $X$ such that $I\mathcal{O}_X$ is invertible and $I=\H^0(X,\mathcal{O}_X(-Z))$.
Case 1
: $(A_n)$ $f= x^2+y^{n+1}+z^2$.
(400,20)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$1$]{}]{} (27,0)[[$E_1$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$1$]{}]{} (57,0)[[$E_2$]{}]{} (60,12) (65,12)[(1,0)[15]{}]{} (84,10)[$\cdots$]{} (100,12)[(1,0)[15]{}]{} (115,18)[[$1$]{}]{} (117,0)[[$E_{n-1}$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (147,0)[[$E_{n}$]{}]{} (145,18)[[$1$]{}]{} (150,12)
Then $Z_0 = \sum_{i=1}^n E_i$ is the fundamental cycle on $X$.
Now suppose that $n=2m$ and $Z=\sum_{i=1}^{n} a_iE_i$ is an Ulrich cycle. Then since $Z_0E_1 = Z_0E_n =-1$ and $Z_0E_i = 0$ for all $2 \le i \le n-1$, we have $a_1=a_n =1$ by Proposition \[Uideal-cond\]. Moreover, as $Z$ is anti-nef, we get two linear system of inequalities: $$\left\{
\begin{array}{rcl}
a_1 & = & 1, \\
a_2 & \le & 2a_1, \\
a_1+a_3 & \le & 2a_2, \\
a_2+a_4 & \le & 2a_3, \\
& \vdots & \\
a_{m-2} + a_{m} & \le & 2a_{m-1}, \\
a_{m-1} + a_{m+1} & \le & 2a_{m}.
\end{array}
\right.
\qquad
\left\{
\begin{array}{rcl}
a_n & = & 1, \\
a_{n-1} & \le & 2a_n, \\
a_n+a_{n-2} & \le & 2a_{n-1}, \\
a_{n-1}+a_{n-3} & \le & 2a_{n-2}, \\
& \vdots & \\
a_{m+3} + a_{m+1} & \le & 2a_{m+2}, \\
a_{m+2} + a_{m} & \le & 2a_{m+1}.
\end{array}
\right.$$ By Lemma \[Fund-linear\], we get $$\begin{aligned}
(a_1,\ldots,a_m,a_{m+1}) &=& (1,2,\ldots,k,k+1,\ldots,k+1), \\
(a_m,a_{m+1},\ldots,a_n) &= & (\ell+1,\ldots,\ell+1,\ell,\ldots, 2,1)\end{aligned}$$ for some $1 \le k, \ell \le m-1$. Then we have $k=\ell$ and $Z=Z_k$. If we put $I_k = \H^0(X,\mathcal{O}_X(-Z_k))$, then one can easily see that $\ell_A(A/I_k) = \frac{-Z_k^2}{2} = k+1$. The other assertion follows from Proposition \[UlrichMod-RDP\].
Similarly, we can prove the assertion in the case of $n=2m+1$.
Case 2
: $(D_n)$ $f= x^2y+y^{n-1}+z^2$.
Now suppose that $n=2m$ and that $Z=\sum_{i=1}^n a_iE_i$ is an Ulrich cycle on $X$. Then $Z_0=E_1 + 2 \sum_{i=2}^{2m-2} E_i + E_{2m-1}+E_{2m}$ is the fundamental cycle on $X$.
(400,30)(-20,0) (10,10)[$Z_0=$]{} (45,18)[[$1$]{}]{} (45,0)[[$E_1$]{}]{} (50,12) (55,12)[(1,0)[20]{}]{} (75,18)[[$2$]{}]{} (75,0)[[$E_2$]{}]{} (80,12) (85,12)[(1,0)[20]{}]{} (105,18)[[$2$]{}]{} (105,0)[[$E_3$]{}]{} (110,12) (115,12)[(1,0)[15]{}]{} (134,10)[$\cdots$]{} (155,12)[(1,0)[15]{}]{} (175,18)[[$2$]{}]{} (160,0)[[$E_{2m-3}$]{}]{} (190,0)[[$E_{2m-2}$]{}]{} (175,12) (180,12)[(1,0)[25]{}]{} (205,18)[[$2$]{}]{} (210,12) (215,14)[(1,1)[11]{}]{} (215,10)[(1,-1)[11]{}]{} (230,26) (230,-2) (226,32)[[$1$]{}]{} (236,18)[[$E_{2m-1}$]{}]{} (226,4)[[$1$]{}]{} (236,-10)[[$E_{2m}$]{}]{}
Since $Z_0 E_2 = -1$ and $Z_0E_i=0$ for every $i\ne 2$, we obtain that $a_2=2$ by Proposition \[Uideal-cond\]. Moreover, as $Z$ is anti-nef, we have the following linear inequalities$:$ $$\left\{
\begin{array}{rcl}
2=a_2 & \le & 2a_1, \\
a_1+a_3 & \le & 2a_2, \\
& \vdots & \\
a_{2m-4} + a_{2m-2} & \le & 2a_{2m-3},
\end{array}
\right.
\qquad
\left\{
\begin{array}{rcl}
a_{2m-3} +a_{2m-1} + a_{2m} & \le & 2a_{2m-2}, \\
a_{2m-2} & \le & 2a_{2m-1}, \\
a_{2m-2} & \le & 2a_{2m}.
\end{array}
\right.$$ If $m=2$, then $(a_1,a_2,a_3,a_4)=
(1,2,1,1)$, $(2,2,1,1)$, $(1,2,2,1)$ or $(1,2,1,2)$. So we may assume that $m \ge 3$. Then $a_1 +a_3 \le 2a_2 =4$ and $a_3 \ge 2$ imply that $a_1 \le 2$, and so $a_1=1$ or $2$.
First we consider the case of $a_1=1$. Then Lemma \[Fund-linear\] yields that $$(a_1,\ldots,a_{2m-2})=(1,2,\ldots,\ell-1,\ell,\ldots,\ell)$$ for some $2 \le \ell \le 2m-2$. If, in addition, $\ell < 2m-2$, then the last three inequalities imply that $a_{2m-1} + a_{2m} \le \ell$, $2 a_{2m-1} \ge \ell$ and $2a_{2m} \ge \ell$. Therefore $\ell$ is even and $\ell = 2k+2$ for some $0 \le k < \frac{n}{2}-2=m-2$. Then $a_{2m-1}=a_{2m} = k+1$. In other words, $Z=Z_k$ and $\ell_A(A/I_k)=k+1$, where $I_k=\H^0(X,\mathcal{O}_X(-Z_k))$ for each $k$.
Otherwise, $\ell=2m-2$. Then the last three inequalities imply that $a_{2m-1} + a_{2m} \le \ell+1=2m-1$, $2 a_{2m-1} \ge 2m-2$ and $2a_{2m} \ge 2m-2$. Therefore $(a_{2m-1},a_{2m})=(m-1,m-1),(m-1,m),(m,m-1)$. That is, $Z = Z_k$ for some $k = m-2,m-1,m$. Also, $\ell_A(A/I_{m-2})=m-1$ and $\ell_A(A/I_{m-1})=\ell_A(A/I_m) = m$, where $I_k = \H^0(X,\mathcal{O}_X(-Z_k))$ for each $k$.
Next, we consider the case of $a_1=2$. Then $a_1=a_2=\cdots =a_{2m-2} = 2$ by Lemma \[Fund-linear\]. Other inequalities imply that $a_{2m-1}=a_{2m} =1$. Then $Z=Z_{m+1}$ and $\ell_A(A/I_{m+1})=2$.
On the other hand, the assertion for Ulrich modules with respect to $I_k$ immediately follows from Proposition \[UlrichMod-RDP\].
Similarly, we can show the assertion in the case of $n=2m+1$.
Case 3
: $(E_6)$ $f= x^3 +y^4+z^2$.
(400,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$1$]{}]{} (25,0)[[$E_2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$2$]{}]{} (55,0)[[$E_3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$3$]{}]{} (83,0)[[$E_4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$2$]{}]{} (115,0)[[$E_5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$1$]{}]{} (145,0)[[$E_6$]{}]{} (150,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (95,26)[[$E_1$]{}]{} (90,34)
Then $Z_0=(2,1,2,3,2,1)=2E_1+E_2+2E_3+3E_4+2E_5+E_6$ is the fundamental cycle on $X$.
Now suppose that $Z=\sum_{i=1}^{6} a_iE_i$ is an Ulrich cycle. Since $Z_0E_1=-1$ and $Z_0E_i=0$ for every $i \ne 1$, we have $a_1=2$ by Proposition \[Uideal-cond\]. In particular, $I = \H^0(X,\mathcal{O}_X(-Z))$ is a simple good ideal with $a_1=2$. (Recall that a good ideal is called [*simple*]{} if it is not the product of two good ideals.) Then we can conclude that $Z=Z_0 = (2,1,2,3,2,1)$ or $Z=Z_1=(2,2,3,4,3,2)$ by [@WY Example 3.6]; see below. The assertion for Ulrich modules follows from the fact that $\ell_A(A/I_k)=k+1$, where $I_k = \H^0(X,\mathcal{O}_X(-Z_k))$ for $k=0,1$.
[**The list of simple good ideals of $(E_6)$**]{}
cycle $\ell_A(A/I)$ $\mu(I)$
----------------- --------------- ----------
$(2,1,2,3,2,1)$ $1$ $3$
$(2,2,3,4,3,2)$ $2$ $3$
$(3,2,4,6,4,2)$ $3$ $4$
$(3,2,4,6,5,3)$ $4$ $4$
$(3,3,5,6,4,2)$ $4$ $4$
$(3,4,5,6,4,2)$ $6$ $4$
$(3,2,4,6,5,4)$ $6$ $4$
cycle $\ell_A(A/I)$ $\mu(I)$
------------------- --------------- ----------
$(4,3,6,8,6,3)$ $6$ $5$
$(4,3,6,8,6,4)$ $7$ $5$
$(4,4,6,8,6,3)$ $7$ $5$
$(5,4,8,10,7,4)$ $10$ $6$
$(5,4,7,10,8,4)$ $10$ $6$
$(6,5,10,12,8,4)$ $15$ $7$
$(6,4,8,12,10,5)$ $15$ $7$
Case 4
: $(E_7)$ $f=x^3+xy^3+z^2$
(400,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$2$]{}]{} (25,0)[[$E_2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$3$]{}]{} (55,0)[[$E_3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$4$]{}]{} (83,0)[[$E_4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$3$]{}]{} (115,0)[[$E_5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$2$]{}]{} (145,0)[[$E_6$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$1$]{}]{} (175,0)[[$E_7$]{}]{} (180,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$2$]{}]{} (95,26)[[$E_1$]{}]{} (90,34)
Then $Z_0=(2,2,3,4,3,2,1)=2E_1+2E_2+3E_3+4E_4+3E_5+2E_6+E_7$ is the fundamental cycle on $X$.
Now suppose that $Z=\sum_{i=1}^{7} a_iE_i$ is an Ulrich cycle. Since $Z_0E_2=-1$ and $Z_0E_i=0$ for every $i \ne 2$, we have $a_2=2$ by Proposition \[Uideal-cond\]. In particular, $I = \H^0(X,\mathcal{O}_X(-Z))$ is a simple good ideal with $a_2=2$. Then we can conclude that $Z=Z_0 = (2,2,3,4,3,2,1)$, $Z_1=(3,2,4,6,5,4,2)$ or $Z_2=(3,2,4,6,5,4,3)$ by [@WY Example 3.7]; see below. The assertion for Ulrich modules follows from the fact that $\ell_A(A/I_k)=k+1$, where $I_k = \H^0(X,\mathcal{O}_X(-Z_k))$ for $k=0,1,2$.
[**The list of simple good ideals of $(E_7)$**]{}
cycle $\ell_A(A/I)$ $\mu(I)$
-------------------- --------------- ----------
$(2,2,3,4,3,2,1)$ $1$ $3$
$(3,2,4,6,5,4,2)$ $2$ $3$
$(3,2,4,6,5,4,3)$ $3$ $3$
$(4,3,6,8,6,4,2)$ $3$ $4$
$(5,3,6,9,7,5,3)$ $4$ $4$
$(6,4,8,12,6,6,3)$ $6$ $5$
cycle $\ell_A(A/I)$ $\mu(I)$
----------------------- --------------- ----------
$(6,4,8,12,6,7,4)$ $7$ $5$
$(7,4,8,12,8,6,3)$ $7$ $5$
$(8,5,10,15,8,8,4)$ $10$ $6$
$(9,6,12,18,15,10,5)$ $15$ $7$
Case 5
: $(E_8)$ $f=x^3+y^5+z^2$
(400,35)(-20,0) (-10,10)[$Z_0=$]{} (25,18)[[$2$]{}]{} (25,0)[[$E_2$]{}]{} (30,12) (35,12)[(1,0)[20]{}]{} (55,18)[[$4$]{}]{} (55,0)[[$E_3$]{}]{} (60,12) (65,12)[(1,0)[20]{}]{} (83,18)[[$6$]{}]{} (83,0)[[$E_4$]{}]{} (90,12) (95,12)[(1,0)[20]{}]{} (115,18)[[$5$]{}]{} (115,0)[[$E_5$]{}]{} (120,12) (125,12)[(1,0)[20]{}]{} (145,18)[[$4$]{}]{} (145,0)[[$E_6$]{}]{} (150,12) (155,12)[(1,0)[20]{}]{} (175,18)[[$3$]{}]{} (175,0)[[$E_7$]{}]{} (180,12) (185,12)[(1,0)[20]{}]{} (205,18)[[$2$]{}]{} (205,0)[[$E_8$]{}]{} (210,12) (90,16)[(0,1)[15]{}]{} (83,38)[[$3$]{}]{} (95,26)[[$E_1$]{}]{} (90,34)
Then $Z_0=(3,2,4,6,5,4,3,2)=3E_1+2E_2+4E_3+6E_4+5E_5+4E_6+3E_7+2E_8$ is the fundamental cycle on $X$.
Now suppose that $Z=\sum_{i=1}^{8} a_iE_i$ is an Ulrich cycle. Since $Z_0E_8=-1$ and $Z_0E_i=0$ for every $i \ne 8$, we have $a_8=2$ by Proposition \[Uideal-cond\]. In particular, $I = \H^0(X,\mathcal{O}_X(-Z))$ is a simple good ideal with $a_8=2$. Then we can conclude that $Z=Z_0 = (3,2,4,6,5,4,3,2)$ or $Z_1=(5,4,7,10,8,6,4,2)$ by [@WY Example 3.8]; see below. The assertion for Ulrich modules follows from the fact that $\ell_A(A/I_k)=k+1$, where $I_k = \H^0(X,\mathcal{O}_X(-Z_k))$ for $k=0,1$.
[**The list of simple good ideals of $(E_8)$**]{}
cycle $\ell_A(A/I)$ $\mu(I)$
---------------------------- --------------- ----------
$(3,2,4,6,5,4,3,2)$ $1$ $3$
$(5,4,7,10,8,6,4,2)$ $2$ $3$
$(6,4,8,12,10,8,6,3)$ $3$ $4$
$(8,5,10,15,12,9,6,3)$ $4$ $4$
$(9,6,12,18,15,12,8,4)$ $6$ $5$
$(10,7,14,20,16,12,8,4)$ $7$ $5$
$(12,8,16,24,20,15,10,5)$ $10$ $6$
$(15,10,20,30,24,18,12,6)$ $15$ $7$
We now recall Example \[7.8\].
1. Let $A=\mathbb{C}[[X,Y,Z]]/(Z^2-XY)$. Then $A$ can regard as the rational double point of type $(A_1)$. So Theorem \[Main-RDP\] implies that ${\mathcal{X}}_A=\{\m\}$ and Ulrich $A$-modules with respect to $\m$ is isomorphic to finite copies of $M_1 \cong (x,z)A$.
2. Let $A=\mathbb{C}[[x,y,z]]/(x^3+y^5+z^2)$. Then $I=(x,y,z^2) \ne \m$ is an Ulrich ideal of colength $2$ and any Ulrich $A$-module with respect to $I$ is isomorphic to a certain indecomposable maximal Cohen-Macaulay $A$-module (say, $M$) of rank $2$. Theorem \[3.2\] yields that $\Omega:=\Syz_A^1(I) \cong \Syz_A^2(A/I)$ is an also Ulrich $A$-module with respect to $I$. Then since $\Omega$ contains $M$ as a direct summand and $\rank_A \Omega = \rank_A A^3 - \rank_A I=2=\rank_A M$, we can conclude that $\Omega \cong M$ itself.
Ulrich modules over rational double points
------------------------------------------
Our theorem \[4.1\] suggests the following question.
\[Umoduleexist\] Let $I$ be an $\m$-primary ideal with $\mu_A(I) > d = \dim A$. Suppose that there exists an Ulrich $A$-module with respect to $I$. Then is $I$ an Ulrich ideal?
The main aim of this subsection is to give an affirmative answer to the question above in the case of two-dimensional rational double points.
\[Umodule-Uideal\] Let $A$ be a two-dimensional rational double point. Let $I$ be an $\m$-primary ideal with $\mu_A(I) > 2$. Then the following conditions are equivalent.
1. $I$ is an Ulrich ideal.
2. There exists an Ulrich $A$-module with respect to $I$.
In what follows, we give a proof of this theorem. First, we prove a general criterion in the higher dimensional case. An ideal $I$ is called *stable* if there exists a minimal reduction $Q$ of $I$ such that $I^2=QI$. Note that if $I$ is stable, then $I^2=Q'I$ for any minimal reduction $Q'$ of $I$.
\[Umodule-multi2\] Let $A$ be a $d$-dimensional hypersurface local domain of $\e_{\m}^0(A)=2$, and let $I$ be an $\m$-primary ideal of $A$ with $\mu_A(I) > d$. If there exists an Ulrich $A$-module with respect to $I$, then $\e_I^0(A)=2 \cdot \ell_A(A/I)$.
If, in addition, $I$ is stable, then it is good.
Suppose that $M$ is an Ulrich $A$-module with respect to $I$. Then $M$ has no free summands. Thus $M$ is an Ulrich $A$-module with respect to $\m$ bacause $\e^0_{\m}(A)=2$ (cf. [@HKuh Corollary 1.4]). Moreover, as $M/IM$ is a free $A/I$-module, we get $$\begin{aligned}
\ell_A(M/IM) & = & \ell_A(A/I) \cdot \mu_A(M) \\
&=& \ell_A(A/I) \cdot \e_\m^0(M) \\
&=& \ell_A(A/I) \cdot \e_\m^0(A) \cdot \rank_A M \\
&=& 2 \cdot \ell_A(A/I) \cdot \rank_A M. \end{aligned}$$ Since this is equal to $\e_I^0(M) = \e^0_{I}(A) \cdot \rank_A M$, we obtain that $\e_{I}^0(A)=2 \cdot \ell_A(A/I)$.
The last assertion follows from the fact that an ideal $I$ of an Artinian Gorenstein local ring $A$ with $\ell(A)=2\ell(I)$ satisfies $(0):I=I$.
As a corollary of Proposition \[Umodule-multi2\], we have the following result. Compare this with Theorem \[Uideal-chara\] below.
\[Umodule-good\] Let $A$ be a two-dimensional rational double point, and let $I$ be an $\m$-primary ideal with $\mu_A(I) > 2$. If there exists an Ulrich $A$-module with respect to $I$, then $I$ is a good ideal.
Let $Q$ be a minimal reduction of $I$, and let $\overline{I}$ denote the integral closure of $I$. Then $\overline{I}^2 = Q \overline{I}$ because $A$ is a two-dimensional rational singularity. Hence we have $I \subseteq \overline{I} \subseteq Q \colon
\overline{I} \subseteq Q \colon I$. On the other hand, Proposition \[Umodule-multi2\] implies that $\e_I^0(A)=2 \cdot \ell_A(A/I)$. This means $\ell_A(I/Q)=\ell_A(Q\colon I/Q)$. Hence $I=\overline{I} = Q \colon I$ and thus $I$ is good.
It suffices to show $(2) \Longrightarrow (1)$. Let $M$ be an Ulrich $A$-module with respect to $I$. Corollary \[Umodule-good\] implies that $I$ is good. According to Corollary \[2.6\], we have only to show $\mu_A(I)=d+1(=3)$. Since $A$ is a hypersurface and $M$ is a maximal Cohen-Macaulay $A$-module without free summand, we have a minimal free presentation $A^t \to A^t \to M \to 0$, which induces an exact sequence $$(A/Q)^t \to (A/Q)^t \xrightarrow{f} M/QM \to 0.$$ As $M/QM=M/IM$ is $A/I$-free, we have $M/QM\cong(A/I)^t$. It is easy to observe that the kernel of $f$ is isomorphic to $(I/Q)^t$. Hence there is a surjection $(A/Q)^t\to(I/Q)^t$, which shows $\mu_A(I/Q)\le1$. Thus $\mu_A(I)=d+1$.
We obtain the following interesting corollary from Theorems \[Umodule-Uideal\] and \[5.1\].
\[dual-RDP\] Let $A$ be a two-dimensional rational double point. If $M$ is an Ulrich $A$-module with respect to some ideal, then so is the dual $M^{*} = \Hom_A(M,A)$.
Assume that $M$ is an Ulrich $A$-module with respect to $I$. Theorem \[Umodule-Uideal\] implies that $I$ is an Ulrich ideal of $A$. Thus $A/I$ is Gorenstein by Corollary \[2.6\]. Applying Theorem \[5.1\] yields that $M^{*}$ is also an Ulrich $A$-module with respect to $I$.
Ulrich ideals of non-Gorenstein rational singularities
------------------------------------------------------
In this subsection, we study Ulrich ideals of two-dimensional non-Gorenstein rational singularities. Notice that the maximal ideal $\m$ is always an Ulrich ideal of such a local ring.
We first show that any Ulrich ideal of a two-dimensional rational singularity is a ideal; see also Corollary \[2.6\]. In order to obtain a characterzation of Ulrich ideals, we need the following definition.
\[Uz\] Let $A$ be a two-dimensional rational singularity, and let $\varphi \colon X \to \Spec A$ be a resolution of singularities of $\Spec A$. Let $\pi^{*}Z_0$ denote the pull-back of the fundamental cycle $Z_0$ on the minimal resoluition to $X$. Then for any anti-nef cycle $Z$ on $X$, we put $$U(Z)= (\varphi^{*}Z_0 \cdot Z)(p_a(Z)-1) + Z^2,$$ where $p_a(Z)$ denotes the arithmetic genus of $Z$; see the remark below.
\[arith\] Let $A$ be a two-dimensional rational singularity and $\varphi \colon X \to \Spec A$ be a resolution of singularities. Then for every effective cycle $0 \ne Y$ on $X$, $p_a(Y) := \frac{Y^2+KY}{2} + 1 \le 0$ (see [@Ar Proposition 1]). Note that $p_a(Y) =0$ implies that $Y$ is connected. Moreover, $p_a(Y+Z)=p_a(Y)+p_a(Z)+YZ-1$ for any effective cycles $Y$ and $Z$.
\[Uideal-chara\] Let $(A,\m)$ be a two-dimensional rational singularity. Let $I$ be an $\m$-primary ideal with $\mu_A(I) > 2$. Then the following conditions are equivaelnt$:$
1. $I$ is an Ulrich ideal.
2. $\e_I^0(A) = (\mu(I)-1)\cdot \ell_A(A/I)$.
3. $I$ is an integrally closed ideal represented on the minimal resolution of singularities $\varphi \colon X \to \Spec A$ such that $I\mathcal{O}_X=\mathcal{O}_X(-Z)$, $I=\H^0(X,\mathcal{O}_X(-Z))$ and $U(Z)=0$.
$(1)\Longleftrightarrow (2)$ follows from Lemma \[2.3\].
$(3) \Longrightarrow (2):$ Any integrally closed ideal $I$ in a two-dimensional rational singularity is stable. Moreover, $U(Z)=0$ means that $\e_{I}^0(A) = (\mu(I)-1)\cdot \ell_A(A/I)$. Thus the assertion immediately follows from this.
$(2) \Longrightarrow (3):$ Since $I$ is an Ulrich ideal by (1), we have that $I=Q\colon I$ for any minimal reduction $Q$ of $I$ by Corollary \[2.6\]. Then as $\overline{I}^2 = Q\overline{I}$, we get $I \subseteq \overline{I} \subseteq Q \colon \overline{I} \subseteq Q \colon I$. Hence $I=\overline{I}$ is integrally closed.
Let $\varphi \colon X \to \Spec A$ be a resolution of singularities such that $I=\H^0(X,\mathcal{O}_X(-Z))$ and $I \mathcal{O}_X = \mathcal{O}_X(-Z)$ is invertible for some anti-nef cycle $Z$ on $X$. Then (2) implies that $U(Z)=0$.
Now suppose that $I$ is *not* represented on the minimal resolution of singularities $\overline{\varphi} \colon \overline{X} \to \Spec A$. Then there exists a contraction $\pi \colon X \to X'$ of a $(-1)$-curve $E$ on $X'$ such that $I$ is not represented on $X'$. Consider the following commutative diagram:
(400,35) (90,25)[$X$]{} (105,28)[(1,0)[40]{}]{} (150,25)[$\overline{X}$]{} (102,20)[(1,-1)[12]{}]{} (115,0)[$X'$]{} (132,10)[(1,1)[12]{}]{} (98,10)[$\pi$]{} (120,32)[$\psi$]{} (142,10)[$\psi'$]{}
Then we may assume that $Z= \pi^{*}Z'+nE$ for some anti-nef cycle $Z'$ on $X'$ and an integer $n \ge 1$. Note that $\psi^{*}Z_0 \cdot E = \pi^{*}Z' \cdot E =0$; see e.g. [@GIW Fact 7.7]. Then $$\begin{aligned}
U(Z)-U(Z') & = & \left(\psi^{*}Z_0 \cdot (\pi^{*}Z'+nE)\right)
(p_a(\pi^{*}Z')+p_a(nE)+\pi^{*}Z'\cdot nE -2) \\
&& + (\pi^{*}Z'+nE)^2- (\psi^{*}Z_0\cdot \pi^{*}Z')\left(p_a(\pi^{*}Z')-1\right)-(\pi^{*}Z')^2 \\
&=& (\psi^{*}Z_0 \cdot \pi^{*}Z')(p_a(nE)-1)+(nE)^2 \\
&=& \left((\psi')^{*}Z_0 \cdot Z' + 2\right) \frac{(nE)^2}{2}
+ \frac{n(K_X \cdot E)}{2}
\left((\psi')^{*}Z_0 \cdot Z' \right).\end{aligned}$$ Since $(\psi')^{*}Z_0 \cdot Z' \le -2$ and $E^2 = K_X \cdot E = -1$, we get $U(Z) > U(Z') \ge 0$. This is a contradiction.
In what follows, we always assume that $\varphi \colon X \to \Spec A$ be the minimal resolution of singularities and $I\mathcal{O}_X = \mathcal{O}_X(-Z)$ is invertible and $I = \H^0(X,\mathcal{O}_X(-Z))$ for some anti-nef cycle $Z$ on $X$. Let $\varphi^{-1}(\m) = \bigcup_i E_i$ denote the exceptional divisor on $X$ with the irreducible components $\{E_i\}_{1 \le i \le r}$. Let $Z_0$ (resp. $K$) denotes the fundamental cycle (resp. the canonical divisor) on $X$. Notice that $Z_0E = EZ_0 \le 0$ and $KE = -E^2 - 2$ for all curves $E$. The next target is to characterize Ulrich cycles in terms of dual graphs. In order to do that, let us begin with the following lemma.
\[Inf-anti\] Let $Z = \sum_{E} k_E E$ and $W = \sum_{E} m_E E$ be anti-nef cycles on $X$. If we put $\inf(Z,W) = \sum_{E} \inf(k_E,m_E) E$, then $\inf(Z,W)$ is also an anti-nef cycle on $X$.
It immediately follows from the definition of anti-nef cycles.
Assume that $Z\ne Z_0$ is an anti-nef cycle on $X$. Then we can find the following anti-nef cycles $Z_1,\ldots,Z_r$ and effective divisors $Y_1,\ldots,Y_r$ such that $0 \le Y_r \le Y_{r-1} \le \cdots \le Y_1 \le Z_0$: $$\left\{
\begin{array}{rcl}
Z=Z_r &=& Z_{r-1} + Y_r, \\
Z_{r-1} & = & Z_{r-2} + Y_{r-1}, \\
&\vdots& \\
Z_2 & = & Z_1 + Y_2, \\
Z_1 & = & Z_0 + Y_1,
\end{array}
\right.$$ where $Z_0$ denotes the fundamental cycle on $X$.
Indeed, we can take an integer $r \ge 1$ such that $Z \not \le r Z_0$ and $Z \le (r+1)Z_0$. Put $Z_i = \inf(Z, (i+1)Z_0)$ for every $i=1,\ldots,r$. Then $Z_1,\ldots,Z_r$ are anti-nef cycles by Lemma \[Inf-anti\]. In particular, $Z_0 \le Z_1 \le Z_2 \le \cdots \le Z_r =Z$. Moreover, if we put $Y_i = Z_i - Z_{i-1}$ for every $i=1,\ldots,r$, then we can obtain the required sequence.
The following lemma plays a key role in the proof of the main theorem in this subsection.
\[Key-Ul\] Let $Z$, $Z'$ be anti-nef cycles on $X$ with $Z' = Z+Y$, where $Y$ is an effective divisor. Then$:$
1. $$\begin{aligned}
U(Z') - U(Z)
&=& (YZ_0)\big\{(p_a(Z)-1)+(p_a(Y)-1)\big\}
+(YZ)(Z'Z_0+2) \\
& & \qquad + (p_a(Y)-1)(ZZ_0+2)-KY. \end{aligned}$$
2. Assume that $0 \ne Y \le Z_0$ and $e=\e_{\m}^0(A) \ge 3$. Then $U(Z')\ge U(Z)$ holds true, and equality holds if and only if $YZ=YZ_0 = p_a(Y) = (Z-Z_0)Z_0 = K(Z_0-Y)=0$.
Since $p_a(Z+Y) = p_a(Z)+p_a(Y)+YZ-1$ by definition, we have $$\begin{aligned}
U(Z')-U(Z) & =&
(ZZ_0+YZ_0) (p_a(Z)-1+p_a(Y)-1+YZ)+(Z^2+2YZ+Y^2)\\
&& - (ZZ_0)(p_a(Z)-1)-Z^2 \\
&=& (YZ_0)\big\{(p_a(Z)-1) + (p_a(Y)-1) \big\} + (YZ)(ZZ_0+YZ_0+2) \\
&& + (p_a(Y)-1)(ZZ_0) + Y^2 \\
& = & (YZ_0)\big\{(p_a(Z)-1) + (p_a(Y)-1) \big\} + (YZ)(Z'Z_0+2) \\
&&+ (p_a(Y)-1)(ZZ_0+2) - KY, \end{aligned}$$ where the last equality follows from $2(p_a(Y)-1) = KY+Y^2$.
\(2) Assume that $Y \le Z_0$. As $X \to \Spec A$ is the minimal resolution, we have that $KY \le KZ_0$ because $KE \ge 0$ for all curves $E$ on $X$. Since $Z_0$ is anti-nef and $Z-Z_0$, $Y$ are effective, we get $$Z'Z_0 + 2 = (Z-Z_0)Z_0 + YZ_0 + (Z_0^2+2) \le Z_0^2 +2= -e+2 < 0.$$ Moreover, $p_a(Z_0)=0$ implies that $$(p_a(Y)-1)(ZZ_0+2)-KY
= p_a(Y)(ZZ_0+2) -(Z-Z_0)Z_0 -K(Y-Z_0) \ge 0$$ and equality holds if and only if $p_a(Y)=(Z-Z_0)Z_0=K(Y-Z_0)=0$.
Note that $YZ_0$, $YZ \le 0$ and $p_a(Z)-1 + p_a(Y)-1 < 0$. Hence $U(Z') \ge U(Z)$ and equality holds if and only if $YZ_0=YZ=0$ and $p_a(Y)=(Z-Z_0)Z_0=K(Y-Z_0)=0$.
The main result in this subsection is the following theorem, which enables us to determine all Ulrich ideals of a two-dimensional (non-Gorenstein) rational singularity. For an effective divisor $Z$ on $X$, we write $Z=\sum_{E} Z_E E$, where $Z_E$ is a nonnegative integer.
\[All-Uideal\] Let $(A,\m)$ be a two-dimensional rational singularity with $e=\e_\m^0(A)\ge 3$, and let $\varphi \colon X \to \Spec A$ be the minimal resolution of singularities. Set $Z_0 = \sum_{E} n_E E$, the fundamental cycle on $X$. Let $Z$ be an anti-nef cycle on $X$ with $I\mathcal{O}_X=\mathcal{O}_X(-Z)$ and $I=\H^0(X,\mathcal{O}_X(-Z))$. Then the following conditions are equivalent$:$
1. $I$ is an Ulrich ideal, that is, $Z$ is an Ulrich cycle on $X$.
2. 1. There exist a sequence of anti-nef cycles $Z_1,\ldots,Z_r$ and a sequence of effective cycles $Y_r \le \cdots \le Y_1 \le Z_0$ for some $r \ge 1$ such that $$\left\{
\begin{array}{rcl}
Z=Z_r &=& Z_{r-1} + Y_r, \\
Z_{r-1} & =& Z_{r-2} + Y_{r-1}, \\
& \vdots & \\
Z_1 & = & Z_0 + Y_1.
\end{array}
\right.$$
2. $Y_iZ_{i-1} = p_a(Y_i) = K(Z_0-Y_i) =0$ for every $i=1,\ldots,r$.
3. There exist a sequence of anti-nef cycles $Z_1,\ldots,Z_r$ and a sequence of effective cycles $Y_r \le \cdots \le Y_1 \le Z_0$ for some $r \ge 1$ such that $$\left\{
\begin{array}{rcl}
Z=Z_r &=& Z_{r-1} + Y_r, \\
Z_{r-1} & =& Z_{r-2} + Y_{r-1}, \\
& \vdots & \\
Z_1 & = & Z_0 + Y_1.
\end{array}
\right.$$
Moreover, the following conditions are satisfied.
1. $\{E \,|\, E^2 \le -3\}$ is contained in $\Supp(Y_1)$.
2. $\Supp(Y_i)$ is given as one of the connected components of $\{E\,|\, EZ_{i-1}=0\}$ in $\{E\,|\,EZ_0=0\}$.
3. $Y_i$ is the fundamental cycle on $\Supp(Y_i)$.
4. $(Z_0)_E=(Y_i)_E$ for every $E$ with $E^2 \le -3$.
When this is the case, if, in addition, we put $I_i=\H^0(X,\mathcal{O}_X(-Z_i))$, then $I_i$ is an Ulrich ideal such that $$\m = I_0 \supseteq I_1 \supseteq \cdots \supseteq I_r = I \quad
\text{and} \quad \ell_A(I_{i-1}/I_i) =1.$$
Take a sequence as in (1).
$(1) \Longrightarrow (2):$ Lemma \[Key-Ul\] implies that $$0 = U(Z) = U(Z_r) \ge U(Z_{r-1}) \ge \cdots \ge U(Z_1) \ge U(Z_0)=0.$$ Hence all $Z_i$ are Ulrich cycles and $$Y_iZ_{i-1} = Y_iZ_0 = p_a(Y_i) = (Z_i-Z_0)Z_0 = K(Z_0-Y_i)=0$$ for every $i=1,\ldots,r$.
By the Riemann-Roch formula, we have that $\ell_A(A/I_i) = - \frac{Z_i^2+KZ_i}{2} = 1 - p_a(Z_i)$ for each $i$. Then $$\begin{aligned}
\ell_A(I_{i-1}/I_i)
&=&
p_a(Z_{i-1})-p_a(Z_i) \\
&=& p_a(Z_{i-1})- (p_a(Z_{i-1})+p_a(Y_i)+Y_iZ_{i-1}-1) =1. \end{aligned}$$
If $E^2 \le -3$, then $KE=-E^2-2 > 0$. Thus $K(Z_0-Y_1) =0$ implies that $(Z_0)_E = (Y_1)_E$ for every $E$ with $E^2 \le -3$. In particular, $\Supp(Y_i) \supseteq \{E\,|\, E^2 \le -3\}$. On the other hand, $Y_iZ_0=0$ implies that $\Supp(Y_i) \subseteq \{E\,|\, EZ_0=0\}$ because $Z_0$ is an anti-nef cycle.
$(2) \Longrightarrow (3):$ Fix $i$ with $1 \le i \le r$. Since $Z_{i-1}$ is anti-nef and $Y_iZ_{i-1}=0$, a similar argument to the above yields that $\{E \,|\, E^2 \le -3\} \subseteq \Supp(Y_i) \subseteq
\{E \,|\,EZ_{i-1}=0\}$.
As $p_a(Y_i)=0$, $\Supp(Y_i)$ is connected by Remark \[arith\]. Moreover, $\Supp(Y_i)$ is one of the connected components of $\{E\,|\,EZ_{i-1}=0\}$. Indeed, if there exists a curve $E \notin \Supp(Y_i)$ such that $EE'>0$ for some $E' \in \Supp(Y_i)$, then $EZ_{i-1}< 0$ since $EY_i \ge 1$ and $EZ_{i-1}+EY_i = EZ_i \le 0$.
Claim
: $Y_i$ is the fundamental cycle on $\Supp(Y_i)$.
Take an $E \in \Supp(Y_i)$. As $Y_iZ_{i-1}=0$, we have $EZ_{i-1} =0$. If $EY_i > 0$, then $EY_i = EZ_i \le 0$. This is a contradiction. Hence $EY_i\le 0$. Namely, $Y_i$ is anti-nef. Moreover, if $Y_i$ is not the fundamental cycle on $\Supp(Y_i)$, then we know that $p_a(Y_i) \le -1$. This contradicts the assumption $p_a(Y_i)=0$. Hence $Y_i$ must be the fundamental cycle on $\Supp(Y_i)$.
$(3) \Longrightarrow (1):$ $(c)$ implies that $p_a(Y_i)=0$. $(b)$ means that $Y_iZ_{i-1} =0$. Hence $Y_iZ_0=0$. Note that $Y_iZ_0=Y_{i-1}Z_0=\cdots =Y_1Z_0=0$ yield $(Z_i-Z_0)Z_0=0$. $(d)$ implies that $K(Z_0-Y_i)=0$. Therefore $U(Z)=U(Z_r)=\cdots = U(Z_1)=U(Z_0)=0$, as required.
The following assertion does not hold true without the assumption that $A$ is rational; see Example \[2.2\].
\[Gor-Ul\] Let $A$ be a two-dimensional rational singularity. Let $I$ be an Ulrich ideal of $A$. Then $A/I$ is Gorenstein.
We may assume that $A$ is not Gorenstein, that is, $e=\e_\m^0(A) \ge 3$. Then by Theorem \[All-Uideal\], we can find a sequence of Ulrich cycles $Z_1,\ldots,Z_r$ and effective divisors $Y_1,\ldots,Y_r \le Z_0$ satisfying all conditions in Theorem \[All-Uideal\] such that $$\left\{
\begin{array}{rcl}
Z=Z_r & = & Z_{r-1}+Y_r, \\
Z_{r-1} &=& Z_{r-2} + Y_{r-1}, \\
& \vdots & \\
Z_1 & = & Z_0 + Y_1.
\end{array}
\right.$$ Then $Z \le (r+1)Z_0$ and $Z \not \le r Z_0$. In particular, $\m^{r} \not \subseteq I$ and $\m^{r+1} \subseteq I$.
Claim
: There exists a minimal set of generators $\{u_1,\ldots,u_s,t\}$ such that $I=(u_1,\ldots,u_s,t^{r+1})$.
Set $I_{r-1} = \H^0(X, \mathcal{O}_X(-Z_{r-1}))$. Then $I_{r-1}$ is also an Ulrich ideal. So we may assume that we can write $I_{r-1} = (u_1,\ldots,u_s,t^{r})$ for some minimal set of generators of $\m$. Since $\m (u_1,\ldots,u_s) \subseteq I$ and $\m^r \not \subseteq I$, we have that $t^r \notin I$. Hence by $\ell_A(I_{r-1}/I) = 1$, we can choose an element $a_i \in A$ such that $u_i - a_i t^r \in I$ for every $i$. By replacing $u_i$ with $u_i - a_it^r$, we may assume that $I'=(u_1,\ldots,u_r,t^{r+1}) \subseteq I$. As $\ell_A(I_{r-1}/I')=1$ and $I \ne I_{r-1}$, we can conclude that $I=I'$, as required.
\[cyclic\] Let $A$ be a two-dimensional cyclic quotient singularity with $e = \e_\m^0(A) \ge 3$. That is, $A = k[[x,y]]^G$ where $k$ is an algebraically closed field of characteristic $0$ and $G$ is a cyclic group with generator $$g= \left[
\begin{array}{cc}
\varepsilon_n & 0 \\
0 & \varepsilon_n^q
\end{array}
\right]$$ for some integer $n,q$ with $n \ge q \ge 1$, $n \ge 2$ and $(q,n)=1$.
In this case, the exceptional set is a chain of rational curves and there is some $E_i$ with $E_i^2 \le -3$ and $E_iK > 0$. Then $Z_0E_i<0$ for such $E_i$. Thus we can not take $Y_1$ satisfying the conditions (3)(a)(b) in Theorem \[All-Uideal\]. Hence the maximal ideal is the only Ulrich ideal of $A$.
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\[abc\] Let $k$ be an algebraically closed field of characteristic zero. Let $a \ge b \ge c \ge 2$. If we set $A=k[[T,sT^a,s^{-1}T^b,(s+1)^{-1}T^c]]$, then it is a two-dimensional rational singularity with $\e_{\m}^0(A)=3$ and $$A \cong
k[[t,x,y,z]]/(xy-t^{a+b},xz-t^{a+c}-zt^a,yz-yt^c-zt^b).$$ Then $I=(t^m,x,y,z)$ is an Ulrich ideal of colength $m$ for every $m$ with $1 \le m \le c$.
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(200,60)(-40,0) (-5,10)[$Z_1=$]{} (30,7) (25,13)[[$1$]{}]{} (35,7)[(1,0)[19]{}]{} (55,13)[[$2$]{}]{} (60,7) (65,7)[(1,0)[18]{}]{} (85,4)[[$-3$]{}]{} (83,15)[[$2$]{}]{} (90,7) (97,7)[(1,0)[18]{}]{} (115,13)[[$2$]{}]{} (120,7) (125,7)[(1,0)[19]{}]{} (145,13)[[$1$]{}]{} (150,7) (90,13)[(0,1)[13]{}]{} (83,33)[[$2$]{}]{} (90,29) (90,34)[(0,1)[13]{}]{} (83,55)[[$1$]{}]{} (90,51)
(200,60)(-40,0) (-5,10)[$Z_2=$]{} (30,7) (25,13)[[$1$]{}]{} (35,7)[(1,0)[19]{}]{} (55,13)[[$2$]{}]{} (60,7) (65,7)[(1,0)[18]{}]{} (85,4)[[$-3$]{}]{} (83,15)[[$3$]{}]{} (90,7) (97,7)[(1,0)[18]{}]{} (115,13)[[$2$]{}]{} (120,7) (125,7)[(1,0)[19]{}]{} (145,13)[[$1$]{}]{} (150,7) (90,13)[(0,1)[13]{}]{} (83,33)[[$2$]{}]{} (90,29) (90,34)[(0,1)[13]{}]{} (83,55)[[$1$]{}]{} (90,51)
Set $a=b=c=2$. Then $$A \cong
k[[t,x,y,z]]/(xy-t^4,xz-t^4-zt^2,yz-yt^2-zt^2).$$ Furthermore, if we put $I=(x,y-z,t^2,z)A$ and $Q=(x,y-z)A$, then $I^2=QI$ and $\e_{I}^0(A)=(\mu(I)-1) \ell_A(A/I)=6$. Hence $I$ is an Ulrich ideal with $\ell_A(A/I)=2$.
(200,40)(-40,0) (55,13)[[$1$]{}]{} (60,7) (65,7)[(1,0)[18]{}]{} (85,4)[[$-3$]{}]{} (83,15)[[$2$]{}]{} (90,7) (97,7)[(1,0)[18]{}]{} (115,13)[[$1$]{}]{} (120,7) (90,13)[(0,1)[13]{}]{} (83,33)[[$1$]{}]{} (90,29)
[HKuh]{}
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[^1]: 2010 [*Mathematics Subject Classification.*]{} 13A30, 13H10, 13H15, 14B05, 16G50
[^2]: [*Key words and phrases.*]{} Ulrich ideal, Ulrich module, numerical semigroup ring, finite CM–representation type, rational singularity
[^3]: This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 20540050/22540047/22540054/23540059, JSPS Grant-in-Aid for Young Scientists (B) 22740008/22740026 and by JSPS Postdoctoral Fellowships for Research Abroad
|
---
abstract: 'Thermoelectric effects, such as the generation of a particle current by a temperature gradient, have their origin in a reversible coupling between heat and particle flows. These effects are fundamental probes for materials and have applications to cooling and power generation. Here we demonstrate thermoelectricity in a fermionic cold atoms channel, ballistic or diffusive, connected to two reservoirs. We show that the magnitude of the effect and the efficiency of energy conversion can be optimized by controlling the geometry or disorder strength. Our observations are in quantitative agreement with a theoretical model based on the Landauer-Büttiker formalism. Our device provides a controllable model-system to explore mechanisms of energy conversion and realizes a cold atom based heat engine.'
author:
- 'Jean-Philippe Brantut'
- Charles Grenier
- Jakob Meineke
- David Stadler
- Sebastian Krinner
- Corinna Kollath
- Tilman Esslinger
- Antoine Georges
title: A thermoelectric heat engine with ultracold atoms
---
=1
In general, heat and particle transport are coupled processes [@Onsager:1931aa]. This coupling leads to thermoelectric effects, such as a Seebeck voltage drop in a conductor subject to a thermal gradient. These effects are important for probing elementary excitations in materials, for example, giving access to the sign of charge carriers [@ashcroft2002physique]. Moreover, they have practical applications to refrigeration, and power generation from waste-heat recovery [@Goldsmid; @snyder2008com]. Recently, there has also been interest in thermoelectric effects in nano- and molecular-scale electronic devices [@entin_2010; @sanchez_2011]. The progress in modeling solid-state physics with cold atoms [@Esslinger:2010aa; @Bloch:2012aa] raises the question whether thermoelectricity can be observed in such a controlled setting [@Grenier:2012tg; @Kim:2012aa], where set-ups analogous to mesoscopic devices were realized [@Albiez:2005aa; @Ramanathan:2011aa; @Brantut:2012aa]. Whilst the thermodynamic interplay between thermal and density collective modes has been seen in a second sound experiment [@Sidorenkov:2013aa], thermoelectric effects have so far not been investigated.
Here, we demonstrate a cold atoms device in which a particle current is generated by a temperature bias. We prepare a mesoscopic channel connecting two atomic reservoirs having equal particle numbers. Heating one of the reservoirs establishes a temperature bias and the compressible cloud forming the hot reservoir expands. Hence, one naively expects an initial particle flow from the cold denser side to the hot. In contrast, we observe the opposite effect: a net particle current initially directed from the hot to the cold side. This is a direct manifestation of the intrinsic thermoelectric power of the channel. The temperature bias leads to a current of high-energy particles from hot to cold and a current of low-energy particles from cold to hot. In our channel, particles are transported at a rate which increases with energy, leading to an asymmetry between the high- energy and low-energy particle currents. This results in a total current from hot to cold, which overcomes the thermodynamic effect of the reservoirs. Hence, work is performed by carrying atoms from lower to higher chemical potential, and our system can be regarded as a cold-atoms based heat engine.
![ Concept of the experiment. A: A quasi-two dimensional channel connects two atomic reservoirs. A gate beam intersects with the channel and blocks particle and heat transport. A heating beam traverses the left reservoir and heats it in a controlled way. B-D: Line-sums along $y$ of the density in the hot (red) and cold (blue) reservoir at different evolution times (0.01, 0.98 and 3.97s) after time-of-flight. E: $T_h$ (red) and $T_c$ (blue) as a function of time. Dashed line : $\bar{T}$ at the initial time. F: $\Delta N/N_{tot}$ as a function of time. In the channel $\nu_z$ was set to $3.5\,$kHz with a disorder of average strength $542\,$nK (see text). []{data-label="fig:concept"}](fig1.pdf)
A schematic view of the experimental setup is shown in figure \[fig:concept\]A. It is based on our previous work on conduction processes in Fermi gases [@Brantut:2012aa; @Stadler:2012kx]. Initially we prepare $N_{\mathrm{tot}}=3.1(4)\cdot10^5$ weakly interacting $^6$Li atoms at a temperature of $250(9)\,$nK in an elongated trap, where the Fermi temperature of the cloud is $T_F=931(44)\,$nK. Using a repulsive laser beam (not shown) having a nodal line at its center, the cloud is then separated into two identical reservoirs connected by a channel. Tuning the power of the beam allows to adjust the trap frequency $\nu_z$ in the channel up to $10$kHz. We then raise a gate potential in the channel, preventing any energy or particle exchange between the reservoirs. A controlled heating of the left reservoir is performed, increasing its temperature by typically $200$nK. Afterwards, the gate potential is removed abruptly and the system evolves for a variable time. For the observations, the gate potential is raised again, the power of the laser beam creating the channel is ramped to zero in $400$ms, and each reservoir is left to equilibrate independently for $100$ms [@materialsandmethods].
We take absorption images of the cloud and measure the temperatures $T_h$ ($T_c$) and atom number $N_h$ ($N_c$) in the hot (cold) reservoir using finite temperature Fermi fits, as indicated in figure \[fig:concept\]B-D. This allows us to reconstruct the time evolution of the number imbalance $\Delta N = N_c - N_h$ and temperature bias $\Delta T = T_c -
T_h$. Figure\[fig:concept\] E and F show typical results. The temperatures equilibrate fast, similar to the equilibration of atom numbers observed in the case of pure atomic flow [@Brantut:2012aa]. In contrast, the atom-number difference in figure \[fig:concept\]F starts at zero, first grows fast, and then decreases back to zero. This transient atomic current is the fingerprint of thermoelectricity.
To explain quantitatively our observations, we model the channel as a linear circuit element having conductance $G$, thermal conductance $G_T$, and thermopower $\alpha_{ch}$. In this framework, the particle and entropy currents flowing in the channel are given by : $$\label{eq:lin_response}
\left(
\begin{array}{c}
I_N\\
I_S
\end{array}
\right)
=
-G\left(\begin{array}{cc}
1&\alpha_{ch}\\
\alpha_{ch}&L+\alpha_{ch}^2
\end{array}
\right)
\left(
\begin{array}{c}
\mu_c-\mu_h\\
T_c-T_h
\end{array}
\right)\,.$$ In this expression, $I_N = \partial\Delta N/\partial t$ and $I_S =
\partial \Delta S/\partial t$, with $\Delta S = S_c-S_h$. Further $\mu_h$, $\mu_c$ are the chemical potentials of the hot and cold reservoirs, $L=G_T/\bar{T}G$ is the Lorenz number [@ashcroft2002physique] of the channel, and $\bar{T}=(T_h + T_c)/2$. Combining eq. with the thermodynamics of the reservoirs leads to the equation for the time evolution of $\Delta N$ and $\Delta T$ : $$\label{eq:temp_imb}
\tau_0
\frac{d}{dt}\left(
\begin{array}{c}
\Delta N\\
\Delta T
\end{array}
\right)
=
-\left(\begin{array}{cc}
1 &-\kappa(\alpha_r-\alpha_{ch})\\
-\frac{\alpha_r-\alpha_{ch}}{\ell\kappa} &
\frac{L+(\alpha_r-\alpha_{ch})^2}{\ell}
\end{array}\right)
\left(
\begin{array}{c}
\Delta N\\
\Delta T
\end{array}
\right)\,.$$ Here, $\kappa=\left. \frac{\partial N}{\partial
\mu}\right|_{T}$, $C_N=\left. T\,\frac{\partial S}{\partial
T}\right|_{N}$ and $\alpha_r = \left. \frac{\partial S}{\partial
N}\right|_{T}$ are the compressibility, specific heat and dilatation coefficient of each reservoir, calculated at the average temperature $\bar{T}$ and particle number $(N_c+N_h)/2$. $\ell
= \frac{C_N}{\kappa \bar{T}}$ is an analogue of the Lorenz number for the reservoirs, and $\tau_0 = \kappa G^{-1}$ is the particle transport timescale, analogous to a capacitor’s discharge time [@Brantut:2012aa]. Eq. shows that the thermoelectric response results from the competition between the entropy transported through the channel, described by $\alpha_{ch}$, and the entropy created by removing one atom from one reservoir and adding it to the other, described by $\alpha_r$. The channel properties $G,G_T$ and $\alpha_{ch}$ are calculated using the Landauer-Büttiker formalism [@RevModPhys.71.S306; @Buttiker:1988:SEC:49387.49388; @materialsandmethods].
![ Thermoelectric response of a ballistic channel for various confinements. A and B: Time evolution of $\Delta N/N_{tot}$ for $\nu_z = 3.5\,$kHz (A) and $\nu_z = 9.3\,$kHz (B), compared to theory (solid line, see text and [@materialsandmethods]). C and D: Timescale $\tau_0$ and maximum thermoelectric response as a function of confinement. Symbols are the fitted values from experiment and the solid line is the theoretical prediction. []{data-label="fig:ballistic"}](fig2.pdf)
The geometry of the channel influences its thermoelectric properties [@Heremansdresselhaus]. We measured the transient imbalance and the temperature evolution for various confinements in a ballistic channel. Two examples for $\nu_z=3.5$ and $9.3$kHz are presented in figure \[fig:ballistic\]A and B. The evolution of both $\Delta N$ and $\Delta T$ is faithfully described by the theoretical model, which does not involve any adjustable parameter [@materialsandmethods]. As shown in figure \[fig:ballistic\]C, the dynamics of both temperature and atom number evolution is slowing down with increasing $\nu_z$, as expected since the number of conducting modes is reduced. The experimental values are extracted from fitting the experimental data with the theoretical model with $\tau_0$ as the only free parameter [@materialsandmethods]. They agree well with the theoretical predictions and independent measurements performed with a pure particle number imbalance [@materialsandmethods].
At the same time the amplitude of the transient imbalance increases with stronger confinement. This can be qualitatively understood by noting that the energy dependence of the density of states in the channel is enhanced with increasing trap frequencies (see [@materialsandmethods]). We define the thermoelectric response $\mathcal{R} = (\Delta N /N_{\mathrm{tot}}) /
(\Delta T_0/T_F)$ where $\Delta T_0$ is the initial temperature difference $\Delta T_0$ [@materialsandmethods]. In figure \[fig:ballistic\]D we present the maximum thermoelectric response $\mathcal{R}_{\mathrm{max}}$ which displays an approximately linear increase with $\nu_z$.
![ Thermoelectric response in the ballistic to diffusive crossover. A: Time evolution of $\Delta N/N_{tot}$ for increasing disorder strength, for a fixed confinement of $\nu_z = 3.5\,$kHz. Solid lines: theory. B: Fitted timescale $\tau_0$ (black) and $\mathcal{R}_{max}$ (red) as a function of disorder strength for the data set shown in A. C: Rescaled time evolution of $\mathcal{R}$ (see text) in the regime of strong disorder from $\bar{V} = 542\,$nK to $1220\,$nK and fixed $\nu_z = 4.95\,$kHz. Black line: theoretical calculation. Here, $\mathcal{R}$ depends only on the overall timescale, and all the curves collapse after rescaling the time axis. D: Comparing $\mathcal{R}_{\mathrm{max}}$ as a function of timescale for the diffusive (gray points) and ballistic case (open circles). []{data-label="fig:diffusive"}](fig3.pdf)
We now investigate the effects of disorder on the thermoelectric properties of the channel. We project a random potential of adjustable strength $\bar{V}$ onto the channel in the form of a blue detuned laser speckle pattern [@materialsandmethods]. Figure \[fig:diffusive\]A presents the time evolution of $\Delta N/N_{tot}$ for increasing disorder, for fixed $\nu_z = 3.5\,$kHz. First we observe that the time scale $\tau_0$ of the transport process increases: the resistance increases as the channel crosses over from ballistic to diffusive. In addition to this slowdown, $\mathcal{R}_{\mathrm{max}}$ increases from $0.17(8)$ without disorder to $0.55(16)$ for a strong disorder of $1.1\,\mu$K.
For the strongest disorder, we observe (Fig. \[fig:diffusive\]B) that the thermoelectric response saturates, while the timescale $\tau_0$ keeps increasing, indicating the continuous increase of resistance with disorder. To further investigate this point, we performed experiments for a fixed confinement in the channel of $4.95\,$kHz, and several large disorder strengths ranging from $542$ to $1220$nK. As illustrated in Fig. \[fig:diffusive\]C, we find that, in this regime, the full data-set collapses to a single curve, provided the time axis is rescaled by the timescale $\tau_0$ extracted from an independent atomic conduction experiment [@Brantut:2012aa; @materialsandmethods]. This shows that thermoelectricity is independent of the actual resistance, as expected from the fact that thermopower is a ratio of linear response coefficients. This makes thermopower less sensitive than resistance to the details of the conductor, a fact widely used in condensed matter physics [@Mokashi:2012aa].
The effect of disorder can be described by extending our theoretical model [@materialsandmethods], introducing an energy-dependent transparency of the constriction. This transparency involves the energy-dependent mean-free path in the channel, product of the particle velocity and scattering time. At strong disorder, the scaling of the scattering time with disorder is modelled assuming that the energy-dependence of the scattering time can be neglected. When expressed as a function of $t/\tau_0$, a unique theoretical scaling curve independent of $\bar{V}$ is predicted for the thermoelectric response, which describes well the experimental data (Fig. \[fig:diffusive\]C). The time-scale $\tau_0$ is the only adjustable parameter in this strong disorder regime. A fit to the experimental data allows one to extract the dependence of $\tau_0$ on disorder strength [@materialsandmethods].
Extrapolating this dependence into the weak disorder regime, the resulting theoretical curves for the transient particle imbalance accurately describe the data over the entire range of disorder strengths (Fig.\[fig:diffusive\]A).
Confinement and disorder are two independent ways of influencing the thermoelectric properties. To compare their respective merits, we display $\mathcal{R}_{\mathrm{max}}$ as a function of $\tau_0$ in Fig \[fig:diffusive\]D. This shows that disorder is more efficient than confinement to increase thermoelectricity. For the largest time scales, we observe a $\sim 3$-fold increase in the diffusive case compared to the ballistic one. This is due to the stronger energy-dependence of transmission in the diffusive compared to the ballistic case.
![ Our system as a heat engine. A: Evolution of the hot (red) and cold (blue) reservoir in the $\mu$-$N$plane for $\nu_z = 3.5\,\mathrm{kHz}$ and $\bar{V}=542\,\mathrm{nK}$. Experiments: symbols, Theory: solid lines. The black arrows indicate the direction of time and the sum of the enclosed areas yields the total work. B, D, F: Efficiency, power and dimensionless figure of merit of the channel in the ballistic case, as a function of confinement. C, E, G: The same quantities as a function of disorder strength for $\nu_z = 3.5\,$kHz. Orange symbols: experiments; Black symbols: theory (theoretical error bars estimated from the uncertainties on the input parameters). []{data-label="fig:efficiency"}](fig4.pdf)
In the experiment, a controlled exchange of heat between a hot and a cold reservoir is used to produce a directed current, i.e. work. This motivates an analysis in terms of heat engines. To do so, we evaluate the work, efficiency, and power of the process. The area enclosed in the $\mu-N$ plane (Fig. \[fig:efficiency\]A) represents the work $W = 1/2\int_0^\infty dt
(\mu_c-\mu_h)(t) I_N(t)$ produced during the evolution, which we evaluate from $N_{c,h}$ and $T_{c,h}$. Similarly, we evaluate the heat associated to the transport process $Q = -1/2\int_0^\infty dt
(T_c-T_h)(t) I_S(t)$ [@materialsandmethods].
We then introduce the relative efficiency $\eta=W/Q$ [@Goldsmid; @materialsandmethods]. For a reversible process $\eta=1$. We find that $\eta$ is largest for configurations where the thermoelectric response is largest (Fig \[fig:efficiency\]B and C). The large value of the efficiency suggests that the channel is a very good thermoelectric material. Generally, the efficiency increases as the dynamics slows down, since the thermodynamic processes become closer to reversibility. Therefore, a complementary criteria to evaluate the merits of the various thermoelectric configurations is the cycle-averaged power [@Callen:Thermodynamics], estimated by $W/\tau_0$. As shown in figure \[fig:efficiency\]D and E, the power goes down for the largest resistances, since the increase in work is overcompensated by the slow-down of the process.
We now focus on the channel, independently of the reservoirs. We use the transport coefficients extracted using our model to estimate the dimensionless figure of merit $ZT= \alpha_{ch}^2 /L$ [@Goldsmid] of the channel (Fig \[fig:efficiency\]F and G). This number is related to the efficiency achievable with a channel operating at maximum power, and is used as a criterion for engineering thermoelectric devices [@DiSalvo:1999aa]. For the largest thermoelectric response observed, we infer $ZT=2.4$, which is among the largest values observed in any solid-state material [@snyder2008com].
Our experiment demonstrates thermoelectric effects in quantum gases and shows that thermopower is a sensitive observable in this context. We have used it as a probe for the ballistic or diffusive nature of transport, independently from resistance. A high efficiency and figure of merit compared to many materials are found due to the absence of phonon contributions to the heat conductivity [@Goldsmid]. Our technique can be straightforwardly generalised to interacting systems, where thermoelectric properties are of fundamental interest [@Behnia:2004aa; @Zhang:2011aa; @Micklitz:2012aa]. The reversed operation of our device leads in principle to cooling by the Peltier effect. This may be useful to cool quantum gases to low entropy, needed to explore strongly correlated fermions in lattices.
The Zürich team acknowledges fruitful discussions with J. Blatter at the initial stage of the experiment and CK thanks M. Büttiker for helpful discussions. We acknowledge financing from NCCR MaNEP and QSIT of the SNF, the ERC Project SQMS, the FP7 Project NAME-QUAM, the ETHZ Schrödinger chair, the DARPA-OLE program, ANR (FAMOUS), and ETHZ. J.P.B. is partially supported by the EU through a Marie Curie Fellowship.
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Materials and Methods {#materials-and-methods .unnumbered}
=====================
Details on the experimental setup {#details-on-the-experimental-setup .unnumbered}
---------------------------------
### Preparation of the clouds {#preparation-of-the-clouds .unnumbered}
The preparation of the clouds follows the method used in our previous studies [@Brantut] with minor modifications. The $^6$Li atoms are prepared in an incoherent mixture of the lowest and third hyperfine states, and placed in a homogeneous magnetic field of 388G, where the scattering length is about $-800$a$_0$. The cloud is then evaporatively cooled in a hybrid magnetic and optical dipole trap at $1064$nm, with a waist of $21\,\mu$m as in [@Brantut]. To avoid anharmonicity and thus allow accurate thermometry, the cloud is then transferred in a dipole trap propagating along the same direction with a waist of $70\,\mu$m. We use a final laser power of $800$mW for this trap, yielding transverse trap frequencies of $414\,$Hz, with negligible ellipticity. At this laser power, the trap depth is $6.3\mu$K, ensuring that no further evaporation takes place during the measurements.
After the transfer into the final dipole trap, the magnetic field is increased up to 552G, where the scattering length is about $-100$a$_0$. At this magnetic field, the longitudinal trap frequency originating from the curvature of the field is $23.5\,$Hz, measured using the dipole oscillations of a cloud.
### Trap configuration {#trap-configuration .unnumbered}
Like in [@Brantut], we superimpose on the cloud a laser beam at $532$nm having a TEM$_{01}$-like mode profile [@Meyrath:2005aa], propagating along the $x$ direction. This creates the channel at the center of the trap, connected on both sides to the reservoirs. The waist of this laser beam along the long direction of the cloud ($y$ direction) is $30\,\mu$m, about ten times shorter than the cloud. Varying the power of this laser beam allows to tune the confinement along the $z$ direction in the channel while leaving the reservoirs unchanged.
We create disorder by illuminating the channel with a speckle pattern created by a laser at $532$nm and propagating along the $z$ direction. The speckle pattern is centered on the channel and has an envelope with a $1/e^2$ radius of $36.1$ and $40.8\,\mu$m along the $x$ and $y$ directions. The correlation length of the intensity distribution is $370\,$nm ($1/e$ radius of the autocorrelation function evaluated using a Gaussian fit). The disorder strength is the disorder-averaged potential at the centre of the pattern, evaluated from the total laser power and the polarisability of lithium.
The gate beam, which is used to isolate the two reservoirs from each other, is created using a laser beam at $532\,$nm propagating also along the $z$ direction and focused onto the center of the channel, with waists of $43.3$ and $2.6\,\mu$m along the $x$ and $y$ directions. When turned on, the height of the potential hill is larger than $30\,\mu$K, thereby ensuring the complete isolation of each reservoir.
The heating beam consists in a laser beam at $767\,$nm propagating along the $z$ direction with a small angle so that it is focused in one of the reservoirs with a waist of $4.95(2)\,\mu$m. Once the cloud has been separated in two independent reservoirs by the channel and the gate beam (see the main text for the description of the sequence), the heating beam is ramped up and sinusoidally modulated at $660$Hz for $1$s. The laser beam is then switched off before the gate beam is removed.
Systematics and error bars {#systematics-and-error-bars .unnumbered}
--------------------------
Temperature, $\Delta N/N_{\mathrm{tot}}$ and their errors are inferred from 2-dimensional Fermi fits performed on typically three averaged pictures (see, for example fig. 1 in the main text and fig. \[fig:fig1\_som\] here). We separately fit the two reservoirs and checked that it gives consistent results with a fit on a full cloud that has twice the atom number of one reservoir. We also checked that the atom number inferred from simply counting and the Fermi fit is the same within our measurement uncertainty. The position of the gate beam defines the separation of the two reservoirs. In order to avoid effects from the gate beam on the Fermi fits we exclude a band of $\pm 4 $ pixels around the center. The center position has to be carefully adjusted. A misalignment by 1 pixel gives a systematic shift of $\Delta N/N_{\mathrm{tot}} \lesssim 0.02$.
Throughout the text, the error bars represent one standard deviation. The errors on the total atom number and $T_F$ indicated in the main text are the standard deviation evaluated between all the different data sets displayed in the paper. Within a given set, corresponding to one choice of disorder or confinement, the fluctuations of atom number are at most 5%, the temperature fluctuations are below 8%. The error bars displayed for the time constant $\tau_0$ are fitting errors and for $\mathcal{R}$ they are the combination of the error in $\Delta N/N_{\mathrm{tot}}$ and the initial temperature difference $\Delta T_0$.
The error bars on the experimentally measured efficiencies and power displayed in fig. 4 in the main text are coming from gaussian error propagation of the involved quantities $\mu, N, T, S, \tau_0$. The error bars on the theoretically expected efficiencies, power and $ZT$ result from the propagation of the uncertainties on the experimentally determined parameters.
![Averaged images of the two reservoirs after the transport process has taken place and after time of flight. Images A,B,C,D are taken for evolution times of $0.01,\,0.37,\,0.98,\,3.97$ s respectively, where A,C,D correspond line sums shown in fig. 1B,C,D in the main text. Image B is taken where $\Delta N/N_{\mathrm{tot}}$ is maximal. The thin region without atoms is the position of the gate beam. The white box in picture A indicates the part that is excluded for the fitting. The schematic view on the right side illustrates the actual configuration of the experiment.[]{data-label="fig:fig1_som"}](figS1.pdf){width="60.00000%"}
Response to initial temperature imbalance {#response-to-initial-temperature-imbalance .unnumbered}
-----------------------------------------
We have repeated our measurements on the disordered channel having $4.95$kHz confinement and speckle power $542$ nK with various temperature bias. The maximum atom number imbalance observed during the time evolution is presented on figure \[fig:linResp\_amp\_ts\]. The linear increase of the imbalance with heating supports that our measurements remain in the linear response regime, despite the relatively large temperature bias.
![Evolution of the maximal relative imbalance as a function of the initial temperature difference, in blue (left axis). The black curve is a linear fit. The amplitude of the thermoelectric effect remains linear for large values of the initial temperature difference. Timescale vs initial temperature difference, in red (right axis).[]{data-label="fig:linResp_amp_ts"}](figS2.pdf){width="60.00000%"}
In addition to the amplitude, a supplementary check on the timescale has been performed, and is also reported in fig. \[fig:linResp\_amp\_ts\]. It shows that the timescale remains constant over a wide range of initial temperature difference, indicating the linear behavior. Only for the largest values of $\Delta T_0$ one sees a inclination towards shorter particle timescales, which we interprete as a sign of a deviation from linear response. The values of $\frac{\Delta T_0}{T_F}$ considered in the main text are between $0.18$ and $0.25$, a range in which we expect our linear response model to be valid.
Theoretical model {#theoretical-model .unnumbered}
-----------------
The transport setup under consideration is depicted in Fig.1 in the main text. Two reservoirs of fermionic atoms are connected along the $y$-direction by a constriction. The reservoirs are described as three-dimensional noninteracting trapped Fermi gases, in a harmonic potential in the $x-z$ directions, and half-harmonic in the $y$-direction. The reservoir state is characterized by the temperature $T_{h,c}$ and their chemical potential $\mu_{h,c}$, where $h,c$ label the hot and cold reservoir, respectively. The particle number $N(T,\mu)$ and entropy $S(T,\mu)$ of each reservoir are given through the grand-canonical equation of state. It is convenient to introduce the average of a quantity $X$ defined by $\bar{X}=\frac 1 2 (X_h+X_c)$ and its difference $\Delta X= (X_c-X_h)$. The channel is oriented along direction $y$ and a harmonic confinement is present in its transverse ($x-z$) direction. We take the temperature and chemical potential of the channel to be the average temperature and chemical potential of the reservoirs. In the following we describe in more detail how to obtain the transport equation (eq (2) in the main text) for this setup within linear response.
### Model for the reservoirs {#model-for-the-reservoirs .unnumbered}
In the described reservoir configuration, the energy levels are given by $\varepsilon =
\hbar\omega_x(n_x+1/2)+\hbar\omega_z(n_z+1/2)+\hbar\omega_y(2n_y+3/2)$, where $n_i$ labels the energy level and $\omega_i$ the trapping frequency along direction $i$. The Fermi energy of the reservoirs is typically $E_F \simeq 930\,$nK and their temperature $T/T_F \simeq 0.25$, such that we are in the limit where $E_F, k_BT \gg \text{max}(\omega_x,\omega_y, \omega_z)$. Hence, one can neglect the discrete structure of the energy levels and the thermodynamic properties of the reservoirs, the compressibility $\kappa$, the dilatation coefficient $\gamma$ and the heat capacity $C_{N}=\left.
\frac{\partial S}{\partial T}\right|_{N}$, are well captured by the following formulae, with the density of states $g_{r} (\varepsilon)=
\frac{\varepsilon^2}{4(\hbar\omega_x\hbar\omega_y\hbar\omega_z)}$ : $$\begin{aligned}
\label{eq:thcoeffres_1}
\kappa =\left. \frac{\partial N}{\partial \mu}\right|_T &=&
\int_0^\infty
d\varepsilon
g_{r}(\varepsilon)\left(-\frac{\partial f}{\partial \varepsilon}\right)\\
\gamma =\left.\frac{\partial
N}{\partial T}\right|_\mu=\left.\frac{\partial
S}{\partial\mu}\right|_T&=& \int_0^\infty d\varepsilon
g_{r}(\varepsilon) \left(\varepsilon-\mu\right)\left(-\frac{\partial
f}{\partial
\varepsilon}\right)
\label{eq:thcoeffres_2}\\
\label{eq:thcoeffres_3}
\frac{C_N}{T} + \frac{\gamma^2}{\kappa} &=& \int_0^\infty d\varepsilon
g_{r}(\varepsilon) \left(\varepsilon-\mu\right)^2\left(-\frac{\partial
f}{\partial \varepsilon}\right)\,,\end{aligned}$$ where $f(\varepsilon) = \frac{1}{1+e^{\beta(\epsilon-\mu)}}$ is the Fermi-Dirac distribution. The thermodynamic coefficients are calculated at the average temperature $\bar{T}$, particle number $(N_c+N_h)/2$ and chemical potential $(\mu_c+\mu_h)/2$. In the main text, the dilatation properties of the gas are represented by the reservoir contribution $\alpha_r = \frac{\gamma}{\kappa}$ to the total thermopower.
### Model for the channel {#model-for-the-channel .unnumbered}
The channel is modeled by a linear circuit element at the average temperature $\bar{T}$ and chemical potential $\bar{\mu}$. Its linear transport coefficients are given by the following expressions : $$\begin{aligned}
\label{eq:conductance}
G &= \frac{1}{h}\int_0^\infty d\varepsilon\, \Phi(\varepsilon)
\left(-\frac{\partial f}{\partial \varepsilon}\right)\\
\label{eq:thermopower}
T\alpha_{ch}G &= \frac{1}{h}\int_0^\infty d\varepsilon\,
\Phi(\varepsilon)
\left(\varepsilon-\mu\right)\left(-\frac{\partial f}{\partial
\varepsilon}\right)\\
\label{eq:th_conductance}
\frac{G_T}{T}+G\alpha_{ch}^2 &= \frac{1}{h}\int_0^\infty
d\varepsilon\,
\Phi(\varepsilon) \left(\varepsilon-\mu\right)^2\left(-\frac{\partial
f}{\partial
\varepsilon}\right)\,\end{aligned}$$ where $\Phi$ is the transport function of the channel, which plays for the channel a role equivalent to that of the density of states for the reservoirs. $G$ is the conductance, $G_T$ the thermal conductance and $\alpha_{ch}$ the thermoelectric response of the channel. A simple interpretation of $\Phi (\varepsilon)$ is the number of channels available for a particle having an energy $\varepsilon$, since at zero-temperature its value at the Fermi energy is directly related to the conductance $G
(T=0K) = \frac{\Phi(E_F)}{h}$.
For free particles of mass $M$ propagating along the $y$-direction and harmonically confined in the transverse ($x-z$) direction, it is given by : $$\begin{aligned}
\Phi (\varepsilon) & = &
\sum_{n_z=0}^{\infty}\sum_{n_x=0}^{\infty}\int_0^\infty
dk_y\,\frac{\hbar k_y}{M}\mathcal{T}(k_y)
\delta\left(\varepsilon-\hbar\omega_x(n_x+1/2)-\hbar\omega_z(n_z+1/2)-\frac
{\hbar^2k_y^2}{2M}\right)\label{eq:transport_eq}\\
{} & = & \sum_{n_z=0}^{\infty}\sum_{n_x=0}^{\infty}
\mathcal{T}(\varepsilon-\hbar\omega_x(n_x+1/2)-\hbar\omega_z(n_z+1/2))
\vartheta(\varepsilon-\hbar\omega_x(n_x+1/2)-\hbar\omega_z(n_z+1/2))\,,\end{aligned}$$ where $\mathcal{T}$ is the transmission probability which depends on the momentum or energy along the $y$ direction in the first and second line, respectively. The difference between the various transport regimes is contained in the (energy or momentum dependent) transmission probability $\mathcal{T}$.\
In , the energy conservation condition states that a particle entering the channel will distribute its energy between kinetic (propagation with a certain momentum along $y$) and confinement (populating a tranverse mode along $x$ and $z$). The energy dependence of $\Phi$ close to the chemical potential is responsible for a particle-hole asymmetry which enhances the value of the thermopower $\alpha_{ch}$. This effect is larger when the energy dependence is stronger. This is for example the case when the conduction regime goes from ballistic to diffusive, or when the confinement increases.\
In the ballistic case, the transmission probability of the channel is equal to one for all energies. For ballistic conduction, the transport function is in good approximation equal to $\Phi(\varepsilon) \simeq
\frac{1}{2}\left(\frac{\varepsilon}{\hbar\omega_x}+1\right)\left(\frac{
\varepsilon } {
\hbar\omega_z}+1\right)$. Since a larger confinement induces a steeper energy dependence of the transport function $\Phi$ around the chemical potential, the resulting thermopower of the channel $\alpha_{ch}$ is a growing function of $\omega_z$ and $\omega_x$.
In the diffusive case, the constrain that the channel has to obey Ohm’s law leads to the following effective transmission probability for a channel of length $\mathcal{L}$ [@datta1997electronic] : $$\label{eq:transmission probability_general}
\mathcal{T}(\varepsilon,\bar{V}) =
\frac{l(\varepsilon,\bar{V})}{\mathcal{L}+l(\varepsilon,\bar{V})}\,,$$ where $l(\epsilon,\bar{V}) = \tau_s(\bar{V},\varepsilon)v(\varepsilon)$ is the energy dependent mean free path, $\tau_s$ being the scattering time, $v$ the velocity of the carriers, and the speckle power $\bar{V}$. We assume that the scattering time has no energy dependence, and is dominated by its dependence on the speckle power $\bar{V}$.
In the regime of low speckle power, where the scattering time is expected to be very long, the mean free path becomes typically larger than the size of the system $l \gg \mathcal{L}$, and the transmission probability tends towards one, as expected in the ballistic regime.
In contrast, in the regime of strong speckle power, the mean free path is expected to be very small, leading to an effective form of the transmission probability $$\label{eq:effective}
\mathcal{T}(\varepsilon,\bar{V}) \simeq \frac{\tau_s(\bar{V})
v(\varepsilon)}{\mathcal{L}}\,.$$ This corresponds to the solution of the Boltzmann equation [@Ashcroft]. In this situation the resulting thermopower $\alpha_{ch}$ is independent of the scattering time and thus of the details of the speckle potential since it is the ratio of equations and . This prediction is confirmed by the experimental measurements which are independent of the speckle power for strong disorder (see main text). In this diffusive regime, the timescale is given by : $$\label{eq:TS_diffusive}
\tau_0^{-1} =
\frac{4}{3}\frac{\nu_x\nu_y}{\nu_z}\frac{F[3/2,\bar{\mu}/k_B\bar{T}]}{F[1,
\bar{\mu}/k_B\bar{T}]}\frac{\tau_s\sqrt{2k_B\bar{T}/M}}{\mathcal{L}}\,$$ where $F[n,x]$ is a Fermi-Dirac integral [@AStegun]. The result is then used to find the behaviour of the scattering time $\tau_s$ as a function of speckle power with the ansatz $\tau_s = A(\bar{V})^{-B}$. Fitting this form to the time scale at strong speckle gives an exponent $B = 1.51 \pm 0.02$ ,$A = 6.1 \pm
0.25\,$(a.u.) and $A=6.8\pm0.25\,$(a.u.) at $\nu_z=3.5$kHz and $\nu_z=4.95$kHz respectively. This form for the scattering time is then extrapolated for lower speckle powers.
### Evolution of the particle imbalance and temperature difference {#evolution-of-the-particle-imbalance-and-temperature-difference .unnumbered}
Using the linear response equations given in the main text and the properties of the reservoirs given by , and , we derive equations ruling the time evolution of the particle and temperature imbalance: $$\tau_0\frac{d}{dt}
\left(
\begin{array}{c}
\Delta N/\kappa\\
\Delta T
\end{array}
\right)
=
-\underline{\Lambda}
\left(
\begin{array}{c}
\Delta N/\kappa\\
\Delta T
\end{array}
\right) ,
\underline{\Lambda}=\begin{pmatrix}
1 & -\alpha\\
-\frac{\alpha}{\ell} & \frac{L+\alpha^2}{\ell}
\end{pmatrix}\,,
\label{eq:Transport_equation}$$ with $\tau_0 = \kappa/G$. The effective thermoelectric (Seebeck) coefficient is $\alpha\equiv\alpha_r-\alpha_{ch}$ in which the competition between the reservoir contribution and the contribution of the channel is evident.
In the absence of any thermoelectric effect ($\alpha=0$), the time constants for particle and thermal relaxation are $\tau_0$ and $\tau_0
L/\ell$, respectively. At low temperature (typically below $T/T_F=0.1$), the ratio $L/\ell$ tends to one, and the timescales for heat and particle transport are equal, as a consequence of the Wiedemann-Franz law [@Ashcroft].
The direct integration of the set of equations provides the time evolution of the particle imbalance and the temperature difference : $$\begin{aligned}
\label{eq:particle_number_result}
(N_c-N_h)(t) &= \left\lbrace
\frac{1}{2}\left[e^{-t/\tau_-}+e^{-t/\tau_+}\right]
+\left[1-\frac{L+\alpha^2}{\ell}\right]
\frac{e^{-t/\tau_-}-e^{-t/\tau_+}}{2(\lambda_+-\lambda_-)}
\right\rbrace \Delta N_0
+\frac{\alpha\kappa}{\lambda_+-\lambda_-}\left[e^{-t/\tau_-}-e^{-t/\tau_+}
\right ]\Delta T_0\\
(T_c-T_h)(t) &= \left\lbrace
\frac{1}{2}\left[e^{-t/\tau_-}+e^{-t/\tau_+}\right]
+\left[\frac{L+\alpha^2}{\ell}-1\right]\frac{e^{-t/\tau_-}-e^{-t/\tau_+}}{
2(\lambda_+-\lambda_-)}\right\rbrace \Delta
T_0+\frac{\alpha}{\ell\kappa(\lambda_+-\lambda_-)}\left[e^{-t/\tau_-}-e^{
-t/\tau_+}\right]\Delta N_0
\label{eq:temperature_result}\end{aligned}$$ The initial temperature difference and particle imbalance are denoted by $\Delta
T_0$ and $\Delta N_0$, respectively. The inverse time-scales $\tau_\pm^{-1}=\tau_0^{-1}\lambda_\pm$ are given by the eigenvalues of the transport matrix $\underline{\Lambda}$ $$\label{eq:timescales}
\lambda_\pm =
\frac{1}{2}\left(1+\frac{L+\alpha^2}{\ell}\right)\pm\sqrt{\frac{\alpha^2}{
\ell} +\left(\frac{1}{2}-\frac{L+\alpha^2}{2\ell}\right)^2}\,.$$
All the effective transport coefficients are ratios that depend only on the variable $\frac{\mu}{k_BT}$.
### Fitting procedure {#fitting-procedure .unnumbered}
We have used the solution and to the transport equations in order to extract the time scale $\tau_0$ from the experimental data. For each value of the parameters (confinement along the $z$-direction, disorder strength and temperature), two experimental sequences have been performed :
(1) A ’thermoelectric’ sequence, with dominating initial temperature difference which is prepared by the procedure described in the main text. The resulting evolution is a coupled evolution of the particle and temperature differences as described in the main text.
(2) A ’decay’ sequence, with initial particle imbalance, at constant temperature equal to the final temperature in sequence (1).
The time scale shown in the main text is extracted from the first sequence. However, in order to check consistency we compare in fig. \[fig:TS\] the time scales extracted from the two sequences. The two independent fits agree very well also in the ballistic case with the parameter free theoretical calculations.
![Timescale $\tau_0$ extracted from a fit of the particle number imbalance in sequence (1) (in blue) and sequence (2) (in red), vs confinement in a ballistic channel (left panel), and vs speckle power at fixed confinement $\nu_z = 3.5\,$kHz (right panel). The black line in the left panel corresponds to an ab initio prediciton for the timescale from the expressions and .[]{data-label="fig:TS"}](figS3a.pdf "fig:"){width="45.00000%"} \[fig:TSB\]
![Timescale $\tau_0$ extracted from a fit of the particle number imbalance in sequence (1) (in blue) and sequence (2) (in red), vs confinement in a ballistic channel (left panel), and vs speckle power at fixed confinement $\nu_z = 3.5\,$kHz (right panel). The black line in the left panel corresponds to an ab initio prediciton for the timescale from the expressions and .[]{data-label="fig:TS"}](figS3b.pdf "fig:"){width="45.00000%"} \[fig:TSD\]
### Efficiency {#efficiency .unnumbered}
In this section we will derive the expression for the efficiency employed in the main text. In order to do this, we use the conservation of the total energy and particle number present in the setup, i.e. $$\begin{aligned}
dE_{tot}=d(E_{\mathrm{ch}}+E_c+E_h)&=0\\
dN_{tot}=d(N_c+N_h)&=0\,,\end{aligned}$$ where the subscript $\mathrm{ch}$ denotes quantities in the channel. For the particle conservation, we have assumed that in the channel the particles do not accumulate, i.e. $dN_{\mathrm{ch}}=0$. Reexpressing the change of the energy by the change of entropy and particle number in each part of the system $ dE_x = T_xdS_x+\mu_xdN_x$ and using particle number conservation, we obtain $$dE_{tot}= \left(\mu_h-\mu_c\right)dN_h +\left(T_{\mathrm{ch}}dS_{\mathrm{ch}}+T_cdS_c+T_hdS_h\right)=0.$$ Reordering and the derivation with respect to time leads to an expression for the total change of entropy per time given by $$\frac{d(S_{\mathrm{ch}}+S_h+S_c)}{dt} = -\frac{\Delta T}{2\bar{T}}(\dot{S}_h
- \dot{S}_c)-\frac{\Delta \mu}{2\bar{T}}(\dot{N}_h
- \dot{N}_c) = -\frac{\Delta T}{2\bar{T}}I_S-\frac{\Delta
\mu}{2\bar{T}}I_N\,.$$ Here we used the average temperature $\bar{T}=\frac 1 2 (T_c+T_h)=T_{\mathrm{ch}}$.
Thus, the efficiency defined by the ratio of the work to irreversible heat $$\label{eq:efficiency_result}
\eta = -\frac{\int (\dot{N_c}-\dot{N_h})\cdot(\mu_c-\mu_h)dt}{\int
(\dot{S_c}-\dot{S_h})\cdot(T_c-T_h)dt}$$ measures the efficiency relative to a reversible process : $\eta=1$ when the evolution is reversible.
Using the solution for particle imbalance and temperature difference yields the following expression as a function of the effective transport coefficients : $$\eta = \frac{-\alpha\alpha_r}{\ell+L+\alpha^2-\alpha\alpha_r}\,.$$
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|
\#1[$\, #1\,$]{}
[**Acceleration at Relativistic Shocks\
in Gamma-Ray Bursts**]{}
[**Matthew G. Baring$^{1,2}$\
**]{} [*$^{1}$Laboratory for High Energy Astrophysics, NASA/GSFC, Greenbelt, MD 20771, USA\
$^{2}$Universities Space Research Association*]{}
[**Abstract\
**]{}
Most recent extragalactic models of gamma-ray bursts consider the expansion of a relativistic blast wave, emanating from a solar-mass type progenitor, into the surrounding interstellar medium as the site for their activity. The popular perception is that the optical afterglows result from the external shock interface, while the prompt transient gamma-ray signal arises from multiple shocks internal to the expansion. This paper illustrates a number of acceleration properties of relativistic and ultrarelativistic shocks that pertain to GRB models, by way of a standard Monte Carlo simulation. Computations of the spectral shape, the range of spectral indices, and the energy gain per shock crossing are presented, as functions of the shock speed and the type of particle scattering.
Introduction {#intro.sec}
============
The gamma-ray burst (GRB) field has burgeoned in the last few years, and particularly after the discovery (e.g. van Paradijs, et al. 1997; Costa, et al. 1997; Frail, et al. 1997) of optical, radio and X-ray transient counterparts, and the subsequent identification of a cosmological redshift via atomic absorption features in the 8th May 1997 afterglow (Metzger et al. 1997). Theoretical interpretations abound, mostly focusing on some variation of a blast wave expansion impacting on the interstellar medium (ISM) surrounding the burst (e.g. Meszaros & Rees 1993). Supersonic blast wave impact upon the surrounding ISM guarantees relativistic shock formation, basically a relativistic version of supernova remnants; the dissipation of the ram pressure kinetic energy via diffusive particle acceleration in the shock is the commonly-invoked means of converting the bulk motion into a viable supply of energy for radiative purposes. Principal quantities generally appearing in radiation emission models of GRBs include the Lorentz factor of the shock, and the spectral index of the electron (and perhaps ion) population. Furthermore, modelling of ultra-high energy cosmic ray production by GRBs (e.g. Waxman 1995; Vietri 1995) requires knowledge of the mean energy gain a particle experiences in traversing the shock. To date, GRB models of both the transient gamma-ray event and fireball afterglows have invoked only the crudest notions of Fermi acceleration. The aim of this presentation is to probe these shock acceleration properties, which are pertinent to GRB blast wave models, using results from a Monte Carlo simulation of diffusive acceleration.
The Monte Carlo technique we employ here has been described in detail in numerous expositions (Ellison, Jones & Eichler 1981; Jones and Ellison 1991; Baring, Ellison & Jones 1993; Ellison, Baring, & Jones 1996). The simulation technique is a kinematic model. Particles are injected upstream and allowed to convect into the shock, colliding with postulated scattering centers (presumably magnetic irregularities) along the way. As they diffuse between the upstream and downstream regions, they continually gain energy. An important property of the model is that it treats thermal particles like accelerated ones, making no distinction between them. Hence, as the accelerated particles start off as thermal ones, this technique automatically injects particles from the thermal population into the acceleration process. One valuable consequence of this unified treatment is that modification of the shock hydrodynamics by the accelerated population can easily be incorporated. Such non-linear hydrodynamics are omitted in the present [*test-particle*]{} application, though they will probably be an important aspect of the GRB acceleration problem given their relevance to the modelling of supernova remnant emission (e.g. Baring et al. 1999).
The test-particle results presented here use a guiding-center version of the Monte-Carlo technique, older than our latest codes which compute particle gyro-orbits exactly rather than just track the center of gyration. The guiding-center approach, which is detailed in Baring, Ellison & Jones (1993), is often expedient, and is entirely appropriate to plane-parallel shock applications (where the field lies along the shock normal, i.e. ), which form the focus here. This method is precisely that implemented in Ellison, Jones & Reynolds’ (1990, hereafter EJR90) treatment of parallel relativistic shocks, thereby providing a principal motivation for adhering to a similar approach. The updated code replicates results obtained in EJR90. Following EJR90, both large angle scattering (LAS) and pitch angle diffusion (PAD) will be implemented here. For LAS, the mean-free path in the fluid frame is constrained to be proportional to a particle’s gyroradius , in accord many previous expositions (e.g. EJR90, Baring, Ellison & Jones 1993, hereafter BEJ93; Ellison, Baring, & Jones 1996), while for PAD we set to be independent of , following EJR90.
Results {#results.sec}
=======
The primary purpose of this paper is to extend the work of EJR90 to ultrarelativistic shocks, and to explore spectral properties of results from our simulation in the context of gamma-ray bursts. Representative ion distributions obtained in the rest frame of the shock are depicted in Figure 1 for . The notations used are () for the Lorentz factor of the upstream (downstream) flow speed () in the shock rest frame, and for the velocity compression ratio in this frame. The value is chosen to mimic expectations from an ultrarelativistic gas, though other values are possible due to possible mildly-relativistic nature of the downstream gas. The upstream gas temperature and magnetic field are kept sufficiently low to maintain the strength of high Mach number shocks and reduce the number of parameters to which results are sensitive.
The left panel of Figure 1 illustrates how a smooth power-law spanning many decades can be obtained at energies not too far in excess of thermal ones. The spectrum is somewhat steeper, though not markedly so, than the canonical particle distribution for strong non-relativistic shocks. For slower shock speeds, the PAD-generated power-laws are steeper still (e.g. Kirk & Schneider 1987; EJR90), while for faster shocks appropriate to GRB scenarios, the spectral index saturates around the value of 2.2 when , as found by Bednarz & Ostrowski (1998). Index determinations are listed in Table 1, and while here we replicate Bednarz & Ostrowski’s (1998) asymptotic behaviour, we also find that monotonically decreases with , in contradiction to their findings in the range around . Since the simulation reproduces the analytic results of Kirk & Schneider (1987) at of considerably higher than 2.2, our findings of monotonicity appear plausible.
[cccc]{}\
& & PAD & LAS\
& & () & ()\
\
\
2.29 & 0.9 & 2.34 & 1.81\
3 & 0.9428 & 2.23 & 1.59\
5 & 0.9798 & 2.22 & 1.49\
9 & 0.9938 & 2.20 & 1.41\
27 & 0.9993 & 2.19 & -\
81 & 0.9999 & 2.18 & -\
Table 1: Asymptotic spectral indices for plane-parallel () relativistic shocks of various Lorentz factors and compression ratio for the cases of pitch angle diffusion (PAD), and large angle scattering (LAS). Simulational uncertainties in predicting are typically of the order of 1–2% for PAD, and twice that for LAS. Spectra for the case are exhibited in Figure 1.
The right panel of the Figure depicts a spectrum obtained for LAS with similar parameters. Two striking features emerge: (i) that the asymptotic power-law is generated only at much higher energies than in the PAD case, and (ii) when it is obtained, it is much flatter than for PAD. Both are essentially due to the prompt removal of particles by single large angle scatterings from the narrow Lorentz cone of directions for which the ions can remain upstream of the shock. PAD and LAS generate different angular distributions ahead of a relativistic shock, and the fact that these result in dissimilar spectral indices is widely understood (e.g. see EJR90 and references therein). The spectral structures above the thermal peak for LAS correspond to contributions from successive shock crossings in a manner somewhat like the structure seen at mildly suprathermal energies in non-relativistic shocks (e.g. BEJ93). These structures start out flat due to a kinematic spread induced by LAS, and then slowly steepen and merge into the power-law continuum. The trend for LAS is again that for faster shocks, declines (see Table 1). However, any putative saturation could not be demonstrated numerically, due to difficulties in generating good statistics at energies well in excess of eV.
0.0truecm 0.5truecm
The maximum energy of particles in the downstream region for 1,3,5 and 7 shock-crossings are also depicted as vertical dashed lines in Figure 1. For LAS, it is clear that this scales as , and energy amplification per shock crossing that is widely (and erroneously) quoted in GRB model literature. This represents the maximum amplification, and the [*mean amplification is much less*]{}, declining with increasing energy. This contention follows immediately from the tendency of the spectral plateaux to steepen with energy, given that the probability of convection downstream of the shock drops with increasing energy. For PAD, even the maximum amplification falls far short of , and saturates to a factor of order unity at high energies. This can be seen as follows. For a downstream particle of speed (in the shock frame) that crosses upstream, its velocity angle relative to the shock normal must satisfy (). This yields a range of possible upstream fluid-frame Lorentz factors given by , beamed in a narrow cone around . Pitch angle diffusion gradually widens this distribution till , at which point the ions convect downstream again with a shock frame Lorentz factor . It is then trivial to determine that (), a result noted by Gallant & Achterberg (1999).
The implications of these results for gamma-ray burst modelers are the following. First, liberal use of amplification factors in shock crossings when estimating maximum energies obtainable in relativistic shocks is inappropriate. This impacts contentions (Waxman 1995; Vietri 1995) that GRBs can generate ultra-high energy cosmic rays, as do reductions in acceleration times seen when (e.g. see EJR90, and references therein). The disparate nature of the spectral indices between PAD and LAS cases is also of concern to the GRB community. While PAD yields a narrow range of indices more-or-less commensurate with inferences from both prompt gamma-ray emission and delayed X-ray/optical afterglows, the structured, flat LAS spectra may be at odds with GRB data. Furthermore, given the broad dynamic range, huge differences would arise in flux predictions for different wavebands. While the results presented here are for protons, one expects similar spectral properties for (and hence the concerns for emission from) electrons, since they too are relativistic when and hence readily resonate with Alfvén and whistler modes. Bednarz & Ostrowski (1996) argue in favor of PAD operating in shocks with . We believe the situation not to be transparent. The definition of PAD in the context of relativistic shocks is effectively that angular deflections are substantially within the Lorentz loss-cone of half-angle . As soon as deflections exceed this small value, spectra from our simulations quickly flatten to reproduce the LAS ones exhibited here. Hence the critical issue as to whether PAD or LAS operates in GRBs is contingent upon the typical magnitude of particle deflections in field turbulence associated with relativistic shocks. This nontrivial question will be the subject of future investigation.
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Bednarz, J. & Ostrowski, M. 1996, [$\,$283, 447.]{}\
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---
abstract: 'We study the partition function of the ${\cal N}=6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the $U(N_1)\times U(N_2)$ lens space matrix model exactly. The result can be expressed as a product of $q$-deformed Barnes $G$-function and a generalization of multiple $q$-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing $N_2$ to $-N_2$. The answer is given by ${\rm min}(N_1,N_2)$-dimensional integrals and generalizes the “mirror description” of the partition function of the ABJM theory, [*i.e.*]{} the ${\cal N}=6$ supersymmetric $U(N)_k\times U(N)_{-k}$ CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for $|N_1-N_2|>k$ in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples.'
author:
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Hidetoshi Awata$^a$[^1], Shinji Hirano$^b$[^2], and Masaki Shigemori$^c$[^3]\
$^{a}$[[*Department of Mathematics*]{}]{}\
[[*Nagoya University*]{}]{}\
[[*Nagoya 464-8602, Japan*]{}]{}\
$^b$[[*Department of Physics*]{}]{}\
[[*Nagoya University*]{}]{}\
[[*Nagoya 464-8602, Japan*]{}]{}\
$^c$[[*Kobayashi-Maskawa Institute*]{}]{}\
[[*for the Origion of Particles and the Universe*]{}]{}\
[[*Nagoya University*]{}]{}\
[[*Nagoya 464-8602, Japan*]{}]{}
date:
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title: '[**[The Partition Function of ABJ Theory]{}**]{}'
---
Introduction
============
There has recently been remarkable progress in applications of the localization technique [@Witten:1988ze] to supersymmetric gauge theories, notably in dimensions $D\ge 3$: In $D=4$ the Seiberg-Witten prepotential of ${\cal N}=2$ supersymmetric QCD [@Nekrasov:2002qd] was directly evaluated, and the partition functions and BPS Wilson loops of the ${\cal N}=2$ (and $2^{*}$) and ${\cal N}=4$ supersymmetric Yang-Mills theories (SYM) were reduced to eigenvalue integrals of the matrix model type [@Pestun:2007rz], providing, in particular, a proof of the earlier results on a Wilson loop in the ${\cal N}=4$ SYM [@Erickson:2000af; @Drukker:2000rr]. In $D=3$ similar results were obtained for the partition functions and BPS Wilson loops of ${\cal
N}=2$ supersymmetric Chern-Simons-matter (CSM) theories [@Kapustin:2009kz; @Hama:2011ea], including the ${\cal N}=6$ superconformal theories constructed by Aharony, Bergman, Jafferis and Maldacena (ABJM) [@Aharony:2008ug][@Aharony:2008gk]. More recently, the localization technique was further applied to the partition functions of $5$-dimensional SYM with or without matter [@Hosomichi:2012ek; @Kallen:2012va; @Kim:2012av].
The localization method, resulting in the eigenvalue integrals of the matrix model type, allows us to obtain various exact results at strong coupling of supersymmetric gauge theories. In particular, these results provide useful data for the tests of the AdS/CFT correspondence [@Maldacena:1997re] in the case of superconformal gauge theories. For instance, the precise agreement of the $N^{3/2}$ scaling between the free energy of the ABJ(M) theory [@Drukker:2010nc; @Marino:2011nm; @triSE] and its AdS$_4$ dual [@Aharony:2008ug][@Aharony:2008gk] is an important landmark that shows the power of the localization method in the context of AdS/CFT. Rather remarkably, exact agreements were also found in [@Jafferis:2012iv] between the $N^{5/2}$ scaling of 5d superconformal theories and that of their AdS$_6$ duals [@Brandhuber:1999np]. Furthermore, the tantalizing $N^3$ scaling of maximally supersymmetric 5d SYM was found in [@Kim:2012av; @Kallen:2012zn] in line with the conjecture on $(2,0)$ 6d superconformal theory compactified on $S^1$ [@Douglas:2010iu], despite thus far a lack of the precise agreement with its AdS$_7$ dual. It should, however, be noted that the utility of the localization method, unlike the integrability [@Beisert:2010jr], is limited to a class of supersymmetric observables, such as the partition function and BPS Wilson loops. On the other hand, the localization method has an advantage over the integrability in that it can provide exact results at strong coupling beyond the large $N$ limit, where the integrability has not been as powerful.
In this paper we focus on the partition function of the ABJ theory, [*i.e.*]{}, the ${\cal N}=6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ CSM theory, which generalizes the equal rank $N_1=N_2$ case of the ABJM theory [@Aharony:2008gk]. Over the past few years there has been considerable progress in the study of the partition function and Wilson loops of the ABJM theory, whereas the ABJ case has not been as much understood. The ABJ generalization, for instance, has an important new feature, the Seiberg duality, which, however, lacks a full understanding. Besides being a generalization, it has recently been conjectured that the ABJ theory at large $N_2$ and $k$ with $N_2/k$ and $N_1$ fixed finite is dual to the ${\cal N}=6$ parity-violating Vasiliev higher spin theory on AdS$_4$ with $U(N_1)$ gauge symmetry [@Chang:2012kt]. Thus a better understanding of the ABJ theory may provide valuable insights into the relation between higher spin particles and strings. It is therefore worth studying the partition function of the ABJ theory in great detail.
As mentioned above, the partition function of the ABJM theory has been well studied. In the large $N$ limit, the planar free energy has been computed, revealing the aforementioned $N^{3/2}$ scaling [@Drukker:2010nc; @Marino:2011nm; @triSE]. In fact, the result in [@Drukker:2010nc; @Marino:2011nm] is exact in ’t Hooft coupling $\lambda=N/k$ and, in particular, confirms a gravity prediction of the AdS radius shift in [@Bergman:2009zh]. The planar result is not limited to the ABJM case; Drukker-Mariño-Putrov’s results include the partition function and Wilson loops of the ABJ theory, and the ABJ version of the radius shift [@Aharony:2009fc] is also confirmed. In the meantime, beyond the large $N$ limit, the $1/N$ corrections of the ABJM partition function were summed up to all orders by solving the holomorphic anomaly equations of [@Bershadsky:1993cx; @Drukker:2010nc; @Drukker:2011zy] at large $\lambda$ in the type IIA regime $k\gg 1$, and the result turned out to be simply an Airy function [@Fuji:2011km].[^4] Subsequently, Mariño and Putrov developed a more elegant approach, the Fermi gas approach, without making any use of the matrix model techniques or the holomorphic anomaly equations, to compute directly the partition functions of ${\cal N}=3$ and ${\cal N}=2$ CSM theories including the ABJM theory [@Marino:2011eh; @Marino:2012az]. They found, in particular, a universal Airy function behavior for the ${\cal N}=3$ theories at large $N$ in the small $k$ M-theory regime. These non-planar results were reaffirmed by numerical studies in the case of the ABJM theory [@Hanada:2012si]. Furthermore, the Fermi gas approach was applied to the Wilson loops, exhibiting again the Airy function behavior [@Klemm:2012ii]. Meanwhile, a number of exact computations of the ABJM partition function were carried out for various values of $N$ and $k$ [@Okuyama:2011su; @Hatsuda:2012hm; @Putrov:2012zi]. It should also be noted that the nonperturbative effects ${\cal O}(e^{-N})$ of the M- and D-brane type can be systematically studied both in the matrix model [@Drukker:2011zy] and the Fermi gas approaches [@Marino:2011eh].
In the unequal rank $N_1\ne N_2$ case of the ABJ theory, the Fermi gas approach thus far has not been applicable, and the study of finite $N_1$ and $N_2$ corrections to the ABJ partition function has not been as much developed as in the ABJM case. In this paper, we wish to lay the ground for the study of the ABJ partition function at finite $N_1$ and $N_2$. To this end, we first compute the partition function of the L(2,1) lens space matrix model [@Marino:2002fk; @Aganagic:2002wv] exactly. By making use of the relation between the lens space and the ABJ matrix models [@Marino:2009jd], we map the lens space partition function to that of the ABJ matrix model by analytically continuing $N_2$ to $-N_2$. With our particular prescription of the analytic continuation, the final answer for the ABJ partition function is given by ${\rm min}(N_1,N_2)$-dimensional integrals and generalizes the “mirror description” of the partition function of the ABJM theory [@Kapustin:2010xq]. Our result may thus serve as the starting point for the ABJ generalization of the Fermi gas approach. Meanwhile, we test our prescription against perturbative expansions as well as the Seiberg duality conjecture of [@Aharony:2008gk] and find that our final answer perfectly meets the expectations.
The rest of the paper is organized as follows: In Section \[outline\_mainresults\] we outline our strategy for the calculations of the ABJ partition function and summarize the main result at each pivotal step of the computations. Most of the computational details are relegated to rather extensive appendices. In Section \[Examples\] we present a few simple examples of our results in order to elucidate otherwise rather complicated general results. In Section \[Checks\] we state the result of perturbative and nonperturbative checks that we carried out and illustrate with a few simple examples how they were actually done. Section \[CandD\] is devoted to the conclusions and the discussions.
The outline of calculations and main results {#outline_mainresults}
============================================
We are going to compute the partition function of the $U(N_1)_k\times U(N_2)_{-k}$ ABJ theory in the matrix model form [@Kapustin:2009kz; @Hama:2011ea] obtained by the localization technique [@Pestun:2007rz]: $$\begin{aligned}
Z_{\rm ABJ}(N_1,N_2)_k={\cal N}_{\rm ABJ}\int\prod_{i=1}^{N_1}{d\mu_i\over 2\pi}\prod_{a=1}^{N_2}{d\nu_a\over 2\pi}
{\Delta_{\rm sh}(\mu)^2\Delta_{\rm sh}(\nu)^2\over\Delta_{\rm ch}(\mu,\nu)^2}
e^{-{1\over 2g_s}\left(\sum_{i=1}^{N_1}\mu_i^2-\sum_{a=1}^{N_2}\nu_a^2\right)}\ ,
\label{ABJMM}\end{aligned}$$ where the $\Delta_{\rm sh}$ factors are the one-loop determinants of the vector multiplets $$\begin{aligned}
\Delta_{\rm sh}(\mu)=\prod_{1\le i<j\le N_1}\left( 2\sinh\left({\mu_i-\mu_j\over 2}\right)\right)\ ,\quad
\Delta_{\rm sh}(\nu)=\prod_{1\le a<b\le N_2}\left( 2\sinh\left({\nu_a-\nu_b\over 2}\right)\right)\ ,\end{aligned}$$ and the $\Delta_{\rm ch}$ factor is the one-loop determinant of the matter multiplets in the bi-fundamental representation $$\begin{aligned}
\Delta_{\rm ch}(\mu,\nu)=\prod_{i=1}^{N_1}\prod_{a=1}^{N_2}\left(2\cosh\left({\mu_i-\nu_a\over 2}\right)\right)\ .\end{aligned}$$ The string coupling $g_s$ is related to the Chern-Simons level $k\in\mathbb{Z}_{\neq 0}$ by g\_s=[2ik]{} , and the factor ${\cal N}_{\rm ABJ}$ in front is the normalization factor [@Marino:2011nm] \_[ABJ]{}:=[i\^[-[2]{}(N\_1\^2-N\_2\^2)]{}N\_1!N\_2!]{} ,:=k . Note that, because of the relation $$\begin{aligned}
Z_{\rm ABJ}(N_2,N_1)_{k}
=Z_{\rm ABJ}(N_1,N_2)_{-k}
\ ,
\label{N1N2swap}\end{aligned}$$ we can assume $N_1\le N_2$ without loss of generality.
The outline of calculations
---------------------------
Before going into the details of calculations, we shall first lay out our technical strategy: We adopt the idea employed in the large $N$ analysis of the ABJ(M) matrix model in [@Drukker:2010nc; @Marino:2011nm]. Namely, instead of performing the integrals in (\[ABJMM\]) directly,
we first compute the partition of the L(2,1) lens space matrix model [@Marino:2002fk; @Aganagic:2002wv] $$\begin{aligned}
\hspace{-.2cm}
Z_{\rm lens}(N_1,N_2)_k={\cal N}_{\rm lens}\int\prod_{i=1}^{N_1}{d\mu_i\over 2\pi}\prod_{a=1}^{N_2}{d\nu_a\over 2\pi}
\Delta_{\rm sh}(\mu)^2\Delta_{\rm sh}(\nu)^2\Delta_{\rm ch}(\mu,\nu)^2
e^{-{1\over 2g_s}\left(\sum_{i=1}^{N_1}\mu_i^2+\sum_{a=1}^{N_2}\nu_a^2\right)}
\label{lensMM}\end{aligned}$$ with the normalization factor \_[lens]{}=[i\^[-[2]{}(N\_1\^2+N\_2\^2)]{}N\_1!N\_2!]{} .
then analytically continue $N_2$ to $-N_2$ to obtain the partition function of ABJ theory [@Marino:2009jd]
Z\_[ABJ]{}(N\_1,N\_2)\_k=\_[0]{}[C]{}(N\_2, ) Z\_[lens]{}(N\_1,-N\_2+)\_k ,
where the proportionality constant is given in terms of the Barnes $G$-function $G_2(z)$, (N\_2,)=(2)\^[-N\_2]{}[G\_2(N\_2+1)G\_2(-N\_2+1+)]{} .
A key observation is that the partition function (\[lensMM\]) of the lens space matrix model is *a sum of Gaussian integrals* and can thus be calculated *exactly* in a very elementary manner. The analytic continuation $N_2\to -N_2$, on the other hand, is ambiguous and not as straightforward as one might expect. We find the appropriate prescription for the analytic continuation in two steps:
In the first step we propose a natural prescription that correctly reproduces, after a generalized $\zeta$-function regularization, the known perturbative expansions in the string coupling $g_s$. The resulting expression, however, is a formal series that is non-convergent and singular when $k$ is an even integer.
To circumvent these issues, in the second step, we introduce an integral representation which renders a formal series perfectly well-defined.
In other words, the integral representation (A) implements a generalized $\zeta$-function regularization automatically and (B) provides an analytic continuation in the complex parameter $g_s$ for the formal series.
As we will see later, the final answer in the integral representation passes perturbative as well as some nonperturbative tests and generalizes the “mirror description” [@Kapustin:2010xq] of the partition function of the ABJM theory to the ABJ theory.
The main results {#mainresults}
----------------
We present, without much detail of derivations, the main result at each step of the outlined calculations. Most of the technical details will be given in the appendices.
### $\bullet$ The lens space matrix model {#bullet-the-lens-space-matrix-model .unnumbered}
As emphasized above, the lens space partition function (\[lensMM\]) is a sum of Gaussian integrals and can be calculated exactly: $$\begin{aligned}
Z_{\text{lens}}(N_1,N_2)_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
\left({g_s\over 2\pi}\right)^{N\over 2}
q^{-{1\over 3}N(N^2-1)}
\notag\\
&\qquad
\times \sum_{(\cN_1,\,\cN_2)}
\prod_{C_j<C_k} (q^{C_j}-q^{C_k})
\prod_{D_a<D_b} (q^{D_a}-q^{D_b})
\prod_{C_j,D_a} (q^{C_j}+q^{D_a})\ ,\label{lensAns1}\end{aligned}$$ where $$\begin{aligned}
q:=e^{-g_s}=e^{-{2\pi i \over k}}, \qquad N=N_1+N_2.\label{qgs}\end{aligned}$$ The symbol $(\cN_1,\,\cN_2)$ denotes the partition of the numbers $(1, 2, \cdots, N)$ into two groups $\cN_1=(C_1, C_2, \cdots, C_{N_1})$ and $\cN_2=(D_1, D_2, \cdots, D_{N_2})$ where $C_i$’s and $D_a$’s are ordered as $C_1<\cdots<C_{N_1}$ and $D_1<\cdots<D_{N_2}$. The computation proceeds in two steps: (1) Gaussian integrals and (2) sums over permutations. The detailed derivation can be found in Appendix \[Appendix\_LensMM\].
As advertised, the result (\[lensAns1\]) can be written as a product of $q$-deformed Barnes $G$-function and a generalization of multiple $q$-hypergeometric function: $$\begin{aligned}
\hspace{-.4cm}
Z_{\text{lens}}(N_1,N_2)_k
&=
i^{-{\kappa \over 2}(N_1^2+N_2^2)}
\left({g_s\over 2\pi}\right)^{N\over 2}
q^{-{1\over 6}N(N^2-1)}(1-q)^{\half N(N-1)}G_2(N+1;q)\,S(N_1,N_2)\ ,
\label{lensAns2}\end{aligned}$$ where $$\begin{aligned}
S(N_1,N_2)= \sum_{(\cN_1,\,\cN_2)}
\prod\limits_{C_j<D_a}{q^{C_j}+q^{D_a}\over q^{C_j}-q^{D_a}}
\prod\limits_{D_a<C_j}{q^{D_a}+q^{C_j}\over q^{D_a}-q^{C_j}}\ .
\label{Sfunction}\end{aligned}$$ The $q$-deformed Barnes $G$-function $G_2(z;q)$ is defined in Appendix \[Appendix\_q-Analogs\] and, as will be elaborated later, $S(N_1,
N_2)$ is a generalization of multiple $q$-hypergeometric function. Recalling that $q=e^{-g_s}$, it is rather fascinating to observe that the string coupling $g_s$ is not only the loop-expansion parameter in quantum mechanics but also a quantum deformation parameter of special functions.
In Section \[Examples\] we will give simple examples of the lens space partition function in order to elucidate the $q$-hypergeometric structure.
### $\bullet$ The ABJ theory {#bullet-the-abj-theory .unnumbered}
The next step in our strategy is the analytic continuation $N_2\to -N_2$ which maps the partition function of the lens space matrix model to that of the ABJ theory. For this purpose, we find it convenient to work with the second expression of the lens space partition function (\[lensAns2\]). Our claim is that the analytic continuation yields the following expression for the ABJ partition function in a formal series $$\begin{aligned}
Z_{\rm ABJ}(N_1, N_2)_k&= i^{-{\kappa \over 2}(N_1^2+N_2^2)}
(-1)^{\half N_1(N_1-1)}\, 2^{-N_1}\,
\left({g_s\over 2\pi}\right)^{N_1+N_2\over 2} (1-q)^{M(M-1)\over 2}
G_2(M+1;q)\nn\\
&\quad\times
{1\over N_1!}\sum\limits_{s_1,\dots,s_{N_1}\ge 0}
(-1)^{s_1+\cdots+s_{N_1}}
\prod\limits_{j=1}^{N_1}{(q^{s_j+1})_{M}\over (-q^{s_j+1})_{M}}
\prod\limits_{j<k}^{N_1}
{(1-q^{s_k-s_j})^2\over (1+q^{s_k-s_j})^2}\ .
\label{ABJpart_formalsum}\end{aligned}$$ where we defined $M=N_2-N_1$ (for $N_2>N_1)$ and $(a)_n$ is a shorthand notation for the $q$-Pochhammer symbol $(a;q)_n$ defined in Appendix \[Appendix\_q-Analogs\]. We used an $\epsilon$-prescription in continuing $N_2$ to $-N_2$, as explained in detail in Appendix \[Appendix\_AC\].
However, as noted above, there are in principle multiple ways to continue $N_2$ to $-N_2$. It thus requires a particular prescription to fix this ambiguity. Our prescription is to continue $N_2$ to $-N_2$ with $S(N_1, N_2)$ written in the form $$\begin{aligned}
S(N_1, N_2)=\gamma(N_1, N_2)\Psi(N_1, N_2)
\label{gammaPsi}\end{aligned}$$ where $$\begin{aligned}
\gamma(N_1,N_2)&=
(-1)^{\half N_1(N_1-1)}
\prod_{j=1}^{N_1-1}{(-q)_j^2\over (q)_j^2}
\prod_{j=1}^{N_1}
{(-q^j)_{N_2}(-q^j)_{-N_1-N_2}\over (q^j)_{N_2}(q^j)_{-N_1-N_2}}\ ,\label{gamma}
\\
\Psi(N_1,N_2)&=
{1\over N_1!}\sum_{s_1,\dots,s_{N_1}\ge 0}
(-1)^{s_1+\cdots+s_{N_1}}
\prod_{j=1}^{N_1}{(q^{s_j+1})_{-N_1-N_2}\over (-q^{s_j+1})_{-N_1-N_2}}
\prod_{1\le j<k\le N_1}
{(q^{s_k-s_j})_1^2\over (-q^{s_k-s_j})_1^2}\ .
\label{Psi}\end{aligned}$$ As will be explained more in detail in Appendix \[Appendix\_AnalCont\], there are a number of ways to express $S(N_1, N_2)$ that could yield different results after the analytic continuation: The range of the sum in (\[Sfunction\]) runs from $1$ to $N=N_1+N_2$. In order to make sense of analytic continuation in $N_2 (> N_1)$, the finite sum (\[Sfunction\]) is extended to the infinite sum (\[Psi\]). In fact, the summand for $s_i>N-1$ in (\[gammaPsi\]) vanishes after an appropriate regularization. Now the point is that these vanishing terms could yield nonvanishing contributions after the analytic continuation. Clearly, the way to extend the finite sums to infinite ones is not unique, and this is where the ambiguity lies.
Our guideline for the correct prescription is to successfully reproduce the perturbative expansions in $g_s$. Indeed, it can be checked that the formal series (\[ABJpart\_formalsum\]) has the correct perturbative expansions, as we will discuss more in Section \[PE\].
### $\bullet$ The integral representation {#bullet-the-integral-representation .unnumbered}
As alluded to in the outline, the result (\[ABJpart\_formalsum\]) is not the final answer. It is a formal series that is non-convergent and singular when $k$ is an even integer. It can be rendered perfectly well-defined by introducing an integral representation: Specifically, our final answer for the analytic continuation is $$\begin{aligned}
Z_{\rm ABJ}(N_1, N_2)_k&=i^{-{\kappa \over 2}(N_1^2+N_2^2)}
(-1)^{\half N_1(N_1-1)}\, 2^{-N_1}\,
\left({g_s\over 2\pi}\right)^{N_1+N_2\over 2} (1-q)^{M(M-1)\over 2}
G_2(M+1;q)\nn\\
&\quad\times
{1\over N_1!}\prod_{j=1}^{N_1}\left[{-1\over 2\pi i}
\int_{I} {\pi \, ds_j\over \sin(\pi s_j)}\right]
\prod_{j=1}^{N_1}{(q^{s_j+1})_{M}\over (-q^{s_j+1})_{M}}
\prod_{1\le j<k\le N_1}
{(1-q^{s_k-s_j})^2\over (1+q^{s_k-s_j})^2}\ ,
\label{ABJpart_integral}\end{aligned}$$ where $M=N_2-N_1$ (for $N_2>N_1$) and the integration range $I=\left[-i\infty-\eta, +i \infty-\eta\right]$ with $\eta>0$. We note that there is a subtlety in the choice of $\eta$: For example, when the string coupling $g_s$ takes the actual value of our interest, ${2\pi i\over k}$ with an integer $k$, as we will elaborate in Section \[SD\], the parameter $\eta$ should be varied so that the partition function remains analytic in $k$, as one decreases the value of $k$ from the small coupling regime $|g_s|=|2\pi i/k|\ll 1$.
Although we lack a first principle derivation of the integral representation, we can give heuristic arguments as follows: First, this integral representation “agrees” with the formal series (\[ABJpart\_formalsum\]) order by order in the perturbative $g_s$-expansions. The integrals could be evaluated by considering the closed contours $C_j$ composed of the vertical line $I$ and the infinitely large semi-circle $C_j^{\infty}$ on the right half of the complex $s_j$-plane, if the contribution from $C_j^{\infty}$ were to vanish; see Figure \[contour1\]. In the $g_s$-expansions, the poles would only come from the factors $1/\sin(\pi s_j)$ and are at $s_j=n_j\in \mathbb{Z}_{\ge 0}$. Thus the residue integrals would correctly reproduce (\[ABJpart\_formalsum\]). In actuality, however, the contribution from $C_j^{\infty}$ does not vanish, and thus this argument is heuristic at best; we will see precisely how the $g_s$-expansions work in an example in section \[PE\]. We note that, to the same degree of imprecision, the integral representation (\[ABJpart\_integral\]) can be thought of as the Sommerfeld-Watson transform of .[^5]
![*“The integration contour” $C_j=I+C_j^{\infty}$ for the perturbative ABJ partition function: The only perturbative (P) poles are indicated by red “$+$”. See text for detail.*[]{data-label="contour1"}](contourA.pdf){height="2.5in"}
Second, as implied in the first point, the integral representation (\[ABJpart\_integral\]) provides a “nonperturbative completion” for the formal series (\[ABJpart\_formalsum\]). In fact, nonperturbatively, there appear additional poles from the factors $1/(-q^{s_j+1})_M$ and $1/(1+q^{s_k-s_j})^2$ in the contour integrals. They are located at $s_j=-{(2n+1)\pi i\over g_s}-m$ and $s_j=-{(2n+1)\pi i\over g_s}+s_k$ with $n\in\mathbb{Z}$ and $m=1,\cdots,
M$, as shown in Figure \[NPpoles\]. Their residues are of order $e^{1/g_s}$. Hence these can be regarded as nonperturbative (NP) poles, whereas the previous ones are perturbative (P) poles. Again, these statements are rather heuristic, and we will see how precisely P and NP poles contribute to the contour integral in Section \[SD\].
![*The nonperturbative (NP) poles are added and indicated by blue “$\times$”. The left panel corresponds to the complex $g_s$ case. The right panel is the actual case of our interest $g_s=2\pi i/k$. (Shown is the case $k=3$ and $M=3$.)*[]{data-label="NPpoles"}](contourB.pdf "fig:"){height="2.2in"} ![*The nonperturbative (NP) poles are added and indicated by blue “$\times$”. The left panel corresponds to the complex $g_s$ case. The right panel is the actual case of our interest $g_s=2\pi i/k$. (Shown is the case $k=3$ and $M=3$.)*[]{data-label="NPpoles"}](contourC.pdf "fig:"){height="2.2in"}
A few remarks are in order:
As promised, there is no issue of convergence in the expression (\[ABJpart\_integral\]). It is also well-defined in the entire complex $q$-plane. The integrand becomes singular for $q=e^{-2\pi i/k}$ with even integer $k$ as in the formal series (\[ABJpart\_formalsum\]). However, this merely represents pole singularities and yields finite residue contributions.
It should be noted that our main result (\[ABJpart\_integral\]) lacks a first principle derivation. It thus requires [*a posteriori*]{} justification. On this score, as stressed and will be discussed more in Section \[PE\], the integral representation (\[ABJpart\_integral\]) correctly reproduces the perturbative expansions; moreover, it automatically implements a generalized $\zeta$-function regularization needed in the perturbative expansions of the infinite sum (\[ABJpart\_formalsum\]). Meanwhile, a successful test of the Seiberg duality conjectured in [@Aharony:2008gk] provides evidence for our proposed nonperturbative completion. We will explicitly show a few nontrivial examples of the Seiberg duality at work in Section \[SD\].
In the ABJM limit ($M=0$), the integral representation (\[ABJpart\_integral\]) coincides with the “mirror description” of the ABJM partition function found in [@Kapustin:2010xq]. This provides a further support for our prescription and implies that we have found a generalization of the “mirror description” in the case of the ABJ theory. Our finding may thus serve as the starting point for the generalization of the Fermi gas approach developed in [@Marino:2011eh] to the ABJ theory.
One of the ABJ conjectures is that the ${\cal N}=6$ $U(N_1)_k\times U(N_1+M)_{-k}$ theory with $M>k$ may not exist as a unitary theory [@Aharony:2008gk]. It is further expected that the supersymmetries are spontaneously broken in this case [@Bergman:1999na] (see also [@Hashimoto:2010bq]). A manifestation of this conjecture is that the partition function (\[ABJpart\_integral\]) vanishes when $M > k$ because (1-q)\^[[M(M-1)2]{}]{}G\_2(M+1;q)=\_[j=1]{}\^[M-1]{}(q)\_j=0q=e\^[-[2ik]{}]{} . Note that the $q$-deformed Barnes $G$-function $G_2(M+1;q)$ is precisely a factor that appears in the partition function of the $U(M)_k$ Chern-Simons theory. We thus expect that this property is not peculiar to the ${\cal N}=6$ CSM theories but holds for CSM theories with less supersymmetries as long as they contain the $U(M)_k$ CS theory as a subsector.[^6]
Examples {#Examples}
========
In this section we present a few simple examples of the lens space and ABJ partition functions in order to get the feel of the expressions found in the previous section. In particular, these examples clarify the appearance of $q$-hypergeometric functions in the lens space partition function and how they are mapped to in the ABJ partition function. We also provide a simplest example of the exact ABJ partition function.
### $\bullet$ The CS matrix model {#bullet-the-cs-matrix-model .unnumbered}
The first example is the simplest case, the $N_1=0$ or $N_2=0$ case, which corresponds to the Chern-Simons matrix model. From (\[lensAns2\]) one immediately finds for the $U(M)_k$ CS theory that Z\_[CS]{}(M)\_k=Z\_[lens]{}(M,0)\_k= i\^[-[M(M-1)2]{}]{} |k|\^[-[M2]{}]{} q\^[-[M(M\^2-1)6]{}]{}(1-q)\^[M(M-1)]{}G\_2(M+1;q) . \[CSMM\] Note that this takes the more familiar form [@Marino:2004uf; @Tierz:2002jj] (without the level shift) if one uses the formula i\^[-[M(M-1)2]{}]{}(1-q)\^[M(M-1)]{}G\_2(M+1;q)=q\^[[M(M\^2-1)12]{}]{}\_[j=1]{}\^[M-1]{}(2)\^[M-j]{} . It should now be clear that the $q$-deformed Barnes $G$-function is a contribution from the $U(|N_1-N_2|)_k$ pure CS subsector in the $U(N_1)_k\times U(N_2)_{-k}$ theory.
### $\bullet$ The lens space matrix model {#bullet-the-lens-space-matrix-model-1 .unnumbered}
The next simplest example is the $N_1=1$ case studied in detail in Appendix \[Appendix\_N1equalto1\]. From (\[lensAns2\]) together with (\[mckp3Sep12\]) and (\[evuj11Sep12\]), the $U(1)_k\times U(N_2)_{-k}$ lens space partition function yields $$\begin{aligned}
Z_{\text{lens}}(1,N_2)_k
= \,&
i^{-{\kappa\over 2}(N_2^2+1)}
\left({g_s\over 2\pi}\right)^{N_2+1\over 2}
q^{-{N_2(N_2+1)(N_2+2)\over 6}}(1-q)^{N_2(N_2+1)\over 2}G_2(N_2+2;q)\nn\\
&\times {(-q)_{N_2}\over (q)_{N_2}}\, _2\phi_1\left({q^{-N_2},-q\atop -q^{-N_2}} ; q,-1\right)\ ,
\label{lens1N2}\end{aligned}$$ where the special function $\, _2\phi_1=\Phi(1,N_2)$ is a $q$-hypergeometric function [@Gasper-Rahman] whose definition is given in Appendix \[Appendix\_q-Analogs\]. Intriguingly, the whole function $S(1,N_2)$ in the second line is essentially an orthogonal $q$-polynomial, the continuous $q$-ultraspherical (or Rogers) polynomial [@Koekoek-Swarttouw], and very closely related to Schur Q-polynomials [@Rosengren:2006].
The next example is the $N_1=2$ case discussed in detail in Appendix \[Appendix\_N1equalto2\]. In parallel with the previous case, from (\[lensAns2\]) together with (\[kgmb7Nov12\]) and (\[gyiq5Sep12\]), one finds the $U(2)_k\times U(N_2)_{-k}$ lens space partition function $$\begin{aligned}
\hspace{-.3cm}
Z_{\text{lens}}(2,N_2)_k
&=
i^{-{\kappa\over 2}(N_2^2+4)}
\left({g_s\over 2\pi}\right)^{N_2+2\over 2}
q^{-{(N_2+1)(N_2+2)(N_2+3)\over 6}}(1-q)^{(N_2+1)(N_2+2)\over 2}G_2(N_2+3;q)\nn\\
&\times {(-q)_{N_2}(-q^2)_{N_2}\over (q)_{N_2}(q^2)_{N_2}}\,
\Phi^{2:2;4}_{2:1;3}\!\left(
\begin{array}{c@{~}c@{~}c@{~}c@{~}c}
q^{-N_2},-q^2&:&q^{-N_2-1},-q&;& q^2,q^2, -q,-q \\
-q^{-N_2},q^2&:& -q^{-N_2-1} &;& -q^2,-q^2, q
\end{array}
;~q ~;1,-1\right)\ ,
\label{lens2N2}\end{aligned}$$ where the special function $\Phi^{2:2;4}_{2:1;3}=\Phi(2,N_2)$ is a double $q$-hypergeometric function defined in Section 10.2 of [@Gasper-Rahman].
As promised, these examples elucidate that the function $S(N_1,N_2)$ defined in (\[Sfunction\]) is a generalization of multiple $q$-hypergeometric function.
### $\bullet$ The ABJ theory {#bullet-the-abj-theory-1 .unnumbered}
We now present the ABJ counterpart of the previous two examples. Although we have placed great emphasis on the $q$-hypergeometric structure of the lens space partition function, we have not found a way to take full advantage of this fact in understanding the ABJ partition function thus far.
In the meantime, as mentioned in the previous section and discussed in great detail in Appendix \[Appendix\_AC\], we find the expression (\[moge23Sep12\]) more convenient for performing the analytic continuation $N_2\to -N_2$ than the $q$-hypergeometric representation (\[fvrs7Sep12\]). The end result is presented in (\[ABJpart\_integral\]). In the case of the $U(1)_k\times U(N_2)_{-k}$ ABJ partition function, one finds $$\begin{aligned}
Z_{\rm ABJ}(1, N_2)_k=\,&{1\over 2} \,q^{{1\over 12}N_2(N_2-1)(N_2-2)}
|k|^{-{N_2+1\over 2}}\prod_{j=1}^{N_2-2}\left(2\sin{\pi j\over |k|}\right)^{N_2-1-j}
\nn\\
&\hspace{0.1cm}\times \left[{-1\over 2\pi i}
\int_{I} {\pi \, ds\over \sin(\pi s)}
\prod_{l=1}^{N_2-1}\tan\left({(s+l)\pi\over |k|}\right)\right]\ .
\label{ABJN11}\end{aligned}$$ Similarly, the $U(2)_k\times U(N_2)_{-k}$ ABJ partition function yields $$\begin{aligned}
Z_{\rm ABJ}(2, N_2)_k
=\,&
-{1\over 8}q^{{1\over 12}(N_2-1)(N_2-2)(N_2-3)}
|k|^{-{N_2+2\over 2}}\prod_{j=1}^{N_2-3}\left(2\sin{\pi j\over |k|}\right)^{N_2-2-j}
\nn\\
&\times
\prod_{j=1}^{2}\left[{-1\over 2\pi i}
\int_{I} {\pi \, ds_j\over \sin(\pi s_j)}
\prod_{l=1}^{N_2-2}\tan\left({(s_j+l)\pi\over |k|}\right)\right]
\tan^2\left({(s_2-s_1)\pi \over |k|}\right)\ .
\label{ABJN12}\end{aligned}$$ Note that the $U(N_1)_k\times U(N_2)_{-k}$ ABJ theory with finite $N_1$ and large $N_2$ and $k$ is conjectured to be dual to ${\cal N}=6$ parity-violating Vasiliev higher spin theory on $AdS_4$ with $U(N_1)$ gauge symmetry [@Giombi:2011kc; @Chang:2012kt]. It would thus be very interesting to study the large $N_2$ and $k$ limit of the $N_1=1$ and $2$ partition functions [@WIP1]. It may shed some lights on the understanding of the ${\cal N}=6$ parity-violating Vasiliev theory on $AdS_4$.[^7]
Finally, we provide a simplest example of the exact ABJ partition function, [*i.e.*]{}, the $U(1)_k\times U(2)_{-k}$ case. The integral in (\[ABJN11\]) can be carried out by applying a similar trick to the one used in [@Okuyama:2011su]. This yields $$\begin{aligned}
Z_{\rm ABJ}(1, 2)_k=\,&{1\over 2} \,
|k|^{-{3\over 2}}\times
\begin{cases}
\half\left[\sum_{n=1}^{|k|-1}(-1)^{n-1}\tan\bigl({\pi n\over |k|}\bigr)+|k|(-1)^{|k|-1\over 2}\right] & \quad(k={\rm odd})\ ,\\[1ex]
\sum_{n=1}^{|k|-1}(-1)^{n-1}\left({1\over 2}-{n\over k}\right)\tan\bigl({\pi n\over |k|}\bigr) & \quad(k={\rm even})\ .
\end{cases}\end{aligned}$$ It may be worth noting that the formal series (\[ABJpart\_formalsum\]) for the $U(1)_k\times U(2)_{-k}$ theory, albeit nonconvergent, can be expressed in a closed form after a regularization: $$\begin{aligned}
Z_{\rm ABJ}(1, 2)_k=\,&{1\over 2} \,i^{-\kappa}\,
|k|^{-{3\over 2}}\left[\half -{2\over\log q}\left(\log\left(1+q^2\over 1+q\right)+\psi_q(1)-2\psi_{q^2}(1)+\psi_{q^4}(1)\right)\right]\ ,\end{aligned}$$ where $\psi_q(z)$ is a $q$-digamma function defined in Appendix \[Appendix\_q-Analogs\], and we used the regularization $\sum_{s=0}^{\infty}(-1)^s=\half$. This expression is, however, not well-defined for $q$ a root of unity and hence an integer $k$. On the other hand, this exemplifies the fact that the integral representation (\[ABJpart\_integral\]) provides an analytic continuation of the formal series (\[ABJpart\_formalsum\]) in the complex $q$-plane.
Checks {#Checks}
======
As mentioned in Section \[outline\_mainresults\], our main result (\[ABJpart\_integral\]) lacks a first principle derivation. It thus requires [*a posteriori*]{} justification. In this section we show that our prescription passes perturbative as well as nonperturbative tests. We have, however, been unable to prove it in generality. Although our checks are on a case-by-case basis, we have examined several nontrivial cases that provide convincing evidence for our claim.[^8]
Perturbative expansions {#PE}
-----------------------
The perturbative expansion of the lens space free energy is presented in [@Aganagic:2002wv]. In Appendix \[Appendix\_PE\], we extend their result to the order ${\cal O}(g_s^8)$. We would like to see if the perturbative expansions of both (\[ABJpart\_formalsum\]) and (\[ABJpart\_integral\]) correctly reproduce this result with the replacement $N_2$ by $-N_2$. We have checked the cases $N_1=1$ and $N_2$ up to $8$, $N_1=2$ and $N_2$ up to $5$, $N_1=3$ and $N_2$ up to $5$, and $N_1=4$ and $N_2$ up to $4$, to the order ${\cal O}(g_s^8)$ and found perfect agreements with the result in Appendix \[Appendix\_PE\]. These checks are straightforward, and we will not spell out all the details. Instead, we describe only the essential points in the calculations and illustrate with a simple but nontrivial example how the checks were done in detail.
### The formal series {#the-formal-series .unnumbered}
In the case of the formal series (\[ABJpart\_formalsum\]), as remarked in the previous section, the perturbative expansion is correctly reproduced after the generalized $\zeta$-function regularization: \_[s=0]{}\^(-1)\^ss\^n={
[ll]{} [Li]{}\_[-n]{}(-1)=(2\^[n+1]{}-1)(-n) =-[2\^[n+1]{}-1n+1]{}B\_[n+1]{} &(n1) ,\
1+[Li]{}\_0(-1) = =-B\_1 &(n=0) ,
. \[genZetareg\] where ${\rm Li}_s(z)$ is the polylogarithm and $B_n$ are the Bernoulli numbers. We show the detail of the $(N_1,N_2)=(2,3)$ example to illustrate how the generalized $\zeta$-function regularization yields the correct perturbative expansion to the order ${\cal O}(g_s^4)$. In this case there are two infinite sums involved. Now, recall that the summand is a function of $q=\exp\left(-g_s\right)$. Expanding it as a power series in $g_s$ and using the regularization (\[genZetareg\]), one finds $$\begin{aligned}
\hspace{-.2cm}
\text{The 2nd line of (\ref{ABJpart_formalsum})}&={g_s^4\over 32}\left(
\text{Li}^2_{-3,-1}\!-\!2 \text{Li}^2_{-2,-2}\!+\!\text{Li}^2_{-1,-3}\right)\!-\!{g_s^6\over 384}\left(3 \text{Li}^2_{-5,-1}\!-\!10 \text{Li}^2_{-4,-2}\!+\!14 \text{Li}^2_{-3,-3}\right.\nn\\
&\left.-10 \text{Li}^2_{-2,-4}+3 \text{Li}^2_{-1,-5}\right)
+{g_s^8\over 23040}\left(33 \text{Li}^2_{-7,-1}-154 \text{Li}^2_{-6,-2}+336 \text{Li}^2_{-5,-3}\right.\nn\\
&\left.-430 \text{Li}^2_{-4,-4}+336 \text{Li}^2_{-3,-5}-154 \text{Li}^2_{-2,-6}+33 \text{Li}^2_{-1,-7}\right)+{\cal O}(g_s^{10})\nn\\
&=-\frac{1}{512}g_s^4-\frac{19}{12288}g_s^6-\frac{137}{81920}g_s^8+{\cal O}(g_s^{10})\ ,\end{aligned}$$ where we abbreviated the product ${\rm Li}_{-n_1}(-1){\rm Li}_{-n_2}(-1)$ by ${\rm Li}^2_{-n_1,-n_2}$. This yields F\_[ABJ]{}(2,3)=Z\_[ABJ]{}(2,3)= +g\_s\^2+g\_s\^4+[O]{}(g\_s\^6) in agreement with the result in Appendix \[Appendix\_PE\] with the replacement $N_2$ by $-N_2$. Note also that the tree contribution, the first logarithmic term, is in a precise agreement with (\[nimf2Nov12\]).
### The integral representation {#the-integral-representation .unnumbered}
The integral representation (\[ABJpart\_integral\]) does not require any regularization. Instead, the generalized $\zeta$-function regularization (\[genZetareg\]) is automatically implemented by the integral -[12i]{}\_I[ds(s)]{}s\^n =-[2\^[n+1]{}-1n+1]{}B\_[n+1]{} , where $n\ge 0$. It follows immediately from this fact that = at all orders in the $g_s$-expansions. Hence the integral representation correctly reproduces the perturbative expansions.
The Seiberg duality {#SD}
-------------------
As emphasized before, the integral representation (\[ABJpart\_integral\]) provides a “nonperturbative completion” for the formal series (\[ABJpart\_formalsum\]). A way to test this claim is to see if the Seiberg duality conjectured in [@Aharony:2008gk] holds.[^9] This duality is an equivalence between the two ABJ theories; schematically, U(N\_1)\_kU(N\_1+M)\_[-k]{} = U(N\_1+|k|-M)\_kU(N\_1)\_[-k]{} . \[Seibergduality\] We are going to show, in the simple but nontrivial case of $N_1=1$, that the partition functions of the dual pairs agree up to a phase. In fact, a proof of the Giveon-Kutasov duality including the ${\cal N}=6$ case was proposed in [@Kapustin:2010mh], which assumed one conjecture to be proven. In particular, their conjecture gives a formula for the phase differences of the dual pairs. We will explicitly confirm their claim in our examples below.
### Seiberg duality for $\boldsymbol{N_1=1}$ {#seiberg-duality-for-boldsymboln_11 .unnumbered}
For $N_1=1$, the duality relation reads $$\begin{aligned}
U(1)_k\times U(N_2)_{-k} = U(1)_{-k}\times U(2+|k|-N_2)_k\ .
\label{1N2Seibergduality}\end{aligned}$$ In this case, we can actually prove that the integral representation indeed gives identical results for the dual pair, up to a phase. Let us rewrite the $(1, N_2)$ partition function given in in the following form $$\begin{aligned}
Z_{\rm ABJ}(1, N_2)_k&=(2|k|)^{-1} Z_{\rm CS}^0(N_2-1)_k\, I(1,N_2)_k\,
e^{i\theta(1,N_2)_k}\ .\label{ABJN11(2)}\end{aligned}$$ Here, $Z_{\rm CS}^0(M)_k$ is the Chern-Simons (CS) partition function $$\begin{aligned}
Z_{\rm CS}^0(M)_k=|k|^{-{M\over 2}}
\prod_{j=1}^{M-1} \left(2\sin{\pi j\over |k|}\right)^{M-j}
\label{CSMM0}\end{aligned}$$ which is essentially the same as up to a phase due to difference in the framing [@Marino:2011nm]. Moreover, $$\begin{aligned}
I(1,N_2)_k:&=-{1\over 2\pi i}
\int_{I} {\pi \, ds\over \sin(\pi s)}
\prod_{l=1}^{N_2-1}\tan\left({(s+l)\pi\over |k|}\right)\ ,
\label{1N2integral}\\
\theta(1,N_2)_k&: =-{\pi\over 6k}N_2(N_2-1)(N_2-2).\end{aligned}$$ We can show that $Z_{\rm CS}^0(N_2-1)_k$ and $I(1,N_2)_k$ are separately invariant under the Seiberg duality while the phase factor $e^{i\theta(1,N_2)_k}$ gives a phase that precisely agrees with the one given in [@Kapustin:2010mh].
First, the invariance of $Z_{\rm CS}^0(N_2-1)_k$ is nothing but the level-rank duality of the CS partition function, which means the identity $Z_{\rm CS}^0(M)_k=Z_{\rm CS}^0(|k|-M)_k$.[^10] It is straightforward to see that this implies that $Z_{\rm CS}^0(N_2-1)_k$ is invariant under the Seiberg duality . Second, the phase difference between the dual theories is $$\begin{aligned}
\theta(1,N_2)_{k}-\theta(1,2+|k|-N_2)_{-k}
=
\pi \left[\kappa\left(-\frac{1}{6}k^2-\frac{1}{2}N_2^2+N_2-\frac{1}{3}\right)+\frac{1}{2}k(N_2-1)\right].\end{aligned}$$ One can show that this phase difference is exactly the same as the one given in [@Kapustin:2010mh].
Now let us move on to the most nontrivial part, [*i.e.*]{}, the invariance of the integral under the Seiberg duality. One can show that, despite appearances, the integrand is actually the same function for the dual theories up to a shift in $s$. Therefore, the contour integral gives the same answer for the duals, if the contour is chosen appropriately. As explained in section \[mainresults\], the integrand has perturbative (P) poles coming from $\pi\over \sin(\pi s)$ and non-perturbative (NP) poles coming from the product factor $\prod_l \tan$. Although the integrand remains the same under the Seiberg duality, the interpretation of its poles gets interchanged; [*i.e.*]{}, a P pole in the original theory is interpreted as a NP pole in the dual theory, and [*vice versa.*]{} We will see this explicitly in examples below, relegating the general proof to Appendix \[Appendix\_SD\].
The integrand of is an antiperiodic (periodic) function with $s\cong s+|k|$ for odd (even) $k$, and the P and NP poles occur on the real $s$ axis in bunches with this periodicity. The prescription for the contour is to take it to go to the left of one of such bunches. In Appendix \[Appendix\_SD\], we show that this means that $$\begin{aligned}
\eta=\begin{cases}
0_+ & \qquad \text{if}\quad {|k|\over 2}-N_2+1\ge 0,\\
-{|k|\over 2}+N_2-1+0_+ & \qquad \text{if}\quad {|k|\over 2}-N_2+1\le 0.\\
\end{cases}
\label{etaPrescription}\end{aligned}$$ This is required for the Seiberg duality to work, but it is also necessary for the ABJ partition function to be analytic in $k$, which is clearly the case for the original expression . In the weak coupling regime $|g_s|=|2\pi i/k|\ll 1$, the NP poles are far away from the origin (distance $\sim 1/|g_s|\sim |k|$) and we can safely take $\eta=0_+$. However, as we decrease $|k|$ continuously, the NP poles come closer to the origin and, eventually, at some even $|k|$, one of the NP poles that was in the $s>0$ region reaches $s=0$. As we further decrease $|k|$ continuously, this NP pole enters the $s<0$ region. In order for the partition function to be analytic in $k$, one needs to increase the value of $\eta$ so that this NP pole does not move across the contour $I$ but stays to the right of it.
### Odd $\boldsymbol{k}$ case {#odd-boldsymbolk-case .unnumbered}
The integral (\[1N2integral\]) for odd $k$ is equal to the following contour integral $$\begin{aligned}
I(1,N_2)_{k}=-{1\over 4\pi i}
\int_{C} {\pi \, ds\over \sin(\pi s)}
\prod_{l=1}^{N_2-1}\tan\left({(s+l)\pi\over |k|}\right)\ ,\end{aligned}$$ where the integral contour $C$ is given by $C=I\cap I_{i\infty}\cap
I_k\cap I_{-i\infty}$ (clockwise), where the contour $I_k$ is parallel to $I$ and shifted by $|k|$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity; see Figure \[contour2\]. Note that the anti-periodicity of the integrand allows us to write the line integral (\[1N2integral\]) as a closed contour integral, but the contour is different from the tentative contour shown for the sake of sketchy illustration in Figures \[contour1\] and \[NPpoles\]. By summing up pole residues inside $C$, one finds $$\begin{aligned}
I(1,N_2)_{k}
&={1\over 2}
\biggl[
\sum_{n=0}^{|k|-N_2}(-1)^n\prod_{j=1}^{N_2-1}\tan{\pi(n+j)\over |k|}
-|k|(-1)^{|k|-1\over 2}\sum_{n=1}^{N_2-1}
(-1)^n\prod_{\substack{j=1\\ (\neq n)}}^{N_2-1}
\tan{\pi({k\over 2}-n+j)\over |k|}
\biggr].\label{mgzs23Nov12}\end{aligned}$$ The first term comes from P poles and the second from NP poles. Although we prove the Seiberg duality in Appendix \[Appendix\_SD\], it is quite nontrivial that gives the same value for the dual pair .
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![*The integration contour $C=I\cap I_{i\infty}\cap I_k\cap ![*The integration contour $C=I\cap I_{i\infty}\cap I_k\cap
I_{-i\infty}$ (clockwise) and poles, for various values of $k,N_2$. (a) and (b) are Seiberg duals of each other and so are (c) and (d). The contour $I_k$ is parallel to $I$ and shifted by $k$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity. “$+$” (red) denotes the P pole and “$\times$” (blue) the NP pole. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. The choices of the parameter $\eta$ for the contour $I$ are $\eta=0_+$ for (a) and (c), $\eta=\half+0_+$ for (b), and $\eta=1+0_+$ for (d). \[contour2\]*](poles_spc_odd1.pdf "fig:"){height="5cm"} I_{-i\infty}$ (clockwise) and poles, for various values of $k,N_2$. (a) and (b) are Seiberg duals of each other and so are (c) and (d). The contour $I_k$ is parallel to $I$ and shifted by $k$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity. “$+$” (red) denotes the P pole and “$\times$” (blue) the NP pole. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. The choices of the parameter $\eta$ for the contour $I$ are $\eta=0_+$ for (a) and (c), $\eta=\half+0_+$ for (b), and $\eta=1+0_+$ for (d). \[contour2\]*](poles_spc_odd2.pdf "fig:"){height="5cm"}
\(a) $|k|=5,N_2=3$ \(b) $|k|=5,N_2=4$
\[2ex\] ![*The integration contour $C=I\cap I_{i\infty}\cap I_k\cap ![*The integration contour $C=I\cap I_{i\infty}\cap I_k\cap
I_{-i\infty}$ (clockwise) and poles, for various values of $k,N_2$. (a) and (b) are Seiberg duals of each other and so are (c) and (d). The contour $I_k$ is parallel to $I$ and shifted by $k$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity. “$+$” (red) denotes the P pole and “$\times$” (blue) the NP pole. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. The choices of the parameter $\eta$ for the contour $I$ are $\eta=0_+$ for (a) and (c), $\eta=\half+0_+$ for (b), and $\eta=1+0_+$ for (d). \[contour2\]*](poles_spc_even1.pdf "fig:"){height="5cm"} I_{-i\infty}$ (clockwise) and poles, for various values of $k,N_2$. (a) and (b) are Seiberg duals of each other and so are (c) and (d). The contour $I_k$ is parallel to $I$ and shifted by $k$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity. “$+$” (red) denotes the P pole and “$\times$” (blue) the NP pole. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. The choices of the parameter $\eta$ for the contour $I$ are $\eta=0_+$ for (a) and (c), $\eta=\half+0_+$ for (b), and $\eta=1+0_+$ for (d). \[contour2\]*](poles_spc_even2.pdf "fig:"){height="5cm"}
\(c) $|k|=4,N_2=2$ \(d) $|k|=4,N_2=4$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Let us look at this in more detail in the following case: $$\begin{aligned}
U(1)_5\times U(3)_{-5} = U(4)_5\times U(1)_{-5}\ .\end{aligned}$$ Using the above formulas, we obtain the partition functions of this dual pair which can be massaged into $$\begin{aligned}
Z_{\rm ABJ}(1,3)_5&={1\over 50}\sin{\pi\over 5}\biggl[\underbrace{\tan{2\pi\over 5}\left(2\tan{\pi\over 5}+\tan{2\pi\over 5}\right)}_\text{P}\underbrace{-10\cot{\pi\over 5}}_\text{NP}\biggr]e^{-{\pi i\over 5}}\ ,\label{Z135}\\
Z_{\rm ABJ}(1,4)_{-5}&={1\over 50}\sin{\pi\over 5}\biggl[\underbrace{-10\cot{\pi\over 5}}_\text{P}\underbrace{+\tan{2\pi\over 5}\left(2\tan{\pi\over 5}+\tan{2\pi\over 5}\right)}_\text{NP}\biggr]e^{4\pi i\over 5}\ .\label{Z145}\end{aligned}$$ These two indeed agree up to a phase and the phase difference agrees with the conjecture made in [@Kapustin:2010mh]. Observe that the contributions from the P and NP poles are interchanged under the duality. See Figure \[contour2\](a), (b) for the structure of the P and NP poles in the two theories. For discussion on the pole structure in more general cases, we refer the reader to Appendix \[Appendix\_SD\].
### Even $\boldsymbol{k}$ case {#even-boldsymbolk-case .unnumbered}
The even $k$ case is technically a little more tricky. Using a trick similar to the one used in [@Okuyama:2011su], the integral (\[1N2integral\]) for even $k$ can be shown to be equal to the following contour integral $$\begin{aligned}
I(1,N_2)_{k}=-{1\over 2\pi i}
\int_{C} {\pi \, ds\over \sin(\pi s)}\left(a-{s\over k}\right)\
\prod_{l=1}^{N_2-1}\tan\left({(s+l)\pi\over k}\right),\end{aligned}$$ where $a$ is an arbitrary constant. For ${|k|\over
2}-N_2+1\ge 0$, we can evaluate this by summing over pole residues and obtain $$\begin{aligned}
&I(1,N_2)_{k}
=\left(\sum_{n=0}^{{|k|\over 2}-N_2}+\sum_{n={|k|\over 2}}^{|k|-N_2}\right)
\left(a-{n\over |k|}\right)(-1)^n\prod_{j=1}^{N_2-1}\tan{\pi(n+j)\over |k|}\notag\\
& \quad
+\sum_{n=1}^{N_2-1}
(-1)^{{|k|\over 2}-n}
\left[-\left(a-{1\over 2}+{n\over |k|}\right)\sum_{\substack{j=1\\(j\neq n)}}^{N_2-1}
{2\over \sin{2\pi \left({|k|\over 2}-n+j\right)\over |k|}}
+{1\over \pi}
\right]\prod_{\substack{j=1\\ (j\neq n)}}^{N_2-1}\tan{\pi(n+j)\over |k|}.\label{Ievenk}\end{aligned}$$ The first line comes from P poles which are simple, while the second line comes from double poles created by simple NP and P poles sitting on top of each other. We note also that, despite its appearance, this expression does not depend on the constant $a$. The expression of $I(1,N_2)_k$ for ${|k|\over 2}-N_2+1\le 0$ is more lengthy and we do not present it, because the Seiberg duality proven in Appendix \[Appendix\_SD\] guarantees that it can be obtained from .
Let us study in detail the following duality U(1)\_4U(2)\_[-4]{} = U(4)\_4U(1)\_[-4]{} . The partition functions of this dual pair yield $$\begin{aligned}
Z_{\rm ABJ}(1,2)_4&={1\over 32}\biggl[\underbrace{1}_\text{P}\underbrace{-{2\over \pi}}_\text{P+NP}\biggr]\ ,\label{Z124}\\
Z_{\rm ABJ}(1,4)_{-4}&={1\over 32}\biggl[\underbrace{-{2\over \pi}}_\text{P+NP}\underbrace{+1}_\text{NP}\biggr]e^{\pi i}\ .\label{Z144}\end{aligned}$$ These two agree up to a phase. The phase difference is again in agreement with [@Kapustin:2010mh]. The pole structure of the two theories is shown in Figure \[contour2\](c), (d). In the above, “P+NP” means the contribution from a double pole that comes from P and NP poles on top of each other. Again, the contributions from the P and NP poles are interchanged under the duality. Actually, in the even $k$ case, there is a subtlety in interpreting simple poles as P or NP, but for details we refer the reader to Appendix \[Appendix\_SD\].
Conclusions and discussions {#CandD}
===========================
In this paper, we have studied the partition function of the ABJ theory, [*i.e.*]{}, the ${\cal N}=6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ Chern-Simons-matter theory dual to M-theory on $AdS_4\times S^7/Z_k$ with a discrete torsion or type IIA string theory on $AdS_4\times CP^3$ with a NS-NS $B_2$-field turned on [@Aharony:2008gk]. More concretely, we have computed the ABJ partition function (\[ABJMM\]) and found the expression (\[ABJpart\_integral\]) in terms of ${\rm min}(N_1,N_2)$-dimensional integrals as opposed to the original $(N_1+N_2)$-dimensional integrals. This generalizes the “mirror description” of the partition function of the ABJM theory [@Kapustin:2010xq] and may serve as the starting point for the ABJ generalization of the Fermi gas approach [@Marino:2011eh]. We have taken an indirect approach: Instead of performing the eigenvalue integrals in (\[ABJMM\]) directly, we have first calculated the partition function of the L(2,1) lens space matrix model (\[lensMM\]) *exactly* and found the expression (\[lensAns2\]) as a product of $q$-deformed Barnes $G$-function and a generalization of multiple $q$-hypergeometric function. We have then performed the analytic continuation $N_2\to -N_2$ of the lens space partition function to obtain the ABJ partition function. As checks we have shown that our main result (\[ABJpart\_integral\]) correctly reproduces perturbative expansions and in the $N_1=1$ case, [*i.e.*]{}, for the $U(1)_k\times U(N_2)_{-k}$ theories, the Seiberg duality indeed holds. In particular, we have uncovered that the perturbative and nonperturbative contributions to the partition function are interchanged under the Seiberg duality and derived, in the $N_1=1$ case, the formula for the phase difference of dual-pair partition functions conjectured in [@Kapustin:2010mh]. It is also worth remarking that the ABJ partition function (\[ABJpart\_integral\]) vanishes for $|N_1-N_2|>k$ in line with the conjectured supersymmetry breaking [@Bergman:1999na].
As commented before, we note, however, that the analytic continuation is ambiguous and we have adopted a particular prescription that required [*a posteriori*]{} justification. Especially, our prescription involves an intermediate step, namely an infinite sum expression, (\[ABJpart\_formalsum\]) which is non-convergent and becomes singular for an even integer $k$. Although the integral representation (\[ABJpart\_integral\]) provides a regularization and an analytic continuation of the formal series (\[ABJpart\_formalsum\]) in the complex $q$-plane, it would be better if we could render every step of the calculation process well-defined. In this connection, it is somewhat dissatisfying that the $q$-hypergeometric structure enjoyed by the lens space partition function becomes obscured after the analytic continuation to the ABJ partition function. It might be that there is a better way to perform the analytic continuation that manifestly respects the $q$-hypergeometric structure and directly yields a finite sum expression for an integer $k$ without passing to the integral representation.
Although the successful test of the Seiberg duality for the $U(1)_k\times U(N_2)_{-k}$ theories provides compelling evidence for our prescription, a general proof is clearly desired. In this regard, we note, as discussed in Section \[SD\], that the Seiberg duality acts on the $U(|N_1-N_2|)_k$ CS factor and the integral part separately. Namely, apart from a phase factor, the CS and the integral parts are respectively invariant under the duality, where the invariance of the former follows from the level-rank duality. Thus the general proof amounts to showing the invariance of the integral part, [*i.e.*]{}, the second line of (\[ABJpart\_integral\]). We leave this proof for a future work.
Following this work, there are a few more immediate directions to pursue: It is straightforward to generalize our computation of the partition function to Wilson loops [@Drukker:2008zx; @Chen:2008bp; @Rey:2008bh; @Drukker:2009hy; @Cardinali:2012ru]. Indeed, we can proceed almost in parallel with the case of the partition function for the most part including the analytic continuation, although the computation becomes inevitably more involved. We hope to report on our progress in this direction in the near future [@WIP2]. It may also be possible to apply our method to more general CSM theories with fewer supersymmetries, provided that a similar analytic continuation works. Meanwhile, we have stressed in the introduction that this work may have significance to the study of higher spin theories, especially, in connection to the recent ABJ triality conjecture [@Chang:2012kt]. As mentioned towards the end of Section \[Examples\], it is in fact feasible to analyze the $U(1)_k\times U(N_2)_{-k}$ and $U(2)_k\times
U(N_2)_{-k}$ partition functions at large $N_2$ and $k$ [@WIP1]. This may shed lights on the understanding of the ${\cal N}=6$ parity-violating Vasiliev theory on $AdS_4$. In particular, for the $U(1)_k\times U(N_2)_{-k}$ theory, the fact that the Seiberg duality separately acts on the $U(N_2-1)$ CS and the integral parts seems to suggest that it is only the integral part that may be dual to the vector-like Vasiliev theory.
Last but not least, it is most important to gain, if possible, new physical and mathematical insights into the microscopic description of M-theory through all these studies. Although the ABJ(M) theory is a very useful and practical description of maximally supersymmetric 3d conformal field theories, the construction by Bagger-Lambert and Gustavsson based on a 3-algebra [@Bagger:2006sk; @Gustavsson:2007vu] is arguably more insightful, suggesting potentially a new mathematical structure behind quantum membrane theory. What we envisage in this line of study is to search for a way to reorganize the ABJ(M) partition function in terms of the degrees of freedom that might provide an intuitive understanding of the $N^{3/2}$ scaling and suggest hidden structures behind the microscopic description of M-theory such as 3-algebras.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Oren Bergman, Hiroyuki Fuji, Yoichi Kazama, Sanefumi Moriyama, Vasilis Niarchos, Keita Nii, Kazutoshi Ohta, Shuichi Yokoyama, Xi Yin, and Tamiaki Yoneya for comments and discussions. The work of HA was supported in part by Grant-in-Aid for Scientific Research (C) 24540210 from the Japan Society for the Promotion of Science (JSPS). The work of SH was supported in part by the Grant-in-Aid for Nagoya University Global COE Program (G07). The work of MS was supported in part by Grant-in-Aid for Young Scientists (B) 24740159 from the Japan Society for the Promotion of Science (JSPS).
$\boldsymbol{q}$-analogs {#Appendix_q-Analogs}
========================
Roughly, a $q$-analog is a generalization of a quantity to include a new parameter $q$, such that it reduces to the original version in the $q\to
1$ limit. In this appendix, we will summarize definitions of various $q$-analogs and their properties relevant for the main text.
#### $\boldsymbol{q}$-number:
For $z\in\bbC$, the $q$-number of $z$ is defined by $$\begin{aligned}
[z]_q:={1-q^z\over 1-q},\qquad \end{aligned}$$
#### $\boldsymbol{q}$-Pochhammer symbol:
For $a\in\bbC$, $n\in\bbZ_{\ge 0}$, the $q$-Pochhammer symbol $(a;q)$ is defined by $$\begin{aligned}
(a;q)_n&:= \prod_{k=0}^{n-1}(1-aq^k)
=(1-a)(1-aq)\cdots(1-aq^{n-1})
={(a;q)_\infty\over (aq^n;q)_\infty}.\end{aligned}$$ For $z\in\bbC$, $(a;q)_z$ is defined by the last expression: $$\begin{aligned}
(a;q)_z&:=
{(a;q)_\infty\over (aq^z;q)_\infty}
=\prod_{k=0}^\infty{1-aq^k\over 1-aq^{z+k}}
.\label{maux20Sep12}\end{aligned}$$ This in particular means $$\begin{aligned}
(a;q)_{-z}&={1\over (aq^{-z};q)_z}.\end{aligned}$$ For $n\in \bbZ_{\ge 0}$, $$\begin{aligned}
(a;q)_{-n}&={1\over (aq^{-n};q)_n}={1\over \prod_{k=1}^n(1-a/q^k)}.\end{aligned}$$
Note that the $q\to 1$ limit of the $q$-Pochhammer symbol is not the usual Pochhammer symbol but only up to factors of $(1-q)$: $$\begin{aligned}
\lim_{q\to 1}{(q^a;q)_n\over (1-q)^n}&=
a(a+1)\dots(a+n-1).\end{aligned}$$
We often omit the base $q$ and simply write $(a;q)_\nu$ as $(a)_\nu$.[^11]
Some useful relations involving $q$-Pochhammer symbols are $$\begin{aligned}
(a)_\nu&={(a)_z\over (aq^\nu)_{z-\nu}}=(a)_z (aq^z)_{\nu-z},\label{hcey4Sep12}\\
(q)_\nu&=(1-q)^{\nu}\Gamma_q(\nu+1),\label{kjzv20Sep12}\\
(q^\mu)_\nu&={(q)_{\mu+\nu-1}\over (q)_{\mu-1}} =(1-q)^\nu{\Gamma_q(\mu+\nu)\over \Gamma_q(\mu)},\\
(aq^\mu)_\nu&
=(aq^\mu)_{z-\mu}(aq^z)_{\mu+\nu-z}
={(aq^\mu)_z\over (aq^{\mu+\nu})_{z-\nu}}
={(aq^z)_{\mu+\nu-z}\over (aq^z)_{\mu-z}},
\label{jwdu20Sep12}\end{aligned}$$ where $\mu,\nu,z\in\bbC$ and $\Gamma_q(z)$ is the $q$-Gamma function defined below. For $n\in\bbZ$, we have the following formulae which “reverse” the order of the product in the $q$-Pochhammer symbol: $$\begin{aligned}
(aq^z)_n&=(-a)^n q^{zn+\half n(n-1)}(a^{-1}q^{1-n-z})_n,\label{jzmp20Sep12}\\
(\pm q^{-n})_n&=(\mp 1)^n q^{-\half n(n+1)}(\pm q)_n.\label{hcit4Sep12}\end{aligned}$$ If $\nu=n+\epsilon$ with $|\epsilon|\ll 1$, the correction to this is of order $\cO(\epsilon)$: $$\begin{aligned}
(aq^z)_{n+\epsilon}&=(-a)^nq^{zn+{1\over 2}n(n-1)}(a^{-1}q^{1-n-z})_n(1+\cO(\epsilon)),\qquad a\neq 1.\end{aligned}$$ Here we assumed that $a\neq 1$ and $a-1\gg \cO(\epsilon)$,
#### $\boldsymbol{q}$-factorials:
For $n\in\bbZ_{\ge 0}$, the $q$-factorial is given by $$\begin{aligned}
[n]_q!:= [1]_q[2]_q\cdots[n]_q={(q)_n\over (1-q)^n},\qquad [0]_q!=1,
\qquad
[n+1]_q!=[n]_q[n-1]_q!~.\end{aligned}$$
#### $\boldsymbol{q}$-Gamma function:
For $z\in\bbC$, the $q$-Gamma function $\Gamma_q(z)$ is defined by $$\begin{aligned}
\Gamma_q(z+1)&:= (1-q)^{-z}\prod_{k=1}^\infty {1-q^{k}\over
1-q^{z+k}}.\end{aligned}$$ The $q$-Gamma function satisfies the following relations: $$\begin{aligned}
\Gamma_q(z)&
=(1-q)^{1-z}{(q)_\infty\over (q^z)_\infty}
=(1-q)^{1-z}(q)_{z-1},\\
\Gamma_q(z+1)&=[z]_q\Gamma_q(z),\\
\Gamma_q(1)&=\Gamma_q(2)=1,\qquad
\Gamma_q(n)=[n-1]_q! \quad (n\ge 1).\end{aligned}$$ The behavior of $\Gamma_{q}(z)$ near non-positive integers is $$\begin{aligned}
\Gamma_q(-n+\epsilon)
&={(-1)^{n+1}(1-q)q^{\half n(n+1)}\over \Gamma_q(n+1)\,\log q}
{1\over \epsilon}+\cdots,\qquad
\Gamma_q(n+1)=[n]_q!~,\quad $$ where $n\in\bbZ_{\ge 0}$, and $\epsilon\to 0$. As $q\to 1$, this reduces to the formula for the ordinary $\Gamma(z)$, $$\begin{aligned}
\Gamma(-n+\epsilon)&={(-1)^{n}\over \Gamma(n+1)}{1\over \epsilon}+\cdots,\qquad
\Gamma(n+1)=n!~.\end{aligned}$$
#### $\boldsymbol{q}$-Barnes $\boldsymbol{G}$ function:
For $z\in\bbC$, the $q$-Barnes $G$ function is defined by [@Nishizawa] $$\begin{aligned}
G_2(z+1;q)&:=(1-q)^{-{1\over 2}z(z-1)}
\prod_{k=1}^\infty
\biggl[\biggl({1-q^{z+k}\over 1-q^k}\biggr)^k(1-q^k)^z
\biggr].\end{aligned}$$ Some of its properties are $$\begin{gathered}
G_2(1;q)=1,\qquad G_2(z+1;q)=\Gamma_q(z)G_2(z),\\
G_2(n;q)
=\prod_{k=1}^{n-1}\Gamma_q(k)
=\prod_{k=1}^{n-2} [k]_q!
=(1-q)^{-\half(n-1)(n-2)}\prod_{j=1}^{n-2}(q)_j
=\prod_{k=1}^{n-2}[k]_q^{n-k-1},\\
\prod_{1\le A<B\le n}(q^A-q^B)=
q^{{1\over 6}n(n^2-1)}(1-q)^{\half n(n-1)}G_2(n+1;q).
\label{ehwt6Nov12}\end{gathered}$$ The behavior of $G_2(z;q)$ near non-positive integers is $$\begin{aligned}
G_2(-n+\epsilon;q)
={(-1)^{\half (n+1)(n+2)}G_2(n+2;q)\,(\log q)^{n+1}
\over q^{{1\over 6}n(n+1)(n+2)} (1-q)^{n+1}}\epsilon^{n+1}+\cdots,
\label{ih4Nov12}\end{aligned}$$ where $n\in \bbZ_{\ge 0}$, and $\epsilon\to 0$. As $q\to 1$, this reduces to the formula for the ordinary $G_2(z)$, $$\begin{aligned}
G_2(-n+\epsilon)
&=(-1)^{\half n(n+1)}G_2(n+2)\epsilon^{n+1}+\cdots.
\label{il4Nov12}\end{aligned}$$
#### $\boldsymbol{q}$-digamma and $\boldsymbol{q}$-polygamma functions
The $q$-digamma function $\psi_q(z)$ and $q$-polygamma function $\psi^{(n)}_q(z)$, $n\in\bbZ_{\ge 0}$, are defined by $$\begin{aligned}
\psi_q(z):=\partial_z \ln \Gamma_q(z),\qquad
\psi^{(n)}_q(z):=\partial_z^n \psi_q(z)=
\partial_z^{n+1}\ln \Gamma_q(z).\end{aligned}$$ From the definition of $\Gamma_q(z)$, it straightforwardly follows that $$\begin{aligned}
\psi_q(z)&=-\log(1-q)+\sum_{n=0}^\infty {q^{n+z}\over 1-q^{n+z}}\ln q,\qquad
\psi^{(1)}_q(z)=\sum_{n=0}^\infty {q^{n+z}\over (1-q^{n+z})^2}\ln^2 q.\end{aligned}$$
#### $\boldsymbol{q}$-hypergeometric function (basic hypergeometric series):
The $q$-hypergeometric function, or the basic hypergeometric series with base $q$, is defined by [@Gasper-Rahman] $$\begin{aligned}
_{r}\phi_s\left({a_1,\dots,a_r\atop b_1\dots,b_s};q,z\right)
:=\sum_{n=0}^\infty
{(a_1)_n\cdots (a_r)_n\over (q)_n (b_1)_n\cdots (b_s)_n}
\left[(-1)^nq^{n\choose 2}\right]^{1+s-r}z^n.\end{aligned}$$ In particular, for $r=k+1,s=k$, $$\begin{aligned}
_{k+1}\phi_k\left({a_1,\dots,a_{k+1}\atop b_1\dots,b_k};q,z\right)
=\sum_{n=0}^\infty {(a_1)_n\cdots (a_{k+1})_n\over (b_1)_n\cdots (b_k)_n}
{z^n\over (q)_n}.\end{aligned}$$
Lens space matrix model {#Appendix_LensMM}
=======================
The partition function for the lens space matrix model was defined in . Here, we explicitly carry out the integral and write the result in a simple closed form as given in , . The following computation can be thought of as a generalization of the matrix integration technique using Weyl’s denominator formula (see for example [@Aganagic:2002wv; @Marino:2011nm]), explicitly worked out.
First, we note that the 1-loop determinant part can be reduced to a single Vandermonde determinant by shifting the integration variables as $\mu_j\to
\mu_j-{i\pi\over 2},\nu_a\to\nu_a+{i\pi\over 2}$, as follows: $$\begin{aligned}
\begin{split}
&\Delta_{\rm sh}(\mu)\Delta_{\rm sh}(\nu)\Delta_{\rm ch}(\mu,\nu)\\
&=
\prod_{j<k} e^{-{\mu_j+\mu_k\over 2}}(e^{\mu_j}-e^{\mu_k})
\prod_{a<b} e^{-{\nu_a+\nu_b\over 2}}(e^{\nu_a}-e^{\nu_b})
\prod_{j,a} e^{-{\mu_j+\nu_k\over 2}}(e^{\mu_j}+e^{\nu_a})\\
&\to
\prod_{j<k} e^{-{\mu_j+\mu_k\over 2}}(e^{\mu_j}-e^{\mu_k})
\prod_{a<b} e^{-{\nu_a+\nu_b\over 2}}(e^{\nu_a}-e^{\nu_b})
\prod_{j,a}
e^{-{i\pi\over 2}}e^{-{\mu_j+\nu_k\over 2}}(e^{\mu_j}-e^{\nu_a})
\\
&=
e^{-{i\pi\over 2}N_1N_2-{N-1\over 2}(\sum_j \mu_j+\sum_a \nu_a)}\Delta(\mu,\nu),
\end{split}\end{aligned}$$ where $N:= N_1+N_2$ and $\Delta(\mu,\nu)$ is the Vandermonde determinant for $(\mu_j,\nu_a)$ which can be evaluated as $$\begin{aligned}
\Delta(\mu,\nu)
&:=
\prod_{j<k} (e^{\mu_j}-e^{\mu_k})
\prod_{a<b} (e^{\nu_a}-e^{\nu_b})
\prod_{j,a} (e^{\mu_j}-e^{\nu_a})\notag\\
&=\sum_{\sigma\in S_{N}} (-1)^\sigma
e^{ \sum_{j=1}^{N_1} (\sigma(j)-1)\mu_j
+\sum_{a=1}^{N_2}(\sigma(N_1+a)-1)\nu_a }.
\label{itho1Nov12}\end{aligned}$$ Here, $S_N$ is the permutation group of length $N$ and $(-1)^\sigma$ is the signature of $\sigma\in S_N$. Because each term in is an exponential whose exponent is linear in $\mu_j,\nu_a$, the integral in is trivial Gaussian. After carrying out the $\mu_i,\nu_a$ integrals and massaging the result a little bit, we obtain $$\begin{aligned}
\begin{split}
Z_{\text{lens}}(N_1,N_2)_k&
=\cN_{\text{lens}}(-1)^{{1\over 2}N_1(N_1+1)+{1\over 2}N_2(N_2+1)+N_1N_2}
e^{-{g_s\over 6}N(N+1)(N+2)}
\left({g_s\over 2\pi}\right)^{N\over 2}
Z_{\text{lens}}^0,\\
Z_{\text{lens}}^0&:=
\sum_{\sigma,\tau\in S_{N}}(-1)^{\sigma+\tau}
e^{g_s\sum_{A=1}^N\sigma(A)\tau(A)
+{i\pi\over 2}\left(\sum_{A=1}^{N_1}-\sum_{A=N_1+1}^{N_1+N_2}\right)(\sigma(A)+\tau(A))}.
\end{split}
\label{lrdr25Aug12}\end{aligned}$$ Note that the summation over $\tau$ in can be written in terms of a determinant as $$\begin{aligned}
Z_{\text{lens}}^0(N_1,N_2)_k&
=\sum_\sigma (-1)^\sigma
e^{
{i\pi\over 2}\left(\sum_{j=1}^{N_1}-\sum_{j=N_1+1}^{N_1+N_2}\right)\sigma(j)}\det W(\sigma),\label{lreb25Aug12}\\
W(\sigma)_{AB}&:=\begin{cases}
e^{(g_s\sigma(A)+{i\pi\over 2})B}& \qquad (1\le A\le N_1),\\
e^{(g_s\sigma(A)-{i\pi\over 2})B}& \qquad (N_1+1\le A\le N).
\end{cases}\end{aligned}$$ The matrix $W$ is essentially a Vandermonde matrix and its determinant can be evaluated using the formula $$\begin{aligned}
\det[(x_A)^B]=\biggl(\prod_{A=1}^N x_A\biggr)^N \prod_{1\le A<B\le N}
(x_A^{-1}-x_B^{-1})\end{aligned}$$ as follows: $$\begin{aligned}
\det W(\sigma)
&=e^{Ng_s\sum_{A=1}^N \sigma(A)+{i\pi\over 2}N(N_1-N_2)}
\prod_{j<k}(e^{-g_s\sigma(j)-{i\pi\over 2}}-e^{-g_s\sigma(k)-{i\pi\over 2}})\notag\\
&\qquad\qquad\times
\prod_{a<b}(e^{-g_s\sigma(a)+{i\pi\over 2}}-e^{-g_s\sigma(b)+{i\pi\over 2}})
\prod_{j,a}(e^{-g_s\sigma(j)-{i\pi\over 2}}-e^{-g_s\sigma(a)+{i\pi\over 2}})
\notag\\
&=e^{{i\pi\over 4}(N_1(N_1+1)-N_2(N_2+1)-2N_1N_2)}
e^{{g_s\over 2} N^2(N+1)}
\prod_{j<k}(e^{-g_s\sigma(j)}-e^{-g_s\sigma(k)})
\notag\\
&\qquad\qquad\times
\prod_{a<b}(e^{-g_s\sigma(a)}-e^{-g_s\sigma(b)})
\prod_{j,a}(e^{-g_s\sigma(j)}+e^{-g_s\sigma(a)}).\end{aligned}$$ Plugging this into and , the expression for $Z_{\text{lens}}$ is $$\begin{aligned}
Z_{\text{lens}}(N_1,N_2)_k
&=\cN_{\text{lens}}\left({g_s\over 2\pi}\right)^{N\over 2}
(-1)^{\half N_1(N_1+1)}
q^{-{1\over 3}N(N^2-1)}
\sum_{\sigma\in S_N} (-1)^{\sigma+
\sum_{A=1}^{N_1}\sigma(A)}\notag\\
&\qquad
\times\prod_{j<k}(q^{\sigma(j)}-q^{\sigma(k)})
\prod_{a<b}(q^{\sigma(a)}-q^{\sigma(b)})
\prod_{j,a}(q^{\sigma(j)}+q^{\sigma(a)})\label{jims29Aug12}\end{aligned}$$ where $q=e^{-g_s}$.
We can rewrite in a simpler form as follows. $\sigma$ is a permutation of length $N=N_1+N_2$. Let us take its first $N_1$ entries $\sigma(1),\sigma(2),\dots, \sigma(N_1)$, permute them into increasing order, and call them $C_1,\dots,C_{N_1}$ ($C_1<\cdots<C_{N_1}$). Similarly, we take the last $N_2$ entries $\sigma(N_1),\dots, \sigma(N)$, permute them into increasing order, and call them $D_1,\dots,D_{N_2}$ ($D_1<\cdots<D_{N_2}$). Let the signature for the permutation to take $(C_1,\dots,C_{N_1})$ to $(\sigma(1),\dots,\sigma(N_1))$ be $(-1)^C$ and the signature for the permutation to take $(D_1,\dots,D_{N_2})$ to $(\sigma(N_1+1),\dots,\sigma(N))$ be $(-1)^D$. Namely, $$\begin{aligned}
(-1)^C:= \sign\begin{pmatrix}
C_1&\dots &C_{N_1}\\
\sigma(1)&\dots &\sigma(N_1)
\end{pmatrix},\qquad
(-1)^D:=\sign\begin{pmatrix}
D_1&\dots &D_{N_2}\\
\sigma(N_1+1)&\dots &\sigma(N)
\end{pmatrix}.\label{kznq29Aug12}\end{aligned}$$ Then the factors in can be rewritten as $$\begin{aligned}
\prod_{j<k}(q^{\sigma(j)}-q^{\sigma(k)}) =(-1)^C \prod_{C_j<C_k}(q^{C_j}-q^{C_k})
,\quad
\prod_{a<b}(q^{\sigma(a)}-q^{\sigma(b)}) =(-1)^D \prod_{D_a<D_b}(q^{D_a}-q^{D_b})\label{ldve29Aug12}.\end{aligned}$$ These relations are easy to see by looking at the left hand side as Vandermonde determinants. Also, note that $$\begin{aligned}
\sign\begin{pmatrix}
1&\dots&N_1&N_1+1&\dots&N\\
C_1&\dots&C_{N_1}&D_1&\dots&D_{N_2}
\end{pmatrix}
=(-1)^{\half N_1(N_1+1)+\sum_{A=1}^{N_1}\sigma(A)}.\label{kzkp29Aug12}\end{aligned}$$ This is seen as follows. First, let us permute $(C_1,\dots, C_{N_1})$ into $(C_{N_1},\dots,C_1)$, which gives $(-1)^{\half N_1(N_1-1)}$. Next, let us permute $(C_{N_1},\dots, C_1,D_1,\dots, D_{N_2})$ into $(1,\dots,N)$, starting by moving $C_{N_1}$ to the correct position. For this, $C_{N_1}$ commute through other $C_{N_1}-1$ numbers to its right, giving $(-1)^{C_{N_1}-1}$. Next, we move $C_{N_1-1}$ to the correct position, which gives $(-1)^{C_{N_1-1}-1}$. We keep doing this until we get $(1,\dots,N)$. In the end, we obtain $(-1)^{\sum_{j=1}^{N_1} (C_j-1)}=(-1)^{\sum_{A=1}^{N_1}
\sigma(A)-N_1}=(-1)^{\sum_{A=1}^{N_1} \sigma(A)+N_1}$. Combining this with the previous factor, we obtain . Eqs. and mean that $$\begin{aligned}
(-1)^\sigma=(-1)^{C+D+\half N_1(N_1+1)+\sum_{A=1}^{N_1}\sigma(A)}.\label{louk29Aug12}\end{aligned}$$ Plugging into and using , we obtain the following nice concise formula for the partition function for the lens space matrix model: $$\begin{aligned}
Z_{\text{lens}}(N_1,N_2)_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
\left({g_s\over 2\pi}\right)^{N\over 2}
q^{-{1\over 3}N(N^2-1)}
\notag\\
&\qquad
\times \sum_{(\cN_1,\,\cN_2)}
\prod_{C_j<C_k} (q^{C_j}-q^{C_k})
\prod_{D_a<D_b} (q^{D_a}-q^{D_b})
\prod_{C_j,D_a} (q^{C_j}+q^{D_a}),\label{ihsh28Aug12}\end{aligned}$$ which is the expression presented in . Here, $\sum_{(\cN_1,\,\cN_2)}$ means summation over different ways to decompose $\{1,2,\dots,N_1+N_2\}$ into two disjoint sets $\cN_1$ and $\cN_2$ with $\#\cN_1=N_1$, $\#\cN_2=N_2$. Their elements are $$\begin{aligned}
\cN_1&=\{C_1,C_2,\dots,C_{N_1}\},\qquad C_1<C_2<\dots<C_{N_1},\\
\cN_2&=\{D_1,D_2,\dots,D_{N_2}\},\qquad D_1<D_2<\dots<D_{N_2}.\end{aligned}$$
Note that, using the identity , Eqn. can also be rewritten as $$\begin{aligned}
Z_{\text{lens}}(N_1,N_2)_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
\left({g_s\over 2\pi}\right)^{N\over 2}
q^{-{1\over 6}N(N^2-1)}(1-q)^{\half N(N-1)}G_2(N+1;q)\,S(N_1,N_2),
\label{gpmz3Sep12}
\\
S(N_1,N_2)&= \sum_{(\cN_1,\,\cN_2)}
\prod\limits_{C_j<D_a}{q^{C_j}+q^{D_a}\over q^{C_j}-q^{D_a}}
\prod\limits_{D_a<C_j}{q^{D_a}+q^{C_j}\over q^{D_a}-q^{C_j}},
\label{msmi31Aug12}\end{aligned}$$ which is the expression presented in .
Analytic continuation to ABJ matrix model {#Appendix_AnalCont}
=========================================
Here, we will obtain the ABJ matrix model partition function by analytically continuing the lens space matrix model partition function under $N_2\to -N_2$.
Normalization {#Appendix_Norm}
-------------
It has been shown [@Marino:2009jd] that the partition functions for the lens space and ABJ theories agree order by order in perturbation theory upon analytic continuing in the rank as $N_2\to -N_2$. Our strategy is to apply this analytic continuation to the lens space partition function to obtain the exact expression for the ABJ partition function. However, in order to analytically continue the partition functions, not just their perturbative expansion, we must properly normalize them, which is what we discuss first.
Because we already know [@Marino:2009jd] that the analytic continuation works perturbatively, all we have to do is to match the tree level part of the partition function. In the weak coupling limit $g_s\to 0$, the lens space partition function reduces to $$\begin{aligned}
Z_{\text{lens,tree}}
&:=
Z_{\text{lens}}(g_s\to 0)\notag\\
&=
2^{2N_1N_2}\cN_{\text{lens}}
\int \prod_{j} {d\mu_j\over 2\pi}
\prod_{a}{d\nu_a\over 2\pi}
\prod_{j<k}(\mu_j-\mu_k)^2
\prod_{a<b}(\nu_a-\nu_b)^2
e^{-{1\over 2g_s}(\sum_j \mu_j^2+\sum_a \nu_a^2)}.\label{fyi3Nov12}\end{aligned}$$ This is essentially the product of two copies of Gaussian matrix model partition function: $$\begin{aligned}
Z_{\text{lens,tree}}=
i^{-{\kappa\over 2}(N_1^2+N_2^2)} {2^{2N_1N_2}\over N_1! N_2!}
V(N_1,g_s)V(N_2,g_s),\label{niev2Nov12}\end{aligned}$$ where $V(n,g_s)$ is the $U(n)$ Gaussian matrix model integral, $$\begin{aligned}
V(n,g_s):=
\int \prod_{j=1}^{n}{d\lambda_j\over 2\pi}
\Delta(\lambda)^2
e^{-{1\over 2g_s}\sum_{j=1}^n \lambda_j^2},\qquad
\Delta(\lambda)=\prod_{1\le j<k\le n}(\lambda_j-\lambda_k).\label{jrqv3Sep12}\end{aligned}$$ $V(n,g_s)$ can be computed explicitly as [@Mehta] $$\begin{aligned}
V(n,g_s)
= g_s^{n^2\over 2}(2\pi)^{-{n\over 2}}G_2(n+2),
\label{juji2Nov12}\end{aligned}$$ where $G_2(z)$ is the (ordinary) Barnes function. In the present case we have $g_s={2\pi i\over k}$ and the integral is the Fresnel integral. Similarly, the ABJ partition function reduces in the weak coupling limit to $$\begin{aligned}
Z_{\text{ABJ,tree}}
\equiv
Z_{\text{ABJ}}(g_s\to 0)
=
i^{-{\kappa\over 2}(N_1^2-N_2^2)} {2^{-2N_1N_2}\over N_1! N_2!}
V(N_1,g_s)V(N_2,-g_s).\label{gxyb3Nov12}\end{aligned}$$ Note that $$\begin{aligned}
&V(n,-g_s)
=
(-g_s)^{n^2\over 2}(2\pi)^{-{n\over 2}}G_2(n+2)
= i^{-\kappa n^2} g_s^{n^2\over 2}(2\pi)^{-{n\over 2}}G_2(n+2)
= i^{-\kappa n^2}V(n,g_s).
\label{gyjn3Nov12}\end{aligned}$$ In the second equality, we used the fact that, because $g_s=2\pi i/k$, the Gauss integrals we are doing are actually Fresnel integrals and therefore $$\begin{aligned}
(\pm g_s)^{\half}=\sqrt{2\pi\over |k|}\,i^{\pm {\kappa\over 2}}.
\label{gzzx3Nov12}\end{aligned}$$ Using , the tree level ABJ partition function can be written as $$\begin{aligned}
Z_{\text{ABJ,tree}}
=
i^{-{\kappa\over 2}(N_1^2+N_2^2)} {2^{-2N_1N_2}\over N_1! N_2!}
V(N_1,g_s)V(N_2,g_s).\label{fhsv3Nov12}\end{aligned}$$ Looking at and , one may think that $Z_{\text{lens}}$ is analytically continued to $Z_{\text{ABJ}}$ under $N_2\to -N_2$. However, this does not work because $N_2!=\Gamma(N_2+1)$ and $V(N_2,g_s)$ do not transform in the right way under $N_2\to -N_2$.
To find the correct way to normalize partition function, we observe that the Gaussian matrix model can be thought of as coming from gauge fixing the “ungauged” Gaussian matrix integral, $$\begin{aligned}
\Vh(n,g_s)
:=
\int d^{n^2}\!M\, e^{-{1\over 2g_s}\tr M^2}
=(2\pi g_s)^{n^2\over 2}
,
\label{lbym22Sep12}\end{aligned}$$ to the eigenvalue basis. Our claim is that it is such ungauged matrix integrals that should be used for analytic continuation between lens space and ABJ theories. Let us make this statement more precise. Note that the relation between the ungauged Gaussian matrix integral and its gauge-fixed version , is $$\begin{aligned}
\Vh(n,g_s)
={(2\pi)^{\half n(n+1)}\over G_2(n+2)} V(n,g_s).\label{jtqh3Sep12}\end{aligned}$$ Based on this observation, we define the ungauged partition function for the lens space theory as follows: $$\begin{aligned}
\Zh_{\text{lens}}(N_1,N_2)_k
&:=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
{(2\pi)^{\half N_1(N_1+1)+\half N_2(N_2+1)}\over G_2(N_1+2)G_2(N_2+2)}
\int
\prod_{j=1}^{N_1} {d\mu_j\over 2\pi}
\prod_{a=1}^{N_2}{d\nu_a\over 2\pi}
\notag\\
&\qquad\qquad\times
\Delta_{\rm sh}(\mu)^2
\Delta_{\rm sh}(\nu)^2
\Delta_{\rm ch}(\mu,\nu)^2
e^{-{1\over 2g_s}(\sum_j \mu_j^2+\sum_a \nu_a^2)}
\\
&=
{(2\pi)^{\half N_1(N_1+1)+\half N_2(N_2+1)}
\over G_2(N_1+1) G_2(N_2+1) }
Z_{\text{lens}}(N_1,N_2),\end{aligned}$$ where we used the relation $G_2(n+2)=n!\, G_2(n+1)$. The weak coupling limit ($g_s\to 0,k\to\infty$) of this is $$\begin{aligned}
\Zh_{\text{lens}}(N_1,N_2)_{k\to \infty}
=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
2^{2N_1N_2}(2\pi g_s)^{N_1^2+N_2^2\over 2},
\label{nimf2Nov12}\end{aligned}$$ which does not involve $G_2$ or $N_2!$. In a similar manner, we define the ungauged partition function for the ABJ theory by $$\begin{aligned}
\Zh_{\text{ABJ}}(N_1,N_2)_k
&:=
i^{-{\kappa\over 2}(N_1^2-N_2^2)}
{(2\pi)^{\half N_1(N_1+1)+\half N_2(N_2+1)}\over G_2(N_1+2)G_2(N_2+2)}
\int
\prod_{j=1}^{N_1} {d\mu_j\over 2\pi}
\prod_{a=1}^{N_2}{d\nu_a\over 2\pi}
\notag \\
& \qquad\qquad \times
\Delta_{\rm sh}(\mu)^2
\Delta_{\rm sh}(\nu)^2
\Delta_{\rm ch}(\mu,\nu)^{-2}
e^{-{1\over 2g_s}(\sum_j \mu_j^2-\sum_a \nu_a^2)}
\\
&=
{(2\pi)^{\half N_1(N_1+1)+\half N_2(N_2+1)} \over G_2(N_1+1) G_2(N_2+1) }
Z_{\text{ABJ}}(N_1,N_2).\end{aligned}$$ The weak coupling limit of this is $$\begin{aligned}
\Zh_{\text{ABJ}}(N_1,N_2)_{k\to\infty }
=
i^{-{\kappa\over 2}(N_1^2-N_2^2)}
2^{-2N_1N_2}(2\pi g_s)^{N_1^2+N_2^2\over 2}.\label{nimv2Nov12}\end{aligned}$$ By comparing and , we find that the tree level partition functions are related simply as $$\begin{aligned}
\Zh_{\text{lens,tree}}(N_1,-N_2)_k=\Zh_{\text{ABJ,tree}}(N_1,N_2)_k.\end{aligned}$$ Therefore, including the perturbative part, we expect that the full partition functions satisfy $$\begin{aligned}
\Zh_{\text{lens}}(N_1,-N_2)_k=\Zh_{\text{ABJ}}(N_1,N_2)_k.\label{fjjc11Dec12}\end{aligned}$$ We will see that this indeed holds in explicit examples.
In terms of $\Zh_{\text{lens}}$, our result for the lens space partition function can then be written as $$\begin{aligned}
\Zh_{\text{lens}}(N_1,N_2)_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2^2)}
(2\pi)^{N_1^2+N_2^2\over 2}g_s^{N_1+N_2\over 2}
\notag\\
&\qquad\qquad
\times
q^{-{1\over 6}N(N^2-1)}(1-q)^{\half N(N-1)}
B(N_1+N_2,N_1,N_2)\,S(N_1,N_2),\label{mnyq3Sep12}\end{aligned}$$ where we defined $$\begin{aligned}
B(l,m,n):= {G_2(l+1;q)\over G_2(m+1)G_2(n+1)}.\end{aligned}$$ Recall that $G_2(z)$ has zeros at $z=0,-1,-2,\dots$. Therefore, $B(l,m,n)$ for $l,m,n\in\bbZ$ is finite if $m,n\ge 0$ but can be divergent if $m\le 0$ or $n\le 0$.
In going from the lens space matrix model to the ABJ matrix model, we flipped the sign of the quadratic term for $\nu_a$. However, we could have flipped the sign of the quadratic term for $\mu_j$. This implies a simple relation between $\Zh_{\text{lens}}(N_1,-N_2)_{k}$ and $\Zh_{\text{lens}}(N_2,-N_1)_{k}$. The relation is $$\begin{aligned}
\Zh_{\text{lens}}(N_1,-N_2)_k
=\Zh_{\text{lens}}(N_2,-N_1)_{-k}.\label{lcqb22Sep12}\end{aligned}$$ Here we have $-k$ on the right hand side because flipping the sign of the quadratic term in $\mu_j$, not $\nu_a$, will change the sign of $g_s\to -g_s$ in perturbative expansion. In view of the relation , this is nothing but .
Analytic continuation {#Appendix_AC}
---------------------
We would like to analytically continue $\Zh_{\text{lens}}(N_1,N_2)_k$ in $N_2$. The explicit expression for $\Zh_{\text{lens}}(N_1,N_2)_k$ is given by . In particular, we are interested in continuing $N_2$ to a negative integer $-N_2'$ where $N_2'\in
\bbZ_{>0}$. However, this is not so simple because Barnes $G_2(z)$ vanishes for negative integral $z$ and hence $B(N_1+N_2,N_1,N_2)$ in diverges at $N_2=-N_2'$. To deal with this situation, let us analytically continue $N_2$ to $$\begin{aligned}
N_2=-N_2'+\epsilon,\qquad N_2'\in\bbZ_{>0},\qquad |\epsilon|\ll 1\end{aligned}$$ and send $\epsilon\to 0$ at the end of the computation. Using the behavior of $G_2(z;q),G_2(z)$ near negative integral $z$ given in and , one can show that $B(N_1+N_2,N_1,N_2)$ diverges as $\epsilon\to 0$ as $$\begin{aligned}
&B(N_1+N_2,N_1,N_2)=B(N_1-N_2'+\epsilon,N_1,-N_2'+\epsilon)
\notag\\[2ex]
&\qquad
=
\begin{cases}
(-1)^{\half N_2'(N_2'-1)}
B(N_1-N_2',N_1,N_2')
\,\epsilon^{-N_2'}
& (N_2'\le N_1),\\[2ex]
(-1)^{N_1N_2'+\half N_1(N_1+1)}q^{{1\over 6}(N_1-N_2')((N_1-N_2')^2-1)}&\\
\qquad\qquad \times
(1-q)^{N_1-N_2'}g_s^{-N_1+N_2'}
B(N_2'-N_1,N_1,N_2')
\,\epsilon^{-N_1}
& (N_1\le N_2'),
\end{cases}
\label{iamq4Sep12}\end{aligned}$$ where we only kept the leading term. Therefore, in order for the entire $\Zh_{\text{lens}}$ to remain finite as $\epsilon\to 0$, the function $S(N_1,-N_2'+\epsilon)$ should vanish as $$\begin{aligned}
S(N_1,-N_2'+\epsilon)
\sim\begin{cases}
\epsilon^{N_2'}& (N_2'\le N_1)\\
\epsilon^{N_1} & (N_1\le N_2')
\end{cases}
~~
=\epsilon^{\min(N_1,N_2')}.\label{mvqa4Sep12}\end{aligned}$$
In the following, we will explicitly carry out analytic continuation of $S(N_1,N_2)$ and find that it indeed behaves as . We will begin with the simple cases with $N_1=1,2$ to get the hang of it, and then move on to the general $N_1$ case.
### $\boldsymbol{N_1=1}$ {#Appendix_N1equalto1}
The simplest case is $N_1=1$, for which gives $$\begin{aligned}
S(1,N_2)
&= \sum_{C=1}^{N_2+1}
\prod_{C<a}{q^{C}+q^{a}\over q^{C}-q^{a}}
\prod_{a<C}{q^{a}+q^{C}\over q^{a}-q^{C}}
= \sum_{C=1}^{N_2+1}
\prod_{j=1}^{N_2-C+1}{1+q^j\over 1-q^j}
\prod_{j=1}^{C-1}{1+q^j\over 1-q^j}\\
&=
\sum_{C=1}^{N_2+1}
{(-q)_{N_2-C+1} \over (q)_{N_2-C+1}}
{(-q)_{C-1} \over (q)_{C-1}}
=\sum_{n=0}^{N_2}
{(-q)_{N_2-n} \over (q)_{N_2-n}}
{(-q)_{n} \over (q)_{n}},
\label{gjqa20Sep12}\end{aligned}$$ where $n=C-1$. $(a)_n$ is the $q$-Pochhammer symbol defined in Appendix \[Appendix\_q-Analogs\]. We want to analytically continue this expression in $N_2$. The explicit $N_2$ dependence of the sum range seems to be an obstacle, but it can be circumvented by the following observation: as a function of $z$, $(q)_z$ has poles of order 1 at $z\in \bbZ_{<0}$, while $(-q)_z$ has no poles. Therefore, the summand in vanishes unless $0\le n\le
N_1$, and we can actually extend the range of summation as $$\begin{aligned}
S(1,N_2)
&= \sum_{n=0}^{\infty}
{(-q)_{N_2-n} \over (q)_{N_2-n}}
{(-q)_{n} \over (q)_{n}}.\label{lnuo31Aug12}\end{aligned}$$ This expression can be analytically continued to complex $N_2$, including negative integers.[^12]
We can rewrite in different forms which we will find more convenient. First, using and , one can show that $$\begin{aligned}
S(1,N_2) = \beta(1,N_2)\,\Phi(1,N_2),\label{mckp3Sep12}\end{aligned}$$ where $$\begin{aligned}
\beta(1,N_2):= {(-q)_{N_2}\over (q)_{N_2}},\qquad
\Phi(1,N_2):=\sum_{n=0}^\infty
{(-1)^n}{(q^{-N_2})_n (-q)_n\over (-q^{-N_2})_n (q)_n}
=
\, _2\phi_1\left({q^{-N_2},-q\atop -q^{-N_2}} ; q,-1\right).\label{evuj11Sep12}\end{aligned}$$ This expression is useful because the relation to $q$-hypergeometric function is manifest. The $q$-hypergeometric function $_2\phi_1$ is defined in Appendix \[Appendix\_q-Analogs\]. In addition, this way of writing $S$ is useful because it splits it into $\beta$ which vanishes for negative integral $N_2\in\bbZ_{<0}$ and $\Phi$ which is finite for all $N_2\in\bbZ$. It is easy to see that the first factor $\beta$ vanishes for negative $N_2=-N_2'\in\bbZ_{<0}$: $$\begin{aligned}
\beta(1,-N_2')
= {(-q)_{-N_2'}\over (q)_{-N_2'}}
= {(q^{1-N_2'})_{N_2'}\over (-q^{1-N_2'})_{N_2'}}
= {(1-q^{1-N_2'})\cdots (1-q^0)\over
(1+q^{1-N_2'})\cdots (1+q^0)}=0.
\label{kjhl20Sep12}\end{aligned}$$ However, we are actually setting $N_2=-N_2'+\epsilon$ and we have to keep track of how fast this vanishes as $\epsilon\to 0$. $\beta(1,-N_2'+\epsilon)$ involves ${(\pm q)_{-N_2'+\epsilon}}$ which, using with $z=-1+\epsilon$ and , can be rewritten as $$\begin{aligned}
(\pm q)_{-N_2'+\epsilon}
&=(\mp 1)^{N_2'-1}q^{-\half N_2'(N_2'+1)}{(\pm q)_{-1+\epsilon}\over (\pm q)_{N_2'-1}}.\label{msfw4Sep12}\end{aligned}$$ The behavior of $(\pm q)_{-1+\epsilon}$ can be seen, using the definition , as follows: $$\begin{aligned}
(q)_{-1+\epsilon}
&
={(1-q)(1-q^2)\cdots\over (1-q^\epsilon)(1-q^{1+\epsilon})\cdots}
=-{1\over \epsilon\ln q},
\quad
(-q)_{-1+\epsilon}
={(1+q)(1+q^2)\cdots\over (1+q^\epsilon)(1+q^{1+\epsilon})\cdots}
={1\over 2},
\label{mbuy20Sep12}\end{aligned}$$ where we kept only leading terms. We will do this kind of manipulation to extract $\epsilon\to 0$ behavior over and over again below, but we will not present the details henceforth. So, the behavior of $\beta(1,N_2)$ near integral $N_2$ is $$\begin{aligned}
\beta(1,N_2+\epsilon)=
\begin{cases}
{(-q)_{N_2}\over (q)_{N_2}}
& (N_2>0),
\\
(-1)^{N_2'} {\epsilon \ln q\over 2} {(q)_{N_2'-1}\over (-q)_{N_2'-1}}
& (N_2=-N_2'<0).
\end{cases}
\label{gnlw21Sep12}\end{aligned}$$ The $\cO(\epsilon)$ behavior for $N_2<0$ is the correct one to cancel the divergence of $B$ that we saw in , . On the other hand, the second factor $\Phi$ in is finite for all $N_2\in\bbZ$. For $N_2>0$, $(q^{-N_2})_n$ becomes zero for $n\ge N_2+1$ and the sum reduces to a finite sum. For $N_2=-N_2'<0$, the sum $(q^{-N_2})_n=(q^{N_2'})_n$ is non-vanishing for all $n\ge 0$.
There is another useful expression for $S(1,N_2)$. Using $q$-Pochhammer formulas, we can show that $$\begin{aligned}
S(1,N_2)=\gamma(1,N_2)\,\Psi(1,N_2),\label{jyes7Nov12}\end{aligned}$$ where $$\begin{aligned}
\gamma(1,N_2):= {(-q)_{N_2}(-q)_{-N_2-1}\over (q)_{N_2}(q)_{-N_2-1}},\qquad
\Psi(1,N_2):=\sum_{s=0}^\infty
(-1)^s{(q^{s+1})_{-N_2-1}\over (-q^{s+1})_{-N_2-1}}
\label{gowu21Sep12}\end{aligned}$$ and we relabeled $n\to s$. This expression is useful because some symmetries are more manifest, as we will see later in the $N_1\ge 2$ cases. At the same time, however, $\Psi$ is slightly harder to deal with for $N_2>0$ than $\Phi$, because $(q^{s+1})_{-N_2-1}={1\over
(q^{s-N_2})_{N_2+1}}$ can diverge. So, in this way of writing $S$, we should introduce $\epsilon$ even for $N_2>0$ and set $N_2\to
N_2+\epsilon$. Just as we did for $\beta$, we can evaluate $\gamma(1,N_2)$ near integral $N_2$ and the result is $$\begin{aligned}
\gamma(1,N_2+\epsilon)
=
(-1)^{N_2}\,{\epsilon \ln q\over 2} \qquad
\qquad \text{for all $N_2\in\bbZ$.}
\label{gowz21Sep12}\end{aligned}$$ For $N_2<0$, this just cancels the $\epsilon^{-1}$ divergence from $B$ given in , while $\Psi$ is finite. For $N_2>0$, for which $B$ is finite, the $\epsilon$ coming from is canceled by $\Psi$ which goes as $\epsilon^{-1}$ in this case. In more detail, for $N_2>0$, it is only the $0\le s\le N_2$ terms in $\Psi$ that behave as $\epsilon^{-1}$ and cancel against $\gamma\sim \epsilon$, whereas the $s>N_2$ terms are finite and vanish when multiplied by $\gamma\sim\epsilon$. This is a complicated way to say that, in the sum , only $0\le s\le N_2$ terms contribute.
Introduction of all these quantities may seem unnecessary complication, but this will become useful in more general $N_1\ge 2$ cases discussed below. How various quantities behave as $\epsilon\to 0$ is summarized in Table \[table:epsilonBehavior1\].
$B$ $\beta$ $\Phi$ $\gamma$ $\Psi$ $S=\beta \Phi=\gamma\Psi$ $\Zh\propto B S$
--------- ----------------- ------------ -------- ------------ ----------------- --------------------------- ------------------
$N_2>0$ finite finite finite $\epsilon$ $\epsilon^{-1}$ finite finite
$N_2<0$ $\epsilon^{-1}$ $\epsilon$ finite $\epsilon$ finite $\epsilon$ finite
: *The $\epsilon\to 0$ behavior of various quantities for $N_1=1$. Although $B$ and $S=\beta\Phi=\gamma\Psi$ can be individually singular, the partition function $\Zh\propto BS$ is always finite. \[table:epsilonBehavior1\]*
Now we are ready to present the expression for the analytically continued partition function $\Zh_{\text{lens}}$ for $N_1=1$ and $N_2=-N_2'<0$. Combining and , and using , we obtain the expression for the ABJ partition function $\Zh_{\text{ABJ}}(1,N_2')_k=\Zh_{\text{lens}}(1,-N_2')_k$: $$\begin{aligned}
\Zh_{\text{ABJ}}(1,N_2')_k
&=
i^{-{\kappa\over 2}(1+N_2'^2)}
(2\pi)^{1+N_2'^2\over 2}g_s^{1+N_2'\over 2}
(1-q)^{(N_2'-1)(N_2'-2)\over 2}
{G_2(N_2';q)\over 2G_2(N_2'+1)}
\Psi(1,-N_2'),
\label{gwsv10Sep12}\end{aligned}$$ where $$\begin{aligned}
\Psi(1,-N_2')&=\sum_{s=0}^\infty
(-1)^s{(q^{s+1})_{N_2'-1}\over (-q^{s+1})_{N_2'-1}}.\end{aligned}$$
### $\boldsymbol{N_1=2}$ {#Appendix_N1equalto2}
For $N_1=2$, the general formula gives the following expression for $S$: $$\begin{aligned}
S(2,N_2)&=
\sum_{1\le C_1<C_2\le N_2+2}
\prod_{a=C_1+1}^{C_2-1}{q^{C_1}+q^{a}\over q^{C_1}-q^a}
\prod_{a=C_2+1}^{N_2+2}{q^{C_1}+q^{a}\over q^{C_1}-q^a}
\prod_{a=C_2+1}^{N_2+2}{q^{C_2}+q^{a}\over q^{C_2}-q^a}\notag\\
&\qquad\qquad\qquad\qquad
\times
\prod_{a=1}^{C_1-1}{q^a+q^{C_1}\over q^a-q^{C_1}}
\prod_{a=1}^{C_1-1}{q^a+q^{C_2}\over q^a-q^{C_2}}
\prod_{C_1+1}^{C_2-1}{q^a+q^{C_1}\over q^a-q^{C_1}}\\
&=
\sum_{1\le C_1<C_2\le N_2+2}
{(-q)_{C_2-C_1-1}\over (q)_{C_2-C_1-1}}
{(-q^{C_2-C_1+1})_{N_2-C_2+2}\over (q^{C_2-C_1+1})_{N_2-C_2+2}}
{(-q)_{N_2-C_2+2}\over (q)_{N_2-C_2+2}}
\notag\\
&\qquad\qquad\qquad\qquad
\times
{(-q)_{C_1-1}\over (q)_{C_1-1}}
{(-q^{C_2-C_1+1})_{C_1-1}\over (q^{C_2-C_1+1})_{C_1-1}}
{(-q)_{C_2-C_1-1}\over (q)_{C_2-C_1-1}}.\label{hndk21Sep12}\end{aligned}$$ Just as we did for the $N_1=1$ case, we want to analytically continue this expression by eliminating the explicit $N_2$ dependence of the sum range by extending it. However, this turns out to be a non-trivial issue and, in particular, the way to do it is not unique. Before discussing it, let us first consider rewriting $S$ in different forms.
First, just as in the $N_1=1$ case, we can rewrite $S$ in a form closely related to $q$-hypergeometric functions. Namely, $$\begin{aligned}
S(2,N_2)&=\beta(2,N_2)\,\Phi(2,N_2),\label{kgmb7Nov12}\end{aligned}$$ where $$\begin{aligned}
\begin{split}
\beta(2,N_2)&={(-q)_{N_2}(-q^2)_{N_2}\over (q)_{N_2}(q^2)_{N_2}},\\
\Phi(2,N_2)&=\sum_{n_1,n_2}
(-1)^{n_2}
{(-q)_{n_1} (q^{-N_2-1})_{n_1}\over (q)_{n_1} (-q^{-N_2-1})_{n_1}}
{(-q)_{n_2}^2 (q^2)_{n_2}^2\over (q)_{n_2}^2 (-q^2)_{n_2}^2}
{(q^{-N_2})_{{n_1}+{n_2}}(-q^2)_{{n_1}+{n_2}}\over (-q^{-N_2})_{{n_1}+{n_2}}(q^2)_{{n_1}+{n_2}}}
\end{split}
\label{gyiq5Sep12}\end{aligned}$$ and $C_1-1=n_1,C_2-C_1-1=n_2$. The original range of summation corresponds to $n_1\ge 0,n_2\ge 0,n_1+n_2\le N_2$, but we did not specify the range here for the reason mentioned above. This expression is the analogue of the $N_1=1$ relation ; $\beta$ diverges for $N_2<0$ while $\Phi$ is finite for both $N_2>0$ and $N_2<0$. $\Phi$ has the same form as the double $q$-hypergeometric function defined in [@Gasper-Rahman], if the summation were over $n_1,n_2\ge 0$.
The second expression for $S$ is $$\begin{aligned}
S(2,N_2)&=\gamma(2,N_2)\,\Psi(2,N_2),\end{aligned}$$ where $$\begin{aligned}
\gamma(2,N_2)&=
-
{(-q)_1^2\over (q)_1^2}
{(-q)_{N_2}(-q^2)_{N_2}\over (q)_{N_2}(q^2)_{N_2}}
{(-q)_{-N_2-2}(-q^2)_{-N_2-2}\over (q)_{-N_2-2}(q^2)_{-N_2-2}},\notag\\
\Psi(2,N_2)&=\sum_{s_1,s_2} (-1)^{s_1+s_2}
{(q^{s_1+1})_{-N_2-2} \over (-q^{s_1+1})_{-N_2-2}}
{(q^{s_2+1})_{-N_2-2} \over (-q^{s_2+1})_{-N_2-2}}
{(q^{s_2-s_1})_1^2 \over (-q^{s_2-s_1})_1^2}\label{hnsz21Sep12}\end{aligned}$$ and $s_1=C_1-1,s_2=C_2-1$. This expression is the analogue of . The original range of summation corresponds to $0\le s_1<s_2\le N_2+1$.
Now let us discuss the issue of the sum range. For the purpose of studying when the summand vanishes, the $\beta\Phi$ expression is convenient, because $\beta$ just cancels the divergence of $B$ while $\Phi$ is always finite. So, all we need to know is when the summand in $\Phi$ vanishes. Note that, when regularized, $(q^m)_n$ with $m,n\in\bbZ$ has the following behavior: $$\begin{aligned}
\begin{split}
n>0: & \quad
(q^m)_n = (1-q^m)\cdots (1-q^{m+n-1})
= \begin{cases}
\cO(\epsilon) & m\le 0 ~\text{and} ~ m+n-1\ge 0,\\
\cO(1) &\text{otherwise},
\end{cases}
\\
n<0: & \quad
(q^m)_n = {1\over (q^{m+n})_{-n}}
={1\over (1-q^{m+n})\cdots (1-q^{m-1})}\\
&\qquad \qquad \qquad \qquad \quad
= \begin{cases}
\cO(\epsilon^{-1}) & m+n\le 0 ~\text{and} ~ m-1\ge 0,\\
\cO(1) &\text{otherwise}.
\end{cases}
\end{split}\label{hdrp8Nov12}\end{aligned}$$ Here, regularizing $(q^m)_n$ means to replace $N_2$ entering $m,n$ by $N_2+\epsilon$. Furthermore, when $n_1,n_2<0$, we must regularize the summand in by setting $n_1\to n_1+\eta$, $n_2\to n_2+\eta$ with $\eta\to 0$. In this case, we must replace $\cO(\epsilon)$ in by $\cO(\epsilon,\eta)$ and $\cO(\epsilon^{-1})$ by $\cO(\epsilon^{-1},\eta^{-1})$. Using this, it is straightforward to determine the range of $(n_1,n_2)$ for which the summand in $\Phi$ remains non-vanishing after setting $\epsilon,\eta\to 0$.
![*The regions that can contribute to $\Phi(1,N_2)$. (a), (b): For $(n_1,n_2)\in\bbZ^2$ in the shaded regions (denoted by dots), the summand in $\Phi(1,N_2)$ in is $\cO(1)$. Outside the shaded regions, the summand is $\cO(\epsilon,\eta)$ and vanishes as $\epsilon,\eta\to 0$. (c): the $N_2=-1$ case special and the summand is non-vanishing only on the dots. \[nonvanishing\_n1n2\]* ](nonvanishing_n1n2.pdf){height="2.9in"}
In Figure \[nonvanishing\_n1n2\], we described the regions in the $(n_1,n_2)$ plane in which the summand appearing in $\Phi(1,N_2)$ is non-vanishing. Figure \[nonvanishing\_n1n2\](a) shows that, for $N_2>0$, the summand is non-vanishing in the original range of summation, $n_1\ge 0,n_2\ge 0,n_1+n_2\le N_2$ (region I), as it should be. We would like to extend the range in order to eliminate the $N_2$ dependence and thereby analytically continue $\Phi(1,N_2)$ to negative $N_2$. The requirements for the extension are
1. The range specification does not involve $N_2$,
2. For $N_2>0$, it reproduces the original result .
Clearly, there are more than one ways to extend the range satisfying these requirements. One simple way would be to take $n_1\ge0,n_2\ge 0$ as the extended range. For $N_2>0$, this reduces to region I and reproduces the original result, while for $N_2<0$ this sums over region I in Figure \[nonvanishing\_n1n2\](b). (We consider $N_2\le -2$, since $N_2=-1$ is rather exceptional as one can see in Figure \[nonvanishing\_n1n2\](c). The latter case will be discussed later.) Another possible extension is $n_2\ge 0$. This also reproduces the original result for $N_2>0$, but for $N_2<0$ this sums over not only regions I but also IIA and IIB.
Therefore, the way to analytically continue $\Phi(1,N_2)$ is ambiguous and, mathematically, any such choices are good (ignoring the fact that the sum may not be convergent and is only formal). Namely, the data for discrete $N_2\in\bbZ_{>0}$ is not enough to uniquely determine the analytic continuation for all $N_2\in\bbC$. Additional input comes from the physical requirement that it reproduce the known ABJ results for $N_2<0$. Furthermore, for $N_2=-1$, $\Zh_{\text{lens}}(2,-1)_k$ is expected to be related to $\Zh_{\text{lens}}(1,-2)_{-k}$ by the relation .
Here we simply present the prescription which satisfies these physical requirements. The explicit checks are done in the main text where it is shown that its perturbative expansions agree with the known ABJ result and, when exact non-perturbative expressions for the ABJ matrix integral are known, it reproduces them. Moreover, the fact that the prescription reproduces the relation between $\Zh_{\text{lens}}(2,-1)_k$ and $\Zh_{\text{lens}}(1,-2)_{-k}$ is shown for general $N_1$ below.
The key observation to arrive at such a prescription is that, as we can see from Figure \[nonvanishing\_n1n2\](a), the summand is non-vanishing not only in the original region I but also in region IV. The meaning of this is easier to see in the $\gamma\Psi$ representation in terms of $s_1,s_2$. In Figure \[nonvanishing\_s1s2\], we presented the same diagram as Figure \[nonvanishing\_n1n2\] but on the $(s_1,s_2)$ plane.
![*The regions that can contribute to $\Phi(1,N_2)$. These are the same as Figure \[nonvanishing\_n1n2\], but plotted for $(s_1,s_2)$ instead. \[nonvanishing\_s1s2\]* ](nonvanishing_s1s2.pdf){height="2.6in"}
As we can see from the Figure, the non-vanishing regions have the symmetry $$\begin{aligned}
s_1\leftrightarrow s_2.\label{hnla21Sep12}\end{aligned}$$ Actually, as we can immediately see from the explicit expression for $\Psi$ given in , this is a symmetry of the summand, not just its non-vanishing regions. Therefore, it is natural to relax the ordering constraint $s_1<s_2$ in the original range and sum over both regions I and IV, after dividing by $2$. If $s_1=s_2$, the summand in automatically vanishes. Namely, we can write $\Psi$ as $$\begin{aligned}
\Psi(2,N_2)={1\over 2}\sum_{s_1,s_2=0}^\infty (-1)^{s_1+s_2}
{(q^{s_1+1})_{-N_2-2} \over (-q^{s_1+1})_{-N_2-2}}
{(q^{s_2+1})_{-N_2-2} \over (-q^{s_2+1})_{-N_2-2}}
{(q^{s_2-s_1})_1^2 \over (-q^{s_2-s_1})_1^2}.\label{glud9Nov12}\end{aligned}$$ Here we have extended the sum range so that $s_1,s_2$ run to infinity, which is harmless in the $N_2>0$ case.
Our prescription is that we use the expression even for $N_2=-N_2'<0$. As we can see from Figure \[nonvanishing\_s1s2\](b), this sums over regions I and IVA. As we have been emphasizing, it is by no means clear at this point that this is the right prescription. The justification is given in the main text where it is shown that this is consistent with all known results. One can also show that the other possible prescriptions, such as $n_1\ge
0,n_2\ge 0$, which covers region I, and $n_2\ge 0$, which covers regions I, IIA and IIB, would not reproduce the known results and hence are not correct.
If we set $N_2\to N_2+\epsilon$, the behavior of $\gamma$ is $$\begin{aligned}
\gamma(2,N_2+\epsilon)
=
\left({\epsilon\ln q\over 2}\right)^2
\qquad\text{for all $N_2\in\bbZ$.}
\label{irrq21Sep12}\end{aligned}$$ Substituting this and into , we finally obtain the expression for the ABJ partition function $\Zh_{\text{ABJ}}(2,N_2')_k=\Zh_{\text{lens}}(2,-N_2')_k$: $$\begin{aligned}
\Zh_{\text{ABJ}}(2,N_2')_k
&=i^{-{\kappa\over 2}N_2'^2}
(2\pi)^{2+{N_2'^2\over 2}}g_s^{1+{N_2'\over 2}}
(1-q)^{\half(N_2'-2)(N_2'-3)}
{G_2(N_2'-1;q)\over 4G_2(N_2'+1)}
\Psi(2,-N_2'),
\label{jekb21Sep12}\end{aligned}$$ where it is assumed that $N_2'\ge 2$ and $\Psi(1,-N_2')$ is given simply by setting $N_2=-N_2'$ in : $$\begin{aligned}
\Psi(2,-N_2')={1\over 2}\sum_{s_1,s_2=0}^\infty (-1)^{s_1+s_2}
{(q^{s_1+1})_{N_2'-2} \over (-q^{s_1+1})_{N_2'-2}}
{(q^{s_2+1})_{N_2'-2} \over (-q^{s_2+1})_{N_2'-2}}
{(q^{s_2-s_1})_1^2 \over (-q^{s_2-s_1})_1^2}.\end{aligned}$$
The above formula is valid for $N_2'\ge 2$ but not for $N_2'=1$. This case is important, because $(N_1,N_2')=(2,1)$ is related to $(N_1,N_2')=(1,2)$ by and therefore the summation over two variables $s_1,s_2$ should truncate to a sum with one variable; this provides a further check of our prescription. We will discuss this more generally below, where we discuss general $N_1$.
### General $\boldsymbol{N_1}$ {#Appendix_GenN1}
With the $N_1=1,2$ cases understood, the prescription for general $N_1$ is straightforward to establish, although computations get cumbersome. Much as in the $N_1=1,2$ cases, the general expression for $S$ in can be rewritten in the following form: $$\begin{aligned}
S(N_1,N_2)
&=
\sum_{1\le C_1<\cdots<C_{N_1}\le N}
\prod_{j=1}^{N_1} \Biggl\{\Biggl[\prod_{k=j}^{N_1-1}\prod_{a=C_k+1}^{C_{k+1}-1}{q^{C_j}+q^a\over q^{C_j}-q^a}\Biggr]
\prod_{a=C_{N_1}+1}^{N}{q^{C_j}+q^a\over q^{C_j}-q^a}\notag\\
&\qquad\qquad\qquad\qquad\qquad
\times
\prod_{a=1}^{C_1-1}{q^a+q^{C_j}\over q^a-q^{C_j}}
\Biggl[\prod_{k=1}^{j-1}\prod_{a=C_k+1}^{C_{k+1}-1}{q^a+q^{C_j}\over q^a-q^{C_j}}\Biggr]
\Biggr\}
\notag\\
&=
\sum_{1\le C_1<\cdots<C_{N_1}\le N}
\prod_{j=1}^{N_1} \Biggl\{\Biggl[\prod_{k=j}^{N_1-1}{(-q^{C_k-C_j+1})_{C_{k+1}-C_k-1}\over (q^{C_k-C_j+1})_{C_{k+1}-C_k-1}}\Biggr]
{(-q^{C_{N_1}-C_j+1})_{N-C_{N_1}}\over (q^{C_{N_1}-C_j+1})_{N-C_{N_1}}}\notag\\
&\qquad\qquad\qquad\qquad
\times
{(-q^{C_j-C_1+1})_{C_1-1}\over (q^{C_j-C_1+1})_{C_1-1}}
\Biggl[\prod_{k=1}^{j-1}{(-q^{C_j-C_{k+1}+1})_{C_{k+1}-C_k-1}\over (q^{C_j-C_{k+1}+1})_{C_{k+1}-C_k-1}}\Biggr]
\Biggr\}.\label{mskn15Sep12}\end{aligned}$$ In expressions such as this, it is understood that $\sum_{l=a}^b ...=0$ and $\prod_{l=a}^b ...=1$ if $a>b$.
Again, we can rewrite this in the $\beta \Phi$ and $\gamma \Psi$ representations. The $\beta\Phi$ representation is $$\begin{aligned}
&S(N_1,N_2)=\beta(N_1,N_2)\,\Phi(N_1,N_2),\label{fvrs7Sep12}\\
&\beta(N_1,N_2):=\prod_{j=1}^{N_1}{(-q^{N_1-j+1})_{N_2}\over (q^{N_1-j+1})_{N_2}},\\
&\Phi(N_1,N_2):=
\sum_{n_1,\dots,n_{N_1}}
(-1)^{\sum_{l=1}^{N_1}(N_1-l+1)n_l}\notag\\
&\quad\times
\left[\prod_{j=1}^{N_1-1}\prod_{k=j}^{N_1-1}
{( q^{k-j+1})_{n^{}_{j+1,k}}(-q^{k-j+1})_{n^{}_{j+1,k+1}}^2( q^{k-j+1})_{n^{}_{j+2,k+1}} \over
(-q^{k-j+1})_{n^{}_{j+1,k}}( q^{k-j+1})_{n^{}_{j+1,k+1}}^2(-q^{k-j+1})_{n^{}_{j+2,k+1}}}
\right]
\notag\\
&\quad\times
\left[\,\prod_{j=1}^{N_1}
{( q^{N_1-j+1})_{n^{}_{j+1,N_1}}( q^{-N_1-N_2+j})_{n^{}_{1,j}} \over
(-q^{N_1-j+1})_{n^{}_{j+1,N_1}}(-q^{-N_1-N_2+j})_{n^{}_{1,j}}}
\right]
\left[\prod_{k=0}^{N_1-1}
{(-q^{k+1})_{n^{}_{1,k+1}}( q^{k+1})_{n^{}_{2,k+1}} \over
( q^{k+1})_{n^{}_{1,k+1}}(-q^{k+1})_{n^{}_{2,k+1}} }
\right],\label{hapn6Sep12}\end{aligned}$$ where we defined $n_1=C_1-1$, $n_j=C_j-C_{j-1}-1$ ($j=2,\dots,N_1$), and $n_{a,b}:= \sum_{l=a}^b n_l$. Furthermore, we define $n_{N_1+1,b}=n_{a,N_1+1}=0$. The original sum range $1\le
C_1<\dots<C_{N_1}\le N$ corresponds to $n_j\ge 0$ ($j=1,\dots,N_1$), $n_1+\cdots+n_{N_1}\le N_2$, but we did not specify it in for the same reason as in the $N_1=2$ case. $\Phi$ has the form of the multi-variable generalization of $q$-hypergeometric functions, discussed [*e.g.*]{} in [@Exton]. When we analytically continue by $N_2\to -N_2'+\epsilon$, $\beta(N_1,-N_2'+\epsilon)$ goes to zero, while $\Phi(N_1,-N_2')$ remains finite. The behavior of $\beta$ as $\epsilon\to 0$ is $$\begin{aligned}
&\beta(N_1,-N_2'+\epsilon)
=\begin{cases}
\left(-{\epsilon\log q\over 2}\right)^{N_2'}(-1)^{\half N_2'(N_2'-1)}
\prod_{j=1}^{N_2'-1}{(q)_j\over (-q)_j}
\prod_{j=N_1-N_2'}^{N_1-1}{(q)_j\over (-q)_j}
& (N_2'\le N_1),
\\[2ex]
\left(-{\epsilon\log q\over 2}\right)^{N_1}
(-1)^{N_1N_2'+\half N_1(N_1+1)}
\prod_{j=1}^{N_1-1}{(q)_j\over (-q)_j}
\prod_{j=N_2'-N_1}^{N_2'-1}{(q)_j\over (-q)_j}
& (N_1\le N_2').
\end{cases}
\label{euun7Sep12}\end{aligned}$$
On the other hand, the $\gamma \Psi$ representation is $$\begin{aligned}
S(N_1,N_2)&=\gamma(N_1,N_2)\, \Psi(N_1,N_2),\label{moge23Sep12}\\
\gamma(N_1,N_2)&=
(-1)^{\half N_1(N_1-1)}
\prod_{j=1}^{N_1-1}{(-q)_j^2\over (q)_j^2}
\prod_{j=1}^{N_1}
{(-q^j)_{N_2}(-q^j)_{-N_1-N_2}\over (q^j)_{N_2}(q^j)_{-N_1-N_2}},
\label{fxmk24Sep12}
\\
\Psi(N_1,N_2)&=
{1\over N_1!}\sum_{s_1,\dots,s_{N_1}=0}^{\infty}
(-1)^{s_1+\cdots+s_{N_1}}
\prod_{j=1}^{N_1}{(q^{s_j+1})_{-N_1-N_2}\over (-q^{s_j+1})_{-N_1-N_2}}
\prod_{1\le j<k\le N_1}
{(q^{s_k-s_j})_1^2\over (-q^{s_k-s_j})_1^2},
\label{fudw24Sep12}\end{aligned}$$ where $s_j:= C_j-1, j=1,\dots,N_1$. The original sum range corresponds to $0\le s_1<\cdots<s_{N_1}\le N-1$. However, because of the $s_j\leftrightarrow s_k$ symmetry of this expression, we can forget about the ordering constraints and let $s_j$ run freely, if one divides the expression by $N_1!$, which we have already done above. Furthermore, just as in the $N_1=2$ case, we can safely remove the upper bound in the summation for $N_2>0$. Our prescription for analytic continuation to $N_2<0$ is to use this same expression , by setting $N_2=-N_2'+\epsilon$ with $\epsilon\to 0$.
The behavior of $\gamma(N_1,N_2)$ near integral $N_2$ can be shown to be $$\begin{aligned}
\gamma(N_1,N_2+\epsilon)
= (-1)^{N_1N_2+N_1} \left(-{\epsilon\ln q\over 2}\right)^{N_1}
\qquad \text{for all $N_2\in\bbZ$.}
\label{fxom24Sep12}\end{aligned}$$ By substituting and into , we obtain the expression for the ABJ partition function $\Zh_{\text{ABJ}}(N_1,N_2')_k=\Zh_{\text{lens}}(N_1,-N_2')_k$: $$\begin{aligned}
\Zh_{\text{ABJ}}(N_1,N_2')_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2'^2)}
(-1)^{\half N_1(N_1-1)}
2^{-N_1}(2\pi)^{N_1^2+N_2'^2\over 2}
g_s^{N_1+N_2'\over 2}
\notag\\
&\qquad\qquad\times
(1-q)^{\half(N_2'-N_1)(N_2'-N_1-1)}
B(N_2'-N_1,N_1,N_2)
\Psi(N_1,-N_2')
\notag\\
&=
i^{-{\kappa\over 2}(N_1^2+N_2'^2)}
(-1)^{\half N_1(N_1-1)}
2^{-N_1}(2\pi)^{N_1^2+N_2'^2\over 2}
g_s^{N_1+N_2'\over 2}
\notag\\
&\qquad\qquad\qquad
\times
{\prod_{j=1}^{N_2'-N_1-1}(q)_j\over G_2(N_1+1)G_2(N_2'+1)}
\Psi(N_1,-N_2'),
\label{nijj23Sep12}\end{aligned}$$ where we assumed that $N_2'\ge N_1$ and $$\begin{aligned}
\Psi(N_1,-N_2')=
{1\over N_1!}\sum_{s_1,\dots,s_{N_1}= 0}^\infty
(-1)^{s_1+\cdots+s_{N_1}}
\prod_{j=1}^{N_1}{(q^{s_j+1})_{N_2'-N_1}\over (-q^{s_j+1})_{N_2'-N_1}}
\prod_{1\le j<k\le N_1}
{(1-q^{s_k-s_j})^2\over (1+q^{s_k-s_j})^2}.\label{kadc9Nov12}\end{aligned}$$
The above expression is valid only for $N_2'\ge N_1$. If $N_2'<N_1$, then the summation in over $N_1$ variables should reduce to that of $\Zh_{\text{lens}}(N_2',-N_1)_k$ over $N_2'$ variables to be consistent with the symmetry . Let us see how this works by setting $N_2'\to N_2'-\epsilon$ in . Because of and , only terms that diverge as $\sim \epsilon^{-(N_1-N_2')}$ in the $s$-sum survive. Divergences can appear from $$\begin{aligned}
(q^{s_j+1})_{N_2'-N_1-\epsilon}
={1\over (q^{s_j+1+N_2'-N_1-\epsilon})_{N_1-N_2'}}
={1\over (1-q^{s_j+1+N_2'-N_1-\epsilon})\cdots (1-q^{s_j-\epsilon})},\end{aligned}$$ where we are keeping only the leading term. For this to give a divergent $(\sim \epsilon^{-1})$ contribution, it should be that $s_j+1+N_2'-N_1\le 0$, namely, $s_j\le N_1-N_2'-1$ (this is impossible for $N_1\le N_2'$). Because $s_1,\dots,s_{N_1}$ should be different from one another, the most singular case we can have is when $\{s_1,\dots,s_{N_1}\}\supset \{0,1,\dots,N_1-N_2'-1\}$. In this case, we have precisely $\cO(\epsilon^{-(N_1-N_2')})$. Concretely, let us set $$\begin{aligned}
s_j&=
\begin{cases}
j-1 &\quad (1\le j\le N_1-N_2'),\\
N_1-N_2'+s'_{j-N_1+N_2'}&\quad
(N_1-N_2'+1\le j\le N_2')
\end{cases}\label{uy10Nov12}\end{aligned}$$ with $s_j'\ge 0$ and multiply the result by a combinatoric factor ${N_1\choose N_1-N_2'}\cdot(N_1-N_2')!={N_1!\over N_2'!}$. By substituting these into and massaging the result, we can show $$\begin{aligned}
&\Psi(N_1,-N_2'+\epsilon)
=(-1)^{N_1N_2'+N_1}\left(-{2\over \epsilon \ln q}\right)^{N_1-N_2'}\label{vk10Nov12}\\
&\qquad
\times
{1\over N_2'!}\sum_{s_1',\dots,s_{N_2'}'= 0}^\infty\!\!
(-1)^{s_1'+\cdots+s_{N_2'}'}
\prod_{j=1}^{N_2'}{(q^{s_j'+1})_{N_1-N_2'}\over (-q^{s_j'+1})_{N_1-N_2'}}
\prod_{1\le j<k\le N_2'}{(q^{s_k'-s_j'})_1^2\over (-q^{s_k'-s_j'})_1^2}
\qquad (N_2'\le N_1).\notag\end{aligned}$$ Namely, the summation over $N_1$ variables $s_1,\dots,s_{N_1}$ correctly reduced to summation over $N_2'$ variables $s'_1,\dots,s'_{N_2'}$, and the $\epsilon$ dependence of $\Psi$, combined with $\gamma\sim
\epsilon^{N_1}$, is the correct one to cancel the divergence of $B\sim
\epsilon^{-N_2'}$ (see ). So, for $N_2'<N_1$, the expression for the ABJ partition function $\Zh_{\text{ABJ}}(N_1,N_2')_k=\Zh_{\text{lens}}(N_1,-N_2')_k$ is $$\begin{aligned}
\Zh_{\text{lens}}(N_1,-N_2')_k
&=
i^{-{\kappa\over 2}(N_1^2+N_2'^2)}(-1)^{\half N_2'(N_2'-1)}
2^{-N_2'}(2\pi)^{N_1^2+N_2'^2\over 2}g_s^{N_1+N_2'\over 2}
q^{-{1\over 6}(N_1-N_2')((N_1-N_2')^2-1)}
\notag\\
&\qquad\qquad
\times
(1-q)^{\half(N_1-N_2')(N_1-N_2'-1)}
B(N_1-N_2',N_1,N_2')
\Psi(N_2',-N_1)
\notag\\
&=
i^{-{\kappa\over 2}(N_1^2+N_2'^2)}(-1)^{\half N_2'(N_2'-1)}
2^{-N_2'}(2\pi)^{N_1^2+N_2'^2\over 2}g_s^{N_1+N_2'\over 2}
\notag\\
&\qquad\qquad
\times
{ q^{-{1\over 6}(N_1-N_2')((N_1-N_2')^2-1)}
\prod_{j=1}^{N_1-N_2'-1}(q)_j\over G_2(N_1+1)G_2(N_2'+1)}
\Psi(N_2',-N_1),
\label{xl10Nov12}\end{aligned}$$ where $$\begin{gathered}
\Psi(N_2',-N_1)
={1\over N_2'!}\sum_{s_1',\dots,s_{N_2'}'=0}^\infty\!\!
(-1)^{s_1'+\cdots+s_{N_2'}'}
\prod_{j=1}^{N_2'}{(q^{s_j'+1})_{N_1-N_2'}\over (-q^{s_j'+1})_{N_1-N_2'}}
\prod_{1\le j<k\le N_2'}{(q^{s_k'-s_j'})_1^2\over (-q^{s_k'-s_j'})_1^2}
\qquad (N_2'\le N_1).
\notag\end{gathered}$$
Using the explicit expressions and , It is straightforward to show that the relation between $\Zh_{\text{lens}}(N_1,-N_2')_k$ and $\Zh_{\text{lens}}(N_2',-N_1)_{-k}$ holds.
In Table \[table:epsilonBehaviorGeneral\], we present a summary of how various quantities behave as $\epsilon\to 0$ for various values of $N_2$.
range of $N_2$ $B$ $\beta$ $\Phi$ $\gamma$ $\Psi$ $S=\beta \Phi=\gamma\Psi$ $\Zh\propto B S$
--------------------------- -------------------- ------------------- -------- ------------------ ----------------------- --------------------------- ------------------
$N_2>0$ finite finite finite $\epsilon^{N_1}$ $\epsilon^{-N_1}$ finite finite
$N_2<0$, $0<N_2'\le N_1$ $\epsilon^{-N_2'}$ $\epsilon^{N_2'}$ finite $\epsilon^{N_1}$ $\epsilon^{N_2'-N_1}$ $\epsilon^{N_2'}$ finite
$N_2<0$, $N_1\le N_2'$ $\epsilon^{-N_1}$ $\epsilon^{N_1}$ finite $\epsilon^{N_1}$ finite $\epsilon^{N_1}$ finite
: *The $\epsilon\to 0$ behavior of various quantities for general $N_1$. If $N_2<0$, we define $N_2'=-N_2$. \[table:epsilonBehaviorGeneral\]*
The perturbative free energy {#Appendix_PE}
============================
In this appendix, we present the free energy of the lens space matrix model computed by perturbative expansion, up to eight loop order ${\cal
O}(g_s^8)$: $$\begin{gathered}
F_{\rm lens}(N_1,N_2)-F^{\rm tree}_{\rm lens}(N_1,N_2)=
g_s\biggl(\frac{N_1^3}{12}+\frac{N_1^2N_2 }{4} +\frac{ N_1N_2^2}{4}+\frac{N_2^3}{12} -\frac{N_1}{12}-\frac{N_2}{12}\biggr)\\
+g_s^2\left(\frac{N_1^4}{288}+\frac{N_1^3N_2 }{48} +\frac{N_2^2 N_1^2}{16} +\frac{N_2^3
N_1}{48}+\frac{N_2^4}{288} -\frac{N_1^2}{288}+\frac{N_1N_2}{48}-\frac{N_2^2}{288}\right)\\
+g_s^4\biggl(-\frac{N_1^6}{86400}-\frac{ N_1^5N_2}{7680}-\frac{N_1^4N_2^2 }{1536}-\frac{5 N_1^3 N_2^3}{1152}-\frac{N_1^2N_2^4 }{1536}-\frac{N_1N_2^5 }{7680}-\frac{N_2^6}{86400} \\
+\frac{N_1^4}{34560}+\frac{7 N_1^3N_2 }{4608}-\frac{ N_1^2N_2^2 }{768}+\frac{7 N_1N_2^3 }{4608}+\frac{N_2^4}{34560}-\frac{N_1^2}{57600}-\frac{N_1N_2}{960}-\frac{N_2^2}{57600}\biggr)\\
+g_s^6\biggl(\frac{N_1^8}{10160640}+\frac{ N_1^7N_2}{645120}+\frac{N_1^6N_2^2 }{92160}+\frac{N_1^5N_2^3 }{92160}+\frac{7 N_1^4N_2^4}{9216}+\frac{N_1^3N_2^5 }{92160}
+\frac{N_1^2N_2^6 }{92160}+\frac{ N_1N_2^7}{645120}+\frac{N_2^8}{10160640}\\
-\frac{N_1^6}{2177280}+\frac{ N_1^5N_2}{92160}-\frac{N_1^4N_2^2}{2304} +\frac{N_1^3N_2^3 }{27648}-\frac{ N_1^2N_2^4}{2304}+\frac{ N_1N_2^5}{92160}-\frac{N_2^6}{2177280}\\
+\frac{N_1^4}{1451520}+\frac{N_1^3N_2}{11520}+\frac{N_1^2N_2^2}{3840} +\frac{N_1N_2^3}{11520}+\frac{N_2^4}{1451520}-\frac{N_1^2}{3048192}-\frac{N_1N_2}{12096}-\frac{N_2^2}{3048192}\biggr)\\
+g_s^8\biggl(
-\frac{N_1^{10}}{870912000}-\frac{17 N_1^9N_2 }{743178240}-\frac{17 N_1^8N_2^2}{82575360}-\frac{N_1^7N_2^3 }{774144}+\frac{97N_1^6N_2^4 }{4423680}-\frac{2821 N_1^5N_2^5 }{14745600} +\frac{97 N_1^4N_2^6}{4423680}\\
-\frac{N_1^3N_2^7 }{774144}-\frac{17 N_1^2N_2^8 }{82575360}-\frac{17 N_1 N_2^9}{743178240}-\frac{N_2^{10}}{870912000}+\frac{N_1^8}{116121600}+\frac{29 N_1^7 N_2}{123863040} -\frac{259 N_1^6 N_2^2}{17694720}\\
+\frac{937 N_1^5 N_2^3}{8847360}+\frac{53 N_1^4 N_2^4}{442368}+\frac{937 N_1^3 N_2^5}{8847360}-\frac{259 N_1^2N_2^6}{17694720}+\frac{29N_1 N_2^7}{123863040}
+\frac{N_2^8}{116121600}-\frac{N_1^6}{41472000} \\
+\frac{853 N_1^5N_2 }{58982400}-\frac{1487 N_1^4N_2^2}{11796480}-\frac{83N_2^3 N_1^3}{1769472} -\frac{1487N_1^2 N_2^4 }{11796480}+\frac{853 N_1 N_2^5}{58982400}-\frac{N_2^6}{41472000}+\frac{N_1^4}{34836480}\\
-\frac{23 N_1^3N_2 }{37158912}+\frac{325N_1^2 N_2^2}{3096576} -\frac{23N_1 N_2^3 }{37158912}+\frac{N_2^4}{34836480}-\frac{N_1^2}{82944000}-\frac{17 N_1N_2}{1382400}-\frac{N_2^2}{82944000}
\biggr)\ .\nn
\label{feoi21Sep12}\end{gathered}$$ This perfectly agrees with the result in [@Aganagic:2002wv] to the order presented there. Meanwhile, we have explicitly checked that the perturbative free energy of the ABJ matrix model is indeed related to the lens space free energy by F\_[ABJ]{}(N\_1,N\_2)= F\_[lens]{}(N\_1,-N\_2) , including the tree contribution with the normalization discussed in Appendix \[Appendix\_Norm\].
The Seiberg duality {#Appendix_SD}
===================
In this Appendix, we show that the $(1,N_1)$ ABJ partition function $Z_{\rm ABJ}(1,N_1)_k$ given in is invariant under the Seiberg duality up to a phase. Because in the main text we have shown that $Z_{\rm CS}^0(N_2-1)_k$ is invariant and that the phase factor precisely agrees with the one given in [@Kapustin:2010mh], all that remains to be shown is the invariance of the integral $I(1,N_2)_k$ defined in .
As claimed in the main text, for Seiberg dual pairs, we can show that the integrand appearing in $I(1,N_2)_k$ is the same up to a shift in $s$. More precisely, the claim to be proven is that the integrand $$\begin{aligned}
f_{N_2}(s):={\pi\over \sin(\pi s)}\prod_{j=1}^{N_2-1}
\tan{\pi(s+j)\over |k|}\label{kvze1Dec12}\end{aligned}$$ has the following property: $$\begin{aligned}
f_{N_2}(s)=f_{\Nt_2}\!\!\left(s-{|k|\over 2}+N_2-1\right),\qquad
\Nt_2:=2+|k|-N_2.\label{jhyp27Nov12}\end{aligned}$$ Therefore, as long as we take the prescription for the contour, $I(1,N_2)_k$ defined by the contour integral remains the same.
Note that, if two meromorphic functions $f(s)$ and $g(s)$ have poles and zeros at the same points and with the same order, then they must be equal to each other up to an overall constant. This can be shown as follows. If $z=\alpha$ is a pole or a zero, we can write $f(s)=a(z-\alpha)^n,g(s)=b(z-\alpha)^n$ near $z=\alpha$ by the assumption. This means that $f'/f=g'/g =n(z-\alpha)^{-1}$ near $z=\alpha$. Now, recall that Mittag-Leffler’s theorem in complex analysis states that, if two functions have poles at the same points and if the singular part of the Laurent expansion around each of them is the same, then the two functions are identical. So, because $f'/ f$ and $g'/ g$ share poles and residues, they must be identical. This means that $f(s)=cg(s)$ with a constant $c$. In the present case, it is easy to show that the overall scale of $f_{N_2}(s)$ and $f_{\Nt_2}(s-{|k|\over
2}+N_2-1)$ is the same asymptotically, because both tend to ${2\pi
i^{N_2-2}e^{-\pi \sigma}}$ for $s=i\sigma$, $\sigma\to +\infty$. So, in order to show that these two functions are equal, we only have to show that they share poles and zeros.
So, let us compare the poles and zeros of the two functions $f_{N_2}(s)$ and $f_{\Nt_2}(s-{|k|\over 2}+N_2-1)$. Recall the expression for $f_{N_2}(s)$ given by . First, ${\pi\over \sin(\pi
s)}$ gives simple poles at $s\in \bbZ$ (P poles) but no zero. On the other hand, $\tan{\pi (s+j)\over |k|}$ gives simple poles at $s=|k|(p+\half)-j$, $p\in\bbZ$ (NP poles), and simple zeros at $s=|k|q-j$, $q\in\bbZ$ (NP zeros). Using this data, we can find the pole/zero structure of the two functions as we discuss now. We should consider odd and even $k$ cases separately,
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![*The pole/zero structure of the integrand function for odd $k$. “$+$” (red) denotes the P pole, “$\times$” (blue) the NP pole, and “$\bullet$” (green) the NP zero. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. If the original theory is in case (a) ${|k|\over 2}-N_2+1\ge 1$, then the Seiberg dual is in case (b) ${|k|\over 2}-N_2+1\le 1$, and [*vice versa.*]{} In the figure, actual Seiberg dual theories are shown. In the upper panels, all P poles coming from ${\pi\over \sin \pi s}$ and all NP poles and NP zeros coming from $\prod_j \tan$ are shown. In the lower panels, P poles and NP zeros that cancel each other are removed. We see that the actual poles are the same in the dual theories (a) and (b), with P and NP poles interchanged. \[poles\_gen\_odd\]*](poles_gen_odd1.pdf "fig:"){height="5.6cm"} ![*The pole/zero structure of the integrand function for odd $k$. “$+$” (red) denotes the P pole, “$\times$” (blue) the NP pole, and “$\bullet$” (green) the NP zero. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. If the original theory is in case (a) ${|k|\over 2}-N_2+1\ge 1$, then the Seiberg dual is in case (b) ${|k|\over 2}-N_2+1\le 1$, and [*vice versa.*]{} In the figure, actual Seiberg dual theories are shown. In the upper panels, all P poles coming from ${\pi\over \sin \pi s}$ and all NP poles and NP zeros coming from $\prod_j \tan$ are shown. In the lower panels, P poles and NP zeros that cancel each other are removed. We see that the actual poles are the same in the dual theories (a) and (b), with P and NP poles interchanged. \[poles\_gen\_odd\]*](poles_gen_odd2.pdf "fig:"){height="5.6cm"}
- ${|k|\over 2}-N_2+1\ge 1$ (shown is the $N_2=3,|k|=7$ case). - ${|k|\over 2}-N_2+1\le 1$ (shown is the $N_2=6,|k|=7$ case).
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Odd $\boldsymbol{k}$:
For odd $k$, $f_{N_2}(s)$ has poles but no zeros. All poles are simple poles and they can be divided into two groups: $$\begin{aligned}
\begin{split}
\rm P &:\qquad s=0,\dots,|k|-N_2,\\
\rm NP&:\qquad s={|k|\over 2}-N_2+1,\dots,{|k|\over 2}-1,
\end{split}\label{itpg27Nov12}\end{aligned}$$ where periodicity $s\cong s+|k|$ is understood; see Figure \[poles\_gen\_odd\]. Note that this is valid even for ${|k|\over 2}-N_2+1<0$, for which some of the poles are at $s<0$. P means poles coming from $\pi \over \sin\pi s$ while NP means poles coming from $\prod_j\tan$. Some of the P poles got canceled by NP zeros and reduced to regular points. NP poles are not canceled. P and NP poles never collide, because the former are at integral $s$ while the latter are at half-odd-integral $s$.
means that $f_{\Nt_2}(s)$ has simple poles at $$\begin{aligned}
\begin{split}
\rm P&:\qquad s=0,\dots,|k|-\Nt_2=0,\dots,-2+N_2,\\
\rm NP&:\qquad s={|k|\over 2}-N_2+1,\dots,{|k|\over 2}-1
=-{|k|\over 2}+N_2-1,\dots,{|k|\over 2}-1,
\end{split}\end{aligned}$$ which in turn means that $f_{\Nt_2}(s+N_2-{|k|\over 2}-1)$ has simple poles at $$\begin{aligned}
\begin{split}
\rm P&:\qquad s={|k|\over 2}-N_2+1,\dots,{|k|\over 2}-1,\\
\rm NP&:\qquad s=0,\dots,|k|-N_2;
\end{split}\end{aligned}$$ This is the same as , with P and NP interchanged. This proves the identity for odd $k$. Figure \[poles\_gen\_odd\] shows the explicit pole/zero structure in the specific case of $U(1)_7\times U(3)_{-7}=U(1)_{-7}\times U(6)_{7}$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![*The pole/zero structure of the integrand function for even $k$. See Figure \[poles\_gen\_odd\] for explanation of the symbols. In the figure, actual Seiberg dual theories are shown. In the upper panels, all P poles coming from ${\pi\over \sin \pi s}$ and all NP poles and NP zeros coming from $\prod_j \tan$ are shown. In the lower panels, poles that are canceled by NP zeros are removed. If a P pole, a NP pole and a NP zero all collide, the resulting simple pole is interpreted as a NP pole. The surviving poles are the same in the dual theories (a) and (b), with P and NP poles interchanged. \[poles\_gen\_even\]* ](poles_gen_even1.pdf "fig:"){height="5.1cm"} ![*The pole/zero structure of the integrand function for even $k$. See Figure \[poles\_gen\_odd\] for explanation of the symbols. In the figure, actual Seiberg dual theories are shown. In the upper panels, all P poles coming from ${\pi\over \sin \pi s}$ and all NP poles and NP zeros coming from $\prod_j \tan$ are shown. In the lower panels, poles that are canceled by NP zeros are removed. If a P pole, a NP pole and a NP zero all collide, the resulting simple pole is interpreted as a NP pole. The surviving poles are the same in the dual theories (a) and (b), with P and NP poles interchanged. \[poles\_gen\_even\]* ](poles_gen_even2.pdf "fig:"){height="5.1cm"}
- ${|k|\over 2}-N_2+1\ge 1$ (shown is the $N_2=3,|k|=8$ case). - ${|k|\over 2}-N_2+1\le 1$ (shown is the $N_2=7,|k|=8$ case).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
#### Even $\boldsymbol{k}$:
Also for even $k$, the function $f_{N_2}(s)$ has poles but no zeros. Some of the poles are simple while others are double. Let us think of a double pole as made of two simple poles on top of each other. Then there are two groups of simple poles, as follows: $$\begin{aligned}
\begin{split}
\rm P&:\qquad s=0,\dots,|k|-N_2,\\
\rm NP&:\qquad s={|k|\over 2}-N_2+1,\dots,{|k|\over 2}-1,
\end{split}\label{kabs27Nov12}\end{aligned}$$ where $s\cong s+|k|$ is again implied; see Figure \[poles\_gen\_even\]. For $k$ even, NP zeros can cancel P poles and NP poles, and it becomes ambiguous whether we should call a particular pole P or NP. This happens in the ${|k|\over 2}-N_2+1<0$ case, where a P pole, a NP pole and a NP zero all can be at the same point. When this happens, we think of the P pole getting canceled by the NP zero, and group the remaining simple pole into NP, as we did above. This is arbitrary, but it is a unique choice for which the structure becomes identical to the odd $k$ case, .
Because is the same as the odd $k$ case, , the rest goes exactly the same, and we conclude that $f_{N_2}(s)$ and $f_{\Nt_2}(s+N_2-{|k|\over 2}-1)$ are identical, with P and NP interchanged. Figure \[poles\_gen\_even\] shows the explicit pole/zero structure in the specific case of $U(1)_8\times U(3)_{-8}=U(1)_{-8}\times U(7)_{8}$.
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[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: There remains an unresolved mismatch in the $1/N^2$ correction to the AdS radius shift between the field theory [@Drukker:2011zy; @Fuji:2011km] and the gravity dual [@Bergman:2009zh]. On the other hand, quite recently, a one-loop quantum gravity test of the ABJM conjecture was done successfully [@Bhattacharyya:2012ye].
[^5]: We thank Yoichi Kazama and Tamiaki Yoneya for pointing this out to us.
[^6]: We thank Vasilis Niarchos for discussions on this point.
[^7]: Since higher spin theories are inherently dual to vector models [@Sundborg:2000wp; @Sezgin:2002rt; @Klebanov:2002ja], the ABJ theory apparently contains more degrees of freedom than higher spin fields [@Aharony:2011jz]. Those extra degrees of freedom are the large $N_2$ dual of the $U(N_2)$ Chern-Simons theory and thus topological closed strings [@Gopakumar:1998ki]. It is then plausible to expect that the higher spin partition function is given by the ratio $Z_{\rm ABJ}/Z_{\rm
CS}$. We thank Hiroyuki Fuji and Xi Yin for related discussions.
[^8]: We also recall that, in the ABJM case $N_1=N_2$, the expression (\[ABJpart\_integral\]) reproduces the “mirror description” of the ABJM partition function [@Kapustin:2010xq]. Furthermore, for simple cases such as $(N_1,N_2)=(1,2),(1,3)$, it is possible to explicitly carry out the ABJ matrix integral and check that it agrees with the expression (\[ABJpart\_integral\]) for all $k$.
[^9]: This duality is a special case of the Giveon-Kutasov duality of ${\cal N}=2$ CS theories [@Giveon:2008zn] that is further generalized to theories with fundamental and adjoint matter by Niarchos [@Niarchos:2008jb].
[^10]: A proof of the level-rank duality can be found [*e.g.*]{} in Appendix B of [@Kapustin:2010mh].
[^11]: We will not use the symbol $(a)_\nu$ to denote the usual Pochhammer symbol.
[^12]: We did not make $n$ to run over the entire $\bbZ$ because it would give $S=0$. Namely, including $n\in
\bbZ_{<0}$ would exactly cancel the contribution from $n\in \bbZ_{\ge
0}$. Showing this requires to regularize the sum, for example, by $n\to
n+\eta$ for $n\in \bbZ_{<0}$ with $\eta\to 0$.
|
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title: Hyperspectral Image Classification and Clutter Detection via Multiple Structural Embeddings and Dimension Reductions
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title
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---
address: |
Chennai Mathematical Institute\
Plot H1, SIPCOT IT Park, Siruseri\
Kelambakkam 603103, India
author:
- 'B. Ravinder'
title: Bases for local Weyl modules in type $C$
---
|
---
abstract: 'We give a signed fundamental domain for the action on ${\mathbb{C}}^*\times {\mathbb{R}}_+^{n-2}$ of the totally positive units ${E(k)_+}$ of a number field $k$ of degree $n$ and having exactly one pair of complex embeddings. This signed fundamental domain, built of $k$-rational simplicial cones, is as convenient as a true fundamental domain for the purpose of studying Dedekind zeta functions. However, while there is no general construction of a true fundamental domain, we construct a signed fundamental domain from any set of fundamental units of $k$.'
address: 'Instituto de Matemáticas, Facultad de Ciencias, Universidad de Valparaíso, Gran Bretaña 1091, 3er piso, Valparaíso, Chile'
author:
- Milton Espinoza
title: Signed Shintani cones for number fields with one complex place
---
[^1] [^2]
Introduction
============
Motivated by the study of special values of $L$-functions over totally real number fields, Shintani introduced in 1976 [@Sh1] a geometric method that allowed him to write any partial zeta function of a totally real number field as a finite sum of certain Dirichlet series, which can be considered as a natural generalization of the Hurwitz zeta function. Later [@Sh2] Shintani extended these results to general number fields. In order to enunciate Shintani’s geometric method, fix a number field $k$ with $r$ real embeddings and $s$ pairs of complex embeddings ([[*[i.$\,$e. ]{}*]{}]{}$[k:{\mathbb{Q}}]=2s+r$), and let $E(k)$ be its group of units. Given a complete set $\tau_i:k\rightarrow{\mathbb{C}}$ $(1\leq i\leq s+r)$ of embeddings of $k$, $$\label{incrustaciones}
\underbrace{\tau_1,\overline{\tau}_1,\tau_2,\overline{\tau}_2,\dots,\tau_s,\overline{\tau}_s}_{\mathrm{complex \ embeddings}},\underbrace{\tau_{s+1},\tau_{s+2},\dots,\tau_{s+r}}_{\mathrm{real \ embeddings}},$$ we can consider $k\subset{\mathbb{C}}^{s}\times{\mathbb{R}}^{r}$ by identifying $x\in k$ with $$\big(x^{(1)},x^{(2)},\dots,x^{(s+r)}\big)\in{\mathbb{C}}^s\times{\mathbb{R}}^r,$$ where $x^{(i)}:=\tau_i(x)$. Put $${E(k)_+}:=E(k)\cap \big({\mathbb{C}}^s\times {\mathbb{R}}_+^{r}\big) \qquad \mathrm{and} \qquad k_+:=k\cap\big(({\mathbb{C}}^*)^s\times {\mathbb{R}}_+^{r}\big),$$ where ${\mathbb{R}}_+^{r}:=(0,\infty)^r$. Then the group ${E(k)_+}$ of totally positive units of $k$ acts on $({\mathbb{C}}^*)^s\times {\mathbb{R}}_+^{r}$ by component-wise multiplication, where $({\mathbb{C}}^*)^s:=({\mathbb{C}}\smallsetminus\{0\})^s$. On the other hand, if $v_1,v_2,\dots,v_d\in {\mathbb{C}}^s\times{\mathbb{R}}^r$ ($1\leq d\leq 2s+r$) is a set of ${\mathbb{R}}$-linearly independent vectors, we shall call $$C(v_1,v_2,\dots,v_d):=\{t_1v_1+t_2v_2+\dots+t_dv_d \ | \ t_i>0\}$$ the *$d$-dimensional simplicial cone generated by* $v_1,v_2,\dots,v_d$.
Shintani proved [@Sh2 Proposition 2] that there exists a finite set $\{C_j \ | \ j\in J\}$ of simplicial cones, all with generators in $k_+$ ([[*[i.$\,$e. ]{}*]{}]{}$k$-rational), such that $$({\mathbb{C}}^*)^s\times {\mathbb{R}}_+^{r} = \bigcup_{j\in J} \ \bigcup_{\varepsilon\in{E(k)_+}}\varepsilon C_j \qquad (\mathrm{disjoint \ union}).$$ Equivalently, the finite disjoint union $\bigcup_{j\in J}C_j$ is a fundamental domain for the action of ${E(k)_+}$ on $({\mathbb{C}}^*)^s\times {\mathbb{R}}_+^{r}$. Note that this result does not provide any description of the cones involved.
When $k$ is a totally real number field, Colmez proved [@Co1][@Co2] the existence of special units $\eta_1,\eta_2,\dots,\eta_{r-1}\in{E(k)_+}$ such that if we put $$f_{1,\sigma}:=1 \qquad \mathrm{and} \qquad f_{j,\sigma}:=\eta_{\sigma(1)}\eta_{\sigma(2)}\dots\eta_{\sigma(j-1)} \quad (2\leq j\leq r),$$ for $\sigma$ in the symmetric group $S_{r-1}$, then the finite disjoint union $$\label{Colmezcones}
\big\{C_\sigma:=C(f_{1,\sigma},...,f_{r,\sigma}) \ | \ \sigma\in S_{r-1}\big\}$$ (together with some boundary faces of the $C_\sigma$) is a fundamental domain of ${\mathbb{R}}_+^r$ under the action of the group $U$ generated by the $\eta_i$. Unfortunately, we do not know of any practical algorithm for finding these special units when $r\geq 4$.[^3]
In 2012, Díaz y Díaz and Friedman [@DF1] removed this obstruction by considering signed fundamental domains. More precisely, if $\eta_1,...,\eta_{r-1}$ is any set of independent units in ${E(k)_+}$, then the Colmez cones $C_\sigma$, together with some boundary faces, form a signed fundamental domain for the action on ${\mathbb{R}}_+^r$ of the group $U$ generated by the $\eta_i$, [[*[i.$\,$e. ]{}*]{}]{}$$\label{ecuacionvirtual}
\sum_{\substack{w_\sigma=+1\\ \sigma\in S_{r-1}}} \, \sum_{z\in
C_\sigma\cap U \cdot x } w_\sigma\ +\ \sum_{\substack{w_\sigma=-1\\
\sigma\in S_{r-1}}} \,
\sum_{z\in C_\sigma\cap U \cdot x } w_\sigma \ = \ 1\qquad\big( x\in {\mathbb{R}}_+^r\big),$$ where all sums are over finite sets of cardinality bounded independently of $x$, and $w_\sigma=\pm 1$ is a sign associated to the cone $C_\sigma$.[^4] These signed fundamental domains are as convenient as true fundamental domains for computing partial zeta functions, but they have the advantage of being explicitly constructed from any set of independent units $\eta_1,...,\eta_{r-1}\in{E(k)_+}$. To prove their result, Díaz y Díaz and Friedman used topological degree theory on the quotient manifold ${\mathbb{R}}_+^r/{E(k)_+}$. In the following points we give an overview of such proof, as it helps to understand the present work.
[**.**]{} [=1.4em]{}
Consider the multiplicative action of $U$ on half-lines $L\subset {\mathbb{R}}_+^r\cup\{0\}$ with initial point at the origin. Parameterize each $L$ by $y_L\in{\mathbb{R}}_+^{r-1}$, where $$\{(y_L,1)\}=L\cap\{x\in{\mathbb{R}}_+^r\,|\,x^{(r)}=1\}.$$ Then the group $\widetilde{U}:=\langle\widetilde{\eta}_1,\dots,\widetilde{\eta}_{r-1}\rangle$ acts on ${\mathbb{R}}_+^{r-1}$ by multiplication, where $$\widetilde{\eta}_i\in{\mathbb{R}}_+^{r-1} \quad (1\le i\le r-1), \qquad\qquad \widetilde{\eta}_i^{(j)}:=\eta_i^{(j)}/\eta_i^{(r)} \quad (1\le j\le r-1).$$
For each $\sigma\in S_{r-1}$, let $c_\sigma\subset{\mathbb{R}}_+^{r-1}$ be the set of parameters of half-lines going through the Colmez cone $C_\sigma$ (see ), [[*[i.$\,$e. ]{}*]{}]{}, $c_\sigma$ is the intersection of $C_\sigma$ with the hyperplane $\{x\in{\mathbb{R}}_+^r\,|\,x^{(r)}=1\}$. If $\{(c_\sigma,w_\sigma)\}_{\sigma\in S_{r-1}}$ is a signed fundamental domain for the action of $\widetilde{U}$ on ${\mathbb{R}}_+^{r-1}$, then $\{(C_\sigma,w_\sigma)\}_{S_{r-1}}$ is a signed fundamental domain for the action of $U$ on ${\mathbb{R}}_+^r$.
Let $I^{r-1}:=[0,1]^{r-1}$ be the unit hypercube of $r-1$ dimensions, and consider the usual simplicial decomposition of $I^{r-1}$ into $(r-1)!$ simplices, $$I^{r-1}=\bigcup_{\sigma\in S_{r-1}}D_\sigma, \qquad\qquad D_\sigma:=\{y\in I^{r-1} | y^{(\sigma(r-1))}\le\dots\le y^{(\sigma(1))}\}.$$ There exist two continuous functions $f,f_0:I^{r-1}\to{\mathbb{R}}_+^{r-1}$ such that
(a) $f$ is a piecewise affine map that maps $D_\sigma$ onto the closure of $c_\sigma$ for each $\sigma\in S_{r-1}$. The function $f_0$ maps $I^{r-1}$ onto the closure of a fundamental domain for the action of $\widetilde{U}$ on ${\mathbb{R}}_+^{r-1}$; this fundamental domain is easy to describe but it is not of the form we want.
(b) $f$ and $f_0$ induce homotopic functions $F,F_0:\widehat{T}\to T$ between two tori; $\widehat{T}=I^{r-1}/\!\sim$, with $y\sim y+e_i$ whenever $y,y+e_i\in I^{r-1}$, where $e_i$ is the $i^{\mathrm{th}}$ standard basis vector of ${\mathbb{R}}^{r-1}$; and $T={\mathbb{R}}_+^{r-1}/\widetilde{U}$. Moreover, $F_0$ is a homeomorphism of (global) topological degree $\deg(F_0)=\deg(F)=\pm1$.
Equation , with $C_\sigma$, $U$ and ${\mathbb{R}}_+^r$ replaced respectively by $c_\sigma$, $\widetilde{U}$ and ${\mathbb{R}}_+^{r-1}$, follows from interpreting the left hand side as a sum of local degrees of $F$ divided by $\deg(F)$ (local-global principle of topological degree theory). Hence $\{(c_\sigma,w_\sigma)\}_{\sigma\in S_{r-1}}$ is a signed fundamental domain for the action of $\widetilde{U}$ on ${\mathbb{R}}_+^{r-1}$, and the main result of [@DF1] follows from point 2.
When $k$ is not totally real, our knowledge of explicit fundamental domains is very limited. There are some examples in a paper [@RS] of Sczech and Ren, who found explicit cones to give numerical evidence of their refinement of Stark’s conjecture over complex cubic number fields. A more general approach can be found in [@Ok], where explicit cones are presented for the field given by the polynomial $X^3+kX-1$. We know of no results for non totally real fields of degree four or more.
The aim of this work is extend the results of [@DF1] to number fields $k$ having exactly one complex place. In extending the topological approach of [@DF1], we find two obstructions. The first one is that we have to choose some elements in $k_+$ to generate $[k:{\mathbb{Q}}]$-dimensional cones together with the given units, unlike the totally real case where the given units provide all the generators for the $r$-dimensional Colmez cones since the rank of the unit group is $r-1$. The other obstruction is that ${{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$ is a non-convex set; this restricts our choice of generators for the cones, which are convex subsets, and also adds considerable technical difficulty to the use of topological degree theory, because there is no obvious way to construct homotopies having the properties we need in a non-convex set. After overcoming these obstructions, our proof will follow the same lines of [@DF1] described in the above overview.
To get an idea of our construction, suppose that $k$ is a complex cubic number field, and that $\varepsilon=(\varepsilon^{(1)},\varepsilon^{(2)})\in{\mathbb{C}}^*\times{\mathbb{R}}_+$ is a totally positive unit of $k$ of infinite order. Put $\widetilde{\varepsilon}:=\varepsilon^{(1)}/\varepsilon^{(2)}\in{\mathbb{C}}^*$ and assume $|\widetilde{\varepsilon}|>1$; as in [@DF1], in order to get a signed fundamental domain (built of simplicial cones) for the action of $\langle\varepsilon\rangle$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+$, it is sufficient to find a signed fundamental domain (built of triangles) for the action of $\langle\widetilde{\varepsilon}\rangle$ on ${\mathbb{C}}^*$. For each $\ell=0,1,2$, choose $\alpha_\ell\in{\mathbb{C}}^*$ such that $\alpha_\ell/|\alpha_\ell|={\mathrm{exp}}(2\pi i\ell/3)$, and let $\Delta$ be the triangle with vertices $\alpha_0$, $\alpha_1$, $\alpha_2$. We can order the vertices of $\Delta$ and $\widetilde{\varepsilon}\Delta$ by ordering their arguments in $[0,2\pi)$ counterclockwise; of course this depends on $\widetilde{\varepsilon}$. Suppose we get $\alpha_0 < \widetilde{\varepsilon}\alpha_2 < \alpha_1 < \widetilde{\varepsilon}\alpha_0 < \alpha_2 < \widetilde{\varepsilon}\alpha_1;$ if we put $$\begin{aligned}
&V_1=\{\widetilde{\varepsilon}\alpha_0 , \alpha_2 , \widetilde{\varepsilon}\alpha_1\}, \qquad
&&V_2=\{\alpha_2 , \widetilde{\varepsilon}\alpha_1 , \alpha_0\}, \qquad
&&V_3=\{\widetilde{\varepsilon}\alpha_1 , \alpha_0 , \widetilde{\varepsilon}\alpha_2\}, \\
&V_4=\{\alpha_0 , \widetilde{\varepsilon}\alpha_2 , \alpha_1\}, \qquad
&&V_5=\{\widetilde{\varepsilon}\alpha_2 , \alpha_1 , \widetilde{\varepsilon}\alpha_0\}, \qquad
&&V_6=\{\alpha_1 , \widetilde{\varepsilon}\alpha_0 , \alpha_2\},\end{aligned}$$ then the triangle $\Delta_\ell$ with vertices $V_\ell$ does not contain the origin for each $\ell=1,\dots,6$, since its vertices lie in a convex subset of ${\mathbb{C}}^*$. Looking at $\Delta_1$, we deduce that there is a unique $d\in{\mathbb{Z}}$ such that $\arg(\widetilde{\varepsilon})$, $2\pi d-2\pi/3$, and $\arg(\widetilde{\varepsilon})+2\pi/3$ lie in an interval of length less than $\pi$, where $\arg(z)$ represents the argument of $z\in{\mathbb{C}}^*$ in the range $[-\pi,\pi)$. Consider the following elements of ${\mathbb{R}}^2$: $$\begin{aligned}
&\phi_{\alpha_2}=(0,d-1/3), &&\phi_{\alpha_0}=(0,d), &&\phi_{\alpha_1}=(0,d+1/3), &&\overline{\phi}_{\alpha_2}=(0,d+2/3),\\
&\phi_{\widetilde{\varepsilon}\alpha_0}=(1,0), &&\phi_{\widetilde{\varepsilon}\alpha_1}=(1,1/3), &&\phi_{\widetilde{\varepsilon}\alpha_0}=(1,2/3), &&\overline{\phi}_{\widetilde{\varepsilon}\alpha_0}=(1,1);\end{aligned}$$ also put $$\begin{aligned}
&V'_1=\{\phi_{\widetilde{\varepsilon}\alpha_0},\phi_{\alpha_2},\phi_{\widetilde{\varepsilon}\alpha_1}\}, \qquad
&&V'_2=\{\phi_{\alpha_2},\phi_{\widetilde{\varepsilon}\alpha_1},\phi_{\alpha_0}\}, \qquad
&&V'_3=\{\phi_{\widetilde{\varepsilon}\alpha_1},\phi_{\alpha_0},\phi_{\widetilde{\varepsilon}\alpha_0}\}, \\
&V'_4=\{\phi_{\alpha_0},\phi_{\widetilde{\varepsilon}\alpha_0},\phi_{\alpha_1}\}, \qquad
&&V'_5=\{\phi_{\widetilde{\varepsilon}\alpha_0},\phi_{\alpha_1},\overline{\phi}_{\widetilde{\varepsilon}\alpha_0}\}, \qquad
&&V'_6=\{\phi_{\alpha_1},\overline{\phi}_{\widetilde{\varepsilon}\alpha_0},\overline{\phi}_{\alpha_2}\};\end{aligned}$$ and let $\Delta'_\ell$ be the triangle with vertices $V'_\ell$ for each $\ell=1,\dots,6$. If $D$ is the union of all the $\Delta'_\ell$, then $D$ is the closure of a fundamental domain for ${\mathbb{R}}^2$ under the translation action of its subgroup ${\mathbb{Z}}^2$, and the $\Delta'_\ell$ form a simplicial decomposition of $D$. Thus we can define a piecewise affine map $f:D\to{\mathbb{C}}^*$ by $$f(\overline{\phi}_{\widetilde{\varepsilon}\alpha_0})=\widetilde{\varepsilon}\alpha_0, \qquad f(\overline{\phi}_{\alpha_2})=\alpha_2, \qquad f(\phi_v)=v \quad (\text{for each vertex $v$ of $\Delta$ and $\widetilde{\varepsilon}\Delta$}).$$ Now, the set $\mathcal{F}=\{z\in{\mathbb{C}}^*\,|\,1\le|z|<|\widetilde{\varepsilon}|\}$ is an obvious fundamental domain for the action of $\langle\widetilde{\varepsilon}\rangle$ on ${\mathbb{C}}^*$, and the function $f_0:D\to {\mathbb{C}}^*$ defined by $$f_0(t,\theta)=\widetilde{\varepsilon}^t{\mathrm{exp}}(2\pi i\theta) \qquad\qquad \big((t,\theta)\in D\big),$$ has image the closure of $\mathcal{F}$; here complex powers are defined by the principal branch of the logarithm. One verifies that $f$ and $f_0$ are homotopic through the homotopy $$g_\lambda(t,\theta)=\lambda f(t,\theta)+(1-\lambda)f_0(t,\theta) \qquad\qquad (\lambda\in I, \ (t,\theta)\in D);$$ this homotopy is well defined because $f(\Delta'_\ell)$ and $f_0(\Delta'_\ell)$ are contained in a (same) convex subset of ${\mathbb{C}}^*$ for each $\ell=1,\dots,6$. Furthermore, $f$, $f_0$, and $g_\lambda$ descend to continuous maps between the tori $\widehat{T}=D/\!\sim$ and $T={\mathbb{C}}^*/\langle\widetilde{\varepsilon}\rangle$, where $\sim$ identifies points of $D$ lying in the same orbit with respect to the translation action of ${\mathbb{Z}}^2$ on ${\mathbb{R}}^2$. This means that the maps between $\widehat{T}$ and $T$ induced by $f$ and $f_0$ are homotopic. From this point forward, our proof follows the same lines of [@DF1]. Note that in this case $\alpha_0$, $\alpha_1$, and $\alpha_2$ are not necessarily elements coming from $k_+$; this is a minor problem which will be solved by choosing elements of $k_+$ not “too far” from the $\alpha_\ell$.
We are very grateful to the referees for helping us to improve the exposition of this article and for encouraging us to enhance this introduction with an overview of our construction.
The signed fundamental domain
=============================
From now on we assume $r:=\mathrm{rank}\big({E(k)_+}\big)=[k:{\mathbb{Q}}]-2>0$. Fix a set of independent units $\varepsilon_1,\dots,\varepsilon_r\in{E(k)_+}$, and let $V\subset{E(k)_+}$ be the subgroup they generate. Following Colmez [@Co1], define $$\label{ftsigma}
f_{t,\sigma} := \varepsilon_{\sigma(1)} \varepsilon_{\sigma(2)}\cdots\;
\varepsilon_{\sigma(t-1)}=\prod_{j=1}^{t-1}
\varepsilon_{\sigma(j)}\qquad\ (1\le t\le r+1,
\ \,\sigma\in S_r).$$ For $t=1$ we mean $f_{1,\sigma}:=1=(1,1,\dots,1)\in {\mathbb{C}}^*\times{\mathbb{R}}^r_+$. Thus $f_{t,\sigma}\in{E(k)_+}\subset{\mathbb{C}}^*\times{\mathbb{R}}^r_+$. Define $$\label{xitt'sigma}
\xi_\sigma(t,t'):=\tau_1(f_{t,\sigma}^{-1}f_{t',\sigma})\in{\mathbb{C}}^* \qquad\qquad (1\le t,t'\le r+1,\ \,\sigma\in S_r),$$ where $\tau_1$ is a fixed complex embedding of $k$ (see ). When $t=r+1$ in , we will write $$\label{xitsigma}
\xi_\sigma(t'):=\xi_\sigma(r+1,t') \qquad\qquad (1\le t'\le r+1,\ \,\sigma\in S_r).$$ Note that for all $1\le t,t',t''\le r+1$ and all $\sigma\in S_r$ we have $$\label{xitsigmaprop}
\xi_\sigma(t,t')^{-1}=\xi_\sigma(t',t), \qquad \xi_\sigma(t,t')\cdot\xi_\sigma(t'',t)=\xi_\sigma(t'',t'), \qquad \xi_\sigma(t)\cdot\xi_\sigma(t')^{-1}=\xi_\sigma(t',t).$$
Let $\arg(z)$ be the argument in the interval $[-\pi,\pi)$ of the nonzero complex number $z$. For a fixed integer $N\ge3$, let ${\mathrm{m}}={\mathrm{m}}_N:{\mathbb{C}}^*\rightarrow {\mathbb{Z}}$ be the function defined by $$\label{m}
{\mathrm{m}}(z):=\left\lceil\frac{-N\arg(z)}{2\pi} \right\rceil \qquad (z\in{\mathbb{C}}^*), \qquad -\frac{N}{2}<{\mathrm{m}}(z)\leq \left\lceil\frac{N}{2}\right\rceil,$$ where the ceiling function $\lceil \ \rceil:{\mathbb{R}}\to {\mathbb{Z}}$ satisfies $x\leq\lceil x\rceil<x+1$. Then, for $\sigma\in S_r$ and $t,t'\in\{1,\dots,r+1\}$, consider the next three conditions $$\begin{aligned}
& {\mathrm{m}}(\xi_\sigma(t,t'))\equiv{\mathrm{m}}(\xi_\sigma(t'))-{\mathrm{m}}(\xi_\sigma(t)) \ ({\mathrm{mod} \ }N), \label{orden1}\\
& {\mathrm{m}}(\xi_\sigma(t,t'))+{\mathrm{m}}(\xi_\sigma(t',t))\equiv1 \ ({\mathrm{mod} \ }N), \label{orden2}\\
& t'<t. \label{orden3}\end{aligned}$$ We shall say that $t\prec_\sigma t'$ if and only if the pair $(t,t')$ satisfies condition , and at least one of the conditions and . In Proposition \[ordentotal\] we will prove, for $\sigma\in S_r$, that the relation $\prec_\sigma$ is a strict total order on the set $\{1,\dots,r+1\}$. Also, in Lemma \[mprop\] we will prove that ${\mathrm{m}}(\xi_\sigma(t,t'))$ is congruent modulo $N$ to either ${\mathrm{m}}(\xi_\sigma(t'))-{\mathrm{m}}(\xi_\sigma(t))$ or ${\mathrm{m}}(\xi_\sigma(t'))-{\mathrm{m}}(\xi_\sigma(t))+1$.
Finally, let $\widetilde{S}_r$ be the product of sets $$S_r\times\{1,\dots,r+1\}\times\{0,\dots,N-1\}$$ with cardinality $\big([k:{\mathbb{Q}}]-1\big)!\cdot N$.
The seven-step algorithm
------------------------
With the above conventions and definitions, the following seven steps produce a signed fundamental domain of Díaz y Díaz–Friedman type (see ) for the action of the group $V$ on ${\mathbb{C}}^*\times{\mathbb{R}}^r_+$.
[**.**]{} [=1.4em]{}
Fix an integer $N\geq 3$, and consider the function ${\mathrm{m}}={\mathrm{m}}_N$ defined in .
For each $\sigma\in S_r$, order the set $\{1,\dots,r+1\}$ using the strict total order $\prec_\sigma$ defined by conditions , and .
For each $\sigma\in S_r$, let $\rho_\sigma\in S_{r+1}$ be the unique permutation such that $$\label{rosigma}
\rho_\sigma(r+1)\prec_\sigma\rho_\sigma(r)\prec_\sigma\dots\prec_\sigma\rho_\sigma(2)\prec_\sigma\rho_\sigma(1).$$
For each $t\in{\mathbb{Z}}$, choose and fix an element $\alpha_t=\alpha(t)\in k_+$ such that $$\alpha_t=\alpha_{t'} \quad \mathrm{if} \quad t\equiv t'({\mathrm{mod} \ }N), \qquad
\arg\!\left(\alpha_t^{(1)}\cdot{\mathrm{exp}}\left(-2\pi i t/N\right)\right)\in\left(\frac{-\pi}{2N},\frac{\pi}{2N}\right).$$
Let $\mu=(\sigma, q, n)\in \widetilde{S}_r$. For $t\in\{1,\dots,r+1\}$, write $$\label{ftsigmaqj}
f_{t,\mu}=f(t,\sigma,q,n):=\begin{cases}
f_{t,\sigma}\cdot\alpha\big({\mathrm{m}}(\xi_\sigma(t))+n\big) & \mathrm{if} \ t\nprec_\sigma\rho_\sigma(q), \\
f_{t,\sigma}\cdot\alpha\big({\mathrm{m}}(\xi_\sigma(t))+n+1\big) & \mathrm{if} \ t\prec_\sigma\rho_\sigma(q),
\end{cases}$$ and for $t=r+2$ write $$\label{ftsigmaqj2}
f_{t,\mu}=f(t,\sigma,q,n):=f_{\rho_\sigma(q),\sigma}\cdot\alpha\!\Big({\mathrm{m}}\!\Big(\xi_\sigma\big(\rho_\sigma(q)\big)\Big)+n+1\Big).$$
For $\mu=(\sigma,q,n)\in \widetilde{S}_r$, define $w_{\mu}=\pm 1$ or $0$ as $$\label{wsigmaqn}
w_{\mu}:=\frac{\mathrm{sgn}(\sigma)
\cdot\mathrm{sign}\big(\!\det(f_{1,\mu}\,,\,f_{2,\mu}\,,\,
\dots\,,\,f_{r+2,\mu})\big)
}{\mathrm{sign}\big(\!\det({\mathrm{Log}}\,\,\varepsilon_1,
{\mathrm{Log}}\,\,\varepsilon_2,\dots,{\mathrm{Log}}\,\,\varepsilon_r)\big)},$$ where $\mathrm{sgn}(\sigma)$ is the usual signature ([[*[i.$\,$e. ]{}*]{}]{}$\pm1$) of the permutation $\sigma\in S_r$, $${\mathrm{Log}}\,\,\varepsilon_i\in {\mathbb{R}}^r\qquad \mathrm{with} \qquad\big( {\mathrm{Log}}\,\,\varepsilon_i \big)^{(j)}:= \log \, |\varepsilon_i^{(j)}|
\quad \, (1\le j\le r),$$ the $f_{i,\mu}$ are regarded as elements of ${\mathbb{R}}^{r+2}$ by the map $$\label{Psi}
(z,x^{(1)},\dots,x^{(r)})\mapsto\big({\mathrm{Re}}(z),{\mathrm{Im}}(z),x^{(1)},\dots,x^{(r)}\big) \qquad (z\in{\mathbb{C}},\,x^{(i)}\in{\mathbb{R}}),$$ and $\,\mathrm{sign}\big(\!\det(v_1,v_2,\dots,v_\ell)\big)$ is the sign of the determinant of the $\ell\times \ell$ real matrix whose columns are the $v_i$.
For each $\mu\in \widetilde{S}_r$ with $w_{\mu}\not=0$, consider the real hyperplanes $$\label{hiperplano}
H_{i,\mu}:=\sum_{\substack{1\le t\le r+2\\ t\not=i}}{\mathbb{R}}\cdot f_{t,\mu} \qquad\qquad (1\le i\le r+2),$$ each of which separates ${\mathbb{C}}\times{\mathbb{R}}^r$ into two disjoint half-spaces, ${\mathbb{C}}\times{\mathbb{R}}^r=H_{i,\mu}^+\cup H_{i,\mu} \cup H_{i,\mu}^-$, where $H_{i,\mu}^+$ is the half-space containing $f_{i,\mu}$. Then define $C_{\mu}=C_\mu(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r)$ by $$\begin{aligned}
\label{Csigmaqn}
\nonumber
C_{\mu}&:={\mathbb{R}}_{1,\mu}\cdot f_{1,\mu}+{\mathbb{R}}_{2,\mu}\cdot f_{2,\mu}+ \cdots+{\mathbb{R}}_{r+2,\mu}\cdot f_{r+2,\mu},\\
{\mathbb{R}}_{i,\mu}&:=\begin{cases}
[0,\infty) & \mathrm{if} \ e_{r+2}\in H_{i,\mu}^+, \\
(0,\infty) & \mathrm{if} \ e_{r+2}\in H_{i,\mu}^-,
\end{cases} \qquad\qquad(1\le i\le r+2),\end{aligned}$$ with $e_{r+2}:=[0,0,\dots,0,1]\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$.
*Some remarks*. The choice $N=3$ in the first step of the algorithm generates the minimum number of cones, namely $([k:{\mathbb{Q}}]-1)!\cdot3$. Also note that $N$, as well as the $\alpha_t$ chosen in the fourth step, are not included in the posterior notation since they remain fixed along the whole algorithm. In step five, we clearly have $f_{t,\mu}\in k_+\subset{\mathbb{C}}^*\times{\mathbb{R}}^r_+$ for all $t\in\{1,\dots,r+2\}$. In step six, note that the absolute value of the determinant in the denominator of is half of the regulator of the independent units $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r$, and so is non-zero. Also, in the following when identify ${\mathbb{C}}\times {\mathbb{R}}^\ell = {\mathbb{R}}^{\ell+2}$ as an ${\mathbb{R}}$-vector space, we will be referring to the isomorphism with $r=\ell$. Finally, the definitions given in the seventh step of the algorithm make sense since if $w_\mu\not=0$, then each closed cone $\overline{C}_{\mu}:=\sum_{t=1}^{r+2}{\mathbb{R}}_{\ge0}\cdot f_{t,\mu}$ has a non-empty interior; furthermore, in Lemma \[lema49\] we will prove that $e_{r+2}$ cannot lie in any of the $H_{i,\mu}$.
We will call the above algorithm the *seven-step algorithm* (7SA). It produces our main result.
\[Main\] Let $k$ be a number field with $r>0$ real embeddings, and exactly one pair of conjugate complex embeddings. Suppose that the units $\varepsilon_1,\dots,\varepsilon_r$ generate a subgroup $V$ of finite index in the group of totally positive units of $k$. Then the signed cones $\big\{(C_{\mu}, w_{\mu})\big\}_{ w_{\mu}\not=0}$ defined in and give a signed fundamental domain for the action of $V$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+^r:=\big({\mathbb{C}}\smallsetminus\{0\}\big)\times (0,\infty)^r$. That is, $$\label{Basic}
\sum_{\substack{w_{\mu}=+1\\ \mu\in \widetilde{S}_r}} \, \sum_{z\in
C_{\mu}\cap V \cdot x } w_{\mu}\ +\ \sum_{\substack{w_{\mu}=-1\\ \mu\in \widetilde{S}_r}} \,
\sum_{z\in C_{\mu}\cap V \cdot x } w_{\mu} \ = \ 1\qquad\big( x\in {{\mathbb{C}}^*\times{\mathbb{R}}_+^r}),$$ where all sums are over finite sets of cardinality bounded independently of $x$.
Corollaries of Theorem 1
------------------------
If $w_{\mu}\not=-1$ for all $\mu\in \widetilde{S}_r$, then each orbit $V\cdot x$ must intersect only one of the $C_{\mu}$’s, and only once at that. Hence
Suppose that $w_{\mu}\not=-1$ for all $\mu\in \widetilde{S}_r$, then $\cup_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} C_{\mu}$ is a true fundamental domain for the action of $V$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+^r$.
The next corollary shows that a signed fundamental domain is as convenient as a true one for dealing with partial zeta functions associated to $k$.[^5] Fix an integral ideal $\mathfrak{f}$ of $k$, and put $\mathfrak{f}\infty$ the formal product of $\mathfrak{f}$ with all the infinite places of $k$. Let $\zeta_\mathfrak{f}(\overline{\mathfrak{a}},s):=\sum_{\mathfrak{b}\in\overline{\mathfrak{a}}}\mathrm{N}\mathfrak{b}^{-s}$ (${\mathrm{Re}}(s)>1$) be the Dedekind partial zeta function attached to a ray class $\overline{\mathfrak{a}}$ modulo $\mathfrak{f}\infty$ represented by the integral ideal $\mathfrak{a}$. Here $\mathfrak{b}$ runs over all integral ideals in $\overline{\mathfrak{a}}$, and $\mathrm{N}$ is the absolute norm.
\[zetaparcial\] Suppose $\varepsilon_1,\dots,\varepsilon_r$ generate the group ${E(k)_+}^{\mathfrak{f}}$ of totally positive units of $k$ that are congruent to $\mathrm{1}$ mod $\mathfrak{f}$, and suppose we have chosen $\alpha_0,\dots,\alpha_{N-1}\in\mathfrak{a}^{-1}\mathfrak{f}$ in the fourth step of the 7SA. Then $$\zeta_\mathfrak{f}(\overline{\mathfrak{a}},s)=\mathrm{N}\mathfrak{a}^{-s}
\sum_{\substack{\mu\in \widetilde{S}_r\\w_{\mu}\not=0 }} w_{\mu}\sum_{x\in
R_\mathfrak{f}(\mathfrak{a},C_{\mu})}\zeta_\mathfrak{f}(C_{\mu},x,s)\qquad\qquad\big({\mathrm{Re}}(s)>1\big),$$ where $\zeta_\mathfrak{f}(C_{\mu},x,s)$ is the Shintani zeta function $$\begin{aligned}
\zeta_\mathfrak{f}(C_{\mu},x,s)&:=\sum_{n_1,\dots,n_{r+2}=0}^\infty\, \Big|x^{(1)}+\sum_{t=1}^{r+2} n_tf_{t,\mu}^{(1)}
\Big|^{-2s}\cdot\prod_{j=2}^{r+2} \Big(x^{(j)}+\sum_{t=1}^{r+2} n_tf_{t,\mu}^{(j)}
\Big)^{-s},
\\
R_\mathfrak{f}(\mathfrak{a},C_{\mu}) &:=\Big\{ x\in 1+\mathfrak{a}^{-1}\mathfrak{f} \ \big| \ x=\sum_{t=1}^{r+2} y_t f_{t,\mu},
\ y_t\in I_{t,\mu} \Big\},
\\
I_{t,\mu}&:= \begin{cases}[0,1)\ & \mathrm{if}\ e_{r+2}\in H_{t,\mu}^+ ,\\
(0,1]\ & \mathrm{if}\ e_{r+2}\in H_{t,\mu}^-.
\end{cases}\end{aligned}$$
Examples
========
In this section we show three examples of signed fundamental domains obtained by using the 7SA. Our numerical results are up to an error less than $10^{-28}$.
Cubic case
----------
Let $k={\mathbb{Q}}(\gamma)$, where $\gamma^3+\gamma^2-1=0$. Then the discriminant of the complex cubic number field $k$ is $-23$. Let $$\varepsilon_1=\gamma=\frac{1}{\gamma^2+\gamma}=\big[(-0{.}8774...)+(-0{.}7448...)i \, , \, 0{.}7548...\big] \ \in \ {E(k)_+}.$$ In the next two examples, we give signed fundamental domains for the action of $\langle \varepsilon_1\rangle$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+$.
### Example 1
If $N=3$, $\alpha_0=1$, $\alpha_1=2\gamma^2+2\gamma+1$, and $\alpha_2=2\gamma+1$, one verifies that $\arg\!\big(\alpha_0^{(1)}\big)=0,$ $$\begin{aligned}
\arg\!\big(\alpha_1^{(1)}\cdot{\mathrm{exp}}(-2\pi i/3)\big)=-0{.}2424..., \quad \text{and} \quad
\arg\!\big(\alpha_2^{(1)}\cdot{\mathrm{exp}}(-4\pi i/3)\big)=0{.}0545...\end{aligned}$$ lie in the interval $(-\pi/6,\pi/6)=(-0{.}5235...,0{.}5235...)$. Following steps 1, 2 and 3, we have $${\mathrm{m}}\!\big(\xi_{(1)}(1)\big)=-1, \qquad {\mathrm{m}}\!\big(\xi_{(1)}(2)\big)=0,\qquad {\mathrm{m}}\!\big(\xi_{(1)}(2,1)\big)=-1,\qquad {\mathrm{m}}\!\big(\xi_{(1)}(1,2)\big)=2,$$ where $(1)\in S_1$ is the identity permutation; hence $2\prec_{(1)}1$, and $\rho_{(1)}$ is the identity permutation of $S_2$. Now, using and we compute $$\begin{aligned}
&f_{1,(1),2,0}=2\gamma+1, && f_{2,(1),2,0}=\gamma, &&f_{3,(1),2,0}=\gamma+2,\\
&f_{1,(1),1,0}=2\gamma+1, &&f_{2,(1),1,0}=\gamma+2, &&f_{3,(1),1,0}=1,\\
&f_{1,(1),2,1}=1, &&f_{2,(1),2,1}=\gamma+2, &&f_{3,(1),2,1}=2\gamma^2+\gamma,\\
&f_{1,(1),1,1}=1, &&f_{2,(1),1,1}=2\gamma^2+\gamma, &&f_{3,(1),1,1}=2\gamma^2+2\gamma+1,\\
&f_{1,(1),2,2}=2\gamma^2+2\gamma+1, &&f_{2,(1),2,2}=2\gamma^2+\gamma, &&f_{3,(1),2,2}=\gamma,\\
&f_{1,(1),1,2}=2\gamma^2+2\gamma+1, &&f_{2,(1),1,2}=\gamma, &&f_{3,(1),1,2}=2\gamma+1.\end{aligned}$$ Then we compute the $w_\mu$ using , with ${\mathrm{Log}}\ \varepsilon_1=0{.}1405...$; $$\begin{aligned}
&\det\big(f_{1,(1),2,0},f_{2,(1),2,0},f_{3,(1),2,0}\big)=0, &&w_{(1),2,0}=0,\\
&\det\big(f_{1,(1),1,0},f_{2,(1),1,0},f_{3,(1),1,0}\big)=0, &&w_{(1),1,0}=0,\\
&\det\big(f_{1,(1),2,1},f_{2,(1),2,1},f_{3,(1),2,1}\big)=-4{.}7958..., &&w_{(1),2,1}=-1,\\
&\det\big(f_{1,(1),1,1},f_{2,(1),1,1},f_{3,(1),1,1}\big)=4{.}7958..., &&w_{(1),1,1}=+1,\\
&\det\big(f_{1,(1),2,2},f_{2,(1),2,2},f_{3,(1),2,2}\big)=4{.}7958..., &&w_{(1),2,2}=+1,\\
&\det\big(f_{1,(1),1,2},f_{2,(1),1,2},f_{3,(1),1,2}\big)=4{.}7958...., &&w_{(1),1,2}=+1.\end{aligned}$$ Finally, the following equations allow us to determine the ${\mathbb{R}}_{i,\mu}$ defined in step 7. $$\begin{aligned}
e_3&=(-0{.}3681...)f_{1,(1),2,1}+(0{.}3898...)f_{2,(1),2,1}+(0{.}0155...)f_{3,(1),2,1}\\
&=(0{.}0216...)f_{1,(1),1,1}+(-0{.}2344...)f_{2,(1),1,1}+(0{.}3898...)f_{3,(1),1,1}\\
&=(0{.}4114...)f_{1,(1),2,2}+(-0{.}2561...)f_{2,(1),2,2}+(-0{.}0216...)f_{3,(1),2,2}\\
&=(0{.}1553...)f_{1,(1),1,2}+(-0{.}2778...)f_{2,(1),1,2}+(0{.}2561...)f_{3,(1),1,2}.\end{aligned}$$ Therefore, the cones of the signed fundamental domain are $$\begin{aligned}
&C_{(1),2,1}=\{t_1+t_2(\gamma+2)+t_3(2\gamma^2+\gamma) \ | \ t_1>0, \ t_2\geq 0, \ t_3\geq 0\},\\
&C_{(1),1,1}=\{t_1+t_2(2\gamma^2+\gamma)+t_3(2\gamma^2+2\gamma+1) \ | \ t_1\geq 0, \ t_2> 0, \ t_3\geq 0\},\\
&C_{(1),2,2}=\{t_1(2\gamma^2+2\gamma+1)+t_2(2\gamma^2+\gamma)+t_3\gamma \ | \ \ t_1\geq 0, \ t_2>0, \ t_3>0\},\\
&C_{(1),1,2}=\{t_1(2\gamma^2+2\gamma+1)+t_2\gamma+t_3(2\gamma+1) \ | \ \ t_1\geq 0, \ t_2> 0, \ t_3\geq 0\}.\end{aligned}$$ Figure \[Figure1\] represents the intersection of the plane $\{(z,1) \ | \ z\in{\mathbb{C}}\}\subset {\mathbb{C}}\times{\mathbb{R}}$ with the signed fundamental domain. The blue region indicates the cones with $w_\mu$ positive, the red region indicates the cone with $w_\mu$ negative, and the purple region represents the intersection of two cones with opposite signs.
![Signed fundamental domain for the action of $\langle\gamma\rangle$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+$, where $\gamma^3+\gamma^2-1=0$, $\alpha_0=1$, $\alpha_1=2\gamma^2+2\gamma+1$, and $\alpha_2=2\gamma+1$.[]{data-label="Figure1"}](Figure1.pdf)
### Example 2
If $N=3$, $\alpha_0=1$, $\alpha_1=\gamma^2+\gamma$, and $\alpha_2=\gamma$, then the 7SA gives $$C_{(1),1,1}=\{t_1+t_2\gamma^2+t_3(\gamma^2+\gamma) \ | \ \ t_1\geq 0, \ t_2> 0, \ t_3\geq 0\},$$ $w_{(1),1,1}=+1$, and $w_\mu=0$ for all $\mu\in \widetilde{S}_r$ with $\mu\not=\big((1),1,1\big)$. Therefore, in this case the 7SA gives a true fundamental domain for the action of $\langle\gamma\rangle$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+$.
Quartic case
------------
Let $k={\mathbb{Q}}(\gamma)$, where $\gamma^4+\gamma-1=0$. Then the discriminant of $k$ is $-283$. Let $$\varepsilon_1=\gamma^2=\frac{1}{\gamma^3+\gamma^2+1} \qquad \mathrm{and} \qquad \varepsilon_2=\gamma^2+1=\frac{1}{\gamma^3-\gamma+1}$$ be two independent totally positive units of $k$, with $$\gamma=\big[(0{.}2481...)+(-1{.}0339...)i \, , \, -1{.}2207... \, , \, 0{.}7244...\big] \ \in \ {\mathbb{C}}\times{\mathbb{R}}^2.$$
### Example 3
If $N=3$, $\alpha_0=1$, $\alpha_1=\gamma^2-\gamma+1$, and $\alpha_2=\gamma^2+\gamma$, then the 7SA gives the signed fundamental domain for the action of $\langle\varepsilon_1,\varepsilon_2\rangle$ on ${\mathbb{C}}^*\times{\mathbb{R}}_+^2$ with $$\begin{aligned}
C_{(1),2,0}=\{&t_1(\gamma^2-\gamma+1)+t_2\gamma^2+t_3(-2\gamma^3+3\gamma^2-3\gamma+2)+t_4(\gamma^2+\gamma) \ |\\
&t_1\ge0, \ t_2>0, \ t_3>0, \ t_4\ge0\},\\
C_{(1),3,1}=\{&t_1(\gamma^2+\gamma)+t_2(-\gamma^3+\gamma^2-\gamma+1)+t_3(-2\gamma^3+3\gamma^2-3\gamma+2)+t_4 \ |\\
&t_1\ge0, \ t_2>0, \ t_3\ge0, \ t_4\ge0\},\\
C_{(12),1,0}=\{&t_1(\gamma^2+\gamma)+t_2(\gamma^3+\gamma^2+1)+t_3(-2\gamma^3+3\gamma^2-3\gamma+2)+t_4(\gamma^2+1) \ |\\
&t_1>0, \ t_2\ge0, \ t_3>0, \ t_4\ge0\},\\
C_{(12),3,1}=\{&t_1(\gamma^2+\gamma)+t_2(\gamma^2+1)+t_3(-2\gamma^3+3\gamma^2-3\gamma+2)+t_4 \ |\\
&t_1>0, \ t_2\ge0, \ t_3>0, \ t_4>0\},\end{aligned}$$ and $w_{(1),2,0}=w_{(1),3,1}=w_{(12),1,0}=w_{(12),3,1}=+1$. The rest of the $w_\mu$ are 0. So, as in the previous example, this signed fundamental domain is actually a true one.
Construction of $f$ {#constructionoff}
===================
As in [@DF1], we will prove Theorem \[Main\] by interpreting the left-hand side of as a sum of local degrees of a certain continuous map $F:\widehat{T}\to T$ between a standard $(r+1)$-torus $\widehat{T}$ and an $(r+1)$-torus $T$. Using a basic result in algebraic topology, this sum of local degrees equals the global degree of $F$. We will compute this global degree by proving that $F$ is homotopic to an explicit homeomorphism $F_0$, whose degree can be easily computed. Our contribution lies in the construction of a piecewise affine map $f$, which we will use to define $F$.
The argument at the complex embedding
-------------------------------------
As we said in the Introduction, the non-convexity of ${{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$ is an obstruction to deal with. To bypass this obstruction, we will divide ${{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$ into certain convex regions using the argument at the complex place. For $N\in{\mathbb{N}}$ (with $N\geq 3$ and fixed), we define the regions $$\label{semiplanos}
{\mathcal{S}}_t={\mathcal{S}}_{t,N}:={\mathrm{exp}}\left(2\pi it /N\right)\cdot{\mathcal{S}}_0\subset{\mathbb{C}}^* \qquad\qquad (t\in{\mathbb{Z}}),$$ where $${\mathcal{S}}_0={\mathcal{S}}_{0,N}:=\left\{z\in{\mathbb{C}}^* \ \left| \ \arg(z)\in [-\pi/2N,5\pi/2N) \right.\right\}.$$ Since $N\ge3$ the ${\mathcal{S}}_t$ are convex, and their union for $t\in{\mathbb{Z}}$ is ${\mathbb{C}}^*$. Also ${\mathcal{S}}_t={\mathcal{S}}_{t'}$ if and only if $t\equiv t'({\mathrm{mod} \ }N)$.
Now we define the “windmill arms” ${\mathcal{A}}_t$ by $$\label{aspas}
{\mathcal{A}}_t={\mathcal{A}}_{t,N}:={\mathrm{exp}}\left(2\pi it/N\right)\cdot{\mathcal{A}}_0 \qquad\qquad (t\in{\mathbb{Z}}),$$ where $${\mathcal{A}}_0={\mathcal{A}}_{0,N}:=\left\{z\in{\mathbb{C}}^* \ \left| \ \arg(z)\in [-\pi/2N,\pi/2N] \right.\right\}.$$ Since ${\mathcal{A}}_0$ and the interior $\stackrel{\circ}{{\mathcal{A}}}_1$ of ${\mathcal{A}}_1$ are contained in ${\mathcal{S}}_0$, we have $$\label{aspasemiplano}
{\mathcal{A}}_t\subset{\mathcal{S}}_t, \qquad\qquad \stackrel{\circ}{{\mathcal{A}}}_{1+t}\subset{\mathcal{S}}_t, \qquad\qquad {\mathcal{A}}_{1+t}\not\subset{\mathcal{S}}_t\qquad\qquad (t\in{\mathbb{Z}}).$$ Figure \[Figure2\] shows the windmill arms $\mathcal{A}_t$ in the case $N=3$.
![$\mathcal{A}_t$ for $N=3$.[]{data-label="Figure2"}](Figure2.pdf)
Before continuing the study of the regions described above, we need some elementary properties of the function ${\mathrm{m}}:{\mathbb{C}}^*\to{\mathbb{Z}}$ defined in . In the following, all the congruences ($\equiv$) will be modulo $N$.
\[mprop\] Let $z,u,v,w\in{\mathbb{C}}^*$. Then the following hold.
1. ${\mathrm{m}}(zw)$ is congruent to either ${\mathrm{m}}(z)+{\mathrm{m}}(w)$ or ${\mathrm{m}}(z)+{\mathrm{m}}(w)-1$.
2. If ${\mathrm{m}}(z)+{\mathrm{m}}(z^{-1})\equiv0$, then ${\mathrm{m}}(zw)\equiv{\mathrm{m}}(z)+{\mathrm{m}}(w)$.
3. $\arg\!\Big(z\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(z)/N\big)\Big)$ lies in $\big[0,\,2\pi/N\big)$.
4. If ${\mathrm{m}}(u^{-1}v)+{\mathrm{m}}(v^{-1}w)\equiv{\mathrm{m}}(w)-{\mathrm{m}}(u)$, then ${\mathrm{m}}(u^{-1}w)\equiv{\mathrm{m}}(w)-{\mathrm{m}}(u)$.
5. We have ${\mathrm{m}}(v^{-1}u)+{\mathrm{m}}(vu^{-1})\equiv0$ if the following four equations hold; $$\begin{aligned}
&{\mathrm{m}}(u^{-1}w)+{\mathrm{m}}(uw^{-1})\equiv0, &&{\mathrm{m}}(u^{-1}v)\equiv{\mathrm{m}}(v)-{\mathrm{m}}(u),\\
&{\mathrm{m}}(u^{-1}w)\equiv{\mathrm{m}}(w)-{\mathrm{m}}(u), &&{\mathrm{m}}(v^{-1}w)\equiv{\mathrm{m}}(w)-{\mathrm{m}}(v).\end{aligned}$$
First note that for all $x,y\in{\mathbb{R}}$ and all $\ell'\in{\mathbb{Z}}$ we have $$\lceil x+\ell'\rceil=\lceil x\rceil+\ell' \qquad \mathrm{and} \qquad \lceil x\rceil + \lceil y\rceil -1\leq \lceil x+y\rceil\leq \lceil x\rceil + \lceil y\rceil.$$ Thus (i) follows easily from , and from these two properties of the ceiling function.
To prove (ii), first note that ${\mathrm{m}}(zw)$ is congruent to either ${\mathrm{m}}(z)+{\mathrm{m}}(w)-1$ or ${\mathrm{m}}(z)+{\mathrm{m}}(w)$ (by (i)). Suppose ${\mathrm{m}}(zw)\equiv{\mathrm{m}}(z)+{\mathrm{m}}(w)-1$. Using (i), we have that ${\mathrm{m}}(w)={\mathrm{m}}(zwz^{-1})$ is congruent to either ${\mathrm{m}}(zw)-{\mathrm{m}}(z)$ or ${\mathrm{m}}(zw)-{\mathrm{m}}(z)-1$ (since $-{\mathrm{m}}(z)\equiv{\mathrm{m}}(z^{-1})$), and so congruent to either ${\mathrm{m}}(w)-1$ or ${\mathrm{m}}(w)-2$, which is absurd since $N\ge3$.
Let us prove (iii). Using the identity $\lceil x+\ell'\rceil=\lceil x\rceil+\ell'$ $(x\in{\mathbb{R}}, \ \ell'\in{\mathbb{Z}})$, we have that ${\mathrm{m}}\!\Big(z\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(z)/N\big)\Big)$ is congruent to $\left\lceil\frac{-N}{2\pi}\big(\arg(z)+2\pi{\mathrm{m}}(z)/N\big)\right\rceil$, and so congruent to 0. In general, if $w'\in{\mathbb{C}}^*$ is such that ${\mathrm{m}}(w')\equiv 0$, then we have that $\lceil-N\arg(w')/2\pi-Nq\rceil=0$ for some $q\in{\mathbb{Z}}$. But this is equivalent to $0\leq\arg(w')+2\pi q<2\pi/N$, so $q=0$. Therefore, we have proved (iii).
If ${\mathrm{m}}(u^{-1}v)+{\mathrm{m}}(v^{-1}w)+{\mathrm{m}}(u)$ is congruent to ${\mathrm{m}}(u^{-1}w)+1+{\mathrm{m}}(u)$, then it is congruent to either ${\mathrm{m}}(w)+2$ or ${\mathrm{m}}(w)+1$ by (i), but this is absurd since $N\ge 3$. Hence, ${\mathrm{m}}(u^{-1}w)+{\mathrm{m}}(u)\equiv{\mathrm{m}}(w)$.
To prove (v), suppose ${\mathrm{m}}(v^{-1}u)+{\mathrm{m}}(vu^{-1})\equiv1$. Using (ii) we have that $${\mathrm{m}}(wv^{-1})\equiv{\mathrm{m}}(u^{-1}w\cdot v^{-1}u)\equiv{\mathrm{m}}(u^{-1}w)+{\mathrm{m}}(v^{-1}u),$$ but while the left-hand side of these congruences is congruent to ${\mathrm{m}}(w)-{\mathrm{m}}(v)$, the right-hand side is congruent to ${\mathrm{m}}(w)-{\mathrm{m}}(v)+1$, which is absurd. Therefore, from (i) we have ${\mathrm{m}}(v^{-1}u)+{\mathrm{m}}(vu^{-1})\equiv0$, since ${\mathrm{m}}(1)=0$.
Next we give necessary and sufficient conditions for some inclusion relations of the regions ${\mathcal{A}}_t$ and ${\mathcal{S}}_t$. These conditions are based on modular arithmetic, and they allow us to relate ${\mathcal{A}}_t$ and ${\mathcal{S}}_t$ with the relation $\prec_\sigma$ defined by conditions , and .
\[aspadentrosemiplano\] Let $z\in{\mathbb{C}}^*$ and let $t,k\in{\mathbb{Z}}$. Then the following hold.
1. $z\cdot{\mathcal{A}}_t\subset{\mathcal{S}}_k$ if and only if ${\mathrm{m}}(z)\equiv t-k$.
2. $z\cdot{\mathcal{A}}_t={\mathcal{A}}_k$ if and only if ${\mathrm{m}}(z)+{\mathrm{m}}(z^{-1})\equiv0$ and ${\mathrm{m}}(z)\equiv t-k$.
To prove (i), first assume $z\cdot{\mathcal{A}}_t\subset{\mathcal{S}}_k$. From it is clear that $z\cdot{\mathcal{A}}_t={\mathrm{exp}}\big(i\arg(z)\big)\cdot{\mathcal{A}}_t$. Thus, from we get $${\mathrm{exp}}\big(i\arg(z)\big)\cdot{\mathrm{exp}}(2\pi it/N)\cdot{\mathrm{exp}}(-2\pi ik/N)\cdot w' \ \in \ {\mathcal{S}}_0$$ for all $w'\in{\mathcal{A}}_0$. Putting $w'={\mathrm{exp}}(-\pi i/2N)$ and then $w'={\mathrm{exp}}(\pi i/2N)$, we see that there exist $q,q'\in{\mathbb{Z}}$ such that $$\begin{aligned}
\nonumber
&-\pi/2N\leq\arg(z)+2\pi t/N-2\pi k/N-\pi/2N+2\pi q<5\pi/2N,\\
\nonumber
&-\pi/2N\leq\arg(z)+2\pi t/N-2\pi k/N+\pi/2N+2\pi q'<5\pi/2N.\end{aligned}$$ This implies that $$\begin{aligned}
\nonumber
&-3/2<-N\arg(z)/2\pi-t+k-Nq\leq0,\\
\nonumber
&-1<-N\arg(z)/2\pi-t+k-Nq'\leq 1/2.\end{aligned}$$ If $t-k\not\equiv {\mathrm{m}}(z)$, the above would imply that ${\mathrm{m}}(z)-t+k$ is congruent to both $\pm 1$, which is absurd since $N\geq 3$. Conversely, suppose $t-k\equiv {\mathrm{m}}(z)$. Let $w'\in{\mathcal{A}}_0$. Using Lemma \[mprop\] (iii) and $\arg(w')\in[-\pi/2N,\pi/2N]$, we have for some $t'\in{\mathbb{Z}}$ $$\arg\!\Big(z\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(z)/N\big)\cdot w'\Big)+2\pi t' \ \in \ [-\pi/2N,5\pi/2N) \ \subset \ [-\pi,\pi),$$ so $t'=0$. Then, using , , and that $t\equiv {\mathrm{m}}(z)+k$, we get that $$\begin{aligned}
z\cdot{\mathcal{A}}_t=z\cdot{\mathrm{exp}}\left(2\pi it/N\right)\cdot{\mathcal{A}}_0=z\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(z)/N\big)\cdot{\mathrm{exp}}(2\pi ik/N)\cdot{\mathcal{A}}_0\end{aligned}$$ is contained in ${\mathrm{exp}}(2\pi ik/N)\cdot{\mathcal{S}}_0={\mathcal{S}}_k$. This concludes the proof of (i).
From , we have $z\cdot{\mathcal{A}}_t={\mathcal{A}}_k$ if and only if $\arg(z)+2\pi(t-k)/N=2\pi q$ for some $q\in{\mathbb{Z}}$. So if $z\cdot{\mathcal{A}}_t={\mathcal{A}}_k$, then $$-N\arg(z)/2\pi=t-k+Nq \qquad \mathrm{and} \qquad -N\arg(z^{-1})/2\pi=k-t+Nq'$$ for some $q,q'\in{\mathbb{Z}}$. Hence, ${\mathrm{m}}(z)+{\mathrm{m}}(z^{-1})\equiv0$, and ${\mathrm{m}}(z)\equiv t-k$, using definition . Conversely, suppose ${\mathrm{m}}(z)+{\mathrm{m}}(z^{-1})\equiv0$, and ${\mathrm{m}}(z)\equiv t-k$. Since $\lceil x\rceil+\lceil -x\rceil$ equals either 0 or 1 depending on whether $x\in{\mathbb{Z}}$ or $x\in{\mathbb{R}}\smallsetminus{\mathbb{Z}}$ respectively, we see that ${\mathrm{m}}(z)+{\mathrm{m}}(z^{-1})\equiv0$ implies $-N\arg(z)/2\pi\in{\mathbb{Z}}$. Hence, ${\mathrm{m}}(z)\equiv t-k$ implies $\arg(z)+2\pi(t-k)/N=2\pi q$ for some $q\in{\mathbb{Z}}$.
Let $\sigma\in S_r$. For $(t,t')\in{\mathbb{Z}}\times{\mathbb{Z}}$ with $1\leq t,t'\leq r+1$, consider: $$\begin{aligned}
\xi_\sigma(t,t')\cdot{\mathcal{A}}_{{\mathrm{m}}(\xi_\sigma(t'))}&\subset {\mathcal{S}}_{{\mathrm{m}}(\xi_\sigma(t))},\label{1.2}\\
\xi_\sigma(t,t')\cdot{\mathcal{A}}_{{\mathrm{m}}(\xi_\sigma(t'))}&\not={\mathcal{A}}_{{\mathrm{m}}(\xi_\sigma(t))},\label{1.1}\\
t'&<t.\label{2.2}\end{aligned}$$ Using Lemma \[aspadentrosemiplano\], and the definition of $\prec_\sigma$ (see conditions , and ), we get that $t\prec_{\sigma}t'$ if and only if $(t,t')$ satisfies condition , and at least one of the conditions and . Now we prove that $\prec_\sigma$ is a strict total order on the set $\{\ell\in{\mathbb{Z}}; 1\leq \ell\leq r+1\}$.
\[ordentotal\] For each $\sigma\in S_r$, the relation $\prec_\sigma$ is a strict total order on the set $\{\ell\in{\mathbb{Z}}; 1\leq \ell\leq r+1\}$.
As $\sigma$ remains fixed along the proof, we will exclude it from the notation; furthermore, we will write $M(t,t'):={\mathrm{m}}(\xi_\sigma(t,t'))$ and $M(t):={\mathrm{m}}(\xi_\sigma(t))$ for any $1\le t, t'\le r+1$.
*Transitivity*. Suppose $t\prec t'$ and $t'\prec t''$. Using condition , we have $$M(t,t')+M(t',t'')\equiv M(t'')-M(t).$$ Then putting $u=\xi(t)$, $v=\xi(t')$, $w=\xi(t'')$ in Lemma \[mprop\] (iv), and using , the last congruence implies condition for $(t,t'')$.
Now suppose $(t,t'')$ satisfies neither nor . From (i) and we have $M(t,t'')+M(t'',t)\equiv0$. Also, we have that the pairs $(t,t')$, $(t',t'')$, and $(t,t'')$ satisfy condition . Thus we have satisfied the hypotheses of Lemma \[mprop\] (v) with $u=\xi(t)$, $v=\xi(t')$, and $w=\xi(t'')$, and also with $u=\xi(t'')$, $v=\xi(t')$, and $w=\xi(t)$. Then $M(t,t')+M(t',t)\equiv0$ and $M(t',t'')+M(t'',t')\equiv0$, so $t'<t$ and $t''<t'$, which contradicts $t''\ge t$. Therefore, $(t,t'')$ must satisfy at least one of the conditions and .
*Trichotomy law*. Suppose $t'\not=t$ (say $t'<t$). If $(t,t')$ does not satisfy condition , then $M(t,t')$ is congruent to $M(t')-M(t)+1$ by using Lemma \[mprop\] (i) with $z=\xi(t,t')$ and $w=\xi(t)$. Also we have $M(t,t')+M(t',t)\equiv1$ by using Lemma \[mprop\] (ii) with $z=\xi(t,t')$ and $w=\xi(t)$. Combining these two congruences we get that $(t',t)$ satisfies conditions and , so $t'\prec t$. If $(t,t')$ satisfies , then $t\prec t'$.
Now if $t\prec t'$ and $t'\prec t$, condition for $(t,t')$ and $(t',t)$ implies that the pairs $(t,t')$ and $(t',t)$ do not satisfy , so $t<t'$ and $t'<t$, which is absurd. Also, it is clearly impossible that $t=t'$ and $t\prec t'$.
\[Cor0.5\] Let $\sigma\in S_r$, and let $(t, t')\in{\mathbb{Z}}\times{\mathbb{Z}}$ with $1\leq t, t'\leq r+1$. If $t\prec_\sigma t'$, then $\xi_\sigma(t',t)\cdot\stackrel{\circ}{{\mathcal{A}}}_{1+{\mathrm{m}}(\xi_\sigma(t))}\subset {\mathcal{S}}_{{\mathrm{m}}(\xi_\sigma(t'))}$.
Again $\sigma$ remains fixed along the proof, so we will use the notation adopted in the proof of Proposition \[ordentotal\].
Using and Lemma \[mprop\] (i), we have that $M(t',t)$ is congruent to either $M(t)-M(t')$ or $M(t)-M(t')+1$. If $M(t',t)\equiv M(t)-M(t')$, then $M(t',t)+M(t,t')\equiv0$ by trichotomy. Hence, Lemma \[aspadentrosemiplano\] (ii) implies that $\xi(t',t)\cdot{\mathcal{A}}_{1+M(t)}={\mathcal{A}}_{1+M(t')}$, and so $$\xi(t',t)\cdot\stackrel{\circ}{{\mathcal{A}}}_{1+M(t)}=\stackrel{\circ}{{\mathcal{A}}}_{1+M(t')}\subset
{\mathcal{S}}_{M(t')}$$ by . On the other hand, if $M(t',t)\equiv M(t)-M(t')+1$, then Lemma \[aspadentrosemiplano\] (i) implies that $\xi(t',t)\cdot{\mathcal{A}}_{1+M(t)}\subset {\mathcal{S}}_{M(t')}$.
The order $\prec_\sigma$ depends on the permutation $\sigma\in S_r$ by definition. In general, we are not interested in studying the behavior of $\prec_\sigma$ with respect to $\sigma$, except in the following case.
\[lema42\] Let $\sigma\in S_r$. Define $\widetilde{\sigma}\in S_r$ by putting $\widetilde{\sigma}(1):=\sigma(r)$, and $\widetilde{\sigma}(j):=\sigma(j-1)$ for each $2\leq j\leq r$. Consider the set $$B_\sigma:=\left\{1\leq t\leq r+1 \ \left| \ {\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\cdot\xi_\sigma(t)\big) \equiv
{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+{\mathrm{m}}(\xi_\sigma(t))\right.\right\}$$ and its complement $B_\sigma^c\subset\{1,\dots,r+1\}$. Then for any $t,t'\in\{1,\dots,r\}$ we have
1. $f_{t+1,\widetilde{\sigma}}=\varepsilon_{\sigma(r)}\cdot f_{t,\sigma}$, $\xi_{\widetilde{\sigma}}(t+1)=\varepsilon_{\sigma(r)}^{(1)}\cdot \xi_\sigma(t)$, $\xi_{\widetilde{\sigma}}(t+1,t'+1)=\xi_\sigma(t,t').$
2. If $t,t'\in B_\sigma$ or $t,t'\in B_\sigma^c$, then $t\prec_\sigma t'$ if and only if $t+1\prec_{\widetilde{\sigma}}t'+1$.
3. If $t\in B_\sigma$ and $t'\in
B_\sigma^c$, then $t\prec_\sigma t'$ and $t'+1\prec_{\widetilde{\sigma}}t+1$.
Note that since $\xi_\sigma(r+1)=1$ and ${\mathrm{m}}(1)=0$, we have that $r+1\in B_\sigma$ for all $\sigma\in S_r$, so $B_\sigma\not=\varnothing$. The fact that $f_{t+1,\widetilde{\sigma}}=\varepsilon_{\sigma(r)}\cdot f_{t,\sigma}$ follows easily from the definition of $f_{t,\sigma}$, and from the definition of $\widetilde{\sigma}$. Moreover, since $f_{r+1,\sigma}$ does not depend on $\sigma$, we obtain (i) from the definition of $\xi_\sigma(t)$, and from .
Let us prove (ii). Using (i), we have ${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1,t'+1))\equiv{\mathrm{m}}(\xi_\sigma(t,t'))$, so it is clear that $(t+1,t'+1)$ satisfies for $\widetilde{\sigma}$ if and only if $(t,t')$ satisfies for $\sigma$. But $(t+1,t'+1)$ satisfies for $\widetilde{\sigma}$ if and only if $(t,t')$ satisfies for $\sigma$. Now if $t,t'\in B_\sigma$, we obtain $${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))\equiv{\mathrm{m}}(\xi_\sigma(t'))-{\mathrm{m}}(\xi_\sigma(t))$$ by using (i), so we have that $(t+1,t'+1)$ satisfies for $\widetilde{\sigma}$ if and only if $(t,t')$ satisfies for $\sigma$. If $t,t'\in B_\sigma^c$, the proof follows analogously, noting that $$\label{Bsigmac}
\ell\in B_\sigma^c \ \Longleftrightarrow \ {\mathrm{m}}\big(\xi_{\widetilde{\sigma}}(\ell+1)\big) \equiv
{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+{\mathrm{m}}(\xi_\sigma(\ell)) -1.$$
To prove (iii), first we will prove $t\prec_\sigma t'$ by contradiction; suppose $t'\prec_\sigma t$. Since $t\in B_\sigma$ and $t'\in B_\sigma^c$ we have $${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))\equiv{\mathrm{m}}(\xi_\sigma(t))-{\mathrm{m}}(\xi_\sigma(t'))+1,$$ where the right-hand side is congruent to ${\mathrm{m}}(\xi_\sigma(t',t))+1$ by using that $(t',t)$ satisfies condition for $\sigma$, and so congruent to ${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1,t+1))+1$ by using (i). But this contradicts Lemma \[mprop\] (i) since $N\ge3$.
Finally, we will prove that $t\prec_\sigma t'$ implies $t'+1\prec_{\widetilde{\sigma}}t+1$. For the sake of contradiction, suppose that $t+1\prec_{\widetilde{\sigma}} t'+1$ and $t\prec_\sigma t'$. Thus condition implies that ${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))$ is congruent to ${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1,t'+1))$, and so congruent to ${\mathrm{m}}(\xi_\sigma(t,t'))$ by (i). On the other hand, since $t\in B_\sigma$ and $t'\in B_\sigma^c$ we have $${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))\equiv{\mathrm{m}}(\xi_\sigma(t))-{\mathrm{m}}(\xi_\sigma(t'))+1,$$ where the right-hand side is congruent to $1-{\mathrm{m}}(\xi_\sigma(t,t'))$ since $(t,t')$ satisfies condition for $\sigma$. Hence $${\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))\equiv1-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t'+1))+{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1)),$$ a contradiction. Therefore we have proved (iii).
Domain of $f$
-------------
The aim of this section is to define the domain of the functions $F$ and $F_0$ mentioned at the beginning of section \[constructionoff\]. As we have anticipated in the introduction, this domain is a $(r+1)$-torus $\widehat{T}=D/\!\sim$, where $D$ is the closure of a fundamental domain for ${\mathbb{R}}^{r+1}$ under the translation action of its subgroup ${\mathbb{Z}}^{r+1}$, and $\sim$ identifies points of $D$ lying in the same ${\mathbb{Z}}^{r+1}$-orbit.
Note that $\xi_\sigma(r+1)=1$, so ${\mathrm{m}}(\xi_\sigma(r+1))=0$ for all $\sigma\in S_r$. It follows that $(r+1,t)$ satisfies condition and for all $t\in\{1,\dots,r\}$ and $\sigma\in S_r$.
Putting $z=\xi_\sigma(t)$ in Lemma \[mprop\] (iii), we have that $$\arg\!\Big(\xi_\sigma(t)\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(\xi_\sigma(t))/N\big)\Big) \in [0, 2\pi/N)$$ for all $t\in\{1,\dots,r+1\}$ and $\sigma \in S_r$, so there is a unique $d_{t,\sigma}\in{\mathbb{Z}}$ such that $$\begin{aligned}
\sum_{j=1}^t \arg\!\big(\varepsilon_{\sigma(j-1)}^{(1)}\big) - \sum_{j=1}^{r+1}\arg\!\big(\varepsilon_{\sigma(j-1)}^{(1)}\big) &+
\frac{2\pi}{N}{\mathrm{m}}(\xi_\sigma(t)) + 2\pi d_{t,\sigma} \ \in \ \Big[0, \ \frac{2\pi}{N}\Big) \label{dtsigma},\end{aligned}$$ where $\sum_{j=1}^{r+1}\arg\!\big(\varepsilon_{\sigma(j-1)}^{(1)}\big)$ is independent of $\sigma$, and the summands corresponding to $j=1$ are $0$ by definition. Note that in $$\label{fijoporrho}
\rho_\sigma(r+1)=r+1.$$
Considering , and the permutation $\rho_\sigma\in S_{r+1}$ defined in , we can make the following definition.
\[phi\] For $t\in\{1,\dots,r+1\}$, $\mu=(\sigma,q,n)\in \widetilde{S}_r$, and $j\in{\mathbb{Z}}$, we let $$a(t,\sigma,j):=\frac{1}{N}\big({\mathrm{m}}(\xi_\sigma(t))+j\big)+d_{t,\sigma} \ \in \ {\mathbb{R}}.$$ Also, we define $\phi_{t,\mu}$ and $\phi_{r+2,\mu} \ \in \ {\mathbb{R}}^{r+1}$ by putting $$\begin{aligned}
&\phi_{t,\mu}=\phi(t,\sigma,q,n):= \begin{cases}\sum\limits_{j=1}^te_{\sigma(j-1)}\ +\ a(t,\sigma,n)\cdot e_{r+1}\ & \mathrm{if}\ t\not\prec_\sigma\rho_\sigma(q)
,\\
\sum\limits_{j=1}^te_{\sigma(j-1)}\ +\ a(t,\sigma,n+1)\cdot e_{r+1}\ & \mathrm{if}\ t\prec_\sigma\rho_\sigma(q),
\end{cases}\\
&\phi_{r+2,\mu}=\phi(r+2,\sigma,q,n):=\sum_{j=1}^{\rho_\sigma(q)}e_{\sigma(j-1)}\ +\ a(\rho_\sigma(q),\sigma,n+1)\cdot e_{r+1}.\end{aligned}$$ Here, $e_{\sigma(0)}=e_0:=0$ by definition, and $\{e_i\}_{i=1}^{r+1}$ is the usual basis of ${\mathbb{R}}^{r+1}$.
\[phiind\] For each $\mu=(\sigma,q,n)\in \widetilde{S}_r$, the set $\{\phi_{t,\mu}\}_{t=1}^{r+2}$ is affinely independent.
Recall that a subset $\{w_i\}_{i=1}^n$ of a real vector space $V$ is affinely independent if and only if the set $\{w_i-w_j\}_{\substack{1\leq i\leq n\\ i\not=j}}$ is ${\mathbb{R}}$-linearly independent for some fixed $j\in\{1,\dots,n\}$.
We will prove Lemma \[phiind\] by showing that $\{\phi_{t,\mu}-\phi_{r+2,\mu}\}_{t=1}^{r+1}$ is linearly independent in ${\mathbb{R}}^{r+1}$. Of course, the set $\{e_{\sigma(t-1)}\}_{t=1}^{r+1}$ is affinely independent in ${\mathbb{R}}^{r+1}$. This implies that the set $\{v_{t}\}_{\substack{1\leq
t\leq r+1\\ t\not= \rho_\sigma(q)}}$ is linearly independent, where $v_t$ equals $\sum_{j=1}^te_{\sigma(j-1)} - \sum_{j=1}^{\rho_\sigma(q)}e_{\sigma(j-1)}$, with the product $v_{t}e_{r+1}^T=0$ (here, $e_{r+1}^T$ denote the transpose of $e_{r+1}$). Then, the determinant of the $(r+1)\times(r+1)$ matrix $$\label{19}
(v_{t} \ ; \ \phi_{\rho_\sigma(q),\mu}-\phi_{r+2,\mu})_{\substack{1\leq t\leq r+1\\ t\not= \rho_\sigma(q)}},$$ whose columns are the vectors $v_{t}$ and $\phi_{\rho_\sigma(q),\mu}-\phi_{r+2,\mu}$, is zero because $$\label{1.16x}
\phi_{\rho_\sigma(q),\mu}-\phi_{r+2,\mu}=-\frac{1}{N}\cdot e_{r+1},$$ by Definition \[phi\]. Furthermore, $$\begin{aligned}
\nonumber
\phi_{t,\mu}-\phi_{r+2,\mu}:= \begin{cases}v_{t} + \big(a(t,\sigma,n)-a(\rho_\sigma(q),\sigma,n+1)\big)\cdot e_{r+1}\ & \mathrm{if}\ \rho_\sigma(q)\prec_\sigma t ,\\
v_{t} + \big(a(t,\sigma,n+1)-a(\rho_\sigma(q),\sigma,n+1)\big)\cdot e_{r+1}\ & \mathrm{if}\ t\prec_\sigma\rho_\sigma(q),
\end{cases}\end{aligned}$$ for all $1\leq t\leq r+1$ with $t\not=\rho_\sigma(q)$. Hence, we can transform the matrix in into the matrix $(\phi_{t,\mu}-\phi_{r+2,\mu})_{t=1}^{r+1}$ using elementary operations. Therefore, the set $\{\phi_{t,\mu}-\phi_{r+2,\mu}\}_{t=1}^{r+1}$ is linearly independent in ${\mathbb{R}}^{r+1}$.
The above lemma implies that every non-empty subset of $\{\phi_{t,\mu}\}_{t=1}^{r+2}$ is affinely independent in ${\mathbb{R}}^{r+1}$.
Now we establish some notation. If $w_1,\dots,w_\ell$ are elements of a real vector space $W$, then the (closed) polytope they generate is the set of convex sums $$P=P(w_1,\dots,w_\ell):=\Big\{ w\in W \Big|\,w=\sum_{t=1}^{\ell} b_tw_t,\ \
b_t\ge0,\ \ \sum_{t=1}^{\ell}b_t=1 \Big\}$$ ($P(\varnothing):=\varnothing$). In general, if $w=\sum_{t=1}^{\ell} b_tw_t$, $b_t\in{\mathbb{R}}$, and $\sum_{t=1}^{\ell}b_t=1$, then the $b_t$ are called barycentric coordinates of $w$ with respect to the set $w_1,\dots,w_\ell$. If $w_1,\dots,w_\ell$ is affinely independent, the barycentric coordinates of $w$ are uniquely determined by $w$, so we can write $b_t=b_t(w)$.
For any $\sigma\in S_r$ and any $t\in\{1,\dots,r+1\}$, let $$v_t:=\sum\limits_{j=1}^te_{\sigma(j-1)}+v_t^{(r+1)}e_{r+1}\,, \qquad\qquad w_t:=\sum\limits_{j=1}^te_{\sigma(j-1)}+w_t^{(r+1)}e_{r+1}.$$ If $x,y\in{\mathbb{R}}^{r+1}$ are vectors such that $x^{(j)}=y^{(j)}$ $(1\leq j\leq r)$, then we claim that $$\label{bari=}
x=\sum_{t=1}^{r+1} b_tv_t, \quad y=\sum_{t=1}^{r+1} b'_tw_t, \quad \sum_{t=1}^{r+1}b_t=\sum_{t=1}^{r+1}b'_t=1 \qquad \Longrightarrow \qquad b_t=b'_t$$ for any $t\in\{1,\dots,r+1\}$. To prove , it is easy to verify that $$b_{r+1}+\dots+b_{j+1}=b'_{r+1}+\dots+b'_{j+1} \qquad\qquad (1\leq j\leq r)$$ by multiplying $x$ and $y$ by $e_{\sigma(j)}^T$ since $x^{(j)}=y^{(j)}$ $(1\leq j\leq r)$. Hence, $b_t=b'_t$ $(2\leq j\leq r+1)$, and also $b_1=b'_1$ since the sum of the barycentric coordinates is 1.
Using the above notation, we define the polytopes $$\label{P}
P_{1,\mu}=P_{1}(\sigma,q,n):=P\Big(\{\phi_{t,\mu}\}_{t=1}^{r+1}\Big), \qquad
P_{2,\mu}=P_{2}(\sigma,q,n):=P\Big(\{\phi_{t,\mu}\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_\sigma(q)}}\Big)$$ for $\mu=(\sigma,q,n)\in \widetilde{S}_r$. The next lemma will allow us to give an alternative description of these polytopes.
\[lema8\] Let $\mu=(\sigma,q,n)\in \widetilde{S}_r$. If $x^{(1)},\dots,x^{(r)}\in[0,1]$ satisfy $x^{(\sigma(1))}\geq\dots\geq x^{(\sigma(r))}$, there exist unique $y_1, y_2\in{\mathbb{R}}$ such that $(x^{(1)},\dots,x^{(r)},y_1)\in P_{1,\mu}$ and $(x^{(1)},\dots,x^{(r)},y_2)\in P_{2,\mu}$. Furthermore, such $y_1$ and $y_2$ satisfy $y_1\leq y_2$.
Put $b_1:=1-x^{(\sigma(1))}$, $b_{t}:=x^{(\sigma(t-1))}-x^{(\sigma(t))}$ $(2\leq t\leq r)$, and $b_{r+1}:=x^{(\sigma(r))}$. Clearly, $b_t\geq 0$ for all $1\leq t\leq r+1$, and also $\sum_{t=1}^{r+1}b_t=1$. Thus $v:=\sum_{t=1}^{r+1}b_t\phi_{t,\mu} \in P_{1,\mu}$, and we can check that $ve_{\sigma(j)}^T=x^{(\sigma(j))}$ for each $1\leq j\leq r$. Putting $y_1:=ve_{r+1}^T$, uniqueness follows from . Analogously, existence and uniqueness of $y_2$ follow from .
Now let us prove the last statement of Lemma \[lema8\]. Using , note that the vectors $v_{y_1}:=(x^{(1)},\dots,x^{(r)},y_1)$ and $v_{y_2}:=(x^{(1)},\dots,x^{(r)},y_2)$ have equal barycentric coordinates with respect to the vertices $\{\phi_{t,\mu}\}_{t=1}^{r+1}$ and $\{\phi_{t,\mu}\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_\sigma(q)}}$ respectively. Since these sets differ only in the elements $\phi_{\rho_\sigma(q),\mu}$ and $\phi_{r+2,\mu}$, we deduce that $v_{y_2}-v_{y_1}$ equals $b(\phi_{r+2,\mu}-\phi_{\rho_\sigma(q),\mu})=\frac{b}{N}e_{r+1},$ where $b$ is a barycentric coordinate of $v_{y_2}$ with respect to $\{\phi_{t,\mu}\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_\sigma(q)}}$. Therefore, $(v_{y_2}-v_{y_1})e_{r+1}^T=y_2-y_1\geq 0$.
Since both $y_1$ and $y_2$ in Lemma \[lema8\] depend on $\mu\in \widetilde{S}_r$ and on $x^{(1)},\dots,x^{(r)}\in[0,1]$, we shall write $y_{i,\mu}(x)=y_{i}(\sigma,q,n)(x):=y_{i}$ for any $i=1,2$, $\mu=(\sigma,q,n)\in \widetilde{S}_r$, and $x\in[0,1]^r\times{\mathbb{R}}$ such that $x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))}$.
From the proof of Lemma \[lema8\], note that $v=v(x)=\sum_{t=1}^{r+1}b_t(x)\phi_{t,\mu}$ is continuous in $x$ since each of the $b_t=b_t(x)$ is continuous in $x$. Therefore, $y_1(x)=v(x) e_{r+1}^T$ is also continuous. The same holds for $y_2(x)$.
\[Dsigmaqn\] For $\mu=(\sigma,q,n)\in \widetilde{S}_r$, define $$D_{\mu}=D(\sigma,q,n):=\left\{x\in[0,1]^r\times{\mathbb{R}}\left|\begin{array}{c}
x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))},\\
y_{1,\mu}(x)\leq x^{(r+1)}\leq y_{2,\mu}(x).
\end{array}\right.\right\}.$$
Lemma \[lema8\] implies that $y_{i,\mu}$ is an affine function, [[*[i.$\,$e. ]{}*]{}]{}$$(1-t)y_{i,\mu}(x)+ty_{i,\mu}(z)=y_{i,\mu}\big((1-t)x + tz\big)\qquad (i=1,2; \ t\in[0,1]; \ x,z\in D_{\mu}),$$ since $P_{1,\mu}$ and $P_{2,\mu}$ are convex sets. Hence, $D_{\mu}$ is a convex set. Furthermore, $\phi_{t,\mu}\in D_{\mu}$ $(1\leq t\leq r+2)$. In order to verify this, note that $$\phi_{t,\mu}^{(\sigma(j))} = \begin{cases} 1 \ & \mathrm{if}\ j<t ,\\
0 \ & \mathrm{if} \ j\geq t,
\end{cases} \qquad \phi_{r+2,\mu}^{(\sigma(j))} = \begin{cases} 1 \ & \mathrm{if} \ j<\rho_\sigma(q),\\
0 \ & \mathrm{if} \ j\geq \rho_\sigma(q)
\end{cases}\qquad (1\leq j,t\leq r+1),$$ and $$\phi_{t,\mu}^{(r+1)}=y_{1,\mu}(\phi_{t,\mu}), \qquad\qquad \phi_{r+2,\mu}^{(r+1)}=y_{2,\mu}(\phi_{r+2,\mu}) \qquad\qquad (1\leq t\leq r+1)$$ since $\phi_{t,\mu}\in P_{1,\mu}$ and $\phi_{r+2,\mu}\in P_{2,\mu}$. Thus, $P(\phi_{1,\mu},\dots,\phi_{r+2,\mu})\subset D_{\mu}$. On the other hand, each $v\in D_{\mu}$ is contained in the straight line passing through $$\big(v^{(1)},\dots,v^{(r)},y_{1,\mu}(v)\big)\in P_{1,\mu} \qquad\qquad \mathrm{and} \qquad\qquad\big(v^{(1)},\dots,v^{(r)},y_{2,\mu}(v)\big)\in P_{2,\mu},$$ so $$\label{Dsigmaqn2}
P(\phi_{1,\mu},\phi_{2,\mu},\dots,\phi_{r+2,\mu}) = D_{\mu}.$$ Therefore Lemma \[phiind\] implies that the $D_\mu$ are $(r+1)$-simplices.
We shall see next that we can put the $D_{\mu}=D(\sigma,q,n)$ a top one another so that every intersection of two adjacent simplices is an $r$-simplex. We shall do this by fixing $\sigma\in S_r$ and varying $q$ and $n$. More precisely, fix $\sigma\in S_r$. From and Definition \[phi\] we have $$\label{identidadesphi}
\phi(t,\sigma,q-1,n)=\phi(t,\sigma,q,n) \qquad \mathrm{and} \qquad \phi(\rho_{\sigma(q)},\sigma,q-1,n)=\phi(r+2,\sigma,q,n)$$ $(1\leq t\leq r+1, \ t\not=\rho_\sigma(q); \ 2\leq q\leq r+1; \ 0\leq n\leq N-1)$. Also $$\phi(t,\sigma,r+1,n+1)=\phi(t,\sigma,1,n) \qquad \mathrm{and} \qquad \phi(\rho_{\sigma}(1),\sigma,r+1,n+1)=\phi(r+2,\sigma,1,n)$$ $(1\leq t\leq r+1, \ t\not=\rho_{\sigma}(1); \ 0\leq n\leq N-2)$. Thus we have $$\begin{aligned}
&P_{1}(\sigma,q-1,n)=P_{2}(\sigma,q,n) &&(2\leq q\leq r+1; \ 0\leq n\leq N-1),\\
&P_{1}(\sigma,r+1,n+1)=P_{2}(\sigma,1,n) &&(0\leq n\leq N-2).\end{aligned}$$ Applying Lemma \[lema8\] to the last two identities, we get the following chain of inequalities for any $x\in [0,1]^r\times{\mathbb{R}}$ such that $x^{(\sigma(1))}\geq\dots\geq x^{(\sigma(r))}$. $$\begin{aligned}
\nonumber
&y_{1}(\sigma,r+1,0)(x)\leq y_{1}(\sigma,r,0)(x)\leq\dots\leq y_{1}(\sigma,1,0)(x)\leq \\
\nonumber
&y_{1}(\sigma,r+1,1)(x)\leq y_{1}(\sigma,r,1)(x)\leq\dots\leq y_{1}(\sigma,1,1)(x)\leq \\
\nonumber
&\vdots\\
&y_{1}(\sigma,r+1,N-1)(x)\leq y_{1}(\sigma,r,N-1)(x)\leq\dots\leq y_{2}(\sigma,1,N-1)(x). \label{25}\end{aligned}$$ Note that this chain ends with $y_{2}(\sigma,1,N-1)(x)$, the only “link” of the chain indexed by 2 instead 1. However, we can index $y_{2}(\sigma,1,N-1)(x)$ by 1 since $$\begin{aligned}
\nonumber
P_{2}(\sigma,1,N-1)=P\Big(\big\{\phi(t,\sigma,1,N-1)\big\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_{\sigma(1)}}}\Big)
&=P\Big(\big\{e_{r+1}+\phi(t,\sigma,r+1,0)\big\}_{t=1}^{r+1}\Big)\\
&=e_{r+1}+P_{1}(\sigma,r+1,0), \label{politoposciclicos}\end{aligned}$$ so $y_{2}(\sigma,1,N-1)(x)=1+y_{1}(\sigma,r+1,0)(x)$, where $y_{1}(\sigma,r+1,0)(x)$ is the first link of the chain. Hence, in the following we shall write $$\label{ysigmaqn}
y_{\mu}(x):=y_{1,\mu}(x) \qquad (\mu\in \widetilde{S}_r; \ x\in[0,1]^{r}\times{\mathbb{R}}\ \mathrm{with} \ x^{(\sigma(1))}\geq\dots\geq x^{(\sigma(r))}).$$
Considering the above, put $$\label{D}
D:=\bigcup_{\sigma\in S_r}\triangle_\sigma=\bigcup_{\mu\in \widetilde{S}_r} D_{\mu},$$ where $$\triangle_\sigma:=\left\{x\in [0,1]^r\times{\mathbb{R}}\left|\begin{array}{c}
x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))},\\
y(\sigma,r+1,0)(x)\leq x^{(r+1)}\leq 1+y(\sigma,r+1,0)(x).
\end{array}\right.\right\}.$$ We are interested in three properties of $D$. In the first place, $D$ is a finite union of compact sets, and so is a compact subset of ${\mathbb{R}}^{r+1}$. Secondly, $D$ is the topological closure of a fundamental domain for ${\mathbb{R}}^{r+1}$ under the translation action of its subgroup ${\mathbb{Z}}^{r+1}$. Indeed, the set $$\label{D2}
\mathfrak{D}:=\bigcup_{\sigma\in S_r}\mathfrak{D}_\sigma,$$ $$\mathfrak{D}_\sigma:=\left\{x\in [0,1)^r\times{\mathbb{R}}\left|\begin{array}{c}
x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))},\\
y(\sigma,r+1,0)(x)\leq x^{(r+1)}< 1+y(\sigma,r+1,0)(x).
\end{array}\right.\right\},$$ is such a fundamental domain. The quotient space $$\label{torocanonico}
\widehat{T}:=D/\sim$$ is homeomorphic to the standard $(r+1)$-torus ${\mathbb{R}}^{r+1}/{\mathbb{Z}}^{r+1}$, where $\sim$ is the identification of elements in the same orbit with respect to this action.
Finally, we will show that the $D_{\mu}$ form a simplicial decomposition of $D$. Since we have , we only need to verify that the $D_{\mu}$ intersect each other in faces. Recall that a face of a polytope $P$ is the polytope generated by a subset of only its vertices. Now we need some technical remarks.
\[Lema21previos\] Let $\mu=(\sigma,q,n),\, \mu'=(\sigma',q',n')\in \widetilde{S}_r$. Put $$\mathfrak{B}_{\sigma,\sigma'}:=\{1\}\cup\Big\{2\le t\le r+1 \ \Big| \ \{\sigma(j-1) | 2\leq j\leq t\}=\{\sigma'(j-1) | 2\leq j\leq t\}\Big\}.$$ Then the following hold.
1. If $w\in{\mathbb{R}}^r$ satisfies $w^{(\sigma(1))}\geq\dots\geq w^{(\sigma(r))}$ and $w^{(\sigma'(1))}\geq\dots\geq
w^{(\sigma'(r))}$, then $w^{(\sigma(j))}=w^{(\sigma'(j))}$ for all $1\leq j\leq r$.
2. Let $v\in[0,1]^r\times{\mathbb{R}}$ with $v^{(\sigma(1))}\geq\dots\geq v^{(\sigma(r))}$ and $v^{(\sigma'(1))}\geq\dots\geq v^{(\sigma'(r))}$. If $$v=\sum_{t=1}^{r+1}b_t\phi_{t,\mu} \qquad {\text or} \qquad v=(b_{\rho_{\sigma}(1)}/N)e_{r+1}+\sum_{t=1}^{r+1}b_t\phi(t,\sigma,1,N-1)$$ for some $b_t\in{\mathbb{R}}$, then $b_t=0$ whenever $t\not\in\mathfrak{B}_{\sigma,\sigma'}$.
3. If $t, t'\in \mathfrak{B}_{\sigma,\sigma'}$, then $t\prec_\sigma t'$ if and only if $t\prec_{\sigma'} t'$.
4. If $t\in\mathfrak{B}_{\sigma,\sigma'}$, then $a(t,\sigma,\ell)=a(t,\sigma',\ell)$ for any $\ell\in{\mathbb{Z}}$.
To prove (i), for the sake of contradiction suppose that $w^{(\sigma(j))}\not=w^{(\sigma'(j))}$ for some $j\in\{1,\dots,r\}$ (say $w^{(\sigma(j))}<w^{(\sigma'(j))}$). Since $w^{(\sigma(j))}<w^{(\sigma'(j))}\leq w^{(\sigma'(i))}$ for all $i\in\{1,\dots,j\}$, there are at least $j$ coordinates of $w$ greater than $w^{(\sigma(j))}$. But this contradicts $w^{(\sigma(1))}\ge\dots\geq w^{(\sigma(j))}$, which implies that there are at most $j-1$ of such coordinates.
Let us prove (ii). If $t\in\{2,\dots,r+1\}$ is such that $\{\sigma(j-1) \ | \ 2\leq j\leq t\}\not=\{\sigma'(j-1) \ | \ 2\leq j\leq t\}$ (note that $t\not=r+1$) then there exists $j\in\{2,\dots,t\}$ such that $\sigma(j-1)=\sigma'(i)$ with $i\in\{t,\dots,r\}$. This implies that $$1\geq\dots\geq v^{(\sigma'(j-1))}\geq\dots\geq v^{(\sigma'(t-1))}\geq\dots\geq v^{(\sigma(j-1))}\geq\dots\geq 0.$$ Since $v^{(\sigma'(j-1))}=v^{(\sigma(j-1))}$ (by (i)), we have $v^{(\sigma'(t-1))}=v^{(\sigma'(t))}$. Hence, $v^{(\sigma(t-1))}=v^{(\sigma(t))}$, and so we conclude $b_t=0$ from the identity $$v^{(\sigma(\ell))}=ve_{\sigma(\ell)}^T=b_{r+1}+\dots +b_{\ell+1} \qquad\qquad (1\le\ell\le r).$$
To show (iii), note that $\xi_\sigma(t)=\xi_{\sigma'}(t)$ and $\xi_\sigma(t')=\xi_{\sigma'}(t')$ for all $t,t'\in\mathfrak{B}_{\sigma,\sigma'}$. Thus, assertion (iii) follows from conditions , , and .
Assertion (iv) follows directly from and from Definition \[phi\].
\[Lema21a\] Let $\mu=(\sigma,q,n),\,\mu'=(\sigma',q',n')\in \widetilde{S}_r$. If $A\subset\{\phi_{t,\mu}\}_{t=1}^{r+1}$ and $A'\subset\{\phi_{t,\mu'}\}_{t=1}^{r+1}$, then $P(A)\cap P(A') = P(A\cap A')$. Also, this assertion remains valid if we replace both (or one of the) sets $\{\phi_{t,\mu}\}_{t=1}^{r+1}$ and $\{\phi_{t,\mu'}\}_{t=1}^{r+1}$ by $\{\phi(t,\sigma,1,N-1)\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_\sigma(1)}}$ and $\{\phi(t,\sigma',1,N-1)\}_{\substack{1\leq t\leq r+2\\
t\not=\rho_{\sigma'}(1)}}$ respectively.
First note that in any case $A \cap A' \subset P(A)\cap P(A')$, and so $P(A \cap A') \subset P(A)\cap P(A')$ since $P(A)\cap P(A')$ is a convex set. Thus we have only to prove the reverse inclusion. Suppose that $P(A)\cap P(A')\not=\varnothing$ (otherwise the inclusion is obvious). Take $v\in P(A)\cap P(A')$ and then expand it in its barycentric coordinates with respect to $A$ and $A'$: $v=\sum_{t=1}^{r+1}b_t\phi_{t,\mu}=\sum_{t=1}^{r+1}b'_t\phi_{t,\mu'}$, where $b_t, b'_t\geq 0$; $\sum_{t=1}^{r+1}b_t=\sum_{t=1}^{r+1}b'_t=1$; $b_t=0$ if $\phi_{t,\mu}\not\in A$, and $b'_t=0$ if $\phi_{t,\mu'}\not\in
A'$. From , we know that $P(A)\subset D_{\mu}$ and $P(A')\subset D_{\mu'}$, so $v^{(\sigma(1))}\geq\dots\geq v^{(\sigma(r))}$ and $v^{(\sigma'(1))}\geq\dots\geq v^{(\sigma'(r))}$. Hence, using and Lemma \[Lema21previos\] (ii) we have $$\label{27}
\sum_{t\in\mathfrak{B}_{\sigma,\sigma'}} b_t(\phi_{t,\mu}-\phi_{t,\mu'})=0.$$ Without loss of generality we can assume $n\leq n'$. First suppose $n<n'$; we claim that $v\in P(A\cap A')$. Indeed, Definition \[phi\] and Lemma \[Lema21previos\] (iv) imply that $(\phi_{t,\mu}-\phi_{t,\mu'})e_{r+1}^T\leq 0$ for all $t\in\mathfrak{B}_{\sigma,\sigma'}$. Therefore, $b_t=b'_t>0$ implies that $\phi_{t,\mu}=\phi_{t,\mu'}$ by , and then $v\in P(A\cap A')$. If $n=n'$, for the sake of contradiction suppose that there exist $t,t'\in\mathfrak{B}_{\sigma,\sigma'}$ such that $$\label{29}
\big(\phi_{t,\mu}-\phi(t,\sigma',q',n)\big)e_{r+1}^T< 0\qquad \mathrm{and} \qquad
\big(\phi_{t',\mu}-\phi(t',\sigma',q',n)\big)e_{r+1}^T> 0.$$ Thus, $t\not= t'$. If $t\prec_\sigma t'$, we have that $t\prec_\sigma t'\prec_\sigma \rho_\sigma(q)$ by Definition \[phi\], Lemma \[Lema21previos\] (iv), and the second inequality of . That is, $\big(\phi_{t,\mu}-\phi(t,\sigma',q',n)\big)e_{r+1}^T\geq 0$, which contradicts the first inequality of . If $t'\prec_\sigma t$ we have $t'\prec_{\sigma'} t\prec_{\sigma'}\rho_{\sigma'}(q')$ by Lemma \[Lema21previos\] (iii), Definition \[phi\], Lemma \[Lema21previos\] (iv), and the first inequality of . Therefore, $\big(\phi_{t',\mu}-\phi(t',\sigma',q',n)\big)e_{r+1}^T\leq 0$, which contradicts the second inequality of . Thus $v\in P(A\cap A')$, as in the case $n<n'$.
To prove the last part of the lemma, let us verify the case $A\subset\{\phi(t,\sigma,1,N-1)\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_\sigma(1)}}$ and $A'\subset\{\phi_{t,\mu'}\}_{t=1}^{r+1}$. Let $v\in P(A)\cap
P(A')$. Expanding $v$ in barycentric coordinates, $$v=(b_{\rho_{\sigma}(1)}/N)e_{r+1}+\sum_{t=1}^{r+1}b_t\phi(t,\sigma,1,N-1)=\sum_{t=1}^{r+1}b'_t\phi_{t,\mu'}.$$ Using and Lemma \[Lema21previos\] (ii), we conclude that $\sum_{t\in\mathfrak{B}_{\sigma,\sigma'}} b_t\delta_t=0$, where $$\delta_t := \begin{cases} \phi(t,\sigma,1,N-1)-\phi_{t,\mu'} \ & \mathrm{if} \quad t\not=\rho_{\sigma}(1) ,\\
\phi(r+2,\sigma,1,N-1)-\phi_{\rho_{\sigma}(1),\mu'} \ & \mathrm{if} \quad t=\rho_{\sigma}(1).
\end{cases}$$ In both cases, Definition \[phi\] and Lemma \[Lema21previos\] (iv) show that $\delta_te_{r+1}^T\geq 0$. Therefore, $\delta_t=0$ whenever $b_t>0$, which implies that $v\in P(A\cap A')$.
The case $A\subset\{\phi_{t,\mu}\}_{t=1}^{r+1}$ and $A'\subset\{\phi(t,\sigma',1,N-1)\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_{\sigma'}(1)}}$ follows as the previous one by symmetry. Finally, the case $A\subset\{\phi(t,\sigma,1,N-1)\}_{\substack{1\leq t\leq r+2\\ t\not=\rho_{\sigma}(1)}}$ and $A'\subset\{\phi(t,\sigma',1,N-1)\}_{\substack{1\leq
t\leq r+2\\ t\not=\rho_{\sigma'}(1)}}$ follows by using , , and the first part of the lemma.
\[Lema21b\] Let $\mu=(\sigma,q,n),\,\mu'=(\sigma',q',n')\in \widetilde{S}_r$. Then there exist $\widehat{q}\in\{1,\dots,r+1\}$ and $\widehat{n}\in\{0,\dots,N-1\}$ such that $y(\sigma,q,n)(v)$ equals either $y(\sigma',\widehat{q},\widehat{n})(v)$ or $1+y(\sigma',r+1,0)(v)$ for all $v\in [0,1]^r\times{\mathbb{R}}$ satisfying $v^{(\sigma(1))}\geq\dots\geq v^{(\sigma(r))}$ and $v^{(\sigma'(1))}\geq\dots\geq v^{(\sigma'(r))}$. Moreover, $y(\sigma,r+1,0)(v)=y(\sigma',r+1,0)(v)$ for all such $v$.
Consider the following three cases.
*Case 1*. Suppose $t\prec_\sigma\rho_\sigma(q)$ for all $t\in \mathfrak{B}_{\sigma,\sigma'}$, and $n\leq N-2$. Using Definition \[phi\], Lemma \[Lema21previos\] (iv) and , we have $\phi_{t,\mu}=\phi(t,\sigma',r+1,n+1)$ for all $t\in\mathfrak{B}_{\sigma,\sigma'}$ ($\widehat{q}=r+1$ and $\widehat{n}=n+1$) since $t\not\prec_{\sigma'} r+1$ for all $t\in\{1,\dots,r+1\}$.
*Case 2*. Suppose $t\prec_\sigma\rho_\sigma(q)$ for all $t\in \mathfrak{B}_{\sigma,\sigma'}$, and $n=N-1$. Since $a(t,\sigma,N)=1+a(t,\sigma,0)$, we get that $\phi_{t,\mu}=e_{r+1}+\phi(t,\sigma',r+1,0)$ for all $t\in\mathfrak{B}_{\sigma,\sigma'}$ by proceeding as in the previous case.
*Case 3*. If there exists $t\in\mathfrak{B}_{\sigma,\sigma'}$ with $t\not\prec_\sigma \rho_\sigma(q)$, put $t_0:=\min_{t\in\mathfrak{B}_{\sigma,\sigma'}}\{t\not\prec_\sigma \rho_\sigma(q)\}$. That is, $$t\not\prec_\sigma\rho_\sigma(q), \quad t\in\mathfrak{B}_{\sigma,\sigma'}, \quad t\not=t_0 \qquad \Longrightarrow \qquad t_0\prec_\sigma t$$ (note that $\mathfrak{B}_{\sigma,\sigma'}\not=\varnothing$ since $1\in\mathfrak{B}_{\sigma,\sigma'}$). Define $\widehat{q}:=(\rho_{\sigma'})^{-1}(t_0)$. If $t\prec_\sigma\rho_\sigma(q)$ with $t\in\mathfrak{B}_{\sigma,\sigma'}$, then $t\prec_\sigma t_0=\rho_{\sigma'}(\widehat{q})$. If $t\not\prec_\sigma\rho_\sigma(q)$ with $t\in \mathfrak{B}_{\sigma,\sigma'}$, then $t\not\prec_\sigma t_0$ (by the definition of $t_0$). Using Lemma \[Lema21previos\] (iii) and Lemma \[Lema21previos\] (iv), we have $\phi_{t,\mu}=\phi(t,\sigma',\widehat{q},n)$ for all $t\in\mathfrak{B}_{\sigma,\sigma'}$ ($\widehat{n}=n$). In particular, $\phi(t,\sigma,r+1,0)=\phi(t,\sigma',r+1,0)$ for all $t\in\mathfrak{B}_{\sigma,\sigma'}$ ($t_0=\widehat{r+1}=r+1$).
On the other hand, if $v\in{\mathbb{R}}^{r+1}$ satisfies $v^{(\sigma(1))}\geq\dots\geq v^{(\sigma(r))}$ and $v^{(\sigma'(1))}\geq\dots\geq v^{(\sigma'(r))}$, we can write $\big(v^{(1)},\dots,v^{(r)},y_{\mu}(v)\big)$ as $\sum_{t\in\mathfrak{B}_{\sigma,\sigma'}}b_t\phi_{t,\mu}$ by , Lemma \[lema8\], and Lemma \[Lema21previos\] (ii). Therefore, the lemma follows from the three cases above.
\[cadenas=\] Recall and the chain of inequalities , $$\begin{aligned}
\nonumber
&y_{1}(\sigma,r+1,0)(x)\leq y_{1}(\sigma,r,0)(x)\leq\dots\leq y_{1}(\sigma,1,0)(x)\leq \\
\nonumber
&y_{1}(\sigma,r+1,1)(x)\leq y_{1}(\sigma,r,1)(x)\leq\dots\leq y_{1}(\sigma,1,1)(x)\leq \\
\nonumber
&\vdots\\
&y_{1}(\sigma,r+1,N-1)(x)\leq y_{1}(\sigma,r,N-1)(x)\leq\dots\leq y_{2}(\sigma,1,N-1)(x).\end{aligned}$$ From Lemma \[Lema21b\], applied to $v\in [0,1]^r\times{\mathbb{R}}$ such that $v^{(\sigma(1))}\geq\dots\geq v^{(\sigma(r))}$ and $v^{(\sigma'(1))}\geq\dots\geq v^{(\sigma'(r))}$, we have that the links of both chains for $v$ (with $\sigma$ and $\sigma'$) are the same. This fact allows us to finish the proof of the simplicial decomposition of $D$.
\[intersecciondesimplices\] Let $\mu=(\sigma,q,n),\, \mu'=(\sigma',q',n')\in \widetilde{S}_r$. Then $$\label{32}
D_{\mu}\cap D_{\mu'}=P\big(\{\phi_{t,\mu}\}_{t=1}^{r+2}\cap \{\phi_{t,\mu'}\}_{t=1}^{r+2}\big),$$ and so $D=\cup_{\mu\in \widetilde{S}_r}D_\mu$ is a simplicial decomposition of $D$.
We have only to verify that the left-hand side of is contained in the right-hand side, as the other inclusion is obvious by . Let $v\in D_{\mu}\cap D_{\mu'}$. From Definition \[Dsigmaqn\], we have the next four possibilities: $$\begin{aligned}
&y_{1,\mu}(v)=v^{(r+1)}<y_{2,\mu}(v), &&y_{1,\mu}(v)<v^{(r+1)}=y_{2,\mu}(v),\\
&y_{1,\mu}(v)=v^{(r+1)}=y_{2,\mu}(v), &&y_{1,\mu}(v)<v^{(r+1)}<y_{2,\mu}(v).\end{aligned}$$ In the first three cases, Lemma \[lema8\] and Remark \[cadenas=\] imply that $v$ lies in the intersection of two polytopes, as in Lemma \[Lema21a\], and so we have the desired inclusion.
Now suppose $y_{1,\mu}(v)<v^{(r+1)}<y_{2,\mu}(v)$. In this case, $v$ lies on the straight line passing through the points $\big(v^{(1)},\dots,v^{(r)},y_{1,\mu}(v)\big)$ and $\big(v^{(1)},\dots,v^{(r)},y_{2,\mu}(v)\big)$. Then using Remark \[cadenas=\], Lemma \[lema8\], and Lemma \[Lema21a\], we have that these two points lie in $P\big(\{\phi_{t,\mu}\}_{t=1}^{r+2}\cap \{\phi_{t,\mu'}\}_{t=1}^{r+2}\big)$. Therefore, we have the desired inclusion by convexity.
The piecewise affine map $f$
----------------------------
Now we construct the piecewise affine map $f$ mentioned at the beginning of this section. In Proposition \[Escher\], we shall define $f$ as a function on $D$ (see ) that descends to the quotient $\widehat{T}$ described in .
Consider the function $$\begin{aligned}
\nonumber
&\ell:\big({\mathbb{C}}\times{\mathbb{R}}^r\big)\smallsetminus\{x^{(r+1)}=0\}\longrightarrow{\mathbb{C}}\times{\mathbb{R}}^{r-1},\\
\label{ell}
&\ell(x):=\left(\frac{x^{(1)}}{x^{(r+1)}},\frac{x^{(2)}}{x^{(r+1)}},\dots,\frac{x^{(r)}}{x^{(r+1)}}\right)\in{\mathbb{C}}\times{\mathbb{R}}^{r-1},\end{aligned}$$ valid for any $x=(x^{(1)},\dots,x^{(r+1)})\in{\mathbb{C}}\times{\mathbb{R}}^r$ with non-vanishing last coordinate $x^{(r+1)}$. We define $\widetilde{V}:=\ell(V)=\langle\widetilde{\varepsilon_1},\dots,\widetilde{\varepsilon_r}\rangle$, where $V:=\langle\varepsilon_1,\dots,\varepsilon_r\rangle$ and $\widetilde{\varepsilon_j}:=\ell(\varepsilon_j)$ $(1\leq j\leq r)$. Here the $\varepsilon_j$ are totally positive independent units of $k$, as in Theorem \[Main\]. It is clear that $\widetilde{V}$ acts on $\ell\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^r}\big)={\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}$ by component-wise multiplication.
Let $\mu=(\sigma,q,n)\in \widetilde{S}_r$. For each $t\in{\mathbb{Z}}$, choose $\alpha_t=\alpha(t)\in k\cap\left({\mathbb{C}}^*\times{\mathbb{R}}^r_+\right)$ as in the fourth step of the 7SA. From , we can readily verify that $$\label{alfaaspas}
\alpha_t^{(1)} \, \in \ \stackrel{\circ}{{\mathcal{A}}_t} \qquad (t\in{\mathbb{Z}}).$$ Since the set $\{\phi_{t,\mu}\}_{t=1}^{r+2}$ is affinely independent by Lemma \[phiind\], we can define $A_{\mu}=A(\sigma,q,n):{\mathbb{R}}^{r+1}\to {\mathbb{C}}\times{\mathbb{R}}^{r-1}$ as the unique affine map such that $$\label{afin}
A_{\mu}(\phi_{t,\mu}):=\varphi_{t,\mu} \qquad\qquad (1\leq t\leq r+2),$$ where $\varphi_{t,\mu}:=\ell(f_{t,\mu})$ and $f_{t,\mu}$ is defined by $$\begin{aligned}
\label{ftmu}
&f_{t,\mu}=f(t,\sigma,q,n):=\begin{cases}
f_{t,\sigma}\cdot\alpha(Na(t,\sigma,n)) & \mathrm{if} \ t\nprec_\sigma\rho_\sigma(q), \\
\nonumber
f_{t,\sigma}\cdot\alpha(Na(t,\sigma,n+1)) & \mathrm{if} \ t\prec_\sigma\rho_\sigma(q),
\end{cases} \qquad (1\leq t\leq r+1)\\
&f_{r+2,\mu}=f(r+2,\sigma,q,n):=f_{\rho_\sigma(q),\sigma}\cdot\alpha(Na(\rho_\sigma(q),\sigma,n+1)).\end{aligned}$$ Except for minors changes in notation, this definition of $f_{t,\mu}$ is the one given in and . In fact, it is easy to verify that $$\label{aym}
a(t,\sigma,j+1)-a(t,\sigma,j)=1/N, \qquad Na(t,\sigma,j)\equiv {\mathrm{m}}(\xi_\sigma(t))+j \qquad (j\in{\mathbb{Z}})$$
At first sight, we do not know if the image of the map $A_{\mu}$ restricted to $D_{\mu}$ is contained in ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ or not. This issue will be important when we define the function $f$ by using the $A_{\mu}$. The next lemma answers this question, and will prove important in working with homotopies later. For its proof we shall use the following property of affine maps. Let $W$ and $W'$ be two real vector spaces. If $w\in W$ has barycentric coordinates $b_i$ $(1\le i\le \ell)$ with respect to $w_1,\dots,w_\ell$, and $A:W\to W'$ is an affine map with $A(w_i)=p_i$ $(1\le i\le \ell)$, then the same $b_i$ are also barycentric coordinates for $A(w)$ with respect to $p_1,\dots,p_\ell$. Therefore, using definition , $$\label{prop.afin}
v=\sum_{t=1}^{r+2}b_t\phi_{t,\mu}, \quad b_t\in{\mathbb{R}}, \quad \sum_{t=1}^{r+2}b_t=1 \qquad \Longrightarrow \qquad A_\mu(v)=\sum_{t=1}^{r+2}b_t\varphi_{t,\mu}.$$
\[afindentrosemiplano\] Let $\mu=(\sigma,q,n)\in \widetilde{S}_r$. Then, for any $t\in\{1,\dots,r+2\}$, we have $$A_{\mu}(D_\mu) \ \subset \ (f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)})\times {\mathbb{R}}_+^{r-1} \ \subset \ {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}.$$
We note two properties of the map $\ell$ defined in . If $x\in{\mathbb{C}}^*\times{\mathbb{R}}_+^{r}$, then $$\label{prop.ell}
\arg\!\big(\ell(x)^{(1)}\big)=\arg(x^{(1)}) \qquad \mathrm{and} \qquad \ell(x)^{(j)}\in{\mathbb{R}}_+ \qquad (2\leq j\leq r).$$ In particular, these properties are satisfied by $x=f_{t,\mu}$, for any $t\in{\mathbb{Z}}$.
To prove the lemma, first we shall study three cases for $A_\mu(\phi_{t,\mu})$ $(1\le t\le r+2)$.
*Case 1*. Suppose $1\leq t\leq r+1$ and $t\prec_\sigma\rho_\sigma(q)$. From , we have $$\label{50a}
A_{\mu}(\phi_{t,\mu}):=\ell(f_{t,\sigma}\cdot\alpha_{Na(t,\sigma,n+1)}).$$ Since $t\prec_\sigma\rho_\sigma(q)$, Corollary \[Cor0.5\] implies that $\xi_\sigma(\rho_\sigma(q),t)\cdot\stackrel{\circ}{{\mathcal{A}}}_{1+{\mathrm{m}}(\xi_\sigma(t))}\subset{\mathcal{S}}_{{\mathrm{m}}(\xi_\sigma(\rho_\sigma(q)))}$. Multiplying this inclusion by $\tau_1(f_{\rho_\sigma(q),\sigma})\cdot{\mathrm{exp}}(2\pi in/N)$, and using , we get $$f_{t,\sigma}^{(1)}\cdot\stackrel{\circ}{{\mathcal{A}}}_{Na(t,\sigma,n+1)}\subset f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}.$$ Then, using , and , the last inclusion implies that $A_{\mu}(\phi_{t,\mu})$ lies in $(f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)})\times {\mathbb{R}}_+^{r-1}$.
*Case 2*. Suppose that $1\leq t\leq r+1$ and $t\not\prec_\sigma\rho_\sigma(q)$. From , $$\label{50b}
A_{\mu}(\phi_{t,\mu}):=\ell(f_{t,\sigma}\cdot\alpha_{Na(t,\sigma,n)}).$$ Since $t\not\prec_\sigma\rho_\sigma(q)$, we have that (respectively ) implies $$\xi_\sigma(\rho_\sigma(q),t)\cdot{\mathcal{A}}_{{\mathrm{m}}(\xi_\sigma(t))}\subset {\mathcal{S}}_{{\mathrm{m}}(\xi_\sigma(\rho_\sigma(q)))}$$ whenever $\rho_\sigma(q)\prec_\sigma t$ (respectively $t=\rho_\sigma(q)$). Multiplying the last inclusion by $\tau_1(f_{\rho_\sigma(q),\sigma})\cdot{\mathrm{exp}}(2\pi in/N)$, and using , we get $$f_{t,\sigma}^{(1)}\cdot{\mathcal{A}}_{Na(t,\sigma,n)}\subset f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}.$$ Then, using , and , from the last inclusion we have that $A_{\mu}(\phi_{t,\mu})$ lies in $\big(f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}\big)\times {\mathbb{R}}^{r-1}$.
*Case 3*. Finally, if $t=r+2$, we have $$\label{50c}
A_{\mu}(\phi_{t,\mu}):=\ell(f_{\rho_\sigma(q),\sigma}\cdot\alpha_{Na(\rho_\sigma(q),\sigma,n+1)})$$ (see and ). From , note that $\stackrel{\circ}{{\mathcal{A}}}_{1+n+{\mathrm{m}}(\xi_\sigma(\rho_\sigma(q)))}\subset {\mathcal{S}}_{{n+{\mathrm{m}}(\xi_\sigma(\rho_\sigma(q)))}}$. Multiplying this inclusion by $f_{\rho_\sigma(q),\sigma}^{(1)}$, and using , we have $$f_{\rho_\sigma(q),\sigma}^{(1)}\cdot\stackrel{\circ}{{\mathcal{A}}}_{Na(\rho_\sigma(q),\sigma,n+1)}\subset
f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}.$$ Then, from and , we get that $A_{\mu}(\phi_{t,\mu})$ lies in $\big(f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}\big)\times {\mathbb{R}}^{r-1}$.
The lemma follows from , , and the three previous cases by the convexity of $(f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)})\times {\mathbb{R}}_+^{r-1},$ a product of convex sets.
Given $\sigma\in S_r$, define $\widetilde{\sigma}\in S_r$ by $\widetilde{\sigma}(1):=\sigma(r)$, and $\widetilde{\sigma}(j):=\sigma(j-1)$ for each $j\in\{2,\dots,r\}$. Recall the set of integers $$B_\sigma:=\left\{1\leq t\leq r+1 \ \left| \ {\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\cdot\xi_\sigma(t)\big) \equiv
{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+{\mathrm{m}}(\xi_\sigma(t)) \right.\right\}$$ defined in Lemma \[lema42\]. From , for each $t\in\{1,\dots,r\}$, there exists $\kappa_{t,\sigma}'\in{\mathbb{Z}}$ such that $$\label{ktsigma'}
{\mathrm{m}}\big(\xi_\sigma(t)\varepsilon_{\sigma(r)}^{(1)}\big)=\begin{cases} {\mathrm{m}}(\xi_\sigma(t))+{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\kappa_{t,\sigma}'N \
&\mathrm{if} \ t\in B_\sigma\\
{\mathrm{m}}(\xi_\sigma(t))+{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\kappa_{t,\sigma}'N-1 \ &\mathrm{if} \ t\in B_\sigma^c\end{cases}.$$ On the other hand, and the definition of $\widetilde{\sigma}$ imply that $$2\pi d_{t+1,\widetilde{\sigma}}-2\pi d_{t,\sigma}+\arg\!\big(\varepsilon_{\sigma(r)}^{(1)}\big)
+\frac{2\pi}{N}{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))-\frac{2\pi}{N}{\mathrm{m}}(\xi_\sigma(t)) \ \in \ \left(-\frac{2\pi}{N} \ , \ \frac{2\pi}{N}\right)$$ for any $t\in\{1,\dots,r\}$. Multiplying the last expression by $-N/2\pi$, we conclude that $$-Nd_{t+1,\widetilde{\sigma}}+Nd_{t,\sigma}-\frac{N}{2\pi}\arg\!\big(\varepsilon_{\sigma(r)}^{(1)}\big)-{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))
+{\mathrm{m}}(\xi_\sigma(t)) \ \in \ \left(-1 \ , \ 1\right).$$ Using , the ceiling function $\lceil \ \rceil$, and dividing by $N$, we get $$d_{t+1,\widetilde{\sigma}}-d_{t,\sigma}-\frac{1}{N}{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\frac{1}{N}{\mathrm{m}}(\xi_{\widetilde{\sigma}}(t+1))
-\frac{1}{N}{\mathrm{m}}(\xi_\sigma(t)) \ \in \ \left\{0,-\frac{1}{N}\right\}.$$ Since $N\geq 3$, Lemma \[lema42\] (i) and imply that $$\label{revelacion}
d_{t+1,\widetilde{\sigma}}-d_{t,\sigma}+\kappa_{t,\sigma}'=0 \qquad\qquad (1\leq t\leq r, \ \sigma\in S_r).$$
\[antesdeEscher\] Let $\mu=(\sigma,q,n)\in \widetilde{S}_r$ be such that $\rho_\sigma(q)\not=r+1$, and define $\widetilde{\sigma}\in S_r$ by $\widetilde{\sigma}(1):=\sigma(r)$, and $\widetilde{\sigma}(j):=\sigma(j-1)$ for each $j\in\{2,\dots,r\}$. Then there exist $\kappa_{\mu}\in {\mathbb{Z}}$, $\widetilde{q}\in\{1,\dots,r+1\}$, and $\widetilde{n}\in\{0,\dots,N-1\}$, such that we have $$\begin{aligned}
\label{caso1escher}
&\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}=\phi_{t+1,\widetilde{\mu}}\in D_{\widetilde{\mu}} \qquad\qquad (1\le t\le r),\\
\label{caso2escher}
&\phi_{r+2,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}=\phi_{r+2,\widetilde{\mu}}\in D_{\widetilde{\mu}},\end{aligned}$$ with $\widetilde{\mu}:=(\widetilde{\sigma},\widetilde{q},\widetilde{n})\in \widetilde{S}_r$.
Let $\widetilde{q}:=(\rho_{\widetilde{\sigma}})^{-1}\big(1+\rho_\sigma(q)\big)$. We will divide the proof into two cases according to whether $\rho_\sigma(q)\in B_\sigma$ or not.
**Case 1.** Suppose $\rho_\sigma(q)\in B_\sigma$. Choose $\kappa_{\mu}\in{\mathbb{Z}}$ so that $n-{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\kappa_{\mu} N$ lies in $\{0,\dots,N-1\}$. Let $\widetilde{n}:=n-{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\kappa_{\mu} N$, and $\widetilde{\mu}:=(\widetilde{\sigma},\widetilde{q},\widetilde{n})\in \widetilde{S}_r$. From Definition \[phi\], and , we have for each $t\in\{1,\dots,r\}$ that $$\label{58}
a(t+1,\widetilde{\sigma},\widetilde{n})-a(t,\sigma,n)
=\begin{cases} \kappa_{\mu} \ &\mathrm{if} \ t\in B_\sigma,\\
\kappa_{\mu}-1/N \ &\mathrm{if} \ t\in B_\sigma^c.\end{cases}$$ To prove in this case, fix $t\in\{1,\dots,r\}$.
\(i) If $t\in B_\sigma$ and $t\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (ii), and imply $$\label{59}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n}+1)e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}$$ (recall $a(t,\sigma,n+1)=a(t,\sigma,n)+1/N$ and $a(t+1,\widetilde{\sigma},\widetilde{n}+1)=a(t+1,\widetilde{\sigma},\widetilde{n})+1/N$).
\(ii) If $t\in B_\sigma$ and $t\not\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (ii), and imply $$\label{60}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n})e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$
\(iii) If $t\in B_\sigma^c$ and $t\not\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (iii), and imply $$\label{61}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n}+1)e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$ Note that Lemma \[lema42\] (iii) implies that $t$ can only satisfy one of the above three assumptions. Hence follows from , and .
Let us prove . Using Definition \[phi\], , the definition of $\widetilde{q}$, and , $$\phi_{r+2,\mu}+e_{\sigma(r)}+\kappa_\mu e_{r+1}
=\phi_{\rho_\sigma(q)+1,\widetilde{\mu}}+(1/N)e_{r+1}
=\phi_{r+2,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$ Thus, we have proved Lemma \[antesdeEscher\] when $\rho_\sigma(q)\in B_\sigma$.
**Case 2.** Suppose $\rho_\sigma(q)\in B_\sigma^c$. Choose $\kappa_{\mu}\in{\mathbb{Z}}$ so that $n-{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+1+\kappa_{\mu} N$ lies in $\{0,\dots,N-1\}$. Note that the definition of $\kappa_{\mu}$ here differs from the one given in case 1. Let $\widetilde{n}:=n+1-{\mathrm{m}}\big(\varepsilon_{\sigma(r)}^{(1)}\big)+\kappa_{\mu} N$, and $\widetilde{\mu}:=(\widetilde{\sigma},\widetilde{q},\widetilde{n})$. Then, Definition \[phi\], and imply for each $t\in\{1,\dots,r\}$ that $$\label{alturacaso2}
a(t+1,\widetilde{\sigma},\widetilde{n})-a(t,\sigma,n)=\begin{cases} \kappa_{\mu}+1/N \ &\mathrm{if} \ t\in B_\sigma\\
\kappa_{\mu} \ &\mathrm{if} \ t\in B_\sigma^c\end{cases}.$$ To prove , fix $t\in\{1,\dots,r\}$.
\(i) If $t\in B_\sigma$ and $t\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (iii), and imply $$\label{59caso2}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n})e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$
\(ii) If $t\in B_\sigma^c$ and $t\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (ii), and imply $$\label{60caso2}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n}+1)e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$
\(iii) If $t\in B_\sigma^c$ and $t\not\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , Lemma \[lema42\] (ii), and imply $$\label{61caso2}
\phi_{t,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\sum_{i=1}^{t+1}e_{\widetilde{\sigma}(i-1)}+a(t+1,\widetilde{\sigma},\widetilde{n})e_{r+1}
=\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}.$$ Again, Lemma \[lema42\] (iii) implies that $t$ can only satisfy one of the above three assumptions. Hence, follows from , and .
Finally, using Definition \[phi\], , the definition of $\widetilde{q}$, and , $$\phi_{r+2,\mu}+e_{\sigma(r)}+\kappa_{\mu}e_{r+1}
=\phi_{\rho_\sigma(q)+1,\widetilde{\mu}}+(1/N)e_{r+1}
=\phi_{r+2,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}},$$ which finishes the proof of Lemma \[antesdeEscher\].
We now construct our piecewise affine map $f$ with domain $D:=\cup_{\mu\in \widetilde{S}_r}D_\mu$.
\[Escher\] There exists a continuous map $f:D\to {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ with the following properties:
1. If $x\in D_{\mu}$, then $f(x)=A_{\mu}(x)$, where $A_\mu$ was defined in .
2. If $x\in D$ and $x+e_{r+1}\in D$, then $f(x+e_{r+1})=f(x)$.
3. If $x\in D$ and $x+e_j+\beta e_{r+1}\in D$ for some standard basis vector $e_j$ of ${\mathbb{R}}^{r+1}$ distinct from $e_{r+1}$, and some $\beta\in{\mathbb{Z}}$, then $$f(x+e_j+\beta e_{r+1})=\widetilde{\varepsilon}_j\cdot f(x) \qquad\qquad (1\leq j\leq r).$$
To prove the existence of $f$ and (i), we only need to show that if $x\in D_{\mu}\cap D_{\mu'}$, then $A_{\mu}(x)=A_{\mu'}(x)$. Suppose $v=\phi_{t,\mu}$ ($1\leq t\leq r+2$) is a vertex of $D_{\mu}$. Then, using Definition \[phi\], , , and , we have $$\label{verticeind}
A_{\mu}(v)=\ell\big(\alpha(Nv^{(r+1)})\big)\cdot\prod_{j=1}^r\widetilde{\varepsilon}_j^{v^{(j)}}.$$ Since the last expression is independent of $\mu$, we have $A_{\mu}(v)=A_{\mu'}(v)$ whenever $v$ is a vertex of $D_{\mu}$ and of $D_{\mu'}$. But Proposition \[intersecciondesimplices\] implies that $D_{\mu'}\cap D_{\mu}$ is a $d$-simplex (for some $1\leq d\leq r$) whose $d+1$ vertices are also vertices of $ D_{\mu}$ and of $D_{\mu'}$. An affine map on a $d$-simplex is uniquely determined by its values on the $d+1$ vertices, so $A_{\mu}(x)=A_{\mu'}(x)$ for all $x\in D_{\mu'}\cap D_{\mu}$.
To prove (ii), note that shows us that $x\in P_{1}(\sigma,r+1,0)\subset D(\sigma,r+1,0)$ for some $\sigma\in S_r$. If we write $x$ in its barycentric coordinates with respect to the vertices of $P_{1}(\sigma,r+1,0)$, $x=\sum_{t=1}^{r+1}b_t\phi(t,\sigma,r+1,0)$, $b_t\geq 0$, $\sum_{t=1}^{r+1}b_t=1$, then using Definition \[phi\], $$\begin{aligned}
e_{r+1}+x &= \sum_{t=1}^{r+1}b_t\big(e_{r+1}+\phi(t,\sigma,r+1,0)\big)\\
&= b_{\rho_\sigma(1)}\phi(r+2,\sigma,1,N-1)+\sum_{\substack{1\leq t\leq r+1\\ t\not=\rho_\sigma(1)}}b_t\phi(t,\sigma,1,N-1) \ \in \ D(\sigma,1,N-1),\end{aligned}$$ where the last equality follows from . Hence, using (i), , , and , $$\begin{aligned}
f(x) &= A(\sigma,r+1,0)(x) = \sum_{t=1}^{r+1}b_t\varphi(t,\sigma,r+1,0)\\
&= b_{\rho_\sigma(1)}\varphi(\rho_\sigma(1),\sigma,r+1,0)+\sum_{\substack{1\leq t\leq r+1\\t\not=\rho_\sigma(1)}}b_t\varphi(t,\sigma,r+1,0)\\
&= b_{\rho_\sigma(1)}\varphi(r+2,\sigma,1,N-1)+\sum_{\substack{1\leq t\leq r+1\\t\not=\rho_\sigma(1)}}b_t\varphi(t,\sigma,1,N-1)\\
&= A(\sigma,1,N-1)(x+e_{r+1})=f(x+e_{r+1}).\end{aligned}$$
Now let us prove (iii). Since $x\in D$ and $x+e_j+\beta e_{r+1}\in D$, we have that $x\in D_{\mu}$ for some $\mu=(\sigma,q,n)\in \widetilde{S}_r$, where we can suppose $\sigma(r)=j$ because $x^{(j)}=0$ (see Definition \[Dsigmaqn\]). Writing $x$ in barycentric coordinates with respect to $D_{\mu}$: $x=\sum_{t=1}^{r+2}b_t\phi_{t,\mu}$, $b_t\geq 0$, $\sum_{t=1}^{r+2}b_t=1$, note that $$0=x^{(j)}=x^{(\sigma(r))}=xe_{\sigma(r)}^T=\begin{cases} b_{r+2}+b_{r+1} \ &\mathrm{if} \ \rho_\sigma(q)=r+1\\
b_{r+1} \ &\mathrm{if} \ \rho_\sigma(q)\not=r+1\end{cases}.$$ We will divide the proof into two cases, depending on whether $\rho_\sigma(q)=r+1$ or not. First suppose that $\rho_\sigma(q)\not=r+1$. Since $\sum_{t=1}^{r+2}b_t=1$ and $b_{r+1}=0$, Lemma \[antesdeEscher\] implies that there exist $\kappa_{\mu}\in {\mathbb{Z}}$ and $\widetilde{\mu}=(\widetilde{\sigma},\widetilde{q},\widetilde{n})\in \widetilde{S}_r$ such that $$\begin{aligned}
x+e_j+\kappa_{\mu}e_{r+1} &= b_{r+2}(\phi_{r+2,\mu}+e_j+\kappa_{\mu}e_{r+1}) +\sum_{t=1}^rb_t(\phi_{t,\mu}+e_j+\kappa_{\mu}e_{r+1}) \\
&=b_{r+2}\phi_{r+2,\widetilde{\mu}} +\sum_{t=1}^rb_t\phi_{t+1,\widetilde{\mu}} \ \in \ D_{\widetilde{\mu}}\subset D,\end{aligned}$$ where $\widetilde{\sigma}(1):=\sigma(r)$ and $\widetilde{\sigma}(j):=\sigma(j-1)$ ($2\leq j\leq r$). Moreover, since $x+e_j+\beta e_{r+1}\in D$, Definition implies that $\kappa_\mu\in\{\beta-1,\,\beta,\,\beta+1\}$. Therefore, using (ii), $$\label{betaykappa}
f(x+e_j+\beta e_{r+1})=f(x+e_j+\kappa_{\mu}e_{r+1}).$$ Then, (i), , and show that $$\label{Escher2}
f(x+e_j+\kappa_{\mu}e_{r+1})=A_{\widetilde{\mu}}(x+e_j+\kappa_{\mu}e_{r+1})= b_{r+2}\varphi_{r+2,\widetilde{\mu}}+\sum_{t=1}^rb_t\varphi_{t+1,\widetilde{\mu}}.$$ On the other hand, putting $v=\phi_{r+2,\mu}$, we can use to compute $$\begin{aligned}
\nonumber
&\varphi_{r+2,\widetilde{\mu}}=:A_{\widetilde{\mu}}(\phi_{r+2,\widetilde{\mu}})=A_{\widetilde{\mu}}(\phi_{r+2,\mu}+e_j+\kappa_\mu e_{r+1})\\
\nonumber
&=\ell\big(\alpha(Nv^{(r+1)}+N\kappa_\mu)\big)\cdot\prod_{1\leq i\leq r}\widetilde{\varepsilon}_i^{(v^{(i)}+e_j^{(i)})}=\widetilde{\varepsilon}_j\cdot\ell\big(\alpha(Nv^{(r+1)})\big)\cdot\prod_{1\leq
i\leq r}\widetilde{\varepsilon}_i^{v^{(i)}}\\
&=\widetilde{\varepsilon}_j\cdot\varphi_{r+2,\mu}=\widetilde{\varepsilon}_j\cdot A_\mu({\phi_{r+2,\mu}}).\label{Escher3}\end{aligned}$$ Analogously, putting $w=\phi_{t,\mu}$ ($1\leq t\leq r$), $$\begin{aligned}
\nonumber
&\varphi_{t+1,\widetilde{\mu}}=:A_{\widetilde{\mu}}(\phi_{t+1,\widetilde{\mu}})=A_{\widetilde{\mu}}(\phi_{t,\mu}+e_j+\kappa_\mu e_{r+1})\\
\nonumber
&=\ell\big(\alpha(Nw^{(r+1)}+N\kappa_\mu)\big)\cdot\prod_{1\leq i\leq r}\widetilde{\varepsilon}_i^{(w^{(i)}+e_j^{(i)})}=\widetilde{\varepsilon}_j\cdot\ell\big(\alpha(Nw^{(r+1)})\big)\cdot\prod_{1\leq
i\leq r}\widetilde{\varepsilon}_i^{w^{(i)}}\\
&=\widetilde{\varepsilon}_j\cdot\varphi_{t,\mu}=\widetilde{\varepsilon}_j\cdot A_\mu({\phi_{t,\mu}}) \qquad (1\leq t\leq r).\label{Escher4}\end{aligned}$$ Thus, , , , and (i) imply that the right-hand side of equals $$\begin{aligned}
\widetilde{\varepsilon}_j\cdot \Big(b_{r+2}A_\mu({\phi_{r+2,\mu}})+\sum_{t=1}^rb_tA_\mu({\phi_{t,\mu}})\Big)=\widetilde{\varepsilon}_j\cdot
A_\mu(x)=\widetilde{\varepsilon}_j\cdot f(x).\end{aligned}$$ This, together with proves Proposition \[Escher\] in this case.
Now suppose that $\rho_\sigma(q)=r+1$. Then, $q=r+1$ since $\rho_\sigma(r+~1)=r+1$ (see ). Note that $b_{r+2}=b_{r+1}=0$ since $b_{r+2}+b_{r+1}=0$ and $b_{r+2}, b_{r+1}\geq 0$. Here, if $x\in D_{\mu}=D(\sigma,r+1,n)$, then $x\in D(\sigma,r,n)$ since we have $\phi(t,\sigma,r+1,n)=\phi(t,\sigma,r,n)$ ($1\leq t\leq r$) by . Therefore, the proof reduces to the case $\rho_\sigma(q)\not=r+1$.
Proof of Theorem \[Main\]
=========================
Maps descending to tori
-----------------------
Now we define the maps $F$ and $F_0$ mentioned at the beginning of section \[constructionoff\]. Recall the $(r+1)$-torus $\widehat{T}$, defined in by identifying the elements of $$D:=\bigcup_{\sigma\in S_r}\left\{x\in {\mathbb{R}}^{r+1} \left|\begin{array}{c}
1\geq x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))}\geq 0,\\
y(\sigma,r+1,0)(x)\leq x^{(r+1)}\leq 1+y(\sigma,r+1,0)(x).
\end{array}\right.\right\}$$ that lie in the same orbit with respect to the action on ${\mathbb{R}}^{r+1}$ of the subgroup ${\mathbb{Z}}^{r+1}$. Also recall that the set $$\mathfrak{D}:=\bigcup_{\sigma\in S_r}\left\{x\in {\mathbb{R}}^{r+1} \left|\begin{array}{c}
1> x^{(\sigma(1))}\geq x^{(\sigma(2))}\geq\dots\geq x^{(\sigma(r))}\geq 0,\\
y(\sigma,r+1,0)(x)\leq x^{(r+1)}< 1+y(\sigma,r+1,0)(x).
\end{array}\right.\right\}$$ defined in is a fundamental domain for this action. Proposition \[Escher\] means that the piecewise affine map $f:D\to{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ descends to a continuous map $F$ between $\widehat{T}$ and the quotient $\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ coming from the action of $\widetilde{V}:=\langle \widetilde{\varepsilon}_1,\dots,\widetilde{\varepsilon}_r\rangle$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ by component-wise multiplication. More precisely, $F:\widehat{T}\to \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ is defined by the commutative diagram $$\label{diagramaF}
\begin{CD}
D @>f>> {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\\
@VV\widehat{\pi}V @VV\pi V\\
\widehat{T} @>F>{\phantom{\simeq}}> \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}
\end{CD},$$ where $\widehat{\pi}$ and $\pi$ are the natural quotient maps, and $f$ was defined in Proposition \[Escher\].
There is another function between $\widehat{T}$ and $\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ that naturally comes from a function on $D$. We define $f_0:D\to {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ by $$\label{f0}
\big(f_0(x)\big)^{(j)}:=\begin{cases} \big(\widetilde{\varepsilon}_1^{(1)}\big)^{x^{(1)}}\dots \big(\widetilde{\varepsilon}_r^{(1)}\big)^{x^{(r)}}\cdot{\mathrm{exp}}(2\pi i
x^{(r+1)}) \ &\mathrm{if} \ j=1,\\
\big(\widetilde{\varepsilon}_1^{(j)}\big)^{x^{(1)}}\dots \big(\widetilde{\varepsilon}_r^{(j)}\big)^{x^{(r)}} \ &\mathrm{if} \ 2\leq j\leq r,\end{cases}$$ for all $x=(x^{(1)},\dots,x^{(r+1)})\in D$, where powers are defined using the branch of the argument in $[-\pi,\pi)$. From , it is clear that $f_0$ is a continuous map that satisfies properties (ii) and (iii) of Proposition \[Escher\]. Thus, $f_0$ descends to a continuous map $F_0:\widehat{T}\to \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ defined by the commutative diagram $$\label{diagramaF0}
\begin{CD}
D @>f_0 >> {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\\
@VV\widehat{\pi}V @VV\pi V\\
\widehat{T} @>F_0>{\phantom{\simeq}}> \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}
\end{CD},$$ where $\widehat{\pi}$ and $\pi$ are again the natural quotient maps, and $f_0$ was defined in .
Let us write, using upper and lower case to distinguish the slightly different domains, $$\begin{aligned}
\label{LOG}
&{\mathrm{LOG}}:{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\to {\mathbb{R}}^r,\quad\big({\mathrm{LOG}}(x)\big)^{(j)}:=\log|x^{(j)}| \qquad (1\leq j\leq r),\\
\label{Log}
&{\mathrm{Log}}:{{\mathbb{C}}^*\times{\mathbb{R}}_+^r}\to {\mathbb{R}}^r,\quad\big({\mathrm{Log}}(x)\big)^{(j)}:=\log|x^{(j)}| \qquad (1\leq j\leq r).\end{aligned}$$ Note that the function in is used in the sixth step of the 7SA.
\[reguladorconsigno\] Let ${\mathrm{LOG}}$ and ${\mathrm{Log}}$ be the functions defined respectively in and . Then $$\label{lema48}
\det\!\big({\mathrm{LOG}}(\widetilde{\varepsilon}_1),\dots,{\mathrm{LOG}}(\widetilde{\varepsilon}_r)\big)=(r+2)\det\!\big({\mathrm{Log}}(\varepsilon_1),\dots,{\mathrm{Log}}(\varepsilon_r)\big).$$ In particular, none of the determinants in vanish, both have the same sign, and $\Lambda:=\sum_{j=1}^r{\mathbb{Z}}\cdot{\mathrm{LOG}}(\widetilde{\varepsilon}_j)\subset{\mathbb{R}}^r$ is a full lattice.
We follow [@DF1 Lemma 19]. Using the identity $|\varepsilon_j^{(1)}|^2\cdot\prod_{i=2}^{r}\varepsilon_j^{(i)}=1/\varepsilon_j^{(r+1)}$ $(1\leq j\leq r)$, reduces to showing $r+2=\det(I_r+B_r)$, where the $r\times r$ matrices $I_r$ and $B_r$ are, respectively, the identity and the matrix whose first column has only entries equal to 2 and all the other entries are 1’s. But $\det(\lambda I_r-B_r)=\lambda^{r-1}\big(\lambda-(r+1)\big)$, using the eigenvalues 0 and $r+1$ of $B_r$. Setting $\lambda=-1$ concludes the proof.
We will soon show that $f_0(\mathfrak{D})$ is a fundamental domain for the action of $\widetilde{V}$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$, but we first make some geometric remarks. If $\ell(\varepsilon)=\widetilde{\varepsilon}\in \widetilde{V}$ satisfies $|\widetilde{\varepsilon}^{(j)}|=1$ for all $1\leq j\leq r$ (recall $r>0$), then $|\varepsilon^{(1)}|=\varepsilon^{(2)}=\dots=\varepsilon^{(r+1)}$. Since $1=|N_{k/{\mathbb{Q}}}(\varepsilon)|$, we have then also $|\varepsilon^{(r+1)}|=1$. Since $\varepsilon^{(r+1)}>0$, we see that $$\label{unidadnormauno}
\varepsilon\in V, \quad \ell(\varepsilon)=\widetilde{\varepsilon}, \quad |\widetilde{\varepsilon}^{(j)}|=1 \quad (1\leq j\leq r) \qquad \Longrightarrow \qquad
\varepsilon=1.$$
If $\nu:{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\to {\mathbb{R}}_+^r$ is defined by the formula $\big(\nu(x)\big)^{(j)}:=|x^{(j)}|$ $(1\leq j\leq r)$, then Lemma \[reguladorconsigno\] implies that $(\nu\circ f_0)(\mathfrak{D})$ is a fundamental domain for the action of $\nu(\widetilde{V})$ on ${\mathbb{R}}_+^r$. Since the exponential ${\mathrm{exp}}(2\pi i x^{(r+1)})$ in , restricted to $\mathfrak{D}$, runs over the unit circle exactly once, it is clear that each $x\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ is in the orbit of some $\widetilde{x}\in f_0(\mathfrak{D})$ under the action of $\widetilde{V}$. Furthermore, if there are two such elements $\widetilde{x}, \widetilde{y}\in f_0(\mathfrak{D})$, then $\nu(\widetilde{x}) \ \mathrm{and} \ \nu(\widetilde{y})\in (\nu\circ f_0)(\mathfrak{D})$ belong to the same orbit under the action of $\nu(\widetilde{V})$, which implies that $\nu(\widetilde{x})=\nu(\widetilde{y})$. But this means that $\widetilde{x}^{(j)}=\widetilde{y}^{(j)}$ for $j\geq 2$, and $|\widetilde{x}^{(1)}|=|\widetilde{y}^{(1)}|$, which implies $x=y$ (see ). Therefore, $f_0(\mathfrak{D})$ is a fundamental domain of ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ under the action of $\widetilde{V}$, and $F_0:\widehat{T}\to \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ is surjective.
We now prove that $F_0$ is a homeomorphism. If $f_1:{\mathbb{R}}^r\to {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ is defined by $$\big(f_1(x)\big)^{(j)}:=\big(\widetilde{\varepsilon}_1^{(j)}\big)^{x^{(1)}}\dots \big(\widetilde{\varepsilon}_r^{(j)}\big)^{x^{(r)}} \qquad (1\leq j\leq r),$$ then the composition $({\mathrm{LOG}}\circ f_1):{\mathbb{R}}^r\to {\mathbb{R}}^r$ is a homeomorphism that satisfies $$\label{f1inyectiva}
({\mathrm{LOG}}\circ f_1)(x)=({\mathrm{LOG}}\circ f_0)(x,b) \qquad \big(x\in[0,1]^{r}, \ b\in{\mathbb{R}}, \ (x,b)\in D\big).$$ Hence, definition and imply that $f_0$ is injective on $\mathfrak{D}$. Now take two elements $x,y\in\mathfrak{D}$, and denote by $[x], [y]$ their respective cosets in $\widehat{T}$. If $F_0([x])=F_0([y])$, using we have $f_0(x)=u\cdot f_0(y)$ for some $u\in \widetilde{V}$. But $f_0(y)$ and $u\cdot f_0(y)$ lie in $f_0(\mathfrak{D})$, which implies that $u=1$ since $f_0(\mathfrak{D})$ is a fundamental domain for the action of $\widetilde{V}$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$. Hence, $x=y$ since $f_0$ is injective on $\mathfrak{D}$. Therefore, $F_0:\widehat{T}\to \big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ is a continuous bijective map on a compact set. So $F_0$ is a homeomorphism, and the quotient $$\label{toro}
T:=\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$$ is an $(r+1)$-torus.
Now let us prove a result for $f_0$ analogous to Lemma \[afindentrosemiplano\], which will allow us to define a homotopy between $F$ and $F_0$. Recall that $$\big(f_0(x)\big)^{(1)}=\big|\widetilde{\varepsilon}_1^{(1)}\big|^{x^{(1)}}\dots \big|\widetilde{\varepsilon}_r^{(1)}\big|^{x^{(r)}}\cdot{\mathrm{exp}}\big(\omega(x)i\big)\qquad (x\in D),$$ where $\omega:{\mathbb{R}}^{r+1}\to {\mathbb{R}}$ is the ${\mathbb{R}}$-linear map $$\label{omega}
\omega(x):=2\pi x^{(r+1)}+\sum_{j=1}^{r}\arg\!\big(\widetilde{\varepsilon}_j^{(1)}\big)x^{(j)}.$$
\[f0dentrosemiplano\] Let $\mu=(\sigma,q,n)\in \widetilde{S}_r$. Then $$f_0(D_{\mu}) \ \subset \ \big(f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}\big)\times {\mathbb{R}}_+^{r-1} \ \subset \ {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}.$$
Put $\theta_{t,\sigma}:=\arg\!\Big(\xi_\sigma(t)\cdot{\mathrm{exp}}\big(2\pi i{\mathrm{m}}(\xi_\sigma(t))/N\big)\Big)$. We claim $$\label{claim}
t\prec_\sigma t' \quad \Longrightarrow \quad \theta_{t,\sigma}\le\theta_{t',\sigma}\qquad (\text{for all $t, t'\in\{1,\dots,r+1\}$ and $\sigma\in S_r$}).$$ For the sake of contradiction, suppose $t\prec_\sigma t'$ and $\theta_{t,\sigma}>\theta_{t',\sigma}$. From , there exists $q\in{\mathbb{Z}}$ such that $$\frac{-N}{2\pi}(\theta_{t',\sigma}-\theta_{t,\sigma})=\frac{-N\arg(\xi_\sigma(t,t'))}{2\pi}-{\mathrm{m}}(\xi_\sigma(t'))+{\mathrm{m}}(\xi_\sigma(t))+Nq \ \in \ (0, 1).$$ Evaluating the ceiling function at the last expression we contradict condition , and so we have proved .
In proving Lemma \[f0dentrosemiplano\], implies that we have only to worry about the first coordinate of the elements in $f_0(D_{\mu})$. We only have to study three cases for $\omega(\phi_{t,\mu})$ with $t\in\{1,\dots,r+2\}$.
*Case 1*. If $1\leq t\leq r+1$ with $t\prec_\sigma\rho_\sigma(q)$, then Definition \[phi\], , and imply $$\omega(\phi_{t,\mu})- \omega(\phi_{\rho_\sigma(q),\mu})=\theta_{t,\sigma}-\theta_{\rho_\sigma(q),\sigma}+2\pi/N \ \in \ (0,2\pi/N]$$ since $\arg\!\big(\varepsilon_{j}^{(1)}\big)=\arg\!\big(\widetilde{\varepsilon}_{j}^{(1)}\big)$ for all $1\leq j\leq r$.
*Case 2*. If $1\leq t\leq r+1$ with $\rho_\sigma(q)\prec_\sigma t$, then Definition \[phi\], , and imply $$\omega(\phi_{t,\mu})- \omega(\phi_{\rho_\sigma(q),\mu})=\theta_{t,\sigma}-\theta_{\rho_\sigma(q),\sigma} \ \in \ [0,2\pi/N)$$
*Case 3*. If $t=\rho_\sigma(q)$ or $t=r+2$, $$\omega(\phi_{t,\mu})- \omega(\phi_{\rho_\sigma(q),\mu}) \ \in \ \{0, 2\pi /N\}.$$
Therefore, using the linearity of $\omega$, and the convexity of $D_{\mu}$ and $[0,2\pi/N]$, the above three cases allow us to claim that $\omega(D_{\mu})$ is contained in $$[\omega(\phi_{\rho_\sigma(q),\mu}) \ , \ \omega(\phi_{\rho_\sigma(q),\mu})+2\pi/N]
\ \subset \ [\omega(\phi_{\rho_\sigma(q),\mu})-\pi/2N \ , \ \omega(\phi_{\rho_\sigma(q),\mu})+5\pi/2N).$$ Thus, the proof follows from definition , and from the identity $$\omega(\phi_{\rho_\sigma(q),\mu})=2\pi a(\rho_\sigma(q),\sigma,n)+\sum_{j=1}^{\rho_\sigma(q)-1}\arg\!\big(\widetilde{\varepsilon}_{\sigma(j)}^{(1)}\big).$$
The next lemma summarizes the properties of $F$ and $F_0$ that we shall use later.
\[homotopicas\] Let $F,F_0:\widehat{T}\to T$ be the functions defined by the diagrams and , where $T:=\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$. Then $F$ is homotopic to $F_0$, and $F_0$ is a homeomorphism between the $(r+1)$-tori $\widehat{T}$ and $T$.
By the discussion following Lemma \[reguladorconsigno\], we have only to show that $F$ and $F_0$ are homotopic. For $x\in D$ and $\lambda\in [0,1]$, consider $f_\lambda(x):=\lambda f(x)+(1-\lambda)f_0(x)\in {\mathbb{C}}\times {\mathbb{R}}_+^{r-1}$. Since $x\in D$, there exists $\mu=(\sigma,q,n)\in \widetilde{S}_r$ such that $x\in D_\mu$. Using Lemma \[afindentrosemiplano\] and Lemma \[f0dentrosemiplano\], we have that $f(x)^{(1)}$ and $f_0(x)^{(1)}$ lie in $f_{\rho_\sigma(q),\sigma}^{(1)}\cdot{\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}\subset {\mathbb{C}}^*$. But ${\mathcal{S}}_{Na(\rho_\sigma(q),\sigma,n)}$ is a convex set, so $$f_\lambda(x):=\lambda f(x)+(1-\lambda)f_0(x)\subset {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}.$$ Hence, we can define $f_\lambda:D\to {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ by $f_\lambda(x)=\lambda f(x)+(1-\lambda)f_0(x)$. Clearly, $(\lambda,x)\mapsto f_\lambda(x)$ is continuous.
Suppose $x\in D$ and $x+e_{r+1}\in D$. Then, using Lemma \[Escher\] (ii) and , we have $$f_\lambda(x+e_{r+1})=(1-\lambda)f_0(x+e_{r+1})+\lambda f(x+e_{r+1})
=(1-\lambda)f_0(x)+\lambda f(x)
=f_\lambda(x).$$
Now suppose $x\in D$, and $x+e_j+\beta e_{r+1}\in D$ for some standard basis vector $e_j$ of ${\mathbb{R}}^{r+1}$ distinct from $e_{r+1}$, and some $\beta\in{\mathbb{Z}}$. Then, using Lemma \[Escher\] (iii) and , we have $$\begin{aligned}
f_\lambda(x+e_j+\beta e_{r+1})&=(1-\lambda)f_0(x+e_j+\beta e_{r+1})+\lambda f(x+e_j+\beta e_{r+1})\\
&=(1-\lambda)\widetilde{\varepsilon}_j f_0(x)+\lambda \widetilde{\varepsilon}_j f(x)=\widetilde{\varepsilon}_j f_\lambda(x).\end{aligned}$$ Therefore, $f_\lambda$ descends to a homotopy $F_\lambda:\widehat{T}\to T$ between $F_0$ and $F$.
We end this section with some computations which we will need when we determine the local and global degrees of $F$ and $F_0$.
Consider ${\mathbb{C}}\times{\mathbb{R}}^{r-1}={\mathbb{R}}^{r+1}$ as a real vector space. For $\mu\in \widetilde{S}_r$, let $L_\mu:{\mathbb{R}}^{r+1}\to {\mathbb{R}}^{r+1}$ be the linear part of the affine map $A_\mu$ defined in . That is, $L_\mu$ is the unique ${\mathbb{R}}$-linear map such that $A_\mu-L_\mu$ is constant. Then, $$\label{detafin}
\mathrm{sign}\big(\!\det(L_{\mu})\big)=(-1)^{r+1}\mathrm{sgn}(\sigma)
\cdot\mathrm{sign}\big(\!\det(f_{1,\mu},f_{2,\mu},
\dots,f_{r+2,\mu})\big),$$ where $\det(L_{\mu})$ is the determinant of $L_\mu$, and $\det(f_{1,\mu},f_{2,\mu},
\dots,f_{r+2,\mu})$ is the determinant of the $(r+2)\times (r+2)$ matrix having columns $f_{i,\mu}$.
On the other hand, if $P$ is an interior point of the set $D$ defined in , then $$\label{detf0}
\mathrm{sign}\Big(\!\det\!\big(\mathrm{d}f_{0_P}\big)\Big)=
(-1)^{r+1}\mathrm{sign}\Big(\!\det\!\big({\mathrm{LOG}}(\widetilde{\varepsilon}_1),\dots,{\mathrm{LOG}}(\widetilde{\varepsilon}_r)\big)\Big),$$ where $\det\!\big(\mathrm{d}[f_0]_P\big)$ is the Jacobian at $P$ of the function $f_0:D\to {\mathbb{R}}^{r+1}$ defined in , ${\mathrm{LOG}}:{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\to{\mathbb{R}}^r$ is defined by $\big({\mathrm{LOG}}(x)\big)^{(j)}:=\log|x^{(j)}|$ ($1\leq j\leq r$), and $\det\!\big({\mathrm{LOG}}(\widetilde{\varepsilon}_1),\dots,{\mathrm{LOG}}(\widetilde{\varepsilon}_r)\big)$ is the determinant of the $r\times r$ matrix having columns ${\mathrm{LOG}}(\widetilde{\varepsilon}_i)$.
First let us prove . We have $$L_\mu(x)=A_\mu(x+\phi_{r+2,\mu})-A_\mu(\phi_{r+2,\mu})\qquad(x\in{\mathbb{R}}^{r+1}).$$ Using Definition , we have $$\label{63}
L_\mu(\phi_{t,\mu}-\phi_{r+2,\mu})=\varphi_{t,\mu}-\varphi_{r+2,\mu}\qquad (1\leq t\leq r+1).$$ Now we compute the values of $L_{\mu}$ on the standard basis $\{e_j\}_{1\leq j\leq r+1}$ of ${\mathbb{R}}^{r+1}$. Since $\phi_{r+2,\mu}-\phi_{\rho_\sigma(q),\mu}=(1/N)e_{r+1}$, putting $t=\rho_\sigma(q)$ in we conclude that $$\label{64}
L_\mu(e_{r+1})=N(\varphi_{r+2,\mu}-\varphi_{\rho_\sigma(q),\mu}).$$ From Definition \[phi\], we have $$e_{\sigma(t)}=(\phi_{t+1,\mu}-\phi_{r+2,\mu})-(\phi_{t,\mu}-\phi_{r+2,\mu})-(\phi_{t+1,\mu}^{(r+1)}-\phi_{t,\mu}^{(r+1)})e_{r+1}\qquad (1\leq t\leq r).$$ Using and we have then $$\label{65}
L_\mu(e_{\sigma(t)})=\varphi_{t+1,\mu}-\varphi_{t,\mu}-N(\phi_{t+1,\mu}^{(r+1)}-\phi_{t,\mu}^{(r+1)})(\varphi_{r+2,\mu}-\varphi_{\rho_\sigma(q),\mu})\qquad(1\leq t\leq r).$$
Let $P_{\overline{\sigma}}:{\mathbb{R}}^{r+1}\to{\mathbb{R}}^{r+1}$ be the linear map determined by $P_{\overline{\sigma}}(e_t):=e_{\overline{\sigma}(t)}$, where $\overline{\sigma}\in S_{r+1}$ is defined by $\overline{\sigma}(r+1):=r+1$, and $\overline{\sigma}(t):=\sigma(t)$ for each $1\leq t\leq r$. Note that $\mathrm{sgn}(\sigma)=\mathrm{sgn}(\overline{\sigma})=\det(P_{\overline{\sigma}})$. We have already proved that $$\label{66}
\mathrm{sgn}(\sigma)\cdot\det(L_\mu)=\det(L_\mu\circ P_{\overline{\sigma}})=\det\!\big(L_\mu(e_{\sigma(1)}),\dots,L_\mu(e_{\sigma(r)}),L_\mu(e_{r+1})\big).$$ By and , we get that the right-hand side of equals $$N\det\!\big(\varphi_{2,\mu}-\varphi_{1,\mu}\,,\,\varphi_{3,\mu}-\varphi_{2,\mu}\,,\,\dots\,,\,\varphi_{r+1,\mu}-\varphi_{r,\mu}\,,\,\varphi_{r+2,\mu}-\varphi_{\rho_\sigma(q),\mu}\big)$$ using elementary column operations. Adding the first column above to the second, then the second to the third, and so on until adding the $(r-1)$-th column to the $r$-th, we find that $\mathrm{sgn}(\sigma)\cdot\det(L_\mu)$ equals $$\begin{aligned}
N\det\!\big(\varphi_{2,\mu}-\varphi_{1,\mu}\,,\,\varphi_{3,\mu}-\varphi_{1,\mu}\,,\,\dots\,,\,\varphi_{r+1,\mu}-\varphi_{1,\mu}\,,\,\varphi_{r+2,\mu}-\varphi_{\rho_\sigma(q),\mu}\big),\end{aligned}$$ Adding the column $\varphi_{\rho_\sigma(q),\mu}-\varphi_{1,\mu}$ above to the last one, we obtain $$\label{67}
\mathrm{sgn}(\sigma)\cdot\det(L_\mu)=
N\det\!\big(\varphi_{2,\mu}-\varphi_{1,\mu}\,,\,\dots\,,\,\varphi_{r+1,\mu}-\varphi_{1,\mu}\,,\,\varphi_{r+2,\mu}-\varphi_{1,\mu}\big).$$ Since $\varphi_\mu:=\ell(f_{t,\mu})\in {\mathbb{C}}\times{\mathbb{R}}^{r-1}={\mathbb{R}}^{r+1}$, the $(r+1)\times(r+1)$ determinant in is related to the $(r+2)\times(r+2)$ determinant in the right-hand side of by the identity $$\mathrm{sign}\big(\det(w_1,\dots,w_{r+2})\big)=
(-1)^{r+1}\mathrm{sign}\Big(\!\det\!\big(\ell(w_2)-\ell(w_1),\dots,\ell(w_{r+2})-\ell(w_{1})\big)\Big),$$ valid for any $w_i\in {\mathbb{C}}\times{\mathbb{R}}^r={\mathbb{R}}^{r+2}$ with $w_i^{(r+1)}>0$ ($1\leq i\leq r+2$). [^6] Combining this with , we get formula .
To prove , consider $\widetilde{f}_0:D\to {\mathbb{R}}^{r+1}$ defined by $$\widetilde{f}_0(x):=\Big(\big|f_0(x)^{(1)}\big| \, , \, \omega(x) \, , \, f_0(x)^{(2)} \, , \, \dots \, , \, f_0(x)^{(r)}\Big) \qquad
(x\in D),$$ where $\omega$ is the ${\mathbb{R}}$-linear map defined in . To compute $\mathrm{sign}\big(\!\det\!(\mathrm{d}[f_0]_P)\big)$, consider the change of coordinates $$\begin{aligned}
&\mathcal{C}:{\mathbb{R}}_+\times\big(\omega(P)-\pi\,,\,\omega(P)+\pi\big)\times{\mathbb{R}}^{r-1}\rightarrow {\mathbb{R}}^{r+1},\\
&\mathcal{C}(R,\vartheta,x^{(1)},\dots,x^{(r-1)}):=(R\cos\vartheta,R\sin\vartheta,x^{(1)},\dots,x^{(r-1)}).\end{aligned}$$ Hence, $f_0=\mathcal{C}\circ \widetilde{f}_0$ in some neighborhood of $P$, and $\det\!\big(\mathrm{d}[\mathcal{C}]_{\widetilde{f}_0(P)}\big)=\big|f_0(P)^{(1)}\big|$. Computing the corresponding partial derivatives, we obtain $$\begin{aligned}
\nonumber
&\det\!\big(\mathrm{d}[f_0]_P\big)=\det\!\big(\mathrm{d}[\mathcal{C}\circ
\widetilde{f}_0]_P\big)=\det\!\big(\mathrm{d}[\mathcal{C}]_{\widetilde{f}_0(P)}\big)\cdot\det\!\big(\mathrm{d}[\widetilde{f}_0]_P\big)=\\
\nonumber
&(-1)^{r+1}2\pi\cdot\det\!\big({\mathrm{LOG}}(\widetilde{\varepsilon}_1),\dots,{\mathrm{LOG}}(\widetilde{\varepsilon}_r)\big)\cdot\big|f_0(P)^{(1)}\big|^2\cdot\prod_{i=2}^{r}f_0(P)^{(i)},\end{aligned}$$ which proves formula .
Degree computations
-------------------
The properties used here concerning topological degree theory are summarized in [@DF1 Proposition 21]. Recall the $(r+1)$-tori $\widehat{T}:=D/\sim$ and $T:=\big({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big)/\widetilde{V}$ defined respectively in and . Also recall the commutative diagrams $$\label{diagramas}
\begin{CD}
D @>f_0 >> {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\\
@VV\widehat{\pi}V @VV\pi V\\
\widehat{T} @>F_0>\simeq> T
\end{CD},
\qquad
\qquad \begin{CD}
D @>f>> {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\\
@VV\widehat{\pi}V @VV\pi V\\
\widehat{T} @>F>{\phantom{\simeq}}> T
\end{CD},$$ defining $F_0$ and $F$. In the following, fix an orientation of the real vector space ${\mathbb{C}}\times{\mathbb{R}}^{r-1}={\mathbb{R}}^{r+1}$, and use it to fix orientations in $\widehat{T}$ and $T$. Since $\widehat{\pi}:D\to \widehat{T}$ restricted to $\stackrel{\circ}{D}$ is a local homeomorphism, and the tori are connected and oriented, we orient $\widehat{T}$ by declaring $\widehat{\pi}$ an orientation-preserving map. Here, the open set $\stackrel{\circ}{D}\,\subset{\mathbb{R}}^{r+1}$ has the induced orientation. Thus, the local degree of $\widehat{\pi}$ at any point of $\stackrel{\circ}{D}$ is $+1$. To orient $T$, give the induced orientation to ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\subset{\mathbb{C}}\times{\mathbb{R}}^{r-1}={\mathbb{R}}^{r+1}$, and orient $T$ by declaring $\pi:{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\to T$ a local homeomorphism of local degree $+1$.
### Global degree {#gradoglobal}
Let $F:\widehat{T}\to T$ be the map defined in . The degree $\deg(F)$ of $F$ is defined since $F$ is a continuous map between compact oriented manifolds. We shall prove that $$\label{signogradoglobal}
\deg(F)=(-1)^{r+1}\mathrm{sign}\big(\det({\mathrm{Log}}\ \varepsilon_1 \ ,\dots, \ {\mathrm{Log}}\ \varepsilon_r)\big).$$ To verify this formula, note that the homotopy in Lemma \[homotopicas\] shows that $\deg(F)=\deg(F_0)$ [@DF1 Proposition 21 (6)]. So we have only to prove that $\deg(F_0)$ is given by the right-hand side of . Since $F_0$ is a homeomorphism between connected manifolds, $\deg(F_0)$ equals the local degree $\mathrm{locdeg}_{\widehat{\pi}(P)}(F_0)$ of $F_0$ at $\widehat{\pi}(P)$ for any $P\in\,\stackrel{\circ}{D}$. Thus, $\deg(F_0)=\mathrm{locdeg}_{\widehat{\pi}(P)}(F_0)$ for all $P\in\,\stackrel{\circ}{D}$. From , we have $ F_0\circ \widehat{\pi}=\pi\circ f_0$, and $f_0$ is a local homeomorphism around $P$. Then for $P$ in the interior $\stackrel{\circ}{D}$ of $D$, $$\deg(F_0)=\mathrm{locdeg}_{\widehat{\pi}(P)}(F_0)={\mathrm{locdeg}}_P(f_0) \qquad (P\in\,\stackrel{\circ}{D})$$ by [@DF1 Proposition 21 (7)] since $\pi$ and $\widehat{\pi}$ has local degree +1. The local degree at $P$ of the local diffeomorphism $f_0$ is given by [@DF1 Proposition 22]. Therefore, follows from .
### Local degree
The local degree of $F:\widehat{T}\to T$ can be easily computed at points where $F$ is a local diffeomorphism. If $x$ is an interior point of the simplex $D_{\mu}$, and $w_{\mu}\not=0$, then the local degree ${\mathrm{locdeg}}_{\widehat{\pi}(x)}(F)$ of $F$ at $\widehat{\pi}(x)$ is defined, and $$\label{gradolocalfacil}
{\mathrm{locdeg}}_{\widehat{\pi}(x)}(F)=v_{\mu}:=(-1)^{r+1}\mathrm{sgn}(\sigma)\cdot\mathrm{sign}\big(\det(f_{1,\mu}, \ \dots, \ f_{r+2,\mu})\big).$$ To verify this formula, using we have $ F\circ \widehat{\pi}=\pi\circ f$. Since $f$ restricted to $D_{\mu}$ is the bijective affine map $A_{\mu}$ whenever $w_{\mu}\not=0$ (see [@DF1 Lemma 15]), it is clear that $f$ is a local diffeomorphism around $x$. But $\widehat{\pi}$ and $\pi$ are local diffeomorphisms of degree $+1$, so $F$ is a local diffeomorphism around $\widehat{\pi}(x)$. Then ${\mathrm{locdeg}}_{\widehat{\pi}(x)}(F)={\mathrm{locdeg}}_x(f)$. Finally, using [@DF1 Proposition 22], we have that follows from .
Preliminary results {#resultadosprevios}
-------------------
The next lemma shows that the vector $[0,0,\dots,0,1]\in{\mathbb{C}}\times{\mathbb{R}}^{r}$ cannot lie in any of the $H_{i,\mu}$ ($\mu\in \widetilde{S}_r$), as we mentioned in the remarks after the 7SA (see ). As always, we suppose $r>0$.
\[lema49\] Let $v_1,v_2,\dots,v_\ell\in k$ with $\ell<[k:{\mathbb{Q}}]=r+2$, let $\tau_j:k\to {\mathbb{C}}$ the $r+2$ distinct embeddings of $k$ into ${\mathbb{C}}$ (with $\tau_1$ and $\tau_{r+2}$ the non-real embeddings, $\overline{\tau}_1=\tau_{r+2}$), and define $\widetilde{J}:k\to {\mathbb{C}}\times{\mathbb{R}}^{r}$ by $\big(\widetilde{J}(v)\big)^{(j)}:=\tau_j(v)$ for $v\in k$ and $1\leq j\leq r+1$. Then $e_{r+2}:=[0,0,\dots,0,1]\in {\mathbb{C}}\times{\mathbb{R}}^r$ does not lie in the ${\mathbb{R}}$-subspace $${\mathbb{R}}\cdot\widetilde{J}(v_1)+{\mathbb{R}}\cdot \widetilde{J}(v_2)+\dots+{\mathbb{R}}\cdot \widetilde{J}(v_\ell) \ \subset
\ {\mathbb{C}}\times{\mathbb{R}}^r.$$
Suppose $e_{r+2}$ lies in ${\mathbb{R}}\cdot\widetilde{J}(v_1)+\dots+{\mathbb{R}}\cdot \widetilde{J}(v_\ell)$. This means that there are scalars $c_j\in{\mathbb{R}}$ such that $$[0,0,\dots,0,1]=c_1\widetilde{J}(v_1)+\dots+c_\ell\widetilde{J}(v_\ell)\in {\mathbb{C}}\times{\mathbb{R}}^r.$$ Using the definition of $\widetilde{J}$, we have $c_1\tau_1(v_1)+\dots+c_\ell\tau_1(v_\ell)=0$ in the first coordinate of the last equation (recall $r>0$). Then, taking the complex conjugate, $$0=\overline{c_1\tau_1(v_1)+\dots+c_\ell\tau_1(v_\ell)}=c_1\tau_{r+2}(v_1)+\dots+c_\ell\tau_{r+2}(v_\ell).$$ Hence, if we define $J:k\to {\mathbb{C}}^{r+2}$ by $\big(J(v)\big)^{(1)}:=\tau_1(v)$, $\big(J(v)\big)^{(2)}:=\tau_{r+2}(v)$, and $\big(J(v)\big)^{(j)}:=\tau_{j-1}(v)$ for $v\in k$ and $3\leq j\leq r+2$, we have that the vector $[0,0,\dots,0,1]\in{\mathbb{C}}^{r+2}$ can be written as $c_1J(v_1)+\dots+c_\ell J(v_\ell)$. But [@DF1 Lemma 9] shows that $[0,0,\dots,0,1]\in{\mathbb{C}}^{r+2}$ cannot lie in ${\mathbb{C}}\cdot J(v_1)+\dots+{\mathbb{C}}\cdot J(v_\ell)$, so we have a contradiction.
We will prove that $\{C_{\mu},w_{\mu}\}_{w_{\mu}\not=0}$ (see and ) is a signed fundamental domain for the action of $V$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$ by showing that $\{C_{\mu},w_{\mu}\}_{w_{\mu}\not=0}$ is related to a signed fundamental domain for the action of $\widetilde{V}$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$. For $\mu=(\sigma,q,n)\in
\widetilde{S}_r$, we define $$\begin{aligned}
c_{\mu}&:= \Big\{y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\ \big| \ y=
\sum_{t=1}^{r+2}b_t\varphi_{t,\mu},\ \,\sum_{t=1}^{r+2}b_t=1,
\ \,b_t\in J_{t,\mu} \Big\} ,\\
\nonumber
& \varphi_{t,\mu}:=\ell(f_{t,\mu}),\quad \quad
J_{i,\sigma}:=
\begin{cases} [0,1] &\mathrm{ if\ } e_{r+2}\in H_{t,\mu}^+,\\
(0,1] &\mathrm{ if\ } e_{r+2}\in H_{t,\mu}^-,
\end{cases}\end{aligned}$$ for each $w_{\mu}\not=0$ and $t\in\{1,\dots,r+2\}$, where $f_{t,\mu}\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$ is defined in . The closure of $c_\mu$ in ${\mathbb{C}}\times{\mathbb{R}}^{r-1}$ is $$\overline{c}_{\mu}=P(\varphi_{1,\mu},\dots,\varphi_{r+2,\mu})=f(D_{\mu})=A_{\mu}(D_{\mu}),$$ where $f$ is the function defined in Proposition \[Escher\].
\[relaciondominios\] If $\{c_{\mu},w_{\mu}\}_{w_{\mu}\not=0}$ satisfies $$\label{75}
\sum_{\substack{\mu\in \widetilde{S}_r\\ w_{\mu}\not=0}}
\,\sum_{z\in c_{\mu}\cap\widetilde{V}\cdot y}w_{\mu}=1\qquad\qquad\big( y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\big),$$ where the cardinality of $c_{\mu}\cap\widetilde{V}\cdot y$ is bounded independently of $y$, then $\{C_{\mu},w_{\mu}\}_{w_{\mu}\not=0}$ is a signed fundamental domain for the action of $V$ on ${{\mathbb{C}}^*\times{\mathbb{R}}_+^r}$.
The proof in [@DF1 Proposition 10] works in our case as it only involves the underlying real vector space structure.
Define $$\label{B}
\mathcal{B}:=\bigcup_{\mu\in \widetilde{S}_r}\mathcal{B}_\mu, \qquad\qquad \mathcal{B}_\mu:=\bigcup_{\widetilde{\varepsilon}\in \widetilde{V}}\widetilde{\varepsilon}\cdot\partial\overline{c}_{\mu},$$ where $\partial\overline{c}_{\mu}$ is the boundary of $c_\mu$ in ${\mathbb{C}}\times{\mathbb{R}}^{r-1}$. Note that $\overline{c}_\mu\subset\mathcal{B}$ when $w_\mu=0$, for then $\overline{c}_\mu$ coincides with its boundary $\partial\overline{c}_{\mu}$.
Now define $J_\mu(y)\subset \widetilde{V}$ as $$J_\mu(y):=\{\widetilde{\varepsilon}\in \widetilde{V} \,|\, \widetilde{\varepsilon}\cdot y\in c_\mu \}\qquad\qquad (y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}; \ \mu\in \widetilde{S}_r).$$ Then we have the following lemma.
\[aproximaciongradoslocales\] For any $y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ and $\mu\in \widetilde{S}_r$, there exists $T_\mu(y)\in(0,1)$ such that $T_\mu(y)\le t<1$ implies $J_\mu(y)=J_\mu(ty)$ and $ty\not\in\mathcal{B}_\mu$.
Again the proof in [@DF1 Lemma 25] applies verbatim to our case.
End of the proof
----------------
From Lemma \[relaciondominios\], to establish Theorem \[Main\], we have to prove , and that for any $\mu\in \widetilde{S}_r$ the set $c_{\mu}\cap\widetilde{V}\cdot y$ is bounded independently of $y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$. The last part follows using the surjective group homomorphism ${\mathrm{LOG}}:{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}\to{\mathbb{R}}^r$ defined in . Indeed, since ${\mathrm{LOG}}(\overline{c}_{\mu})$ is compact and ${\mathrm{LOG}}(\widetilde{V})$ is a lattice, we have only to show that there are no two (distinct) elements $u,\, v\in \widetilde{V}$ such that ${\mathrm{LOG}}(u\cdot y)={\mathrm{LOG}}(v\cdot y)$. But ${\mathrm{LOG}}(u\cdot y)={\mathrm{LOG}}(v\cdot y)$ implies $|(uv^{-1})^{(j)}|=1$ for all $1\leq j\leq r$. Therefore, since $\widetilde{V}=\ell(V)$, we have that implies $u=v$.
Note that the above also implies that $J_\mu(y)=\{\widetilde{\varepsilon}\in \widetilde{V} \,|\, \widetilde{\varepsilon}\cdot y\in c_\mu \}$ is finite (possibly empty) for all $\mu\in \widetilde{S}_r$ and $y\in {{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$. Furthermore, we have $$\label{cardinaldeJ}
\sum_{z\in c_\mu\cap \widetilde{V}\cdot y}1= \mathrm{Card}\big(J_\mu(y)\big)\qquad (y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}, \ \mu\in \widetilde{S}_r).$$
Now we prove at a point $y\in({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}})\smallsetminus\mathcal{B}$, where $\mathcal{B}$ was defined in . Let $\alpha:=\pi(y)\in T\smallsetminus\pi(\mathcal{B})$. Since $\deg(F)=\pm1\not=0$, we have that $F$ is surjective (see [@DF1 Proposition 21 (3)]). Let $\delta\in F^{-1}(\alpha)\subset \widehat{T}$, and suppose $x\in D$ satisfies $\widehat{\pi}(x)=\delta$. Then $\alpha=F\big(\widehat{\pi}(x)\big)=\pi\big(f(x)\big)$. If $x\in\partial D_\mu$ for some $\mu\in \widetilde{S}_r$, then $f(x)\in f(\partial D_\mu)\subset\partial \overline{c}_\mu\subset\mathcal{B}$, contradicting $\alpha\not\in\pi(\mathcal{B})$. Thus, $x\not\in\partial D_\mu$ for any $\mu \in \widetilde{S}_r$. Similarly, $x\not\in D_\mu$ for any $\mu \in \widetilde{S}_r$ such that $w_\mu=0$. Since $w_\mu\not=0$, the map $f=A_\mu$ gives a bijection between the interior of $D_\mu$ and the interior of $\overline{c}_\mu$. It follows that $f$ is a local homeomorphism in a neighborhood of $x$, as are $\widehat{\pi}$ and $\pi$. Hence $F$ is a local homeomorphism in a neighborhood of $\delta$. Thus, $\delta=\widehat{\pi}(x)$ with $x$ in the interior $\stackrel{\circ}{D}_\mu$ of some $D_\mu$, and $w_\mu\not=0$. Moreover, as $\widehat{\pi}$ restricted to $\stackrel{\circ}{D}$ is a bijection onto its image, there is a unique point $x\in\widehat{\pi}^{-1}(\delta)$. Also, $f(x)$ is in the interior of $\overline{c}_\mu$, so $f(x)\in c_\mu$.
Now we calculate as in [@DF1] using , the invariance of the degree under homotopy, and the local-global principle of topological degree theory[^7] (see [@DF1 Proposition 21 (6) and (9)]), $$\begin{aligned}
\deg(F)&=\sum_{\delta\in F^{-1}(\alpha)}{\mathrm{locdeg}}_\delta(F)=\sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{\substack{x\in D_\mu\\ \widehat{\pi}(x)\in F^{-1}(\alpha)}}{\mathrm{locdeg}}_{\widehat{\pi}(x)}(F)\\
&= \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{\substack{x\in D_\mu\\ F(\widehat{\pi}(x))=\alpha}}v_\mu = \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{\substack{x\in D_\mu\\ \pi(f(x))=\pi(y)}}v_\mu = \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{\substack{x\in D_\mu\\ f(x)\in \widetilde{V}\cdot y}}v_\mu\\
&= \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{z\in c_\mu\cap \widetilde{V}\cdot y}v_\mu = \deg(F)\sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}} \ \sum_{z\in c_\mu\cap \widetilde{V}\cdot y}w_\mu,\end{aligned}$$ since $v_\mu=\deg(F)w_\mu$ by , and . On dividing both sides by $\deg(F)=\pm 1$, follows for $y\in({{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}})\smallsetminus\mathcal{B}$.
We can now prove for any $y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$. Lemma \[aproximaciongradoslocales\] shows the existence of $y_0=y_0(y)\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ such that $J_\mu(y_0)=J_\mu(y)$ and $y_0\not\in\mathcal{B}_\mu$ for all $\mu\in \widetilde{S}_r$. Thus $y_0\not\in\mathcal{B}:=\cup_\mu\mathcal{B}_\mu$. In particular, we know that holds for $y_0$. Hence, using , $$\begin{aligned}
1 &= \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}}\sum_{z\in c_\mu\cap \widetilde{V}\cdot y_0}w_\mu = \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}}w_\mu\mathrm{Card}\big(J_\mu(y_0)\big)\\
&= \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}}w_\mu\mathrm{Card}\big(J_\mu(y)\big) = \sum_{\substack{\mu\in \widetilde{S}_r\\ w_\mu\not=0}}\sum_{z\in c_\mu\cap \widetilde{V}\cdot y}w_\mu.\end{aligned}$$
[XXX]{} P. Colmez, [*[Résidu en $s = 1$ des fonctions zêta $p$-adiques]{}*]{}, Invent. Math. [**[91]{}**]{} (1988), 371–389.
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F. Diaz y Diaz and E. Friedman, [*[Signed fundamental domains for totally real number fields]{}*]{}, Proc. London Math. Soc. (to appear) (2013), available at http://arxiv.org/abs/1303.3989.
F. Diaz y Diaz and E. Friedman, [*[Colmez cones for fundamental units of totally real cubic fields]{}*]{}, J. Number Th. [**132**]{} (2012), 1653–1663.
J. Neukirch, [*Algebraic number theory*]{}, Grundlehren der mathematischen Wissenschaften [**[322]{}**]{}, Berlin: Springer-Verlag (1999).
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T. Ren and R. Sczech, [*[A refinement of Stark’s conjecture over complex cubic number fields]{}*]{}, J. Number Th. [**[129]{}**]{} (2009), 831–857.
T. Shintani, [*[On evaluation of zeta functions of totally real algebraic number fields at non-positive integers]{}*]{}, J. Fac. Sci. Univ.Tokyo, Sec. IA [**[23]{}**]{} (1976), 393–417.
T. Shintani, [*[A remark on zeta functions of algebraic number fields]{}*]{}, Automorphic Forms, Representation Theory and Arithmetic (Bombay Colloquium 1979), Springer, Berlin Heidelberg New York, 1981.
[^1]: This work was partially supported by the Chilean FONDECYT grants 1085153 and 1110277
[^2]: I would like to address special thanks to my advisor, Eduardo Friedman
[^3]: See [@DF2] for the cubic case $r=3$.
[^4]: In fact, in [@Co1], the special units $\eta_i$ are characterized by the condition $w_\sigma=+1$ for all $\sigma\in S_{r-1}$.
[^5]: Its proof coincides with that of [@DF1 Corollary 6], so we omit it.
[^6]: To prove this identity, start with the matrix $(w_1,\dots,w_{r+2})$, divide the $i^{\mathrm{th}}$ column ([[*[i.$\,$e. ]{}*]{}]{}$w_i$) by $w_i^{(r+1)}$ for all $1\le i\le r+2$. This makes no change in the sign of the determinant as $w_i^{(r+1)}>0$. Now subtract the first column from each of the other columns and expand by the last row.
[^7]: Note that $F^{-1}(\alpha)$ is finite since $c_\mu\cap\widetilde{V}\cdot y$ is finite, and since the map $f=A_\mu$ gives a bijection between the interior of $D_\mu$ and the interior of $\overline{c}_\mu$ for all $y\in{{\mathbb{C}}^*\times{\mathbb{R}}_+^{r-1}}$ and $\mu\in \widetilde{S}_r$.
|
---
abstract: 'We describe [*Spitzer*]{} images of a sample of dwarf and low surface brightness galaxies, using the high sensitivity and spatial resolution to explore the morphologies of dust in these galaxies. For the starbursting dwarf UGC10445, we present a complete infrared spectral energy distribution and modeling of its individual dust components. We find that its diffuse cold (T=19K) dust component extends beyond its near-infrared disk and speculate that the most plausible source of heating is ultraviolet photons from starforming complexes. We find that the mass of T=19K dust in UGC10445 is surprisingly large, with a lower limit of 3$\times$10$^6$M$_{\odot}$. We explore the implications of having such a high dust content on the nature and evolution of the galaxy.'
author:
- 'J. L. Hinz, M. J. Rieke, G. H. Rieke, P. S. Smith, K. Misselt, M. Blaylock, and K. D. Gordon'
title: Dust in Dwarfs and Low Surface Brightness Galaxies
---
Introduction
============
Low surface brightness galaxies (LSBGs) have been assumed to have little to no dust. Their low metallicities imply that their dust to gas ratios should be systematically lower than in their high surface brightness counterparts (Bell et al. 2000). IRAS detected only two LSBGs, a further indication that dust is less important in these galaxies; this was reinforced by observations in which multiple distant galaxies are seen through LSB disks (O’Neil et al. 1997; Holwerda et al. 2005). Likewise, dust has been assumed to be an unimportant component of dwarf galaxies. Dwarfs also have low metallicities, and an explanation for this is the loss of metals and dust due to hot galaxian winds driven by supernovae (Mac Low & Ferrara 1999), where the smaller gravitational potential well of dwarfs allows for the escape of most of the metals and dust (Hogg et al. 2005).
However, infrared (IR) and millimeter observations have shown that dust in dwarfs can be retained, with up to 80% of the total dust mass comprised of a cold component (Galliano et al. 2003; 2005; Madden et al. 2005). The cold dust, much like the H[i]{} gas, has been shown in some cases to spread beyond the optical extent (Tuffs & Popescu 2005). Building on early [*Spitzer*]{} observations of dwarf galaxies (Rosenberg et al. 2006), we present IRAC and MIPS observations of a sample of dwarfs and LSBGs, concentrating on results regarding dust in one dwarf galaxy.
Observations and Data Reduction
===============================
The observations described here are from a guaranteed time observer program (P.I.D. 62; M. Rieke, PI). IRAC images and MIPS photometry mode data were obtained for all galaxies in the sample (see Table 1). IRAC images were reduced with the standard Spitzer Science Center data pipeline; MIPS data were reduced using the Data Analysis Tool (DAT; Gordon et al. 2005).
MIPS images of two example LSBGs are shown in Figure 1. These galaxies appear to have detections at all three wavelengths, implying that some dust must be present. We find that, in general for our sample, galaxies displaying extended emission at 24$\micron$, indicating active star formation, also tend to have detections at the other MIPS wavelengths. However, those LSBGs with no detection or only point-like emission at 24$\micron$ do not appear to have detectable emission at 70 and 160$\micron$. The galaxies with little or no detection at 24$\micron$ are generally the large diffuse spirals such as Malin1 as opposed to more compact structures such as UGC6879.
[lcccccc]{} Galaxy & Morphological & Distance &\
& Type & \[km s$^{-1}$\] &\
UGC5675 & Sm & 1102\
UGC6151 & Sm & 1331\
UGC6614 & (R)SA(r)a & 6351\
UGC6879 & SAB(r)d & 2383\
UGC9024 & S & 2323\
UGC10445 & SBc & 963\
Malin1 & S & 24750\
The closest and brightest of our sample, the starbursting dwarf UGC10445, has the most easily accessible ancillary data, and we present more detailed results for this galaxy alone. We used circular apertures to calculate flux densities at IRAC and MIPS wavelengths for UGC10445 and combined these with $H$ and $K$-band photometry (de Jong & van der Kruit 1994), IRAS fluxes, and a 170$\micron$ flux from the ISO Serendipity Survey (Stickel et al. 2004) to produce the spectral energy distribution (SED) shown in Fig. 2.
The 160$\micron$ emission for UGC10445 (see Fig. 2) remains well above background longer and extends out further than all the other wavelengths presented. This extended emission is not the result of resolution differences: all wavelengths are convolved with a kernel that transforms images to the 160$\micron$ resolution.
Modeling
========
We model the emission by dust in UGC10445, as represented by the SED in Fig. 2, with a modified Planck function three-component dust model: a PAH component, a warm silicate component (T=50K), and a cool silicate component (19K). We estimate the dust masses to be $\sim2\times10^3$M$_{\odot}$ for the warm component and $\sim3\times10^6$M$_{\odot}$ for the T=19K material. This value is a lower limit to the cool dust mass, as we are not sensitive to dust colder than 19K.
Popescu et al. (2002) propose that cold dust in galaxies is heated by the diffuse nonionizing ultraviolet (UV) radiation produced by young stars, with a small contribution from the optical radiation produced by old stars. Although there is little UV flux past 1$\farcm$5 for UGC10445, the flux needed to heat the dust grains to T=19K is not large. A simple $\nu F_{\nu}$ comparison of FUV and 160$\micron$ luminosities indicates that the quantities are approximately equal for the galaxy. Dust providing a modest level of visual extinction would have sufficient optical depth in the UV to power the cold dust emission through absorption of diffuse UV radiation.
Using a value of the H[i]{} mass from the literature (Lee et al. 2002), the H[i]{} gas mass to dust mass ratio of UGC10445 is 500, which has implications for the history of the galaxy. If we take the yield in heavy elements through stellar processes to be 0.002 (Kuzio de Naray et al. 2004), the rotation velocity to be 65kms$^{-1}$ (Lee et al. 2002), and assume that 50% of the metals are retained in the gravitational well of the galaxy (Garnett 2002), it follows that at least 3$\times$10$^9$ M$_\odot$ of stars must have formed to produce the 3$\times$10$^6$M$_\odot$ of dust observed at 160$\micron$. If the near-IR output is from the old stellar population left from this long duration star formation, we can calculate a $K$-band stellar mass-to-light (M/L$_{*,K}$) ratio and retrieve the mass of stars necessary to create the total dust mass. We select a M/L$_{*,K}$ of 0.33 (Bell & de Jong 2001) which, using the $K$-band magnitude (de Jong & van der Kruit 1994), leads to a total stellar mass of 3.9$\times$10$^9$M$_\odot$. The current star formation rate (e.g., van Zee 2000) would require $\geq$26Gyr to form this mass of stars. Therefore, the current star formation rate of UGC10445 must be below the typical star forming rate over its lifetime.
Summary
=======
Based on the observations of LSBGs and dwarfs in the sample, our preliminary conclusions are that large diffuse LSBGs such as Malin1 contain no cool dust detectable by [*Spitzer*]{}, while LSBGs or dwarfs with modest amounts of star formation visible at 24$\micron$ have corresponding emission at 160$\micron$. One explanation for this is that dust in the outer reaches of the galaxies may have to be heated by UV photons escaping from H[ii]{} regions before being detectable. Additionally, surprisingly large amounts of dust (T=19K) are shown to exist in at least one dwarf galaxy (UGC10445) with an extended, diffuse cool dust component reaching out beyond its near-IR disk.
We thank J. Lee, P. Knezek, T. Pickering, C. Tremonti, C. Popescu, and R. Tuffs for helpful discussions. This work is based on observations made with [*Spitzer*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407. Support for this work was provided by NASA through Contract Numbers 1255094 and 1256318 issued by JPL/Caltech.
Bell, E. F., Barnaby, D., Bower, R. G., de Jong, R. S., Harper, D. A., Hereld, M., Loewenstein, R. F., & Rauscher, B. J. 2000, , 312, 470 Bell, E. F., & de Jong, R. S. 2001, , 550, 212 de Jong, R. S., & van der Kruit, P. C. 1994, , 106, 451 Galliano, F., et al. 2003, A&A, 407, 159 Galliano, F., et al. 2005, A&A, 434, 867 Garnett, D. R. 2002, , 581, 1019 Gordon, K. D., et al. 2005, , 117, 503 Hogg, D. W., et al. 2005, ApJ, 624, 162 Holwerda, B. W., Gonzalez, R. A., Allen, R. J., & van der Kruit, P. C. 2005, , 129, 1396 Kuzio de Naray, R., McGaugh, S. S., & de Block, W. J. G. 2004, , 335, 887 Lee, J. C., Salzer, J. J., Impey, C., Thuan, T. X., & Gronwall, C. 2002, , 124, 3088 Mac Low, M. & Ferrara, A. 1999, ApJ, 513, 142 Madden, S. C. et al. 2005, astro-ph/0510086 O’Neil, K., Bothun, G. D., & Impey, C. D. 1997, Bulletin of the American Astronomical Society, 29, 1398 Popescu, C. C., Tuffs, R. J., V[" o]{}lk, H. J., Pierini, D., & Madore, B. F. 2002, , 567, 221 Rosenberg, J. L., Ashby, M. L. N., Salzer, J. J., & Huang, J.-S. 2006, , 636, 742 Stickel, M., Lemke, D., Klaas, U., Krause, O., & Egner, S. 2004, , 422, 39 Tuffs, R. J. & Popescu, C. C. 2005, API COnf. Proc. 761: The Spectral Energy Distributions of Gas-Rich Galaxies, 761, 344 van Zee, L. 2000, , 119, 2757
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---
abstract: 'We study gravitational waves generated during the inflationary epoch in presence of a decaying cosmological parameter on a 5D geometrical background which is Riemann flat. Two examples are considered, one with a constant cosmological parameter and the second with a decreasing one.'
address: |
$^1$ Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, (7600) Mar del Plata, Argentina.\
$^2$ Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).\
$^3$ Departamento de Física, Universidade Federal da Paraíba,\
Caixa Postal 5008, 58059-970 João Pessoa, Pb, Brazil.
author:
- '$^{1}$Silvina Paola Gomez Martínez [^1] , $^{3}$José Edgar Madriz Aguilar[^2] and $^{1,2}$Mauricio Bellini[^3]'
title: Gravitational waves during inflation in presence of a decaying cosmological parameter from a 5D vacuum theory of gravity
---
Introduction
============
Gravitational waves has been subject of attention since long time ago [@alpha]. Under the standard model of cosmology plus the theory of inflation, it is very natural to predict the existence of the background of gravitational waves [@1]. Among the primordial perturbations generated during inflation, there were basically normal scalar part and tensor part. The primordial scalar perturbations provided the seeds of large scale structure which then had gradually formed today’s galaxies, which is being tested in current observations of cosmic microwave background (CMB). The tensor perturbations have escaped out of the horizon during inflation, and thus can be completely conserved to form the relic of background gravitational waves, which carries the information of the very early universe. In this sense the tensor perturbations are very significant for studies of very early universe. Their amplitude is related to the energy scale of inflation and they are potentially detectable via observations of $B$-mode polarization in the cosmic microwave background if the energy scale of inflation is larger than $\sim 3 \times 10^{15}$ GeV [@2; @3; @4; @5]. Such a detection would be very important to test inflation. The direct detection of GW is one of the most exciting scientific goals because it would improve our understanding of laws governing the early universe and provide new means to observe it. The most sensitive detectors which are already operating, under construction or being planned, are based on optical interferometers [@delta]. In particular, gravitational wave perturbations will be measured in future by the Planck satellite, which is designed to produce high-resolution temperature and polarization maps of CMB. If no primordial GW are detected by CMB, a direct-detection experiment to understand the simplest form of inflation must have a sensitivity improved by two to three orders of magnitude over current plans [@*L]. The basal mechanism of generation of primordial gravitational waves in cosmology has been discussed in [@..]. There are two main 4D formalisms developed in the literature; the coordinate based approach of Bardeen [@bar] and the covariant formalism [@cov].
The idea that our universe is a 4D space-time embedded in a higher dimensional has been a topic of increased interest in several branches of physics, and in particular, in cosmology. This idea has generated a new kind of cosmological models that includes quintessential expansion. In particular, theories on which is considered only one extra dimension have become quite popular in the scientific community. Among these theories are counted the braneworld scenarios [@bw], the Space-Time-Matter (STM) theory [@STM] and all noncompact Kaluza-Klein theories.
In this work we shall study the evolution of gravitational waves on the early universe, which is governed by a (decaying) cosmological parameter $\Lambda(t)$, from a 5D vacuum state defined on a $4+1$ Riemann flat spacetime. A decaying cosmological parameter can be introduced in a geometrical manner through the 5D background line element [@M1] $$\label{eq1}
dS_{b}^{2}=\psi^{2}\frac{\Lambda (t)}{3}dt^{2}-\psi^{2}e^{2\int
\sqrt{\Lambda/3}\,dt}dr^2 - d\psi^{2},$$ where $dr^{2}=\delta _{ij}dx^{i}dx^{j}$, being $\lbrace x^{i} \rbrace =\lbrace x,y,z\rbrace$ the local cartesian coordinates. In addition $t$ and $\psi$ are the time-like and fifth space-like local coordinates respectively. Adopting a natural unit system ($\hbar =c=1$), the fifth coordinate $\psi$ has spatial units whereas the cosmological parameter $\Lambda (t)$ has units of $(length)^{-2}$. The background metric in (\[eq1\]) is Riemann-flat, $R^{A}\,_{BCD}=0$, describing perfectly a 5D geometric vacuum.\
The usual approach would consist to consider the tensor perturbed line element obtained from (\[eq1\]) $$\label{eq2}
dS^{2}=\psi^{2}\frac{\Lambda (t)}{3}dt^{2}-\psi^{2}e^{2\int
\sqrt{\Lambda/3}\,dt}(\delta _{ij}+Q_{ij})dx^{i}dx^{j}-d\psi ^{2},$$ being $Q_{ij}(t,\vec{r},\psi)$ the transverse traceless tensor denoting the tensor fluctuations with respect to the metric background (\[eq1\]), and thereby it satisfies $tr(Q_{ij})=Q^{i}\;_{i}=0$ and $Q^{ij}_{;i}=0$. The 3D spatial components of the metric (\[eq2\]) can be written as $g_{ij}=-\psi^{2}{\rm exp}\left[2\int \sqrt{\Lambda/3}\,dt\right](\delta _{ij}+Q_{ij})$, so that its contravariant components can be linearly approximated by $g^{ij}\simeq -\psi ^{-2} {\rm exp}\left[-2\int \sqrt{\Lambda/3}\,dt\right](\delta ^{ij}-Q^{ij})$.\
Under this approach the dynamics obeyed by the tensor fluctuations $Q_{ij}$, is obtained using the 5D linearized Einstein´s equations in the 5D vacuum $\delta R_{AB}=0$. However, as it is well-known solely the space-space components contribute for tensor fluctuations, so the former expression reduces simply to $$\label{eq3}
\delta R_{ij}=0.$$ Hence, after some algebra, one obtain the dynamics of the tensor fluctuations $Q_{ij}$ is determined by $$\label{eq4}
\ddot{Q}_{ij} + \left[3\sqrt{\frac{\Lambda}{3}}-
\frac{1}{2}\frac{\dot{\Lambda}}{\Lambda}\right]\dot{Q}_{ij}-\frac{\Lambda}{3} \ e^{-2\int \sqrt{\Lambda/3}\,dt} \nabla^2_r
Q_{ij}-\frac{\Lambda}{3}\left[4\psi Q_{ij,\psi}+\psi^{2}Q_{ij,\psi,\psi}\right]=0,$$ where $(,)$ denotes the partial derivative while the dot is denoting ordinary derivation with respect the cosmic time $t$.\
The 5D tensor modes
====================
In this letter we shall consider a different approach, which, give us the dynamics (\[eq4\]), but from the the action $I=- \int
d^4x \ d\psi \ ^{(5)} L$, given by the background gravitational Lagrangian, plus free tensorial fields, $Q_{ij}$ ($A$, $B$ run from $0$ to $4$ and $i$, $j$ from $1$ to $3$) $$^{(5)} L = \sqrt{\left|\frac{^{(5)} g}{^{(5)} g_0}\right|}
\left[ \frac{^{(5)} {\cal R}}{16\pi G} + \frac{M^2_p }{2} g^{AB}
Q^{ij}_{\, ;A} Q_{ij;B}\right],$$ being $(;)$ the covariant derivative. Furthermore, $^{(5)} {\cal
R}=0$ is the 5D background Ricci scalar and $^{(5)} g = \psi^8
\Lambda e^{6 \int\sqrt{\Lambda/3} dt}/3$ is the determinant of the background covariant metric tensor $g_{AB}$. Following the usual quantization process for $Q_{ij}(t,\vec{r},\psi)$ we demand that the next commutation relation must be satisfied $$\label{com}
\left[Q_{ij}(t,\vec{r},\psi),\frac{\partial L}{\partial
Q_{lm,t}}(t,\vec{r'},\psi ')\right]=i \delta^l_i \ \delta^m_j \
g^{tt} M^2_p \sqrt{\left|\frac{^{(5)}g}{^{(5)}g_0}\right|}
\left(\frac{\psi_0}{\psi}\right)^3
e^{-\int\left[3\sqrt{\frac{\Lambda}{3}} -
\frac{\dot\Lambda}{2\Lambda}\right] dt} \times
\delta^{(3)}(\vec{r}-\vec{r'})\delta(\psi - \psi ').$$
On the other hand, $^{(5)} g_0 \equiv ^{(5)} g[\psi=\psi_0,
\Lambda_0\equiv \Lambda(t=t_0)]$, being $\psi_0$ and $t_0$ some constants to be specified. In this work we are interested to study the dynamics of $Q_{ij}$. We express the functions $Q_{ij}$ as a Fourier expansion of the form $$\label{eq5}
Q^{i}\,_{j}(t,\vec{r},\psi)=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k_{r}\,dk_{\psi}\sum _{\alpha}\,^{(\alpha)} e^{i}\,_{j}\left[a_{k_{r}k_{\psi}}^{(\alpha)}
e^{i\vec{k}_{r}\cdot\vec{r}}\zeta _{k_{r}k_{\psi}}(t,\psi)+a_{k_{r}k_{\psi}}^{(\alpha)\,\,\dagger}
e^{-i\vec{k}_{r}\cdot\vec{r}}\zeta _{k_{r}k_{\psi}}^{*}(t,\psi)\right],$$ with $\alpha $ counting the number of polarization degrees of freedom, the asterisk $(*)$ denoting complex conjugate and the creation $a_{k_{r}k_{\psi}}^{(\alpha)\,\,\dagger}$ and annihilation $a_{k_{r}k_{\psi}}^{(\alpha)}$ operators satisfying the algebra $$\label{eq6}
\left[a_{k_{r}k_{\psi}}^{(\alpha)},a_{k'_{r}k'_{\psi}}^{(\alpha ')\,\,\dagger}\right]=g^{\alpha\alpha '}\delta %%@
^{(3)}(\vec{k}_{r}-\vec{k}'_{r})\delta (\psi -\psi '),\qquad \left[a_{k_{r}k_{\psi}}^{(\alpha)},a_{k'_{r}k'_{\psi}}^{(\alpha %%@
')}\right]=\left[a_{k_{r}k_{\psi}}^{(\alpha)\,\,\dagger},a_{k'_{r}k'_{\psi}}^{(\alpha ')\,\,\dagger}\right]=0.$$ The polarization tensor $^{(\alpha)}e_{ij}$ obeys $$\label{eq7}
^{(\alpha)}e_{ij}=\,^{(\alpha)}e_{ji},\quad ^{(\alpha)}
e_{ii}=0,\quad k^{i}\,^{(\alpha)}e_{ij}=0,\quad ^{(\alpha)}e_{ij}(-\vec{k}_r)=\,^{(\alpha)}e_{ij}^{*}(\vec{k}_r).$$ Inserting (\[eq5\]) in (\[eq4\]) we obtain that the dynamics of the 5D tensor modes $\zeta _{k_{r} m}(t,\psi)$ is given by $$\label{eq8}
\ddot{\zeta}_{k_{r}k_{\psi}}+\left[3\sqrt{\frac{\Lambda}{3}}-\frac{1}{2}
\frac{\dot{\Lambda}}{\Lambda}\right]
\dot{\zeta}_{k_{r}k_{\psi}}+\left[\frac{\Lambda}{3}k_{r}^{2}\,e^{-2\int
\sqrt{\Lambda/3}\,dt}-\frac{\Lambda}{3}\left(4\psi\frac{\partial}{\partial
\psi} +\psi
^{2}\frac{\partial^2}{\partial\psi^2}\right)\right]\zeta_{k_{r}k_{\psi}}=0.$$ Decomposing the tensor modes $\zeta _{k_{r}k_{\psi}}(t,\psi)$ into Kaluza-Klein modes $$\label{eq9}
\zeta _{k_{r}k_{\psi}}(t,\psi)\sim\xi _{k_r}(t)\Theta _{m}(\psi),$$ equation (\[eq8\]) yields $$\begin{aligned}
\label{eq10}
\ddot{\xi}_{k_r}+\left(3\sqrt{\frac{\Lambda}{3}}-\frac{1}{2}
\frac{\dot{\Lambda}}{\Lambda}\right)\dot{\xi}_{k_r}+\left[\frac{\Lambda}{3
}\,e^{-2\int \sqrt{\Lambda/3}\,dt}k_{r}^{2}+m^{2}\frac{\Lambda}{3}\right]\xi _{k_r}&=&0,\\
\label{eq11} \psi^{2}\frac{d^{2}\Theta
_m}{d\psi^2}+4\psi\frac{d\Theta _{m}}{d\psi}+m^{2}\Theta _{m}&=&0,\end{aligned}$$ where the parameter $m^2=\left(k_{\psi} \psi\right)^2$ is related with the squared of the KK-mass measured by a class of observers in 5D. Using the transformation $\xi _{k_r}(t)=exp\,[-(1/2)\int
(3\sqrt{\Lambda/3}-(\dot{\Lambda}/2\Lambda))\,dt]\,\chi _{k_r}(t)$ and $\Theta_m(z) = e^{-3/2 z} L_m(z)$, with $z={\rm
ln}(\psi/\psi_0)$, in eqs. (\[eq10\]) and (\[eq11\]), respectively, we obtain $$\begin{aligned}
&& \ddot{\chi}_{k_r}+\left[\frac{\Lambda}{3}\,e^{-2\int \sqrt{\Lambda/3}\,dt} k_{r}^{2}-
\frac{1}{4}\sqrt{\frac{3}{\Lambda}}\dot{\Lambda}+\frac{1}{4}
\frac{\ddot{\Lambda}}{ \Lambda}-\frac{5}{16}\frac{\dot{\Lambda}^2}{\Lambda^2}
+\frac{3}{4}\sqrt{\frac{\Lambda}{3}}\frac{\dot{\Lambda}}{\Lambda}+\left( \frac{m^2}{3}-\frac{3}{4}\right)\Lambda \right]\chi %%@
_{k_r}=0, \label{eq12}\\
&& \frac{d^2 L_m(z)}{dz^2} + \left[m^2 - \frac{9}{4}\right] L_m(z) =0.
\label{eq12'}\end{aligned}$$ This way, given a cosmological parameter $\Lambda (t)$, the temporal evolution of the tensor modes $\xi _{k_r}(t)$ in 5D is determined by solutions of (\[eq12\]). Once a solution for $\xi _{k_r}(t)$ is obtained, it should satisfy the algebra (\[com\]). This can be made guaranteeing that such a solution obey $$\label{eq13}
\chi_{k_{r}}\dot{\chi}_{k_{r}}^{*}-
\dot{\chi}_{k_{r}}\chi_{k_{r}}^{*}=i, \qquad \left| L_m\right|^2=1.$$ On the other hand, note that equation (\[eq11\]) is exactly the same as the one obtained in [@Ed]. Therefore about the behavior of the modes with respect the fifth coordinate we can say that for $m>3/2$ the KK-modes are coherent on the ultraviolet sector (UV), described by the modes $$\label{eq14}
k^2_r > \left\{\frac{3}{2\Lambda} \frac{d}{dt}\left[ 3
\sqrt{\frac{\Lambda}{3}} - \frac{\dot\Lambda}{2\Lambda}\right]
-\frac{3}{4\Lambda}
\left(3\sqrt{\frac{\Lambda}{3}}-\frac{\dot\Lambda}{2\Lambda}\right)^2
- m^2\right\} \ e^{2\int \sqrt{\frac{\Lambda}{3}} dt} >0.$$ However, for $m<3/2$ those modes are unstable and diverge at infinity. The modes with $m=0$ and $m > 3/2$ comply with the conditions (\[eq13\]), so that they are normalizable.\
Effective 4D dynamics
=====================
To describe the 4D dynamics we can make a foliation on $\psi=\psi_0$ on the line element (\[eq1\]), such that the effective 4D background metric holds: $\left.dS^2\right|_{eff} =
ds^2$, where $$\label{61}
ds^2 =
\psi^2_0 \frac{\Lambda(t)}{3} dt^2 - \psi^2_0 e^{2 \int
\sqrt{{\Lambda\over 3}}dt} dr^2.$$ In this section we shall study the dynamics of the 4D tensor-fluctuations $h_{ij}(t,\vec{r})\equiv
Q_{ij}(t,\vec{r},\psi=\psi_0)$, making emphasis on the long wavelength section, which describes this field on cosmological scales. The effective 4D action $^{(4)} I$ is ($\alpha$, $\beta$ run from $0$ to $3$) $$\label{act}
^{(4)} I = -\int d^4 x \sqrt{\left|\frac{^{(4)} g}{^{(4)}
g_0}\right|} \left. \left[ \frac{^{(4)} {\cal R}}{16\pi G} +
\frac{M^2_p }{2} g^{\alpha\beta} Q^{ij}_{\, ;\alpha}
Q_{ij;\beta}\right] \right|_{\psi=\psi_0},$$ where $^{(4)} {\cal R} = 12/\psi^2_0$ is the effective 4D Ricci scalar valuated on the metric (\[61\]), such that we obtain an equation of state which describes an effective 4D vacuum: ${\rm p}
= -\rho$, being ${\rm p}$ and $\rho$ the pressure and the energy density, respectively. Hence, the metric (\[eq1\]) could be considered as an extension of the Ponce de Leon one[@PdL], which, on a foliation $\psi=\psi_0$, also describes an effective 4D vacuum dominated expansion.
The effective 4D equation of motion for the 4D tensor-fluctuations is $$\label{eq17}
\ddot{h}^i_j + \left[3\sqrt{\frac{\Lambda}{3}} -
\frac{\dot\Lambda}{2\Lambda} \right] \dot{h}^i_j -
\frac{\Lambda}{3} e^{-2\int\sqrt{\frac{\Lambda}{3}} dt} \nabla^2_r
h^i_j - \left. \frac{\Lambda}{3} \left[\frac{4}{\psi}
\frac{\partial Q^i_j}{\partial \psi} + \psi^2 \frac{\partial^2
Q^i_j}{\partial\psi^2} \right]\right|_{\psi= \psi_0} =0.$$ Using the eq. (\[eq11\]), we obtain $$\label{eq18}
\ddot{h}^i_j + \left[3\sqrt{\frac{\Lambda}{3}} -
\frac{\dot\Lambda}{2\Lambda} \right] \dot{h}^i_j -
\frac{\Lambda}{3} e^{-2\int\sqrt{\frac{\Lambda}{3}} dt} \nabla^2_r
h^i_j + \frac{m^2 \Lambda}{3} h^i_j =0.$$ After make the transformation $h^i_j(t,\vec r) = e^{-1/2 \int \left[
3\left({\Lambda\over 3}\right)^{1/2}-{\dot\Lambda\over 2\Lambda}\right] dt}
\chi^i_j(t,\vec r)$, we obtain $$\label{eq19}
\ddot{\chi}^i_j - \frac{\Lambda}{3}
e^{-2\int\sqrt{\frac{\Lambda}{3}} dt} \nabla^2_r \chi^i_j - \left[
\frac{1}{4} \sqrt{\frac{3}{\Lambda}} \dot\Lambda + \frac{1}{4}
\frac{\ddot\Lambda}{\Lambda} - \frac{5}{16}
\frac{\dot\Lambda^2}{\Lambda^2} + \frac{3}{4}
\sqrt{\frac{\Lambda}{3}} \frac{\dot\Lambda}{\Lambda} + \left(
\frac{m^2}{3} - \frac{3}{4} \right) \Lambda \right] \chi^i_j =0,$$ such that it is possible to make a Fourier expansion for $\chi^i_j$ $$\label{eq5'}
\chi^{i}\,_{j}(t,\vec{r})=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k_{r}\,dk_{\psi}\sum _{\alpha=+,\times}\,^{(\alpha)} e^{i}\,_{j}\left[a_{k_{r}k_{\psi}}^{(\alpha)}
e^{i\vec{k}_{r}\cdot\vec{r}}\chi _{k_{r}k_{\psi}}(t,\psi)+a_{k_{r}k_{\psi}}^{(\alpha)\,\,\dagger}
e^{-i\vec{k}_{r}\cdot\vec{r}}\chi _{k_{r}k_{\psi}}^{*}(t,\psi)\right] \delta(k_{\psi}-k_{\psi_0}),$$ where we require that the modes $\chi_{k_r}\equiv \chi_{k_r k_{\psi_0}}$ satisfy the commutation relation $$\label{cr1}
\left[\chi_{k_{r}}(t,\vec{r}),\dot{\chi}_{k_r}(t,\vec{r'})\right]=i\delta^{(3)}(\vec{r}-\vec{r'}).$$ Using (\[eq5’\]) this expression reads $$\label{mod}
\chi_{k_{r}} \dot\chi^*_{k_{r}} - \chi^*_{k_{r}} \dot\chi_{k_{r}}=i,$$ which is the condition for the modes to be normalizable on the UV-sector. Inserting (\[eq5’\]) in (\[eq19\]) we obtain the dynamical equation for the $k_{r}$-modes $$\label{deq1}
\ddot{\chi}_{k_{r}}+\left[\frac{\Lambda}{3}e^{-2\int\sqrt{\frac{\Lambda}{3}}dt}k_{r}^{2}-\left(\frac{1}{4}\sqrt{\frac{3}{\Lambda}}%%@
\dot{\Lambda}+\frac{1}{4}\frac{\ddot{\Lambda}}{\Lambda}-\frac{5}{16}\frac{\dot{\Lambda}^{2}}{\Lambda^2}+\frac{3}{4}\sqrt{ \frac{ %%@
\Lambda}{3}}\frac{\dot{\Lambda}}{\Lambda}+\left(\frac{m^{2}}{3}-\frac{3}{4}\right)\Lambda\right)\right]\chi _{k_r}=0.$$ This way for a given $\Lambda (t)$ corresponds a normalized solution for the $k_{r}$-modes by solving (\[deq1\]). Once obtained a normalized solution of (\[deq1\]), we will be able of dealing with the spectrum on super Hubble scales. The amplitude of the 4D tensor metric fluctuations $<h^{2}>=<0|h^{i}\,_{j}\,h_{i}\,^{j}|0>$ on the IR-sector is given by $$\label{deq2}
\left<h^{2}\right> =\frac{4}{\pi^2}e^{-\int \left[
3\left({\Lambda\over 3}\right)^{1/2}-{\dot\Lambda\over 2\Lambda}\right] dt}\int _{0}^{\epsilon %%@
k_H}\frac{dk_{r}}{k_r}k_{r}^{3}\left[\chi _{k_r}(t)\chi _{k_r}^{*}(t)\right]_{IR},$$ where $\epsilon=k_{max}^{IR}/k_p \ll 1$ is a dimensionless parameter, being $k_{max}^{IR}=k_{H}(t_i)$ the wave-number related to the Hubble radius at the time $t_{i}$. This time corresponds at the time when the gravitational modes re-enter to the horizon. In addition, $k_p$ is the Planckian wave-number. Clearly, in order to obtain an explicit spectrum it is necessary to specify first a functional form for $\Lambda (t)$. Some illustrative examples will be studied in the next section.
Examples
========
In order to illustrate the formalism developed in the previous section, along the present section we study a pair of interesting examples. The first one contemplating a constant cosmological parameter $\Lambda=3H_{0}^{2}$, and the second one considering a decaying $\Lambda (t)=3p^{2}/t^{2}$.
Case $\Lambda=3 H^2_0$
----------------------
The simplest example results of taking the cosmological parameter $\Lambda$ to be a constant, and in particular $\Lambda = 3 H^2_0$. In this particular case the equation of motion for the modes $\chi_{k_{r}}$ becomes $$\label{modd}
\ddot\chi_{k_r} + \left[H^2_0 e^{-2 H_0 t} k^2_r-
\left(m^2 - \frac{9}{4}\right) H^2_0 \right] \chi_{k_r} =0.$$ The general solution for this equation is $$\label{sol}
\chi_{k_r}(t) = A_1 \ {\cal H}^{(1)}_{\nu}\left[k_r e^{-H_0t}\right]
+ A_2 \ {\cal H}^{(2)}_{\nu}\left[k_r e^{-H_0 t}\right],$$ where $A_{1}$ and $A_{2}$ are integration constants. After the Bunch-Davies normalization [@BD] the normalized solution reads $$\label{so1}
\chi_{k_r}(t) = \frac{i}{2}\sqrt{\frac{\pi}{H_{0}}}
{\cal H}^{(2)}_{\nu}\left[k_r e^{-H_0 t}\right],$$ with $\nu = (1/2)\sqrt{4m^2 - 9}$. This solution is stable for $m^2 > 9/4$, on scales $k^2_r > \left(m^2 - {9\over 4}\right) e^{2 H_0 t} >0$. Now considering the asymptotic expansion for the Hankel function ${\cal H}_{\nu}^{(2)}[y]\simeq (-i/\pi)\Gamma (\nu)[y/2]^{-\nu}$ in (\[so1\]), the amplitude of the 4D tensor metric fluctuations (\[deq2\]) on cosmological (super Hubble) scales gives $$\label{deq3}
\left<h^{2}\right>_{SH}
=\frac{2^{2\nu}}{\pi^{3}(3-2\nu)}\frac{\Gamma^{2}(\nu)}{H_{0}}e^{-(3-2\nu)H_{0}t}\epsilon ^{3-2\nu}k_{H}^{3-2\nu},$$ where $k_{H}(t)\sim \, e^{H_{0}t}$. Hence, the gravitational spectrum ${\cal P}_{g}(k_r)$ takes the form $$\label{deq4}
{\cal P}_{g}(k_r)=
\left.\frac{2^{2\nu}}{\pi^{3}}\frac{\Gamma^{2}(\nu)}{H_0}e^{-(3-2\nu)H_{0}t}k_{r}^{3-2\nu}\right|_{k_r=\epsilon
k_{H}}$$ We can see from (\[deq4\]) that for $m\simeq 3/\sqrt{2}>(3/2)$, the gravitational spectrum ${\cal P}_{\nu}(k_r)$ is nearly scale invariant and consequently the tensor spectral index becomes $n_{T}\equiv 3-2\nu\simeq 0$ in this case.
Case $\Lambda = 3 p^2/t^2$
--------------------------
Another interesting case appears considering a decaying cosmological parameter $\Lambda = 3 p^2/t^2$, with the restriction $\dot\Lambda <0$. In this case the equation of motion for the modes $\chi_{k_r}(t)$ results to be $$\label{ecuac}
\ddot\chi_{k_r} + \left\{ k^2_r p^2 t^{2p}_0 t^{-2(p+1)} -
\left[\left(m^2-\frac{9}{4}\right)p^2 -\frac{9}{4} p +
\frac{1}{4}\right] t^{-2} \right\} \chi_{k_r} =0,$$ where $M^2(t) = \left[\left(m^2 - {9\over 4}\right) p^2 - {9\over
4} p + {1\over 4} \right] t^{-2}$ can be interpreted as an effective squared term of mass. The permitted values for $m$ should be $$\label{m}
\frac{9}{4} < m^2 \leq \frac{9}{4} \left(\frac{9}{4} +1\right),$$ for which $$\label{p}
\frac{9 - \sqrt{117 - 16 m^2}}{2(4 m^2-9)} \leq p \leq \frac{9 +
\sqrt{117 - 16 m^2}}{2(4 m^2-9)}.$$ The general solution of (\[ecuac\]) can be expressed in terms of Bessel functions as $$\label{ecu}
\chi_{k_r}(t) = B_1 \left[\frac{y(t)}{2}\right]^{-\mu/(2p)}t^{(1-\mu)/2}\Gamma\left(1+\frac{\mu}{2p}\right){\cal %%@
J}_{\mu}[y(t)]+B_{2}\left[\frac{y(t)}{2}\right]^{\mu/(2p)}t^{(1+\mu)/2}\Gamma\left(1-\frac{\mu}{2p}\right){\cal J}_{-\mu}[y(t)],$$ where $\mu = \sqrt{4p^{2}m^{2}-9p(p+1)+2}$ and $y(t)=k_{r}(t_{0}/t)^{p}$. In general this expression is not normalizable. However there exist some particular solutions of (\[p\]) that are normalizable. A particular case that yields a normalizable solution results by taking $m=\pm [1/(2p)]\sqrt{9p(p+1)-1}$. In this case the dynamical equation for the quantum gravitational modes reduces to $$\label{deq5}
\ddot{\chi} _{k_r}+k_{r}^{2}p^{2}t_{0}^{2p}t^{-2(p+1)}\chi _{k_r}=0,$$ whose general solution is $$\label{deq6}
\chi _{k_r}(t)=\sqrt{t}\left\{C1{\cal H}_{\omega}^{(1)}\left[k_{r}\left(\frac{t_0}{t}\right)^{p}\right]+C_{2}{\cal %%@
H}_{\omega}^{(2)}\left[k_{r}\left(\frac{t_0}{t}\right)^{p}\right]\right\},$$ being $C_{1}$ and $C_{2}$ integration constants, and $\omega
=1/(2p)$. The Bunch-Davies normalized solution is then $$\label{deq7}
\chi _{k_r}(t)=\frac{i}{2}\sqrt{\frac{1}{\pi p}}\,{\cal H}_{\omega}^{(2)}\left[k_{r}\left(\frac{t_0}{t}\right)^{p}\right].$$ In this case the amplitude of the 4D tensor metric fluctuations on super-Hubble scales ($k_r \gg k_H$) reads $$\label{deq8}
\left<h^{2}\right>_{SH} =\frac{2^{2\omega}}{p\,\pi %%@
^{5}}\frac{\Gamma^{2}(\omega)}{3-2\omega}\left(\frac{t_0}{t}\right)^{(3-2\omega)p+1}\epsilon^{3-2\omega}k_{H}^{3-2\omega}.$$ Therefore the gravitational spectrum ${\cal P}_{g}(k_{r})$ is in this case $$\label{deq9}
{\cal P}_{g}(k_r)=\left.\frac{2^{2\omega}}{p\,\pi^{5}}\Gamma^{2}(\omega)\left(\frac{t_0}{t}\right)^{(3-2\omega)p+1}
k_{r}^{3-2\omega} \right|_{k_r=\epsilon k_H}\,.$$ Clearly, for $p\simeq 1/3$ (that corresponds to $m \simeq
3\sqrt{3}/2 > 3/2$), the spectrum is nearly scale invariant i.e. $n_{T}\equiv 3-2\omega \simeq 0$.
Final Comments
==============
In this letter we have studied the emergence of gravitational waves in the early universe, which is considered as dominated by a decaying cosmological parameter, from a 5D Riemann flat background metric, on which we define the 5D vacuum. Our approach is different to others, because we consider gravitational waves as originated by a physical field $Q_{ij}$, but not a tensorial linearized fluctuation of the metric. Therefore, it is possible to deal with $Q_{ij}$ as a quantum traceless tensorial field with null divergence.
The effective 4D dynamics of GW, $h_{ij} = Q_{ij}(\psi = \psi_0)$, can be viewed as induced by the foliation of the fifth coordinate. In this letter we have worked two examples: [**a)**]{} In the case with constant cosmological parameter, $\Lambda = \Lambda_0$, we obtain that the KK mass of the gravitons should be $m \simeq
3/\sqrt{2}$ to obtain a nearly scale invariant tensorial power spectrum of $\left<h^2\right>_{SH}$. This value of $m$ corresponds to an effective 4D mass $M \simeq 3H_0/2 = \sqrt{3\Lambda}/2$. Note that this mass can be related to the Einstein mass $m_E$, as it is considered in[@wess]: $M \simeq {3\over 2 G m_E}$. [**b)**]{} The case where the cosmological parameter decrease with time is the more interesting. In this case the gravitons are normalizable only when their effective 4D mass $M$ become null. We obtain that the spectrum of the squared fluctuations of gravitational waves are scale invariant for KK masses with values $m\simeq 3\sqrt{3}/2$.
.2cm
**[Acknowledgements]{}**
.2cm SPGM acknowledges UNMdP for financial support. MB acknowledges CONICET and UNMdP (Argentina) for financial support. JEMA acknowledges CNPq-CLAF and UFPB for financial support.\
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[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
[^3]: E-mail address: [email protected]
|
---
abstract: 'Arm locking is a proposed laser frequency stabilization technique for the Laser Interferometer Space Antenna (LISA), a gravitational-wave observatory sensitive in the milliHertz frequency band. Arm locking takes advantage of the geometric stability of the triangular constellation of three spacecraft that compose LISA to provide a frequency reference with a stability in the LISA measurement band that exceeds that available from a standard reference such as an optical cavity or molecular absorption line. We have implemented a time-domain simulation of a Kalman-filter-based arm locking system that includes the expected limiting noise sources as well as the effects of imperfect a priori knowledge of constellation geometry on which the design is a based. We use the simulation to study aspects of the system performance that are difficult to capture in a steady-state frequency domain analysis such as frequency pulling of the master laser due to errors in estimates of heterodyne frequency. We find that our implementation meets requirements on both the noise and dynamic range of the laser frequency with acceptable tolerances and that the design is sufficiently insensitive to errors in the estimated constellation geometry that the required performance can be maintained for the longest continuous measurement intervals expected for the LISA mission.'
author:
- 'J.I. Thorpe'
- 'P. Maghami'
- 'J. Livas'
bibliography:
- 'Thorpe041811.bib'
title: Time Domain Simulations of Arm Locking in LISA
---
Introduction
============
The Laser Interferometer Space Antenna[@Bender_98; @Jennrich_09] is a planned facility for observing gravitational radiation in the milliHertz frequency band, a regime rich in astrophysical sources. The LISA measurement concept[@Jennrich_09] calls for laser interferometry to be used to measure fluctuations in the distance between freely-falling test masses contained within spacecraft separated by $\sim5\times10^{9}\,\mbox{m}$ with a precision of $\sim10\times10^{-12}\,\mbox{m}$, or $\sim10\,\mbox{pm}$. The interferometric measurements are performed as a series of one-way measurements between pairs of spacecraft (SC) and then combined using a technique known as Time Delay Interferometry (TDI)[@Armstrong_99; @Shaddock_03] to form observables that suppress laser frequency noise while retaining gravitational wave signals.
The capability of TDI to reject laser frequency noise is chiefly limited by imperfect knowledge of the absolute light travel times between the spacecraft (often referred to as the “arm lengths”), which is expected to have an accuracy of $\sim1\,\mbox{m}$ or $\sim 3\,\mbox{ns}$. With this level of arm length accuracy, the contribution of laser frequency noise in the TDI observables will satisfy the allocated equivalent path length noise of $2.5\,\mbox{pm}/\sqrt{\mbox{Hz}}$ so long as the input laser frequency noise does not exceed a level of
$$\tilde{\nu}_{pre-TDI}(f)=\left(282\,\frac{\mbox{Hz}}{\sqrt{\mbox{Hz}}}\right)\cdot\sqrt{1+\left(\frac{2.8\,\mbox{mHz}}{f}\right)^{4}},\label{eq:preTDI-req}$$
where the Fourier frequency $f$ ranges over the LISA measurement band, $0.1\,\mbox{mHz}\leq f\leq0.1\,\mbox{Hz}$. The expected free-running noise level of the LISA lasers in the measurement band is roughly $10\,\mbox{kHz}/\sqrt{\mbox{Hz}}\cdot\left(1\,\mbox{Hz}/f\right)$, which exceeds the requirement in (\[eq:preTDI-req\]) by more than four orders of magnitude at the low end of the LISA band. Consequently, the lasers must be stabilized using an external frequency reference. A number of candidate stabilization schemes have been studied and determined to be viable from a noise performance perspective[@Shaddock_09; @Thorpe10]. The current focus is on evaluating other aspects of each candidate scheme such as complexity, robustness to implementation errors, and operational constraints so that the most effective design can be selected.
Two of the candidate schemes rely on arm locking, which utilizes the existing LISA science signals to derive a frequency reference from the geometry of the constellation. In one scheme, arm locking is the sole method employed to stabilize the laser frequency where in the other it is combined with another stabilization method in a hybrid system. We focus on the latter case in this paper.
Since its original introduction[@Sheard_03], the arm locking concept has been refined[@Sutton_08; @McKenzie_09] leading to improvements in its expected performance. Arm locking has also been studied in a number of hardware models[@Sheard_05; @Marin_05; @Wand_09], which have helped to identify potential implementation issues that were not readily apparent from frequency-domain studies of arm locking. Chief among these was the discovery that the inability to predict the heterodyne frequencies of the LISA science signals due to imperfect knowledge of the inter-spacecraft Doppler shifts leads to “frequency pulling” of the arm locked laser. If not properly mitigated, this frequency pulling can be so severe that the laser exceeds its dynamic range in a matter of hours.
In this paper we present the results from a series of time-domain simulations of arm locking. The goal is to combine the attractive features of the frequency domain models (realistic noise sources, orbit models, etc.) with those of the hardware models (sensitivity to transients, non-linearities, etc.). The rest of the paper is organized as follows. In section \[sec:Background\] we describe the problem of arm locking in LISA, defining the relevant signals. In section \[sec:Doppler\] we briefly review the frequency-pulling effect. We discuss our particular arm locking design and the technical details of our simulation in section \[sec:Simulations\] and present results in section \[sec:Results\].
\[sec:Background\]Arm Locking Model
===================================
To maintain the focus on the arm locking dynamics, we make a few simplifications in our model of the LISA interferometry. The first is that we only consider the interferometric measurements made between different SC (the “long-arm” signals) and ignore the additional measurements made between the SC and the proof mass (the “short-arm” signals). While a combination of both signals are needed to reach the $\sim10\,\mbox{pm}$ sensitivity levels required for detecting gravitational waves, the $\sim\mbox{nm}$ sensitivity level of the long-arm signals is more than sufficient for frequency stabilization at the level of (\[eq:preTDI-req\]).
A second simplification is that we model the two lasers on board the master SC as a single laser. In reality one of the lasers will be phase locked to the other using a high-gain phase lock loop. The residual noise in this phase lock loop is expected to be far below the other noise sources considered in this paper. Readers interested in additional detail on the LISA interferometric measurement concept should consult one of the many overview papers[@Shaddock_08; @Jennrich_09; @Thorpe10].
Notation
--------
Our notation system is an adaptation of that used by McKenzie, et al.[@McKenzie_09]. The most notable difference is that we represent signals as fluctuations in *frequency* rather than fluctuations in *phase* and replace $\phi$ with $\nu$ to reflect this. The three LISA spacecraft (SC) are labeled $SC_{i},\: i=1,2,3$ and it is assumed that $SC_{1}$ is the master SC. The many different frequency signals are labeled with both an alphabetic and a numeric subscript. The alphabetic subscript refers to the physical nature of the signal while the numeric subscript refers to the SC involved in producing the signal. In a two digit numeric subscript, the first digit indicates the receiving SC while the second refers to the transmitting SC. For example, $\nu_{S13}$ denotes the shot noise on the photoreceiver on board $SC_1$ that is receiving signals from $SC_3$. Appendix \[sec:notationKey\] contains a table summarizing the notation used in this paper and, where possible, the corresponding notation in [@McKenzie_09].
A single LISA laser link
------------------------
Figure \[fig:AL\_diagram\] shows a schematic of the LISA constellation. We begin our analysis of the arm locking signal chain with Laser 1 on $SC_{1}$, which produces light with a frequency $\nu_{O1}$. This frequency is a combination of the intrinsic frequency noise of the laser, $\nu_{L1}$, and the control signal provided by the arm locking loop. As the laser departs $SC_{1}$ in the direction of $SC_{3}$, it picks up a Doppler shift due to the motion of $SC_{1}$. The magnitude of this Doppler shift is $\lambda^{-1}\overrightarrow{V}_{1}\cdot\hat{\eta}_{13}$, where $\lambda$ is the wavelength of the laser, $\overrightarrow{V}_{1}$ is the velocity of $SC_{1}$, and $\hat{\eta}_{13}$ is the unit vector along the path from $SC_{1}$ to $SC_{3}$. For the purposes of calculating Doppler shifts, we make the assumption that $\hat{\eta}_{ij} = - \hat{\eta}_{ji}$ even though the rotation of the constellation causes these angles to differ on the $\sim\mu$radian level [@Jennrich_09].
![\[fig:AL\_diagram\]Schematic of frequency signals relevant to arm locking. See section \[sec:Background\] of the text for details and Appendix \[sec:notationKey\] for a key to notation. Adapted from Figure 1 of [@McKenzie_09]. ](schematic.eps){width="12cm"}
The laser then experiences a delay of $\tau_{13}$ on the order of $5\times10^{9}\,\mbox{m}/c\approx17\,\mbox{s}$ as it travels to $SC_{3}$. At $SC_{3}$, the signal picks up another Doppler term due to the motion of $SC_{3}$. At the photoreceiver on $SC_3$ it is interfered with the local laser (with frequency $\nu_{O3}$) to generate an electrical heterodyne signal with frequency
$$\nu_{H31}(t)=\nu_{O3}(t)-\nu_{O1}(t-\tau_{13})-\lambda^{-1}\left[\overrightarrow{V}_{1}(t-\tau_{13})-\overrightarrow{V}_{3}(t)\right]\cdot\hat{\eta}_{13}+\nu_{S31}(t),\label{eq:nuH31}$$
where $\nu_{S31}$ is a shot noise contribution due to the low light level of the received beam. A device we will refer to as the “frequency meter” (although it is more commonly called a phase meter [@Shaddock_06]) is used to measure the frequency of the heterodyne signal. The first step is digitization, which introduces a noise term due to fluctuations in the frequency of the oscillator used to drive the digitizers. To first order, this clock noise is additive with a spectral density that is proportional to the instantaneous heterodyne frequency,
$$\tilde{\nu}_{C31}(f)\equiv\nu_{H31}\cdot \tilde{y}_{3}(f).\label{eq:nuC31_def}$$
Here $\tilde{y}_{3}(f)$ represents the spectrum of fractional frequency fluctuations of the clock on board $SC_3$. The frequency meter also measures the frequency of the heterodyne signal relative to some model signal $\nu_{M31}$. The model signal can be used to remove the slow drift of the heterodyne frequency caused by time-varying Doppler shifts or to impose a constant frequency offset in a phase-lock loop. The output of the frequency meter is given by
$$\nu_{A31}(t)=\nu_{H31}(t) - \nu_{M31}(t) + \nu_{C31}(t).\label{eq:nuA31}$$
Doppler Shifts
--------------
The LISA SC will experience relative velocities along their lines of sight of several meters per second, resulting in Doppler shifts of several MHz. These Doppler shifts are approximately constant in the LISA measurement band and it is convenient to remove them prior to implementing arm locking. We begin by separating the SC velocity terms into a deterministic term arising from the SC orbits ($\overrightarrow{V}_{Oi}$) and a stochastic term arising from attitude jitter of the SC ($\delta\overrightarrow{V}_{i}$)
$$\overrightarrow{V}_{i}(t)=\overrightarrow{V}_{Oi}(t)+\delta\overrightarrow{V}_{i}(t).\label{eq:Vel_split}$$
This in turn leads to two Doppler contributions in the heterodyne signals $\nu_{Hij}$, an orbital motion term ($\nu_{Dij}$) and a SC jitter term ($\nu_{Jij}$) given by
$$\nu_{Dij}(t)=\lambda^{-1}\left[\overrightarrow{V}_{Oi}(t)-\overrightarrow{V}_{Oj}(t-\tau_{ji})\right]\cdot\hat{\eta}_{ij},\label{eq:DopOrbitFreq}$$
$$\nu_{Jij}(t)=\lambda^{-1}\left[\delta\overrightarrow{V}_{i}(t)-\delta\overrightarrow{V}_{j}(t-\tau_{ji})\right]\cdot\hat{\eta}_{ij}.\label{eq:JitterNoise}$$
With these signal definitions, the heterodyne signal on board $SC_{3}$ can be written
$$\nu_{H31}(t)=\nu_{O3}(t)-\nu_{O1}(t-\tau_{13}+\nu_{D31}(t)+\nu_{J31}(t)+\nu_{S31}(t)\label{eq:nuH31_linear}$$
Phase locking on $SC_2$ and $SC_3$
----------------------------------
Arm locking requires that the slave SC ($SC_2$ and $SC_3$ in our example) operate in a transponder mode, returning to the master SC a copy of the light field that was received. This is accomplished by using a phase-lock loop to control the lasers on board the slave SC. Using $SC_3$ in Figure \[fig:AL\_diagram\] as an example, the controller $G_3$ adjusts the frequency of Laser 3 to minimize the output of the frequency meter, $\nu_{A31}$. In the Laplace domain, the output of Laser 3 will be
$$\nu_{O3}(s)=\frac{G_{3}}{1+G_{3}}\left[\nu_{M31}+\nu_{O1}(s)e^{-s\tau_{13}}-\nu_{D31}(s)-\nu_{J31}(s)-\nu_{S31}(s)-\nu_{C31}(s)\right]+\frac{1}{1+G_{3}}\nu_{L3}(s).\label{eq:SC3_PLL}$$
Under the assumption of a high-bandwidth phase lock loop, $G_{3}\gg1$, this simplifies to
$$\nu_{O3}(s)\approx\nu_{M31}+\nu_{O1}(s)e^{-s\tau_{13}}-\nu_{D31}(s)-\nu_{J31}(s)-\nu_{S31}(s)-\nu_{C31}(s).\label{eq:SC3_PLL_ideal}$$
Formation of the arm locking error signal
-----------------------------------------
The signal from Laser 3 is transmitted back to $SC_{1}$, picking up a Doppler contribution form $SC_{3}$, a time delay $\tau_{31}$, and a Doppler contribution from $SC_{1}$. At $SC_{1}$ it is interfered with Laser 1 on a photoreceiver, generating shot noise $\nu_{S13}$. Fluctuations in the heterodyne signal are measured by a frequency meter, which subtracts a model $\nu_{M13}(t)$ and adds a clock noise $\nu_{C13}$ to produce the main science signal for the $SC_{1}-SC_{3}$ arm, $\nu_{A13}(t)$. Using (\[eq:SC3\_PLL\_ideal\]) to replace $\nu_{O3}$, $\nu_{A13}$ can be written as
$$\begin{aligned}
\nu_{A13}(t) & = & \left[\nu_{O1}(t)-\nu_{O1}(t-\tau_{13}-\tau_{31})\right]+\left[\nu_{J13}(t)+\nu_{J31}(t-\tau_{31})\right]\nonumber \\
& + & \left[\nu_{S13}(t)+\nu_{S31}(t-\tau_{31})\right]+\left[\nu_{C13}(t)+\nu_{C31}(t-\tau_{31})\right]\nonumber \\
& + & \left[\nu_{D13}(t)+\nu_{D31}(t-\tau_{31})\right]-[\nu_{M13}(t)+\nu_{M31}(t-\tau_{31})].\label{eq:nu_A13_time}\end{aligned}$$
The second to last bracketed term in (\[eq:nu\_A13\_time\]) represents the deterministic part of the heterodyne signal. The model signal in the frequency meter, $\nu_{M13}(t)$, can be used to remove this term, leaving behind a (hopefully small) residual error term, $\nu_{E13}(t)$,
$$\nu_{M13}(t)=\nu_{D13}(t)+\nu_{D31}(t-\tau_{31})-\nu_{M31}(t-\tau_{31})-\nu_{E13}(t).\label{eq:nuM13}$$
These residual errors lead to frequency pulling of the master laser. Section \[sec:Doppler\] presents estimates for the size of these errors. With the deterministic terms (mostly) removed, $\nu_{A13}$ can be represented in the Laplace domain as
$$\begin{aligned}
\nu_{A13}(s) & = & \nu_{O1}(s)\left[1-e^{-s(\tau_{13}+\tau_{31})}\right]+\left[\nu_{J13}(s)+\nu_{J31}(s)e^{-s\tau_{31}}\right]+\left[\nu_{C13}(s)+\nu_{C31}(s)e^{-s\tau_{31}}\right]\nonumber \\
& + & \left[\nu_{S13}(s)+\nu_{S31}(s)e^{-s\tau_{31}}\right]+\nu_{E13}(s).\label{eq:nuA13_s}\end{aligned}$$
The $SC_{1}-SC_{2}$ arm produces a signal, $\nu_{A12}(s)$, that is analogous to $\nu_{A13}(s)$. The arm locking sensor is a linear combination of these two signals that is used to estimate the Laser 1 fluctuations, $\nu_{O1}$, so that they can be suppressed in a feedback loop. The output of the arm locking sensor, labeled $\nu_{B1}$ in Figure \[fig:AL\_diagram\] is given by
$$\nu_{B1}=\textbf{S}\left[\begin{array}{c}\nu_{A12}\\\nu_{A13}\end{array}\right],\label{eq:nuB1def}$$
where $\textbf{S}$ is the arm locking sensor vector that describes the specific linear combination of the two individual arm signals. For example, the “common-arm” sensor, uses the sensor vector $\textbf{S}_{+}\equiv[\frac{1}{2},\:\frac{1}{2}]$. Table I in [@McKenzie_09] provides expressions for several arm locking sensors that have been studied in the literature.
Noise Levels {#subsec:noiseLevels}
------------
### Intrinsic Laser Frequency noise
For this work we assume a pre-stabilized, frequency tunable laser source with a frequency noise spectral density in the LISA band of
$$\tilde{\nu}_{L}(f)=\left(800\,\frac{\mbox{Hz}}{\sqrt{\mbox{Hz}}}\right)\cdot\sqrt{1+(\frac{2.8\,\mbox{mHz}}{f})^{4}}\quad 0.1\,\mbox{mHz}\leq f \leq 0.1\,\mbox{Hz}.\label{eq:MZnoise}$$
This is representative of the noise-floor of the Mach-Zehnder interferometer stabilization system[@Steier_09] that will fly on LISA Pathfinder [@Armano_09], a LISA technology demonstrator mission.
### Shot Noise
Shot noise is uncorrelated at each detector and has an equivalent frequency noise spectrum of
$$\tilde{\nu}_{S}(f)=\sqrt{\frac{hc}{\lambda P_{rec}}}\left(\frac{\mbox{Hz}}{\sqrt{\mbox{Hz}}}\right)\left(\frac{f}{1\,\mbox{Hz}}\right),\label{eq:ShotNoise}$$
where $\lambda=1064\,\mbox{nm}$ is the laser wavelength and $P_{rec}\sim100\,\mbox{pW}$ is the received power. For these numbers, (\[eq:ShotNoise\]) gives $\tilde{\nu}_{S}=43\,\mu\mbox{Hz}/\sqrt{\mbox{Hz}}\cdot(f/1\,\mbox{Hz})$.
### Clock Noise
The spectral density of the fractional frequency variations of the SC clocks are estimated to be (Table IV of [@McKenzie_09])
$$\tilde{y}(f)=2.4\times10^{-12}/\sqrt{f}.\label{eq:ClockNoiseLevel}$$
While LISA will employ a clock-transfer scheme to correct for differential clock noise between the SC[@Klipstein_06], we assume that that correction takes place in post processing on the ground and is not applied to the arm locking error signals on board the SC.
### Spacecraft Jitter\[sub:Spacecraft-motion\]
The spacecraft jitter noise model is based on simulations of the drag-free control performance of LISA[@Maghami03]. The jitter can be divided into two orthogonal components in the plane of the LISA constellation that are independent. Each of these has a position jitter in the LISA measurement band of
$$\delta\tilde{x}(f)=2.5\,\mbox{nm}/\sqrt{\mbox{Hz}}\quad 0.1\,\mbox{mHz}\leq f \leq 0.1\,\mbox{Hz}.\label{eq:JitterNoiseLevel}$$
Note that due to the fact that the interior angle of the constellation is not $90\,\mbox{deg}$, the spacecraft jitter contributions from $SC_{1}$ will be partially correlated in the frequency meter signals $\nu_{A12}(t)$ and $\nu_{A13}(t)$. Finally, we note that it would in principle be possible to remove the spacecraft jitter by including the “short-arm” interferometers in both the phase-lock error signals in the far SC and the arm locking error signals in the master SC. This would reduce the jitter from $\sim\mbox{nm}$ to $\sim\mbox{pm}$ in the LISA band. As with the clock noise correction, this would require additional on-board processing and is not necessary to reach the pre-TDI noise requirement specified in (\[eq:preTDI-req\]).
\[sec:Doppler\]Laser frequency pulling and Heterodyne estimation
================================================================
In section \[sec:Background\], we explained how Doppler shifts arising from the SC orbits enter the long-arm frequency meter signals and how the deterministic parts of the signals are removed using models of the heterodyne frequency. If we take the expression for the main science signal of the $SC_{1}-SC_{3}$ arm, $\nu_{A13}$, as expressed in (\[eq:nuA13\_s\]) and consider only the terms resulting from the laser frequency, $\nu_{O1}$, and the errors in the heterodyne estimate, $\nu_{E13}$, the result is$$\nu_{A13}(s)=\nu_{O1}(s)\left[1-e^{-s(t-\tau_{13}-\tau_{31})}\right]+\nu_{E13}(s).$$ If we take the low frequency limit, $s\rightarrow0$, we find that the first term vanishes. In other words, the signal in $\nu_{A13}$ is insensitive to fluctuations in laser frequency at zero frequency. The second term, however, is unaffected. The situation is obviously the same for $\nu_{A12}$ and also for any arm locking sensor formed as a linear combination of $\nu_{A12}$ and $\nu_{A13}$. If the arm locking controller has any gain at zero frequency, it will cause the laser to ramp in an attempt to zero out the heterodyne error terms. This laser frequency pulling can be mitigated by reducing the arm locking loop gain below the LISA measurement band, although care must be taken to ensure that sufficient gain is still present within band. The rate of pulling is proportional to the error in the estimate of the heterodyne frequency, hence the design requirements of the control filter will be driven by the accuracy with which the heterodyne frequency can be estimated.
As pointed out by [@McKenzie_09], it is convenient to combine the heterodyne models from the individual arms into a common and differential component. For the case where $SC_{1}$ is the master SC, the common and differential heterodyne models are $$\begin{aligned}
\nu_{M+}(t) & \equiv & \nu_{M12}(t)+\nu_{M13}(t)\nonumber \\
\nu_{M-}(t) & \equiv & \nu_{M12}(t)-\nu_{M13}(t)\label{eq:CommDiffDop}\end{aligned}$$ It is also expected that arm locking may require periodic re-acquisition, either because of an external disturbance (e.g. pointing of the high-gain antenna) or because some component of the arm locking system is in a non-desirable operating range (e.g. laser near a longitudinal mode transition, arms close to equal, etc.). Consequently, the heterodyne frequency only needs to be estimated for periods on the order of weeks. For such periods, it is appropriate to use a quadratic model,
$$\nu_{Mx}(t)\approx\nu_{0x}+\gamma_{0x}t+\alpha_{0x}t^{2},\:\:x=(+,-)\label{eq:DopTaylor}$$
Expected Doppler
----------------
Although the models for the heterodyne signals include both Doppler shifts and intentional frequency offsets, the Doppler shifts provide the only source of uncertainty. There are a number of realizations of the LISA orbits that can be used to derive expected Doppler frequencies. All exhibit a primary frequency of $\sim 1\,\mbox{yr}^{-1}$ with harmonics of various amplitudes. There is also a secular component that tends to degrade the constellation (higher Doppler shifts, larger arm length mismatches, etc.) as the mission progresses. In Figure \[fig:Doppler\] we plot each of the six Doppler parameters in (\[eq:DopTaylor\]) resulting from an orbital solution by Hughes [@Hughes_08] that was optimized to minimize the average Doppler frequency in each arm. Each plot contains three traces, one for each possible choice of master SC.
\[subsec:DoppMeth\]Doppler Estimation Methods
---------------------------------------------
A number of methods have been proposed for determining the Doppler frequency. One method is to use the orbital ephemeris, such as the one plotted in Figure \[fig:Doppler\] to predict the Doppler. With periodic updates to the ephemeris from ranging data taken during normal SC down-link operations, the ephemeris velocities can be expected to be accurate to $3\,\mbox{mm}/\mbox{s}$ [@Thornton05]. One issue is that the measured velocity is the projected component along the line of sight between Earth and the SC. Transverse velocities are not directly measured but are still constrained by the orbital model. Consequently one would expect that the errors in Doppler estimation could differ by a large amount between different inter-SC links.
Another method for estimating the Doppler frequency is to differentiate the active ranging signal that is used to determine the absolute link lengths for the TDI algorithm. Unlike ground tracking, this method directly measures the velocity along the inter-SC link. Ranging is expected to have position accuracy of $\sim1\,\mbox{m}$ or better over averaging periods of $\sim1000\,\mbox{s}$ [@Esteban_09]. This suggests that velocities could be measured to $\sim\mbox{mm}/\mbox{s}$ accuracy, corresponding to Doppler frequency errors on the order of $\sim\mbox{kHz}$. Additional processing such as longer averaging, Kalman filtering, or combination with an orbital model may allow for further improvements [@Heinzel11]. In all cases, the processing (including the determination of range from the pseudo-random code) would take place on ground. Consequently there would be some delay before the updated Doppler model could be uploaded to the SC.
McKenzie, et al. [@McKenzie_09] proposed a simple method for determining the heterodyne frequency directly from the frequency meter data itself. If we consider the expression for the main science signals (\[eq:nu\_A13\_time\]), all of the terms contain mean-zero stochastic processes with the exception of the Doppler terms. Applying a simple averaging filter to this signal can suppress the noise terms to reveal the heterodyne frequency. This simple algorithm relies only on information from the master SC and could be easily implemented on board. Table \[tab:DopErr\] gives the errors in the Doppler coefficients estimated by McKenzie, et al. assuming a MZ stabilized laser with a frequency noise spectrum given by (\[eq:MZnoise\]) and a $200\,\mbox{s}$ averaging time.
Parameter $\nu_{0+}$ $\nu_{0-}$ $\gamma_{0+}$ $\gamma_{0-}$ $\alpha_{0+}$ $\alpha_{0-}$
----------- ----------------- ------------------- -------------------- --------------------- --------------- ---------------
Error $45\,\mbox{Hz}$ $0.51\,\mbox{Hz}$ $2.2\,\mbox{Hz/s}$ $0.02\,\mbox{Hz/s}$ \* \*
: \[tab:DopErr\]Doppler errors from $200\,\mbox{s}$ averaging of science signal with Mach-Zehnder stabilized laser frequency noise given by (\[eq:MZnoise\]). Adapted from Table III of [@McKenzie_09]. A \* indicates the error was greater than the expected signal and hence the measurement is not used.
Arm Locking Simulations\[sec:Simulations\]
==========================================
\[sub:Design\]Sensor Design
---------------------------
As mentioned in the introduction, a number of arm locking variants have been proposed. They differ in the way the science signals from the two arms extending from the master SC are combined to form an error signal (and of course the matching controller design that completes the control system). In the language of (\[eq:nuB1def\]), the sensor vector $\textbf{S}$ differs for each design. The general goal in designing the sensor vector is to make the transfer function from laser frequency noise to arm locking sensor output as simple as possible with flat amplitude and phase responses. Ideally, $|P(f)|\approx1$ and $\partial \angle P(f)/\partial f = 0$, where
$$P(f)\equiv\frac{\nu_{B1}(f)}{\nu_{O1}(f)}\label{eq:SensorTF}$$
For example, the original proposal for single arm locking, with sensor matrix $\textbf{S}_{single}=[\frac{1}{2},\:\:0]$, has $P_{single}(f)=i\sin(2\pi f\tau)e^{-2\pi if\tau}$, where $\tau=\tau_{12}+\tau_{21}$ is the round-trip light travel time. This transfer function has nulls at frequencies $f_{n}=n/\tau\approx33\,\mbox{mHz}\cdot n,\: n=1,2,3...$, a number of which lie in the LISA measurement band. Since the sensor cannot measure the frequency fluctuations at these frequencies, the control system cannot correct for them. Furthermore, the rapid swings in the transfer function phase at the $f_{n}$ frequencies make it difficult to design a stable controller that extends beyond $f_1$. More sophisticated arm locking sensors, such as the modified dual arm locking sensor (MDALS)[@McKenzie_09] make a careful blend of the two science signals to generate a sensor with a nearly-flat transfer function in the measurement band.
The problem of blending of multiple sensors to generate the best possible measurement of a state variable is a classical problem in control theory. In a previous work[@Maghami_09], we applied Kalman filtering techniques to generate an arm locking sensor. We will briefly review this approach here.
We begin by making a time-invariant, discrete-time linear state space model of the LISA constellation. The state vector represents the time history of the laser frequency noise over the storage time in the arms,
$$\overrightarrow{x}_{k}=\left\{\begin{array}{c}
\nu_{O1}[(k-1)\Delta t] \\
\nu_{O1}[(k-2)\Delta t] \\
\vdots \\
\nu_{O1}[(k-T_{12})\Delta t]
\end{array}\right\}.\label{eq:stateVector}$$
where $\Delta t$ is the discretization time, $k$ is the time index and $T_{1j}\equiv round [(\tau_{1j} + \tau_{j1})/\Delta t]$ is the index corresponding to the round-trip light travel time between $SC_1$ and $SC_j$. We assume without loss of generality that $T_{12} > T_{13}$. Note that the first element in the state vector represents the frequency delayed by a single time step as opposed to the instantaneous frequency.
The state vector is updated using the following relations
$$\overrightarrow{x}_{k+1} = \mathbf{A}\overrightarrow{x}_k+\mathbf{B}\,u_k + \mathbf{\Gamma}\,w_k,\label{eq:stateAB}$$
$$\mathbf{A}=\left[\begin{array}{cccccc}
0 & 0 & 0 & \cdots & \cdots & 0 \\
1 & 0 & 0 & \cdots & \cdots & 0 \\
0 & 1 & 0 &\ddots & \cdots & \vdots \\
0 & 0 & 1 &\ddots & \cdots & \vdots \\
\vdots & \vdots & \ddots &\ddots & 0 & \vdots \\
0 & 0 & \cdots &\cdots & 1 & 0
\end{array}\right]\label{eq:Amatrix}$$
$$\mathbf{B}=\mathbf{\Gamma}=\left[1\:\:0\:\cdots\:0\right]^{T}.\label{Bmatrix}$$
The scalar $u_k$ represents the frequency control signal applied to the laser. in this case it would be the output of the block labeled $G_1$ in Figure \[fig:AL\_diagram\].
$$u_k = \nu_{B1}(k\Delta t) \otimes G_1, \label{eq:udef}$$
.
where $\otimes$ denotes convolution. Similarly, the scalar $w_k$ represents the instantaneous frequency noise applied to the laser at time $k \Delta t$,
$$w_k = \nu_{L1}(k \Delta t).\label{eq:wdef}$$
In words, (\[eq:stateAB\]) says that the laser control signal and the laser noise effect only the first element of the state vector and the remaining elements are determined through simple time delays.
The two frequency meter outputs can be combined to form a two-element measurement vector, $\overrightarrow{y}_k$ which is determined from the following relation,
$$\overrightarrow{y}_k\equiv \left[\begin{array}{c} \nu_{A12}(k\Delta t) \\ \nu_{A13}(k\Delta t) \end{array} \right] = \mathbf{C}\,\overrightarrow{x}_k+\mathbf{D}\,u_k+\mathbf{H}\,w_k+\overrightarrow{n}_k.\label{eq:stateMeas}$$
$$\mathbf{C}=\left[\begin{array}{cccccc}
0 & 0 & \cdots & \cdots & \cdots & -1 \\
0 & \cdots & 0 & -1 & 0 & \cdots \end{array}\right].\label{eq:Cdef}$$
$$\mathbf{D}=\mathbf{H}=\left[ \begin{array}{c} 1 \\ 1 \end{array} \right].\label{Dmatrix}$$
$\mathbf{C}$ is a $2\times T_{12}$ matrix describing how the state vector couples into the frequency meter measurements. All elements of $\mathbf{C}$ are zero with the exception of $(1,\,T_{12})$ and $(2,\,T_{13})$, which are $-1$. This represents the fact that the frequency meter measurement includes a copy of the master laser phase delayed by the round-trip light travel time in the arm. The $2\times 1$ vector $\overrightarrow{n}_k$ represents the noise in each of the two frequency meter signals at the current time step. This includes the shot noise, clock noise, and spacecraft jitter noise contributions. The noise level can be determined from the noise spectra in section \[subsec:noiseLevels\] and the transfer functions to frequency meter outputs in (\[eq:nuA13\_s\]).
Equations (\[eq:stateAB\]) and (\[eq:stateMeas\]) are the classical state-space description of a linear system. Kalman filtering [@Stengel; @Phillips_and_Nagle] is a prescription for generating an optimal estimate of the state vector provided information about the system matrices and noise processes are available. In our case, the state matrices are determined by the arm-lengths (through the definitions of $T_{12}$ and $T_{13}$) and the noise models that determine $w_k$ and $\overrightarrow{n}_k$. The output of the Kalman filter is an estimate of the state vector, $\overrightarrow{x}_k$, the first element of which corresponds to the laser frequency at time $(k-1)\Delta t$. This element is the output of our Kalman-filter based sensor, which we refer to as an Optimal Arm Locking Sensor or OALS.
In Figure \[fig:SensorComp\], we show a comparison of the transfer function from laser frequency noise to sensor output for the single-arm, MDALS, and OALS designs. The round-trip arm lengths were chosen to be $33\,\mbox{s}$ and $32.4\,\mbox{s}$ for all three sensors for direct comparison purposes. There is no specific significance to the choice. The OALS sensor was computed with $\Delta t = 0.1\,\mbox{s}$ and perfect arm-length knowledge was assumed for both MDALS and OALS. When compared with the single-arm sensor, both MDLAS and OALS exhibit much flatter responses in the LISA measurement band. This allows arm-locking systems based on them to achieve more uniform suppression, particularly near the round-trip frequencies. The OALS has less ripple than the MDALS at frequencies below $\sim 100\,\mbox{mHz}$ but reaches a peak in-band ripple around $300\,\mbox{mHz}$ which is similar to that of MDALS. Both sensors can be used to build arm locking systems that meet the LISA performance criteria. The OALS is optimal in the sense that it is generated using optimal control theory techniques. When paired with a suitable controller, we find that the net system performance is similar to that with achieved with the MDALS sensor, which gives us some confidence in that design.
![\[fig:SensorComp\]Transfer function from master laser frequency noise to sensor output for various arm locking sensors: Single arm sensor[@Sheard_03], MDALS[@McKenzie_09], and OALS (this work). For all cases, the round-trip light travel times in the two arms are $33\,\mbox{s}$ and $32.4\,\mbox{s}$.](sensorCompare.eps){width="12cm"}
Controller Design
-----------------
The second component in an arm locking system is a controller, which takes the estimate of the laser frequency provided by the arm locking sensor and generates a frequency tuning command for the laser. The design goals of the controller are to provide sufficient gain within the LISA measurement band to suppress the intrinsic frequency fluctuations of the master laser (\[eq:MZnoise\]) below the levels tolerated by TDI (\[eq:preTDI-req\]). As mentioned in section \[sec:Doppler\], care must also be taken to minimize the controller gain at very low (below measurement band) frequencies to mitigate laser frequency pulling.
The controller is based on a classical lead-lag design. It includes a second-order lead filter at the lower frequencies (break frequency at $0.05\,\mbox{mHz}$) to abate laser pulling due to uncompensated Doppler and Doppler derivative. It also includes a shaping filter and a single-order attenuation filter at $4\,\mbox{Hz}$ to limit the controller action to the LISA band. Figure \[fig:ControllerBode\] contains a Bode plot of the controller.
![\[fig:ControllerBode\]Bode plot of arm locking controller](controller.eps){width="12cm"}
Simulation Design
-----------------
We implemented a discrete-time simulation of arm locking as described in the preceding sections using the SIMULINK software package. Each arm was modeled in a manner consistent with section \[sec:Background\]. The round-trip arm lengths were assumed to be $\tau_{12}+\tau_{21}=33\,\mbox{s}/c$ and $\tau_{13}+\tau_{31}=32.4\,\mbox{s}/c$, where $c$ is the speed of light. The phase lock loops on $SC_{2}$ and $SC_{3}$ were assumed to be perfect ($G_{2}=G_{3}\gg1$) with constant frequency offsets ($\nu_{M21}=10\,\mbox{MHz}$, $\nu_{M31}=15\,\mbox{MHz}$). The Doppler shifts in each arm were modeled as a linearly-varying frequency with the coefficients provided by the orbital model in Figure \[fig:Doppler\] at a time $t=1\,\mbox{yr}$. Doppler errors were linear in time with the coefficients provided in Table \[tab:DopErr\]. The spectrum of intrinsic frequency fluctuations in the laser systems was modified from (\[eq:MZnoise\]) to include two poles at $0.6\,\mu\mbox{Hz}$, limiting the total frequency excursion to $\sim(20\,\mbox{MHz})$ over the maximum simulation period of two weeks. A two pole roll off at $0.5\,\mbox{Hz}$ was added to the spacecraft jitter noise in (\[eq:JitterNoise\]) to model the dynamics of the SC above the measurement band.
The simulation cadence was $500\,\mu\mbox{s}$. System dynamics, noise generators, and the controller operated at this cadence. The OALS was implemented with the designed 10Hz sampling rate, with appropriate downsampling and upsampling filters providing the rate transitions. The OALS filter order was also reduced from the nominal order of 332 to 38 using balanced reduction[@Maciejowski89]. This reduction provides a dramatic increase in simulation speed without changing the behavior in the LISA measurement band.
Results\[sec:Results\]
======================
Component Noise Sources
-----------------------
The first goal of the time-domain simulation was to verify the analytic, frequency-domain model of the arm locking system. Figure \[fig:AL\_noise\_freq\] contains a noise decomposition of the OALS arm locking system derived from an analytic model. As can be seen, the overall noise in the stabilized laser is dominated by residual laser frequency noise, with the other noise sources being nearly four orders of magnitude smaller. Figure \[fig:AL\_noise\_time\] shows a similar plot obtained using the time-domain simulations. To obtain each curve, the simulation was run with all noise sources except the source of interest turned off. In all cases, the Doppler estimation errors were set to zero. The time series were then used to estimate a spectra. The two plots show good agreement over most of the LISA band. The primary differences are a broadening of the sharp spectral features near $f=n/\tau$ and a roll-off at low frequencies in the time-domain plots. Both of these effects are consistent with spectral estimation errors in the logarithmic power spectral density algorithm [@Trobs_06] used to compute the spectra from the time series outputs.
Laser Frequency Pulling
-----------------------
The second goal of the time-domain simulations was to explore phenomena that are not easily treated analytically in the frequency domain. The laser frequency pulling described in section \[sec:Doppler\] is an important example of such a phenomenon. We ran a simulation spanning two full weeks (the expected time between SC maintenance periods) with Doppler estimation errors consistent with our models of the errors in the averaging method. Figure \[fig:long\_run\_time\] shows the results of this simulation. In the top panel, there are two curves plotted: the frequency change of the arm locked system and the frequency change of the intrinsic MZ noise. The first thing to notice is that the arm locked system drifts over approximately $20\,\mbox{MHz}$ over the two week simulation period, well within the expected linear tuning range of the LISA lasers. The second thing to notice is that the frequency drift in the arm locked system is approximately equal to the drift in the intrinsic noise. This is due to the fact that the arm locking loop has no effect below the LISA measurement band. The lower panel plots the difference of the arm locked and intrinsic frequency drifts, which gives an indication as to the level of additional drift generated by the arm locking system. After an initial transient decays over the first few days, the remaining fluctuations are less than $1\,\mbox{MHz}$. This demonstrates that this arm locking design does not produce any significant pulling of the master laser frequency beyond what is already present in the MZ stabilization system.
![\[fig:long\_run\_time\]Top panel: comparison of arm locked laser frequency drift with intrinsic laser frequency drift. Bottom panel: Difference of the two curves in the upper panel giving a rough estimate of the contribution to laser frequency drift from the arm locking system. ](longrun.eps){width="12cm"}
Robustness to Arm Length Errors
-------------------------------
Like the MDALS sensor, the OALS sensor requires some a priori knowledge of the LISA arm lengths. Many of the same techniques described in section \[subsec:DoppMeth\] that can potentially be used for estimating the Doppler frequencies can also be applied to estimate arm lengths. Which technique is most applicable will depend on how sensitive the performance of the arm locking system is to errors in the estimated arm lengths used to compute the sensor. For example, if the maximum tolerable error is $\sim1\,\mbox{m}$ then active ranging is likely the best candidate. If, on the other hand, errors of $\sim10\,\mbox{km}$ are tolerable, it may be possible to compute them from orbital ephemerides on the ground and upload new coefficients to the OALS periodically.
To check the robustness of the OALS to errors in the arm length, we first define the mean and differential arm lengths assuming $SC_1$ is the master SC, $$\tau_{m}\equiv\frac{1}{2}\left[\tau_{12}+\tau_{21}+\tau_{13}+\tau_{31}\right],\label{eq:tau_m_def}$$ $$\delta\tau\equiv\left[\tau_{12}+\tau_{21}-\tau_{13}-\tau_{31}\right],\label{eq:delta_tau_def}$$ We then design an OALS for a specific set of nominal arm lengths, $\tau_m^{(0)}=32.85\,\mbox{s}$ and $\delta\tau^{(0)}=0.3\,\mbox{s}$, corresponding to the constellation geometry at $t\approx1.25\,\mbox{yrs}$ in the orbital solution used in Figure \[fig:Doppler\]. This sensor is used to stabilize an array of arm locking systems with different true arm lengths, which corresponds not only to an error in the determination of the true arm lengths, but also represents the situation where the spacecraft constellation has evolved in time away from the design point. To quantify the effect of arm length errors, we define the figure of merit
$$\Psi_{0}(\tau_m,\delta\tau)\equiv 20\,\log_{10}\left[\underset{f}{\max}\frac{\tilde{\nu}_{pre-TDI}(f)}{\tilde{\nu}_{O1}(f,\tau_m,\delta\tau)}\right]\label{eq:costFcn0}$$
where $\tilde{\nu}_{O1}(f,\tau_m,\delta\tau)$ is the residual noise in the master laser and $\tilde{\nu}_{pre-TDI}(f)$ is the residual noise requirement specified in (\[eq:preTDI-req\]). $\Psi_0$ measures the minimum margin in the LISA measurement band between the arm locking system system and the pre-TDI requirement. Where this minimum is positive, the frequency stabilization is guaranteed to meet performance requirements.
Figure \[fig:armSens\] shows a contour plot of $\Psi_0$ for this example plotted on the $(\tau_m,\delta\tau)$ plane. The design point $(\tau_m^{(0)},\delta\tau^{(0)})$ is indicated by a white diamond. The evolution of $\tau_m$ and $\delta\tau$ due to the LISA orbit near the design is indicated by the dashed line with the grey dots indicating time intervals of 15 days. Figure \[fig:armSens\] shows that positive margin exists for $\sim 20\,\mbox{days}$ prior to the design point and $\sim 100\,\mbox{days}$ afterwards. This is much larger than the expected intervals between maintenance activities, indicating that updating the OALS sensor coefficients will not drive the maintenance schedule of the mission. The existence of a large operating window gives us some confidence that the OALS design is robust enough for the LISA application.
When arm length errors eventually do cause the system performance to degrade to the point where the margin is inadequate, it will be necessary to change the OALS coefficients and possibly which SC is designated as the master. Changing the coefficients can likely be done smoothly without losing lock or degrading system performance. Changing the master SC will require re configuring of the phase lock loops aboard all SC and will result in some down time. This should only be required when $\delta\tau$ for a certain arm combination becomes sufficiently small, likely $1-3$ times per year.
![\[fig:armSens\]Robustness of an example OALS-based arm locking system to errors in arm length versus the mean ($\tau_m$) and differential ($\delta\tau$) arm lengths. The contours show $\Psi_0$, the minimum margin in any given frequency bin within the LISA measurement band. The white diamond marks the design point for the sensor and the dashed line shows the evolution of the arm lengths due to LISA orbital motion near the design point with the grey dots spaced in time by 15 days. The dark red color shows a positive margin between 0 and 5 dB indicating that the performance of the system meets or exceeds the requirement. Evolution of the system is from left to right, indicating that the system has adequate performance from approximately 20 days before the design point to approximately 100 days after.](costMax.eps){width="12cm"}
\[sec:Conclusions\]Conclusions
==============================
Arm Locking is a candidate laser frequency stabilization technique for LISA. We have used a time domain simulation to study a design for an arm locking system based on a Kalman filter optimal blended sensor and a controller which meets the frequency stability requirements for LISA assuming the master laser is pre-stabilized to a level of $800\,\mbox{Hz}/\sqrt{\mbox{Hz}}$. Time domain simulations allowed us to study transient phenomena and the performance of the stabilization system as the conditions of the LISA constellation evolve during the normal orbital motion, including Doppler shifts and imperfectly known arm lengths. The simulations indicate that it is possible to implement arm locking without excessive pulling of the the master laser frequency, and that the arm locking sensor performance is robust against errors in the absolute arm length estimates. This robustness allows the sensor to be periodically updated with pre-computed filter coefficients at intervals that are operationally reasonable. Although our time-domain simulation necessarily included specific sensor and controller designs, the same simulation infrastructure could be applied to study other candidate sensor and controller designs.
We would like to thank Kirk McKenzie for providing the tools needed to compute the MDALS arm-locking performance and Steve Hughes for providing LISA orbital data.
Copyright (c) 2011 United States Government as represented by the Administrator of the National Aeronautics and Space Administration. No copyright is claimed in the United States under Title 17, U.S. Code. All other rights reserved.
Key to Notation\[sec:notationKey\]
==================================
Symbol Description Correspondence in [@McKenzie_09]
-------------------------------- ---------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------
$G_{i}(s)$ Controller transfer function on $SC_{i}$ $G_{i}(s)$
$\textbf{S}$ Arm locking sensor vector $\textbf{S}$
$\overrightarrow{V}_{i}$ total velocity of $SC_{i}$ N/A
$\overrightarrow{V}_{Oi}$ orbital component of $SC_{i}$ velocity N/A
$\delta\overrightarrow{V}_{i}$ jitter component of $SC_{i}$ velocity $\left(\delta\overrightarrow{V}_{i}-\delta\overrightarrow{V}_{j}\right)\cdot\hat{\eta}_{ij}=\frac{\partial}{\partial t}\Delta X_{ij}$
$\tilde{y}_{i}(f)$ fractional frequency fluctuations of $SC_{i}$ clock $\tilde{y}_{i}(f)$
$\hat{\eta}_{ij}$ unit vector from $SC_{i}$ to $SC_{j}$ N/A
$\lambda$ wavelength of lasers $\lambda$
$\nu_{Aij}$ output of frequency meter $ij$ $\frac{\partial}{\partial t}\phi_{Aij}$
$\nu_{B1}$ output of arm locking sensor $\frac{\partial}{\partial t}\phi_{B1}$
$\nu_{Cij}$ clock noise generated by frequency meter $ij$ $\frac{\partial}{\partial t}\phi_{Cij}$
$\nu_{Dij}$ Orbital Doppler shift measured by frequency meter $ij$ N/A
$\nu_{E1j}$ Error in heterodyne model $\nu_{DE1j}$
$\nu_{Hij}$ heterodyne signal at photoreceiver $ij$ N/A
$\nu_{Jij}$ Spacecraft jitter Doppler shift at measurement $ij$ $\frac{\partial}{\partial t}\phi_{Xij}$
$\nu_{Li}$ intrinsic frequency noise of Laser $i$ $\frac{\partial}{\partial t}\phi_{Li}$
$\nu_{Mij}$ Heterodyne model signal on frequency meter $ij$ $\Delta_{i1}\:\:i=2,3$
$\nu_{M+(-)}$ model of common (differential) component of heterodyne signals on $SC_{1}$ N/A
$\nu_{0+(-)}$ constant part of $\nu_{M+(-)}$ $\nu_{0+(-)}$
$\gamma_{0+(-)}$ linear part of $\nu_{M+(-)}$ $\gamma_{0+(-)}$
$\alpha_{0+(-)}$ quadratic part of $\nu_{M+(-)}$ $\alpha_{0+(-)}$
$\nu_{Oi}$ frequency output of Laser $i$ $\frac{\partial}{\partial t}\phi_{Oi}$
$\nu_{Sij}$ shot noise at photoreceiver $ij$ $\frac{\partial}{\partial t}\phi_{Sij}$
$\tau_{ij}$ light travel time from $SC_{i}$ to $SC_{j}$ $\tau_{ij}$
: \[tab:NotationTable\]Partial key to notation and comparison with [@McKenzie_09]
|
---
abstract: 'In a two-dimensional quantum wire in a perpendicular magnetic field with a smooth embedded repulsive scattering potential we find in the multimode conductance resonances caused by bound states with negative binding energy. These resonances are the counterexamples to well known dips in the conductance and evanescent states caused by quasi-bound states of attractive scattering centers in the wire.'
author:
- Vidar Gudmundsson
- 'Chi-Shung Tang'
- Andrei Manolescu
title: |
A state with negative binding energy induced by\
coherent transport in a quantum wire
---
Coherent quantum transport of electrons through quantum wires has been measured and modeled by many groups during the last one and half decade. The conductance quantization and deviations from it have been a focus of many researchers. Early on, it was discovered that an attractive scatterer can block a conduction channel totally for a narrow range of energy just at the end of a conduction plateau or step.[@Bagwell90:10354; @Gurvitz93:10578] This sharp dip structure was analyzed to be caused by a total reflection of the incoming electron wave in one channel due to a quasi-bound state originating from the evanescent mode in the next higher energy subband.
Without an external constant magnetic field nothing corresponding to this blocking can occur in a quantum wire with an embedded repulsive scatterer as it has no quasi-bound states in contrast to the attractive scatterer. In an external magnetic field this argument does not hold as can be inferred by a publication of Laughlin.[@Laughlin83:3383]
In this publication we will show that, indeed, the transport properties of a quantum wire with an embedded repulsive scatterer in a constant external magnetic field can exhibit signs of quasi-bound states. We will demonstrate that these states can reveal their presence both by resonance transmission peaks and by dips in the conductance indicating resonant backscattering processes.
We consider a multi-mode transport of electrons along the $x$-direction through a two-dimensional quantum wire defined by a parabolic confinement in the $y$-direction with the characteristic energy $E_0 = \hbar\Omega_0$. The electrons incident from the left ($x\rightarrow -\infty$) impinge on a smooth Gaussian scattering potential $V_{sc}=V_0 \exp{(-\beta r^2 )}$ Together the magnetic field ${\bf B}=B{\hat{\bf z}}$ and the parabolic confinement define a natural length scale $a_w=\sqrt{\hbar/(m^*)\Omega_w}$, where $\Omega_w=\sqrt{\omega_c^2+\Omega_0^2}$, with the cyclotron frequency $\omega_c=eB/(m^*c)$, is the natural frequency of the quantum wire in a magnetic field. The length $a_w$ can be considered as an effective magnetic length in the wire system. A mixed momentum-coordinate presentation of the wave functions $\Psi_E(p,y)=\int dx\:\psi_E(x,y)e^{-ipx}$ and expansion in channel modes $\Psi_E(p,y)=\sum_n\varphi_n(p)\phi_n(p,y)$ in terms of the eigenfunctions for the pure parabolically confined wire in magnetic field leads to a coupled set of Lippman-Schwinger integral equations in momentum space. These equations are then transformed into integral equations for the $T$-matrix in order to facilitate numerical evaluation.[@Gurvitz95:7123; @Gudmundsson05:BT] The conductance is evaluated according to the Landauer-B[ü]{}ttiker formalism together with the scattering wave functions from the $T$-matrix.[@Gudmundsson05:BT]
We consider a broad parabolic quantum wire with confinement energy $\hbar\Omega_0 = 1$ meV. At a vanishing magnetic field $B = 0$ this energy corresponds to $a_w = 33.7$ nm. We select a fairly narrow but smooth scatterer in the center of the wire with $\beta = 1\times 10^{-2}$ nm$^{-2}$, equivalent to $\beta a_w^2 = 11.4$ at $B = 0$, or in other words the scattering potential has reached $e^{-1}$ at $r\approx 10$ nm, close to the value of the effective Bohr radius in GaAs $a_0^* = 9.79$ nm. In the numerical calculations we have included at least 13 channel modes in the wire and use an unevenly spaced grid in momentum space to apply a repeated 4-point Gaussian integration to the integral equations for the T-matrix. Numerical accuracy is assured through comparison to solutions obtained with a larger basis set.
In order to understand better the results for a wire with a repulsive scatterer we first show in Fig. \[Fig\_1\] results for a wire with an [*attractive*]{} scatterer.
![(Color online) (a) The conductance in units of $G_0=2e^2/h$ of a broad wire with one embedded narrow potential well at the center $x=0$, as function of $X=E/(\hbar\Omega_w)+1/2$. (b) The probability density of the scattering state corresponding to an incident state with $X=1.996$ and $n=1$. $E_0=\hbar\Omega_0=1.0$ meV, $V_0=-8$ meV, $\beta = 10^{-2}$ nm$^{-2}$, and for GaAs $a_0=9.79$ nm. $a_w=29.34$ nm at $B=0.5$ T, and $a_w=14.68$ nm at $B=3.0$ T.[]{data-label="Fig_1"}](Fig_1.eps){width="48.00000%"}
To compare conductance curves for different values of $B$ we show them as functions of $X = E/(\hbar\Omega_w)+1/2$. The integer part of the parameter $X$ counts how many channels in the wire are open for transport for an incoming electron with propagating energy $E$. At low magnetic field, $B = 0.5$ T, dips are seen in the conductance curve just before $X$ assumes integer values. These well known dips can in a perturbative picture be explained as being caused by a backscattering of the incoming electron in subband $n$ by an evanescent state in subband $n+1$ that the scattering potential has lowered into the gap between the subbands just below the $n+1$ subband. The probability density of the electronic state in the first subband with $X=1.996$ is shown in Fig.\[Fig\_1\](b). On the incident side, the left side, we see the interference pattern of the incoming and outgoing $n=1$ state, and on the right side there is only the evanescent probability.
For a higher magnetic field, $B = 3$ T, the Lorentz force corresponding to the kinetic energy necessary to reach the evanescent state is large enough to press the electron into one side of the wire far enough from its center suppressing the overlap between the incoming and the evanescent state. As a result there is no dip found, and the only clear deviation from perfect conductance is in the beginning of each conductance step where the Lorentz force is small enough to still allow for an encounter between the electron and the scattering potential.
This together with the curious fact noted by Laughlin[@Laughlin83:3383] that two electron restricted to a plane perpendicular to a constant magnetic field can form a bound state with negative binding energy leads us to the following train of thought: If two electrons form a bound state, so can also one do around a smooth potential hill. The potential hill in the middle of a quantum wire will lift at least one state higher into the band where it originated. This quasi-bound state just above the band minimum might influence the scattering at high magnetic field just in the beginning of a new conductance step. In a flat two-dimensional system the state would be a true bound state, but in a quantum wire there is always an equipotential line along the wire edge with the same energy as the bound state guaranteeing at least a vanishingly small overlap between the edge and the bound state.
The parameters chosen earlier in this paper for the wire and the scattering potential proved fruitful in the search for a bound state of an electron around a hill. In Fig. \[Fig\_2\](a) we show the conductance of a wire with an embedded potential hill, a repulsive scatterer. A fine structure is visible at the beginning of each conductance step. In Fig. \[Fig\_2\](b) we focus in on the beginning of the second step for two close values of $B$.
![(Color online) The conductance in units of $G_0=2e^2/h$ of a broad wire with one embedded narrow potential hill at the center $x=0$, as function of $X=E/(\hbar\Omega_w)+1/2$ (a), and the same repeated for a narrow range of $X$. $E_0=\hbar\Omega_0=1.0$ meV, $V_0=+8$ meV, $\beta = 10^{-2}$ nm$^{-2}$.[]{data-label="Fig_2"}](Fig_2.eps){width="48.00000%"}
The arrows point at a transmission resonance at $X = 2.054$ and a dip at $X = 2.242$ for $B = 3.0$ T. We shall also analyze a dip at the beginning of the first step at $X = 1.242$ for an electron entering the wire in the $n = 1$ channel mode at $B = 2.0$ T. The corresponding probability densities for the electronic states are seen in Fig. \[Fig\_3\].
![(Color online) Corresponding to special values of $X$ in Fig. \[Fig\_2\] the probability density of the scattering state corresponding to an incident state with $X=1.242$, $n=1$, and $B=2.0$ T (a), $X=2.054$, $n=2$, and $B=3.0$ T (b), and $X=2.242$, $n=2$, and $B=3.0$ T (c). $E_0=\hbar\Omega_0=1.0$ meV, $V_0=+8$ meV, $\beta = 10^{-2}$ nm$^{-2}$.[]{data-label="Fig_3"}](Fig_3.eps){width="48.00000%"}
The kinetic energy of incoming electrons close to the transmission peak at $X = 2.054$ is low leading to a small Lorentz force acting on them. The electrons thus encounter the scattering potential and there is some backscattering, though small. The probability density in Fig. \[Fig\_3\](b) and the corresponding transmission peak in Fig. \[Fig\_2\] shows us that at $X = 2.054$ the electrons come into resonance with a quasi-bound state with binding energy of approximately $-0.29$ meV. This low energy and the potential curvature of the wire give the probability density a shape that reflects both the structure of the resonant edge state and the circular scattering potential. The double hump structure of the incoming and the transmitted wave are signs of the $n = 2$ mode or channel, the $n = 1$ channel is further away from the center of the wire as is discussed below.
At higher kinetic energy for the incoming electrons the Lorentz force is large enough to allow most of them to bypass the potential resulting in almost perfect conductance, except for the dip at $X = 2.242$ where the electrons come into resonance with a higher lying quasi-bound state that causes some of them to be backscattered. The probability density of this state is displayed in Fig. \[Fig\_3\](c). It is long lived judging from the narrowness of the dip and the strength of the probability density that subdues almost the incoming and the reflected wave on the color scale used here. The binding energy of this state is $-1.18$ meV and the smaller coupling to the edge states makes its symmetry much closer to the circular symmetry of the scatterer. The double ring structure of this state suggests that there are more quasi-bound states with simpler structure and larger negative binding energy but their still smaller coupling to the edge states makes them invisible to our calculation.
A simple estimate of the binding energy of the scattering potential neglecting the confinement of the wire and using the semiclassical quantization condition of Bohr places the value of $-1.18$ meV in between values obtained by assuming the quantum numbers 2 and 2+1/2. This is in accordance with the double hump structure we see in the probability density in Fig. \[Fig\_3\](c).
It should be stated here that the probability densities for the $n = 1$ states corresponding to the $n = 2$ states in Fig. \[Fig\_3\](b) and (c) at $B = 3$ T are straight edge states with their maxima at $y \approx 8a_w$. So, we are observing a scatterer with radius approximately $0.8a_w$, the bound state in Fig. \[Fig\_3\](b) has a radius of approximately $3a_w$, and the distance from the bound state to the edge is approximately $5a_w$ ($a_w = 14.7$ nm here). So, indeed, our scatterer is narrow but smooth leading to quasi-bound states of simple structure.
In Fig. \[Fig\_3\](a) is the probability density corresponding to to the dip at $X=1.242$ in the first subband. We can faintly see the incoming and the reflected waves. Here $B = 2.0$ T and due to the greater magnetic length or $a_w$ than for $B = 3.0$ T the symmetry of the quasi-bound state is influenced by the edge states. The resonance dip for this structure is deeper at $B = 2.1$ T (see Fig. \[Fig\_2\](a)) but the faint probability for the incoming and the outgoing wave at $B = 2.0$ T shows us the effective width of the wire here for this $n=1$ state.
The resonance at $B = 3$ T in Fig. \[Fig\_2\](a) corresponding to the state in Fig. \[Fig\_3\](c) is narrow reflecting a long lived state. At a lower magnetic field $B = 2$ T the corresponding resonance is broad indicating a relatively short lived resonance state producing it. Exactly this can be verified by looking at Fig. \[Fig\_4\],
 $E_0=\hbar\Omega_0=1.0$ meV, $V_0=+8$ meV, $\beta = 10^{-2}$ nm$^{-2}$.[]{data-label="Fig_4"}](Fig_4.eps){width="48.00000%"}
where the incoming and outgoing waves in the $n=2$ channel are clearly seen and one can infer from the structure of the probability density that semiclassically speaking the electrons undergoe few reflections before being returned.
We have demonstrated that quasi-bound states of a repulsive potential hill in a quantum wire in an external magnetic field can leave their fingerprints on the conductance of the system for suitably selected parameters. A small coupling to the edge states gives a long lived quasi-bound state with the symmetry of the scatterer, but the small coupling may hinder the state in affecting the conductance of the system. The narrow but smooth potential hill in an external magnetic field behaves like a “quantum peg” to which the electrons can be hooked for some time in contrast to the more familiar picture of the quasi-bound state of a quantum dot where one imagines the electron rattling for a short time in a bowl-like structure, the dot.
The use of a smooth scattering potential with feature lengths comparable to the effective Bohr radius in the system has allowed us to make visible the effects of fairly simple “low-lying” quasi-bound states at intermediate strengths of magnetic field $B\approx 3$ T. Clearly the effects of these quasi-bound states on the conductance are smaller than the effects of the corresponding states of an attractive scatterer so they are probably not easy to detect experimentally.
Here we have focused our view on the bound state of a smooth repulsive scatterer in a quantum wire in a perpendicular magnetic field as a complimentary system to the wire with an embedded attractive scatterer, and have compared the effects of these two scatterers on the conductance of the system. A similar, system has been considered analytically in the extreme quantum limit where a one band limit could be applied by Jain and Kivelson in order to explain the break down of the quantum Hall effect in narrow constrictions.[@Jain88:1542] Takagaki and Ferry used a hard wall version of both a quantum strip and a scatterer to explore with a tight-binding formalism how a circulating edge channel around an antidot leads to an Aharonov-Bohm-type oscillation of the conductance as the magnetic field normal to the strip is varied.[@Takagaki93:8152] They assume the diameter of the antidot is large compared to the characteristic wavelength of the electrons and focus their attention on the magnetic coupling of the the electron waves through the constriction on each side of the antidot. Corresponding experimental system with two large antidots has been investigated by Gould et al.[@Gould95:11213]
We expect that a larger hard wall scatterer may also produce quasi-bound states (with edge state character around the scatterer) that could influence the conductance of the system, but we believe that those states might produce effects in the conductance that are harder to find due to their very low binding energy and closeness to the beginning of a conductance plateau.
The research was partly funded by the Research and Instruments Funds of the Icelandic State, the Research Fund of the University of Iceland, and the National Science Council of Taiwan. C.S.T. acknowledges the computational facility supported by the National Center for High-performance Computing of Taiwan.
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|
---
abstract: 'The proton elastic form factor ratio is accessible in unpolarized Rosenbluth-type experiments as well as experiments which make use of polarization degrees of freedom. The extracted values show a distinct discrepancy, growing with $Q^2$. Three recent experiments tested the proposed explanation, two-photon exchange, by measuring the positron-proton to electron-proton cross section ratio. In the results, a small two-photon exchange effect is visible, significantly different from theoretical calculation. Theory at larger momentum transfer remains untested. This paper discusses the possibilities for future measurements at larger momentum transfer.'
author:
- 'Jan C. Bernauer'
bibliography:
- 'nstar.bib'
date: 'Received: date / Accepted: date'
title: 'Two-Photon Exchange: Future experimental prospects '
---
Introduction
============
Proton elastic form factors have been studied intensively with electron-proton scattering using unpolarized beams and target. The experiments produced data over an extensive range of (negative) four-momentum transfers, $Q^2$. Via the so-called Rosenbluth separation technique, the two elastic form factors were separated. More recently, experiments exploiting the polarization of beam or target measured the form factor ratio directly. While the former see rough agreement with scaling, i.e., a more or less constant ratio even for large $Q^2$, the latter show a roughly linear fall-off of the ratio. Figure \[figratio\] shows a selection of the available data and recent fits.
![\[figratio\]The proton form factor ratio $\mu G_E/G_M$, determined via Rosenbluth-type (gray points, from [@litt; @bartel; @andivahis; @walker; @christy; @qattan]) and polarization-type (black points, from [@gayou; @punjabi; @jones; @puckett10; @paolone; @puckett12]) experiments. While the former show a constant ratio, the latter indicate a linear downward trend. Curves represent phenomenological fits [@bernauer13], to either the Rosenbluth-type world data set alone (light gray curves) or to all data (dark gray curves).](ratio.pdf){width="\textwidth"}
The form factors encode the distribution of charge and magnetization in the proton and this “form factor ratio puzzle” is a limiting factor in their precise determination. It is therefore of importance to resolve this puzzle.
Two-photon exchange
===================
Blunden et al. [@Blunden:2003sp] suggested that hard two-photon exchange (TPE), neglected in standard radiative corrections, could be an important effect in Rosenbluth-type experiments, and that an inclusion of TPE might resolve the discrepancy. Two-photon exchange corresponds to a group of diagrams in the second order Born approximation of lepton scattering, namely those where two photon lines connect the lepton and proton. While the “soft” case, when one of the photons has negligible momentum, is included in the standard radiative corrections, like ref. [@MoTsai; @Maximon2000], to cancel infrared divergences from other diagrams, the “hard” part, where both photons can carry considerable momentum, is not. The exact division in “soft”and “hard” is arbitrary and depends on the specific radiative correction used.
Theoretical calculations
------------------------
Current theoretical calculations can be roughly divided into two groups: hadronic calculations, e.g. [@Blunden:2017nby], which are believed valid for $Q^2$ from 0 up to a couple of GeV$^2$, and GPDs based calculations, e.g. [@Afanasev:2005mp], valid from a couple of GeV$^2$ and up.
Phenomenological extraction
---------------------------
The amount of data available for the form factor ratio allows for an extraction of the expected TPE size. In [@bernauer13], the authors built a model based on the following assumptions:
- TPE is the dominany source of the difference.
- TPE affects the Rosenbluth-type experiments and leaves polarization data unchanged. This is good approximation as the effect of TPE on the cross section is magnified in the Rosenbluth separation to a substantially larger effect on $G_E$ for $Q^2>>0$.
- The effect is roughly linear in $\epsilon$. This is supported by the fact that no strong deviations from a straight line have been found in Rosenbluth separations so far.
- The effect vanishes for forward scattering, i.e., for $\epsilon=1$.
- For $Q^2\rightarrow 0$, TPE is given by the Feshbach Coulomb correction [@McKinley:1948zz]. Modern theoretical calculations have the same limit.
Assuming a correction of the form $1+\delta_{TPE}$ to the cross section, with $$\delta_{TPE}=\delta_\mathsf{Feshbach}+a(1-\epsilon)\ln{(1+b*Q^2)},\label{eqfesh}$$ the authors could fit the combined world data set with excellent $\chi^2$. This extraction will be used in the following to predict the size of the effect.
Current status
==============
Three contemporary experiments have tried to measure the size of TPE, based at VEPP-3 [@Rachek:2014fam], Jefferson Lab (CLAS, [@Adikaram:2014ykv]) and DESY (OLYMPUS, [@Henderson:2016dea]). The next-order correction to the elastic lepton-proton cross section contains terms corresponding to the product of the diagrams of one-photon and two-photon exchange. These terms change sign when switching between $e^-$ and $e^+$. Therefore, the size of TPE can be determined by measuring the ratio of positron to electron scattering: $R_{2\gamma}=\frac{\sigma_{e^+}}{\sigma_{e^-}}\approx 1+2\delta_{TPE}$.
![\[figdiff\] Difference of the data of the three recent TPE experiments [@Rachek:2014fam; @Adikaram:2014ykv; @Henderson:2016dea] to the calculation in [@Blunden:2017nby] (left) and the phenomenological prediction from [@Bernauer:2013tpr] (right).](q2_blunden.pdf "fig:"){width="50.00000%"}![\[figdiff\] Difference of the data of the three recent TPE experiments [@Rachek:2014fam; @Adikaram:2014ykv; @Henderson:2016dea] to the calculation in [@Blunden:2017nby] (left) and the phenomenological prediction from [@Bernauer:2013tpr] (right).](q2_bernauer.pdf "fig:"){width="50.00000%"}
In Fig. \[figdiff\] the difference of the data of the three experiments to the calculation by Blunden et al. [@Blunden:2017nby] and the phenomenological prediction by Bernauer et al. [@Bernauer:2013tpr] is shown. The three data sets are in good agreement which each other, and appear about 1% lower than the calculation. The prediction appears closer for most of the $Q^2$ range, but is above the data for the largest available $Q^2$. This is worrisome, as this coincides with the opening of the divergence in the fits in Fig. \[figratio\]. It might be an indication for an additional effect beyond TPE driving the discrepancy.
No hard TPE is ruled out by the data. The experiments agree with the phenomenological prediction with a reduced $\chi^2$ of 0.68. Compared to that, the theoretical calculation (red. $\chi^2$ of 1.09) is significantly worse, and the large normalization shifts to achieve this value rules them out at 99.6% confidence level.
The existing data show that TPE exists, but is small the in the covered region. Hadronic calculations are close, but can not explain the data perfectly. The calculations based on GPDs are only valid at higher $Q^2$ and are so far not tested by any experiment.
For a more in-depth review, see [@Afanasev:2017gsk]. Without a resolution of the puzzle and a test of TPE at larger $Q^2$, the extraction of reliable form factor information is impossible, especially from the high precision, large $Q^2$ measurements which are part of the Jefferson Lab 12 GeV program. Clearly, new data are needed. In the following, we will discuss experimental possibilities.
Future experiments
==================
Effect size and figure of merit
-------------------------------
As can be seen from Eq. \[eqfesh\], the size of TPE scales linearly with $1-\epsilon$, but only weakly with $Q^2$. The strongest signal is therefore at small $\epsilon$ and large $Q^2$. The cross section, however, drops fast exactly for the same kinematic conditions. We can construct a figure of merit to find the optimal kinematics: the ratio of expected deviation of $R_{2\gamma}$ from 1 and the expected uncertainty. $$FOM=\frac{\left|R_{2\gamma}-1\right|}{\sqrt{\Delta^2_\mathsf{stat.}+\Delta^2_\mathsf{syst.}}}$$ Here, the total uncertainty is split into a statistical and a systematical part. For the following, we assume a 1% systematic error.
Positron beams of the relevant energies are rare. Two possible sites for such an experiment are DESY and Jefferson Lab.
Measurement at Jefferson Laboratory
-----------------------------------
Jefferson Lab is evaluating the construction of a positron source for CEBAF. We assume that such a source would enable CEBAF to deliver up to of unpolarized positrons impinging on a 10 cm liquid hydrogen target, which yields an instantaneous luminosity of $\mathcal{L}=\SI{2.6}{\per\pico\barn\per\second}$.
For this paper, we investigated the measurement possibilities of Hall A and C. The main spectrometers of Hall A and the HMS spectrometer in Hall C can easily be used. SHMS in Hall C is limited to forward angles, and thus $\epsilon\approx1$, if used as a lepton spectrometer, but could be used to detect the protons instead. BigBite in Hall A is limited in the maximum momentum and thus minimal angle. However, because of the large acceptance, measurements at very low values of $\epsilon$ are possible. Figure \[figfomjlab\] shows the figure of merit for two days of beam per species, with the smaller-acceptance spectrometers represented by the left figure and BigBite by the right figure.
![\[figfomjlab\]Figures of merit as a function of $\epsilon$, for various $Q^2$, for days of beam per species at Jefferson lab. Left: small acceptance spectrometers, right: BigBite.](fomjlab.pdf "fig:"){width="50.00000%"}![\[figfomjlab\]Figures of merit as a function of $\epsilon$, for various $Q^2$, for days of beam per species at Jefferson lab. Left: small acceptance spectrometers, right: BigBite.](fombb.pdf "fig:"){width="50.00000%"}
Measurement at DESY
-------------------
DESY currently investigates a new test beam facility which would make TPE measurements with a beam possible. The proposed facility size and schedule constraints indicate non-magnetic calorimetric detectors as ideal, such as the those designed and built for PANDA. We assume five detector elements covering 10 msr each. The beam impinges on a 10 cm liquid hydrogen target. The left part of Fig. \[figdesy\] shows the FOM plot for 30 days per species. With a 2.85 GeV beam, the experiment could test TPE up to a $Q^2$ of about 6 GeV$^2$ with more than 5$\sigma$. The projected errors for such a measurement are shown on the right.
![\[figdesy\]Left: figure of merit as a function of $\epsilon$, for various $Q^2$, for 30 days of beam per species at DESY. Right: expected statistical error of data points and predicted effect size. ](fomdesy.pdf "fig:"){width="50.00000%"}![\[figdesy\]Left: figure of merit as a function of $\epsilon$, for various $Q^2$, for 30 days of beam per species at DESY. Right: expected statistical error of data points and predicted effect size. ](desypredict.pdf "fig:"){width="50.00000%"}
Conclusion
==========
The discrepancy in the form factor ratio is a serious limitation in the exact determination of the proton form factors and must be studied further in a dedicated program. The proposed test beam area at DESY could host an experiment to investigate TPE at larger momentum transfers on a short time scale. At Jefferson Lab, an upgraded CEBAF would make more precise experiments at even larger momentum transfers possible. This would test both hadronic and GPD-based theoretical calculations of TPE. Even if both calculations would be found lacking, the data would allow a phenomenological model precise enough to analyze contemporary and future form factor measurements.
Acknowledgments
===============
This work was supported by the Office of Nuclear Physics of the U.S. Department of Energy, grant No. DE-FG02-94ER40818.
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abstract: 'We obtain, for the first time, an analytic theory of the forward stimulated Brillouin scattering instability of a spatially and temporally incoherent laser beam, that controls the transition between statistical equilibrium and non-equilibrium (unstable) self-focusing regimes of beam propagation. The stability boundary may be used as a comprehensive guide for inertial confinement fusion designs. Well into the stable regime, an analytic expression for the angular diffusion coefficient is obtained, which provides an essential correction to a geometric optic approximation for beam propagation.'
author:
- 'Pavel M. Lushnikov$^{1,2}$ and Harvey A. Rose$^1$'
date: 'November 19, 2003'
title: Instability Versus Equilibrium Propagation of Laser Beam in Plasma
---
Laser-plasma interaction has both fundamental interest and is critical for future experiments on inertial confinement fusion (ICF) at the National Ignition Facility (NIF)[@Lindl1995]. NIF’s plasma environment, in the indirect drive approach to ICF, has hydrodynamic length and time scales of roughly millimeters and 10 ns respectively, while the laser beams that traverse the plasma, have a transverse correlation length, $l_c$, of a few microns, and coherence time $T_c$ of roughly a few ps. These microscopic fluctuations induce corresponding small-scale density fluctuations and one might naively expect that their effect on beam propagation to be diffusive provided self-focusing is suppressed by small enough [@RoseDuBois1993] $T_c$, $T_c \ll
l_c/c_s$, with $c_s$ the speed of sound. However, we find that there is a collective regime of the forward stimulated Brillouin scattering [@SchmittAfeyan1998] (FSBS) instability which couples the beam to transversely propagating low frequency ion acoustic waves. The instability has a finite intensity threshold even for very small $T_c$ and can cause strong non-equilibrium beam propagation (self-focusing) as a result.
We present for the first time, an analytic theory of the FSBS threshold in the small $T_c$ regime. In the stable regime, an analytic expression for the beam angular diffusion coefficient, $D$, is obtained to lowest order in $T_c$, which is compared with simulation. $D$ may be used to account for the effect of otherwise unresolved density fluctuations on beam propagation in a geometric optic approximation. This would then be an alternative to a wave propagation code [@StillBergerEtAl2000], that must resolve the beam’s correlation lengths and time, and therefore is not a practical tool for exploring the large parameter space of ICF designs. Knowledge of this FSBS threshold may be used as a comprehensive guide for ICF designs. The important fundamental conclusion is, for this FSBS instability regime, that even very small $T_c$ may not prevent significant self-focusing. It places a previously unknown limit in the large parameter space of ICF designs.
We assume that the beam’s spatial and temporal coherence are linked as in the induced spatial incoherence [@LehmbergObenschain1983] method, which gives a stochastic boundary condition at $z=0$ ($z$ is the beam propagation direction ) for the various Fourier transform components [@comment2], $\hat E$, of the electric field spatial-temporal envelope, $E$, $$\begin{aligned}
\label{phik}
\hat E({\bf k },z=0,t)=|\hat E({\bf k})|\exp\Big [ i\phi_{\bf
k}(t)\Big ], \nonumber \\
\Big \langle \exp i\Big [\phi_{\bf
k}(t)-\phi_{{\bf k}'}(t')\Big ]\Big \rangle =\delta_{{\bf k
k}'}\exp\Big (-|t-t'|/T_c\Big).\end{aligned}$$ The amplitudes, $|\hat E({\bf
k})|$, are chosen to mimic that of actual experiments, as in the idealized “top hat” model of NIF optics: $$\begin{aligned}
\label{tophat}
|\hat E({\bf k})|=const, \ k<k_m; \ |\hat E({\bf k})|=0, \ k>k_m,\end{aligned}$$ with $1/l_c\equiv k_m\simeq k_0/(2F)$, $F$ the optic $f/\#$, and the average intensity, $ \Big \langle I\Big \rangle \equiv \Big \langle |E|^2\Big \rangle
=I_0$ determines the constant. At electron densities, $n_e$, small compared to critical, $n_c$, and for $F^2\gg 1$, $E$ satisfies [@comment3] $$\label{Eeq1}
\Big (i\frac{\partial}{\partial
z}+\frac{1}{2k_0}\nabla^2-\frac{k_0}{2}\frac{n_e}{n_c}\rho
\Big )E=0, \ \nabla=(\frac{\partial}{\partial x},\frac{\partial}{ \partial
y}).$$ $k_0$ is $\simeq$ the laser wavenumber in vacuum. The relative density fluctuation, $\rho = \delta n_e/n_e$, absent plasma flow and thermal fluctuations which are ignored here, propagates acoustically with speed $c_s$: $$\label{neq1}
(R_0^{\rho\rho})^{-1} \ln (1+\rho)\equiv \Big (\frac{\partial^2}{\partial
t^2}+2\tilde\nu\frac{\partial}{\partial t}-c_s^2\nabla^2 \Big )\ln (1+\rho)=c_s^2 \nabla^2 I.$$ $\tilde \nu$ is an integral operator whose Fourier transform is $\nu k c_s$, where $\nu$ is the Landau damping coefficient. $E$ is in thermal units defined so that in equilibrium the standard $\rho=\exp (-I_0)-1$ is recovered. The physical validity of Eqs. $(\ref{Eeq1}),(\ref{neq1})$ as a model of self-focusing in plasma has been discussed before [@KawSchmidtWilcox1973; @SchmittOng1983; @Schmitt1988]. If $n_e/n_c$ is taken constant, there are 3 dimensionless parameters for $\rho\ll 1$: $\nu$,$ \, \tilde I_0\equiv
(k_0/k_m)^2(n_e/n_c)I_0/\nu, \,$ and $\tilde T_c\equiv k_mc_sT_c$.
Since Eqn. $(\ref{Eeq1})$ is linear in $E$, it may be decomposed, at any $z$, into a finite sum, $E=\sum_{j}E_{{\bf m}_j}({\bf
x},z,t)$, where each term has a typical wavevector ${\bf m}_j:$ $E_{{\bf m}_j}({\bf x},z=0,t)\sim \exp(i{{\bf m}_j}\cdot {\bf
x})$. Cross terms $E_{{\bf m}_j}E^*_{{\bf m}_{j'}}, \ {\bf
m}_j\neq {{\bf m}_j'}$, in the intensity, vary on the times cale $\tilde T_c$ so that their effect on the density response, Eq. $(\ref{neq1})$, is suppressed for $\tilde T_c\ll 1$ (see detailed discussion in [@RoseDuBoisRussell1990]). Similar consideration may be applied to general media with slow nonlinear response, including photorefractive media [@Segev1997]. Then the rhs of Eq. $(\ref{neq1})$ can be approximated as $$\begin{aligned}
\label{nFeq1}
c_s^2 \nabla^2 I=c_s^2 \nabla^2\sum\limits_{j}|E_{{\bf m}_j}|^2=c_s^2 \nabla^2
\int d{\bf v} F({\bf x},{\bf v},z,t). \\
\label{Fdef1}
F({\bf x},{\bf v},z,t)=\int d{\bf r}\sum\limits_{j j'}\delta_{{\bf m}_j {\bf m}_{j'}}\nonumber \\
\times E_{{\bf m}_j}({\bf x}-{\bf r}/2,z,t)E^*_{{\bf m}_{j'}}({\bf
x}+{\bf r}/2,z,t)e^{i{\bf v}\cdot {\bf r}}/(2\pi)^2\end{aligned}$$ is a variant of the Wigner distribution function which satisfies, as follows from Eq. $(\ref{Eeq1})$, $$\begin{aligned}
\label{Fteq1}
\frac{\partial F}{\partial z} +2{\bf v}\cdot \frac{\partial F}{\partial {\bf
x}}-\frac{i}{\pi^2}\int\Big[\hat\rho\big(-2[{\bf v}-{\bf v}'],z,t\big)\times \nonumber \\
\exp\big (-2i[{\bf v}-{\bf v}']\cdot {\bf x}\big )
-\hat\rho\big(2[{\bf v}-{\bf v}'],z,t\big)\times \nonumber \\
\exp\big (2i[{\bf v}-{\bf v}']\cdot {\bf x}\big )\Big ]F({\bf x},{\bf
v}',z,t)d{\bf v}'=0,\end{aligned}$$ with boundary value $F({\bf x},{\bf v},z=0,t)\equiv F_0({\bf
v})=|\hat E({\bf v})|^2$. Here the unit of $x$ is $(1/k_0)\sqrt{n_c/n_e}$ and that of $z$ is $(2/k_0)n_c/n_e$. Zero density fluctuation, $\rho=\partial \rho/\partial t=0,$ is an equilibrium solution of (4), $(\ref{nFeq1})$ and $(\ref{Fteq1})$, whose linearization admits solutions of the form, $\delta \rho
\sim e^{\lambda z}\exp i({\bf k}\cdot {\bf x}-\omega t)$, for real $\bf k$ and $\omega$, with $$\begin{aligned}
\label{lambeq1}
{\tilde \lambda \equiv}k_0\lambda/k_m^2=\frac{\tilde k(i\tilde I_0-2f)}{2\tilde I_0}
\left [\frac{f^2\tilde k^2-if\tilde I_0\tilde k^2-\tilde I_0^2}
{f(f-i\tilde I_0)} \right ]^{1/2}, \nonumber \\
f\equiv \frac{\omega^2-k^2c_s^2+2i\nu\omega kc_s}{2i\nu k^2c_s^2}, \ \tilde k\equiv \frac{k}{k_m}.\end{aligned}$$ Here and below we assume that the principle branches of square and cubic roots are always chosen so that the branch cut in the complex plane is on the negative axis and values of square root and cubic root are positive for positive values of their arguments. The real part of $\lambda$, $\lambda_r\equiv
Re(\lambda)$ has a maximum, as a function of $\omega$, close to resonance, $\omega=\pm k c_s [1+O(\nu)]$. Below we calculate all quantities at resonance $\omega=\pm k c_s$ because analytical expressions are much simpler in that case. $\lambda_r(k)$ has a maximum, $\lambda_{max}=k_m^2\tilde \lambda_{max}/k_0>0$, at $k
\equiv k_{max}$, $$\begin{aligned}
\label{kmax}
k_{max}/k_m= \tilde I_0 \sqrt{7(3\, \tilde I_0^2-2)2^{2/3}c^{-1}+8-2^{1/3}c}\times\nonumber \\
\big [3^{1/2}2(1+\tilde I_0^2)^{1/2}\big ]^{-1}, \nonumber \\
c=(c_1+c_2)^{1/3}, \quad c_1=-40 +
225 \tilde I_0^2 -27\tilde I_0^4, \nonumber \\ c_2=-3i(\tilde
I_0^2+4)\sqrt{27-60\tilde I_0^2-81\tilde I_0^4},\end{aligned}$$ Modes with $k> k_{cutoff}$ are stable ($\lambda_r<0$), with $k_{cutoff}
=k_m\tilde I_0^2(1+\tilde I_0^2)^{-1/2}/2,$ which defines a wavenumber-dependent FSBS threshold.
As $\tilde
I_0 \to 0,$ at fixed $k$, $k_0 \lambda_r \to -k^2/\tilde I_0,$ recovering the $\delta(z)$ behavior of density response function $R^{\rho\rho}_0$ in $(\ref{neq1})$. If $k_m$ is set to zero, the coherent forward stimulated Brillouin scattering (FSBS) convective gain rate [@SchmittAfeyan1998] is recovered in the paraxial wave approximation. Unlike the static response, $\lambda(k,\omega=0),$ which is stable [@comment4] for all $k$ for small enough $I_0 $, the resonant response remains unstable at small $k$ [@comment5] since as $\tilde I_0\to 0,
\ \tilde \lambda_{max}\to 0.024\tilde I_0^5 $ and $k_{cutoff}\to
k_m \tilde I_0^2/2$.
Since the FSBS instability peaks near $\omega=\pm kc_s,$ one expects an acoustic-like peak to appear in the intensity fluctuation power spectrum, $|I(k,\omega)|^2$, for $k$ less than $k_{cutoff}$ as in the simulation ($f/8,\;{{n_e} \mathord{\left/
{\vphantom {{n_e} {n_c}}} \right. \kern-\nulldelimiterspace}
{n_c}}=0.1$) results shown in figure 1.
The fraction of power in this acoustic peak, ${{\int\limits_{{{2kc_s} \mathord{\left/ {\vphantom {{2kc_s} 3}}
\right. \kern-\nulldelimiterspace} 3}<\left| \omega
\right|<{{4kc_s} \mathord{\left/ {\vphantom {{4kc_s} 3}} \right.
\kern-\nulldelimiterspace} 3}} {\left| {I\left( {k,\omega }
\right)} \right|^2d\omega }} \mathord{\left/ {\vphantom
{{\int\limits_{{{2kc_s} \mathord{\left/ {\vphantom {{2kc_s} 3}}
\right. \kern-\nulldelimiterspace} 3}<\left| \omega
\right|<{{4kc_s} \mathord{\left/ {\vphantom {{4kc_s} 3}} \right.
\kern-\nulldelimiterspace} 3}} {\left| {I\left( {k,\omega }
\right)} \right|^2d\omega }} {\int_{-\infty }^{+\infty } {\left|
{I\left( {k,\omega } \right)} \right|^2d\omega }}}} \right.
\kern-\nulldelimiterspace} {\int_{-\infty }^{+\infty } {\left|
{I\left( {k,\omega } \right)} \right|^2d\omega }}} \qquad,$ increases significantly as $\tilde I_0$ passes through its threshold value for a particular $k$, as shown in figure 2.
There is no discernible difference in shape between $|E(k,\omega,z)|^2$ at $z=0$, where it is $\propto 1/\big
[1+(\omega T_c)^2\big ]$, and at finite $z$, for small $T_c$.
If $\tilde \lambda _{\max }\ll 1$, i.e., $\tilde I_0\lesssim 1,$ then the FSBS growth length, ${1 \mathord{\left/ {\vphantom {1 {\lambda
_{\max }}}} \right. \kern-\nulldelimiterspace} {\lambda _{\max
}}}$, is large compared to the (vacuum) $z$ correlation length, $\propto{{k_0} \mathord{\left/ {\vphantom {{k_0} {k_m^2}}}
\right. \kern-\nulldelimiterspace} {k_m^2}}$, and it is found, for small $T_c$, that a quasi-equilibrium is attained: various low order statistical moments are roughly constant over the simulation range once $k_m^2z/k_0\gtrsim 5$, as seen in figure 3.
A true equilibrium cannot be attained since $\langle k^2\rangle
\equiv \langle|\nabla E|^2\rangle/I_0$ grows due to scattering from density fluctuations as in figure 4.
A dimensionless diffusion coefficient, $\tilde D\equiv (k_0 /k_m^4)
\frac{d}{dz}\langle k^2\rangle,$ (proportional to the rate of angulare diffusion) may be extracted from the data of figure 4 by fitting a smooth curve to $\langle k^2\rangle$ for $5<k_m^2
z/k_0 <76$, and evaluating its slope, extrapolated to $z=0$. This yields a diffusion coefficient of 4.4E-04.
$\tilde D$ may be compared to the solution of the stochastic Schroedinger equation (SSE) [@BalRyzhik2002] with a self-consistent random potential [@Zakharov], $\rho$, whose covariance, $C^{\rho\rho}$ ($C^{\rho\rho}$ is a quadratic functional of $F(k)$) is evaluated as follows [@Moody2000]. Take $E$ as given by Eqn. $(\ref{Eeq1})$ with $\rho$ set to $0$ since it goes to zero with $\tilde T_c$, and use it in Eqn. $(\ref{neq1})$, with $\ln(1+\rho)\to \rho$, to evaluate $C^{\rho\rho}$. This is consistent only if $\tilde I_0< 1,$ so that the density responce is stable except at small $k/k_m$. It follows, to leading order in $\tilde T_c$, that the SSE prediction for $\tilde D,$ for the top hat spectrum, $$\label{Deq1}
\tilde D_{SSE}=\nu \tilde T_c\tilde I_0^2/68.8\ldots,$$ has the value 3.2E-04 for the parameters of Fig. 4. Note that $\tilde D_{SSE}$ is proportional to $\langle \rho^2 \rangle$ and the roughly $20\%$ increase of $\langle \rho^2
\rangle$ over its perturbative evaluation (see figure 3) used in the SSE accounts for about $1/2$ of the difference between $\tilde D$ and $\tilde D_{SSE}$.
We find that $\tilde D$ depends essentially on the spectral form, $\langle {| {\hat E( k)}|^2} \rangle =F( k )$ , e.g., for Gaussian $F(k)$ with the same value of $\langle k^2 \rangle$, $D_{Gaussain}\approx 3D_{top\;hat}$. A numerical example of this dependence is found in figures 4 and 5. $\tilde D$ changes by $40\%$ over $5<k_m^2 z/k_0 <76$, because $F(k)$ changes significantly as seen in figure 5.
In this sense, for NIF relevant boundary conditions, angular diffusion is an essential correction to the geometrical optics model, which (absent refraction) has constant $F(k)$.
Eqn. (10) implies that ${d \mathord{\left/ {\vphantom {d {dz}}}
\right. \kern-\nulldelimiterspace} {dz}}\left\langle {\left( {{k \mathord{\left/ {\vphantom {k {k_m}}} \right. \kern-\nulldelimiterspace} {k_m}}} \right)^2} \right\rangle \propto {1 \mathord{\left/ {\vphantom {1 {k_m}}} \right. \kern-\nulldelimiterspace} {k_m}}$, while $\lambda _{\max }\propto {1 \mathord{\left/ {\vphantom {1 {k_m^8}}} \right. \kern-\nulldelimiterspace} {k_m^8}}$. If the diffusion length is smaller than the FSBS growth length, then propagation, which effectively increases $k_m$, will reinforce this ordering. This stability condition may be expressed as $\tilde D>\tilde \lambda _{\max }$, or qualitatively as [@comment6] $$\label{Deq2}
\nu \tilde T_c>\tilde I_0^3.$$ This is a global condition, as opposed to the wavenumber dependent threshold, $ k_{cutoff}( {\tilde I_0} )$. However, even if Eqn. (11) is violated, it is not until $k_{cutoff}\approx
1.5k_m$, so that the peak of the density fluctuation spectrum is unstable, that FSBS has a strong effect. For these larger $I_0$ values a quasi-equilibrium is not attained, and it is more useful to consider an integral measure, $\triangle (\langle k^2
\rangle,z)\equiv\langle k^2 \rangle(z)-\langle k^2 \rangle(0)$, of the change in beam angular divergence, rather than the differential measure, $D$. $\triangle/\tilde I_0^2$ is shown in Fig. 6, normalized to unity at $\tilde I_0=0.61$.
Note that we have not observed significant departure from Gaussian $E$ fluctuations for $\tilde I_0<2$ for the parameters of figure 6, which is consistent with the absence of self-focusing.
Therefore in this regime the effect of FSBS is benign, and perhaps useful for NIF design purposes: correlation lengths decrease, at an accelerated pace compared to SSE for $\tilde I_0 \sim 1$, with $z$, while electric field fluctuations stay nearly Gaussian. As a result [@Afeyan], the intensity threshold for other instabilities (e.g., backscatter SBS) increases [@RoseDuBois1994]. If $\tilde I_0>4$, there are large non-Gaussian fluctuations of $E$, which indicates strong self-focusing.
In conclusion, well above the FSBS threshold we observe strong self-focusing effects, while well below threshold beam propagation is diffusive in angle with essential corrections to geometric optics. In an intermediate range of intensities the rate of angular diffusion increases with propagation. In the weak and intermediate regimes, the diffusion results in decreasing correlation lengths which could be beneficial for NIF.
One of the author (P.L.) thanks E.A. Kuznetsov for helpful discussions.
Support was provided by the Department of Energy, under contract W-7405-ENG-36.
J.D. Lindl, Phys. Plasma [**2**]{}, 3933 (1995).
It is also assumed that intensity fluctuations which self-focus on a time scale $<T_c$ are not statistcally significant. See H. A. Rose and D. F. DuBois, Physics of Fluids [**B5**]{}, 3337(1993).
A. J. Schmitt and B. B. Afeyan, Phys. Plasmas [**5**]{}, 503 (1998).
C. H. Still, et al., Phys. Plasmas [**7**]{}, 2023 (2000).
R. H. Lehmberg and S. P. Obenschain, Opt. Commun. [**46**]{}, 27 (1983).
Fourier transform is in the $xy$ plane with ${\bf x}\to (x,y), \ {\bf k}\to(k_x, k_y)$.
This requires that the speed of light, $c \gg L_z / T_c$, where $L_z$ is the $z$ correlation length.
P. K. Kaw, G. Schmidt and T. W. Wilcox, Phys. Fluids [**16**]{}, 1522 (1973).
A. J. Schmitt and R. S. B. Ong, J. Appl. Phys [**54**]{}, 3003 (1983).
A. J. Schmitt, Phys. Fluids [**31**]{}, 3079 (1988).
H. A. Rose, D. F. DuBois and D. Russell, Sov. J. Plasma Phys. [**16**]{}, 537 (1990).
D.N. Christodoulides, T.H. Coskun, M. Mitchell and M. Segev, PRL [**78**]{}, 646 (1997).
The precise condition depends on $F_0$: see Ref. \[11\] and H. A. Rose and D. F. DuBois, Phys. Fluids B [**4**]{}, 252 (1992). The first derivation of an analogous result for the case of the modulational instabiity of a broad Langmuir wave spectrum was done by A. A. Vedenov and L. I. Rudakov, Soviet Physics Doklady [**9**]{}, 1073 (1965); Doklady Akademii Nauk SSR [**159**]{}, 767 (1964).
If $k$ is constrained by finite beam size effects to be, $e.g.$, $>1/$(beam diameter), then stability is regained for small enough $I_0$.
See, $e.g.$, G. Bal, G. Papanicolaou and L. Ryzhik, Nonlinearity [**15**]{}, 513 (2002).
This may be viewed as a special case of the wave kinetic Eq. \[see e.g. V.E. Zakharov, V.S. Lvov, and G. Falkovich, [*Kolmogorov Spectra of Turbulence I: Wave turbulence*]{} (Springer-Verlag, New York, 1992)\].
It is assumed that the density fluctuations are only due to the beam itself, in contrast to the experimental configuration found in J. D. Moody, et al., Phys. Plasma [**7**]{}, 2114 (2000).
If collisonal absorption is included in Eq. (3), with rate $\kappa$, then for $T_c\to 0$, the stability condition is $\lambda_{max}< \kappa$.
B. Afeyan (private comm. 2003) has reached somewhat similar conclusions in the context of self-focusing.
H. A. Rose and D. F. DuBois, Phys. Rev. Lett. [**72**]{}, 2883 (1994).
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---
abstract: 'Neumann-Lara and Urrutia showed in 1985 that in any set of $n$ points in the plane in general position there is always a pair of points such that any circle through them contains at least $\tfrac{n-2}{60}$ points. In a series of papers, this result was subsequently improved till $\tfrac{n}{4.7}$, which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about $j$-facets of sets of points in $\mathbb{R}^3$, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least $\tfrac{n}{4.7}$ points.'
author:
- |
Pedro A. Ramos Raquel Viaña\
Departamento de Matemáticas\
Universidad de Alcalá\
Alcalá de Henares, Spain\
title: |
Depth of segments and circles through points\
enclosing many points: a note[^1].
---
Introduction
============
The problem that we address in this work was proposed by Neumann-Lara and Urrutia in [@nlu], where the following result is shown: given a set $P$ of $n$ points in the plane in general position – no three of them are collinear and no four of them are cocircular – there is always a pair of points $p,q\in P$ such that every circle through $p$ and $q$ contains at least $\left\lceil\tfrac{n-2}{60}\right\rceil$ other points of $P$. In a series of papers [@hrw; @bssu; @h] this bound was slightly improved and, shortly afterwards, Edelsbrunner et al. [@ehss], by using techniques related to the complexity of higher order Voronoi diagrams, showed a bound of $(\tfrac{1}{2}-\tfrac{1}{\sqrt{12}})n+O(1) \approx \tfrac{n}{4.7}$, which is the best currently known lower bound for the problem. Regarding the upper bound, in [@hrw] Hayward et al. constructed a set of $4m$ points such that for any two of them there are circles passing through them and containing less than $m$ points. Therefore, this example shows that $\lceil\tfrac{n}{4}\rceil-1$ is an upper bound for the problem. In the same paper, the authors study the problem for sets of points in convex position, and give a bound of $\lceil\tfrac{n}{3}\rceil-1$, which is also shown to be tight. Urrutia [@u] has conjectured that $\tfrac{n}{4}$ is, up to perhaps an additive constant, the tight bound for the general problem.
In this note we give an alternative proof of the result by Edelsbrunner et al., transforming the problem from circles in the plane to planes in the space. We introduce the concept of depth of a segment in a set of points $P\subset\mathbb{R}^3$ and, by using known results about the number of $j$-facets, we show that there is always a pair of points such that every circle through them has, both inside and outside, at least $\tfrac{n}{4.7}$ points. Furthermore, we propose a new conjecture about the maximal number of segments with depth $k$ that a set of points in convex position can have, which implies a stronger version of the original conjecture.
Transforming the problem
========================
We use the well known transformation which maps the point $p=(p_x,p_y)\in\mathbb{R}^2$ to the point $\hat{p}=(p_x,p_y,p_x^2+p_y^2)\in\mathbb{R}^3$ in the paraboloid $z=x^2+y^2$. Among the useful properties of this transformation (see, for instance, [@e]) we will use the next one:
Given three non collinear points $p,q,r\in\mathbb{R}^2$, a point $s$ is inside the circle through them if and only if point $\hat{s}$ is bellow the plane defined by $\hat{p},\hat{q},\hat{r}\in\mathbb{R}^3$.
Therefore, the original problem is transformed into this one: given a set of $n$ points in the paraboloid $z=x^2+y^2$, show that there exist a pair of points such that any plane passing through them leaves bellow at least $\lceil \tfrac{n}{4}\rceil - 1$ points. This motivates the following definition:
Given a set of points $P\subset\mathbb{R}^3$ and two points $p,q\in P$, the [*depth*]{} of segment $pq$ is defined as the smallest integer $k$ such that any plane through $p$ and $q$ has on each side at least $k$ points of $P$.
We observe that segments with depth zero are the edges of the convex hull and we are interested in showing that any set of points has segments with “high depth”.
We recall that, given points $p,q,r\in P$, the (oriented) triangle $pqr$ is a $j$-facet of $P$ if it has exactly $j$ points on the positive side of its affine hull. Therefore, if $pqr$ is a $j$-facet, its edges have depth at most $j$. A subset $T\subset P$ is a $k$-set if it has $k$ points and the sets $T$ and $P\smallsetminus T$ can be separated by a plane. The number of $j$-facets of a set of points in $\mathbb{R}^d$ is related to the number of $(j\pm d)$-sets and obtaining tight bounds for these quantities, even for $d=2$, is a famous open problem. The number of $(\leq j)$-facets is much better understood. In order to state the result, we need some notation.
Let $e_j(P)$ be the number of $j$-facets of $P$ and let $E_j(P)=\sum_{i=0}^j e_i(P)$ be the number of $(\leq j)$-facets. In [@w] Welzl shows the following:
Let $P\subset\mathbb{R}^3$ be a set of $n$ points in general position. Then, $$E_j(P)\leq 2\Big[\binom{j+2}{2}\,n - 2\,\binom{j+3}{3} \Big] \qquad \text{if $\,\,0\leq 2j \leq n-4$.}$$ Furthermore, the bound is tight and is achieved if and only if the set $P$ is in convex position.
Because for a set of points in convex position $E_j(P)$ is known, the following result follows immediately:
\[cor:ej\] Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position. Then, $$e_j(P)=E_j(P)-E_{j-1}(P)=2(j+1)n-2(j+1)(j+2) \qquad \text{if $\,\,0\leq 2j \leq n-4$.}$$
Next we use this result to bound the number of segments with depth at most $j$ for a set of points in convex position. We denote by $s_j(P)$ the number of segments of $P$ with depth $j$ and by $S_j(P)=\sum_{i=0}^j s_i(P)$ the number of segments with depth at most $j$.
\[p:T\_j\] Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position. Then, $$S_j(P) \leq 3(j+1)n-3(j+1)(j+2) \qquad \text{if $\,\,0\leq 2j \leq n-4$.}$$
Let $j$ be such that $0\leq 2j \leq n-4$. We claim that if $pq$ is a segment with depth at most $j$, then it is contained in at least two $j$-facets of $P$. In order to prove the claim, consider first the case when the depth is smaller than $j$ and let $\pi$ be an oriented plane passing through $p$ and $q$ and having less than $j$ points in the positive side (denoted $\pi^+$ in Figure \[fig1\]). Because in the negative side of $\pi$ there are more than $\lceil\tfrac{n}{2}\rceil$ points, if we rotate the plane around $pq$ in a direction we find, before having rotated 180º, a point $r$ such that the plane $\pi_1$ passing through $p$, $q$ and $r$ leaves on the positive side exactly $j$ points of $P$ and, therefore, $pqr$ (oriented conveniently) is a $j$-facet of $P$. In the same way, if we rotate plane $\pi$ in the opposite direction, we find another point $s$ and, thus, another $j$-facet containing segment $pq$. Finally, if the depth of $pq$ is $j$, we observe that the first point that we find when the plane rotates must be in the negative side of the plane and thus it defines a $j$-facet.
![Illustration for the proof of Proposition \[p:T\_j\].[]{data-label="fig1"}](fig1)
Because each $j$-facet has 3 edges, it follows that $2S_j(P)\leq 3e_j(P)$ and, from Corollary \[cor:ej\] we get $$S_j(P) \leq \frac{3}{2}\, e_j(P) = 3(j+1)n-3(j+1)(j+2) \qquad \text{for $\,0\leq 2j \leq n-4$.}$$
We are ready to show the main result of this paper.
In a set $P\subset\mathbb{R}^3$ of $n$ points in convex position there exist segments with depth at least $$\Bigl(\frac{1}{2}-\frac{1}{\sqrt{12}}\Bigr)\,n + O(1) \approx \frac{n}{4.7}.$$
Because $n$ determine $\binom{n}{2}$ segments, while $S_j(P)$ is smaller than $\binom{n}{2}$ there must be segments with depth bigger than $j$. Therefore, from Proposition \[p:T\_j\] we get $$3(j+1)n-3(j+1)(j+2) = \binom{n}{2},$$ whose smaller solution is $$j=\frac{n-3}{2}-\Bigl( \frac{(n-2)^2-1}{12} \Bigr)^{1/2} = \Bigl(\frac{1}{2}-\frac{1}{\sqrt{12}}\Bigr)\,n + O(1).$$
Finally, if we apply this result to the original problem of circles passing through pairs of points, we obtain immediately the following result:
Let $P$ be a set of $n$ points in the plane in general position. There always exists a pair of points $p,q\in P$ such that every circle through $p$ and $q$ has, both inside and outside, at least $$\Bigl(\frac{1}{2}-\frac{1}{\sqrt{12}}\Bigr)\,n + O(1) \approx \frac{n}{4.7}$$ points of $P$.
A new conjecture
================
We propose a new conjecture which has arisen during our study of this problem.
\[c2\] Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position and let $s_j(P)$ be the number of segments with depth $j$. Then, $$s_j(P) \leq 3n-8j-6 \qquad \text{if $\,\,0\leq j \leq \lceil\tfrac{n}{4}\rceil - 1$.}$$
Of course, the result is obvious (with equality) for $j=0$ and it is easy to give an almost tight bound for $j=1$:
Let $P\subset\mathbb{R}^3$ be a set of $n$ points in convex position. Then, $$s_1(P) \leq 3n-12.$$
A segment $uv$ has depth one if and only if it is not an edge of the convex hull of $P$, denoted by $\conv (P)$, but there exists a point $p\in P$ such that $uv$ is an edge of $\conv(P\smallsetminus\{p\})$. If we denote by $\delta(p)$ the number of vertices adjacent to $p$ in $\conv (P)$, the number of new edges in $\conv (P\smallsetminus\{p\})$ is exactly $\delta(p)-3$. Therefore, $$\label{eq:s1}
s_1(P) \leq \sum_{p\in P} (\delta(p)-3) = 3n-12.$$
The inequality in (\[eq:s1\]) is strict if there is a segment $uv$ with depth one and points $p$ and $q$ such that $uv$ is an edge both of $\conv (P\smallsetminus\{p\})$ and $\conv (P\smallsetminus\{q\})$. In this situation, we say that segment $uv$ is generated by two points. It is easy to see that a segment with depth one cannot be generated by more than two points. Therefore, Conjecture \[c2\] for $s_1(P)$ is equivalent to show that there are always at least two segments generated by two points.
In the following we construct a set $P\subset\mathbb{R}^3$ such that $s_j(P) = 3n-8j-6$ for every $j=0,\ldots,\tfrac{n}{4} - 1$, thus showing that the bound in Conjecture \[c2\] would be tight. Consider the arc of circle $C=\{(x,y,z)\in\mathbb{R}^3\,|\,x^2+z^2=1,y=0,x>0.99\}$ and rotate it $45º$ counterclockwise around the $x$ axis. Let $n=4m$, put points $C_p=\{p_1,\ldots,p_m\}$ in $C$ and perturb them slightly to achieve general position. Now construct points $C_q$ and $C_r$ by rotating $C_p$ around the $z$ axis, $120º$ and $240º$, respectively. Finally, consider the arc $C'=\{(x,y,z)\in\mathbb{R}^3\,|\,x^2+z^2=1,y=0,z>0.99\}$ and put the rest of the points, $C_s=\{s_1,\ldots,s_m\}$, near $C'$ but slightly perturbed to achieve general position. The convex hull of $P=C_p\cup C_q \cup C_r\cup C_s$ is shown in Figure \[fig2\].a (top view) and Figure \[fig2\].b (bottom view).
![Construction reaching $s_j(P) = 3n-8j-6$ for $j=0,\ldots,\tfrac{n}{4} - 1$.[]{data-label="fig2"}](ej-bound){width="80.00000%"}
The fact that $s_j(P)=4n-8j-6$ for $j=0,\ldots,\tfrac{n}{4}-1$ can be easily checked taking into account the following simple observations:
- A segment $s$ has depth $j$ if it is in the convex hull of $P\smallsetminus T$ for some $j$-set $T$ and it is not in the convex hull of $P\smallsetminus S$ for any $k$-set $S$ with $k<j$.
- Given $T\subset P$ with $|T|<n/4$, the convex hull of $P'=P\smallsetminus T$ has “the same structure” as $\conv (P)$, i.e., consecutive points in each of the chains are adjacent, the first point in $C_s'$ is adjacent to all the points in $C_r'$ and $C_p'$, and so on.
We conclude the note stating a direct implication of the previous conjecture. Because $$\sum_{j=0}^{\lfloor\tfrac{n}{4}\rfloor -2} (3n-8j-6) \leq \binom{n}{2}-(n+2),$$ Conjecture \[c2\] would imply:
\[c3\] For every set of $n$ points in the plane in general position, there are always $n+2$ pairs of points such that any circle through them has, both inside and outside, at least $\lfloor\tfrac{n}{4}\rfloor - 1$ points.
Acknowledgements
================
I would like to thank Julian Pfeiffle for his constructions using Polymake and Boris Aronov, Imre Bárány, David Orden, and Micha Sharir for helpful discussions.
[9]{}
A. Andrzejak, B. Aronov, S. Har-Peled, R. Seidel, and E. Welzl. Results on $k$-sets and $j$-facets via continuous motion arguments. In [*Proc. 14th Annu. ACM Sympos. Comput. Geom.*]{}, (1998), p. 192–199.
I. Bárány, J. H. Schmerl, S. J. Sidney, and J. Urrutia. A Combinatorial Result About Points and Balls in Euclidean Space, [*Discrete Comput. Geom.*]{}, [**4**]{} (1989), p. 259–262.
H. Edelsbrunner. [*Algorithms in Combinatorial Geometry*]{}, Springer-Verlag, 1987.
H. Edelsbrunner, N. Hasan, R. Seidel, and X. J. Shen. Circles Through Two Points that Always Enclose Many Points, [*Geometriae Dedicata*]{}, [**32**]{} (1989), p. 1–12.
R. Hayward, A Note on the Circle Containment Problem, [*Discrete Comput. Geom.*]{}, [**4**]{} (1989), p. 263–264.
R. Hayward, D. Rappaport, and R. Wenger. Some Extremal Results on Circles Containing Points, [*Discrete Comput. Geom.*]{}, [**4**]{} (1989), p. 253–258.
V. Neumann-Lara, and J. Urrutia. A Combinatorial Result on Points and Circles in the Plane, [*Discrete Math.*]{}, [**69**]{} (1988), p. 173–178.
J. Urrutia. Some Open Problems. LATIN 2002, Cancún, México Abril 3-6, Lecture Notes in Computer Science (Springer) 2286 (2002), p. 4-11.
E. Welzl. Entering and leaving $j$-facets. [*Discrete Comput. Geom.*]{}, [**25**]{} (2001), p. 351–364.
[^1]: Partially supported by CAM grant S-0505/DPI/0235-02. Part of this work was done while the author was visiting the Mathematical Sciences Research Institute.
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abstract: 'This paper is concerned with the Cauchy problem of the one-dimensional free surface equation of shallow water wave, we obtain local well-posedness of the free surface equation of shallow water wave in Sobolev spaces. In addition, we also derive a wave-breaking mechanism for strong solutions.'
address:
- 'School of Mathematics, Northwest University, Xi’an 710127, China'
- 'School of Mathematics, Northwest University, Xi’an 710069, China'
author:
- MiaoMiao Dang
- Zhouyu Li
title: 'well-posedness of a kind of the free surface equation of shallow water wave'
---
[*Keywords:*]{} Local Well-posedness; Wave-breaking; Shallow water. 0.2cm
[*AMS Subject Classification (2000):*]{} 35G25, 35Q58\
Introduction
============
For one-dimensional surfaces, the water waves equations read in the following nondimensionalized form $$\label{CH-0}
\begin{cases}
\mu\partial^{2}_x\Phi+\partial_z\Phi^{2}=0 \; & \text{in} \; \Omega_{t},\\
\partial_z\Phi=0 \quad &\text{at} \; z=-1,\\
\partial_t\eta-\displaystyle\frac{1}{\mu}(-\mu\partial_x\eta\partial_x\Phi+\partial_z\Phi)=0 \quad & \text{at}\; z=\varepsilon\eta,\\
\partial_t\Phi+\displaystyle\frac{\varepsilon}{2}(\partial_{x}\Phi)^{2}+\displaystyle\frac{\varepsilon}{2\mu}(\partial_{z}\Phi)^{2}=0 \quad & \text{at} \; z=\varepsilon\eta,
\end{cases}$$ where $\varepsilon$ and $\mu$ are two dimensionless parameters defined as $$\varepsilon=\displaystyle\frac{a}{h},\quad \mu=\displaystyle\frac{h^{2}}{\lambda^{2}},$$ and $h$ is the mean depth, $a$ is the typical amplitude and $\lambda$ the typical wavelength of the waves under consideration. Where $x\longmapsto \varepsilon\eta(t, x)$ parameterizes the elevation of the free surface at time $t$, $\Omega_{t}=\{(x, z), -1<z<\varepsilon\eta(t, x)\}$ is the fluid domain delimited by the free surface and the flat bottom $\{z=-1\}$, and where $\Phi(t, \cdot)$ (defined on $\Omega_{t}$) is the velocity potential associated to the flow (that is, the two-dimensional velocity field $v$ is given by $v=(\partial_{x}\Phi,\partial_t\Phi)^{T})$.
Making assumptions on the respective size of $\varepsilon$ and $\mu$, one is led to derive (simpler) asymptotic models from . In the shallow-water scaling ($\mu\ll1$), one can derive the so-called Green-Naghdi (GN) equations (see [@Green1976] for the derivation, and [@Samaniego1] for a rigorous justification), without any smallness assumption on $\varepsilon$ (that is, $\varepsilon= O(1)$). $$\label{CH-1}
\begin{cases}
\eta_{t}+[(1+\varepsilon\eta)u]_{x}=0, \\
u_{t}+\eta_{x}+\varepsilon uu_{x}=\displaystyle\frac{\mu}{3}\displaystyle\frac{1}{1+\varepsilon\eta}[{1+\varepsilon\eta}^{3}(u_{xt}+\varepsilon uu_{x}-\varepsilon u^{2}_{x})]_{x},
\end{cases}$$ where $u(t, x)=\displaystyle\frac{1}{1+\varepsilon\eta}\int^{\varepsilon\eta}_{-1}\partial_{x}\Phi(t, x, z)dz$ denotes vertically averaged horizontal component of the velocity.
Because of the complexity of water-waves problem, they are often replaced for practical purposes by approximate asymptotic systems. The most prominent examples are the GN equations, which is a widely used model in coastal oceanography.
A recent rigorous justification of the GN model was given by [@Li2006] in 1D and for flat bottoms, which is based on the energy estimates and the proof of the well-posedness for the GN equations and the water wave problem , and by Alvarez-Samaniego and Lannes [@Samaniego2] allowed losses of derivatives in this energy estimate and therefore construct a solution by a Nash-Moser iterative scheme and proved the well-posedness of the GN equation in 1D and 2D and discuss the problem of their validity as asymptotic models for the water-waves equations.
Recently, Gui and Liu [@Gui-Liu-Sun2016] consider the 1-D R-CH equation is well-posed and show that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation.
In this paper, we consider the well-posedness of the free surface equation, which approximate solutions consistent with the GN equation.
The family of equations $$\label{CH-2}
\eta_{t}+\eta_{x}+\displaystyle\frac{3}{2}\varepsilon\eta\eta_{x}-\displaystyle\frac{3}{8}\varepsilon^{2}\eta^{2}\eta_{x}+\displaystyle\frac{3}{16}\varepsilon^{3}\eta^{3}\eta_{x}
+\mu(\alpha\eta_{xxx}+\beta\eta_{xxt})=\varepsilon\mu(\gamma\eta\eta_{xxx}+\delta\eta_{x}\eta_{xx}),$$ for the evolution of the surface elevation can be used to construct an approximate solution consistent with the GN equations(see [@Constantic2009]). Where $\varepsilon, \mu, \alpha, \beta, \gamma$ and $\delta$ are constants.
Let $q\in \R$ and assume that $$\alpha=q,\beta=q-\displaystyle\frac{1}{6},\gamma=-\displaystyle\frac{3}{2}q-\displaystyle\frac{1}{6},\delta=-\displaystyle\frac{9}{2}q-\displaystyle\frac{5}{24}$$ especially, choosing $q=\displaystyle\frac{1}{12},\mu=12,\varepsilon=1$ the equation reads $$\label{CH-3}
\eta_{t}+\eta_{x}+\displaystyle\frac{3}{2}\eta\eta_{x}-\displaystyle\frac{3}{8}\eta^{2}\eta_{x}+\displaystyle\frac{3}{16}\eta^{3}\eta_{x}
+\eta_{xxx}-\eta_{xxt}=-\displaystyle\frac{7}{2}\eta\eta_{xxx}-7\eta_{x}\eta_{xx}.$$ In this paper, we will investigate well-posedness of the Cauchy problem of the equivalent form of the free surface equation : $$\label{1.4}
\begin{cases}
& \partial_{t}\eta-\partial_{x}\eta-\frac{7}{2}\eta\partial_{x}\eta=(1-\partial_{x}^{2})^{-1}\partial_{x}
\big(-2\eta-\frac{5}{2}\eta^{2}+\frac{7}{4}(\partial_{x}\eta)^{2}+\frac{1}{8}\eta^{3}-\frac{3}{64}\eta^{4}\big),\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \forall \, t > 0,\, x \in \R,\\
&\eta|_{t=0}=\eta_{0},\qquad \qquad \forall \, x\in \R,
\end{cases}$$ and show the wave-breaking phenomenon. Our main results are stated as follows.
\[thm-main-1\] Suppose that $\eta_{0}\in H^{s}(\R)$ with $s>\frac{3}{2},$ then there exist a positive time $T>0$ such that, the equation has a unique strong solution $\eta\in{\mathcal{C}}([0,T];H^{s})\cap\mathcal{C}^{1}([0,T]; H^{s-1})$ and the map $\eta_{0}\mapsto \eta$ is continuous from a neighborhood of $\eta_{0}$ in $H^{s}$ into $\eta \in {\mathcal{C}}([0,T];H^s)\cap{\mathcal{C}}^{1}([0,T]; H^{s-1}).$ Moreover, the energy $$\label{lem-1}
H=H(\eta)=\frac{1}{2}\int_{\R}(\eta^{2}+\eta_{x}^{2})dx,$$ is independent of the existence time $t\in[0,T).$
\[thm-main-2\] Let $\eta_{0}\in H^{s}$ be as in Theorem \[thm-main-1\] with $s>\frac{3}{2}.$ Let $\eta$ be the corresponding solution to . Assume $T_{\eta_{0}}^{\ast}>0$ is the maximal existence time. Then $$\label{CH-y1}
T_{\eta_{0}}^{\ast}<\infty\Rightarrow\int_{0}^{T_{\eta_{0}}^{\ast}}\|\partial_{x}\eta(\tau)\|_{L^{\infty}}d\tau=\infty.$$
[**Remark:**]{} The blow-up criterion implies that the lifespan $T_{\eta_{0}}^{\ast}$ does not depend on the regularity index $s$ of the initial data $\eta_{0}.$
\[thm-main-3\] Let $\eta_{0}\in H^{s}(\mathbb{R})$ with $s>\frac{3}{2}$, $C_{0}=\frac{1}{2}+3\|\eta_{0}\|_{H^{1}}^{2}+\frac{3}{16}\|\eta_{0}\|_{H^{1}}^{3}+\frac{3}{32}\|\eta_{0}\|_{H^{1}}^{4}$. Assume that the initial value $\eta_{0}$ satisfies that $\eta_{0x}(x_{0})>\sqrt{\frac{2}{7}C_{0}}$ with the point $x_{0}$ defined by $\eta_{0x}(x_{0})=\inf\limits_{x\in\R}\eta_{0x}(x)$. Then the corresponding solution to the blows up in finite time in the following sense: there exists a $T_{0}$ with $0<T_{0}\leq\displaystyle\frac{2}{7(1-\sigma)\eta_{0x}(x_{0})}$ such that $$\label{CH-4}
\begin{split}
\limsup_{t\rightarrow T_{0}}(\inf_{x\in\R}\eta_{x}(t,x))=+\infty,
\end{split}$$ where $\sigma\in(0, 1)$ such that $\sqrt{\sigma}\eta_{0x}(x_{0})=\sqrt{\frac{2}{7}C_{0}}.$
This paper is organized as follows. In Section 2, we collect some elementary facts and inequalities which will be used later. Section 3 is devoted to the local well-posedness of the free surface system . Finally, using the transport equation theory, we can give the wave-breaking phenomenon Theorem \[thm-main-3\] in Section 4.
Let us complete this section with the notations we are going to use in this context.
[**Notations:**]{} Let $A, B$ be two operators, we denote $[A,B]=AB-BA,$ the commutator between $A$ and $B$. We shall denote by $(a, b)$ (or $(a, b)_{L^2}$) the $L^2(\R)$ inner product of $a$ and $b$, and $\int\cdot dx\triangleq\int_{\mathbb{R}}\cdot dx$.
We always denote the Fourier transform of a function $u$ by $\hat{u}$ or $\cF(u)$. For $s \in \mathbb{R}$, we denote the pseudo-differential operator $\Lambda^s:=(1-\Delta)^{\frac{s}{2}}$ with the Fourier symbol $(1+|\xi|^2)^{\frac{s}{2}}$. Note that if $g(x)=\frac{1}{2}e^{-|x|},$ $x\in\R,$ then $(1-\partial_{x}^{2})^{-1}f=g\ast f$ for all $f\in L^{2}(\R),$ where $\ast$ denotes the spatial convolution. To simplify the notations, we shall use the letter $C$ to denote a generic constant which may vary from line to line.
For $X$ a Banach space and $I$ an interval of $\R,$ we denote by ${\mathcal{C}}(I;\,X)$ the set of continuous functions on $I$ with values in $X,$ for $q\in[1,+\infty],$ the notation $L^q(I;\,X)$ stands for the set of measurable functions on $I$ with values in $X,$ such that $t\longmapsto\|f(t)\|_{X}$ belongs to $L^q(I).$
Preliminaries
=============
In this section, we will give some elementary facts and useful lemmas which will be used in the next section.
Let us first recall some basic facts about the regularizing operator called a mollifier, see [@Majda2002] for more details. Given any radial function $$\begin{aligned}
\rho(|x|)\in \mathcal{C}_0^\infty(\mathbb{R}^N), \quad \rho\geq0, \quad\int_{\mathbb{R}^N}\rho dx=1,\end{aligned}$$ define the mollification $\mathcal{J}_\varepsilon u$ of $u\in L^p(\mathbb{R}^N)$, $1\leq p\leq \infty$, by $$\begin{aligned}
\label{mollifier-1}
(\mathcal{J}_\varepsilon u)(x)=\varepsilon^{-N}\int_{\mathbb{R}^N}\rho(\displaystyle\frac{x-y}{\varepsilon})u(y) dy,
\quad \varepsilon>0.\end{aligned}$$
Mollifiers have several well-known properties:\
(i). $\mathcal{J}_\varepsilon u$ is a $\mathcal{C}^\infty$ function;\
(ii). for all $u\in \mathcal{C}^0(\mathbb{R}^N)$, $\mathcal{J}_\varepsilon u\rightarrow u$ uniformly on any compact set $\Omega$ in $\mathbb{R}^N$ and $\|\mathcal{J}_\varepsilon u\|_{L^\infty}\leq\|u\|_{L^\infty}$;\
(iii). mollifiers commute with distribution derivatives, $D^{\alpha}\mathcal{J}_\varepsilon u=\mathcal{J}_\varepsilon D^\alpha u$;\
(iv). for all $u\in H^m(\mathbb{R}^N)$, $\mathcal{J}_\varepsilon u$ converges to $u$ in $H^m$ and the rate of convergence in the $H^{m-1}$ norm is linear in $\varepsilon$: $\lim_{\varepsilon\rightarrow0}\|\mathcal{J}_\varepsilon u-u\|_{H^m}=0, \, \|\mathcal{J}_\varepsilon u-u\|_{H^{m-1}}\leq C\varepsilon\|u\|_{H^m}$;\
(v). for all $u\in H^m(\mathbb{R}^N)$, $k\in\mathbb{Z}^+\cup \{0\}$, and $\varepsilon>0$, $
\|\mathcal{J}_\varepsilon u\|_{H^{m+k}}\leq\frac{C(m, k)}{\varepsilon^k}\|u\|_{H^m}, \, \|\mathcal{J}_\varepsilon u\|_{L^\infty}\leq\frac{C(k)}{\varepsilon^{\frac{N}{2}+k}}\|u\|_{L^2}.
$
\[lem-Lions-Aubin’s\] Assume $X\subset E\subset Y$ are Banach spaces and $X\hookrightarrow\hookrightarrow E$. Then the following embeddings are compact:
(i)$\left\{\varphi:\varphi\in L^q([0, T]; X), \displaystyle\frac{\partial \varphi}{\partial t}\in L^1([0, T]; Y)\right\}\hookrightarrow\hookrightarrow
L^q([0, T]; E)\quad if \quad 1\leq q\leq\infty$;
(ii)$\left\{\varphi:\varphi\in L^\infty([0, T]; X), \displaystyle\frac{\partial \varphi}{\partial t}\in L^r([0, T]; Y)\right\}\hookrightarrow\hookrightarrow
{\mathcal{C}}([0, T]; E)\quad if \quad 1\leq r\leq\infty$.
\[lem-Calculus inequalities\] Let $s>0$. Then the following two estimates are true:
\(i) $\|uv\|_{H^s(\R)}\leq C\|u\|_{H^s(\R)}\|v\|_{H^s(\R)} \quad \mbox{for all} \quad s>\frac{1}{2}$;\
(ii) $\|[\Lambda^s, u]v\|_{L^2(\R)} \leq C(\|u\|_{H^s}\|v\|_{L^\infty(\R)}+\|\grad u\|_{L^\infty(\R)}\|v\|_{H^{s-1}(\R)})$,
where all the constants $C$s are independent of $u$ and $v$.
\[lem-1-D Moser-type estimates\] The following estimates holds:\
(i)For $s\geq0$, $$\|fg\|_{H^s(\R)}\leq C\{\|f\|_{H^s(\R)}\|g\|_{L^\infty(\R)}+\|g\|_{H^s(\R)}\|f\|_{L^\infty(\R)}\}.$$ (ii))For $s_{1}\leq\frac{1}{2},s_{2}>\frac{1}{2}$ and $s_{1}+s_{2}>0$, $$\|fg\|_{H^{s_{1}}(\R)}\leq C\|f\|_{H^{s_{1}}(\R)}\|g\|_{H^{s_{2}}(\R)}.$$
To study the wave-breaking criterion of the system , we need the following lemma on the transport equation (especially taking the space dimension $d = 1$).
\[lem-Transport equation theory\] Suppose that $s>-\frac{d}{2}$. Let $\upsilon$ be a vector field such that $\nabla\upsilon$ belongs to $L^{1}([0,T];H^{s-1})$ if $s>1+\frac{d}{2}$ or to $L^{1}([0,T];H^{\frac{d}{2}}\bigcap L^{\infty})$otherwise. Suppose also that $f_{0}\in H^{s},F\in L^{1}([0,T];H^{s})$ and that $f\in L^{\infty}([0,T];H^{s})\bigcap \mathcal{C}([0,T];S^{'})$solves the d-dimensional linear transport equations $$\label{lem-2.4}
\begin{cases}
\partial_{t}f+\upsilon \cdot \nabla f= F,\, \forall \, t>0, \, x \in \R^d,\\
f|_{t=0}=f_{0}.
\end{cases}$$ Then $f\in \mathcal{C}([0,T];H^{s})$. More precisely, there exists a constant C depending only on $s,p$ and $d$, and such that the following statements hold:
(i)If $s\neq1+\frac{d}{2}$, $$\|f\|_{H^s}\leq \|f_{0}\|_{H^s}+\int_{0}^{t}\|F(\tau)\|_{H^s}d\tau+c\int_{0}^{t}V^{'}\|F(\tau)\|_{H^s}d\tau,$$ or hence $$\|f\|_{H^s}\leq e^{CV(t)}(\|f_{0}\|_{H^s}+\int_{0}^{t}e^{-CV(t)}\|F(\tau)\|_{H^s}d\tau),$$ with $V(t)=\int_{0}^{t}\|\nabla\upsilon(\tau)\|_{H^{\frac{d}{2}}\bigcap L^{\infty}}d\tau$ if $s<1+\frac{d}{2}$ and $V(t)=\int_{0}^{t}\|\nabla\upsilon(\tau)\|_{H^{s-1}}d\tau$ else.
(ii))If $f=\upsilon$, then for all $s>0,$ the estimates both above hold with $V(t)=\int_{0}^{t}\|\nabla\upsilon(\tau)\|_{L^{\infty}}d\tau.$
We also need the following lemma about the boundness of the operator $(1-\partial_x^2)^{-1}$.
\[2-5\] Let $m\in\R$ and $f$ be an $S^{m}$-multiplier(that is, $f:\R^{d}\rightarrow \R$ is smooth and satisfies that for all multi-index $\alpha,$ there exists a constant $C_{\alpha}$ such that $\forall\xi\in\R^{d}, |\partial^{\alpha}f(\xi)|\leq C_{\alpha}(1+|\xi|)^{m-|\alpha|}).$ Then for all $s\in\R$ and $1\leq p,r\leq\infty,$ the operator $f(D)$ is continuous from $H^{s}$ to $H^{s-m}$, that is $\|f(D)(u)\|_{H^{s-m}}\leq C\|(u)\|_{H^{s}}$.
Local well-posedness
====================
This section is devoted to the proof of the local well-posedness of the system :
The proof is based on the energy method. We divide it into four steps. [**Step 1: Construction of smooth approximate solution**]{}
We introduce the following approximate system of $$\label{CH-7}
\begin{cases}
\partial_{t}\eta^{\varepsilon}-\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}-\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon}\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon} \!\!\!\!
&=\mathcal{J_\varepsilon}(1-\partial_{x}^{2})^{-1}\partial_{x}[-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}\\
&\quad+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}],\\
\eta^{\varepsilon}|_{t=0}=\eta^{\varepsilon}_{0},
\end{cases}$$ where $\mathcal{J_\varepsilon}$ denotes mollifier operator. The regularized equation reduces to an ordinary differential system: $$\label{CH-8}
\begin{cases}
\partial_{t}\eta^{\varepsilon}=\partial_{x}\eta^{\varepsilon}+\frac{7}{2}\mathcal{J_\varepsilon}[\mathcal{J_\varepsilon}\eta^{\varepsilon}\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}]+
\mathcal{J_\varepsilon}(1-\partial_{x}^{2})^{-1}\partial_{x}[-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}\\
\quad\quad+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}],\\
\eta^{\varepsilon}|_{t=0}=\eta^{\varepsilon}_{0}.
\end{cases}$$
The classical Picard Theorem ensures that the has a unique smooth solution $\eta^{\varepsilon} \in \mathcal{C}([0,T_{\varepsilon}]; H^s(\mathbb{R}))$ for some $ T_{\varepsilon}>0$.
[**Step 2: Uniform estimates to the approximate solutions**]{}
Applying the operator $\Lambda^s$ to the system and then taking the $L^2$ inner product, we get $$\label{CH-9}
\begin{split}
\frac{1}{2}\frac{d}{dt}\|\Lambda^s \eta^{\varepsilon}\|_{L^2}^2
&=\int_{\R} \Lambda^s \mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\cdot \Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}dx+\int_{\R} \Lambda^s [\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}(\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon})]\cdot \Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}dx\\
&\quad+\int_{\R} \mathcal{J_\varepsilon}\Lambda^{s-2} \partial_{x}[-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}
+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}\\
&\quad+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}]\cdot\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}dx=:I_1+I_2+I_3,
\end{split}$$ and $$I_1=\int_{\R} \Lambda^s \mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\cdot \Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon} dx=\frac{1}{2}\int_{\R}\partial_{x}(\Lambda^s\mathcal{J_\varepsilon}\eta^{\varepsilon})^{2}dx=0,$$ we may get from a standard commutator’s process that $$\label{CH-10}
\begin{split}
I_2&=\int_{\R}(\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon})\partial_{x}\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}\cdot\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}dx+\int_{\R}[\Lambda^s,(\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon})]\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\cdot\Lambda^s\mathcal{J_\varepsilon}\eta^{\varepsilon}dx\\
&=:I_{2.1}+I_{2.2}.
\end{split}$$ For $I_{2.1}$ we get by integration by parts that $$\label{CH-11}
\begin{split}
|I_{2.1}|&\leq \frac{1}{2}\|\partial_{x}(\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon})\|_{L^{\infty}}\|\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^2}^2\leq C\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^{\infty}}\|\Lambda^s\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^2}^2\\
&\leq C\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^{\infty}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}},
\end{split}$$ thanks to Hölder’s inequality and commutator estimate, we infer that $$\label{CH-12}
\begin{split}
|I_{2.2}|&\leq \|[\Lambda^s,(\frac{7}{2}\mathcal{J_\varepsilon}\eta^{\varepsilon})]\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{L^2}\|\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^2}\\
& \leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}(\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}\|\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{L^{\infty}}+\|\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{L^{\infty}}\|\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{H^{s-1}})\\
& \leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}(\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}\|\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{L^{\infty}}+\|\mathcal{J_\varepsilon}\partial_{x}\eta^{\varepsilon}\|_{L^{\infty}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}})\\
& \leq C\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^{\infty}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}.
\end{split}$$ Substituting and into $I_{2}$ leads to $$\label{CH-13}
\begin{split}
& |I_2|\leq C\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^{\infty}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}.
\end{split}$$
Thanks to the Sobolev embedding theorem $H^{s}\hookrightarrow L^{\infty}(for s>\frac{1}{2})$, we know that $$\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^{\infty}}\leq C\|\partial_{x}\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s-1}}\leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}},$$ where $s>\frac{3}{2}$.
Which along with implies that $$\label{CH-14}
\begin{split}
& |I_2|\leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{3}_{H^{s}},
\end{split}$$ because $H^{s-1}(s-1>\frac{1}{2})$ is a Banach algebra, we get from the Sobolev embedding inequality that $$\begin{split}
|I_3| &\leq \|\mathcal{J_\varepsilon}\Lambda^{s-2} \partial_{x}[-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}
+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}]\|_{L^2}\|\Lambda^s \mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{L^2}\\
&\leq C\|\partial_{x}[-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}
+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}]\|_{H^{s-2}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}\\
&\leq C\|-2\mathcal{J_\varepsilon}\eta^{\varepsilon}-\frac{5}{2}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}
+\frac{7}{4}\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}+\frac{1}{8}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}-\frac{3}{64}\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}\|_{H^{s-1}}\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}},
\end{split}$$ hence, $$\label{CH-15}
\begin{split}
|I_3| &\leq C[\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s-1}}+\|\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{2}\|_{H^{s-1}}+\|\mathcal{J_\varepsilon}(\partial_{x}\eta^{\varepsilon})^{2}\|_{H^{s-1}}\\
&\quad+\|\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{3}\|_{H^{s-1}}+\|\mathcal{J_\varepsilon}(\eta^{\varepsilon})^{4}\|_{H^{s-1}}]\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}\\
& \leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}(1+2\mathcal{J_\varepsilon}\|\eta^{\varepsilon}\|_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{3}_{H^{s}}).
\end{split}$$ Substituting and into leads to $$\label{CH-16}
\begin{split}
\frac{d}{dt}\|\eta^{\varepsilon}\|^{2}_{H^{s}}&\leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}(1+2\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{3}_{H^{s}})+C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{3}_{H^{s}}\\
& \leq C\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}(1+3\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{2}_{H^{s}}+\|\mathcal{J_\varepsilon}\eta^{\varepsilon}\|^{3}_{H^{s}}).
\end{split}$$
Therefore, by the bootstrap argument, we may get that, there is a positive time $T$ ($\leq T_{\varepsilon}$) independent of $\varepsilon$ such that for all $\varepsilon>0$, $$\label{CH-17}
\begin{split}
\sup_{0\leq t\leq T}\|\eta^{\varepsilon}(x,t)\|_{H^s}\leq C\|\eta_{0}\|_{H^s},
\end{split}$$ which along with implies that $$\label{CH-18}
\{\eta^{\varepsilon}(x,t)\}_{n\in N} \quad \mbox{is uniformly bounded in} \quad {\mathcal{C}}([0, T]; H^s(\mathbb{R})).$$ Furthermore, there holds $$\label{CH-19}
\{\partial_{t}\eta^{\varepsilon}(x,t)\}_{n\in N} \quad \mbox{is uniformly bounded in} \quad {\mathcal{C}}([0, T]; H^{s-1}(\mathbb{R})).$$
[**Step 3: Convergence**]{}
With ,, and , the Aubin-Lions’s compactness lemma ensures that there exist a subsequence of $\{\eta^{\varepsilon}(x,t)\}_{\varepsilon>0}$ converges to some limit $\eta(x,t)$ on $[0, T]$ which solves , moreover, there holds $$\label{CH-20}
\eta(x,t) \in {\mathcal{C}}([0,T];H^s(\mathbb{R}))\cap{\mathcal{C}}^{1}([0, T]; H^{s-1}(\mathbb{R})).$$
[**Step 4: Uniqueness of the solution**]{}
Let $\eta^{1}$ and $\eta^{2}$ be two solutions of with the same initial data and satisfy . We denote $\eta^{1, 2}:=\eta^1-\eta^2$, then $\eta^{1, 2}$ satisfies $$\begin{cases}
\begin{matrix}
\partial_{t}\eta^{1, 2} = \qquad & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\partial_{x}\eta^{1, 2}+\frac{7}{2}\eta^{1}\partial_{x}\eta^{1, 2}+\frac{7}{2}\eta^{1,2}\partial_{x}\eta^{ 2}+(1-\partial_{x}^{2})^{-1}\partial_{x}\big(-2\eta^{1,2} & \\
&\!\!\! -\frac{5}{2}\eta^{1,2}(\eta^{2}+\eta^{1})
+\frac{7}{4}\partial_{x}\eta^{1,2}(\partial_{x}\eta^{1} +\partial_{x}\eta^{2})+\frac{1}{8}\eta^{1,2}((\eta^{1})^{2} &
\\
& +\eta^{1}\eta^{2}+(\eta^{2})^{2})
-\frac{3}{64}\eta^{1,2}(\eta^{2}+\eta^{1})((\eta^{1})^{2}+(\eta^{2})^{2})\big) & \quad
\forall \, t > 0,\, x \in \R,\\
\eta^{1, 2}|_{t=0}=0 & & \quad \forall \, x \in \R.
\qquad \; \,
\end{matrix}
\end{cases}$$ Thanks to transport equation theory, we have $$\label{CH-21}
\begin{split}
&e^{-C\int_{0}^{t}\|\partial_{x}\eta^{1}(\tau)\|_{H^{s-1}}d\tau}\|\eta^{1,2}(t)\|_{H^{s-1}}\\
&\quad\leq\|\eta_{0}^{1,2}\|_{H^{s-1}}+C\int_{0}^{t}e^{-C\int_{0}^{t}\|\partial_{x}\eta^{1}(\tau^{'})\|_{H^{s-1}}d\tau^{'}}
\times(\|\frac{7}{2}\eta^{1,2}\partial_{x}\eta^{2}\|_{H^{s-1}}\\
&\quad+\|(1-\partial_{x}^{2})^{-1}\partial_{x}\{-2\eta^{1,2}-\frac{5}{2}\eta^{1,2}(\eta^{2}+\eta^{1})
+\frac{7}{4}\partial_{x}\eta^{1,2}(\partial_{x}\eta^{1}+\partial_{x}\eta^{2})\\
&\quad+\frac{1}{8}\eta^{1,2}((\eta^{1})^{2}+\eta^{1}\eta^{2}+(\eta^{2})^{2})-\frac{3}{64}\eta^{1,2}(\eta^{2}+\eta^{1})((\eta^{1})^{2}+(\eta^{2})^{2})\}\|_{H^{s-1}})d\tau.
\end{split}$$ For $s>1+\frac{1}{2}, H^{s-1}$ is an algebra, so we know that $$\label{CH-22}
\begin{split}
\|\frac{7}{2}\eta^{1,2}\partial_{x}\eta^{2}\|_{H^{s-1}}
\leq C\|\eta^{1,2}\|_{H^{s-1}}\|\partial_{x}\eta^{2}\|_{H^{s-1}}
\leq C\|\eta^{1,2}\|_{H^{s-1}}\|\eta^{2}\|_{H^{s}}.
\end{split}$$ On the other hand, from Lemma \[lem-1-D Moser-type estimates\] and Lemma \[2-5\], we get $$\label{CH-23}
\begin{split}
&\|(1-\partial_{x}^{2})^{-1}\partial_{x}(-2\eta^{1,2}-\frac{5}{2}\eta^{1,2}(\eta^{2}+\eta^{1}))\|_{H^{s-1}}\\
&\leq C
\|\eta^{1,2}\|_{H^{s-1}}(1+\|\eta^{2}\|_{H^{s-1}}+\|\eta^{1}\|_{H^{s-1}})\leq C\|\eta^{1,2}\|_{H^{s-1}}(1+\|\eta^{2}\|_{H^{s}}+\|\eta^{1}\|_{H^{s}}),
\end{split}$$ and $$\label{CH-24}
\begin{split}
&\|(1-\partial_{x}^{2})^{-1}\partial_{x}(\frac{7}{4}\partial_{x}\eta^{1,2}(\partial_{x}\eta^{1}+\partial_{x}\eta^{2}))\|_{H^{s-1}}\\
&\leq C\|\partial_{x}\eta^{1,2}\|_{H^{s-2}}(\|\partial_{x}\eta^{1}\|_{H^{s-1}}+\|\partial_{x}\eta^{2}\|_{H^{s-1}})\leq C\|\eta^{1,2}\|_{H^{s-1}}(\|\eta^{1}\|_{H^{s}}+\|\eta^{2}\|_{H^{s}}).
\end{split}$$ Similarly, we have $$\label{CH-25}
\begin{split}
&\|(1-\partial_{x}^{2})^{-1}\partial_{x}(\frac{1}{8}\eta^{1,2}((\eta^{1})^{2}+\eta^{1}\eta^{2}+(\eta^{2})^{2}))\|_{H^{s-1}}\\
&\quad\leq C\|\eta^{1,2}\|_{H^{s-1}}(\|\eta^{1}\|_{H^{s-1}}^{2}+\|\eta^{1}\|_{H^{s-1}}\|\eta^{2}\|_{H^{s-1}}
+\|\eta^{2}\|_{H^{s-1}}^{2}))\\
&\quad\leq C\|\eta^{1,2}\|_{H^{s-1}}(\|\eta^{1}\|_{H^{s}}^{2}+\|\eta^{1}\|_{H^{s}}\|\eta^{2}\|_{H^{s}}
+\|\eta^{2}\|_{H^{s}}^{2}),
\end{split}$$ and $$\label{CH-26}
\begin{split}
&\|(1-\partial_{x}^{2})^{-1}\partial_{x}(-\frac{3}{64}\eta^{1,2}(\eta^{2}+\eta^{1})((\eta^{1})^{2}+(\eta^{2})^{2}))\|_{H^{s-1}}\\
&\quad\leq C\|\eta^{1,2}\|_{H^{s-1}}(\|\eta^{1}\|_{H^{s-1}}+\|\eta^{2}\|_{H^{s-1}})
(\|\eta^{1}\|_{H^{s-1}}^{2}+\|\eta^{2}\|_{H^{s-1}}^{2})\\
&\quad\leq C\|\eta^{1,2}\|_{H^{s-1}}(\|\eta^{1}\|_{H^{s}}+\|\eta^{2}\|_{H^{s}})
(\|\eta^{1}\|_{H^{s}}^{2}+\|\eta^{2}\|_{H^{s}}^{2}).
\end{split}$$ Similarly, we get $$\label{CH-27}
\begin{split}
&\|(1-\partial_{x}^{2})^{-1}\partial_{x}\{-2\eta^{1,2}-\frac{5}{2}\eta^{1,2}(\eta^{2}+\eta^{1})-\frac{7}{4}\partial_{x}\eta^{1,2}(\partial_{x}\eta^{1}
+\partial_{x}\eta^{2})\\
&\quad+\frac{1}{8}\eta^{1,2}((\eta^{1})^{2}+\eta^{1}\eta^{2}+(\eta^{2})^{2})-\frac{3}{64}\eta^{1,2}(\eta^{2}+\eta^{1})((\eta^{1})^{2}+(\eta^{2})^{2})\}\|_{H^{s-1}}\\
&\leq C\|\eta^{1,2}\|_{H^{s-1}}(1+\|\eta^{1}\|_{H^{s}}+\|\eta^{2}\|_{H^{s}}+\|\eta^{1}\|_{H^{s}}^{2}+\|\eta^{2}\|_{H^{s}}^{2}+\|\eta^{1}\|_{H^{s}}\|\eta^{2}\|_{H^{s}}\\
&\quad+(\|\eta^{1}\|_{H^{s}}+\|\eta^{2}\|_{H^{s}})
(\|\eta^{1}\|_{H^{s}}^{2}+\|\eta^{2}\|_{H^{s}}^{2}))\\
&\leq C\|\eta^{1,2}\|_{H^{s-1}}(1+\|\eta^{1}\|_{H^{s}}^{3}+\|\eta^{2}\|_{H^{s}}^{3}).
\end{split}$$ Therefore, from to , with Young’s inequality, we obtain $$\label{CH-28}
\begin{split}
&e^{-C\int_{0}^{t}\|\partial_{x}\eta^{1}(\tau)\|_{H^{s-1}}d\tau}\|\eta^{1,2}(t)\|_{H^{s-1}}\\
&\leq \|\eta_{0}^{1,2}\|_{H^{s-1}}+C\int_{0}^{t}e^{-C\int_{0}^{t}\|\partial_{x}\eta^{1}(\tau^{'})\|_{H^{s-1}}d\tau^{'}}\|\eta^{1,2}(t)\|_{H^{s-1}}(1+\|\eta^{1}\|_{H^{s}}^{3}+\|\eta^{2}\|_{H^{s}}^{3})d\tau.
\end{split}$$ Hence, applying the Gronwall’s inequality, we reach $$\label{CH-29}
\begin{split}
\|\eta^{1}(t)-\eta^{2}(t)\|_{H^{s-1}}\leq\|\eta_{0}^{1,2}\|_{H^{s-1}} e^{-C\int_{0}^{t}1+\|\eta^{1}\|_{H^{s}}^{3}+\|\eta^{2}\|_{H^{s}}^{3}d\tau}.
\end{split}$$ With $\|\eta_{0}^{1,2}\|_{H^{s-1}}=0$, we get that $\|\eta^{1}(t)-\eta^{2}(t)\|_{H^{s-1}}\equiv0$, which implies that $\eta^{1}\equiv\eta^{2},\forall x\in \R, \,t \in[0,T].$
Based on the argument of the proof of uniqueness, we may readily get the map $\eta_{0}\mapsto \eta$ is continuous from a neighborhood of $\eta_{0}$ in $H^{s}$ into $\eta(x,t) \in {\mathcal{C}}([0,T];H^s)\bigcap{\mathcal{C}}^{1}([0, T]; H^{s-1}).$
Therefore, from Step 1 to Step 4, we complete the proof of Theorem \[thm-main-1\].
Weave-breaking criteria
=======================
In this section, attention is turned to investigating conditions of wave breaking.
With Theorem \[thm-main-1\] in hand, we are now ready to complete the proof of the wave-breaking. The proof of Theorem \[thm-main-2\] strongly depends on Lemma \[lem-Transport equation theory\] on the localization analysis for the transport equation.
Applying the operator $\Lambda^s$ to the first equation of system and then taking the $L^2$ inner product, we get $$\label{CH-30}
\begin{split}
&\frac{1}{2}\frac{d}{dt}\|\Lambda^s \eta\|_{L^2}^2
=\int_{\R} \Lambda^s \partial_{x}\eta\cdot \Lambda^s \eta dx+\int_{\R} \Lambda^s \partial_{x}\eta(\frac{7}{2}\eta)\cdot \Lambda^s \eta dx\\
&\quad+\int_{\R} \Lambda^{s-2} \partial_{x}[-2\eta-\frac{5}{2}\eta^{2}
+\frac{7}{4}(\partial_{x}\eta)^{2}+\frac{1}{8}\eta^{3}-\frac{3}{64}\eta^{4}]\cdot\Lambda^s \eta dx=:I_1+I_2+I_3,
\end{split}$$ and $$I_1=\int_{\R} \Lambda^s \partial_{x}\eta\cdot \Lambda^s \eta dx=\frac{1}{2}\int_{\R}\partial_{x}(\Lambda^s\eta)^{2}dx=0,$$ we may get from a standard commutator’s process that $$\label{CH-31}
\begin{split}
I_2&=\int_{\R}(\frac{7}{2}\eta)\partial_{x}\Lambda^s \eta\cdot\Lambda^s \eta dx+\int_{\R}[\Lambda^s,(\frac{7}{2}\eta)]\partial_{x}\eta\cdot\Lambda^s\eta dx=:I_{2.1}+I_{2.2}.
\end{split}$$ For $I_{2.1}$ we get by integration by parts that $$\label{CH-32}
\begin{split}
|I_{2.1}|&\leq \frac{1}{2}\|\partial_{x}(\frac{7}{2}\eta)\|_{L^{\infty}}\|\Lambda^s \eta\|_{L^2}^2\leq C\|\partial_{x}\eta\|_{L^{\infty}}\|\Lambda^s\eta\|_{L^2}^2\leq C\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|^{2}_{H^{s}},
\end{split}$$ thanks to Hölder’s inequality and commutator estimate, we infer that $$\label{CH-33}
\begin{split}
|I_{2.2}|&\leq \|[\Lambda^s,(\frac{7}{2}\eta)]\partial_{x}\eta\|_{L^2}\|\Lambda^s \eta\|_{L^2} \leq C\|\eta\|_{H^{s}}(\|\eta\|_{H^{s}}\|\partial_{x}\eta\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}}\|\partial_{x}\eta\|_{H^{s-1}})\\
& \leq C\|\eta\|_{H^{s}}(\|\eta\|_{H^{s}}\|\partial_{x}\eta\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|_{H^{s}}) \leq C\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|^{2}_{H^{s}}.
\end{split}$$ Substituting and into $I_{2}$ leads to $$\label{CH-34}
\begin{split}
& |I_2|\leq C\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|^{2}_{H^{s}}.
\end{split}$$ Because $H^{s-1}(s-1>\frac{1}{2})$ is a Banach algebra, we get from the Sobolev embedding inequality that $$\label{CH-35}
\begin{split}
\quad\quad\quad|I_3|& \leq \|\Lambda^{s-2} \partial_{x}[-2\eta-\frac{5}{2}\eta^{2}
+\frac{7}{4}(\partial_{x}\eta)^{2}+\frac{1}{8}\eta^{3}-\frac{3}{64}\eta^{4}]\|_{L^2}\|\Lambda^s \eta\|_{L^2}\\
& \leq C\|\partial_{x}[-2\eta-\frac{5}{2}\eta^{2}
+\frac{7}{4}(\partial_{x}\eta)^{2}+\frac{1}{8}\eta^{3}-\frac{3}{64}\eta^{4}]\|_{H^{s-2}}\|\eta\|_{H^{s}}\\
& \leq C\|-2\eta-\frac{5}{2}\eta^{2}
+\frac{7}{4}(\partial_{x}\eta)^{2}+\frac{1}{8}\eta^{3}-\frac{3}{64}\eta^{4}\|_{H^{s-1}}\|\eta\|_{H^{s}}.
\end{split}$$
Hence, $$\label{CH-35}
\begin{split}
|I_3|& \leq C[\|\eta\|_{H^{s-1}}+\|\eta^{2}\|_{H^{s-1}}+\|(\partial_{x}\eta)^{2}\|_{H^{s-1}}+\|\eta^{3}\|_{H^{s-1}}+\|\eta^{4}\|_{H^{s-1}}]\|\eta\|_{H^{s}}\\
& \leq C[\|\eta\|^{2}_{H^{s}}+\|\eta\|_{L^{\infty}}\|\eta\|^{2}_{H^{s}}+\|\partial_{x}\eta\|_{L^{\infty}}\|\eta\|^{2}_{H^{s}}+\|\eta\|^{2}_{L^{\infty}}\|\eta\|^{2}_{H^{s}}+\|\eta\|^{3}_{L^{\infty}}\|\eta\|^{2}_{H^{s}}]\\
& \leq C\|\eta\|^{2}_{H^{s}}(1+\|\eta\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}}+\|\eta\|^{2}_{L^{\infty}}+\|\eta\|^{3}_{L^{\infty}}),
\end{split}$$ substituting and into leads to $$\label{CH-36}
\begin{split}
&\frac{d}{dt}\|\eta\|^{2}_{H^{s}}\leq C\|\eta\|^{2}_{H^{s}}(1+\|\eta\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}}+\|\eta\|^{2}_{L^{\infty}}+\|\eta\|^{3}_{L^{\infty}}).
\end{split}$$ Thanks to Gronwall’s inequality, one can see $$\|\eta(t)\|^{2}_{H^{s}}\leq\|\eta_{0}\|^{2}_{H^{s}}e^{C\int_{0}^{t}(1+\|\eta\|_{L^{\infty}}+\|\partial_{x}\eta\|_{L^{\infty}}+\|\eta\|^{2}_{L^{\infty}}+\|\eta\|^{3}_{L^{\infty}})d\tau},$$ using the Sobolev embedding theorem $H^{s}\hookrightarrow L^{\infty}$(for $s>\frac{1}{2})$, we get from Theorem \[thm-main-1\] that $$\|\eta(t)\|_{L^{\infty}}\leq C\|\eta_{0}\|_{H^{1}},$$ which implies that $$\|\eta(t)\|^{2}_{H^{s}}\leq\|\eta_{0}\|^{2}_{H^{s}}e^{C\int_{0}^{t}(1+C_{1}+\|\partial_{x}\eta\|_{L^{\infty}}d\tau)},$$ where $C_{1}=C_{1}(\|\eta_{0}\|_{H^{1}}).$
Therefore, if the maximal existence time $T_{\eta_{0}}^{\ast}<\infty$ satisfies $\int_{0}^{T_{\eta_{0}}^{\ast}}\|\partial_{x}\eta(\tau)\|_{L^{\infty}}d\tau<\infty,$ then it implies that $$\limsup_{t\rightarrow T_{\eta_{0}}^{\ast}}(\|\eta(t)\|_{H^{s}})<\infty,$$ contradicts the assumption on the maximal existence time $T_{\eta_{0}}^{\ast}<\infty$. Hence, the proof of Theorem \[thm-main-2\] is complete.
\[lem-4.2\] Let $\eta_0\in H^{s}(\R)$ with $s>\frac{3}{2},$ and let $T>0$ be the maximal existence time of the solution $\eta$ to the with initial data $\eta_{0}.$ Then the corresponding solution blows up in finite time if and only if $$\label{CH-p}
\begin{split}
\lim_{t\rightarrow T^{-}}\sup_{x\in\R}\eta_{x}(t,x)=+\infty.
\end{split}$$
Applying a simple density argument, we only need to show that Lemma \[lem-4.2\] holds with some $s\geq3.$ Here we assume $s=3$ to prove the above Lemma.\
Multiplying the equation by $\eta$ and integrating by parts, we get $$\label{CH-a}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta^{2}+\eta_{x}^{2})dx=0.
\end{split}$$
On the other hand, multiplying equation by $\eta_{xx}$ and integrating by parts, we get $$\label{CH-b}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta_{x}^{2}+\eta_{xx}^{2})dx=3\int_{\R}\big(\eta\eta_{x}\eta_{xx}(1-\frac{1}{4}\eta
+\frac{1}{8}\eta^{2})+\frac{7}{2}\eta_{x}\eta_{xx}^{2}\big) dx.
\end{split}$$
Therefore, combining with , one can see that $$\label{CH-c}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta^{2}+2\eta_{x}^{2}+\eta_{xx}^{2})dx=3\int_{\R}\big(\eta\eta_{x}\eta_{xx}(1-\frac{1}{4}\eta
+\frac{1}{8}\eta^{2})+\frac{7}{2}\eta_{x}\eta_{xx}^{2}\big) dx.
\end{split}$$
From interpolation inequality we know that $$\label{CH-b1}
\begin{split}
3\int_{\R}\eta\eta_{x}\eta_{xx}dx&=-\frac{3}{2}\int_{\R}\eta_{x}^{3}dx\leq\frac{3}{2}\|\eta_{x}\|^{3}_{L^{3}(\R)}
\leq\frac{3}{2}C\|\eta_{x}\|^{\frac{5}{2}}_{L^{2}(\R)}\|\eta_{xx}\|^{\frac{1}{2}}_{L^{2}(\R)}\\
&\leq\|\eta_{xx}\|^{2}_{L^{2}(\R)}+C\|\eta_{x}\|^{\frac{10}{3}}_{L^{2}(\R)},
\end{split}$$ $$\label{CH-b2}
\begin{split}
-\frac{3}{4}\int_{\R}\eta^{2}\eta_{x}\eta_{xx} dx&=\frac{3}{4}\int_{\R}\eta\eta_{x}^{3} dx\leq\frac{3}{4}\|\eta\|_{L^{\infty}(\R)}\|\eta_{x}\|^{3}_{L^{3}(\R)}\leq\frac{3}{8}\|\eta\|^{2}_{H^{1}(\R)}+\frac{3}{8}\|\eta_{x}\|^{6}_{L^{3}(\R)}\\
&\leq\frac{3}{8}\|\eta\|^{2}_{H^{1}(\R)}+\frac{3}{8}C\|\eta_{x}\|^{5}_{L^{2}(\R)}\|\eta_{xx}\|_{L^{2}(\R)}\\
&\leq\frac{3}{8}\|\eta_{0}\|^{2}_{H^{1}(\R)}
+\frac{3}{32}C\|\eta_{x}\|^{10}_{L^{2}(\R)}+\|\eta_{xx}\|^{2}_{L^{2}(\R)},
\end{split}$$ $$\label{CH-b3}
\begin{split}
\frac{3}{8}\int_{\R}\eta^{3}\eta_{x}\eta_{xx} dx&=-\frac{9}{16}\int_{\R}\eta^{2}\eta_{x}^{3} dx\leq\frac{9}{32}\|\eta_{0}\|^{4}_{H^{1}(\R)}+\frac{9}{32}\|\eta_{x}\|^{6}_{L^{3}(\R)}\\
&\leq\frac{9}{32}\|\eta_{0}\|^{4}_{H^{1}(\R)}+\|\eta_{xx}\|^{2}_{L^{2}(\R)}+\frac{9}{128}C\|\eta_{x}\|^{10}_{L^{2}(\R)}.
\end{split}$$
Assume that $T<+\infty$ and there exists $M>0$ such that $$\label{CH-d}
\begin{split}
\eta_{x}(t,x)\leq M, \quad \forall(t,x)\in[0,T)\times\R.
\end{split}$$
Therefore, from to , we obtain $$\label{CH-e}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta^{2}+2\eta_{x}^{2}+\eta_{xx}^{2})dx&\leq 3\|\eta_{xx}\|_{L^{2}}^{2}+C_{2}+\int_{\R}\frac{21}{2}\eta_{x}\eta_{xx}^{2}dx\\
&\leq(3+\frac{21}{2}M)\|\eta_{xx}\|_{L^{2}}^{2}+C_{2},\\
\end{split}$$ where $C_{2}=C\|\eta_{0}\|_{H^{1}(\R)}^{\frac{10}{3}}+\frac{3}{8}\|\eta_{0}\|_{H^{1}(\R)}^{2}
+C\|\eta_{0}\|_{H^{1}(\R)}^{10}+\frac{9}{32}\|\eta_{0}\|_{H^{1}(\R)}^{4}.$
Applying Gronwall’s inequality to yields for every $t\in[0,T)$ $$\label{CH-f}
\begin{split}
\|\eta(t)\|_{H^{2}(\R)}^{2}\leq (2\|\eta_{0}\|_{H^{2}(\R)}^{2}+C_{2}T)e^{(3+\frac{21}{2}M)T}.
\end{split}$$
Differentiating equation with respect to $x$, and multiplying the result equation by $\eta_{xxx},$ then integrating by parts, we have $$\label{CH-g}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta_{xx}^{2}+\eta_{xxx}^{2})dx=&-\frac{15}{2}\int_{\R}\eta_{x}\eta_{xx}^{2}dx+\frac{15}{4}\int_{\R}\eta\eta_{x}\eta_{xx}^{2}dx
-\frac{9}{4}\int_{\R}\eta\eta_{x}^{3}\eta_{xx}dx\\
&-\frac{45}{16}\int_{\R}\eta^{2}\eta_{x}\eta_{xx}^{2}dx+\frac{35}{4}\int_{\R}\eta_{x}\eta_{xxx}^{2}dx,
\end{split}$$ we only need to know $$\label{CH-g1}
\begin{split}
-\frac{9}{4}\int_{\R}\eta\eta_{x}^{3}\eta_{xx}dx\leq\frac{9}{16}\|\eta_{x}\|_{L^{5}}^{5}\leq\frac{9}{16}C\|\eta_{x}\|_{L^{2}}^{\frac{7}{2}}\|\eta_{xx}\|_{L^{2}}^{\frac{3}{2}}
\leq\|\eta_{xx}\|_{L^{2}}^{2}+C\|\eta_{0}\|_{H^{1}(\R)}^{14},
\end{split}$$ which implies that $$\label{CH-g2}
\begin{split}
\frac{d}{dt}\int_{\R}(\eta_{xx}^{2}+\eta_{xxx}^{2})dx&\leq\frac{15}{2}M\int_{\R}\eta_{xx}^{2}dx+\frac{15}{4}\|\eta\|_{L^{\infty}}M\int_{\R}\eta_{xx}^{2}dx
+\|\eta_{xx}\|_{L^{2}}^{2}\\
&+C\|\eta_{0}\|_{H^{1}(\R)}^{14}
+\frac{45}{16}\|\eta\|^{2}_{L^{\infty}}M\int_{\R}\eta_{xx}^{2}dx+\frac{35}{4}M\int_{\R}\eta_{xxx}^{2}dx\\
&\leq(1+C_{3}M)\int_{\R}(\eta_{xx}^{2}+\eta_{xxx}^{2})dx+C_{4},
\end{split}$$ where $C_{3}=\frac{35}{4}+\frac{15}{4}\|\eta_{0}\|_{H^{1}(\R)}+\frac{45}{16}\|\eta_{0}\|_{H^{1}(\R)}^{2},$ $C_{4}=C\|\eta_{0}\|_{H^{1}(\R)}^{14}$ and we have used the assumption . Hence, applying Gronwall’s inequality implies that $$\label{CH-h}
\begin{split}
\int_{\R}(\eta_{xx}^{2}+\eta_{xxx}^{2})dx\leq(\|\eta_{0xx}\|_{H^{1}(\R)}^{2}+C_{4}T)e^{(1+C_{3}M)T},
\end{split}$$ together with yields for every $t\in[0,T)$ $$\label{CH-i}
\begin{split}
\|\eta(t)\|_{H^{3}(\R)}^{2}\leq[2\|\eta_{0}\|_{H^{3}(\R)}^{2}+(C_{2}+C_{4})T]e^{[4+(\frac{21}{2}+C_{3})M]T},
\end{split}$$ which contradicts the assumption the maximal existence time $T<+\infty$.
Conversely, the Sobolev embedding theorem $H^{s}(\R)\hookrightarrow L^{\infty}(\R)($with $s>\frac{1}{2})$ implies that if holds, the corresponding solution blows up in finite time, which completes the proof of Lemma \[lem-4.2\].
\[lem-4.3\] Let $T>0$ and $\eta\in\mathcal{C}^{1}([0,T);H^{2}(\R)).$ Then for $\forall t\in[0,T),$ there exists at least one point $\xi(t)\in\R$ with $$M(t)=\inf_{x\in \R}(\eta_{x}(t,x))=\eta_{x}(t,\xi(t)).$$ The function $M(t)$ is absolutely continuous on $(0,T)$ with $$\frac{dM(t)}{dt}=\eta_{tx}(t,\xi(t)) \quad a.e. \quad on \quad(0,T).$$
The technique used here is inspired from [@Gui2011]. Similar to the proof of Lemma \[lem-4.2\], we assume $s=3$ to prove Theorem \[thm-main-3\], now we consider the Lagrangian scale of the initial value problem $$\left\{
\begin{array}{l}
\displaystyle\frac{\partial q}{\partial t}=\eta(t,q), \ \ \ \ \ \ \ \ \ \ \ \forall 0<t<T,x\in \R,\\
q(0,x)=x, \ \ \ \ \ \ \ \ \ \ \ \ \ \forall x\in \R,
\end{array}
\right.$$ where $\eta\in \mathcal{C}([0,T);H^{s})$ with $s>\frac{3}{2}$, and $T>0$ being the maximal time of existence. A direct calculation also yields $q_{tx}(t,x)=\eta_{x}(t,q(t,x))q_{x}(t,x)$. Hence for $t>0,x\in \R$, we have $$q_{x}(t,x)=e^{\int_{0}^{t}\eta_{x}(\tau,q(\tau,x))}d\tau>0,$$ which implies that $q(t,\cdot): \R\rightarrow \R$ is a diffeomorphism of the line for every $t\in[0,T)$. By Lemma \[lem-4.3\], we know that $\xi(t)$ \[with $t\in[0,T)$\] exists such that $$M(t):=\eta_{x}(t,\xi(t))=\inf_{x\in \R}(\eta_{x}(t,x))\quad \forall t\in[0,T).$$ And then $$\label{CH-37}
\eta_{xx}(t,\xi(t))=0\quad for \quad a.e.\quad t\in[0,T).$$ On the other hand, since $q(t,\cdot):\R\rightarrow\R$ is the diffeomorphism for every $t\in[0,T),$ there exists $x_{1}(t)\in \R$ such that $$q(t,x_{1}(t))=\xi(t)\quad \forall t\in[0,T).$$
Differentiating both sides of the first equation of with respect to $x$, and we get $$\label{CH-38}
\begin{split}
\eta_{tx}=&\eta_{xx}+\frac{7}{2}\eta_{x}^{2}+\frac{7}{2}\eta\eta_{xx}+(- 2g_{x}\ast\eta_{x})+(-5g_{x}\ast\eta\eta_{x})\\
&+\frac{7}{2}g_{x}\ast\eta_{x}\eta_{xx}+\frac{3}{8}g_{x}\ast\eta^{2}\eta_{x}
+(-\frac{3}{16}g_{x}\ast\eta^{3}\eta_{x}).
\end{split}$$ Given $x\in\R,$ let $$M(t)=\eta_{x}(t,\xi(t)), \quad \forall t\in[0,T),$$ we have $$\label{CH-39}
\frac{d}{dt}M(t)=\frac{7}{2}M^{2}(t)+f(t,\xi(t)),$$ for $t\in[0,T),$ where $f$ represents the function
$$\label{CH-40}
\begin{split}
f(t,\xi(t))=&\big((-2g_{x}\ast\eta_{x})+(-5g_{x}\ast\eta\eta_{x})\\
&+\frac{3}{8}g_{x}\ast\eta^{2}\eta_{x}
+(-\frac{3}{16}g_{x}\ast\eta^{3}\eta_{x})\big)(t,\xi(t)).
\end{split}$$
We use the fact that $H(\eta)$ is the conservation law of the . On the other hand, the continuous embedding of $H^{1}(\R)$ into $L^{\infty}(\R)$, applying Young’s inequality and $g(x)=\frac{1}{2}e^{-|x|}$ lead to $$\label{CH-41}
\begin{split}
-2g_{x}\ast\eta_{x}(t,\xi(t))&\geq-2|g_{x}\ast\eta_{x}|\geq -2\|g_{x}\|_{L^{2}}\|\eta_{x}\|_{L^{2}}\geq-\|\eta_{x}\|_{L^{2}}\\
&\geq-(\frac{1}{2}+\frac{1}{2}\|\eta_{x}\|_{L^{2}}^{2})\geq-(\frac{1}{2}+\frac{1}{2}\|\eta_{0}\|_{H^{1}}^{2}),
\end{split}$$ and $$\label{CH-42}
-5g_{x}\ast\eta\eta_{x}(t,\xi(t))\geq-5|g_{x}\ast\eta\eta_{x}|\geq-5\|g_{x}\|_{L^{\infty}}\|\eta\eta_{x}\|_{L^{1}}\geq-\frac{5}{2}\|\eta_{0}\|_{H^{1}}^{2}.$$ Similarly, we have $$\label{CH-43}
\begin{split}
|\frac{3}{8}g_{x}\ast\eta^{2}\eta_{x}|(t,\xi(t))&\leq\frac{3}{8}\|\eta^{2}\eta_{x}\|_{L^{1}}\leq\frac{3}{16}\|\eta\|_{L^{\infty}}\int_{\R}\eta^{2}+\eta_{x}^{2}dx\\
&\leq\frac{3}{16}\|\eta_{0}\|_{H^{1}}\int_{\R}\eta_{0}^{2}+\eta_{0x}^{2}dx\leq\frac{3}{16}\|\eta_{0}\|_{H^{1}}^{3},
\end{split}$$ so we know that $$\label{CH-44}
\frac{3}{8}g_{x}\ast\eta^{2}\eta_{x}(t,\xi(t))\geq-\frac{3}{16}\|\eta_{0}\|_{H^{1}}^{3},$$ and $$\label{CH-45}
-\frac{3}{16}g_{x}\ast\eta^{3}\eta_{x}(t,\xi(t))\geq-\frac{3}{16}|g_{x}\ast\eta^{3}\eta_{x}|\geq-\frac{3}{16}\|\eta^{3}\eta_{x}\|_{L^{1}}
\geq-\frac{3}{32}\|\eta_{0}\|_{H^{1}}^{4}.$$
Combining to , we get $$f(t,\xi(t))\geq-(\frac{1}{2}+3\|\eta_{0}\|_{H^{1}}^{2}+\frac{3}{16}\|\eta_{0}\|_{H^{1}}^{3}+\frac{3}{32}\|\eta_{0}\|_{H^{1}}^{4}),$$ then we have $$\label{CH-47}
\frac{d}{dt}M(t)\geq\frac{7}{2}M^{2}(t)-C_{0},$$ where $C_{0}=\frac{1}{2}+3\|\eta_{0}\|_{H^{1}}^{2}+\frac{3}{16}\|\eta_{0}\|_{H^{1}}^{3}+\frac{3}{32}\|\eta_{0}\|_{H^{1}}^{4}.$\
By the assumption $M(0)=\eta_{0x}(x_{0})>\sqrt{\frac{2}{7}C_{0}},$ we have $M^{2}(0)>\frac{2}{7}C_{0}.$ We now claim that is true for any $t\in[0,T).$ In fact, assuming the contrary would, in view of $M(t)$ being continuous, ensure the existence of of $t_{0}\in[0,T)$ such that $M^{2}(t)>\frac{2}{7}C_{0}$ for $t\in[0,t_{0})$ but $M^{2}(t_{0})=\frac{2}{7}C_{0},$ combining this with would give $$\frac{d}{dt}M(t)\geq 0\quad a.e.\quad on \quad[0,t_{0}).$$ Since $M(t)$ is absolutely continuous on $[0,t_{0}],$ an integration of this inequality would give the following inequality and we get the contradiction $$M(t_{0})>M(0)=\eta_{0x}(x_{0})>\sqrt{\frac{2}{7}C_{0}},$$ this proves the previous claim.
Using this together with and the absolute continuity of the function $M(t),$ we see that $M(t)$ is strictly increasing on $[0,T).$ Therefore, choose that $\sigma\in(0,1)$ such that $\sqrt{\sigma}M(0)=\sqrt{\frac{2}{7}C_{0}},$ then we get from that $$\frac{d}{dt}M(t)\geq\frac{7}{2}M^{2}(t)-\frac{7}{2}\sigma M^{2}(0)\geq\frac{7(1-\sigma)}{2}M^{2}(t)\quad a.e.\quad on \quad[0,T).$$ Since $M$ is locally Lipschitz on $[0,T)$ and strictly positive, it follows that $\frac{1}{M}$ is locally Lipschitz on $[0,T).$ This gives $$\label{CH-48}
\frac{d}{dt}\Big(\frac{1}{M(t)}\Big)=-\frac{1}{M^{2}(t)}\frac{d}{dt}M(t)\leq-\frac{7(1-\sigma)}{2}\quad a.e.\quad on \quad[0,T).$$ Integration of this inequality yields $$\frac{1}{M(t)}-\frac{1}{M(0)}\leq-\frac{7(1-\sigma)}{2}t,\quad t\in[0,T).$$ Since $M(t)>0$ on $[0,T),$ we get the maximal existence time $T\leq\frac{2}{7(1-\sigma)M(0)}<\infty.$ Moreover, thanks to $M(0)=\eta_{0x}(x_{0})>0$ again, implies that $$\eta_{x}(t,\xi(t))=M(t)\geq\frac{\eta_{0x}(x_{0})}{1-\frac{7(1-\sigma)}{2}t\eta_{0x}(x_{0})}\rightarrow+\infty,$$ as $t\rightarrow\frac{2}{7(1-\sigma)\eta_{0x}(x_{0})}$. Therefore, thanks to Lemma \[lem-4.2\], we complete the proof of Theorem \[thm-main-3\].
[**Acknowledgments.**]{} The author would like to thank professor Guilong Gui for his valuable comments and suggestions. The authors are partially supported by the National Natural Science Foundation of China under the grants 11571279, 11331005, and 11601423.
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|
---
abstract: 'Reliable uncertainty estimation is crucial for perception systems in safe autonomous driving. Recently, many methods have been proposed to model uncertainties in deep learning-based object detectors. However, the estimated probabilities are often uncalibrated, which may lead to severe problems in safety-critical scenarios. In this work, we identify such uncertainty miscalibration problems in a probabilistic LiDAR 3D object detection network, and propose three practical methods to significantly reduce errors in uncertainty calibration. Extensive experiments on several datasets show that our methods produce well-calibrated uncertainties, and generalize well between different datasets.'
author:
- 'Di Feng$^{1,2}$, Lars Rosenbaum$^1$, Claudius Gläser$^1$, Fabian Timm$^1$, Klaus Dietmayer$^2$'
bibliography:
- 'bibliography.bib'
title: 'Can We Trust You? On Calibration of a Probabilistic Object Detector for Autonomous Driving [^1] [^2] [^3]'
---
[^1]: $^1$ Robert Bosch GmbH, Corporate Research, Driver Assistance Systems and Automated Driving, 71272 Renningen, Germany.
[^2]: $^2$ Institute of Measurement, Control and Microtechnology, Ulm University, 89081 Ulm, Germany.
[^3]: We thank our colleagues Florian Faion and Florian Drews for their suggestions and inspiring discussions. We also thank Bill Beluch for reading the script. The video to this paper can be found at <https://youtu.be/pH5qT11vmyM>.
|
---
author:
- |
[**M. Naghdi [^1]** ]{}\
*Department of Physics, Faculty of Basic Sciences*,\
*University of Ilam, Ilam, West of Iran.*
title: |
**A Truncation of 11-Dimensional Supergravity\
for Fubini-Like Instantons in AdS$_4$/CFT$_3$**
---
**Abstract**
From a consistent Kaluza-Klein truncation of 11-dimensional supergravity over $AdS_4 \times CP^3 \ltimes S^1/Z_k$, with a general 4-form ansatz, we arrive at a set of equations and solutions for the included fields. In particular, we have a bulk equation for a self-interacting (pseudo)scalar with arbitrary mass. By computing the energy-momentum tensors of the associated Einstein equations, to include the backreaction, and setting them to zero, we solve the resulting equations with the bulk one and get solutions corresponding to marginal and marginally relevant deformations of the boundary CFT$_3$, which break the conformal symmetry. These bulk (pseudo)scalars are $SU(4) \times U(1)$-singlet and the corresponding solutions break all supersymmetries and parity because of the associated (anti)M-branes wrapping around specific and mixed internal and external directions. As a result and according to AdS$_4$/CFT$_3$ duality rules, we would realize the boundary counterpart in three-dimensional Chern-Simon $U(N)$ field theories with matters in fundamental representations. In particular, we build a $SO(4)$-invariant Fubini-like instanton solution by setting a specific boundary Lagrangian. The resulting solution is used to describe the dynamics of thin-wall bubbles that cause instability and big crunch singularities in the bulk because of the unboundedness of the boundary double-hump potential from below. Relations to mass-deformed ABJM model, $O(N)$ vector models and scale invariance breaking are also discussed. Meanwhile, we evaluate corrections for the background actions because of the bulk and boundary instantons.
Introduction
============
In a few recent studies, we have considered Kaluza-Klein reductions of 11-dimensional (11D) supergravity (SUGRA) over $AdS_4 \times S^7/Z_k$ when the internal space $S^7/Z_k$ is an $U(1)$ bundle on $CP^3$. As a result, we found localized or partially localized objects in the bulk of AdS$_4$, and explored their boundary CFT$_3$ holographic duals according to the well-known AdS/CFT correspondence rules (see [@Aharony99] as an original review). Following the special truncation considered in [@Me7] in probe approximation, here we include also the backreaction so that the truncation would be consistent.
In fact, after considering a general 4-form ansatz of the 11D SUGRA and facing our setups with those in [@Gauntlett03], where a more general 4-form ansatz and truncation are analyzed, we obtain the equations of motion (EOM’s) for the involved (pseudo)scalars in Euclideanized Anti-de Sitter space ($EAdS_4$) and get some solutions including the backreaction. Next, we take a specific 4-form ansatz and obtain a second-order nonlinear partial deferential equation for an included (pseudo)scalar that could be massive, massless or tachyonic and is self-interacting. To address the backreaction, we compute the energy-momentum (EM or stress) tensors of the Einstein equations because of the bulk flux turned on, and get a new set of scalar equations, in external and internal spaces, which must in turn be satisfied to insure that the resulting solution or object does not backreact on the background geometry. By solving them together with the main bulk equation, we see that to have instantons we have to take an exactly marginal or a marginally relevant deformation on the boundary [^2].
The resulting (pseudo)scalars are singlets of $SU(4) \times U(1) \equiv H$ (as the isometry group of the whole internal space) and, at least because of the associated (anti)M-branes wrapping around the mixed directions, break all supersymmetries (SUSY’s) ($\mathcal{N}=6 \rightarrow 0$) according to the intersection rules of M-branes, for instance, in [@Bergshoeff96]. Meanwhile, we notice that a reasonable way to justify the SUSY breaking and get the needed singlet operators on the boundary theory, is to swap the fundamental representations (reps) of $SO(8)\equiv G$ (as the isometry group of $S^7$) for supercharges, fermions and scalars; and then discuss how we can get the wished $H$-singlets under the branching $G \rightarrow H$. In addition, the mass term in the bulk action breaks the scale invariance (SI) and that, although the bulk solution including the backreaction is scale-invariant in leading order, a relevant or mass term beside a marginal term in the boundary Lagrangian breaks the conformal $SO(4,1)$ symmetry.
Indeed, as a dual description for the bulk solution including the backreaction, we first consider a scalar Lagrangian with a marginal deformation term (or a triple-trace deformation of a relevant dimension-one operator) plus a so-called mass term that meet our needs well; see [@Elitzur05]. Next, we make instanton solutions for the massless case and argue that they should be $SO(4)$-invariant on $S^3$ (or $SO(3,1)$ on 3D de Sitter space-time $dS_3$ after the Lorenz analytical continuation), cause vacuum instability and are dual to big crunch singularities in the bulk. The solution’s moduli $a$ and $\vec{u}_0$ mark size and location of the boundary instanton and of a thin-wall bubble that may nucleate everywhere in the bulk. The boundary duals might be realized in 3D Chern-Simon (CS) $U(N)$ and $O(N)$ vector models, although we continue to use the former model [^3]. Especially, we focus on an $U(1)$ part of the ABJM quiver gauge group with matters in fundamental reps of $SU(4)$, because of symmetry arguments and correspondence rules.
The organization of this paper is as follows: In section 2, we introduce the background and a general 4-form ansatz of 11D SUGRA. Then, we obtain its corresponding equations and solutions in $EAdS_4$ space, without and with including the backreaction, in Appendix A from facing our ansatz with that in [@Gauntlett03]. In section 3, we use a special version of the general ansatz, which results in an interesting bulk equation; and to get the solutions including the backreaction, we first compute the associated EM tensors of the Einstein equations in Appendix B (with some useful formulas used in computations in subappendix B.1) and then write the EOM’s by zeroing both the external and internal components of the stress tensors. As a result, we obtain the main solution and conditions arisen from solving the latter equations and the main one in the bulk simultaneously in subsection 3.1. In subsection 3.2, we evaluate the correction to the background 11D action based on the solution including the backreaction. In section 4, we discuss the symmetries of the bulk setup and solutions and argue how they can help to fulfill the state/operator correspondence and find the correct boundary counterparts. Section 5 deals with the field theory duals of the main bulk solution including the backreaction. There, we present a suitable dual Lagrangian, find a plain solution and interpret implications of it. In particular, we discuss the relation of the latter setup to a massive deformation of the ABJM model and other issues such as SI breaking in subsection 5.2. In section 6 is a summary and more comments especially on the instability and false vacuum decay because of the instanton.
The background and Genaral Ansatz {#sec02A}
=================================
The background we consider is $$\label{eq01A}
\begin{split}
& ds^2_{11D}= {R_{AdS}^2}\, ds^2_{AdS_4} + R_7^2\, ds^2_{S^7/Z_k}, \\
& G_4^{(0)}= d\mathcal{A}_3^{(0)} = 3\, R_{AdS}^3\, \mathcal{E}_4 = N \mathcal{E}_4,
\end{split}$$ where the geometry is $AdS_4 \times S^7/Z_k$ of 11D SUGRA with $S^7$ as a $S^1$ fiber-bundle on $CP^3$, and the 4-form ansatz is that in ABJM [@ABJM]. The general 4-form ansatz we are considering here is $$\label{eq01}
\begin{split}
G_4 = & \left( 3 f_1 + R_{AdS} \ast_4(df_2 \wedge \mathcal{A}_3^{(0)}) \right) \wedge J^2 - R_{AdS}^{-1} \left( df_3 - f_4\, \ast_4 \mathcal{A}_3^{(0)} \right) \wedge J \wedge e_7 \\
& + \frac{1}{4 R_{AdS}^3} \ast_4(df_5 \wedge \ast_4 \mathcal{A}_3^{(0)}) \wedge J + \frac{3}{16 R_{AdS}^5} \ast_4 df_6 \wedge e_7 + \frac{3}{64 R_{AdS}^3} f_7\, \mathcal{E}_4,
\end{split}$$ where $f_1, f_2, \ldots$ are scalar functions in the external space, $R=R_7= 2 R_{AdS}$ is the $AdS$’s radius of curvature, $\mathcal{E}_4$ is the unit-volume 4-form on $AdS_4$, the 2-form $J=d \omega$ is the Kähler form on $CP^3$, and $e_7 = (d\acute{\varphi}+\omega)$ with $\acute{\varphi}$ as the fiber coordinate. For the general 4-form anstaz (\[eq01\]), we have derived the equations and conditions arising from satisfying the Bianchi identity ($dG_4=0$) and the Euclidean EOM $$\label{eq01B}
d \ast_{11} G_4 - \frac{i}{2}\, G_4 \wedge G_4=0$$ in [@Me7]. However, it is also interesting to discuss the backreaction, that is considering the Einstein’s equations as well; and of course we have done it in Appendix \[Appendix.AA\], where equations and solutions without and with including the backreaction are analysed in accordance with computations in [@Gauntlett03] where a more general ansatz is employed.
Special 4-Form Ansatz and Solutions {#sec03}
===================================
We employ a special 4-form ansatz from (\[eq01\]), made of the plain forms $e_7, J, \mathcal{E}_4$, whose clear form reads $$\label{eq11}
\frac{\tilde{G}_4}{(2 R_{AdS})^4} = 8\, \bar{f}_1 J^2 - 2\, df_3 \wedge J \wedge e_7 + \frac{3}{8} f_7\, \mathcal{E}_4,$$ where $f_1 N =\bar{f_1}$; and the resulting EOM becomes $$\label{eq12}
\ast_4 d \left(\ast_4 d\bar{f}_1 \right) - \frac{4}{R^2} \left(1 \pm 3\, \bar{C}_7 \right) \bar{f}_1 - 2 \times 192\, \bar{f}_1^3=0,$$ where for different values of $\bar{C}_7$, towers of tachyonic, massless (with $\bar{C}_7 =\frac{1}{3}$) and massive (pseudo)scalars in the bulk of $EAdS_4$ are possible.
It is interesting to compare the ansatz (\[eq11\]) with (2.5) in [@Gauntlett03]. As a result, we see that with $U = V = \chi = A_1 = B_1 = B_2 = 0$ and $$\label{eq13a}
f = 6\, f_7, \quad h= 4\, R^4 \bar{f}_1, \quad dh= - R^4\, df_3,$$ the formalisms match, and the counterpart of (2.23) in [@Me7] is $$\label{eq13b}
f = \frac{6}{R^7} \left(\epsilon + h^2 \right), \quad \epsilon =\pm\, \bar{C}_7\, R^6,$$ which comes from the equation (B.11) of [@Gauntlett03]. In particular, we read from (B.13) $$\label{eq13c}
\ast_4 d (\ast_4 dh)- (16 + 24 \epsilon)\, h - 2 \times 12\, h^3 =0,$$ which is the same as (\[eq12\]) up to some scaling and notice that $R=1$ is set. Then, from the latter equation, we read $m^2 R_{AdS}^2=-2$ with $\epsilon = -1$ (skew-whiffed) and $m^2 R_{AdS}^2=10$ with $\epsilon = 1$ (Wick-rotated) [^4], which were already discussed in [@Me6] and [@Me7] respectively.\
To continue, we note that considering only the EOM (\[eq12\]) means working in probe approximation, that is excluding the backreaction, for which we presented some solutions in [@Me7] with dual descriptions. Here we include the backreaction as well.
Solutions Including the Backreaction {#subsec03.01}
------------------------------------
To get solutions including the backreaction, we should first compute the stress-energy tensors of the replying Einstein equations. The details of such computations are given in Appendix \[Appendix.A0\], where (\[eq20a\]), (\[eq20b\]) and (\[eq20c\]) come from zeroing the external, internal and seventh components of the EM tensors, respectively. Next, by combining the latter equations with the main one (\[eq12\]), we arrive at $$\label{eq23a}
\Box_4 \bar{f}_1 =0,$$ $$\label{eq23b}
\Box_4 \bar{f}_1 + \left(8 \pm 12\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2}=0,$$ $$\label{eq23c}
\Box_4 \bar{f}_1 + \frac{8}{3 R^2} \bar{f}_1=0,$$ respectively, which are conditions imposed on the (pseudo)scalar from including the backreaction on the background geometry of the external and internal spaces. In other words, if we find solutions to these equations, the corresponding objects (e.g. instantons as topological objects) do not backreact on the background geometry. However, satisfying (\[eq23a\]), (\[eq23b\]) and (\[eq23c\]) simultaneously results in $\bar{f}_1=0$; Still, we may discuss each of them separately. In particular, the solution $$\label{eq24a}
\bar{f}_1(u,\vec{u}) = c_{8} + \frac{c_{9}\, u^3}{\left[u^2 + (\vec{u}-\vec{u}_0)^2 \right]^3}$$ of (\[eq23a\]), which corresponds to a *marginal* deformation with the boundary operator $\Delta_+ =3$, does not backreact on the external space geometry.
On the other hand, satisfying (\[eq23b\]) and (\[eq23c\]) simultaneously, which means taking the backreaction in the whole internal space into account, again results in the trivial solution $\bar{f}_1=0$; but if we take the massless solution (\[eq24a\]), which is in turn realized with $\bar{C}_7 = \frac{1}{3}$ (given that $(1 \pm 3\, \bar{C}_7)={m^2} {R_{AdS}^2}$) in the skew-whiffed version of (\[eq12\]), in the internal $CP^3$ space equation (\[eq23b\]), we will have $$\label{eq24b}
\Delta_{\pm} = \frac{3}{2} \pm \frac{\sqrt{5}}{2},$$ which corresponds to a *marginally relevant* operator with $\Delta_+ < 3$ (and the same behavior for \[eq23c\]); That means for this solution does not have any backreaction on the background metric, one must take a marginally relevant deformation in the corresponding boundary theory; We return to this interesting case when discussing dual solutions in section \[sec.5\]. [^5]
Correction to the Action {#subsec03.02}
------------------------
In is also interesting to compute the 11D action correction based on the solution including the backreaction. To this end, we use the 11D SUGRA action in Euclidean space as $$\label{eq25}
S_{11D}^E = -\frac{1}{2 \kappa_{11}^2} \left[ \int d^{11}x \, \sqrt{g} \, \mathcal{R}_{11} + \frac{1}{2} \int \left(\tilde{G}_4 \wedge \ast_{11} \tilde{G}_4 - \frac{i}{3} \tilde{\mathcal{A}}_3 \wedge \tilde{G}_4 \wedge \tilde{G}_4 \right) \right],$$ where $\mathcal{R}_{11}$ is the Ricci scalar and $\kappa_{11}^2 = 8\pi \mathcal{G}_{11}$ with $\kappa_{11}$ as the gravitational constant. To evaluate the correction, we should employ (\[eq11\]) for $\tilde{G}_4$ and its 11D dual as $$\label{eq26a}
\frac{\tilde{G}_7}{(2 R_{AdS})^7} = \bar{f}_1\, \mathcal{E}_4 \wedge J \wedge e_7 + \ast_4 df_3 \wedge
J^2 + f_7\, J^3 \wedge e_7,$$ and that $$\label{eq26b}
\tilde{G}_4 = d\tilde{\mathcal{A}}_3, \quad \tilde{\mathcal{A}}_3 = \tilde{\mathcal{A}}_3^{(0)} + (8\, R^8) (\bar{f}_1\, J \wedge e_7), \quad \tilde{G}_4^{(0)} = d\tilde{\mathcal{A}}_3^{(0)} = \frac{3}{8} R^4 f_7\, \mathcal{E}_4.$$ Next, plug (\[eq26a\]) and (\[eq26b\]) into (\[eq25\]) together with $$\label{eq27}
df_3 = -4\, d\bar{f}_1, \quad f_7 = +i\, 32\, R\, \bar{f}_1^2 \pm i\, \frac{\bar{C}_7}{R},$$ which in turn come from (\[eq13a\]) and (\[eq13b\]), the right part of the action for us (the second and third terms) becomes $\bar{S}_{11D}^E = S_0 + S_{11}^{modi.}$, where $$\label{eq28a}
S_0 = \frac{9}{R^2\, \kappa_{11}^2} \texttt{vol}_4 \wedge \texttt{vol}_7, \quad \texttt{vol}_7 = \frac{R^7}{3!} \int J^3 \wedge e_7 = \frac{\pi^4\, R^7}{3\, k},$$ is from the ABJM background realized with $\bar{C}_7=1$, and $$\label{eq28b}
S_{11}^{modi.} = \frac{3\, R^4}{2 \kappa_{11}^2}\, \texttt{vol}_7 \int \bigg(8\, R^2\, \bar{f}_1^2\, \mathcal{E}_4 + 32\, d\bar{f}_1 \wedge \ast_4 d\bar{f}_1 + 384\, R^2\, \bar{f}_1^4\, \mathcal{E}_4 \bigg).$$ Then, by putting the solution (\[eq24a\]) with $c_8=0$ in the latter action and taking (see [@Me6]) $$\label{eq29}
\mathcal{E}_4 = \frac{1}{u^4}\, dx \wedge dy \wedge dz \wedge du, \quad \kappa_{11}^2 = \frac{16\, \pi^5}{3} \sqrt{\frac{R^9}{3\, k^3}},$$ we arrive at, the correction in the unit 7D internal volume, $$\label{eq28c}
S_{corr.} = \frac{9\, c_9^2}{32\, \pi^3} \sqrt{3\, k^3\, R^3} \left[\frac{35}{48} \frac{1}{\epsilon^6} + \frac{4199}{8192} \frac{c_9^2}{\epsilon^{12}}\right],$$ as the finite part of the action, where $\epsilon>0$ is a cutoff parameter used instead of $u=0$ to evade the infinity of integrals with respect to (wrt) $u$.
Dual Symmetries and Correspondence {#sec.04}
==================================
We first remind that the ansatz (\[eq11\]) and the (pseudo)scalars therein are singlets of $SU(4) \times U(1)$ and so, we look if we can find the wished singlet (pseudo)scalars in the spectrum of the involved 11D SUGRA over $AdS_4 \times CP^3 \ltimes S^1/Z_k$. This task was already done in [@Me7], where we considered three massive (pseudo)scalars. But, for the solution including the backreaction, the (pseudo)scalars should be almost massless and so, we should look whether we can find any singlet massless (pseudo)scalar in the spectrum or not.
To this end, we first note that the massless multiplet ($n=0$) of $G$ (as the isometry group of $S^7$) includes a graviton ($\textbf{1}$), a gravitino ($\textbf{8}_s$), 28 spin-1 fields ($\textbf{28}$), 56 spin-$\frac{1}{2}$ fields ($\textbf{56}_{s}$), 35 scalars ($\textbf{35}_{v}$) of $0_1^+$ emerging from the external ingredients ($\mathcal{A}_{\mu \nu \rho}$), and 35-pseudoscalars ($\textbf{35}_{c}$) of $0_1^-$ emerging from the internal ingredients ($\mathcal{A}_{m n p}$). In the higher KK multiplets ($n>0$), the massless pseudoscalar sets in $\acute{\textbf{840}}_{s}$ of $0_1^-$ with $n=2$ while the massless scalar sets in $\textbf{1386}_{v}$ of $0_1^+$ with $n=4$ of $G$ (see, for instance, [@FreedmanNicolai], [@NilssonPope] and [@Biran]), and there is not any $H$-singlet under the branching $G\rightarrow H$.
On the other hand, we read from the ansatz structure that it breaks all SUSY’s because of the mixed internal directions around which the associated (anti)M-branes wrap– see also [@DuffNilssonPope84] that states the solutions with 4-form components all in the internal space break SUSY’s and parity; and as a result, the boundary duals might be realized in CS-matter $U(N)$ and $O(N)$ vector models. Thus, a consistent way to meet this need is to swap the fundamental reps $\textbf{8}_s$, $\textbf{8}_c$ and $\textbf{8}_v$ of $SO(8)$. On it, after swapping $\textbf{s} \leftrightarrow \textbf{v}$ with $\textbf{c}$ rep fixed (that means exchanging supercharges(spinors) with scalars while keeping the fermions unchanged), we have the rep $\textbf{1386}_{s}$ whose $U(1)$-neutral reps under the branching read $$\label{eq30}
\textbf{1386}_{s} \rightarrow \textbf{1}_{0}\oplus \acute{\textbf{20}}_{0} \oplus \textbf{105}_{0} \oplus \textbf{336}_{0} \oplus ...\ ,$$ which include an $H$-singlet mode. On the other hand, for the massless pseudoscalar of the original model ($\acute{\textbf{840}}_{s}$), there is not any singlet under $H$ even after both swappings.
As another point, we note that the ansatz breaks the inversion symmetry and thus conformal transformation of $SO(4,1)$ (as isometry of $EAdS_4$) besides the fact that the mass term in the bulk equation (\[eq12\]) breaks the SI; and as a result, we argue that our solution must be $SO(4)$ invariant; see [@Me7] for more details. It is also notable that although the resulting equation (\[eq23a\]) and main solution (\[eq24a\]) preserve full conformal symmetry, the marginally relevant deformation breaks the SI as we will discuss in the next section.
Boundary Field Theory Duals {#sec.5}
===========================
For the general ansatz (\[eq01\]), the scalar profiles of the forms (\[eq03a\]) and (\[eq03d2\]) are already discussed in [@Me4] and [@N] respectively and so, we do not pay more attention to them. Instead, we focus on the duals for the solutions including the backreaction.\
Indeed, from the bulk description with backreaction in subsection \[subsec03.01\], we see that the dual boundary operator have to be for an exactly marginal or a marginally relevant deformation. Although in [@Me4], [@Me5] and [@Me3] we studied samples of marginal operators and deformations, here we concentrate on a special sample of the (exactly) marginal deformation $\Delta_+=3$, valid as an approximation for another case too, and study aspects of it. In our formalism, we make this operator from the singlet (pseudo)scalar after the swapping $\textbf{8}_s \leftrightarrow \textbf{8}_v$ of the original supersymmetric theory. On the other hand, besides breaking SUSY’s, the $H$-singlet states break the even-parity symmetry of the ABJM model, which might in turn be understood through fractional M2-brane [@ABJ] associated with the probe (anti)M5-brane wrapping around $R^3 \times S^3/Z_k$. As a result and after gauging, we remain with just the diagonal $U(1)$ part of the quiver gauge group $SU(N)_k \times SU(M)_{-k}$, for which we set $A_i^-=0$, as it is for the boundary baryonic symmetry under which our modes are singlet [^6]. In other words, the fundamental fields are neutral to the diagonal $U(1)$ that couples to $A_i^+ \equiv (A_i + \hat{A}_i)$ while $A_i^-$ acts as the baryonic symmetry and, since our (pseudo)scalars are neutral, we assign zero to it.
By the way, we could consider the marginal deformation as a triple-trace deformation [^7] of the operator $\mathcal{O}_1 \sim \texttt{tr}(y \bar{y})$ used in [@Me5]. The most recognized case with the latter operator is the $O(N)$ vector models made of it and its multi-trace deformations. The interesting case is the tri-critical model, which includes just the kinetic term and a term proportional to a triple-trace deformation of it; see for instance [@Elitzur05]. With respect to the bulk studies, as a reasonable proposal consistent with our discussions, we match the bulk to boundary field as $\bar{f}_1 \rightarrow \varphi^2$ that in turn means the bulk instantons are square of the boundary ones [@deharo06], in leading order of course. As a result, from the EOM (\[eq12\]), we can take a dual Lagrangian for the boundary theory. Fortunately, a form for such an effective Lagrangian is already known (see [@Elitzur05] and [@deHaro2]) and follows our wishes. In fact, next to the CS term (here we continue to use the $U(N)$ model), we can consider [^8] $$\label{eq31}
\mathcal{L}_{3}^{eff.} = \frac{1}{2} (\partial_i \varphi)^2+ \frac{1}{16} \mathcal{R}_3\, \varphi^2 - \frac{\lambda_6}{6 N^2}\, \varphi^6,$$ where $\mathcal{R}_3$ is the boundary 3D scalar curvature that is $\frac{6}{R_0^2}$ for the three-sphere ($S^3$) of the radius $R_0$ [^9]. Such a Lagrangian is actually used (with $\lambda_6 > 0$) as a dual to describe the dynamics of Coleman-de Luccia large-expanding vacuum bubbles of $AdS_4$ in thin-wall approximation; see [@Barbon1003] and [@Maldacena010] for related discussions. In our setup, these shells might emerge from the special (anti)M5-brane wrapping that results in domain-walls interpolating among the bulk vacua; see also [@Bena]. In other words, note that with $\lambda_6 > 0$, we have an unbounded potential from below signaling instability near the potential extrema, which in turn describes the bulk big crunch singularities. Indeed, the $O(3,1)$ invariant bubble solution includes an open and infinite Friedmann-Lemaître-Robertson-Walker universe inside $AdS_4$ space-time that collapses to a big crunch singularity. A field theory dual for the latter is obtained from a marginal triple-trace deformation of the ABJM model in [@CrapsHertog]; see also [@BzowskiHertog] for a recent related study.
An Explicit SO(4)-Invariant Solution {#subsec.5.1}
------------------------------------
Because the classical solutions for the EOM from the Lagrangian with different $\mathcal{R}_3$’s could be related by conformal transformations, to have a simple analytical solution, we consider a massless instanton on $S^3_\infty$(the three-sphere at infinity, $R_0\rightarrow \infty$) that is in turn equivalent to the solution on $R^3$ and so, we set the second term of (\[eq31\]) to zero for now and provide complementary discussions in the next subsection [^10]. Equally and to comply with our formalism of using the ABJM boundary action, we set the fermions to zero with keeping only the $U(1)_{diag}$ part of the gauge group beside the scalar part; That is $$\label{eq32}
\mathcal{\bar{L}}_{3} = \mathcal{L}_{CS}^+ - \texttt{tr} (\partial_i Y_A^{\dagger}\, \partial^i Y^A ) - V_{bos},$$ where $\mathcal{L}_{CS}^+$ is the CS Lagrangian associated with $A_i^+$ and $V_{bos}$ is the bosonic potential of ABJM; see, for instance, [@Me4]. Then, to make a proper solution, we take the following ansatz $$\label{eq33}
Y^A = i\, \varphi(r)\, S^A, \quad Y_A^\dagger = \varphi(r)\, S_A^\dagger, \quad S^A= S^B S_B^\dagger S^A - S^A S_B^\dagger S^B,$$ where the presence of $i$ factor and using different $Y^A$ and $Y_A^\dagger$ are because of being in Euclidean space; and further set $Y^3=Y^4=0$. A clear solution for $S^1$ and $S^2$ reads [@Gomiz], [@Terashima] $$\label{eq33a}
(S_1^\dagger)_{m,n} = \sqrt{m-1}\, \delta_{m,n}, \quad (S_2^\dagger)_{m,n} = \sqrt{N-m}\, \delta_{m+1,n}.$$ Thereupon, the scalar EOM of $Y_A^\dagger$ becomes $$\label{eq34}
\partial_i \partial^i \varphi(r) + \frac{12 \pi^2}{k^2}\, \varphi(r)^5 =0,$$ from which a solution reads $$\label{eq34a}
\varphi(r)= \sqrt{\frac{k}{2 \pi}}\, \left( \frac{a}{a^2 + (\vec{u}-\vec{u}_0)^2} \right)^{1/2},$$ which is Fubini-like [@Fubini1] and the $O(4)$-invariant one we have been looking for. Meanwhile, we notice that by the ansatz (\[eq33\]), the gauge field equations are satisfied trivially. As a commentary, note that because of the SI in its UV fixed-point, the model admits an infinite family of instantons responsible to form the bulk vacuum bubbles, which in turn break the $AdS_4$ isometry to $SO(3,1)$. The broken generators, with four free parameters $a$ and $\vec{u}_0$ marking the size and position of the boundary instanton respectively, act to translate the bubbles around in the bulk 4D volume. Further, the finite contribution for the action on $S^3_\infty$ based on the solution (\[eq34a\]), after linearization of $S^A$ matrices, reads $$\label{eq34b}
\int_0^\infty \frac{r^2}{(a^2 + r^2)^2}\, dr = \frac{\pi}{4\, a} \ \ \Rightarrow \ \ S_{b} \cong -\frac{\pi}{2 k\, \lambda^2},$$ where in the last term on the RHS we have used the large $N$ normalization coefficient $1/N^4$ for the marginal deformation term and $\lambda= N/k$.
Still, a subtle point is that as our bulk solutions are $H$-singlets, how we can use the ansatz (\[eq33\]). To resolve this, we first note that $Y^A$ and $Y_A^\dagger$ may be considered as independent degrees of freedom in Euclidean space, which signals the parity breaking as well. Next, we notice that with $Y^3=Y^4=0$, one of the $SU(2)$’s is considered as a spectator and then, from one of the two complex scalar fields of the remaining $SU(2)$, we may write $$\label{eq35}
Y \equiv Y_1^\dagger + Y^1, \quad Y^\dagger \equiv Y_2^\dagger + Y^2;$$ Last, wrt the swapping $\textbf{s} \leftrightarrow \textbf{v}$ and with a linear combination of these, we can build the singlet scalars as $$\label{eq35a}
y = \varphi(r) (S_1^\dagger S^2 - S^1 S_2^\dagger), \quad \bar{y} = i \varphi(r) (S^1 S_2^\dagger - S_1^\dagger S^2).$$
Relation to Mass Deformation and SI Breaking {#subsec.5.2}
--------------------------------------------
An ansatz like (\[eq33\]) is already used as a tool to make the fuzzy $S^3$ solutions of the mass-deformed ABJM model; see [@Gomiz], [@Terashima] and [@HanakiLin]. In fact, the mass deformation of the original ABJM model, associated with turning the bulk flux on, is done through a relevant operator, $\mathcal{L}_{mass}=\mu^2 \texttt{tr}(Y^A Y_A^\dagger)$, which breaks the global R-symmetry $SU(4)$ down into $SU(2) \times SU(2) \times U(1)$ while breaks the conformal symmetry $SO(3,2)$ entirely. The vacua of the mass-deformed model have interpretations as fuzzy three-spheres. In other words, the M5-branes are understood as M2-branes puffing into a fuzzy sphere near the M5-brane core. The fuzzy-funnels solutions describing M2-M5 brane intersections have interpretations in the field theory on M2-branes as domain-walls. It is interesting that the second term of the Lagrangian (\[eq31\]) has a similar structure to this mass deformation term.
There also are other interesting points and discussions with the Lagrangian and our setup. First note that the potentials of $O(N)$ vector models in three dimensions are renormalizable in the large $N$ limit; and by including relevant and marginal terms up to the so-called $\varphi^6$, the potentials are stable for $0\leq g_6 \leq g_6^c =(4 \pi)^2$ and unstable for $g_6 > g_6^c$ and $g_6<0$ because of the potential unbounded from below–note that $g_6 \equiv -\lambda_6$ and that the sign of $g_6$ controls the behavior of the system in the classical case. There is the tri-critical model just with $g_6 (\varphi^2)^3$ for which the beta function is zero at $N\rightarrow \infty$, while $1/N$ corrections break the SI and a massless Goldstone boson (dilaton) is appeared as a dynamical bound-state of $(\vec{\varphi}.\vec{\varphi})$–we remind that for any finite and positive $g_6$, the operator becomes irrelevant quantum mechanically. In the next-to-leading order of the $1/N$ expansion, the dilaton gets a tachyonic mass and the spontaneously broken phase becomes unstable. In other words, in infinite $N$ limit, there is the UV fixed-point $g_6^c \cong 158$ and by increasing $g_6$, one reaches to the UV fixed-point $g_6^* = 192$, which in turn is an instability region where non-perturbative effects are dominated [@Bardeen1984]; see also [@Elitzur05]. As a result, the nonzero vacuum expectation value $\langle \varphi^2 \rangle \neq 0$ is equivalent to the massive deformation discussed above or a relevant deformation by the $\Delta_{-}=1$ operator. In addition, this marginal plus relevant deformations are consistent with the marginally relevant operator we have got in (\[eq24b\]) by including the backreaction.
Further, according to the discussions in [@RabinoviciSmolkin011], in the presence of CS term, only the dilaton causes a bounded potential and for $g_6 > g_6^c$ just the neutral states under $U(1)$ are formed; and this is the unstable phase we have met as well. Still, a more interesting discussion is in [@BardeenMoshe014], where spontaneous breaking of SI in 3D $U(N)_k$ CS theories coupled to a scalar in fundamental rep is studied in the large $N$ limit. By adding a self-interacting term like $\lambda_6 (y^\dagger y)^3$, they have shown that there is a massive phase for a critical combination of $\lambda=N/k$ (the t’Hooft coupling) and $\lambda_6$; Indeed, for the tri-critical model, there is a spontaneous SI breaking for $\lambda^2 + \frac{\lambda_6}{8 \pi^2}=4$. Near the critical phase, the $U(N)$-singlet massive bound-state plays role as a pseudo-dilaton. Meanwhile, this phase of the SI breaking is dual to the parity broken version of the Vasiliev’s HS theory in $AdS_4$ bulk [@Vasiliev01].
Final Comments
==============
In this study, we first considered a general 4-form ansatz of 11D supergravity over $AdS_4 \times CP^3 \ltimes S^1/Z_k$ and after comparing its structure with that in [@Gauntlett03], analyzed the resulting equations and solutions, especially by including the backreaction. Next, making use of a special 4-form ansatz with the same settings, we obtained a particular nonlinear EOM, from the identity and equation of the 4-form, that included all massive, massless and tachyonic modes for an involved (pseudo)scalar in $EAdS_4$. Then, we referred to the Einstein equations and got the equations (\[eq20a\]), (\[eq20b\]), (\[eq20c\]) from setting the bulk energy-momentum tensors to zero. Then, by solving them simultaneously with the bulk equation (\[eq12\]), we obtained an instanton solution in subsection \[subsec03.01\], from the consistent truncation, corresponding to an exactly marginal or a marginally relevant deformation of the boundary theory. After that, we evaluated a correction to the main background supergravity action in subsection \[subsec03.02\]. After an analysis of dual symmetries, we finally discussed the corresponding field theory counterparts, solutions and some related points.
In fact, for the boundary dual of the bulk solution with considering the backreaction, we employed a scalar Lagrangian including a marginal deformation term, which could be considered as a triple-trace deformation of a dimension-one operator, plus a relevant mass term of the latter operator, like those used in the standard $O(N)$ and $U(N)$ vector models. Then, considering the correspondence rules, we built a $SO(4)$-invariant Fubini-like instanton solution. The condition to break scale invariance, except by a boundary mass term, bulk interpretations of the solutions and other related issues were also addressed. In particular, we noticed that the marginal deformation in the CFT$_+$ fixed-point triggers an instability of CFT$_3$ on $S^3$ (or $dS_3$ in Lorentzian signature) [^11].
As the last comment, we notice that because of the special (anti)M-brane wrapping, there are domain-wall flows corresponding to thin-wall bubbles of $AdS_-$ (for the true vacuum) that expand exponentially within $AdS_+$ (for the false vacuum) and show Bose-Einstein condensations on $dS_3$ space-time. In the case that the condenses lead to the bulk crunches, the instability dynamics is described by finite $N$ corrections including formation and collisions of multi-bubbles; see [@Elitzur05] and [@Barbon1003]. In other words, a $AdS_-$ thin-wall bubble expanding within $AdS_+$ is equivalent to a flow between the UV (CFT$_+$) and IR (CFT$_-$) fixed points. Arriving the bubbles in a finite time to the boundary has a CFT interpretation as rolling in the potential of the unstable marginal boundary operator. Meantime, it is argued in [@BarbonRabinovici011] that the crunches are associated with *negative energy falls* at least for marginally relevant operators. One should also note that although the constant-field arrangements $\bar{\varphi}=0$ and $\bar{\varphi} = \pm \left[{3}/{(4 R_0^2\, \lambda)} \right]^{1/4}$ are the local minimum and maximums of the potential $V(\varphi)=\frac{3}{8\, R_0^2}\, \varphi^2 - \frac{\lambda_6}{6 N^2}\, \varphi^6$ in (\[eq31\]) respectively, and have *bounce* nature, the bubble dynamic is however described by a field similar to $\varphi(r)$ in (\[eq34a\]). For discussions on *generalized Fubini instantons* and *oscillating Fubini instantons* of double-hump potentials of this type, see [@BumHoonLee014A] and [@BumHoonLee014].\
Solutions from Matching with [@Gauntlett03] {#Appendix.AA}
===========================================
In this Appendix, we discuss the equations, solutions and backreaction issue for the general ansatz (\[eq01\]) briefly. To this end, we first face the formalism and reduction here for the metric and 4-form with $ds^2$ and $G_4$ given, respectively, in the equations (2.4) and (2.5) of [@Gauntlett03] [^12] that read $$\label{eq02a}
ds_{11}^2 = ds_4^2 + e^{2 U} ds^2 ({K E_6}) + e^{2 V} \left( \eta + A_1 \right)^2,$$ where $ds_4^2$ and $ds^2 ({K E_6})$ (a Kähler-Einstein metric) are equivalent to our full $AdS_4$ and $CP^3$ metric respectively, $U, V$ are scalar fields and $A_1$ is an 1-form on the external 4D space, and $$\label{eq02b}
\begin{split}
G_4^{(1)} = & 2 h\, J \wedge J + H_1 \wedge J \wedge \left( \eta + A_1 \right) + H_2 \wedge J + H_3 \wedge \left( \eta + A_1 \right) + f\, \texttt{vol}_4 \\
& + \sqrt{3}\, \left(\chi_1 \wedge \Omega + \chi\, \left( \eta + A_1 \right) \wedge \Omega + c.c. \right),
\end{split}$$ where $h, f$ are real scalars, $H_r\, (r=1,2,3,)$ are $r$-forms, $\chi_1$ is a complex 1-form and $\chi$ is a complex scalar all on the external 4D space, and $\Omega$ is the complex $(3,0)$ form on $CP^3$. To adjust the formalisms, we should first take $e_7 \rightarrow 2 e_7 \equiv \eta$ and as a result $J \rightarrow 2J$, and set $U = V = \chi_1 = \chi = A_1 = B_1=0$. Next, from comparing the ansatzs and satisfying the Bianchi identity $d G_4=0$, we have $$\label{eq03a}
H_1=dh=0, \quad f_1= f_2= c_1 + c_2\, u^3, \quad h=\frac{3}{2} c_1, \quad df_3 = f_4\, \ast_4 \mathcal{A}_3^{(0)},$$ where $c_1, c_2, \ldots$ are constants of integration, $u$ is the horizon coordinate in the Poincar$\acute{e}$ metric $$\label{eq03aa}
ds^2_{EAdS_4} = \frac{1}{u^2} \left(du^2 + dx_i ^2 \right),$$ with $i=(1,2,3)$ for $(x,y,z)=\vec{u}$ respectively, and $$\label{eq03b}
f= \pm \frac{6}{R^7} \epsilon, \quad \texttt{vol}_4 =\frac{R^4}{16} \mathcal{E}_4,$$ where $\epsilon = f_7 = - \frac{i}{3} R^3$, with $i$ for being in Euclidean space [^13] and the lower sign for the skew-whiffed solutions in general, and $$\label{eq03c}
H_2 = \frac{2}{R^3} \ast_4(df_5 \wedge \ast_4 \mathcal{A}_3^{(0)}), \quad H_3 = \frac{3}{R^5} \ast_4 df_6,$$ where $H_2 = 2 B_2$, $H_3 = dB_2$ [^14]. Note also that to satisfy the Bianchi identity $dH_3 =0$ and $dH_2 =2 H_3$, the conditions read [^15] $$\label{eq03d}
f_5=-\frac{3}{2 R} f_6, \quad \partial_i \partial^i f_5 = 0$$ with $r=\sqrt{x_i x^i}$ and the solution $$\label{eq03d2}
f_5(r)=c_3 + \frac{c_4}{r}.$$ It is also notable that to satisfy the EOM of $G_4$, an extra condition is setting $c_1=0$.
Now, we try to present solutions including the backreaction. To do this, from the action (2.10) of [@Gauntlett03] we read $$\label{eq04}
S_{4E} = \int d^4x\, \sqrt{g_4}\, R^7 \left(- \left(\mathcal{R}_4-\frac{10}{3} \Lambda \right) + \frac{3}{4 R^4} H_{\mu \nu}\, H^{\mu \nu} + \frac{1}{12 R^2} H_{\mu \nu \rho}\, H^{\mu \nu \rho} \right),$$ where $\mathcal{R}_4$ is the scalar curvature of $EAdS_4$ and $\Lambda = - \frac{12}{R^2}$ is the cosmological constant, and that with (\[eq03c\]), we have $$\label{eq05}
H_{\mu \nu}\, H^{\mu \nu} = \frac{32}{R^8}\, u^2 (\partial_i f_5)(\partial^i f_5), \quad H_{\mu \nu \rho}\, H^{\mu \nu \rho} = \frac{384}{R^{10}}\, u^2 (\partial_i f_5)(\partial^i f_5).$$ As a result, the (pseudo)scalar equation from the action is (\[eq03d\]) that in turn satisfies $$\label{eq06}
d( R^2 \ast_4 H_3) + 6 \ast_4 H_2 =0, \quad d( R^3 \ast_4 H_2)=0,$$ which are the (pseudo)scalar equations from (B.9)-(B.13) of [@Gauntlett03].
On the other hand, from the 11D Einstein equation $$\label{eq07}
\mathcal{R}_{MN}= \frac{1}{6} \left(\frac{3}{3!} {G}_{MPQR}\, {G}_N^{PQR} - \frac{1}{4!} g_{MN}\, {G}_{PQRS}\, {G}^{PQRS} \right),$$ we read the equation $$\label{eq08}
(\partial_i f_5)(\partial^i f_5) + {C}_0\, \frac{R^{10}}{u^2} =0,$$ with ${C}_0=\frac{2}{7}, \frac{1}{2}$ and $-\frac{2}{5}$ corresponding to $\mathcal{R}_{\mu \nu}$, $\mathcal{R}_{mn}$ and $\mathcal{R}_{77}$ from (B.19), (B.21) and (B.22) of [@Gauntlett03], respectively. Then, a solution from (\[eq08\]) is also realized as (\[eq03d\]) with $$\label{eq09}
c_4 = c_5\, \frac{r^2}{u},$$ where $c_5$ has different values for different $C_0$’s. It is noteworthy that the solution is structurally similar to the (pseudo)scalar $m^2 =+4$ asymptotic solution near the boundary ($u = 0$), associated with the non-normalizable mode $\Delta_- = -1$ of course. It should also be noted that it is not possible to have a unique solution including the backreaction on the whole background geometry; In fact, to have just one solution (one $c_5$), we must consider the backreaction in one part of 11D space (e.g. the external space) and neglect it on the remaining parts.
Details of Computing the Stress Tensors {#Appendix.A0}
=======================================
Although we could discuss on the backreaction according to relations (B.19-22) of [@Gauntlett03] directly, we do it in our own way in this Appendix. For the Einstein equation $$\label{eq14a}
\mathcal{R}_{MN} - \frac{1}{2} g_{MN} \mathcal{R} = 8 \pi \mathcal{G}_{11} T_{MN}^{\tilde{G}_4},$$ we consider $$\label{eq14b}
T_{MN}^{\tilde{G}_4} = \frac{1}{4!} \left[4 \tilde{G}_{MPQR} \tilde{G}_N^{PQR} - \frac{1}{2} g_{MN} \tilde{G}_{PQRS} \tilde{G}^{PQRS} \right],$$ where we use the capital indices $M, N,...$ for the 11D space-time directions and small indices $m, n,...$ for the 6D internal $CP^3$ space, and the Greek indices $\mu, \nu,....$ for the 4D external $AdS_4$ space.
We compute the external and internal components of the above EM tensor for the ansatz (\[eq11\]) wrt its dual 7-from, which in components read $$\label{eq15a}
\tilde{G}_{PQRS} \equiv \hat{c}_1\ \tilde{G}_{m n p q} + \hat{c}_2\ \tilde{G}_{\mu m n 7} + \hat{c}_3\ \tilde{G}_{\mu \nu \rho \sigma},$$ $$\label{eq15b}
\tilde{G}_{PQRSTUV} \equiv \bar{f}_1\ \tilde{G}_{\mu \nu \rho \sigma m n 7} + \tilde{G}_{\mu \nu \rho m n p q 7} + f_7\ \tilde{G}_{m n p q r s 7},$$ respectively, where we use $e_7$ as the seventh vielbein (i.e. as a coordinate base) and that $$\label{eq16a}
\begin{split}
& \tilde{G}_{m n p q} = 6\, F_{m n p q},\quad \mathcal{F}_{m n p q} = J_{[{m n}} J_{{p q}]} = \frac{1}{3} \left(J_{m n}J_{p q} - J_{p n}J_{m q} - J_{q n}J_{p m}\right), \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tilde{G}_{\mu m n 7} = (\partial_\mu f_3)\, J_{m n} = - \tilde{G}_{7 m n \mu}, \quad \tilde{G}_{\mu \nu \rho \sigma} = \varepsilon_{\mu \nu \rho \sigma}, \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \hat{c}_1 = 8\, R^4\, \bar{f}_1, \quad \hat{c}_2 = -2\, R^4 \quad \hat{c}_3 = \frac{3}{8} R^4\, f_7,
\end{split}$$ $$\label{eq16b}
\begin{split}
\tilde{G}_{\mu \nu \rho \sigma m n 7} & = (105)\, \varepsilon_{\mu \nu \rho \sigma}\, J_{m n}, \quad \tilde{G}_{\mu \nu \rho m n p q 7} = (210)\, A_{\mu \nu \rho}\, \mathcal{F}_{m n p q}, \quad \tilde{G}_{m n p q r s 7} = \frac{7!}{8}\, \mathcal{F}_{m n p q r s}, \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ A_{\mu \nu \rho}=\sqrt{g_4}\ g^{\sigma \sigma} \varepsilon_{\sigma \mu \nu \rho}\, \partial^{\sigma} f_3, \\
\mathcal{F}_{m n p q r s} & = J_{[{m n}} J_{p q} J_{{r s}]} = \frac{1}{15} (J_{m q} J_{n s} J_{p r} + J_{m s} J_{n p} J_{q r} - J_{m s} J_{n q} J_{p r} - J_{m r} J_{n p} J_{q s} + J_{m r} J_{n q} J_{p s} \\
& + J_{m p} J_{n r} J_{q s}- J_{m p} J_{n s} J_{q r} - J_{m q} J_{n r} J_{p s} + J_{m n} J_{q r} J_{p s} + J_{n p} J_{m q} J_{r s} + J_{n r} J_{p q} J_{m s} \\
& + J_{m n} J_{p q} J_{r s} - J_{m n} J_{q s} J_{p r} - J_{n q} J_{m p} J_{r s} - J_{n s} J_{p q} J_{m r} ).
\end{split}$$
To continue, we write $$\label{eq17a}
\tilde{G}_{PQRS} \tilde{G}^{PQRS} = \hat{c}_1\, \tilde{G}_{m n p q} \tilde{G}^{m n p q} + 12\, \hat{c}_2\, \tilde{G}_{\mu m n 7} \tilde{G}^{\mu m n 7} + \hat{c}_3\, \tilde{G}_{\mu \nu \rho \sigma} \tilde{G}^{\mu \nu \rho \sigma},$$ $$\label{eq17b}
\tilde{G}_{\mu PQR} \tilde{G}_\nu^{PQR} = 3\, \hat{c}_2\, \tilde{G}_{\mu m n 7} \tilde{G}_\nu^{m n 7} + \hat{c}_3\, \tilde{G}_{\mu \rho \sigma \delta} \tilde{G}_\nu^{\rho \sigma \delta},$$ $$\label{eq17c}
\tilde{G}_{m PQR} \tilde{G}_n^{PQR} = \hat{c}_1\, \tilde{G}_{m p q r} \tilde{G}_n^{p q r} + 6\, \hat{c}_2\, \tilde{G}_{m p 7 \mu} \tilde{G}_n^{p 7 \mu},$$ $$\label{eq17d}
\tilde{G}_{7 PQR} \tilde{G}_7^{PQR} = 3\, \hat{c}_2\ \tilde{G}_{7 m n \mu} \tilde{G}_7^{m n \mu},$$ in which the numerical coefficients are for permutations of indices. As a result, we obtain $$\label{eq18a}
\tilde{G}_{PQRS} \tilde{G}^{PQRS} = 96 \left[\frac{8}{R^7} \bar{f}_1^2 + \frac{1}{2 R^5} (\partial_\mu f_3)(\partial^\mu f_3) + \frac{3}{8 R^7} f_{7}^2\right],$$ $$\label{eq18b}
\tilde{G}_{\mu PQR} \tilde{G}_\nu^{PQR} = \frac{9}{R^7} f_{7}^2\ g_{\mu \nu} + \frac{12}{R^5} (\partial_\mu f_3)(\partial_\nu f_3),$$ $$\label{eq18c}
\tilde{G}_{m PQR} \tilde{G}_n^{PQR} = \frac{128}{R^7} \bar{f}_1^2\ g_{m n} + \frac{4}{R^5} (\partial_\mu f_3)(\partial^\mu f_3)\ g_{m n} ,$$ $$\label{eq18d}
\tilde{G}_{7 PQR} \tilde{G}_7^{PQR} = \frac{12}{R^5} (\partial_\mu f_3)(\partial^\mu f_3)\ g_{7 7},$$ with a $4!$ coefficient for all terms. In obtaining the latter results, we have used the differential-geometry formulas in Appendix \[Appendix.A1\] together with permutations of the indices depended on their locations on the forms besides the following relations for the Kähler form on $CP^3$: $$\label{eq19}
\begin{split}
& \ \ \ \ J_{m n} = - J_{n m}, \quad J_{m n} J^{m n} = 6, \quad g_{m p} g_n^{\ p} = g_{m n}, \\
& J_{m n} J_{p q} J_{r s}\ \varepsilon^{\acute{m} n p q r s} = 8 g_m^{\ \acute{m}}, \quad J_{m n} J_{p q} J_{r s}\ \varepsilon^{m n p q r s} =48.
\end{split}$$
Then, plugging (\[eq18a\]) with (\[eq18b\]), (\[eq18c\]) and (\[eq18d\]) back into (\[eq14b\]), making use of (\[eq13b\]), taking traces and using the Euler-Lagrange equation, we finally arrive at $$\label{eq20a}
\Box_4 \bar{f}_1 - \left(2 \pm 6\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 192\, \bar{f}_1^3=0,$$ $$\label{eq20b}
\Box_4 \bar{f}_1 - \left(1 \pm 9\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 288\, \bar{f}_1^3=0,$$ $$\label{eq20c}
\Box_4 \bar{f}_1 - \left(-1 \pm 3\, \bar{C}_7 \right) \frac{\bar{f}_1}{R^2} - 96\, \bar{f}_1^3=0,$$ respectively.
Some Useful Formulas {#Appendix.A1}
--------------------
The main differential-geometry relations, in order to do the computations of Appendix \[Appendix.A0\], in general D-dimensions read $$\label{eq21a}
\tilde{G}_\alpha = \frac{1}{\alpha!} \tilde{G}_{R_1 R_2 ... R_\alpha} dX^{R_1 R_2 ... R_\alpha}, \quad dX^{R_1 R_2 ... R_\alpha} \equiv dX^{R_1} \wedge dX^{R_2} \wedge ... \wedge dX^{R_\alpha},$$ $$\label{eq21b}
\ast_D \tilde{G}_\alpha = \frac{\sqrt{g_D}}{(D- \alpha)!\ \alpha!} \varepsilon_{R_1 R_2 ... R_{\alpha} R_{\alpha +1} ... R_{D- \alpha}} \tilde{G}^{R_1 R_2 ... R_\alpha} dX^{R_{\alpha +1} ... R_{D- \alpha}},$$ $$\label{eq21c}
\tilde{G}_\alpha \wedge \tilde{H}_\beta = \frac{1}{\alpha!} \frac{1}{\beta!} \tilde{G}_{R_1 R_2 ... R_\alpha} \tilde{H}_{S_1 S_2 ... S_\beta}\ dX^{R_1 R_2 ... R_\alpha S_1 S_2 ... S_\beta},$$ $$\label{eq21d}
\tilde{G}_\alpha \wedge \ast_D \tilde{G}_\alpha = \frac{\sqrt{g_D}}{\alpha!} \tilde{G}_{R_1 R_2 ... R_D} \tilde{G}^{R_1 R_2 ... R_D}\ dX^{1 2 ... D},$$ $$\label{eq21e}
dX^{1 2 ... D} = \frac{1}{D!}\varepsilon_{R_1 R_2 ... R_D} dX^{R_1 R_2 ... R_D},$$ $$\label{eq21f}
\varepsilon^{R_1 R_2 ... R_\alpha S_1 S_2 ... S_{D-\alpha}}\ \varepsilon_{R_1 R_2 ... R_\alpha T_1 T_2 ... T_{D-\alpha}} = \alpha! (D-\alpha)!\ \delta^{[S_1...}_{T_1 ...} \delta^{S_{D-\alpha}]}_{T_{D-\alpha}},$$ for the $\alpha$-form $\tilde{G}$ and $\beta$-form $\tilde{H}$. It is also notable that we have used $$\label{eq22}
\tilde{G}^{PQRS} = \frac{1}{\sqrt{g_{11}}\ 7!}\, \varepsilon^{PQRS R_1 R_2 ... R_7} \tilde{G}_{R_1 R_2 ... R_7},$$ as a result of (\[eq21b\]) with $D=11$.
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[^1]: E-Mail: [email protected]
[^2]: It is noteworthy that the exactly marginal solution, which is obtained by setting the bulk stress tensors to zero, might be attributed to a bulk (pseudo)scalar profile with $SO(4,1)$ symmetry–we have already analysed such massless modes in [@Me4].
[^3]: The ABJM model [@ABJM] in large $k$ reduces to a 3D $O(N)$ vector model and on that basis, a marginal triple-trace deformation with a well-known Ultra-Violet (UV) fixed-point at $g_6^*=192$ is studied in [@CrapsHertog]; we return to this issue in subsection \[subsec.5.2\].
[^4]: Remember that with $\eta = 2 e_7$, the modes in (\[eq13c\]) match with those in (\[eq12\]).
[^5]: It is also interesting to consider solutions for each of the equations (\[eq23a\]), (\[eq23b\]) and (\[eq23c\]) separately, in the same way done in subsection (2.4) of [@Me7] for the equation (\[eq12\]).
[^6]: It should be noted that the singlet sections of $U(N)$ and $O(N)$ CS vector models, with complex and real (pseudo)scalars in fundamental reps, are dual to the non-minimal and minimal Vasiliev higher-spin (HS) bulk theories (see [@Giombi01] for a review), respectively, with parity breaking scheme [@Aharony1110], [@Minwalla1207]. As a result, our setups may be recast in that framework; For instance, our bulk massive modes may be considered as arisen from some loop corrections or interactions in the minimal HS model or as fluctuations about the minimal solution of $m^2 R_{AdS}^2=-2$.
[^7]: It is notable that with mixed boundary conditions, corresponding to multi-trace deformations, the effective action reads ${\Gamma}_{eff.} [\alpha] = S_{on}[\alpha] + \int d^3 \vec{u}\ \tilde{f}(\alpha)$, where $S_{on}$ is the bulk on-shell action and $\tilde{f}(\alpha)$ is a function of the local operator $\mathcal{{O}}_{\Delta_-}$ with which the action is deformed. In general, the mixed boundary conditions lead to conformal field theories only if $\tilde{f}(\alpha) \sim \alpha^{3/\Delta_-}$ or $\beta = f_0\, \alpha^{({3}/{\Delta_-})-1}$, where $\alpha$ and $\beta$ act as vacuum expectation value and source for the operator $\Delta_-$ and conversely for $\Delta_+$, and different values of $f_0$ correspond to various points along the lines of marginal deformations. \[ftn.10.\]
[^8]: Note that we use the metric signature $(-, +, +,\ldots +)$ for both gravity and field theories; and after Wick-rotation, we reach to the fully positive signature metric along with $t_M \rightarrow i t_E$ and $e^{-i S_M} \rightarrow e^{-S_E}$, where $S_E$ is the positive Euclidean action.
[^9]: In fact, wrt the footnote \[ftn.10.\] and fact that the solution (\[eq24a\]) could be interpreted as a marginal triple-trace deformation, we can read the holographic effective action $\tilde{\Gamma}_{eff.} [\alpha]$, from the bulk analysis as shown in [@deHaro2], which agrees with (\[eq31\]).
[^10]: An analysis with a Lagrangian like (\[eq31\]), when the boundary is defined on $S^3$, is presented in [@SmolkinTurok].
[^11]: Note also that multi-trace deformations in general destabilize $AdS_4$ vacua, see [@HertogMaeda01], although the triple-trace deformation here preserves the conformal invariance in leading order.
[^12]: Note that our ansatz (\[eq01\]) is a general one that could be constructed from the scalar functions in the external space next to the given forms $\omega, e_7$ and $\mathcal{A}_3^{(0)}$ of the background solution, while $G_4$ in [@Gauntlett03] includes more ingredients. Except for this overall likeness, we have our own objectives and to succeed them, we study the bulk modes, equations, solutions and other related topics as well as their field theory duals in details, none of which has been covered in [@Gauntlett03].
[^13]: One should note that with $G_4 \rightarrow i G_4$, the Euclidean equation (\[eq01B\]) goes to that in [@Gauntlett03]; and that with $\epsilon =1$ in [@Gauntlett0912], the matching term is just the ABJM background solution $G_4^{(0)}$.
[^14]: It is remarkable that, according to the discussion after the equation (2.18) in [@Gauntlett03], the massive ($m^2 R_{AdS}^2=12$) 2-form $B_2$ could be dualised to a pseudoscalar ($a$) with $\Delta_+ = 5$, which might in turn be identified with $f_5$ we have considered in [@Me7].
[^15]: Note that if we set $f_5$=0 in the main ansatz (\[eq01\]), we have a massless (pseudo)scalar in $AdS_4$ that is already studied in [@Me4].
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abstract: |
This report deals with the design of handover schemes for radio access networks (RAN) in 5G networks, using programmable data plane switches. The network architecture is expected to be a centralized cloud infrastructure, connected via a backhaul network to many edge-computing clouds that are closer to the end-user. Some of the network services can be implemented in edge devices to improve network performance.
In 5G networks, the C-RAN architecture splits the Base Band Unit (BBU) into Central and Distributed Units (CU and DU). This structure has created a mid-haul Network, connecting CUs and DUs. The mid-haul network has created a dataplane challenge that does not exist in traditional distributed RANs – the need for efficient connections between the CUs and DUs. Traditional encapsulation techniques can be used to transport packets across the CU and DU. However, the recent advancements in dataplane programmability can be used to enhance the system performance. In this report, we show how P4 switches can be used to parse the packets between DU, CU, and Back Haul (Core Network) for potential system improvements. In particular, we consider the scenario of mobile handover, that arises when a user moves between different cells in the mobile network. The proposed protocol is called *SMARTHO*, illustrating a smart handover.
*Programming Protocol-Independent Packet Processors (P4)* is a programming language designed to support specification and programming the forwarding plane behavior of network switches/routers. With P4 switches, the protocol designer can define customized packet headers, parsing of headers, and defining new match-action routines. In *SMARTHO*, we use P4 Switches to intervene in the handover process for fixed-path mobile users. Such users could be those in a train, drones, devices with high-degree of predictable mobility, etc. A resource pre-allocation scheme that reserves resources before the UE reaches a future cell, is proposed. The solution is implemented using a P4-based switch introduced between the CU and the DU. The P4 switch is used to spoof the behavior of User Equipment (UE) and perform the resource allocation in advance. This is expected to reduce the handover time as the user moves along its path.
The proposed SMARTHO framework is implemented in the *mininet* emulation environment and in a reconfigurable hardware environment using NetFPGA-SUME boards. For Mininet based simulation, we used virtual hosts connected using P4 switches, using the P4 behavior model (P4BMv2) software switch. User and control traffic is also generated to simulate the mobile traffic and measure the HO performance. User traffic is represented using ICMP ping packets over a tag. The results show a handover response time improvement of 18% for a tandem of two HOs and 25% for a tandem of three HOs. For testbed implementation, we used NetFPGA-SUME boards as P4 switches. The Xilinx SDNet tool-chain is used to compile P4 programs directly to NetFPGA-SUME. Raw data packets are generated using the *scapy* tool. The handover time was measured to be approximately 50 milliseconds in the experiments conducted.
author:
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title: |
**TECHNICAL REPORT**\
Design and Implementation of SMARTHO – A Network Initiated Handover mechanism in NG-RAN, on P4-based Xilinx NetFPGA switches
---
Programmable Data Plane, P4 language, Prototype, Mininet Emulation, Mobility Management, 5G Networks, Next Generation-Radio Access Network (NG-RAN), Handover Mechanism.
Introduction
============
This report deals with improving handover performance in 5G Wireless networks, using the programmable data plane switch paradigm. A large number of operators are now evaluating Next-Generation RAN (NG-RAN) as a way to meet future service requirements. NG-RAN is an enhancement to the earlier Cloud-RAN (C-RAN) architecture that is fully-centralized and fixed, but not adaptive to network traffic. Part of this work was published as a short paper [@CNSM18] and as a M.S. (by Research) Thesis at Indian Institute of Technology Madras, Chennai, INDIA [@PhaniThesis].
In the NG-RAN architecture, real-time (RT) functions are deployed near the antenna site to manage air interface resources, by the Distributed Units (DU). At the same time, non-real-time (NRT) control functions are hosted centrally in the Central Unit. This split functionality is now part of the 3GPP specification[@3gpp38401]. The services offered by the CU and DU can be virtualized in software and placed in Commercial off-the-shelf (COTS) servers, using Network Function Virtualizaton (NFV) [@giannoulakis2014applications; @hawilo2014nfv; @abdelwahab2016network; @costa2015sdn].
In this report, we design a solution for handling mobile device handover, using programmable data-plane switches based on P4 programming language[@bosshart2014p4]. P4-based switches are used to parse the packets and to invoke additional actions defined by the protocol designer. These actions can be made to perform simple forwarding or can aid functional behaviour of the system.
In particular, we propose a Smart Handover (SMARTHO) process for fixed-path mobile devices, such as LTE users in a train, drones, predictable mobility devices, etc. is considered. In particular, the handover is considered for Intra-CU HO from one Radio Head (RH) to another RH in a different DU, but connected to the same CU. This scenario is shown in Figure \[5gintracuho1\]. A resource allocation scheme that reserves resources ahead of the UE in its path is proposed. The solution is implemented using a P4-based switch introduced between the CU and the DU. We use the P4 switch to spoof the behaviour of User Equipment (UE) and perform the resource allocation in advance. Using an implementation based on Mininet and P4BM software switch, it is seen that the proposed method results in an 18% and 25% improvement in the sequence of two and three handovers, respectively. A prototype of the mechanism has also been implemented in a reconfigurable hardware environment using Xilinx NetFPGA-SUME boards, using the P4 Programmable Data Plane (PDP) language [@bosshart2014p4; @netfpgap4; @p4].
We have considered the Intra-CU handover in this report; however, this idea can be applied to other HO processes specified in 3GPP [@3gpp38401].
![Intra-CU Handover.[]{data-label="5gintracuho1"}](Diagrams/IntraCUHO.pdf){width="0.7\linewidth"}
Background {#backgr}
==========
This section presents the relevant background material.
5G NG-RAN
---------
There are several service dimensions in 5G networks [@3gpp22891], including support for massive Machine-Type Communications (mMTC), enhanced Mobile Broadband (eMBB), and Ultra-Reliable Low-Latency Communications (UR-LCC) services. Each service has very different performance requirements and traffic profiles. To serve these new markets and to increase revenues substantially, operators need highly scalable and flexible networks. A large number of operators are now evaluating Next-Generation RAN (NG-RAN) as a way to meet future service requirements. From the initial days of deploying Cloud-RAN[@checko2015cloud], which was business oriented to save operational costs, the focus has now evolved to meet the future complex and varied service requirements.
### C-RAN and NG-RAN
The traditional C-RAN architecture is fully-centralized and fixed, which is not adaptive to the movable traffic and the advanced software defined networking concepts. As a result, it is urgent to improve the friable capability of C-RANs. This led the research community to work on functional split options in C-RAN. In FluidNet[@fluidnet], the novel concept of re-configurable fronthaul is proposed, to flexibly support one-to-one and one-to-many logical mappings between Base Band Units (BBUs) and Radio Resource Heads (RRHs) to perform proper transmission strategies. R-FFT[@thyagaturu2018r] proposed IFFT/FFT the PHY layer split, which would reduce the fronthaul bitrate requirements and enable statistical multiplexing. An optimal functional split is discussed by wang et al., team[@wang2017interplay]. The technical report[@3gpp38801] sets out various options for the RAN and its interfaces to the core network.
In the NG-RAN architecture, real-time (RT) functions are deployed near the antenna site to manage air interface resources, while non-real-time (NRT) control functions are hosted centrally to coordinate transmissions across the coverage area. In NG-RAN, this is being formalized with the Centralized Unit (CU) and Distributed Unit (DU) functional split. This functional architecture is now native to the 3GPP specification[@3gpp38401].
Architectural Principles of CU and DU Split {#archprin}
-------------------------------------------
The implementation of the NG-RAN architecture and its subsequent deployment in the network depends on the functional split between distributed radio and centralized control, called the DU-CU split. The DU will process low-level radio protocol and real-time services while the CU will process non-real-time radio protocols.
3GPP has recognized eight different split options[@3gpp38801]. Of these option-2 and option-3, are the most widely discussed two splits. In option-2 the function split will have “Radio Resources Control” (RRC), “Packet Data Convergence Protocol” (PDCP) in the CU. DU will perform the low-level stack of “Radio Link Control” (RLC), “Media Access Control” (MAC), while the physical layer and RF will be in Remote Radio Unit (RRU).
In the option-3 split, low RLC (a partial function of RLC), MAC, physical layer are in DU. PDCP and high RLC (the other partial function of RLC) are in the CU. These split options are discussed in 3GPP status meeting[@3gppcudusplit]. The services of CU and DU can be virtualized and put in Commercial off-the-shelf (COTS) servers, these virtualized network nodes or Virtual Network Functions (VNFs) can be realized with a network architectural concept called Network Function Virtualization (NFV)[@giannoulakis2014applications]. NFV offers a new way to design, deploy and manage virtual network nodes. It also enables us to decouple suppliers hardware and software business models, opening new innovations and opportunities for SW integrators.
The management and operational aspects of NG-RAN with CU and DU splits would be easy to handle using NFV. There are several research papers which already attempted in virtualizing mobile network functions [@hawilo2014nfv; @abdelwahab2016network; @costa2015sdn].
Intra-CU Handover
-----------------
In a wireless network, user equipment (UE) handover from one cell to another cell is an important aspect of mobility management. In this report, we consider intra-DU handover within a single CU. Typically, there are 3 phases in a handover (HO) process: Preparation Execution and Completion.
The preparation phase deals primarily with resource allocation for the UE in the next DU. In this phase, the Measurement Report (MR) message from the Source\_DU will be transmitted to the CU, which would select the Target\_DU for the HO. The CU will send the HO request (UE Context Request), containing Target-DU-ID, UE context info & UE History Information. When the Target\_DU receives the HO request, it begins handover preparation to ensure seamless service provision for the UE. The Target\_DU would respond with setting up Access Stratum (AS) security keys, uplink bearers connecting to the backhaul, reserve Radio Resource Control (RRC) resources to be used by the mobile device over the radio link and allocates Cell-Radio Network Temporary Identifier. Once the resources are allocated by the Target\_DU, a response message called the “UE context setup response” is sent to the CU.
Once handover preparation between the two DUs (Source\_DU and Target\_DU) is completed, the execution phase will start to have the UE perform a handover. The Source\_DU instructs the UE to perform a handover by sending RRC Connection Reconfiguration message that includes all the information needed to access the Target\_DU. The Target\_DU sends an Uplink RRC Transfer message to the CU to convey the received RRCConnectionReconfigurationComplete message. Then, downlink packets are sent to the UE. Also, uplink packets are sent from the UE, which are forwarded to the CU through the Target\_DU.
In the final completion phase, the CU sends the UE context release command to the Source\_DU which would release all the bearers from CU to Source\_DU.
In this report, we deal with the preparation phase, by proposing a advanced resource allocation scheme along a set of pre-defined DU nodes. The proposed design, working model and elements involved in SMARTHO are discussed in Section \[Design\].
Related Work
------------
There are several papers that deal with handover procedures involving high mobility. We focus on works dealing with handover support for fixed-path mobile users, such as those on a train.
In [@trainseamless], a dual\_link HO scheme is studied for wireless Mobile communication in high\_speed rails. Here, an extra antenna is used, one for handover and other for data communication with the base station. In [@raildistantenna], a radio-over-fibre based approach has been proposed to provide communications inside long tunnels using distributed antenna systems, and performing HO over these antennae. In [@multitunnelmobility], a multiple-tunnel based approach with multiple interfaces and a modified “Hierarchical Mobile IPv6” (HMIPv6) Mobility Management method, is considered. In, [@li2016mobility], mobility prediction based handover with RAN-Cache has been studied for HetNets.
In [@seamlessHOLTEWIFI], a variant of Proxy Mobile Internet Protocol (PMIP) is developed to reduce ping-pong (PP) events and handover failures. In [@mihprotocol], vertical handover is considered by introducing a layer between MAC and PHY layers; this extra layer performs the handover across different technologies.
A measurement of LTE performance on high velocity environment is studied in[@lteperformanceonltevel]. Some papers have studied approaches on the Time To Trigger (TTT) for handover. The work in [@mobperfhetnets] showed that a lower value of TTT for HO would decrease the handover failure, but would increased the ping-pong effect. The work [@handoverinmobility] suggested that handover margin is more appropriate than TTT to adjust handover timing, in response to the change in mobility conditions. In [@lterailtriggeropt], the relation between the TTT and the position of high speed train was investigated. The work in [@zheng2008performance] presents an integrated HO algorithm in LTE networks, while a Received Signal Strength (RSS) based TTT algorithm has been studied in [@anas2007performance].
In all above papers dealing with fixed-path user mobility, pre-allocation of resources along the path have not been considered. In this report, We attempt this approach with the use of programmable data-plane entities.
Programmable Data Plane Switches
--------------------------------
The recent Software Defined Networking networking paradigm (SDN) and associated protocols and implementations such as OpenFlow Protocol[@martinez2015next] and Open VSwitch (OVS)[@pfaff2015design] allow programmability in the data plane. However, these are are not protocol independent. When these switches are used in mobile networks where protocol stack largely differ from the standard protocols, the forwarding behaviour would be limited to encapsulation/tunneling mechanisms. The strict parsers and forwarding routines can help improve the forwarding behaviour[@hommes2017optimising; @macdavid2017concise; @chourasia2015sdn], but would not aid in adding new system functions.
Programming Protocol independent Packet Parsers (P4) provides is an upcoming framework for realizing programmable data-plane switches [@bosshart2014p4]. P4 switches are expected to perform better than traditional L2-L3/ Open Flow switches due to the additional functionality enabled. For instance, we show that a simple tag based forwarding approach over an IP-based encapsulation mechanism is showing 27% improvement using a P4 behaviour model (P4BM) software switch. Hence, this report considered the use of P4-based switches for improving handover performance in future wireless networks.
Proposed SMARTHO Framework {#Design}
==========================
This section presents the details of the proposed Smart Handover (SMARTHO) mobility management framework.
SMARTHO Architecture and Components
-----------------------------------
This section presents the architecture, components and message exchanges involved in SMARTHO model. 3GPP has already discussed the NG-RAN architecture[@3gpp38401]. For SMARTHO, we introduce programmability into the data plane without changing the existing architectural framework.
![Proposed SMARTHO Framework.[]{data-label="smarthofrm"}](Diagrams/SMARTHOConnections.pdf){width="0.6\linewidth"}
The main components of the proposed CU and DU architecture are COTS compute servers, P4 switches, and a Network Controller. The compute servers will implement the functions of CU and DU, P4 switches, and Network Controller. The interconnections and components of SMARTHO framework are shown in Figure \[smarthofrm\].
The network controller at the CU (CU\_Controller) will store the “UE Mobility Information” and the “UE Context Information”. The network controller at the DU (DU\_Controller) will store the RRC Connection Reconfiguration (RRCCR) message. The P4 switches will process the messages from processing units and perform the SMARTHO process, by sending appropriate instruction messages to CU and DU Controllers.
The first handover of a given UE will set the UE context information in the CU\_Controller. After the first HO is completed, the SMARTHO initiation will happen which automates the subsequent handovers. The P4 switches in CU (CU\_P4) and DU (DU\_P4) will send the instruction messages to CU\_Controller to access the mobility information and DU\_P4 switches to store the RRCCR message respectively.
These P4 switches can be hardware switches[@benavcek2017line] or a virtual switch[@P4Software]. Placement of P4 switches in CU and DU can impact the routing performance of the system. A study of this aspect is not in the scope of this report. Hence, without loss of generality, we assume all the P4 switches are at the access layer connected directly to servers and controller as shown in Figure \[CUDUArchitecture\].
![Topology interconnecting CU and DUs.[]{data-label="CUDUArchitecture"}](Diagrams/CU_Architecture.pdf){width="0.6\linewidth"}
Modified Handover Sequence
--------------------------
![Sequence diagram of Intra-CU Handover.[]{data-label="5gseqdiaintracuho"}](Diagrams/Smart_HO_Vertical.pdf){width="0.7\linewidth"}
The entire 3GPP process with P4 switches in CU with sequence of messages is shown in Figure \[5gseqdiaintracuho\]. In the first handover, P4 switches will parse the incoming packets and negotiate with local storage at CU to determine if the UE is having a fixed path. If so, after the completion of first HO, the P4-switch will generate “UE Context Setup Request” message and forward it to the Target HO entities, on behalf of the UE. This is referred to a Smart Handover (SMARTHO) in this report. This action will trigger the HO preparation phase, even before UE reaches the specified HO points, as shown in Figure \[5gintracuho1\].
This would make all the Target HO entities to reserve resources and respond to CU with appropriate “UE Context Setup Response”. The “UE Context Setup Response” message would be saved at Source\_DU and can be later forwarded by the P4-switch as a response to the UE MR. By this spoofing approach, we parallelize the HO preparation phase, which will improve the performance of the handover process.
Architecture and Design of P4 switches {#designofp4}
--------------------------------------
There are several switch architectures such as Pisces[@shahbaz2016pisces] and Portable Switch Architecture (PSA)[@p416psa] that support protocol independent switches. In this report, we use the Very Simple Switch (VSS) Architecture[@P416]. VSS has basic programming blocks needed for protocol independent switch, which are sufficient to implement the SMARTHO process.
The programming blocks of VSS are: (i) Parser; (ii) Match-Action Pipeline; and (iii) De-parser. The parser is a Finite State Machine (FSM), which either accepts or rejects the packet. For every packet the P4 switch receives, it will parse the packets and would extract the header information. The header information obtained is used in the Match-Action Pipeline to invoke a necessary action routine in Match-Action control block. The De-parser will reconstruct the packet, putting back the extracted content of the header with necessary modifications, if needed.
Next generation mobile networks have a complex packet structure. Designing a parser for entire packet structure would overload the functionality of the P4 switch, increasing the complexity of the parser. Also, the structure of the packets for mobile networks would depend on the state information. P4 switches are not scalable to parse such packets as of now. To simplify this process, we design a *tag*-based approach to identify necessary packets for SMARTHO. The tag will be added by the processing units or controller. The P4-switches in the SMARTHO model handles three types of packets:
1. User packets of the 5G system: These packets are ICMP packets encapsulated over the tag, the forwarding is done using tag information.
2. Control packets for HO: In case of Intra CU HO, the entire HO process has twelve control messages exchanging, shown in Figure \[5gseqdiaintracuho\]. These packets have to be identified and will be sent to P4 switches or controller for processing.
3. Instruction packets: These packets will either instruct the P4 switch to initiate specific methods in Match-Action control block or the controller to store/retrieve the data.
Custom Data Structures
----------------------
Three special data structures have been defined to store the necessary state information: Mobility Table (MT), Controller Cache (CC) and RRC Table (RRCT). MT and CC will reside in CU\_Controller and RRCT will reside in DU\_Controller. The details are given below.
\[DSforarc\] A data structure is defined to store the necessary information needed for the SMARTHO process. We define three data structures Mobility Table (MT), Controller Cache (CC) and RRC Table (RRCT). MT and CC will reside in CU\_Controller and RRCT will reside in DU\_Controller
### Mobility Table (MT)
MT stores the mobility information of the UE. With the details in MT, P4 switch will identify the Target\_DU for the next HO. The controller would use MT information to trigger the SMARTHO-Initiation (discussed in the Section \[SMARTHOInit\]) at an appropriate time. Every MT entry contains:
- UE-ID: Identification of the user equipment
- Source DU ID: The source DU global identification
- Target DU ID: The next target DU global identification for the current Source DU ID
- Time Interval: Appropriate time interval after which the SMARTHO process is triggered.
### Controller Cache (CC)
The UE Context Information is retrieved from the message “UE Context Setup Request”, which is triggered from CU processing unit. This information thus retrieved is stored in CU Controller Cache (CC). CU\_P4 switch forwards the “UE Context Setup Request” to CU Controller as shown in Figure \[SMARTHOStep1\] to update the UE context information in CC. Every CC entry contains:
- UE-ID: Identification of the user equipment
- UE-AMBR: Aggregated Max Bit Rate
- UE-Security-Algorithm: Encryption algorithm used by UE
- Security-Base-Key: Base key to encryption keys
### RRC Table (RRCT)
RRCT will store the final HO preparation message (UEModReq/RRCCR) at DU\_Controller. The DU\_P4 switch will instruct the DU\_Controller to store the UEModReq message. The RRCT contains all the fields of UEModReq message, as shown below.
- UE-ID: Identification of the user equipment
- Target DU ID: Aggregated Max Bit Rate
- Bearer information: Bearer ID allocated by the Target\_DU
- Security-Algorithm: Security algorithm at the Target DU
All the three types of packets are encoded with the respective tags. The differentiation is done based on the extracted tag and examining the valid/invalid bit[@P416]. The parser in P4 switch should be indicated about the appropriate tag, for this we use, Ethernet-Type from Ethernet header. IEEE802.3 has assigned EtherType 0x0101-0x01FF as experimental, we can use any of these for indication of tag header. The parser routine of the P4 switch in CU is shown in Algorithm \[parsercup4\].
*header\_union* Tag{ *FrwdTag* t1; *CntrlTag* t2; *InstTag* t3;} *struct* Parsed\_packet { ethernet; tag;} **parser** { $state$ start packet.$extract$(hdr.ethernet); $transition$ $select$(hdr.ethernet.etherType) $16w0$x$0101$ : parse\_inst\_tag; $16w0$x$0102$ : parse\_cntrl\_tag; : parse\_frwd\_tag; $state$ parse\_inst\_tag packet.$extract$(hdr.tag.t3); $transition$ $accept;$ $state$ parse\_cntrl\_tag packet.$extract$(hdr.tag.t2); $transition$ $accept;$ $state$ parse\_frwd\_tag packet.$extract$(hdr.tag.t1); $transition$ $accept;$ }
Implementation Details {#smarthoimpl}
======================
The HO preparation is a resource allocation phase, in the case of fixed path mobile devices the resource allocation can be done a priori. The idea is to preset all the subsequent HOs with appropriate timing delays based on the first HO request.
The preparation phase for the second Intra-CU HO is done before the UE reaches the second Intra-CU HO point. The P4 switch initiates the preparation phase for the Second Intra CU HO, i.e., CU\_P4 switch along with CU\_Controller spoofs the UE and sends a “UE Message Setup Request” to the Target\_DU. When the UE reaches the vicinity of the second HO point, UE will trigger the MR message to Source\_DU; subsequently, the Source\_DU\_P4 will respond with the RRCCR message.
![Operation of handover process, using P4 switches.[]{data-label="5gintracuho2"}](Diagrams/IntraCUHO2.pdf){width="0.7\linewidth"}
As described earlier, we perform the HO preparation phase in advance of the UE movement, in order to decrease the overall HO time. Figure \[5gintracuho2\] presents the working details of SMARTHO, with a sequence of three Intra-CU handover (HO) points. The operation of SMARTHO has three phases: SMARTHO-Data Setup, SMARTHO-Initiation and SMARTHO-Completion, as described below.
![Trigger sequence of SMARTHO.[]{data-label="SMARTHOStep1"}](Diagrams/SMARTHODataSetup){width="0.6\linewidth"}
Data Setup
----------
The current context of the UE has to be retrieved, before the start of the SMARTHO process. The context information of UE can be retrieved from the “UE Context Setup Request” message, which is exchanged between CU and Target\_DU as shown in Figure \[5gseqdiaintracuho\]. The UE context information is updated in the data table CC.
This message is sent to the CU\_P4 switch. The CU\_P4 switch can identify the control packets for HO, this can be done by changing the code at CU part, to send the HO message “UE Context Setup Request” with tag value $0$x$03$, as discussed in Section \[designofp4\]. The CU\_P4 will identify the tag and execute a routine to send the message set\_ue\_context to the CU\_Controller, which will store the UE context information in CC, as shown in Figure \[SMARTHOStep1\]. The set\_ue\_context contains the UE identifier, Aggregate Maximum Bit Rate (AMBR) for the UE, and other relevant information.
**control** $inout$ $metadata$ meta, $table$ etherforward $key$ = hdr.ether.dst\_addr : $exact$; $actions$ = ether\_port\_forward; operation\_drop; $const$ default\_action = operation\_drop(); action cu\_controller\_forward() $standard\_metadata.egress\_spec$ $= controller\_port$; $table$ source\_gnb\_controllerforward $key$ = hdr.ue\_context.src\_gnb\_addr : $exact$; $actions$ = prepare\_\_port\_forward; operation\_drop; $const$ default\_action = operation\_drop(); *$apply$*{ if (hdr.tag.$isValid()$) if(hdr.ue\_context.$isValid()$) cu\_controller\_forward(); else source\_gnb\_controller\_forward.$apply()$; else etherforward.$apply()$; } }
The P4 switch at CU identifies the set\_ue\_context message and forwards it to the CU\_Controller, this is shown at a high level in Algorithm \[cup4\]. Once the CU\_Controller receives the set\_ue\_context message, it updates its CC using a packet sniffer at the controller.
![Initiation sequence of SMARTHO.[]{data-label="SMARTHOStep2"}](Diagrams/SmarthoInitiation2.pdf){width="0.6\linewidth"}
SMARTHO - Initiation {#SMARTHOInit}
--------------------
\[VARIABLES\][Variables]{}[EndVariables]{} [ ]{} The initiation of the SMARTHO process is shown in Figure \[SMARTHOStep2\]. The Source\_gNB\_DU sends the “UE Context Release Complete” message with a tag value of $0$x$0c$ to the CU\_P4. This switch parses the packet and identifies the message with the tag value and initiates the process of SMARTHO. This is done by sending the smartho\_init message to the CU\_Controller with a tag value of $0$x$02$. The purpose of the smartho\_init message is to retrieve the address of Target\_gNB\_DU from MT for the next HO and *delay* information of the UE. This delay value is used to hold the process before starting the preparation phase.
The CU\_Controller runs a packet sniffer at the ingress port. When a smartho\_init message is received, the sniffer runs a background process. This will send the smartho\_trigger message to the CU\_P4 switch with a tag value of $0$x$02$ as shown in Algorithm \[smarthoInit\]. The smartho\_trigger message is sent after a particular delay value, as discussed later in Section \[timinganalysis\].
The smartho\_trigger message is the basis to send the spoofed “UE Context Setup Request” message for the next HO to the Target\_gNB\_DU. This will initiate the HO preparation phase for the subsequent HO.
$mobility\_tag=2$ mobility\_details\[\]=query\_mobility\_table(ue\_id,src\_du) context\_details\[\]=query\_controller\_cache(ue\_id) $delay$(mobility\_details\[time\_interval\]) ether=Ether(dst\_addr, type=0x0101) tag=Tag(mobility\_tag) context\_info=create\_header(context\_details) ue\_context\_req\_pkt = ether/tag/context\_info srp1(ue\_context\_req\_pkt, iface=“eth”)
![Completion sequence of SMARTHO.[]{data-label="SMARTHOStep3"}](Diagrams/SMARTHOCompletion.pdf){width="0.6\linewidth"}
SMARTHO Completion
------------------
The final phase of SMARTHO is to handover the UEModReq/RRCCR message as a response to UE MR, as shown in Figure \[SMARTHOStep3\]. The UEModReq/RRCCR message that is sent from CU to Source\_DU is intercepted by the Source\_DU\_P4 switch. This would instruct the Source\_DU\_Controller to store UEModReq/RRCCR message. This message contains the UEModReq information that is updated in the RRCT of DU\_Controller. Algorithm \[ducontroller\] and the P4 code segment shown in Algorithm \[dup4\] present the details of this operation.
When a UE sends the MR to Source\_DU, the Source\_DU would respond with “Uplink RRC Transfer message” to CU. The DU\_P4 switch intercepts this message and instructs the controller to get the UEModReq/RRCCR message which is forwarded to UE as shown in Figure \[SMARTHOStep3\].
$store\_rrc\_tag=15$ $mr\_uplink\_rrc\_tag=1$ rrc\_packet\_data\[\]=extract\_packet\_content(packet) query load rrct rrc\_packet\_data\[\] query uemod\_reqmsg=get rrct(packet.ue) srp1(uemod\_reqmsg, iface=“eth”) $\text{sniff(iface="eth",prn=DATA\_UPDT)}$
$ue\_context\_tag=1$ $mobility\_tag=2$ **parser** { $state$ start { packet.$extract$(hdr.ethernet); $transition$ $select$(hdr.ethernet.etherType) { $0x0101$ : parse\_inst; $default$ : $accept$; } } $state$ parse\_inst { packet.$extract$(hdr.tag); $transition$ $select$(hdr.tag.tag\_value) { $0x01$ : parse\_ue\_context; $default$ : $accept$; } } $state$ parse\_ue\_context { packet.$extract$(hdr.ue\_context); $transition accept;$ } }
Delay Estimation for Early Resource Allocation {#timinganalysis}
----------------------------------------------
The UE context setup is done by the Target\_DU before allocating the resources, as described earlier. Once the UE context set-up is done at the T\_DU, the T\_DU waits for the “Random Access Procedure”. If this is not received before timer expiry, the “UE Context Release Request” will be initiated to release all the necessary bearers. The timer expiry is triggered based on the user inactivity or by policy controls[@ralfkreherltesignaling]. In SMARTHO, the advanced allocation of resources would be wasted. Hence, an appropriate delay has to be put before SMARTHO Initiation.
To estimate the delay ($t_{delay}$) to initiate the SMARTHO process, we need three inputs: (i) Estimated arrival of Measurement Report (MR) for next HO ($t_{MR}$); (ii) Total Response time for HO preparation ($t_{prep\_HO}$); (iii) Trigger time, for “UE Context Release Request” by T\_DU ($t_{trig}$)
Using Machine Learning techniques with the features such as traffic intensity at switches, history information and so on, we can predict the estimated arrival time of the MR message.
The HO preparation time ($t_{prep\_HO}$) would include the processing times of CU and DU cloud units and processing times of routers connecting CU, DU and RRH. For estimating this we model the system as a simple network of queues. We assume that the packet arrival process at a UE is Poisson; service time is exponential; and routers have limited buffer capacity. We model the routers as a $M/M/1/B$ queue, and the CU and DU entities as $M/M/1$. We model the system as a tandem of $M/M/1/B$ and $M/M/1$ queuing system. The variables are shown in Table \[Variables\_queuing\_model\].
[|l|l|]{} $t_{pd\_sDU\_CU}$ & Propagation delay from Source\_DU to CU\
$t_{pd\_tDU\_CU}$ & Propagation delay from Target\_DU to CU\
$t_{pc\_cd}$ &
-------------------------------------
Expected response time at CU and DU
in HO preparation phase
-------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$t_{pc\_rt}$ &
---------------------------------------------------
Expected delay by routers in HO preparation phase
---------------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$n^{r\_sd}$ &
----------------------------------------------
number of routers between RRH and Source\_DU
----------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$n^{r\_td}$ &
----------------------------------------------
number of routers between RRH and Target\_DU
----------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$n^{sd\_cu}$ &
---------------------------------------------
number of routers between Source\_DU and CU
---------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$n^{td\_cu}$ &
---------------------------------------------
number of routers between Target\_DU and CU
---------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$n$ &
------------------------------------------------------------------
total number of routers between RRH, Source\_DU,
Target\_DU and CU. Each router indexed as $x \epsilon \{1...n\}$
$ n = n^{td\_cu}+n^{sd\_cu}+n^{r\_sd}+n^{r\_sd}$
------------------------------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$B_x$ &
--------------------------------------------
Buffer size in router $x$, present between
CU and DU, $ x \epsilon {1,2,...n}$
--------------------------------------------
: Variables in queuing model[]{data-label="Variables_queuing_model"}
\
$\lambda_{x}$ & packet arrival rates in router $x$\
$\mu_{x}$ & router $x$ processing rates in\
$E[r_{x}]$ & expected response time of router $x$\
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For $M/M/1/B_x$ system, the response time is given by:
$$\label{eq:1}
E[r_{x}] = \frac{\lambda_x}{\mu_x-\lambda_x} + \frac{B_x\lambda_x^{B_x+1}}{\mu_x(\mu_x^{B_x}-\lambda_x^{B_x})}$$
For CU and DU as $M/M/1$, the steady-state response time is given by: $$\label{eq:2}
E[r_{X}] = \frac{1}{\mu_{X}-\lambda_{X}} \text{ where, $X\varepsilon$\{CU,S\_DU,T\_DU\}}$$ The total time taken for HO preparation is processing the Context Requests and Measurement report. The four messages indexed 2,3,4,5 shown in Figure \[5gseqdiaintracuho\] are HO preparation messages.\
The processing time taken by the routers in HO preparation phase ($t_{proc\_{rt}}$) is, $$\begin{aligned}
t_{proc\_{rt}} = 2\left(\sum_{x=1}^{n^{sd\_cu}}E[r_{x}] + \sum_{x=1}^{n^{td\_cu}}E[r_{x}]\right)\end{aligned}$$ The processing time taken by the CU and DU in HO preparation phase ($t_{proc\_{cd}}$) is, $$\begin{aligned}
t_{proc\_{cd}} = 2*\left(E[r_{S\_DU}] + E[r_{T\_DU}]\right)\end{aligned}$$ Total time taken for HO preparation is, $$\begin{aligned}
\small
t_{prep\_HO} = 2*t_{pd\_sDU\_CU} + 2*t_{pd\_tDU\_CU} + t_{pc\_{rt}} + t_{pc\_{cd}}\end{aligned}$$ The trigger time ($t_{trig}$) will include the trigger time and uplink transfer time, approximated as:
$$\begin{aligned}
t_{trig} = \text{trigger time}+t_{pd\_tDU\_CU}+\sum_{x=1}^{n^{td\_cu}}E[r_{x}])\end{aligned}$$
$$\begin{aligned}
t_{delay} = t_{MR}-(t_{prep\_HO}-t_{trig})\end{aligned}$$
This value of delay of $t_{delay}$ is used an approximate value during SMARTHO initiation, described earlier in Section \[SMARTHOInit\].
Implementation in Mininet Emulator
==================================
The proposed SMARTHO framework was implemented in the mininet emulation environment [@mininet], where mininet-based hosts emulate the CU and DU. Mininet hosts are connected using P4 switches, developed using the P4 behaviour model (P4BM) with VSS model architecture, [@P4Software]. Raw data packets are created using the *scapy* tool[@scapy], that sends a continuous sequence of raw data packets from one host to another. User and control traffic are also generated to simulate the mobile traffic and measure the HO performance. User traffic is represented using ICMP ping packets over a tag. The measurement of IP and tag based forwarding is done on user traffic. Control traffic is generated to simulate the HO procedure, packets are created with customized headers containing UE identification, over the tag. The tag of the control packet is also used as the identification for the HO message.
![Network topology for Performance study of SMARTHO.[]{data-label="simarch"}](Diagrams/SimulationArchitecture.pdf){width="0.6\linewidth"}
Comparison of Tag and IP-based forwarding
-----------------------------------------
In order to study tag- and IP-based forwarding, user traffic is sent among the hosts. P4 switches between these hosts parse the packets and either forward the packet or execute the SMARTHO process. This kind of tag-based approach is already investigated by Fayazbakhsh et al. [@fayazbakhsh2014enforcing], where they used the tag for origin binding.
The comparison results are shown in Figure \[tagperf\], with hosts separated by one, two or four intermediate switches. The metrics measured are the average response time and drop count of the packets. As seen, tag-based forwarding performs better than IP based forwarding. Consider Figure \[tagdropperf\_4\] and x-axis range of (20,60) parallel ping process. Here, it is clearly seen that tag-based forwarding is showing much lower packet drops when the number of hops increase.
In specialized environments such mobile networks, which are not connected to the Internet until the Packet Gateway, a tag-based forwarding approach is better. The tag-based identification of packets makes the P4 parser simple, allowing innovations in other aspects too, such as network slicing.
![Performance of Intra-CU HOs in tandem.[]{data-label="singHOperfintand"}](results/Graph_1/bargraph_solid_state.pdf){width="0.6\linewidth"}
Performance of SMARTHO handover
-------------------------------
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For this study, a mininet environment as shown in Figure \[simarch\] was created. We considered a tandem of Intra-CU handovers, sending user packets between the RRH and CU. User packets are generated as parallel ping process in RRH to simulate varying arrival rates. The inter-arrival time between Intra-CU HO was exponential. The HO procedure begins at the RRH by sending MR message to Source\_DU as shown in the simulation architecture. The HO time is measured from the moment RRH has sent the MR message to the Source\_DU, to the RRCCR message received at RRH indicating the HO is completed.
Figure \[singHOperfintand\] presents the performance for handover time on a single UE. The graph shows the total time spent for handover. As seen, the SMARTHO process performs better than the traditional HO process. There is no improvement of HO response time with single HO, this is because the SMARTHO process will perform the data setup in first HO and automates the subsequent HOs. Improvement of 18% for two tandem HOs and 25% for three tandem HOs is achieved and this improvement will increase as the tandem of HOs increases. This is because the overall time spent on HOs will proportionally decrease as the HO preparation phase is done in advance for all the subsequent HOs.
In the next study, we increased the intensity of HO requests, with multiple UEs requesting handovers. Figure \[SMARTHO\_Resp1\] presents the response time. The results show that the SMARTHO process performs better than the traditional HO process, with higher improvements with increase in the number of transmit nodes.
Figure \[SMARTHO\_Resp2\] presents the drop percentage of the HOs, where a handover is considered dropped when the response exceeds a threshold. It is observed that the proposed SMARTHO process is better when the number of intermediate nodes is higher.
Xilinx NetFPGA based prototype testbed
======================================
This section presents the details of the proof-of-concept prototype implementation of the proposed SMARTHO architecture using Xilinx NetFPGAs. The implementation details of the testbed, its architecture, and evaluation results, challenges faced in the development and the performance results of the testbed is also discussed.
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Xilinx NetFPGA-SUME based prototype
-----------------------------------
Xilinx NetFPGA-SUME boards [@zilberman2014netfpga] were used as P4 switches for the testbed implementation. The NetFPGA-SUME boards (Figure \[fig:netfpgasume\]) enable researchers to prototype high-performance applications in hardware. We used Xilinx SDNet toolchain [@netfpgap4], which simplifies the design of packet processing data planes that target FPGA hardware. The overall prototype system is shown in Figure \[sysconnection\].
Four hosts are needed to emulate the behavior of Intra\_CU\_HO, shown earlier in Figure \[simarch\]. The testbed set-up has three Intel-Xeon, 2.6 GHz i7 core CPU with 64 GB RAM for Source\_DU, Target\_DU, and CU. For RRH we used Intel Core i7, 32GB RAM. Systems are integrated with 10G Ethernet and NetFPGA-SUME switch, as shown in Figure \[sysconnection\]. SFP+ fiber-optic LC connector ports are fixed to NetFPGA-SUME and 10G Ethernet boards, and boards are connected with LC 50/125 optical fibers as shown in Figure \[sysconnection2\].
The NetFPGA-SUME board is installed in the host PCI-e slot, as shown in Figure \[fig:netfpgasumesyscon\]. NetFPGA\_Sume boards have four SFP+ 10Gbps ports, Xilinx refers to these interfaces as nf0, nf1, nf2, and nf3 where nf0 is the port closest to the link lights on the board. Loading driver modules (summe\_riffa), the ports on NetFPGA-SUME is recognized, as shown in Figure \[fig:portsofnetfpgasume\]. These network interfaces are the means by which the host machine can communicate with the dataplane in the FPGA.
![Four ports of NetFPGA after installation of PCI-e drivers.[]{data-label="fig:portsofnetfpgasume"}](Diagrams/testbed_pics/ports_netfpgasume.png){width="\textwidth"}
For traffic generation, a 10Gbps Ethernet card is used at the RRH. The systems are interconnected using nf1 and nf2 ports. Listed below are the port connections for the testbed.
- nf1 port of RRH is connected to nf1 port of Source\_DU
- nf2 port of Source\_DU connected to nf1 port of Target\_DU
- nf2 port of Target\_DU is connected to nf1 port of CU
Overall, three optical LC 50/125 optical fibres and six SFP+ fiber-optic LC connector ports are used.
Working model {#HOtestbed}
-------------
For simulating mobile traffic, we used the Scapy tool [@scapy] at the RRH. The Scapy tool generates raw data packets that mimic the control messages for HO. The Scapy programs are executed at RRH and custom packets are created to emulate the control messages of Intra\_CU\_HO. The format of the header is shown in Procedure \[smarthoheader\]. These custom packets defined in Scapy were used as mobile HO control messages. By invoking Smartho(2,4) a packet is created with control\_information as 2 and forwarding\_tag as 4. The forwarding\_tag field is used to set the destination port, control\_information will represent a HO message as shown in Figure \[contrlmessages\], i.e., Smartho(1,4) represent the measurement report, Smartho(2,4) represents the uplink RRC measurement message, and so on.
**class** $"Smartho"$, *IntField* *IntField* *def*
The Xilinx SDNet tool provides the metadata list to configure the destination port and to know the source port. There is also one bit for each of the interfaces (nf0, nf1, nf2, and nf3) in the src\_port and dst\_port fields (bits 1, 3, 5, and 7). So for example, if the data-plane wants to send a packet up to the host and have it arrive on the nf0 Linux network interface then it must set bit 1 of the dst\_port field (e.g., dst\_port = 0b00000010). The destination port can be set by the P4 program with variable sume\_metadata.dst\_port. When sume\_metadata.dst\_port is set as one the packets are egress to nf0 port, for SMARTHO, all the systems are connected using nf1 and nf2 ports alone i.e., sume\_metadata.dst\_port should be either set to 4 or 16.
There are three operations performed by the NetFPGA-SUME switches in SMARTHO testbed implementation:
**Change of control information:** Control message received at the host (discussed in Section \[HOtestbed\]), is changed as per the next sequence message, as shown in Figure \[contrlmessages\] & Figure \[contrlmessages2\].
**Setting the sume\_metadata.dst\_port:** Extract the value of forwarding\_tag\_port and set this to sume\_metadata.dst\_port. This would set the egress port for the control message.
**Change the forwarding\_tag\_port:** We used look\_up\_table to change the forwarding\_tag\_port. The look\_up\_table is statically set and does an exact match with control message and src\_port information put together. Since the architecture setup is static, the look\_up\_table is loaded at compile time. Based on the control\_message and sume\_metadata.src\_port the forwarding port is decided. This forwarding information will be extracted and updated to the header forwarding\_tag\_port. The forwarding\_tag\_port will then be used to set the egress port by the next host.
### Traditional HO process
For traditional HO, the sequence of messages that is exchanged in the testbed is shown in Figure \[contrlmessages\]. This emulation represents the complete traditional HO procedure. The custom function Smartho(x,y) would add a custom header over Ethernet (Procedure \[smarthoheader\] shows the high-level packet contents). The initial control message (Measurement report) Smartho(1,4) is sent from RRH, to Source\_DU to trigger the HO process.
![Control message simulating Traditional HO approach on testbed.[]{data-label="contrlmessages"}](Diagrams/testbed_pics/set_up_control_msgs.pdf){width="85.00000%"}
In the Figure \[contrlmessages\], the exchange of sequence of HO messages is shown. In the preparation phase the Measurement Report (MR) message from the RRU would be transmitted to the CU. CU would select the Target\_DU for the HO by sending HO request (UE Context Request - Smartho(3,4)). In reality, the UE Context Request should contain Target-DU-ID, UE context information & UE History Information. In testbed we emulated the process creating a custom header with context information as three. Upon Target\_DU receiving the HO request it begins handover preparation to ensure seamless service provision for the UE. The Target\_DU would respond with setting up Access Stratum (AS) security keys, uplink bearers connecting to the backhaul, reserve Radio Resource Control (RRC) resources to be used by the mobile device over the radio link and allocates Cell-Radio Network Temporary Identifier. This was not implemented in Target\_DU hosts. To emulate this, we add a 2 ms delay after sending a Smartho(3,4) message. HO execution and completion phase are emulated as data exchange as shown in Figure \[contrlmessages\] between the hosts.
![Screenshot showing the results of traditional HO time, with one HO.[]{data-label="trad_1"}](Diagrams/testbed_pics/Traditional_Single_HO.png){width="90.00000%"}
![Screenshot showing the results of traditional HO time, with two HOs in tandem.[]{data-label="trad_2"}](Diagrams/testbed_pics/Traditional_two_ho.png){width="90.00000%"}
For simulation of more than one tandem of HOs we sent Smartho(1,4) multiple times from RRH, i.e., after the first HO is completed, RRH would resend Smartho(1,4) to Source\_DU to simulate the subsequent HO. Figure \[trad\_1\] & Figure \[trad\_2\] shows the screenshot of traditional HO with single and two HOs in tandem.
### SMARTHO process
For a single HO, the SMARTHO process does not show any difference from the traditional HO approach. This was discussed in Section \[smarthoimpl\].
![Control message simulating SMARTHO HO behaviour on testbed.[]{data-label="contrlmessages2"}](Diagrams/testbed_pics/set_up_control_msgs2.pdf){width="90.00000%"}
The control messaging for SMARTHO process with the tandem of two HOs is shown in Figure \[contrlmessages2\]. Here, after first HO is completed, the subsequent HOs are performed from the HO execution phase. In the testbed, we did not implement the controller part, i.e., after the first HO messages are executed, the subsequent HOs will send message Smartho(6,4) as a reply for message Smartho(1,4). Figure \[smartho\_2\] shows the screenshot of SMARTHO for two-HOs in tandem; similar screenshot is shown in Figure \[smartho\_3\] to demonstrate SMARTHO for three-HOs in tandem.
![Screenshot showing the results of HO time with proposed SMARTHO approach, with two HOs in tandem.[]{data-label="smartho_2"}](Diagrams/testbed_pics/Smartho_two_ho.png){width="90.00000%"}
![Screenshot showing the results of HO time with proposed SMARTHO approach, with two HOs in tandem.[]{data-label="smartho_3"}](Diagrams/testbed_pics/Smartho_three_ho.png){width="80.00000%"}
![Performance results from the prototype implementation.[]{data-label="perfres"}](results/Graph_FPGA/bargraph_solid_state.pdf){width="70.00000%"}
[P[1.0in]{}\*[8]{}[P[1.0in]{}]{}]{}\
& &\
(lr)[2-5]{}(lr)[6-9]{} Number of HOs in Tandem & Traditional Approach & SMARTHO & Traditional Approach & SMARTHO\
1,000 & 49,556 & 49,393 & 49.556 & 49.393\
2,000 & 99,824 & 99,356 & 49.912 & 49.678\
3,000 & 149,693 & 149,305 & 49.897 & 49.768\
4,000 & 213,782 & 203,911 & 53.445 & 50.977\
5,000 & 251,660 & 251,264 & 50.332 & 50.252\
Figure \[perfres\] presents the performance of SMARTHO and the Traditional HO approaches, obtained using the prototype. The experiments are performed on a single UE. In future work, we can increase the intensity of HOs by sending HO requesting messages in parallel, from multiple UEs, to emulate the behavior of LTE-R nodes. We see that as the number of HOs increases, the total time taking to complete all the HOs is also increased. It is seen that handover in both approaches takes around 50 milliseconds. Since we measure HO per UE, the difference is not much, and also the ‘Time per HO’ is almost the same in both Traditional and SMARTHO. This is not exactly what was expected; however, the knowledge and expertise gained in implementing this in the P4 environment is significant. In future work, we will continue to investigate the performance bottlenecks and identify coding changes to improve the overall delay. These experiments can also be extended to multiple UEs in future work.
Using the above testbed experiments, we have demonstrated the feasibility of implementation of SMARTHO in a P4-based programmable dataplane switch.
Conclusions
===========
In this report, we have presented the use P4-based dataplane switches to improve handover efficiency, in a wireless network. The proposed approach has been studied using a Mininet implementation. The experimental results show that the proposed SMARTHO approach does have benefits over the traditional handover process. The handover mechanism was implemented on a Xilinx NetFPGA based P4 switch and the system’s working was demonstrated.
As part of future work, the Tag-based approach can be considered for supporting network slicing and virtualization techniques. Further detailed experiments with multiple UEs and varying loads can also be conducted.
### Acknowledgments {#acknowledgments .unnumbered}
We thank Mr. Karthik Karra, Dr. Manikantan Srinivasan and Dr. C.S. Ganesh (IIT Madras) for sharing their insights and feedback. This work was supported by an IIT Madras IRDA-2017 award and by a DST-FIST grant (SR/FST/ETI-423/2016) from Government of India (2017–2022).
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[Phanindra Palagummi]{} is currently with Microsoft, Hyderabad, India. He received the M.S. (by Research) degree in Computer Science and Engineering from Indian Institute of Technology Madras, Chennai, INDIA in 2019; and the B.Tech. degree in Computer Science and Engineering from Nova College of Engineering and Technology, affiliated to JNTU, in 2010. His research interests include computer networking.
[Krishna M. Sivalingam]{} is an Institute Chair Professor in the Department of CSE, IIT Madras, Chennai, INDIA, where he was also Head of the Department from 2016 till 2019. Previously, he was a Professor in the Dept. of CSEE at University of Maryland, Baltimore County, Maryland, USA from 2002 until 2007; with the School of EECS at Washington State University, Pullman, USA from 1997 until 2002; and with the University of North Carolina Greensboro, USA from 1994 until 1997. He has also conducted research at Lucent Technologies’ Bell Labs in Murray Hill, NJ, and at AT&T Labs in Whippany, NJ. He received his Ph.D. and M.S. degrees in Computer Science from State University of New York at Buffalo in 1994 and 1990 respectively; and his B.E. degree in Computer Science and Engineering in 1988 from Anna University’s College of Engineering Guindy, Chennai (Madras), India. While at SUNY Buffalo, he was a Presidential Fellow from 1988 to 1991.
His research interests include wireless networks, optical wavelength division multiplexed networks, and performance evaluation. His work has been supported by several sources including AFOSR, DST India, DOT India, IBM, NSF, Cisco, Intel, Tata Power Company and Laboratory for Telecommunication Sciences. He holds three patents in wireless networks and has published several research articles including more than seventy journal publications. He has co-edited a book on Next Generation Internet Technologies in 2010; on Wireless Sensor Networks in 2004; on optical WDM networks in 2000 and 2004. He is serving or has served as a member of the Editorial Board for journals including IEEE Networking Letters, ACM Wireless Networks Journal, IEEE Transactions on Mobile Computing, and Elsevier Optical Switching and Networking Journal. He has served as Editor-in-Chief of Springer Photonic Network Communications Journal and EAI Endorsed Transactions on Future Internet.
He is a Fellow of IEEE, a Fellow of INAE and an ACM Distinguished Scientist.
|
---
abstract: 'We systematically study the preinflationary dynamics of the spatially flat Friedmann-Lemaitre-Robertson-Walker universe filled with a single scalar field that has the generalized $\alpha-$attractor potentials, in the framework of loop quantum cosmology, in which the big bang singularity is replaced generically by a non-singular quantum bounce due to purely quantum geometric effects. The evolution can be divided into two different classes, one is dominated initially (at the quantum bounce) by the kinetic energy of the scalar field, and one is not. In both cases, we identify numerically the physically viable initial conditions that lead to not only a slow-roll inflationary phase, but also enough $e$-folds to be consistent with observations, and find that the output of such a viable slow-roll inflationary phase is generic. In addition, we also show that in the case when the evolution of the universe is dominated initially by the kinetic energy of the scalar field (except for a very small set in the phase space), the evolution before reheating is aways divided into three different phases: [*bouncing, transition and slow-roll inflation*]{}. This universal feature does not depend on the initial conditions of the system nor on the specific potentials of the scalar field, as long as it is dominated initially by the kinetic energy of the scalar field at the bounce. Moreover, we carry out phase space analyses for the models under consideration and compare our results with the power-law and Starobinsky potentials.'
author:
- 'M. Shahalam$^1$ [^1]'
- 'M. Sami$^2$ [^2]'
- 'Anzhong Wang$^{1,3}$ [^3]'
title: 'Preinflationary dynamics of $\alpha-$attractor in loop quantum cosmology'
---
Introduction {#sec:intro}
============
In the early 1980’s, the cosmic inflation stood out as a popular paradigm for resolving various problems in the standard model of cosmology, such as the horizon and flatness, etc. It also explains the origin of inhomogeneities that are observed in the cosmic microwave background and the formation of the large scale structure of the universe [@guth1981]. During the last three decades, a wide range of inflationary models have been proposed, including conformal attractors [@conformal], $\alpha-$attractors [@alpha; @alpha1; @alpha2; @alpha3; @alpha4], Starobinsky and the chaotic inflation in supergravity, which is known as Goncharov and Linde (GL) model [@staro1980; @staro1; @staro2; @staro3; @staro4; @GL]. These models provide very similar cosmological predictions with respect to the significant differences in their potentials, and have an excellent fit with the current observations. According to Planck 2015 results [@Planck2015], in the case of a single field inflation, the potentials of $\alpha-$attractors and Starobinsky are consistent with the observations, while the quadratic potential is not equally favorable.
Despite the triumph of the standard inflationary models, which are based on the classical theory of general relativity (GR), their past is inadequate due to the existence of a big bang singularity. All scalar field models of inflation suffer from this initial and inevitable singularity [@borde1994; @borde2003]. Clearly, with this it is difficult to know when and how to set the initial conditions. Moreover, to be consistent with the current observations, the universe should have expanded at least 60 $e$-folds during the inflation. Meanwhile, in a large class of inflationary models, it is often more than 70 $e$-folds [@martin2014]. However, in these models the size of the current universe is smaller than the Planck at the onset of inflation. Consequently, the semi-classical treatments are questionable during inflation. This is the so-called trans-Planckian problem [@martin2001; @berger2013].
To address the above issues, in this paper we shall study the preinflationary dynamics of the generalized $\alpha-$attractor model in the context of loop quantum cosmology (LQC), in which the big bang singularity is generically replaced by a quantum bounce [@agullo2013a; @agullo2013b; @agullo2015; @ashtekar2011; @ashtekar2015; @barrau2016], and investigate whether following the quantum bounce a desired slow-roll inflation generically exists or not [@ashtekar2010; @psingh2006; @zhang2007; @chen2015; @bolliet2015; @schander2016; @bolliet2016; @Bonga2016; @Mielczareka].
In the literature, there are mainly two distinct approaches for the preinflationary universe, the dressed metric [@agullo2013b; @metrica; @metricb; @metricc] and the deformed algebra [@algebraa; @algebrab; @algebrac; @algebrad; @algebrae; @algebraf]. Although both approaches give rise to the same set of dynamical equations in the case of the background evolution of the universe, their perturbations are different [@bolliet2016]. The corresponding non-Gaussianities were also studied both numerically [@agullo15; @ABS17] and analytically [@ZWKCS18] recently, and found that it is consistent with current observations.
However, in this paper since we are mainly concerned with the background evolution of the universe, the results to be presented in this work will be valid to both approaches. Keeping this in mind, we shall compare our results with the power-law and Starobinsky potentials obtained in [@psingh2006; @chen2015; @Bonga2016; @alam2017; @Tao2017a; @Tao2017b]. In particular, we shall show that, when the kinetic energy of the inflaton initially dominates at the bounce (except for a very small set in the phase space), the evolution of the universe before reheating can be divided universally into three different phases [@alam2017; @Tao2017a; @Tao2017b]: [*bouncing, transition and slow-roll inflation*]{}. During these phases, the evolutions of both background and linear perturbations of the universe are all known analytically [@Tao2017a; @Tao2017b]. In the small exceptional region of the phase space, we find that the potential energy first evolutes almost as a constant in the bouncing phase, but oscillating afterward, in contrast to the rest of regions in which the kinetic energy of the inflaton dominates the evolution of the universe at the quantum bounce. As a result, in this exceptional region, a slow-roll inflation is not resulted.
The rest of the paper is organized as follows. In Sec. \[sec:EOM\], we briefly discuss the basic equations of the background evolution of the universe in the framework of LQC. In Sec. \[sec:alpha\], we examine the generalized $\alpha-$attractor model, and shall divide it into three models, namely $T$, $E$ and $\alpha-$attractor with $n=2$ in the sub-sections \[sec:Tmodel\], \[sec:Emodel\] and \[sec:n=2\], respectively. These sub-sections are devoted to the detailed analysis of the background evolution in the framework of the positive inflaton velocity (PIV, $\dot\phi > 0$) and negative inflaton velocity (NIV, $\dot\phi < 0$), and also in the form of kinetic energy dominated (KED) and potential energy dominated (PED) cases at the bounce. The phase portraits for the models under consideration are presented in Sec. \[sec:phase\]. In Sec. \[sec:compare\], we compare our results with the ones obtained previously for the power-law and Starobinsky potentials. Our main conclusions are summarized in Sec. \[sec:conc\].
Before turning to the next section, it is interesting to note that pre-inflationary universe has been also studied recently in the framework of loop quantum gravity (LQG) [@yang2009; @DL17; @adlp; @lsw2018a; @lsw2018b; @agullo18] by using Thiemann’s quantization scheme for the Lorentz part of the Hamiltonian [@thiemann], and among other things, it was shown that the resolution of the big bang singularity (replaced by a quantum bounce) is robust, although the details near the bounce depend on the ways to regularize the Hamiltonian [@lsw2018a; @lsw2018b]. When the kinetic energy of the inflaton dominates at the bounce, the evolution of the universe before reheating can be also divided universally into three different phases, [*bouncing, transition and slow-roll inflation*]{} [@lsw2018a; @lsw2018b]. During these phases, the evolution of the background of the universe is also known analytically [@Tao2017a; @Tao2017b].
We would also like to note that recently inflation with different potentials have been studied in Einstein’s theory of gravity and string-inspired models [@HISY; @BG15; @sahni18; @SW08; @nozari], and various interesting results were obtained. In addition, in the framework of quantum reduced loop gravity (QRLG), its Hamiltonian ($H_{QRLG}$) is almost identical to the Hamiltonian of LQC ($H_{LQC}$), except for a leading term [@QRLG1; @QRLG2]. If this term is zero, then $H_{QRLG}$ exactly coincides with $H_{LQC}$. Similar to LQC, bounce occurs in QRLG. If we use the modified Friedmann equation of QRLG, then we can also obtain three different phases, bouncing, transition and the slow-roll inflation. However, the background dynamics would not be exactly the same as in LQC due to the dependence on the parameters of QRLG. Yet, the modified Friedmann equation of group field theory (GFT) is also almost the same as in LQC, except with the last term of energy $E_{j0}$ [@GFT1; @GFT2], The geometric interpretation of $E_{j0}$ is not transparent, but its effect on the dynamics is as following: (a) For $E_{j0}=0$, the effective dynamics is same as in LQC. (b) For $E_{j0}>0$, the bounce will take place at a higher space-time curvature. (c) For $E_{j0}<0$, the bounce will occur at a lower space-time curvature.
The issue of estimating the duration of the slow-roll inflation with effective isotropic, anisotropic and Bianch I Universe in LQC have been investigated in [@LQC1; @LQC2; @LQC3], in which it was found that the probability distribution function during the slow-roll inflation is peaked at the values of e-folds which are consistent with observations. Moreover, the duration of the slow-roll inflation does not depend crucially on the modified background evolution [@LQC4]. A new related study for the probability of inflation has been discussed in [@LQC5], in which the existence of the quantum bounce affects the probability of inflation.
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[l l cc l]{}\
Model & $\alpha$ && Slow-roll inflation depending on the range of $\phi_B$ &\
\
& & KED (SR) & Existence of KED & PED (SR)\
& & (except subset) & subset (NSR) &\
\
$T$ & $0 < \alpha < 4.3 \times 10^9 $ & All & Yes & $-$\
& $4.3 \times 10^9 \leq \alpha $ & All & Yes & All\
\
$E$ & $0 < \alpha < 0.02$ & All & No (for $\dot{\phi}>0$) & All\
& & & Yes (for $\dot{\phi}<0$) &\
& $0.02 \leq \alpha < 0.6$ & All & Yes & None\
& $ 0.6 \leq \alpha $ & All & Yes & All\
\
$n=2$ & $0 < \alpha < 0.1$ & All & No (for $\dot{\phi}>0$) & All\
& & & Yes (for $\dot{\phi}<0$) &\
& $0.1 \leq \alpha < 2.4$ & All & Yes & None\
& $2.4 \leq \alpha $ & All & Yes & All\
\
\[tab:n012\_alpha\]
\[tab:n012\_alpha\_dphi\]
\[tab:n0\_dphip\]
Background evolution {#sec:EOM}
====================
In the framework of LQC, the modified Friedmann equation in a spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background is written as [@ashtekar2006] $$\begin{aligned}
H^2=\frac{8 \pi}{3 m_{Pl}^2}~\rho \Big{(}1-\frac{\rho}{\rho_c}\Big{)},
\label{eq:Hub}\end{aligned}$$ where $H=\dot{a}/a$ is the Hubble parameter, $\rho=\dot{\phi}^2/2+V(\phi)$ represents the energy density of the inflaton field, and $V(\phi)$ is the potential of the field. The dot denotes a derivative with respect to the cosmic time $t$, $m_{Pl}$ is the Planck mass and $\rho_c$ is the critical energy density that corresponds to the maximum value of energy density, and is given by $\rho_c \simeq 0.41 m_{pl}^4$ [@Meissne; @Domagala].
The conservation equation in the context of LQC remains the same as in the classical theory $$\begin{aligned}
\dot{\rho}+3H(\rho+p)=0.
\label{eq:conser}\end{aligned}$$ Here, $p$ denotes the pressure of the matter field. Eq.(\[eq:conser\]) gives the Klein-Gordon equation for a single scalar field $$\begin{aligned}
\ddot{\phi}+3H \dot{\phi}+ \frac{dV(\phi)}{d\phi}=0.
\label{eq:ddphi}\end{aligned}$$ Eq.(\[eq:Hub\]) tells that at $\rho=\rho_c$, $H=0$ that means quantum bounce occurs at $\rho=\rho_c$. The numerical evolution of the background with the bouncing phase has been extensively studied in the literature. One of the main results is that a desired slow-roll inflation is achieved [@psingh2006; @Mielczarek; @zhang2007; @chen2015; @alam2017; @Tao2017a; @Tao2017b; @ashtekar2011]. Keeping this in mind, we shall study “bounce and slow-roll" with the generalized $\alpha-$attractor model (see Sec. \[sec:alpha\]).
Let us first examine the evolution equations for a general potential $V(\phi)$. We numerically solve Eqs.(\[eq:Hub\]) and (\[eq:ddphi\]) with the initial values of $a(t)$, $\phi(t)$ and $\dot{\phi}(t)$ at a specific time. A natural option of the time is at the bounce $t=t_B$, for which we have $$\begin{aligned}
\rho &=& \rho_c = \frac{1}{2}\dot{\phi}^2(t_B)+V(\phi(t_B)), \nonumber\\
\dot{a}(t_B)&=&0,
\label{eq:bounce}\end{aligned}$$ from which we find $$\begin{aligned}
\dot{\phi}(t_B) &=& \pm \sqrt{2 \Big{(} \rho_c - V(\phi(t_B)) \Big{)}}.
\label{eq:bounce2}\end{aligned}$$ Without loss of the generality, we can always choose $$\begin{aligned}
a(t_B) &=& 1.
\label{eq:bounce3}\end{aligned}$$ Hereafter, we shall read $\phi(t_B)$ and $\dot{\phi}(t_B)$ as $\phi_B$ and $\dot{\phi}_B$. From Eq.(\[eq:bounce2\]), one can clearly see that for a given potential, the initial values will be uniquely identified by $\phi_B$ only. Subsequently, we shall consider two cases: (a) PIV: $\dot{\phi}_B > 0$ and (b) NIV: $\dot{\phi}_B < 0$.
Second, we introduce the following quantities that are essential for this paper [@alam2017; @Tao2017a; @Tao2017b].
\(1) The equation of state (EoS) $w(\phi)$ for the inflaton field is defined as $$\begin{aligned}
w(\phi) = \frac{\dot{\phi}^2/2-V(\phi)}{\dot{\phi}^2/2+V(\phi)}.
\label{eq:w}\end{aligned}$$ In the slow-roll inflationary phase, $w(\phi)\simeq-1$.
To differentiate the initial conditions for being dominated by the kinetic energy (KE) or potential energy (PE) at the bounce, we also introduce the quantity $w^B$, so that $$w^B \equiv w(\phi) \Big{\vert}_{\phi=\phi_B}
= \begin{cases} > 0, \qquad \text{KE} > \text{PE} \\
= 0, \qquad \text{KE}=\text{PE} \\
< 0, \qquad \text{KE} < \text{PE} \end{cases}
\label{eq:wb}$$
\(2) The slow-roll parameter $\epsilon_H$, that is expressed in terms of the Hubble parameter and its derivatives, $$\begin{aligned}
\epsilon_H = - \frac{\dot{H}}{H^2}.
\label{eq:epsilon}\end{aligned}$$ During the slow-roll inflation, $\epsilon_H \ll 1$.
\(3) The number of $e$-folds $N_{inf}$ during the slow-roll inflation is given by $$\begin{aligned}
N_{inf} = ln \Big{(} \frac{a_{end}}{a_i} \Big{)} = \int_{t_i}^{t_{end}} H(t) dt \nonumber \\
= \int_{\phi_i}^{\phi_{end}} \frac{H}{\dot{\phi}} d\phi \simeq \int_{\phi_{end}}^{\phi_i} \frac{V}{V_{\phi}} d\phi,
\label{eq:Ninf}\end{aligned}$$ where $a_i$ ($a_{end}$) represents the expansion factor when the inflation starts (ends), i.e. $\ddot{a}(t_i) \gtrsim 0$ and $w(\phi_{end})=-1/3$.
\(4) Using Eqs.(\[eq:Hub\]) and (\[eq:ddphi\]), we obtain an analytical expression of the scale factor $a(t)$ during the bouncing phase. In this phase, if the potential is very small compared to the kinetic energy, then Eqs.(\[eq:Hub\]) and (\[eq:ddphi\]) become $$\begin{aligned}
&& H^2 = \frac{8 \pi}{3 m_{Pl}^2}~\frac{1}{2}\dot{\phi^2} \Big{(}1-\frac{\dot{\phi^2}}{2\rho_c}\Big{)},\nonumber\\
&& \ddot{\phi}+3H \dot{\phi}=0.
\label{eq:Hreduce}\end{aligned}$$ We solve the above equations analytically, and find [@alam2017; @Tao2017a; @Tao2017b] $$\begin{aligned}
\dot{\phi} &=& \pm \sqrt{2 \rho_c} \left( \frac{a_B}{a(t)} \right)^3,\nonumber \\
a(t) &=& a_B \left( 1+ \delta \frac{t^2}{t_{Pl}^2} \right)^{1/6},
\label{eq:a}\end{aligned}$$ where $t_{Pl}$ denotes the Planck time, and $\delta = {24 \pi \rho_c}/{m_{Pl}^{4}}$ is a dimensionless parameter.
\(5) We define a quantity $r_w$, that is the ratio between the kinetic and potential energies, $$\begin{aligned}
r_{w} &\equiv & \frac{\frac{1}{2}\dot{\phi}^2}{V(\phi)}.
\label{eq:rw}\end{aligned}$$ Following Eq.(\[eq:rw\]), one can define $r_{w}^c$ that corresponds to $N_{inf} \simeq 60$ during the slow-roll inflation, $$r_{w}^c \equiv \frac{\frac{1}{2}\dot{\phi}^2}{V(\phi)} \Big{\vert}_{N_{inf}\simeq 60}
= \begin{cases} r_{w}^c > r_{w} \qquad N_{inf} < 60 \\
r_{w}^c = r_{w} \qquad N_{inf}\simeq 60 \\
r_{w}^c < r_{w} \qquad N_{inf} > 60. \end{cases}
\label{eq:rwc}$$ In the following section, we shall discuss the generalized $\alpha-$attractor model in the context of PIV and NIV at the quantum bounce.
$\alpha-$attractor model {#sec:alpha}
========================
In this section, we shall study “bounce and slow-roll" with the generalized $\alpha-$attractor model in the framework of LQC. Let us consider the following form of the potential [@alam2018; @linder15]: $$\begin{aligned}
V(\phi) &=& \alpha c^2 \frac{\left[\tanh \left( \frac{\phi }{\sqrt{6\alpha }}\right)\right] ^2}{\left[ 1+\tanh \left( \frac{\phi }{\sqrt{6\alpha }}\right) \right]^{2n}},
\label{eq:potGen}\end{aligned}$$ where the parameters $\alpha$ and $c$ have the dimensions of $M_{Pl}^2$ and $M_{Pl}$, and $M_{Pl}=m_{Pl}/\sqrt{8 \pi}$ is the reduced Planck mass. The parameter $n$ takes the values $n = 0,1,2,3...$ For large field values ($\phi \rightarrow \infty$), the generalized $\alpha-$attractor potential becomes flatten, and for small field values ($\phi \rightarrow 0$), it behaves as a quadratic one. For different values of $n$, Eq.(\[eq:potGen\]) gives the following forms of the potentials.\
For $n=0$, we have $$\begin{aligned}
V(\phi) &=& \alpha c^2 \left[ \tanh \left( \frac{\phi }{\sqrt{6\alpha }}\right) \right]^2.
\label{eq:Tpot}\end{aligned}$$ In the literature, Eq.(\[eq:Tpot\]) is known as $T-model$ [@alpha; @alpha2; @alpha3], and also represents GL model for $\alpha=1/9$ [@GL]. We find values of $\alpha$ and $c$ that are consistent with the Planck 2015 results for inflationary universe [@Planck2015]. Here, we write only those values which shall be used in figures and tables. However, one can also obtain other combinations (see Appendix A) $$\begin{aligned}
\alpha &=& 10 m_{Pl}^2, \qquad~~~ c = 1.8 \times 10^{-5} m_{Pl}\nonumber \\
\alpha &=& 10^{10} m_{Pl}^2, \qquad c = 8.2 \times 10^{-6} m_{Pl}.
\label{eq:Talphac}\end{aligned}$$
For $n=1$, we have $$\begin{aligned}
V(\phi) &=& \frac{\alpha c^2}{4} \left(1-e^{-\sqrt{\frac{2}{3\alpha}}\phi} \right)^2.
\label{eq:Epot}\end{aligned}$$ This is called $E-model$ (generalization of the Starobinsky model) [@alpha1]. Eq.(\[eq:Epot\]) corresponds to the Starobinsky model when $\alpha=1$ [@staro1980]. For the numerical evolution, combinations of $\alpha$ and $c$ that are in agreement with the Planck data [@Planck2015], are given as (see Appendix A) $$\begin{aligned}
\alpha &=& 0.1 m_{Pl}^2, \qquad c = 3.3 \times 10^{-4} m_{Pl},\nonumber \\
\alpha &=& 5 m_{Pl}^2, \qquad~~ c = 4.9 \times 10^{-5} m_{Pl}.
\label{eq:Ealphac}\end{aligned}$$
For $n=2$, we have $$\begin{aligned}
V(\phi) &=& \alpha c^2 \frac{\left[\tanh \left( \frac{\phi }{\sqrt{6\alpha }}\right)\right]^2}{\left[ 1+\tanh \left( \frac{\phi }{\sqrt{6\alpha }}\right) \right]^{4}}.
\label{eq:n2pot}\end{aligned}$$ Values of $\alpha$ and $c$ that are compatible with the Planck data [@Planck2015] are given by (see Appendix A) $$\begin{aligned}
\alpha &=& 0.5 m_{Pl}^2, \qquad c = 2.9 \times 10^{-4} m_{Pl},\nonumber \\
\alpha &=& 5 m_{Pl}^2, \qquad~~ c = 9.4 \times 10^{-5} m_{Pl}.
\label{eq:n2alphac}\end{aligned}$$ Here, we are interested in the dynamics of the inflaton field having potentials (\[eq:Tpot\]), (\[eq:Epot\]) and (\[eq:n2pot\]). Numerically, we shall solve Eqs.(\[eq:Hub\]), (\[eq:ddphi\]) with (\[eq:Tpot\]), (\[eq:Epot\]) and (\[eq:n2pot\]), and examine whether following the quantum bounce, a desired slow-roll inflation exists or not.
Before proceeding, let us first consider the inflationary potentials (\[eq:Tpot\]), (\[eq:Epot\]) and (\[eq:n2pot\]) that are shown in Fig. \[fig:pot\]. The predictions of Eq.(\[eq:Tpot\]) with (\[eq:Epot\]) and (\[eq:n2pot\]) are similar but not identical as the main difference is in the shape of the potentials. The potential (\[eq:Tpot\]) is symmetric about $\phi=0$, whereas potentials (\[eq:Epot\]) and (\[eq:n2pot\]) are not symmetric in nature.
Second, we present the range of $\alpha$ (depending on $\phi_B$) having inflationary/non-inflationary phase for the models under consideration in Table \[tab:n012\_alpha\]. Following this, we choose some values of $\alpha$ in each case and use them to draw the figures with different values of $\phi_B$, and its corresponding range for inflationary/non-inflationary phase with PIV and NIV are shown in Table \[tab:n012\_alpha\_dphi\].
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T-model {#sec:Tmodel}
-------
Let us discuss some characteristics of $T-model$ \[Eq.(\[eq:Tpot\])\]. We show the evolution of potential (\[eq:Tpot\]) vs the scalar field in Fig. \[fig:pot\]. This potential asymptotically approaches a plateau for large field values, and as the field approaches the origin, it is oscillating. The potential is symmetric with respect to $\phi=0$.
We numerically solve Eqs.(\[eq:Hub\]) and (\[eq:ddphi\]) with $T-model$. Here, we only consider the case $\dot{\phi}_B>0$ (PIV) because the initial conditions for $T-model$ at the bounce have symmetry $(\phi_B,\dot{\phi}_B) \rightarrow (-\phi_B,-\dot{\phi}_B)$, and the results for $\dot{\phi}_B<0$ (NIV) can be easily obtained by using the above symmetry. Further, initial conditions can be categorized into two sub-cases, namely, KED and PED at the bounce.
For this model, we choose two values of $\alpha$, $\alpha=10m_{Pl}^2$ and $10^{10}m_{Pl}^2$, and the corresponding values of the parameter $c$ are given by Eq.(\[eq:Talphac\]). In the case of $\alpha=10m_{Pl}^2$, only KED initial conditions are possible at the quantum bounce. To get both KED and PED initial conditions at the bounce, $\alpha$ should be large like $10^{10}m_{Pl}^2$ as the potential contains $c^2$ term that is very small, as can be seen from Eq.(\[eq:Talphac\]).
First, we numerically evolve $T-model$ with the background given by Eqs.(\[eq:Hub\]) and (\[eq:ddphi\]) for $\alpha=10 m_{Pl}^2$. The results for a set of KED initial conditions at the bounce are presented in Fig. \[fig:n0alpha10\_dphp\], in which the scale factor $a(t)$, the EoS $w(\phi)$ and the slow-roll parameter $\epsilon_H$ are shown for the same set of $\phi_B$. In the future evolution of $w(\phi)$ and $\epsilon_H$, we obtain inflationary and non-inflationary phases. This means, in the entire parameter space of the KED initial conditions, we also have a small subset that does not provide inflationary phase, see Fig. \[fig:n0alpha10\_dphp\] and Table \[tab:n012\_alpha\_dphi\].
From the top panels of Fig. \[fig:n0alpha10\_dphp\], one can clearly see that the desired slow-roll inflationary phase is obtained for the chosen initial values of $\phi_B/m_{Pl}=-5, 4, 10$. In this region, $a(t)$ grows exponentially, $w(\phi) \simeq -1$ and $\epsilon_H \ll 1$. For NIV ($\dot{\phi}_B<0$), one can obtain the same results with the replacement of $\phi_B$ by $-\phi_B$ \[i.e. $\phi_B/m_{Pl}=5, -4,-10$\].
From the curves of $w(\phi)$ (top panel of Fig. \[fig:n0alpha10\_dphp\]), we notice that the evolution of the universe before reheating can be split up into three different phases, namely bouncing, transition and slow-roll [@alam2017; @Tao2017a; @Tao2017b]. During the bouncing phase, the kinetic energy remains dominant, and $w(\phi) \simeq +1$. In the transition phase, $w(\phi)$ decreases drastically from $+1$ $(t/t_{Pl} \simeq 10^3)$ to $-1$ $(t/t_{Pl} \simeq 10^4)$. This transition phase is slightly short in comparison with the other two phases. In the slow-roll phase, $w(\phi)$ is close to $-1$, and remains so until the end of the slow-roll inflation. During the bouncing phase, it is remarkable to note that the evolution of $a(t)$ (top panel of Fig. \[fig:n0alpha10\_dphp\]) is independent for a wide range of initial values of $\phi_B$, and exhibits the compatible behavior with the analytical solution (\[eq:a\]).
The entire range of KED initial conditions is from $-\infty$ to $+\infty$. In this range most of initial values provide inflationary phase. However, there is a small subset that does not give inflationary phase, see Table \[tab:n012\_alpha\_dphi\]. Total number of $e$-folds $N_{inf}$ during the inflationary phase can be obtained for different values of $\phi_B$, and the range for $\dot{\phi_B}>0$ is given as (See Table \[tab:n012\_alpha\_dphi\]) $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-\infty, -3.1) \cup (-1.61, +\infty)\nonumber\\
&& ~~~~~~~~~~~~~ \rightarrow \text{slow-roll}~(N_{inf}>0), \nonumber\\
&& -3.1 < \phi_B \leq -1.6 \rightarrow \text{no slow-roll inflation}.
\label{eq:TNphiB}\end{aligned}$$ To be consistent with the Planck data [@Planck2015], at least 60 $e$-folds are needed during the slow-roll inflation, and to obtain it one has to require (see Table \[tab:n0\_dphip\]) $$\begin{aligned}
\frac{\phi_B}{m_{Pl}} & \in & (-\infty, -4.9) \cup (1.1, +\infty).
%&& \rightarrow N_{inf} \gtrsim 60
\label{eq:TN60phiB}\end{aligned}$$ In the case of initial conditions with $\dot{\phi}_B<0$, we use the symmetry $(\phi_B,\dot{\phi}_B) \rightarrow (-\phi_B,-\dot{\phi}_B)$, then the constraints are $$\begin{aligned}
\frac{\phi_B}{m_{Pl}} & \in & (-\infty, 1.61) \cup (3.1, +\infty) \rightarrow \text{slow-roll}~(N_{inf}>0)\nonumber\\
&& 1.6 \leq \phi_B < 3.1 \rightarrow \text{no slow-roll}, \nonumber\\
\frac{\phi_B}{m_{Pl}} & \in & (-\infty, -1.1) \cup (4.9, +\infty) \rightarrow N_{inf} \gtrsim 60.
\label{eq:TNphiBsym}\end{aligned}$$ From Table \[tab:n0\_dphip\], one notices that the number of $e$-folds $N_{inf}$ grows as the absolute values of $\phi_B$ increase, which implies that an absolute large value of $\phi_B$ can produce more number of $e$-folds. The similar results for power-law potentials were obtained in [@alam2017].
Next, we study $T-model$ with $\alpha=10^{10}m_{Pl}^2$. The results are displayed in Fig. \[fig:n0alpha10p10\_dphp\] for $\dot{\phi}_B>0$. Here, we use a large value of $\alpha$ to get both KED and PED initial conditions at the quantum bounce. In Fig. \[fig:n0alpha10p10\_dphp\], we show the evolution of the scale factor $a(t)$, EoS $w(\phi)$ and slow-roll parameter $\epsilon_H$, and show the inflationary and non-inflationary phases of the universe. Top, middle and bottom panels of Fig. \[fig:n0alpha10p10\_dphp\] are obtained for the different sets of initial conditions of $\phi_B$ that correspond to KED with slow-roll (Top), without slow-roll (Middle) and PED with slow-roll (Bottom). From the top and middle panels of Fig. \[fig:n0alpha10p10\_dphp\], we conclude that the KED initial conditions have a subset that does not provide slow-roll inflation phases. The range of the above subset is given in Table \[tab:n012\_alpha\_dphi\].
Let us compare Top and bottom panels of Fig. \[fig:n0alpha10p10\_dphp\] that are obtained for KED and PED initial conditions. In top panels, the evolution of $a(t)$ exhibits the universal feature which is consistent with the analytical solution (\[eq:a\]). The evolution of $w(\phi)$ shows three different phases, namely bouncing, transition and slow-roll. In bottom panels, the universal feature of $a(t)$ is lost, and the bouncing phase no longer exists, though the slow-roll inflation $w(\phi)\simeq -1$ can still be achieved.
The range of $\phi_B$ that provides inflationary and non-inflationary phases is given by (see Table \[tab:n012\_alpha\_dphi\]): $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-\phi_{max}, -3.21) \cup (-1.8, \phi_{max}), \nonumber\\
&&~~~~~~~~~~~~~ \rightarrow \text{slow-roll}~(N_{inf}>0), \nonumber\\
&& -3.2 \leq \phi_B < -1.8 \rightarrow \text{no slow-roll inflation,}
\label{eq:TNphiB2}\end{aligned}$$ where $$\begin{aligned}
\phi_{max} \simeq \sqrt{6 \alpha} \arctan \text{h} \left( \sqrt{\frac{\rho_c}{\alpha c^2}} \right) \simeq 2.56 \times 10^5 m_{Pl}.
\label{eq:TNphimax2}\end{aligned}$$ To obtain at least 60 $e$-folds during the slow-roll inflationary phase, one has to require (see Table \[tab:n0\_dphip\]): $$\begin{aligned}
\frac{\phi_B}{m_{Pl}} \in (-\phi_{max}, -5.1) \cup (1.05, \phi_{max}).
%&& \rightarrow N_{inf} \gtrsim 60
\label{eq:TN60phiB2}\end{aligned}$$
For $\dot{\phi}_B<0$, the same results can be obtained with the symmetry $(\phi_B,\dot{\phi}_B) \rightarrow (-\phi_B,-\dot{\phi}_B)$, and are given by $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-\phi_{max}, 1.8) \cup (3.21, \phi_{max}), \nonumber\\
&&~~~~~~~~~~~~ \rightarrow \text{slow-roll} ~(N_{inf}>0)\nonumber\\
&& 1.8 < \phi_B \leq 3.2 \rightarrow \text{no slow-roll}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (-\phi_{max}, -1.05) \cup (5.1, \phi_{max})\nonumber\\
&&~~~~~~~~~~ \rightarrow N_{inf} \gtrsim 60.
\label{eq:TNphiB2sym}\end{aligned}$$ As mentioned in the case of $\alpha=10m_{Pl}^2$, here also one can get more e-folds for the large absolute values of $\phi_B$, see Table \[tab:n0\_dphip\].
\[tab:n1\_dphip\]
\[tab:n1\_dphin\]
Finally, we consider the evolutions of the kinetic and potential energies in Fig. \[fig:n0rho\], and pay particular attention on the case in which a slow-roll inflationary phase is absent, although the kinetic energy of the inflaton still dominates the evolution of the universe at the bounce. Left panel is plotted for KED initial condition with $\phi_B=5 m_{Pl}$. In the bouncing phase, KE dominates the evolution whereas PE remains sub-dominant. As time increases, KE decreases until the transition phase in which KE falls below the PE, and thereafter, PE dominates and remains so for most of the time of the evolution, during which the slow-roll inflation is resulted. Middle panel is shown for a value ($\phi_B=-2.4 m_{Pl}$) of a subset of KED initial conditions. In this case, PE is sub-dominant initially and remains so during the entire evolution. It never overtakes the KE. As a result, a slow-roll inflationary phase is absent. Right panel exhibits the PED case where PE dominates generically during the whole process, and gives rise to a slow-roll inflationary phase for a long period.
E-model {#sec:Emodel}
-------
In this subsection, we study the features of $E-model$ \[Eq.(\[eq:Epot\])\]. The $E-model$ potential is displayed in Fig. \[fig:pot\]. This kind of potentials is bounded below by zero i.e. $V(\phi) \geq 0$. On the positive side ($\phi \rightarrow \infty$), the potential (\[eq:Epot\]) achieves a finite value $V(\phi) \rightarrow \alpha c^2/4$, whereas in the negative side ($\phi \rightarrow -\infty$) it diverges. Hence, this potential is asymmetric. In LQC, the total energy density can not exceed the value of $\rho_c$. Therefore, the critical energy density constrains the initial values of $\phi_B$ as $(\phi_{min}, \infty)$, where $$\begin{aligned}
\phi_{min} &\simeq & \sqrt{6 \alpha} \arctan \text{h} \left( \frac{\sqrt{\rho_c}}{\sqrt{\alpha c^2}-\sqrt{\rho_c}} \right)\nonumber\\
&\simeq & -3.64 m_{Pl}~ \text{for}~ \alpha=0.1 m_{Pl}^2,\nonumber\\
&\simeq & -25.65 m_{Pl}~ \text{for}~ \alpha=5 m_{Pl}^2.
\label{eq:Ephimin}\end{aligned}$$ The $E-model$ reduces to the Starobinsky model for $\alpha=1$. Here, we shall not discuss the Starobinsky model as the evolutions and the phase space analysis have been already studied in detail in [@Tao2017a; @Tao2017b; @Bonga2016]. Hence, in this sub-section, we shall investigate $E-model$ with different values of $\alpha$ ($\alpha\not=1$).
From Eq.(\[eq:Ephimin\]), one can obtain $\phi_{min}$ for the given value of $\alpha$ and $c$. First, let us work with $\alpha=0.1 m_{Pl}^2$ and $c=3.3 \times 10^{-4} m_{Pl}$ \[Eq.(\[eq:Ealphac\])\]. In this case, we have $\phi_{min} \simeq -3.64 m_{Pl}$. Numerically, we examine the whole range of the inflaton field in order to identify the initial conditions that can give rise to the slow-roll inflation. We find the KED (PED) evolution exists in a very long (narrow) range and given by (see, Table \[tab:n012\_alpha\_dphi\]):
$\bullet$ For $\dot{\phi}_B>0$, we have $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-1.51, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.52 < \phi_B \leq -1.5 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -3.52 \rightarrow \text{KED=PED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -3.53) \rightarrow \text{PED (NSR)},
\label{eq:EphiBP}\end{aligned}$$ where $\phi_{min}$ is given by Eq.(\[eq:Ephimin\]).
$\bullet$ For $\dot{\phi}_B<0$, we obtain $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (2.4, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.51 < \phi_B < 2.4 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -3.51 \rightarrow \text{KED=PED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -3.52) \rightarrow \text{PED (NSR)}.
\label{eq:EphiBN}\end{aligned}$$
The results of the background evolution for KED and PED initial conditions are shown in Figs. \[fig:n1alpha01\_dphp\] and \[fig:n1alpha01\_dphn\], with $\dot{\phi}_B>0$ and $\dot{\phi}_B<0$, respectively. In both figures, the evolutions of $a(t)$, $w(\phi)$ and $\epsilon_H$ are obtained numerically for the same set of the initial values of $\phi_B$. In the case of KED initial conditions, the evolution of $a(t)$ is universal during the bouncing phase as it does not depends on the form of the potentials nor on the initial values of $\phi_B$, and can be well approximated by the analytical solution (\[eq:a\]). This is mainly due to the fact that the potential remains very small in comparison with the kinetic one during the whole bouncing phase, and its effects on the evolution of the background is almost negligible. From the evolution of $w(\phi)$, one can see that the background evolution is divided into three different phases: bouncing, transition and slow-roll. The period of transition phase is very small in comparison with the other two. In the bouncing phase, $w(\phi) \simeq +1$, while in the transition phase it suddenly decreases from $+1~ (t/t_{Pl} \approx 10^3)$ to $-1 ~ (t/t_{Pl} \approx 10^4)$. In the slow-roll inflationary phase, it is very close to $-1$ until the end of the slow-roll inflation. In the KED case, we also have a subset that does not provide the slow-roll inflation, which is shown clearly in the middle panels of Figs. \[fig:n1alpha01\_dphp\] and \[fig:n1alpha01\_dphn\]. The range of this subset is presented in Table \[tab:n012\_alpha\_dphi\]. In the case of PED initial conditions, the universality of the scale factor $a(t)$ is lost, and the bouncing phase does not exist any more, and the slow-roll inflationary phase can not be obtained. See the bottom panels of Figs. \[fig:n1alpha01\_dphp\] and \[fig:n1alpha01\_dphn\].
In Tables \[tab:n1\_dphip\] and \[tab:n1\_dphin\], we display the initial values of $\phi_B$ that provide the desired slow-roll inflation, from which one can see that, for the successful inflation at least 60 $e$-folds are needed and to obtain this, the values of $\phi_B$ should be in the range of $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-0.38, +\infty) \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B>0, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (3.15, +\infty) \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B<0,
\label{eq:ENphiB}\end{aligned}$$ within which, Tables \[tab:n1\_dphip\] and \[tab:n1\_dphin\] exhibit that $N_{inf}$ grows as $\phi_B$ increases.
Next, we work with $\alpha=5 m_{Pl}^2$ and $c=4.9 \times 10^{-5} m_{Pl}$ \[Eq.(\[eq:Ealphac\])\]. In this case, $\phi_{min} \simeq -25.65 m_{Pl}$. We numerically search the entire range of $\phi_B$, and find the initial values that can lead to the slow-roll inflation. Here, KED (PED) evolution has large (small) range and given by (see, Table \[tab:n012\_alpha\_dphi\]), $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-24.70, -3.11) \cup (-1.4, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.1 \leq \phi_B < -1.4 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -24.71 \rightarrow \text{KED=PED (SR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -24.72) \rightarrow \text{PED (SR)},
\label{eq:E5phiBP}\end{aligned}$$ for $\dot{\phi}_B>0$, and $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-24.69, 1.4) \cup (3, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& 1.4 < \phi_B < 3 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -24.7 \rightarrow \text{KED=PED (SR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -24.71) \rightarrow \text{PED (SR)}
\label{eq:E5phiBN}\end{aligned}$$ for $\dot{\phi}_B<0$.
We show the results of the background evolution for KED and PED initial conditions in Figs. \[fig:n1alpha5\_dphp\] and \[fig:n1alpha5\_dphn\], with $\dot{\phi}_B>0$ and $\dot{\phi}_B<0$, respectively. In both figures, we show the evolutions of $a(t)$, $w(\phi)$ and $\epsilon_H$ for the same set of initial values of $\phi_B$. In the bouncing phase, the numerical evolution of $a(t)$ is compatible with the analytical solution (\[eq:a\]) in the case of KED initial conditions whereas such a universality disappears in the PED case. From the evolution of $w(\phi)$, we obtain three different phases, namely bouncing, transition and slow-roll inflation in the KED case, while in the PED case the bouncing and transition phases no longer exist, though the slow-roll inflation can still be achieved.
In this case, the entire range of $\phi_B$ (except for a small subset of KED) lead to the slow-roll inflation as shown in Table \[tab:n012\_alpha\_dphi\]. However, this is not possible in the case of $\alpha=0.1 m_{Pl}^2$ and the Starobinsky model [@Tao2017a; @Tao2017b; @Bonga2016] where the PED and a subset of KED initial conditions do not provide the slow-roll inflation.
In Tables \[tab:n1\_dphip\] and \[tab:n1\_dphin\], we present the initial values of $\phi_B$ that lead to the slow-roll inflation, from which one can clearly see that, in order to get at least 60 $e$-folds during the slow-roll inflation, one has to require $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min},-5.35) \cup (0.76, +\infty), \nonumber\\
&& ~~~~~~~~~~~~ \rightarrow N_{inf} \gtrsim 60,
\label{eq:E5NphiBP}\end{aligned}$$ for $\dot{\phi}_B>0$, and $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min},-2.2) \cup (4.42, +\infty), \nonumber\\
&& ~~~~~~~~~~~~~~ \rightarrow N_{inf} \gtrsim 60
\label{eq:E5NphiBN}\end{aligned}$$ for $\dot{\phi}_B<0$. In the above ranges, $N_{inf}$ grows as $\phi_B$ increases.
Finally, we compare the numerical evolutions of the KE and the PE, which are shown in Fig. \[fig:n1rho\]. Left panels (top and bottom) correspond to KED case at the bounce. Initially KE dominates and PE sub-dominates. As the evolution arrives in the transition phase both the energies become comparable. Soon PE becomes dominant, whereby a slow-roll inflation is resulted. Middle panels (top and bottom) are displayed for a subset of KED initial conditions where the slow-roll inflation cannot be obtained as the PE remains sub-dominant. Right panels (top and bottom) is for the PED case. In top right ($\alpha=0.1 m_{Pl}^2$), the slow-roll inflation is not possible as the PE remains sub-dominant throughout the whole evolution, while the bottom right ($\alpha=5 m_{Pl}^2$) provides the slow-roll inflation.
It is remarkable to note that the $E-model$ with small values of $\alpha$ (like $ \alpha=0.1 m_{Pl}^2$ etc.) does not provide a slow-roll inflation for the entire range of $\phi_B$. More preciously, PED and a subset of KED initial conditions do not lead to the slow-roll inflation. Though, a large range of KED initial values give rise to the slow-roll inflation. Such results are consistent with the Starobinsky model [@Tao2017a; @Tao2017b; @Bonga2016]. However, when the $E-model$ has large values of $\alpha$ (like $ \alpha=5 m_{Pl}^2$), the whole range of initial values of $\phi_B$ (except a subset of KED initial conditions) produces the slow-roll inflation phase.
$\alpha-$attractor model with $n=2$ {#sec:n=2}
-----------------------------------
We now turn to consider the $\alpha$-attractor model with $n=2$ \[Eq.(\[eq:n2pot\])\]. The evolution of potential (\[eq:n2pot\]) is shown in Fig. \[fig:pot\]. Similar to $E-model$, the potential (\[eq:n2pot\]) is bounded below by zero ($V(\phi) \geq 0$), and gives finite value $V(\phi)\simeq 2.8 \times 10^{-9} m_{Pl}^4$ (for $\alpha=5m_{Pl}^2 $) at $\phi \rightarrow +\infty$, whereas it shows divergence at $\phi \rightarrow -\infty$. Therefore, this is an asymmetric potential. In LQC, the maximum energy density is $\rho_c$ that constrains the initial values of $\phi_B$ as $(\phi_{min}, \infty)$, and $\phi_{min}$ is given as $$\begin{aligned}
\phi_{min} &\simeq & \sqrt{6 \alpha} \arctan \text{h} \left[\sqrt{\mu^2-1}-\mu \right]\nonumber\\
&\simeq & -4.08 m_{Pl}~ \text{for}~ \alpha=0.5 m_{Pl}^2, \nonumber\\
&\simeq & -12.88 m_{Pl}~ \text{for}~ \alpha=5 m_{Pl}^2,
\label{eq:n2phimin}\end{aligned}$$ where $$\begin{aligned}
\mu = 1-\sqrt{\frac{\alpha c^2}{4\rho_c}}.\end{aligned}$$ Let us solve the background equations (\[eq:Hub\]) and (\[eq:ddphi\]) with (\[eq:n2pot\]) numerically for $\alpha=0.5 m_{Pl}^2$ and $c=2.9 \times 10^{-4} m_{Pl}$. In this case, $\phi_{min} \simeq -4.08 m_{Pl}$. Similar to the $T-$ and $E-models$, the initial conditions are divided into two sub-classes; KED and PED, see Table \[tab:n012\_alpha\_dphi\], and is given by $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-1.39, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.93 < \phi_B \leq -1.4 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -3.93 \rightarrow \text{KED=PED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -3.94) \rightarrow \text{PED (NSR)},
\label{eq:n2phiBP}\end{aligned}$$ for $\dot{\phi}_B>0$, and $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (2.4, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.92 < \phi_B < 2.4 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -3.92 \rightarrow \text{KED=PED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -3.93) \rightarrow \text{PED (NSR)},
\label{eq:n2phiBN}\end{aligned}$$ for $\dot{\phi}_B<0$, where $\phi_{min}$ is given by Eq.(\[eq:n2phimin\]). The numerical results are illustrated in Figs. \[fig:n2alpha05\_dphp\] and \[fig:n2alpha05\_dphn\] for a set of KED and PED initial conditions with $\dot{\phi}_B>0$ and $\dot{\phi}_B<0$, respectively. The explanation of these figures is quite similar to the case of $E-model$. Therefore, we shall not repeat again. Here, we shall discuss the rest of results for the model (\[eq:n2pot\]). In Tables \[tab:n2\_dphip\] and \[tab:n2\_dphin\], we demonstrate the different inflationary parameters. Looking at both tables, the range of $\phi_B$ that is restricted to produce enough $e-$folds for the desired slow-roll inflation, is given by $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-0.3, +\infty) \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B>0, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (3.23, +\infty) \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B<0,
\label{eq:n2N60phiB}\end{aligned}$$ within which, one can infer that the number of $e-$folds grows as the values of $\phi_B$ increase as shown in Tables \[tab:n2\_dphip\] and \[tab:n2\_dphin\].
In the case of $\alpha=5 m_{Pl}^2$ and $c=9.4 \times 10^{-5} m_{Pl}$, the numerical results are displayed in Figs. \[fig:n2alpha5\_dphp\] and \[fig:n2alpha5\_dphn\]. The range of $\phi_B$ is given as follows (see Table \[tab:n012\_alpha\_dphi\]): $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-12.4, -3.4) \cup (-1.4, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& -3.4 < \phi_B < -1.4 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -12.41\rightarrow \text{KED=PED (SR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -12.42) \rightarrow \text{PED (SR)},
\label{eq:n2phiB2P}\end{aligned}$$ for $\dot{\phi}_B>0$, and $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (-12.39, 1) \cup (2.7, +\infty) \rightarrow \text{KED (SR)}, \nonumber\\
&& 1 < \phi_B < 2.7 \rightarrow \text{subset of KED (NSR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} = -12.4\rightarrow \text{KED=PED (SR)}, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -12.41) \rightarrow \text{PED (SR)},
\label{eq:n2phiB2N}\end{aligned}$$ for $\dot{\phi}_B<0$, where $\phi_{min}$ is given by Eq.(\[eq:n2phimin\]). To obtain enough $e-$folds for the desired slow-roll inflarion, the range of $\phi_B$ requires as (see Tables \[tab:n2\_dphip\] and \[tab:n2\_dphin\]): $$\begin{aligned}
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -6) \cup (0.45, +\infty), \nonumber\\
&& ~~~~~~~~~~~~ \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B>0, \nonumber\\
&& \frac{\phi_B}{m_{Pl}} \in (\phi_{min}, -3.6) \cup (4.04, +\infty), \nonumber\\
&& ~~~~~~~~~~~~ \rightarrow N_{inf} \gtrsim 60 ~\text{for}~ \dot{\phi}_B<0.
\label{eq:n2N60phiB2}\end{aligned}$$ In the above range, $N_{inf}$ grows as the absolute value of $\phi_B$ increases that are displayed in Tables \[tab:n2\_dphip\] and \[tab:n2\_dphin\].
\[tab:n2\_dphip\]
\[tab:n2\_dphin\]
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Finally, we show the numerical evolutions of KE and PE in Fig. \[fig:n2rho\]. The explanation of this figure is similar to Fig. \[fig:n1rho\] of $E-model$.
Phase space analysis and the desired slow-roll inflation {#sec:phase}
========================================================
In this section, we study the phase space analysis for the models considered in the last sections. Let us first examine the symmetric $T-model$ with two different values of $\alpha$. In the case of $\alpha=10 m_{Pl}^2$, the range of initial conditions having slow-roll/no slow-roll inflation, and consistent with observations are presented in Eqs.(\[eq:TNphiB\]), (\[eq:TN60phiB\]) and (\[eq:TNphiBsym\]). Fig. \[fig:Tport\] exhibits the evolution of few trajectories in $(\phi/m_{Pl}, \dot{\phi}/m_{Pl}^2)$ plane starting from the quantum bounce (boundary curve without arrows). As mentioned in the subsection \[sec:Tmodel\], the initial data surface is not compressed: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \rightarrow \pm \infty$. The dashed (blue) trajectories demonstrate the non-inflationary phase, while the solid (blue) ones show the inflationary phase but do not lead to the desired slow-roll inflation as they do not produce sufficient $e-$folds. Only red trajectories exhibit the desired slow-roll inflation which are compatible with observations. Similarly, dashed and solid (blue) parts of the initial surface correspond to non-inflationary and a subset of inflationary phase that is not consistent with observations, whereas red part is compatible with observations. One can clearly see that the region of the non-inflationary phase and the part which does not provide the desired slow-roll inflation are almost negligible in comparison with the whole initial phase. Thus, a substantial fraction of the initial conditions generate a desired slow-roll inflation. For $\alpha=10^{10} m_{Pl}^2$, the range of initial conditions that are compatible with observations or not are presented in Eqs.(\[eq:TNphiB2\]), (\[eq:TN60phiB2\]) and (\[eq:TNphiB2sym\]), and the phase portrait is shown in Fig. \[fig:Tport\]. In this case, the initial data surface is compact: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \rightarrow \pm 2.56 \times 10^5 m_{Pl}$ \[see Eq.(\[eq:TNphimax2\])\]. The rest is the same as in the case of $\alpha=10 m_{Pl}^2$. Note that, a small portion of the full initial conditions are shown in Fig. \[fig:Tport\].
Next, we carry out the phase analysis for $E-model$ with $\alpha=0.1 m_{Pl}^2$ and $5 m_{Pl}^2$. The phase portraits are shown in Fig. \[fig:Eport\]. Let us first consider $\alpha=0.1 m_{Pl}^2$. In this case, the initial surface is semi compact: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \in (-3.64, \infty)$. Fig. \[fig:Eport\] shows that the trajectories starting from the bounce represent the slow-roll (red ones correspond to enough $e-$folds that are consistent with observations while solid blue ones are not) and without slow-roll inflation (dashed blue). Here, PED and a subset of KED initial conditions do not lead to the slow-roll inflation (blue; dashed and solid lines), while KED initial values (except a small subset) provide (red). In a similar way, the dashed and solid (blue) parts of the bounce (boundary curve) display the region of non-inflationary and inflationary phases (not compatible with observations), and the red part denotes the desired slow-roll inflation phase that is consistent with observations. The range of initial conditions are presented in Eqs.(\[eq:EphiBP\]), (\[eq:EphiBN\]) and (\[eq:ENphiB\]). Second, we take $\alpha=5 m_{Pl}^2$, here also the bounce surface is semi compact: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \in (-25.65, \infty)$. The rest is the same as in the case of $\alpha=0.1 m_{Pl}^2$ except PED initial conditions. Here, one can obtain the desired slow-roll inflation with the KED (except a subset) and the PED initial values. However, it is not possible for the $\alpha=0.1 m_{Pl}^2$ case. The range of initial conditions is shown in Eqs.(\[eq:E5phiBP\]), (\[eq:E5phiBN\]), (\[eq:E5NphiBP\]) and (\[eq:E5NphiBN\]). Notice that, a small portion of the whole initial conditions is displayed in Fig. \[fig:Eport\].
Finally, we investigate the $\alpha-$attractor model with $n=2$. The phase portraits are presented in Fig. \[fig:n2port\] for $\alpha=0.5 m_{Pl}^2$ (left) and $\alpha=5 m_{Pl}^2$ (right). In this case, the bouncing phase is also semi-finite: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $; $\phi_B \in (-4.08, \infty)$ for $\alpha=0.5 m_{Pl}^2$ and $\phi_B \in (-12.88, \infty)$ for $\alpha=5 m_{Pl}^2$. Similar to $E-model$, here also one value of $\alpha$ leads to the desired slow-roll inflation for both KED and PED initial conditions whereas it is not feasible for another value. In Fig. \[fig:n2port\], only a small part of initial values is shown. However, the entire range of initial conditions is given by Eqs.(\[eq:n2phiBP\])$-$(\[eq:n2N60phiB2\]).
Comparison with the power-law and the Starobinsky potentials {#sec:compare}
============================================================
In the literature, a large number of inflationary models have been studied that can be consistent with observations. In the case of a single field inflation, Planck 2015 results demonstrate that the quadratic potential is not favored compared to the power-law \[$V(\phi) \propto \phi^n$ with $n<2$\], the Starobinsky and $\alpha-$attractor models [@Planck2015]. Therefore, in this section, we shall compare our results with these known models [@alam2017; @Bonga2016].
Let us first consider the results of the $T-model$ with the power-law and Starobinsky potentials. In the case of power-law potential, both KED and PED initial conditions produce the desired slow-roll inflation, and are consistent with observations in terms of the number of $e-$folds [@alam2017]. In the case of $T-model$ (with $\alpha=10^{10} m_{Pl}^2$), there is a small subset of KED initial conditions that does not generate the slow-roll inflation, and in terms of the number of $e-$folds, both KED (except for a very small subset) and PED initial values are consistent with observations, while the Starobinsky potential is observationally compatible only for KED (except for a very small subset) initial values and not for PED ones [@Bonga2016].
Next, we consider the results of the $E-model$ with the power-law and Starobinsky potentials. In the case of power-law potential, both KED and PED initial conditions are compatible with observations, whereas the Starobinsky inflation is consistent only for KED ones. In the case of $E-model$, there is a subset of KED initial values that corresponds to the non-inflationary phase. For $\alpha=0.1 m_{Pl}^2$, our results are consistent with the Starobinsky model as both models (Starobinsky and $E-model$ with $\alpha=0.1 m_{Pl}^2$) lead to the desired slow-roll inflationary phase only for KED (except for a very small subset) initial values and not for PED ones. However, if we consider large values of $\alpha$ (say $\alpha=5 m_{Pl}^2$), the scenario will be different as in this case both KED (except a very small subset) and PED initial conditions lead to the desired slow-roll inflation and consistent with present observations in terms of the number of $e-$folds.
Finally, we consider the obtained results of $\alpha-$attractor model with $n=2$. Similar to $E-model$, here also the $\alpha-$model with $n=2$ is consistent with the Starobinsky model for small values of $\alpha$ (say $\alpha=0.5 m_{Pl}^2$), but for large values of $\alpha$ (say $\alpha=5 m_{Pl}^2$), both KED (except for a small subset) and PED initial values provide the desired slow-roll inflation, and compatible with the current observations as they all produce enough $e-$folds.
Conclusions {#sec:conc}
===========
In the context of LQC, in this paper we have systematically investigated the preinflationary dynamics of the $T$, $E-models$ and $\alpha-$attractor with $n=2$ for various cases (PIV and NIV, also KED and PED). Our analysis bears resemblance with the study of the scalar field dynamics for the $\alpha-$attractor effective potential on RS brane with time like extra dimension [@shtanov]. We have chosen these models as they are favored by the Planck 2015 data [@Planck2015].
In particular, we have first performed the detailed numerical analysis of the background evolution of the universe for $T-model$ with $\alpha=10m_{Pl}^2$ and $10^{10}m_{Pl}^2$. Due to the symmetry of $T-model$ potential, we have chosen only PIV at the quantum bounce. Further, initial conditions are divided into the KED and PED cases at the bounce. In the case of $\alpha=10m_{Pl}^2$, we have only KED initial conditions during the entire bouncing phase. However, to obtain both KED and PED initial values at the bounce, $\alpha$ should be very large (say $\alpha=10^{10}m_{Pl}^2$). The numerical results for $T-model$ are shown in Figs. \[fig:n0alpha10\_dphp\] and \[fig:n0alpha10p10\_dphp\], in which the scale factor $a(t)$, EoS $w(\phi)$ and slow-roll parameter $\epsilon_H$ are shown for the same set of initial values of $\phi_B$. In the evolution of $w(\phi)$ and $\epsilon_H$, we have obtained inflationary and non-inflationary phases for the KED case. This implies that a small subset exists in which it does not give inflation, see Figs. \[fig:n0alpha10\_dphp\], \[fig:n0alpha10p10\_dphp\] and Table \[tab:n012\_alpha\_dphi\]. In the case of KED initial conditions (except for a very small a subset), the universe is always divided into three distinct phases prior to the reheating: [*bouncing, transition and the slow-roll inflation*]{}. In the bouncing phase, the evolution of the background is independent not only of the wide ranges of initial values but also of the potentials. Specially, the numerical evolution of the expansion factor $a(t)$ has shown the universal feature and well approximated by the analytical solution (\[eq:a\]), see upper panels of Fig. \[fig:n0alpha10p10\_dphp\]. During this phase, the EoS stays pegged at unity, $w(\phi) \simeq +1$. Though, in the transition phase, it decreases quickly from $w(\phi) \simeq +1$ to $w(\phi) \simeq -1$. The span of the transition phase is very short in comparison with other two phases. Afterwards, the universe enters the slow-roll inflationary phase, where $\epsilon_H$ is large initially, but soon decreases to almost zero, by which the slow-roll inflation starts, as exhibited in the upper panels of Fig. \[fig:n0alpha10p10\_dphp\]. During the slow-roll inflation, we also obtained the number of $e$-folds that is displayed in Table \[tab:n0\_dphip\]. In the case of PED initial values, the universality of the scale factor $a(t)$ is lost, and the bouncing phase no longer exists. However, the slow-roll inflation can still be acquired for a long period, and correspondingly one can obtain a large number of $e$-folds, as shown in the bottom panels of Fig. \[fig:n0alpha10p10\_dphp\] and Table \[tab:n0\_dphip\].
Next, we have investigated the evolution of the background for $E-model$ with $\alpha=0.1m_{Pl}^2$ and $5m_{Pl}^2$. This model is not symmetric. Therefore, we have examined both PIV and NIV: the numerical evolution of the background is divided into the form of KED and PED initial conditions at the bounce. In LQC, the total energy density can not be larger than $\rho_c$. We have found that the KED evolution has a large range of $\phi_B$ than the PED ones, see Table \[tab:n012\_alpha\_dphi\]. The numerical evolutions of $a(t)$, $w(\phi)$ and $\epsilon_H$ for $E-model$ with $\alpha=0.1m_{Pl}^2$ and $5m_{Pl}^2$ are shown in Figs. \[fig:n1alpha01\_dphp\], \[fig:n1alpha01\_dphn\], \[fig:n1alpha5\_dphp\] and \[fig:n1alpha5\_dphn\]. In the case of $\alpha=0.1m_{Pl}^2$, the entire range of $\phi_B$ does not give rise to the slow-roll inflation. In other words, a large range of KED (except a small subset) initial conditions provide the slow-roll inflation whereas a small subset of KED and the whole range of PED initial conditions do not. Similar results for the Starobinsky model were shown in [@Bonga2016]. Although, in the case of $\alpha=5m_{Pl}^2$, both KED (except a small subset) and PED initial values provide slow-roll inflation as shown in Figs. \[fig:n1alpha5\_dphp\], \[fig:n1alpha5\_dphn\] and Table \[tab:n012\_alpha\_dphi\]. We have also found the number of $e$-folds which is exhibited in Table \[tab:n1\_dphip\] and \[tab:n1\_dphin\].
Then, we have considered the background evolution of the $\alpha-$attractor model with $n=2$ for $\alpha=0.5m_{Pl}^2$ and $5m_{Pl}^2$. This model is also asymmetric, and the total energy density can not exceed $\rho_c$. The numerical results are shown in Figs. \[fig:n2alpha05\_dphp\], \[fig:n2alpha05\_dphn\], \[fig:n2alpha5\_dphp\] and \[fig:n2alpha5\_dphn\]. Similar to the $E-model$, here also, for small values of $\alpha$ (say $\alpha=0.5m_{Pl}^2$ ), we do not get slow-roll inflation for a small subset of KED and the entire range of PED initial conditions, while a large range of KED initial values produces the slow-roll inflation. Though, for large values of $\alpha$ (say $\alpha=5m_{Pl}^2$ ), both KED (except a small subset) and PED initial values are capable to produce the slow-roll inflationary phase. We have also obtained $N_{inf}$’s that are displayed in Tables \[tab:n2\_dphip\] and \[tab:n2\_dphin\]. Looking at both tables, physically viable initial conditions are identified that are consistent with the Planck data [@Planck2015].
Finally, we have presented the phase space analysis for the above three models. For $T-model$ with $\alpha=10 m_{Pl}^2$, the quantum bounce is not compact: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \rightarrow \pm \infty$ whereas for $\alpha=10^{10} m_{Pl}^2$, it is compact: $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $ and $\phi_B \rightarrow \pm 2.56 \times 10^5 m_{Pl}$. In the case of $E-model$ and $\alpha-$attractor with $n=2$, the initial surface is semi-finite: for $E-model$ with $\alpha=0.1 m_{Pl}^2$ and $\alpha=5 m_{Pl}^2$, it is $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $; $\phi_B \in (-3.64, \infty)$ and $\phi_B \in (-25.65, \infty)$, respectively, while for $\alpha-$attractor with $n=2$, this is given as $| \dot{\phi}_B | < 0.91 m_{Pl}^2 $; $\phi_B \in (-4.08, \infty)$ for $\alpha=0.5 m_{Pl}^2$ and $\phi_B \in (-12.88, \infty)$ for $\alpha=5 m_{Pl}^2$. The phase portraits for these models are shown in Figs. \[fig:Tport\], \[fig:Eport\] and \[fig:n2port\], where dashed blue curves correspond to the cases without slow-roll inflationary phase and solid curves (red and blue) provide slow-roll inflation. However, only the red curves are observationally consistent with the Planck 2015 data, not the blue ones [@Planck2015].
Acknowledgements {#acknowledgements .unnumbered}
================
M.S. would like to thank T. Zhu for fruitful discussions. A.W. is supported in part by the National Natural Science Foundation of China (NNSFC) with the Grants Nos. 11375153 and 11675145.
Appendix A: Some Physical Quantities {#sec:Append .unnumbered}
====================================
From Eq.(\[eq:Ninf\]), one finds $$\begin{aligned}
N_{inf} \simeq \int_{\phi_{end}}^{\phi_*} \frac{V(\phi)}{V'({\phi})} d\phi,
\label{eq:Ninf2}\end{aligned}$$ where $\phi_*$ and $\phi_{end}$ are the values of the inflaton field at the onset and end of the slow-roll inflation.
The slow-roll parameter $\epsilon_V$ is defined as $$\begin{aligned}
\epsilon_V = \frac{M_{Pl}^2}{2} \left(\frac{V'(\phi)}{V(\phi)}\right)^2,
\label{eq:ev}\end{aligned}$$ where $M_{Pl}=m_{Pl}/\sqrt{8 \pi}$. At the end of the slow-roll inflation, $\epsilon_V=1$. Hence, one can obtain $\phi_{end}$ from Eq.(\[eq:ev\]).
During the slow-roll inflation, $\dot{\phi}^2 \ll V(\phi)$ then Eq.(\[eq:Hub\]) becomes $$\begin{aligned}
H_*{^2} \simeq \frac{8 \pi}{3 m_{Pl}^2}~V(\phi_*).
\label{eq:HubSR}\end{aligned}$$ The upper bound on $H_*$ during the slow-roll inflation is given by [@Planck2015] $$\begin{aligned}
\frac{H_*}{M_{Pl}} < 3.6 \times 10^{-5} ~~ (\text{95 \% Confidence level}).
\label{eq:H*}\end{aligned}$$ In our analysis, we shall use $H_*{/M_{Pl}}=3.0 \times 10^{-5}$. Putting the value of $H_*{/M_{Pl}}$ in Eq.(\[eq:HubSR\]), one can get $\phi_*$.
Substituting the values of $\phi_*$ and $\phi_{end}$ with $N_{inf}=60$ in Eq.(\[eq:Ninf2\]), we obtain the values of $\alpha$ and $c$ for $T$, $E-models$ and $\alpha-$attractor model with $n=2$.
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[^1]: E-mail address: [email protected]
[^2]: E-mail address: [email protected]
[^3]: E-mail address: Anzhong$\[email protected]
|
---
abstract: |
Martian bow shocks, the solar wind interacting with an unmagnetized planet, are studied. We theoretically investigated how solar parameters, such as the solar wind dynamic pressure and the solar extreme ultraviolet (EUV) flux, influence the bow shock location, which is still currently not well understood. We present the formula for the location of the bow shock nose of the unmagnetized planet. The bow shock location, the sum of the ionopause location and bow shock standoff distance, is calculated in the gasdynamics approach. The ionopause location is determined using thermal pressure continuity, i.e., the solar wind thermal pressure equal to the ionospheric pressure, according to tangential discontinuity. The analytical formula of the ionopause nose location and the ionopause profile around the nose are obtained. The standoff distance is calculated using the empirical model. Our derived formula shows that the shock nose location is a function of the scale height of ionosphere, the dynamic pressure of the solar wind and the peak ionospheric pressure. The theoretical model implies that the shock nose location is more sensitive to the solar EUV flux than solar wind dynamics pressure. Further, we theoretically show that the bow shock location is proportional to the solar wind dynamic pressure to the power of negative C, where C is about the ratio of the ionospheric scale height to the distance between bow shock nose and the planet center. This theory matches the gasdynamics simulation and is consistent with the spacecraft measurement result by Mars Express [\[]{}Hall, et al. (2016) J. Geophys. Res. Space Physics, 121, 11,474-11,494[\]]{}.
Keywords: bow shock of the unmagnetized planet, standoff distance, ionosphere, ionopause, gasdynamics theory
author:
- 'I-Lin Yeh$^{1,2}$, Sunny W.Y. Tam$^{2}$, Po-Yu Chang$^{2}$'
bibliography:
- 'bs\_ref.bib'
title: Theoretical investigation of the bow shock location for the solar wind interacting with the unmagnetized planet
---
[UTF8]{}[bsmi]{}
$^{1}$Department of Physics, UC San Diego, USA
$^{2}$Institute of Space and Plasma Sciences, National Cheng Kung University, Taiwan
Introduction
============
In the nowadays space physics research, more research focuses on the solar wind interacting with the magnetized planet than the unmagnetized planet or weakly-magnetized planet. However, the study of Martian bow shock is recently a hot topic due to the growing interests in the exploration of Mars, a weakly-magnetized planet. Furthermore, more and more data from the spacecraft measurement are available. A theoretical model of the bow shock nose position has been derived in this work.
There are three types of interaction between the solar wind and an obstacle in space: (1) solar wind interacting with the magnetized obstacle like Earth. (2) solar wind interacting with the unmagnetized obstacle with an atmosphere like Mars and Venus. (3) solar wind interacting with the unmagnetized obstacle without atmosphere like moon. Figure \[fig:Schematic-of-solar\] is a schematic of these three types of interaction.
![Schematic of solar wind past the (a) Earth (b) Moon (c) Mars and Venus. Courtesy of Ref. [@spreiter1970solar].\[fig:Schematic-of-solar\]](Graphics_paper/Solar_wind_flow_past_earth_moon_venus){width="0.9\columnwidth"}
The detached bow shock is formed because of the supersonic solar wind and the deflection of the incident solar wind flow by the magnetosphere or ionosphere. Since the moon has neither ionosphere nor magnetosphere, no bow shock is formed in the solar wind interacting with the moon.
Formation of the bow shock in plasma interaction with Mars, unmagnetized planets with an atmosphere (Fig. \[fig:Schematic-unma\]), is as follows. First, the ionization by solar EUV radiation in the atmosphere forms an ionospheric obstacle, acting as a conductor. The boundary of the ionosphere is called ionopause. Then the solar-wind plasma with its frozen-in field flows at a supersonic velocity toward the conducting obstacle, resulting in the appearance of the bow shock.
![Schematic of the solar wind interaction with an unmagnetized planet with an atmosphere. Courtesy of Ref.[@kivelson1995introduction]. \[fig:Schematic-unma\] ](Graphics_paper/solar_wind_interaction){width="0.7\columnwidth"}
Martian bow shock has been detailedly studied by spacecraft measurement and numerical simulation. Martian bow shock is formed by the interaction between the solar wind and the Martian ionosphere. Recently, the first measurement study[@vogt2015ionopause] of the ionopause from the mission Mars Atmosphere and Volatile EvolutioN (MAVEN, 2014-present) was released in 2015. This mission will provide us a deeper understanding of the Martian bow shock. The shape of the bow shock is often modeled using the least-squares fitting of an axisymmetric or non-axisymmetric conic section[@farris1994determining; @trotignon2006martian] with the data from spacecraft measurement. On the other hand, the theoretical model of the planetary bow shock location and shape can be seen in the review paper by Spreiter[@spreiter1970solar; @spreiter1966hydromagnetic; @spreiter1995location], Slavin[@slavin1981solar; @slavin1983solar] and Verigin[@verigin2003planetary]. An efficient computational model for determining the global properties of the solar wind past a planet based on axisymmetric magnetohydrodynamics was proposed by Spreiter[@spreiter1980new]. The specific study of the magnetohydrodynamics simulation for the solar wind interaction with Mars can be seen in Ref.[@spreiter1992computer] and Ref.[@ma2002three].
However, how the factors influence the location of the Martian bow shock is not well understood. The main factors impacting the bow shock position are the solar wind dynamic pressure $P_{dyn}=\rho_{\infty}v_{\infty}^{2}$ and solar EUV flux $l_{euv}$. According to the fitting results (Fig. \[fig:The-response-of\]) from the data of Mars Express Analyser of Space Plasma and EneRgetic Atoms (ASPERA-3)[@hall2016annual], it is shown that the bow shock location ($r_{s}$) reduces in altitude with increasing solar wind dynamic pressure in the relation $r_{s}\propto P_{dyn}^{-0.02}$ and increases in altitude with increasing solar EUV flux in the relation $r_{s}\propto0.11\,l_{EUV}$. It means that the bow shock position is more sensitive to the solar EUV flux than the solar wind dynamic pressure.
![The response of the location of the Martian bow shock with the solar parameters. (a) Bow shock location against solar EUV radiance. (b) Bow shock location against solar wind dynamic pressure. Courtesy of [@hall2016annual].\[fig:The-response-of\]](Graphics_paper/Annual_variations_loc-par.jpeg){width="0.9\columnwidth"}
Other parameters controlling the Martian bow shock location are the intense localized Martian crustal magnetic fields[@vignes2002factors], the magnetosonic Mach number[@edberg2010magnetosonic], the interplanetary magnetic fields and the convective electric field[@edberg2008statistical]. In this thesis, we will mainly focus on the dependence of the solar wind dynamics pressure and the ionospheric pressure, which is dependent on the solar EUV radiation.
It is not well understood how the location of the Martian bow shock is influenced by the solar parameters such as solar wind dynamic pressure and EUV flux. In this thesis work, we study the location of the bow shock generated from the interaction between the solar wind and unmagnetized planet in the theoretical aspect. The formula for the shock nose location as a function of the solar wind dynamic pressure and EUV flux will be presented and compared with the spacecraft measurement. Past studies by others are all related to observation, but this study provides a theory to explain the spacecraft measurement results. On the other hand, we are not going to study fine structure in the transition region of a shock and the shock microphysics, even though it is more theoretically fascinating. The dissipation mechanism for the shock will not influence the location of the bow shock, so we use the ideal hydrodynamics formulation throughout this work. Note that the thickness of the discontinuity surface is zero under the ideal hydrodynamics description.
The analytical theory is essential because the relationship between each physical quantities can be directly known in the formula. However, in simulation, to know the results from different conditions requires different runs, which is very numerically intensive especially for multi-scale and multi-physics simulation. This work will be beneficial for both space physics and laboratory astrophysics research. In laboratory astrophysics research.
In this paper, we will focus on theoretically determining the location of the nose of the bow shock for the solar wind past the unmagnetized planet. The formula for shock nose location is developed and compared to the results from the hydrodynamics simulation and spacecraft measurement. In section \[sec:Determination-of-the\], we will give a detailed derivation of the formula for the shock nose position. The comparison of our formula and the spacecraft measurement results will be shown in section \[subsec:Comparison-theory\]. In section \[sec:Conclusion\], the conclusion of the thesis will be given.
Determination of the location of the bow shock nose \[sec:Determination-of-the\]
================================================================================
We theoretically investigate the bow shock location as a function of the solar wind and the ionospheric conditions, such as solar wind dynamic pressure $\rho_{\infty}v_{\infty}^{2}$, ionospheric scale length $H$, ionospheric peak pressure $P_{M,i}$ and the location of the ionospheric peak pressure $r_{M,i}$. The equation of bow shock location is derived and will be used to design the future experiments. We only focus on the nose location of the bow shock but not the whole shape profile of the bow shock. The shape of the bow shock and the ionopause is assumed to be symmetric around the x-axis.
The schematic of the bow shock and the obstacle boundary (ionopause) is shown in the Fig. \[fig:Definition-of-varaibles\]. We use the following symbols to represent the geophysical quantities in the report: $r_{o}$ is nose positions of the obstacle, $r_{s}$ is nose positions of the bow shock, $r_{M,i}$ is the location inside ionosphere where maximal thermal pressure occurs, $\Delta$ is the bow shock standoff distance, i.e., the distance between ionopause nose position and bow shock nose position.
![Definition of each variable used in the report \[fig:Definition-of-varaibles\]](Graphics_paper/shock_coordinate){width="0.7\columnwidth"}
The goal is to determine the shock nose location $r_{s}$: $$r_{s}=r_{o}+\Delta.\label{eq:rs=00003Dro+delta}$$ Ionopause nose location $r_{o}$ is calculated using the continuity of the thermal pressure by tangential discontinuity[@spreiter1970solar; @spreiter1966hydromagnetic; @verigin2003planetary]; bow shock standoff distance is calculated by the empirical formula[@farris1994determining; @verigin2003planetary].
We first introduce the hydrodynamics boundary conditions in subsection \[subsec:Hydrodynamics-boundary-condition\]. The derivation of the ionopause nose location and the radius of curvature at ionopause nose are in subsection \[subsec:Ionopause-(obstacle-boundary)\]. The standoff distance formula is introduced in subsection \[subsec:Bow-shock-standoff\]. Finally, the formula of the bow shock nose location is shown in subsection \[subsec:shock location\]. The comparison of the theory and the observation results will be given in next section (section \[subsec:Comparison-theory\]).
Hydrodynamics boundary condition\[subsec:Hydrodynamics-boundary-condition\]
---------------------------------------------------------------------------
Hydrodynamics formulation is used throughout this thesis. Ideal magnetohydrodynamics equations are $$\begin{aligned}
\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\,\vec{v} & =0,\nonumber \\
\rho(\frac{\partial\vec{v}}{\partial t}+\vec{v}\cdot\nabla v) & =-\nabla P+\frac{1}{\mu_{0}}(\nabla\times\vec{B})\times\vec{B},\nonumber \\
\frac{\partial\vec{B}}{\partial t} & =\nabla\times\vec{v}\times\vec{B},\nonumber \\
\frac{\partial P}{\partial t}+\vec{v}\cdot\nabla P & =-\gamma\,P\:\nabla\cdot\vec{v},\label{eq:Ideal_mhd}\end{aligned}$$ where $p$ is the pressure, $\rho$ is the mass density, $\vec{v}$ is the velocity and $\vec{B}$ is the magnetic field. The first one is the continuity equation, the second is the momentum equation, the third is Faraday’s law and the last is the entropy conservation equation, or adiabatic equation. Here we assume the gas follows the polytropic condition and adiabatic process.
Magnetohydrodynamics equations can be reduced to hydrodynamics equations under the condition that the magnetic pressure term is much smaller than the thermal pressure term in the right-hand side of the momentum equation (second equation in Eq. \[eq:Ideal\_mhd\]) $$|\frac{\frac{1}{\mu_{0}}(\nabla\times\vec{B})\times\vec{B}}{\nabla P}|\approx\frac{B^{2}/2\mu_{0}}{P}=1/\beta,\label{eq:beta}$$ where plasma beta $\beta$ is defined as thermal pressure divided by magnetic pressure. For the condition of solar wind past the unmagnetized planet, $\beta$ is much larger than 1 in both space and laboratory, so we can neglect the force term containing the magnetic field in the momentum equation, reducing the magnetohydrodynamics formulation to pure hydrodynamics formulation $$\begin{aligned}
\frac{\partial\rho}{\partial t}+\nabla\cdot\rho\,\vec{v} & =0,\nonumber \\
\rho(\frac{\partial\vec{v}}{\partial t}+\vec{v}\cdot\nabla v) & =-\nabla p,\nonumber \\
\frac{\partial P}{\partial t}+\vec{v}\cdot\nabla P & =-\gamma\,P\:\nabla\cdot\vec{v.}\label{eq:hydro_equ}\end{aligned}$$ Throughout the paper, we will use hydrodynamics formulation instead of magnetohydrodynamics due to the high beta condition. The first one is the mass conservation equation, the second is the momentum equation for ideal fluid, or Euler equation and the third is the adiabatic equation.
The boundary condition for steady-state ideal hydrodynamics[@landau1987fluid] is $$\begin{aligned}
\left[\rho v_{n}\right] & =0\nonumber \\
\left[P+\rho v_{n}^{2}\right] & =0\nonumber \\
\left[\rho v_{n}\vec{v_{t}}\right] & =0\nonumber \\
\left[v_{n}(\frac{\rho v^{2}}{2}+\frac{\gamma\,P}{\gamma-1})\right] & =0,\label{eq:hydro_BC}\end{aligned}$$ where the subscript $n$ and $t$ are the normal direction and tangential direction, respectively. The brackets mean the difference of the quantity between both sides of the boundary. The Eq. \[eq:hydro\_BC\] indicates the continuity of the mass flux, momentum flux and energy flux. Note that the discontinuity surfaces of the ionopause and the bow shock are zero thickness under the description of the dissipationless ideal (magneto)hydrodynamics[@landau1987fluid; @zel2012physics].
In our study, we are interested in two types of boundary:
- Tangential discontinuity[@spreiter1970solar; @spreiter1966hydromagnetic; @landau1987fluid] at the ionopause $$\begin{aligned}
\left[\rho\right]\neq & 0\nonumber \\
\left[\vec{v_{t}}\right]\neq & 0\nonumber \\
v_{n}= & 0\nonumber \\
\left[P\right]= & 0.\label{eq:tangential discontinuity}\end{aligned}$$ The normal velocity is zero in the tangential discontinuity. We will utilize the continuity of the thermal pressure to determine the location and the radius of curvature at the ionopause nose. Furthermore, we can observe that there is a density jump across the ionopause according to tangential discontinuity.
- Shock waves[@spreiter1970solar; @spreiter1966hydromagnetic; @landau1987fluid] at the bow shock front $$\begin{aligned}
\left[\rho v_{n}\right]= & 0\nonumber \\
\left[\vec{v_{t}}\right]= & 0\nonumber \\
\left[P+\rho v_{n}^{2}\right]= & 0\nonumber \\
\left[\frac{v_{n}^{2}}{2}+\frac{\gamma\,P/\rho}{\gamma-1}\right]= & 0.\label{eq:shock}\end{aligned}$$ For our purpose of the study of the global phenomenon like bow shock position, the ideal fluid description is enough. We are not going to study microphysics such as the shock formation mechanism, so the dissipation process of the shock will not be discussed throughout the thesis. In general, the shock in the space is formed in a collisionless magnetized environment and the shock dissipation mechanism is the wave-particle interaction.
Overview of the theory
----------------------
In this subsection, we have an overview of all the theories which are used for calculating the location of the bow shock nose in terms of the solar wind and the ionospheric conditions. Fig. \[fig:Thermal-pressure-along\] shows the variation of the thermal pressure along the stagnation streamline.
![Thermal pressure along the stagnation streamline. \[fig:Thermal-pressure-along\] ](Graphics_paper/Pressure_variation){width="0.7\columnwidth"}
- \(a) Solar wind
- In the region of the solar wind, the thermal pressure can be expressed as a function of the dynamic pressure and the sonic Mach number, i.e., $P_{\infty}=\rho_{\infty}v_{\infty}^{2}/(M_{\infty}^{2}\,\gamma)$.
- \(a) – (b) Bow shock
- At the bow shock, momentum flux conservation in the normal shock relation is used
$$P_{\infty}+\rho_{\infty}v_{\infty}^{2}=P_{s}+\rho_{s}v_{s}^{2}.\label{eq:pcons}$$
Note that the entropy increases across the shock.
- \(b) Magnetosheath
- Within the magnetosheath, the plasma follows the process of the isentropic compression, i.e., the combination of the energy conservation of the compressible flow (Bernoulli equation)
$$\frac{2\gamma}{\gamma-1}P+\rho\,v^{2}={\rm constant},\label{eq:ber}$$
and the adiabatic relation
$$P\,V^{\gamma}={\rm constant,}\label{eq:adia}$$
where $V$ is the volume. We can observe that the sum of the thermal pressure and the dynamic pressure is conserved before and after shock, but not conserved along the stagnation streamline within the magnetosheath. Therefore, the value of $P+\rho v^{2}$ before shock is not the same as that at stagnation point.
- (b)-(c) Ionopause
- At the ionopause, the thermal pressure is continuous according to tangential discontinuity.
- \(c) Upper ionosphere
- Within the upper ionosphere, the hydrostatic equilibrium (the balance between the gravity force and the pressure gradient) is assumed.
Ionopause (obstacle boundary) \[subsec:Ionopause-(obstacle-boundary)\]
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Ionopause, the boundary of the ionosphere, is the location of the thermal pressure balance according to tangential discontinuity.[@spreiter1970solar; @verigin2003planetary]. We first investigate the pressure variation on the center line, then the ionopause nose location $r_{o}$ and the radii of curvature at the ionopause nose $R_{o}$.
### Thermal pressure at the ionopause\[subsec:Thermal-pressure-ionospause\]
Ionopause profile is determined by the thermal pressure continuity at both sides of the ionopause according to tangential discontinuity. Here we discuss the thermal pressure at both sides of the ionopause respectively.
- Thermal pressure at the inner side of the ionopause
The thermal pressure in the ionosphere is assumed to be spherical symmetric and at hydrostatic equilibrium[@spreiter1970solar] in equivalence to the balance between pressure gradient and gravity force. So the thermal pressure inside the ionosphere can be expressed as $$P_{i}(r)=P_{M,i}\ {\rm exp}(\frac{r_{M,i}-r}{H}),\label{eq:exp of pion-1}$$ where $P_{i}(r)$ is the pressure inside the ionosphere, $r_{M,i}$ is the location inside ionosphere where peak thermal pressure $P_{M,i}$ occurs and $H=k_{B}T/mg$ is the scale height in which $m=1.67\times10^{-24}g$ is the mass for a singly ionized hydrogen, $k_{B}$ is Boltzmann’s constant and $T$ is the absolute temperature for plasma and assumed to be constant inside the ionosphere.
- Thermal pressure at the outer side of the ionopause
We use Rayleigh pitot tube formula[@spreiter1966hydromagnetic; @spreiter1992computer; @landau1987fluid] to obtain the thermal pressure just outside the ionosphere as a function of the solar wind dynamic pressure. Rayleigh pitot tube formula is used for the stagnation pressure at the blunt body nose with a detached bow shock. It is derived in two steps: (1) applying the hydrodynamic normal shock jump condition to get the downstream thermal pressure; (2) applying the isentropic compression to determine the thermal pressure at the stagnation point with Bernoulli’s law on the stagnation streamline within the magnetosheath. The rigorous derivation is shown in Appendix. The Rayleigh pitot tube formula is given as $$P_{o}=P_{\infty}M_{\infty}^{2}(\frac{\gamma+1}{2})^{(\gamma+1)/(\gamma-1)}\frac{1}{\left[\gamma-(\gamma-1)/(2M_{\infty}^{2})\right]^{1/(\gamma-1)}},\label{eq:Rayleigh}$$ where $P_{o}$ is the thermal pressure at the ionopause nose, $P_{\infty}$ is the thermal pressure of the solar wind, $M_{\infty}$ is the sonic Mach number of the solar wind and $\gamma$ is the specific heat ratio. Then, we plug $M_{\infty}=\frac{v_{\infty}}{\sqrt{\gamma p_{\infty}/\rho_{\infty}}}$ into the Rayleigh pitot tube formula, the relationship between thermal pressure at the ionopause $P_{o}$ as a function of solar wind dynamic pressure $\rho_{\infty}v_{\infty}^{2}$ can be expressed as $$P_{o}=k\rho_{\infty}v_{\infty}^{2},\label{eq:ionos pres}$$ where $$k=(\frac{\gamma+1}{2})^{(\gamma+1)/(\gamma-1)}\frac{1}{\gamma\left[\gamma-(\gamma-1)/(2M_{\infty}^{2})\right]^{1/(\gamma-1)}}.\label{eq:k}$$ For $\gamma=5/3$ and $M_{\infty}\gg1$, this relation can be simplified to $k=0.88$.
### Nose position of the ionopause $r_{o}$
The formula of the ionopause nose position $r_{o}$ is determined by the thermal pressure continuity ( Fig. \[fig:pressure balance\]) at the ionopause according to tangential discontinuity:
$$P_{o}=P_{i}(r_{o}).\label{eq:pi=00003Dpinf}$$
![The schematic of the ionopause, where the ionsperic pressure is equal to solar wind thermal pressure.\[fig:pressure balance\]](Graphics_paper/ionosphere_pressure){width="0.7\columnwidth"}
By solving Eq. \[eq:pi=00003Dpinf\] with the expression of the thermal pressure at the both side of the ionopause (Eq. \[eq:exp of pion-1\] and Eq. \[eq:ionos pres\]), the formula of the nose position of the bow shock $r_{o}$ can be derived as $$r_{o}=r_{M,i}+H\ {\rm ln}(\frac{P_{M,i}}{k\rho_{\infty}v_{\infty}^{2}}).\label{eq:exp of ro}$$
The derived equation of the nose position (Eq. \[eq:exp of ro\]) of the ionopause is reasonable: the shorter the scale height $H$ or the larger the dynamic pressure of the solar wind $\rho_{\infty}v_{\infty}^{2}$, then ionopause closer to planet surface.
### Radius of curvature at ionopause nose $R_{o}$\[subsec:Radius-of-curvature\]
In this section, we analytically calculate the radius of curvature at ionopause nose $R_{o}$ by solving the ionopause profile equation near the ionopause nose. Since the ionopause is symmetric at x-axis, the ionopause profile can be expressed as $x=x(y)$ . We do the Taylor expansion at $y=0$ of the ionopause profile $x=x(y)$, then we can get the equation of the ionopause profile at the vicinity of the ionopause nose $$x(y)=x(0)+(y-0)\,x'(0)+\frac{1}{2}(y-0)^{2}x"(0)+...\label{eq:expansion}$$ Note that on the ionopause profile, $x(0)=r_{o}$ and $x'(0)=0$. Furthermore, by the definition of the radius of curvature $R(y)=|\frac{(1+x'(y))^{3/2}}{x"(y)}|$, the radius of curvature at ionopause nose ($y=0$) can be written as $R_{o}=-1/x"(0)$. Thus, the equation of the ionopause profile near the ionopause nose can be reduced to a quadratic equation $$x=r_{o}-\frac{1}{2R_{o}}y^{2}.\label{eq:expansion with Ro}$$ Here we neglect the third and higher order term of the Taylor expansion.
The whole ionopause profile can be determined by the thermal pressure continuity at the ionopause according to tangential discontinuity, that is, the thermal pressure is equal at the outer side and the inner side of the ionopause. $$k\rho_{\infty}v_{\infty}^{2}\,{\rm cos^{2}}\psi=P_{i}(r),\label{eq:therbal}$$ where $\psi$ is the angle between $v_{\infty}$ and the normal to ionopause, which is shown in Fig. \[fig:psi\].
![Element of the ionopause and the coordinate. Modified figure from Ref. [@spreiter1970solar]. \[fig:psi\] ](Graphics_paper/psi){width="0.7\columnwidth"}
The left hand side of the equation is the thermal pressure approximated at the outer side of the ionopause deviated from the nose position and the right hand side is the ionospheric pressure we introduced in section \[subsec:Thermal-pressure-ionospause\]. Since $k\rho_{\infty}v_{\infty}^{2}$ is the ionospheric pressure at the ionopause nose $P_{i}(r_{o})$, the equation of the ionopause profile (Eq.\[eq:therbal\]) can be reduced to
$$P_{i}(r_{o})\,{\rm cos^{2}}\psi=P_{i}(r).\label{eq:preba}$$
By the geometric relation, $cos^{2}\psi$ can be expressed as $$cos^{2}\psi=\left(\frac{dy}{ds}\right)^{2}=\frac{\left(r\,d\theta\:{\rm cos}\theta+dr\,{\rm sin}\theta\right)^{2}}{dr^{2}+(r\,d\theta)^{2}}.\label{eq: cospsi}$$ We substitute the cosine relation in Eq.\[eq: cospsi\] into the pressure continuity equation at the ionopause (Eq.\[eq:preba\]), we get $$P_{i}(r_{o})\frac{\left({\rm cos}\theta+(\frac{dr}{r\,d\theta})\,{\rm sin}\theta\right)^{2}}{\left(\frac{dr}{r\,d\theta}\right)^{2}+1}=P_{i}(r).\label{eq: pres_balance_polar}$$ Then we solve for $dr/r\:d\theta$ to obtain $$\frac{dr}{r\,d\theta}=\frac{-P_{i}(r_{o})\,{\rm sin}2\theta+2\sqrt{P_{i}(r)P_{i}(r_{o})-P_{i}^{2}(r)}}{2(P_{i}(r_{o}){\rm sin^{2}}\theta-P_{i}(r))}.\label{eq: dr/rdtheta}$$ This is the differential equation for the ionopause profile, which can be solved numerically[@spreiter1970solar] with the initial condition $r\left(\theta=0\right)=r_{o}$. Note that the ionopause is symmetric about the $x=0$ axis ($\theta=0$), so the first-order derivative at the ionopause nose is zero, that is, $$\frac{1}{r(0)}\frac{dr}{d\theta}(0)=\frac{1}{r_{o}}\frac{dr}{d\theta}(0)=0.\label{eq:derir0}$$ In the second term of the numerator in the ionopause profile differential equation (Eq.\[eq: dr/rdtheta\]), it contains a square root. The value of the quantity inside the square root must be equal or larger than zero, or the square root term will become imaginary, which is physically unallowable. So, we can get $$P_{i}(r)P_{i}(r_{o})-P_{i}^{2}(r)\geq0,\label{eq:ppo}$$ where $P_{i}(r)$ is the ionospheric pressure exponentially decaying outward because of hydrostatic equilibrium. Then we can obtain that the ionopause profile must follow the condition $$r\geq r_{o}.\label{eq:rlargerro}$$ The equality occurs at the ionopause nose.
For our purpose of deriving the radius of curvature at the ionopause nose, we only have to focus on the vicinity of the ionopause nose, i.e., the region $\theta\rightarrow0$ and $r\rightarrow r_{o}$. Furthermore, at the ionospause nose, $dr/rd\theta$ can be approximated by $dx/dy.$ The differential equation for the ionopause profile (Eq. \[eq: dr/rdtheta\]) at the vicinity of the ionopause nose can be simplified to
$$\begin{aligned}
\frac{dx}{dy} & =-\sqrt{\frac{P_{i}(r_{o})}{P_{i}(r)}-1}\nonumber \\
& =-\sqrt{\frac{P_{i}(r_{o})}{P_{i}(r)}}\sqrt{1-\frac{P_{i}(r)}{P_{i}(r_{o})}}\nonumber \\
& \simeq-\sqrt{1-\frac{P_{i}(r)}{P_{i}(r_{o})}}.\label{eq:dxdy}\end{aligned}$$
Now we express $P_{i}(r)$ in Taylor series at $r=r_{o}$, then the right hand side of the Eq. \[eq:dxdy\] can be rewritten as $$\begin{aligned}
-\sqrt{1-\frac{P_{i}(r)}{P_{i}(r_{o})}} & =-\sqrt{-(r-r_{o})\frac{P_{i}'(r_{o})}{P_{i}(r_{o})}-\frac{1}{2}(r-r_{o})^{2}\frac{P_{i}"(r_{o})}{P_{i}(r_{o})}-...}\nonumber \\
& =-\sqrt{-(r-r_{o})\frac{P_{i}'(r_{o})}{P_{i}(r_{o})}}\sqrt{1+\frac{1}{2}(r-r_{o})\frac{P_{i}"(r_{o})}{P_{i}'(r_{o})}+...}\nonumber \\
& \simeq-\sqrt{-(r-r_{o})\frac{P_{i}'(r_{o})}{P_{i}(r_{o})}}\left(1+\frac{1}{4}(r-r_{o})\frac{P_{i}"(r_{o})}{P_{i}'(r_{o})}+...\right).\label{eq:sqrt1-p}\end{aligned}$$ Thus, now the differential equation for the ionopause profile near the ionopause nose is $$\frac{dx}{dy}=-\sqrt{-(r-r_{o})\frac{P_{i}'(r_{o})}{P_{i}(r_{o})}}\left(1+\frac{1}{4}(r-r_{o})\frac{P_{i}"(r_{o})}{P_{i}'(r_{o})}+...\right)\label{eq:dxdyf}$$
Also, by $x\rightarrow r_{o}$and $y\rightarrow0$ at the vicinity of the ionopause nose, $r-r_{o}$ can be approximated by
$$\begin{aligned}
r-r_{o} & =\sqrt{x^{2}+y^{2}}-r_{o}\nonumber \\
& =\sqrt{\left[r_{o}+\left(x-r_{o}\right)\right]^{2}+y^{2}}-r_{o}\nonumber \\
& =\sqrt{r_{o}^{2}+2\,r_{o}(x-r_{o})+(x-r_{o})^{2}+y^{2}}-r_{o}\\
& =r_{o}\sqrt{1+2\,\frac{x-r_{o}}{r_{o}}+\frac{\left(x-r_{o}\right)^{2}}{r_{o}^{2}}+\frac{y^{2}}{r_{o}^{2}}}-r_{o}\nonumber \\
& \simeq r_{o}(1+\frac{x-r_{o}}{r_{o}}+\frac{\left(x-r_{o}\right)^{2}}{2\,r_{o}^{2}}+\frac{y^{2}}{2\,r_{o}^{2}})-r_{o}\\
& =x-r_{o}+\frac{\left(x-r_{o}\right)^{2}}{2\,r_{o}}+\frac{y^{2}}{2\,r_{o}}.\label{eq:r-ro-1}\end{aligned}$$
By substituting the equation of the ionopause profile near the ionopause nose (Eq.\[eq:expansion with Ro\]) in to Eq.\[eq:r-ro-1\], we get $$\begin{aligned}
r-r_{o} & =-\frac{1}{2R_{o}}y^{2}+\frac{\left(-\frac{1}{2R_{o}}y^{2}\right)^{2}}{2\,r_{o}}+\frac{y^{2}}{2\,r_{o}}\label{eq:r-ro2}\\
& \simeq\frac{y}{2}^{2}\left(\frac{1}{r_{o}}-\frac{1}{R_{o}}\right)\label{eq:r-ro3}\end{aligned}$$ In Eq.\[eq:r-ro2\], the second term at the right hand side is negligible as $y\rightarrow0$ since it is of the order $y^{4}$ and the other two terms are of the order $y^{2}$.
We plug the $r-r_{o}$ relation (Eq. \[eq:r-ro3\]) into the differential equation of the ionopause profile (Eq. \[eq:dxdyf\]), then integrate the differential equation $$\sideset{}{_{r_{o}}^{x}}\int dx=\sideset{-}{_{0}^{y}}\int dy\,\sqrt{\frac{-1}{2}\frac{P_{i}'(r_{o})}{P_{i}(r_{o})}\left(\frac{1}{r_{o}}-\frac{1}{R_{o}}\right)}\left(y+\frac{1}{8}\left(\frac{1}{r_{o}}-\frac{1}{R_{o}}\right)\frac{P_{i}''(r_{o})}{P_{i}'(r_{o})}\,y^{3}+...\right).\label{eq:integration}$$ Therefore we obtain the formula of the ionopause profile at the vicinity of the ionopause: $$x=r_{o}-\sqrt{-\frac{P_{i}'(r_{o})}{2P_{i}(r_{o})}\left(\frac{1}{r_{o}}-\frac{1}{R_{o}}\right)}\left(\frac{y^{2}}{2}+\frac{1}{32}\left(\frac{1}{r_{o}}-\frac{1}{R_{o}}\right)\frac{P_{i}''(r_{o})}{P_{i}'(r_{o})}\,y^{4}+...\right).\label{eq:ionopause_eq}$$
By comparing Eq. \[eq:ionopause\_eq\] and Eq. \[eq:expansion with Ro\], finally, we get the equation of the radius of curvature at the ionopause nose $$R_{o}=\sqrt{-\frac{2P_{i}(r_{o})}{P_{i}'(r_{o})}\frac{R_{o}r_{o}}{R_{o}-r_{o}}}.\label{eq:exp of Ro}$$ Eq.\[eq:exp of Ro\] can be rearranged as a quadratic equation $$R_{o}^{2}-r_{o}\,R_{o}-\sqrt{-\frac{2P_{i}(r_{o})}{P_{i}'(r_{o})}}\,r_{o}=0.\label{eq:quadraRo}$$ Then we solve it and get $$R_{o}=\frac{r_{o}\pm\sqrt{r_{o}^{2}+4\,\sqrt{-\frac{2P_{i}(r_{o})}{P_{i}'(r_{o})}}\,r_{o}}}{2}.\label{eq:Ro}$$ We take the plus sign in the nominator since $R_{o}$ will be negative if we take the minus sign, which is not physically allowable. Thus, we obtain the expression of the radius of curvature at the ionopause nose $$R_{o}=\frac{r_{o}+\sqrt{r_{o}^{2}+4\,\sqrt{-\frac{2P_{i}(r_{o})}{P_{i}'(r_{o})}}\,r_{o}}}{2}.\label{eq:FinalRo}$$
With the ionospheric pressure $P_{i}(r)$ of the form in Eq. \[eq:exp of pion-1\], the radius of curvature at the ionopause nose can be expressed as $$R_{o}=\frac{r_{o}+\sqrt{r_{o}^{2}+8\,H\,r_{o}}}{2},\label{eq:radi_ionop}$$ where $H$ is the scale height of the ionosphere. The expression of the radius of curvature at the ionopause nose from our calculation is the same as that from the Table A1 in Verigin *et al.* (2003) [@verigin2003planetary]. Thus, by plugging Eq.\[eq:radi\_ionop\] into Eq.\[eq:expansion with Ro\], we can get the ionopause profile near the ionopause nose $$x=r_{o}-\frac{1}{r_{o}+\sqrt{r_{o}^{2}+8\,H\,r_{o}}}\,y^{2}.\label{eq:ionopa_eq}$$ In the derivation of the radius of curvature at the ionopause nose and the ionopause profile near the ionopause nose, we made many assumptions to get it. We have shown that the assumption we made above is all valid by verifying our analytical results with the simulation results, which will be given in Section \[subsec:Verification-of-the\].
This equation of the radius of the curvature at the ionopause (Eq.\[eq:radi\_ionop\]) tells that $$R_{o}\geq r_{o}.\label{eq:Ro>=00003Dro}$$ The equality occurs as the ionospheric scale height $H$ is close to zero. By the equation of the ionopause nose location (Eq.\[eq:exp of ro\]), we get that $r_{o}$ is equal to $r_{M,i}$ when $H$ is zero. Combining the above relation, we found that $$R_{o}\rightarrow r_{M,i}\;{\rm as\;H\rightarrow0}.\label{eq:ro->rmi}$$ This relation means that if the ionospheric pressure decreases very sharply outward, the radius of the ionopause nose is about the distance between the location of the ionospheric peak pressure and the planet center. Furthermore, we can observe that $R_{o}$ is smaller as the dynamic pressure of the solar wind is larger since $r_{o}$ is smaller. These results are physically reasonable.
Bow shock standoff distance $\Delta$\[subsec:Bow-shock-standoff\]
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The standoff distance of the bow shock $\Delta$ is determined by the empirical model [@farris1994determining; @spreiter1966hydromagnetic; @verigin2003planetary; @seiff1962recent]. This empirical model is supported by gasdynamics experiment and observations of the flow past the planets[@slavin1983solar]. The standoff distance is expressed by the empirical model $$\frac{\Delta}{R_{o}}=\alpha\frac{\rho_{\infty}}{\rho_{s}},\label{eq:standoff}$$ where $\rho_{\infty}$and $\rho_{s}$ are the mass density before and after shock, respectively and $$\alpha\approx0.87$$ is the empirical coefficient from [@farris1994determining; @verigin2003planetary]. The bow shock nose is farther from the obstacle as the radius of curvature of the obstacle nose is larger (Fig. \[fig:detached\]). Bow shock can touch the obstacle only when the leading end of the obstacle is pointed.
![Schematic of the detached shock. Courtesy of [@landau1987fluid]\[fig:detached\] ](Graphics_paper/detached){width="0.7\columnwidth"}
The density ratio across the shock is related to solar wind Mach number and the specific heat ratio $$\frac{\rho_{\infty}}{\rho_{s}}=\frac{(\gamma-1)M_{\infty}^{2}+2}{(\gamma+1)M_{\infty}^{2}}.\label{eq:ratio}$$ Thus, in the condition of the solar wind Mach number much larger than 1, the standoff distance can be expressed as $$\Delta=\alpha\ R_{o}\epsilon,\quad M_{\infty}\gg1,\label{eq:exp of Delta}$$ where $\epsilon=\frac{\gamma-1}{\gamma+1}$. By substituting the Eq. \[eq:radi\_ionop\] into Eq. \[eq:exp of Delta\], we get $$\begin{aligned}
\Delta & =\alpha\epsilon\frac{r_{o}+\sqrt{r_{o}^{2}+8\,H\,r_{o}}}{2},\quad M_{\infty}\gg1,\label{eq:deltaro}\\
& =\frac{1}{2}\,r_{o}\,\alpha\,\epsilon\left(1+\sqrt{1+8\,H/r_{o}}\right),\quad M_{\infty}\gg1.\nonumber \end{aligned}$$ According to this relation, we can obtain the value of the standoff distance if we have the ratio of $H$ (scale height of the ionosphere) to $r_{o}$ (the distance between ionopause and the center of the planet).
The formula of the bow shock nose location\[subsec:shock location\]
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Thus, combining the results above, the shock nose location can be written as
$$\begin{aligned}
r_{s} & =r_{o}+\Delta,\label{eq: shock nose location}\\
& =r_{o}+\frac{1}{2}\,\alpha\,\epsilon\left(r_{o}+\sqrt{r_{o}^{2}+8\,H\,r_{o}}\right),\quad M_{\infty}\gg1,\nonumber \end{aligned}$$
where $$r_{o}=r_{M,i}+H\ {\rm ln}(\frac{P_{M,i}}{k\rho_{\infty}v_{\infty}^{2}}),\label{eq:oexp}$$ , scale height $H=k_{B}T/(mg)$ and $\epsilon=\frac{\gamma-1}{\gamma+1}$.
Note that this equation is only valid for the sonic Mach number of the solar wind much larger than 1. As we can see in the shock nose equation (Eq. \[eq: shock nose location\]): the shorter the scale height $H$ or the larger the dynamic pressure of the solar wind $\rho_{\infty}v_{\infty}^{2}$, bow shock nose is closer to the planet. This result is reasonable and intuitive.
Comparison with the numerical simulation and spacecraft measurement results\[subsec:Comparison-theory\]
=======================================================================================================
The comparison of our formula of the bow shock location with the numerical simulation and the spacecraft measurement results is given in this section.
Verification of the analytical form of the radius of curvature by simulation\[subsec:Verification-of-the\]
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In this subsection, we verify the analytical results of the radius of curvature from the Section \[subsec:Radius-of-curvature\] by the numerical simulation. In our calculation, we analytically solve the differential equation of the ionopause profile (Eq. \[eq: dr/rdtheta\]) near the ionopause nose ($\theta\rightarrow0$) with the initial condition $r\left(\theta=0\right)=r_{o}$ to the get the equation of the ionopause profile near the ionopause nose (Eq. \[eq:ionopa\_eq\]). The radius of curvature at the ionopause nose is also obtained in Eq. \[eq:radi\_ionop\]. In the derivation of the equation of the ionopause profile near the ionopause nose and the radius of curvature at the ionopause nose, we made many assumptions, so we will verify that the assumptions are valid by numerical simulation.
We use the function *NDSolve* in the software *Mathematica* to solve the differential equation of the ionopause profile (Eq. \[eq: dr/rdtheta\]) to get the ionopause profile $r=r(\theta)$. We first compare the numerical result with our analytical result of the ionopause profile near the ionopause nose (Eq. \[eq:ionopa\_eq\]). Fig. \[fig:Ionopause-profile-calculated\] shows the comparison of the ionopause profile from analytical theory and numerical simulation. We can observe that, from both the simulation and analytical results, the ionopause follows the rule that $$r\geq r_{o},\label{eq:rlargerro-1}$$ which we show in the section \[subsec:Radius-of-curvature\]. Also, we can see that the ionopause profile from analytical theory and numerical simulation match well near the nose position ($y\rightarrow0$), which is reasonable since the analytical result is calculated under the approximation that $\theta\rightarrow0$ in polar coordinate or $y\rightarrow0$ in Cartesian coordinate.
![Ionopause profile calculated from analytical theory (Eq. \[eq:ionopa\_eq\]) and numerical simulation. The analytical results is calculated under the approximation that $\theta\rightarrow0$ in polar coordinate or $y\rightarrow0$ in Cartesian coordinate. \[fig:Ionopause-profile-calculated\]](Graphics_paper/pro22){width="0.8\columnwidth"}
We then compare the numerical results with our analytical result of the radius of curvature at the ionopause nose (Eq. \[eq:radi\_ionop\]). In the numerical simulation, we numerically solve the differential equation of the ionopause profile (Eq. \[eq: dr/rdtheta\]) to get the ionopause profile $r=r(\theta)$ using the function *NDSolve* in *Mathematica* and then calculate the radius of curvature at $\theta=0$ in polar coordinate using the formula $$R(\theta)=\frac{\left(r^{2}(\theta)+r'^{2}(\theta)\right)^{3/2}}{|r^{2}(\theta)+2\,r'^{2}(\theta)-r(\theta)r''(\theta)|}.\label{eq:radi_polar}$$ The comparison results are shown in Fig. \[fig:radius-of-curvature\]. We can see that the difference between analytical and numerical results is smaller than 1 percent. Thus, our analytical form of the radius of curvature at the ionopause nose (Eq. \[eq:radi\_ionop\]) is verified. And we can say that the assumptions we made in the derivation are valid.
![Radius of curvature at the ionopause nose from analytical theory (Eq. \[eq:radi\_ionop\]) and numerical simulation. The thick line is the analytical theory (Eq. \[eq:radi\_ionop\]) and the red dots are simulation results. The cases with $H/r_{o}=0.01,\,0.1,\,0.3,\,0.5,\,0.7,\,0.8,\,1.0$ are considered. \[fig:radius-of-curvature\]](Graphics_paper/R-H){width="0.8\columnwidth"}
Comparison with hydrodynamics simulation
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We now want to compare our formula of the bow shock nose with the simulation and spacecraft measurement results. Our derived formula of the bow shock location is $r_{s}=r_{o}+\Delta$ and the standoff distance is $$\frac{\Delta}{r_{o}}=\frac{1}{2}\,\alpha\,\epsilon\left(1+\sqrt{1+8\,H/r_{o}}\right),\quad M_{\infty}\gg1,\label{eq:sta_dt}$$ where $\epsilon=\frac{\gamma-1}{\gamma+1}$. In Fig. \[fig:Comparison-of-the\], we compare our formula of the standoff distance with the nonlinear gasdynamics simulation result for the bow shock profile, which is referred to the paper by Spreiter *et al.*, 1970[@spreiter1970solar].
![Comparison of the derived standoff distance formula (Eq. \[eq:sta\_dt\]) with the gasdynamics simulation results from *Spreiter et al.*, 1970[@spreiter1970solar]. The red dots are the simulation results; the black line is Eq. \[eq:sta\_dt\]. The cases with $H/r_{o}=0.01,\,0.1,\,0.2,\,0.25,\,0.5,\,0.75,\,1.0$ are considered.\[fig:Comparison-of-the\]](Graphics_paper/Comparison_with_sim){width="0.7\columnwidth"}
As we can observe in the comparison, our derived formula (Eq. \[eq:sta\_dt\]) and the simulation results match well and both show that the standoff distance becomes larger with the increasing scale heights of the ionosphere. We can conclude that the theory is validated by the simulation results. Note that this formula of the bow shock standoff distance is only applied for the solar wind interacting with the unmangetized planet and the Mach number of the solar wind solar wind must be much larger than 1. The other assumption in this theory is that the ionosphere of the unmagnetized planet is in hydrostatic condition, resulting in the thermal pressure exponentially decaying outward.
Comparison with spacecraft measurements in Martian bow shock
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We investigate the influence of the shock location from solar parameters, which control the $P_{dyn}$ (dynamic pressure of the solar wind), $P_{M,i}$ (peak pressure of the ionosphere), $H=k_{B}T/mg$ (scale height of the ionosphere). Our derived formula of the shock nose location is $$r_{s}=r_{o}+\frac{1}{2}\,r_{o}\,\alpha\,\epsilon\left(1+\sqrt{1+8\,H/r_{o}}\right),\quad M_{\infty}\gg1,\label{eq:rs}$$ where $\epsilon=\frac{\gamma-1}{\gamma+1}$ and $$r_{o}=r_{M,i}+H\ {\rm ln}(\frac{P_{M,i}}{k\,P_{dyn}}).\label{eq:oexp-1}$$
According to the spacecraft measurement[@vogt2015ionopause] (Fig. \[fig:The-response-of\]), the measurement data shows that the Martian bow shock location increases linearly with the increasing EUV flux, i.e., $$r_{s}\propto0.11\,l_{EUV},\label{eq:rspropleuv}$$ but it reduces through a power law relationship with solar wind dynamic pressure, i.e., $$r_{s}\propto p_{dyn}^{-0.02}.\label{eq:obser}$$ We can observe that our relation in Eq. \[eq:rs\] and the spacecraft measurement results in Eq. \[eq:obser\] both shows that the increasing solar EUV flux and decreasing solar wind dynamic pressure will increase the bow shock location. The increasing solar EUV flux will cause the $P_{M,i}$ (peak pressure of the ionosphere) increase via increasing the ionization rate. Furthermore, the increasing solar EUV flux will let the temperature increase, i.e., a larger scale height $H$. Since the dynamics pressure $P_{dyn}$ term is in the logarithm in our relation in Eq. \[eq:rs\], we suggest that the variation of dynamics pressure has less impact on the shock location than the EUV flux, which controls the scale height $H$ and ionospheric peak pressure $P_{M,i}$. Thus, our derived formula is qualitatively consistent with the spacecraft measurement results that the shock nose location is more sensitive to the solar EUV flux than solar wind dynamics pressure.
The power-law dependence of Martian bow shock location on solar wind dynamics pressure according to spacecraft measurement results (Eq. \[eq:obser\]) can be rewritten $$\frac{d\,r_{s}}{r_{s}}=-C\,\frac{d\,P_{dyn}}{P_{dyn}}.\label{eq:Pdyn-rs}$$ where $$C=0.02.$$
Now we derive the mathematical expression of the $C$ using our formula of the shock nose location (Eq. \[eq:rs\]) and then compare it with the spacecraft measurement results. In the Martian condition, the scale height $H$ is much shorter than the distance between the Martian ionopause nose and the Mars center $r_{o}$, so the square root term in our formula of the bow shock location (Eq. \[eq:rs\]) can be expanded, then it can be reduced to
$$\begin{aligned}
r_{s} & \simeq r_{o}+\frac{1}{2}\,r_{o}\,\alpha\,\epsilon\,(1+1+4\frac{H}{r_{o}}),\label{eq:rsexpan1}\\
& =r_{o}(1+\alpha\,\epsilon)+2\,\alpha\,\epsilon\,H,\label{eq:rsexpan2}\\
& =(1+\alpha\,\epsilon)\,r_{M,i}+(1+\alpha\,\epsilon)\,H\,{\rm ln}\left(\frac{P_{M,i}}{k\,P_{dyn}}\right)+2\,\alpha\,\epsilon\,H.\label{eq:rs-expan}\end{aligned}$$
We are interested in the bow shock location $r_{s}$dependence on $H,\;P_{M,i},\;P_{dyn}$. The total variation of the bow shock location $r_{s}$ is written as
$$d\,r_{s}=\left[\left(1+\alpha\,\epsilon\right){\rm ln}\left(\frac{P_{M,i}}{k\,P_{dyn}}\right)+2\,\alpha\,\epsilon\right]\,d\,H+\left[\left(1+\alpha\,\epsilon\right)\,\frac{H}{P_{M,i}}\right]\,d\,P_{M,i}+\left[-\left(1+\alpha\,\epsilon\right)\,\frac{H}{P_{dyn}}\right]\,d\,P_{dyn}.\label{eq:drs}$$
The bow shock location $r_{s}$ dependence on the solar wind dynamics pressure $P_{dyn}$ is
$$\frac{d\,r_{s}}{r_{s}}=-(1+\alpha\,\epsilon)\,\frac{H}{r_{s}}\,\frac{d\,P_{dyn}}{P_{dyn}},\label{eq:drs/rs}$$
where we assume the $H\;{\rm and}\;P_{M,i}$ are fixed. Thus, we get the expression of $C$ as $$C=(1+\alpha\,\epsilon)\,\frac{H}{r_{s}}.\label{eq:C}$$ $C$ is approximately the ratio between ionospheric scale length and the distance between bow shock nose location and the planet center. According to spacecraft measurement results in the Martian environment [@vogt2015ionopause], the ionospheric scale height $H$ is about 100 km and the distance between the Martian bow shock location between the Mars center is about 2.5 Mars radius (8473 km). Also, with $\alpha=0.87$ and $\epsilon=0.25$, the value of $C$ calculated from Eq. \[eq:C\] is 0.014, which is at the same order of the results from spacecraft measurement results ($C=0.02$). Thus, in terms of the bow shock location dependence on the solar wind dynamics pressure, our formula is verified by the spacecraft measurement results. This formula can be used for the future spacecraft measurement prediction.
Conclusion\[sec:Conclusion\]
============================
It is not well understood how the solar parameters, such as solar wind dynamics pressure, solar wind EUV flux and ionospheric pressure, influence the location of the Martian (unmagnetized planet) bow shock. The location of the bow shock produced from the solar wind interacting with the unmagnetized planet has been theoretically investigated. The formula for the location of the bow shock produced from the solar wind interacting with the unmagnetized planet is presented. The bow shock location is the sum of the ionopause location and standoff distance. The whole calculation is based on the gasdynamics formulation since the magnetic effect can be neglected in space environment for our purpose. We determine the ionopause nose location using thermal pressure continuity according to tangential discontinuity. The standoff distance of the bow shock produced by the supersonic plasma jet with sonic Mach number much larger than 1 past the obstacle is given in the empirical formula that the standoff distance is proportional to the radius of curvature at obstacle leading end. The formula of the shock nose position was derived and showed that the shock nose location increases with the increasing scale height of ionosphere, the decreasing dynamic pressure of the solar wind and the increasing peak pressure of the ionosphere. Furthermore, we derived the equation of the ionopause profile around the nose. Our derived theory is consistent with the numerical simulation. The derived analytical form of the radius of curvature at the obstacle leading end is verified by the numerical simulation. Also, the derived formula of the bow shock location for the unmagnetized planet is consistent with the results from the gasdynamics simulations.
Furthermore, we derived that the relation between the unmagnetized planet bow shock location and solar wind dynamics pressure, which is $r_{s}\propto p_{dyn}^{-C},$ where the constant $C$ is roughly the ratio between ionospheric scale length and the distance between bow shock nose location and the The constant $C$ calculated from the Martian parameter with our formula is at the same order as the results from the spacecraft measurements by Mars Express[@hall2016annual]. Also, our derived formula is qualitatively consistent with the spacecraft measurement results that the shock nose location is more sensitive to the solar EUV flux than solar wind dynamics pressure.
Since our model only provide the relation between bow shock location and the solar wind dynamics pressure, in order to have a more thorough comparison of our theory with the measurement results, we have to study how the solar EUV flux controls the ionospheric pressure, which is related to the photoionization from EUV radiation. On the other hand, throughout this work, we neglect the effect of the magnetic field for simplicity. In fact, Mars has some local magnetic field, which can influence the bow shock nose location. The magnetic field also influences the formation mechanisms of the bow shock and the ionopause. In the realistic condition, the interaction between the interplanetary magnetic field and the ionosphere will generate the “induced magnetosphere” and “magnetic pile-up boundary”. Furthermore, the magnetic draping effect will occur. Although the detailed study of how important the interplanetary magnetic field plays the role on controlling the bow shock location is yet to be done, our model with no magnetic field effect can now accurately predict the bow shock nose location and its relationship with the solar wind dynamics pressure, which has been verified by both numerical simulation and spacecraft measurement results.
Acknowledgment
==============
I.L.Yeh acknowledges support through MOST grant No. and the helpful discussion with Frank C.Z. Cheng and Ling-Hsiao Lyu. I.L. Yeh did all the calculation and simulation and wrote the paper draft. P.Y. Chang came up with this project and reviewed the draft. Sunny W.Y. Tam reviewed the draft and provided the insights on the comparison with the theory and spacecraft measurement results.
|
---
abstract: 'The velocity basis of the Poincaré group is used in the direct product space of two irreducible unitary representations of the Poincaré group. The velocity basis with total angular momentum $j$ will be used for the definition of relativistic Gamow vectors.'
address:
- |
The University of Texas at Austin\
Austin, Texas 78712\
[email protected]
- |
The University of Texas at Austin\
Austin, Texas 78712\
[email protected]
author:
- 'A. Bohm'
- 'H. Kaldass'
title: Relativistic Partial Wave Analysis Using the Velocity Basis of the Poincaré Group
---
Introduction
============
Resonances are obtained in the scattering of two (or more) elementary particles, and quasistationary states decay into a two (or many) particle system with masses $m_{i}$ and spins $s_{i}$, $i=1,\,2\cdots$. Relativistic resonances and decaying states are therefore described in the direct product space of two irreducible representation spaces of the Poincaré group ${\cal H}={\cal H}_{1}(m_{1},s_{1})\otimes{\cal H}_{2}(m_{2},s_{2})$. Non-relativistic resonances and decaying states have been described by Gamow vectors [@our]. Gamow vectors are characterized by a value of angular momentum $j$ in the center-of-mass frame and by a complex energy $z_{R}=\left(E_{R}-i\frac{\Gamma}{2}\right)$, representing resonance energy $E_{R}$ and lifetime $\frac{\hbar}{\Gamma}$. They are generalized eigenvectors in a Rigged Hilbert Space $\Phi\subset{{\cal H}}\subset\Phi^{\times}$ of the self-adjoint Hamiltonian $H$ with complex eigenvalue $z_{R}$ [@our]. Relativistic resonances and unstable particles are characterized by their spin (total angular momentum in the center-of-mass frame of the decay products) and the value ${{\mathsf{s}}}={{\mathsf{s}}}_{R}
\equiv \left(M_{R}-i\frac{\Gamma}{2}\right)^{2}$ of the invariant mass squared ${{\mathsf{s}}}=(p_{1}+p_{2})^{2}=(E^{2}-{\bbox}{p}^{2})$ where $M_{R}$ is the resonance mass and $\frac{\hbar}{\Gamma_{R}}$ is its lifetime. We want to find relativistic Gamow vectors which are generalized eigenvectors of the total mass operator $M^{2}=P_{\mu}P^{\mu}=(P_{1\mu}+P_{2\mu})(P_{1}^{\mu}+P_{2}^{\mu})$ with complex eigenvalue ${{\mathsf{s}}}_{R}$ and with spin $j$. These must be obtained from the direct product space ${\cal H}_{1}(m_{1},s_{1})\otimes{\cal H}_{2}(m_{2},s_{2})$.
Eigenspaces of $M^{2}$ with real values of invariant mass ${{\mathsf{s}}}$ and total angular momentum $j$ are obtained by the relativistic partial wave analysis [@aW60; @hJ62; @aM62] using the Wigner basis, i.e., using momentum eigenvectors $|{\bbox}{p}_{i},s_{3i}(m_{i},s_{i}){\rangle}$ in the spaces ${\cal H}_{i}$ and eigenvectors $|{\bbox}{p},j_{3}({{\mathsf{s}}},j){\rangle}$ of $P_{\mu}=P_{1\mu}+P_{2\mu}$ in the direct product space $\cal H$.
In distinction to the non-relativistic case, in the relativistic case Lorentz transformations intermingle energy and momenta. If one wants to make an analytic continuation of ${{\mathsf{s}}}$ from the values $(m_{1}+m_{2})^{2}\leq {{\mathsf{s}}}<\infty$ to the complex values ${{\mathsf{s}}}_{R}$ (of the pole position in the second sheet of the relativistic $S$-matrix $S_{j}({{\mathsf{s}}})$) this will also lead to complex momenta. To restrict the unwieldy set of complex momentum representations [@cm] we want to construct complex mass representations of the Poincaré group $\cal P$ whose momenta are “minimally complex” in the sense that though $p_{\mu}$ and $m$ are complex, the $4$-velocities ${\hat{p}}_{\mu}\equiv \frac{p_{\mu}}{m}$ remain real. This can be carried out because, as explained in section $2$, the $4$-velocity eigenvectors $|{\bbox}{{\hat{p}}},j_{3}({{\mathsf{s}}},j){\rangle}$ provide as valid basis vectors for the representation space of $\cal P$ as the usual momentum eigenvectors. Moreover, they are more useful for physical reasoning than the momenta eigenvectors, because the $4$-velocities seem to fulfill to rather good approximation “velocity super-selection rules” which the momenta do not [@velocityvectors]. Therefore we will use the velocity basis $|{\bbox}{{\hat{p}}}_{i},s_{3i}(m_{i},s_{i}){\rangle}$ for the relativistic partial wave analysis and obtain the Clebsch-Gordan coefficients of the Poincaré group for the velocity basis. This is done in section $3$ for $s_{1}=s_{2}=0$, which applies to the case of $\pi^{+}\pi^{-}$ in the final state. This gives the velocity eigenvectors $|{\bbox}{{\hat{p}}},j_{3} ({{\mathsf{s}}},j){\rangle}$ of the direct product space ${\cal H}=\sum_{j=0}^{\infty}
\int_{(m_{1}+m_{2})^{2}}^{\infty}d\mu({{\mathsf{s}}}){\cal H}({{\mathsf{s}}},j)$ from which we obtain the four-velocity scattering states $|{\bbox}{{\hat{p}}},j_{3} ({{\mathsf{s}}},j)^{\pm}{\rangle}$ using the Lippmann-Schwinger equation as e.g., done in [@sW95]. The relativistic Gamow vectors $|{\bbox}{{\hat{p}}},j_{3}({{\mathsf{s}}}_{R},j)^{\pm}{\rangle}$ will be obtained in a subsequent paper from the scattering states by analytic continuation. In the Appendix, we derive the Clebsch-Gordan coefficients for the velocity basis of $\cal P$ for the general case.
Velocity Basis of the Poincaré Group
====================================
We denote the ten generators of the unitary representation ${\cal U} (a, \Lambda)$ of $(a, \Lambda)\in \cal{P}$, by $$\label{generators}
P^{\mu},\, J^{\mu\nu}{\hspace{1cm}} \mu,\nu\,=\, 0,1,2,3\, .$$ The standard choice of the invariant operators and of a complete set of commuting observables (c.s.c.o.) is $$\begin{aligned}
\nonumber
M^{2}=P_{\mu}P^{\mu}\,&,&\quad W=-w_{\mu}\,w^{\mu},\\
\label{csco}
P_{i}\,(i=1,2,3)\,&,&\quad S_{3}=M^{-1}
{\cal U}(L(p))\,w_{3}\,{\cal U}^{-1}(L(p))\, ,\end{aligned}$$ here $$\label{w}
w_{\mu}=\frac{1}{2}\,\epsilon_{\mu \nu \rho \sigma}\, P^{\nu}
J^{\rho \sigma}\, ,$$ $M^{-1}$ is the inverse square root of the positive definite operator $P^{\mu}P_{\mu}$, and ${\cal U}(L(p))$ is the representation of the boost that depends upon the parameters $p_{\mu}\,(\mu=0,1,2,3)$, which are the eigenvalues of the operators $P_{\mu}$. Only three of these parameters are independent in an irreducible representation, because of the relation $m^{2}=p_{\mu}p^{\mu}$. The standard boost (“rotation free”) matrix $L^{\mu}_{.\, \nu}(p)$ is given by $$\label{standardboost}
L^{\mu}_{.\,\nu}(p)=
\bordermatrix{ &\nu=0 &\nu=n \cr
\mu=0&\frac{p^{0}}{m} &-\frac{p_{n}}{m} \cr
\mu=m&\frac{p^{m}}{m} &\delta^{m}_{n}-
\displaystyle{\frac{\frac{p^{m}}{m}\,
\frac{p_{n}}{m}}
{1+\frac{p^{0}}{m}}}\cr }\, .$$ Note that $p_{\mu}=\eta_{\mu\,\nu}p^{\nu}$ and we use the metric $\eta_{\mu\,\nu}=\scriptstyle{\left(
\begin{array}{cccc}
1 & & & 0 \\
&-1 & & \\
& & -1 & \\
0 & & & -1
\end{array}\right)}\, $ [^1]. It has the property that $$\label{standardmomentum}
L^{-1}(p)^{\mu}_{.\,\nu}p^{\nu}=
\left(\begin{array}{c}
m\\
0\\
0\\
0
\end{array}\right)\, .$$ One feature shown in (\[standardboost\]) which we want to make use of, is that the boost $L^{\mu}_{.\,\nu}(p)$ does not depend upon $p$ but only upon the 4-velocity $\frac{p}{m}\equiv {\hat{p}}$. The complete basis system in the irreducible representation space ${\cal H}(m^{2},j)$ which consists of eigenvectors of the c.s.c.o. (\[csco\]) is the Wigner basis usually denoted as $$\label{wignerbasis}
|{\bbox}{p},j_{3}(m,j)\rangle \, .$$ It has the transformation property under the translation $(a,{\rm I})$ and the Lorentz transformation $(0,\Lambda)$ :
\[poincaretransformations\] $$\label{translation}
{\cal U}(a,{\rm I})|{{\bbox}p},j_{3}\rangle=
e^{ip^{\mu}a_{\mu}}|{{\bbox}p},j_{3}
\rangle$$ $$\label{lorentztransformation}
{\cal U}(0,\Lambda)|{{\bbox}p},\xi\rangle=\sum_{\xi^{'}}|
{\bbox}{\Lambda p},
\xi^{'}\rangle D_{\xi^{'}\xi}({\cal R}(\Lambda,p))\, ,$$ where $\cal{R}$ is the Wigner rotation $$\label{wignerrotation}
{\cal R}(\Lambda,p)=L^{-1}(\Lambda p)\Lambda L(p)\, .$$ The Wigner rotation depends upon the $10$ parameters of $\Lambda$ and upon the parameters ${\hat{p}}^{\mu}=\frac{p^{\mu}}{m}\, .$ In an UIR there are $3$ independent ${\hat{p}}^{\mu}$ and : $$\label{boost}
|p,j_{3}\rangle={\cal U}(L(p))|{{\bbox}p}={{\bbox}0},
j_{3}\rangle \, ,$$
where we have omitted the fixed values $m\, j$ as we shall often do in an UIR. Every vector (of a dense subspace of physical states) of ${\cal H} (m,j)$ can be written according to Dirac’s basis vector decomposition as
\[diracbasisvectors\] $$\label{dirac}
\phi=\int \, d\mu\left({\bbox}{p}\right)\sum_{\xi}|{\bbox}{p},\xi
\rangle
\langle {{\bbox}p},\xi\,|\,\phi\rangle \, ,$$ where one has many arbitrary choices for the measure. It is usually chosen to be given by $$\label{measure}
d\mu\left({\bbox}{p}\right)=\rho\left({\bbox}{p}\right)d^{3}
{\bbox}{p}\, ,$$ where one can choose any (measurable) function $\rho$, in particular a smooth function. The choice of $\rho$ is connected to the “normalization” of the Dirac kets through : $$\label{normalization}
\langle \xi^{'},{\bbox}{p'}\,|\,{\bbox}{p},\xi \rangle
=\frac{1}{\rho\left({\bbox}{p}\right)}\, \delta^{3}
({\bbox}{p}-{{\bbox}p'})\,
\delta_{\xi \xi'} \, .$$ One convention[^2] for $\rho$ is the Lorentz invariant measure : $$\label{invariantmeasure}
\rho\left({\bbox}{p}\right)=\frac{1}{2E({\bbox}{p})}
\, ,\quad\text{where }
E\left({\bbox}{p}\right)=\sqrt{m^{2}+{{\bbox}{p}}^{\,2}}\, .$$
The mathematically precise form of the Dirac decomposition is the Nuclear Spectral Theorem for the complete system of commuting (essentially self-adjoint) operators. It is the same as (\[diracbasisvectors\]), however with well defined mathematical quantities. The state vectors $\phi$ in (\[dirac\]) must be elements of a dense subspace $\Phi$ of the representation space ${{\cal H}}$ of an UIR : $$\phi \in \Phi \subset {{\cal H}}(m,j)\,;$$ and the basis vectors $|{\bbox}{p},\xi\rangle \in \Phi^{\times}$ are elements of the space of antilinear functionals on $\Phi$ which fulfill the condition :
$$\label{generalizedeigenvector}
\langle P_{i} \psi \,|\, {\bbox}{p}, \xi \rangle=p_{i}\,
\langle \psi \,|\,
{\bbox}{p}, \xi
\rangle \quad\text{for every }\psi \in \Psi\, .$$
This condition means the $|{\bbox}{p},\xi{\rangle}$ are generalized eigenvectors of $P_{i}$, which is also written as $$P_{i}^{\times}|{\bbox}{p},\xi{\rangle}=p_{i}\,|{\bbox}{p},\xi{\rangle}\, ,$$
where $P_{i}^{\times}$ is an extension of $P_{i}^{\dagger}(=P_{i})$; and the “component of $\phi$ along the basis vector $|{\bbox}{p},\xi\rangle$”, the $\langle {\bbox}{p},\xi\,|\,\phi \rangle={\langle}\phi\,|\,{\bbox}{p},\xi{\rangle}^{*}
$, are antilinear continuous functionals $F(\phi)=\langle {\bbox}{p},\xi|\phi\rangle^{*}$ on the space $\Phi$.
The space $\Phi$ is a dense nuclear subspace of $\cal{H}$ [@aB73]. (E.g., $\Phi$ could be chosen to be the subspace of differentiable vectors of $\cal{H}$ equipped with a nuclear topology defined by the countable number of norms : $||\phi||_{p}=\sqrt{(\phi,(\Delta+1)^{p}\phi)}$, where $\Delta=\sum_{\mu} P_{\mu}^{2}+\sum_{\mu \, \nu}\frac{1}{2}J^{2}_{\mu \, \nu}
$ is the Nelson operator [@eN59]. But it could also be chosen as another dense nuclear subspace of $\cal{H}$.) The three spaces form a Gel’fand triplet, or Rigged Hilbert Space $$\label{RHS}
\Phi \, \subset \, {\cal H} \, \subset \, \Phi^{\times}$$ and the bra-ket $< \, | \, >$ is an extension of the scalar product $( \, , \, )$. The $\langle {\bbox}{p}, \xi\,|\,\phi \rangle
={\langle}\phi\,|\,{\bbox}{p}, \xi{\rangle}^{*}$ are the Wigner momentum wavefunctions.
The Wigner kets (\[wignerbasis\]) are not the only basis system of ${\cal H}(m,j)$ that one can use to expand every vector $\phi \in \Phi$. For every different choice of c.s.c.o.in the enveloping algebra $\cal{E}({\cal P})$ (the algebra generated by $P_{\mu}$, $J_{\mu \, \nu}$) one obtains a different system of basis vectors; in this way one can obtain e.g., Lorentz basis (eigenvectors of the Casimir operators of $SO(3,1)_{J_{\mu \nu}}$ [@hJ62; @aB73]), or the spinor basis (whose Fourier transforms are the relativistic fields [@sW95]) etc. We want to choose still another basis system, which is similar to the Wigner basis except that it is a basis of eigenvectors of the $4$-velocity operator $\hat{P}_{\mu}\equiv P_{\mu}M^{-1}$ rather than the momentum operator $P_{\mu}$.
With the 4-velocity operator, one defines the operators $$\label{what}
\hat{w}_{\mu}=\frac{1}{2}\,\epsilon_{\mu \nu \rho \sigma}
\hat{P}^{\nu}J^{\rho \sigma}=w_{\mu}M^{-1} \, ,$$ and the spin tensor $$\Sigma_{\mu \, \nu}=\epsilon_{\mu \nu \rho \sigma}\hat{P}^{\rho}
\hat{w}^{\sigma}\,.$$ The c.s.c.o. is then given by $$\label{cscohat}
\hat{P}_{m},\quad S_{3},\quad \hat{W}=-\hat{w}_{\mu} \hat{w}^{\mu}
=\frac{1}{2}\Sigma_{\mu \nu}\Sigma^{\mu \nu},\quad M^{2}\, ,$$ and we denote its generalized eigenvectors by $$\label{wignerbasishat}
|{\bbox}{{\hat{p}}},j_{3};{{\mathsf{s}}}=m^{2},j{\rangle}\, ,$$ where ${\hat{p}}_{\mu}=\frac{p_{\mu}}{m}$ are the eigenvalues of $\hat{P}_{\mu}$.
The basis vector expansion for every $\phi \in \Phi$ with respect to the basis system (\[wignerbasishat\]) is given by
\[diracbasisvectorhat\] $$\label{dirachat}
\phi=\sum_{j_{3}}\int \, \frac{d^{3}{\hat{p}}}{2 {\hat{p}}^{0}}
\,|{\bbox}{{\hat{p}}}, j_{3}\rangle \langle j_{3}, {\bbox}{{\hat{p}}}
\,|\, \phi \rangle \, ,$$ where we have chosen the invariant measure $$\begin{aligned}
\label{measurehat}
d\mu({\bbox}{{\hat{p}}}) &=& \frac{d^{3}{\hat{p}}}{2{\hat{p}}^{0}}
= {\frac{1}{m^{2}}} \, {\frac{d^{3}p}{2 E({\bbox}{p})}}\\ \nonumber {\hat{p}}^{0} &=& \sqrt{1+{\bbox}{{\hat{p}}}^{2}} \, .\end{aligned}$$ As a consequence of (\[measurehat\]), the $\delta$-function normalization of these velocity-basis vectors is $$\begin{aligned}
\nonumber
\langle \xi , {\bbox}{{\hat{p}}}\,|\,{\bbox}{{\hat{p}}'}, \xi' \rangle
&=& 2 {\hat{p}}^{0} \delta^{3}({\bbox}{{\hat{p}}}-{\bbox}{{\hat{p}}'})
\, \delta_{\xi \xi'}\\
\label{normalizationhat}
&=& 2 p^{0} m^{2} \delta^{3}
({\bbox}{p}-{\bbox}{p'})\,
\delta_{\xi \xi'} \, .\end{aligned}$$
Mathematically, every c.s.c.o. is equally valid. But, for a given physical problem one c.s.c.o. may be more useful than another. For instance a c.s.c.o. that contains physically distinguished observables (e.g., observables whose eigenstates happen to appear predominantly in nature) is more useful for calculations in physics than the c.s.c.o. whose eigenvectors are very different from physical eigenstates. Two different c.s.c.o.’s lead to different basis systems, whose vectors can be expanded with respect to each other. But this expansion is usually very complicated and intractable, for which reason the choice of the physically right c.s.c.o. is very important for each particular physical problem. This is the reason for which the Lorentz basis of the Poincaré group is pretty useless for physics, because the Casimir operators of $SO(3,1)$ are not important observables as compared to the momentum. However, the two c.s.c.o. (\[csco\]) and (\[cscohat\]) are not even different in an irreducible representation of ${\cal P}$, since its operators differ only by a factor of the operator $M$, which is an invariant. The basis systems (\[wignerbasis\]) and (\[wignerbasishat\]) are therefore the same, i.e., their values differ by a normalization-phase factor $N(p,j_{3})$ $$\label{basisrelation}
|\,{\bbox}{{\hat{p}}}, j_{3} \, (m,j)\rangle
=|\,{{\bbox}p},j_{3} \, (m,j) \rangle \, N(p, j_{3}).$$ The Poincaré transformations (\[poincaretransformations\]) act on the basis vectors (\[basisrelation\]) in the following way
\[poicaretranformationshat\] $$\begin{aligned}
&{\cal U}(a,{\rm I})|{\bbox}{{\hat{p}}},j_{3}\rangle=
e^{im{\hat{p}}^{\mu}a_{\mu}}|{\bbox}{{\hat{p}}}, j_{3}\rangle \label{translationhat}\\
&{\cal U}(L({\hat{p}}))|{\bbox}{{\hat{p}}}={\bbox}{0},j_{3}
\rangle=|{\bbox}{{\hat{p}}},j_{3}\rangle \, .
\label{boosthat}\end{aligned}$$
The distinction between the basis vectors $|\,{\bbox}{p}, \xi \rangle$ and $|\,{\bbox}{{\hat{p}}}, \xi \rangle$ becomes important if one does not have an unitary irreducible representation of $\cal{P}$ but a representation with many different values for $(m^{2},j)$, e.g., ${\cal H}=\sum_{m^{2},j}
\oplus {\cal H}(m,j)$. Then one has besides the observables (\[generators\]), additional observables $X_{\alpha}$ (generators of an intrinsic symmetry group or a spectrum generating group) and an additional system of commuting observables : $$\label{commutingobservables}
B=B_{1},B_{2}, \cdots , B_{N}$$ whose eigenvalues, $b=(b_{1},b_{2}, \cdots ,b_{N})$, characterize the elementary particles described by ${\cal H}(m,j)={\cal H}^{b}(m,j)$ [^3]. In order that (\[csco\]) and (\[commutingobservables\]) combine into a c.s.c.o., the operators $B$ have to commute with $M^{2}, P_{\mu}, W \text{ and }S_{3}$. If also the other observables $X_{\alpha}$, which change the particle species number $b$, commute with $M^{2}, P_{\mu}, W \text{ and }S_{3}$, then the combination of (\[csco\]) and (\[commutingobservables\]) gives a useful c.s.c.o. However, if the $X_{\alpha}$ do not commute with $M^{2}$ (i.e., the particle species number changing operators $X_{\alpha}$ transform also from one mass eigenstate to another mass eigenstate changing also the mass $m_{b}$ into $m_{b'}$) then the $X_{\alpha}$ will also not commute with $P_{\mu}$, $\left[ X_{\alpha}, P_{\mu}\right]
\neq 0$. In this case, it may still happen [@velocityvectors] that a “velocity superselection rule” holds : $$\label{velocitysuperselectionrule}
\left[ X_{\alpha}, \hat{P}_{\mu} \right] =0\quad (\text{or at least }
\left[ X_{\alpha},\hat{P}_{\mu} \right]\approx 0)\, .$$ Then combination of (\[commutingobservables\]) with (\[cscohat\]), i.e., the $$\label{cscohat2}
\hat{P}_{i},\, \hat{w}_{3},\, \hat{W},\, M^{2},\, B_{1},\cdots,\,B_{N}$$ will form a useful c.s.c.o., but the combination of (\[csco\]) with (\[commutingobservables\]) will not. The generalized eigenvectors of (\[cscohat2\]), $|{\bbox}{{\hat{p}}},\xi,b,m,j
\rangle $, will then be a much more useful basis system for every $\phi \in \Phi \subset {\cal H}=\sum \oplus {\cal H}^{b}(m,j)$ than the corresponding momentum eigenvectors. Using the eigenvectors of (\[cscohat2\]), we have the Dirac basis vector expansion : $$\label{dirachat2}
\phi=\sum_{m,b}\sum_{j,\xi}\int \frac{d^{3}{\hat{p}}}{2\, {\hat{p}}^{0}}
|\,{\bbox}{{\hat{p}}},\xi,b,m,j\rangle
\langle j,m,b,\xi, {\bbox}{{\hat{p}}}\,|\,
\phi \rangle \text{ for every } \phi \in \Phi\, .$$ The momentum eigenvectors $|{\bbox}{p},\xi,b \ldots \rangle$ may either not exist (if $\left[ B, P_{\mu} \right] \not= 0$), or if they do exist, they are not useful because the $X_{\alpha}$ change the value of $p$, which then becomes a function of $b$, $p=p_{b}$. As a consequence, quantities like form factors depend upon $b$ through $p$. In contrast, using the velocity eigenvectors $|{\bbox}{{\hat{p}}},\xi,b,\cdots\rangle$ under the assumption (\[velocitysuperselectionrule\]) will lead to form factors with universal (independent of $b$) dependence upon the four-velocity. This was the original motivation for the introduction of the velocity-basis vectors $|{\bbox}{{\hat{p}}},\xi,b,\cdots\rangle$ [@velocityvectors].
The subject of the present work is the description of relativistic decaying states by representations of the Poincaré group, combining Wigner’s idea [@eW39] of the description of stable relativistic particles by an UIR of $\cal P$, with Gamow’s idea of describing decaying particles by eigenvectors with complex energy. Therefore, we need in the rest frame basis vectors with complex energy, i.e., the $m$ (and the ${{\mathsf{s}}}=m^{2}$) in (\[wignerbasis\]) or in (\[wignerbasishat\]) has to be continued to complex values e.g., to ${{\mathsf{s}}}=(M_{R}-i\Gamma/2)^{2}$. This will result in a continuation of the momenta $p_{\mu}$ to complex values as well and can lead to an enormous complication of the Poincaré group representations (see e.g., [@cm]). We want to do this analytic continuation in the invariant mass ${{\mathsf{s}}}$ such that the $p_{\mu}$ are continued to complex values in such a way that the ${\hat{p}}_{\mu}=\frac{p_{\mu}}{\sqrt{{{\mathsf{s}}}}}$ remain real. Then, we obtain a smaller class of complex mass representations of $\cal P$ which are as similar in property as possible to Wigner’s UIR $(m,j)$. These are the minimally complex-mass representations which we shall denote by $({{\mathsf{s}}},j)$.
For this minimal analytic continuation to be possible, it must be compatible with the boost (\[boost\]) and (\[boosthat\]). The crucial observation is that the boosts $L(p)$ are in fact, according to (\[standardboost\]) only functions of ${\hat{p}}_{\mu}=\frac{p_{\mu}}{\sqrt{{{\mathsf{s}}}}}\,$; $L(p)=L({\hat{p}})$. As a consequence, the operators representing the boost ${\cal U}(L(p))={\cal U}(L({\hat{p}}))$ are functions of the real parameters ${\hat{p}}$ and not of complex parameters $p$. This means they are the same operator functions in all the subspaces of the direct sum $\sum_{m_{b},j}\oplus{\cal H}(m_{b},j)$ and of the continuous direct sum $$\label{directsum}
\sum_{j,n}\int_{m_{0}^{2}}^{m_{1}^{2}}\oplus
{\cal H}^{n}({{\mathsf{s}}},j)d\mu({{\mathsf{s}}})$$ of the irreducible representations $$\label{irrep}
{\cal H}({{\mathsf{s}}},j),\qquad
{{\mathsf{s}}}=p_{\mu}p^{\mu}=E-{\bbox}{p}^{2}\,.$$ If we consider in (\[directsum\]) only (continuous) direct sums with the same value for $j=j_{R}$ then ${\cal U}(\Lambda)$ for any Lorentz transformation $\Lambda$ is, according to (\[wignerrotation\]), the same operator function of the $6$ parameters which are given by the three ${\hat{p}}^{m}$ or the three $v^{m}$ : $$\label{uandv}
\left(
\begin{array}{c}
{\hat{p}}^{0} \\
{\hat{p}}^{m}
\end{array}
\right)=\left(
\begin{array}{c}
\left( 1-\frac{{\bbox}{v}^{2}}{c^{2}} \right)^{-\frac{1}{2}} \\
\left( 1-\frac{{\bbox}{v}^{2}}{c^{2}} \right)^{-\frac{1}{2}}v^{m}
\end{array}\right)$$ and the three rotation angles (e.g., Euler angles in the rest frame). The analytic continuation in ${{\mathsf{s}}}$ can therefore be accomplished without affecting the Lorentz transformations. The Lorentz transformations in the minimally-complex mass representation are represented unitarily by the same operators ${\cal U}(\Lambda)$ as in Wigner’s UIR $(m,j_{R})$. At rest, on $|{\bbox}{0},j_{3}\, ({{\mathsf{s}}},j_{R}){\rangle}$, only the time translations of $\cal P$ will be represented non-unitarily for complex values of ${{\mathsf{s}}}$. And using (\[boosthat\]) only the label ${{\mathsf{s}}}$ in the velocity basis $|{\bbox}{{\hat{p}}},j_{3}\, ({{\mathsf{s}}},j_{R}){\rangle}$ is complex. The basis vector decomposition (\[dirachat2\]) using the velocity basis, $$\label{dirachat3}
\phi= \sum_{j_{3}}\int d\mu({{\mathsf{s}}})\int d\mu ({\bbox}{{\hat{p}}})
|\,{\bbox}{{\hat{p}}},j_{3}({{\mathsf{s}}},j)\rangle
\langle ({{\mathsf{s}}},j)j_{3},{\bbox}{{\hat{p}}}\,|\,
\phi\rangle
\quad\text{for } \phi \in \Phi \subset {\cal H}({{\mathsf{s}}},j)\, ,$$ is therefore more suitable than (\[diracbasisvectors\]) that uses the momentum basis, because ${\bbox}{{\hat{p}}}$ is independent of ${{\mathsf{s}}}$ while ${\bbox}{p}=\sqrt{{{\mathsf{s}}}}{\bbox}{{\hat{p}}}$ is not. If we deform the contour of integration for ${{\mathsf{s}}}$ from the real axis as in (\[directsum\]) into the complex ${{\mathsf{s}}}$-plane then the integral over $d\mu({\bbox}{{\hat{p}}})$ in (\[dirachat3\]) remains unaffected.
Relativistic Kinematics for (two-particle) Resonance Scattering
===============================================================
Continuous direct sums like (\[directsum\]) appear in the case of scattering experiments of two relativistic particles like e.g., the process
\[resonancescattering\] $$\label{twoelectronsresonance}
e^{+}\, e^{-} \rightarrow \rho^{0} \rightarrow
\pi^{+}\, \pi^{-} \, ,$$ or the more theoretical process $$\label{twopionsresonance}
\pi^{+}\,\pi^{-} \rightarrow \rho^{0} \rightarrow
\pi^{+}\, \pi^{-} \, .$$
These processes predominantly happen in the $j^{P}=1^{-}$ partial amplitude if the $\rho$-meson mass region is selected for the invariant mass square $$\label{invariantmass}
{{\mathsf{s}}}=(p_{1}+p_{2})^{2}=E_{\rho}^{2}+{\bbox}{p}_{\rho}^{2}\, ,\,\,
E_{\rho}=E_{1}+E_{2}\, ,\,\,
{\bbox}{p}_{\rho}={\bbox}{p_{1}}+{\bbox}{p_{2}}\, ,$$ where $p_{1}$ and $p_{2}$ are the momenta of the two pions $\pi^{+}$,$\pi^{-}$ [^4]. The relativistic one particle states are given by an irreducible representation space ${\cal H}^{n_{i}}(m_{i},s_{i})$ of the Poincaré group $\cal{P}$. The independent, *interaction-free* two-particle states (or $n$ particle states)— like the $\pi^{+}\, \pi^{-}$ system in (\[twopionsresonance\])— are given by the direct product of the irreducible representation spaces ${\cal H}(m_{1},s_{1})$ and ${\cal H}(m_{2},s_{2})$ : ${\cal H}^{n_{1}}(m_{1},s_{1})
\otimes{\cal H}^{n_{2}}(m_{2},s_{2})\equiv {\cal H}$. Empirical evidence suggests that the resonances in processes like (\[resonancescattering\]) appear in one partial amplitude with a given value of resonance spin $j_{R}$ (e.g., $j_{\rho}^{P}=1^{-}$). Therefore, the first problem is the reduction of the direct product ${\cal H}(m_{1},s_{1})\otimes{\cal H}(m_{2},s_{2})$ into a direct sum of ${\cal H}^{n}({{\mathsf{s}}},j)$; the second problem is how to go from the free two-particle system to the interacting two-particle system.
The first problem has been solved in general [@aW60; @hJ62; @aM62] $$\label{reduction}
{\cal H}\equiv {\cal H}^{n_{1}}(m_{1},s_{1})\otimes
{\cal H}^{n_{2}}(m_{2},s_{2})
=\int_{(m_{1}+m_{2})^{2}}^{\infty}d\mu({{\mathsf{s}}})\sum_{nsl}\sum_{j}
\oplus{\cal H}^{nsl}({{\mathsf{s}}},j)\,.$$ The sums in (\[reduction\]) extend over $$j=
\begin{array}{ccccl}
0 & 1 & \cdots & \text{ if } & s_{1}+s_{2}=\text{ integer }\\
1/2 & 3/2 & \ldots & \text{ if } & s_{1}+s_{2}=\text{ half integer }
\end{array}\, ,$$ and the degeneracy indices $(l,s)$ for a given $j$ are summed over $$\begin{aligned}
\nonumber
s&=&s_{1}+s_{2}\, ,\, s_{1}+s_{2}-1\, ,\, \ldots |s_{1}-s_{2}|\\
\nonumber l&=&j+s\, ,\, j+s-1\, ,\, j+s-2\, ,\, \ldots j-s \, .\end{aligned}$$ Here $j$ represents the total angular momentum of the combined $\pi^{+}\pi^{-}$ system; one of these values will be the resonance spin $j_{R}$. The degeneracy indices $(s,l)$ for each fixed value of $j$ are the total spin angular momentum and the total orbital angular momentum of the two $\pi$, respectively. The quantum number $n$ is summed over all channel numbers that can be obtained by combining the species numbers $n_{1}$ and $n_{2}$ of the two $\pi$.
Instead of the invariant mass square ${{\mathsf{s}}}=p_{\mu}\,p^{\mu}=E^{2}-
{\bbox}{p}^{2}$ that we have used in (\[reduction\]) one often uses $w=\sqrt{{{\mathsf{s}}}}$, the invariant mass or the energy in the center of mass system of the two particles $n_{1},
n_{2}$ [@aW60; @hJ62; @aM62]. The choice of the measure $$\eqnum{\ref{reduction}$a$}
d\mu({{\mathsf{s}}})=\rho({{\mathsf{s}}})d{{\mathsf{s}}},\quad (\text{or if one uses $w$, of }
d\mu(w)=\rho(w)dw)$$ depends upon the normalization of the system of generalized basis vectors of (\[reduction\]). We shall use $$\label{choice}
\eqnum{\ref{reduction}$b$}
\rho({{\mathsf{s}}})=1,\text{ and then } \rho(w)=2w$$ if we label the basis by $w$ so that we do not change the “normalization” of the kets. The resonance space will be related (but will not be identical) to a subspace of (\[reduction\]) with a definite value of angular momentum $j$ (e.g., $j=j_{3}^{P}=1^{-}$ in case of the $\rho$-resonance of (\[resonancescattering\])). This is based on empirical evidence; resonances appear in one particular partial amplitude with a particular value of resonance spin $j=j_{R}$ (though it may happen that there are more than one resonance in the same partial amplitude, but at different resonance energy ${{\mathsf{s}}}_{R_{1}},\,{{\mathsf{s}}}_{R_{2}},\,\cdots$). We will therefore single out a particular subspace $$\label{subspace}
{\cal H}^{nls}=\int_{(m_{1}+m_{2})^{2}}^{\infty}
d{{\mathsf{s}}}\oplus {\cal H}^{nls}({{\mathsf{s}}},j)$$ with definite degeneracy or/and channel quantum numbers $\eta=ls,$ $n$.
The reduction (\[reduction\]) is usually done using the Wigner momentum kets (\[wignerbasis\]) in which the Clebsch-Gordan coefficients are given by [@aW60; @hJ62; @aM62] : $$\label{cg}
\langle\, p_{1}s_{13}\,p_{2}s_{23}\,[m_{1}s_{1}, m_{2}s_{2}]\,|\,
pj_{3}\,[wj],\eta\,\rangle,
\text{ where $\eta$ now denotes }\eta=n,l,s\,.$$ For the reasons mentioned above we want to work with the 4-velocity eigenkets $|\hat{p},j_{3}\,[w,j],\eta\,\rangle$ which are eigenvectors of the operators $$\label{operators}
\hat{P}_{\mu}=({P}^{(1)}_{\mu}+{P}^{(2)}_{\mu})M^{-1},\,\,
M^{2}=(P^{(1)}_{\mu}+P^{(2)}_{\mu})
(P^{(1)\mu}+P^{(2)\mu})$$ with eigenvalues $$\label{eigenvalues}
\hat{p}^{\mu}=
\left(
\begin{array}{c}
\hat{E}=\frac{p^{0}}{w}=\sqrt{1+{\bbox}{\hat{p}}^{2}}={\hat{p}}^{0}\\
{\bbox}{\hat{p}}=\frac{{\bbox}{p}}{w}
\end{array} \right)
\text{ and eigenvalues }
w^{2}={{\mathsf{s}}}\, .$$ In here $\hat{P}^{(i)}_{\mu}$ are the 4-velocity operators in the one particle spaces ${\cal H}^{n_{i}}(m_{i},s_{i})$ with eigenvalues $\hat{p}^{i}_{\mu}=\frac{p^{i}_{\mu}}{m_{i}}$. The Clebsch-Gordan coefficients are the transition coefficients $\langle\hat{p}_{1}\hat{p}_{2}\,s_{13}s_{23}\,[m_{1}s_{1},m_{2}s_{2}]\,
|\,\hat{p}j_{3}\,[wj],\eta\,\rangle$ between the direct product basis $$\label{directproductbasis}
|\hat{p}_{1}s_{13}\,m_{1}s_{1}\rangle\otimes
|\hat{p}_{2}s_{23}\,m_{2}s_{2}\rangle
\equiv
|\hat{p}_{1}\hat{p}_{2}\,s_{13}s_{23}\,[m_{1}s_{1},m_{2}s_{2}]\,\rangle$$ and the angular momentum basis $|{\hat{p}}j_{3}\,[wj],\eta\,{\rangle}$.
To obtain the Clebsch-Gordan coefficients, one follows the same procedure as given in the classic papers [@aW60; @hJ62; @aM62] for the Clebsch-Gordan coefficients (\[cg\]). This will be done in the Appendix, where the general case will be discussed. Here we shall restrict ourselves to the special case $s_{1}=0,s_{2}=0$ to avoid the inessential complications due to the $SO(3)$ Clebsch-Gordan coefficients for the angular momentum couplings $s_{1}\otimes s_{2}\rightarrow s$, $s\otimes l \rightarrow j$ and the occurrence of the Wigner rotations $R(L^{-1}(\hat{p}),{\hat{p}}_{i})$ of the inverse boost $L^{-1}(\hat{p})$ which will enter in (\[cg\]). Also for the process (\[twopionsresonance\]) this is sufficient, since $s_{{\pi}^{+}}=s_{{\pi}^{-}}=0$. There is no degeneracy of the angular momentum basis vectors in this case and $|{\hat{p}}j_{3}\,[wj]\,{\rangle}$ is given in terms of (\[directproductbasis\]) by $$\begin{aligned}
\label{expansionhat}
&|\,\hat{p}j_{3}\,[wj]\,\rangle=\int\frac{d^{3}\hat{p}_{1}}{2\hat{E}_{1}}
\frac{d^{3}\hat{p}_{2}}{2\hat{E}_{2}}
\,|\,\hat{p}_{1}\hat{p}_{2}[m_{1}m_{2}]\,\rangle\,\langle\,
\hat{p}_{1}\hat{p}_{2}[m_{1}m_{2}]\,|\,\hat{p}j_{3}\,[wj]\,\rangle\\
\nonumber
&\text{for any }(m_{1}+m_{2})^{2}\leq w^{2}<\infty\qquad j=0,1,\cdots\end{aligned}$$ The choice of the measure $\frac{d^{3}\hat{p}_{i}}{2\hat{E}_{i}({\bbox}{{\hat{p}}}_{i})}
=\frac{d^{3}p_{i}}{m_{i}^{2}2E_{i}}$ is the same as (\[dirachat\]).
From the 4-translation invariance (conservation of $4$-momentum) it follows that the Clebsch-Gordan is of the form $$\label{deltafour}
\langle\,\hat{p}_{1}\hat{p}_{2}\,|\,\hat{p}j_{3}\,[wj]\,\rangle
=\delta^{4}(p-r)\langle\!\langle\,\hat{p}_{1}\hat{p}_{2}\,|\,\hat{p}j_{3}
\,[wj]\,\rangle\!\rangle\, ,\quad \text{where } r\equiv p_{1}+p_{2}\, .$$ The reduced matrix element in the center-of-mass is in analogy to the non-relativistic case given by [@aB93] $$\label{reducedelement}
\langle\!\langle\,\hat{p}_{1}^{cm}\hat{p}_{2}^{cm}\,|\,
{\bbox}{0}j_{3}\,[wj]\,\rangle\!\rangle=Y_{jj_{3}}({\bbox}{e})
\tilde\mu_{j}(w,m_{1},m_{2})\, ,$$ where $\tilde{\mu}_{j}(w,m_{1},m_{2})$ is a function of $w$ (or ${{\mathsf{s}}}$) which depends upon our choice of “normalization” for the basis vectors $|{\hat{p}}j_{3}\,[wj]\,{\rangle}$ in (\[expansionhat\]). The equations (\[deltafour\]) and (\[reducedelement\]) are combined into $$\begin{aligned}
\label{cgseries}
&\langle\,\hat{p}_{1}\hat{p}_{2}\,|\,\hat{p}j_{3}\,[wj]\,\rangle
=2\hat{E}({\bbox}{{\hat{p}}})\delta^{3}({\bbox}{p}-{\bbox}{r})
\delta(w-\epsilon)Y_{jj_{3}}({\bbox}{e})\mu_{j}(w,m_{1},
m_{2})\\
\nonumber &\text{ with }\epsilon^{2}=r^{2}=(p_{1}+p_{2})^{2} \, ,\end{aligned}$$ where again $\mu_{j}(w,m_{1},m_{2})$ is a function that fixes the $\delta$-function “normalization” of $|{\hat{p}}j_{3}\,[wj]\,{\rangle}$. The unit vector ${\bbox}{e}$ in (\[reducedelement\]) is chosen to be in the c.m. frame the direction of ${\bbox}{\hat{p}}_{1}^{cm}=-\frac{m_{2}}{m_{1}}\,{\bbox}{\hat{p}}_{2}
^{cm}$. In general it is obtained from the relative “4-momentum” $q_{\mu}$ of Michel and Wightman [@aW60] by $e_{i}=L^{-1}(p)^{.\,\mu}_{i}q_{\mu}$. The $\mu_{j}(w,m_{1},m_{2})$ and $\tilde\mu_{j}(w,m_{1},m_{2})$ are some weight functions which are determined from the required “normalization” of the 4-velocity kets (\[expansionhat\]). Since for a fixed value of $[wj]$ these generalized eigenvectors are the basis of the irreducible representation space ${\cal H}(w,j)$ of the Poincaré group, we want them to be normalized like (\[measurehat\]), which in (\[expansionhat\]) has been already assured by the choice of the invariant measure $\frac{d^{3}{\hat{p}}_{i}}{2\hat{E}_{i}}$. Therefore, in analogy to (\[normalizationhat\]), we take for the normalization of the basis vectors (\[expansionhat\]) to be $$\begin{aligned}
\label{normalizationchoice}
\langle\,\hat{p}'j_{3}'\,[w'j']\,|\,\hat{p}j_{3}\,[wj]\,\rangle\,
=2\hat{E}({\bbox}{{\hat{p}}})\delta^{3}({\bbox}{\hat{p}'}
-{\bbox}{\hat{p}})\delta_{j_{3}'j_{3}}\delta_{j'j}\delta({{\mathsf{s}}}-{{\mathsf{s}}}')\, ,\\
\nonumber
\text{ where }\hat{E}({\bbox}{{\hat{p}}})=\sqrt{1+{\bbox}{{\hat{p}}}^{2}}
=\frac{1}{w}\sqrt{w^{2}+{\bbox}{p}^{2}}\equiv\frac{1}{w}E({\bbox}{p},w)\, .\end{aligned}$$ The $\delta$-function normalization $\delta({{\mathsf{s}}}'-{{\mathsf{s}}})=\frac{1}
{2w}\delta(w-w')$ in (\[normalizationchoice\]) is a consequence of the choice (\[choice\]) for the measure. After we have chosen the normalization as in (\[normalizationchoice\]), one determines the weight function $\mu_{j}(w,m_{1},m_{2})$ using (\[expansionhat\]). The result is : $$\label{weight}
\left| \mu_{j}(w,m_{1},m_{2})
\right|^{2}=\frac{2m_{1}^{2}m_{2}^{2}w^{2}}
{\sqrt{\lambda(1,(\frac{m_{1}}{w})^{2},(\frac{m_{2}}{w})^{2})}}\, ,$$ where $\lambda$ is defined by [@aW60]: $$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+bc+ac)\,.$$ Except for the normalization factor $\mu$, which follows from our chosen normalization (\[normalizationchoice\]), the values of the Clebsch-Gordan coefficients (\[cgseries\]) is quite obvious [^5]. It expresses momentum conservation and the only factor that one may be puzzled about is that it should be consistent with the 4-velocity normalization expressed by the $\delta^{3}({\bbox}{\hat{p}'}-{\bbox}{\hat{p}})$, ${\bbox}{\hat{p}}=\frac{{\bbox}{p}}{w}$ in (\[normalizationchoice\]). Therewith, we have obtained by (\[expansionhat\]) with (\[directproductbasis\]) and (\[cgseries\]) a system of basis vectors for the space (\[reduction\]) (with $s_{1}=s_{2}=0$) which is the representation space of scattering processes like (\[twopionsresonance\]). As expected, the basis vectors are outside the Hilbert space; $|\,\hat{p}j_{3}\,[wj]\,\rangle\in\Phi^{\times}\supset{\cal H}\supset\Phi$. They have definite values of angular momentum $j$ and invariant mass $w\equiv\sqrt{{{\mathsf{s}}}}$ [^6]; we shall define the Gamow vectors (describing $\rho^{0}$) in terms of linear combinations of these c.m.-energy eigenvectors with a definite value of $j$. However, since the resonances form and decay under the influence of an interaction and the $|\hat{p}j_{3}\,[wj]\,\rangle$ are interaction-free eigenvectors of the “free-particle” Hamiltonian $$\label{freehamiltonian}
K=P_{1}^{0}+P_{2}^{0}$$ we have to go from the free-particle basis vectors (\[expansionhat\]) to the interaction-basis vectors. This can be done in analogy to the non-relativistic case and may be justified in two ways :\
1) One assumes that the time translation generator for the interaction system has two terms ([@sW95] Ch.3), $H_{0}$ and the interaction $V$ $$\label{interactionhamiltonian}
H=H_{0}+V$$ in such a way that to each eigenvector of $H_{0}$ with eigenvalue $E=w\sqrt{1+{{\bbox}{{\hat{p}}}}^{2}}$, $$\label{freeeigenvalue}
H_{0}\,|\,\hat{p}j_{3}\,[wj]\,\rangle=E\,|\,\hat{p}j_{3}\,[wj]\,\rangle\,,$$ there correspond eigenvectors of $H$ with the same eigenvalue $$\label{interactioneigenvalue}
H|\hat{p}j_{3}\,[wj]^{\pm\, int}\,\rangle=E\,|\,\hat{p}j_{3}\,[wj]^{\pm\, int }
\,\rangle \, .$$ Since vectors are not completely defined by the requirement that they be eigenvectors of an operator with a given eigenvalue (but may differ by a phase factor (phase shifts) or unitary transformation (S-matrix) in case of degeneracy) we have added the additional label int. This additional specification of the eigenvectors can be chosen in a variety of ways that are connected with the spaces $\Phi$ that one admits, i.e., with initial and final boundary conditions (as explained for the non-relativistic case in [@aB93]). Since (\[interactionhamiltonian\]) may be a questionable hypothesis in relativistic physics a second justification does not make use of the existence of the Hamiltonian splitting (\[interactionhamiltonian\]).\
2) One assumes the existence of an $S$-operator and of M[ø]{}ller operators $\Omega^{+}$ and $\Omega^{-}$. $\Omega^{+}$ transforms non-interacting states $\phi^{in}$ which are prepared by an apparatus far away from the interaction region into exact state vectors $\phi^{+}$, $$\label{freein}
\Omega^{+}\phi^{in}=\phi^{+}\, ,\quad
\phi^{+}(t)=e^{-iHt}\phi^{+}\, ,$$ which evolve with the exact time-evolution operator $H$. $\Omega^{-}$ transforms observables $|\psi^{out}\rangle\langle\psi
^{out}|$ registered by the detector placed far away from the interaction region into the vectors $\psi^{-}$ which evolve with the exact $H$ in the interaction region : $$\label{freeout}
\Omega^{-}\psi^{out}=\psi^{-}\, ,
\quad
\psi^{-}(t)=e^{iHt}\psi^{+}\, ,$$ where $t$ is the time in the c.m. frame. The basis vectors for the free-particle space and the interaction-basis vectors are then assumed to be related by [^7] $$\label{moeller}
|\,\hat{p}j_{3}\,[wj]^{\pm}\,\rangle=\Omega^{\pm}|\,\hat{p}j_{3}\,[wj]\,\rangle
\, .$$ If (\[interactionhamiltonian\]) also holds then the symbol $\Omega^{\pm}$ at the center-of-mass is given by the solution of the Lippmann-Schwinger equation $$\label{interactioninout}
|{\bbox}{0}j_{3}\,[wj]^{\pm}\,\rangle=\left(1+\frac{1}{w-H\pm i\epsilon}V\right)
|{\bbox}{0}j_{3}\,[wj]\,\rangle\, .$$ The vectors $|\hat{p}j_{3}\,[wj]^{\pm}\,\rangle$ are obtained from the basis vectors at rest $|{\bbox}{0}j_{3}\,[wj]^{\pm}\,\rangle$ by the boost (rotation-free Lorentz transformation) ${\cal U}(L(\hat{p}))$ whose parameters are the $\hat{p}^{m}$ and whose generators are the interaction-incorporating observables $$\label{interactionobservables}
P_{0}=H,\quad P^{m},\quad J_{\mu\nu} \, ,$$ i.e., the exact generators of the Poincaré group ([@sW95] section $3.3$). These vectors (\[interactioninout\]), which for a fixed value of $[wj]$ span an irreducible representation space of the Poincaré group with the “exact generators”, will be used for the definition of the relativistic Gamow vectors. The values of $j$ and ${{\mathsf{s}}}=w^{2}$ are $j=\text{ integer }$ (for $s_{1}=s_{2}=0$ otherwise also half integer) and $(m_{1}+m_{2})^{2}\leq {{\mathsf{s}}}<\infty$. The value of $j$ will be fixed and represents the resonance spin; the same we do with parity and the degeneracy quantum numbers ($n,\,\eta$). The values of ${{\mathsf{s}}}$ we shall continue from the physical values into the complex plane of the relativistic $S$-matrix.
We are grateful for some helpful correspondence with L. Michel. Support from the Welch Foundation is gratefully acknowledged.
Reduction of the Direct Product of Two One-Particle UIR of $\cal P$$\,^8$ {#reduction-of-the-direct-product-of-two-one-particle-uir-of-cal-p8 .unnumbered}
=========================================================================
We discuss here the reduction of the direct product of two one-particle irreducible representation spaces of the Poincaré group $[m_{1},s_{1}]\otimes[m_{2},s_{2}]$ into a continuous direct sum of irreducible representation (irrep) spaces $[{{\mathsf{s}}},j]$ of invariant mass squared ${{\mathsf{s}}}$ and spin $j$. This has been done in [@aW60; @hJ62; @aM62] using the Wigner basis systems of momentum eigenvectors. Here we shall do it using the $4$-velocity basis vectors of the Poincaré group $\cal P$ and obtain the Clebsch-Gordan coefficients of $\cal P$ for the velocity basis. For the one particle spaces, we choose the c.s.c.o. (\[cscohat\]) with the generalized eigenvectors (\[wignerbasishat\]). Thus, the one particle spaces ${{\cal H}}(m,j)$ are labeled by the mass $m$ and the spin $j$ of the particle. In analogy to the case of one-particle, a two-particle irrep space is labeled by the square of the total invariant mass ${{\mathsf{s}}}=(p_{1}+p_{2})^{2}$ and the total angular momentum $j$ of the two particles. The two-particle irrep space is denoted by ${{\cal H}}^{\eta}_{n}({{\mathsf{s}}},j)$, where $\eta$ is a degeneracy label and $n$ is a particle species label. Thus the reduction problem is written as $$\label{a:reduction}
{{\cal H}}(m_{1},s_{1})\otimes {{\cal H}}(m_{2},s_{2})
=\sum_{j\eta}\int_{(m_{1}+m_{2})^{2}}^{\infty}
\oplus {{\cal H}}^{\eta}_{n}({{\mathsf{s}}},j) d{{\mathsf{s}}}\, .$$ As in (\[wignerbasishat\]), the two-particle basis vectors of ${{\cal H}}^{\eta}_{n}({{\mathsf{s}}},j)$ have as the only continuous variables the total four velocity of the two particles and the square of the total invariant mass of the two particles. These basis vectors are denoted by : $$\label{a:basishat}
|{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n{\rangle}$$ with the normalization : $$\label{a:normalizationhat}
{\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}=2{\hat{p}}_{0}\,\delta_{nn'}\delta_{jj'}\delta_{{\sigma}{\sigma}'}
\delta_{\eta\eta'}
\delta^{3}({\bbox}{{\hat{p}}-{\hat{p}}'})\delta({{\mathsf{s}}}-{{\mathsf{s}}}')\,,$$ where ${\sigma}$ is the three-component of the total angular momentum $j$. We denote the basis vectors of ${{\cal H}}(m_{1},s_{1})\otimes
{{\cal H}}(m_{2},s_{2})$ by : $$\label{a:directproductbasis}
|\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}]\,{\rangle}\otimes
|\,{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}]\,{\rangle}\equiv
|\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}]\,{\rangle}\,,$$ where ${\sigma}_{1}\, ,{\sigma}_{2}$ are the three-components of the spins $s_{1}\, ,s_{2}$ respectively. In order to obtain the Clebsch-Gordan coefficients, $$\label{a:clebschgordon}
{\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}]\,|\,
{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta, n\,{\rangle}\, ,$$ of the reduction (\[a:reduction\]), we start by relabeling the basis vectors in (\[a:directproductbasis\]) by using ${{\mathsf{s}}}$, ${\hat{p}}$ and the unit vector ${\bbox}{\hat{n}}=\frac{{\bbox}{p_{1}}-{\bbox}{p_{2}}}{|{\bbox}{p_{1}}-{\bbox}{p_{2}}|}$ as continuous parameters (we note that both sets, $\{{\hat{p}}_{1},\, {\hat{p}}_{2}\}$ and $\{{\hat{p}},\, {\bbox}{\hat{n}},\,{{\mathsf{s}}}\}$ consist of six independent parameters). Thus, we can write : $$\label{a:relabel}
|\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],\,{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}]\,{\rangle}\equiv
|\,{\hat{p}}\,{\bbox}{\hat{n}}\,{{\mathsf{s}}},\,{\sigma}_{1}[m_{1}s_{1}]{\sigma}_{2}[m_{2}s_{2}]\,{\rangle}\,.$$ In the rest frame of both particles, i.e., for ${\hat{p}}=
{\hat{p}}_{R}=\scriptstyle{\left(
\begin{array}{cccc}
1 \\ 0 \\0 \\0
\end{array}
\right)}\, , $ we can expand the unit vector ${\bbox}{\hat{n}}$ in terms of orbital angular momentum basis vectors : $$\label{a:sh}
|{\bbox}{\hat{n}}{\rangle}=\sum_{ll_{3}}|ll_{3}{\rangle}{\langle}ll_{3}|{\bbox}{\hat{n}}{\rangle}=\sum_{ll_{3}}|ll_{3}{\rangle}Y^{*}_{ll_{3}}({\bbox}{\hat{n}})\, .$$ We can further use the angular momentum Clebsch-Gordan coefficients to combine the two spins, $s_{1}$ and $s_{2}$, to give a total spin $s$ with three component $\mu$, which in turn is added to the orbital angular momentum $l$ with three component $l_{3}$ to form a total angular momentum $j$ with three component ${\sigma}$. This gives the basis vector for the two-particle irrep space $$\label{a:basistwoparticles}
|\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]ls;\,m_{1}s_{1},m_{2}s_{2}{\rangle}\, .$$ Thus, the degeneracy label $\eta$ in (\[a:basishat\]) designates the total spin $s$ and the total orbital angular momentum $l$ of both particles; and the masses $m_{1},\,m_{2}$ and spins $s_{1},\,s_{2}$ of both particles are included in the particle species label $n$.
Thus, (\[a:reduction\]) can be rewritten in more details as : $$\label{a:reductiondetails}
{{\cal H}}(m_{1},s_{1})\oplus {{\cal H}}(m_{2},s_{2})
=\sum_{jls}\int_{(m_{1}+m_{2})^{2}}^{\infty}
\oplus {{\cal H}}_{n}^{ls}({{\mathsf{s}}},j)d{{\mathsf{s}}}\, ,$$ $$\begin{aligned}
\nonumber
\text{where }\quad s&=&|s_{1}-s_{2}|,\,|s_{1}-s_{2}|+1,\,
\cdots\, ,s_{1}+s_{2}\\
\nonumber
j &=& |l-s|,\,|l-s|+1,\,\cdots\, ,l+s \, .\end{aligned}$$ With (\[a:relabel\]) and (\[a:sh\]), we deduce that in the rest frame, the Clebsch-Gordan coefficients of (\[a:reductiondetails\]) are given by : $$\begin{aligned}
\nonumber
{\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n\,|\,
{\hat{p}}_{R}{\sigma}[{{\mathsf{s}}}j],\eta,n'\,{\rangle}&=& 2N_{n}({{\mathsf{s}}}) \delta_{nn'}
\theta({{\mathsf{s}}}-(m_{1}+m_{2})^{2})
\delta^{3}({\bbox}{p_{1}+p_{2}})
\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})\\ \label{a:cgrest}
&& \times \sum_{l_{3}\mu}C_{s_{1}s_{2}}(s\mu,{\sigma}_{1}{\sigma}_{2})
C_{sl}(j{\sigma},\mu l_{3})Y_{ll_{3}}({\bbox}{\hat{n}})\, ,\end{aligned}$$ where $N_{n}({{\mathsf{s}}})$ is a normalization factor. Having obtained the Clebsch-Gordan coefficients in the rest frame (\[a:cgrest\]), we can use the boost operator (\[boosthat\]) to obtain the Clebsch-Gordan coefficients in a general frame [^8]: $$\begin{aligned}
\nonumber
{\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n\,|\,
{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n'\,{\rangle}&=& {\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n\,|\,
{\cal U}(L(p))\,|\,{\hat{p}}_{R}{\sigma}[{{\mathsf{s}}}j]\eta,n'\,{\rangle}\\
\nonumber
&=&2{\hat{p}}_{0}N_{n}({{\mathsf{s}}})\delta_{nn'}\theta({{\mathsf{s}}}-(m_{1}+m_{2})^{2})
\delta^{3}({\bbox}{p-p_{1}-p_{2}})\\
\nonumber
&& \times \delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})
\sum_{{\sigma}_{1}'{\sigma}_{2}'}D^{s_{1}*}_{{\sigma}_{1}'{\sigma}_{1}}(
R(L^{-1}(p),p_{1}))D^{s_{2}*}_{{\sigma}_{2}'{\sigma}_{2}}(
R(L^{-1}(p),p_{2}))\\ \label{a:cg}
&&\times \sum_{l_{3}\mu}C_{s_{1}s_{2}}(s\mu,{\sigma}_{1}'{\sigma}_{2}')
C_{sl}(j{\sigma},\mu l_{3})\, Y_{ll_{3}}({\bbox}{e})\, ,\end{aligned}$$ where $R(\lambda,p)$ is the Wigner rotation given in (\[wignerrotation\]) and $${\bbox}{e}=\frac{\overrightarrow{L^{-1}(p)(p_{1}-p_{2})}}
{\left| \overrightarrow{L^{-1}(p)(p_{1}-p_{2})} \right| }
\, .$$ The normalization factor $N_{n}({{\mathsf{s}}})$ depends upon our normalization choice (\[a:normalizationhat\]). Before discussing how to obtain it, let us first introduce the following notations :
\[a:notations\] $$\begin{aligned}
&\Gamma(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s\mu)
=\sum_{{\sigma}_{1}'{\sigma}_{2}'}
D^{s_{1}*}_{{\sigma}_{1}'{\sigma}_{1}}(R(L^{-1}(p),p_{1}))
D^{s_{2}*}_{{\sigma}_{2}'{\sigma}_{2}}(R(L^{-1}(p),p_{2}))
C_{s_{1}s_{2}}(s\mu,{\sigma}_{1}'{\sigma}_{2}')\, ,\\
&Y_{j{\sigma}ls}({\bbox}{e},\mu)=\sum_{l_{3}}C_{sl}(j{\sigma},\mu l_{3})
Y_{ll_{3}}({\bbox}{e})\, .\end{aligned}$$
With the above notations, (\[a:cg\]) is written as $$\begin{aligned}
\nonumber
{\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n\,|\,
{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,\, n'{\rangle}&=& 2{\hat{p}}_{0}N_{n}({{\mathsf{s}}})\,\delta_{nn'}\delta^{3}({\bbox}{p-p_{1}-p_{2}})
\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})\\ \label{a:cg2}
&& \times\sum_{\mu}\Gamma(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s\mu)
Y_{j{\sigma}ls}({\bbox}{e},\mu)
\,.\end{aligned}$$ In order to obtain the normalization factor $N_{n}({{\mathsf{s}}})$, we insert a complete set of basis vectors (\[a:directproductbasis\]) in ${\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}$ and use (\[a:cg2\]). Upon doing so, we obtain : $$\begin{aligned}
\nonumber
{\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}&=&\sum_{n''{\sigma}_{1}{\sigma}_{2}}
\int\frac{d^{3}{\hat{p}}_{1}}{2{\hat{p}}_{1}^{0}}\frac{d^{3}{\hat{p}}_{2}}{2{\hat{p}}_{2}^{0}}
{\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],
{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n''\,{\rangle}\\
\nonumber
&&\times {\langle}\,{\hat{p}}_{1}{\sigma}_{1}[m_{1}s_{1}],{\hat{p}}_{2}{\sigma}_{2}[m_{2}s_{2}],n''\,|
\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}\\
\nonumber
&=& (2{\hat{p}}_{0})^{2}|N_{n}({{\mathsf{s}}})|^{2} \delta_{nn'}
\delta^{3}({\bbox}{p-p'}) \delta({{\mathsf{s}}}-{{\mathsf{s}}}')\\
\nonumber
&& \times \sum_{{\sigma}_{1}{\sigma}_{2}}\sum_{\mu\mu'}\int
\frac{d^{3}{\hat{p}}_{1}}{2{\hat{p}}_{1}^{0}}
\frac{d^{3}{\hat{p}}_{2}}{2{\hat{p}}_{2}^{0}}
\delta^{3}({\bbox}{p-p_{1}-p_{2}})\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})\\
\label{a:cg3}
&& \times \Gamma^{*}(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s'\mu')
\Gamma(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s\mu)
Y^{*}_{j'{\sigma}'\eta'}({\bbox}{e},\mu')Y_{j{\sigma}\eta}({\bbox}{e},\mu)\, .\end{aligned}$$ Using the unitarity of the rotation matrices : $$\sum_{{\sigma}}D^{*j}_{{\sigma}'{\sigma}}D^{j}_{{\sigma}''{\sigma}}=\delta_{{\sigma}'{\sigma}''}$$ and the identity $$\sum_{{\sigma}_{1}{\sigma}_{2}}C_{s_{1}s_{2}}(s\mu,{\sigma}_{1}{\sigma}_{2})
C_{s_{1}s_{2}}(s'\mu',{\sigma}_{1}{\sigma}_{2})=\delta_{ss'}\delta_{\mu\mu'}\, ,$$ we find that $$\label{a:gammaidentity}
\sum_{{\sigma}_{1}{\sigma}_{2}}\Gamma^{*}(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s'\mu')
\Gamma(s_{1}{\sigma}_{1},s_{2}{\sigma}_{2},s\mu)
=\delta_{ss'}\delta_{\mu\mu'}\, .$$ With the identity (\[a:gammaidentity\]), (\[a:cg3\]) can be written as : $$\begin{aligned}
\nonumber
{\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,p{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}&=&
(2{\hat{p}}_{0})^{2}|N_{n}({{\mathsf{s}}})|^{2}\delta_{nn'}
\delta^{3}({\bbox}{p-p'})
\delta({{\mathsf{s}}}-{{\mathsf{s}}}')\delta_{ss'}
\sum_{\mu l_{3}l_{3}'}C_{sl'}(j'{\sigma}',\mu l_{3}')
C_{sl}(j{\sigma},\mu l_{3})\\ \label{a:cg4}
&& \times \int
\frac{d^{3}{\hat{p}}_{1}}{2{\hat{p}}_{1}^{0}}
\frac{d^{3}{\hat{p}}_{2}}{2{\hat{p}}_{2}^{0}}
\delta^{3}({\bbox}{p-p_{1}-p_{2}})\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})
Y^{*}_{l'l_{3}'}({\bbox}{e})Y_{ll_{3}}({\bbox}{e})\,.\end{aligned}$$ In order to solve the integration in (\[a:cg4\]), namely $$\begin{aligned}
\nonumber
I &=& \int
\frac{d^{3}{\hat{p}}_{1}}{2{\hat{p}}_{1}^{0}}
\frac{d^{3}{\hat{p}}_{2}}{2{\hat{p}}_{2}^{0}}
\delta^{3}({\bbox}{p-p_{1}-p_{2}})\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})
Y^{*}_{l'l_{3}'}({\bbox}{e})Y_{ll_{3}}({\bbox}{e})\\
\label{a:integration}
&=& \frac{1}{m_{1}^{2}m_{2}^{2}}\int
\frac{d^{3}p_{1}}{2p_{1}^{0}}
\frac{d^{3}p_{2}}{2p_{1}^{0}}
\delta^{3}({\bbox}{p-p_{1}-p_{2}})\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})
Y^{*}_{l'l_{3}'}({\bbox}{e})Y_{ll_{3}}({\bbox}{e})\, ,\end{aligned}$$ we perform the change of variables (as in equation $(4.9)$ in [@aW60]) : $$\begin{aligned}
\nonumber
p_{1}=\frac{({{\mathsf{s}}}+m_{1}^{2}-m_{2}^{2})}{2{{\mathsf{s}}}}r+
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{2\sqrt{{{\mathsf{s}}}}}q \\
\label{a:changeofvariables}
p_{2}=\frac{({{\mathsf{s}}}-m_{1}^{2}+m_{2}^{2})}{2{{\mathsf{s}}}}r-
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{2\sqrt{{{\mathsf{s}}}}}q \end{aligned}$$ where $$\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2(ab+ac+bc)\, .$$ With these new variables, we find that
\[a:newvariables\] $$\delta(p_{1}^{2}-m_{1}^{2})\delta(p_{2}^{2}-m_{2}^{2})
\delta^{3}({\bbox}{p}-{\bbox}{p_{1}}-{\bbox}{p_{2}})\delta({{\mathsf{s}}}-(p_{1}+p_{2})^{2})
=\frac{4{{\mathsf{s}}}^{3/2}}{\lambda^{3/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}\frac{1}{2p_{0}}
\delta(q^{2}+1)\delta(r.q)\delta^{4}(r-p)$$ $$d^{4}p_{1}d^{4}p_{2}=\frac{\lambda^{2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{16{{\mathsf{s}}}^{2}}
d^{4}rd^{4}q$$ and $${\bbox}{e}=\overrightarrow{L^{-1}(p)q}\, .$$
Using (\[a:newvariables\]), the integration (\[a:integration\]) becomes : $$\label{a:integration1}
I=\frac{1}{m_{1}^{2}m_{2}^{2}}\frac{1}{2p_{0}}
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{4\sqrt{{{\mathsf{s}}}}}
\int d^{4}q\delta(q^{2}+1)\delta(p.q)Y_{l'l_{3}'}^{*}\left(\overrightarrow
{L^{-1}(p)q}\right)
Y_{ll_{3}}\left(\overrightarrow{L^{-1}(p)q}\right)\, .$$ Performing the change of variable $e=L^{-1}(p)q$ in (\[a:integration1\]), we obtain : $$\begin{aligned}
\nonumber
I &=& \frac{1}{m_{1}^{2}m_{2}^{2}}\frac{1}{2p_{0}}
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{8{{\mathsf{s}}}}
\int d\Omega({\bbox}{e})Y_{l'l_{3}'}^{*}({\bbox}{e})Y_{ll_{3}}({\bbox}{e})\\
\label{a:integration2}
&=& \frac{1}{m_{1}^{2}m_{2}^{2}}\frac{1}{2p_{0}}
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{8{{\mathsf{s}}}}
\delta_{ll'}\delta_{l_{3}l_{3}'}\,.\end{aligned}$$ Using (\[a:integration2\]) and the identity $$\sum_{\mu l_{3}}C_{sl}(j'{\sigma}',\mu l_{3})
C_{sl}(j{\sigma},\mu l_{3})=\delta_{jj'}\delta_{{\sigma}{\sigma}'}\, ,$$ (\[a:cg4\]) finally becomes : $$\label{a:cg5}
{\langle}\,{\hat{p}}'{\sigma}'[{{\mathsf{s}}}'j']\eta',n'\,|\,{\hat{p}}{\sigma}[{{\mathsf{s}}}j]\eta,n\,{\rangle}=(2{\hat{p}}_{0})|N_{n}({{\mathsf{s}}})|^{2}\frac{1}{m_{1}^{2}m_{2}^{2}}
\frac{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}{8{{\mathsf{s}}}^{3}}
\delta_{nn'}\delta_{jj'}\delta_{{\sigma}{\sigma}'}\delta_{\eta\eta'}
\delta^{3}({\bbox}{{\hat{p}}-{\hat{p}}'})\delta({{\mathsf{s}}}-{{\mathsf{s}}}')\, .$$ Comparing (\[a:cg5\]) with (\[a:normalizationhat\]), we find that : $$\nonumber
|N_{n}({{\mathsf{s}}})|^{2}=\frac{8m_{1}^{2}m_{2}^{2}{{\mathsf{s}}}^{3}}
{\lambda^{1/2}({{\mathsf{s}}},m_{1}^{2},m_{2}^{2})}\, .$$
[99]{}
A. Bohm. *Quantum Mechanics*, first edition, (Springer, 1979); in particular Chapter XXI of the third edition (Springer, 1994); A. Bohm. and M. Gadella, *Dirac Kets, Gamow Vectors, and Gel’fand Triplets*, Lecture Notes in Physics, Vol. 348 (Springer, Berlin, 1989); A. Bohm, S. Maxson, M. Loewe, M. Gadella, Physica A **236**, 485 (1997). A. S. Wightman, *Lectures on Invariance in Relativistic Quantum Mechanics* (Les Houches, 1960). H. Joos, Fortschr. Physik **10**, 65 (1962). A. J. Macfarlane, Rev. Mod. Phys. **34**, 41 (1962). E. G. Beltrametti and G. Luzzatto, Nuovo Cimento **36**, 1217 (1965); A. O. Barut in *Lectures in Theoretical Physics* Vol. VII A, eds. W. E. Brittin and A. O. Barut, (The University Of Colorado Press, Boulder, 1965), p. 121; S. S. Sannikov, Soviet J. Nuclear Phys. **4**, 416 (1967); M. Comi, L. Lanz, L. A. Lugiato, G. Ramella, J. Math. Phys. [**16**]{} 910, 1975. The use of velocity eigenvectors has been suggested as early as $1965$ by J. Werle, *On a symmetry scheme described by non-Lie algebra*, ICPT preprint, Trieste, 1965, unpublished. It has been incorporated in the spectrum-generating group approach for the mass and spin spectrum, A. Bohm, Phys. Rev. **175**, 1767 (1968). It has been used to relate the form factors for different decays to universal form factors for an $SU(3)$ multiplet in A. Bohm, Phys. Rev. D **13**, 2110 (1976), and A. Bohm, J. Werle, Nucl. Phys. B **106**, 165 (1976). As “dynamical stability group of $P_{\mu}/M$” the same idea has been suggested by H. van Dam and L. C. Biedenharn, Phys. Rev. D **14**, 405, (1976). Velocity eigenvectors were reintroduced around $1990$ as states of the Heavy Quark Effective Theory, again to reduce the number of independent form factors: A. F. Falk, H. Georgi, B. Grinstein, M. B. Wise, Nucl. Phys. B **343**, 1 (1990), H. Georgi *Proceedings of the Theoretical Advanced Study Institute*, eds. R. K. Ellis et al. (World Scientific, 1992) and reference thereof. In this approach one often gets the impression that the use of velocity basis vectors leads to approximate results. This, however is not the case if one takes into consideration that velocity eigenkets do not represent physical states, but that physical state vectors are obtained from the kets as “continuous superpositions” using the right measure in the integration. If this is done, the values of observable quantities do not depend upon whether one uses the velocity basis or the momentum basis, only that the use of the velocity basis often provides a more practical means of computation, by leading to form factors that do not depend upon the mass. S. Weinberg. *The Quantum Theory of Fields*, Vol. 1, (Cambridge University Press, 1995). E. P. Wigner, Ann. Math. (2) **40**, 149 (1939). A. Bohm in *Studies in Mathematical Physics*, ed. A. O. Barut, (Reidel, 1973), p. 197; A. Bohm, J. Math. Phys. **8**, 1557 (1967) (Appendix B); B. Nagel in [*Studies in Mathematical Physics*]{}, ed. A. O. Barut, (Reidel, 1973), p. 135; A. Bohm, M. Gadella and S. Wickramasekara in [*Generalized Functions, Operator Theory, and Dynamical Systems*]{}, eds. I. Antoniou and G. Lumer, (Chapman and Hall/CRC, London, 1999), p. 202. E. Nelson, Ann. Math. (2) [**[70]{}**]{}, 572 (1959); E. Nelson, W. F. Stinespring, Amer. J. Math. [**81**]{}, 547 (1959). See e.g., A. Bohm. *Quantum Mechanics*, third edition, (Springer, 1993) : Section XVI.1.
[^1]: Some of the references we use here have different convention, e.g., $\eta_{\mu\, \nu}\rightarrow -\eta_{\mu\, \nu}$ [@sW95], and $L^{-1}\rightarrow L(p)$ [@hJ62].
[^2]: This is the convention of [@eW39; @aW60; @hJ62; @aM62], but not of [@sW95]
[^3]: The quantum numbers $b$ are called the particle species numbers in [@sW95].
[^4]: Though our discussions apply with obvious modifications to the general case of $$1+2+3+\cdots \rightarrow R_{i}\rightarrow 1^{'}+2^{'}+3^{'}+\cdots$$ these generalizations lead to enormously more complicated equations. For the sake of simplicity, we shall therefore consider a resonance scattering process like (\[resonancescattering\]).
[^5]: A formula like (\[cgseries\]) is also given and explained in section $3.7$ of [@sW95] which for $s=0,\, s_{1}=s_{2}=0$ agrees with (\[cgseries\]) except for the normalization factor (\[weight\]). For $s\ne 0,\, s_{i}\ne 0$, see Appendix.
[^6]: Written in terms of Hilbert spaces, $d\mu({{\mathsf{s}}})$ means Lebesgue integrations. However, within the RHS mathematics, one can choose for $\langle\, \phi\,|\,\hat{p}j_{3}\,[wj]\,\rangle$ a smooth function and use Riemann integration and assign to each vector a well defined value $w$ (not just up to a set of measure zero)
[^7]: In non-relativistic scattering off a fixed target one assumes that the $|{\bbox}{p}^{+}\rangle$ related by (\[moeller\]) to the $|{\bbox}{p}\rangle$ are not eigenvectors of ${\bbox}{P}$ since $[V,P_{i}]\ne 0$.
[^8]: Formula $(3.7.5)$ in [@sW95], which corresponds to (\[a:cg\]) but for different choices of basis and normalizations, is missing the rotation matrices factors that appear in the Clebsch-Gordan coefficients away from the rest frame, as exhibited in (\[a:cg\]).
|
---
abstract: 'We propose a nonlocal strain measure for use with digital image correlation (DIC). Whereas the traditional notion of compatibility (strain as the derivative of the displacement field) is problematic when the displacement field varies substantially either because of measurement noise or material irregularity, the proposed measure remains robust, well-defined and invariant under rigid body motion. Moreover, when the displacement field is smooth, the classical and nonlocal strain are in agreement. We demonstrate, via several numerical examples, the potential of this new strain measure for problems with steep gradients. We also show how the nonlocal strain provides an intrinsic mechanism for filtering high frequency content from the strain profile and so has a high signal to noise ratio. This is a convenient feature considering image noise and its impact on strain calculations.'
address:
- 'Richard B. Lehoucq, Computational Mathematics, Sandia National Laboratories, P.O. Box 5800; Albuquerque, New Mexico 87185. *E-mail address:* [[email protected]]{}, *Phone:* (505) 845-8929'
- 'Phillip H. Reu, Sensing & Imaging Technologies, Sandia National Laboratories, P.O. Box 5800; Albuquerque, New Mexico 87185. *E-mail address:* [[email protected]]{}, *Phone:* (505) 284-8913'
- 'Correspondence to: Daniel Z. Turner, Multiscale Science, Sandia National Laboratories, P.O. Box 5800; Albuquerque, New Mexico 87185. *E-mail address:* [[email protected]]{}, *Phone:* (505) 845-7443'
author:
- 'R. B. Lehoucq'
- 'P. L. Reu'
- 'D. Z. Turner'
bibliography:
- 'Master\_References.bib'
title: A Nonlocal Strain Measure for Digital Image Correlation
---
Introduction
============
Digital image correlation has revolutionized material characterization by means of non-contact, full-field displacement measurement [@Sutton; @Bruck; @Hild; @Chu] including several advancements towards applying DIC to characteristically difficult problems, such as extended DIC [@Roux; @McNeill; @Rethore; @Rethore2; @Poissant]. The reader is referred to the paper [@Reu1] for an overview of DIC and its applications. An important part of this process involves the mapping of displacement data with material or constitutive models via an appropriate measure of strain. There are a number of challenges to making this connection including the effects of image noise, the appropriateness of strain measures in the context of discontinuities (across pixels), and data loss from curve fitting or filtering techniques. We propose a new nonlocal strain measure that alleviates many of these issues.
Several strain measures have been proposed for DIC (for a review see [@Pan; @Grama] and the reference therein). A common theme among existing strain measures involves the use of finite difference approximations of spatial partial derivatives. Although these methods have been used effectively for a variety of complex problems, there is a growing awareness of their deficiencies for problems involving cracks or strongly heterogeneous materials. Further, finite difference based approaches are sensitive to noise and often struggle to remain invariant under rigid body motion. As an alternative to these approaches, we propose a strain measure that is built on the structure of a recently proposed nonlocal vector calculus that exploits integral operators to calculate a strain.
Nonlocal vector calculus involves operators that do not use partial derivatives, but rather integrals. A comprehensive introduction to nonlocal vector calculus is given in [@Du] based upon preliminary work given in [@GunzburgerNL]. This formalism is applied to a nonlocal diffusion equation in [@DuDiff] and the peridynamic Navier equation [@DuNL]. In the present work, we employ these ideas to construct a robust strain measure for DIC that provides many advantages over the classical strain measure. In particular, the nonlocal strain measure intrinsically filters high frequency noise from the displacement data and is well-defined even for locations at which a classical derivative may not exist.
Nonlocal Strain
===============
We motivate the nonlocal strain measure by considering the one-dimensional problem. We then introduce the nonlocal strain measure for two and three dimensions. In so doing, we provide a systematic basis for computing a well-defined, discrete strain from noisy, discrete displacement data. In contrast, conventional finite difference approximations are notoriously sensitive to noisy data and inevitably lead to a tradeoff between smoothness and accuracy.
One dimension
-------------
When a function $f$ is continuous at $x$, then we have the well-known relationship
$$\begin{aligned}
\int_{-\infty}^{\infty} f(y) \, \delta(y - x) \; \text{d}y &= f(x)\,, \label{eq:fZero}
\intertext{where $\delta(x)$ is the Dirac delta (generalized) function. Probably less well-known is the relationship}
\int_{-\infty}^{\infty} f(y) \, \delta^\prime(y - x) \; \text{d}y &= -f^\prime(x)\,. \label{eq:antisymZero}\end{aligned}$$
when $f^\prime$ is continuous at $x$. This is easily established by integrating by parts the lefthand side to obtain $$\begin{aligned}
\int_{-\infty}^{\infty} f(y) \, \delta^\prime(y - x) \; \text{d}y & = f(x) \, \delta(x)\bigg|_{-\infty}^{\infty} - \int_{-\infty}^{\infty} f^\prime(y) \, \delta(y - x) \; \text{d}y \\
& = -f^\prime(x)\,,
\end{aligned}$$ where we used and the formal property $$\delta(x) =
\begin{cases}
0 & x \neq 0\\
\text{undefined} & x = 0
\end{cases}\,.$$ By selecting $f=1$ in and , we obtain two useful relations $$\int_{-\infty}^{\infty} \delta(y - x) \; \text{d}y = 1\,, \quad \int_{-\infty}^{\infty} \delta^\prime(y - x) \; \text{d}y = 0\,.$$ In so many words, $\delta(x)$ and $\delta^\prime(x)$ are even and odd functions. More generally, the former and latter functions are symmetric and antisymmetric about the origin, respectively.
The above discussion suggests a definition for the nonlocal derivative of $f$ at $x$ as $$\label{nl-deriv-1d}
-\int_{-\infty}^{\infty} f(y) \, \alpha_\epsilon(y - x) \; \text{d}y\, \text{ where } \int_{-\infty}^{\infty} \alpha_\epsilon(y - x) \; \text{d}y = 0\,.$$ The function $\alpha_\epsilon$ is an integrable approximation to $\delta^\prime(x)$ representing the kernel of the integral operator and the parameter $\epsilon$ is a length scale associated with the approximation. This length scale can, for example, denote the nonzero region of compact support for $\alpha_\epsilon$. The following example clarifies the procedure used to construct an appropriate $\alpha_\epsilon$ and the role of $\epsilon$. Consider the function
\[eq:HatFunction\] $$\begin{aligned}
\phi_\epsilon (x) & = \begin{cases}
\displaystyle 2\frac{x + \gamma \epsilon}{\gamma \, \epsilon^2} \quad &-\gamma \,\epsilon < x < 0\\[1.5ex]
\displaystyle 2 \frac{(1-\gamma)\,\epsilon -x }{(1-\gamma)\epsilon^2}\quad &0 < x < (1-\gamma) \,\epsilon \\
0 & \quad \text{otherwise}
\end{cases}
\intertext{ with derivative}
\phi^\prime_\epsilon (x) & = \begin{cases}
\displaystyle \frac{2}{\gamma \, \epsilon^2}\quad &-\gamma \,\epsilon < x < 0 \\[1.5ex]
\displaystyle - \frac{2 }{(1-\gamma)\,\epsilon^2}\ \quad &0 < x < (1-\gamma) \,\epsilon \\
0 & \quad \text{otherwise}
\end{cases}\end{aligned}$$ where $\gamma$ is a dimensionless skew parameter satisfying the constraint $$\begin{aligned}
0 < \gamma < 1\,. \label{gamma-constraint}\end{aligned}$$ A simple calculation reveals that $$\label{eq:SimpleCalcs}
\int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} \phi_\epsilon(x) \, dx =1\,, \quad \int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} \phi^\prime_\epsilon(x) \, dx = 0\quad \text{for all} \quad \epsilon>0\,,$$ when $\gamma$ is greater than $0$ and strictly less than $1$. Otherwise, the properties no longer hold. Therefore as $\epsilon \to 0$ $$\begin{aligned}
\int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} f(y) \, \phi_\epsilon(x-y) \, dy & \to f(x) \\
- \int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} f(y) \, \frac{\phi^\prime_\epsilon(x-y)}{2} \, dy & \to f^\prime(x)\,,
\end{aligned}$$
where the limits hold if $f$ is differentiable at $x$. We remark that if $f$ and $\phi_\epsilon$ have units of length and per length, respectively, then the nonlocal derivative is a dimensionless quantity.
Plots of this kernel and its derivative are shown in Figure \[fig:Phi\]. The plots indicate that $\gamma$ skews the function $\phi$ into the right half of the plane. This is a useful feature when $x$ is near the end point of an interval over which approximations to $f^\prime(x)$ are needed.
![(a) Kernel function $\phi_\epsilon(x)$ and (b) its derivative.[]{data-label="fig:Phi"}](Phi.pdf)
Selecting $f(x)=ax+b$ and using a midpoint quadrature rule, which is exact for linear functions, grants $$\begin{aligned}
\int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} \big( ay+b\big) \, \phi_\epsilon(x-y) \, dy & = ax+b\,, \\
-\int_{-\gamma \,\epsilon}^{(1-\gamma)\epsilon} \big( ay+b\big)\, \frac{\phi^\prime_\epsilon(x-y)}{2} \, dy & = a\,,
\end{aligned}$$ so that the derivative of a linear function is computed exactly with a simple quadrature rule. Alternatively, these two relations can be established via integration by parts.
The advantage of the nonlocal derivative is that it provides a systematic and robust basis for approximating $f^\prime(x)$ given smooth *or irregular data*, avoiding the characteristic sensitivity of finite difference approximations to noisy data.
Two and three dimensions
------------------------
Define the nonlocal gradient of a vector function $\mathbf{f}$ as $$\begin{aligned}
\tilde{\nabla}\, \mathbf{f}(\mathbf{x}) &:= -\int_{\mathbb{R}^n} \mathbf{f}(\mathbf{y}) \otimes\boldsymbol{\alpha}_\epsilon(\mathbf{y} - \mathbf{x}) \, \text{d}\mathbf{y} \label{nl-deriv-nd}
\intertext{where $ \mathbf{x}\otimes\mathbf{y}$ denotes the dyadic product of $ \mathbf{x}$ and $ \mathbf{y}$, $\boldsymbol{\alpha}_\epsilon$ is the kernel of the integral operator approximating $\nabla\delta(\mathbf{x})$ satisfying}
\mathbf{0} &= \int_{\mathbb{R}^n} \boldsymbol{\alpha}_\epsilon(\mathbf{y} - \mathbf{x}) \, \text{d} \mathbf{y}\,, \notag\end{aligned}$$ and $n=1,2,3$ (when $n=1$, then and coincide). The parameter $\epsilon$ is a length scale associated with the approximation. This length scale can, for example, denote the nonzero region of compact support for $\boldsymbol{\alpha}_\epsilon$. The nonlocal derivative is a variation of the corresponding deformation gradient tensor $\mathbf{\bar{F}}$ proposed in [@Silling1 p.180] and also the weighted nonlocal gradient proposed in [@Du pp.520-527]. This weighted gradient assumed that $\boldsymbol{\alpha}_\epsilon(\mathbf{y} - \mathbf{x})+\boldsymbol{\alpha}_\epsilon(\mathbf{x} - \mathbf{y})=\mathbf{0}$, a sufficient, but not necessary, condition to satisfy the second integral in . As discussed in the one-dimensional case, if $\mathbf{f}$ and $\boldsymbol{\alpha}_\epsilon$ have units of length and per volume per length, then the nonlocal gradient is a dimensionless quantity.
The determination of an appropriate $\boldsymbol{\alpha}_\epsilon$ follows as in the one-dimensional case. As an example, let the scalar function $$\psi(\mathbf{x}) = \psi(x_1,\, x_2) = \phi_{1,\epsilon}(x_1)\, \phi_{2,\epsilon}(x_2)\,,$$ where $\phi_{1,\epsilon}\,,\phi_{2,\epsilon}$ are the one-dimensional functions given by with skew parameters $0<\gamma_1<1$, and $0<\gamma_2<1$, respectively (see Figure \[fig:psi\]).
![Plot of $\psi(\mathbf{x})$.[]{data-label="fig:psi"}](PsiNoContours.pdf)
Therefore $$\boldsymbol{\alpha}_\epsilon(\mathbf{x}) = \nabla \psi(\mathbf{x})
= \bigg(\frac{\partial \phi_{1,\epsilon}(x_1)}{\partial x_1}\phi_{2,\epsilon}(x_2) \,, \frac{\partial \phi_{2,\epsilon}(x_2)}{\partial x_2 }\phi_{1,\epsilon}(x_1) \bigg)\,.$$ A plot of the components of $\boldsymbol{\alpha}_\epsilon(\mathbf{x})$ is shown in Figure \[fig:GradPsi\].
![Plot of the components of $\boldsymbol{\alpha}_\epsilon(\mathbf{x})$, (a) $\nabla_{x_1} \psi(\mathbf{x})$ (b) $\nabla_{x_2} \psi(\mathbf{x})$[]{data-label="fig:GradPsi"}](GradPsiCombinedNoContours.pdf)
By construction, we see that $$\int_{\mathbb{R}^2} \psi(\mathbf{x})\,\text{d} \mathbf{x} = 1\,, \text{ and } \int_{\mathbb{R}^2} \nabla \psi(\mathbf{x}) \,\text{d} \mathbf{x} = \mathbf{0}\,.$$
Homogenous deformation
----------------------
Selecting the function $\mathbf{f} = \mathbf{A\,x} + \mathbf{c}$, where $\mathbf{A}$ and $\mathbf{c}$ are a constant tensor and vector, results in $$\begin{aligned}
\tilde{\nabla}\, (\mathbf{A\,x} + \mathbf{c}) &= \tilde{\nabla}\, \mathbf{A\,x} + \tilde{\nabla}\,\mathbf{c} \\
& = -\mathbf{A}\int_{\mathbb{R}^2} \mathbf{y} \otimes\boldsymbol{\alpha}_\epsilon(\mathbf{y} - \mathbf{x}) \, \text{d}\mathbf{y} + \mathbf{c} \otimes \int_{\mathbb{R}^2} \boldsymbol{\alpha}_\epsilon(\mathbf{y} - \mathbf{x}) \, \text{d}\mathbf{y}\\
&= -\mathbf{A}\int_{\mathbb{R}} \int_{\mathbb{R}} (y_1,\, y_2 ) \\
&\qquad \; \otimes\bigg(\frac{\partial \phi_{1,\epsilon}(y_1-x_1)}{\partial y_1} \phi_{2,\epsilon}(y_2 - x_2) \,, \frac{\partial \phi_{2,\epsilon}(y_2-x_2)}{\partial y_2 } \phi_{1,\epsilon}(y_1 - x_1) \bigg) \\
& \qquad \quad \text{d} y_1\, \text{d} y_2\\
&= -\mathbf{A} \big( -\mathbf{I}\big) = \mathbf{A}\,,
\end{aligned}$$ where $\mathbf{I}$ denotes the unit tensor and the fourth equality follows from integrating by parts each of the four components of the tensor.
In particular, the above derivation demonstrates that $$\label{nl-grad-linear-trans}
\tilde{\nabla}\, (\mathbf{A\,x}) = \mathbf{A} \tilde{\nabla}\, \mathbf{x} = \mathbf{A}\,,$$ so that $\tilde{\nabla\tilde}$ commutes with a constant tensor exactly as does the classical gradient operator. Moreover, when $\mathbf{A}=\mathbf{I}$ then $\tilde{\nabla}\, \mathbf{x} = \mathbf{I}$, or in words, the nonlocal gradient of the identity vector map is the identity tensor.
Definition of nonlocal strain {#sec:nl-str}
-----------------------------
Consider the nonlocal deformation gradient defined as
$$\begin{aligned}
\tilde{\mathbf{F}} & := \mathbf{I} + \tilde{\nabla}\mathbf{u} \,. \label{nl-def-grad}
\intertext{The nonlocal strain tensor, $\tilde{\mathbf{E}}$, is then defined as}
\label{eq:NonlocalStrain}
\tilde{\mathbf{E}} & := \frac{1}{2}\big( \tilde{\mathbf{F}}^T\tilde{\mathbf{F}} -\mathbf{I} \big)\,.\end{aligned}$$
Rigid body motions and finite rotations
---------------------------------------
For this nonlocal strain measure to be of meaningful value, the property $\tilde{\mathbf{F}}^T\tilde{\mathbf{F}} = \mathbf{I}$ must hold for any motion that does not deform the body. In other words, the nonlocal strain should be invariant under rigid body motion defined by the mapping
$$\begin{aligned}
\zeta(\mathbf{x}) &= \mathbf{R}\, \mathbf{x} + \mathbf{c} \label{urb}
\intertext{with rigid body displacement}
\mathbf{u}_{rb}(\mathbf{x}):= \zeta(\mathbf{x}) &- \mathbf{x} = \big(\mathbf{R} -\mathbf{I}\big)\, \mathbf{x} + \mathbf{c} \,,
\intertext{where $\mathbf{R}$ is a rotation tensor, i.e., $\mathbf{R}^T \mathbf{R}=\mathbf{I}$ with positive determinant, and $\mathbf{c}$ is a constant vector. The corresponding nonlocal deformation gradient is}
%
\mathbf{I} + \tilde{\nabla}\mathbf{u}_{rb}
&= \mathbf{I} + \big(\mathbf{R}-\mathbf{I}\big) = \mathbf{R}\,,\end{aligned}$$
where we used for the first equality so that $$\big( \mathbf{I} + \tilde{\nabla}\mathbf{u}_{rb} \big)^T\big( \mathbf{I} + \tilde{\nabla}\mathbf{u}_{rb} \big) = \mathbf{I}$$ as required for the nonlocal strain to be invariant under rigid body motion.
Nonlocal strain over a bounded domain
-------------------------------------
Let $\Omega$ represent a bounded open domain partitioned into $M$ non-overlapping subdomains $\Omega_i$ with boundary $\Gamma$ such that $$\begin{aligned}
\Omega = \overset{M}{\underset{i = 1}{\bigcup}} \; \Omega_i \;.
\end{aligned}$$ In the context of DIC, $\Omega_i$ represents the decomposition of data as either pixels or collections of pixels with an associated area.
Let $\mathbf{u}_i$ denote the displacement over $\Omega_i$. Since the value of $\mathbf{u}_i$ is constant over $\Omega_i$, the displacement vector field over $\Omega$ is $$\begin{aligned}
\label{dispuO}
\mathbf{u}(\mathbf{x}) = \sum_{j=1}^M \mathbf{u}_i \, \mathds{1}_{\Omega_j}(\mathbf{x})\end{aligned}$$ where the indicator function $\mathds{1}_{\Omega_j}$ is given by $$\mathds{1}_{\Omega_j}(\mathbf{x}) :=
\begin{cases}
1 & \mathbf{x} \in \Omega_j \,,\\
0 & \mathbf{x} \notin \Omega_j\,.
\end{cases}$$ Inserting the DIC displacement into results in
\[nl-discrete-deriv-nd\] $$\begin{aligned}
\tilde{\nabla}\, \mathbf{u} (\mathbf{x}) & = - \sum_{j=1}^M \mathbf{u}_j \otimes \int_{\Omega_j} \nabla \psi(\mathbf{y} - \mathbf{x}) \, \text{d}\mathbf{y}\ \qquad \mathbf{x} \in \Omega\,,
\intertext{or equivalently,}
\tilde{\nabla}\, \mathbf{u} & = - \sum_{j=1}^M \mathbf{u}_j \otimes \tilde{\nabla}\,\mathds{1}_{\Omega_j} \text{ over } \Omega\,.\end{aligned}$$
These two relations explain that the function $\nabla \psi(\mathbf{x})$ need not be evaluated directly, but rather only its integral over the subdomain $\Omega_j$ is needed. This is an important result, given that $\nabla \psi(\mathbf{x})$ is not defined at the origin or along the axes.
A special case occurs when $\mathbf{c} = \mathbf{u}_1=\mathbf{u}_2=\cdots=\mathbf{u}_M$ so that $\mathbf{u}(\mathbf{x}) = \mathbf{c} \, \mathds{1}_{\Omega}(\mathbf{x})$ and $$\begin{aligned}
\tilde{\nabla}\, (\mathbf{c} \, \mathds{1}_{\Omega}) & = \mathbf{c} \, \tilde{\nabla}\, \mathds{1}_{\Omega}\,.
%\intertext{where}
%\tilde{\nabla}\, \mathds{1}_{\Omega}(\mathbf{x}) & = - \mathbf{1} \otimes \int_{\Omega} \nabla \psi(\mathbf{y} - \mathbf{x}) \, \text{d}\mathbf{y}\, \qquad \mathbf{x} \in \Omega\,.
\intertext{If we suppose that $\Omega$ is a region such that for any $\mathbf{x} \in \Omega$, there exists skew parameters $\gamma_1$ and $\gamma_2$ that satisfy the constraint \eqref{gamma-constraint} (the support of $\psi(\mathbf{x})$ is contained within $\Omega$) then}
\tilde{\nabla}\, \mathds{1}_{\Omega}(\mathbf{x}) & \equiv \mathbf{0} \qquad \mathbf{x} \in \Omega\,.\end{aligned}$$ For example, if $\Omega$ is a rectangle, then the above conditions can be fulfilled since either $\gamma_1$ or $\gamma_2$ (or both) can be selected appropriately when $\mathbf{x} $ is close to the boundary, $\Gamma$, of $\Omega$.
Discrete approximation
----------------------
The invariance under rigid body motion for the discrete approximation to $\tilde{\mathbf{E}}$ also holds assuming a sufficiently accurate quadrature rule is used to integrate $ \nabla \psi$ and $\mathbf{u}$. This can be easily verified. Consider a square, two-dimensional domain of size $L \times L$, where $L = 100$ units, with a synthetic rigid body displacement field of the form $$\mathbf{R} = \left[ \begin{array}{cc}
\text{cos}\;\theta & -\text{sin}\;\theta \\
\text{sin}\;\theta & \text{cos}\;\theta \end{array} \right] \; , \qquad \mathbf{c} = \mathbf{0} \; ,$$ where $\theta = 45$ degrees. The domain is discretized into cells (pixels) of size one unit $\times$ one unit. Computing the nonlocal strain according to for various values of $\epsilon$ results in the strain values given in Table \[tab:RBMError\], which shows that the strain is indeed negligible.
$\epsilon$ $\tilde{\mathbf{E}}_{11}$ $\tilde{\mathbf{E}}_{12}$ $\tilde{\mathbf{E}}_{21}$ $\tilde{\mathbf{E}}_{22}$
------------ --------------------------- --------------------------- --------------------------- ---------------------------
10 5.9e-15 6.9e-15 6.9e-15 1.1e-14
8 5.3e-15 6.7e-15 6.7e-15 1.2e-14
6 7.0e-15 9.1e-15 9.1e-15 1.7e-14
4 9.9e-15 1.0e-14 1.0e-14 1.8e-14
2 2.0e-14 2.0e-14 2.0e-14 2.7e-14
: $L_2$-norms of the nonlocal strain for various support sizes\[tab:RBMError\]
As an aside, when the displacement is given by , care needs to be taken to align the discontinuity of $\nabla \psi$ over the pixel containing the origin with a pixel boundary for the quadrature involving $\nabla \psi$ since by the displacement is constant over $\Omega_j$. We also remark that the nonlocal strain at a point depends upon all the values of the displacement field surrounding the point. The quadrature for the discrete approximation of the nonlocal strain is akin to using an extremely high order differencing scheme, except that the stencil is over the full area of the nonlocal support, not just along the coordinate directions aligned with the positive and negative $x_1$ and $x_2$ axes (as it is in finite difference difference approximations to the classical gradient).
Virtual strain guage
--------------------
For the purpose of comparison, we include here a brief description of how strain is typically calculated in the context of DIC. There are a number of nuances related to the various methods available, but in general, they fall into two categories: those that apply finite differencing to the displacement data and those that that use polynomial smoothing. In either case, filtering may also be applied. In the case of local polynomial smoothing, a polynomial function is fit to the displacement data over a particular region. This region is defined by the strain window size, which can be considered as a regularization parameter for a given polynomial basis. It can be shown that this process effectively acts as a low pass filter. The resulting strain measure is highly sensitive to the virtual strain gauge (VSG) size (and the subset size), where the VSG size is a function of the strain window and step size. For small VSG sizes, the resulting strain field will be more accurate, but contain a large amount of noise. Conversely, a large VSG size limits the amount of noise, but tends towards less accuracy. In the numerical examples below, we perform a comparison between the VSG approach and the nonlocal strain measure to illustrate the differences between the two.
Numerical Results
=================
In this section we demonstrate the performance of the nonlocal strain for a number of examples including verification problems and results for images taken from experiments.
Non-fully-differentiable function on a bounded domain
-----------------------------------------------------
The nonlocal strain, as detailed in §\[sec:nl-str\], is well defined within the domain including at pixel interfaces and for fields that are continuous but not differentiable. To illustrate these two features, consider a square domain of size $L \times L$ units with a displacement profile of the form $$\begin{aligned}
u_1(\mathbf{x}) &= \begin{cases}
x_1 & 0 < x_1 \leq L/2 \,\\
L - x_1 & L/2 < x_1 < L\,
\end{cases} \\
u_2(\mathbf{x}) &= \begin{cases}
x_2 & 0 < x_2 \leq L/2 \,\\
L - x_2 & L/2 < x_2 < L\,
\end{cases}
\end{aligned}$$ Note that $\mathbf{u}(\mathbf{x})$ is not differentiable along the lines $x_1 = L/2$ and $x_2 = L/2$. The displacement profile is shown in Figure \[fig:HatDisp\]. The four components of the nonlocal strain tensor are shown in Figure \[fig:HatStrain\]. The nonlocal strain does not suffer from boundary effects if the support of $\boldsymbol{\alpha}_\epsilon$ is weighted properly towards the interior of the domain using the skew parameters $\gamma_1$ and $\gamma_2$. It is also important to point out that in this example the discontinuity lies on the interface between two subdomains $\Omega_j$.
![(a) Non-fully-differentiable displacement data $u_1(\mathbf{x})$ (b) $u_2(\mathbf{x})$[]{data-label="fig:HatDisp"}](HatDisp.pdf)
![Components of the nonlocal strain tensor applied to the non-fully-differentiable displacement field (a) $\tilde{\mathbf{E}}_{11}$ (b) $\tilde{\mathbf{E}}_{12}$ (c) $\tilde{\mathbf{E}}_{21}$ (d) $\tilde{\mathbf{E}}_{22}$[]{data-label="fig:HatStrain"}](HatStrain.pdf)
Nonlocal strain and noisy data
------------------------------
This numerical example demonstrates the effectiveness of the nonlocal strain measure for noisy data. The vector displacement profile for this example was generated by adding noise (with mean 0 and standard deviation 5%) to a smooth function over a square domain of size 100 by 100 pixels. The displacement profile is given as $$\label{noisy-u}
\begin{aligned}
u_1(\mathbf{x}) &= \text{sin}\left( \frac{a}{2\pi} x\right)\text{cos}\left( \frac{a}{2\pi} y\right) + \sigma_{n1} \\
u_2(\mathbf{x}) &= \text{cos}\left( \frac{a}{2\pi} x\right)\text{sin}\left( \frac{a}{2\pi} y\right) + \sigma_{n2} \; ,
\end{aligned}$$ where $a=1/5$ and $\sigma_n$ is the amount of noise. Figure \[fig:NoisyDisp\] shows the noisy displacements plotted over the domain.
![(a) Noisy displacement data $u_1(\mathbf{x})$ (b) $u_2(\mathbf{x})$[]{data-label="fig:NoisyDisp"}](NoisyDisp.pdf)
For the purpose of comparison, Figure \[fig:NoisyStrain\] shows the nonlocal strain component $\tilde{\mathbf{E}}_{11}$ computed for the noisy displacement data vs. the classical strain component $\mathbf{E}_{11}$ for the vector field with the noise suppressed, i.e., $\sigma_{n1} = \sigma_{n2} = 0$.
The nonlocal strain was computed using a support size, $\epsilon = 20$ units. This example demonstrates the robustness of the nonlocal strain measure in the context of noisy data and that the nonlocal strain is in agreement with the classical strain. Similar results hold for the other nonlocal strain components, although not shown here.
![(a) Classical strain component $\mathbf{E}_{11}$ computed by taking the derivative of the displacement profile with no noise (b) nonlocal strain component $\tilde{\mathbf{E}}_{11}$ computed for the noisy displacement profile.[]{data-label="fig:NoisyStrain"}](NoisyStrainXX.pdf)
The error in each component of the nonlocal strain is shown in Figure \[fig:NoisyStrainError\].
![Error in the nonlocal strain components (a) $\tilde{\mathbf{E}}_{11}$ (b) $\tilde{\mathbf{E}}_{22}$ (c) $\tilde{\mathbf{E}}_{12}$.[]{data-label="fig:NoisyStrainError"}](NoisyStrainError.pdf)
DIC Challenge: synthetic strain concentrations of various periods and amplitudes
--------------------------------------------------------------------------------
This example shows the nonlocal strain measure as applied to one of the DIC Challenge [@DICChallenge] image sets with synthetic pixel displacements applied to induce strain concentrations of various periods and amplitudes. The data used in this example represent Sample 15 from the DIC Challenge images. The details of how the images were constructed are given on the DIC Challenge website. The reference image is `Ref.tif` and the deformed image is `P200_K50.tif`. The correlation parameters used are given in Table \[tab:Sample15Params\]. A step size of one pixel was used to enable evaluating the VSG and nonlocal strain measures with small support or small strain window size. It should be pointed out that the subset solutions are not independent at this step size [@Ke].
The displacement profile for a vertical line drawn through the images at $x = 1000$ pixels is shown in Figure \[fig:Sample15Disp\]. Figure \[fig:Sample15Strain\] shows the nonlocal strain results for this data set for various values of $\epsilon$ compared to the VSG method with various strain window sizes. To compute the VSG results, the Ncorr [@Ncorr] software was used.
As expected, for larger values of $\epsilon$ the high frequency content of the nonlocal strain measure is effectively filtered out, but note the overall structure of the strain profile is preserved (multiple sub-peaks in the main strain concentration are captured). Using a larger strain window size also smooths the result, but the structure is not preserved. Reported in Figure \[fig:Sample15Strain\] is the maximum strain computed for each method, for each nonlocal support or strain window size. Taking the single pixel nonlocal support value of the maximum strain as the most accurate, the loss in accuracy going to a support size of 20 pixels is 9.4%, whereas the loss in accuracy of the VSG method is 17.0%, suggesting that smoothing the strain data with the VSG method leads to greater loss in accuracy than using the nonlocal strain.
Parameter Value
----------------------------------------------------- ---------------------
Subset size 25
Step size 1
Interpolation Bicubic
Matching criterion SSD
Virtual strain gauge size Varies from 1 to 20
Test function support, $\epsilon_1$ or $\epsilon_2$ Varies from 1 to 20
: Correlation parameters used for Sample 15 images\[tab:Sample15Params\]
![Displacement profile of the DIC Challenge, Sample 15 data set for a vertical line drawn through the images at $x = 1000$ pixels.[]{data-label="fig:Sample15Disp"}](Sample15Disp.pdf)
![Nonlocal strain calculated for the DIC Challenge, Sample 15 images. (a) $\epsilon = 1$ pixel, strain window size = 1 (b) $\epsilon = 5$ pixels, strain window size = 5 (c) $\epsilon = 20$ pixels, strain window size = 20.[]{data-label="fig:Sample15Strain"}](Sample15Strain.pdf)
DIC Challenge: experimental data for a plate with a hole being loaded in tension
--------------------------------------------------------------------------------
This example involves computing the nonlocal strain from images taken of a steel plate with a small hole being loaded in tension. The images were obtained from the DIC Challenge website and represent Sample 12. Table \[tab:PlateHoleParams\] lists the parameters used in the correlation to obtain the displacement values.
Parameter Value
----------------------------------------------------- ----------------------------
Subset size 35
Step size 1
Interpolation Bilinear
Matching criterion SSD
Test function support, $\epsilon_1$ or $\epsilon_2$ Varies from 5 to 50 pixels
: Correlation parameters for the plate with a hole images\[tab:PlateHoleParams\]
Note that a step size of one was chosen to introduce the maximum amount of high frequency content into the displacement solution. It is well known that increasing the step size leads to a smoother strain profile, but decreases accuracy as it introduces artificial dissipation of the displacement gradients. Using a virtual strain gauge approach to calculate the strain inevitably leads to a tradeoff between accuracy and smoothness. Large strain gradients cannot be captured if too much smoothing is introduced, leading to poor accuracy. Conversely, resolving steep gradients leads to a highly oscillatory solution.
Figure \[fig:PHStrain\] shows the strain calculated using equation with the test function of equation for varying sizes of the test function support, $\epsilon$. It can be seen from Figure \[fig:PHStrain\] that $\epsilon$ acts as a filter for high frequency content, but does not introduce dissipation, leading to a *smooth and accurate strain profile*.
![Plot of the nonlocal strain for a plate with a hole being loaded in tension for various values of $\epsilon$ (the width of the operator’s support). The left image shows color contours of the principle strain as calculated using a virtual strain gauge in VIC 2D. Note that increasing $\epsilon$ has the effect of filtering high frequency noise in the displacement solution.[]{data-label="fig:PHStrain"}](PHStrain.pdf)
Conclusions
===========
We have introduced a new, nonlocal measure of strain for use in digital image correlation that is both noise filtering and appropriate for discontinuous displacement fields. With regard to noise, as opposed to curve fitting processes that discard data, the nonlocal process incorporates the full data set, but diminishes the effect of outliers by a distributed weighting of values that has a smoothing effect. In this way, the nonlocal strain maintains high fidelity data content without oscillations. Another feature of the nonlocal strain measure is that it gives the same result as the classical strain measure for differentiable displacement fields, but also provides a meaningful value when the displacement field is discontinuous (in which case the classical strain is not defined). We have also shown that this strain measure is invariant under rigid body motion, which is necessary to prevent non-physical strains from arising due to large rotations or motion without deformation. Through a number of academic and experiment-based numerical examples we have demonstrated the effectiveness of this strain measure for both problems with analytic solutions and problems of engineering relevance. The results reveal that the nonlocal strain measure provides smoothing with less loss of accuracy than the VSG method. Ultimately, this work provides a framework to connect material characterization with experiments in a way that more fully incorporates data content and is robust enough to treat data with noise and discontinuities.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.
This work was also supported in part by the Institute for Structural Engineering at Stellenbosch University. Their support is gratefully acknowledged.
|
---
abstract: 'The Magellanic Stream ($\approx 2\times 10^9 \Msun\; [d/55\kpc]^2$) encircling the Galaxy at a distance $d$ is arguably the most important tracer of what happens to gas accreting onto a disk galaxy. Recent observations reveal that the Stream’s mass is in fact dominated (3:1) by its ionised component. Here we revisit the origin of the mysterious recombination emission observed along much of its length that is overly bright ($\sim 150-200\mR$) for the known Galactic ultraviolet (UV) background ($\approx 20-40\mR\; [d/55\kpc]^{-2}$). In an earlier model, we proposed that a slow shock cascade was operating along the Stream due to its interaction with the extended Galactic hot corona. We find that, for a smooth coronal density profile, this model can explain the bright emission if the coronal density satisfies $2 \times 10^{-4} < (n / \pcc) < 4 \times 10^{-4}$ at $d = 55$ kpc. But in view of updated parameters for the Galactic halo and mounting evidence that most of the Stream must lie far beyond the Magellanic Clouds ($d>55\kpc$), we revisit the shock cascade model in detail. At lower densities, the gas is broken down by the shock cascade but mostly mixes with the hot corona without significant recombination. At higher densities, the hot coronal mass (including the other baryonic components) exceeds the baryon budget of the Galaxy. If the emission arises from the shock cascade, the upper limit on the smooth coronal density constrains the Stream’s mean distance to $\lesssim 75$ kpc. If, as some models indicate, the Stream is even further out, either the shock cascade is operating in a regime where the corona is substantially mass-loaded with recent gas debris, or an entirely different ionization mechanism is responsible.'
author:
- 'Thor Tepper-García, Joss Bland-Hawthorn, and Ralph S. Sutherland'
bibliography:
- '/Users/tepper/references/complete.bib'
title: 'The Magellanic Stream: break up and accretion onto the hot Galactic corona'
---
Introduction {#sec:intro}
============
The Galaxy is surrounded by a vast amount of neutral gas in the form of high-velocity clouds [HVC; @oor70a]. Formally, these are neutral gas structures at a Galactic latitude $\vert b \vert > 30^{\circ}$ having kinematic properties not consistent with the overall Galactic rotation [@wak01a]. We now recognize that many of these make up the Magellanic Stream [MS; @die71a; @wan72a; @mat74a], roughly $2\times 10^9 \Msun\; [d/55\kpc]^2$ of gas [@fox14a] that has been stripped from the Magellanic Clouds (MCs), two dwarf galaxies in orbit around the Galaxy at a mean distance $d \approx 55$ kpc [@wal12a; @gra14a].
The Magellanic Stream is ideal to study the environment of the Galaxy. Radio ( 21cm) surveys show that the Stream extends for 200$^\circ$ across the Southern Galactic Hemisphere [@nid10a], and absorption line measurements towards distant quasars indicate a cross-section of roughly one quarter of the whole sky [@fox14a]. Dynamical models [@bes07a; @gug14a] constrained by accurate measurements of the proper motions of the MCs [@kal13a] agree that they move on a highly eccentric orbit, and that the MS spans a wide range in Galactocentric distance, from its source in the MCs system at roughly $55\kpc$ to $ 80 - 150 \kpc$ above the South Galactic Pole (SGP) all the way to the tip of the tail.
Given its relative proximity, the MS has been observed across the electromagnetic spectrum. Beyond , it has been detected in molecular [@ric01b] and ionized [@lu94a; @fox05b; @sem03a] gas. A shadowing experiment aimed at measuring the coronal soft X-ray emission discovered that the emission is enhanced in the direction of the MS [@bre09a]. Recombination optical emission () was detected for the first time by @wei96a and later confirmed by others [@rey98a; @put04a; @mad12a]. But despite repeated attempts, to date, no stars have been discovered at any location along the Stream [e.g. @ost97a]. The data obtained by the recently completed Hubble Space Telescope (HST) Cosmic Origins Spectrograph (COS) UV absorption survey of the MS [@fox13a; @ric13a; @fox14a] indicates that the Stream is dominated by ionized gas, as was first proposed by [@bla07a]. These data collectively suggest the existence of a strong interaction between the Stream gas and the hot halo [or corona; @spi56a] of the Galaxy.
Radiative hydrodynamic models [the ’shock cascade’; @bla07a] indicate that the MS-halo interaction may be strong enough to explain observed disruption of the Stream [@nid10a] and its high ionization fraction. At the same time, the presence of coherent and strong enough magnetic fields [@put98a] may stabilise the gas against severe ablation and provide thermal insulation to inhibit total evaporation of the neutral clouds [@mcc10b]. Whether such a shielding mechanism is operating all along the Stream is currently unknown. But without it, the Stream is likely to evaporate and mass-load the Galactic halo with a substantial amount of baryons [@mcc15b].
Therefore, if we are to understand the complex environment of galaxies and how gas settles into galaxies [@hei09b], we need to explain first the observed properties of the Magellanic Stream within the framework of a multiphase hydrodynamical model. Elucidating the mechanism behind the bright spots of emission observed along the Magellanic Stream has proven to be particularly challenging [see @bla07a]. To date, there have been two competing models [see also @kon01a]: (i) the slow shock cascade discussed above; (ii) a new interpretation invoking a powerful flare of UV radiation from the Galactic Centre (GC) powered by the accretion of material onto the central black hole in Sgr A$^*$ [@bla13a].
The GC flare model is inspired by two important circumstances: 1) The discovery of the $\gamma$-ray emitting bubbles discovered by the [*Fermi*]{} satellite extending roughly 50$^\circ$ ($10 \kpc$) from the GC [@su10a]; 2) The observation that the brightest optical emission along the Stream is confined to a cone with half-angle $\theta_{1/2} \approx 25^\circ$ roughly centred on the SGP [@mad12a]. The GC flare model has found support from the timescales and energy budget required to ionise the Stream, which are consistent with the results from jet-driven numerical models of the [*Fermi*]{} bubbles [@guo12a]. More recently, [@fox14a] have discovered that the ionisation levels over the SGP require an energetically harder ionising spectrum than elsewhere along the Stream, with the exception of a localised region near the LMC.
The shock cascade model, on the other hand, explains the observation that the brightest detections lie at the leading edges of the clouds that make up the MS [@wei02a]. However, this model may fail to produce the observed emission levels [*if*]{} the distance to the Stream at the SGP significantly exceeds the traditional view of $d = 55 \kpc$, as indicated by most orbit calculations for the Magellanic Clouds over the past five years [@bes12a; @gug14a]. The shock cascade model is strongly dependent on the density structure of the Galactic hot halo, and it assumes that the coronal density smoothly declines with Galactocentric distance as $\propto r^{-2}$. But it now appears that both of these assumptions may be false justifying our efforts to revisit the shock cascade model.
The goal of this study is to investigate the strength of the recombination () emission produced by the interaction of the MS gas with the Galactic corona exploring a range of Galactocentric distances and different halo parameters, i.e., adopting different density profiles and temperatures of the gas sitting at rest in a fixed dark-matter (DM) potential. Note that throughout the paper we assume a flat, dark-energy- and matter (baryonic and cold dark-matter; CDM) dominated Universe, and a cosmology defined by the set of parameters (relevant to this work) $h = 0.7$, $\Omega_m = 0.3$, and $\Omega_{\Lambda} = 0.7$.
A model of the Galactic halo {#sec:halo}
============================
The density and temperature structure of the Galactic corona is largely determined by the underlying gravitational potential. The potential, in turn, is determined by the three main components of the Galaxy: the stellar bulge, the stellar and gaseous disc, and the DM halo. Given the mass of the bulge ($\sim 10^{10} \Msun$) and the disc [$\sim10^{11} \Msun$; @kaf14a], and their size, these components are expected to dominate the Galactic potential only at $r \lesssim 5 \kpc$ and at $r \lesssim 15 \kpc$, respectively. In other words, with exception of the inner $\sim 5 \kpc$ [@weg15a], the DM halo dominates the Galactic potential at all distances, with a similar contribution from the disc at scales comparable to its length. The distance to the nearest point of the MS is believed to be $d \approx 55 \kpc$, which is the average distance to the LMC [$\approx 50 \kpc$; @wal12a], and the SMC [$\approx 60 \kpc$; @gra14a]. Consequently, in modelling the interaction of the Magellanic Stream with the Galactic corona, it is safe to ignore the contributions to the potential from the bulge and the disc, and to focus instead on the DM halo only.
The Galactic DM halo {#sec:dmhalo}
--------------------
We model the DM halo of the Galaxy assuming it is well described by a single-component isothermal sphere. We opt for such a model given its solid physical foundation, with properties that can be derived from first principles starting from a few basic assumptions [see e.g. @bin08a].
The [*scale-free*]{} potential, $W$, and the [*scale-free*]{} density, $y$, of an isotropic isothermal (DM) sphere are defined by [@kin66a]: $$\begin{aligned}
\frac{ 1 }{ x^2 } \frac{ d }{ dx } \left( x^2 \frac{ dW }{ dx}\right) & = -9 ~y_{ \textnormal{\sc iso} }(x) \, , \label{eq:ode1} \\
y_{ \textnormal{\sc iso} }(x) & = \exp\left[ W(x)\right] \, . \label{eq:ode2} \end{aligned}$$ where $x$ is a scale-free coordinate.
This system of equations has no analytic solution, but it can be integrated numerically to values $x \ll 1$ to (nearly) arbitrary precision. Appropriate boundary conditions are, for example, the requirement that both the potential and the force vanish at the origin, i.e. $W(0) \equiv 0$ and $W'(0) \equiv 0$, respectively, which implies $y_{ \textnormal{\sc iso} }(0) = 1$. Note that $W(x) \leq 0$ for $x \geq 0$.
The connection between the scale-free quantities and their physical counterparts is given by $$\label{eq:iso}
r = r_c x \, ; \quad \rho(r) = \rho_c y_{ \textnormal{\sc iso} }(x) \, ; \quad \psi(r) = \sigma^2 W(x) \, .$$ Here, $\rho_c$, $r_c$, and $\sigma$ are the central density, the core radius, and the constant velocity dispersion, respectively. A solution corresponding to a particular physical system is obtained by fixing two of these three parameters (or any other two independent physical quantities of the system, for that matter); the third parameter is tied to the other two through the relation $r_c^2 = 9 \sigma^2 / 4 \pi G \rho_c$ [@kin66a].
{width="48.00000%"} {width="48.00000%"}
Alternatives to the isothermal sphere as a viable choice to describe a self-gravitating system of collisionless, DM particles, i.e. a DM halo include the @nav96a [NFW] model, $$\label{eq:nfw}
y_{ \textnormal{\sc nfw } }(r) = (r / r_s)^{-1} (1 + r / r_s)^{-2} \, ,$$ where $r_s$ is a characteristic scale length; and the @ein65a profile, $$\label{eq:ein}
y_{ \textnormal{\sc ein} }(r) = \exp{ \left\{ - \frac{ 2 }{ \alpha } \left[ \left( \frac{ r }{ r_s } \right)^{ \alpha } - 1 \right] \right\} }$$ where $\alpha$ is a free parameter. Regardless of the model adopted, if the density profile $\rho(r)$ is known, a general formalism can be applied which allows us to calculate scale-free quantities [q.v. @ste02b] on a case-by-case basis to describe a particular physical system.
In order to specify our isothermal DM halo, and to compare its properties to the NFW and Einasto models, we proceed as follows. Based on the results of @kaf14a, who assume the dark halo of the Galaxy to be of NFW type, we fix the virial mass $\Mvir$ and the concentration $\xvir \! \equiv \! \rvir\ \! / r_c$ of the NFW dark halo to $\Mvir \! = 10^{12} \Msun$, and $\xvir = 15$, respectively. Our adopted value for the virial mass implies[^1] $\rvir \approx 260 \kpc$ and $\vvir \equiv (G \!\Mvir / \!\rvir\ \!)^{1/2} \approx 130 \kms$, respectively. Then we calculate the scale parameters for the isothermal and the Einasto DM halos by requiring that the virial mass and the [*physical*]{} potential at the virial radius in each case match the corresponding values for the NFW dark halo. Table \[tab:dmhalo\] summarises our assumed values (in bold font face) and lists the derived values of the relevant scaling parameters for each of the three halo models. The potential and corresponding mass for each model are shown in Figure \[fig:dmhalo\]. As can be seen, the relevant properties of an isothermal DM halo are very similar to the NFW and Einasto models. We note that the virial temperature is very similar across models, which is of relevance for the discussion in later sections.
[lccc]{} & Isothermal & Einasto$^f$ & NFW \
\
\[$10^{12}$ \] & [**1.00**]{} & [**1.00**]{} & [**1.00**]{}$^a$ \
\[\] & 259 & 259 & 259 \
\[\] & 129 & 129 & 129 \
& 433 & 13.8 & [**15.0**]{}$^a$ \
$\psi_{vir} ~[10^{15} \cm^2 \s^{-2}]$ & [**-1.11** ]{} & [**-1.11**]{} & -1.11 \
$r_{s} ~[\!\kpc]$$^b$ & 0.60 & 18.8 & 17.3 \
$v_{s} ~[\!\kms]$$^c$ & 90.1 & 82.4 & 104 \
$M_{h}(\leq\!\!\rvir) ~[10^{10} \Msun]$$^d$ & 1.41 & 3.03 & 1.32 \
$T_{\textnormal{\sc dm}} ~[10^5 \K]$$^e$ & 5.81 & 4.85 & 7.73 \
{width="48.00000%"} {width="48.00000%"}
The Galactic corona {#sec:gashalo}
-------------------
It has long been known that the potential well of the Galaxy is filled with a diffuse, hot, gaseous component [the corona; @spi56a]. However, the origin of this gas, its thermodynamic state, its physical properties, its extension and hence its total mass, are still unknown. A recent attempt to constrain the density structure of the corona implies a total gas mass of the Galaxy around $\sim10^{10} \Msun$ within $\rvir$ [@mil13a; @mil15a], which is consistent with other estimates [e.g. @gat13a]. This important result relies on the assumptions that the corona is smooth, in collisional ionisation equilibrium, and isothermal, with a temperature of $2 \times 10^{6} \K$. While the gaseous halo of the Galaxy is most likely not smooth nor strictly isothermal, the mean temperature of the gas inferred from its associated X-ray emission appear remarkably uniform across the sky [@hen13e; @hen14e].
Given this circumstance, we model the hot halo of the Galaxy as a single-phase, [*smooth*]{}, spherically symmetric component consisting of an ideal gas at a constant temperature $T_{h}$, in hydrostatic equilibrium with the DM potential $\psi$ of an isothermal sphere. We further assume that the self-gravity of the gas is negligible, which is justified in the case of the Galaxy given that the inferred gas mass of the hot halo is on the order of $10^{-2}$ the mass of the DM halo [@sut98a]. Under these assumptions, the total particle density of the corona is given by $n(r) = n_0 \exp[ \psi(r) / a^2 ]$. Here, $a^2 = k T_{h} / \mu m_u$ is the isothermal sound speed, $\mu$ is the mean molecular weight, $m_u$ is the atomic mass unit, and $k$ is Boltzmann’s constant.
The virial ‘temperature’ of the dark matter halo follows from the equivalence $a^2 \equiv \sigma^2$, $$\label{eq:tdm}
T_{\textnormal{\sc dm}} \equiv \frac{ \mu \, m_u }{ k } \sigma^2 \, .$$ It worth emphasising that the above is is merely an [*equivalent*]{} temperature, as $\mu \, m_u$ is not literally the DM particle mass (which is currently unknown). Table \[tab:dmhalo\] lists the virial temperatures of the different models.
It is straightforward to show that under these assumptions the distribution of the hot gas in the potential is effectively governed by the [*thermal ratio*]{} [@cav76a][^2] $$\label{eq:tau}
\tratio \equiv \frac{ T_{\textnormal{\sc dm}} }{ T_{h} } = \sigma^2 / a^2 \, ,$$ such that $$\label{eq:partdens}
n(r) = n_0 \exp{\left[ \!\tratio W(r / r_c) \right]} \, ,$$ The total gas mass within $r$ for a particular value of follows from the integral of $n$ over the appropriate volume, $$\label{eq:gasm}
M_{h}(r) =4 \pi \, \mu ~m_u \int_0^r \!\!\! n(x) x^2 dx \, .$$
{width="48.00000%"} {width="48.00000%"}\
{width="48.00000%"} {width="48.00000%"}
Clearly, the density field of gas in hydrostatic equilibrium with a fixed isothermal potential will be different for different gas temperatures. If the gaseous halo is ‘hotter’ than the DM particles, i.e. $\tratio < 1$, then the density falls off more gently with radius. Values $\tratio > 1$ on the other hand reflect the fact that the gas has cooled [*below*]{} the virial temperature of the DM halo, and is hence more concentrated. Large values thus imply lower densities in the outer region of the halo, which in turn leads to weaker hydrodynamic interactions (at a fixed distance).
Since the the virial temperature of our DM halo model is fixed, we control the temperature of the halo gas by varying . In the following we consider values of $\tratio$ in the range $[0.5, \, 1.5]$ only, and choose the set of values $\tratio \in \{1.5, \, 1.0, \, 0.75, \, 0.5 \}$ as representative of this range. These values imply gas temperatures $T_{h} \sim 10^6 \K$ (see Table \[tab:gashalo\]), consistent with estimates of the temperature of the Galactic hot halo [@sno00a]. We ignore values of $\tratio < 0.5$, since these yield overly shallow density profiles, which are inconsistent with observations (see Figure \[fig:dens\]). Similarly, we ignore values of $\tratio > 1.5$ because these imply gas temperature which are too low compared to observations (see Section \[sec:ic\]).
We compare in Figure \[fig:dmhalo2\] the density profiles of gas at rest in different DM potentials, all scaled to a fiducial value $n = 2 \times 10^{-4} \pcc$ at 55 kpc (see below and Section \[sec:ic\]). For a given , the gas density profile within a NFW DM potential (as defined by the respective parameters in Table \[tab:dmhalo\]) is steeper compared to the isothermal DM potential, but shallower than the Einasto DM halo. The difference in the density profile across the models reflects the difference in the potential in the radial range of interest. In contrast, the gas mass enclosed within for all three models and a given are comparable (with exception of the mass for the Einasto model with = 0.5 which is well above $10^{11} \Msun$). Therefore, of all three models at a fixed , the isothermal sphere halo leads to [*the highest gas density at any given distance*]{} beyond 55 kpc for roughly the same gas mass, and hence to the [*strongest*]{} hydrodynamic interaction in the outer halo.
The different density profiles of gas sitting in an isothermal DM halo for our adopted values of (equation \[eq:partdens\]), scaled to $n = 2 \times 10^{-4} \pcc$ at 55 kpc, are shown in the top-left panel of Figure \[fig:dens\]. For comparison, we include there a set of values of the halo density at various Galactocentric distances obtained from observations using a variety of methods (see Section \[sec:ic\]). It is reassuring that all our models are fairly consistent with these measurements.
The top-right panel of Figure \[fig:dens\] displays the mass enclosed within a given radius (equation \[eq:gasm\]) for each of the models shown in the top-left panel. As a consistency check, we compare the model masses to the upper limit on the Galactic baryion budget within set by the universal mean baryon-to-total mass ratio $f_b \equiv \Omega_b / \Omega_m$. The most recent estimates of the baryon and cold dark matter mass densities $\Omega_b h^2 = 0.02205 \pm 0.00028$ and $\Omega_c h^2 = 0.1199 \pm 0.0027$ imply $f_b \approx 0.16$. This value, together with the total mass of the Galaxy ($M_{tot} \sim 10^{12} \Msun$) allow for a maximum gas mass of the Galaxy within below $10^{11} \Msun$. Clearly, all our models result in masses within which are below this limit, indicated by the grey hatched area in the right panel of Figure \[fig:dens\]. Note that a density profile shallower than the = 0.5 model, or a value of $n(55 \kpc)$ significantly higher than our fiducial value, would result in a gas mass largely inconsistent with these constraints. We include in this figure the range of hot halo masses inferred from observations by @mil15a, comparable to the masses estimated by others [e.g @gat13a], and the somewhat lower values inferred by [@sal15a]. Note that these estimates are all directly comparable to our model results since in all cases a smooth, monotonically decreasing density profile has been assumed. All our models, with exception of the = 0.5 model, predict masses within that are consistent with the inferred mass.
Thus, the model = 0.5 appears to be marginally consistent both with the mean density of the Galactic corona at large distances, and with the constraint on the total gas mass of the Galaxy. The model = 1.5, although compatible with these constraints, appears too concentrated to be a plausible description of the Galaxy’s halo. In contrast, the models = 0.75 and = 1 both display the best performance in terms of both the density profile and the gas mass enclosed within the virial radius of the Galaxy, although the former model yields a slightly hotter and more massive gas halo. In addition, these models reproduce by construction the relevant properties of the Galactic DM halo. Thus. we consider these models in particular provide a fully self-consistent and well founded description of the Galactic hot halo, despite its idealized nature.
$T_{h}$ \[$10^6 \K$\] $M_{h}(\!\rvir)$ \[$10^{10} \Msun$\]
--- ----------------------- ---------------------------------------- --------
0 5 1.16 3.36
0 75 0.77 2.06
1 0 0.58 1.41
1 5 0.39 2.33
: Isothermal gaseous halo properties[]{data-label="tab:gashalo"}
Numerical experiment {#sec:exp}
====================
We simulate the passage of a stream of gas emulating the Magellanic Stream in its orbit through the Galactic hot halo by means of a 3-dimensional (3D) ‘wind-tunnel experiment’, expanding on the work by @bla07a. In brief, we place a warm and essentially neutral, [*fractal*]{} gas cloud at a distance initially at rest with respect to the computational volume, and exposed it to a hot wind at a constant temperature $T_h$ (for a fixed ) and constant density $n$ (for a fixed and ) flowing with velocity $\vec{v}_h$ under a fixed impact angle $\vartheta$ with respect to the gas cloud (see below). The warm gas is assumed to be initially in a state of pseudo-equilibrium with the hot gas, defined by the mean cloud-to-halo density ratio (or overdensity) $\eta \equiv \rho_{w} / \rho_h$, and the mean cloud-to-halo pressure ratio $\xi \equiv P_{w} / P_h$ (see equation \[eq:tempw\]).
Code {#sec:cod}
----
We choose for our experiment the high-resolution, multi-phase, shock-capturing hydrodynamic grid-based code [<span style="font-variant:small-caps;">fyris alpha</span>]{} [@sut10a], especially developed for astrophysical applications. The code solves the fluid dynamic equations in one, two, and three dimensions as required. It has been shown to be fast, robust and accurate when compared to similar codes, and it performs well when subject to a standard suite of test cases as developed by @lis03a. A unique feature of the [<span style="font-variant:small-caps;">fyris alpha</span>]{} code is that it includes non-equilibrium cooling through time-dependent ionisation calculations. In addition, the code allows for the use of a variable equation of state (EoS) through a variable adiabatic index $\gamma$ and / or a variable mean molecular weight $\mu$. These features are essential due to the large difference in the relevant time-scales which determine the physical state of multi-phase gas, as well as the large range of densities encountered in these type of simulations.
Observational constraints, initial conditions, and set-up {#sec:ic}
---------------------------------------------------------
The proper motion of the Large Magellanic Cloud (LMC) has recently been measured using [*HST*]{} data, yielding an orbital velocity $v_{\textnormal{\sc lmc}} = 321 \pm 24 \kms$ [@kal13a]. We adopt the high-end value and set the speed of the hot wind to $v_h = 350 \kms$. Also, we adopt a value for the impact angle $\vartheta = 24^\circ$, such that the (shear) velocity of the hot wind is given by $\vec{v}_h = (v_h \cos \vartheta, \, v_h \sin \vartheta, \, 0) = (320, \, 141, 0) \kms$. Note that our results are fairly insensitive to the adopted value of $\vartheta$, as long as $\vartheta \gg 0$ and $\vartheta \ll \pi / 2$, which is supported by the believe that the orbit of the Magellanic Stream is likely neither radial nor tangential with respect to the gaseous Galactic halo. We will assess the impact of this plausible, albeit arbitrary, choice on our results when dealing with virtual observations in Section \[sec:results\] below.
The Stream’s mean metallicity away from the MCs is now well constrained to $Z \approx 0.1 \Zsun$ [@ric13a; @fox14a]. In contrast, the metallicity of the halo gas is still uncertain, although cosmological simulations [@ras09a] and pulsar dispersion measures towards the LMC [@mil15a] both suggest that it is likely in the range $Z \sim 0.1 \Zsun - 0.3 \Zsun$ far away from the disk. We choose a value for the metallicity of the halo of $Z = 0.1 \Zsun$, which is consistent with the mean value observed in external galaxies similar to the MW [NGC 891; @hod13a].
Parameter Value Remarks
--------------------------- -------------------- ------------------------------------------
$(n_x, n_y, n_z)$ $(432, 216, 216)$ Grid dimensions
$(x, y, z)$ \[kpc\] $(18, 9, 9)$ Physical dimensions
$\delta x$ \[pc\] 42 Spatial resolution (approximate)
$T_{h} ~[\!\K]$ $10^6$ Halo gas temperature$^a$
$T_{w} ~[\!\K]$ $10^3$ Initial Stream gas temperature$^b$
$M_{w}(\!\HI) ~[\!\Msun]$ $10^7$ Initial Stream neutral gas mass
$\eta$ $100$ Initial ratio of cloud : halo density
$\xi$ $0.1$ Initial ratio of cloud : halo pressure
$n(55 \kpc) ~[\!\pcc]$ $2 \times 10^{-4}$ Total particle density at 55 kpc
$Z_{h} ~[\!\Zsun]$ 0.1 Halo gas metallicity
$Z_{w} ~[\!\Zsun]$ 0.1 Stream’s metallicity
$X$ 0.7154 Hydrogen mass fraction
$Y$ 0.2703 Helium mass fraction
$\vartheta ~[^{\circ}]$ 24 Impact angle
$\Delta v ~[\!\kms]$ 200 Velocity range of emission spectra
with pixel size $\delta v = 2 \kms$.
: Relevant simulation parameters / initial conditions[]{data-label="tab:sims"}
Only gas clouds that are not overly dense and which have low pressure support with respect to the ambient medium will be disrupted in realistic timescales of $\sim 100 \Myr$ [@bla09b]. We adopt $\eta = 100$ and $\xi = 0.1$. With these parameters fixed, the initial temperature of the warm gas phase is set by the temperature of the hot gas phase through $$\label{eq:tempw}
T_{w} = \left( \frac{ \mu_{w} }{ \mu_h } \right) \left( \frac{ \xi }{ \eta } \right) T_h \, ,$$ where it should be noted that the mean molecular weight will be generally different in each phase.
A key parameter of the models is the normalisation of the density profile at the [*canonical*]{} distance of the Stream above the SGP ($d = 55 \kpc$). Although still uncertain, different lines of evidence indicate that it is likely in the range of $10^{-5} \pcc - 10^{-3} \pcc$ at $20 \kpc\ \lesssim r \lesssim 100 \kpc$. For example, @bli00a estimate a lower limit on the mean halo density of $n \approx 2.4 \times 10^{-5} \pcc$ out to $d \leq 250 ~\kpc$ based on the assumption that the gas-poor dwarf spheroidals orbiting the Galaxy have been stripped from their gas by the ram-pressure exerted by the hot halo. Along the same line, and combining observations with (2D) hydrodynamic simulations, @gat13a have inferred a range of halo densities $n \approx (1 - 4) \times 10^{-4} \pcc$ at $50 ~\kpc < d < 100 ~\kpc$. @sta02a have found that the gas clouds at the tail of the MS are likely in pressure equilibrium with the hot halo, and using this they have put an upper limit on the halo density of $10^{-3} \pcc$ and $3 \times 10^{-4} \pcc$ at a distance $z = 15 ~\kpc$ and $z = 45 ~\kpc$ from the Galactic plane, respectively. @and10a infer a range for the mean halo density of $n \approx (6 - 10) \times 10^{-4}$ out to the LMC ($d \approx 50 \kpc$) based on dispersion measures of LMC pulsars. However, the stripping of the LMC’s disc requires a somewhat lower value of $n(48.2 \pm 3 \kpc) = (1.1 \pm 0.44) \times 10^{-4} \pcc$ [@sal15a].
Based on these results, we adopt a fiducial value $n(55 \kpc) = 2 \times 10^{-4} \pcc$, consistent with @bla07a. As shown in Figure \[fig:dens\], this choice leads to models for the Galactic corona that largely agree with the results from observations over a broad range in distances. In this respect, we consider both $n(55 \kpc)$ and to be well constrained by observation. Note that the models are completely defined by the value of $\tau$, given that all the other parameters are either fixed or they depend on $\tau$ (Table \[tab:sims\]).\
We run all simulations in a rectangular box of comoving size $18 \times 9 \times 9 \kpc^3$, using a fixed grid composed of $432 \times 216 \times 216$ cells. These settings imply a spatial resolution of ${\delta x}= ( 9 / 216) \kpc\ \approx 42 \pc$. The fragment of gas representing the Magellanic Stream is initially constrained to a cylinder $18 \kpc$ in length and $2 \kpc$ in diameter, and whose axis of symmetry runs parallel to the $x$ axis of the coordinate system defined by the box. The impact angle is defined with respect to the $x$-axis of this cylinder. In this setup, the $x$-axis coincides with the Magellanic longitude, $l_M$, whereas any of $y$ or $z$ run along the Magellanic latitude, $b_M$ [@wak01a; @nid08a]. The simulated Magellanic Stream consists of an gas distribution initially at temperature $T_{w} \sim 10^3 \K$ (see equation \[eq:tempw\]); a mean initial hydrogen particle density $n \sim 10^{-3} \pcc$; and a total neutral gas mass $M_{w}(\!\HI) \sim 10^7 \Msun$. The initial warm gas density field corresponds to the density of a fractal medium described by a Kolmogorov turbulent power spectrum $P(k) \propto k^{-5/3}$, with a minimum wavenumber $k_{min} = 8$ (relative to the grid) corresponding to a spatial scale of 2.25 [q.v. @sut07a], comparable to the typical size of clouds in the Stream. The gas cloud is assumed to be at a fixed distance from the Galactic Centre in the direction of the SGP. The temperature $T_h$ and the density $n$ of the hot wind are set according to the value of $\tau$ and the distance as given by equations and , respectively.
For each $\tau \in \{0.5, \, 0.75, \, 1.0, \, 1.5 \}$, we consider a set of Galactocentric distances $r_{\textnormal{\sc ms}} \in \{55, \, 75, \, 100, \, 125, \, 150 \} \kpc$ which together span the range of plausible orbits of the MS above the SGP [@gug14a]. This yields 20 models. Each model is run for a total (simulation) time of $320 \Myr$, starting from $t_{sim} = 0 \Myr$, assuming free boundary conditions. The simulation output for a given set of values $\{\tau, \, \rms, \, t_{sim} \}$ – in steps of $\Delta t_{sim} = 10 \Myr$ – consists of a series of datacubes containing information about the - and densities, $n_{ \HI }$ and $n_{\!\HII }$, respectively; the gas temperature $T$, and the gas velocity $\vec{v} = (v_x, \, v_y, \, v_z)$. Using this information, we compute the emission and the column density of each cell, and from these the surface brightness and total column density along the for each snapshot.
Emission line spectra
---------------------
We compare the emission of the gas in our simulations to Fabry-Pérot observations along the Magellanic Stream [@bla13a and references therein], and complement these with results on the associated column density measurements.
Given the one-to-one correspondence between the 21cm emission (i.e., brightness temperature) $T_B$ and the column density of a parcel of optically thin gas [e.g. @dic90b], we use the total column density along the () as a proxy for the corresponding 21cm emission, i.e. we [*define*]{} the 21cm intensity to be $I_{\!\HI} \equiv \NHI$.
The emission is computed using $$\label{eq:mua}
\mua = \mua^{(shock)} + \mua^{(phot)} \, .$$ The first term accounts for the ionisation that results from slow shocks produced by the collision of the trailing cloud gas with the leading gas ablated by the interaction with the hot halo, and is given by $$\label{eq:hacell}
\mua^{(shock)} = [1 + (Y / 4 X)] ~K_{R} \; \alphabha \; \!\! \int \! (n_{\rm HII})^2 ~ds\, ,$$ where $\alphabha(T)$ is the effective recombination coefficient (equation \[eq:peqeff\]), $K_R \approx 1.67 \times 10^{-4} \cm^2 ~{\rm s} \mR$,[^3] and the factor $[1 + (Y / 4 X)]$ accounts for the conversion of electron density to particle density. Adopting a hydrogen and helium mass fractions $X = 0.7154$ and $Y = 0.2703$, respectively [@asp09a solar bulk composition],[^4] assuming the gas is fully ionised ($\!\nH \approx n_{\!\HII}$), it follows that $[1 + (Y / 4 X)] \approx 1.09$.[^5]
The second term in equation accounts for the ionising effect of the cosmic ultraviolet background radiation (UVB). The emission along the of gas in photoionisation equilibrium with the UVB radiation field is $$\label{eq:haphot}
\mua^{(phot)} = \frac{ 1 }{ 4 \pi } \Gamma_{\!\HI} \; \!\! \int \! (f_{\!\Ha} \; n_{\!\HI}) ~ds \, .$$ where $f_{\!\Ha}(T)$ gives the fraction of recombinations that produce an photon, and $f_{\!\Ha}(10^4 \K) \approx 0.45$ (equation \[eq:fha\]). The recombination rate, $\Gamma_{\!\HI}$, is related to the total ionising photon flux $\Phi_i$ through $$\label{eq:ionflux}
\Phi_i = 1.59 \times 10^4 ~{\rm photon} \psc \ps ~\left( \frac{\gamma + 3}{4 \gamma} \right) ~\left( \frac{ \Gamma_{\!\HI} }{ 10^{-13} \ps} \right) \, .$$ We adopt $\Gamma_{\!\HI} = 10^{-13} \ps$ [appropriate for $z = 0$; @wey01a] and $\gamma = 1.8$ [@shu99b], corresponding to an ionising flux $\Phi_i \sim 10^4$ photons PS. . If we used instead the most recent estimate $\Gamma_{\!\HI} = 4.6 \times 10^{-14} \ps$ [@shu15a], the flux would be lower by roughly a factor 2.
To mimic radiation transfer effects, we limit the depth (along the in any direction) of the gas ionised by the UVB to a maximum value defined by the condition that the column recombination ($n_e \, n_{\!\HII} \, \alpha_B$) equals the incident ionising photon flux ($\Phi_i$). This condition is equivalent to restricting the ionising effect of the UVB to a column of neutral gas $\sim 10^{17} \psc$ (see Appendix \[sec:pi\]). The effect of the cosmic UVB is to produce an ionisation skin around the cloud featuring an surface brightness at a level of roughly 5 mR. Again, if we used instead $\Gamma_{\!\HI} = 4.6 \times 10^{-14} \ps$, this value would decrease to roughly 2 mR. It is important to mention that this approach is not entirely self-consistent with our simulations because the ionising effect of the UVB is not included at runtime, and because it assumes photoionisation equilibrium. Also, we ignore for the moment the contribution of the Galactic ionising field, which would produce an additional mean signal of $21 \, \zeta \, (d / 55 \kpc)^{-2} \mR$ [$\zeta \approx 2$; @bla13a].
To allow for a faithful comparison with observations, we map the simulation data onto observed space by projecting the simulation volume along a given axis, so as to mimic the projection of the observed emission along the Stream onto the plane of the sky. We choose, for convenience, an axis parallel to one side of the simulation box, and perpendicular to the Stream’s main axis, i.e. the $y$-axis. However, we will address the potential bias introduced by this choice by comparing the results obtained by projecting along all three orthogonal axes, $x$, $y$ and $z$.
Each across the projected datacube thus corresponds to a pencil-beam spectrum. and 21cm pencil-beam spectra along the chosen projection axis are constructed by computing the intensity for each cell in the simulation volume. The intensity and the 21cm emission intensity observed at velocity $v_m$ (in the rest-frame of the gas at $t_{sim} = 0$) of a parcel of gas at cell $\mathbf{n} \equiv (i, j, k)$ with bulk velocity $v_{\mathbf{n}}$ are, respectively, $$\begin{aligned}
\label{eq:spec}
I_{ \textnormal{\Ha}} (v_m - v_{\mathbf{n}}, \mathbf{n}) & = \mua(\mathbf{n}) \; \phi_{\mathbf{n}}(v_m - v_{\mathbf{n}}) \, , \notag \\
I_{\!\HI} (v_m - v_{\mathbf{n}}, \mathbf{n}) & = \NHI(\mathbf{n}) \; \phi_{\mathbf{n}}(v_m - v_{\mathbf{n}}) \, ,\end{aligned}$$ where the normalised line profile , with .
We adopt a spectral range in terms of velocity of $\Delta v = 200 \kms$ which corresponds to the spectral range provided by the WHAM spectrometer. The velocity scale is given with respect of the initial rest-frame of the gas, such that emission spectrum spans the range $[-100, +100] \kms$, with a pixel size $\delta v = 2 \kms$.
At each beam position (or ‘pointing’), we compute the intensity-weighted average of all cells within the beam, resulting in a single spectrum per pointing. We adopt a low resolution beam of $\onedeg$ diameter on the sky, identical to the resolution provided by the WHAM spectrometer.
We choose a rectangular (rather than a circular) beam, which allows for a full coverage of the projected image, and which greatly simplifies the scanning procedure.[^6] Given that a beam with a diameter of $2 \theta$ subtends a solid angle $\Omega_{beam} \approx \pi \left( \theta / 2 \right)^2$ (provided that $\theta \ll 1$), the linear size of a square subtending a solid angle $\Omega_{beam}$ at a distance relative to the angular dimension of a single cell in our simulation, $\delta \Omega_V \approx ( {\delta x}/ \rms )^2$, is roughly ${\delta l} = (\pi / 4 )^{1/2} (\!\rms / \delta x ) ~\theta$. The emission within a beam pointing at each velocity $v_m$, i.e. the beam spectrum, is then $$\overline{ I }_{ \textnormal{X} } = [{\delta l}]^{-2} \sum_{[{\delta l}]} \sum_{[{\delta l}]} I_{ \textnormal{X} } \, ,$$ where the sum extends over all cells within $\Omega_{beam}$, and $X \in \{\!\HI, \HII \}$. Here, the notation $[k]$ indicates the largest odd integer smaller than or equal to $k$.
The average of the emission within each beam pointing can effectively be obtained by overlaying a rectangular grid on the projected image with a cell size equal the solid angle subtended by a circular beam of diameter $2 \theta$ at that distance, and computing the arithmetic mean within each new cell. We choose the origin of the matrix to be shifted by half a beam size in each direction ($xy$) to avoid the uncertainties associated with the simulation volume’s boundaries.
It is important to emphasise that the pixelation of the simulation volume and the position of the beam pointings with respect to the projected datacube are rather arbitrary. Also, the latter is also generally different for each adopted distance . This results from the fact that for a computational volume with fixed comoving size and fixed grid, and a beam of fixed angular size, an increasing fraction of the gas that represents the Stream will be sampled by the beam with increasing distance. Because of this, and also to avoid a bias in the resulting emission introduced by potentially bright features induced by chance alignments (rather than due to intrinsically bright gas blobs), [*prior*]{} to computing the spatial average within each beam pointing we smooth each 2D spatial slice at each velocity bin using a circular Gaussian kernel with a full width at half maximum of half the beam size and a total width of (i.e., truncated at) the size of the beam. According to the approach described above, the projected intrinsic emission map from our simulation is smoothed at each given distance using a Gaussian kernel with FWHM of $[{\delta l} / 2]$ pixel and a total width of $[{\delta l}]$ pixel.
Finally, in order to take into account the instrumental line broadening, we convolve each spatially averaged spectrum using a Gaussian kernel[^7] with a , which roughly corresponds to the resolution of the WHAM spectrograph [@rey98a]. The surface brightness and 21cm emission maps are simply obtained from the 3D spectral datacube by simply integrating each spectrum along the velocity coordinate (equations \[eq:spec\]).
A selected example illustrating the result of the above procedure is shown in Figure \[fig:hamap\] (see also Figure \[fig:kin\]). As we show below (see Section \[sec:ion\]; Figure \[fig:ionfracevol\]), the to mass ratio in this snapshot roughly matches to the corresponding ratio observed in the Magellanic Stream [$\sim3$; @fox14a]. The effect of the ionising cosmic UVB is apparent: all the Stream gas is lit up and emitting at level of $\sim 5 \mR$. But there are brighter spots which are a consequence the shock cascade, whereby the trailing clouds collide with the material ablated by hydrodynamic instabilities from the leading gas, thus being shock ionised [q.v. @bla07a]. Interestingly, while the bright spots seem to closely track the high parcels of gas, the converse is not true, with high appearing with no correspondent strong emission. However, these differences become less apparent, although they remain, as a result of the beam smearing. The most dramatic effect of the latter is the dilution of the and signals with respect to the brightest levels seen at the (intrinsic) resolution of the simulation by a significant factor. Only the brightest spots in would observable with a Fabry-Pérot interferometer for reasonable integration times, leading to a significant fraction of the ionised gas mass falling below the detection threshold.
\
\
Results {#sec:results}
=======
Gas emission
------------
We follow the evolution of the gas emission over a period of 320 in all 20 models. To illustrate these results, we adopt the model = 1 as our standard model. The top panels of Figure \[fig:haevol\] shows the evolution of the / emission at four representative distances = 55 , 75 , 100 , and 150 . The middle panels show the corresponding result for all models = 0.5, 0.75, 1.0, and = 1.5, at a fixed distance = 55 kpc. Note that the comparison of the results for a fixed at different distances allows to assess the effect of the density on the / emission for a fixed temperature, while the comparison of models with different at a fixed distance of 55 kpc (and thus a fixed density) helps us explore the effect of the temperature.
We find that the gas ionises quickly ($\sim 50 \Myr$), and after $\sim 300 \Myr$, the column density has decreased uniformly both with time, dropping by nearly an order of magnitude. This can be understood as a consequence of the increasing ionisation of the gas due to the interaction with the hot halo gas. The [*detected*]{} column density of the gas farther out is also lower with respect to the [*intrinsic*]{} gas density, which is an effect of the beam dilution. Note that a beam of diameter samples a region of roughly twice (three times) the size at = 100 kpc (150 ) with respect to = 55 kpc.
Similarly, the highest emission – this is, the [*maximum*]{} value at each given time – comes from the gas which is closest. In contrast to the behaviour of the density, in this case the effect is governed by the increase of halo gas density with distance, and the corresponding strength of the hydrodynamic interaction leading to the shock cascade. What is surprising is that even at the lowest Galactocentric distance of = 55 kpc, the emission never exceeds $\sim 40 \mR$; and it barely reaches 10 mR at 75 kpc. At even larger distances, $\rms\ \gtrsim 100 \kpc$, the emission is dominated by the recombination of the gas ionized by the cosmic UV background. Note that often we do find in our simulation pixels with $\mua > 30 \mR$, and occasionally on the order of $\sim 100 \mR$, but their strong signal is washed out as a result of the beam smearing (Figure \[fig:hamap\]; see also Section \[sec:modvars\]). The similarity in emission across models with different shows that these results are insensitive to variations in the halo gas temperature by factors of a few. This indicates that the emission is dominated by the gas ionised through cloud-cloud collisions that trigger the shock cascade, rather than the gas ionised by the interaction with the hot halo. The insignificance of the halo gas temperature together with the fact that the virial temperature across the DM halo models presented previously is very similar (Table \[tab:dmhalo\]) makes the choice of the DM halo model irrelevant, as long as the corresponding gas density profiles are comparable.
We see that the maximum level of emission at any reasonable distance is comparable to, or even less than, the emission induced by the Galactic ionising starlight $\sim 20 - 40 \mR \; [d / 55 \kpc]^{-2}$. This is the reason for us to ignore this component in our models, since it would otherwise outshine the emission produced by the shock cascade. Taking the contribution of the Galactic UV into account would elevate the emission at 55 kpc to $ \sim 70 \mR$, and to $ \sim 30 \mR$ at 75 kpc, which are significantly lower than the Stream’s emission observed over the SGP and at the tail of the Stream at $l_M \approx 260^{\circ}$, respectively [@bla13a]. It is however unclear at this point how the ionising effect of the Galactic UV included self-consistently at runtime would affect these limits.
Given that observations performed with the WHAM instrument typically reach a sensitivity of $\gtrsim 30 \mR$, the Stream gas in our models would be essentially undetectable (ignoring for the moment the contribution which results from the Galactic ionising field). In contrast, and considering that the sensitivity of e.g the GASS survey is roughly $\NHI = 1.6 \times 10^{18} \psc$, the gas at distances $d \lesssim 100 ~\kpc$ would be bright in 21cm, and marginally detectable at $d \sim 150 ~\kpc$, even after 300 Myr.
We find that much of the gas dislodged from the main body of our model Stream is low density material that mixes rapidly with the halo gas, thereby being heated (and thus ionised) to temperatures well above $10^5 \K$, which are on the order of the temperature expected for turbulent mixing [@beg90a]. At these temperatures, the emissivity drops by nearly two orders of magnitude with respect to its value $10^4 \K$ (see equation \[eq:peqeff\]), and the ionised gas becomes thus practically invisible in . Only the gas ionised by cloud-cloud collisions remains at relatively low temperatures ($\sim 10^4 \K$), and recombines quickly, thus providing the strongest signal. However, the fraction of warm ionised gas is very low overall, and thus is the corresponding signal. Hence, only a mechanism such as slow shocks which is able to ionise a significant fraction of the gas without increasing its temperature far above $10^4 \K$ will lead to significant emission.\
As the reader may recall, all the above results correspond to virtual observations where the simulation cube has been projected along the $y$-axis. Given that this choice is somewhat arbitrary, we calculate the corresponding results for projections along all three orthogonal axes for model = 1 at 55 kpc. The outcome of this exercise is summarized in the bottom panels of Figure \[fig:haevol\]. We have checked that the results are essentially the same for all other models. The column density is consistently highest when observed along the $x$-axis, given that this axis coincides with the axis of symmetry of the initially cylindrical gas configuration in our setup, and the traverses a larger path across the cloud. The projections along the $y$ and $z$ axes yield nearly identical results, as expected from the symmetry of the initial cloud structure.
In our experiment, the projection along the $x$-axis is equivalent to observe the Stream ’face-on’. Hence, one would naïvely expect that the emission should be highest when projecting along this direction. But surprisingly, the intensity in our models is very similar regardless of the projection, being only slightly stronger when viewing the gas cloud face-on. This implies that the choice of projection axis to compute virtual observations is essentially irrelevant. Moreover, since different projections effectively imply significant variations in the impact angle, our choice of a particular value for $\vartheta$ turns to be irrelevant as well, as far as the intensities are concerned.
Gas kinematics
--------------
The kinematics of the warm neutral and ionised gas phases as traced by and emission provide a insight into the mechanism ionising the Stream. We explore this using the spectra of our model identified by = 1 and = 55 at $t_{sim} = 170 \Myr$. Note that the results, with exception of the emission strength, are virtually identical for all other models. An example of a strong and 21cm emission lines is shown in the top panel of Figure \[fig:kin\]. These correspond to the pointing with the brightest emission shown in the bottom panel of Figure \[fig:hamap\]. The integrated strength of the line is indicated in each case in the top-left corner. For reference, the typical sensitivity of WHAM ($30 \mR$) translates into a spectral sensitivity of $0.5 \mR / \kms$, assuming a typical line width of 30 . Similarly, the sensitivity of the GASS survey ($\NHI = 1.6 \times 10^{18} \psc$) corresponds to a spectral sensitivity of $10^{17} \psc / \kms$. Thus, while the emission in this case is marginally above the WHAM detection threshold, the corresponding 21cm signal would be comfortably detected in a survey similar to GASS. We find typical line widths of $20 - 30 \kms$ (FWHM), which are consistent with observations [@put03b].
We study the difference in the kinematics of the warm neutral and ionised gas by comparing the velocity centroid of the emission to the 21cm emission, distinguishing between pointings ‘on’ and ‘off’ the clouds. In this context, ‘on’ (‘off’) means that the 21cm emission is above (below) the GASS detection limit $\NHI = 1.6 \times 10^{18} \psc$. Note that the line centroid corresponds to the intensity-weighted mean velocity. In addition, we flag those pointings where the emission is above the level expected from ionisation by the UBV (Figure \[fig:kin\], bottom panel). We do not find a significant difference between the velocities of the warm neutral and ionised gas phases; their respective velocity centroids agree within $\pm 5 \kms$, as observed [@put03b; @bar15a]. There is a tendency for the lines to have higher slightly higher velocity centroids. This arises from the higher line asymmetry resulting from a more extended emission along the . It is interesting that we barely find any emission above 5 mR [*detached*]{} from 21cm emission. Also, the velocity of this strong emission is generally low ($v \lesssim 20 \kms$), with a tendency for the strongest emission to have the lowest velocities (not shown), indicating that the strong emission is physically associated to the gas. This coincidence in both velocity and physical space of the / signal is – recall the weak dependence of the emission on the halo temperature – another characteristic signature of the shock cascade.
Gas ionisation timescales {#sec:ion}
-------------------------
It has recently been inferred that the mass of the ionised gas kinematically associated to the Magellanic Stream is roughly 3 times larger than its mass [@fox14a]. Here, we briefly explore the evolution of the ionised and warm-neutral gas mass fractions, using the and masses in as a proxy for the ionised and warm-neutral gas phases, respectively. Again, we focus on the results obtained from our models characterised by = 0.50, 0.75, 1.00 and = 55 . Note that the results are qualitatively the same for all other models. Figure \[fig:ionfracevol\] shows the evolution of the individual mass fractions $f_X \equiv M_X / M_{tot}$, where $X \in \{ \HI, \HII \}$ and $M_{tot} = M_{\!\HI} + M_{\!\HII}$, as well as the evolution of the to mass ratio. We include for reference the inferred value of the ratio : $= 3$. We find that the gas evolves on a typical timescale of 100 , which is consistent with the time estimate which results from assuming that the cloud disrupt by the action of Kelvin-Helmholtz instabilities and our adopted (initial) value of $\eta$. After this period, the ionisation effect of the halo-cloud and cloud-cloud interaction leads to a reduction of the warm neutral gas mass by half. The : mass ratio increases rapidly with time, and the inferred : mass ratio of 3 is reached after $\sim170 \Myr$.
Since our simulations assume free boundary conditions, it is difficult to quantify with precision how much of the neutral gas is ionised and how much simply escapes the simulation volume. Nonetheless, we estimate that only a negligible fraction of the warm-neutral gas is lost by $t_{sim} \lesssim 270 \Myr$. Therefore, the ionisation of the gas due to interactions with the hot halo gas, and due to cloud-cloud interactions lead to a strong evolution of the mass fractions in the neutral and ionised phases. The relatively short survival timescale implies the requirement for a continuous replenishment of gas from the MC to the Stream [@bla07a].\
Note that, for the ease of discussion, in the following we shall refer to the set of models discussed in the last sections, defined by $n(55 \kpc) = 2 \times 10^{-4} \pcc$, as the ‘standard’ model set.
Conservative departures from the standard models {#sec:modvars}
------------------------------------------------
\
Given the results of the last sections, our standard set of models appear to indicate that the shock cascade [*fails*]{} to produce the mean level of emission [$\sim160 \mR$; @bla13a] observed along the Magellanic Stream. But there are two factors that deserve closer consideration. First, a halo density at any given distance within [*higher*]{} than implied by our standard models could enhance the onset and development of hydrodynamic instabilities (Kelvin-Helmholtz), thus promoting the shock cascade and the resulting emission. Secondly, an increase in the beam resolution would certainly diminish the smearing effect on bright spots which have characteristic sizes significantly smaller than the beam. Consider, for instance, that a beam with a diameter of 10’ samples a region which is nearly ten times smaller than a beam at = 55 kpc. Indeed, the brightest observations along the Stream have been obtained with spectroscopy over smaller apertures (3’-10’) than the WHAM survey [e.g. @put03b].
Therefore, it is important to extend the parameter space of our study, in terms of both the halo density and the adopted beam size. We now [*increase*]{} the normalisation of the halo gas density at 55 kpc by a factor of 2, i.e. $n(55 \kpc) = 4 \times 10^{-4} \pcc$, but with initial conditions and set up which are otherwise identical in every aspect to the standard models. These shall be referred to as the ’extended’ models. In addition to increasing the density, we produce a new set of virtual observations for the standard models, adopting a smaller beam size (i.e. a higher resolution) with a diameter of $\theta = 10'$ (rather than ), and two sets of virtual observations for the extended models, adopting either a low () or a high (10’) resolution beam. We shall refer to these as the ‘low-resolution’ and ‘high-resolution’ models, respectively, keeping in mind that is not the actual hydrodynamical model, but the virtual observation, to which the resolution refers.
Note that the density profiles implied by the extended models are fairly consistent with observations (Figure \[fig:dens\], bottom-left panel), although there is no model which agrees with the data over the full range in distance. However, the enclosed gas mass in the halo within resulting from each of these density profiles is larger than the the mean range of masses inferred from observations, and - with exception of the = 1 extended model – they are all inconsistent with the mass limit imposed by the universal baryon-to-total-mass fraction (Figure \[fig:dens\], bottom-right panel). Hence, all the extended models but the = 1 model, may be deemed ’unphysical’. Nonetheless, it is still of interest to explore the intensity in these type of models, as will be discussed later.
Each of the models in either the standard or the extended set is run for a total simulation time of $t_{sim} = 320 \Myr$, and a virtual observation of the intensity of the gas at the appropriate resolution is produced every $\Delta t_{sim} = 10 \Myr$. Therefore, for each model and at each time step we obtain a whole new distribution of intensities. In order to deal with the overwhelming amount of information, and to make a meaningful comparison between models, we opt for the following approach: Since the total time lapse $t_{sim}$, and the choice of output time step are somewhat arbitrary, for each model we single out the snapshot at which the maximum intensity anywhere in the gas (i.e. at any beam pointing) is largest. This is further justified by the fact that the maximum emission does not evolve strongly with time (Figure \[fig:haevol\]). Note that the snapshot thus selected will in general be different for each model. For this particular snapshot and model, we also obtain the value of the emission at the 90 percentile level of the corresponding distribution. We then assess the performance of each model simply by comparing both the maximum emission to the mean level of emission ($\sim 160 \mR$) observed along the Magellanic Stream. The result of these approach applied to both the standard and the extended models is collected in Figure \[fig:modvars\].
There, the top panels correspond to the standard models, and the bottom panels show the results for the extended set; the left (right) panels correspond to the low (high) resolution cases. In any panel, each ‘data point’ corresponds to a particular model identified by {$n(55 \kpc)$, $\theta$, , }, and it consists of a symbol (circle or cross), and two numerical values. The value to the top-right of a given symbol (in parentheses) indicates the [*maximum*]{} emission (in mR), while the value to the bottom-left indicates the value at the 90 percentile level. A circle indicates whether the maximum intensity exceeds 160 mR; a cross signals failure to do so. The circle diameter is roughly proportional to the maximum intensity in each case. In addition, we have greyed hatched the parameter space corresponding to models that are deemed ’unphysical’ as per the above discussion. As a guide, note that the series of numbers in parentheses, i.e. the maximum intensity, shown on the top-left panel for model = 1 correspond to the results shown in the top-right panel of Figure \[fig:haevol\] at 220 Myr (ignoring the 150 kpc series). Similarly, the maximum intensities at = 55 kpc for all models correspond to the results shown in the bottom-right panel of Figure \[fig:haevol\] at 170 Myr = 0.5, and at 220 Myr for all the other models.
Apparently, both a higher density and an improved resolution enhance the emission, but in different ways. On the one hand, increasing the density shifts the overall intensity towards the high-end. This can be seen by comparing the maximum intensity (and the 90 percentile) between the standard and the extended models, which are roughly a factor 2 - 3 higher in the latter. Increasing the resolution, on the other hand, boosts only the maximum emission, without significantly affecting the distribution of intensities as a whole, as can be judged by comparing the values at the 90 percentile level.
The standard models at low resolution (top-left panel) fail dramatically at any distance in matching the Stream’s mean emission. Both the standard model at high resolution (top-right) and the extended model at low resolution (bottom-left) result in intensities in the 100 mR regime only in the near field at $55 \kpc$. In this sense, both a higher halo density model, and virtual observations with a high resolution are equally crucial factors in pushing the emission towards higher levels. It is worth mentioning at this point that the original model by [@bla07a] - an instance of a ‘standard’ model in our terminology – implicitly assumed an infinite resolution, and was therefore capable of reproducing emission at levels of a few hundred milli-Rayleigh at 55 kpc.
The extended, high resolution models (bottom-right) are the most promising of all the models considered here. In the near field, the shock cascade in all these models results in intensities which reach, or even exceed, the highest levels of $\sim 700 \mR$ observed along the Magellanic Stream over the SGP. In the far field ($\!\rms\ \sim 75 - 100 \kpc$), two of the models ( = 0.5 and 0.75) produce emission consistent with the mean emission of $\sim 160 \mR$ observed along the Stream. However, the success of these models comes at a cost. The increase in the halo density is accompanied by an increase in the halo mass. This makes all but the = 1 model be [*inconsistent*]{} with the limit on the gas mass of the Galaxy imposed by the cosmic fraction of baryons relative to the total mass. Given that the isothermal sphere model yields an upper limit on the halo gas density at any distance and for a given gas mass, the situation is even more unfortunate for any other reasonable DM halo model.
Discussion
==========
Within the context of the ‘shock cascade’ model, we have shown that the interpretation of the MS optical emission ($\approx 100 - 200$ mR) away from the Magellanic Clouds may still work for the updated parameters of the Galaxy under a narrow set of conditions. Conventionally, the MS was assumed to be on a circular orbit at the midpoint of the LMC and SMC ($d\approx 55$ kpc). At this distance, for a smooth halo density profile, as long as the coronal halo density satisfies $2 \times 10^{-4} < (n / \pcc) < 4 \times 10^{-4}$, the shock cascade generates sufficient emission to explain the observations. The upper limit on density ensures that the mass of the corona (when including the other baryonic components) does not exceed the baryonic mass budget of the Galaxy.
In recent years, the first accurate measurements of the MC’s proper motions, combined with a smaller estimate of the Galaxy’s total mass, have led to a major revision of their binary orbit about the Galaxy. A highly elliptic orbit is now favoured by most researchers, which pushes the Stream’s mean distance further out than the conventional assumption. For a smooth halo, the lower density limit of the above range can occur at $d \approx 75$ kpc without violating the constraint on the baryonic mass budget. If the Stream’s mean distance (especially over the SGP) happens to exceed this limit, then we are forced to either reject the model, or consider more complex density distributions for the Galactic corona (see below).
An alternative interpretation of the Stream’s optical emission has recently been put forward in the context of the Galaxy’s nuclear activity. The energetic bubbles observed with the Fermi-LAT in gamma rays [@su10a] indicate that a powerful event has taken place at the nucleus in the recent past. If the gamma rays are produced through inverse Compton upscattering of soft photons, this event can be dated to 1-3 Myr ago [@guo12a]. Within the context of this model, @bla13a show that the Stream emission can also be explained by accretion-disk driven ionization for Stream distances of 100 kpc or more over the poles. The recent discovery of high ionization species (e.g. , ) over the SGP [@fox15a] may lend further support to this model. On the other hand, @bla07a provide diagnostics of slow shocks (e.g. Balmer decrement) that are likely to be observable along the Magellanic Stream in future observing campaigns. If enhanced Balmer decrements ($\!\Ha/ \!\Hb \gtrsim 3$) are confirmed along the Stream, then some variant of the shock cascade model may be needed.
There is now increasing evidence that the CGM of low redshift galaxies is multi-phase, with a comparable fraction of baryons both in a hot and a warm phase [@wer14a]. Modern simulations of the CGM also suggest that the hot halos of galaxies are likely to be heavily structured, at least during a major phase of gas accretion. While CDM accretion may be isotropic on average, individual events involving massive systems are not, as clearly demonstrated by the Magellanic Clouds in orbit around the Galaxy, or the Sgr dwarf which extends through much of the halo [@iba94a]. The mass of this system was probably comparable to the LMC and may well have retained gas before being tidally disrupted. A more recent accretion event is attested by the massive stream, the Smith Cloud ($\gtrsim 2\times 10^6$), that is presently being stripped and ablated by the corona [@bla98a; @loc08a; @nic14b].
We may need therefore to consider the possibility that the corona is inhomogeneous rather than smooth. This would allow for significant density variations along different directions and at different distances, without violating constraints on the total baryonic mass of the Galaxy. In turn, based on our extended models, this would restore the shock cascade as a viable model to explain the emission, allowing the Stream to lie at $\sim 100$ kpc the SGP as predicted by dynamical models [e.g. @gug14a]. In this scenario, we may envisage the strong optical emission along the Stream as a result of the Stream’s gas colliding with high-density debris of past accretion events scattered along its orbit. At this point, however, it is not clear what a suitable model for an inhomogeneous hot halo might be. We will address this in future work.
If confirmed, the larger distance to the Stream of 100 kpc would lift its mass to roughly $8 \times 10^9 \Msun$, comparable to the total coronal gas mass. If it all breaks down, it would roughly double the mass of the corona, at least for a while. We thus speculate that the halo of the Galaxy is substantially mass-loaded with gas lost by smaller accreted systems. The interaction with the hot halo may prevent this gas from cooling sufficiently to condense and ‘rain’ down on the disc. Such a process is analogous to the meteorologic phenomenon know as [*virga*]{}, a type of atmospheric precipitation that evaporates while dropping and thus fails to reach the ground. The heavy halo may thus serve as a huge reservoir, from which gas may eventually be forced out by the strong interaction at the disc-halo interface [e.g. @mar10b].
We thank the anonymous referee who made a number of excellent suggestions which significantly improved the presentation of our results. TTG acknowledges financial support from the Australian Research Council (ARC) through a Super Science Fellowship and an Australian Laureate Fellowship awarded to JBH.
recombination coefficient {#sec:harec}
==========================
We describe the temperature dependence of the hydrogen [*total*]{} case B recombination coefficient, $\alpha_{B}$, and of the effective recombination coefficient, $\alphabha $, with the generic fitting formula [@peq91a]: $$\label{eq:peqeff}
\alpha (T) = \alpha_0 \times \, \frac{ (1 + c) (T / 10^4 K)^b}{1 + c ~(T / 10^4 K)^d} \, .$$ The parameter values appropriate in each case are, respectively, $\alpha_0 = 2.585 \times 10^{-13} \cm^3 \ps$, $b = -0.6166$, $c = 0.6703$, and $d = 0.5300$, and $\alpha_0 = 1.169 \times 10^{-13} \cm^3 \ps$, $b = -0.648$, $c = 1.315$, and $d = 0.523$. Note that this formula is accurate to two percent in the range $40 \K < T < 2 \times 10^4 \K$.
Photoionisation-induced emission {#sec:pi}
=================================
We include in our model the contribution of the cosmic UV background (UVB) ionising radiation, which leads to a low, but non-negligible level of emission along the Stream. For simplicity, we assume that the gas is highly ionised ($n_{\rm H} \approx n_{\!\HII}$); and photoionisation [*equilibrium*]{}, which implies that the ionisation and recombination events balance each other: $$\label{eq:pieq}
(n_e ~n_{\!\HII} ) \; \alpha_B = n_{\!\HI} \; \Gamma_{\!\HI} \, ,$$ Here, $\Gamma_{\!\HI}$ is the photoionisation rate in units of photon per atom per second; $J_{\nu}$ is the angle-averaged specific intensity of the UVB; $\nu_{ \textnormal{\sc ll} } \approx 3.29 \times 10^{15} \Hz$ is the minimum photon frequency required to ionise hydrogen; and $\sigma_{\rm H}(\nu)\approx \sigma_0 \; (\nu / \nu_{ \textnormal{\sc ll} })^3$ is the hydrogen ionisation cross section with $\sigma_0 = 6.3 \times 10^{-18} \cm^{2}$.
The emission induced by the metagalactic ionising radiation field along the is thus given by equation , where the fraction of recombinations that produce an photon is $$\label{eq:fha}
f_{\!\Ha}(T) \equiv \frac{ \alpha_{B}^{ (\textnormal{\Ha}) } (T)}{ \alpha_B(T) } \approx 0.452 ~g(T)\, ,$$ Here, $g$ is a monotonically decreasing function of temperature (see equation \[eq:peqeff\]) $$\label{eq:gt}
g(T) = \left( \frac{ 1 + c }{ 1 + c' } \right) \left[ \frac{ 1 + c' ~(T / 10^4 K)^{d'} } { 1 + c ~(T / 10^4 K)^{d} } \right] (T / 10^4 K)^{b - b'} \, ,$$ which satisfies $g(10^4 \K) \equiv 1$, and $g \in (0.6, \, 1.3)$ for $T \in [10^3, \, 10^6] \K$.
In general, $f_{\!\Ha}$ (through $T$) and $n_{\!\HI}$ both vary along the . However, an estimate of the signal resulting from the ionisation by the cosmic UVB of gas at can nevertheless be obtained assuming the gas temperature to be uniform along the . In this case, and inserting the appropriate numerical values we get $$\mua^{(phot)} \approx 452 \mR ~\left( \frac{ \NHI }{ 10^{18} \psc} \right) ~\left( \frac{ \Gamma_{\!\HI} }{ 10^{-12} \ps} \right) \, ,$$ where $\NHI$ is the integral of $n_{\!\HI}$ along the . Hence, for an ionisation rate $\Gamma_{\!\HI} \sim 10^{-13} \ps$ at $z = 0$ and gas at $10^4 \K$ with $\NHI \sim 10^{17} \psc$, the signal resulting from the ionisation by the cosmic UVB of gas at $10^4 \K$ is roughly 5 mR.
Since we are not performing proper radiative transfer calculations, we limit the depth (along the ) of the gas ionised by the UVB to a value $L_{max}$ defined by the condition that the column recombination equals the incident ionising photon flux $\psi_i$: $$\label{eq:pieq2}
\alpha_B \int_{0}^{L_{max}} \!\!\!\!\!\!\!\! (n_e ~n_{\!\HII} ) ~ds \stackrel{!}{=} \psi_i \, ,$$ where the ionising photon flux (in photons PS. ) is given by equation . This condition implies that all the ionising photons be absorbed within a depth $L_{max}$, assuming the gas has been exposed to the (uniform) UV radiation field long enough to reach ionisation equilibrium [which is well justified in the case of the Stream; see @bla13a their Appendix]. Note that this condition is equivalent to restricting the ionising effect of the UVB to a column of gas $\sim 10^{17} \psc$. Indeed, using equations and , the condition becomes (using $\gamma = 1.8$) $$\int_{0}^{L_{max}} \!\!\!\!\!\!\!\! n_{\!\HI} ~ds = \frac{ 2 }{ 3 } \NHI({\rm LL}) \, . \notag$$
where $\NHI({\rm LL}) \equiv \sigma_0^{-1} \approx 1.6 \times 10^{17} \psc$.\
[^1]: The viral mass and the virial radius are linked to one another through the relation $$\label{eq:rvir}
\rvir = \left( \frac{3 \Mvir}{ 4 \pi \Delta_c \overline{\rho}_{m} } \right)^{1/3} \, .$$ We adopt a value for the cosmic mean matter density $\overline{\rho}_{m} \approx 2.76 \times 10^{-30} \g \pcc$ and $\Delta_c \approx 337$. Note that we define the virial radius at $z = 0$ [cf. @shu14a].
[^2]: Note that the designation of thermal ratio by the Greek letter $\beta$ is widespread in the literature. Here, we adopt its original designation.
[^3]: 1 milli-Rayleigh (mR) corresponds to [@bak76a], or $2.41 \times 10^{-4} \erg \psc \ps \psr$ at .
[^4]: For our adopted metallicity of 10 percent the solar value, $\Zsun = 0.0142$, the contribution of heavy elements to the electron density can be neglected.
[^5]: We assume that helium is only singly ionised, given that the ionisation energy of is comparatively high [$E \approx 54.4 ~{\rm eV}$; @kra14a].
[^6]: A alternative arrangement consisting of a tightly packed array of circular windows [which conveys equal weight to every pixel within the beam; see e.g., @haf03a their Figure 3] would yield essentially the same results.
[^7]: Note that the LSF of the WHAM spectrometer is only poorly approximated by a Gaussian [@tuf97a]. However, given the typical / 21cm line widths (20 - 40 ), this approximation hardly affects our results.
|
---
abstract: 'We consider a Two-Higgs-Doublet Model (2HDM) constrained by the condition that assures cancellation of quadratic divergences up to the leading two-loop order. Regions in the parameter space consistent with existing experimental constraints and with the cancellation condition are determined. The possibility for CP violation in the scalar potential is discussed and regions of $\tgb-M_{H^\pm}$ with substantial amount of CP violation are found. The model allows to ameliorate the little hierarchy problem by lifting the minimal scalar Higgs boson mass and by suppressing the quadratic corrections to scalar masses. The cutoff originating from the naturality arguments is therefore lifted from $\sim 0.6\tev$ in the Standard Model to $\gsim 2.5 \tev$ in the 2HDM, depending on the mass of the lightest scalar.'
address:
- '$^1$ Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Hoża 69, PL-00-681 Warsaw, Poland'
- '$^2$ Department of Physics and Technology, University of Bergen, Postboks 7803, N-5020 Bergen, Norway'
author:
- 'B. Grzadkowski$^1$, P. Osland$^2$'
title: 'Tuned Two-Higgs-Doublet Model'
---
Introduction
============
This project aims at extending the Standard Model (SM) in such a way that there would be no quadratic divergences up to the leading order at the two-loop level of the perturbation expansion. The quadratic divergences were first discussed within the SM by Veltman [@Veltman:1980mj], who, adopting dimensional reduction [@Siegel:1979wq], found the following quadratically divergent one-loop contribution to the Higgs boson ($h$) mass $$\delta^{\rm (SM)} \mhs = \frac{\Lams}{\pi^2 v^2}\left[\frac32
\mts-\frac18\left(6\mws+3\mzs\right) - \frac38 \mhs \right],
\label{hcor}$$ where $ \Lam$ is a UV cutoff and $v \simeq 246 \gev $ denotes the of the scalar doublet. The issue of quadratic divergences was then investigated further in [@Osland:1992ay] and [@Einhorn:1992um].
Within the SM precision measurements require a light Higgs boson, therefore the correction (\[hcor\]) exceeds the mass itself even for small values of $ \Lam $, e.g. for $\mh = 130 \gev$ one obtains $\delta^{\rm (SM)}
\mh^2 \simeq \mh^2$ already for $\Lam \simeq 600 \gev$. On the other hand, if we assume that the scale of new physics is widely separated from the electro-weak scale, then constraints that emerge from analysis of operators of dimension 6 require $\Lam \gsim$ a few TeV. The lesson from this observation is that regardless of what physics lies beyond the SM, some amount of fine tuning is necessary; either we tune to lift the cutoff above $\Lam \simeq 600 \gev$, or we tune when precision observables measured at LEP are fitted. Tuning both in corrections to the Higgs mass and in LEP physics is, of course, also a viable alternative which we are going to explore below. So, we will look for new physics in the TeV range which will allow to lift the cutoff implied by quadratic corrections to $\mhs$ to the multi-TeV range [*and*]{} which will be consistent with all the experimental constraints—both require some amount of tuning. Note that within the SM the requirement $\delta^{\rm (SM)} \mhs = 0$ implies an unrealistic value of the Higgs boson mass $\mh \simeq
310~\gev$.
Here we are going to argue that the Two-Higgs-Doublet Model (2HDM) in certain region of its parameter space can soften the little hierarchy problem both by suppressing quadratic corrections to scalar masses [*and*]{} it allows to lift the central value for the lightest Higgs mass.
The Two-Higgs-Doublet Model {#non-IDM}
============================
In order to accommodate CP violation we consider here a 2HDM with softly broken $\zBB_2$ symmetry which acts as $\Phi_1\to -\Phi_1$ and $u_R\to -u_R$ (all other fields are neutral). The scalar potential then reads $$\begin{aligned}
V(\phi_1,\phi_2) &=& -\frac12 \left\{m_{11}^2\phi_1^\dagger\phi_1
+ m_{22}^2\phi_2^\dagger\phi_2 + \left[m_{12}^2 \phi_1^\dagger\phi_2
+ \hc \right]\right\}
+ \frac12 \lam_1 (\phi_1^\dagger\phi_1)^2
+ \frac12 \lam_2 (\phi_2^\dagger\phi_2)^2
\nonumber \\
&&
+ \lambda_3(\phi_1^\dagger\phi_1)(\phi_2^\dagger\phi_2)
+ \lambda_4(\phi_1^\dagger\phi_2)(\phi_2^\dagger\phi_1)
+ \frac12\left[\lambda_5(\phi_1^\dagger\phi_2)^2 + \hc\right]
\label{2HDMpot}\end{aligned}$$ The minimization conditions at $\langle \phi_1^0 \rangle = v_1/\sqrt{2}$ and $\langle \phi_2^0 \rangle = v_2/\sqrt{2}$ can be formulated as follows: m\_[11]{}\^2= v\_1\^2\_1+v\_2\^2(\_[345]{}-2), m\_[22]{}\^2=v\_2\^2\_2+v\_1\^2(\_[345]{}-2), \[min\] where $\lambda_{345}\equiv \lam_3+\lam_4+{{\rm Re\thinspace}}\lam_5$ and $\nu\equiv {{\rm Re\thinspace}}m_{12}^2/(2v_1v_2)$.
Quadratic divergences {#one-loop}
---------------------
At the one-loop level the cancellation of quadratic divergences for the scalar Green’s functions at zero external momenta ($\Gamma_i$, $i=1,2$) in the 2HDM type II model implies [@Newton:1993xc] $$\begin{aligned}
\Gamma_1\equiv \frac32 \mw^2 + \frac34 \mz^2
+ \frac{v^2}{2}\left( \frac32 \lam_1 + \lam_3 + \frac12 \lam_4 \right)
- 3 \frac{\mb^2}{\cbb^2} = 0,
\label{qdcon1_mod2}\\
\Gamma_2\equiv\frac32 \mw^2 + \frac34 \mz^2
+ \frac{v^2}{2}\left( \frac32 \lam_2 + \lam_3 + \frac12 \lam_4 \right)
-3 \frac{\mt^2}{\sbb^2} = 0,
\label{qdcon2_mod2}\end{aligned}$$ where $v^2\equiv v_1^2+v_2^2$, $\tan\beta\equiv v_2/v_1$ and we use the notation: $s_\theta \equiv \sin\theta$ and $c_\theta\equiv \cos\theta$. Note that when $\tan\beta$ is large, the two quark contributions can be comparable. In the type II model the mixed, $\phi_1-\phi_2$, Green’s function is not quadratically divergent.
The quartic couplings $\lambda_i$ can be expressed in terms of the mass parameters and elements of the rotation matrix needed for diagonalization of the scalar masses (see, for example, Eqs. (3.1)–(3.5) of [@ElKaffas:2007rq]). Therefore, for a given choice of $\alpha_i$’s, the squared neutral-Higgs masses $M_{1}^2$, $M_{2}^2$ and $M_3^2$ can be determined from the cancellation conditions (\[qdcon1\_mod2\])–(\[qdcon2\_mod2\]) in terms of $\tgb$, $\mu^2$ and $M_{H^\pm}^2$. It is worth noticing that scalar masses resulting from a scan over $\alpha_i$, $M_{H^\pm}$ and $\tgb$ exhibit a striking mass degeneracy in the case of large $\tan\beta$: $M_1 \simeq M_2 \simeq M_3
\simeq \mu^2+4\mb^2$.
At the two-loop level the leading contributions to quadratic divergences are of the form of $ \Lam^2 \ln \Lam$. They could be determined adopting a method noticed by Einhorn and Jones [@Einhorn:1992um], so that the cancellation conditions for quadratic divergences up to the leading two-loop order read: \_1+\_1=0 \_2+\_2=0 \[2-loop-con\] with \_1 &=& \[ 9 g\_2 \_[g\_2]{} + 3 g\_1 \_[g\_1]{} + 6\_[\_1]{} + 4 \_[\_3]{} + 2 \_[\_4]{}\]()\
\_2 &=& \[ 9 g\_2 \_[g\_2]{} + 3 g\_1 \_[g\_1]{} + 6\_[\_2]{} + 4 \_[\_3]{} + 2 \_[\_4]{} -24 g\_t \_[g\_t]{}\]() where $\beta$’s are the appropriate beta functions while $\bar \mu$ is the renormalization scale. Hereafter we will be solving the conditions (\[2-loop-con\]) for the scalar masses $M_i^2$ for a given set of $\alpha_i$’s, $\tgb$, $\mu^2$ and $M_{H^\pm}^2$. For the renormalization scale we will adopt $v$, so $\bar\mu=v$. Then those masses together with the corresponding coupling constants, will be used to find predictions of the model for various observables which then can be checked against experimental data.
![\[Fig:allowed-2500-300-400-500\] Two-loop allowed regions in the $\tan\beta$–$M_{H^\pm}$ plane, for $\Lam=2.5\tev$, for $\mu=300, 400, 500\gev$ (as indicated). Red: positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. ](fig2a.eps "fig:"){width="5.cm"} ![\[Fig:allowed-2500-300-400-500\] Two-loop allowed regions in the $\tan\beta$–$M_{H^\pm}$ plane, for $\Lam=2.5\tev$, for $\mu=300, 400, 500\gev$ (as indicated). Red: positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. ](fig2b.eps "fig:"){width="5.cm"} ![\[Fig:allowed-2500-300-400-500\] Two-loop allowed regions in the $\tan\beta$–$M_{H^\pm}$ plane, for $\Lam=2.5\tev$, for $\mu=300, 400, 500\gev$ (as indicated). Red: positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. ](fig2c.eps "fig:"){width="5.cm"}
Allowed regions {#sec:allowed}
---------------
In order to find phenomenologically acceptable regions in the parameter space we impose the following experimental constraints: the oblique parameters $T$ and $S$, $B_0-\bar{B}_0$ mixing, $B\to X_s \gamma$, $B\to \tau \bar\nu_\tau X$, $B\to D\tau \bar\nu_\tau$, LEP2 Higgs-boson non-discovery, $R_b$, the muon anomalous magnetic moment and the electron electric dipole moment (for details concerning the experimental constraints, see refs. [@Grzadkowski:2009bt; @ElKaffas:2007rq; @WahabElKaffas:2007xd]). Subject to all these constraints, we find allowed solutions of (\[2-loop-con\]). For instance, imposing all the experimental constraints we find allowed regions in the $\tan\beta$–$M_{H^\pm}$ plane as illustrated by the red domains in the $\tan\beta$–$M_{H^\pm}$ plane, see Fig. \[Fig:allowed-2500-300-400-500\] for fixed values of $\mu$. The allowed regions were obtained scanning over the mixing angles $\alpha_i$ and solving the two-loop cancellation conditions (\[2-loop-con\]). Imposing also unitarity in the Higgs-Higgs-scattering sector [@Kanemura:1993hm; @Akeroyd:2000wc; @Ginzburg:2003fe] (yellow regions), the allowed regions are only slightly reduced. Requiring that also experimental constraints are satisfied the green regions are obtained.
For parameters that are consistent with unitarity, positivity, experimental constraints and the two-loop cancellation conditions (\[2-loop-con\]), we show in Fig. \[Fig:2-loop-masses-2500\] scalar masses resulting from a scan over $\alpha_i$, $M_{H^\pm}$ and $\tgb$. As we have noticed for the one-loop spectrum, large $\tan\beta$ implies similar scalar masses. This is indeed what is being observed in Fig. \[Fig:2-loop-masses-2500\] also for the two-loop case. The allowed solutions “peak” around $M_{H^\pm}\sim \mu$ with $20 \lsim \tan\beta \lsim 50$.
![ Two-loop distributions of allowed masses $M_2$ vs $M_1$ (left panels) and $M_3$ vs $M_2$ (right) for $\Lam=2.5\tev$, resulting from a scan over the full range of $\alpha_i$, $\tan\beta \in (0.5,50)$ and $M_{H^\pm} \in
(300,700)\gev$, for $\mu=300, 400, 500~{\rm GeV}$. Red: Positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. []{data-label="Fig:2-loop-masses-2500"}](fig4a.eps "fig:"){width="12cm"} ![ Two-loop distributions of allowed masses $M_2$ vs $M_1$ (left panels) and $M_3$ vs $M_2$ (right) for $\Lam=2.5\tev$, resulting from a scan over the full range of $\alpha_i$, $\tan\beta \in (0.5,50)$ and $M_{H^\pm} \in
(300,700)\gev$, for $\mu=300, 400, 500~{\rm GeV}$. Red: Positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. []{data-label="Fig:2-loop-masses-2500"}](fig4b.eps "fig:"){width="12cm"} ![ Two-loop distributions of allowed masses $M_2$ vs $M_1$ (left panels) and $M_3$ vs $M_2$ (right) for $\Lam=2.5\tev$, resulting from a scan over the full range of $\alpha_i$, $\tan\beta \in (0.5,50)$ and $M_{H^\pm} \in
(300,700)\gev$, for $\mu=300, 400, 500~{\rm GeV}$. Red: Positivity is satisfied; yellow: positivity and unitarity both satisfied; green: also experimental constraints satisfied at the 95% C.L., as specified in the text. []{data-label="Fig:2-loop-masses-2500"}](fig4c.eps "fig:"){width="12cm"}
CP violation {#cpv}
------------
Here we will verify the possibility of having CP violation in the scalar potential (\[2HDMpot\]), subject to the two-loop cancellation of quadratic divergences (\[2-loop-con\]). In order to parametrize the magnitude of CP violation we adopt the $U(2)$-invariants introduced by Lavoura and Silva [@Lavoura:1994fv] (see also [@Branco:2005em]). However here we use the basis-invariant formulation of these invariants $J_1$, $J_2$ and $J_3$ as proposed by Gunion and Haber [@Gunion:2005ja]. As is proven there (theorem \#4) the Higgs sector is CP-conserving if and only if ${{\rm Im\thinspace}}J_i=0$ for all $i$. In the basis adopted here the invariants read [@Grzadkowski:2009bt]: $$\begin{aligned}
{{\rm Im\thinspace}}J_1&=&-\frac{v_1^2v_2^2}{v^4}(\lambda_1-\lambda_2){{\rm Im\thinspace}}\lambda_5,
\label{Eq:ImJ_1} \\
{{\rm Im\thinspace}}J_2&=&-\frac{v_1^2v_2^2}{v^8}
\left[\left((\lambda_1-\lambda_3-\lambda_4)^2-|\lambda_5|^2\right) v_1^4
+2(\lambda_1-\lambda_2) {{\rm Re\thinspace}}\lambda_5 v_1^2v_2^2\right.\nonumber\\
&&\hspace*{1.2cm}\left.
-\left((\lambda_2-\lambda_3-\lambda_4)^2-|\lambda_5|^2\right) v_2^4\right]
{{\rm Im\thinspace}}\lambda_5,\label{Eq:ImJ_2} \\
{{\rm Im\thinspace}}J_3&=&\frac{v_1^2v_2^2}{v^4}(\lambda_1-\lambda_2)
(\lambda_1+\lambda_2+2\lambda_4){{\rm Im\thinspace}}\lambda_5.
\label{Eq:ImJ_3}\end{aligned}$$ It is seen that there is no CP violation when ${{\rm Im\thinspace}}\lambda_5=0$, see [@Grzadkowski:2009bt] for more details.
As we have noted earlier, $\tgb$ above $\sim 40$ implies approximate degeneracy of scalar masses. That could jeopardize the CP violation in the potential since it is well known that the exact degeneracy $M_1=M_2=M_3$ results in vanishing invariants ${{\rm Im\thinspace}}J_i$ and no CP violation (exact degeneracy implies ${{\rm Im\thinspace}}\lambda_5=0$). Using the one-loop conditions (\[qdcon1\_mod2\])–(\[qdcon2\_mod2\]) one immediately finds that $\lambda_1-\lambda_2=4(\mb^2/\cbb^2-\mt^2/\sbb^2)/v^2$, which implies $${{\rm Im\thinspace}}J_1 = 4\, {{\rm Im\thinspace}}\lambda_5 (\cbb^2\mt^2-\sbb^2 \mb^2)/v^2
=-4\, {{\rm Im\thinspace}}\lambda_5 \left(\mb/v\right)^2 +
{\cal{O}}\left({{\rm Im\thinspace}}\lambda_5/\tgb^2\right)
\label{imj1}$$ Note that if $\tgb$ is large then ${{\rm Im\thinspace}}J_1$ is suppressed not only by ${{\rm Im\thinspace}}\lambda_5 \simeq 0$ (as caused by $M_1\simeq M_2\simeq M_3$) but also by the factor $(\mb^2/v^2)$, as implied by the cancellation conditions (\[qdcon1\_mod2\])–(\[qdcon2\_mod2\]). The same suppression factor appears for ${{\rm Im\thinspace}}J_3$. The case of ${{\rm Im\thinspace}}J_2$ is more involved, however when $\mb^2/v^2$ is neglected all the invariants (\[Eq:ImJ\_1\])–(\[Eq:ImJ\_3\]) have the same simple asymptotic behavior for large $\tgb$: $${{\rm Im\thinspace}}J_i \sim {{\rm Im\thinspace}}\lambda_5/\tgbs$$ Those conclusions qualitatively remain also at the two-loop level. For a quantitative illustration we plot in Fig. \[Fig:imj-2500-300-500\] maximal values of the invariants in the $\tan\beta$–$M_{H^\pm}$ plane with all the necessary constraints imposed, looking for regions which still allow for substantial CP violation. At high values of $\tan\beta$ these invariants are of the order of $10^{-3}$, in qualitative agreement with the discussion above. It is worth noticing that the corresponding invariant in the SM; ${{\rm Im\thinspace}}Q = {{\rm Im\thinspace}}(V_{ud} V_{cb} V_{ub}^\star V_{cd}^\star)$ [@Bernreuther:2002uj] is of the order of $\sim 2\times 10^{-5} \sin \delta_{KM}$ ($V_{ij}$ and $\delta_{KM}$ are elements of the CKM matrix and CP-violating phase, respectively). Therefore the model considered here allows for CP violation at least two orders of magnitude larger than in the SM.
![ Absolute values of the imaginary parts of the $U(2)$-invariants $|{{\rm Im\thinspace}}J_i|$ at the two-loop level for $\Lam=2.5\tev$, for $\mu=500~{\rm GeV}$ (top) and $\mu=300~{\rm GeV}$ (bottom). The color coding in units $10^{-3}$ is given along the right vertical axis.[]{data-label="Fig:imj-2500-300-500"}](fig6a.eps "fig:"){width="16cm"} ![ Absolute values of the imaginary parts of the $U(2)$-invariants $|{{\rm Im\thinspace}}J_i|$ at the two-loop level for $\Lam=2.5\tev$, for $\mu=500~{\rm GeV}$ (top) and $\mu=300~{\rm GeV}$ (bottom). The color coding in units $10^{-3}$ is given along the right vertical axis.[]{data-label="Fig:imj-2500-300-500"}](fig6b.eps "fig:"){width="16cm"}
Summary {#sum}
=======
The goal of this project was to build a minimal realistic model which would ameliorate the little hierarchy problem through suppression of the quadratic divergences in scalar boson mass corrections and through lifting the mass of the lightest Higgs boson. It has been shown that it could be accomplished within the Two-Higgs-Doublet Model type II. Phenomenological consequences of requiring no quadratic divergences in corrections to scalar masses were discussed. The 2HDM type II was analyzed taking into account the relevant existing experimental constraints. Allowed regions in the parameter space were determined. An interesting scalar mass degeneracy was noticed for $\tgb \gsim 40$. The issue of possible CP violation in the scalar potential was discussed and regions of $\tgb-M_{H^\pm}$ with substantial strength of CP violation were identified. The cutoff implied by the naturality arguments is lifted from $\sim
600\gev$ in the SM up to at least $\gsim 2.5 \tev$, depending on the mass of the lightest scalar. In order to accommodate a possibility for dark matter a scalar gauge singlet should be added to the model.
References {#references .unnumbered}
==========
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|
---
abstract: 'We investigate the parameter space of two-field inflation models where inflation terminates via a first-order phase transition causing nucleation of bubbles. Such models experience a tension from the need to ensure nearly scale invariant density perturbations, while avoiding a near scale-invariant bubble size distribution which would conflict observations. We perform an exact analysis of the different regimes of the models, where the energy density of the inflaton field ranges from being negligible as compared to the vacuum energy to providing most of the energy for inflation. Despite recent microwave anisotropy results favouring a spectral index less than one, we find that there are still viable models that end with bubble production and can match all available observations. As a by-product of our analysis, we also provide an up-to-date assessment of the viable parameter space of Linde’s original second-order hybrid model across its full parameter range.'
author:
- Marina Cortês
- 'Andrew R. Liddle'
bibliography:
- 'firstorder.bib'
title: 'Viable inflationary models ending with a first-order phase transition'
---
Introduction {#intro}
============
One of the open questions in inflationary cosmology is the mechanism by which inflation came to an end. The current literature is dominated by two paradigms, violation of slow roll bringing inflation to an end while the field is still evolving, and a second-order phase transition of hybrid inflation type. However, Guth’s original (but unsuccessful) proposal [@guth] invoked a first-order phase transition whereby inflation ended by nucleation of bubbles of true vacuum. First-order transitions have subsequently experienced bursts of popularity. In the late 1980s, La and Steinhardt [@la_steinhardt] initiated intensive investigation of ‘extended inflation’ models, where modifications to Einstein gravity allowed bubble nucleation to complete in single-field inflation. A few years later those models were struggling in face of observations, and focus instead returned to Einstein gravity, now in a two-field context with one rolling and one tunnelling field [@linde; @adams_freese; @copeland_al], although see [@notari1; @notari2].
In addition to the usual quantum fluctuation mechanism, first-order inflation models produce density perturbations through the bubble collisions and subsequent thermalization. The spectrum of bubble sizes produced must be far from scale invariance to avoid clear violation with observed microwave anisotropies — the largest of the bubbles would otherwise be blatantly visible [@liddle_wands91; @liddle_wands92; @griffiths_al]. This requirement is typically at odds with the need to maintain scale invariance in the spectrum produced by quantum fluctuations, a tension sufficient to exclude extended inflation variants except in extremely contrived circumstances [@liddle_wands]. The purpose of this paper is to investigate whether the strengthened constraints of the post Wilkinson Microwave Anisotropy Probe (WMAP) era have eliminated the Einstein gravity first-order models too and, by implication, assess whether it is plausible that voids exist below current detection limits.
In Guth’s original model, with one field, the inflaton must remain in the metastable vacuum long enough to allow for sufficient $e$-folds of inflation but in this case inflation never ends, the bubbles never thermalize and the transition doesn’t complete. Introduction of a second field allows a time-dependent nucleation rate, permitting enough inflation to occur while the nucleation rate is low and a successful end when the rate rises to high enough values. This idea was proposed independently by Linde [@linde] and, in more detail, by Adams and Freese [@adams_freese] under the name ‘double-field inflation’.
Typically the second field, which is trapped in the metastable vacuum, also provides most of the energy density for inflation, although this depends on the particular values of parameters chosen. In that regime, the usual prediction is for a blue spectrum of density perturbations, $n_{\rm S}>1$. In the last few years the trend in cosmic microwave background (CMB) observations has been a tightening of the confidence limits around a central value $n_{\rm S}$ smaller than one, disfavouring this regime. Since our goal is to investigate the general viability of this type of model we will probe the entire parameter space, including the intermediate region where the contributions of each field to the energy density are comparable, making no approximations based on inflaton or the false vacuum domination.
As stated above one expects these models to run into difficulty with recent observations closing in on a nearly scale invariant scalar spectrum. CMB anisotropies observations place constraints on the maximum size of bubbles that survive from a first-order phase transition, at the time when scales of cosmological interest leave the horizon. In turn this places a strong upper limit on the nucleation rate at this time, after which it must rise sufficiently to complete the transition and provide a graceful exit for inflation. In order to meet these two requirements the field must proceed swiftly along the potential, what, in light of observations, places the model under stress.
$\begin{array}{c c}
\epsfxsize=8.5cm
\epsffile{2nd_order.eps} &
\epsfxsize=8.5cm
\epsffile{1st_order.eps}\\
\mbox{\bf (a)} & \mbox{\bf (b)}
\end{array}$
The first-order model
=====================
We consider throughout a fairly general form of the potential for a first-order phase transition, given by Copeland et al [@copeland_al]. $$\begin{aligned}
\label{pot}
V(\phi,\psi)&=&\frac{1}{4}\lambda(M^4+\psi^4)+ \frac{1}{2}\alpha M^2
\psi^2 -\frac{1}{3}\gamma M \psi^3 \nonumber\\
&& + \frac{1}{2} m^2 \phi^2 + \frac{1}{2} \lambda' \phi^2 \psi^2 \,.\end{aligned}$$ This extends the simplest second-order hybrid inflation model by addition of the cubic term for the $\psi$ field. As in conventional hybrid inflation, one envisages that initially the inflaton field $\phi$ is displaced far from its minimum, and the auxiliary field $\psi$ is then held in a false vacuum state by its coupling to the inflaton. Perturbations are generated during this initial phase as $\phi$ rolls slowly along the flat direction. The dynamics in this region are pretty much those of single-field slow-roll inflation, though the auxiliary field $\psi$ may provide most of the energy density for inflation, see Fig. \[2pot\].
In a model where the phase transition is second-order, shown in Fig. \[2pot\]a, the false vacuum becomes unstable after $\phi$ passes a certain value, $\phi_{\rm inst}$, and the fields evolve classically to their true vacuum (here producing topological defects as causally separated regions make independent choices as to which minimum to finish in). Although not the main topic of this paper, we explore current constraints on this model in the Appendix.
In the first-order case, shown in Fig. \[2pot\]b, if the parameters in Eq. (\[pot\]) are chosen appropriately, a second minimum develops once the field evolves past a point of inflection, $\phi_{\rm
infl}$. At this point bubbles of the true vacuum begin to nucleate and expand at the speed of light. The percolation rate is initially very small as the vacuum energies are comparable, but as $\phi$ approaches zero the interaction between the fields triggers a steep rise in the bubble production. Inflation ends when the nucleation rate reaches high enough values that the bubbles percolate and thermalize. In this case there is only one true vacuum and hence no topological defects. The channel in which the field rolls after tunnelling is much too steep to sustain any inflation within the bubbles.
$\begin{array}{c c c}
\epsfxsize=5.6cm
\epsffile{pot1.eps} &
\epsfxsize=5.6cm
\epsffile{pot2.eps} &
\epsfxsize=5.6cm
\epsffile{pot3.eps} \\
\mbox{\bf (a)} & \mbox{\bf (b)} & \mbox{\bf (c)}
\end{array}$
For large values of $\phi$ there is only one minimum of the potential, and in the $\psi$ direction the potential looks like Fig. \[3reg\]a. However if $\gamma^2 > 4\alpha \lambda$, a second minimum develops after $\phi$ reaches a point of inflection $$\phi_{\rm infl}^2=M^2 \frac{\gamma^2-4 \alpha \lambda}{4 \lambda'}\,,$$ as in Fig. \[3reg\]b. The presence of the cubic term in the potential then breaks the degeneracy between the two minima, making it possible for the field to tunnel to the newly formed minimum. It is this second minimum that eventually becomes the true vacuum and the $\psi$ field begins to tunnel once the transition becomes energetically favourable, Fig. \[3reg\]c.
As mentioned in the previous section the quantum generation of perturbations occurs away from this minimum, while the inflaton is rolling in the $\phi$ direction, and we consider horizon exit to occur around 55 $e$-folds before the end of inflation [@liddle_leach]. This evolution of $\phi$ is a crucial feature of the model since it is the introduction of a time dependence in the tunneling rate that will allow the phase transition to complete, bringing inflation to an end.
The rate at which the bubbles nucleate is given by the percolation parameter (the number of bubbles generated per unit time per unit volume), $$p=\frac{\Gamma}{H^4}\,.$$ In the limit of zero temperature (taken because the transition occurs during inflation) the nucleation rate of bubbles can be approximated by [@callan_coleman], $$\label{perc}
p=\frac{\lambda M^4}{4H^4}\exp(-S_{\rm E}) \,,$$ where $S_{\rm E}$ is the four-dimensional Euclidean action. $S_{\rm
E}$ was obtained for first-order transition quartic potentials by Adams [@adams], who fitted the result as $$S_{\rm E}= \frac{4 \pi^2}{3 \lambda}(2-\delta)^{-3}(\alpha_1
\delta+\alpha_2 \delta^2+\alpha_3 \delta^3) \,,
\label{4act}$$ where $\alpha_1= 13.832 ,~\alpha_2=-10.819,~\alpha_3=2.0765$, and $\delta$ is a monotonic increasing function of $\phi^2$, $$\label{delta}
\delta=\frac{9 \lambda \alpha}{\gamma^2}+\frac{9 \lambda \lambda'
\phi^2}{\gamma^2 M^2} \,.$$ The allowed range has $0<\delta<2$ (outside this range solutions correspond to energetically disallowed transitions).
The transition to the true vacuum is complete once the percolation reaches unity, (one bubble per Hubble time per Hubble volume), allowing the bubbles of the true vacuum to coalesce.
However in the most general case inflation need not end through bubble nucleation. If the potential is too steep slow-roll is violated before bubbles thermalize and inflation ends before the transition completes. In this case the precise mechanism which completes the transition is irrelevant given that it occurs after inflation ends, and for our purposes the scenario is indistinguishable from the single-field case. (In this paper we do not consider gravitational waves produced via bubble collisions, but these may provide a further observable [@hogan; @kosowsky_al; @huber_konstandin; @caprini_al] that can ultimately be used to constrain this type of model.)
The distinction between the two possibilities is given by the two values of the field, that at which the nucleation rate reaches unity, and that which makes (violation of slow-roll), where $\epsilon$ is the usual slow roll parameter defined in Eq. (\[eps\]). Inflation ends by whichever value of $\phi$ is reached first, $$\phi_{\rm end}=\max(\phi_{\rm \epsilon},\phi_{\rm crit}) \,.$$
Inflationary dynamics
=====================
Regimes
-------
Two different regimes can be distinguished, regarding which field we wish to have dominate the energy density. In the usual hybrid inflation regime the energy density of the potential is dominated by the false vacuum $\lambda M^4 \gg m^2 \phi^2$, which provides the energy for inflation. In the opposite regime, in which the inflaton dominates the energy density, the dynamics rapidly approach those of single-field inflation since, as we will see, slow-roll violation occurs sooner.
Working in either of these two regimes would allow us to simplify some of the expressions governing the dynamics during inflation, such as the number of $e$-folds and the slow-roll parameters, Eqs. (\[Ne\]), (\[eps\]) and (\[eta\]), and to proceed via an analytical treatment instead of a numerical one. However our purpose here is to probe the dynamics of the full $n_{\rm S}-r$ parameter space, ($r$ is the tensor-to-scalar ratio given by Eq. (\[r\])) so as to determine whether there still remain models consistent with CMB observations. Hence we also include the intermediate regime in our analysis, where the energy densities of the two fields are comparable, particularly when the transition between slow roll violation and bubble nucleation occurs. For this reason we will retain the full form of the potential and proceed through numerical calculations.
Field dynamics
--------------
In order to specify the dynamics of each model we begin by finding the field value, $\phi_{\rm max}$, at which inflation ends so we need to determine $\phi_{\rm \epsilon}$ and $\phi_{\rm crit}$. $\phi_{\rm \epsilon}$ is obtained by evaluating the first slow-roll parameter for our potential and taking it to unity, [^1] $$\epsilon \equiv \frac{m_{\rm Pl}^2}{16 \pi}
\left(\frac{V'}{V}\right)^2 = \frac{m^4 \phi^2 \, m_{\rm
Pl}^2}{\pi (\lambda M^4 + 2 m^2 \phi^2)^2}\approx 1 \,.$$ Inverting for $\phi$ yields, $$\label{eps}
\phi^2_{\epsilon} = \frac{m^2 m_{\rm Pl}^2 \pm m m_{\rm Pl} \sqrt{m^2
m_{\rm Pl}^2 -8\pi\lambda M^4}-4\pi\lambda M^4}
{8 \pi m^2} \,,$$ and we take the largest value of $\phi$. Note that the solution exists only for large values of $m$, where $m^2 m_{\rm Pl}^2 > 8 \pi \lambda
M^4$.
To determine $\phi_{\rm crit}$ we need to find the value at which the percolation parameter reaches unity, $p_{\rm crit} \sim 1$. Solving Eq. (\[perc\]), we get $$S_{\rm crit}\sim \ln \frac{\lambda M^4}{4 p_{\rm crit} H^4}
\label{S_cr} \,,$$ where $S_{\rm crit}$ is given by Eq. (\[4act\]).
Inverting Eq. (\[S\_cr\]) yields a value for $\phi_{\rm crit}$ (only one of the three roots lies in the allowed range) and in turn this allows us to determine $\phi_{\rm end}$, and, by comparison with $\phi_{\rm \epsilon}$, the mechanism by which inflation ends.
Knowing $\phi_{\rm end}$ we can calculate the value of the field at horizon exit, $\phi_{\rm 55}$. In this model $\phi$ rolls towards its minimum at $\phi=0$ so $\phi_{\rm 55} > \phi_{\rm end}$. Using the expression for the number of $e$-folds between two field values $\phi_1$ and $\phi_2$ we get, $$N(\phi_1,\phi_2)\equiv \ln \frac{a_2}{a_1}
\sim -\frac{8 \pi}{m_{\rm Pl}^2}
\int_{\phi_1}^{\phi_2}\frac{V}{V'} \, d\phi \,.$$ For $\phi_1= \phi_{55}$ and $\phi_2= \phi_{\rm end}$, and substituting for $V$, we have $$\label{Ne}
N(\phi_{55},\phi_{\rm end})= 2 \pi \lambda \frac{M^4}{m^2 m_{\rm Pl}^2}
\ln \frac{\phi_{55}}{\phi_{\rm end}} + \frac{2\pi}{m_{\rm
Pl}^2}(\phi_{55} ^2 - \phi_{\rm end}^2) \,,$$ where we make no assumptions on the relative size of the two masses and retain both terms. Substitution of $\phi_{\rm end}$ yields $\phi_{\rm 55}$ and now we can calculate the scalar spectral index, $n_{\rm S}$, and the tensor-to-scalar ratio, $r$, at horizon exit, by use of their expressions in terms of the usual slow-roll parameters, $$\begin{aligned}
\label{nS}
n_{\rm S}-1&=&-6 \epsilon + 2 \eta \,;\\
r&=& 16 \epsilon \,, \label{r}\end{aligned}$$ where $\epsilon$ is given by Eq. (\[eps\]), and $\eta$ is $$\label{eta}
\eta \equiv \frac{m_{\rm Pl}^2}{8 \pi} \frac{V''}{V}
=\frac{m^2 \, m_{\rm Pl}^2}{2\pi (\lambda M^4 + 2 m^2 \phi^2)} \,,$$ where the last equality is obtained by substitution of the potential.
At this point we can locate the model in the $n_{\rm S}-r$ plane and determine its position in relation to WMAP5 confidence limits [@komatsu].
Choosing parameters
-------------------
Throughout we set the self-interaction and coupling constants, $\lambda$ and $\lambda '$ respectively, equal to unity. We are then left with two constants, $\alpha$ and $\gamma$, and requiring the energy density of the true vacuum to be zero fixes one of these in terms of the other. We will fix $\alpha$ in terms of $\gamma$ but the reverse option could just as well be taken.
The CMB amplitude normalization can be used to relate the two masses. We use this to fix the mass of the light field $\phi$ and then we are left with only two undetermined parameters: the energy of the false vacuum, $M$, and the constant $\alpha$. For each value of $\alpha$, varying $M$ fully determines the dynamics of the fields, and describes a trajectory in the $n_{\rm S}-r$ plane shown in Fig. \[nr1\].
Each line is composed of two branches which correspond to the two solutions of the WMAP normalization, and converge for large values of $M \sim 2.7 \times 10^{-3} m_{\rm Pl}$. For values of $M$ larger than this there is no solution to the amplitude normalization hence no viable models. This can be seen also in Fig. \[2m\] which illustrates how the two different approximation schemes converge to a common behaviour and cease to exist after a certain value of $M$ (c.f. Fig. 1 of Ref. [@copeland_al]).
The right-hand branch in Fig. \[nr1\] corresponds to the lower branch in Fig. \[2m\] and to the smaller value of $m$ from the WMAP normalization. In this branch the approximate relation $M \sim
m^{2/5}$ (in Planck units) holds and the false vacuum dominates. The dynamics are indistinguishable in the $n_{\rm S}-r$ plane when $M <
10^{-4} m_{\rm Pl}$. We start with the typical slightly blue tilted spectrum and negligible tensor fraction. As $m$ continues to increase so does the deviation from $n_{\rm S} \sim 1$ until the approximate relation between the two masses breaks down and we have the inflaton playing a more significant role in the relative contribution of the two fields. At this point we observe a turn in the $n_{\rm S}-r$ plane, and the solution enters the intermediate region of comparable field energy densities.
Despite this we still observe inflation ending by bubble nucleation throughout this branch, from small values of $M$ to the maximum at $M
\sim 2.7 \times 10^{-3} m_{\rm Pl}$.
In the opposite branch, on the left-hand side, the model starts inside the WMAP5 95% confidence contour, well inside the inflaton dominated regime. Similarly to the other branch we observe an initial period where there is little dependence on the false vacuum energy, corresponding to the plateau on Fig. \[2m\], and the dynamics are very well approximated by those of standard single-field inflation with a $\phi^2$ potential, well known to satisfy WMAP5 data.
This regime breaks down as the false vacuum energy increases and eventually we recover the regime where the phase transition triggers the end of inflation before the violation of slow roll, meaning we are again in the bubble production scenario. The interesting results here draw from the fact that the transition occurs inside the WMAP5 95% confidence contour, making these viable models even away from false vacuum domination. Fig. \[zoom\] is a zoom of this region showing the field mass, $M$, at which the transition to bubble nucleation occurs, $M \sim 5\times 10^{-4} m_{\rm Pl}$, still allowed by the 95% confidence limits.
{width="80.00000%"}
Three constraints
=================
In the previous section we looked at constraints in the $n_{\rm S}-r$ plane. By specifying a value for $\alpha$, one of our two free parameters $(M,\alpha)$, the CMB normalization then allows us to recover a trajectory in this plane and assess where the density perturbations are compatible with WMAP5 data. We now compute other constraints on the scenario, in the $M-\alpha$ plane.
Model consistency
-----------------
We begin with the requirement that $M$ be not larger than an upper limit above which, for a particular choice of couplings, the transition does not complete ($\phi_{\rm crit}$ does not exist). We call this the model consistency constraint, which translates to a relation for the value of $\phi_{\rm crit}$, coming from the requirement that there exists a solution of Eq. (\[delta\]) for $\delta$. Because of the constant term in Eq. (\[delta\]) this is an additional requirement to $0<\delta<2$.
Since we have chosen to set $m$ by the CMB normalization this can be translated into an excluded region in the $(M,\alpha)$ plane (although alternatively we could have expressed it in terms of a region in $(M,m)$, by having $\alpha$ specified by the CMB normalization instead). This yields the region below the upper (blue) curve in Fig. \[3constr\]. We see that specifying a value for the false vacuum density imposes an upper limit on the coupling $\alpha$ (alternatively on the inflaton mass, $m$) in order for the model to have the possibility to complete the phase transition.
Big bubble constraint
---------------------
We adopt here a fairly crude criterion to judge whether the bubbles are compatible with observations, which is that any bubbles produced at the end of inflation and expanded to astrophysical sizes must, during the epoch of recombination, have a comoving size not larger than $20h^{-1}{\rm Mpc}$ [@liddle_wands91]. This corresponds to a maximum filling fraction at that time of $10^{-5}$, and puts an upper bound on the percolation rate of bubbles at the time the scales we observe today left the horizon: $$\left(\frac{\Gamma}{H^4}\right)_{55} \leq 10^{-5} \,.$$ With our form for the action Eq. (\[4act\]) and choice of potential this becomes $$S_{55}\sim -2.9 +4 \ln{\frac{m_{\rm Pl}}{\lambda^{1/4} M}}+ 11.5 \,.$$ This gives us the region between the short dashed (black) lines in Fig. \[3constr\].
{width="80.00000%"}
WMAP constraint
---------------
We can similarly place constraints on the $(M,\alpha)$ plane, by considering the 95% confidence limit resulting from the WMAP5 $n_{\rm
S}-r$ plane when tensors are included. $$n_{\rm S} \lesssim 1.05 \,.
\label{nSwmap}$$ Inverting Eq. (\[nSwmap\]) gives us an upper limit on $M$ in terms of $\alpha$, resulting in the region left of the long dashed (red) line in Fig. \[3constr\].
We also see from Fig. \[3constr\] that this constraint is opposed to that coming from the CMB maximum bubble size requirement, as we argued in Section \[intro\]. Big bubbles at last scattering put an upper limit on the nucleation rate at horizon crossing while CMB constraints on the spectral tilt put a lower bound on the nucleation rate, from the requirement that $n_{\rm S}$ is not too distant from scale invariance.
Nevertheless, a region of parameter space survives all constraints.
Conclusions
===========
Our principal conclusion is that there do remain Einstein gravity models of first-order inflation which are compatible with observations, despite the increasing tension between the need for a scale-invariant primordial spectrum and the suppression of large-scale bubbles. We have exhibited a particular class of model and found the parameter region where the first-order model is viable. Its predictions for $n_{\rm S}$ and $r$ are similar to the simple $m^2
\phi^2$ slow-roll inflation model, though a little further from scale-invariance.
In this paper we have imposed a relatively simple constraint on the bubbles, and have then assumed that their impact on the CMB is negligible as far as constraints on the primordial perturbations are concerned. A more detailed treatment would combine the two perturbation sources and refit to the CMB data, which may lead to some modification to the outcome in regimes where the bubble production is close to the observational limit. For models where the bubbles are safely within the observational limits this is not an issue.
This paper demonstrates that we are still some way from having a clear view as to how the inflationary period of the Universe may have ended. The literature contains three different mechanisms — violation of slow-roll, a second-order instability during slow-roll, and bubble nucleation — and we have shown that the last (and least popular) of these remains a viable option. First-order models are of phenomenological interest as the bubble spectrum is an additional source of inhomogeneity that could be considered in matching high-precision observations. The bubble collisions may also generate detectable gravitational waves [@hogan; @kosowsky_al; @huber_konstandin; @caprini_al]. There is therefore an ongoing need to refine understanding of the nature of perturbations induced by a primordial bubble spectrum.
M.C. was supported by FCT (Portugal) and by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. A.R.L. was supported by STFC (UK). We thank Andy Albrecht, Katie Freese, Andrei Linde, and Eric Linder for discussions and comments.
{width="80.00000%"}
The second-order model {#sec2nd}
======================
Although not part of our main study, the full parameter range of the second-order hybrid inflation model [@linde_2nd_I; @linde_2nd_II; @copeland_al] is easily studied using the machinery we have used for the first-order case. The second-order model also uses the energy density of an auxiliary field to raise the energy scale for inflation without endangering slow roll. The phase transition in this case is continuous, with the $\psi$ field rolling down to the true vacuum (see Fig. \[2pot\](a)). There are no bubbles now and hence no bubble constraint; we just have to consider whether the usual perturbations are compatible with WMAP5 data. Furthermore since now there is no cubic term to break the degeneracy between the two minima, there is the possibility of topological defect formation at the end of inflation, as different regions in space roll towards one or the other minimum. However we do not consider their possible impact here.
The dynamics are closely related to those in the first-order case. The critical point where the phase transition completes is a point of instability $\phi_{\rm inst}$, after which $\psi=0$ becomes unstable and starts to roll. The potential for this case is a particularization of the first-order potential Eq. (\[pot\]) with $\lambda=-\alpha$ and $\gamma=0$, and becomes, $$\label{pot2}
V(\phi,\psi)= \frac{1}{4} \lambda(\psi^2-M^2)^2+ \frac{1}{2} m^2
\phi^2 + \frac{1}{2} \lambda' \phi^2 \psi^2 \,.$$
Apart from the expression determining $\phi_{\rm inst}$, we can retain most of the expressions from the first-order model and build a similar picture in the $n_{\rm S}-r$ plane. We present this in Fig. \[nr2\], again for $\lambda = \lambda'=1$. We see that the false vacuum dominated regime, which has $n_{\rm S}>1$ and negligible $r$, lies entirely outside the WMAP5 allowed region, as does the main curve of the intermediate regime. Only once the trajectory heads towards the slow-roll limit does it become compatible with observations. At $M\sim9 \times10^{-4} m_{\rm Pl}$ the models cross the WMAP5 95% contour and at $M\sim5 \times10^{-4} m_{\rm Pl}$ inflation ends through slow roll violation instead of a phase transition.
[^1]: The field $\psi$ sits in the false vacuum during the inflationary phase, since this is the only minimum available to $\psi$ in this region of the potential. This happens regardless of the means to ending inflation, so $\psi$ is set to zero throughout this section.
|
---
abstract: |
In [@05] B. Ebanks and H. Stetk[æ]{}r obtained the solutions of the functional equation $f(xy)-f(\sigma(y)x)=g(x)h(y)$ where $\sigma$ is an involutive automorphism and $f,g,h$ are complex-valued functions, in the setting of a group $G$ and a monoid $M$. Our main goal is to determine the complex-valued solutions of the following more general version of this equation, viz $f(xy)-\mu(y)f(\sigma(y)x)=g(x)h(y)$ where $\mu:
G\longrightarrow \mathbb{C}$ is a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in G$. As an application we find the complex-valued solutions $(f,g,h)$ on groups of the equation $f(xy)+\mu(y)g(\sigma(y)x)=h(x)h(y)$.
author:
- Bouikhalene Belaid and Elqorachi Elhoucien
title: A class of functional equations on monoids
---
Introduction
============
We recall that a semigroup $S$ is a set equipped with an associative operation. We write the operation multiplicatively. A monoid is a semigroup $M$ with identity element that we denote $e$. A function $\chi$ : $S\longrightarrow
\mathbb{C}$ is said to be multiplicative if $\chi(xy)=\chi(x)\chi(y)$ for all $x,y\in S.$\
Let $S$ be a semigroup and $\sigma$ : $G\longrightarrow G$ an involutive homomorphism, that is $\sigma(xy)=\sigma(x)\sigma(y)$ and $\sigma(\sigma(x))=x$ for all $x,y\in G.$\
The complex-valued solutions of the following variant of d’Alembert’s functional equation $$\label{eq1}
f(xy)+f(\sigma(y)x)=2f(x)f(y),\;x,y\in S.$$ was determined by Stetkær [@007]. They are the functions of the form $$f(x)=\frac{\chi+\chi\circ\sigma}{2},$$ where $\chi$ : $G\longrightarrow
\mathbb{C}$ is multiplictive.\
In the same year, Ebanks and Stetkær [@05] obtained on monoids the complex-valued solutions of the following d’Alembert’s other functional equation $$\label{eq2}
f(xy)-f(\sigma(y)x)=g(x)h(y),\; x,y\in S.$$ This functional equation contains, among others, an equation of d’Alembert [@01; @02; @03] $$\label{eq3}
f(x+y)-f(x-y)=g(x)h(y),\;x,y\in \mathbb{R}$$ the solutions of which are known on abelian groups, and a functional equation $$\label{eq4}
f(x+y)-f(x+\sigma(y))=g(x)h(y),\;x,y\in G$$ studied by Stetkær \[12, Corollary III.5\] on abelian groups $G$.\
There are various ways of extending functional equations from abelian groups to non-abelian groups. The $\mu$-d’Alembert functional equation $$\label{eq5}
f(xy)+\mu(y)f(x\tau(y))=2f(x)f(y),\; x,y\in G$$ which is an extension of d’Alembert functional equation $$\label{eq6}
f(xy)+f(x\tau(y))=2f(x)f(y),\; x,y\in G,$$ where $\tau$ is an involutive of $G$, is closely related to pre-d’Alembert functions. It occurs in the literature. See Parnami, Singh and Vasudeva [@parnami], Davison \[5, Proposition 2.11\], Ebanks and Stetkær [@st2], Stetkær \[15, Lemma IV.4\] and Yang \[16, Proposition 4.2\]. The functional equation (\[eq5\]) has been treated systematically by Stetkær [@st3]. The non-zero solutions of (\[eq5\]) are the normalized traces of certain representations of $G$ on $\mathbb{C}^{2}$. Davison proved this via his work [@davison] on the pre-d’Alembert functional equation on monoids.\
The variant of Wilson’s functional equation $$\label{eq77777}
f(xy)+\mu(y)f(\tau(y)x)=2f(x)g(y),\; x,y\in G;$$ $$\label{eq7}
f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\; x,y\in G$$ with $\mu\neq 1$ was recently studied on groups by Elqorachi and Redouani [@elqorachi].\
The complex-valued solutions of equation (\[eq77777\]) with $\tau(x)=x^{-1}$ and $\mu(x)=1$ for all $x\in G$ are obtained on groups by Ebanks and Stetkær [@st2].\
The present paper complements and contains the existing results about (\[eq2\]) by finding the solutions $f,g,h$ of the extension $$\label{eq8}
f(xy)-\mu(y)f(\sigma(y)x)=g(x)h(y),\; x,y\in S$$of it to monoids that need not be abelian, because on non-abelian monoids the order of factors matters and involutions and involutive automorphisms differ.\
As in [@05] one of the main ideas is to relate the functional equation (\[eq2\]) to a sine substraction law on monoids. In our case we need the solutions of the following version of the sine subtraction law $$\label{eq9}\mu(y)f(x\sigma(y))=f(x)g(y)-f(y)g(x), \; x,y\in
S.$$ These results are obtained in Theorem 2.1. We need also the solutions of equation (\[eq7\]) on monoids. There are not in the literature, but we derived them in Theorem 3.2. In section 4 we obtain on of the main results about (\[eq8\]). Furthermore, as an application we find the complex-valued solutions $(f,g,h)$ of the functional equation$$\label{EQ1}
f(xy)+\mu(y)g(\sigma(y)x)=h(x)h(y),\; x,y\in G$$ on groups and monoids in terms of multiplicative and additive functions.
Notation and preliminary matters
---------------------------------
Throughout this paper $G$ denotes a group and $S$ a semigroup. A monoid is a semigroup $M$ with an identity element that we denote e. The map $\sigma : S\longrightarrow S$ denotes an involutive automorphism. That it is involutive means that $\sigma(\sigma(x)) =
x$ for all $x \in S$. The mapping $ \mu : S
\longrightarrow\mathbb{C}$ is a multiplicative function such that $
\mu(x\sigma(x))= 1$ for all $x \in S$. If $\chi: S \longrightarrow
\mathbb{C}$ is a multiplicative function such that $\chi\neq0$, then $I_{\chi}=\{x\in S\; |\; \chi(x)=0\}$ is either empty or a proper subset of $S$. Furthermore, $I_{\chi}$ is a two-sided ideal in $S$ if not empty and $S\backslash I_{\chi}$ is a subsemigroup of $S$. If $S$ is a topological space, then we let $\mathcal{C}(S)$ denote the algebra of continuous functions from $S$ into $\mathbb{C}$.\
For later use we need the following results. The next proposition corresponds to Lemma 3.4 in \[8\].
Let $S$ be a semigroup, and suppose $f,g
:S\longrightarrow\mathbb{C}$ satisfy the sine addition law $$f(xy)=f(x)g(y)+f(y)g(x), \; x,y\in S$$ with $f\neq 0$. Then there exist multiplicative functions $\chi_{1},\chi_{2}: G\longrightarrow \mathbb{C}$ such that $$g=\frac{\chi_{1}+\chi_{2}}{2}.$$ Additionally we have the following\
i) If $\chi_{1}\neq \chi_{2}$ , then $f = c(\chi_{1}-\chi_{2})$ for some constant $c \in \mathbb{C}\setminus\{0\}$.\
ii) If $\chi_{1}=\chi_{2}$, then letting $\chi:=\chi_{1}$ we have $g=\chi$. If $S$ is a semigroup such that $S=\{xy\in :
x, y\in S\}$ (for instance a monoid), then $\chi\neq 0$.\
If $S$ is a group, then there is an additive function $A: S\longrightarrow\mathbb{C}$, $A\neq 0$, such that $f=\chi A$.\
If $S$ is a semigroup which is generated by its squares, then there exists an additive function $A: S\setminus I_{\chi}\longrightarrow
\mathbb{C}$ for which $$f(x)=\left\lbrace
\begin{array}{l}
\chi(x)A(x)\; \; for \;x\in S\setminus I_{\chi}\\
0 \; \; \;\; for \; x\in I_{\chi}.
\end{array}\right.$$ Furthermore, if $S$ is a topological group, or if $S$ is a topological semigroup generated by its squares, and $f, g \in
\mathcal{C}(S)$, then $ \chi_{1}, \chi_{2}, \chi\in \mathcal{C}(S)$. In the group case $A \in \mathcal{C}(S)$ and in the second case $A
\in \mathcal{C}(S\setminus I_{\chi})$.
$\mu$-sine subtraction law on a group and on a monoid
=====================================================
In this section we deal with a new version of the sine subtraction law $$\mu(y)k(x\sigma(y))=k(x)l(y)-k(y)l(x), \; x,y\in S,$$ where $k, l$ are complex valued functions and $\mu$ is a multiplicative function. The new feature is the introduction of the function $\mu.$ We shall say that $k$ satisfies the $\mu$-sine subtraction law with companion function $l$. If $S$ is a topological semigroup and $k, l$ satisfy (2.1) such that $k\neq 0$ and $k$ is a continuous function then $l$ is also a continuous function. In the case where $\mu=1$ and $G$ is a topological group, the functional equation (2.1) was solved in \[8\].\
Here we focus exclusively on (2.1), and we include nothing about other extensions of the cosine, sine addition and subtraction laws. The next theorem is the analogue of Theorem 3.2 in \[8\].
Let $G$ be a group and let $\sigma: G\longrightarrow\mathbb{C}$ be a involutive automorphism. Let $\mu: G\longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in
G$. The solution $k, l: G\longrightarrow \mathbb{C}$ of the $\mu$-sine subtraction law (2.1) with $k\neq 0$ are the following pairs of functions, where $\chi: G\longrightarrow
\mathbb{C}\setminus\{0\}$ denotes a character and $c_{1}\in \mathbb{C}$, $c_{2}\in \mathbb{C}\setminus\{0\}$ are constants.\
i) If $\chi\neq \mu\;\chi\circ\sigma$, then $$k=c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2}, \; l=\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}.$$ ii) If $\chi=\mu\;\chi\circ\sigma$, then $$k=\chi A,\;\; l=\chi(1+c_{1}A)$$ where $A: G\longrightarrow \mathbb{C}$ is an additive function such that $A\circ \sigma =-A\neq 0$.\
Furthermore, if $G$ is a topological group and $k \in
\mathcal{C}(G)$, then $l, \chi, \mu\;\chi\circ\sigma, A \in
\mathcal{C}(G)$.
By interchanging $x$ and $y$ in (2.1) we get that $\mu(x)k(y\sigma(x))=-\mu(y)k(x\sigma(y))$ for all $x,y\in G$. By setting $y=e$ we get that $k(x)=-\mu(x)k(\sigma(x))$ for all $x\in
G$. Using this, equation (2.1) and the fact that $\mu(x\sigma(x))=1$ we get for all $x,y\in G$ that $$\begin{aligned}
&&
k(x)[l(y)-\mu(y)l(\sigma(y))]-k(y)[l(x)-\mu(x)l(\sigma(x))]\\&&=k(x)l(y)-\mu(y)k(x)l(\sigma(y))-k(y)l(x)+\mu(x)k(y)l(\sigma(y))\\
&&=\mu(y)k(x\sigma(y)+\mu(xy)k(\sigma(x))l(\sigma(y))-\mu(xy)k(\sigma(y))l(\sigma(x))\\
&=&\mu(y)k(x\sigma(y))+\mu(xy)[k(\sigma(x))l(\sigma(y))-k(\sigma(y))l(\sigma(x))]\\&&=
\mu(y)k(x\sigma(y))+\mu(xy)\mu(\sigma(y))k(\sigma(x)y)\\&&=
\mu(y)k(x\sigma(y))+\mu(x)k(\sigma(x)y)\\&&=
mu(y)k(x\sigma(y))-\mu(x)\mu(\sigma(x)y)k(x\sigma(y))\\&&=
\mu(y)k(x\sigma(y))-\mu(y)k(x\sigma(y))\\&&= 0.\end{aligned}$$ So that we get for all $x,y\in G$ that $$k(x)[l(y)-\mu(y)l(\sigma(y))]=k(y)[l(x)-\mu(x)l(\sigma(x))].$$ By using some ideas from \[8\] we let $l^{+}(x)=\frac{l(x)+\mu(x)l(\sigma(x))}{2}$ and $l^{-}(x)=\frac{l(x)-\mu(x)l(\sigma(x))}{2}$ for all $x\in G$. We have $l=l^{+}+l^{-}$, $l^{+}(\sigma(x))=\mu(\sigma(x))l^{+}(x)$ and $l^{-}(\sigma(x))=-\mu(\sigma(x))l^{-}(x)$ for all $x\in G$. From (2.2) we have for all $x,y\in G$ that $k(x)l^{-}(y)=k(y)l^{-}(x)$. Since $k\neq 0$ there exists an $x_{0}\in G$ such that $k(x_{0})\neq 0$. Thus we have $l^{-}(y)=\frac{l^{-}(x_{0})}{k(x_{0})}k(y)=ck(y)$ for all $y\in G$. Then $l^{-}=ck$. On the other hand by replacing $l$ by $l^{+}+l^{-}$ in (2.1) and by using $k(x)l^{-}(y)=k(y)l^{-}(x)$ we get that $$\mu(y)k(x\sigma(y))=k(x)l^{+}(y)-k(y)l^{+}(x), \; x,y\in G$$ Replacing $y$ by $\sigma(y)$ in (2.3) we get that $$k(xy)=k(x)l^{+}(y)+k(y)l^{+}(x), \; x,y\in G.$$ According to Proposition 1.1 we have\
either i) or ii) $k=c_{1}(\chi_{1}-\chi_{2})$ and $l^{+}=\frac{\chi_{1}+\chi_{2}}{2}$ where $c_{1}\in
\mathbb{C}\setminus\{0\}$. Since $k(x)=-\mu(x)k(\sigma(x))$ and $l^{+}(x)=\mu(x)l^{+}(\sigma(x))$ for all $x\in G$ we get that $c_{1}(\chi_{1}-\chi_{2})=-c_{1}\mu(\chi_{1}\circ
\sigma-\chi_{2}\circ \sigma)$ and $\mu(\chi_{1}\circ
\sigma+\chi_{2}\circ \sigma)=\chi_{1}+\chi_{2}$. Then $\chi_{2}=\mu\chi_{1}\circ \sigma$, $l^{+}=\frac{\chi_{1}+\mu\chi_{1}\circ \sigma}{2}$ and $k=c_{1}\frac{\chi_{1}-\mu\chi_{1}\circ \sigma}{2}$. Since $l^{-}
=ck=cc_{1}\frac{\chi_{1}-\mu\chi_{1}\circ
\sigma}{2}=c_{2}\frac{\chi_{1}-\mu\chi_{1}\circ \sigma}{2}$ where $c_{2}\in \mathbb{C}\setminus\{0\}$. By using the fact $l=l^{-}+l^{+}$ we get (i).\
ii) we have $l^{+}=\chi$ and $k=\chi A$. Since $l^{+}(\sigma(x))=\mu(\sigma(x))l^{+}(x)$ for all $x\in G$ we get that $\chi=\mu\chi\circ \sigma$. From $k(x)=-\mu(x)k(\sigma(x))$ for all $x\in G$ we get that $A=-A\circ \sigma$. By using the fact $l=l^{-}+l^{+}=c\chi A+\chi=\chi(cA+1)$. This completes the proof.
In the next proposition we extend Theorem 2.1 to monoids.
Let $M$ be a monoid and let $\sigma: M\longrightarrow M$ be a involutive automorphism. Let $\mu: M\longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in
M$. The solution $k, l: M\longrightarrow \mathbb{C}$ of the $\mu$-sine subtraction law (2.1) with $k\neq 0$ are the following pairs of functions, where $\chi: M\longrightarrow \mathbb{C}\setminus\{0\}$ denotes a multiplicative function and $\chi(e)=1$ and $c_{1}\in \mathbb{C}$, $c_{2}\in \mathbb{C}\setminus\{0\}$ are constants :\
i) If $\chi\neq \mu\;\chi\circ\sigma$, then $$k=c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}, \; l=\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2}.$$ ii) If $\chi=\mu\;\chi\circ\sigma$ and $S$ is generated by its squares, then $$k(x)=\chi(x) A(x),\;\; l(x)=\chi(x)(1+c_{1}A(x))\; \forall x\in M\backslash I_{\chi}$$ $$l(x)=k(x)=0, \;\;\;\; \forall x\in I_{\chi}$$ where $A: M\longrightarrow \mathbb{C}\setminus\{0\}$ is an additive function such that $A\circ \sigma =- A\neq 0$.\
Furthermore, if $M$ is a topological monoid, and $k \in
\mathcal{C}(M)$, then $l, \chi, \mu\chi\circ \sigma \in
\mathcal{C}(M)$ and $A \in \mathcal{C}(M\backslash I_{\chi})$.
By the same way in Proposition 3.6 in \[8\] and by using Theorem 2.1 and the Proposition 1.1 we get the proof.
A variant of Wilson’s functional equation on monoids
====================================================
The solutions of the functional equation (\[eq7\]) on groups are obtained in [@elqorachi]. More precisely, we have the following theorem.
Let $G$ be a group, let $\sigma$ : $G\longrightarrow G$ a homomorphism such that $\sigma\circ\sigma =
I$, where $I$ denotes the identity map, and $\mu$ : $G\longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x)) = 1$ for all $x\in G$. The solutions $f, g$ of the functional equation (\[eq7\]) are the following pairs of functions, where $\chi$ : $G\longrightarrow \mathbb{C}$ denotes a multiplicative function and $c,\alpha\in \mathbb{C}^{\ast}$\
(i) $f = 0$ and $g$ arbitrary.\
(ii) $g = \frac{\chi+\mu \chi\circ \sigma}{ 2}$ and $f = \alpha g$.\
(iii) $g = \frac{\chi+\mu \chi\circ \sigma}{ 2}$ and $f = (c + \alpha/2 )\chi-(c -\alpha/2
)\chi\circ \sigma$ with $(\mu-1)\chi = (\mu-1)\chi\circ \sigma$.\
(iv) $g = \chi$ and $f=\chi(a+\alpha)$, where where $\chi=\mu\;\chi\circ\sigma$ and $a$ is an additive map which satisfies $a\circ\sigma+ a = 0$.
We shall now extend this result to monoids.
Let $M$ be a monoid, let $\sigma$ : $M\longrightarrow M$ a homomorphism such that $\sigma\circ\sigma =
I$, where $I$ denotes the identity map, and $\mu$ : $M\longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x)) = 1$ for all $x\in M$. The solutions $f, g$ of the functional equation (\[eq7\]) are the following pairs of functions, where $\chi$ : $M\longrightarrow \mathbb{C}$ denotes a multiplicative function and $c,\alpha\in \mathbb{C}^{\ast}$\
(i) $f = 0$ and $g$ arbitrary.\
(ii) $g = \frac{\chi+\mu \chi\circ \sigma}{ 2}$ and $f = \alpha g$.\
(iii) $g = \frac{\chi+\mu \chi\circ \sigma}{ 2}$ and $f = (c + \alpha/2 )\chi-(c -\alpha/2
)\chi\circ \sigma$ with $(\mu-1)\chi = (\mu-1)\chi\circ \sigma$.\
(iv) $g = \chi$, $f_e(xy)=f_e(x)\chi(y)+f_e(y)\chi(x)$ for all $x,y\in S$ and with $f=f_e +f(e)\chi$. Furthermore, if $M$ is a monoid which is generated by its squares, then $\chi=\mu\;\chi\circ\sigma$, there exists an additive function $a$ : $M\backslash I_{\chi}\longrightarrow \mathbb{C}$ for which $a\circ\sigma+ a =
0$, $$f(x)=\left\lbrace
\begin{array}{l}
\chi(x)(a(x)+\alpha)\; \; for \;x\in M\setminus I_{\chi}\\
0 \; \; \;\; for \; x\in I_{\chi}
\end{array}\right.$$ Indeed, If $M$ is a topological group, or $M$ is a topological monoid generated by its squares, $f,g,\mu\in
C(M)$, and $\sigma$ : $M\longrightarrow M$ is continuous, then $\chi\in C(M)$. In the group case $a\in C(M)$ and in the second case $a\in C(M\backslash I_{\chi})$.
Verifying that the stated pairs of functions constitute solutions consists of simple computations. To see the converse, i.e., that any solution $f,g$ of (\[eq7\]) is contained in one of the cases below, we will use \[8, Lemma 3.4\] and \[9, Theorem 3.1\]. All, except the last paragraphs of part (iv) and the continuity statements, is in Theorem 3.1 in [@elqorachi]. Now, we assume that $M$ is a monoid generated by its squares. We use the notation used in the proof of Theorem 3.1 in [@elqorachi], in particular for the last paragraphs of part (iv) we have $$\label{equation1}
f_e(xy)=f_e(x)\chi(y)+f_e(y)\chi(x)$$ for all $x,y\in S$ and with $f=f_e +f(e)\chi$. So, from \[9, Lemma 3.4\] we get $f(x)=0+f(e)\chi(x)=0+f(e)0=0$ if $x\in I_{\chi}$ and $f(x)=\chi(x)(a(x)+f(e))$ if $x\in S\backslash I_{\chi}$ and where $a$ is an additive function of $S\backslash I_{\chi}$.\
Now, we will verify that $\chi=\mu\;\chi\circ\sigma$ and $a\circ\sigma=-a$. Since $f,g$ are solution of equation (\[eq7\]) we have $$\label{equation2}
f(xy)+\mu(y)f(\sigma(y)x)=2f(x)\chi(y)$$ for all $x,y\in M.$ By using the new expression of $f$ and the fact that $I_\chi$ is an ideal, we get after an elementary computation that $f(xy)=f(yx)$ for all $x,y\in M$. So, equation (\[equation2\]) can be written as follows $$\label{equation3}
f(xy)+\mu(y)f(x\sigma(y))=2f(x)\chi(y),\;x,y\in
M.$$ By replacing $y$ by $\sigma(y)$ in (\[equation3\]) and multiplying the result obtained by $\mu(y)$ we get $$\label{equation4}
f(xy)+\mu(y)f(x\sigma(y))=2f(x)\mu(y)\chi(\sigma(y)),\;x,y\in
M.$$ Finally, by comparing (\[equation3\]), (\[equation4\]) and using $f\neq 0$ we get $\chi(y)=\mu(y)\chi(\sigma(y))$ for all $y\in M.$\
By Substituting the expression of $f$ into (\[equation3\]) and using $\chi(y)=\mu(y)\chi(\sigma(y))$ and $\mu(y\sigma(y))=1$ for all $y\in M$, we find after reduction that $\chi(x)\chi(y)[a(y)+a(\sigma(y))]=0$ for all $x,y\in M\backslash
I_\chi$. Since $\chi\neq 0$ we get $a(y)+a(\sigma(y))=0$ for all $y\in M\backslash
I_\chi$.\
For the topological statement we use \[13, Theorem 3.18(d)\]. This completes the proof.
Solutions of (\[eq8\]) on groups and monoids
============================================
In this section we solve the functional equation (\[eq8\]) on monoids. In the next proposition we show that if $(f,g,h)$ is a complex-valued solution of equation (\[eq8\]), then $h$ satisfies the $\mu$-sine subtraction law.
Let $M$ be a monoid, let $\sigma$ be a involutive automorphism on $S$, let $\mu$ be a multiplicative function on $M$ such that $\mu(x\sigma(x))=1$ for all $x\in M.$ Suppose that $f,g,h:
M\longrightarrow \mathbb{C}$ satisfy the functional equation (\[eq8\]). Suppose also that $g\neq 0$ and $h\neq 0$. Then\
i) $h(\sigma(x))=-\mu(\sigma(x))h(x))$ for all $x\in M$.\
ii) $h(xy)=h(yx)$ for all $x,y\in M$.\
3i) $h$ satisfies the $\mu$-sine subtraction law (2.1).\
4i) If $g(e)=0$, then $g=bh$ for some $b\in \mathbb{C}\setminus\{0\}$.\
5i) If $g(e)\neq 0$, then $h$ satisfies the $\mu$-sine subtraction law with companion function $\frac{g}{g(e)}$.\
Moreover, if $M$ is a topological monoid, and $h\neq 0$ is continuous, then the companion function is also continuous.
We follow the path of the proof of Proposition 3.1 in \[8\].\
By substituent $(x,yz), (\sigma(y),\sigma(z)x)$ and $(z,\sigma(xy))$ and $(z,\sigma(xy))$ in (\[eq8\]) we obtain $$f(xyz)-\mu(yz)f(\sigma(yz)x)=g(x)h(yz),$$ $$f(\sigma(yz)x)-\mu(\sigma(z)x)f(z\sigma(xy))=g(\sigma(y))h(\sigma(z)x),$$ $$f(z\sigma(xy))-\mu(\sigma(xy))f(xyz)=g(z)h(\sigma(xy)).$$ By multiplying (4.1) by $\mu(\sigma(xy))$ we obtain that $$\mu(\sigma(xy))f(xyz)-\mu(\sigma(x)z)f(\sigma(yz)x)=\mu(\sigma(xy))g(x)h(yz).$$ By adding (4.3) and (4.4) we obtain $$f(z\sigma(xy))-\mu(\sigma(x)z)f(\sigma(yz)x)=g(z)h(\sigma(xy))+\mu(\sigma(xy))g(x)h(yz).$$ By multiplying (4.5) by $\mu(\sigma(z)x)$ we obtain $$\mu(\sigma(z)x)f(z\sigma(xy))-f(\sigma(yz)x)=\mu(\sigma(z)x)g(z)h(\sigma(xy))+\mu(\sigma(zy))g(x)h(yz).$$ By adding (4.6) and (4.2) we obtain $$0=g(\sigma(y))h(\sigma(z)x)+\mu(\sigma(z)x)g(z)h(\sigma(xy))+\mu(\sigma(zy))g(x)h(yz).$$ Setting $x_{0}$ such that $g(x_{0})\neq 0$ and the fact that $\mu(x\sigma(x))=\mu(x)\mu(\sigma(x))=1$ for all $x\in M$, we get that $$h(yz)=\mu(y)g(\sigma(y))l(z)+g(z)l_{1}(y), \; y,z\in M$$ where $l, l_{1}$ are complex valued functions on $M$. Using (4.8) in (4.7) we obtain for all $x,y,z\in M$ $$g(x)g(\sigma(y))\{l_{1}(\sigma(z))+\mu(\sigma(z))l(z)\} +
g(x)g(z)\{\mu(\sigma(z))l(\sigma(y))+\mu(\sigma(zy))l_{1}(y)\}
+$$ $$g(z)g(\sigma(y))\{\mu(\sigma(z))l(x)+\mu(\sigma(z)x)l_{1}(\sigma(x))\} =0.$$ Putting $x=x_{0}$, $y=\sigma(x_{0})$, the equation (4.9) becomes $$l_{1}(\sigma(z))+\mu(\sigma(z))l(z)=c\mu(\sigma(z))g(z),\; z\in
M,$$ where $c\in \mathbb{C}$ is a constant. By putting (4.10) in (4.9) we obtain $3\mu(\sigma(z))cg(x)g(\sigma(y))g(z)=0$ for all $x,y,z\in S$. Since $g\neq 0$ and $\mu(\sigma(z))\neq 0$ it follows that $c=0$ and then $l_{1}(\sigma(z))=-\mu(\sigma(z))l(z)$ for all $z\in S$. So that equation (4.8) becomes $$h(yz)=\mu(y)g(\sigma(y))l(z)-\mu(y)g(z)l(\sigma(y)),\; x,y,z\in S$$ Replacing $(y,z)$ by $(\sigma(z),\sigma(y))$ in (4.11) we obtain $$h(\sigma(zy))=\mu(\sigma(zy))(\mu(y)g(z)l(\sigma(y))-\mu(y)g(\sigma(y))l(z))=-\mu(\sigma(zy))h(yz).$$ From which we obtain by putting $y=e$ that $$h(\sigma(z))=-\mu(\sigma(z))h(z), \;z\in M.$$ From (4.12) and (4.13) we get that $h$ a central function i.e. $h(yz)=h(zy)$ for all $y,z\in M$.\
Next we consider two cases :\
First case : Suppose $g(e)=0$. Let $z=e$ and $x=x_{0}$ in (4.7) give that $$h(y)=-c\mu(y)g(\sigma(y)), \; y\in M$$ for some $c\in \mathbb{C}\setminus\{0\}$. By replacing $y$ by $\sigma(y)$ in (4.14) we obtain that $h(\sigma(y))=-c\mu(\sigma(y))g(y)$ for all $y\in M$. By using (4.14) we get that $g=\frac{1}{c}h=bh$ where $b=\frac{1}{c}$ and that $$g(\sigma(z))=-\mu(\sigma(z))g(z), \;z\in
S.$$ Using (4.15) in (4.11) and setting $m=-bl$ we get $$\begin{aligned}
h(yz)&=&\mu(y)g(\sigma(y))l(z)-\mu(y)g(z)l(\sigma(y))\\
&=&-\mu(y)\mu(\sigma(y))g(y)l(z)-\mu(y)g(z)l(\sigma(y))\\
&=&-g(y)l(z)-\mu(y)g(z)l(\sigma(y))\\
&=&-bh(y)l(z)-\mu(y)bh(z)l(\sigma(y))\\
&=&h(y)m(z)+\mu(y)h(z)m(\sigma(y)).\end{aligned}$$ By replacing $z$ by $\sigma(z)$ we obtain $$h(y\sigma(z))=h(y)(m\circ \sigma)(z)+\mu(y)h(\sigma(z))m(\sigma(y)), \; y,z\in M.$$ By multiplying (4.16) by $\mu(z)$ and by setting $n(z)= (m\circ
\sigma)(z)\mu(z)$ we obtain the $\mu$-sine subtraction law with the companion function $n$ $$\mu(z)h(y\sigma(z))=h(y)n(z)-h(z)n(y).$$ This ends the first case.\
Second case : Suppose $g(e)\neq 0$. Then we obtain from (4.7) with $x=e$ that $$h(yz)=[\mu(y)g(\sigma(y))h(z)+g(z)h(y)]/g(e).$$ Interchanging $y$ and $z$ in (4.18) we get $$h(zy)=[\mu(z)g(\sigma(z))h(y)+g(y)h(z)]/g(e).$$ By replacing $z$ by $\sigma(z)$ (4.19) and multiplying (4.19) by $\mu(z)$ we get $$\mu(z)h(\sigma(z)y)=h(y)\frac{g}{g(e)}(z)-\frac{g}{g(e)}(y)h(z), \; y,z\in M.$$ Since $h$ is central it follows that $h$ satisfies the $\mu$-sine subtraction law with the companion function $\frac{g}{g(e)}$ $$\mu(z)h(y\sigma(z))=h(y)\frac{g}{g(e)}(z)-\frac{g}{g(e)}(y)h(z), \; y,z\in M.$$
In the next two theorems we obtain the solutions of equation (1.9) by using our results for the $\mu$-sine subtraction law. We will follow the method used in \[8\].\
Let $\mathcal{N}_{\mu}(\sigma,S)$ be the nullspace given by $$\mathcal{N}_{\mu}(\sigma,S)=\{\theta: S\longrightarrow\mathbb{C} \; :\theta(xy)-\mu(y)\theta(\sigma(y)x)=0, \;x,y\in G\}.$$ In the next theorem we consider the group case
Let $G$ be a group, let $\sigma$ be a involutive automorphism on $G$, let $\mu:G \longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in G.$ Suppose that $f, g, h: G \longrightarrow \mathbb{C}$ satisfy functional equation (1.9). Suppose also that $g \neq 0$ and $h \neq 0$. Then there exists a character $\chi$ of $G$, constants $c$, $c_1$, $c_{2}\in \mathbb{C}$, and a function $\theta\in \mathcal{N}_{\mu}(\sigma,S)$ such that one of the following holds\
i) If $\chi\neq \mu\;\chi\circ\sigma$, then $$h=c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}, \; g=\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2},$$ $$f=\theta+\frac{c_{1}}{2}[c\frac{\chi-\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi+\mu\;\chi\circ\sigma}{2}]$$ ii) If $\chi=\mu\;\chi\circ\sigma$, then $$h=\chi A,\;\; g=\chi(c+c_{2}A), f=\theta+\chi A(\frac{c}{2}+\frac{c_{2}}{4}A).$$ where $A: G\longrightarrow \mathbb{C}\setminus\{0\}$ is an additive function such that $A\circ \sigma =-A\neq 0$.\
Conversely, the formulas of (i) and (ii) define solutions of (1.9).\
Moreover, if $G$ is a topological group, and and $f, g, h \in
\mathcal{C}(G)$, then $\chi, \mu\;\chi\circ\sigma, A, \theta\in
\mathcal{C}(G)$, while $A \in \mathcal{C}(G)$.
The proof of the theorem 4.2 will be integrated into that of theorem 4.3 in which we consider the monoid case
Let $M$ be a monoid which is generated by its squares, let $\sigma$ be an involutive automorphism on $M$, let $\mu:M \longrightarrow
\mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in M.$ Suppose that $f, g, h: M
\longrightarrow \mathbb{C}$ satisfy functional equation (1.9). Suppose also that $g \neq 0$ and $h \neq 0$. Then there exists a multiplicative function $\chi: M\longrightarrow \mathbb{C}$ $\chi\neq 0$, constants $c$, $c_{1}$, $c_{2}\in \mathbb{C}$, and a function $\theta\in \mathcal{N}_{\mu}(\sigma,S)$ such that one of the following holds\
i) If $\chi\neq \mu\;\chi\circ\sigma$, then $$h=c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}, \; g=\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2},$$ $$f=\theta+\frac{c_{1}}{2}[c\frac{\chi-\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi+\mu\;\chi\circ\sigma}{2}]$$ ii) If $\chi=\mu\;\chi\circ\sigma$, then $h(x)=g(x)=0$ and $f(x)=\theta(x)$ for $x\in I_{\chi}$, and $$h(x)=\chi(x)A(x),\;\; g(x)=\chi(x)(c+c_{2}A(x)), f(x)=\theta(x)+\chi(x) A(x)(\frac{c}{2}+\frac{c_{2}}{4}A(x))$$ for $x\in M\setminus I_{\chi}$ where $A: M\setminus I_{\chi}\longrightarrow \mathbb{C}\setminus\{0\}$ is an additive function such that $A\circ \sigma =-A\neq 0$.\
Furthermore, if $M$ is a topological monoid and $k \in C(M)$, then $l$, $\chi$, $\mu\chi\circ \sigma$, $A\in \mathcal{C}(M)$. Conversely, the formulas of (i) and (ii) define solutions of (1.9).\
Moreover, if $M$ is a topological monoid, and $f, g, h \in
\mathcal{C}(M)$, then $\chi, \mu\;\chi\circ\sigma, A, \theta\in
\mathcal{C}(M)$, while $A\in \mathcal{C}(M\setminus I_{\chi})$.
According to Proposition 4.1 we have two following cases:\
First case : Suppose that $g(e)=0$, then $h$ satisfies the $\mu$-sine subtraction law and $g=bh$ where $b\in
\mathbb{C}\setminus\{0\}$. According to Theorem 2.1 and Proposition 2.2 we get (for $M$ is a group or a monoid) that if $\chi\neq
\mu\chi\circ \sigma$, then $h=c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}$ where $c_{1}\in
\mathbb{C}$. Since $g=bh$, then $g=c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2}$ where $c_{2}\in
\mathbb{C}$. Subsisting $g$ and $h$ in (3.1) we get for all $x,y\in
M$ $$\begin{aligned}
f(xy)-\mu(y)f(\sigma(y)x)&=&g(x)h(y)\\
&=&\frac{c_{1}c_{2}}{4}[\chi(x)-\mu(x)\chi(\sigma(x))][\chi(y)-\mu(y)\chi(\sigma(y))]\\&=&
\frac{c_{1}c_{2}}{4}[\chi(xy)-\mu(y)\chi(\sigma(y)x)-\mu(x)\chi(\sigma(\sigma(y)x))\\&&+\mu(xy)\chi(\sigma(xy))].\end{aligned}$$ Let $\theta=f-\frac{c_{1}c_{2}}{4}(\chi+\mu\;\chi\circ\sigma)$ we have $\theta\in \mathcal{N}_{\mu}(\sigma,S)$ and $f=\theta+\frac{c_{1}}{2}[c\frac{\chi-\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi+\mu\;\chi\circ\sigma}{2}]$ with $c=0$.\
Now, if $\chi= \mu\chi\circ \sigma$.\
When $M$ is a group then we get from Theorem 2.1 that $h=\chi A$ where $A$ is an additive function such that $A\circ \sigma=-A\neq
0$. Since $g=bh$, then $g=b\chi A=c_{2}\chi A$. Subsisting $g$ and $h$ in (3.1) we get $$\begin{aligned}
f(xy)-\mu(y)f(\sigma(y)x)&=&g(x)h(y)c_{2}\chi(x)A(x)\chi(y)A(y)\\
&=&\frac{c_{2}}{4}[\chi(y)A(xy)^{2}-\chi(\sigma(y))A(\sigma(y)x)^{2}].\end{aligned}$$ Let $\theta=f-c_{2}\frac{\chi A^{2}}{4}$. Then $\theta\in \mathcal{N}_{\mu}(\sigma,S)$.\
When $M$ is a monoid, by Proposition 2.2 and and by the same way as in \[6\] we have $\theta=\theta_{1}\cup \theta_{2}$ where where $\theta_{1}(x)= f(x)-c_{2}\frac{\chi(x) A^{2}(x)}{4}$ on $M\setminus I_{\chi}$ and $\theta_{2}(x)=f(x)$ on $I_{\chi}$.\
Second case : Suppose $g(e)\neq 0$. We have $h$ and $\frac{g}{g(e)}$ play the role of $k$ and $l$ respectively in Theorem 2.1 or in Proposition 2.2. If $\chi\neq \mu\;\chi\circ\sigma$, then $h=c_{1}\frac{\chi-\mu\;\chi\circ\sigma}{2}$ where $c_{1}\in
\mathbb{C}$ and $g=c\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_{2}\frac{\chi-\mu\;\chi\circ\sigma}{2}$. By the same way as in \[9\] we get that $\theta=f-c_{1}\frac{[(c+c_{2})\chi-(c-c_{2})\chi\circ\sigma]}{4}\in\mathcal{N}_{\mu}(\sigma,S)$.\
Finally, if $g(e)\neq 0$ and $\chi=\mu\;\chi\circ\sigma$ we get the remainder by the same way as in \[9\].
Applications: Solutions of equation (\[EQ1\]) on groups and monoids
===================================================================
In this section, we use the results obtained in the previous paragraph to solve the functional equation (\[EQ1\]) on groups and monoids. We proceed as follows to reduce the equation to the functional equation (\[eq7\]) and (\[eq8\]) so that we can apply Theorem 3.1, Theorem 3.2, Theorem 4.2. and Theorem 4.3.
Let $G$ be a group, and $\sigma$ an homomorphism involutive of $G$. Let $\mu:G \longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in
G$. Suppose that the functions $f, g, h: G \longrightarrow
\mathbb{C}$ satisfy the functional equation (\[EQ1\]). Suppose also that $f+g\neq0.$ Then there exists a character $\chi$ of $G$, constants $\alpha\in \mathbb{C}^{\ast}$, $c_1$, $c_{2}\in
\mathbb{C}$, and a function $\theta\in \mathcal{N}_{\mu}(\sigma,S)$ such that one of the following holds\
(a) If $\chi\neq
\mu\;\chi\circ\sigma$, then $f=\frac{1}{2}[(1+\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}+2c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}+\theta]$; $g=\frac{1}{2}[(1-\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}-\theta]$ and $h=\frac{1}{\alpha}[\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}]$.\
(b) If $\chi= \mu\;\chi\circ\sigma$ then $f=\frac{c_2\chi(2+A)+\theta+\chi
A c_2(1+\frac{A}{4})}{2}$; $g=\frac{c_2\chi(2+A)-\theta-\chi
A c_2(1+\frac{A}{4})}{2}$ and $h=\frac{1}{\alpha}c_2\chi(2+A).$
Let $f,g,h$ : $G\longrightarrow \mathbb{C}$ satisfy the functional equation (\[EQ1\]). The case $g=-f$ was treated in Theorem 3.1 and Theorem 3.2. From now, on we assume that $f+g\neq 0$. Let $h_o:=\frac{h-\mu h\circ \sigma}{2}$ respectively $h_e:=\frac{h+\mu h\circ \sigma}{2}$ denote the odd, respectively even, part of $h$ with respect to $\mu$ and $\sigma$.\
Setting $x=e$ in (\[EQ1\]) gives us $$\label{equation51}
f(y)+\mu(y)g(\sigma(y))=h(e)h(y)$$ for all $y\in G.$ Taking $y=e$ in (\[EQ1\]) and using $\mu(e)=1$ we find $$\label{equation52}
f(x)+g(x)=h(e)h(x)$$ for all $x\in G.$ So, by comparing (\[equation51\]) with (\[equation52\]) we get $$\label{equation53}
g(x)=\mu(x)g(\sigma(x)),\;x\in G.$$ We note that $(f+g)(xy)+\mu(y)(f+g)(\sigma(y)x)=f(xy)+\mu(y)g(\sigma(y)x)+g(xy)+\mu(y)f(\sigma(y)x)=h(x)h(y)+g(xy)+\mu(y)f(\sigma(y)x)$. By using (\[equation53\]) we have $g(xy)=\mu(xy)g(\sigma(x)\sigma(y))$, then we get $g(xy)+\mu(y)f(\sigma(y)x)=\mu(y)f(\sigma(y)x)+\mu(xy)g(\sigma(x)\sigma(y))=\mu(y)[f(\sigma(y)x)+\mu(x)g(\sigma(x)\sigma(y))]=\mu(y)h(x)h(\sigma(y))$. Which implies that $$\label{equation540}
(f+g)(xy)+\mu(y)(f+g)(\sigma(y)x)=2h(x)h_e(y)$$ for all $x,y\in G.$ From (\[equation52\]) and the assumption that $f+g\neq
0$ we get $h(e)\neq 0$. So, equation (\[equation540\]) can be written as follows $$\label{equation541}
(f+g)(xy)+\mu(y)(f+g)(\sigma(y)x)=2(f+g)(x)\frac{h_e(y)}{h(e)}$$ for all $x,y\in G.$\
On the other hand by using similar computation used above, we obtain $$\label{equation542}
(f-g)(xy)-\mu(y)(f-g)(\sigma(y)x)=2h(x){h_o(y)}=(f+g)(x)\frac{2h_o(y)}{h(e)}$$ for all $x,y\in G$. We can now apply Theorem 3.1, Theorem 3.2, Theorem 4.2 and Theorem 4.3.\
If $h_o=0$, then $f-g\in \mathcal{N}(\sigma,G)$, so there exists $\theta\in \mathcal{N}_{\mu}(\sigma,G)$ such that $f-g=\theta.$ Since $f,g$ satisfy (\[equation541\]) then from Theorem 3.1 we get the only possibility $f+g=\alpha^{2}\frac{\chi+\mu\;\chi\circ\sigma}{2}$ and $h=\alpha\frac{\chi+\mu\;\chi\circ\sigma}{2}$ for some character $\chi$ : $G\longrightarrow \mathbb{C}$ and a constant $\alpha\in \mathbb{C}$ and we deduce that $f=\frac{1}{2}[\theta+\alpha^{2}(\frac{\chi+\mu\;\chi\circ\sigma}{2})]$; $g=\frac{1}{2}[-\theta+\alpha^{2}(\frac{\chi+\mu\;\chi\circ\sigma}{2})]$. We deal with case (i).\
So during the rest of the proof we will assume that $h_o\neq0.$ the function $f+g,$ $h_o$ are solution of equation (\[equation542\]) with $f+g\neq 0$ and $h_o\neq 0$, so we know from Theorem 4.2 that there are only the following $2$ possibilities:\
(i) $f-g=\theta+\frac{c_1}{2}[c\frac{\chi-\mu\;\chi\circ\sigma}{2}+c_2\frac{\chi+\mu\;\chi\circ\sigma}{2}]$; $f+g=\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}$ for some character $\chi$ on $G$ such that $\chi\neq\mu\;\chi\circ\sigma$, $\theta\in \mathcal{N}_{\mu}(\sigma,G)$ and constants $c,c_1,c_2\in
\mathbb{C}$. So, we have $g=\frac{1}{2}[(1-\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}+(c_2-\frac{cc_1}{2})\frac{\chi-\mu\;\chi\circ\sigma}{2}-\theta]$; $f=\frac{1}{2}[(1+\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}+(c_2+\frac{cc_1}{2})\frac{\chi-\mu\;\chi\circ\sigma}{2}+\theta]$. Since $g=\mu g\circ\sigma$, then we have $c_2=\frac{cc_1}{2}$ and $f$, $g$ are as follows: $f=\frac{1}{2}[(1+\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}+2c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}+\theta]$; $g=\frac{1}{2}[(1-\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}-\theta]$. We deal with case (a).\
(ii) $f-g=\theta+\chi A(\frac{c}{2}+\frac{c_2}{4}A)$; $f+g=\chi(c+c_2A)$, where $\chi$ is a character on $G$ such that $\chi=\mu\;\chi\circ\sigma$, $A$ is an additive map on $G$ such that $A\circ\sigma=-A$, $\theta\in \mathcal{N}_{\mu}(\sigma,G)$ and $c,c_2\in
\mathbb{C}$. So, we get $g=\frac{\chi(c+c_2A)-\theta-\chi
A(\frac{c}{2}+\frac{c}{4})A}{2}$. Since $g=\mu g\circ\sigma$ so, we have $c=2c_2$. Consequently we have $g=\frac{c_2\chi(2+A)-\theta-\chi
A c_2(1+\frac{A}{4})}{2}$; $f=\frac{c_2\chi(2+A)+\theta+\chi
A c_2(1+\frac{A}{4})}{2}$, where $A$ : $G \longrightarrow \mathbb{C}$ is an additive function such that $A\circ\sigma(x)=-A(x)$ for all $x\in G$. We deal with case (b).
Let $M$ be a monoid which is generated by its squares, let $\sigma$ an involutive automorphism on $M$. Let $\mu:M\longrightarrow \mathbb{C}$ be a multiplicative function such that $\mu(x\sigma(x))=1$ for all $x\in
M$. Suppose that $f, g, h: M \longrightarrow \mathbb{C}$ satisfy the functional equation (\[EQ1\]). Suppose also that $f+g\neq0.$ Then there exists a character $\chi$ of $M$, constants $\alpha\in
\mathbb{C}^{\ast}$, $c_1$, $c_{2}\in \mathbb{C}$, and a function $\theta\in \mathcal{N}_{\mu}(\sigma,M)$ such that one of the following holds\
(a) If $\chi\neq \mu\;\chi\circ\sigma$, then $f=\frac{1}{2}[(1+\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}+2c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}+\theta]$; $g=\frac{1}{2}[(1-\frac{c_1c_2}{2})\frac{\chi+\mu\;\chi\circ\sigma}{2}-\theta]$ and $h=\frac{1}{\alpha}[\frac{\chi+\mu\;\chi\circ\sigma}{2}+c_2\frac{\chi-\mu\;\chi\circ\sigma}{2}]$.\
(b) If $\chi= \mu\;\chi\circ\sigma$ then $f(x)=\frac{\theta(x)}{2}$, $g(x)=\frac{-\theta(x)}{2}$ and $h(x)=0$ for all $x\in I_\chi$; $f(x)=\frac{c_2\chi(2+A(x))+\theta(x)+\chi(x)
A c_2(1+\frac{A(x)}{4})}{2}$; $g(x)=\frac{c_2\chi(2+A(x))-\theta(x)-\chi(x)
A c_2(1+\frac{A(x)}{4})}{2}$ and $h(x)=\frac{1}{\alpha}c_2\chi(2+A(x))$ for all $x\in M\backslash I_\chi$ and where $A$ : $M\backslash I_\chi \longrightarrow \mathbb{C}$ is an additive function such that $A\circ\sigma(x)=-A(x)$ for all $x\in M\backslash I_\chi$.
[99]{} Aczél, J., Dhombres, J. Functional equations in several variables. With applications to mathematics, information theory and to the natural and social sciences. Encyclopedia of Mathematics and its Applications, vol. 31. Cambridge University Press, Cambridge, 1989 d’Alembert, J. Recherches sur la courbe que forme une corde tendue mise en vibration, I. Hist. Acad. Berlin 1747(1747), 214-219 d’Alembert, J.: Recherches sur la courbe que forme une corde tendue mise en vibration, II. Hist. Acad. Berlin 1747(1747), 220-249 d’Alembert, J. Addition au Mémoire sur la courbe que forme une corde tendue mise en vibration. Hist. Acad. Berlin 1750(1750), 355-360 Davison, Thomas M.K. D’Alembert’s functional equation on topological monoids, Publ. Math. Debrecen 75(2009), 41-66 Ebanks, B., Stetkær, H. On Wilson’s functional equations, Aequationes Math., 89(2015), 339-354 Ebanks, B.R., Sahoo, P.K., Sander, W. Characterizations of Information Measures. World Scientific Publishing Co, Singapore, 1998 Ebanks, B.R., Stetk[æ]{}r, H. d’Alembert’s other functional equation on monoids with involution. Aequationes Math. 89(2015), 187-206 Elqorachi, E., Redouani, A. Solutions and stability of a variant of Wilson’s functional equation, arXiv:1505.06512v1 \[math.CA\], 2015 Parnami, J.C, Singh, H., Vasudeva, H.L. On an exponential-cosine functional equation. Period. Math. Hungar. 19(1988), 287-297 Stetk[æ]{}r, H.: On a variant of Wilson’s functional equation on groups, Aequationes Math., 68(2004), 160-176 Stetk[æ]{}r, H. Functional equations on abelian groups with involution. Aequationes Math. 54(1997), 144-172 Stetk[æ]{}r, H. Functional Equations on Groups. World Scientific Publishing Co, Singapore, 2013 Stetkær, H. A variant of d’Alembert’s functional equation, Aequationes Math., 89(2015), 657-662 Stetkær, H. d’Alembert’s functional equation on groups. Recent developments in functional equations and inequalities, pp. 173-191, Banach Center Publ., vol. 99. Polish Acad. Sci. Inst. Math., Warsaw, 2013 Yang, D. The symmetrized Sine addition formula, Aequationes Math., 82(2011), 299-318
Belaid Bouikhalene\
Departement of Mathematics and Informatics\
Polydisciplinary Faculty, Sultan Moulay Slimane university, Beni Mellal, Morocco.\
E-mail : [email protected].\
\
Elhoucien Elqorachi,\
Department of Mathematics,\
Faculty of Sciences, Ibn Zohr University, Agadir, Morocco,\
E-mail: [email protected]
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---
author:
- '[Hermann G. Matthies]{}'
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- '\\thebib/fuq-new.bib'
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title: |
[Analysis of Probabilistic and\
Parametric Reduced Order Models]{}[^1]
---
\[2003/12/01\]
[ `` ]{}
[^1]: Partly supported by the Deutsche Forschungsgemeinschaft (DFG) through SPP 1886 and SFB 880.
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---
abstract: 'Capturing the temporal dynamics of user preferences over items is important for recommendation. Existing methods mainly assume that all time steps in user-item interaction history are equally relevant to recommendation, which however does not apply in real-world scenarios where user-item interactions can often happen accidentally. More importantly, they learn user and item dynamics separately, thus failing to capture their joint effects on user-item interactions. To better model user and item dynamics, we present the Interacting Attention-gated Recurrent Network (IARN) which adopts the attention model to measure the relevance of each time step. In particular, we propose a novel attention scheme to learn the attention scores of user and item history in an interacting way, thus to account for the dependencies between user and item dynamics in shaping user-item interactions. By doing so, IARN can selectively memorize different time steps of a user’s history when predicting her preferences over different items. Our model can therefore provide meaningful interpretations for recommendation results, which could be further enhanced by auxiliary features. Extensive validation on real-world datasets shows that IARN consistently outperforms state-of-the-art methods.'
author:
- Wenjie Pei
- Jie Yang
- Zhu Sun
- Jie Zhang
- Alessandro Bozzon
- 'David M.J. Tax'
bibliography:
- 'sigproc.bib'
title: 'Interacting Attention-gated Recurrent Networks for Recommendation'
---
<ccs2012> <concept> <concept\_id>10002951.10003317</concept\_id> <concept\_desc>Information systems Information retrieval</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003347.10003350</concept\_id> <concept\_desc>Information systems Recommender systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
<ccs2012> <concept> <concept\_id>10002951.10003317</concept\_id> <concept\_desc>Information systems Information retrieval</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003347.10003350</concept\_id> <concept\_desc>Information systems Recommender systems</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is partially funded by the Social Urban Data Lab (SUDL) of the Amsterdam Institute for Advanced Metropolitan Solutions (AMS). This work is partially supported by the SIMTech-NTU Joint Lab on Complex Systems.
|
---
abstract: '*Background*: Continuous Integration (CI) systems are now the bedrock of several software development practices. Several tools such as TravisCI, CircleCI, and Hudson, that implement CI practices, are commonly adopted by software engineers. However, the way that software engineers use these tools could lead to what we call “Continuous Integration Theater”, a situation in which software engineers do not employ these tools effectively, leading to unhealthy CI practices. *Aims*: The goal of this paper is to make sense of how commonplace are these unhealthy continuous integration practices being employed in practice. *Method*: By inspecting 1,270 open-source projects that use TravisCI, the most used CI service, we quantitatively studied how common is to use CI (1) with infrequent commits, (2) in a software project with poor test coverage, (3) with builds that stay broken for long periods, and (4) with builds that take too long to run. *Results*: We observed that 748 ($\sim$60%) projects face infrequent commits, which essentially makes the merging process harder. Moreover, we were able to find code coverage information for 51 projects. The average code coverage was 78%, although Ruby projects have a higher code coverage than Java projects (86% and 63%, respectively). However, some projects with very small coverage ($\sim$4%) were found. Still, we observed that 85% of the studied projects have at least one broken build that take more than four days to be fixed. Interestingly, very small projects (up to 1,000 lines of code) are the ones that take the longest to fix broken builds. Finally, we noted that, for the majority of the studied projects, the build is executed under the 10 minutes rule of thumb. *Conclusions*: Our results are important to an increasing community of software engineers that employ CI practices on daily basis but may not be aware of bad practices that are eventually employed.'
author:
-
-
-
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bibliography:
- 'references.bib'
title: Continuous Integration Theater
---
Continuous Integration, Test coverage, Bad practices
Introduction {#sec:intro}
============
Continuous Integration (CI) is the practice of merging all developer working copies into a shared mainline, several times a day [@Humble:2010]. Although the culture of continuously integrating changes dates from the 70s [@Brooks:1978], CI practices has gained momentum only in the last 10 years, being more widely discussed, employed, and researched. Consequently, CI is nowadays one of the pillars of the software engineering practice, not only in commercial projects, but also in open source projects [@Pinto:SPE:2018; @Reboucas:2017:CII].
The success of CI can be partially accredited to first world class tools that have considerably automated most of the required steps to inspect, integrate, and test source code change in a transparent and straightforward manner. The use of CI tools not only accelerates the software development process (since merging changes become more frequent without reducing software quality [@Vasilescu:2015:QPO]), but software bugs can also be identified earlier and faster [@Vasilescu:2014:CIS].
There are several tools offering support for developers that plan to incorporate the CI practices into their software projects. Such tools include TravisCI, CircleCI, and Hudson. More interestingly, however, is the fact that some of these tools are readily available in social coding environments such as GitHub and GitLab. Essentially, this integration implies that everyone with a GitHub account can gratuitously benefit from the complex pipeline of version control systems, code review systems, and continuous integration tools, with little to no configuration effort. Therefore, it comes as no surprise to state that CI tools (e.g., TravisCI) are highly used and demanded by developers [@Hilton:2016].
However, the sole usage of CI tools does not necessarily imply that a software development team properly adhere to the CI practices. Recent works have shown that the use of automation tools may produce no benefits, unless the development team is willing to change their development culture [@Luz:ESEM:2018]. As an example, according to Fowler [@fowler:CI], one of the CI practices is to promote self-testing builds. However, to have a self-testing code one needs a suite of automated tests that can check a large part of the code base for eventual bugs. Unfortunately, although CI tools re-execute test suites after every new change (i.e., to avoid the introduction of new bugs), CI tools cannot identify whether the software project contains a comprehensive test suite. Therefore, in such a case, the development team will not benefit from the test automation supposed to be promoted by CI tools. Thus, a common misconception that has been acknowledged about CI is that the sole adoption of a CI tool does not imply the proper adherence to CI practices [@Luz:ESEM:2018; @Zhang:FSE:2018]. Indeed, such kind of situation has long been one of the Achilles’ heels of agile. To publicize that is adherent to agile practices aiming to gain some kind of credibility, while under the covers the basic practices are not properly followed [@ELORANTA2016].
In this paper, we investigate a set of CI bad practices. These bad practices are related to the use of CI (1) with infrequent commits on the master branch (i.e., delaying integration), (2) in a software project with poor test coverage (i.e., missing eventual bugs), (3) with builds that remain broken for long periods for time (i.e., blocking new features), and (4) with builds with considerably long durations (i.e., limiting the rapid feedback). These bad practices constitute what is known as the “Continuous Integration Theater”[^1] in the practitioners arena. According to the grey literature:
“Continous Integration Theater describes the illusion of practising continuous integration while\
not really practising it.”[^2].
Although these bad practices are commonly discussed in the grey literature, little research has been devoted to shed some light on the existence of projects performing the CI Theater.
To conduct this investigation, we leverage the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset, which is a comprehensive dataset of data and metadata regarding projects that use [<span style="font-variant:small-caps;">TravisCI</span>]{}. Whenever necessary, we enriched this dataset with data from [<span style="font-variant:small-caps;">Coveralls</span>]{}, which is a third-party service that provides test coverage information. Through a mostly quantitative analysis over 1,270 open source projects and their 534,417 builds, we produce a list of findings regarding CI bad practices that are employed in open source projects, some of which are not always obvious. We now highlight our main findings here.
1. **Infrequent commits are frequent.** We empirically defined the value of our metrics for infrequent commits as 2.36 commits per weekday. We then found that 60% of the studied projects have less than 2.36 commits, suffering from infrequent commits. The size of the project has no influence on the (in)frequency of commits. Large Ruby projects, however, are the most active ones and do not adhere to this rule.
2. **Test coverage could mislead CI results.** We identified 51 projects that we could measure test coverage information. On average, Java projects have 63% of test coverage, whereas Ruby projects have 86%. At the bare minimum, we find one Java project with 4% of test coverage, and one Ruby project with 14% of test coverage. This finding suggests that the report of a CI service could be compromised, since some projects might not place enough care in curating their test coverage (e.g., a passing build may be hiding bugs due to the poor test coverage).
3. **Long to be fixed broken builds.** We observed that 85% of the analyzed projects have at least one build that took more than four days to be fixed. This finding is particularly unfortunate since broken builds that take several days to be fixed may introduce an additional burden (or distrust) on the development team. Interestingly, we observed that large projects (either Java or Ruby) have less instances of long to be fixed broken builds than smaller projects. These long to be fixed builds, on very small projects, are fixed, on average, in 40 days, which is strong smell of the CI theater.
4. **Builds are executed quickly, though.** In order to provide quick feedback, builds should be executed under 10 minutes [@fowler:CI; @Hilton:FSE:2017]. We found only 43 projects that do not adhere to this general rule of thumb. As an exception to this rule, we found 43 very large and complex projects, such as the JRuby (the Ruby implementation for the Java VM) or the Facebook Presto (a distributed SQL query engine for big data), that have builds which take longer than 30 minutes. In spite of these cases, this symptom of the CI theater was hardly observed.
Method {#sec:method}
======
In this section we introduce our research questions (Section \[sec:rqs\], and the approaches we used to gather (Section \[sec:dataset\]), analyze (Section \[sec:analysis\]) data. We also provide a package for help anyone who want to replicate this study(Section \[sec:replication\]).
Research Questions {#sec:rqs}
------------------
In this work we studied the following four important research questions.
**Rationale.** One of the main advantages of CI systems is that they decrease the pain of merging new changes. This relief comes from the practice of merging continuously. However, sometimes software engineers opt not to integrate continuously (e.g., they take too much time working on a separate branch and only after days of work they apply the changes in the master branch). Practitioners have baptized the bad practice of working in silos—either in their local branches or remote branches—as “Continuous Isolation”[^3].
**Rationale.** Test coverage measures how much of a software project is exercised during testing. If a project has a fragile test suite (and consequently a low test coverage), new changes that clearly introduce bugs are potentially not caught during build time. Therefore, CI systems offer little help in software projects that do not carefully build their testing arsenal. Although many criteria were introduced to measure code coverage [@Ammann:2002:ISSRE; @Gligoric:2013:ISSTA; @Atanas:2005:FASE], roughly speaking, test coverage is measured by the number of lines of code exercised by test cases divided by the total number of lines of code.
**Rationale.** Here we sought to investigate how common and how long broken builds stay broken in our dataset. A broken master is particularly undesirable because it may block features from rolling out (i.e., a faulty commit needs to be detected and rolled backed). Notable practitioners, such as Martin Fowler, have suggested that “if the mainline build fails, it needs to be fixed right away” [@fowler:CI], making a broken build an urgent, high priority task. However, if broken builds stay red longer than this, it may suggest that projects maintainers may not be taking into account the build status and, perhaps, releasing software with bugs. Still, if developers work on a faulty master, their productivity may get hampered substantially.
**Rationale.** In this final research question, our intention is to explore how long take the builds in our dataset to process. The whole point of Continuous Integration is to provide rapid feedback. Advocates from the XP practices provide a general rule of thumb suggesting that, for most projects, 10 minutes is an expected metric. According to Fowler, “it’s worth putting in concentrated effort to make \[the ten minutes rule\] happen, because every minute you reduce off the build time is a minute saved for each developer every time they commit.” [@fowler:CI]
Curating the Dataset {#sec:dataset}
--------------------
To conduct this research, we rely mostly on the dataset curated by [<span style="font-variant:small-caps;">TravisTorrent</span>]{} [@Beller:MSR:2017]. This dataset focus on software builds created and reported in the [<span style="font-variant:small-caps;">TravisCI</span>]{}platform, which is one of the most popular CI services nowadays. As of 2017, [<span style="font-variant:small-caps;">TravisCI</span>]{}was reported being present in 50% of the projects hosted on GitHub[^4]. The most recent release of [<span style="font-variant:small-caps;">TravisTorrent</span>]{}is from November, 1st, 2017. More concretely, this dataset stores information about the builds executed, the build logs, how many tests were executed (and which ones failed), etc. Although the last release of the datasets is from 2017, we observed that the dataset contains build information between February 2012 and March of 2016. The initial status of the dataset contains information about 1,283 open source projects, 3,702,595 build jobs, and 3,702,595 commits. We performed three additional filtering steps in the dataset, namely:
- **Removing Not a Number (NaN) records.** When analyzing the dataset, we noticed that there are some inconsistencies between the number of builds and the number of commits. Since the relationship between commit and build is one to one, we found puzzling cases in which there are more commits than builds. When analyzing the dataset, we perceived the existence of some NaN records, which our script computed as zero. We inquired [<span style="font-variant:small-caps;">TravisTorrent</span>]{}documentation, and it informs, in rare cases, that the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}infrastructure does not record a push event for every build confirmation, thus generating the aforementioned data inconsistency. We then removed the rows that have NaN columns.
- **Removing duplicated jobs.** We noted that some projects are configured to test the build against several different configurations (jobs). Since studying different jobs is not part of the scope of this research, we decided to remove duplicated jobs.
- **Removing JavaScript projects.** After performing these two filters, we noted that only four projects were written in JavaScript, namely `zhangkaitao/es`, `dianping/cat`, `palantir/eclipse-typescript`, and `brooklyncentral/clocker`. We opted to do not consider JavaScript projects due to the small sample.
Figure \[fig:filters\] present the quantitative of data left after each filter, as well as the percentage of reduction. In the end, we ended up with 1,270 projects, 534,417 build jobs, and 1,288,431 commits. A reduction of 1% on projects, 85% on build jobs, and 81% on commits compared to the original dataset.
![The impact of applying each filter on the quantitative of data, and the percentage of reduction.[]{data-label="fig:filters"}](images/filters.png){width="50.00000%"}
We used this data to provide answers to **RQ1**, **RQ2**, **RQ3**, and **RQ4**. In particular, for **RQ2**, since [<span style="font-variant:small-caps;">TravisTorrent</span>]{}does not provide coverage information, we have to complement it with data from [<span style="font-variant:small-caps;">Coveralls</span>]{}. The [<span style="font-variant:small-caps;">Coveralls</span>]{}platform tracks the coverage information of software repositories under development on GitHub, GitLab, and BitBucket coding websites. [<span style="font-variant:small-caps;">Coveralls</span>]{}has a fine-grained coverage report, comprising each source code file, and each source code line in the file. [<span style="font-variant:small-caps;">Coveralls</span>]{}also provides an API in which it made available information about the branch, the total coverage, and the change in the coverage in a particular build. The data available on [<span style="font-variant:small-caps;">Coveralls</span>]{}have also be used in other studies (e.g., [@Hilton:2018:ASE]). Since [<span style="font-variant:small-caps;">Coveralls</span>]{}provides an integration with TravisCI, we investigate which projects in the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset were also configured to use the [<span style="font-variant:small-caps;">Coveralls</span>]{}platform. After finding the intersection between these two datasets, we investigate the coverage status on [<span style="font-variant:small-caps;">Coveralls</span>]{}of the last available builds of open source available on [<span style="font-variant:small-caps;">TravisTorrent</span>]{}.
Analysing data {#sec:analysis}
--------------
To help our analysis, we grouped the projects according to their programming language (Ruby and Java) and to their size. In terms of size, we grouped the projects in very small, small, medium, large, and very large. More precisely:
- **Very small**: (less than 1,000 lines of code), 336 projects found;
- **Small**: (more than 1,000 and less than 10,000 lines of code), 622 projects found;
- **Medium**: (more than 10,000 and less than 100,000 lines of code), 261 projects found;
- **Large**: (more than 100,000 and less than 1,000,000 lines of code), 36 projects found.
- **Very large**: (more than 1,000,000 lines of code), only one project found.
One may argue that our sample of small projects should be removed from this study. However, although our set of very small projects might not be mission-critical, they are already configured to use [<span style="font-variant:small-caps;">TravisCI</span>]{}, which makes them valuable for this research. Still, these projects differ from other vary small projects on GitHub that do not use [<span style="font-variant:small-caps;">TravisCI</span>]{}, which may encompass books and classroom projects. Moreover, since we found only one very large project (the `aws/aws-sdk-java` Java project), when we present the distributions grouped according to the project size, we do not show data for this very large group. We used [<span style="font-variant:small-caps;">TravisTorrent</span>]{}information to measure lines of code. According to the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset website, the column “gh\_sloc” refers to the “*Number of executable production source lines of code, in the entire repository*”
Replication package {#sec:replication}
-------------------
For replication purposes, all scripts and code used to deal with the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset are available as a Jupyter notebook[^5].
Results {#sec:results}
=======
In this section we report the results grouped by each research question.
RQ1: How common is running CI in the master branch but with infrequent commits?
-------------------------------------------------------------------------------
In this first research question we are intended to analyze infrequent commits made at the master branch. We start by filtering out the commits made to other branches, resulting in a total of 42,3045 commits in the master branch. These commits lead to 368,886 builds in 1,270 open source projects. Figure \[fig:frequecyPerWeek\] shows the absolute number of commits according to the week day. Our next logical step was to empirically categorize what are infrequent commits. For each group of projects, we studied the frequency of commits per day. We found out a remarkable uniformity, as Table \[tab:commits\] shows.
average median 3rd quartile standard dev.
------------ --------- -------- -------------- ---------------
Very small 2.13 2.0 2.0 2.15
Small 2.32 2.0 2.0 2.61
Medium 2.35 2.0 2.0 2.78
Large 2.91 2.0 3.0 3.38
Very large 2.68 2.0 3.0 1.62
: Information about our dataset (per language)[]{data-label="tab:commits"}
As one could see, the average number of commits per day is between 2.13 (for very small projects) and 2.91 (for large projects). We also noted a very similar commit frequency when considering the programming language used. For instance, for very small, small, medium, large, and very large Ruby projects, the average of commits per day are, respectively, 2.15, 2.37, 2.42, 3.41, and 0.0 commits. In the Java projects, we found an akin finding: the average of commits per day for the very small, small, medium, large, and very large projects are, respectively, 1.68, 2.15, 2.25, 2.35 and 2.68 commits. Figure \[fig:project\_commit\_frequency\] shows the two distributions.
Overall, the average of commits per weekday per day is 2.36 (regardless of the size of the project, programming language, and weekday). We then considered a project with infrequent commits any project with an average lower than 2.36 commits per day. This empirical observed threshold is somehow in line with the grey literature, which suggest that “*CI developers must integrate all their work into trunk (also known as mainline or master) on a regular basis (at least daily).*”[^6] However, when we analyzed how common our studied projects are adhering to this threshold, we found that 748 (59.60%) face from this infrequent commits concern (214 (56,51%) Java and 534 (60,89%) Ruby). Figure \[fig:project\_commit\_frequency\] shows the distribution of commits per day, but now grouping the results in terms of the Ruby and Java programming language. As one can observe, Ruby projects tend to be more active than Java projects (median of commits for Ruby projects is 2.00 and for Java projects it is one). More interestingly, however, is the fact that Java projects have a very stable commit behavior, even when considering projects with different size. In particular, regardless of the size of the Java projects, 50% of them have infrequent commits. This finding is particularly relevant because if developers take too much time to commit to master (e.g., when working locally or on other branches), they may have to deal with merge conflicts more frequently, which not only require substantial effort from them but also hinder software development activities [@Cavalcanti:2017:OOPSLA]
However, this finding does not ring true when considering Ruby projects. Large Ruby projects, in particular, tend to be more active than the other ones.
$
\begin{array}{cc}
\includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/size_project_ruby.eps} & \includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/size_project_java.eps} \\
\end{array}
$
[**RQ1 Summary:** We categorized projects with infrequent commits when they have less than 2.36 commits per day. We found that, in general, $\sim$60% of the projects in our dataset suffer from infrequent commits. In particular, half of the Java (regardless of their size) have infrequent commits, which may hinder software development activities.]{}\
RQ2: How common is running a build with poor test coverage?
-----------------------------------------------------------
In this research question, we are interested to understand the test coverage of our studied projects. If the test coverage is small, it may suggest that the use of [<span style="font-variant:small-caps;">TravisCI</span>]{}is underused, since the potential benefits of running a comprehensive test suite automatically to find bugs is skipped.
From our corpus of 378 Java projects and 877 Ruby projects, we found only 25 Java projects and 58 Ruby projects with [<span style="font-variant:small-caps;">Coveralls</span>]{}information. This reduced our corpus to 83 projects. There is a gotcha, however. The last release of the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset was on 2017. We then applied another filter to select only projects with coverage information during the same period that we had build information. More concretely, we selected the last build record available on [<span style="font-variant:small-caps;">TravisTorrent</span>]{}and tried to match whether [<span style="font-variant:small-caps;">Coveralls</span>]{}had coverage information on the same day of the last build. Since we observed that the relationship between build records on [<span style="font-variant:small-caps;">TravisTorrent</span>]{}and coverage records on [<span style="font-variant:small-caps;">Coveralls</span>]{}is roughly one to one, we provide a grace period: for those projects that we did not find coverage information for the exact same day of the last build, we extended our search to find coverage records over the last seven days prior to the build day. For instance, if the last build information that we have for a given project is on November, 20th 2016, we first search for coverage information on the same day (November, 20th 2016); if no data was found, we search for coverage information until November, 13th 2016. After this process, we ended up with 16 Java projects and 35 Ruby projects. Overall, the average coverage of these projects in the last available build was 78.99% (median: 88.46%). Figure \[fig:codeCoverage\] shows the coverage distribution for these two set of projects.
As we can see, the coverage of these group of projects varied greatly. On one hand, Java projects seem to have much more coverage variation. On average, Java projects have 63.69% of code coverage (median: 73.16%) 3rd quartile: 83.10%, standard deviation: 27.01%), varying from 4.0% at the lowest coverage, up to 98.17% at the highest coverage. The Java project with the lowest coverage is `connectbot/connectbot`. Moreover, we found three additional Java projects with less than 50% of coverage rate, namely: `psi-probe/psi-probe` (24% of coverage), `myui/hivemall` (33% of coverage), and `igniterealtime/Smack` (35% of coverage). For these projects, we conducted a follow up analysis to understand whether their coverage evolved over time. Interestingly, we observed that these projects did not expressed major changes in their level of coverage. For instance, `psi-probe/psi-probe` improved from 24% in 2016 to 35% of coverage in 2019, `myui/hivemall` kept the same coverage level in 2019 as from 2016: 33%, and `igniterealtime/Smack` improved from 35% in 2016 to 38% coverage in 2019. Still, regarding `connectbot/connectbot` which is the Java project with the lowest code coverage (4%), we observed that this project improved its coverage to 34%. In particular, we identified one single commit[^7] that made the coverage jumped from 4% to 29%. When we inspected this particular commit, the commit message suggested that the intention was to “*Create combined coverage target*”. Inspecting the commit changes, we observed that the author of this commit decided to exclude some directories (that may contain code not relevant to this project) from the build process. After applying this commit, the coverage improved 25%. On the Ruby side, however, the landscape is completely different. We observed that, on average, the coverage of the Ruby project is 85.98% (median: 92%, 3rd quartile: 97.10%, standard deviation: 20.93%), varying from 14.83% at the lowest coverage, up to 100% at the highest coverage. More interestingly, however, is the fact that 17 (48%) Ruby projects have coverage greater than 90%.
We hypothesize that this high coverage scenario for Ruby projects is intrinsically related to the characteristics of the Ruby programming language. Since Ruby is a dynamic typed programming language, developers are only aware of eventual bugs caught by the type system during runtime. Therefore, they may have to rely on a good test suite to minimize eventual bugs that may only appear on the fly. On the other hand, Java developers take advantage of static typing, which avoid some class of bugs that could pass through unattended otherwise.
[**RQ2 Summary:** We found 51 projects in our dataset that have records on [<span style="font-variant:small-caps;">Coveralls</span>]{}. Although the overall coverage was 78%, the coverage of Java and Ruby projects differs greatly. The average code coverage of Ruby projects was 86%, whilst for Java projects it was 63%. This suggests that although poor test coverage exist, a significant number of studied projects take care of their code coverage.]{}
RQ3: How common is allowing the build to stay broken for long periods?
----------------------------------------------------------------------
To analyze this research question, we studied the period, in terms of days elapsed, of broken builds. For each broken build, we counted the number of days between the commit that broken the build until the commit that fixed the build. Since practitioners did not have a clear rule of thumb for the maximum duration that a build could stay broken (the grey literature suggest that the build should be fixed right away [@fowler:CI], we took a conservative approach and used the third quartile of the overall duration of broken builds. Therefore, we assume four days as the threshold for this research question (mean: 7 days, 3rd quartile: 4 days, standard deviation: 29 days).
$
\begin{array}{cc}
\includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/days_broken_ruby.eps}
&
\includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/days_broken_java.eps}
\end{array}
$
When we applied this threshold in the dataset, we observed that 1,072 (85.4%) out of the the 1,270 projects have at least one broken build that took more than four days to be fixed. Figure \[fig:days\_broken\] shows the distribution. More concretely, 85.42% of Java projects have at least one long-to-be-fixed broken build (88,48% for Ruby projects). However, the most interesting observation for this set of experiment is related to broken builds according to the size of the projects. In contrast to a natural belief, large projects (which tend to be more complex and difficult to reason about) are the ones that fix a broken build faster. This holds true for large projects written in the two programming languages, and when comparing to every other size of projects. More specifically, large Java projects let build stay broken, on average, for 2 days (median: 0 days, 3rd quartile: 1 days, max: 408 days). For large Ruby project, the average is 1 day (median: 0 days, 3rd quartile: 1 days, max: 140 days). This finding is in sharp contrast to what was found in smaller projects. For instance very small Java projects, on average, let the build to stay broken for 20 days (the average for very small Ruby projects: 21).
One hypothesis for this behavior is that large projects may count with a large workforce of source code contributors that are readily available to fix broken changes. Moreover, large projects may have a very large user base; for this reason, a broken build may impact several users. Conversely, small projects not only may have to rely on a single code contributor [@Avelino:2016:ICPC] but also may not be as popular as large projects (therefore, there might be little rush to fix a broken build since a very small user base would be regularly updating the master).
[**RQ3 Summary:** We observed that 85% of the analyzed projects have at least one build that took more than four days to be fixed. Interestingly, we observed that large projects (either Java or Ruby) have less long broken builds than smaller projects.]{}
RQ4: How common are long running builds?
----------------------------------------
$
\begin{array}{cc}
\includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/duration_build_ruby.eps}
&
\includegraphics[scale=0.4, clip=true, trim= 0px 0px 0px 0px]{images/duration_build_java.eps} \\
\end{array}
$
For this research question, we studied the duration of the 368,886 builds in our dataset. To measure the build duration, we relied on the “tr\_duration” column of the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset that “*The full build duration as returned from the Travis CI API*”. When investigating the data, we perceived several NaN records in this particular column. We then removed all NaN records, which reduced our data set from 368,886 builds to 55,044 builds. This new sample comprehends 261 projects (253 Java and 8 Ruby). Among the Java projects, there are 12 very small, 98 small, 118 medium, and 24 large ones. For Ruby projects, we found 1 very small, 6 small, and 1 large. No medium and very large Ruby projects were found in this regard.
Figure \[fig:build\_duration\] shows the distribution of the builds’ time duration. For this new set of projects, we observed that, on average, the build took 4 minutes and 18 seconds to run (median: 1 minute and 26 seconds, 3rd quatile: 4 median and 42 seconds, standard deviation: 7 minutes and 26 seconds). Moreover, the build of projects written in Java take on average 4 minutes and 25 seconds (median: 1 minutes and 24 seconds, 3rd quartile: 4 minutes and 31 seconds, standard deviation: 7 minutes and 49 seconds), whereas the the build of projects written in Ruby take on average 3 minutes and 40 seconds (Median: 1 Minute and 47 seconds, 3rd quartile: 5 Minute and 43 seconds, standard deviation: 4 minutes and 23 seconds). More interesting to this research, however, are the time duration of builds made in the large projects, either for Ruby or Java projects. We found 43 (16%) out of the 261 projects have at least one build that took longer than 10 minutes. These projects have produced 7,046 long builds out of the 55,044 total ones.
One natural thought is that large projects may take more time to build because they have a more complicated compilation process. To shed some light along these lines, we investigated the build process of some large projects. We found two large Ruby projects with build time longer than 10 minutes. One of these projects is `jruby/jruby`, which is an implementation of the Ruby programming language for the Java Virtual Machine. When we analyzed the build output for this project, we noted it took about 18 minutes to run integration testing suite (which has 3,389 tests). In contrast, we found 43 Java projects with long builds. At the worst case scenario, we found one project, `geoserver/geoserver`, which its build took 59 minutes. When analyzing its [<span style="font-variant:small-caps;">TravisCI</span>]{}configuration file, we observed that it downloads the maven binaries, and execute its for every build. This process requires the CI system to download third-party libraries used in the project during every single build. Other projects with long build duration are `facebook/presto`, `spotify/helios`, and `biojava/biojava`.
[**RQ4 Summary:** We observed that only 16% of the projects do not adhere to the 10 minutes rule of thumb for build duration. However, when considering Java large projects, the landscape changes significantly: 52% of them take more than 10 minutes.]{}
Threats to Validity {#sec:threats}
===================
In a study such as this, there are always several limitations and threats to validity. First, this research was built upon the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset. Although this dataset provide a comprehensive taxonomy about build information of over one thousand of GitHub projects that use [<span style="font-variant:small-caps;">TravisCI</span>]{}, the last release of this dataset was on 2017, and the most recent build was recorded on March, 21st, 2016. Therefore, our results cannot be extended to the build behavior of these projects today. However, due to the scale of our analysis, we do not expect major changes in the main results, if the most recent builds were considered.
Moreover, to answer RQ1 and RQ3 we extract thresholds from our sample and apply them to the sample itself. This approach may produced self-evident conclusions in case of normal or close to normal distributions. However, we still decided to proceed with this strategy since there is no golden standard, in the context of CI, about what is an adequate number for the frequency of commits nor an acceptable period of time for builds to remain broken. We expect that these questions could be revisited in future works.
Further, we used [<span style="font-variant:small-caps;">Coveralls</span>]{}to gather coverage information. Since this is a proprietary third-party service, we have to blindly rely on its output. A possible mitigation plan would be to download, compile, and execute tests for the open-source projects locally. However, this is often non-trivial task (e.g., some projects fail, some projects require manual configuration, etc). Since recent related work is also employing [<span style="font-variant:small-caps;">Coveralls</span>]{}(e.g., [@Hilton:2018:ASE]), we opted to use the [<span style="font-variant:small-caps;">Coveralls</span>]{}infrastructure to gather coverage information for this work as well.
Another threat to validity is related to the amount of NaN (Not a Number) records in our dataset. To avoid influence the results with these NaN records, we decided to removed them all. This decision, however, may also affect some of our findings. For instance, since there were several NaN in the build duration column, we ended up without medium Ruby projects for RQ4. Finally, one may argue that our approach of providing a grace period could introduce bias, since seven days of software development can greatly change the coverage information. To mitigate this concern, we investigate a random sample of 10 projects and we perceived that their coverage do not change much during the period of seven days. The maximum variation recorded was -1.14% (two projects also had zero variation).
Related Work {#sec:relatedwork}
============
There is a recent flow of empirical studies targeting continuous integration systems, in general, and TravisCI, in particular.
Vasilescu and colleagues [@Vasilescu:2014:CIS] performed a quantitative study of over 200 active Github projects. They restricted their search to Java, Ruby, and Python projects. Among the findings, they found that 92% of the selected projects have configured to use Travis-CI, but 45% of them have no associated builds recorded in the Travis database. Differently from Vasilescu and colleagues [@Vasilescu:2014:CIS], our work focuses on analyzing projects that, despite using CI tools, do not actually employ CI practices. We also search for builds involved in several other programming languages.
The study by Hilton and colleagues [@Hilton:2016] aimed at understanding how software developers use CI tools. Through the analysis of CI builds and a survey, they observed that CI is widely adopted in popular projects and reduces the time between releases. Our work complements their work by analyzing broken builds. Although the work of Hilton and colleagues [@Hilton:2016] provide some initial discussion about build breakage, they did not provide an in-depth investigation in this regard.
The work of Vasilescu and colleagues [@Vasilescu:2015:QPO] analyzed historical data of GitHub projects to see the effects of using CI. They observed that CI helped to increase the number of accepted pull requests from core developers. They also found that CI reduces the quantity of rejected pull-requests while maintaining code quality. Our work complements the previous research by showing that not all projects that adopt CI tools actually employ the CI practices.
Beller and colleagues [@Beller:2017:Oops] studied “how central testing really is in Continuous Integration”. They observed that testing is the main reason as to why builds fail in CI. Different to their work, we also focus on the frequency of commits, build duration, and the required time to fix a broken build. Taher Ghaleb and colleagues [@Ghaleb:EMSE:2019] studied the CI builds with long durations. They built a mixed-effects regression model to study 67 GitHub projects with long build durations. Among the observations, the authors highlight that some CI practices can produce a longer build duration. Complementary to the work by Taher Ghaleb and colleagues [@Ghaleb:EMSE:2019], we also study the usage of CI in builds with a long duration. However, we also study other CI usage scenarios, such as the use of CI with infrequent commits and poor testing coverage.
Bernardo and Colleagues [@Bernardo:MSR:2018] empirically studied whether the adoption of Travis-CI is associated with a shorter time to deliver new functionalities to end users (i.e., delivery delay). They found that the adoption of Travis-CI may not always quicken the delivery of software functionalities. However, they observed that adopting CI is usually associated with a higher proportion of functionalities delivered per software release. We complement their work by quantitatively studying unhealthy CI practices. For example, the observation that adopting CI may increase the time to deliver functionalities might be associated with some of the CI bad practices that we have studied in our work (e.g., poor code coverage).
Zhao and colleagues [@Zhao:ASE:2017] empirically investigated the adoption of Travis-CI in a large sample of GitHub projects. They quantitatively compared the CI transition in these projects using metrics such as commit frequency, code churn, pull request closing, and issue closing. In addition, they conducted a survey with a sample of the developers of the studied projects. The survey consisted of three questions related to the adoption of Travis-CI and CI in general. The main observations were: (i) a small increase in the number of merged commits after CI adoption; (ii) a statistically significant decreasing in the number of merge commit churn; (iii) a moderate increase in the number of issues closed after CI adoption; and (iv) a stationary behavior in the number of closed pull requests as well as a longer time to close PRs after the CI Adoption. Contrary to the work performed by Zhao and colleagues [@Zhao:ASE:2017], we studied four scenarios of unhealthy CI usage instead of the impact that the adoption of CI may bring to a software project.
The study by Maartensson and colleagues [@Maartensson:JSEP:2019] investigated the following general question “[*How can the continuous integration and delivery pipeline be designed in order to support all existing stakeholder interests*]{}.” To this end, the authors surveyed practitioners from 10 software development companies which develop large‐scale software‐intensive embedded system.They proposed a conceptual model that shows practitioners how better design a CI pipeline to include test activities that support all the different interests of the involved stakeholders. Differently from their work, we qualitatively study 1,270 open-source projects. Our work can be complementary to the work by Maartensson and colleagues [@Maartensson:JSEP:2019] in the sense that we shed light on some bad CI practices that can be avoided when designing a CI pipeline.
Gallaba and colleagues [@Gallaba:ASE:2018] empirically investigated the noise and heterogeneity that might lurk in CI build data. They found that CI builds may contain breakages that are ignored by developers. Their analyses of Java projects reveal that builds may contain breakages that occur outside the build tool. Instead of studying the possible noise in CI build data, our work focuses on the unhealthy usage of CI. Our work can complement the study of bias in the existing quantitative analyses, since we observe that not all usage of CI can be healthy. For example, many CI projects that have been quantitatively studied in the field may contain unhealthy CI usage scenarios.
Finally, Zampetti and colleagues [@Zampetti:SANER:2019study] studied the interplay between pull request reviews and CI builds. They analyzed a sample of 857 pull requests that incurred in a build breakage when they were submitted. The result of this analysis was a taxonomy of build breakage types that are discussed through pull requests. They also surveyed 13 developers to complement the observations of their qualitative study. 11 out of the 13 respondents highlighted that the build status actually contribute to the decision taken by the pull request reviewer. Also, the respondents mentioned that the majority of reviewers do not accept a pull request if the build is failing. Our study complements the work by Zampetti and colleagues [@Zampetti:SANER:2019study], since we observe that around 60% of our studied projects perform infrequent commits, which makes the merging process harder.
Conclusions {#sec:conclusions}
===========
In this work, we studied four bad practices that comprehends our notion of Continuous Integration Theater, namely (1) performing infrequent commits to the mainline repository, (2) building a project with poor test coverage, (3) allowing the build to remain in a broken state for long periods, and (4) using CI with long duration builds. To perform our empirical study, we leveraged the [<span style="font-variant:small-caps;">TravisTorrent</span>]{}dataset. In addition, whenever necessary, we used [<span style="font-variant:small-caps;">Coveralls</span>]{}to gather the test coverage of our studied projects. Through the study of 1,270 projects, our results reveal that although some bad practices are commonly employed, such as infrequent commits in the master branch (in $\sim$60% of the projects), other bad practices are not as frequent (such as a build taking too long to process).
Our research shows that the ‘CI Theater’ is present, to some extent, in a considerable amount of software projects. This results imply that existing research that analyzes the benefits of CI (e.g., the time to deliver new functionalities) should consider whether the studied projects have also adopted good CI practices. Using projects that perform the ‘CI Theater’ in empirical analyses may introduce some bias in the analyses.
For future work, we plan to extend our analysis to a newer dataset of CI builds to verify whether the 2016 data generates a significant impact our the results. Still, we plan to interview developers to investigate the effects of bad practices on software health. Finally, we plan to enrich the list of bad practices either by asking practitioners other bad practices that they face with CI or by empirically observing developers working with CI.
***Acknowledgments*.** We thank the reviewers for their helpful comments. This research was partially funded by CNPq/Brazil (406308/2016-0) and UFPA/PROPESP.
[^1]: https://www.thoughtworks.com/radar/techniques/ci-theatre
[^2]: https://www.gocd.org/2017/05/16/its-not-CI-its-CI-theatre.html
[^3]: https://medium.com/continuousdelivery/continuous-integration-not-continuous-isolation-d068a756df0f
[^4]: https://github.blog/2017-11-07-github-welcomes-all-ci-tools/
[^5]: https://github.com/wagnernegrao/ci-analysis
[^6]: https://continuousdelivery.com/foundations/continuous-integration/
[^7]: https://github.com/connectbot/connectbot/pull/410/commits/575766a6444
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0.1cm
[**THE GLUEBALL CANDIDATE $\bf\eta(1440)$\
AS $\bf\eta$ RADIAL EXCITATION** ]{}
Abstract\
The Particle Data Group decided to split the $\eta(1440)$ into two states, called $\eta_L$ and $\eta_H$. The $\eta(1295)$ and the $\eta_H$ are supposed to be the radial excitations of the $\eta$ and $\eta'$, respectively. The $\eta_L$ state cannot be accomodated in a quark model; it cannot be a $q\bar q$ state, however, it might be a glueball. In this contribution it is shown that that the $\eta(1295)$ does not have the properties which must be expected for a radially excited state. The splitting of the $\eta(1440)$ is traced to a node in the wave function of a radial excitation. Hence the two peaks, $\eta_L$ and $\eta_H$, originate from one resonance which is interpreted here as first radial excitation of the $\eta$.
Contributed to\
32nd International Conference on High Energy Physics\
August 16 – 22, 2004\
Beijing, China\
Short history of the $\eta (1440)$
==================================
The E/$\iota$ was discovered in 1967 in $p\bar p$ annihilation at rest into $(K\bar K\pi)\pi^+\pi^-$. It was the first meson found in a European experiment and was called E-meson [@Baillon67]. Mass and width were determined to be $M = 1425 \pm 7, \Gamma = 80\pm 10$MeV, with quantum numbers $J^{PC} = 0^{-+}$. In the charge exchange reaction $\pi^- p \to n\rm K\bar K\pi$, using a 1.5 to 4.2GeV/c pion beam [@Dahl:ad], a state was observed with $M = 1420 \pm 20, \Gamma = 60\pm 20$MeV and $J^{PC} = 1^{++}$. Even though the quantum numbers were different, it was still called E-meson.
In 1979 there was a claim [@Stanton:ya] for the $\eta (1295)$ which was later confirmed in other experiments. In 1980 the E–meson was observed [@Scharre:1980zh] in radiative J/$\psi$ decays into $(K\bar K\pi)$ with $M = 1440 \pm 20, \Gamma = 50\pm 30$MeV; the quantum numbers were ‘rediscovered’ [@Edwards:1982nc] to be $J^{PC} = 0^{-+}$. The E–meson was renamed $\iota (1440)$ to underline the claim that it was the $\iota^{\rm st}$ glueball discovered in an experiment. The $\iota (1440)$ is a very strong signal, one of the strongest, in radiative J/$\psi$ decays. The radial excitation $\eta (1295)$ is not seen in this reaction; hence the $\iota (1440)$ must have a different nature. At that time it was proposed (and often still is) to be a glueball. Further studies, in particular by the Obelix collaboration at LEAR [@Nichitiu:2002cj], showed that the $\iota (1440)$ is split into two components, a $\eta_L\to a_0(980)\pi$ with $M = 1405 \pm 5, \Gamma = 56\pm 6$MeV and a $\eta_H\to \rm K^*\bar K +\bar K^*K$ with $M = 1475 \pm 5,
\Gamma = 81\pm 11$MeV: there seem to be 3 $\eta$ states in the mass range from 1280 to 1480 MeV.
The $\eta (1295)$ is then likely the radial excitation of the $\eta$. It is mass degenerate with the $\pi (1300)$, hence the pseudoscalar radial excitations seem to be ideally mixed. Then, the $\bar ss$ partner should have a 240 MeV higher mass. The $\eta_H$ could play this role. The $\eta_L$ does not find $\eta_L$ a slot in the spectrum of $\bar qq$ mesons; the low mass part of the $\iota (1440)$ could be a glueball. This conjecture is consistent with the observed decays. A pure flavor octet $\eta (xxx)$ state decays into $\rm K^*K$ but not into $a_0(980)\pi$. In turn, a pure flavor singlet $\eta
(xxx)$ state decays into $a_0(980)\pi$ but not into $\rm K^*K$. The $\eta_H$, with a large coupling to $\rm K^*K$, cannot possibly be a glueball, whereas the $\eta_L$ with its $a_0(980)\pi$ decay mode can be.
The PDG 2004 supports this interpretation of the pseudoscalar mesons [@Eidelman:2004wy]:\
3.5mm
Two quantitative tests have been proposed to test if a particular meson is glueball–like: the stickiness and the gluiness. The stickiness of a resonance R with mass $m_{\rm R}$ and two–photon width $\Gamma _{{\rm R} \to \gamma\gamma}$ is defined as: $$S_{\rm R} = N_l \left(\frac{m_{\rm R}}{K_{{\rm J}\to\gamma {\rm R}}}\right)^{2l+1}
\frac{\Gamma _{{\rm J}\to\gamma {\rm R}}}{\Gamma _{{\rm R} \to \gamma\gamma}} \ ,$$ where $K_{{\rm J}\to\gamma {\rm R}}$ is the energy of the photon in the J rest frame, $l$ is the orbital angular momentum of the two initial photons or gluons ($l=1$ for $0^-$), $\Gamma _{{\rm J}\to\gamma {\rm R}}$ is the J radiative decay width for R, and $N_l$ is a normalization factor chosen to give $S_{\eta} = 1$. The L3 collaboration determined [@Acciarri:2000ev] this parameter to be $S_{\eta(1440)}=79\pm 26$.
The gluiness ($G$) was introduced [@Close:1996yc; @Paar:pr] to quantify the ratio of the two–gluon and two–photon coupling of a particle and is defined as: $$G = \frac{9\,e^4_q}{2}\,\biggl(\frac{\alpha}{\alpha _s}\biggr)^2 \,
\frac{\Gamma _{{\rm R} \to {\rm gg}}}{\Gamma _{{\rm R} \to \gamma\gamma}} \ ,$$ where $e_q$ is the relevant quark charge. $\Gamma _{{\rm R} \to {\rm
gg}}$ is the two–gluon width of the resonance [R]{}, calculated from equation (3.4) of ref. [@Close:1996yc]. Stickiness is a relative measure, gluiness is a normalised quantity and is expected to be near unity for a $q\bar{q}$ meson. The L3 collaboration determined [@Acciarri:2000ev] this quantity, $G_{\eta(1440)}=41\pm 14$.
These numbers can be compared to those for the $\eta '$ for which $S_{\eta '} = 3.6 \pm 0.3$ and $G_{\eta '} = 5.2 \pm 0.8$ is determined, for $\alpha_s(958 MeV)=0.56\pm0.07$. Also $\eta'$ is ‘gluish’, but much more the $\eta_L$. The $\eta_L$ is the first glueball!
The $\eta (1295)$ and the $\eta (1440)$ in radiative J/$\psi$ decays
====================================================================
Radiative J/$\psi$ decays show an asymmetric peak in the $\eta(1440)$ region therefore both the $\eta_L$ and the $\eta_H$, must contribute to the process. Obvoiusly, radial excitations are produced in radiative J/$\psi$ decays (not only glueballs). The $\eta(1295)$ must therefore also be produced, but it is not - at least not with the expected yield. Is there evidence for this state in other reactions?
At BES, $\eta (1295)$ and $\eta (1440)$ were studied in J/$\psi\to(\rho\gamma)\gamma$ and $\to
(\phi\gamma)\gamma$ [@Bai:2004qj]. The $\eta(1440)$ (seen at 1424MeV) is seen to decay strongly into $\rho\gamma$ and not into $\phi\gamma$. This is not consistent with the hypothesis of $\eta(1475)$ being a $s\bar s$ state. A peak below 1300MeV is assigned to the $f_1(1285)$ even though a small contribution from $\eta(1295)$ cannot be excluded.
The $\eta (1295)$ and the $\eta (1440)$ in $\gamma\gamma$ at LEP
================================================================
Photons couple to charges; in $\gamma\gamma$ fusion a radial excitation is hence expected to be produced more frequently than a glueball. In $\gamma\gamma$ fusion, both electron and positron scatter by emitting a photon. If the momentum transfer to the photons is small, the $e^+$ and $e^-$ are scattered into forward angles (passing undetected through the beam pipe), thus the two photons are nearly real. If the $e^+$ or $e^-$ has a large momentum transfer, the photon acquires mass, and we call the process $\gamma\gamma^*$ collision. Two massless photons couple to the $\eta(1295)$ but not to the $ f_1(1285)$; in this way, a peak at $\sim$1290MeV can be identified as one of the two states. The L3 collaboration studied $\rm\gamma\gamma^*$ and $\gamma\gamma\to K^0_sK^{\pm}\pi^{\mp}$. At low $q^2$, a peak at 1440MeV is seen, it requires high $q^2$ to produce a peak at 1285MeV. A pseudoscalar state is produced also at vanishing $q^2$ while $J^{PC}=1^{++}$ is forbidden for $q^2\to 0$. Hence the structure at 1285MeV is due to $f_1(1285)$ and not due to $\eta(1295)$. There is no evidence for $\eta (1295)$ from $\gamma\gamma$ fusion. The stronger peak contains contributions from $\eta (1440)$ and $f_1(1420)$ [@Acciarri:2000ev]. The coupling of the $\eta$(1440) meson to photons is stronger than that of the $\eta$(1295): the assumption that the $\eta$(1295) is a $(u\bar u+d\bar d)$ radial excitation must be wrong!
The $\eta (1295)$ and $\eta (1440)$ in $p\bar p$ annihilation
=============================================================
The Crystal Barrel collaboration searched for the $\eta (1295)$ and $\eta (1440)$ in the reaction $p\bar p\to\pi^+\pi^-\eta (xxx)$, $\eta (xxx)\to\eta\pi^+\pi^-$. The search was done by assuming the presence of a pseudoscalar state of given mass and width, mass and width are varied and the likelihood of the fit is plotted. Fig. \[escan\] shows such a plot [@Reinnarth]. A clear pseudoscalar resonance signal is seen at 1405MeV. Two decay modes are observed, $a_0(980)\pi$ and $\eta\sigma$ with a ratio $0.6\pm0.1$. We use the notation $\sigma$ for the full $\pi\pi$ S–wave.
A scan for an additional $0^+ 0^{- +}$ resonance provides no evidence for the $\eta (1295)$ but for a second resonance at 1480MeV, see Fig. \[escan\], with $M=1490\pm 15
,\Gamma=74\pm 10$. This is the $\eta_H$. It decays to $a_0(980)\pi$ and $\eta\sigma$ with a ratio $0.16\pm0.10$. This data provides the first evidence for $\eta_H\to\eta\pi\pi$ decays.
-- --
-- --
The phenomena observed in the pseudoscalar sector are confusing: The $\eta (1295)$, the assumed radial excitation of the $\eta$, is only seen in $\pi^- p\to n (\eta\pi\pi)$, not in $p\bar p$ annihilation, nor in radiative J/$\psi$ decay, nor in $\gamma\gamma$ fusion. In all these reactions it should have been observed. There is no reason for it to have not been produced if it is a $\bar qq$ state. On the other hand, we do not expect glueballs, hybrids or multiquark states so low in mass. In the 70’s, the properties of the $a_1(1260)$ were obscured by the so–called Deck effect ($\rho$–$\pi$ re-scattering in the final state). Possibly, $a_0(980)\pi$ re-scattering fakes a resonant–like behavior but the $\eta(1295)$ is too narrow to make this possibility realistic. Of course there is the possibility that the $\eta(1295)$ is mimicked by feed–through from the $f_1(1285)$. In any case, I exclude the $\eta (1295)$ from the further discussion.
The next puzzling state is the $\eta (1440)$. It is not produced as $\bar ss$ state but decays with a large fraction into $\rm K\bar K\pi$ and it is split into two components. I suggest that the origin of these anomalies is due to a node in the wave function of the $\eta(1440)$! This node has an impact on the decay matrix elements calculated by Barnes [*et al.*]{} [@Barnes:1996ff] within the $^3P_0$ model.
$E/\iota$ decays in the $^3P_0$ model
=====================================
The matrix elements for decays of the $\eta (1440)$ as a radial excitation (=$\eta_R$) depend on spins, parities and decay momenta of the final state mesons. For $\eta_R$ decays to $\rm K^*K$, the matrix element is given by $$f_P = \frac{2^{9/2}\cdot 5}{3^{9/2}}\cdot x\left(1-\frac{2}{15}x^2 \right).$$ In this expression, $x$ is the decay momentum in units of 400MeV/c; the scale is determined from comparisons of measured partial widths to model predictions. The matrix element vanishes for $x=0$ and $x^2 = 15/2$, or $p=1$GeV/c. These zeros have little effect on the shape of the resonance.
The matrix element for $\eta_R$ decays to $a_0(980)\pi$ or $\sigma\eta$ has the form $$f_S = \frac{2^{4}}{3^{4}}\cdot \left(1-\frac{7}{9}x^2 +
\frac{2}{27}x^2 \right)$$ and vanishes for $p=0.45$GeV/c. The decay to $a_0(980)\pi$ vanishes at the mass 1440MeV. This has a decisive impact on the shape, as seen in Figure \[node\]. Shown are the transition matrix elements as given by Barnes et al. [@Barnes:1996ff] and the product of the squared matrix elements and a Breit–Wigner distribution with mass 1420MeV and width 60MeV.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[node\] Amplitudes for $\eta(1440)$ decays to $a_0\pi$ (first row), $\sigma\eta$ (second row), and $\rm K^*\bar K$ (third row); the Breit-Wigner functions are shown on the left, then the squared decay amplitudes [@Barnes:1996ff] and, on the right, the resulting squared transition matrix element.](chef_bilder_1.eps "fig:"){width="36.00000%" height="6cm"} ![\[node\] Amplitudes for $\eta(1440)$ decays to $a_0\pi$ (first row), $\sigma\eta$ (second row), and $\rm K^*\bar K$ (third row); the Breit-Wigner functions are shown on the left, then the squared decay amplitudes [@Barnes:1996ff] and, on the right, the resulting squared transition matrix element.](chef_bilder_2.eps "fig:"){width="36.00000%" height="6cm"} ![\[node\] Amplitudes for $\eta(1440)$ decays to $a_0\pi$ (first row), $\sigma\eta$ (second row), and $\rm K^*\bar K$ (third row); the Breit-Wigner functions are shown on the left, then the squared decay amplitudes [@Barnes:1996ff] and, on the right, the resulting squared transition matrix element.](chef_bilder_3.eps "fig:"){width="38.00000%" height="6cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The $\eta(1440)\to a_0(980)\pi$ and $\to\rm K^*K$ mass distributions have different peak positions; at approximately the $\eta_L$ and $\eta_H$ masses. Hence there is no need to introduce the $\eta_L$ and $\eta_H$ as two independent states. One $\eta(1420)$ and the assumption that it is a radial excitation describe the data.
This conjecture can be further tested by following the phase motion of the $a_0(980)\pi$ or $\sigma\eta$ isobar [@Reinnarth]. The phase changes by $\pi$ and not by 2$\pi$, see Fig. \[phase\].
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[phase\] Complex amplitude and phase motion of the $a_0(980)\pi$ isobars in $\rm p\bar p$ annihilation into $4\pi\eta$. In the mass range from 1300 to 1500MeV the phase varies by $\pi$ indicating that there is only one resonance in the mass interval. The $\sigma\eta$ (not shown) exhibits the same behavior [@Reinnarth]. ](beta-1-a0.eps "fig:"){width="50.00000%"} ![\[phase\] Complex amplitude and phase motion of the $a_0(980)\pi$ isobars in $\rm p\bar p$ annihilation into $4\pi\eta$. In the mass range from 1300 to 1500MeV the phase varies by $\pi$ indicating that there is only one resonance in the mass interval. The $\sigma\eta$ (not shown) exhibits the same behavior [@Reinnarth]. ](phase-1-a0.eps "fig:"){width="50.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Conclusions
===========
Summarizing, the results for the radial excitations of pseudoscalar mesons are as follows:
- The $\eta(1295)$ is not a $q\bar q$ meson.
- The $\eta (1440)$ wave function has a node leading to two appearantly different states $\eta_L$ and $\eta_H$.
- There is only one $\eta$ state, the $\eta(1420)$, in the mass range from 1200 to 1500 MeV and not 3!
- The $\eta(1440)$ is the radial excitation of the $\eta$. The radial excitation of the $\eta'$ is expected at about 1800MeV; it might be the $\eta (1760)$.
The following states are most likely the pseudoscalar ground states and radial excitations:
---------- ------------- ----------------- -------------- ---------
$1^1S_0$ $\pi$ $\eta^{\prime}$ $\eta$ K
$2^1S_0$ $\pi(1300)$ $\eta(1760)$ $\eta(1440)$ K(1460)
---------- ------------- ----------------- -------------- ---------
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|
---
abstract: 'In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, and show that the semi-classical Schrödinger equation produces $O({\varepsilon})$ oscillations in the random variable space. With the Gaussian wave packet transform, the original Schrödinger equation is mapped to an [ordinary differential equation (ODE)]{} system for the wave packet parameters coupled with a [partial differential equation (PDE)]{} for the quantity $w$ in rescaled variables. Further, we show that the $w$ equation does not produce ${\varepsilon}$ dependent oscillations, and thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, i.e. simulating the $w$ equation, it is sufficient to use ${\varepsilon}$ independent samples. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm, and hopefully shed light on possible future directions.'
author:
- 'Shi Jin[^1], Liu Liu[^2], Giovanni Russo[^3] and Zhennan Zhou[^4]'
bibliography:
- 'GWPT\_Ref.bib'
title: 'Gaussian wave packet transform based numerical scheme for the semi-classical Schrödinger equation with random inputs [^5]'
---
=1
Introduction
============
In simulating physical systems, which are often modeled by differential equations, there are inevitably modeling errors, imprecise measurements of the initial data or the background coefficients, which may bring about uncertainties to the equation. There has been a growing interest in analyzing such models to understand the impact of these uncertainties, and thus design efficient numerical methods.
When it comes to quantum dynamics, quantifying the effect of uncertainty is even a trickier task. The solution to the Schrödinger equation is a complex valued wave function, whose nonlinear transforms (e.g. position density, flux density) lead to probabilistic measures of the physical observables. Thus, the uncertainty in the Schrödinger equation may or may not result in changes in measurable quantities from the quantum state.
We consider the following semi-classical Schrödinger equation with random inputs $$i{\varepsilon}\partial_{t}\psi^{{\varepsilon}}(t,\mathbf x,\mathbf z)=-\frac{{\varepsilon}^2}{2}\Delta_{\mathbf x}\psi^{{\varepsilon}}(t,\mathbf x,\mathbf z)+V(\mathbf{x},\mathbf z)\psi^{{\varepsilon}}(t,\mathbf x,\mathbf z),
\label{eq:main equationz}$$ $$\label{eq:main equationz0}
\psi^{{\varepsilon}}(0,\mathbf x,\mathbf z)= \psi^{{\varepsilon}}_{\text{in}}(\mathbf x,\mathbf z).$$ Here, ${\varepsilon}\ll 1$ is the semi-classical parameter, which is reminiscent of the scaled Plank constant, and $V(\mathbf{x},\mathbf z)$ is the scalar potential function, which is slow-varying and often used to model the external field. The initial condition $\psi^{{\varepsilon}}_{\text{in}}(\mathbf x,\mathbf z)$ will be assumed to be in the form of a semi-classical wave packet, which is a Gaussian wave packet parameterized by the wave packet position, the wave packet momentum, etc.
The uncertainty is described by the random variable $\bz$, which lies in the random space $I_{\bz}$ with a probability measure $\pi(\bz)d \bz$. We introduce the notation for the expected value of $f(\bz)$ in the random variable $\mathbf z$, $$\langle f \rangle_{\pi(\mathbf z)} = \int f(\bz) \pi(\mathbf z) d \mathbf z.$$ In this paper, we only consider the uncertainty coming from initial data and potential functions, that is the uncertainty is [*classical*]{}. For example, the external classical field, the wave packet position or the wave packet momentum is uncertain, which reflects on the uncertainty in physical observables.
We do not, however, aim to analyze different types of uncertainties in quantum dynamics, but rather, we study how the uncertainty propagates in the semi-classical Schrödinger equation. When ${\varepsilon}\ll 1$, it is well known that the Schrödinger equation is in the high frequency regime, where the solution generates $O({\varepsilon})$ scaled oscillations in space and time. As we shall show in this paper, the solution generically propagates $O({\varepsilon})$ scaled oscillations in the $\bz$ variable as well even if the random variable $\bz$ obeys an ${\varepsilon}$ independent probability distribution. The high frequency of the solution in space, time and uncertainty leads to unaffordable computational cost, which makes conventional numerical methods infeasible. Thus, it is of great interest to design efficient numerical method based on the multiscale nature of the analytical solutions.
In the semi-classical regime, due to the $O(\varepsilon)$ scaled oscillation in the solution to the Schrödinger equation, the wave function $\psi^{\varepsilon}$ does not converge in the strong sense as $\varepsilon\rightarrow0$. The high frequency nature of the wave function of the semi-classical Schrödinger equation also causes significant computation burdens. If one aims for direct simulation of the wave function, one of the best choices is the time splitting spectral method, as analyzed in [@BaoJin] by Bao, Jin and Markowich. See also [@reviewsemiclassical; @SL-TS; @NUFFT; @S-LLG], where the meshing strategy $\Delta t=O(\varepsilon)$ and $\Delta x=O(\varepsilon)$ is sufficient for moderate values of $\varepsilon$. Another advantage of the time splitting methods is that if one is only interested in the physical observables, the time step size can be relaxed to $o(1)$, in other words, independently of $\varepsilon$, whereas one still needs to resolve the spatial oscillations. As we shall show in the paper, when the uncertainty is present, the wave function is also highly oscillatory in the $\bz$ variable, which means the size of samples in random variable grows as ${\varepsilon}\rightarrow 0$ in order to obtain accurate approximations of the quantum dynamics.
There are quite a few approximate methods other than directly simulating the semi-classical Schrödinger equation, which are valid in the limit $\varepsilon\to 0$, such as the level set method and the moment closure method based on the WKB analysis and the Wigner transform, see, for example, [@reviewsemiclassical] for a general discussion. In the past few years, many wave packets based methods have been introduced, which reduce the full quantum dynamics to Gaussian wave packets dynamics [@Heller; @Heller2; @Hagedorn], and thus gain significant savings in computation cost, such as the Gaussian beam method [@Ralston; @EGB; @QianYing], the Hagedorn wave packet approach [@LubichHagedorn; @ZhouHagedorn] and the Frozen Gaussian beam method [@Kay1; @Kay2; @FGALuYang; @FGALuZhou1; @FGALuZhou2]. When random inputs are considered, in theory, one can design numerical methods based on those approximation tools with ${\varepsilon}$ independent samples in $\bz$, however, the approximation errors persist in spite of other potential challenges.
A related work to our current subject is [@NGO-17], where the authors developed the generalized polynomial chaos (gPC)-based stochastic Galerkin method for a class of highly oscillatory transport equations containing uncertainties that arise in semi-classical modeling of non-adiabatic quantum dynamics. Built upon and modified from the nonlinear geometrical optics based method, this scheme can capture oscillations with frequency-independent time step, mesh size as well as degree of the polynomial.
Clearly, an exact reformulation of the semi-classical Schrödinger equation that separates the multiscales in dynamics is desired for efficient simulation. Very recently, Russo and Smereka proposed a new method based on the so-called Gaussian wave packet transform [@GWPT; @GWPT2; @ZR], which reduces the quantum dynamics to Gaussian wave packet dynamics together with the time evolution of a rescaled quantity $w$, which satisfies another equation of the Schrödinger type, in which the modified potential becomes time dependent. We emphasize that the Gaussian wave packet transform is an equivalent reformulation of the full quantum dynamics, and there are no more ${\varepsilon}$ dependent oscillations in the $w$ equation. This motivates us to investigate whether this transform facilitates design of efficient numerical methods for semi-classical Schrödinger equations with random inputs.
In this work, we study the semi-classical limit of the Schrödinger equation with random inputs, which is the Liouville equation with a random force field, and show that the semi-classical Schrödinger equation produces $O({\varepsilon})$ oscillations in the $\bz$ variable in general. However, with the Gaussian wave packet transform, the original Schrödinger equation is mapped to an ODE system for the wave packet parameters coupled with a PDE for the quantity $w$ in rescaled variables, where the ODE system and the $w$ equation also depend on the random variable. Further, we show that the $w$ equation does not produce ${\varepsilon}$ dependent oscillations in the rescaled spatial variable, thus it is more amenable for numerical simulations. We propose multi-level sampling strategy in implementing the Gaussian wave packet transform, where in the most costly part, simulating the $w$ equation, it is sufficient to use ${\varepsilon}$ independent samples, thus the complexity of the whole algorithm has satisfactory scaling behavior as ${\varepsilon}$ goes to $0$. We also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm as well as hopefully shed light on possible future directions.
The rest of the paper is outlined as follows. Section \[sec:2\] discusses the semi-classical limit of the Schrödinger equation and analyzes the regularity of $\psi$ in the random space. In Section \[sec:3\], we introduce the Gaussian wave packet transform and prove that the $w$ equation is not oscillatory in the random space. [Section \[sec:3.5\] briefly discusses the comparison between quantum and classical systems.]{} Section \[sec:4\] shows extensive numerical tests by using the stochastic collocation method to demonstrate the efficiency and accuracy of our proposed scheme. Relations of different numbers of the collocation points needed in each step of the implementation will be explained and studied numerically. Conclusion and future work are given in Section \[sec:5\].
The semi-classical Schrödinger equation with random inputs {#sec:2}
==========================================================
The semi-classical limit with random inputs {#Semi-Limit}
-------------------------------------------
In this part, we investigate the semi-classical limit of the Schrödinger equation with random inputs by the Wigner transform [@Wigner; @reviewsemiclassical; @Bal; @Lions]. Obviously, the potential function $V(\mathbf x,\bz)$ can be decomposed as $$\label{potde}
V(\mathbf x,\bz)=\bar V(\mathbf x)+ N(\mathbf x, \bz),$$ such that $$\langle V \rangle_{\pi} = \bar V, \quad \langle N \rangle_\pi =0.$$
For $f,\, g\in L^{2}(\mathbb{R}^{d})$, the Wigner transform is defined as a phase-space function $$W^{\varepsilon}(f,g)\left(t,\bx,\bm \xi\right) = \frac{1}{(2\pi)^{d}}\int_{\R^{d}}e^{i \mathbf y\cdot \bm \xi}\bar{f}
\left(\bx+\frac{\varepsilon}{2}\mathbf y\right)g\left(\bx-\frac{\varepsilon}{2}\mathbf y\right)d \mathbf y,$$ [where $\bar{f}$ represents the complex conjugate of $f$. ]{}
Recall that $\psi^{\varepsilon}(t,\bx)$ is the exact solution of equation . Denote $W^{\varepsilon}( t, \bx, \bm \xi, \bz)=W^{\varepsilon}(\psi^{\varepsilon},\psi^{\varepsilon}),$ it is possible to prove that $W^{\varepsilon}$ satisfies the Wigner equation [@Wigner; @BaoJin] $$\partial_{t}W^{\varepsilon}+\bm \xi \cdot\nabla_{\bx}W^{\varepsilon}+\Theta[V]W^{\varepsilon}=0,$$ in which $\Theta[V]W^{\varepsilon}$ is a pseudo-differential operator acting on $W^{\varepsilon}$ defined by $$\Theta[V]W^{\varepsilon}:=\frac{i}{(2\pi)^{d}\varepsilon}\int_{\R^{d}}\left(V\left(\bx+\frac{\varepsilon}{2} \bm \alpha\right)-V\left(\bx-\frac{\varepsilon}{2} \bm \alpha\right)\right)\hat{W}^{\varepsilon}(t,\bx,\bm \alpha, \bz)e^{i\bm \alpha\cdot\bm \xi}\,d \bm \alpha,$$ [where $\hat W$ represents the Fourier transform of $W$ with respect to the momentum component $\xi$.]{}
By Weyl’s Calculus (see [@Wigner]) as $\varepsilon\rightarrow0$, the Wigner measure $W^{0}=\lim_{\varepsilon\rightarrow0}W^{\varepsilon}({\color{black}\psi^{\varepsilon}}, {\color{black}\psi^{\varepsilon}})$ satisfies the classical Liouville equation $$W_{t}^{0}+\bm \xi \cdot\nabla_{\bx}W^{0}-\nabla_{\bx}V \cdot\nabla_{\bm \xi}W^{0}=0,$$ with $$W^{0}(t=0,\bx,\bm\xi,\bz)=W_{I}^{0}(\bx,\bm\xi,\bz):=\lim_{\varepsilon\rightarrow0}W^{\varepsilon}(\psi_{0}^{\varepsilon},\psi_{0}^{\varepsilon}).$$ All the limits above are defined in an appropriate weak sense (see [@Wigner; @Lions]).
With the decomposition of the potential as in , the classical Liouville equation becomes $$\label{eq:WigMea}
W_{t}^{0}+\bm \xi \cdot\nabla_{\bx}W^{0}-\nabla_{\bx} \bar V \cdot\nabla_{\bm \xi}W^{0}-\nabla_{\bx} N \cdot\nabla_{\bm \xi}W^{0}=0.$$ This clearly shows, due to the random potential, the bi-characteristics of the Liouville equation contains the random force term with $\langle -\nabla_{\bx} N \rangle_\pi =-\nabla_{\bx} \langle N \rangle_\pi =0$, and the characteristic equations are $$\left\{
\begin{split}
&\dot \bx =\bm \xi, \\
&\dot {\bm \xi} = -\nabla_{\bx} \bar V-\nabla_{\bx} N.
\end{split}
\right.$$ Also, by definition, it is easy to check that $\forall \mathbf z \in I_{\mathbf z}$, $W^{{\varepsilon}}$ is real-valued. To sum up, in the semi-classical limit, the Wigner measure $W^0$ picks up the dependence of the random variable $\mathbf z$ though the initial condition and the vector field $\nabla_{\bx} N$.
Next, we discuss if one can derive the averaged equation to integrate out the random variable $\mathbf z$. In equation , by taking the average with respect to the random variable $\bz$, one gets $$\partial_t \langle W^0 \rangle_\pi +\bm \xi \cdot\nabla_{\bx} \langle W^{0} \rangle_\pi-\nabla_{\bx} \bar V \cdot\nabla_{\bm \xi} \langle W^{0}\rangle_{\pi}- \langle \nabla_{\bx} N \cdot\nabla_{\bm \xi}W^{0} \rangle_\pi.$$ Notice that, in the last term, both $N$ and $W^0$ depend on $\bz$, thus it cannot be directly written as a term involving $\langle W^0 \rangle_\pi$, rather it connects with the covariance of $\nabla_{\bx}N$ and $\nabla_{\bm \xi}W^0$. In fact, $$\begin{aligned}
- \langle \nabla_{\bx} N \cdot\nabla_{\bm \xi}W^{0} \rangle_\pi & = - \langle (\nabla_{\bx} N - \nabla_{\bx} \langle N \rangle_\pi) \cdot ( \nabla_{\bm \xi} W^{0} - \nabla_{\bm \xi} \langle W^0 \rangle_\pi )\rangle_\pi - \nabla_{\bx} \langle N \rangle_\pi \cdot\nabla_{\bm \xi} \langle W^0 \rangle_\pi \\
&= - {\rm Cov}( \nabla_{\bx} N, \nabla_{\bm \xi}W^0), \end{aligned}$$ since $\nabla_{\bx} \langle N \rangle_\pi =0$.
To illustrate the effect of the random potential, we consider the following special case $$\bar V= \frac{1}{2}|\bx|^2, \quad N = -\bx \cdot \mathbf g(\bz).$$ In this case, the Liouville equation simplifies to $$\label{sliou}
W_{t}^{0}+\bm \xi \cdot\nabla_{\bx}W^{0}- \bx \cdot\nabla_{\bm \xi}W^{0}+ \mathbf g(\bz) \cdot\nabla_{\bm \xi}W^{0}=0.$$ Then, if one considers the following change of variables $$\bx=\tilde \bx + \mathbf g (\bz), \quad \bm \xi = \tilde{\bm \xi},$$ then, equation becomes $$\label{sliou2}
W_{t}^{0}+ \tilde{\bm \xi} \cdot\nabla_{\tilde \bx}W^{0}- \tilde{\bx} \cdot\nabla_{\tilde{\bm \xi}}W^{0}=0.$$ And correspondingly, the initial condition becomes $$W^{0}(t=0,\tilde \bx, \tilde{\bm\xi},\bz)=W_{I}^{0}(\tilde \bx + \mathbf g (\bz), \tilde{\bm \xi}).$$ Thus, the average with respect to the random variable $\bz$ can be taken and one gets a closed equation for $\langle W^{0} \rangle_\pi$, $$\label{sliou3}
\partial_t \langle W^{0} \rangle_\pi+ \tilde{\bm \xi} \cdot\nabla_{\tilde \bx} \langle W^{0} \rangle_\pi - \tilde{\bx} \cdot\nabla_{\tilde{\bm \xi}}\langle W^{0} \rangle_\pi=0.$$ and $$\langle W^{0} \rangle_\pi (t=0,\tilde \bx, \tilde{\bm\xi})= \langle W_{I}^{0}(\tilde \bx + \mathbf g (\cdot) ,\tilde{\bm \xi}) \rangle_\pi.$$
This example shows that the randomness in the slow-varying potential changes the transport part of the Liouville equation, although in the averaged equation the transport structure may be even unchanged.
When the potential is random and with a general form of $z$ dependence, the two processes of a) first pushing ${\varepsilon}\to 0$ then taking expected value in $z$ of the classical limit (the Liouville equation); and b) first taking expected value in $z$ on the Wigner equation then letting ${\varepsilon}\to 0$ do not commute. Though numerically, our simulation results seem to suggest the commutation of these two iterated processes. In later section we will make a numerical comparison to show that the expected values of the position density and flux do converge in the $\varepsilon\to 0$ limit.
To conclude this part, we remark that in [@Bal; @AMS98] and subsequent works, the authors have considered apparently a related but fundamentally different model, where the unperturbed system is the semi-classical Schrödinger with fast-varying smaller magnitude and random perturbation in the potential is also fast-varying. And they show that the randomness in that scaling introduces additional scattering terms in the limit equations, while in our case the randomness only persist in the initial data and the force field in the limit equation.
Regularity of $\psi$ in the $\bz$ variable
------------------------------------------
The semi-classical Schrödinger equation is a family of dispersive wave equations parameterized by ${\varepsilon}\ll 1$, and it is well known that the wave equation propagates $O({\varepsilon})$ scaled oscillations in space and time. However, it is not clear yet whether the small parameter ${\varepsilon}$ induces oscillations in the random variable $\mathbf z$.
Here and in subsection \[Reg-W\], we will conduct a regularity analysis of $\psi$ in the random space, which enables us to study the oscillatory behavior of solutions in the random space, which gives guidance on how many collocation points needed in each step of the collocation method should depend on the scaled constant ${\varepsilon}$.
To investigate the regularity of the wave function in the $\bz$ variable, we check the following averaged norm $$\label{energy} ||f||_{\Gamma }:= \left (\int_{I_z}\int_{\mathbb R^3}\left| f(t, \mathbf x, \bz)\right|^2\, d{\mathbf x}\pi(\bz)d{\bz} \right)^{\frac 1 2}.$$ To be more precise, (\[energy\]) denotes the square root of the expected value in $\bz$ of the square of the $L^2(x)$ norm of $f$. We name it $\Gamma$-norm for short.
One first observes that $\forall\, \mathbf z \in I_{\mathbf z}$, $$\frac{\partial}{\partial t} \| \psi^{{\varepsilon}}\|^2_{L^2_{\mathbf x}}(t,\bz)=0,$$ thus $$\frac{d}{dt} \| \psi^{{\varepsilon}}\|_{\Gamma}^2=0,$$ which means the $\Gamma$-norm of the wave function $\psi^{\varepsilon}$ is conserved in time, $$\| \psi^{{\varepsilon}}\|_{\Gamma} (t) = \| \psi^{{\varepsilon}}_{\text{in}}\|_{\Gamma}\,.$$
However, we show in the following that $\psi^{\varepsilon}$ has ${\varepsilon}$-scaled oscillations in $\mathbf z$ even if $V$ and $\psi_{\text{in}}^{{\varepsilon}}$ do not have ${\varepsilon}$-dependent oscillations in $\mathbf z$. We first examine the first-order partial derivative of $\psi^{\varepsilon}$ in $z_1$, and denote $\psi^1= \psi^{\varepsilon}_{z_1}$ and $V^1=V_{z_1}$, then by differentiating the semi-classical schrödinger equation (\[eq:main equationz\]) with respect to $z_1$, one gets $$i {\varepsilon}\psi^1_t =- \frac{{\varepsilon}^2}{2}\Delta_{\mathbf x} \psi^1 + V^1 \psi^{\varepsilon}+ V \psi^1.$$
By direct calculation, $$\begin{aligned}
\frac{d}{dt} \| \psi^{1}\|^2_{\Gamma} &= \int \bigl (\psi^1_t \bar \psi^1 + \psi^1 \bar \psi^1_t \bigr) \pi d \mathbf x d \mathbf z \\
& = \int \bigl(\frac{1}{i{\varepsilon}} V^1 \psi^{\varepsilon}\bar \psi^1 - \frac{1}{i{\varepsilon}} V^1 \psi^1 \bar \psi^{\varepsilon}\bigr) \pi d \mathbf x d \mathbf z \\
& \le \frac{2}{{\varepsilon}} \| \psi^{1}\|_{\Gamma}\, \| V^1 \psi^{{\varepsilon}}\|_{\Gamma}\,,\end{aligned}$$ where the Cauchy-Schwarz inequality and the Jensen inequality are used in the last step, more specifically, $$\begin{aligned}
&\int V^{1}\psi^{{\varepsilon}}\bar\psi^{1} dx \leq \left( \int (V^{1}\psi^{{\varepsilon}})^2 dx \right)^{1/2} \left( \int (\bar\psi^{1})^2 dx\right)^{1/2}, \\
&\int\int V^{1}\psi^{{\varepsilon}}\bar\psi^{1} dx\, \pi(z)dz \leq \left(\int \left(\int V^{1}\psi^{{\varepsilon}}\bar\psi^{1} dx\right)^2 \pi(z)dz\right)^{1/2} \leq ||V^{1}\psi^{{\varepsilon}}||_{\Gamma}\,
||\psi^{1}||_{\Gamma}\,.\end{aligned}$$ Thus $$\frac{d}{dt} \| \psi^{1}\|_{\Gamma} \le\frac{1}{{\varepsilon}} \| V^1 \psi^{{\varepsilon}}\|^2_{\Gamma}\,,$$ For $t=O(1)$, the pessimistic estimate implies $$\| \psi^{1}\|_{\Gamma} =O\bigl({\varepsilon}^{-1}\bigr).$$ Moreover, for $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, denote $|\mathbf k|=\sum_{j=1}^n k_j $, we can similarly conclude that $$\label{est:psiz}
\|\partial_{\mathbf z}^{\mathbf k} \psi^{{\varepsilon}}\|_{\Gamma} =O\bigl( {{\varepsilon}^{-|\mathbf k|}}\bigr).$$
Although the estimates above are apparently pessimistic, we would like to show that the high frequency oscillations in $\mathbf z$ can be seen in the following example. For simplicity, we consider $x$ be one dimensional. If the potential $V$ is quadratic in $x$, it has been shown by Heller in [@Heller] that, $$\label{Heller}
\phi(x,t)=\exp \left[ i \frac{\alpha(t)\bigl(x-q(t)\bigr)^2-p(t)\bigl(x-q(t)\bigr)+ \gamma(t) }{{\varepsilon}}\right]$$ is an exact solution to the semi-classical Schrödinger equation, provided that, $q(t)$, $p(t)$, $\alpha(t)$ and $\gamma(t)$ satisfy the following system of equations $$\label{psys2}
\left\{ \begin{array}{l}
\dot q = p,
\\ \dot p = - V_q(q),
\\ \dot \alpha = -2 \alpha^2 -\frac{1}{2}V_{qq}(q),
\\ \dot \gamma =\frac{1}{2}p^{2}-V(q)+i{\varepsilon}\alpha.
\end{array}\right.$$
Due to the same reason, $$\Phi(t,x,\mathbf z)=\exp \left[ i \frac{\alpha(t,\mathbf z)\bigl(x-q(t,\mathbf z)\bigr)^2-p(t,\mathbf z)\bigl(x-q(t,\mathbf z)\bigr)+ \gamma(t,\mathbf z) }{{\varepsilon}}\right]$$ is an exact solution to equation (\[eq:main equationz\]), when potential $V(x, \mathbf z)$ is quadratic in $x$. Clearly, this specific solution saturates the estimate , which implies, even if initially $\psi^{\varepsilon}$ is smooth in $\mathbf z$, it will pick up ${\varepsilon}$-dependent oscillations in the $\mathbf z$ variables. In Section \[sec:4\], we will also show numerically the ${\varepsilon}$-scaled oscillations of such wave functions in the $\mathbf z$ variable.
To conclude this section, we emphasize the numerical challenges with respect to the random variable $\bz$. Due to the oscillatory behavior in $\mathbf z$, if one applies the generalized polynomial chaos (gPC)-based stochastic methods directly to the semi-classical Schrödinger equation, one needs at least ${\varepsilon}$-dependent basis functions or quadrature points to get an accurate approximation. The stochastic collocation method will be discussed in detail in subsection \[sec:4a\].
The semi-classical Schrödinger equation and the Gaussian wave packet transformation {#sec:3}
===================================================================================
To overcome the numerical burdens in sampling the random variable $z$, we introduce the Gaussian wave packet transformation (abbreviated by GWPT), which has been proven to be a very efficient tool for computing the Schrödinger equation in the high frequency regime. In essence, the GWPT equivalently transforms the highly oscillatory wave equation to [an equation for a new rescaled wave function $w$]{}, thus facilitate the design of efficient numerical methods. [At variance with most methods based on the Gaussian beam or Gaussian wave packet, the GWPT approach is not based on an asymptotic expansion in ${\varepsilon}$, therefore it is equivalent to the original Schrödinger equation for [*all*]{} ${\varepsilon}$.]{} The main goal of this section is to study the [*uniform regularity*]{} of $w$ in the $\bz$ variable, and conclude that the number of basis functions or quadrature points is [*independent of ${\varepsilon}$*]{}, if applying the SG or SC method in our GWPT framework.
Review of the Gaussian wave packet transformation
-------------------------------------------------
First, we briefly summarize the Gaussian wave packet transformation applied to the semi-classical Schrödinger equation with random inputs, which is a natural extension of the GWPT method for the deterministic problem [@GWPT]. Consider the semi-classical Schrödinger equation given by (\[eq:main equationz\])–(\[eq:main equationz0\]). Note that in this work, random inputs are assumed to be classical, thus we only consider the cases when the wave packet position and momentum in the GWPT parameters depend on the random variable $\bz$.
We start by the following [*ansatz*]{} $$\label{ansatz}
\psi(t,\mathbf x,\bz)= \widetilde w(t,\boldsymbol\xi,\bz)\exp \left(g(t,\boldsymbol\xi,\bz)\right):=\widetilde w(t, \boldsymbol\xi, \bz)\exp \left(i\left(\boldsymbol\xi^{T}
\boldsymbol{\alpha}_R\, \boldsymbol\xi + {\mathbf p}^{T}\boldsymbol\xi +\gamma\right)/{\varepsilon}\right),$$ where $\boldsymbol\xi=\mathbf x - \mathbf q$, $\boldsymbol{\alpha}_R$ is a real-valued symmetric matrix and $\gamma$ is a complex-valued scalar.
Denote $\boldsymbol{\alpha}_R={\mathrm{Re}}(\boldsymbol{\alpha})$, $\boldsymbol{\alpha}_I={\mathrm{Im}}(\boldsymbol{\alpha})$ and $\boldsymbol{\alpha}=\boldsymbol{\alpha}_R + i \boldsymbol{\alpha}_I$. Insert the ansatz (\[ansatz\]) into (\[eq:main equationz\]), then $\widetilde w(t, \boldsymbol\xi, \bz)$ satisfies $$\begin{aligned}
\widetilde w_{t} = -2\, \boldsymbol\xi^{T} \boldsymbol{\alpha}_R \nabla_{\boldsymbol\xi} \widetilde w + \frac{i {\varepsilon}}{2} \Delta_{\boldsymbol\xi}\widetilde w
- \frac{i}{{\varepsilon}} \left(U_{r}+2\, \boldsymbol\xi^T \boldsymbol{\alpha}_I^2\, \boldsymbol\xi \right)\widetilde w,\end{aligned}$$ provided $\mathbf p$, $\mathbf q$, $\boldsymbol\alpha$ satisfy the following equations $$\label{psys}
\left\{ \begin{array}{l}
\dot {\mathbf q} = \mathbf p,
\\ \dot {\mathbf p} = -\nabla V(\mathbf q),
\\ \dot {\boldsymbol\alpha} = -2\, \boldsymbol{\alpha}^2 - \frac{1}{2}\, \nabla\nabla V(\mathbf q),
\\ \dot \gamma = \frac{1}{2}\, {\mathbf p}^T \mathbf p - V(\mathbf q) + i{\varepsilon}\, {\rm Tr}(\boldsymbol{\alpha}_R),
\end{array}\right.$$ while $$U_r=V(\boldsymbol\xi + \mathbf q)-V(\mathbf q)-\boldsymbol\xi^T\, \nabla V(\mathbf q)-\frac{1}{2}\, \boldsymbol\xi^T\, \nabla^2 V(\mathbf q)\boldsymbol\xi.$$ At last, introduce the change of variables $\widetilde w(t,\boldsymbol\xi, \bz)=w(t,\boldsymbol\eta, \bz)$, where $$\label{changeV}
\boldsymbol\eta={\mathbf B\boldsymbol\xi}/\sqrt{{\varepsilon}},$$ with $$\dot {\mathbf B} = - 2 \mathbf B\, \boldsymbol{\alpha}_R, \qquad \mathbf B(0)=\sqrt{\boldsymbol{\alpha}_I(0)}\,,$$ then $\boldsymbol{\alpha}_I = {\mathbf B}^T\, \mathbf B$ and $$\label{weq}
w_{t}=\frac{i}{2}\, {\rm Tr}\left({\mathbf B}^T\, \nabla_{\boldsymbol\eta}^2 w\, {\mathbf B}\right) - 2 i\, \boldsymbol\eta^{T}(\mathbf {B}^T)^{-1}\,\boldsymbol{\alpha}_I^2\, \mathbf {B}^{-1}\boldsymbol\eta w+\frac{1}{i{\varepsilon}}\, U_{r}w,$$ Note that in the $\boldsymbol\eta$ variable, $$\frac{1}{i{\varepsilon}}U_{r}=O(\sqrt {\varepsilon}),$$ so the $w$ equation (\[weq\]) is not oscillatory in $\boldsymbol\eta$ nor in $t$. Furthermore, if one drops those $O(\sqrt{{\varepsilon}})$ terms, one expects to recover the leading order Gaussian beam method [@GWPT2].
In our numerical tests, we will only consider the initial data $\psi$ given by a Gaussian wave packet, i.e. $$\label{initial_GWP}
\psi(0,\mathbf x,\bz)=\exp \left(i\left(\boldsymbol\xi^{T}
\boldsymbol{\alpha}\, \boldsymbol\xi + {\mathbf p}^{T}\boldsymbol\xi +\gamma\right)/{\varepsilon}\right),$$ More general initial conditions, (with a numerical support proportional to $\sqrt{\varepsilon}$) can be approximated with the desired accuracy as a superposition of relatively small number of Gaussian wave packets. See [@GWPT Section 2.6] for more details. [Another point of view to shed some light on the GWPT formulation is the following: Direct solution of the Schroedinger equation for a modulated wave-packet requires a lot of grid points, in most of which the wave function is almost zero, while the change of variable allows to work with a fixed computational domain of length $O(1)$, where the non oscillatory transformed wave function $w$ can be resolved with a relatively small number of grid points. ]{}
Regularity of $w$ in the $\bz$ variable {#Reg-W}
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It is well understood that the $w$ equation no longer propagates ${\varepsilon}$-dependent oscillations in space or in time. We show in the following that the $w$ equation is not oscillatory in the $\bz$ variable either. In this section, we assume the spatial variable $x,\,\eta$ is one dimensional to simplify the analysis. Now the $w$ equation (\[weq\]) reduces to $$\label{w-eq1}
w_t = \frac{i}{2} \alpha_I w_{\eta \eta} - 2 i \alpha_I \eta^2 w+ \frac{U_r}{i {\varepsilon}} w,$$ where $$U_r (t,\eta,\mathbf z)= V(q+\sqrt{{\varepsilon}} B^{-1} \eta, \mathbf z ) - V(q,\mathbf z)-\sqrt{{\varepsilon}}B^{-1}\eta\, V_x(q,\mathbf z)- \frac{1}{2}\bigl (\sqrt{{\varepsilon}}B^{-1}\eta\bigr)^2\, V_{xx}(q,\mathbf z),$$ and $B=\sqrt{\alpha_I}$. Observe that $w=w(t,\eta;\mathbf z)$ is a function of independent variables $(t,\eta)$, but it obtains the dependence of the parameter $\mathbf z$ through the coefficients. We emphasize that, although the change of variable is $\mathbf z$ dependent, in the $w$ equation, $\eta$ and $\mathbf z$ are independent variables. This is due to the fact that it is $w=w(t,\eta;\mathbf z)$, not $w=w(t,\eta(\mathbf z);\mathbf z)$. With the random inputs, the Gaussian wave packet transform is straightforward for all $\bz$. Besides the time dependence, the Gaussian wave packet parameters $q$, $p$, $\alpha$, $B$ and $\gamma$ also depend on $\mathbf z$. The smooth components $w$ and $U_r$ depend on $\bf z$. Then, it is not yet clear whether $w$ has ${\varepsilon}$ dependent $z$ derivatives.
We make the assumption that the potential is infinitely smooth with bounded derivatives in both $x$ and $\mathbf z$, namely, for $m\in \N$ and $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, there exists a constant $C_{m,\mathbf k}$ such that, $$\label{assump1}
|\partial_x^m \partial_{\mathbf z}^{\mathbf k}V|\le C_{m,\mathbf k},$$ where $\partial_{\mathbf z}^{\mathbf k}=\partial_{z_1}^{k_1}\cdots \partial_{z_n}^{k_n}$. We also assume that the $w$ equation (\[w-eq1\]) is equipped with an initial condition, $w(0,\eta,\mathbf z)=w_{\text{in}}(\eta,\mathbf z)$, which has a $O(1)$ sized support, satisfying [the following assumption:]{} for $m\in \N$ and $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, there exists an ${\varepsilon}$-independent constant $C_{m,\mathbf k}$ such that, $$\label{assump2}
\| \partial_\eta^m\partial_{\mathbf z}^{\mathbf k} w_{\text{in}}\|_{\Gamma}\le C_{m,\mathbf k}.$$
Our goal is to show that the smoothness in the $\bz$ variable will be preserved in time. Before calculating the regularity of $w$ in the $\bz$ variable, we first show Lemma \[lemma:parameters\] and Lemma \[Newlemma:Ur\], which will be used for the main result of this section namely Theorem \[w\_z\].
\[lemma:parameters\] Assume boundedness condition , and initial data for wave packet parameters satisfying the following: there exists an ${{\varepsilon}}$ independent constant $C_{\mathbf K}$, such that $$|\partial_{\mathbf z}^{\mathbf k} q(0)| \le C_{\mathbf K}, \quad |\partial_{\mathbf z}^{\mathbf k} p(0)| \le C_{\mathbf K}, \quad |\partial_{\mathbf z}^{\mathbf k} \alpha(0) | \le C_{\mathbf K}.$$ Then, there exists an ${{\varepsilon}}$ independent constant $C_{T,\mathbf K}$, such that for $t\in[0,T]$, $$|\partial_{\mathbf z}^{\mathbf k} q(t)| \le C_{T,\mathbf K}, \quad |\partial_{\mathbf z}^{\mathbf k} p(t)| \le C_{T,\mathbf K}, \quad |\partial_{\mathbf z}^{\mathbf k} \alpha(t) | \le C_{T,\mathbf K}.$$
The proof follows standard estimations of the ODE system of the parameters, which we shall omit here. We also remark that we have only listed the parameters needed for showing the regularity properties of the $w$ equation (\[w-eq1\]), but this argument clearly works for other wave packets parameters as well.
\[Newlemma:Ur\] With the boundedness assumptions , for all $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, it is $$\partial_{\bz}^{\bk}U_r=O({\varepsilon}^{\frac 3 2}).$$
Recall that $$\label{Ur}
U_r (t,\eta,\mathbf z)= V(q+\sqrt{{\varepsilon}} B^{-1} \eta, \mathbf z ) - V(q,\mathbf z)-\sqrt{{\varepsilon}}B^{-1}\eta\, V_x(q,\mathbf z)- \frac{1}{2}\bigl (\sqrt{{\varepsilon}}B^{-1}\eta\bigr)^2\, V_{xx}(q,\mathbf z).$$ We observe that, assumption together with the Taylor’s Theorem implies, for $m\in \N$ and $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, $$\begin{gathered}
\label{est:T}
T_{m,\mathbf k}:=\partial_x^m \partial_{\mathbf z}^{\mathbf k}V(q+\sqrt{{\varepsilon}} B^{-1} \eta, \mathbf z )-\partial_x^m \partial_{\mathbf z}^{\mathbf k}V(q, \mathbf z )- \sqrt{{\varepsilon}} B^{-1} \eta \partial_x^{m+1} \partial_{\mathbf z}^{\mathbf k}V(q, \mathbf z )\\
- \frac{1}{2} \bigl (\sqrt{{\varepsilon}}B^{-1}\eta \bigr)^2 \partial_x^{m+2} \partial_{\mathbf z}^{\mathbf k}V(q, \mathbf z )=O({\varepsilon}^{\frac 3 2}).\end{gathered}$$ Thus, it is clear that $|U_r| = O({\varepsilon}^{\frac 3 2})$.
Next, we examine the first order derivative in $z_1$. By direct calculation, $$\begin{aligned}
\partial_{z_1}U_r &= \partial_{z_1}V(q+\sqrt{{\varepsilon}}B^{-1} \eta, \bz) + \partial_{x}V(q+\sqrt{{\varepsilon}}B^{-1}\eta, \bz)\partial_{z_1}(q+\sqrt{{\varepsilon}}B^{-1}\eta) \\
&\quad - (\partial_{z_1}V(q, z) + \partial_{x}V(q, \bz)\partial_{z_1}q) \\
&\quad - \left(\partial_{z_1}( \sqrt{{\varepsilon}} B^{-1}\eta)\partial_x V(q, \bz) + \sqrt{{\varepsilon}}B^{-1}\eta\, \partial_{x z_1}V(q, \bz)+ \sqrt{{\varepsilon}}B^{-1}\eta\, \partial_{xx}V(q, \bz) \partial_{z_1}q \right) \\
&\quad - \frac{1}{2}{\varepsilon}\left( \partial_{z_1}(B^{-1}\eta)^2 \partial_{xx}V(q, \bz) + (B^{-1}\eta)^2 \partial_{xx z_1}V(q, \bz)+ (B^{-1}\eta)^2 \partial_{xx x}V(q, \bz) \partial_{z_1} q\right) \\
& =\partial_{z_1}V(q+\sqrt{{\varepsilon}}B^{-1} \eta, \bz) - \partial_{z_1}V(q, z) \\
& \quad - \sqrt{{\varepsilon}}B^{-1}\eta\, \partial_{x z_1}V(q, \bz) - \frac 1 2 (B^{-1}\eta)^2 \partial_{xx z_1}V(q, \bz) \\
& \quad + \partial_{x}V(q+\sqrt{{\varepsilon}}B^{-1}\eta, \bz)\partial_{z_1}q -\partial_{x}V(q, \bz)\partial_{z_1}q \\
& \quad - \sqrt{{\varepsilon}}B^{-1}\eta\, \partial_{xx}V(q, \bz) \partial_{z_1}q - \frac 1 2 (B^{-1}\eta)^2 \partial_{xx x}V(q, \bz) \partial_{z_1} q \\
& \quad + \left( \partial_{x}V(q+\sqrt{{\varepsilon}}B^{-1}\eta, \bz)- \partial_x V(q, \bz) - \sqrt{{\varepsilon}}B^{-1}\eta\, \partial_{xx}V(q, \bz)\right) \partial_{z_1}(\sqrt{{\varepsilon}}B^{-1}\eta). \end{aligned}$$ Thus, by and Lemma \[lemma:parameters\], we conclude that $\partial_{z_1}U_r =O({\varepsilon}^{\frac 3 2})$.
By induction, one easily sees that $\partial_{\mathbf z}^{\mathbf k} U_r $ is a summation of products of $T_{m,\mathbf k}$ (add up with some other terms). Hence, by and Lemma \[lemma:parameters\], it can be concluded that $\partial_{\mathbf z}^{\mathbf k} U_r =O({\varepsilon}^{\frac 3 2})$, and the lemma follows.
We give the definition of the averaged norm of $w=w(t, \eta, \bz)$, $$||w||_{\mathcal T}^2:= \langle \| w \|_{L^2(\eta)}^2 \rangle_{\pi(\bz)},$$ which is analogous to the $\Gamma$–norm defined in (\[energy\]) for $\psi$ while using $\eta$ variable in the $L^2(\eta)$ norm here. We now present the main theorem of this section:
\[w\_z\] With the boundedness assumptions and conditions on the initial data , the $w$ equation (\[w-eq1\]) preserves the regularity in the following sense: for a fixed $T>0$, $\mathbf k=(k_1,k_2,\cdots,k_n) \in \N^n$, there exists an ${\varepsilon}$-independent constant $M_{T,\mathbf k}$, such that for $0\le t \le T$, $$\label{reg_wz}
|| \partial_{\bz}^{\bk}w||_{\mathcal T}\le M_{T,\mathbf k}\,.$$
The $w$ equation can be written as $$\label{w_t} w_t = \frac{i}{2}\alpha_{I}w_{\eta\eta}- i\tilde U w,$$ where $$\tilde U = 2\alpha_{I}\eta^2 +\frac{U_r}{{\varepsilon}},$$ and $U_r$ given in (\[Ur\]). We first look at the $\mathcal T$-norm of $w$, $$\begin{aligned}
& \frac{d}{d t} ||w||_{\mathcal T}^2 = \int (w \bar w)_t\, d\eta \pi(\bz)d\bz = \int (w_t \bar w+ w \bar w_t)\, d\eta\pi(\bz)d\bz \\
&\qquad\quad = \int \bigl(\frac{i}{2}\alpha_{I}w_{\eta\eta} - i\tilde U w)\bar w + w(-\frac{i}{2}\alpha_{I}\bar w_{\eta\eta} +
i \tilde U \bar w)\bigr)\, d\eta\pi(\bz)d\bz =0, \end{aligned}$$ which implies the averaged norm of $w$ is preserved, $||w||_{\mathcal T}= ||w_{\text{in}}||_{\mathcal T}$.
Differentiating (\[w\_t\]) with respect to $z_1$, by the chain rule, one has $$\partial_t \partial_{z_1}w = \frac{i}{2}\bigl(\partial_{z_1}\alpha_{I}\, w_{\eta\eta}
+ \alpha_{I} w_{\eta\eta z_1}\bigr)
- i \partial_{z_1}\tilde U w - i \tilde U \partial_{z_1}w.$$ By direct calculation (omit the $\mathcal T$-subscript in the norm $||\cdot||_{\mathcal T}$ for notation simplicity), $$\begin{aligned}
\frac{d}{dt} ||\partial_{z_1}w||^2
& = \int\left(\partial_t\partial_{z_1}w\, \partial_{z_1}\bar w + \partial_{z_1}w\, \partial_t\partial_{z_1}\bar w\right) d\eta\pi(\bz)d\bz \\
& = \int \left( \left[\frac{i}{2}\left(\partial_{z_1}\alpha_{I}\, w_{\eta\eta} + \alpha_{I} w_{\eta\eta z_1}\right)
- i \partial_{z_1}\tilde U w - i\tilde U \partial_{z_1}w\right] \partial_{z_1}\bar w \right. \\
&\quad \left. + \partial_{z_1}w \left[-\frac{i}{2}\bigl(\partial_{z_1}\alpha_{I}\, \bar w_{\eta\eta}+ \alpha_{I} \bar w_{\eta\eta z_1}\bigr) + i\partial_{z_1}\tilde U \bar w + i\tilde U \partial_{z_1}\bar w\bigr)\right]\right) d\eta\pi(\bz)d\bz \\
& = \int \left[\frac{i}{2}\partial_{z_1}\alpha_{I}(w_{\eta\eta}\, \partial_{z_1}\bar w - \partial_{z_1}w\, \bar w_{\eta\eta}) + i \partial_{z_1}\tilde U (-w\, \partial_{z_1}\bar w + \partial_{z_1}w\, \bar w)\right] d\eta\pi(\bz)d\bz \\
& \leq ||\partial_{z_1}w||\, ||\partial_{z_1}\alpha_{I}\, w_{\eta\eta\eta}|| + 2 ||\partial_{z_1}\tilde U w||\, ||\partial_{z_1}w|| , \end{aligned}$$ where integration by parts and the Cauchy-Schwarz inequality are used. Thus $$\label{wz_1}\frac{d}{dt} ||\partial_{z_1}w|| \leq \frac{1}{2}||\partial_{z_1}\alpha_{I}\, w_{\eta\eta\eta}|| + ||\partial_{z_1}\tilde U w|| .$$
Clearly, to prove the boundedness of $||\partial_{z_1}w||$, it suffices to show the boundedness of the right hand side of . Fortunately, the right hand side of does not involve the $\mathbf z$ derivative of $w$. The estimates of the $\eta$ derivatives of $w$ are standard, which can be carried out in the following deductive way.
We calculate that $$\begin{aligned}
\label{W_eta}
\begin{split}
\frac{d}{d t} ||w_{\eta}||^2 &= \int \bigl((w_t)_{\eta} \bar w_{\eta}+ w_{\eta} (\bar w_t)_{\eta} \bigr) d\eta\pi(\bz)d\bz \\
&= \int \bigl (-i (\tilde U w)_{\eta} \bar w_{\eta}+ i w_{\eta} (\tilde U \bar w)_{\eta} \bigr) d\eta\pi(\bz)d\bz \\
& \le \left(\int (\tilde U w)_{\eta}^2 \, \bar w_{\eta}^2 \, d\eta\pi(\bz)d\bz\right)^{1/2}
\left((\tilde U \bar w)_{\eta}^2 \, w_{\eta}^2\, d\eta\pi(\bz)d\bz\right)^{1/2} \\
& = 2 ||w_{\eta}||\, ||(\tilde U w)_{\eta}||.
\end{split}\end{aligned}$$ Since $\partial_{\eta}\tilde U= 4 \alpha_{I}\eta + \partial_{\eta}U_r / {\varepsilon}=O(1)$, and by the chain rule, $$|| (\tilde U w)_{\eta}|| = || \tilde U_{\eta} w + \tilde U w_{\eta}||
\leq C_1 ||w|| + C_2 ||w_{\eta}||,$$ where $C_1, C_2>0$ are constants. Thus $$\label{w_eta1}\frac{d}{d t}||w_{\eta}|| \leq C_1 ||w|| + C_2 ||w_{\eta}||.$$ For $t \in [0,T]$, the boundedness of $||w_{\eta}||$ follows from the Grönwall’s inequality and assumption . Similarly, we get $$\begin{aligned}
\frac{d}{d t} ||w_{\eta\eta}||^2 & = \int \bigl ((w_t)_{\eta\eta} \bar w_{\eta\eta}+ w_{\eta\eta} (\bar w_t)_{\eta\eta}\bigr) d\eta\pi(\bz)d\bz \\
&= \int \bigl (-i(\tilde U w)_{\eta\eta} \bar w_{\eta\eta}+ i w_{\eta\eta} (\tilde U \bar w)_{\eta\eta}\bigr) d\eta\pi(\bz)d\bz \\
& \le 2 ||w_{\eta\eta}||\, ||(\tilde U w)_{\eta\eta}||, \end{aligned}$$ where the Cauchy-Schwarz inequality is used again just as in (\[W\_eta\]). By the chain rule, $\exists\, C_1,C_2, C_3 \in \R$ such that $$||(\tilde U w)_{\eta\eta}|| \le C_1 ||w|| + C_2 ||w_{\eta}|| + C_3 ||w_{\eta\eta}||,$$ thus $$\label{w_eta2}
\frac{d}{d t} ||w_{\eta\eta}|| \le C_1 ||w|| + C_2 ||w_{\eta}|| + C_3 ||w_{\eta\eta}||.$$ The boundedness of $||w_{\eta\eta}||$ follows from Grönwall’s inequality and assumption , and similarly for $||w_{\eta\eta\eta}||$. $||\alpha_{I}w_{\eta\eta}||$, $||\tilde U w_{\eta}||$ are also bounded. By Lemma \[Newlemma:Ur\], [$||\partial_{z_1} \tilde U w|| \sim C {\varepsilon}^{\frac 1 2}||w||$.]{} Using Grönwall’s inequality on (\[wz\_1\]), one gets $$||\partial_{z_1} w||\leq C_{T},$$ where $C_{T}$ is a $O(1)$ constant, independent of ${\varepsilon}$. By induction, we obtain $$||\partial_{\bz}^{\bk}w||\leq C_{T},$$ where $\partial_{\bz}^{\bk}=\partial_{z_1}^{k_1}\cdots \partial_{z_n}^{k_n}$. Therefore, we have shown Theorem \[w\_z\].
Quantum and classical uncertainty {#sec:3.5}
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In this section we briefly discuss about quantum and classical uncertainty, and about the comparison between quantum and classical systems, for small values of the rescaled Planck’s constant. For simplicity, we first consider the case with one degree of freedom, $x\in \mathbb{R}$, and scalar random variable $z$.
Moments and expectations
------------------------
A quantum system is completely determined by its wave function $\psi$. For each realisation of the random variable $z$, the quantum system is described by $\psi(x,t,z)$.
The primary physical quantities of interest include the position density, $$\rho(t,x,z) = |\psi(t,x,z)|^2,$$ and the current density, $$j(t,x,z) = \varepsilon\, \text{Im}\left(\overline\psi(t,x,z)\nabla\psi(t,x,z)\right).$$
Some quantities of interest to look at are the mean and standard deviation in $z$. In this way we can define the means: $$\begin{aligned}
\mathbb{E}[\rho](x,t) & = \int\rho(x,t,z) \pi(z)\, dz, \quad \mathbb{E}[j](x,t) = \int j(x,t,z) \pi(z)\, dz
\label{eq:E1}\end{aligned}$$ and variance: $$\begin{aligned}
\mathbb{Var}[\rho](x,t) & = \mathbb{E}[\rho^2]-\mathbb{E}[\rho]^2, \quad
\mathbb{Var}[j](x,t) = \mathbb{E}[j^2]-\mathbb{E}[j]^2.
\label{eq:V1}\end{aligned}$$ The standard deviation will be computed as the square root of the variance: $$\mathbb{SD}[\rho] = \sqrt{\mathbb{Var}[\rho]}, \quad \mathbb{SD}[j] = \sqrt{\mathbb{Var}[j]}.$$
For quantum systems we denote by $<h> = <\psi|h|\psi>$ the expectation value of observable $h$. Such a quantity will in general be a function of time and $z$. For example $$\begin{aligned}
<q> & = <q>(t,z) = <\psi|\hat{q}|\psi> = \int \bar{\psi}(x,t,z)x\psi(x,t,z)\,dx,\\
<p> & = <p>(t,z) = <\psi|\hat{p}|\psi> = - i\varepsilon \int \bar{\psi}(x,t,z)\psi_x(x,t,z)\,dx,\end{aligned}$$ where $\hat{q} = x\cdot$ and $\hat{p} = -i\varepsilon\frac{\partial}{\partial x}$ denote, respectively, the position and momentum operators when the wave function $\psi$ is in the space representation.
Because of the uncertainty in the parameter $z$, such quantities are random variables. It is possible to compute mean and variance of them as a function of time: $$\begin{aligned}
\mathbb{E}[<h>] &= \int <h>(t,z)\pi(z)\,dz,\\
\mathbb{Var}[<h>] & = \mathbb{E}((<h>-\mathbb{E}(<h>))^2) = \int (<h>-\mathbb{E}[<h>])^2\pi(z)\,dz, \end{aligned}$$ where $h$ denotes, for example, $q$ or $p$.
Notice that the average density $\mathbb{E}[\rho]$ and the average current $\mathbb{E}[j]$ can be used to compute an (ensemble) average particle position and momentum, since the two integration processes commute. However, the same is not true for the variance. As we shall see, it is possible to consider the classical limit of $\mathbb{Var}[<x>]$, while is it hard to define such a limit for $\mathbb{Var}[\rho]$.
Classical limit {#CL}
---------------
In classical mechanics, position and momentum of the particle follow Hamilton’s equation $$\dot{q} = {\frac{\partial{H}}{\partial{p}}}, \quad \dot{p} = - {\frac{\partial{H}}{\partial{q}}},$$ subject to some initial condition $q(0,z) = q_0(z),\>p(0,z) = p_0(z)$. As in the quantum case, the uncertainty can be introduced at the level of the initial condition or at in the potential that defines the Hamiltonian: $$H = \frac{p^2}{2m} + V(q,z),$$ where the random parameter $z$ is distributed with a given density $\pi(z)$.
Position and momentum at a given time are therefore function of $z$ as well: $q = q(t,z), \> p = p(t,z)$. If such a density is known, then the probability distribution function (pdf) of the coordinate $q$ can be found by classical techniques to find pdf of a function of a random variable. One which is commonly adopted in the physics community is given by $${{\mathcal P}}_q(x,t) = \int \delta(x-q(t,z))\pi(z)\,dz,
\label{classic_density}$$ where $\delta$ denotes Dirac’s delta, and the integral has to be interpreted in the usual distributional sense. This representation can be interpreted as follows: for each realization or the random variable, a classical particle can be seen as a singular particle density $$\rho_c(x,t,z) = \delta(x-q(t,z)).$$ The probability distribution is then computed by weighting each value of the parameter with its probability density function, thus obtaining expression (\[classic\_density\]).
Assuming the function $q(t,z)$ is monotone in $z$, the integral can be easily computed by substitution, using the inverse function $z = z(t,q)$, yielding $${{\mathcal P}}_q(x) = \pi(z(t,x)) |\partial z/\partial x|.$$ A suitable generalisation is possible in the case $q(t,z)$ is not monotone: $$\label{pdf_q}
{{\mathcal P}}_q(x) = \sum_{z:q(t,z) = x}\frac{\pi(z)}{|\partial q(t,z)/\partial z|}.$$ Likewise, the mean current density distribution can be computed by smoothing the singular current corresponding to a single realization of the parameter $z$ $$j_c(x,t,z) = \rho_c(x,t,z)p(t,z)$$ by the pdf $\pi(z)$, obtaining $$j_c(x,t) = \int \delta(x-q(t,z))p(t,z)\pi(z)\, dz.$$ Using the same argument, such current distribution can be computed as $$j_c(x,t) = \sum_{z:q(t,z) = x}\frac{\pi(z)}{|\partial q(t,z)/\partial z|}p(t,z).$$
The situation with several degrees of freedom or with multivariate distribution is slightly different. Let us denote by $d$ the number of degrees of freedom, and by $m$ the number of random parameters. If $d=m$ then Equation (\[pdf\_q\]) is still valid by interpreting $|\partial q(t,z)/\partial z|$ as the Jacobian of the transformation between $z$ and $q$. If $d>m$, then in general the pdf will be proportional to a Dirac mass on a manifold of dimension $d-m$. If $d<m$ then in general the pdf will still be a function, which can be computed by integration on a manifold of dimension $m-d$. As an example we mention here the case $d=1$, $m=2$. $$\label{pdf_q2}
{{\mathcal P}}_q(x) = \sum_{\Gamma:q(t,z) = x}\int_\Gamma\frac{\pi(z)}{|\nabla_zq(t,z)|}\, d\Gamma,$$ where the sum is performed on all lines $\Gamma$ such that $q(t,z)=x$.
Sometimes one is not interested in the computation of the space distribution of the particle density or the current density, but just in some moments, such as the mean and the variance. They can be computed as $$\begin{aligned}
\mathbb{E}[q] & = \int q(t,z)\pi(z)\,dz,\quad
\mathbb{E}[p] = \int p(t,z)\pi(z)\,dz,\\
\mathbb{Var}[q] & = \mathbb{E}[q^2]-\mathbb{E}[q]^2, \quad
\mathbb{Var}[p] = \mathbb{E}[p^2]-\mathbb{E}[p]^2.\end{aligned}$$
Numerical Simulations {#sec:4}
=====================
The stochastic collocation method {#sec:4a}
---------------------------------
We now briefly review the gPC method [@XK-02]. In its stochastic Galerkin formulation, the gPC-SG approximation has been successfully applied to many stochastic physical and engineering problems, see for instance an overview [@Xiu; @LK].
On the other hand, the stochastic collocation (SC) method [@XH; @GWZ-14] is known as a popular choice for complex systems with uncertainties when reliable, well-established deterministic solvers exist. It is non-intrusive, so it preserves all features of the deterministic scheme, and easy to parallelize [@Xiu; @NTW-08]. The basic idea is as follows. Let $\{\bz_k\}_{k=1}^{N_z}\subset I_{\bz}$ be the set of collocation nodes, $N_z$ the number of samples. For each fixed individual sample $\bz_k$, $k=1,\ldots,N_z$, one applies the deterministic solver to the deterministic equations as in [@GWPT], obtains the solution ensemble for a general function $f(t,x,\bz)$, $f_k(t,x)=f(t,x,\bz_k)$, then adopts an interpolation approach to construct a gPC approximation $$f(t,x,\bz)=\sum_{k=1}^{N_z}f_k(t,x)l_k(\bz),$$ where $l_k(\bz)$ depends on the construction method. The Lagrange interpolation is used here by choosing $l_k(\bz_i)=\delta_{ik}$. With samples $\{\bz_k\}$ and corresponding weights $\{\nu_k\}$ chosen from the quadrature rule, the integrals in $\bz$ are approximated by $$\label{Int-z} \int_{I_{\bz}}f(t,x,\bz)\pi(\bz)d\bz \approx \sum_{k=1}^{N_z}f_k(t,x)\nu_k,$$ where $$\nu_k = \int_{I_{\bz}}l_k(\bz)\pi(\bz) d\bz.$$ [ Note that in practice Lagrange interpolation is used here only in order to construct the weights of a quadrature formula, once the nodes are assigned. ]{}
Considering the structure complexity of the deterministic solver developed in [@GWPT; @GWPT2], we choose the SC rather than gPC-SG in this project due to its simplicity and efficiency in implementation. In numerical simulation, we need to sample the Gaussian wave packet parameters and the $w$ function, while in solution construction, we are interested in the wave equation as well as physical observables which are nonlinear transforms of the wave equation. In this sense, the stochastic collocation method offers great flexibility in computing averages of various quantities in the random space. We will introduce below how the SC and the GWPT method are combined in our numerical implementation.
Numerical implementation
------------------------
We first briefly review the meshing strategy for the GWPT based method in $x$ and $t$ when the random variables are not present. Recall that the GWPT maps the semi-classical Schrödinger equation to the ODE system for the wave packet parameters and the $w$ equation. In numerical simulation, we denote the time step for the ODE system by $\Delta t_1$, the time step for the $w$ equation by $\Delta t_2$ and the spatial grid size for the $w$ equation by $\Delta \eta$. We introduce a spatial mesh in $\bx$ with grid size $\Delta x$, in the final reconstruction step for $\psi$.
Since the $w$ equation does not produce $O({\varepsilon})$ scaled oscillations, $\Delta t_2$ and $\Delta\eta$ can be chosen independently of ${\varepsilon}$. The phase term in the GWPT is computed by solving the ODE system of the Gaussian wave packet parameters, where the ODE system does not contain stiff terms due to ${\varepsilon}$. However, the numerical error in solving the phase term is magnified by a factor of ${\varepsilon}^{-1}$ when constructing the wave function, thus we often need to take $\Delta t_1$ to be $O({\varepsilon}^{1/k})$, where $k$ denotes the accuracy order of the ODE solver. Finally, $\Delta x$ used in the reconstruction step also needs to be $O({\varepsilon})$ in order to resolve small oscillations in the wave function. The interested readers may refer to [@GWPT; @GWPT2] for a more detailed discussion.
We now discuss the sampling strategy for each step of the numerical implementation in the random space. Three sets of collocation points in $\bz$ are used in our numerical tests:
- number of points to solve the ODEs for the wave packet parameters given by the following system (\[psys\_z\]): $N_{z,1}$,\
At each collocation point ${\bz}_k$ ($k=1, \cdots, N_{z,1}$), we have $$\label{psys_z}
\left\{ \begin{array}{l}
\partial_t q(t,{\bz}_k) = p(t,{\bz}_k),
\\[2pt]\partial_t p(t,{\bz}_k) = -\nabla V(q,{\bz}_k),
\\[2pt] \partial_t\alpha(t,{\bz}_k) = -2 (\alpha(t,{\bz}_k))^2 -\frac{1}{2}\nabla\nabla V(q,{\bz}_k),
\\[2pt] \partial_t\gamma(t,{\bz}_k) =\frac{1}{2}(p(t,{\bz}_k))^2 -V(q,{\bz}_k)+i{\varepsilon}\, {\rm Tr}(\alpha_{R}(t,{\bz}_k)).
\end{array}\right.$$
- number of points to solve the $w$ equation (\[w\_t\]): $N_{z,2}$,
- number of points to reconstruct $\psi$: $N_{z,3}$.
The sets of mesh points $N_{z,1}$, $N_{z,2}$ and $N_{z.3}$ used above are denoted by $M_{1}$, $M_{2}$ and $M_{3}$ respectively. The cardinality of the set $M_j$ is $N_{z,j}$ ($j=1,2,3$). [ How we choose these collocation points depends on the distribution of $\bz$. The correspondence between the type of polynomial chaos and their underlying random variables can be found in [@Xiu]. In our numerical tests, if $\bz$ is uniformly distributed on $[-1,1]$, the Legendre-Gauss quadrature nodes and weights are used; if $\bz$ follows the Gaussian distribution, the Gauss-Hermite quadrature rule is applied. ]{}
[ We solve the ODE system (\[psys\_z\]) by using the fourth-order Runge-Kutta method.]{} After computing the wave packet parameters in (\[psys\_z\]) at the mesh points $M_{1}$ in the $z$ direction, cubic spline interpolation is used to get the values of these parameters at the mesh points $M_{2}$, which prepares us to update $w$ by solving (\[w\_t\]) in time at the same mesh points of $z$, i.e., $M_{2}$. In the reconstruction step, for each ${\bz}_k$, $$\label{eq:reconstruct}
\psi(x, t, {\bz}_k)=\widetilde w(\xi, t, {\bz}_k)\exp\left(i\, (\xi^{T} \alpha_R\, \xi +p^{T}\xi+\gamma)/{\varepsilon}\right),$$ with $\xi=x-q$ and ${\bz}_k\in M_3$. Cubic spline interpolation is used to obtain values of wave packet parameters from $M_{1}$ to $M_{3}$, and values of $w$ from $M_{2}$ to $M_{3}$, which is the reconstruction mesh points. Now we have the values of $\psi=\psi(x, t, {\bz}_k)$ at mesh points $\{{\bz}_k\}$, for $k=1, \cdots, N_{z, 3}$. Finally, in order to plot the solution $\psi$ and its physical quantities of interest at a set of fixed physical location, denoted by $X_0$, cubic spline interpolation is used again. Note that in practice one does not need to reconstruct $\psi$: the observables, such as the expectation values for position and momentum, can be computed directly from $w$. See Section 2.4 in [@GWPT].
We now discuss the sampling strategy in $\bz$ for the GWPT method. Although the parameter system does not have ${\varepsilon}$ oscillations in $\bz$, the numerical error in the parameters are magnified by ${\varepsilon}^{-1}$ when reconstructing the wave function as in . Therefore, one expects that $N_{z,1}$ should depend on ${\varepsilon}$ in order to obtain accurate approximation of the wave function, and the dependence is related to the ODE solver used for the parameter system. Due to the regularity property of the $w$ equation in $\bz$, we expect that we can take ${\varepsilon}$ independent numbers of collocation points, and thus it suffices to take $N_{z,2}=O(1)$. Clearly, we need to take $N_{z,3}=O({\varepsilon}^{-1})$ to resolve the oscillation in $\bz$ in the wave function reconstruction.
We remark that, although in the sampling strategy in $z$ we require $N_{z,1}$ and $N_{z,3}$ to be ${\varepsilon}$ dependent, it does not cause much computational burdens, because the $N_{z,1}$ collocation points are only used to the ODE system and the $N_{z,3}$ collocation points are used in the final step. On the other hand, [*in the most expensive part, solving the $w$ equation in time, we only use ${\varepsilon}$ independent numbers of collocation points. Hence, such sampling strategy is desired for the sake of computational efficiency.*]{}
Finally, considering that certain physical observables, such as position density and flux density can be obtained directly from the $w$ function, then $N_{z,1}$ is expected to be independent of ${\varepsilon}$ if one is only interested to capture the correct physical observables. This argument will be verified numerically in the tests of Section \[sec:numtest\].
### Numerical observables
With the obtained $\psi(t,X,{\bz}_k)$ and the corresponding weights $\{\nu_k\}$ for $k=1, \cdots, N_{z, 3}$, chosen from the quadrature rule, one can approximate the integral given by (\[energy\]), $$\label{E_approx} ||\psi||_{\Gamma}^2 =\int_{I_{\bz}}\int \left|\psi(t, \mathbf x, \bz)\right|^2 d{\mathbf x}\pi(\bz)d\bz
\approx \sum_{k=1}^{N_{{z}, 3}}\sum_{i=1}^{N_x} |\psi(x_i, {\bz}_k)|^2\Delta x\, \nu_k.$$
To be consistent with the $\Gamma$-norm we defined in (\[energy\]), we first denote the $L^1(x)$ norm of $j(t,x,\bz_k)$ by $\widetilde j (t,\bz_k)$ for $k=1, \cdots N_{z,3}$, $$\widetilde j (t,\bz_k) = {\varepsilon}\int \text{Im} \left( \overline\psi(t, x, \bz_k)\nabla\psi(t, x, \bz_k)\right) dx \approx
\sum_{i=1}^{N_x} {\varepsilon}\left( \overline\psi(t, x_i, \bz_k)\nabla\psi(t, x_i, \bz_k)\right) \Delta x,$$ then get $\mathbb {E}(\widetilde j)$, $\mathbb {Var}(\widetilde j)$ and $\mathbb{SD}(\widetilde j)$: $$\label{CD} \mathbb E(\widetilde j)\approx \sum_{k=1}^{N_{{z}, 3}} \widetilde j(t,\bz_k) \nu_k, \qquad
\mathbb {Var}(\widetilde j) = \mathbb E(\widetilde j^2) - (\mathbb E(\widetilde j))^2, \qquad
\mathbb{SD}(\widetilde j) = \sqrt{\mathbb{Var}(\widetilde j)}.$$
### Definition of errors
The error in $\psi$ is computed by comparing $\psi$ with the reference solutions obtained from the second-order direct splitting method, where a sufficiently large number of collocation points $N_{z, 4}$ is used. Denote the set of mesh points by $M_{4}$. The error is measured under the averaged norm (\[energy\]), with the discretized form of the approximation shown in (\[E\_approx\]). More precisely, we use the relative error defined by $$\label{Er_psi} \text{Er}[\psi] = \frac{||\psi_{G} - \psi_{D} ||_{\Gamma}}{||\psi_{D} ||_{\Gamma}},$$ where $\psi_{G}$ represents the solution obtained by the GWPT method, and $\psi_{D}$ represents the one obtained from the direct splitting method. To compute $||\psi_{G} - \psi_{D} ||_{\Gamma}$ in (\[Er\_psi\]), one needs values of both $\psi_{G}(t, X_0, z_j)$ and $\psi_{D}(t, X_0, z_j)$ at the same set of mesh points of $z$, denoted by $M_{5}$ and the number of collocation points $N_{z, 5}$. Thus $$||\psi_{G} - \psi_{D}||_{\Gamma}^2 \approx \sum_{j=1}^{N_{{z}, 5}}\sum_{i=1}^{N_x} |\psi_{G}(x_i, {z}_j) - \psi_{D}(x_i, {z}_j)|^2 \Delta x\, \nu_j.$$ Here one needs to find the interpolated values of $\Psi_{G}$ and $\Psi_{D}$ corresponding to the mesh points $M_{5}$, from mesh points $M_{3}$ and $M_{4}$ respectively. We let $M_5=M_3$ for simplicity. All the interpolation in $z$ space refers to the spline interpolation.
To quantify the errors in mean and standard deviation of the current density, we use $$\label{Er_j}
\text{Er}_1 [j] = \left|\frac{{\mathbb E}(\widetilde j_{G} - \widetilde j_{D})}{{\mathbb E}(\widetilde j_{D})}\right|, \qquad
\text{Er}_2 [j] = \left|\frac{\mathbb {SD}(\widetilde j_{G} - \widetilde j_{D})}{\mathbb{SD}(\widetilde j_{D})}\right|,$$ where $\mathbb{E}$ and $\mathbb{SD}$ are calculated by using (\[CD\]). Note that the mass at each collocation point $\bz_k$ namely $\int |\psi(t,x,\bz_k)|^2 dx$ is a conserved quantity with respect to time and a constant. Thus it is not interesting to compute the relative errors of $\rho$ in terms of the $\Gamma$-norm as that for $j$.
Numerical Tests {#sec:numtest}
---------------
[**Part I: Relation between $N_{z,1}$, $N_{z,2}$, $N_{z,3}$, $N_{z,4}$ and ${\varepsilon}$** ]{}
We know from the deterministic problem in [@GWPT] that all the stiffness in time and space of the original Schrödinger equation for $\psi$ associated with very small values of ${\varepsilon}$ is essentially been removed in the equation for $w(\eta,t)$ by the GWP transform. Since we proved in subsection \[Reg-W\] that $w$ equation is not oscillatory in $z$, all the orders of $z$-derivatives of $w$ have a uniform upper bound, thus the number of collocation points used to solve $w$, namely $N_{z,2}$ is expected to be independent of $\varepsilon$.
In Part I of the numerical tests, we will demonstrate for sufficiently large $N_{z,1}$ and $N_{z,3}$ that are proportional to $1/{\varepsilon}$ and $1/\sqrt{{\varepsilon}}$ respectively, one can choose $N_{z,2}$ [*uniformly*]{} with respect to ${\varepsilon}$.\
We put uncertainty in the potential function in Test (a1)–(a3).
[**Test (a1)**]{}
In this test, we assume $z$ a one-dimensional random variable that follows a uniform distribution on $[-1,1]$. Consider the spatial domain $x\in[-\pi, \pi]$ with periodic boundary conditions. The initial data of $\psi$ is $$\label{Psi-IC} \psi(x,0)=A \exp\bigl[(i/{\varepsilon})\bigl(\alpha_0 (x-q_0)^2 + p_0 (x-q_0)\bigr)\bigr],$$ where $q_0=\pi/2, p_0=0, \alpha_0=i$. We name Test (a1-i) and Test (a1-ii) with different potentials: $$\begin{aligned}
& \text {{\bf Test (a1-i)}} \qquad V(x,z)=(1+0.95z)x^2, \\[4pt]
& \text {{\bf Test (a1-ii)}} \qquad V(x, z) = (1-\cos(x))(1+0.9z). \end{aligned}$$ Let $\Delta t_1=0.01$, $\Delta\eta=0.3125$ in the $w$ equation, and $\Delta t_2=2.5\times 10^{-4}$ in the ODEs for solving the parameters $p$, $q$, $\alpha$, $\gamma$. $\Delta t=1/600$ and $\Delta x=2\pi/9600$ in the reference solutions. $T=1$ in Test (a1)–(a3).
![Test (a1-i), $\varepsilon=1/256$. Plot at $x=-0.1316$, $T=1$. []{data-label="TestI-a4"}](Test1_b2-eps-converted-to.pdf){width="1.0\linewidth"}
In Test (a1-i), we choose the random potential $V(x,z)=(1+0.95z)x^2$. The reason we choose potential function quadratic in $x$ is that the form of exact solution (\[Heller\]) is known in this case. Solutions of $\text{Re}(\psi)$, $\rho$, $j$ as a function of $z$ at some physical location $x$ are plotted in Figure \[TestI-a4\]. The plot indicates that 1) the behaviors of $\psi$ or $z$-derivatives of $\psi$ are much more oscillatory than that of $\rho$ and $j$; 2) regarding the relations between the amplitude of $\text{Re}(\psi)$, $\rho$, $j$ and their first and second order partial $z$-derivatives, numerical results seem to suggest that $\text{Re}(\psi)$ increases by $O(1/{\varepsilon})$ each time we differentiate it, and $\rho$, $j$ tend to increase by $O(1/\sqrt{{\varepsilon}})$ as the order of $z$-derivative increases.\
We now compare the trend of maximum values of the $z$-derivatives of $\psi$ and $w$ with respect to different ${\varepsilon}$:
$\varepsilon$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $\frac{1}{512}$
--------------------------- ---------------- ---------------- ----------------- ----------------- ----------------- --
$\max\partial_z Re(\psi)$ $40.5971$ $96.1058$ $228.0143$ $541.3633$ $1.2843e+03$
$\max\partial_z Re(w)$ $0.0618$ $0.0521$ $0.0439$ $0.0370$ $0.0313$
: Test (a1-ii). Comparison of $\max\partial_z Re(\psi)$ and $\max\partial_z Re(w)$. []{data-label="Comp"}
One can observe from Table \[Comp\] that $\max\partial_z \text{Re}(\psi)$ increase much more rapidly than $\max\partial_z \text{Re}(w)$, the former doubles its values as ${\varepsilon}$ decreases to half smaller, the latter slightly changes its values. This demonstrates that while $\psi$ is highly oscillatory in the random space, $w$ is smooth in $z$, and it guides us to choose the following number of collocation points: take sufficiently large $N_{z, 1}=N_{z, 3}=N_{z,4}=500$ and $N_{z, 2}=32$ in Test (a1)–(a3).
![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_RhoM_Eps512-eps-converted-to.pdf "fig:"){width="0.49\linewidth"} ![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_RhoSD_Eps512-eps-converted-to.pdf "fig:"){width="0.49\linewidth"} ![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_CdM_Eps512-eps-converted-to.pdf "fig:"){width="0.49\linewidth"} ![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_CdSD_Eps512-eps-converted-to.pdf "fig:"){width="0.49\linewidth"} ![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_RhoM_CL-eps-converted-to.pdf "fig:"){width="0.49\linewidth"} ![Test (a1-ii). Comparison of the GWPT, DS and Classical method. ${\varepsilon}=\frac{1}{512}$ for the first two rows. []{data-label="I-a1"}](Test1_CdM_CL-eps-converted-to.pdf "fig:"){width="0.49\linewidth"}
${\varepsilon}$ $N_{z,1}$ $N$
----------------- -------- --------- ----------- -------
$\frac{1}{256}$ $30.2$ $40.6$ $400$ $400$
$\frac{1}{512}$ $44.4$ $123.7$ $600$ $600$
$\frac{1}{640}$ $60.2$ $286.3$ $800$ $800$
$\frac{1}{768}$ $70.6$ $450.5$ $900$ $900$
: Comparison of CPU time (in seconds) of using the GWPT and DS with different ${\varepsilon}$. $\Delta t_1$, $\Delta t_2$, $\Delta\eta$ in GWPT are the same as in Test (a1-i). $N_{z,2}=32$, $N_{z,1}=N_{z,3}=N_{z,4}$ and $\Delta t=1/N$, $\Delta x = 2 \pi/6N$ in the DS method. []{data-label="T0"}
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $2.9968e-05$ $3.1235e-07$ $1.0328e-06$
$\frac{1}{64}$ $3.2442e-05$ $3.6868e-07$ $9.5517e-07$
$ \frac{1}{128}$ $3.1316e-05$ $1.5865e-07$ $4.2628e-07$
$\frac{1}{256}$ $3.7924e-05$ $1.6370e-07$ $3.8290e-07$
: Test (a1-ii). Error of $\psi$ and mean and standard deviation of $j$ with respect to different $\varepsilon$. []{data-label="T1"}
$N_{z,2}$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
----------- ------------------- ------------------ ------------------ -- --
$2$ $0.0013$ $3.8978e-07$ $5.4256e-05$
$4$ $1.7219e-05$ $7.4463e-09$ $3.0955e-06$
$8$ $1.3416e-07$ $6.4828e-09$ $4.1892e-08$
$16$ $7.5591e-09$ $4.5480e-10$ $1.4003e-09$
$32$ $4.9371e-10$ $2.9899e-11$ $8.8615e-11$
: Test (a1-ii), ${\varepsilon}=1/256$. Relative errors for solutions computed by increasing $N_{z,2}$. []{data-label="T2"}
Test (a2)–(a3) use the same initial data of $\psi$ as shown in (\[Psi-IC\]).\
[**Test (a2)**]{} Let $z$ follow the normal Gaussian distribution, with the random potential $$V(x, z) = (1-\cos(x))(1+0.9z).$$
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------
$\frac{1}{32}$ $4.8231e-05$ $2.4561e-07$ $1.8024e-05$
$\frac{1}{64}$ $3.0519e-05$ $2.3575e-07$ $8.7407e-06$
$ \frac{1}{128}$ $3.1520e-05$ $2.3147e-07$ $6.1686e-06$
$\frac{1}{256}$ $3.7955e-05$ $2.1877e-07$ $5.3444e-06$
: Test (a2). Error of $\psi$ and mean and standard deviation of $j$ with respect to different $\varepsilon$. []{data-label="T3"}
[**Test (a3)**]{} We let $$\label{V1} V(x, z) = (1-\cos(x))(1+ \varepsilon z),$$ with $z$ followed the normal Gaussian distribution.
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $2.9898e-05$ $4.9752e-08$ $8.3984e-07$
$\frac{1}{64}$ $3.2074e-05$ $1.4983e-07$ $7.9849e-07$
$ \frac{1}{128}$ $3.1146e-05$ $5.9345e-08$ $3.5599e-07$
$\frac{1}{256}$ $3.7461e-05$ $8.1869e-08$ $3.3017e-07$
: Test (a3). Error of $\psi$ and mean and standard deviation of $j$ with respect to different ${\varepsilon}$. []{data-label="T4"}
In Figure \[I-a1\], we see that solutions of the mean and standard deviation of $\rho$ and $j$ obtained from the GWPT match well with the reference solutions calculated from the time splitting spectral method [@BaoJin]. In the last row of Figure \[I-a1\], we compare the mean of $\rho$ and $j$ obtained from the GWPT for several values of ${\varepsilon}$ (lines) with the mean density and current obtained by classical mechanics (dots). One observes a tendency of convergence of solutions by using the GWPT to the classical case as ${\varepsilon}$ becomes smaller from ${\varepsilon}=\frac{1}{32}$ and ${\varepsilon}=\frac{1}{128}$ to ${\varepsilon}=\frac{1}{512}$. However, it is yet to be specified the precise description of the weak convergence, which remains a difficult question, as discussed at the end of subsection \[Semi-Limit\]. [In Table \[T0\], we compare the CPU time of using the GWPT and the time-splitting spectral method [@BaoJin] for Test (a1-ii) at output time $T=0.3$, with various ${\varepsilon}$ values. The experiment was done using MATLAB R2018b on macOS Mojave system with 2.4 GHz Intel Core i5 processor and 8GB DDR3 memory. One can observe that the computational saving of the GWPT becomes more apparent as ${\varepsilon}$ decreases. The efficiency of the GWPT compared to the commonly used time-splitting spectral method is clearly demonstrated. ]{}
In Tables \[T1\], \[T3\] and \[T4\], one observes a uniform accuracy for $\psi$ and $j$ with respect to different small values of ${\varepsilon}$. Thus we conclude that in order to capture $\psi$ and $j$, ${\varepsilon}$-independent $N_{z,2}$ ($N_{z,2}=32$) can be used for all small ${\varepsilon}$, by putting $N_{z,1}$, $N_{z,3}$ and $N_{z,4}$ sufficiently large. In Table \[T2\], fixing ${\varepsilon}$, we see a fast spectral convergence of relative errors between using $N_{z,2}$ and $2N_{z,2}$ for $\psi$ and $j$, which indicates again that small ${\varepsilon}$-independent $N_{z,2}$ can be chosen to obtain accurate values of $\psi$ and $j$.
In the following Test (b) and Test (c), we let the initial data for $\psi$ depend on the random variable $z$ that follows a uniform distribution on $[-1,1]$. Let $N_{z,1}=N_{z,3}=N_{z,4}=500$, $N_{z,2}=32$ in Test (b) and Test (c).\
[**Test (b)**]{} Let $$V(x) = 1- \cos(x), \qquad \psi(x,0, z)=A \exp\bigl[(i/{\varepsilon})\bigl(\alpha_0 (x-q_0)^2 + p_0(x-q_0)\bigr)\bigr].$$ Here we assume $q_0$ random, $$q_0=\frac{\pi}{2}(1+0.5z), \qquad p_0=0, \qquad \alpha_0=i.$$
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
----------------- ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $1.1114e-04$ $2.5535e-07$ $1.5763e-06$
$\frac{1}{64}$ $3.1764e-05$ $7.8356e-08$ $2.4191e-06$
$\frac{1}{256}$ $5.8151e-05$ $1.5883e-07$ $1.9183e-06$
: Test (b). Error of $\psi$ and mean and standard deviation of $j$ with respect to different $\varepsilon$. $T=1$. []{data-label="I-b"}
[**Test (c)**]{} Let $$V(x) = 1- \cos(x), \qquad \psi(x,0, z)=A \exp\bigl[(i/{\varepsilon})\bigl(\alpha_0 (x-q_0)^2 + p_0(x-q_0)\bigr)\bigr],$$ with $q_0$, $p_0$ depend on $z$, $$q_0=\frac{\pi}{2}(1+0.5z), \qquad p_0=0.5z, \qquad \alpha_0=i.$$
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
----------------- ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $1.6159e-04$ $1.8413e-07$ $5.9809e-07$
$\frac{1}{64}$ $2.5781e-05$ $1.1032e-07$ $1.9014e-07$
$\frac{1}{256}$ $2.5840e-05$ $1.1882e-08$ $1.3267e-07$
: Test (c). Error of $\psi$ and mean and standard deviation of $\rho$ and $j$ with respect to different $\varepsilon$. $T=0.5$. []{data-label="I-c"}
Table \[I-b\] and Table \[I-c\] give the same conclusion as Tests (a1)–(a3), that is, ${\varepsilon}$-independent $N_{z,2}$ (and sufficient large $N_{z,1}$, $N_{z,3}$, $N_{z,4}$) can be chosen to get accurate $\psi$ and $j$.
Now we perform a two-dimensional random variable test:\
[**Test (d)**]{} Let the random potential be $$\label{2D-V}
V(x,z)=(1-\cos(x))(1+0.2 z_1+0.7 z_2),$$ where $z_1$, $z_2$ both follow uniform distributions on $[-1,1]$. The initial data of $\psi$ is given by (\[Psi-IC\]). $\varepsilon=0.1$. Use $N_{z,1}=N_{z,2}=N_{z,3}=N_{z,4}=32$. Solutions and errors at time $T=1$ are shown in Figure \[2D\] below.
![Test (d). Comparison of GWPT with DS. []{data-label="2D"}](RhoM_2D-eps-converted-to.pdf "fig:"){width="0.45\linewidth"} ![Test (d). Comparison of GWPT with DS. []{data-label="2D"}](RhoSD_2D-eps-converted-to.pdf "fig:"){width="0.45\linewidth"} ![Test (d). Comparison of GWPT with DS. []{data-label="2D"}](CdM_2D-eps-converted-to.pdf "fig:"){width="0.45\linewidth"} ![Test (d). Comparison of GWPT with DS. []{data-label="2D"}](CdSD_2D-eps-converted-to.pdf "fig:"){width="0.45\linewidth"}
$\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------- ------------------ ------------------ -- --
$4.2501e-04$ $1.2864e-06$ $1.7532e-06$
Figure \[2D\] shows that we can capture accurately the mean and standard deviation of $\rho$ and $j$ when the random potential has a two-dimensional random variable. Since ${\varepsilon}$ in this test is not so small, $N_{z,1}$, $N_{z,2}$, $N_{z,3}$ and $N_{z,4}$ do not have to be very large.\
[**Part II: Comparison between different perturbations in $V(x,z)$ and an error vs. time plot** ]{}
We compare three different orders of perturbations, using ${\varepsilon}$-independent $N_{z,1}$, $N_{z,2}$, $N_{z,3}$ and ${\varepsilon}$-dependent $N_{z,4}$.
We first introduce Test (a4) and compare it with Test (a2) and Test (a3).\
[**Test (a4)**]{} We assume that $z$ follows the normal Gaussian distribution, and the random potential given by $$\label{V2} V(x, z) = (1-\cos(x))(1+\sqrt{{\varepsilon}}z).$$
If one only needs the information of macroscopic quantities such as the current density, whose errors are measured by $\text{Er}_1[j]$ and $\text{Er}_2[j]$ in (\[Er\_j\]), then the collocation points $N_{z, 1}$, $N_{z, 2}$, $N_{z, 3}$ used in the GWPT method can be chosen independently of small $\varepsilon$.
From Tables \[P1\], \[P2\] and \[P3\] below, in which $N_{z,1}=N_{z,2}=N_{z,3}=32$, $N_{z,4}=500$ and $T=1$. Recall that Test (a2) has a $(1+0.9z)$ perturbation in $V(x,z)$; Test (a3) has a $(1+{\varepsilon}z)$ perturbation in $V(x,z)$, and Test (a4) has a $(1+\sqrt{{\varepsilon}}z)$ perturbation in $V(x,z)$. One observes that errors for the mean and standard deviation of $j$ in the three tests are uniformly small with respect to different values of small ${\varepsilon}$. This is usually not the case for $\psi$, whose errors $\text{Er}[\psi]$ increase for smaller values of $\varepsilon$, because of its increasingly oscillatory behavior. However, the errors of $\psi$ in Table \[P1\] are shown to be much smaller than that in Tables \[P2\] and \[P3\], which indicates that if the random perturbation of the potential is relatively small, e.g., of $O({\varepsilon})$ perturbation in the test of Table \[P1\] compared to $O(\sqrt{{\varepsilon}})$ and $O(1)$ perturbation in the tests of Table \[P2\] and \[P3\] respectively, then ${\varepsilon}$-independent $N_{z,1}$, $N_{z,2}$, $N_{z,3}$ can be used to capture $\psi$ accurately.
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $2.8859e-05$ $9.5081e-08$ $4.1857e-07$
$\frac{1}{64}$ $2.9239e-05$ $8.4372e-09$ $3.7980e-07$
$ \frac{1}{128}$ $3.1145e-05$ $5.9345e-08$ $3.5599e-07$
$\frac{1}{256}$ $3.7461e-05$ $8.1869e-08$ $3.3013e-07$
: Test (a3). Error of $\psi$ and mean and standard deviation of $j$ with respect to different ${\varepsilon}$. []{data-label="P1"}
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $5.0359e-04$ $3.2411e-08$ $5.8941e-07$
$\frac{1}{64}$ $0.0020$ $6.2931e-08$ $5.3575e-07$
$ \frac{1}{128}$ $0.0093$ $1.0983e-07$ $5.0308e-07$
$\frac{1}{256}$ $0.0535$ $1.2743e-07$ $4.5854e-07$
: Test (a4). Error of $\psi$ and mean and standard deviation of $j$ with respect to different $\varepsilon$.[]{data-label="P2"}
$\varepsilon$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
------------------ ------------------- ------------------ ------------------ -- --
$\frac{1}{32}$ $0.5842$ $4.2896e-07$ $1.8105e-05$
$\frac{1}{64}$ $1.9324$ $4.6665e-07$ $8.8690e-06$
$ \frac{1}{128}$ $1.8047$ $4.7258e-07$ $6.4676e-06$
$\frac{1}{256}$ $1.6294$ $4.1936e-07$ $4.6450e-06$
: Test (a2). Error of $\psi$ and mean and standard deviation of $j$ with respect to different ${\varepsilon}$. []{data-label="P3"}
[**part (iv):** ]{} Semi-log error plot vs. time
$T$ $\text{Er}[\psi]$ $\text{Er}_1[j]$ $\text{Er}_2[j]$
----------- ------------------- ------------------ ------------------ -- --
$1$ $3.9592e-05$ $1.9887e-07$ $7.6387e-07$
$2^{1/2}$ $6.4139e-05$ $1.1964e-06$ $3.5285e-06$
$2$ $6.3926e-05$ $6.5872e-07$ $3.9316e-06$
$2^{3/2}$ $4.2088e-04$ $3.9568e-06$ $4.9694e-06$
$4$ $9.0106e-05$ $3.1581e-06$ $2.6140e-06$
$2^{5/2}$ $8.2447e-04$ $1.6763e-06$ $3.6021e-05$
$8$ $0.0037$ $1.1940e-05$ $3.0379e-04$
: Test (a4). Error of $\psi$ and mean and standard deviation of $j$ with respect to different time. ${\varepsilon}=1/128$. []{data-label="Time-T"}
![A semi-log plot of the errors versus time shown in Table \[Time-T\]. []{data-label="Time-P"}](Test_Time3-eps-converted-to.pdf){width="0.6\linewidth"}
In Table \[Time-T\], we compute the errors of $\psi$, and mean and standard deviation of $j$ with respect to different output time, with a corresponding semi-log plot shown in Figure \[Time-P\]. One can observe that the overall trend of all the errors increase as time becomes longer. This trend is similar as that in the counterpart deterministic problem [@GWPT].
Conclusion and future work {#sec:5}
==========================
In this paper, we consider random potential or initial data in the semi-classical Schr[ö]{}dinger equation. Based on the Gaussian wave packet transform numerical method studied for the deterministic problem in [@GWPT], we adopt the stochastic collocation method to numerically compute the Schr[ö]{}dinger equation with random inputs. We analyze how the number of collocation points needed at each step depends on the Planck constant ${\varepsilon}$, according to regularity analyses of the solution $\psi$ and $w$ in the random space. A variety of numerical experiments demonstrate our arguments and efficiency of the proposed numerical method.
In the future, we propose to work on high-dimensional random variable test and develop efficient numerical solvers. Multidimensional physical space was considered in [@GWPT2], and we expect to study the counterpart problem with uncertainties. In [@ZR], the semi-classical Schr[ö]{}dinger equation in the presence of electromagnetic field was reformulated by the Gaussian wave packets transform. It is interesting to consider random electromagnetic field and extend the numerical method studied in [@ZR] to the random case.
[^1]: School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China ([email protected])
[^2]: The Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin, Austin, TX 78712, USA ([email protected])
[^3]: Department of Mathematics and Computer Science, University of Catania, Via A.Doria 6, 95125, Catania, Italy ([email protected])
[^4]: Beijing International Center for Mathematical Research, Peking University, Beijing, China ([email protected])
[^5]: S. Jin was supported by NSF grants DMS-1522184 and DMS-1107291: RNMS KI-Net, and NSFC grants No.31571071 and No.11871297. L. Liu was partially supported by the funding DOE–Simulation Center for Runaway Electron Avoidance and Mitigation, project No.DE-SC0016283. G. Russo was partially funded by the ITN-ETN Marie-Curie Horizon 2020 program ModCompShock, Modeling and computation of shocks and interfaces, project ID: 642768 and NSF grant No.DMS-1115252. Z. Zhou was partially supported by a start-up fund from Peking University and NSFC grant No.11801016.
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---
author:
- 'Frithjof Karsch$^,$'
title: 'Determination of Freeze-out Conditions from Lattice QCD Calculations[^1]'
---
Introduction
============
One of the main motivations for the beam energy scan (BES) at RHIC is to explore the QCD phase diagram at non-vanishing baryon chemical potential and to collect evidence for or against the existence of a critical point at a certain pair ($T,\mu_B$) of temperature ($T$) and baryon chemical potential ($\mu_B$) values. Whether or not a phase transition at a parameter set $(T_{cp},\mu_{cp})$ exists is one of the major uncertainties in our understanding of the QCD phase diagram.
In the vicinity of a critical point various thermodynamic quantities will show large fluctuations. However, even if equilibrated, the hot and dense matter created in a heavy ion collision will expand and cool down. Fluctuations of thermodynamic quantities thus, in general will not be characteristic for a specific ($T,\mu_B$) point in the QCD phase diagram. The situation may, however, be different for fluctuations of conserved charges that freeze-out at $(T_{f},\mu_{f})$ and will not change afterwards. For this reason the analysis of event-by-event fluctuations of baryon number, electric charge, and strangeness as well as their higher order cumulants play a central role in the interpretation of thermal conditions created in the BES at RHIC. They provide unique information about the thermal conditions at the time of chemical freeze-out. In fact, this is quite generally the case and is not only restricted to fluctuations in the vicinity of $(T_{cp},\mu_{cp})$. It also is the case at any point on the freeze-out line mapped in the BES. In particular, the cumulants of fluctuations of conserved charges will also provide information on critical behavior at $\mu_B=0$, if the freeze-out points are close to the ”true” chiral phase transition that exists in QCD for vanishing quark mass values and describes a line $T_c(\mu_B)$ in the phase diagram. Whether or not fluctuation observables will be more sensitive to a possibly existing critical endpoint at $(T_{cp},\mu_{cp})$ or, for instance, the chiral transition at $T_c(\mu_B\simeq 0)$ crucially depends on the proximity of the freeze-out parameters $(T_f,\mu_f)$ to the critical region of the corresponding critical points. In fact, the current determination of freeze-out parameters based on Hadron Resonance Gas (HRG) model calculations [@cleymans] and the determination of the QCD crossover and chiral transition lines, which are known from lattice calculations in leading order $(\mu_B/T)^2$ [@curvature; @Fodor_curv], suggest that freeze-out and transition lines differ more as $\mu_B$ increases. This situation is illustrated in Fig. \[fig:phase\].
![Phase diagram of QCD in the space of temperature, baryon chemical potential and light quark mass (left) and the freeze-out line determined from a comparison of ratios of particle yields and hadron resonance gas model calculations (right). Also shown in the right hand figure are results for the chiral phase transition line calculated in lattice QCD to leading order in the square of the baryon chemical potential [@curvature]. \[fig:phase\]](phase_3d_freeze_4 "fig:"){width="50.00000%"}![Phase diagram of QCD in the space of temperature, baryon chemical potential and light quark mass (left) and the freeze-out line determined from a comparison of ratios of particle yields and hadron resonance gas model calculations (right). Also shown in the right hand figure are results for the chiral phase transition line calculated in lattice QCD to leading order in the square of the baryon chemical potential [@curvature]. \[fig:phase\]](crossover_line "fig:"){width="70.00000%"}
Freeze-out parameter
====================
It is common practice in heavy ion phenomenology to determine the chemical freeze-out parameters and their dependence on beam energy by comparing experimentally measured particle yields with HRG model calculations [@cleymans]. In fact, this approach seems to be quite successful and reliable. It is, however, evident that this approach is conceptually unsatisfactory and must fail, when freeze-out happens close to a critical point in the QCD phase diagram where the dependence of thermodynamics on $T$ and $\mu_B$ is more complex than in a HRG. Clearly one eventually wants to compare experimental observables with theoretical predictions based on (equilibrium) QCD. Extracting information on particle yields at finite $T$ directly from QCD is difficult, if not impossible. However, the experimental measurements of fluctuation observables and their higher order cumulants [@STAR], which all probe thermal conditions at freeze-out, and the improved theoretical calculations of fluctuations of conserved charges in lattice regularized equilibrium QCD thermodynamics [@cheng; @Mukherjee] make it now possible to determine freeze-out conditions directly from QCD. We will outline in the following a determination of $T_f$ and $\mu_f$ at different values of the beam energy. For simplicity we ignore possible, small non-zero values of the electric charge and strangeness chemical potentials. We also will ignore complications that may arise from the limited phase space in which fluctuation observables are being analyzed experimentally. Our point here is a conceptual one! We present this discussion for the case of baryon number fluctuations but will later on generalize it to the case of electric charge fluctuations.
The baryon chemical potential at freeze-out
-------------------------------------------
The $n$-th order cumulants of net baryon number fluctuations, $\chi^{B}_{n}$, can be calculated in lattice QCD at vanishing baryon chemical potential as suitable derivatives of the pressure $p/T^4$. For small, non-zero values of $\mu_B$ this allows then to calculate cumulants from a Taylor series expansion in $\mu_B/T$, $$\chi^{B}_{n,\mu} =
\sum_{k=0}^{\infty} \frac{1}{k!}\chi^{B}_{k+n}(T)
\biggl( {\mu_B \over T}\biggr)^k \;\;\;\;\;\;
{\rm with}\;\;\;\;\;\;
\chi^{B}_{n} =\left. \frac{1}{VT^3}
\frac{\partial^n\ln Z}{\partial(\mu_{B}/T)^n}
\right|_{\mu_B=0} \; .$$ Appropriate ratios of these cumulants are related to shape parameters of the probability distribution of net baryon number, i.e., the mean value $M_B$, variance $\sigma_B$, skewness $S_B$ and kurtosis $\kappa_B$. In particular, one has
$$\frac{\sigma_B^2}{M_B} = \frac{\chi_{2,\mu}^{B}}{\chi_{1,\mu}^{B}} ,
\;\; ~~~~
S_B\sigma_B = \frac{\chi_{3,\mu}^{B}}{\chi_{2,\mu}^{B}} ,\;\; ~~~~
\kappa_B \sigma_B^2 = \frac{\chi_{4,\mu}^{B}}{\chi_{2,\mu}^{B}} \; .$$
Let us consider the Taylor expansion for the simplest even-odd ratio of cumulants, $\chi^{B}_{2,\mu} / \chi^{B}_{1,\mu}$. In next to leading order one finds,
$$\frac{\sigma_B^2}{M_B}\equiv \frac{\chi_{2,\mu}^{B}}{\chi_{1,\mu}^{B}} =
\frac{T}{\mu_{B}} \left[\frac{1+
\frac{1}{2}\frac{\chi_{4}^{B}}{\chi_{2}^{B}}(\mu_B/T)^2
+...}{1+
\frac{1}{6}\frac{\chi_{4}^{B}}{\chi_{2}^{B}}(\mu_B/T)^2+...}
\right] \; .$$
A similar relation holds for $\chi^{B}_{3,\mu} / \chi^{B}_{2,\mu}$. To leading order the ratios of even and odd cumulants thus determine directly the ratio of $\mu_B$ and $T$ at the time of freeze-out, $\sigma_B^2/M_B=(T_f/\mu_f)(1+{\cal O}((\mu_f/T_f)^2))$. The coefficient of the next-to-leading order correction is small for all temperatures; current lattice QCD calculations suggest $\chi_{4}^{B}/\chi_{2}^{B} < 1.5$ for all temperatures. Therefore, the systematic errors that arise from ignoring this correction also remains small for a broad range of beam energies covered in the BES at RHIC. In fact, the systematic error is at most 2% at $\sqrt{s_{NN}}=200$ GeV and rises to about 20% at $\sqrt{s_{NN}}=39$ GeV.
[*Even-odd ratios of cumulants are good observables to determine the value of the\
baryon chemical potential at freeze-out.*]{}
We give results for $\mu_{f}/T_f$ based on measurements of $\chi_{2,\mu}^{B}/\chi_{1,\mu}^{B}$ by the STAR collaboration [@STAR] in Table 1. These compare quite well with HRG model calculations.
------ ----------------------------- -------------- -------------
STAR QCD HRG
$\chi_P^{(2)}/\chi_P^{(1)}$ $\mu_f/T_f$ $\mu_f/T_f$
200 5.3(9) 0.190(30)(4) 0.183
63.4 2.35(42) 0.43(8)(3) 0.43
------ ----------------------------- -------------- -------------
: The ratio of baryon chemical potential and temperature at freeze-out determined from measurements of the ratio of squared variance and mean value of net proton number fluctuations by comparing to lattice QCD calculations of corresponding cumulants of net baryon number fluctuations (third column). Results are given for the two largest values of the beam energy scan. The second error in the third column gives an estimate for the systematic error that arises from neglecting next-to-leading order corrections in the Taylor expansion (Eq. (3)). The last column gives the result for $\mu_f/T_f$ obtained by comparing measured particle yields with HRG model calculations.
The freeze-out temperature
--------------------------
While the ratio of even-odd cumulants is most sensitive to the baryon chemical potential, the ratio of even-even cumulants is, at leading order, determined only by $T_f$. For small values of the baryon chemical potential a low order Taylor series thus again is sufficient. E.g., one finds for the ratio of fourth and second order cumulants, $$\kappa_B \sigma_B^2 \equiv \frac{\chi_{4,\mu}^{B}}{\chi_{2,\mu}^{B}} =
\frac{\chi_{4}^{B}(T)}{\chi_{2}^{B}(T)} \left[\frac{1+
\frac{1}{2}\frac{\chi_{6}^{B}(T)}{\chi_{4}^{B}(T)}(\mu_B/T)^2
+...}{1+
\frac{1}{2}\frac{\chi_{4}^{B}(T)}{\chi_{2}^{B}(T)}(\mu_B/T)^2+...}
\right] \; ,$$ where we explicitly point out the $T$-dependence of cumulants at $\mu_B=0$. A potential difficulty in the determination of the freeze-out temperature from measurements of $\kappa_B\sigma_B^2=\chi_{4,\mu}^{B}/ \chi_{2,\mu}^{B}$ is that lattice QCD calculations [@cheng] suggest that this quantity varies rapidly only in the crossover region but shows little variation at low temperature where it stays close to unity. The discretization errors inherent in these calculations are, however, still too large to allow a direct comparison with experimental data. This will change when improved calculations with a better fermion discretization scheme and closer to the continuum limit will be completed [@Mukherjee].
Freeze-out conditions from electric charge fluctuations
=======================================================
The discussion presented in the previous section carries over to fluctuations of other conserved charges, e.g. electric charge or strangeness. The former is of particular interest, as it may soon be accessible experimentally. It will also avoid the problems that arise from the fact that the conserved net baryon number is not accessible directly in a heavy ion experiment. What is measured instead is the fluctuation of proton number, which may change even after freeze-out [@Asakawa]. Also theoretically, electric charge fluctuations allow a more concise determination of freeze-out parameters as $\kappa_Q\sigma_Q^2$ shows a characteristic variation with temperature also in the hadronic phase [@cheng].
[*A measurement of $\chi_{4,\mu}^{Q}/ \chi_{2,\mu}^{Q}$ will allow to determine the freeze-out temperature.*]{}
More precisely, it determines a line $T_f(\mu_B)$ in the QCD phase diagram; the additional measurement of an even-odd cumulant ratio will then fix $\mu_B\equiv\mu_f$ and calculations of further ratios will provide consistency checks. In Fig. \[fig:chi42\](left) we show results for the quadratic fluctuations of net electric charge and compare this to HRG model calculations [@hotQCD]. Preliminary results for $\chi_{4,\mu}^{Q}/ \chi_{2,\mu}^{Q}$ [@BNLBI] are shown in Fig. \[fig:chi42\](right). We stress that the latter require a careful cut-off analysis, which already for quadratic electric charge fluctuations is difficult [@hotQCD].
Fig. \[fig:chi42\] shows that QCD results for quadratic electric charge fluctuations are consistent with HRG model calculations only for temperatures $T\lsim 160$ MeV. Preliminary results for the quartic fluctuations [@BNLBI] suggest that the ratio $\chi_{4}^{Q}/ \chi_{2}^{Q}$ differs strongly from HRG results at $T\simeq 160$ MeV, i.e. $\chi_{4}^{Q}/ \chi_{2}^{Q})_{HRG}\simeq 1.7$, and is less than unity.
Conclusion
==========
Even-odd ratios of cumulants of conserved charge fluctuations allow to determine the baryon chemical potential at freeze-out. A more precise experimental determination of $\kappa_B\sigma_B^2$ as well as $\kappa_Q\sigma_Q^2$ and improved lattice QCD results for these observables will not only allow to determine the freeze-out temperature $T_f$, it will also provide information on the deviation of the freeze-out line $T_f(\mu_B)$ from the crossover line $T_{pc}(\mu_B)$ and the chiral phase transition line $T_{c}(\mu_B)$.
![Quadratic fluctuations of net electric charge at $\mu_B=0$ (left) [@hotQCD] and preliminary results for the ratio of quartic to quadratic fluctuations [@BNLBI]. Results are compared to HRG model calculations at low temperatures (HRG) and an ideal quark gas (SB) at high temperature. For comparison, in a HRG at $T\simeq 160$ MeV, one has $\chi_4^Q/\chi_2^Q\simeq 1.7$. \[fig:chi42\]](chiQ_fK1_exp_fit "fig:"){width="70.00000%"} ![Quadratic fluctuations of net electric charge at $\mu_B=0$ (left) [@hotQCD] and preliminary results for the ratio of quartic to quadratic fluctuations [@BNLBI]. Results are compared to HRG model calculations at low temperatures (HRG) and an ideal quark gas (SB) at high temperature. For comparison, in a HRG at $T\simeq 160$ MeV, one has $\chi_4^Q/\chi_2^Q\simeq 1.7$. \[fig:chi42\]](chiQ4_Q2_highT "fig:"){width="70.00000%"}
[99]{} J. Cleymans, H. Oeschler, K. Redlich and S. Wheaton, Phys. Rev. C [**73**]{}, 034905 (2006). O. Kaczmarek [*et al.*]{}, Phys. Rev. [**D83**]{}, 014504 (2011). G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, JHEP [**1104**]{}, 001 (2011), \[arXiv:1102.1356 \[hep-lat\]\] M. M. Aggarwal [*et al.*]{} (STAR Collaboration), Phys. Rev. Lett. [**105**]{}, 22302 (2010). M. Cheng [*et al.*]{}, Phys. Rev. [**D79**]{}, 074505 (2009). for a recent overview see: S. Mukherjee, J. Phys. G G [**38**]{}, 124022 (2011), \[arXiv:1107.0765 \[nucl-th\]\]. A. Bazavov [*et al.*]{} (HotQCD Collaboration), in preparation. BNL-Bielefeld Collaboration, in preparation. M. Kitazawa and M. Asakawa, arXiv:1107.2755 \[nucl-th\].
[^1]: presented at the International Conference “Critical Point and Onset of Deconfinement - CPOD 2011”, Wuhan, November 7-11, 2011;\
This work has been supported in part by contracts DE-AC02-98CH10886 with the U.S. Department of Energy and the Bundesministerium für Bildung und Forschung under grant 06BI9001.
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---
abstract: 'There is an equation relating numbers of curves on ${\mathbb{F}}_0={\mathbb{P}}^1\times{\mathbb{P}}^1$ satisfying incidence conditions and numbers of curves on ${\mathbb{F}}_2$ satisfying incidence conditions. The purpose of this paper is to give a tropical proof of this equation (for some cases). We use two tropical methods. The first method proves the formula for rational curves. We use induction on the degree and two Kontsevich-type formulas for curves on ${\mathbb{F}}_0$ and on ${\mathbb{F}}_2$. The formula for ${\mathbb{F}}_2$ was not known before and is proved using tropical geometry. The second method proves the formula for small degree and any positive genus and uses lattice paths.'
address:
- ' Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany'
- 'CRCG, Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, D-37073 Göttingen, Germany'
author:
- Marina Franz and Hannah Markwig
title: 'Tropical enumerative invariants of ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$'
---
Introduction
============
In tropical geometry, algebraic curves are replaced by certain balanced piece-wise linear graphs called tropical curves. Tropical geometry has gained lots of attention recently. One of the interesting results is that we can determine numbers of algebraic curves on toric surfaces satisfying incidence conditions by counting the corresponding tropical curves instead (Mikhalkin’s Correspondence Theorem, see [@Mi03]). This is true in particular for the toric surfaces ${\mathbb{F}}_0={\mathbb{P}}^1\times {\mathbb{P}}^1$ and ${\mathbb{F}}_2$. The tropical numbers can be determined using certain lattice paths in the polygon dual to the toric surface (see [@Mi03], theorem 2).
Gromov-Witten invariants can be thought of as “virtual” solutions to enumerative problems. They are deformation invariants, thus they coincide for the two surfaces ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$. For ${\mathbb{F}}_0$, Gromov-Witten invariants are enumerative, i.e. they count curves on ${\mathbb{F}}_0$ satisfying incidence conditions. For ${\mathbb{F}}_2$, they are not, but it is known how they are related to enumerative numbers. Therefore there is an equation relating the enumerative numbers of ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$.
The purpose of this paper is to give a tropical proof of this equation (for some cases), using Mikhalkin’s Correspondence Theorem.
Let us introduce this equation in more details. Let $C$ denote the class of a general section of ${\mathbb{F}}_2$ and $F$ the class of the fiber of ruling. Then the Picard group of ${\mathbb{F}}_2$ is generated by $C$ and $F$. The exceptional curve is linearly equivalent to $C-2F$. The Picard group of ${\mathbb{F}}_0$ is generated by two fibers of ruling which we will denote by $C$ and $F$ as well. We can degenerate ${\mathbb{F}}_2$ to ${\mathbb{F}}_0$ such that the class $aC+bF$ on ${\mathbb{F}}_2$ degenerates to the class $aC+(a+b)F$ on ${\mathbb{F}}_0$. Then for nonnegative $a$, $b$ with $a+b\geq 1$ we have
$$\label{eq1}
{N^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} = \sum_{k=0}^{a-1}{\binom{b+2k}{k}{N^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a-k$},\emph{$b+2k$})}}.$$
where ${N^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ and ${N^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ denote the numbers of nodal irreducible curves of genus $g$ of class $aC+(a+b)F$ in ${\mathbb{F}}_0$ (resp. of class $aC+bF$ in ${\mathbb{F}}_2$) through $4a+2b+g-1$ points in general position. (See [@AB01], theorem 3.1.1, for rational curves and [@Vak00b], section 8.3, for arbitrary genus.)
Let us now introduce the analogous tropical numbers. The polygon corresponding to the divisor class $aC+(a+b)F$ on ${\mathbb{F}}_0$ is a rectangle with vertices $(0,0)$, $(a,0)$, $(a,a+b)$ and $(0,a+b)$, the polygon corresponding to the divisor class $aC+bF$ on ${\mathbb{F}}_{2}$ is a quadrangle with vertices $(0,0)$, $(a,0)$, $(a,b)$ and $(0,2a+b)$.
We consider plane tropical curves dual to these polygons. Thus, we consider plane tropical curves of degree $\Delta_{{\mathbb{F}}_{0}}(a,a+b)$ and $\Delta_{{\mathbb{F}}_{2}}(a,b)$, where $\Delta_{{\mathbb{F}}_{0}}(a,a+b)$ denotes the multiset of the vectors $(-1,0)$ and $(1,0)$ each $a+b$ times and $(0,-1)$ and $(0,1)$ each $a$ times and $\Delta_{{\mathbb{F}}_{2}}(a,b)$ denotes the multiset of the vectors $(-1,0)$ $2a+b$ times, $(0,-1)$ $a$ times, $(1,0)$ $b$ times and $(2,1)$ $a$ times. We denote by ${\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ (resp. ${\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$) the number of irreducible plane *tropical* curves of degree $\Delta_{{\mathbb{F}}_{0}}(a,a+b)$ (resp. $\Delta_{{\mathbb{F}}_{2}}(a,b)$) and genus $g$ through $4a+2b+g-1$ points in general position (see [@Mi03]).
Our central result is the following theorem:
\[thm-main\] The following equation holds for
- nonnegative integers $a$, $b$ with $a+b\geq 1$ and $g=0$, and for
- $0\leq a\leq 2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$:
$$\label{eq1trop}
{\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} = \sum_{k=0}^{a-1}{\binom{b+2k}{k}{\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a-k$},\emph{$b+2k$})}}.$$
Of course this theorem (and even the more general case) is an immediate consequence of equation \[eq1\] and Mikhalkin’s Correspondence theorem which states that ${N^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}={\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ and ${N^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}={\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$. However, we want to give a proof within tropical geometry.
We use two different tropical methods to prove theorem \[thm-main\].
To prove the statement for nonnegative $a$, $b$ with $a+b\geq 1$ and $g=0$, we use induction on the degree and generalizations of Kontsevich’s formula for enumerative numbers on ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$. While Kontsevich’s formula for ${\mathbb{F}}_0$ (see theorem \[kontsevich0\]) was known and can be proved without tropical geometry ([@FP97], section 9), our formula for ${\mathbb{F}}_2$ (see theorem \[kontsevich2\]) is new and was derived using tropical geometry. To derive a Kontsevich-type formula tropically, we compute numbers of curves satisfying point and line conditions and mapping to a special point in tropical $\mathcal{M}_{0,4}$ under the forgetful map. To prove Kontsevich’s formula for ${\mathbb{P}}_2$, one can show that all such curves have a contracted bounded edge and can thus be interpreted as reducible tropical curves ([@GM053]). For ${\mathbb{F}}_2$, this statement is no longer true. Instead, we get a correction-term corresponding to curves that do not have a contracted bounded edge. We show that these curves can also be interpreted as reducible curves in a different way.
To prove theorem \[thm-main\] for $0\leq a\leq 2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$, we use Mikhalkin’s lattice path algorithm to count tropical curves (see theorem 2 of [@Mi03]) and observations about those lattice paths from [@GM052].
Unfortunately, it seems that none of the above methods can be generalized to other cases easily.
The paper is organized as follows. In section \[sec-1\], we prove our tropical Kontsevich formulas for ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$. In section \[sec-2\], we use those formulas to prove theorem \[thm-main\] for $a$, $b$ with $a+b\geq 1$ and $g=0$ using induction. In section \[sec-3\], we prove theorem \[thm-main\] for $0\leq a\leq 2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$ using lattice paths.
We would like to thank Ionut Ciocan-Fontanine and Andreas Gathmann for helpful discussions and Flavia Stan for help with the Mathematica package MultiSum.
Tropical Kontsevich formulas for ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$ {#sec-1}
======================================================================
To derive tropical Kontsevich formulas for ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$, we generalize the ideas of [@GM053]. Let us start by recalling some notations we will use.
\[not\] Let $\Delta=\Delta_{{\mathbb{F}}_{2}}(a,b)$ and let ${\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ denote the space of rational parametrized tropical curves in $\R^2$ of degree $\Delta$, with $\#\Delta+n$ ends all of which are labelled, and $n$ of which are contracted ends (see definition 4.1 of [@GKM07]). Let $$\operatorname{f\/t}:{\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)\rightarrow {\mathcal{M}}_{0,4}$$ denote the forgetful map which forgets all ends but the first $4$ contracted ends (see definition 4.1 of [@GM053]), and $$\operatorname{ev}_i:{\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)\rightarrow \R^2$$ the evaluation at the contracted end labelled $i$ (see definition 3.3 of [@GM053]). Pick two rational functions $\varphi_{A}$ and $\varphi_{B}$ on ${\mathcal{M}}_{0,4}$ (in the sense of [@AR07], definition 3.1) that correspond to abstract tropical curves $\lambda_A$ (resp. $\lambda_B$) where the ends $x_1$ and $x_2$ come together at a vertex (resp. where $x_1$ and $x_3$ come together) and where the length parameter of the bounded edge is very large.
Note that we use the space of parametrized tropical curves with labelled ends here. The reason is that one can show that this space is a tropical fan (proposition 4.7 of [@GKM07]) and that we can thus use the tropical intersection theory from [@AR07]. Since we want to count tropical curves without the labels of the non-contracted ends, we have to divide by a factor of $|G|$, where $G$ is the group of possible permutations of the labels. In the Kontsevich formula we want to prove (theorem \[kontsevich2\]), we sum over all possibilities to split the degree $\Delta=\Delta_{{\mathbb{F}}_{2}}(a,b)$ into two smaller degrees. To be precise, we would have to sum over all possibilities to pick a *labelled* subset of non-contracted ends forming the smaller degrees. This factor together with the factors for labelling the ends in the small degrees exactly cancel with the total factor of $|G|$. In the following, we will therefore neglect the fact that non-contracted ends are labelled.
The difference of $\varphi_{A}$ and $\varphi_{B}$ is globally given by a bounded rational function on ${\mathcal{M}}_{0,4}$. Therefore, the tropical Cartier divisors $[\varphi_{A}]$ and $[\varphi_{B}]$ are rationally equivalent and by lemma 8.5 of [@AR07] their pull-backs $[\operatorname{f\/t}^{\ast}\varphi_{A}]\cdot {\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ and $[\operatorname{f\/t}^{\ast}\varphi_{B}]\cdot{\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ are rationally equivalent as well.
Set $n=\# \Delta$ and choose rational functions $\varphi_{1}$, $\varphi_{2}$, ${\varphi_{3}}_{1}, {\varphi_{3}}_{2}$ $\ldots$, ${\varphi_{n}}_{1}, {\varphi_{n}}_{2}$ on $\R^{2}$ that correspond to tropical curves $L_{1}$ and $L_{2}$ of degree $\Delta_{{\mathbb{F}}_{2}}(1,0)$ and to points $p_{3}, \ldots, p_{n} \in \R^{2}$ in general position. We can set $\varphi_{1}=\max\{x-{p_{1}}_{1},2(y-{p_{1}}_{2}),0\} $ and $\varphi_{2}=\max\{x-{p_{2}}_{1},2(y-{p_{2}}_{2}),0\}$ to get $L_1$ and $L_2$ in this case.
Because of the above, we have $$\label{eqdeg}
\begin{split}
&\deg([\operatorname{ev}_{1}^{\ast}\varphi_{1}\cdot\operatorname{ev}_{2}^{\ast}\varphi_{2}\cdot\prod_{i=3}^n(\operatorname{ev}_{i}^{\ast}{\varphi_{i}}_{1}\cdot\operatorname{ev}_{i}^{\ast}{\varphi_{i}}_{2})\cdot\operatorname{f\/t}^{\ast}\varphi_{A}\cdot{\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)]) \\
=&\deg([\operatorname{ev}_{1}^{\ast}\varphi_{1}\cdot\operatorname{ev}_{2}^{\ast}\varphi_{2}\cdot\prod_{i=3}^n(\operatorname{ev}_{i}^{\ast}{\varphi_{i}}_{1}\cdot\operatorname{ev}_{i}^{\ast}{\varphi_{i}}_{2})\cdot\operatorname{f\/t}^{\ast}\varphi_{B}\cdot{\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta))].
\end{split}$$
\[rem-generalconditions\] Both above expressions are $0$-dimensional tropical intersection products as defined in [@AR07], even if the set-theoretical intersection is higher-dimensional. If we pick the conditions to be general however, the set-theoretical intersection equals the support of the intersection product. That means that the intersection products above equal the sums of tropical curves in ${\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ that satisfy the conditions, i.e. that pass through $L_1$, $L_2$, $p_3,\ldots,p_n$ and map to $\lambda_A$ resp. $\lambda_{B} \in {\mathcal{M}}_{0,4}$ under $\operatorname{f\/t}$, counted with multiplicity. This can be shown analogously to [@MR08], lemma 3.1.
The following lemma will enable us to compute the multiplicity with which we have to count each curve satisfying the conditions in the intersection product:
\[lem-determinante\] Let $X$ be an abstract tropical variety (in the sense of [@AR07], definition 5.12) of dimension $k$ and $\varphi_{1}$, …, $\varphi_{k}$ rational functions on $X$. Moreover, let $P \in X$ be a point in the interior of a cone $\sigma$ of maximal dimension in $X$. Assume that $\varphi_{i}$ is of the form $\varphi = \max\{ \psi_{i} , \chi_{i} \}$ locally around $P$, where $\psi_{i}, \chi_{i}: X \rightarrow \R$ denote ${\mathbb{Z}}$-affine functions with $\psi_{i}(P)=\chi_{i}(P)$. Let $(\psi_{i}-\chi_{i})_L$ denote the linear part of the affine function $(\psi_{i}-\chi_{i})$ and let $A$ be the $(k \times k)$-matrix with entries $((\psi_{i}-\chi_{i})_L(u_{j}))_{i,j}$ for a basis $u_{1}, \ldots, u_{k}$ of the lattice underlying $X$ at $\sigma$. Then the coefficient of $P$ in the intersection product $\varphi_{1}\cdot\ldots\cdot\varphi_{k}\cdot X$ is equal to $\omega(\sigma)\cdot |\det(A)|$.
The computation of the coefficient of $P$ in the intersection product is local around $P$. Thus, we may assume that $X$ is a tropical fan in some vector space $V$ and extend $\sigma$ to the affine vector space $V_{\sigma}$ spanned by $\sigma$. Furthermore, we may consider the rational functions $\varphi_{i}=\max\{\psi_{i}, \chi_{i}\}$ on the whole space $V_{\sigma}$. Moreover, we may replace the rational functions $\max\{\psi_{i},\chi_{i}\}$ by $\max\{\psi_{i},\chi_{i}\}-\chi_{i}=\max\{\psi_{i}-\chi_{i},0\}$ as changing a rational function by a linear function does not affect the intersection product. We define the morphism $g = (\psi_{1}-\chi_{1}, \ldots, \psi_{k}-\chi_{k}): X \rightarrow \R^{k}$. Then we have $\varphi_{i}=g^{\ast}\mu_{i}$ for $\mu_{i}:\R^{k} \rightarrow \R; (a_{1}, \ldots, a_{k}) \mapsto \max\{a_{i},0\}$ and for all $1 \leq i \leq k$. By the projection formula ([@AR07], proposition 4.8) the multiplicity of $P \in X$ in the intersection product $\varphi_{1} \cdot \ldots \cdot \varphi_{k} \cdot \sigma = g^{\ast}\mu_{1} \cdot \ldots \cdot g^{\ast}\mu_{k} \cdot \sigma$ is equal to the multiplicity of $0$ in $\R^{k}$ in the intersection product $g_{\ast}(g^{\ast}\mu_{1} \cdot \ldots \cdot g^{\ast}\mu_{k} \cdot \sigma) = \mu_{1} \cdot \ldots \mu_{k} \cdot g_{\ast}\sigma$. For dimensional reasons the cycle $g_{\ast}\sigma$ is the whole target space $\R^{k}$ with some weight. But this weight is $\omega(\sigma)\cdot|\det(A)|$. Note $\mu_{1} \cdot \ldots \cdot \mu_{k} \cdot \R^{k}$ is the origin with weight $1$. This finishes the proof.
\[rem-choosecoor\] If $\sigma$ is a cone in $\mathcal{M}^{\text{lab}}_{0,n}(\R^2,\Delta)$, it corresponds to a combinatorial type, i.e. a homeomorphism class of a graph plus direction vectors for all edges (see [@GM053], 2.9). We can deform a parametrized tropical curve $(\Gamma,h,x_i)$ within $\sigma$ by changing the length of the bounded edges or translating the image $h(\Gamma)$. Thus a basis for the lattice underlying $\mathcal{M}^{\text{lab}}_{0,n}(\R^2,\Delta)$ at $\sigma$ is given by the position of a root vertex $h(V)$ and the length of all bounded edges. By remark 3.2 of [@GM053], the absolute value of the determinant of the matrix $A$ from lemma \[lem-determinante\] above is independent from the choice of a root vertex and an order of the bounded edges.
\[ex-det\] Assume $\sigma$ is the cone in $\mathcal{M}^{\text{lab}}_{0,4}(\R^2,\Delta_{{\mathbb{F}}_2}(1,0))$ corresponding to the combinatorial type pictured below.
Following remark \[rem-choosecoor\], we choose the position of $h(x_1)$ and the lengths $l_1,\ldots,l_5$ of the bounded edges as coordinates for $\sigma$.
The following curve $C$ inside $\sigma$ (where $h(x_1)=(0,0)$, $l_1=2$, $l_2=\frac{1}{2}$ and $l_3=l_4=l_5=1$) goes through the points $P_1$ (which is cut out by $\max\{x,0\}$ and $\max\{y,0\}$) and $P_2$ (cut out by $\max\{x,1\}$ and $\max\{y,-2\}$) and through $L_1$ (cut out by $\max\{x-3,2y-2,0\}$) and $L_2$ (cut out by $\max\{x+1,2y+3,0\}$) and maps to the abstract tropical curve with $x_1$ and $x_3$ at one vertex and length parameter $1$ under $\operatorname{f\/t}$. Denote by $\lambda$ a rational function on $\mathcal{M}_{0,4}$ that cuts out this curve.
Because $C$ satisfies the conditions, it contributes to the intersection product $$\begin{aligned}
&\operatorname{ev}_1^\ast(\max\{x,0\})\cdot\operatorname{ev}_1^\ast(\max\{y,0\}) \operatorname{ev}_2 ^\ast(\max\{x,1\})\cdot \operatorname{ev}_2^\ast(\max\{y,-2\})\\ \cdot & \operatorname{ev}_3^\ast(\max\{x-3,2y-2,0\})\cdot \operatorname{ev}_4^\ast(\max\{x+1,2y+3,0\})\\\cdot& \operatorname{f\/t}^\ast(\lambda)\cdot\mathcal{M}_{0,4}^{\text{lab}}(\R^2,\Delta_{{\mathbb{F}}_2}(1,0)). \end{aligned}$$ Let us compute the multiplicity with which it contributes using lemma \[lem-determinante\]. Locally at $h(x_3)$, the function $\max\{x-3,2y-2,0\}$ equals $\max\{2y-2,0\}$ and locally at $h(x_4)$, the function $\max\{x+1,2y+3,0\}$ equals $\max\{x+1,2y+3\}$. Hence locally we have $$\begin{aligned}
\operatorname{ev}_3^\ast(\max\{x-3,2y-2,0\})&=\max\{2h(x_3)_y-2,0\} \mbox{ and }\\ \operatorname{ev}_4^\ast(\max\{x+1,2y+3,0\})&=\max\{h(x_4)_x+1,2h(x_4)_y+3\}.\end{aligned}$$ We can rewrite the pullbacks along $\operatorname{ev}_1$ and $\operatorname{ev}_2$ analogously. Locally, $\operatorname{f\/t}$ equals the map that sends a curve in $\sigma$ with coordinates $(h(x_1),l_1,\ldots,l_5)$ to $l_3$. We also have to write the linear part of the evaluation pullbacks in the basis of $\sigma$, i.e. in $h(x_1),l_1,\ldots,l_5$. For this, note first that $$\begin{aligned}
h(x_2)&=h(x_1)+l_1\cdot \binom{1}{0}+l_3\cdot\binom{-1}{-1}+l_5\cdot\binom{0}{-1},\\
h(x_3)&=h(x_1)+l_1\cdot\binom{1}{0}+l_2\cdot\binom{2}{1}\mbox{ and }\\
h(x_4)&=h(x_1)+l_1\cdot \binom{1}{0}+l_3\cdot\binom{-1}{-1}+l_4\binom{-1}{0}.\end{aligned}$$ Thus, the linear part of $2h(x_3)_y-2$ equals $$2h(x_1)_y+2l_2$$ and the linear part of $h(x_4)_x+1-2h(x_4)_y-3$ equals $$h(x_1)_x+l_1-l_3-l_4-2(h(x_1)_y-l_3)=h(x_1)_x-2h(x_1)_y+l_1+l_3-l_4.$$ So, if we plug for example the vector which has a $1$ at $l_2$ and $0$ everywhere else into the linear part of $2h(x_3)_y-2$, we get $2$. If we plug the vector which has a $1$ at $l_4$ and $0$ everywhere else into the linear part of $h(x_4)_x+1-2h(x_4)_y-3$, we get $-1$. Continuing like this, we can see that the matrix $A$ equals: $$\begin {array}{l|ccccccc}
&h(x_1)_x&h(x_1)_y& l_1&l_2&l_3&l_4&l_5\\ \hline
(P_1)_x &1&0&0&0&0&0&0\\
(P_1)_y &0&1&0&0&0&0&0\\
(P_2)_x &1&0&1&0&-1&0&0\\
(P_2)_y &0&1&0&0&-1&0&-1\\
L_1&0&2&0&2&0&0&0\\
L_2&1&-2&1&0&1&-1&0\\
\operatorname{f\/t}&0&0&0&0&1&0&0
\end {array}$$
Since $ |\det(A)|=2$, the curve $C$ contributes with multiplicity $2$ to the intersection product above.
\[def-mult\] Let $C$ be a curve contributing to a $0$-dimensional intersection product as in example \[ex-det\] consisting of evaluation pullbacks and the pullback of a curve in $\mathcal{M}_{0,4}$ under $\operatorname{f\/t}$ (resp. only evaluation pullbacks). Then we denote by $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$ (resp. $\mult_{\operatorname{ev}}(C)$) the multiplicity with which $C$ contributes to the intersection product, which equals the absolute value of the determinant of the linear part of the combined evaluation and forgetful maps, as we have seen in \[lem-determinante\].
\[rem-detmult\] Note that by [@GM053], proposition 3.8, $\mult_{\operatorname{ev}}(C)$ equals the usual multiplicity of a tropical curve as defined in [@Mi03], 4.15, i.e. the multiplicity with which it contributes to the count of ${\mathcal{N}^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$.
In the following, we want to describe both sides of equation \[eqdeg\] in detail. We want to study the set of curves that satisfy the conditions, and their multiplicity. We will see that we can interpret the curves as reducible curves, and count the contributions from each component separately. This will lead to the formula of theorem \[kontsevich2\] we want to prove.
\[rem-string\] Using the notations from \[not\], let $C$ be a tropical curve in ${\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ passing through $L_1$, $L_2$, $p_3,\ldots,p_n$ and mapping to $\lambda_A$ under $\operatorname{f\/t}$ (hence a curve $C$ that contributes to the left hand side of equation \[eqdeg\] with multiplicity $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$). We would like to generalize proposition 5.1 of [@GM053], which states that $C$ has a contracted bounded edge. However, this is not true in the case of ${\mathbb{F}}_2$. We can have curves like the one shown in the following picture (where the length $l$ is very large) which do allow a very large $\mathcal{M}_{0,4}$-coordinate.
Even though those curves fail to have a contracted bounded edge, we can still interpret them as reducible curves by cutting off the part which is far away to the right (in the picture denoted by $S$). The remaining part (in the picture denoted by $C'$) is a reducible curve of degree $\Delta_{{\mathbb{F}}_2}(a-1,b+2)$. The existence of such curves with a very large $\mathcal{M}_{0,4}$-coordinate leads to the second part of the sum in the recursion formula of theorem \[kontsevich2\]. The part $S$ which is far away to the right is called a *string* following [@GM053], definition 3.5.
\[lem-contractededge\] Using the notations from \[not\], let $C \in {\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta) $ be a tropical curve that passes through $L_{1}$, $L_{2}$, $p_{3}$, …, $p_{n}$, maps to $\lambda_A$ under $\operatorname{f\/t}$ and has a non-zero multiplicity $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$. Then **either**
1. $C$ has a contracted bounded edge **or**
2. $C$ contains a string \[case-string\]
(see remark \[rem-string\]) that can be moved to the right.
The beginning of the proof is similar to proposition 5.1 of [@GM053].
We will show that the set of all points $\operatorname{f\/t}(C)$ is bounded in ${\mathcal{M}}_{0,4}$ where $C$ runs over all curves $C \in {\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta)$ with non-zero multiplicities $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$ that satisfy the conditions but have no contracted bounded edge and no string moving to the right as in the picture. By proposition 2.11 of [@NS04] there are only finitely many combinatorial types in ${\mathcal{M}}_{0,n}^{\text{lab}}(\R^2,\Delta) $. Thus, we may restrict ourselves to tropical curves $C $ of a fixed combinatorial type $\alpha$. Furthermore, we may assume the curves corresponding to $\alpha$ are $3$-valent.
Let $C $ be such a curve and let $C'$ be the curve obtained from $C$ by forgetting the first and the second marked point. Then $C'$ has a string $\Gamma'$ which follows analogously to remark 3.7 of [@GM053]. We claim that using the string, we can deform $C'$ in a $1$-parameter family within its combinatorial type without changing the images of the marked points. To see this, assume first that there is a vertex $V$ contained in the string such that the directions of the adjacent edges do not span $\R^2$ (case (A) in the picture below). Then we can change the lengths of the adjacent edges without changing the image inside $\R^2$, in particular without changing the image of any marked point. Next assume that there is no such vertex contained in the string (case (B)). Then we can take one of the ends of the string (which is necessarily non-contracted) and move it slightly in a non-zero direction modulo its linear span. Consider the next vertex $V$ and let $v$ be the adjacent edge not contained in the string. Then $v$ is non-contracted and our moved end will meet the affine span of $v$ at some point $P$. So we change the length of $v$ such that it ends at $P$ (while keeping the position of its second vertex fixed). Then we also move the second edge of the string to $P$ and go on to the next vertex. Continuing like this, we produce a $1$-dimensional deformation of $C'$ that keeps the images of the marked points fixed.
Assume we could deform $C'$ in a more than $1$-dimensional family while keeping the images of the marked points fixed. Then $C'$ moves in an at least $1$-dimensional family with the image point under all evaluations and the forgetful map fixed. Then $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)= 0$, which is a contradiction to our assumption. In particular, we can see that we cannot have more than one string. Note that the edges adjacent to $\Gamma'$ must be bounded since otherwise we would have two strings.
Now we show that the $1$-dimensional deformation of $C'$ is either bounded itself or does not affect the image under $\operatorname{f\/t}$. From this, the statement follows.
First assume that there are bounded edges adjacent to $\Gamma'$ to both sides of $\Gamma'$ as shown in (i). Then the deformation of $C'$ with the combinatorial type and the conditions fixed are bounded to both sides. This means that the lengths of all inner edges are bounded except possibly the edges adjacent to $x_{1}$ and $x_{2}$. This is sufficient to ensure that the image of these curves under $\operatorname{f\/t}$ is bounded in ${\mathcal{M}}_{0,4}$, too.
Now assume that all bounded edges adjacent to $\Gamma'$ are on one side of $\Gamma'$ (say after picking an orientation of $\Gamma'$ on the left side). Denote the direction vectors of the edges of $\Gamma'$ by $v_{1}, \ldots, v_{k}$ and the direction vectors of the adjacent bounded edges by $w_{1}, \ldots, w_{k-1}$. As above, the movement of $\Gamma'$ to the left with the combinatorial type and the conditions fixed is bounded. If one of the directions $w_{i+1}$ is obtained from $w_{i}$ by a right turn, then the edges corresponding to $w_{i}$ and $w_{i+1}$ meet to the right of $\Gamma'$ as shown in (ii). This restricts the movement of $\Gamma'$ to the right with the combinatorial type and the conditions fixed, too, since the edge corresponding to $v_{i+1}$ then receives length $0$. Hence, as above, the image of these plane tropical curves under $\operatorname{f\/t}$ is bounded in ${\mathcal{M}}_{0,4}$ as well. Thus, we may assume that for all $i$, $1 \leq i \leq k-2$ the direction $w_{i+1}$ is either the same as $w_{i}$ or obtained from $w_{i}$ by a left turn as shown in (iii). The balancing condition then ensures that for all $i$ both the directions $v_{i+1}$ and $-w_{i+1}$ lie in the angle between $v_{i}$ and $-w_{i}$. Therefore, all directions $v_{i}$ and $-w_{i}$ lie in the angle between $v_{1}$ and $-w_{1}$. In particular, the string $\Gamma'$ cannot have any self-intersections in $\R^{2}$. We can therefore pass to the local dual picture where the edges dual to $w_{i}$ correspond to a concave side of a polygon whose other two edges are dual to $v_{1}$ and $v_{k}$ as shown in (iv). But note that both $v_{1}$ and $v_{k}$ are outer directions of a plane tropical curve of degree $\Delta$. Thus, $v_{1}$ and $v_{k}$ must be $\binom{-1}{0}$, $\binom{0}{-1}$, $\binom{1}{0}$ or $\binom{2}{1}$. Consequently, their dual edges have direction vectors $\pm\binom{0}{-1}$, $\pm\binom{1}{0}$, $\pm\binom{0}{1}$ or $\pm\binom{-1}{2}$. We have to distinguish two cases
- $v_{1}$ and $v_{k}$ are $\binom{0}{-1}$ and $\binom{2}{1}$, i.e. their dual edges have direction vectors $\pm(-1,0)$ and $\pm(-1,2)$
- $v_{1}$ and $v_{k}$ are not $\binom{0}{-1}$ and $\binom{2}{1}$.
In case (b) the triangles spanned by two of those vectors do not admit any further integer points. Therefore we have $k=2$ and the string consist just of the two unbounded edges corresponding to $v_{1}$ and $v_{2}$ that are connected to the rest of the plane tropical curve by exactly one internal edge corresponding to $w_{1}$. It remains to show that for all possibilities for $v_{1}$ and $v_{2}$ in case (b) the union of the corresponding edges finally becomes disjoint from at least one of the chosen curves $L_{1}$ and $L_{2}$ as the length of the edge corresponding to $w_{1}$ increases. This can be proved by a case-by-case analysis as shown in the following picture:
In case (a) the triangle spanned by the two vectors $\binom{-1}{0}$ and $\binom{-1}{2}$ admits exactly one further integer point.
In the picture, we denote the duals of the vectors $v_i$ and $w_i$ by $\check{v}_i$ and $\check{w}_i$. Thus, in case (a) we may have $k=3$ and the string $\Gamma'$ may consist of the two unbounded edges corresponding to $v_{1}$ and $v_{3}$ and the bounded edge corresponding to $v_{2}$ that is connected to the rest of the plane tropical curve by the two edges corresponding to $w_{1}$ and $w_{2}$. In this case, the movement of the string is indeed not bounded to the right. Then we are in case (\[case-string\]) of lemma \[lem-contractededge\]. This finishes the proof of the lemma.
\[lem-multstring\] Let $C$ be a curve of type (\[case-string\]) of lemma \[lem-contractededge\] then $$\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C) = \mult_{\operatorname{ev}_1}(C_1) \cdot \mult_{\operatorname{ev}_2}(C_2) \cdot 2 \cdot (C_{1} \cdot L_{1})_{x_{1}} \cdot (C_{1} \cdot L_{2})_{x_{2}}$$ where $\mult_{\operatorname{ev}_{i}}(C_i)$ denotes the multiplicity of the evaluation map at the $\#\Delta_{i}-1$ points of $x_{3}, \ldots, x_{n}$ that lie on $C_{i}$ for $i \in \{1,2\}$ and $(C' \cdot C'')_{p}$ denotes the intersection multiplicity of the plane tropical curves $C'$ and $C''$ at the point $p\in C'\cap C''$. Here, $C_1$ and $C_2$ denote the two irreducible components of the part $C'$ of $C$ that we get when cutting off the string $S$ as in remark \[rem-string\].
Since $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$ equals the absolute value of the determinant of the map $\operatorname{ev}\times \operatorname{f\/t}$ in local coordinates, we set up the matrix $A$ for $\operatorname{ev}\times \operatorname{f\/t}$ as in lemma \[lem-determinante\] and compute its determinant. The local coordinates are the position of a root vertex and the length of all bounded edges, respectively the coordinates of the images of the contracted edges and the length coordinate of the bounded edge of the image under $\operatorname{f\/t}$ in ${\mathcal{M}}_{0,4}$. Because of remark \[rem-choosecoor\], the absolute value of the determinant does not depend on the special choice of such coordinates.
There are exactly two bounded edges that connect the string $S$ with the rest of the curve. We denote these bounded edges by $E'$ and $E''$ and the unique bounded edge that is contained in the string by $E$. Their lengths are denoted by $l'$, $l''$ and $l$, respectively.
As the length of the ${\mathcal{M}}_{0,4}$-coordinate is very large and there is no contracted bounded edge, the lengths $l'$, $l$ and $l''$ must count towards the length of the ${\mathcal{M}}_{0,4}$-coordinate. That is, $x_1$ and $x_2$ have to be on one side of those three edges and $x_3$ and $x_4$ on the other. Let us call the part with $x_1$ and $x_2$ $C_1$, and assume without restriction that $E'$ belongs to $C_1$. Put the root vertex on the $E'$-side. Then the columns of the matrix $A$ corresponding to the lengths $l'$ and $l''$ read:
$l'$ $l''$
------------------------------------ --------------------------------------- ----------------------------------------
evaluation at a point behind $E'$ $\begin{array}{c} 0 \\ 0 \end{array}$ $\begin{array}{c} 0 \\ 0 \end{array}$
evaluation at a point behind $E''$ $\begin{array}{c} 1 \\ 0 \end{array}$ $\begin{array}{c} -1 \\ 0 \end{array}$
${\mathcal{M}}_{0,4}$-coordinate 1 1
If we add the column corresponding to the length $l'$ to the column corresponding to the length $l''$, then the column corresponding to the length $l'+l''$ has only one entry $2$ and all other entries $0$. Thus, we get a factor of $2$ and to compute the determinant of the matrix $A$ we may drop both the ${\mathcal{M}}_{0,4}$-row and the column corresponding to the edge $E''$.
Now, we consider the first marked point $x_{1}$. We require that the plane tropical curve $C$ passes through $L_{1}$ at this point. Let $E_{1}$ and $E_{2}$ be the two adjacent edges of $x_{1}$. We denote their common direction vector by $v=\binom{v_{1}}{v_{2}}$ and their lengths by $l_{1}$ and $l_{2}$, respectively. We may assume that the root vertex is on the $E_{1}$-side of $x_{1}$. Assume that both $E_{1}$ and $E_{2}$ are bounded. If $x_{1}$ is contracted to a point on an unbounded edge of $L_{1}$ with direction vector $\binom{u_{1}}{u_{2}}$, then the columns of the matrix corresponding to $l_{1}$ and $l_{2}$, respectively, read
evaluation at ... $l_{1}$ $l_{2}$
---------------------------------------------- --------------------------- ---------
... $x_{1}$ $|u_{2}v_{1}-u_{1}v_{2}|$ $0$
... a point reached via $E_{1}$ from $x_{1}$ $0$ $0$
... a point reached via $E_{2}$ from $x_{1}$ $v$ $v$
Note that there are two rows for each marked point $x_{3}$, …, $x_{n}$ that is reached via $E_{1}$ or $E_{2}$ from $x_{1}$ and there is one row for the marked point $x_{2}$. If we subtract the column corresponding to $l_{2}$ from the column corresponding to $l_{1}$, then we obtain a column with only one non-zero entry. So for the determinant we get $(C_{1} \cdot L_{1})_{x_{1}}$ as a factor and may drop both the row corresponding to $x_{1}$ and the column corresponding to $l_{1}$. Now assume that one of the edges $E_{1}$ and $E_{2}$ is unbounded. Assume that $E_{1}$ is bounded and $E_{2}$ is unbounded. Then there is a column corresponding to $l_{1}$ but no column corresponding to $l_{2}$. The column corresponding to $l_{1}$ has only one non-zero entry and the same argument as above holds. Note that it is not possible that both $E_{1}$ and $E_{2}$ are unbounded. Taking the factor $(C_{1} \cdot L_{1})_{x_{1}}$ into account, we can now forget the marked point $x_1$ and straighten the $2$-valent vertex to produce only one bounded edge out of $E_1$ and $E_2$.
If we forget the marked point $x_{2}$, we obtain a factor of $(C_{1} \cdot L_{2})_{x_{2}}$ in the same way.
Now, we consider again the string $S$. Remember that we split up the plane tropical curve $C$ at this string into the two parts $C_{1}$ and $C_{2}$. We choose the boundary vertex of the bounded edge $E'$ at the $C_{1}$-side as root vertex and denote it by $V$. Then the matrix reads
------------------------ --------- ------------ ---------------- ------------------ ------------
lengths lengths
evaluation at a ... root in $C_{1}$ $l'$ $l$ in $C_{2}$
... point behind $E'$ $I_{2}$ $\ast$ $0$ $0$ $0$
... point behind $E''$ $I_{2}$ $0$ $\binom{1}{0}$ $\binom{-1}{-1}$ $\ast$
------------------------ --------- ------------ ---------------- ------------------ ------------
where $I_2$ denotes the $2$ by $2$ unit matrix. Assume there are $n_1$ marked points besides $x_1$ and $x_2$ on $C_1$ and $n_2$ marked points besides $x_3$ and $x_4$ on $C_2$, where $n_1+n_2=n-4=\#\Delta-4=4a+2b-4$. Assume the degree of $C_1$ is $\Delta_{{\mathbb{F}}_2}(a_1,b_1)$ and the degree of $C_2$ is $\Delta_{{\mathbb{F}}_2}(a_2,b_2)$. Then $a_1+a_2=a-1$ and $b_1+b_2=b+2$ as we observed in remark \[rem-string\]. Since $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)\neq 0$ we must have $n_1=4a_1+2b_1-1$ and $n_2+2=4a_2+2b_2-1$ (because then the curves will be fixed by the points). Thus (after forgetting $x_1$ and $x_2$) $C_1$ has $n_1+4a_1+2b_1=2n_1+1$ unbounded edges and thus $2n_1-2$ bounded edges. Hence $2n_1-2$ length coordinates belong to bounded edges in $C_1$. $C_2$ has $n_2+2+4a_2+2b_2=2n_2+5$ unbounded edges and thus $2n_2+2$ length coordinates belong to $C_2$. Furthermore, there are $n_{1}$ points behind $E'$ and there are $n_{2}+2$ points behind $E''$.
If we add the $l'$-column to the $l$-column and then multiply the $l$-column by $-1$, then we obtain the following matrix whose determinant has the same absolute value as the determinant that we are looking for.
------------------------ --------- ------------ ---------------- ---------------- ------------
lengths lengths
evaluation at a ... root in $C_{1}$ $l'$ $l$ in $C_{2}$
... point behind $E'$ $I_{2}$ $\ast$ $0$ $0$ $0$
... point behind $E''$ $I_{2}$ $0$ $\binom{1}{0}$ $\binom{0}{1}$ $\ast$
------------------------ --------- ------------ ---------------- ---------------- ------------
Note that this matrix has a block form. The block at the top right is a zero block. We denote the top left block of size $2n_{1}$ by $A_{1}$ and the bottom right block of size $2n_{2}+4$ by $A_{2}$. Then, the determinant that we are looking for is $|\det(A_{1})|\cdot|\det(A_{2})|$.
But the matrix $A_{1}$ is the matrix of evaluation at marked points in $C_{1}$ and $A_{2}$ is the matrix of evaluation at marked points in $C_{2}$. Thus, we have $|\det(A_{1})|=\mult_{\operatorname{ev}_1}(C_1)$ and $|\det(A_{2})|=\mult_{\operatorname{ev}_2}(C_2)$.
Together, we have $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C) = \mult_{\operatorname{ev}_{1}}(C_1) \cdot \mult_{\operatorname{ev}_{2}}(C_2) \cdot 2 \cdot (C_{1} \cdot L_{1})_{x_{1}} \cdot (C_{1} \cdot L_{2})_{x_{2}}$.
We are now ready to prove the main result of this section:
\[kontsevich2\] Let $a$ and $b$ be non-negative integers with $a+b\geq1$. Then $$\begin{aligned}
{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}&=\frac{1}{2}\sum\phi_{2_1}(a_{1},b_{1}){\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}\\&\quad+\frac{1}{2}\sum \phi_{{2}_{2}}(a_{1},b_{1}){\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}\end{aligned}$$ where the first sum goes over all $(a_1,b_1)$ and $(a_2,b_2)$ satisfying $$(a_1,b_1)+(a_2,b_2)=(a,b),$$ $0\leq a_1\leq a$, $0\leq b_1 \leq b$ and $(0,0)\neq (a_1,b_1)\neq (a,b)$, and the second sum goes over all $(a_1,b_1)$ and $(a_2,b_2)$ satisfying $$(a_1,b_1)+(a_2,b_2)=(a-1,b+2),$$ $0\leq a_1\leq a-1$ and $0<b_1<b+2$. We use the shortcuts $\phi_{{2}_{1}}(a_{1},b_{1})$ for $$\begin{split}
\phi_{{2}_{1}}(a_{1},b_{1}) &= (2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \quad - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-1}
\end{split}
\label{defphi21}$$ and $\phi_{{2}_{2}}(a_{1},b_{1})$ for $$\begin{split}
\phi_{{2}_{2}}(a_{1},b_{1}) &= 2(2a_{1}+b_{1})(2a_{2}+b_{2})(b_{1}b_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \quad - 2(2a_{1}+b_{1})(2a_{1}+b_{1})(b_{1}b_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-1}.
\end{split}
\label{defphi22}$$
Let $C$ be a curve passing through $L_1$, $L_2$, $p_3,\ldots,p_n$ and mapping to $\lambda_A$ under $\operatorname{f\/t}$, i.e. a curve that contributes to the left hand side of equation \[eqdeg\] because of remark \[rem-generalconditions\]. By lemma \[lem-determinante\] and notation \[def-mult\], it has to be counted with multiplicity $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$. We will show that $C$ can be interpreted as a reducible curve, and that its multiplicity $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)$ can be split into factors corresponding to the irreducible components.
Because of lemma \[lem-contractededge\] we know that $C$ either has a contracted bounded edge or a string that can be moved to the right as in remark \[rem-string\].
If it has a contracted bounded edge, then it is possible that this edge is adjacent to the marked ends $x_1$ and $x_2$. Then the two marked ends are contracted to the same point in $\R^2$, which has to be an intersection point of $L_1$ and $L_2$. Let us call this point $p$. Let $C'$ denote the curve that arises after forgetting the marked point $x_1$. Analogously to 5.5.a) of [@GM053] we can show that $\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)=\mult_{\operatorname{ev}'}(C')\cdot (L_1.L_2)_p$, where $\operatorname{ev}'$ now denotes the evaluation of $x_2$ at a point combined with all other point evaluations and $(L_1.L_2)_p$ denotes the intersection product of $L_1$ and $L_2$ at $p$. Instead of counting those curves $C$, we can hence count curves $C'$ meeting the points $p_3,\ldots,p_n$ and an intersection point of $L_1$ and $L_2$. Since $(L_1.L_2)=2$ by the tropical Bézout’s theorem (4.2 of [@RST03]), we can conclude that those curves $C$ contribute $2{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ to the left hand side of equation \[eqdeg\].
If $x_1$ and $x_2$ are not adjacent to the contracted bounded edge, then there have to be bounded edges to both sides of the contracted bounded edge, since all other marked points have to meet different points. If there are bounded edges on both sides of the contracted bounded edge, we can cut the bounded edge thus producing a reducible curve with two new contracted ends $z_1$ and $z_2$. Let us call the two components $C_1$ and $C_2$. Since $C$ maps to $\lambda_A$ under $\operatorname{f\/t}$, $x_1$ and $x_2$ have to be on $C_1$. Let us call the degree of $C_1$ $\Delta_{{\mathbb{F}}_{2}}(a_1,b_1)$ and the degree of $C_2$ $\Delta_{{\mathbb{F}}_{2}}(a_2,b_2)$, then we must have $(a_1,b_1)+(a_2,b_2)=(a,b)$, $0\leq a_1\leq a$, $0\leq b_1\leq b$ and $(0,0)\neq (a_1,b_1)\neq (a,b)$. Analogously to 5.5.b) of [@GM053], we can forget $x_1$ and $x_2$ thus producing a factor of $(C_1.L_1)_{x_1}\cdot (C_1.L_2)_{x_2}$. (By abuse of notation, we use the $x_i$ here for the point in $\R^2$ to which the end $x_i$ is contracted to.) With the same arguments as in 5.5.b) of [@GM053], we can see that $$\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C)= \mult_{\operatorname{ev}_1}(C_1) \mult_{\operatorname{ev}_2}(C_2) (C_{1} \cdot C_{2})_{z_{1}=z_{2}} (C_1.L_1)_{x_1}\cdot (C_1.L_2)_{x_2},$$ where $\mult_{\operatorname{ev}_1}(C_1)$ denotes the multiplicity of the evaluation at the points on $C_1$. Note that $4a_1+2b_1-1$ of the other marked points have to be on $C_1$. Now instead of counting the curves $C$ with a contracted bounded edge and bounded edges on both sides, we can pick $4a_1+2b_1-1$ of the points $p_5,\ldots,p_n$ ($\binom{4a+2b-4}{4a_1+2b_1-1}$ possibilities) and count curves $C_1$ through those points, and $C_2$ through the remaining points. Again by tropical Bézout’s theorem we have $(C_1.L_1)=(2a_1+b_1)$ choices to attach $x_1$ and $(C_1.L_2)=(2a_1+b_1)$ choices to attach $x_2$, and we have $(C_1.C_2)=(a_1b_2+a_2b_1+2a_1a_2)$ choices to glue $C_1$ and $C_2$ to a possible $C$. Thus those curves contribute $$\sum(2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-1}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})},$$ where the sum goes over all $(a_1,b_1)+(a_2,b_2)=(a,b)$, $0\leq a_1\leq a$, $0\leq b_1\leq b$ and $(0,0)\neq (a_1,b_1)\neq (a,b)$. In the formula we want to prove, we can see this contribution negatively on the right hand side.
Finally, if $C$ has a string as in remark \[rem-string\], then by lemma \[lem-multstring\] we can conclude that $C$ contributes $$\mult_{\operatorname{ev}\times \operatorname{f\/t}}(C) = \mult_{\operatorname{ev}_1}(C_1) \cdot \mult_{\operatorname{ev}_2}(C_2) \cdot 2 \cdot (C_{1} \cdot L_{1})_{x_{1}} \cdot (C_{1} \cdot L_{2})_{x_{2}}.$$ Instead of counting such curves $C$, we can pick $4a_1+2b_1-1$ of the points $p_5,\ldots,p_n$ ($\binom{4a+2b-4}{4a_1+2b_1-1}$ possibilities) and count curves $C_1$ of degree $\Delta_{{\mathbb{F}}_{2}}(a_1,b_1)$ through those points, and $C_2$ of degree $\Delta_{{\mathbb{F}}_{2}}(a_2,b_2)$ through the remaining points, where now $(a_1,b_1)+(a_2,b_2)=(a-1,b+2)$. There are again $(2a_1+b_1)$ possibilities to attach $x_1$ and also $(2a_1+b_1)$ possibilities to attach $x_2$ to $C_1$. There are $b_1b_2$ choices to pick the edges $E'$ and $E''$ to which we can attach the string $S$. Hence those curves contribute $$\sum 2(2a_{1}+b_{1})(2a_{1}+b_{1})(b_1b_2)\binom{4a+2b-4}{4a_{1}+2b_{1}-1}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})},$$ where now the sum goes over all $(a_1,b_1)+(a_2,b_2)=(a-1,b+2)$, $0\leq a_1\leq a-1$, $0< b_1< b+2$ and $(0,0)\neq (a_1,b_1)\neq (a,b)$. In the formula we want to prove, this contribution appears negatively on the right hand side.
Performing the same analysis for the right hand side of equation \[eqdeg\] and collecting the terms to the different sides, the statement follows.
The following formula for ${\mathbb{F}}_0$ can be proved analogously. The proof is easier in fact, since all curves have a contracted bounded edge and the special case of curves having a string that can be moved to the right as in remark \[rem-string\] does not occur here. We therefore skip the proof. For more details, see [@Fra08].
\[kontsevich0\] Let $a$ and $b$ be non-negative integers with $a+b\geq1$. Then $${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}=\frac{1}{2}\sum\phi_{0}(a_{1},a_{1}+b_{1}){\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{1}$},\emph{$a_{1}+b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{2}$},\emph{$a_{2}+b_{2}$})}$$ where the sum goes over all $(a_1,b_1)$ and $(a_2,b_2)$ satisfying $$(a_{1},a_{1}+b_{1})+(a_{2},a_{2}+b_{2})=(a,a+b),$$ $0\leq a_{1} \leq a$, $-a_{1} \leq b_{1} \leq b+a_{2}$ and $(0,0)\neq(a_{1},a_{1}+b_{1})\neq(a,a+b)$. We use the shortcut $\phi_{0}(a_{1},a_{1}+b_{1})$ for $$\begin{split}
\label{defphi0}
\phi_{0}(a_{1}&,a_{1}+b_{1}) = (2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \quad - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})\binom{4a+2b-4}{4a_{1}+2b_{1}-1}.
\end{split}$$
The proof for nonnegative $a$, $b$ with $a+b\geq 1$ and $g=0$ {#sec-2}
=============================================================
Recall the equation from theorem \[thm-main\] we want to prove. Note that for $k=a$ we would obtain a summand $\binom{b+2a}{a}\mathcal{N}^{0}_{{\mathbb{F}}_{2}}(0,(b+2a))$ which is $0$ for all $a$, $b\in{\mathbb{Z}}_{\geq0}$ with $a+b\geq1$ except for $a=0$ and $b=1$. As in this special case the statement still holds, we may add this summand for $k=a$ and deal with the slightly modified equation $${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} = \sum_{k=0}^{a}{\binom{b+2k}{k}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a-k$},\emph{$b+2k$})}}.$$ We will see later that this is useful.
Let us consider the formula for small $a$ in more detail. For $a=0$ and $a=1$ we have ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}={\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ for all $b\in{\mathbb{Z}}_{\geq 0}$. Hence, in these cases the Gromov-Witten invariants are enumerative for ${\mathbb{F}}_{2}$ as well. For $a=2$ and $b=0$ we have ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$0$})}=10$ while the associated Gromov-Witten invariant is ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2$})}=12$. This is the first interesting case. The formula gives a interpretation of the difference $2$ in terms of a deformation of ${\mathbb{F}}_{2}$: $${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2$})} = 12 = 10 + 2\times 1 = {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$0$})} + \binom{2}{1}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$1$},\emph{$2$})} + \underbrace{\binom{4}{2}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$0$},\emph{$4$})}}_{=0} \text{.}$$
We need the following combinatorial indentity involving binomial coefficients for our proof.
\[binomgleichung\] Let $n$, $m$, $k\in\N$. Then $$\sum_{i=0}^{k}(mi+n(k-i)-2i(k-i))\binom{n}{i}\binom{m}{k-i}=2\cdot n\cdot m \cdot \binom{n+m-2}{k-1}.$$
The equation is essentially a consequence of Vandermonde’s identity which states that $$\sum_{i=0}^{k}\binom{n}{i}\binom{m}{k-i}=\binom{n+m}{k}.$$ Using this we have $$\sum_{i=0}^{k}mi\binom{n}{i}\binom{m}{k-i} = \sum_{i=0}^{k}nm\binom{n-1}{i-1}\binom{m}{k-i} = nm\binom{n+m-1}{k-1}$$ and $$\sum_{i=0}^{k}n(k-i)\binom{n}{i}\binom{m}{k-i} = \sum_{i=0}^{k}nm\binom{n}{i}\binom{m-1}{k-i-1} = nm\binom{n+m-1}{k-1}$$ and $$\begin{split}
\sum_{i=0}^{k}(-2i(k-i))\binom{n}{i}\binom{m}{k-i} &= -2\sum_{i=0}^{k}nm\binom{n-1}{i-1}\binom{m-1}{k-i-1} \\
&= -2nm\binom{n+m-2}{k-2}
\end{split}$$ as $i\binom{n}{i}=n\binom{n-1}{i-1}$ and $(k-i)\binom{m}{k-i}=m\binom{m-1}{k-i-1}$ and thus $$\begin{split}
& \sum_{i=0}^{k}(mi+n(k-i)-2i(k-i))\binom{n}{i}\binom{m}{k-i} \\
& \quad = 2nm\left( \binom{n+m-1}{k-1}-\binom{n+m-2}{k-2} \right) \\
& \quad = 2nm\binom{n+m-2}{k-1}
\end{split}$$ where the last equality follows by Pascal’s rule.
We will prove that the statement holds for all integers $a\geq0$ and $b\geq-a$ with $2a+b\geq1$ by induction on $2a+b$. Note that $b$ may be negative. But as $b \geq -a$ and $2a+b \geq 1$ we have $a+b \geq 0$ and hence the left hand side is well defined. On the right hand side we may have negative entries. We define ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ to be $0$ for all $b<0$, $a\geq 0$. In particular, we conclude that the statement holds for all non-negative integers $a$ and $b$ with $a+b\geq1$.
The induction beginning for $a=0$ and $b=1$ resp. $a=1$ and $b=-1$ is straight forward, we need to use the extra summand with $k=a$ however.
Now let $a\geq0$ and $b\geq-a$ be integers such that $2a+b \geq 1$.
We can assume that the statement holds for all integers $a_{i}\geq0$ and $b_{i}\geq-a_{i}$ with $1 \leq 2a_{i}+b_{i}<2a+b$.
First let us consider the left hand side. We know by theorem \[kontsevich0\] that $$2{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} =
\sum \phi_{0}(a_{1},a_{1}+b_{1}){\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{1}$},\emph{$a_{1}+b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{2}$},\emph{$a_{2}+b_{2}$})}$$ where the sum goes over all $(a_1,b_1)$ and $(a_2,b_2)$ satisfying $(a_{1},a_{1}+b_{1})+(a_{2},a_{2}+b_{2})=(a,a+b)$, $0\leq a_{1} \leq a$, $-a_{1} \leq b_{1} \leq b+a_{2}$ and $(0,0)\neq(a_{1},a_{1}+b_{1})\neq(a,a+b)$, and $\phi_{0}(a_{1},a_{1}+b_{1})$ is defined by equation \[defphi0\].
As $2a_{1}+b_{1}<2a+b$ and $2a_{2}+b_{2}<2a+b$ for all $a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ we have by the induction hypothesis that ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{1}$},\emph{$a_{1}+b_{1}$})} = \sum_{i=0}^{a_{1}}{\binom{b_{1}+2i}{i}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}-i$},\emph{$b_{1}+2i$})}}$ and ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a_{2}$},\emph{$a_{2}+b_{2}$})} = \sum_{j=0}^{a_{2}}{\binom{b_{2}+2j}{j}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}-j$},\emph{$b_{2}+2j$})}}$. Hence we have $$\begin{split}
2{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} &=
\sum \phi_{0}(a_{1},a_{1}+b_{1}) \cdot \left(\sum_{i=0}^{a_{1}}{\binom{b_{1}+2i}{i}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}-i$},\emph{$b_{1}+2i$})}}\right) \\
& \qquad \hspace{5em} \cdot \left(\sum_{j=0}^{a_{2}}{\binom{b_{2}+2j}{j}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}-j$},\emph{$b_{2}+2j$})}}\right) \\
&=
\sum_{k=0}^{a}
\sum_{\substack{i=0 \\ j=k-i}}^{k} \sum
\bigg(\binom{b_{1}+2i}{i}\binom{b_{2}+2j}{j}
\phi_{0}(a_{1},a_{1}+b_{1}) \\
& \qquad \hspace{5em} \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}-i$},\emph{$b_{1}+2i$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}-j$},\emph{$b_{2}+2j$})}\bigg)
\end{split}$$
Let us consider the range of $a_{1}$ and $b_{1}$ in the third sum for a fixed $k$ and $i$. As ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}-i$},\emph{$b_{1}+2i$})}=0$ for all $0 \leq a_{1} < i$ and ${\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}-j$},\emph{$b_{2}+2j$})}=0$ for all $0 \leq a_{2} < j$ (i.e. for all $a-0 = a \geq a-a_{2} = a_{1} > a-j = a-k+i$) we can forget about the summands where $0 \leq a_{1} < i$ or $a-k+i < a_{1} \leq a$. As $\binom{b_{1}+2i}{i}=0$ for all $b_{1} < -i$ and $\binom{b_{2}+2j}{j}=0$ for all $b_{2} < -j$ (i.e. for all $b- b_{2} = b_{1} > b+j$) the range of those $b_{1}$ which give a contribution is $-i \leq b_{1} \leq b+j$. We may add summands for $-2i \leq b_{1} < -i$ and $b+j < b_{1} \leq b+2j$ since they are $0$ anyway. Hence we may restrict our attention to those $(a_{1},a_{1}+b_{1})+(a_{2},a_{2}+b_{2})=(a,a+b)$ with $i \leq a_{1} \leq a-k+i$ and $ -2i \leq b_{1} \leq b+2j$ such that $ (0,0)\neq(a_{1},a_{1}+b_{1})\neq(a,a+b)$. With the definitions $$\begin{array}{ll}
a_{1}':=a_{1}-i & \quad a_{2}':= (a-k)-a_{1}'= a_{2}-j \\
b_{1}':=b_{1}+2i & \quad b_{2}':= (b+2k)-b_{1}'= b_{2}+2j
\end{array}$$ this is equivalent to considering all pairs $$(a_{1}',b_{1}')+(a_{2}',b_{2}')=(a-k,b+2k)$$ with $$0 \leq a_{1}' \leq a-k \text{ and } 0 \leq b_{1}' \leq b+2k \text{ such that } (0,0)\neq(a_{1}',b_{1}')\neq(a,b) \text{.}$$ Let the sums in the following equation go over those pairs now. Then
$$\begin{split}
2{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} &= \sum_{k=0}^{a}\sum_{\substack{i=0 \\ j=k-i}}^{k}\sum
\bigg( \binom{b_{1}'}{i}\binom{b_{2}'}{j}
\phi_{0}(a_{1}'+i,a_{1}'+b_{1}'-i) \\
& \quad \hspace{5em} \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}'$},\emph{$b_{1}'$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}'$},\emph{$b_{2}'$})}\bigg) \\
& = \sum_{k=0}^{a}\sum
\bigg(\sum_{\substack{i=0 \\ j=k-i}}^{k}
\binom{b_{1}'}{i}\binom{b_{2}'}{j}
\phi_{0}(a_{1}'+i,a_{1}'+b_{1}'-i) \\
& \quad \hspace{5em} \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}'$},\emph{$b_{1}'$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}'$},\emph{$b_{2}'$})}\bigg)
\end{split}$$
where $$\begin{split}
& \phi_{0}(a_{1}'+i,a_{1}'+b_{1}'-i) \\
& \quad = (2a_{1}'+b_{1}')(2a_{2}'+b_{2}')(a_{1}'b_{2}'+a_{2}'b_{1}'+2a_{1}'a_{2}'\\
& \qquad \hspace{10em} +b_{2}'i+b_{1}'j-2ij)\binom{4a+2b-4}{4a_{1}'+2b_{1}'-2} \\
& \qquad - (2a_{1}'+b_{1}')(2a_{1}'+b_{1}')(a_{1}'b_{2}'+a_{2}'b_{1}'+2a_{1}'a_{2}'\\
& \qquad \hspace{10em} +b_{2}'i+b_{1}'j-2ij)\binom{4a+2b-4}{4a_{1}'+2b_{1}'-1} \\
& \quad =(2a_{1}'+b_{1}')(2a_{2}'+b_{2}')(a_{1}'b_{2}'+a_{2}'b_{1}'+2a_{1}'a_{2}')
\binom{4a+2b-4}{4a_{1}'+2b_{1}'-2} \\
& \qquad \hspace{5em} - (2a_{1}'+b_{1}')(2a_{1}'+b_{1}')(a_{1}'b_{2}'+a_{2}'b_{1}'+2a_{1}'a_{2}')
\binom{4a+2b-4}{4a_{1}'+2b_{1}'-1} \\
& \qquad + (2a_{1}'+b_{1}')(2a_{2}'+b_{2}')(b_{2}'i+b_{1}'j-2ij)
\binom{4a+2b-4}{4a_{1}'+2b_{1}'-2} \\
& \qquad \hspace{5em} - (2a_{1}'+b_{1}')(2a_{1}'+b_{1}')(b_{2}'i+b_{1}'j-2ij)
\binom{4a+2b-4}{4a_{1}'+2b_{1}'-1}
\end{split}$$ by equation (\[defphi0\]) in theorem \[kontsevich0\].
Let us stop here and consider the right hand side of the equation we want to prove. We have by Theorem \[kontsevich2\] $$\begin{split}
& \sum_{k=0}^{a}{\binom{b+2k}{k}2{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a-k$},\emph{$b+2k$})}} \\ =& \sum_{k=0}^{a} \binom{b+2k}{k}
\cdot\left(\sum{\phi_{{2}_{1}}(a_{1},b_{1}) {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}}\right. \\
& \quad \hspace{5em} + \left. \sum{\phi_{{2}_{2}}(a_{1},b_{1}){\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}}\right) \\
\end{split}$$ where the first sum goes over all pairs such that $(a_1,b_1)+(a_2,b_2)=(a-k,b+2k)$ and the second sum goes over all pairs such that $(a_1,b_1)+(a_2,b_2)=(a-(k+1),b+2(k+1))$. We use the shortcuts $\phi_{{2}_{1}}(a_{1},b_{1})$ and $\phi_{{2}_{2}}(a_{1},b_{1})$ as defined in equation \[defphi21\] and \[defphi22\].
Since for $k=0$ the binomial coefficient $\binom{b+2(k-1)}{k-1}$ is $0$ and for $k=a$ there are no $a_{1}$ and $b_{1}$ which satisfy $(a_1,b_1)+(a_2,b_2)=(a-(k+1),b+2(k+1))$ we can merge the two sums and get $$\begin{split}
\sum_{k=0}^{a}
\sum & \bigg(\bigg( \binom{b+2k}{k}\phi_{{2}_{1}}(a_{1},b_{1}) +\binom{b+2(k-1)}{k-1}\phi_{{2}_{2}}(a_{1},b_{1})\bigg) \\
& \quad \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}\bigg)
\end{split}$$ where the sum now goes over all pairs such that $(a_1,b_1)+(a_2,b_2)=(a-k,b+2k)$.
Thus it remains to show that $$\begin{split}
&\sum_{k=0}^{a}\sum\bigg(\sum_{\substack{i=0 \\ j=k-i}}^{k}
\binom{b_{1}}{i}
\binom{b_{2}}{j}
\phi_{0}(a_{1}+i,a_{1}+b_{1}-i)\bigg) \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})} \\
=&
\sum_{k=0}^{a}
\sum \bigg( \binom{b+2k}{k}\phi_{{2}_{1}}(a_{1},b_{1})+\binom{b+2(k-1)}{k-1}\phi_{{2}_{2}}(a_{1},b_{1})\bigg) \\
&
\hspace{8em} \cdot {\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{1}$},\emph{$b_{1}$})}{\mathcal{N}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a_{2}$},\emph{$b_{2}$})}.
\end{split}$$
Therefore we will show that $$\begin{split}
& \sum_{\substack{i=0 \\ j=k-i}}^{k}\binom{b_{1}}{i}\binom{b_{2}}{j}\phi_{0}(a_{1}+i,a_{1}+b_{1}-i) \\
=&
\binom{b+2k}{k}\phi_{{2}_{1}}(a_{1},b_{1})+\binom{b+2(k-1)}{k-1}\phi_{{2}_{2}}(a_{1},b_{1})
\end{split}
\label{toshow2}$$ for all $k\in\{0,...,a\}$ and for all integers $0\leq a_{1}$, $a_{2}\leq a-k$, $0 \leq b_{1}$, $b_{2} \leq b+2k$ with $a_{1}+a_{2}=a-k$, $b_{1}+b_{2}=b+2k$ and $(0,0)\neq(a_{1},b_{1})\neq(a-k,b+2k)$.
We use the identity from Lemma \[binomgleichung\].
It is $$\begin{split}
& \sum_{\substack{i=0 \\ j=k-i}}^{k}\binom{b_{1}}{i}\binom{b_{2}}{j}\phi_{0}(a_{1}+i,a_{1}+b_{1}-i) \\
\stackrel{(\ref{defphi0})}{=} &
\sum_{\substack{i=0 \\ j=k-i}}^{k}\bigg[\binom{b_{1}}{i}\binom{b_{2}}{j} \\
& \quad \hspace{5em} \cdot
\bigg(
(2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1} \\
& \qquad \hspace{5em} + (2a_{1}+b_{1})(2a_{2}+b_{2})(b_{2}i+b_{1}j-2ij)
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})(b_{2}i+b_{1}j-2ij)
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}
\bigg)\bigg]
\end{split}$$ $$\begin{split}
& \quad =
\bigg(\sum_{\substack{i=0 \\ j=k-i}}^{k}\binom{b_{1}}{i}\binom{b_{2}}{j}\bigg) \\
& \quad \hspace{5em} \cdot \bigg(
(2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}\bigg) \\
& \qquad + \bigg(\sum_{\substack{i=0 \\ j=k-i}}^{k}\binom{b_{1}}{i}\binom{b_{2}}{j}(b_{2}i+b_{1}j-2ij)\bigg) \\
& \quad \hspace{5em} \cdot \bigg((2a_{1}+b_{1})(2a_{2}+b_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}
\bigg)
\end{split}$$ $$\begin{split}
& \quad = \binom{b_{1}+b_{2}}{i+j}
\bigg(
(2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}\bigg) \\
& \qquad + \bigg(2b_{1}b_{2}\binom{b_{1}+b_{2}-2}{i+j-1}\bigg)
\bigg((2a_{1}+b_{1})(2a_{2}+b_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \quad \hspace{12em} - (2a_{1}+b_{1})(2a_{1}+b_{1})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}\bigg)
\end{split}$$ $$\begin{split}
& \quad =
\binom{b_{1}+b_{2}}{i+j}
\bigg((2a_{1}+b_{1})(2a_{2}+b_{2})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \qquad \hspace{5em} - (2a_{1}+b_{1})(2a_{1}+b_{1})(a_{1}b_{2}+a_{2}b_{1}+2a_{1}a_{2})
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}\bigg) \\
& \qquad + \binom{b_{1}+b_{2}-2}{i+j-1}
\bigg(2(2a_{1}+b_{1})(2a_{2}+b_{2})b_{1}b_{2}
\binom{4a+2b-4}{4a_{1}+2b_{1}-2} \\
& \quad \hspace{12em} - 2(2a_{1}+b_{1})(2a_{1}+b_{1})b_{1}b_{2}
\binom{4a+2b-4}{4a_{1}+2b_{1}-1}\bigg) \\
& \quad =
\binom{b+2k}{k}\phi_{{2}_{1}}(a_{1},b_{1})+\binom{b+2(k-1)}{k-1}\phi_{{2}_{2}}(a_{1},b_{1})
\end{split}$$ where the last equality follows from equation (\[defphi21\]) and (\[defphi22\]). This completes the proof.
The proof for $0\leq a\leq 2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$ {#sec-3}
===========================================================================
First, we have to introduce another enumerative invariant, namely the numbers ${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ and ${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ of *not necessarily irreducible* plane tropical curves of degree $\Delta_{{\mathbb{F}}_{0}}(a,a+b)$ (resp. $\Delta_{{\mathbb{F}}_{2}}(a,b)$) and genus $g$ through $4a+2b+g-1$ points in general position (see [@Mi03]). Obviously, ${\mathcal{N}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ equals ${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ minus the number of *reducible* curves satisfying the conditions.
By [@Mi03], theorem 2 we can determine the tropical enumerative numbers ${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})}$ and ${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a$},\emph{$b$})}$ both of ${\mathbb{F}}_0$ and ${\mathbb{F}}_2$ by counting $\lambda$-increasing lattice paths of length $4a+2b+g-1$ in the polygon corresponding to the toric surface ${\mathbb{F}}_0$ respectively ${\mathbb{F}}_2$ and the divisor class $aC+(a+b)F$ respectively $aC+bF$. (Each path has to be counted with a certain multiplicity. For more information, see [@Mi03] or [@GM052].) Here, we fix $\lambda$ to be of the form $ \lambda: \R^2 \to \R,\;\;\lambda (x,y) = x-\varepsilon y $, where $ \varepsilon $ is a small irrational number.
We will first show the following modified version of theorem \[thm-main\] and use this later to prove the theorem for $0\leq a \leq 2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$.
\[lem-reducible\] The following equation holds for
- $0\leq a\leq 1$, $b\geq 0$ with $a+b\geq 1$ and any $g\in {\mathbb{Z}}$, and for
- $a=2$, $b\geq 0$ with $a+b\geq 1$ and any $g\geq 0$:
$${\tilde{\mathcal{N}}^{\emph{$g$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$a$},\emph{$a+b$})} = \sum_{k=0}^{a-1}{\binom{b+2k}{k}{\tilde{\mathcal{N}}^{\emph{$g$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$a-k$},\emph{$b+2k$})}}.$$
If $a=0$, then the polygon corresponding to ${\mathbb{F}}_0$ and $bF$ equals the polygon corresponding to ${\mathbb{F}}_2$ and $bF$. It is just a vertical line of integer length $b$. Hence the number of lattice paths agrees, since the polygon in which we count agrees.
If $a=1$, then a path with $2b+3-g$ steps has to miss $g$ lattice points of the polygons dual to $\Delta_{{\mathbb{F}}_2}(1,b)$ resp. $\Delta_{{\mathbb{F}}_0}(1,b+1)$. Since it cannot have any steps of integer length bigger $1$ in the boundary, it looks like the paths in the following picture, where $i+j=g$.
Thus the left hand side of the equation equals $$\sum_{i+j=g}\binom{b+2-i}{j}\binom{b-j}{i}$$ and the right hand side of the equation equals $$\sum_{i+j=g}\binom{b+1-i}{j}\binom{b+1-j}{i}$$ which are both equal to $\binom{2b+2-g}{g}$ because of Vandermonde’s identity.
Let $a=2$ and $g\geq0$. Since $a=2$, the sum on the right hand has only two summands, as indicated in the following picture.
If $g>0$, then no path of length $4a+2b+g-1=2b+g+7$ fits into the second polygon on the right hand side. Hence that summand is $0$ in this case. If $g=0$ then there is exactly one path of length $2b+7$ which fits into the second polygon on the right hand side, and it counts with multiplicity $1$:
Thus we have to show $ {\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2+b$})}={\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$b$})}$ if $g>0$, and ${\tilde{\mathcal{N}}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2+b$})}={\tilde{\mathcal{N}}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$b$})}+b+2.$
Let $\gamma$ be any path in the polygon dual to $\Delta_{{\mathbb{F}}_0}(2,b+2)$. We want to associate a path $\gamma'$ in the polygon dual to $\Delta_{{\mathbb{F}}_2}(2,b)$ to it. First note that by lemma 3.6 and remark 3.7 of [@GM052], each path that does not count $0$ has two sort of steps: some that go down vertically and others that move exactly one column to the right (with a simultaneous move up or down) — see the picture below. (Although stated for a triangle in [@GM052], this statement is with the same arguments true the polygons dual to $\Delta_{{\mathbb{F}}_0}(a,a+b)$ and $\Delta_{{\mathbb{F}}_2}(a,b)$.) We define $\gamma'$ in the following way: $\gamma'$ has two extra steps in the first column, $\gamma'$ coincides with $\gamma$ in the second column, and it has two steps less in the last column:
More precisely, $\gamma'(0):=(0,b+4)$, $\gamma'(1):=(0,b+3)$, $\gamma'(i):=\gamma(i-2)$ for all $i\geq2$ such that the $x$ -coordinate of $\gamma(i)$ is less than or equal to $1$ and $\gamma'(i):=(2,\gamma(i-2)_y-2)$ for all $i$ such that the $x$-coordinate of $\gamma(i)$ is $2$. Note that in the above picture the lattice points which are not images of $\gamma$ respectively $\gamma'$ are drawn white.
It is possible to associate $\gamma'$ to $\gamma$ if $\gamma(2b+g+7-2)$ is in the $x=2$-column (recall that $\gamma$ has $2b+g+7$ steps). Then $\gamma(2b+g+7-2)_y-2\geq 0$. This holds if $g>0$ for any path, and if $g=0$ for any path except the one which takes every step in the first two columns and only one in the last column:
We denote this path by $\gamma_0$, and we always assume $\gamma\neq \gamma_0$ in the following. Note that the multiplicity of $\gamma_0$ is equal to $\mult(\gamma_0)=\binom{b+2}{b+1}=b+2$.
It is too much to hope that the multiplicity of $\gamma$ and $\gamma'$ coincides. Let us compute the multiplicity of both paths. Note first that if $\gamma$ has a step of lattice length bigger $1$ in the $x=0$- or the $x=2$-column, then its multiplicity is $0$ (and the same holds for $\gamma'$), so we do not need to consider it. Let us assume $\gamma$ has $\alpha_i$ steps of lattice length $i$ in the column $x=1$, and let $t:=\sum_{i\geq2}\alpha_i\cdot i$. Denote by $j$ the number of free lattice points on $x=0$, by $i$ the number of free lattice points above the ones taken by $\gamma$, and by $s-j$ the number of free lattice points on $x=2$. Let $r:=b+2$. Then there are $r-\alpha_1-t-i$ free lattice points on $x=1$ below $\gamma$.
In the picture, $r=6$, $j=3$, $i=2$, $t=2$, $\alpha_1=1$ and $s-j=2$.
By [@GM052], proposition 3.8, the multiplicity of $\gamma$ is equal to $$\mult(\gamma)= (I^\alpha)^2 \cdot \binom{r-i-t}{j}\binom{r-j}{i}\binom{r-s+j}{r-\alpha_1-t-i}\binom{\alpha_1+i}{s-j},$$ where $I^\alpha$ is a shortcut for $\prod_i i^{\alpha_i}$.
To see this, note that no step of lattice length bigger $1$ on the column $x=1$ can be part of a parallelogram, since on $x=0$ and $x=2$, only steps of lattice length $1$ are allowed. Thus the binomial factors above count the numbers of ways to arrange parallelograms with edge length $1$ (both above and below $\gamma$), and the factor in front corresponds to the double areas of triangles involving the steps of higher lattice length on $x=1$ (see remark 3.9 of [@GM052]).
Analogously, $$\mult(\gamma')= (I^\alpha)^2 \cdot \binom{r-i-t}{j}\binom{r-j+2}{i}\binom{r-s+j-2}{r-\alpha_1-t-i}\binom{\alpha_1+i}{s-j}.$$
Proposition 3.8 of [@GM052] is only stated for a triangles, but taking remark 3.10 of [@GM052] into account, it can be generalized to polygons dual to $\Delta_{{\mathbb{F}}_0}(a,a+b)$ and $\Delta_{{\mathbb{F}}_2}(a,b)$ with the same arguments.
As already said, in general $\mult(\gamma)$ will not be equal to $\mult(\gamma')$. However, we can take a set of paths $\gamma$ in the rectangle such that the sum of the multiplicities coincides with the sum of the multiplicities of the corresponding paths $\gamma'$ in the dual of $\Delta_{{\mathbb{F}}_2}(2,b)$: We take all paths $\gamma$ such that their values for $\alpha_i$ (for all $i$) and $s$ coincides. That is, we let $i$ vary from $0$ to $r-t-\alpha_i$ and $j$ vary from $0$ to $s$. We denote the set of all those paths by $\Gamma(s,(\alpha_i)_i)$. The sum of the multiplicities of all paths $\gamma$ in the rectangle in $\Gamma(s,(\alpha_i)_i)$ is then equal to $$\label{RHS}
(I^\alpha)^2 \cdot \sum_{j=0}^s \sum_{i=0}^{r-\alpha_1-t} \binom{r-i-t}{j}\binom{r-j}{i}\binom{r-s+j}{r-\alpha_1-t-i}\binom{\alpha_1+i}{s-j}.$$ The sum of the corresponding paths $\gamma'$ in $\Delta(2,b)$ is equal to $$\label{LHS}
(I^\alpha)^2 \cdot \sum_{j=0}^s \sum_{i=0}^{r-\alpha_1-t}
\binom{r-i-t}{j}\binom{r-j+2}{i}\binom{r-s+j-2}{r-\alpha_1-t-i}\binom{\alpha_1+i}{s-j}.$$ Using the Mathematica package MultiSum (see [@We971], respectively [@We972] for more information), we can show that the sum in (\[RHS\]) (neglecting the factor $(I^\alpha)^2$ which coincides for both expressions anyway) — that we will denote by $F(r,(\alpha_i)_i,s)$ from now on — fulfills the following recurrence: $$\begin{aligned}
&(2r-s+2)(\alpha_1+r-t+2)\cdot F(r,(\alpha_i)_i,s) - 2 (r^2+\alpha_1r-tr+4r-\alpha_1 s-2t+4)\cdot \\& F(r+1,(\alpha_i)_i,s+1)-(s+2)(\alpha_1-r+t-2)\cdot F(r+2,(\alpha_i)_i,s+2)=0.\end{aligned}$$ The sum in (\[LHS\]) — denoted by $G(r,(\alpha_i)_i,s)$ — satisfies the same recurrence. As $(\alpha_1-r+t-2)\neq 0$ we only need to check the initial values $r=\alpha_1+t$, $r=\alpha_1+t+1$, $s=0$ and $s=1$ in order to show that the two sums are equal. If $r=\alpha_1+t$, $F(r,(\alpha_i)_i,s)=G(r,(\alpha_i)_i,s)$ is easy to see. If $r=\alpha_1+t+1$, then $F(r,(\alpha_i)_i,s)=G(r,(\alpha_i)_i,s)$ is equivalent to $$\sum_{j=0}^s\binom{\alpha_1+1}{s-j}\binom{\alpha_1}{s}=\sum_{j=0}^s\binom{\alpha_1}{s-j}\binom{\alpha_1+1}{s},$$ which is true by Vandermonde’s identity. If $s=0$, $F(r,(\alpha_i)_i,s)=G(r,(\alpha_i)_i,s)$ is again easy. If $s=1$, both sides can be seen to be equal to $$(\alpha_1+r-t)\binom{2r-1}{r-\alpha_1-t}+\binom{2r-2}{r-\alpha_1-t-1}$$ using Vandermonde’s identity.
Thus $F(r,(\alpha_i)_i,s)=G(r,(\alpha_i)_i,s)$ holds and therefore the sum of the multiplicities of all paths $\gamma$ in $\Gamma(s,(\alpha_i)_i)$ is equal to the sum of multiplicities of the corresponding paths $\gamma'$.
At last, note that any path in the dual of $\Delta_{{\mathbb{F}}_2}(2,b)$ which is not equal to $\gamma'$ for some $\gamma$ in the rectangle counts with multiplicity $0$, since we can not arrange parallelograms as needed (see again proposition 3.8 and remark 3.9 of [@GM052]):
To sum up, if $g>0$ then the number of lattice paths in the rectangle is equal to $$\begin{aligned}
\sum_{s=0}^{b-g+1} \sum_{(\alpha_i)_i} \sum_{\gamma\in \Gamma(s,(\alpha_i)_i)} \mult(\gamma)
=\sum_{s=0}^{b-g+1} \sum_{(\alpha_i)_i} \sum_{\gamma\in \Gamma(s,(\alpha_i)_i)} \mult(\gamma')\end{aligned}$$ and the right hand side covers all paths in $\Delta(2,b)$ which do not count zero. Thus $${\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2+b$})}= \sum_{\gamma} \mult(\gamma) =\sum_{\gamma}\mult(\gamma')={\tilde{\mathcal{N}}^{\emph{g}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$b$})}.$$ If $g=0$ then the number of lattice paths in the rectangle is equal to
$$\begin{aligned}
\mult(\gamma_0)+
&\sum_{s=0}^{b+1} \sum_{(\alpha_i)_i} \sum_{\gamma\in \Gamma(s,(\alpha_i)_i)} \mult(\gamma)\\
=&(b+2)+ \sum_{s=0}^{b} \sum_{(\alpha_i)_i} \sum_{\gamma\in \Gamma(s,(\alpha_i)_i)} \mult(\gamma')$$
and thus $${\tilde{\mathcal{N}}^{\emph{$0$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$2+b$})}={\tilde{\mathcal{N}}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$b$})}+(b+2){\tilde{\mathcal{N}}^{\emph{$0$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$1$},\emph{$b-2$})}.$$
Let $a=0$. Since we have to fit paths with $2b+g-1$ steps inside a line of integer length $b$, we get $g=-b+1$. Hence $g\geq 0$ if and only if $b=1$. This is the only case in which we have an irreducible curve. For $a=0$ and $b=1$, the equation trivially holds.
Let $a=1$. Since we have to fit $2b+g-1$ steps inside the polygons corresponding to $C+bF$ on ${\mathbb{F}}_2$ resp. $C+(b+1)F$ on ${\mathbb{F}}_0$, we have $g\leq 0$. Again, there is only one case in which we get an irreducible curve, namely $g=0$. For $g=0$, there is only one path in both polygons, it counts with multiplicity one on both sides and corresponds to an irreducible curve. Hence also in this case the equation is true.
Let $a=2$. We know that the equation of lemma \[lem-reducible\] is true, and we want to use it in order to deduce the equation of theorem \[thm-main\] by showing that the number of reducible curves on both sides agrees. We use induction on $b$. For $b=0$, we can see easily that there are no reducible curves for both sides, so the equation follows. Now we can assume that the equation is true for any $c<b$. Since $g\geq 0$, there are no reducible curves of degree $\Delta_{{\mathbb{F}}_2}(1,b+2)$ that contribute to the right hand side. How many reducible curves of degree $\Delta_{{\mathbb{F}}_2}(2,b)$ are there? A reducible curve $C$ could either be equal to $(\bigcup_{j=1}^i C_j)\cup C'$, $i\geq 1$, where each $C_j$ is of degree $\Delta_{{\mathbb{F}}_2}(0,1)$ and $C'$ is of degree $\Delta_{{\mathbb{F}}_2}(2,b')$, where $i+b'=b$, or it could be $(\bigcup_{j=1}^i C_j)\cup \tilde{C}_1\cup \tilde{C}_2$, $i\geq 0$, where again each $C_j$ is of degree $\Delta_{{\mathbb{F}}_2}(0,1)$, $\tilde{C}_1$ is of degree $\Delta_{{\mathbb{F}}_2}(1,b_1)$ and $\tilde{C}_2$ is of degree $\Delta_{{\mathbb{F}}_2}(1,b_2)$, with $b_1+b_2+i=b$. In the first case, we have $i\cdot 0+g'-(i+1)+1=g$ if $g'$ is the genus of $C'$, so $g'=g+i$. Since $g'$ is less than or equal to the number of interior points in the polygon dual to $\Delta_{{\mathbb{F}}_2}(2,b')$, it is less than or equal to $b'+1=b-i+1$. Thus $2i\leq b+1-g$ or $i\leq \lfloor \frac{b+1-g}{2} \rfloor=:h$. Hence from this first case we get a contribution of $\sum_{i=1}^{h} {\mathcal{N}^{\emph{$g+i$}}_{\mathbb{F}_{\emph{$2$}}}(\emph{$2$},\emph{$b-i$})}$ of reducible curves. In the second case, we can show analogously that $g_1+g_2=g+i+1$. But since $\tilde{C}_1$ and $\tilde{C}_2$ are of degree $\Delta_{{\mathbb{F}}_2}(1,b_1)$ resp. $\Delta_{{\mathbb{F}}_2}(1,b_2)$, we have $g_1=g_2=0$, hence $g+i+1=0$ which is not possible since $g\geq 0$. Thus by the induction assumption we know that the total contribution of reducible curves on the right hand side equals $\sum_{i=1}^{h} {\mathcal{N}^{\emph{$g+i$}}_{\mathbb{F}_{\emph{$0$}}}(\emph{$2$},\emph{$b-i+2$})}$. It is easy to see following the same arguments that we have the same contribution on the left hand side. Thus theorem \[thm-main\] follows.
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---
abstract: 'We present a high-order discontinuous Galerkin ([<span style="font-variant:small-caps;">dg</span>]{}) solver of the compressible Navier-Stokes equations for cloud formation processes. The scheme exploits an underlying parallelized implementation of the [<span style="font-variant:small-caps;">ader-dg</span>]{} method with dynamic adaptive mesh refinement. We improve our method by a [<span style="font-variant:small-caps;">pde</span>]{}-independent general refinement criterion, based on the local total variation of the numerical solution. While established methods use numerics tailored towards the specific simulation, our scheme works scenario independent. Our generic scheme shows competitive results for both classical <span style="font-variant:small-caps;">cfd</span> and stratified scenarios. We focus on two dimensional simulations of two bubble convection scenarios over a background atmosphere. The largest simulation here uses order 6 and 6561 cells which were reduced to 1953 cells by our refinement criterion.'
author:
- Lukas Krenz
- Leonhard Rannabauer
- Michael Bader
bibliography:
- 'CP032\_bibliography.bib'
title: 'A High-Order Discontinuous Galerkin Solver with Dynamic Adaptive Mesh Refinement to Simulate Cloud Formation Processes '
---
Introduction
============
In this paper we address the resolution of basic cloud formation processes on modern super computer systems. The simulation of cloud formations, as part of convective processes, is expected to play an important role in future numerical weather prediction [@bauer2015quiet]. This requires both suitable physical models and effective computational realizations. Here we focus on the simulation of simple benchmark scenarios [@giraldo2008study]. They contain relatively small scale effects which are well approximated with the compressible Navier-Stokes equations. We use the [<span style="font-variant:small-caps;">ader-dg</span>]{} method of [@dumbser2008unified], which allows us to simulate the Navier-Stokes equations with a space-time-discretization of arbitrary high order. In contrast to Runge-Kutta time integrators or semi-implicit methods, an increase of the order of [<span style="font-variant:small-caps;">ader-dg</span>]{} only results in larger computational kernels and does not affect the complexity of the scheme. Additionally, [<span style="font-variant:small-caps;">ader-dg</span>]{} is a communication avoiding scheme and reduces the overhead on larger scale. We see our scheme in the regime of already established methods for cloud simulations, as seen for example in [@giraldo2008study; @muller2010adaptive; @muller2018strong].
Due to the viscous components of the Navier-Stokes equations, it is not straightforward to apply the [<span style="font-variant:small-caps;">ader-dg</span>]{} formalism of [@dumbser2008unified], which addresses hyperbolic systems of partial differentials equations ([<span style="font-variant:small-caps;">pde</span>]{}s) in first-order formulation. To include viscosity, we use the numerical flux for the compressible Navier-Stokes equations of Gassner et al. [@gassner2008discontinuous]. This flux has already been applied to the [<span style="font-variant:small-caps;">ader-dg</span>]{} method in [@dumbser2010arbitrary]. In contrast to this paper, we focus on the simulation of complex flows with a gravitational source term and a realistic background atmosphere. Additionally, we use adaptive mesh refinement (<span style="font-variant:small-caps;">amr</span>) to increase the spatial resolution in areas of interest. This has been shown to work well for the simulation of cloud dynamics [@muller2010adaptive]. Regarding the issue of limiting in high-order [<span style="font-variant:small-caps;">dg</span>]{} methods, we note that viscosity not only models the correct physics of the problem but also smooths oscillations and discontinuities, thus stabilizing the simulation.
We base our work on the [[ExaHyPE Engine]{}]{} ([www.exahype.eu](www.exahype.eu)), which is a framework that can solve arbitrary hyperbolic [<span style="font-variant:small-caps;">pde</span>]{} systems. A user of the engine is provided with a simple code interface which mirrors the parts required to formulate a well-posed Cauchy problem for a system of hyperbolic [<span style="font-variant:small-caps;">pde</span>]{}s of first order. The underlying [<span style="font-variant:small-caps;">ader-dg</span>]{} method, parallelization techniques and dynamic adaptive mesh refinement are available for simulations while the implementations are left as a black box to the user. An introduction to the communication-avoiding implementation of the whole numerical scheme can be found in [@charrier2018stop].
To summarize, we make the following contributions in this paper:
- We extend the [[ExaHyPE Engine]{}]{} to allow viscous terms.
- We thus provide an implementation of the compressible Navier-Stokes equations. In addition, we tailor the equation set to stratified flows with gravitational source term. We emphasize that we use a standard formulation of the Navier-Stokes equations as seen in the field of computational fluid mechanics and only use small modifications of the governing equations, in contrast to a equation set that is tailored exactly to the application area.
- We present a general <span style="font-variant:small-caps;">amr</span>-criterion that is based on the detection of outlier cells w.r.t. their total variation. Furthermore, we show how to utilize this criterion for stratified flows.
- We evaluate our implementation with standard <span style="font-variant:small-caps;">cfd</span> scenarios and atmospheric flows and inspect the effectiveness of our proposed [<span style="font-variant:small-caps;">amr</span>]{}-criterion. We thus inspect, whether our proposed general implementation can achieve results that are competitive with the state-of-the-art models that rely on heavily specified equations and numerics.
Equation Set
============
The compressible Navier-Stokes equations in the conservative form are given as$$\label{eq:equation-set}
\quad
\pdv{}{t}
\underbrace{
\begin{pmatrix}
{\rho}\\
{\rho \bm{v}}\\
{\rho E}\end{pmatrix}}_{{\bm{Q}}}
+
\divergence{
\underbrace{
\left(
\underbrace{{
\begin{pmatrix}
{\rho \bm{v}}\\
{\bm{v}}\otimes {\rho \bm{v}}+ \bm{I} {p}\\
{\bm{v}}\cdot (\bm{I} {\rho E}+ \bm{I} {p})
\end{pmatrix}
}}_{{{\bm{F}}^{h}}({\bm{Q}})}
+
\underbrace{{
\begin{pmatrix}
0\\
{\bm{\sigma}}({\bm{Q}}, {\gradient{{\bm{Q}}}}) \\
{\bm{v}}\cdot {\bm{\sigma}}({\bm{Q}}, {\gradient{{\bm{Q}}}}) - \kappa \gradient{T}
\end{pmatrix}
}}_{{{\bm{F}}^{v}}({\bm{Q}}, {\gradient{{\bm{Q}}}})}
\right)}_{{\bm{F}}({\bm{Q}}, {\gradient{{\bm{Q}}}})}}
=
\underbrace{
\begin{pmatrix}
{
\notblank{{\rho}\phantom{{\rho}}}{
S_{{\rho}\phantom{{\rho}}}
}{
\bm{S}
}
}\\
{
\notblank{{\rho \bm{v}}}{
S_{{\rho \bm{v}}}
}{
\bm{S}
}
}\\
{
\notblank{{\rho E}}{
S_{{\rho E}}
}{
\bm{S}
}
}
\end{pmatrix}}_{{
\notblank{}{
S_{}
}{
\bm{S}
}
}({\bm{Q}}, \bm{x}, t)}$$ with the vector of conserved quantities ${\bm{Q}}$, flux ${\bm{F}}({\bm{Q}}, {\gradient{{\bm{Q}}}})$ and source ${
\notblank{}{
S_{}
}{
\bm{S}
}
}({\bm{Q}})$. Note that the flux can be split into a hyperbolic part ${{\bm{F}}^{h}}({\bm{Q}})$, which is identical to the flux of the Euler equations, and a viscous part ${{\bm{F}}^{v}}({\bm{Q}}, {\gradient{{\bm{Q}}}})$. The conserved quantities ${\bm{Q}}$ are the density ${\rho}$, the two or three-dimensional momentum ${\rho \bm{v}}$ and the energy density ${\rho E}$. The rows of \[eq:equation-set\] are the conservation of mass, the conservation of momentum and the conservation of energy.
The pressure ${p}$ is given by the equation of state of an ideal gas $$\label{eq:eos}
{p}= (\gamma - 1) \left({\rho E}- \frac{1}{2} \left({\bm{v}}\cdot {\rho \bm{v}}\right) - gz \right).$$ The term $gz$ is the geopotential height with the gravity of Earth $g$ [@giraldo2008study]. The temperature $T$ relates to the pressure by the thermal equation of state $$\label{eq:temperature}
{p}= {\rho}R T,$$ where $R$ is the specific gas constant of a fluid.
We model the diffusivity by the stress tensor $$\label{eq:stress-tensor}
{\bm{\sigma}}({\bm{Q}}, {\gradient{{\bm{Q}}}}) =
\mu
\bigl(
\left(\nicefrac{2}{3} \divergence{{\bm{v}}} \right) -
\left( \gradient{{\bm{v}}} + \gradient{{\bm{v}}}^\intercal \right)
\bigr),$$ with constant viscosity $\mu$. The heat diffusion is governed by the coefficient $$\label{eq:heat-conduction-coeff}
\kappa = \frac{\mu \gamma}{\Pr} \frac{1}{\gamma - 1} R = \frac{\mu c_p}{\Pr},$$ where the ratio of specific heats $\gamma$, the heat capacity at constant pressure $c_p$ and the Prandtl number $\Pr$ depend on the fluid.
Many realistic atmospheric flows can be described by a perturbation over a background state that is in hydrostatic equilibrium
$$\label{eq:hydrostatic-balance}
\pdv{}{z} {\overline{{p}}}{\left (z \right )} = -g {\overline{{\rho}}}(z),$$
i.e. a state, where the pressure gradient is exactly in balance with the gravitational source term ${
\notblank{{\rho \bm{v}}}{
S_{{\rho \bm{v}}}
}{
\bm{S}
}
} = - \bm{k} {\rho}g$. The vector $\bm{k}$ is the unit vector pointing in $z$-direction. The momentum equation is dominated by the background flow in this case. Because this can lead to numerical instabilities, problems of this kind are challenging and require some care. To lessen the impact of this, we split the pressure ${p}= {\overline{{p}}}+ {p}'$ into a sum of the background pressure ${\overline{{p}}}(z)$ and perturbation ${p}'(\bm{x}, t)$. We split the density ${\rho}= {\overline{{\rho}}}+ {\rho}'$ in the same manner and arrive at $$\label{eq:momentum-equation-split}
\pdv{{\rho \bm{v}}}{t}+ \divergence{ \left(
{\bm{v}}\otimes {\rho \bm{v}}+ \bm{I} {p}'
\right)
} + {{\bm{F}}^{v}}_{\rho \bm{v}}=
-g \bm{k} {\rho}'.$$ Note that a similar and more complex splitting is performed in [@muller2010adaptive; @giraldo2008study]. In contrast to this, we use the true compressible Navier-Stokes equations with minimal modifications.
Numerics
========
The [[ExaHyPE Engine]{}]{} implements an [<span style="font-variant:small-caps;">ader-dg</span>]{}-scheme and a [<span style="font-variant:small-caps;">muscl</span>-Hancock]{} finite volume method. Both can be considered as instances of the more general <span style="font-variant:small-caps;">PnPm</span> schemes of [@dumbser2008unified]. We use a Rusanov-style flux that is adapted to [<span style="font-variant:small-caps;">pde</span>]{}s with viscous terms [@gassner2008discontinuous; @fambri2017space]. The finite volume scheme is stabilized with the van Albada limiter [@van1997comparative]. The user can state dynamic [<span style="font-variant:small-caps;">amr</span>]{} rules by supplying custom criteria that are evaluated point-wise. Our criterion uses an element-local error estimate based on the total variation of the numerical solution. We exploit the fact that the total variation of a numerical solution is a perfect indicator for edges of a wavefront. Let $\bm{f}(\bm{x}): \mathbb{R}^{N_\text{vars}} \to \mathbb{R}$ be a sufficiently smooth function that maps the discrete solution at a point $\bm{x}$ to an arbitrary indicator variable. The total variation (<span style="font-variant:small-caps;">tv</span>) of this function is defined by $$\label{eq:tv}
{\operatorname{TV}}\left[ f(\bm{x}) \right] =
\left\Vert
{\int_{{C_{}}} \vert \gradient{f \left( \bm{x} \right)} \vert \dd{\bm{x}}}
\right\Vert_1$$ for each cell. The operator $\Vert \cdot \Vert_1$ denotes the discrete $L_1$ norm in this equation. We compute the integral efficiently with Gaussian quadrature over the collocated quadrature points.
How can we decide whether a cell is important or not? To resolve this conundrum, we compute the mean and the population standard deviation of the total variation of all cells. It is important that we use the method of [@chan1982updating] to compute the modes in a parallel and numerical stable manner. A cell is then considered to contain significant information if its deviates from the mean more than a given threshold. This criterion can be described formally by $$\label{eq:refinement-criterion}
\operatorname{evaluate-refinement}({\bm{Q}}, \mu, \sigma) =
\begin{cases}
\text{refine} & \text{if } {\operatorname{TV}}({\bm{Q}}) \geq \mu + {T_\text{refine}}\sigma, \\
\text{coarsen} & \text{if } {\operatorname{TV}}({\bm{Q}}) < \mu + {T_\text{coarsen}}\sigma, \\
\text{keep} & \text{otherwise}.
\end{cases}$$ The parameters ${T_\text{refine}}> {T_\text{coarsen}}$ can be chosen freely. Chebyshev’s inequality $$\label{eq:chebychev}
\mathbb{P}\bigl(\vert X - \mu \vert \geq c \sigma \bigr) \leq \frac{1}{c^2},$$ with probability $\mathbb{P}$ guarantees that we neither mark all cells for refinement nor for coarsening. This inequality holds for arbitrary distributions under the weak assumption that they have a finite mean $\mu$ and a finite standard deviation $\sigma$ [@wasserman2004all]. Note that subcells are coarsened only if all subcells belonging to the coarse cell are marked for coarsening. In contrast to already published criteria which are either designed solely for the simulation of clouds [@muller2010adaptive] or computationally expensive [@fambri2017space], our criterion works for arbitrary [<span style="font-variant:small-caps;">pde</span>]{}s and yet, is easy to compute and intuitive.
Results {#sec:results}
=======
In this section, we evaluate the quality of the results of our numerical methods and the scalability of our implementation. We use a mix of various benchmarking scenarios. After investigating the numerical convergence rate, we look at three standard <span style="font-variant:small-caps;">cfd</span> scenarios: the Taylor-Green vortex, the three-dimensional Arnold-Beltrami-Childress flow and a lid-driven cavity flow. Finally, we evaluate the performance for stratified flow scenarios in both two and three dimensions.
CFD Testing Scenarios
---------------------
We begin with a manufactured solution scenario which we can use for a convergence test. We use the following constants of fluids for all scenarios in this section: $$\gamma = 1.4, \quad \Pr = 0.7, \quad c_v = 1.0.$$ Our description of the manufactured solution follows [@dumbser2010arbitrary]. To construct this solution, we assume that $$\begin{aligned}
\label{eq:manufactured-solution}
\begin{split}
{p}(\bm{x}, t) &= \nicefrac{1}{10} \cos( \bm{k} \bm{x} - 2 \pi t ) + \nicefrac{1}{\gamma}, \\
{\rho}(\bm{x}, t) &= \nicefrac{1}{2} \sin (\bm{k} \bm{x} - 2 \pi t) + 1, \\
{\bm{v}}(\bm{x}, t) &= \bm{v_0} \sin(\bm{k} \bm{x} - 2 \pi t),
\end{split}
\end{aligned}$$ solves our [<span style="font-variant:small-caps;">pde</span>]{}. We use the constants $\bm{v_0} = \nicefrac{1}{4} \left( 1, 1 \right)^\intercal$, $\bm{k} = \nicefrac{\pi}{5} \left( 1, 1 \right)^\intercal$ and simulate a domain of size $\left[ 10 \times 10 \right]$ for $\SI{0.5}{\s}$. The viscosity is set to $\mu = 0.1$. Note that does not solve the compressible Navier-Stokes equations \[eq:equation-set\] directly. It rather solves our equation set with an added source term which can be derived with a computer algebra system. We ran this for a a combination of orders $1, \ldots, 6$ and multiple grid sizes. Note that by order we mean the polynomial order throughout the entire paper and not the theoretical convergence order. For this scenario, we achieve high-order convergence (\[fig:convergence-test\]) but notice some diminishing returns for large orders.
![\[fig:convergence-test\]Mesh size vs. error for various polynomial orders $P$. Dashed lines show the theoretical convergence order of $P+1$.](CP032_fig1)
After we have established that the implementation of our numerical method converges, we are going to investigate three established testing scenarios from the field of computational fluid mechanics. A simple scenario is the Taylor-Green vortex. Assuming an *incompressible* fluid, it can be written as $$\begin{aligned}
\label{eq:taylor-green}
\begin{split}
{\rho}(\bm{x}, t) &= 1,\\
{\bm{v}}(\bm{x}, t) &= \exp(-2 \mu t)
\begin{pmatrix}
\phantom{-}\sin(x) \cos(y) \\
- \cos(x) \sin(y)
\end{pmatrix}, \\
{p}(\bm{x}, t) &= \exp(-4 \mu t) \, \nicefrac{1}{4} \left( \cos(2x) + \cos(2y) \right) + C.
\end{split}\end{aligned}$$ The constant $C = \nicefrac{100}{\gamma}$ governs the speed of sound and thus the Mach number $\text{Ma} = 0.1$ [@dumbser2016high]. The viscosity is set to $\mu = 0.1$.
We simulate on the domain $[0,2\pi]^2$ and impose the analytical solution at the boundary. A comparison at time $t = 10.0$ of the analytical solution for the pressure with our approximation (\[fig:taylor-green\]) shows excellent agreement. Note that we only show a qualitative analysis because this is not an exact solution for our equation set as we assume compressibility of the fluid. This is nevertheless a valid comparison because for very low Mach numbers, both incompressible and compressible equations behave in a very similarly. We used an [<span style="font-variant:small-caps;">ader-dg</span>]{}-scheme of order $5$ with a grid of $25^2$ cells.
[0.473]{} ![\[fig:cdf-results\]Two-dimensional <span style="font-variant:small-caps;">cfd</span> scenarios](CP032_fig2 "fig:")
[.473]{} ![\[fig:cdf-results\]Two-dimensional <span style="font-variant:small-caps;">cfd</span> scenarios](CP032_fig3 "fig:")
The Arnold-Beltrami-Childress (<span style="font-variant:small-caps;">abc</span>) flow is similar to the Taylor-Green vortex but is an analytical solution for the three-dimensional *incompressible* Navier-Stokes equations [@tavelli2016staggered]. It is defined in the domain $ \left[ -\pi, \pi \right]^3 $ as $$\begin{aligned}
\label{eq:abc-flow}
\begin{split}
{\rho}(\bm{x}, t) &= 1,\\
{\bm{v}}(\bm{x}, t) &= \phantom{-} \exp(-1\mu t)
\begin{pmatrix}
\sin(z) + \cos(y)\\
\sin(x) + \cos(z)\\
\sin(y) + \cos(x)
\end{pmatrix}, \\
{p}(\bm{x}, t) &= -\exp(-2 \mu t) \, \left(\cos(x)\sin(y) + \sin(x)\cos(z) + \sin(z)\cos(y)\right)
+ C.
\end{split}\end{aligned}$$ The constant $C = \nicefrac{100}{\gamma}$ is chosen as before. We use a viscosity of $\mu = 0.01$ and analytical boundary conditions. Our results (\[fig:abc-flow\]) show a good agreement between the analytical solution and our approximation with an [<span style="font-variant:small-caps;">ader-dg</span>]{}-scheme of order $3$ with a mesh consisting of $27^3$ cells at time $t = \SI{0.1}{\s}$. Again, we do not perform a quantitative analysis as the <span style="font-variant:small-caps;">abc</span>-flow only solves our equation set approximately.
![\[fig:cavity-flow\]Our approximation (solid lines) of the lid-driven cavity flow vs. reference solution (crosses) of [@ghia1982high]. The respective other coordinate is held constant at a value of 0.](CP032_fig4)
![\[fig:cavity-flow\]Our approximation (solid lines) of the lid-driven cavity flow vs. reference solution (crosses) of [@ghia1982high]. The respective other coordinate is held constant at a value of 0.](CP032_fig5){width="\textwidth"}
As a final example of standard flow scenarios, we consider the lid-driven cavity flow where the fluid is initially at rest, with ${\rho}= 1$ and $ {p}(\bm{x}) = \nicefrac{100}{\gamma}$. We consider a domain of size $\SI{1}{m} \times \SI{1}{\m}$ which is surrounded by no-slip walls. The flow is driven entirely by the upper wall which has a velocity of $v_x = \SI{1}{\m/\s}$. The simulation runs for $\SI{10}{\s}$. Again, our results (\[fig:cavity-flow\]) have an excellent agreement with the reference solution of [@ghia1982high]. We used an [<span style="font-variant:small-caps;">ader-dg</span>]{}-method of order $3$ with a mesh of size $27^2$.
Stratified Flow Scenarios
-------------------------
Our main focus is the simulation of stratified flow scenarios. In the following, we present bubble convection scenarios in both two and three dimensions. With the constants $$\label{eq:atmosphere-constants}
\gamma = 1.4 ,\quad \Pr = 0.71 ,\quad R = 287.058 ,\quad p_0 = 10^5 \SI{}{\Pa}, \quad g = \SI{9.8}{m/s^2},$$ all following scenarios are described in terms of the potential temperature $${\theta}= T \left( \frac{p_0}{p} \right)^{R/c_p},$$ with reference pressure $p_0$ [@muller2010adaptive; @giraldo2008study]. We compute the initial background density and pressure by inserting the assumption of a constant background energy in \[eq:hydrostatic-balance\]. The background atmosphere is then perturbed. We set the density and energy at the boundary such that it corresponds to the background atmosphere. Furthermore, to ensure that the atmosphere stays in hydrostatic balance, we need to impose the viscous heat flux $$\label{eq:atmosphere-bc}
{{\bm{F}}^{v}}_{{\rho E}} = \kappa \pdv{\overline{T}}{z}.$$ at the boundary [@giraldo2008study]. In this equation, $\overline{T}(z)$ is the background temperature at position $z$, which can be computed from \[eq:hydrostatic-balance,eq:eos\].
Our first scenario is the colliding bubbles scenario [@muller2010adaptive]. We use perturbations of the form $$\label{eq:bubbles-pertubation}
{\theta'}=
\begin{cases}
A & r \leq a, \\
A \exp \left( - \frac{(r-a)^2}{s^2} \right) & r > a,
\end{cases}$$ where $s$ is the decay rate and $r$ is the radius to the center $$\label{eq:radius}
r^2 = \Vert \bm{x} - \bm{x_c} \Vert_2,$$ i.e., $r$ denotes the Euclidean distance between the spatial positions $\bm{x} = (x, z)$ and the center of a bubble $\bm{x_c} = (x_c, z_c)$ – for three-dimensional scenarios $\bm{x}$ and $\bm{x_c}$ also contain a $y$ coordinate.
![\[fig:two-bubbles-ader\]Left: Colliding Bubbles with [<span style="font-variant:small-caps;">ader-dg</span>]{}. Contour values for potential temperature perturbation are $-0.05, 0.05, 0.1, \ldots 0.45$.\
Right: Comparison of small scale structure between order 3 (top) and order 6 (bottom).](CP032_fig6){width="100.00000%"}
We have two bubbles, with constants $$\label{eq:bubbles-values}
\begin{alignedat}{6}
& \text{warm:} \qquad && A = \SI{0.5}{\K}, \quad&& a = \SI{150}{\m}, \quad&& s = \SI{50}{\m}, \quad&& x_c = \SI{500}{\m,} \quad&& z_c = \SI{300}{\m},\\
& \text{cold:} \qquad && A = \SI{-0.15}{\K}, \quad&& a = \SI{0}{\m}, \quad&& s = \SI{50}{\m}, \quad&& x_c = \SI{560}{\m}, \quad&& z_c = \SI{640}{\m}.
\end{alignedat}$$ Similar to [@muller2010adaptive], we use a constant viscosity of $\mu = 0.001$ to regularize the solution. Note that we use a different implementation of viscosity than [@muller2010adaptive]. Hence, it is difficult to compare the parametrization directly. We ran this scenario twice: once without [<span style="font-variant:small-caps;">amr</span>]{} and a mesh of size $\SI{1000/81}{\m} = \SI{12.35}{\m}$ and once with [<span style="font-variant:small-caps;">amr</span>]{} with two adaptive refinement levels and parameters ${T_\text{refine}}= 2.5$ and ${T_\text{coarsen}}= -0.5$. For both settings we used polynomials of order 6. We specialize the [<span style="font-variant:small-caps;">amr</span>]{}-criterion to our stratified flows by using the potential temperature. This resulted in a mesh with cell-size lengths of approx. , , and . The resulting mesh can be seen in \[fig:two-bubbles-ader\].
We observe that the $L_2$ difference between the potential temperature of the [<span style="font-variant:small-caps;">amr</span>]{} run, which uses 1953 cells, and the one of the fully refined run with 6561 cells, is only $1.87$. The relative error is . We further emphasize that our [<span style="font-variant:small-caps;">amr</span>]{}-criterion accurately tracks the position of the edges of the cloud instead of only its position. This is the main advantage of our gradient-based method in contrast to methods working directly with the value of the solution, as for example [@muller2010adaptive]. Overall, our result for this benchmark shows an excellent agreement to the previous solutions of [@muller2010adaptive]. In addition, we ran a simulation with the same settings for a polynomial order of 3. The lower resolution leads to spurious waves (\[fig:two-bubbles-ader\]) and does not capture the behavior of the cloud.
Furthermore, we simulated the same scenario with our [<span style="font-variant:small-caps;">muscl</span>-Hancock]{} method, using $7^2$ patches with $90^2$ finite volume cells each. As we use limiting, we do not need any viscosity. The results of this method (\[fig:two-bubbles-fv\]) also agree with the reference but contain fewer details. Note that the numerical dissipativity of the finite volume scheme has a smoothing effect that is similar to the smoothing caused by viscosity.
For our second scenario, the cosine bubble, we use a perturbation of the form $$\begin{aligned}
\label{eq:cos-pertubation}
{\theta'}&= \begin{cases}
\nicefrac{A}{2} \left[ 1 + \cos(\pi r) \right] & r \leq a, \\
0 & r > a,
\end{cases}\end{aligned}$$ where $A$ denotes the maximal perturbation and $a$ is the size of the bubble. We use the constants $$\label{eq:cosine-bubble}
A = \SI{0.5}{\K}, \quad a = \SI{250}{\m}, \quad x_c = \SI{500}{\m}, \quad z_c = \SI{350}{\m}.$$ For the three-dimensional bubble, we set $y_c = x_c = \SI{500}{\m}$. This corresponds to the parameters used in [@kelly2012continuous][^1]. For the 2D case, we use a constant viscosity of $\mu = 0.001$ and an [<span style="font-variant:small-caps;">ader-dg</span>]{}-method of order 6 with two levels of dynamic [<span style="font-variant:small-caps;">amr</span>]{}, resulting again in cell sizes of roughly $\SI{111.1}{\m}, \SI{37.04}{\m}, \SI{12.34}{\m}$. We use slightly different [<span style="font-variant:small-caps;">amr</span>]{} parameters of ${T_\text{refine}}= 1.5$ and ${T_\text{coarsen}}= -0.5$ and let the simulation run for . Note that, as seen in \[fig:cosine-2d\], our [<span style="font-variant:small-caps;">amr</span>]{}-criterion tracks the wavefront of the cloud accurately. This result shows an excellent agreement to the ones achieved in [@giraldo2008study; @muller2010adaptive].
For the 3D case, we use an [<span style="font-variant:small-caps;">ader-dg</span>]{}-scheme of order 3 with a static mesh with cell sizes of and a shorter simulation duration of . Due to the relatively coarse resolution and the hence increased aliasing errors, we need to increase the viscosity to $\mu = 0.005$. This corresponds to a larger amount of smoothing. Our results (\[fig:cosine-3d\]) capture the dynamics of the scenario well and agree with the reference solution of [@kelly2012continuous].
[0.5]{}
[0.5]{}
Scalability
-----------
All two-dimensional scenarios presented in this paper can be run on a single workstation in less than two days. Parallel scalability was thus not the primary goal of this paper. Nevertheless, our implementation allows us to scale to small to medium scale setups using a combined [<span style="font-variant:small-caps;">mpi</span>]{} + Thread building blocks ([<span style="font-variant:small-caps;">tbb</span>]{}) parallelization strategy, which works as follows: We typically first choose a number of [<span style="font-variant:small-caps;">mpi</span>]{} ranks that ensure an equal load balancing. [[ExaHyPE]{}]{} achieves best scalability for $1, 10, 83, \ldots$ ranks, as our underlying framework uses three-way splittings for each level and per dimension and an additional communication rank per level. For the desired number of compute nodes, we then determine the number of [<span style="font-variant:small-caps;">tbb</span>]{} threads per rank to match the number of total available cores.
We ran the two bubble scenario for a uniform grid with a mesh size of $729 \times 729$ with order 6, resulting in roughly 104 million degrees of freedom (<span style="font-variant:small-caps;">dof</span>), for 20 timesteps and for multiple combinations of [<span style="font-variant:small-caps;">mpi</span>]{} ranks and [<span style="font-variant:small-caps;">tbb</span>]{} threads. This simulation was performed on the SuperMUC-NG system using computational kernels that are optimized for its Skylake architecture. Using a single [<span style="font-variant:small-caps;">mpi</span>]{} rank, we get roughly $4.9$ millions <span style="font-variant:small-caps;">dof</span> updates (${\text{MDOF/s}}$) using two [<span style="font-variant:small-caps;">tbb</span>]{} threads and $20.2 {\text{MDOF/s}}$ using 24 threads (i.e. a half node). For a full node with 48 threads, we get a performance of $12 {\text{MDOF/s}}$. When using 5 nodes with 10 [<span style="font-variant:small-caps;">mpi</span>]{} ranks, we achieve $29.3 {\text{MDOF/s}}$ for two threads and $137.3 {\text{MDOF/s}}$ for 24 threads.
We further note that for our scenarios weak scaling is more important than strong scaling, as we currently cover only a small area containing a single cloud, where in practical applications one would like to simulate more complex scenarios.
Conclusion
==========
We presented an implementation of a [<span style="font-variant:small-caps;">muscl</span>-Hancock]{}-scheme and an [<span style="font-variant:small-caps;">ader-dg</span>]{}-method with [<span style="font-variant:small-caps;">amr</span>]{} for the Navier-Stokes equations, based on the [[ExaHyPE Engine]{}]{}. Our implementation is capable of simulating different scenarios: We show that our method has high order convergence and we successfully evaluated our method for standard <span style="font-variant:small-caps;">cfd</span> scenarios: We have competitive results for both two-dimensional scenarios (Taylor-Green vortex and lid-driven cavity) and for the three-dimensional <span style="font-variant:small-caps;">abc</span>-flow.
Furthermore, our method allows us to simulate flows in hydrostatic equilibrium correctly, as our results for the cosine and colliding bubble scenarios showed. We showed that our [<span style="font-variant:small-caps;">amr</span>]{}-criterion is able to vastly reduce the number of grid cells while preserving the quality of the results.
Future work should be directed towards improving the scalability. With an improved [<span style="font-variant:small-caps;">amr</span>]{} scaling and some fine tuning of the parallelization strategy, the numerical method presented here might be a good candidate for the simulation of small scale convection processes that lead to cloud formation processes.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This work was funded by the European Union’s Horizon 2020 Research and Innovation Programme under grant agreements No 671698 (project ExaHyPE, [www.exahype.eu](www.exahype.eu)) and No 823844 (ChEESE centre of excellence, [www.cheese-coe.eu](www.cheese-coe.eu)). Computing resources were provided by the Leibniz Supercomputing Centre (project pr83no). Special thanks go to Dominic E. Charrier for his support with the implementation in the [[ExaHyPE Engine]{}]{}.
[^1]: We found that the parameters presented in the manuscript of [@kelly2012continuous] only agree with the results, if we use the same parameters as for 2D simulations.
|
---
abstract: 'Zeckendorf’s theorem states that every positive integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{F_n\}$, where we take $F_1=1$ and $F_2=2$; in fact, it provides an alternative definition of the Fibonacci numbers. This has been generalized for any Positive Linear Recurrence Sequence (PLRS), which is, informally, a sequence satisfying a homogeneous linear recurrence with a positive leading coefficient and non-negative integer coefficients. Note these legal decompositions are generalizations of base $B$ decompositions. We investigate linear recurrences with leading coefficient zero, followed by non-negative integer coefficients, with differences between indices relatively prime (abbreviated ZLRR), via two different approaches. The first approach involves generalizing the definition of a legal decomposition for a PLRS found in Koloğlu, Kopp, Miller and Wang. We prove that every positive integer $N$ has a legal decomposition for any ZLRR using the greedy algorithm. We also show that $D_n$, the number of decompositions of $n$, grows faster than $a_n$, implying the existence of decompositions for every positive integer $N$, but uniqueness is lost. The second approach converts a ZLRR to a PLRR that has the same growth rate. We develop the *Zeroing Algorithm*, a powerful helper tool for analyzing the behavior of linear recurrence sequences. We use it to prove a very general result that guarantees the possibility of conversion between certain recurrences, and develop a method to quickly determine whether our sequence diverges to $+\infty$ or $-\infty$, given any real initial values.'
address:
- 'Department of Mathematics, Harvey Mudd College, Claremont, CA 91711'
- 'Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267'
- 'Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267'
- 'Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267'
author:
- 'Thomas C. Martinez'
- 'Steven J. Miller'
- Clay Mizgerd
- Chenyang Sun
title: 'Generalizing Zeckendorf’s Theorem to Homogeneous Linear Recurrences'
---
Introduction and Definitions {#sec:intro}
============================
History and Past Results {#sec:history}
------------------------
The Fibonacci numbers are one of the most well-known and well-studied mathematical objects, and have captured the attention of mathematicians since their conception. This paper focuses on a generalization of Zeckendorf’s theorem, one of the many interesting properties of the Fibonacci numbers. Zeckendorf [@Ze] proved that every positive integer can be written **uniquely** as the sum of non-consecutive Fibonacci numbers (called the *Zeckendorf Decomposition*), where the Fibonacci numbers[^1] are $F_1 = 1, F_2 = 2, F_3 = 3, F_4 = 5, \dots$. This results has been generalized to other types of recurrence sequences. We set some notation before describing these results.
\[def:plrrdefinition\] We say a recurrence relation is a **Positive Linear Recurrence Relation (PLRR)** if there are non-negative integers $L, c_1, \dots, c_L$ such that $$H_{n+1}\ =\ c_1\, H_n + \cdots + c_L\, H_{n+1-L},$$ with $L, c_1$ and $c_L$ positive.
\[def:plrsdefinition\] We say a sequence $\{H_n\}_{n=1}^{\infty}$ of positive integers arising from a PLRR is a **Positive Linear Recurrence Sequence (PLRS)** if $H_1=1$, and for $1 \leq n < L$ we have $$H_{n+1}\ =\ c_1\,H_n + c_2\,H_{n-2} + \cdots + c_n \,H_1 + 1.$$ We call a decomposition $\sum_{i=1}^m a_i H_{m+1-i}$ of a positive integer $N$ (and the sequence $\{a_i\}_{i=1}^m$) **legal** if $a_1>0$, the other $a_i\geq 0$, and one of the following two conditions hold.
- *Condition 1:* We have $m<L$ and $a_i = c_i$ for $1 \leq i \leq m$.
- *Condition 2:* There exists $s\in \{1,\dots,L\}$ such that $$a_1\ =\ c_1, \ \ a_2\ =\ c_2, \ \ \dots, \ \ a_{s-1}\ =\ c_{s-1}, \ \ a_s\ <\ c_s,$$ $a_{s+1}, \dots , a_{s+\ell} = 0$ for some $\ell \geq 0$, and $\{b_i\}_{i=1}^{m-s-\ell}$ (with $b_i = a_{s+\ell+i}$) is legal.
Informally, a legal decomposition is one where we cannot use the recurrence relation to replace a linear combination of summands with another summand, and the coefficient of each summand is appropriately bounded; other authors [@DG; @Ste] use the phrase $G$-ary decomposition for a legal decomposition. For example, if $H_{n+1} = 3H_n + 2H_{n-1} + 4H_{n-2}$, then $H_5 + 3H_4 + 2H_3 + 3H_2$ is legal, while $H_5 + 3H_4 + 2H_3 + 4H_2$ is not (we can replace $3H_4 + 2H_3 + 4H_2$ with $H_5$), nor is $6H_5+2H_4$ (the coefficient of $H_5$ is too large).\
We now state an important generalization, and then describe what object we are studying and our results. See [@BBGILMT; @BM; @BCCSW; @CFHMN; @CFHMNPX; @DFFHMPP; @Ho; @MNPX; @MW; @Ke; @Len] for more on generalized Zeckendorf decompositions and [@GT; @MW] for a proof of Theorem \[thm:genzeckthmforPLRS\].
\[thm:genzeckthmforPLRS\] Let $\{H_n\}_{n=1}^{\infty}$ be a *Positive Linear Recurrence Sequence*. Then
1. there is a unique legal decomposition for each non-negative integer $N \geq 0$, and
2. there is a bijection between the set $\mathcal{S}_n$ of integers in $[H_n,H_{n+1})$ and the set $\mathcal{D}_n$ (of cardinality $D_n$) of legal decompositions $\sum_{i=1}^n a_i\, H_{n+1-i}$.
While this result is powerful and generalizes Zeckendorf’s theorem to a large class of recurrence sequences, it is restrictive in that the leading term must have a positive coefficient. We examine what happens in general to existence and uniqueness of legal decompositions if $c_1=0$. Special cases were studied in [@CFHMN; @CFHMNPX], focusing on the Kentucky and $(s,b)$-Generacci Sequences; the first still had uniqueness of decomposition while the second did not.
\[def:zlrrdefinition\] We say a recurrence relation is an $s$-deep **Zero Linear Recurrence Relation (ZLRR)** if the following properties hold.
1. *Recurrence relation:* There are non-negative integers $s, L, c_1, \dots, c_L$ such that $$\label{eqn:zlrsrecurrence}
G_{n+1}\ =\ c_1 G_n + \cdots + c_s G_{n+1-s} + c_{s+1} G_{n-s}+ \cdots + c_L G_{n+1-L},$$ with $c_1,\dots, c_s = 0$ and $L, c_{s+1}, c_L$ positive.
2. *No degenerate sequences:* Let $S =\{m \mid c_m \neq 0\}$ be the set of indices of positive coefficients. Then $\gcd(S) = 1$.
We impose the second restriction, because studying a sequence like $G_{n+1} = G_{n-1} + G_{n-3}$, where the odd terms and even terms do not interact, is not desirable as such a sequence naturally splits into two separate, independent sequences. Also note that $0$-deep ZLRR’s are just PLRR’s, for which we can study their sequences very well. Notice that we do not define $s$-deep **Zero Linear Recurrence Sequences (ZLRS)**, which requires the definition of initial conditions and legal decompositions because those depend on how we study ZLRR’s. This paper offers two methods: generalizing Zeckendorf’s theorem to $s$-deep ZLRS’s and converting $s$-deep ZLRR’s to PLRR’s.\
However, before we can study the results of the two methods, we develop some important tools that are necessary for both. We do so in Section \[sec:tools\], mainly looking at characteristic polynomials of PLRR’s and $s$-deep ZLRR’s, and relating some properties to each other. We also look at a generalization of Binet expansions of recurrence sequences, which is more pertinent for the second method, that of converting $s$-deep ZLRR’s to PLRR’s.
Main Results {#sec:mainresults}
------------
In Section \[sec:zlrslegal\], we study the first method, generalizing Zeckendorf’s theorem to $s$-deep ZLRS’s. We begin here the initial conditions and legal decompositions.
\[def:sdeepzlrs\] We say a sequence $\{G_n\}_{n=1}^{\infty}$ of positive integers arising from an $s$-deep ZLRR is an $s$-deep **Zero Linear Recurrence Sequence (ZLRS)** if $G_1 = 1$, $G_2 = 2$, …, $G_{s+1} = s+1$ and for $s+2 \leq n \leq L$,
$$\label{eqn:initcondzlrs}
G_n = \begin{cases}n & c_{s+1} \leq s, \\ c_{s+1}\,G_{n-s+1} + c_{s+2}\,G_{n-s+2} + \cdots + c_{n-1}\, G_1+1 & c_{s+1} > s. \end{cases}$$
We call a decomposition $\sum_{i=1}^m a_i\, G_{m+1-i}$ of a positive integer $N$ (and the sequence $\{a_i\}_{i=1}^m$) **legal** if $a_i \geq 0$, and one of the following conditions hold.
- *Condition 1:* We have $a_1 = 1$ and $a_i = 0$ for $2 \leq i \leq m$.
- *Condition 2:* We have $s<m<L$ and $a_i = c_i$ for $1\leq i\leq m$.
- *Condition 3:* There exists $t \in \{s+1,\dots,L\}$ such that $$a_1\ =\ c_1,\ \ a_2\ =\ c_2,\ \ \dots,\ \ a_{t-1}\ =\ c_{t-1},\ \ a_t\ <\ c_t,$$ $a_{t+1}, \dots, a_{t+\ell}=0$ for some $\ell \geq 0$, and $\{b_i\}_{i=1}^{m-t-\ell}$ (with $b_i = a_{t+\ell+i})$ is legal.
The idea behind Condition 1 is if $N$ appears in the sequence, say $N= G_n$, then we allow this to be a legal decomposition. This is necessary for there to be a legal decomposition for $N=1$ for all $s$-deep ZLRS’s.
\[rem:lagonacciexception\] We note one special case for the initial conditions. If $Z_{n+1} = Z_{n-1} + Z_{n-2}$ (a recurrence relation we call the “Lagonaccis” as it has a similar recurrence relation to the Fibonaccis, but the terms “lag" behind and grow slowly), then $Z_1 = 1$, $Z_2 = 2$, $Z_3 = 4$, $Z_4 = 3$, $Z_5 = 6$, and so on.[^2]
Similarly to the initial conditions of a PLRS, we construct our initial conditions in such a way to guarantee existence of legal decompositions. The main idea behind the definition of legal decompositions is if $N$ does not appear in the sequence (i.e., $N \neq G_n$ for any $n\in\mathbb{N}_0$), then for some $m\in\mathbb{N}_0$, $G_m \leq N < G_{m+1}$,[^3] and we **cannot** use $G_m, G_{m-1},\dots,G_{m-s+1}$ in our decomposition of $N$. Let us illustrate this with an example.
\[ex:lagonaccidecomp10\] Consider again the Lagonacci sequence $Z_{n+1} = Z_{n-1} + Z_{n-2}$, with the first terms $$1,\ 2,\ 4,\ 3,\ 6,\ 7,\ 9,\ 13,\ 16,\ \dots,$$ and let us decompose $N = 10$. Since $Z_7 = 9 \leq 10 < 13 = Z_8$, we ***cannot*** use $Z_7=9$ in our decomposition. So, we use the next largest number, $Z_6=7$, and get $10 = 7+3 = Z_6 + Z_4$. This is a legal 1-deep ZLRS decomposition; however, notice that we can also have $10 = 6+4 = Z_5 + Z_3$.
The above example suggests the following questions. *Is uniqueness of decomposition lost for all ZLRS’s? If so, is it lost for finitely many numbers? For infinitely many numbers? For all numbers from some point onward?*\
Our main results for this method are
\[thm:generalzeckZLRS\] Let $\{G_n\}_{n=1}^{\infty}$ be an $s$-deep *Zero Linear Recurrence Sequence.* Then there exists a legal decomposition for each non-negative integer $N\geq 0$.
\[thm:lossofuniqueness1ZLRS\] Let $\{G_n\}_{n=1}^{\infty}$ be an $s$-deep *Zero Linear Recurrence Sequence.* Uniqueness of decomposition is lost for at least one positive integer $N$. Further, the number of legal decompositions grows exponentially faster than the terms of our $s$-deep ZLRS.
The proof for Theorem \[thm:generalzeckZLRS\] is a fairly straightforward strong induction proof. The difficulty arises with the initial conditions, which are split into two cases. The main idea behind proving Theorem \[thm:lossofuniqueness1ZLRS\] relies on comparing the number of legal decompositions that a ZLRS creates to that of a related PLRS (see Definition \[def:fosteredPLRS\]), and showing that the number of legal decompositions grows faster than the term of the ZLRS. We prove many auxiliary results regarding characteristic polynomials to prove Theorem \[thm:generalzeckZLRS\].
\
We now state the main results of the second method, converting ZLRR’s to PLRR’s. We develop a powerful helper tool in analyzing linear recurrences, the **Zeroing Algorithm**; we give a full introduction of how it works in §\[sec:conversion\]. It is worth noting that this method has more uses than that of generalizing Zeckendorf’s theorem. As the first method required specific initial conditions, converting ZLRR’s to PLRR’s requires no specificity of initial conditions. We have yet to formally describe a manner to use this method to obtain meaningful results about decompositions, but our hope is that others can use the Zeroing Algorithm to do so. Before going further, we introduce an object crucial in the study of recurrence relations.
Given a recurrence relation $$\label{eq:recurrence}
a_{n+1}\ =\ c_1 a_n + \cdots + c_k a_{n+1-k},$$ we call the polynomial $$\label{eq:characteristicpolynomial}
P(x) \ = \ x^k - c_1\, x^{k-1}-c_2\, x^{k-2}-\cdots-c_k$$ the *characteristic polynomial* of the recurrence relation. The degree of $P(x)$ is known as the order of the recurrence relation.
We now state results relating to the second approach, which is converting any ZLRR into a PLRR derived from it in the following sense:
\[def:derivedfrom\] We say that a recurrence relation $R_b$ is *derived from* another recurrence relation $R_a$ if $$P_b(x) \ = \ P_a(x)Q(x),$$ where $P_a(x)$ and $P_b(x)$ are the characteristic polynomials of $R_a$ and $R_b$ respectively, as defined by equation , and $Q(x)$ is some polynomial with integer coefficients with $Q(x)$ not being the zero polynomial.
Since the roots of $P_a$ are contained in $P_b$, any sequence satisfying the recurrence relation $R_a$ also satisfies $R_b$, which means that the two recurrence relations yield the same sequence if the initial values of $\{b_n\}_{n=1}^{\infty}$ satisfy the recurrence relation $R_a$. This provides motivation for why the idea of a derived PLRR is relevant.\
To continue, it is pertinent to state an important result, which we prove in Section \[sec:tools\], specifically Lemma \[lem:greatestroot\]: the characteristic polynomial of any PLRR or ZLRR has a unique positive root of multiplicity 1 and magnitude greater than that of any other root. We call this the *principal root* of the characteristic polynomial, and denote it as $r$.\
We now state a main result, which has two important corollaries that guarantee the possibility of conversion between certain linear recurrences; the Zeroing Algorithm itself provides a constructive way to do so.
\[thm:algorithmdivisibility\] Given some PLRR/ZLRR, let $P(x)$ denote its characteristic polynomial, and $r$ its principal root. Suppose we are given an arbitrary sequence of real numbers $\gamma_1,\gamma_2,\dots,\gamma_m$, and define, for $t\le m$, $$\Gamma_t(x)\ := \ \gamma_1\,x^{t-1}+\gamma_2\,x^{t-2}+\cdots+\gamma_{t-1}\,x+\gamma_t.$$ If $\Gamma_m(r)>0$, there exists a polynomial $p(x)$, divisible by $P(x)$, whose first coefficients are $\gamma_1$ through $\gamma_m$, with no positive coefficients thereafter.
Given arbitrary integers $\gamma_1$ through $\gamma_m$ with $\Gamma_m(r)>0$, there is a recurrence derived from $P(x)$ which has first coefficients $\gamma_1$ through $\gamma_m$ with no negative coefficients thereafter.
\[cor: conversionBIG\] Every ZLRR has a derived PLRR.
A natural question of interest that arises in the study of recurrences is the behavior of the size of terms in a recurrence sequence. The Fibonacci sequence behaves like a geometric sequence whose ratio is the golden ratio, and there is an analogous result for general linear recurrence sequences, proven in [@BBGILMT]:
\[thm:binetexpansion\] Let $P(x)$ be the characteristic polynomial of some linear recurrence relation, and let the roots of $P(x)$ be denoted as $r_1,r_2,\cdots,r_j$, with multiplicities $m_1,m_2,\cdots,m_j\ge1$, respectively.\
Consider a sequence $\{a_n\}_{n=1}^{\infty}$ of complex numbers satisfying the recurrence relation. Then there exist polynomials $q_1,q_2,\cdots,q_j$, with $\deg(q_i)\le m_i-1$, such that $$\label{eq:binetexpansion}
a_n\ = \ q_1(n)\,r_1^n+q_2(n)\,r_2^n+\cdots+q_j(n)\,r_j^n.$$
We call the Binet expansion of the sequence $\{a_n\}_{n=1}^{\infty}$, in analogy to the Binet Formula that provides a closed form for Fibonacci numbers.
One might ask that given a PLRR/ZLRR with some real initial values, do the terms eventually diverge to positive infinity or negative infinity? One approach is to compute as many terms as needed for the eventual behavior to emerge; unfortunately, this could be very time-consuming. One could alternately solve for the Binet expansion, which often requires an excessive amount of computation.\
The fact that the characteristic polynomials for PLRR/ZLRR’s have a principal root $r$ allows us a shortcut. Consider the Binet expansion of a ZLRS/PLRS; the coefficient attached to the $r^n$ term, whenever nonzero, indicates the direction of divergence. We develop the following method to determine the sign of this coefficient from the initial values of the recurrence sequence:
\[thm:algorithm determination\] Given a ZLRS/PLRS $\{a_n\}_{n=1}^{\infty}$ with characteristic polynomial $P(x)$ and real initial values $a_1,a_2,\dots,a_k$, consider the Binet expansion of $\{a_n\}_{n=1}^{\infty}$. The sign of the coefficient attached to $r^n$ agrees with the sign of $$Q(x)\ := \ a_1\,x^{k-1}+(a_2-d_2)\,x^{k-2}+(a_3-d_3)\,x^{k-3}+\cdots+(a_k-d_k),$$ evaluated at $x=r$, where $$d_i\ = \ a_1\,c_{i-1}+a_2\,c_{i-2}+\cdots+a_{i-1}\,c_1\ = \ \sum_{j=1}^{i-1}\,a_j\,c_{i-j}.$$
We conclude in §\[sec:conclusion\] with some open questions for future research.
Eventual Behavior of Linear Recurrence Sequences {#sec:tools}
================================================
In this section, we prove important lemmas related to the roots of characteristic polynomials that are used with both methods. In the celebrated Binet’s Formula for Fibonacci numbers, the principal root of its characteristic polynomial (i.e., the golden ratio) determines the behavior of the sequence as nearly geometric, with the golden ratio being the common ratio. We generalize this characterization of near-geometric behavior to more general linear recurrences.
Properties of Characteristic Polynomials
----------------------------------------
We first prove a lemma regarding recurrence relations of the form , with $c_i$ non-negative integers for $1\leq i\leq k$ and $c_k>0$. We first justify the definition of the principal root.
\[lem:greatestroot\] Consider $P(x)$ as in and let $S:=\{m \, \mid \, c_m \ne 0\}$. Then
1. there exists exactly one positive root $r$, and this root has multiplicity $1$,
2. every root $z \in \mathbb{C}$ satisfies $|z|\le r$, and
3. if $\gcd(S) = 1$, then $r$ is the unique root of greatest magnitude.[^4]
By Descartes’s Rule of Signs, $P(x)$ has exactly one positive root of multiplicity one, completing the proof of Part (1).\
Now, consider any root $z\in\mathbb{C}$ of $P(x)$; we have $z^k=c_1z^{k-1}+c_2z^{k-2}+\cdots+c_k$. Taking the magnitude, we have $$\begin{aligned}
|z|^k\ = \ |z^k|\ = \ |c_1z^{k-1}+c_2z^{k-2}+\cdots+c_k|&\ \le\ |c_1z^{k-1}|+|c_2z^{k-2}|+\cdots+|c_k|\nonumber\\
&\ = \ c_1|z|^{k-1}+c_2|z|^{k-2}+\cdots+c_k,\end{aligned}$$ which means $P(|z|)\le0$. Since $P(x)$ becomes arbitrarily large with large values of $x$, we see that there is a positive root at or above $|z|$ by the Intermediate Value Theorem, which completes Part (2).\
Finally, suppose $\gcd(S) = 1$. Suppose for sake of contradiction that a non-positive root $z$ satisfies $|z| = r$; we must have $P(|z|)=0$, which means $$|z^k|\ = \ |c_1\,z^{k-1}+c_2\,z^{k-2}+\cdots+c_k|\ = \ |c_1\,z^{k-1}|+|c_2\,z^{k-2}|+\cdots+|c_k|.$$ This equality holds only if the complex numbers $c_1\,z^{k-1},c_2\,z^{k-2},\dots,c_k$ share the same argument; since $c_k>0$, $z^{k-j}$ must be positive for all $c_j\ne0$. This implies $z^k$, as a sum of positive numbers, is positive as well. Writing $z=|z|\,e^{i\theta}$, we see that the positivity of $z^k=|z|^k\,e^{ik\theta}$ implies $k\theta$ is a multiple of $2\pi$, and consequently, $\theta = 2\pi d / k$ for some integer $d$. We may reduce this to $2\pi d' / k'$ for relatively prime $d',k'$.\
Let $J:=S\cup\{0\}$. Since $z^{k-j}$ is positive for all $j\in J$, we see that $2\pi d'\,(k-j)/k'$ is an integer multiple of $2\pi$, so $k'$ divides $d'\,(k-j)$; as $d'$ and $k'$ are relatively prime we have $k'$ divides $k-j$. Since the elements of $J$ have greatest common divisor 1, so do[^5] the elements of $K:=\{k-j\mid j\in J\}$. Since $k'$ divides every element of $K$, we must have $k'=1$, so $\theta=2\pi d'$ and thus $z$ is a positive root. This is a contradiction, completing the proof of Part (3).
Next, we prove a lemma that sheds light on the growth rate of the terms of a ZLRR/PLRR with a specific set of initial values.
\[lem:monotonicallyincreasinglessstrong\] For a PLRR/ZLRR, let $r$ be the principal root of its characteristic polynomial $P(x)$. Then, given initial values $a_i=0$ for $0\le i\le k-2$, $a_{k-1}=1$, we have $$\lim_{n\to\infty} \frac{a_n}{r^n} \ = \ C,$$ where $C > 0$. Furthermore, the sequence $\{a_n\}_{n=1}^{\infty}$ is eventually monotonically increasing.
Since $r$ has multiplicity $1$, $q_1$ is a constant polynomial. To see geometric behavior, we note that $$\lim_{n\to\infty}\frac{a_n}{r^n}\ = \ \lim_{n\to\infty}q_1(n)\,\left(\frac{r^n}{r^n}\right)+\lim_{n\to\infty}q_2(n)\,\left(\frac{r_2}{r}\right)^n+\cdots+\lim_{n\to\infty}q_j(n)\,\left(\frac{r_j}{r}\right)^n.$$ Since $|r|>|r_i|$ for all $2\le i\le j$, each limit with a $(r_i/r)^n$ term disappears, leaving just $q_1$, which must be positive, since the sequence $a_n$ does not admit negative terms.\
To see that $a_n$ is eventually increasing, consider the sequence $$\begin{aligned}
A_n&\ := \ a_{n+1}-a_n\nonumber\\
&\ \ = \ (q_1r_1-q_1)\,r_1^n+\left(q_2(n+1)\,r_2-q_2(n)\right)\,r_2^n+\cdots+\left(q_j(n+1)\,r_j-q_j(n)\right)\,r_j^n.\end{aligned}$$
A similar analysis shows $$\lim_{n\to\infty}\frac{\left(q_2(n+1)\,r_2-q_2(n)\right)\,r_2^n+\cdots+\left(q_j(n+1)\,r_j-q_j(n)\right)\,r_j^n}{\left(q_1\,r_1-q_1\right)\,r_1^n}\ = \ 0,$$ meaning that the term $(q_1\,r_1-q_1)\,r_1^n$ grows faster than the sum of the other terms; thus $A_n$ is eventually positive as desired.
\[cor:monotonicallyincreasing\] For a PLRR/ZLRR, let $r$ be the principal root of its characteristic polynomial $P(x)$. Then, given initial values satisfying $a_i\geq0$ for $0\le i\le k-1$ and $a_i > 0$ for some $0\le i \le k-1$, we have $$\lim_{n\to\infty} \frac{a_n}{r^n} \ = \ C,$$ where $C > 0$. Furthermore, the sequence $\{a_n\}_{n=1}^{\infty}$ is eventually monotonically increasing. That is, Lemma \[lem:monotonicallyincreasinglessstrong\] can be generalized to any set of non-negative initial conditions that are not all zero.
We first note that the derivation of does not rely on the initial values; any sequence satisfying the recurrence takes on this form.\
Since one of the initial values $a_0,a_1,\cdots,a_{k-1}$ is a positive integer, we know that one of $a_k,a_{k+1},\cdots,a_{2k-1}$ is also a positive integer by the recurrence relation, which forces $a_{n}$ to be at least $a_{n-k}$. Let $k\le i\le 2k-1$ be such that $a_i$ is positive. Consider the sequence $b_n=a_{n+i-k+1}$, which has $b_{k-1}=a_i>0$. By the recurrence relation, we have $b_n\ge a_n$ for all $n$, which would be impossible if the Binet expansion of $b_n$ had a non-positive coefficient attached to the $r^n$ term. Eventual monotonicity thus follows.
A Generalization of Binet’s Formula
-----------------------------------
In general, the Binet expansion of a recurrence sequence is quite unpleasant to compute or work with. However, things become much simpler when the characteristic polynomial has no multiple roots. In that case, we may construct an explicit formula for the $n$th term of the sequence, given a nice set of initial values. Keeping in mind that linear combinations of sequences satisfying a recurrence also satisfy the recurrence, one could construct a formula for the $n$th term given arbitrary initial values.
Consider a ZLRR with characteristic polynomial $P(x)$ that does not have multiple roots, and initial values $a_i=0$ for $0\le i\le k-2$, $a_{k-1}=1$. Then each term of the resulting sequence may be expressed as $$a_n\ = \ c_1\,r_1^n+c_2\,r_2^n+\cdots+c_k\,r_k^n,$$ where the $r_i$ are the distinct roots of $P(x)$, and $c_i=1/P'(r_i)$.
Since each root has multiplicity $1$, the existence of such explicit form follows from the Binet expansion (see Theorem \[thm:binetexpansion\]), so we are left to prove that $c_i = 1/P'(r_i)$. Using the initial values, we see that the $c_i$ are solutions to the linear system $$\begin{pmatrix}
1 & 1 & 1 & \cdots & 1\\
r_1 & r_2 & r_3 & \cdots & r_k\\
r_1^2 & r_2^2 & r_3^2 & \cdots & r_k^2\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
r_1^{k-1} & r_2^{k-1} & r_3^{k-1} & \cdots & r_k^{k-1}
\end{pmatrix}
\begin{pmatrix}
c_1\\
c_2\\
c_3\\
\vdots\\
c_k
\end{pmatrix}\ = \
\begin{pmatrix}
0\\
0\\
0\\
\vdots\\
1
\end{pmatrix}.$$
Denote the matrix by $A$; by Cramer’s rule, we have $c_i = \det(A_i) / \det(A)$, where $A_i$ is the matrix formed by replacing column $i$ of $A$ with the column vector of zeroes and a single 1. Using Laplace expansion, we see that $\det(A_i)=(-1)^{k+i}\det(M_{ki})$, where $M_{ki}$ is the $k,i$ minor matrix of $A$ formed by deleting row $k$ and column $i$. Notice that both $A$ and $M_{ki}$ are Vandermonde matrices, which means we have $$\det(A)\ = \ \prod_{1\,\le \,a\,<\,b\, \le\, k}(r_b-r_a),\ \ \ \ \det(M_{ki})\ = \ \prod_{\substack{1\,\le\, a\,<\,b\,\le\, k\\ a,\,b\,\ne\, i}}(r_b-r_a).$$ We may thus simplify and find $$\begin{aligned}
c_i&\ = \ (-1)^{k+i}\left(\prod_{\substack{1\,\le\, a\,<\,b\,\le \,k\\ a,\,b\,\ne\, i}}(r_b-r_a)\right)\bigg/\left(\prod_{1\,\le\, a\,<\,b\, \le\, k}(r_b-r_a)\right)\nonumber\\
&\ = \ (-1)^{k+i}\bigg/\left(\prod_{\substack{1\,\le\, a\,<\,b\,\le\, k\\ a\, =\, i\text{ or }b\,=\,i}}(r_b-r_a)\right)\nonumber\\
&\ = \ \frac{(-1)^{k+i}}{(r_i-r_1)(r_i-r_2)\cdots(r_i-r_{i-1})(r_{i+1}-r_i)\cdots(r_{k-1}-r_i)(r_k-r_i)}\nonumber\\
&\ = \ \frac{(-1)^{k+i}}{\left(\prod_{j\,=\,1}^{i-1}(r_i-r_j)\right)(-1)^{k-i}\left(\prod_{j\,=\,i+1}^{k}(r_i-r_j)\right)}\nonumber\\
&\ = \ 1\bigg/\prod_{\substack{1\,\le \,j\,\le\, k\\ j\,\ne\, i}}(r_i-r_j).\end{aligned}$$ Note that the product is simply the function $$f(x)\ = \ \prod_{\substack{1\,\le j\,\le \,k\\ j\,\ne\, i}}(x-r_j)$$ evaluated at $x=r_i$. To evaluate this, we may rewrite $$\begin{aligned}
f(r_i)&\ = \ \lim_{x\to\ r_i}f(x)\ = \ \lim_{x\to\ r_i}\prod_{\substack{1\,\le\, j\,\le\, k\\ j\,\ne\, i}}(x-r_j)\nonumber\\
&\ = \ \lim_{x\to\ r_i}\frac{(x-r_i)}{(x-r_i)}\prod_{\substack{1\,\le \,j\,\le\, k\\ j\,\ne\, i}}(x-r_j)\ = \ \lim_{x\to\ r_i}\frac{\prod_{1\,\le\, j\,\le\, k}(x-r_j)}{x-r_i}\nonumber\\
&\ = \ \lim_{x\to\ r_i}\frac{P(x)}{x-r_i},\end{aligned}$$ which equals $P'(r_i)$ by l’Hôpital’s rule. We thus have $c_i = 1/f(r_i) = 1/P'(r_i)$, completing the proof.
ZLRS-Legal Decompositions {#sec:zlrslegal}
=========================
We prove Theorems \[thm:generalzeckZLRS\] and \[thm:lossofuniqueness1ZLRS\] in sections §\[sec:existence\] and §\[sec:uniqueness\], respectively.
Existence {#sec:existence}
---------
To prove the existence of legal decompositions for every integer $N \geq 0$ given any $s$-deep ZLRS, we show that the greedy algorithm always terminates in a legal decomposition through strong induction. At each step the greedy algorithm uses the largest element from a sequence, index-wise. For example, if $G_{10} < N < G_{11}$, then we use $G_{10}$ in our decomposition of $N$, even if $G_9 \geq G_{10}$. We also need to make sure our decomposition is legal. At each step, we use the largest coefficient possible, depending on the coefficients of our $s$-deep ZLRS, and make sure we do not have more terms than is legal. We show how we do this in the proof of Theorem \[thm:generalzeckZLRS\].
Recall that our $s$-deep ZLRS has the form of Equation . We first prove that the greedy algorithm terminates in a legal decomposition for all integers $N$ up to and including the last initial condition. We let the empty decomposition be legal for $N=0$. These are the base cases. There are two cases and a special third case, as it only applies to a specific sequence.\
Case 1: If $c_{s+1} \leq s$, then note the initial conditions are the first $L$ integers. So, by Condition (1), we trivially have a legal decomposition for all of our initial conditions.\
Case 2: If $c_{s+1} > s$, then our initial conditions are specially constructed so that we guarantee existence of legal decompositions. We do so by adding the smallest integer that cannot be legally decomposed by the previous terms. We illustrate this with an example. Let us take 1-deep ZLRS’s of the form $G_{n+1} = c_1\, G_{n-1} + c_2 \,G_{n-2}$, where $c_1 > 1$ and $c_2 > 0$. Then, our initial conditions start with $G_1 = 1$ and $G_2 = 2$. Assuming $G_3 > G_2$, we know all $N$ with $G_2 < N < G_3$ cannot use $G_2 = 2$ in their decomposition, so we can only use $G_1=1$. We also have a restriction of only being able to use $G_1=1$ at most $c_1$ times. So, the first number we cannot legally decompose is $c_1+1$, thus, $G_3 = c_1+1$, which comes from our construction as well. By a similar argument, $G_4 = 2c_1 + c_2 + 1$.\
Case 3 (Special): If our ZLRS is the Lagonaccis, then we must consider the first four terms in our sequence instead of the first three terms. However, since all four integers appear in our sequence ($Z_1=1, Z_2=2, Z_3=4,$ and $Z_4=3$), we still get a trivial legal decomposition for the first four positive integers.\
This is now our inductive step of the induction proof. We now assume that all integers up to and including $N-1$ has a legal decomposition. We now show that $N$ must have a legal decomposition. Let $G_{t} \leq N < G_{t+1}$. There are two cases to consider.\
Case 1: Suppose $N = G_{t}$. Then, trivially, we have a legal decomposition.\
Case 2: Suppose $N > G_{t}$ and let $m \leq N$ be the largest integer created using a legal decomposition involving only summands drawn from $G_t,G_{t-1}, \dots, G_{t-L}$. Suppose $m = a_1G_{t-s-1}$, with $a_1 < c_{s+1}$. We want to show that $N - m$ can be expressed with the remaining terms. To do so, we need $N-m < G_{t-s-1}$. Suppose not. Then $N-m \geq G_{t-s-1}$. However, this implies that we have not used the maximum number of $G_{t-s-1}$’s in our greedy decomposition, which is a contradiction. So, we now have that $N-m<G_{t-s-1}$. By the strong inductive hypothesis, there exists a legal decomposition of $N-m$. We then add $m$ to this legal decomposition to obtain our decomposition of $N$. Since the decomposition for $N-m$ is legal, adding $m$ is keeps our decomposition legal, by Condition (3) of Definition \[def:sdeepzlrs\]. So, we have a legal decomposition for $N$.
Let $c_i$ be the next non-zero constant in our recurrence relation. We then let $m = c_{s+1}G_{t-s-1} + a_iG_{t-s-i}$ with $a_i < c_{s+i}$. We want to show that $N - m$ can be expressed with the remaining terms. To do so, we need $N-m < G_{t-s-i}$. Suppose not. Then $N-m \geq G_{t-s-i}$. However, this implies that we have not used the maximum number of $G_{t-s-i}$’s in our greedy decomposition, which is a contradiction. So, we have that $N-m<G_{t-s-i}$. By the same reasoning as the previous case, we have a legal decomposition for $N$.
We continue this argument, taking the next non-zero constant, adding that on to $m$, until we reach this final case.
Let $m = c_1G_{t} + c_2 G_{t-1} + \cdots + c_{L-1} G_{t+2-L} + (c_L - 1)G_{t+1-L}$. This is the largest possible value $m$ can attain while still being having a legal decomposition. We want to show that $N-m < G_{t-L+1}$. Noting $N < G_{t+1}$, we see that $$\begin{aligned}
N - m &\ = \ N - {\left( c_1\,G_{t} + \cdots + c_{L-1} G_{t+2-L} + (c_L - 1)\,G_{t+1-L} \right)}\nonumber\\
&\ <\ G_{t+1} - {\left( c_1\,G_{t} + \cdots + c_{L-1} G_{t+2-L} + (c_L - 1)\,G_{t+1-L} \right)}\nonumber\\
&\ = \ {\left( c_1\,G_{t} + \cdots + c_{L-1}\, G_{t+2-L} + c_L\, G_{n+1-L} \right)} - {\left( c_1\,G_{t} + \cdots + c_{L-1}\, G_{t+2-L} + (c_L - 1)\,G_{t+1-L} \right)}\nonumber\\
&\ = \ G_{t+1-L}.\end{aligned}$$
Thus $N-m < G_{t+1-L}$, and in every case we attain a legal decomposition for $N$, as desired. Therefore, by strong induction, we attain a legal decomposition for all positive integers $N$ and all $s$-deep ZLRS’s.
Loss of Uniqueness {#sec:uniqueness}
------------------
We now explore the loss of uniqueness of legal decompositions in $s$-deep ZLRS’s, where $s \geq1$. We prove Theorem \[thm:lossofuniqueness1ZLRS\] after introducing some notation.
\[def:fosteredPLRS\] Let $\{G_n\}_{n=1}^{\infty}$ be an $s$-deep ZLRS, with recurrence relation $$G_{n+1} \ = \ c_1\,G_{n} + c_2\,G_{n-1} + \cdots + c_s\,G_{n+1-s} + c_{s+1}\,G_{n-s} + \cdots + c_L\, G_{n+1-L}.$$ We say a sequence $\{H_n\}_{n=1}^{\infty}$ is a ***fostered PLRS*** of $\{G_n\}_{n=1}^{\infty}$ if $\{H_n\}_{n=1}^{\infty}$ is a PLRS of the form $$H_{n+1} \ = \ c_{s+1}\, H_n + c_{s+2} \,H_{n-1} + \cdots + c_L\, H_{n+s+1-L}.$$
The following lemmas prove results concerning the characteristic polynomials of our $s$-deep ZLRS and its fostered PLRS. We define the characteristic polynomial of our $s$-deep ZLRS as $$\label{eqn:zlrspolynomial}
P_Z(x) \ := \ x^L - c_{s+1}\,x^{L-s-1} - c_{s+2}\,x^{L-s-2} - \cdots - c_{L-1}\,x - c_L,$$ and of our fostered PLRS as $$\label{eqn:plrspolynomial}
P_P(x) \ := \ x^{L-s} - c_{s+1}\,x^{L-s-1} - c_{s+2}\,x^{L-s-2} - \cdots - c_{L-1}\,x - c_L.$$
Note that all results of Lemma \[lem:greatestroot\] apply to $P_Z(x)$ and $P_P(x)$, because these polynomials meet the necessary conditions. We now prove a lemma relating the two positive roots of $P_Z(x)$ and $P_P(x)$.
\[lem:largerroot\] Let $r$ be the root of greatest magnitude of $P_Z(x)$ and $w$ be the root of greatest magnitude of $P_P(x)$, defined in equations and , respectively. Then $w>r>1$.
By Lemma \[lem:greatestroot\], $P_Z(x)$ has exactly one positive root $r$, which also has the greatest magnitude. We then see that $P_Z(r) = 0$ implies $$r^L \ = \ c_{s+1}\,x^{L-s-1} + c_{s+2}\,x^{L-s-2} + \cdots + c_{L-1}\,x +c_L\, >\, 1,$$ which implies $r>1$. Notice that $$\begin{aligned}
0 \ = \ P_Z(r) & \ = \ & r^L - c_{s+1}\,r^{L-s-1} - \cdots - c_{L-1}\,r - c_L \nonumber\\ & \ = \ & r^L +(r^{L-s} - r^{L-s})- c_{s+1}\,r^{L-s-1} - \cdots - c_{L-1}\,r - c_L\nonumber \\ &\ = \ & r^L - r^{L-s} + P_P(r).\end{aligned}$$ Since $r>1$, $r^L - r^{L-s} > 0$, which means $P_P(r)<0$. Since $\lim_{x\to\infty}P_P(x) = \infty$, $P_P$ must have a root greater than $r$ by the Intermediate Value Theorem. By Lemma \[lem:greatestroot\], we also know that the root of greatest magnitude, $w$, is positive. So, we find $w>r>1$.
We now prove lemmas giving stronger relations on the roots $w$ and $r$.
Let $w$ and $r$ be defined as in Lemma \[lem:largerroot\]. Then $w^n > r^{n+1}$, for $n \geq \log_{w/r} w$.
By Lemma \[lem:largerroot\], we know that $w>r>1$. Thus $r/w < 1$, and there exists an $n\in \mathbb{Z}^+$ such that $${\left( \frac{r}w \right)}^n w\ <\ 1,$$ which is equivalent to $w^n > r^{n+1}$, as desired. Simple algebra yields $n \geq \log_{w/r}w$.
\[lem:rootpowersinequality\] Let $w$ and $r$ be as defined in Lemma \[lem:largerroot\]. Then, $w^{L-s}>r^L$.
Equivalently, we prove $w^{L-s} - r^L > 0$. We see that $$\begin{aligned}
w^{L-s} - r^L &\ = \ (c_{s+1}\,w^{L-s-1} + \cdots + c_{L-1}\,w + c_L) - (c_{s+1}\,r^{L-s-1} + \cdots + c_{L-1}\,r + c_L)\nonumber\\ &\ = \ c_{s+1}\,(w^{L-s-1} - r^{L-s-1}) + c_{s+2}\,(w^{L-s-2}-r^{L-s-2}) + \cdots + c_{L-1}\,(w-r).\end{aligned}$$ By Lemma \[lem:largerroot\], we know that $w>r>1$, so $w^t > r^t$ for any $t \in \mathbb{N}$. Recall that $c_i \geq 0$ for all $i$, and $c_{s+1}, c_L >0$. Thus $$c_{s+1}\,(w^{L-s-1} - r^{L-s-1}) + c_{s+2}\,(w^{L-s-2}-r^{L-s-2}) + \cdots + c_{L-1}\,(w-r)\ >\ 0,$$ as desired.
\[cor:growthratesroots\] Let $w$ and $r$ be as defined in Lemma \[lem:greatestroot\]. Then $$\label{eqn:greatestrootinequality}
w^{(n(L-s)/L)-1}\ > \ r^n.$$
By Lemma \[lem:rootpowersinequality\], which states $w^{L-s} > r^L$, and Lemma \[lem:largerroot\], which states $w>r>1$, we have $$w^{(L-s)/L} \ > \ r.$$ Thus $w^{(L-s)/L} / r > 1$, which implies there exists $n$ such that[^6] $${\left( \frac{w^{(L-s)/L}}{r} \right)}^n \ > \ w.$$ Through algebraic manipulation, the above is equivalent to , as desired.
We introduce some more notation before combining these results into the proof of Theorem \[thm:lossofuniqueness1ZLRS\]. As usual, we have $\{G_n\}_{n=1}^{\infty}$ as an $s$-deep ZLRS, with $s>1$, and $\{H_n\}_{n=1}^{\infty}$ as our fostered PLRS.
\[def:zlrslegalpotpourri\] We define the five objects that will be studied in the following lemmas and in the proof of Theorem \[thm:lossofuniqueness1ZLRS\].
1. $D_n$: The set of $s$-deep ZLRS legal decompositions for all integers $N < G_{n+1}$. Note that these decompositions use elements of $\{G_1,G_2,\dots,G_n\}$, and we include the empty decomposition in this count.
2. $E_n$: The set of PLRS legal decompositions for all integers $N < H_{n+1}$. Note that these decompositions use elements of $\{H_1,H_2,\dots,H_n\}$, and we include the empty decomposition in this count.
3. A decomposition arising from the recurrence relation $R$ is denoted by $$\label{eqn:decompositiondef}
(a_na_{n-1}\dots a_2a_1)_R \ = \ a_nR_n + a_{n-1}R_{n-1}+\cdots+a_2R_2+a_1R_1.$$ For example, the decomposition $(a_na_{n-1}\dots a_2a_1)_G$ denotes a decomposition in $D_n$.
4. $f_{G}(N)$: the number of legal decompositions of the positive integer $N$ from the ZLRS $\{G_n\}_{n=1}^{\infty}$.
5. $f_{G, \text{ave}}(n) = \frac1{G_{n+1}} \sum_{m=0}^{G_{n+1}-1} f_G(m)$, the average number of decompositions for all integers $N < G_{n+1}$
As previously proved in [@Ho; @Ke; @KKMW; @Len], we know $E_n$ very well. In fact, we know $|E_n| = H_{n+1}$, since there is a unique decomposition for every integer $N<H_{n+1}$, and we count the empty legal decomposition in this. We do not know $D_n$ very well, but we can bound it using relationships to $E_n$. We now provide some relationships between the sizes of $D_n$ and $E_n$.
\[lem:decompsets\] Let $D_n$ and $E_n$ be as defined in Definition \[def:zlrslegalpotpourri\]. Then
1. $|E_n| \geq |D_n|$ for $n \geq 0$, and
2. $|D_n| \geq |E_{{\left\lfloor n(L-s)/L \right\rfloor}}|$ for $n \geq L$.
\
We first prove (1). Recall that we are considering an $s$-deep ZLRS $\{G_n\}_{n=1}^{\infty}$ and its fostered PLRS $\{H_n\}_{n=1}^{\infty}$. Consider a decomposition $(a_na_{n-1}\dots a_2a_1)_G\in D_n$. We show $(a_na_{n-1}\dots a_2a_1)_H\in E_n$ by showing $(a_na_{n-1}\dots a_2a_1)_H$ satisfies the legal PLRS decomposition conditions and represents an integer $N < H_{n+1}$. We first illustrate this with an example. Let us consider the 2-deep ZLRS $G_{n+1} = 4G_{n-2} + 5G_{n-3} + 7G_{n-4}$, which has fostered PLRS $H_{n+1} = 4H_n + 5H_{n-1}+7H_{n-2}$. Consider the decomposition $(0453000440)_G \in D_{10}$. We wish to show $(0453000440)_H \in E_{10}$. As shown in previous papers, such as [@MW], we know that if $(0453000440)_H$ represents the PLRS legal decomposition for $N$, then $N < H_{11}$. We also see that this decomposition follows all conditions laid out in Definition \[def:plrsdefinition\], as all coefficients are appropriately bounded. So, $(0453000440)_H \in E_{10}$.\
We now show $(a_na_{n-1}\dots a_2a_1)_H\in E_n$ by showing $(a_na_{n-1}\dots a_2a_1)_H$ satisfies the legal PLRS decomposition conditions and represents an integer $N < H_{n+1}$. The latter is simple. Suppose $N \geq H_{n+1}$, then we must use $H_{n+1}$ (or a larger term) in our decomposition; however, our decomposition $(a_na_{n-1}\dots a_2a_1)_H$ does not use $H_{n+1}$ (or any larger term) in its decomposition, so we reach a contradiction. Now suppose $(a_na_{n-1}\dots a_2a_1)_H$ did not satisfy the legal PLRS decomposition conditions. Then, for some $i$ and $j$, we have $a_i > c_j$, where $c_j$ is the corresponding non-negative coefficient. However, if this is true, then it is also the case for $(a_na_{n-1}\dots a_2a_1)_G$, meaning $(a_na_{n-1}\dots a_2a_1)_G$ is not an $s$-deep ZLRS legal decomposition, which is a contradiction. Thus $(a_na_{n-1}\dots a_2a_1)_H\in E_n$, implying $|E_n| \geq |D_n|$ for $n \geq 0$.\
We now prove (2). We wish to create in injective function $f: E_{{\left\lfloor n(L-s)/L \right\rfloor}} \to D_n$. We define $f$ as follows: take a decomposition $(a_{{\left\lfloor n(L-s)/L \right\rfloor}}a_{{\left\lfloor n(L-s)/L \right\rfloor}-1}\dots a_2a_1)_H\in E_{{\left\lfloor n(L-s)/L \right\rfloor}}$ and add $s$ zeros in front of the first positive $a_i$, starting from the left. Then move down the decomposition until Condition 1 or the first portion of Condition 2 of Definition \[def:plrsdefinition\] is met. Then move to the next positive $a_i$, and add $s$ zeros, and repeat. Once we finish this process, we add the sufficient number of zeros to the front of the decomposition, such that we have a total of $n$ coefficients. Note that this guarantees an $s$-deep ZLRS legal decomposition, since we always have $s$ zeros between each ‘chunk’ and the coefficients will be appropriately bounded. We illustrate this with a specific example. Take the $4$-deep ZLRS $G_{n+1} = 2G_{n-4} + 3G_{n-5}+ 5G_{n-6}$, which has fostered PLRS $H_{n+1} = 2H_{n} + 3H_{n-1}+ 5H_{n-2}$, and take $n=24$. Note that $L = 7$, so ${\left\lfloor n(L-s)/L \right\rfloor} = {\left\lfloor 24(7-4)/7 \right\rfloor} =10$. Finally, consider the decomposition $(2302320022)_H \in E_{10}$. We see that $$f((2302340022)_H) \ = \ (000000230000023400000022)_G \in D_{24}.$$
We illustrate this procedure with a general example. Take the decomposition in $E_{{\left\lfloor n(L-s)/L \right\rfloor}}$ that uses the most coefficients. This process is shown in Figure \[fig:functionexample\].
(2,2.5) – (12,2.5); (2,3.5) – (12,3.5); (12,2.5) – (12,3.5);
(5.1,2.5) – (5.1,3.5); (8.2,2.5) – (8.2,3.5); (8.9,2.5) – (8.9,3.5); (2,2.5) – (2,3.5);
(2.9,2.5) – (2.9,3.5); (3.3,2.5) – (3.3,3.5); (4.2,2.5) – (4.2,3.5); (6,2.5) – (6,3.5); (6.4,2.5) – (6.4,3.5); (7.3,2.5) – (7.3,3.5); (9.8,2.5) – (9.8,3.5); (10.2,2.5) – (10.2,3.5); (11.1,2.5) – (11.1,3.5);
at (2.45,3) [$C_{S+1}$]{}; at (3.1,3) [...]{}; at (3.75,3) [$C_{L-1}$]{}; at (4.65, 3) [$C_L -1$]{};
at (5.55,3) [$C_{S+1}$]{}; at (6.2,3) [...]{}; at (6.85,3) [$C_{L-1}$]{}; at (7.75, 3) [$C_L -1$]{};
at (8.55,3) [...]{};
at (9.35,3) [$C_{S+1}$]{}; at (10,3) [...]{}; at (10.65,3) [$C_{L-1}$]{}; at (11.55, 3) [$C_L -1$]{};
(2.2,3.6) – (5.2,3.6) node \[black,midway,yshift=15pt\] [$L-s$]{};
(0.25,0) – (4.25,0) node \[black,midway,yshift=-17pt\] [$L$]{};
(0.25,1) – (1.25,1) node \[black,midway,yshift=10pt\] [$s$]{};
(3.2,2.4) – (2.1, 1.1); (6.5,2.4) – (6.5, 1.1);
(10,2.4) – (12, 1.1);
(0,0) – (14,0); (0,1) – (14,1); (14,0) – (14,1);
(4.2,0) – (4.2,1); (8.4,0) – (8.4,1); (9.8,0) – (9.8,1); (0,0) – (0,1);
(0.4,0) – (0.4,1); (0.75,0) – (0.75,1); (1.15,0) – (1.15,1); (2.05,0) – (2.05,1); (2.4,0) – (2.4,1); (3.3,0) – (3.3,1);
(4.6,0) – (4.6,1); (4.95,0) – (4.95,1); (5.35,0) – (5.35,1); (6.25,0) – (6.25,1); (6.6,0) – (6.6,1); (7.5,0) – (7.5,1);
(10.2,0) – (10.2,1); (10.55,0) – (10.55,1); (10.95,0) – (10.95,1); (11.85,0) – (11.85,1); (12.2,0) – (12.2,1); (13.1,0) – (13.1,1);
at (0.2,0.5) [0]{}; at (0.575,0.5) [...]{}; at (0.95,0.5) [0]{}; at (1.6,0.5) [$C_{s+1}$]{}; at (2.225,0.5) [...]{}; at (2.85,0.5) [$C_{L-1}$]{}; at (3.75,0.5) [$C_L-1$]{};
at (4.4,0.5) [0]{}; at (4.775,0.5) [...]{}; at (5.15,0.5) [0]{}; at (5.8,0.5) [$C_{s+1}$]{}; at (6.425,0.5) [...]{}; at (7.05,0.5) [$C_{L-1}$]{}; at (7.95,0.5) [$C_L-1$]{};
at (9.1, 0.5) [...]{};
at (10,0.5) [0]{}; at (10.375,0.5) [...]{}; at (10.75,0.5) [0]{}; at (11.4,0.5) [$C_{s+1}$]{}; at (12.025,0.5) [...]{}; at (12.65,0.5) [$C_{L-1}$]{}; at (13.55,0.5) [$C_L-1$]{};
Now that we have explained the function, and we see that if a decomposition $x \in E_{{\left\lfloor n(L-s)/L \right\rfloor}}$, then $f(x) \in D_n$, we now show this function is injective. Once we show the function is injective, we know that $|D_n| \geq |E_{{\left\lfloor n(L-s)/L \right\rfloor}}|$.\
Consider decompositions $a,b\in E_{{\left\lfloor n(L-s)/L \right\rfloor}}$ where $a = (a_{{\left\lfloor n(L-s)/L \right\rfloor}}a_{{\left\lfloor n(L-s)/L \right\rfloor}-1}\dots a_2a_1)_H$ and $b=(b_{{\left\lfloor n(L-s)/L \right\rfloor}}b_{{\left\lfloor n(L-s)/L \right\rfloor}-1}\dots b_2b_1)_H$ such that $f(a)=f(b)$. We wish to show that $a=b$. Suppose, to the contrary, that $a\neq b$. Then $a_i \neq b_i$ for some $1 \leq i \leq {\left\lfloor n(L-s)/L \right\rfloor}$. However, if this is the case then $f(a) \neq f(b)$, because $f$ does not change the value of $a_i$ or $b_i$. If the decomposition begins with seven 4’s, then it ends with seven 4’s. The only change $f$ makes to the decomposition is the addition of a number of zeros. Next, the relative positioning of each number in the decomposition is left unchanged. For example, if there is a 5 in the decomposition followed by a 4, this will still be true, albeit there may be $s$ zeros between the 5 and 4 once $f$ is applied. From this, we see that $f$ is injective.
Now, we have all of the ingredients to prove Theorem \[thm:lossofuniqueness1ZLRS\].
We note, by definition $$f_{G,\text{ave}}(n) \ = \ \frac1{G_{n+1}}\sum_{m=0}^{G_{n+1}-1} f_G(m) \ = \ \frac{|D_n|}{G_{n+1}}.$$ We first prove the upper bound. By Lemma \[lem:decompsets\] and the definition of $E_n$, we find $$\frac{|D_n|}{G_{n+1}}\ \leq\ \frac{|E_n|}{G_{n+1}} \ = \ \frac{H_{n+1}}{G_{n+1}}\ \approx\ {\left( \frac{w}{r} \right)}^{n+1},$$ where the approximation is justified by Corollary \[cor:monotonicallyincreasing\]. Note that by Lemma \[lem:largerroot\], $w>r$, so $f_{G,\text{ave}}(n)$ is bounded above by $\lambda_1^{n+1}$ where $\lambda_1 = w/r >1$.\
We now prove the lower bound. Again by Lemma \[lem:decompsets\] and the definition of $E_n$, we find $$\frac{|D_n|}{G_{n+1}} \ \geq\ \frac{|E_{{\left\lfloor n(L-s)/L \right\rfloor}}|}{G_{n+1}} \ = \ \frac{H_{{\left\lfloor n(L-s)/L \right\rfloor}+1}}{G_{n+1}} \ \approx\ \frac{w^{{\left\lfloor n(L-s)/L \right\rfloor}+1}}{r^{n+1}}\ \geq\ \frac{w^{n(L-s)/L}}{r^{n+1}} \ = \ {\left( \frac{w^{\frac{n(L-s)}{L(n+1)}}}{r} \right)}^{n+1},$$ where the approximation is justified by Corollary \[cor:monotonicallyincreasing\]. Note that by Corollary \[cor:growthratesroots\], $w^{(n(L-s)/L)}>r^{n+1}$ for sufficiently large $n$. So, $w^{\frac{n(L-s)}{L(n+1)}} > r$ for sufficiently large $n$, since $w>r>1$. Thus, $f_{G,\text{ave}}(n)$ is bounded below by $\lambda_2^{n+1}$ where $\lambda_2 \ = \ w^{\frac{n(L-s)}{L(n+1)}} / r > 1$. This proves Theorem \[thm:lossofuniqueness1ZLRS\].
The Zeroing Algorithm and Applications {#sec:conversion}
======================================
An alternate approach to understanding decompositions arising from ZLRR’s is to see if for every ZLRR one could associate a PLRR with similar behavior: a derived PLRR. In this section, we develop the machinery of the Zeroing Algorithm, which is an extremely powerful tool for understanding recurrence sequences analytically. We prove a very general result about derived recurrences that implies every ZLRS has a derived PLRS.
The Zeroing Algorithm
---------------------
Consider some ZLRS/PLRS with characteristic polynomial $$P(x)\ := \ x^k-c_1\,x^{k-1}-c_2\,x^{k-1}-\cdots-c_k,$$ and choose a sequence of $k$ real numbers $\beta_1,\beta_2,\dots,\beta_{k}$; the $\beta_i$ are considered the input of the algorithm. For nontriviality, the $\beta_i$ are not all zero. We define the *Zeroing Algorithm* to be the following procedure. First, create the polynomial $$\label{eq:zeroalginitial}
Q_0(x)\ := \ \beta_1\,x^{k-1}+\beta_2\,x^{k-2}+\cdots+\beta_{k-1}\,x+\beta_k.$$ Next, for $t\ge1$, define a sequence of polynomials $$Q_{t}(x)\ := \ x\,Q_{t-1}(x)-q(1,t-1)\,P(x),$$ indexed by $t$, where $q(1,t)$ is the coefficient of $Q_t(x)$ at the $x^{k-1}$ term. We *terminate* the algorithm at step $t$ if $Q_t(x)$ does not have positive coefficients.\
To understand the algorithm through linear recurrences, we denote by $q(n,t)$ the coefficient of $Q_t(x)$ at the term $x^{k-n}$, where $n$ ranges from $1$ to $k$. We unravel the recurrence relation on the polynomials, and obtain the following system of recurrence relations $$\begin{aligned}
q(1,t)&\ = \ q(2,t-1)+c_1\,q(1,t-1),\label{coeffrecur}\\
q(2,t)&\ = \ q(3,t-1)+c_2\,q(1,t-1),\nonumber \\
&\ \ \ \vdots \ \nonumber\\
q(k-1,t)&\ = \ q(k,t-1)+c_{k-1}\,q(1,t-1), \nonumber\\
q(k,t)&\ = \ c_k\,q(1,t-1),\nonumber\end{aligned}$$ with initial values $$\begin{aligned}
q(1,0)\ = \ \beta_1, \ \ \ \ q(2,0)\ = \ \beta_2,\ \ \ \ \cdots, \ \ \ \ q(k,0)\ = \ \beta_k.\end{aligned}$$ Note that if $q(1,t)$ through $q(k,t)$ are all non-positive, then so are $q(1,t+1)$ through $q(k,t+1)$; the same holds for nonnegativity.
The sequence $q(1,t)$ satisfies the recurrence specified by the characteristic polynomial $P(x)$. For each $1\le n\le k$, $q(n,t)$ is a positive linear combination of $q(1,t)$ at various stages: $$\begin{aligned}
q(n,t)&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+\cdots+c_k\,q(1,t-(k+1-n))\nonumber\\
&\ = \ \sum_{i=0}^{k-n}\,c_{n+i}\,q(1,t-(i+1)).\end{aligned}$$
We first examine the sequence $q(1,t)$. For $t\ge k$, we have $$\begin{aligned}
q(1,t)&\ = \ c_1\,q(1,t-1)+q(2,t-1)\nonumber\\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+q(3,t-2)\nonumber\\
&\ \ \ \vdots\ \nonumber \\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+\cdots+c_{k-1}\,q(1,t-(k-1))+q(k,t-(k-1))\nonumber\\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+\cdots+c_{k-1}\,q(1,t-(k-1))+c_k\,q(1,t-k),
\end{aligned}$$ which is what we want.\
The latter part can also be proven by unraveling the system of recurrences: we have $$\begin{aligned}
q(n,t)&\ = \ c_n\,q(1,t-1)+q(n+1,t-1)\nonumber\\
&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+q(n+2,t-2)\nonumber\\
&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+c_{n+2}\,q(1,t-3)+q(n+3,t-3)\nonumber\\
&\ \ \ \vdots\ \nonumber\\
&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+\cdots+q(n+(k-n),t-(k-n))\nonumber\\
&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+\cdots+q(k,t-(k-n))\nonumber\\
&\ = \ c_n\,q(1,t-1)+c_{n+1}\,q(1,t-2)+\cdots+c_k\,q(1,t-(k-n+1)),
\end{aligned}$$ as desired.
Now we may prove a very useful result.
Let $r$ be the principal root of $P(x)$. Consider the Binet expansion of the sequence $q(n,t)$ (indexed by $t$) for each $n$. The sign of the coefficient attached to the term $r^t$ agrees with the sign of $Q_0(r)$.
Recall the recurrence relation $Q_{t}(x)=x\,Q_{t-1}(x)-q(1,t-1)\,P(x)$. Evaluating at $x=r$, the $P(x)$ term drops out and we have $Q_{t}(r)=r\,Q_{t-1}(r)$, and iterating this procedure gives $r^t\,Q_0(r)$.
Recalling that $q(n,t)$ is defined to be the coefficient of $Q_t(x)$ at the term $x^{k-n}$, we have $$r^t\,Q_0(r)\ = \ Q_t(r)\ = \ r^{k-1}\,q(1,t)+r^{k-2}\,q(2,t)+\cdots+r\,q(k-1,t)+q(k,t).$$ Note that this means the sequence $Q_t(r)$ satisfies the recurrence specified by $P(x)$ as well. Since each $q(n,t)$ is a positive linear combination of $q(1,t)$ at various stages, they all have the same sign on the coefficient of the $r^t$ term in their explicit expansion as a sum of geometric sequences, and this sign agrees with the sign of the coefficient of $r^t$ in the expansion of $Q_t(r)$. Now we just need to show the sign in $Q_t(r)$ agrees with the sign of $Q_0(r)$.\
Consider the quantity $\lim_{t\to\infty} Q_t(r) / r^t$, which gives the coefficient of the $r^t$ term in $Q_t(r)$. Since $Q_t(r)=r^tQ_0(r)$, we have $$\lim_{t\to\infty}\,\frac{Q_t(r)}{r^t}\ = \ \lim_{t\to\infty}\,\frac{r^t\,Q_0(r)}{r^t}\ = \ Q_0(r)$$ as desired.
We can now establish an exact condition on when the Zeroing Algorithm terminates.
\[thm:algorithmtermination\] Let $Q_0(x)$ be as defined in and let $r$ be the principal root of $P(x)$ . The Zeroing Algorithm terminates if and only if $Q_0(r)<0$.
If $Q_0(r)<0$, then the coefficient of $r^t$ in the expansion of $q(n,t)$ is also negative for each $n$; this means $q(n,t)$ diverges to negative infinity, and that there must be some $t$ when $q(n,t)$ is non-positive for each $n$.\
For the other direction, if $Q_0(r)\ge0$ then suppose, for contradiction, that there is some $t_0$ where $q(n,t_0)\le0$ for all $n$. Then we would have $$r^{t_0}\,Q_0(r)\ = \ Q_{t_0}(r)\ = \ r^{k-1}\,q(1,t_0)+r^{k-2}\,q(2,t_0)+\cdots+r\,q(k-1,t_0)+q(k,t_0)\ \le \ 0,$$ which implies $Q_0(r)\le0$, forcing $Q_0(r)=0$.\
Notice that this equality only occurs when $q(1,t_0)=q(2,t_0)=\cdots=q(k,t_0)=0$. This means for each $n$, $q(n,t)=0$ for all $t>t_0$, so each $q(n,t)$ is identically zero, which contradicts our assumption of non-triviality.
A General Conversion Result
---------------------------
Now that we have developed the main machinery of the Zeroing Algorithm, we could prove a very general result on converting between linear recurrences.
For ease of notation, extend the $\gamma$ sequence by setting $\gamma_i=0$ for $i>m$. We modify the Zeroing Algorithm slightly to produce the desired $p(x)$.
Consider a sequence of polynomials $Q_t(x)$ of degree at most $k-1$, with $$\begin{aligned}
Q_1(x)&\ = \ \gamma_1\,(P(x)-x^k),\nonumber\\
Q_t(x)&\ = \ x\,Q_{t-1}(x)-(q(1,t-1)-\gamma_t)\,P(x)-\gamma_t\,x^k,\end{aligned}$$ where again, $q(n,t)$ denotes the coefficient of $Q_t(x)$ at $x^{k-n}$. Note that after iteration $m$, $\gamma_t=0$ and we have the unmodified Zeroing Algorithm again.
Define $p_t(x):=x^{k}\,\Gamma_t(x)+Q_t(x)$. At each iteration $t$, we have the following:
1. $P(x)$ divides $p_t(x)$,
2. the first $t$ coefficients of $p_t(x)$ are $\gamma_1$ through $\gamma_t$, and
3. $Q_t(r)\,=\,-r^k\,\Gamma_t(r)$.
A straightforward induction argument suffices for all of them.
\(1) We have $$p_1(x)\ = \ x^k\,\gamma_1(x)+Q_1(x)\ = \ x^k\,\gamma_1+\gamma_1\,(P(x)-x^k)\ = \ \gamma_1\,P(x).$$ Assuming $P(x)$ divides $p_t(x)$, we have $$\begin{aligned}
p_{t+1}(x)&\ = \ x^k\,\Gamma_{t+1}(x)+Q_{t+1}(x)\nonumber\\
&\ = \ x^{k}\,(\gamma_1\,x^{t}+\gamma_2\,x^{t-1}+\cdots+\gamma_{t+1})+Q_{t+1}(x)\nonumber\\
&\ = \ x\cdot x^{k}\,(\gamma_1\,x^{t-1}+\gamma_2\,x^{t-2}+\cdots+\gamma_t)+\gamma_{t+1}\,x^k+x\,Q_{t}(x)-(q(1,t)-\gamma_{t+1})\,P(x)-\gamma_{t+1}\,x^k\nonumber\\
&\ = \ x\,(x^{k}\,(\,\gamma_1\,x^{t-1}+\gamma_2\,x^{t-2}+\cdots+\gamma_t)+Q_t(x))-(q(1,t)-\gamma_{t+1})\,P(x)\nonumber\\
&\ = \ x\,p_t(x)-(q(1,t)-\gamma_{t+1})\,P(x),\end{aligned}$$ which is divisible by $P(x)$ by the inductive hypothesis.
\
(2) We first prove that $Q_t(x)$ has degree at most $k-1$. This is certainly true for $Q_1(x)=\gamma_1(P(x)-x^k)$. Assume $Q_t(x)$ as degree at most $k-1$; we then have $$Q_{t+1}(x)\ = \ x\,Q_{t}(x)-(q(1,t)-\gamma_{t+1})\,P(x)-\gamma_{t+1}\,x^k.$$ It is evident that the highest power of $x$ to appear is $x^k$, which has coefficient $$q(1,t)-(q(1,t)-\gamma_{t+1})-\gamma_{t+1}\ = \ 0.$$ From the construction $p_t(x):=x^{k}\,\Gamma_t(x)+Q_t(x)$, now it is evident that the first $t$ coefficients are just those of $\Gamma_t(x)$.
\
(3) We have $$Q_1(r)\ = \ \gamma_1\,(P(r)-r^k)\ = \ -r^k\,\gamma_1.$$ Suppose $Q_t(r)=-r^k\,\Gamma_t(r)$; we have $$\begin{aligned}
Q_{t+1}(r)&\ = \ r\,Q_{t}(r)-(q(1,t)-\gamma_{t+1})\,P(r)-\gamma_{t+1}\,r^k\nonumber\\
&\ = \ r\,(-r^k\,\Gamma_t(r))-\gamma_{t+1}\,r^k\nonumber\\
&\ = \ -r^k\,(r\,\Gamma_t(r)+\gamma_{t+1})\nonumber\\
&\ = \ -r^k\,\Gamma_{t+1}(r).\end{aligned}$$
Now we have $Q_m(r)=-r^m\,\Gamma_m(r)<0$, since $\Gamma_m(r)>0$. Running the Zeroing Algorithm starting with $Q_m(x)$ yields some $Q_{m+t_0}(x)$ that does not have positive coefficients. We see that $p_{m+t_0}(x)=x^k\,\Gamma_{m+t_0}(x)+Q_{m+t_0}(x)$ is divisible by $P(x)$, has first $m+t_0$ coefficients $\gamma_1$ through $\gamma_m$ followed by $t_0$ 0’s, and thus does not have positive coefficients after $\gamma_m$; we may choose $p(x)=p_{m+t_0}(x)$.
Given $\gamma_1=1$ and arbitrary integers $\gamma_2$ through $\gamma_m$ with $\Gamma_m(r)>0$, there is a recurrence derived from $P(x)$ whose characteristic polynomial has first coefficients $\gamma_1$ through $\gamma_m$ with no positive coefficients thereafter.
Take $p(x)$ from Theorem \[thm:algorithmdivisibility\], which has first coefficients $\gamma_1$ through $\gamma_m$. Since $\gamma_1=1$, $p(x)$ is the characteristic polynomial of a linear recurrence. In fact, since $\gamma_2$ through $\gamma_m$ are integers, $p(x)$, and thus the recurrence, has integer coefficients.
\[cor: conversion\] Every ZLRR has a derived PLRR.
Take $m=2$, $\gamma_1=1,\gamma_2=-1$. We thus have $\Gamma_m(r)=r-1>0$, as shown in the section on characteristic polynomials. We can thus find $p(x)$ with first two coefficients $1$, $-1$ with no positive coefficients thereafter; this is the characteristic polynomial of a PLRR.
Note that a ZLRR does not have a unique derived PLRR; the Zeroing Algorithm simply produces a PLRR whose characteristic polynomial takes the coefficients $1$, $-1$, a bunch of $0$’s, and up to $k$ nonzero terms at the end, where $k$ is the degree of the characteristic polynomial of the ZLRR. In fact, for any positive integer $n$ less than the principal root of a ZLRR, there exists a derived PLRR with leading coefficients $1,-n$; this is seen by taking $\gamma_2=-n$ in \[cor: conversion\].
Fast Determination of Divergence Using the Zeroing Algorithm
------------------------------------------------------------
Finally, we have all of the tools necessary to prove our final result, which predicts the direction of divergence of a PLRS/ZLRS using its initial values.
We set $Q_0(x)=Q(x)$ and run the Zeroing Algorithm; we have proved that the sequence $q(1,t)$ follows the linear recurrence and has behavior determined by $Q_0(r)$. Thus, it suffices to show that $q(1,t)$ has the same initial values as $a_t$; explicitly, $q(1,t-1)=a_{t}$ for $1\le t\le k$.
We first notice, from the recurrences on $q(n,t)$ (\[coeffrecur\]), that $$\begin{aligned}
q(1,t)&\ = \ c_1\,q(1,t-1)+q(2,t-1)\nonumber\\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+q(3,t-2)\nonumber\\
&\ \ \ \vdots\nonumber\\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+\cdots+c_t\,q(1,0)+q(t+1,0)\nonumber\\
&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+\cdots+c_t\,q(1,0)+(\alpha_{t+1}-d_{t+1}).\end{aligned}$$
Now we proceed by strong induction. By construction, $q(1,0)=a_1$. For some $t$, assume $q(1,\tau-1)=a_{\tau}$ for all $1\le \tau <t$. We thus have $$\begin{aligned}
q(1,t)&\ = \ c_1\,q(1,t-1)+c_2\,q(1,t-2)+\cdots+c_t\,q(1,0)+(a_{t+1}-d_{t+1})\nonumber\\
&\ = \ (c_1\,a_t+c_2\,a_{t-1}+\cdots+c_t\,a_1)+a_{t+1}-d_{t+1}\nonumber\\
&\ = \ d_{t+1}+a_{t+1}-d_{t+1}\nonumber\\
&\ = \ a_{t+1}\end{aligned}$$ as desired.
Conclusion and Future work {#sec:conclusion}
==========================
We have introduced two distinct ways to consider decompositions arising from ZLRS’s.
- As we saw from the first method, we can define decompositions in such a way that we have existence, but not uniqueness. Is there a different definition such that we have uniqueness, but not existence? Is it possible to have both existence and uniqueness, or can we prove that having both is impossible for ZLRS’s?\
- Using the Zeroing Algorithm, we were able to convert any ZLRR into a PLRR. A natural question to ask is how long does the algorithm take to terminate (see appendices for painfully long conversions). The challenge of this question lies with the fact that every coefficient of $Q_0(x)$ needs to be taken into account; the degree itself is not enough information.\
- Using the Zeroing Algorithm, how can one understand the nature of the Zeckendorf decompositions with ZLRS’s? Does there need to be specific initial conditions? Is there a definition that is at all meaningful?
Some Examples of Running the Zeroing Algorithm
==============================================
Consider the recurrence relation $$H_{n+1}=2H_{n-1}+H_{n-2},$$ which has characteristic polynomial $P(x)=x^3-2x-1$ (principal root $r=(1+\sqrt{5})/2$), where we have the coefficients $c_1=0,c_2=2,c_3=1$. Suppose we are given $\beta_1=3$, $\beta_2=-2$, $\beta_3=-5$; we run the algorithm as follows:
------ ------ ------ ------ ------ ------ ------ -- ---------------------------- -- --
$3$ $-2$ $-5$ $Q_0(x)=3x^2-2x-5$
$-3$ $0$ $6$ $3$
$-2$ $1$ $3$ $Q_1(x)=-2x^2\,+\,x\,+\,3$
$2$ $0$ $-4$ $-2$
$1$ $-1$ $-2$ $Q_2(x)=x^2\,-\,x-2$
$-1$ $0$ $2$ $1$
$-1$ $0$ $1$ $Q_3(x)=-x^2\,-\,0x+1$
$1$ $0$ $-2$ $-1$
$0$ $-1$ $-1$ $Q_4(x)=0x^2\,-\,x-1$
------ ------ ------ ------ ------ ------ ------ -- ---------------------------- -- --
\
We reach termination on step $4$, since $Q_4$ does not have positive coefficients.\
Suppose that given the same recurrence relation, and initial values $a_0=3,a_1=-2,a_3=1$, we wish to determine whether the recurrence sequence diverges to negative infinity.\
Using the method introduced in Theorem \[thm:algorithm determination\], we first determine the values of
$$d_2\ = \ a_1c_1\ = \ 0,\ \ \ \ \ d_3\ =\ a_1c_2 + a_2c_1\ = \ 6,$$ from which we construct $$Q(x)\ = \ a_1x^2+(a_2-d_2)x+(a_3-d_3)=3x^2-2x-5.$$ We have $Q(r)=3r^2-2r-5=3(r+1)-2r-5=r-2<0$, which predicts that $\{a_n\}$ diverges to negative infinity.\
Manually computing the terms gives $$3,\ -2,\ 1,\ -1,\ 0,\ -1,\ -1,\ -2,\ -3,\ -5,\ -8,\ -13,\ \dots,$$ which confirms our prediction.\
List of ZLRR’s and derived ZLRR’s
=================================
1\. Recurrence: $G_{n+1}=G_{n-1}\,+\,G_{n-2}, \ P(x)=x^3\,-\,0\,x^2\,-\,x\,-\,1.$\
-------------- --------------- -------------- ---- ---- ---- -- -- ---------------------------- -- --
$\gamma_1=1$ 0 -1 -1 $Q_1(x)=0x^2-x-1$
-1 0 1 1
$\gamma_2=-1$ -1 0 1 $Q_2(x)=-x^2\,+\,0x\,+\,1$
1 0 -1 -1
$\gamma_3=0$ 0 0 -1 $Q_3(x)=0x^2\,+\,0x-1$
-------------- --------------- -------------- ---- ---- ---- -- -- ---------------------------- -- --
\
Derived PLRR characteristic polynomial: $x^5\,-\,x^4\,-\,0\,x^3\,-\,0\,x^2\,-\,0\,x\,-\,1$, which corresponds to the recurrence $H_{n+1}=H_n\,+\,H_{n-4}$.\
2. Current ZLRR: $G_{n+1} = G_{n-1}\,+\,G_{n-2}\,+\,G_{n-3}$. Current characteristic polynomial: $x^4\,-\,x^2\,-\,x\,-\,1$.\
Derived characteristic polynomial: $x^6\,-\,x^5\,-\,x^2\,-\,1$. Derived PLRR: $H_{n+1}=H_n\,+\, H_{n-3}\,+\,H_{n-5}$.\
3. Current ZLRR: $G_{n+1} = 2\,G_{n-1}\,+\,2\,G_{n-2}$. Current characteristic polynomial: $x^3\,-\,2\,x\,-\,2$.\
Derived characteristic polynomial: $x^5\,-\,x^4\,-\,2\,x\,-\,4$. Derived PLRR: $H_{n+1} = H_n\,+\, 2\,H_{n-3}\,+\,4\,H_{n-4}$.\
4. Current ZLRR: $G_{n+1} = 19G_{n-1} \,+\, 38G_{n-4}$. Current characteristic polynomial: $x^5\,-\,19x^3\,-\,38$.\
Derived characteristic polynomial: $x^{29}\,-\,x^{28}\,-\,310601172680577\,x^4 \,-\,40586681545596725\,x^3\\ \,-\,4277914985538462\,x^2 \,-\,170201741455942\,x \,-\,81203021913963806$.\
Derived PLRR: $H_{n+1} = H_n\,+\,310601172680577\,H_{n-24} \,+\,40586681545596725\,H_{n-25}\\ \,+\,4277914985538462\,H_{n-26} \,+\,170201741455942\,H_{n-27} \,+\,81203021913963806\,H_{n-28}$.\
5. Current ZLRR: $G_{n+1} = 6\,G_{n-1} \,+\, 3\,G_{n-2} \,+\, 5\,G_{n-3}$. Current characteristic polynomial: $x^4\,-\,6\,x^2\,-\,3\,x\,-\,5$.\
Derived characteristic polynomial: $x^{10}\,-\,x^9\,-\,69\,x^3\,-\,1669\,x^2\,-\,722\,x\,-\,1245$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 69\,H_{n-6}\,+\,1669\,H_{n-7}\,+\,722\,H_{n-8}\,+\,1245\,H_{n-9}$.\
6. Current ZLRR: $G_{n+1} = G_{n-2} \,+\, G_{n-3}$. Current characteristic polynomial: $x^4\,-\,x\,-\,1$.\
Derived characteristic polynomial: $x^{20}\,-\,x^{19}\,-\,4\,x^3\,-\,x^2\,-\,1$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 4\,H_{n-16}\,+\,H_{n-17}\,+\,H_{n-19}$.\
7. Current ZLRR:$G_{n+1} = 3\,G_{n-2} \,+\, G_{n-3} \,+\, 3\,G_{n-4}$. Current characteristic polynomial: $x^5\,-\,3\,x^2\,-\,x\,-\,3$.\
Derived characteristic polynomial: $x^{13}\,-\,x^{12}\,-\,14\,x^4\,-\,3\,x^3\,-\,54\,x^2\,-\,4\,x\,-\,39$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 14\,H_{n-8}\,+\,3\,H_{n-9}\,+\,54\,H_{n-10}\,+\,4\,H_{n-11}\,+\,39\,H_{n-12}$.\
8. Current ZLRR: $G_{n+1} = G_{n-2} \,+\, G_{n-19}$. Current characteristic polynomial: $x^{20}\,-\,x^{17}\,-\,1$.\
Derived characteristic polynomial: $x^{358}\,-\,x^{357}\,-\,4000705295\,x^{19} \,-\,7080648306\,x^{18} \,-\,575930712\,x^{17} \,-\,1937068817\,x^{16} \,-\,1082811308\,x^{15} \,-\,92014103\,x^{14} \,-\,2546102784\,x^{13} \,-\,1062101754\,x^{12} \,-\,372938426\,x^{11} \,-\,3264026504\,x^{10} \,-\,996542899\,x^9 \,-\,834914708\,x^8 \,-\,4089249024\,x^7 \,-\,890353375\,x^6 \,-\,1541366894\,x^5 \,-\,5013188421\,x^4 \,-\,759208181x^3\,-\,2567648478\,x^2 \,-\,6018966637\,x\,-\,635668820$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 4000705295\,H_{n-338} \,+\,7080648306\,H_{n-339} \,+\,575930712\,H_{n-340} \,+\,1937068817\,H_{n-341} \,+\,1082811308\,H_{n-342} \,+\,92014103\,H_{n-343} \,+\,2546102784\,H_{n-344} \,+\,1062101754\,H_{n-345} \,+\,372938426\,H_{n-346} \,+\,3264026504\,H_{n-347} \,+\,996542899\,H_{n-348} \,+\,834914708\,H_{n-349} \,+\,4089249024\,H_{n-350} \,+\,890353375\,H_{n-351} \,+\,1541366894\,H_{n-352} \,+\,5013188421\,H_{n-353}\,+\,759208181\,H_{n-354} \,+\,2567648478\,H_{n-355} \,+\,6018966637\,H_{n-356} \,+\,635668820\,H_{n-357} $.\
9. Current ZLRR: $G_{n+1} = G_{n-2} \,+\, G_{n-19}\,+\,G_{n-20}$. Current characteristic polynomial: $x^{21}\,-\,x^{18}\,-\,x\,-\,1$.\
Derived characteristic polynomial: $x^{156}\,-\,x^{155}\,-\,16626\,x^{20} \,-\,6\,x^{19} \,-\,16814\,x^{18} \,-\,4094\,x^{17} \,-\,1037\,x^{16} \,-\,6777\,x^{15} \,-\,5088\,x^{14} \,-\,1849\,x^{13} \,-\,9106\,x^{12} \,-\,6334\,x^{11} \,-\,3060\,x^{10} \,-\,12166\,x^9 \,-\,7932\,x^8 \,-\,4851\,x^7 \,-\,16190\,x^6 \,-\,10031\,x^5 \,-\,7482\,x^4 \,-\,21483\,x^3 \,-\,12839\,x^2 \,-\,11312\,x \,-\,11809$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 16626\,H_{n-135} \,+\,6\,H_{n-136}\,+\,16814\,H_{n-137} \,+\,4094\,H_{n-138}\,+\,1037\,H_{n-139} \,+\,6777\,H_{n-140} \,+\,5088\,H_{n-141} \,+\,1849\,H_{n-142} \,+\,9106\,H_{n-143} \,+\,6334\,H_{n-144} \,+\,3060\,H_{n-145} \,+\,12166\,H_{n-146} \,+\,7932\,H_{n-147} \,+\,4851\,H_{n-148} \,+\,16190\,H_{n-149} \,+\,10031\,H_{n-150} \,+\,7482\,H_{n-151} \,+\,21483\,H_{n-152} \,+\,12839\,H_{n-153} \,+\,11312\,H_{n-154} \,+\,11809\,H_{n-155}$.\
10. Current ZLRR: $G_{n+1} = G_{n-1} \,+\, 2\,G_{n-2} \,+\, 2\,G_{n-4} \,+\, 3\,G_{n-5}$. Current characteristic polynomial: $x^6\,-\,x^4\,-\,2\,x^3\,-\,2\,x\,-\,3$.\
Derived characteristic polynomial: $x^{11}\,-\,x^{10}\,-\,2\,x^5\,-\,2\,x^4\,-\,15\,x^3\,-\,x^2\,-\,7\,x\,-\,15$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 2\,H_{n-5}\,+\,2\,H_{n-6}\,+\,15\,H_{n-7}\,+\,H_{n-8}\,+\,7\,H_{n-9}\,+\,15\,H_{n-10}$.\
11. Current ZLRR: $G_{n+1} = 40\,G_{n-3} \,+\, 52\,G_{n-4}$. Current characteristic polynomial: $x^5\,-\,40\,x\,-\,52$.\
Derived characteristic polynomial: $x^{25}\,-\,x^{24}\,-\,555888384\,x^4 \,-\,1064960000\,x^3 \,-\,519168000\,x^2 \,-\,3308595200\,x \,-\,4535145472$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 555888384\,H_{n-20} \,+\,1064960000\,H_{n-21} \,+\,519168000\,H_{n-22} \\
\,+\,3308595200\,H_{n-23} \,+\,4535145472\,H_{n-24}$.\
12. Current ZLRR: $G_{n+1} = G_{n-8} \,+\, G_{n-9}$. Current characteristic polynomial: $x^{10}\,-\,x\,-\,1$.\
Derived characteristic polynomial: $x^{488}\,-\,x^{487}\,-\,7634770044678\,x^9 \,-\,16848326467063\,x^8 \,-\,25319805215106\,x^7 \,-\,29495744687667\,x^6 \,-\,27304765351108\,x^5 \,-\,19325535741204\,x^4 \,-\,8910253837548\,x^3 \,-\,1049595609091\,x^2 \,-\,321640563521\,x \,-\,1106933774826$.\
Derived PLRR: $H_{n+1} = H_n\,+\,7634770044678\,H_{n-478} \,+\,16848326467063\,H_{n-479} \,+\,25319805215106\,H_{n-480} \,+\,29495744687667\,H_{n-481}\,+\,27304765351108\,H_{n-482} \,+\,19325535741204\,H_{n-483} \,+\,8910253837548\,H_{n-484} \,+\,1049595609091\,H_{n-485} \,+\,321640563521\,H_{n-486} \,+\,1106933774826\,H_{n-487}$.\
13. Current ZLRR: $G_{n+1}=G_{n-2}\,+\,G_{n-4}\,+\,G_{n-6}$. Current characteristic polynomial: $x^7\,-\,x^4\,-\,x^2\,-\,1$.\
Derived characteristic polynomial: $x^{23}\,-\,x^{22}\,-\,x^6 \,-\,6\,x^5 \,-\,x^4 \,-\,6\,x^3 \,-\,x^2 \,-\,3\,x \,-\,2$.\
Derived PLRR: $H_{n+1} = H_n\,+\, H_{n-16}\,+\,6\,H_{n-17}\,+\,H_{n-18}\,+\,6\,H_{n-19}\,+\,H_{n-20}\,+\,3\,H_{n-21}\,+\,2\,H_{n-22}$.\
14. Current ZLRR: $G_{n+1}=3\,G_{n-1}\,+\,5\,G_{n-2}$. Current characteristic polynomial: $x^3\,-\,3\,x\,-\,5$.\
Derived characteristic polynomial: $x^5\,-\,x^4\,-\,2\,x^2\,-\,4\,x\,-\,15$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 2\,H_{n-2}\,+\,H_{n-3}\,+\,15\,H_{n-4}$.\
15. Current ZLRR: $G_{n+1}=G_{n-6}\,+\,G_{n-12}$. Current characteristic polynomial: $x^{13}\,-\,x^6\,-\,1$.\
Derived characteristic polynomial: $x^{572}\,-\,x^{571}\,-\,141734291356872\,x^{12} \,-\,1386240086076478\,x^{11} \,-\,3383864145243271\,x^{10} \,-\,4628373080436668\,x^9 \,-\,4069191511013055\,x^8 \,-\,2094637579574813\,x^7 \,-\,395154232336030\,x^6 \,-\,528518791146011\,x^5 \,-\,1761055564629423\,x^4 \,-\,2792877805797871\,x^3\\ \,-\,2780671348399214\,x^2 \,-\,1681201891412681\,x \,-\,401879825813162$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 141734291356872\,H_{n-559} \,+\,1386240086076478\,H_{n-560}\\ \,+\,3383864145243271\,H_{n-561} \,+\,4628373080436668\,H_{n-562} \,+\,4069191511013055\,H_{n-563}\\ \,+\,2094637579574813\,H_{n-564} \,+\,395154232336030\,H_{n-565} \,+\,528518791146011\,H_{n-566}\\ \,+\,1761055564629423\,H_{n-567} \,+\,2792877805797871\,H_{n-568} \,+\,2780671348399214\,H_{n-569}\\ \,+\,1681201891412681\,H_{n-570} \,+\,401879825813162\,H_{n-571} $.\
16. Current ZLRR: $G_{n+1}=G_{n-9}\,+\,G_{n-10}$. Current characteristic polynomial: $x^{11}-x-1$.\
Derived characteristic polynomial: $x^{665}\,-\,x^{664}\,-\,17581679276200473\,x^{10} \,-\,43065699679149511\,x^9 \,-\,70765959937154578\,x^8 \,-\,91624450164084254\,x^7 \,-\,98016133194347743\,x^6 \,-\,86803369058214690\,x^5 \,-\,61120624939489989\,x^4 \,-\,30036033003931493\,x^3 \,-\,5927897678515792\,x^2 \,-\,271244487735336\,x \,-\,1643001862841472$.\
Derived PLRR: $H_{n+1}\, =\, H_n \,+\, 17581679276200473\,H_{n-654} \,+\, 43065699679149511\,H_{n-655} \,\\+ \,70765959937154578\,H_{n-656}\, +\, 91624450164084254\,H_{n-657} \,+\, 98016133194347743\,H_{n-658}\,\\
+\,86803369058214690\,H_{n-659}\,+\,61120624939489989\,H_{n-660} \,+\,30036033003931493\,H_{n-661}\\ \,+\,5927897678515792\,H_{n-662}\,+\,271244487735336\,H_{n-663} \,+\,1643001862841472\,H_{n-664}$.\
17. Current ZLRR: $G_{n+1}=G_{n-1}\,+\,G_{n-6}$. Current characteristic polynomial: $x^{7}\,-\,x^5\,-\,1$.\
Derived characteristic polynomial: $x^{37}\,-\,x^{36}\,-\,18\,x^6 \,-\,2\,x^5 \,-\,9\,x^4 \,-\,2\,x^3 \,-\,7\,x^2 \,-\,9\,x \,-\,4$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 18\,H_{n-30}\,+\,2\,H_{n-31}\,+\,9\,H_{n-32}\,+\,2\,H_{n-33}\,+\,7\,H_{n-34}\,+\,9\,H_{n-35}\,+\,4\,H_{n-36}$.\
18. Current ZLRR: $G_{n+1}=2G_{n-2}\,+\,3G_{n-3}\,+\,5G_{n-5}$. Current characteristic polynomial: $x^6\,-\,2\,x^3\,-\,3\,x^2\,-\,5$.\
Derived characteristic polynomial: $x^{19}\,-\,x^{18}\,-\,75\,x^5 \,-\,207\,x^4 \,-\,708\,x^3 \,-\,384\,x^2 \,-\,370\,x \,-\,740$.\
Derived PLRR: $H_{n+1} = H_n\,+\, 75\,H_{n-13}\,+\,207\,H_{n-14}\,+\,708\,H_{n-15} \,+\, 384\,H_{n-16}\,+\,370\,H_{n-17}\,+\,740\,H_{n-18}$.\
19. Current ZLRR: $G_{n+1} = G_{n-1}\,+2\,G_{n-2}$. Current characteristic polynomial: $x^3\,-\,x\,-\,2$.\
Derived characteristic polynomial: $x^8\,-\,x^7\,-\,x^2\,-\,x\,-\,6$. Derived PLRR: $H_{n+1}=H_n\,+\, H_{n-5}\,+\,H_{n-6}\,+\,6H_{n-7}$.\
[CFHMNPX]{}
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T. Lengyel, *A Counting Based Proof of the Generalized Zeckendorf’s Theorem*, Fibonacci Quarterly **44** (2006), no. 4, 324-325.
S. J. Miller, D. Nelson, Z. Pan and H. Xu, *On the Asymptotic Behavior of Variance of PLRS Decompositions*, preprint, <https://arxiv.org/pdf/1607.04692.pdf>
S. J. Miller and R. Takloo-Bighash, *An Invitation to Modern Number Theory*, Princeton University Press, Princeton, NJ, 2006.
S. J. Miller and Y. Wang, *From Fibonacci numbers to Central Limit Type Theorems*, Journal of Combinatorial Theory, Series A **119** (2012), no. 7, 1398-1413.
W. Steiner, *Parry expansions of polynomial sequences*, Integers **2** (2002), Paper A14.
E. Zeckendorf, *Représentation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas*, Bulletin de la Société Royale des Sciences de Liège **41** (1972), pages 179-182.
\
[^1]: If we use the standard initial conditions then 1 appears twice and uniqueness is lost.
[^2]: We use $Z_n$ because the Lagonacci’s are easy to study, with interesting cases, usually requiring special attention. For an example of more standard behavior, consider $Y_{n+1} = 2Y_{n-1} + 2Y_{n-2}$, with $Y_1=1$, $Y_2 = 2$, $Y_3=3$, $Y_4 = 6$, …
[^3]: Note that if $4 \leq N < 3$, then $N$ is not an integer, so we reach no contradiction with our special initial condition case.
[^4]: Note that this is Condition 2 from Definition \[def:zlrrdefinition\], thus met by all $s$-deep ZLRS’s.
[^5]: Observe that $k$ is in both $J$ and $K$. Suppose, for contradiction, that some $q>1$ divides every element of $K$; then, every element of $\{k-\kappa\mid\kappa\in K\}=J$ is divisible by $q$, which is impossible.
[^6]: In fact, this statement is true for $n > \log(w) / \log(w^{(L-s)/L} / r)$.
|
---
abstract: 'Two recent papers[@xmri1; @xmri2] revealed that in our Galaxy, there are very extreme-mass-ratio inspirals composed by brown dwarfs and the supermassive black hole in the center. The event rates they estimated are very considerable for space-borne detectors in the future. However, there are also much more plunge events during the formation of the insprialing orbits. In this work, we calculate the gravitational waves from compact objects (brown dwarf, primordial black hole and etc.) plunging into or being scattered by the center supermassive black hole. We find that the signal-to-noise ratio of this burst gravitational waves are quite large for space-borne detectors. The event rates are estimated as $\sim$ 0.01 in one year for the Galaxy. If we are lucky, this kind of very extreme-mass-ratio bursts (XMRBs) will offer a unique chance to reveal the nearest supermassive black hole and nuclei dynamics. Inside 10 Mpc, the event rate can be as large as 4 per year and the signal is strong enough for space-borne detectors, then we have a good chance to probe the nature of neighboring black holes.'
author:
- |
Wen-Biao Han$^{1,2}$[^1], Xing-Yu Zhong$^{1,2}$[^2], Xian Chen$^{3,4}$[^3], Shuo Xin $^{5,1}$[^4]\
$^1$Shanghai Astronomical Observatory, Shanghai, 200030, China\
$^2$School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China\
$^3$Astronomy Department, School of Physics, Peking University, Beijing 100871, China\
$^4$Kavli Institute for Astronomy and Astrophysics at Peking University, Beijing 100871, China\
$^5$Tongji University, Shanghai, 200092, China
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: 'Very-Extreme-mass-ratio gravitational wave bursts in the Galaxy and neighbors for space-borne detectors'
---
\[firstpage\]
gravitational waves – space-borne detectors – extreme-mass-ratio bursts
introduction
============
After one century, the gravitational wave (GW) which is predicted by Einstein’s gravitational theory, has been detected by advanced LIGO (aLIGO) and advanced Virgo (AdV) more than 10 events in the O1 and O2 run [@gw15a; @gw15b; @gw17a; @gw17b]. The success of the GW detection, excites the plans of space-borne interferometers with arm-length about million kilometers. The Laser Interferometer Space Antenna (LISA [@lisa]), Taiji [@hu2017the] and Tian-Qin [@luo2016tianqin] planned to launch in 2030s, will focus the observation band from 0.1 milli-Hertz to 1 Hz. Extreme-Mass-Ratio Inspiral (EMRI) [@second_editor_1; @emris], e.g. a stellar mass compact object (1-10 $M_{\odot}$) orbiting around a supermassive black hole (SMBH), is a promising source of GW signal at these space-borne detectors’ band [@second_editor_1; @second_editor_2; @lisal3; @second_editor_3]. In particular, there are also extreme-mass-ratio bursts (EMRBs) which are produced when a compact object passes through periapsis on a highly eccentric orbit about a much more massive object [@rubbo06; @hopman07; @toonen09; @berry13a; @berry13b]. The event rate of this kind of burst sources is 0.2/year in 100 Mpc, and the signal-to-noise ratio is up to a few tens based on the analysis by [@berry13b].
Recently, two groups independently reported that in our Galactic center, LISA will see a few of very extreme mass-ratio inspirals. The mass-ratio is about $10^{-8}$ [@xmri1; @xmri2]. This kind of sources called as X-MRIs [@xmri1] are composed by brown dwarfs inspiraling into the SMBH. The event rates they estimated are quite high, could be more than 10 once the LISA becomes to observe.
However, there are much more brown dwarfs with unbounded orbits comparing with the bounded ones (X-MRIs). If the event rate is considerable for LISA, these brown dwarfs with plunge orbits will collide with the SMBH and produce burst GWs. In the present paper, we call this kind of GW sources as very-extreme-mass-ratio-bursts (XMRBs). In our Galaxy, the signal-to-noise(SNR) of XMRBs will be as large as a few of thousands. XMRBs in the Galactic center will be very easy to be found in the LISA’s observation. The GWs of XMRBs will give as unique change to know the nature of the central SMBH. In this paper, we firstly calculate the waveforms and SNRs of some typical XMRBs, then we show that the event rate of this kind of GW bursts is about $10^{-2}$ per year in our Galaxy and about 4 per year inside 10 Mpc valume. Finally, the expectancy of detection of XMRBs in our Galaxy and neighbor galaxies by LISA, Taiji and Tianqin will be addressed in the end.
Orbits and Waveforms of XMRBs
=============================
An XMRB is composed by a compact objects like brown dwarf or primordial black hole with $10^{-2} \sim 10^{-1} m_{\odot}$, and the central black hole has a mass about $4\times 10^{6} m_{\odot}$ if in our Galaxy. So the small objects can be safely treated as test particles. We now discuss the unbounded orbits of a test particle plunging into or scattered by a Schwarzschild black hole, the geodesics are governed by the following equations (with geometric unit: $c=G=1$)
$$\begin{aligned}
(\frac{dr}{dt})^2&=E^2-(1-\frac{2M}{r})(1+\frac{L^2}{r^2})\cdot {(1-\frac{2M}{r})}^{2}\cdot E^{-2}\\
\frac{d\varphi}{dt}&=\frac{L}{r^2}\cdot(1-\frac{2M}{r})\cdot E^{-1}\end{aligned}$$
The initial velocity is assumed much less than the velocity of light, then the energy of particle $E=1$ in the unit of the mass itself, the valid potential energy $U^2=(1-\frac{2M}{r})(1+\frac{L^2}{r^2})$ will have a peak value. This value is equal to 1 when $L=4$. So if $0\leqslant L\leqslant 4$ the particle will plunge into the horizon, and if $L>4$, the particle will be scattered. Figure \[orbit\] shows the plunging and scattered orbits with $L<4$ and $L>4$ respectively.
![The orbits of compact objects with different angular momentum starting from 99 $M$ to the central supermassive black hole.[]{data-label="orbit"}](orbit.pdf){width="\linewidth"}
For the plunging case, we use a formalism based on the Teukolsky equation [@Teukolsky] to compute the GW. The numerical method in frequency-domain was developed in [@SN; @Teukolsky; @1996; @Hughes; @2000; @MST_r; @MST_c; @han10; @han17] (and references inside). The perturbation field $\psi_{4}$, decomposed in frequency domain,
$$\begin{aligned}
\psi_{4}=\frac{1}{(r-iacos\theta)^4}\int^{\infty}_{-\infty}d\omega\sum\limits_{lm}R_{lm\omega}(r)_{-2}S^{a\omega}_{lm}(\theta)e^{-i\omega t+im\phi}\end{aligned}$$
where The function $_{-2}S^{a\omega}_{lm}(\theta)$ is a spin-weighted spheroidal harmonic, it can be computed via eigenvalue [@Hughes; @2000] or continuous fraction methods [@Leaver; @1985]. The radial function $R_{lm\omega}(r)$ obeys the Teukolsky equation $$\begin{aligned}
\Delta^{2}\frac{d}{dr}(\frac{1}{\Delta}\frac{R_{lm\omega}}{dr})-V(r)R_{lm\omega}(r)=-\mathcal{T}_{lm\omega}(r) \,,\end{aligned}$$ where $\mathcal{T}_{lm\omega}(r)$ is the source term, and the potential is: $
V(r)=-\frac{K^2+4i(r-M)K}{\Delta}+8i\omega r+\lambda
$\
where $\Delta =r^2-2Mr+a^2$, $K=(r^2+a^2)\omega-ma$, $\lambda\equiv\varepsilon_{lm}-2am\omega+a^2w^2-2$, the number $\varepsilon_{lm}$ is the eigenvalue of the spheroidal harmonic.
Using the Green function method [@Green], we can obtain the solution of the Teukolsky equation with a purely outgoing property at infinity and a purely ingoing property at the horizon: $$\begin{aligned}
R_{lm\omega}(r)=&\frac{1}{2i\omega C^{trans}_{lm\omega}B^{inc}_{lm\omega}}\{R^{\infty}_{lm\omega}(r)\int^{r}_{r_{+}}dr'R^{H}_{lm\omega}T_{lm\omega}\Delta^{-2}
\notag
\\
&+R^{H}_{lm\omega}(r)\int^{\infty}_{r_{+}}dr'R^{\infty}_{lm\omega}T_{lm\omega}\Delta^{-2}\}\end{aligned}$$ The asymptotic behavior of this solution near horizon and infinity is $$\begin{aligned}
R_{lm\omega}(r\to r_{+})&=\frac{B^{trans}_{lm\omega}\Delta^{2}e^{-iPr^{*}}}{2i\omega C^{trans}_{lm\omega} B^{inc}_{lm\omega}}\int^{\infty}_{r_{+}}dr'R^{\infty}_{lm\omega}T_{lm\omega}\Delta^{-2}\notag
\\&\equiv Z^{\infty}_{lm\omega}\Delta^{2}e^{-iP r^{*}}
\\R_{lm\omega}(r\to \infty)&=\frac{r^{3}e^{i\omega r^{*}}}{2i\omega B^{inc}_{lm\omega}}\int^{\infty}_{r_{+}}dr'R^{H}_{lm\omega}T_{lm\omega}\Delta^{-2}\notag
\\
&\equiv Z^{H}_{lm\omega}r^{3}e^{i\omega r^{*}}\end{aligned}$$ where $P=\omega-ma/2Mr_{+}$, and $r^{*}$ is the tortoise coordinate.
In general, because the homogeneous solution will diverge near the infinity, we cannot get solutions directly from Teukolsky equation with any kind of accuracy. To solve this problem, we can convert the equation to the Sasaki-Nakamura equation [@SN]: $$\begin{aligned}
\frac{d^2X_{lm\omega}}{d{r^{*}}^2}-F(r)\frac{dX_{lm\omega}}{d{r^{*}}}-U(r)X_{lm\omega}=0\end{aligned}$$ and use the transform rlue [@t-s2; @t-s1] from the Sasaki-Nakamura function to the Teukolsky function: $$\begin{aligned}
R^{H,\infty}_{lm\omega}=\frac{1}{\eta}[(\alpha+\frac{\beta_{,r}}{\Delta})\chi^{H,\infty}_{lm\omega}-\frac{\beta}{\Delta}\chi^{H,\infty}_{lm\omega,r}]\end{aligned}$$ where$\chi^{H,\infty}_{lm\omega}=X^{H,\infty}_{lm\omega}\Delta/\sqrt{r^2+a^2}$; $\alpha$, $\beta$, $\eta$ and the potentials $F(r)$, $U(r)$ can be found in [@t-s1]. By this way, we can calculate the solutions of the homogeneous Teukolsky equation.
The $\psi_{4}$ is related to the amplitude of the GW at infinity as $$\begin{aligned}
\psi_{4}(r\to\infty)\to \frac{1}{2}(\ddot{h}_{+}-i\ddot{h}_{\times}) \,.\end{aligned}$$ The gravitational waveform, observed from distance $R$, latitude angle $\theta$ and azimuthal angle $\phi$, is then given by $$\begin{aligned}
h_{+}-i h_{\times}=\frac{2}{R}\sum\limits_{lm}\int^{\infty}_{-\infty}d\omega \frac{1}{\omega^{2}}Z^{H}
_{lm\omega-2}S^{a\omega}_{l m}(\theta)e^{i(m\phi-\omega[t-r^{*}])}\end{aligned}$$ Now, we compute the waveforms for the XMRBs with angular momentums $L=0$, $L=1$, $L=2$, $L=3$, because $L<4$, all these XMRBs will plunge into the black hole directly. These waveforms are calculated by the frequency-domain Teukolsky equation we mentioned above.
![Sensitivity curves (LISA, Taiji and Tianqin) and XRMB’s GW amplitude spectral density with the different angular momentum $L=0$, $L=1$, $L=2$, $L=3$. The mass of the plunging object is $0.1 m_\odot$.[]{data-label="f-waveform"}](PSD_L=0_01.pdf "fig:"){width="0.49\linewidth"} ![Sensitivity curves (LISA, Taiji and Tianqin) and XRMB’s GW amplitude spectral density with the different angular momentum $L=0$, $L=1$, $L=2$, $L=3$. The mass of the plunging object is $0.1 m_\odot$.[]{data-label="f-waveform"}](PSD_L=1_01.pdf "fig:"){width="0.49\linewidth"} ![Sensitivity curves (LISA, Taiji and Tianqin) and XRMB’s GW amplitude spectral density with the different angular momentum $L=0$, $L=1$, $L=2$, $L=3$. The mass of the plunging object is $0.1 m_\odot$.[]{data-label="f-waveform"}](PSD_L=2_01.pdf "fig:"){width="0.49\linewidth"} ![Sensitivity curves (LISA, Taiji and Tianqin) and XRMB’s GW amplitude spectral density with the different angular momentum $L=0$, $L=1$, $L=2$, $L=3$. The mass of the plunging object is $0.1 m_\odot$.[]{data-label="f-waveform"}](PSD_L=3_01.pdf "fig:"){width="0.49\linewidth"}
Figure \[f-waveform\] shows the very strong GW burst signals produced by the XMRBs in the Galactic center, and the frequency is around $10^{-2}$ Hz, corresponding to the most sensitive frequency band of the space-borne detectors. We can see not only the (2,2) modes but also the (3,3) (4,4) and (5,5) ones are strong enough comparing to the sensitivity curves of LISA, Taiji, and Tianqin.
![The time-domain waveforms of the different harmonic modes for the $L=0$ case.[]{data-label="t-waveform_L=0"}](L=0_22waveform.pdf "fig:"){width="0.49\linewidth"} ![The time-domain waveforms of the different harmonic modes for the $L=0$ case.[]{data-label="t-waveform_L=0"}](L=0_33waveform.pdf "fig:"){width="0.49\linewidth"} ![The time-domain waveforms of the different harmonic modes for the $L=0$ case.[]{data-label="t-waveform_L=0"}](L=0_44waveform.pdf "fig:"){width="0.49\linewidth"} ![The time-domain waveforms of the different harmonic modes for the $L=0$ case.[]{data-label="t-waveform_L=0"}](L=0_55waveform.pdf "fig:"){width="0.49\linewidth"}
Figure \[t-waveform\_L=0\] shows the time-domain waveform $h_{+}$ obtained by the inverse Fourier transform, the duration of these signals are around 20 minutes for the modes (2,2), (3,3), (4,4) and (5,5).
The plunging signals are very strong, and very important to detect the structure of the central black hole. However, there are also scattered orbits, while the angular momentum $L$ is larger than 4. Because the particle is far away from the central SMBH in its entire orbit, we use the quadrupole formula to get the time-domain waveform [@peters; @1963; @peters; @1964], then obtain the frequency-domain waveform by the Fourier transform.
![time-domain and frequency-domain waveforms for scattered orbits with angular momentum $L=5$, $L=8$, the mass of the plunging object is still $0.1 m_\odot$.[]{data-label="58"}](GW_L=58.pdf "fig:"){width="0.49\linewidth"} ![time-domain and frequency-domain waveforms for scattered orbits with angular momentum $L=5$, $L=8$, the mass of the plunging object is still $0.1 m_\odot$.[]{data-label="58"}](PSD_58.pdf "fig:"){width="0.49\linewidth"}
Figure \[58\] shows the signal which is produced by the scattered particle will exist several hours. Of course , it is much weaker than the plunging one, but if in our Galaxy, we will see this kind of signals can still be detected. For quantitatively demonstrating the strength of the XMRBs, we compute the signal-to-noise ratios (SNR) of these signals [@SNR] . $$\begin{aligned}
{\rm SNR}^2=4Re\int_{0}^{\infty}\frac{|\tilde{h}(f)|^2}{S_n(f)}df\end{aligned}$$
0.05in
$L$ 5 8
-------------- ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ------ ----- ----
$l=m$ 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 2
LISA 6468 4430 3193 2392 6477 4435 3197 2392 6495 4444 3202 2397 6527 4457 3211 2401 280 20
TaiJi 4784 735 3381 1876 4793 4793 2475 1879 4807 3390 539 2479 4830 3400 2484 1886 167 12
TianQin 1965 1445 1091 851 1970 1450 1091 851 1970 1450 1095 851 1984 1454 1095 851 51 4
\[tablesnr\]
From Table \[tablesnr\], we can find that the SNRs of the plunging sources are very high. However for the scattered sources, the SNR is much lower than the plunging one. For the $L=5$ and $L=8$, the SNRs are 280, 167, 51 and 20, 12, 4 correspond to LISA, Taiji and Tianqin if in our Galaxy. Therefore, if for the neighboring galaxies, the signals of these scattered sources will be too weak to detect. However, for the plunging burst signals, even in 10 Mpc distance, the signal of (2,2) mode may still be detected by LISA if the plunging objects have a little heavier mass.
Event rates of XMRBs in our Galaxy
===================================
We can estimate the event of these burst using the formula [@xmri1], $$\begin{aligned}
\Gamma\simeq \frac{N}{T_{\rm rlx}
\ln(\theta_{\rm lc}^{-2})}\label{rate}\end{aligned}$$ where $N$ is the number of brown dwarfs, $T_{\rm rlx}$ is the relaxation timescale due to star-star scattering (two-body relaxation), and $\theta^2_{\rm lc}$ is the solid angle of the loss cone within which a brown dwarf will plunge into the SMBH. Since the brown dwarfs are normally coming from elongated orbits, we can calculate $\theta_{\rm lc}$ with $L/L_c$, where $L=8$ is the maximum angular momentum in our simulation which still leads to a significant SNR, and $L_c$ is the angular momentum of a circular orbit with the same energy of the plunging brown dwarf.
We note that the exact value of $\Gamma$ depends on the distance from the SMBH. However, in the limit $\theta_{\rm lc}\ll 1$, the majority of the stars in the loss cone are coming from the influence radius, where the enclosed stellar mass becomes comparable to the mass of the SMBH [@LC2013]. Therefore, we can estimate $\Gamma$ using the values at the influence radius of the SMBH in the Galactic Center, about $3$ pc, and the corresponding relaxation timescale is $T_{\rm rlx}\sim10^9$ years [@G2010]. To derive $N_*$ we follow the assumption in Amaro-Soeane 2019 about the initial mass function of stars and we find that there are about $N\sim3\times10^{6}$ brown dwarfs within the influence radius. With these consideration we find that $\Gamma\sim10^{-3}\,{\rm yr^{-1}}$.
Equation (\[rate\]) is derived under the assumption that the nuclear star cluster around the SMBH is spherical. However, observations showed that the stellar distribution in the central $8$ pc of the Galactic Center is triaxial [@FK2017]. In such a potential, the loss cone is refilled mainly by stars on chaotic orbits and the loss-cone filling rate can be orders of magnitude higher than the rate due to two-body relaxation [@MP2004]. For this reason, we think it possible that the event rate of the XMRBs in the Galactic Center could reach $\Gamma\sim10^{-2}\,{\rm yr^{-1}}$. A careful modeling of the XMRB rate in a triaxial potential is needed to better quantify the event rate.
However, the event rate estimation above is in our Galaxy. If we consider inside 10 Mpc volume, considering the number density of $10^5 m_\odot$ is 0.1/Mpc$^3$ [@Marconi2004], the number of SMBHs inside 10 Mpc is about four hundreds. For this distance, from the Table \[tablesnr\], the SNR can achieve at $\sim 10$ for the dominant (2,2) modes. Therefore, if we take into account the neighbor galaxies inside 10 Mpc, the event rate can arrive at 4 per year, this makes sense for space-borne detectors like LISA. The detection of such kind of plunging signals will reveal the nature of black holes.
Conclusions
============
The very extreme-mass-ratio sources in our Galaxy is very meaningful for GW space detection, because of the extremely small mass ratio $\sim 10^{-8}$, the gravitational self-force of the small body can be ignored , together with the high SNR, the spacetime of the central SMBH can be figured out very precisely.
In the present work, we propose a kind of GW sources named as XMRBs in our Galaxy and neighboring galaxies with considerable event rates. In contrast with the Galactic inspiral sources [@xmri1; @xmri2], we consider compact objects such as brown dwarfs and primordial black s plunging into or being scattered by the central supermassive black hole at Sgr A\* or by the nearby SMBHs. Because both the plunging or scattering of small objects into the black hole produce transient GW signals (comparing with insprialing sources), we call them as very extreme-mass-ratio bursts (XMRBs). The frequency of this source is around $10^{-3}\sim 10^{-2}$ Hz and it corresponds to the most sensitive frequency band of the space-borne detectors.
The Galactic inspiraling objects stay outside of the innermost stable circular orbit of the SMBHs, but the plunging objects collide with the BH horizon directly, then the latter can produce GW signals carrying the direct information of horizon. Our calculation show that the SNR is around $10^2\sim 10^3$ for the plunging XMRBs, and $\sim$ 20 for the scattering ones with angular momentum up to $L = 8$ for the sources in our Galaxy. For the source at 10 Mpc, the SNR can still be as large as $\sim 10$. The signals are strong enough and event rate we estimated can be arrive at 0.01 per year for the plunging sources in the Galaxy (the event rate is not sensitive with angular momentum). If we are lucky, this kind of sources are very important for observing the nearest SMBH. However, in 10 Mpc volume, there are a few hundreds of galaxies [@Marconi2004], then the event rate may arrive at $\sim 4$ per year and the SNR of plunge source is still large enough for the space-borne detectors. This makes sense for the future space-borne detectors and the detection of the nature of black holes.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by NSFC No.11273045.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail:[email protected]
[^4]: E-mail: [email protected]
|
---
author:
- 'Ana Ovcharova[^1] on behalf of the ATLAS Collaboration'
title: 'Track-Based Alignment of the Inner Detector of ATLAS'
---
Introduction {#intro}
============
The ATLAS Inner Detector (ID), shown in Fig. \[fig:1\], is composed of the Pixel, the Semiconductor Tracker (SCT) and the Transition Radiation Tracker (TRT). The ID is designed to achieve the momentum and vertex resolutions required for high-precision measurements [@ATLASPaper]. To ensure that the resulting requirements on track reconstruction are met, the position and orientation of each active detector element must be known with sufficient accuracy such that track parameter resolution is degraded by less than 20% of the design values. The following is an outline of the procedure, results and some of the challenges of the alignment of the ID.
Alignment strategy {#sec:1}
==================
The alignment is derived by minimizing track residuals which are defined as the difference between the expected and the measured hit positions. The $\chi^{2}$ function to be minimized is given by: $$\chi^{2}=\sum_{Trks} \vec r(\boldsymbol{\vec\tau}, \vec a)^{T} V^{-1} \vec r(\boldsymbol{\vec\tau}, \vec a),$$ where $V$ is the hit covariance matrix and $\vec r(\boldsymbol{\vec\tau}, \vec a)$ is the vector of track residuals, which are a function of both the track parameters, $\boldsymbol{\vec\tau}$, and the alignment constants, $\vec a$. ID alignment implements two flavors of $\chi^{2}$-based algorithms: the Global $\chi^{2}$ and the Local $\chi^{2}$. In the Global $\chi^{2}$ approach, a simultaneous minimization with respect to all track parameters and alignment constants is done. This approach ensures that full correlation between alignable objects intersected by a common track is retained. In the Local $\chi^{2}$ minimization, module correlations are discarded, rendering alignment less computationally intensive. However, it is necessary to perform multiple iterations to reach convergence. The alignment uses isolated high-$p_{T}$ tracks to reduce the impact of pattern recognition ambiguities and of multiple scattering. Both collision and cosmic ray tracks are used to maximize long-distance correlations between detector elements. ID alignment is staged at several levels of granularity, corresponding to the hierarchy of its mechanical structure. Table 1 shows the substructures and algorithms used at different levels in the Autumn 2010 alignment. The numbers in the DoF column represent the product of the number of substructures and the allowed degrees of freedom for each. For example, at Level 2, the Pixel half shells were allowed all three rotations and three translations, while the endcaps only two translations and one rotation [@AlignPaper]. In the latest alignment, discussed in Sec. 4, the Global $\chi^{2}$ was used at all levels but the wire-by-wire alignment of the TRT (approximately 700,000 DoF). The latter used the Local $\chi^{2}$ approach due to computational restrictions.
[llclcl]{} Level & Structures & \# DoF & Method\
& PIX: whole & &\
Level 1 & SCT: barrel + 2 endcaps & 41 & Global $\chi^2$\
& TRT: barrel + 2 endcaps & &\
& PIX: half shells + disks & &\
Level 2 & SCT: layers + disks & 852& Global $\chi^2$\
& TRT: modules + wheels & &\
& PIX: modules & &\
Level 3 & SCT: modules & 722104 & Local $\chi^2$\
& TRT: wires & &\
Alignment performance
=====================
The Autumn 2010 alignment was the first to use 7 TeV collision data in addition to pixel module wafer deformation input from the production survey. It was also the first wire-by-wire TRT alignment. The impact of these improvements is evident in the reduced width of the residual distributions in the barrel sections of all sub-detectors, the Pixel, SCT and TRT, shown in Figs. \[fig:2\], \[fig:3\] and \[fig:4\], respectively. Similar trends are observed in the endcap regions, where the large track statistics used in this alignment were particularly advantageous.[@AlignPaper]
Weak modes and constrained alignment
====================================
There exist systematic detector deformations that cannot be detected using the outlined approach as they retain the helical form of tracks at the expense of biasing the track parameters. They are commonly referred to as “weak modes” and can be identified by examining the kinematics of resonance decays such as $Z\rightarrow\mu\mu$, $J/\Psi\rightarrow\mu\mu$ and $K_{S}\rightarrow \pi\pi$. The bias introduced in the track momenta by such misalignments violates the symmetries inherent in these decays and thereby results in unexpected dependences of the reconstructed invariant mass on various kinematic observables. A striking example is the dependence of the Z invariant mass on the $\phi$ track parameter of the positive muon, see Fig. \[fig:5\]. The approach to correct such misalignments is to constrain some parameters during the alignment, thereby, minimizing the possibility of retaining biases. Some examples of useful constraints are: momentum measurements by the Muon Spectrometer, vertex position constraint and the calorimeter derived constraint. The calorimeter derived constraint, or $E/p$ constraint, uses the fact that the calorimeter response for positrons and electrons should be the same. Differences between the ratio of energy to momentum measurement between electrons and positrons in $Z\rightarrow ee$ or $W\rightarrow e\nu$ decays can then be attributed to mismeasurement of the momentum in the tracker and used to obtain corrections to the reconstructed track momenta in bins of azimuthal angle and pseudorapidity. During alignment, the track momenta are then constrained to the corrected value. The $E/p$ correction has resulted in the latest significant improvement in the alignment as evidenced by the increase in the $Z$ invariant mass resolution shown in Fig. \[fig:6\].
Run-by-run alignment monitoring
===============================
Significant changes of the detector alignment occur due to external factors. Some of the identified causes include temperature changes and magnet ramping. The time-ordered global shifts of selected substructures in the direction transverse to the beam pipe are shown in Fig. \[fig:7\]. The largest changes observed are less than 10 $\mu$m. To monitor and better understand this behavior, the Level 1 alignment constants are now recomputed on a run-by-run basis.
Additionally, as resonances have been shown to be a powerful probe in uncovering weak-mode misalignments and, thereby, momentum biases, plots of the reconstructed mass as a function of various kinematic variables and the mass itself (as in Figs. \[fig:5\] and \[fig:6\]) are also produced automatically for every run as a part of the ATLAS data quality monitoring.
Conclusion and outlook
======================
The current implementation of the alignment procedure has been shown to be effective and well suited for the challenges posed by the alignment of the ATLAS ID. The next step is to evaluate the systematics caused by residual misalignments. It has already been seen that resonances are a powerful handle for tackling this problem and ongoing studies will soon provide quantitative measures of any remaining biases.
ATLAS Collaboration, *The ATLAS Experiment at the CERN Large Hadron Collider*, JINST 3 S08003 (2008). ATLAS Collaboration, *Alignment of the ATLAS Inner Detector Tracking System with 2010 LHC proton-proton collisions at $\sqrt s$ = 7 TeV*, ATLAS-CONF-2011-012, https://cdsweb.cern.ch/record/1334582 (2011).
[^1]:
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---
abstract: 'We propose a scheme for distillation of free bipartite entanglement from bipartite bound-entangled states. The crucial element of our scheme is an ancillary system that is coupled to the initial bound-entangled state via appropriate weak measurements. We show that in this protocol free entanglement can be always generated with nonzero probability by using a single copy of the bound-entangled state. We also derive a lower bound on the entanglement cost of the protocol and conclude that, on average, applying weaker measurements results in relatively higher values of free entanglement as well as lower costs.'
author:
- 'S. Baghbanzadeh'
- 'A. T. Rezakhani'
title: 'Distillation of free entanglement from bound-entangled states using weak measurements'
---
Introduction
============
Entanglement [@QuantEnt] is a physical resource that plays a leading role in performing quantum computation and quantum information processing [@Nielsen:book]. This is also helpful in understanding relevant properties of many-body quantum systems. It has been shown [@Osterloh; @wsls] that a singularity in the entanglement profile of the ground state of a many-body system, even a change in the type of its entanglement from bound (free) to free (bound) [@BEQPT], can be accompanied by quantum phase transitions.
Bound-entangled states are states from which no pure entanglement can be extracted only by local operations and classical communications (LOCC) [@BE; @Alber]. However, recent studies unveil the usefulness of these states in some protocols such as secure quantum key distribution [@Hor-key], remote quantum information concentration [@remote], quantum data hiding [@hiding], channel discrimination [@chdis], and reducing communication complexity [@compelexity]. It was also indicated [@act; @Masanes] that application of any bound-entangled state along with some free-entangled state can increase the teleportation power of the free-entangled state. Preparation of some particular bipartite and multipartite bound-entangled states is now possible in nuclear magnetic resonance [@NMR], optical [@Smolin1; @several; @unconditional; @BeW; @deGaussification], as well as ion-trapped systems [@decoherence]. Moreover, bound entanglement can naturally arise, e.g., in the XY spin model through applying an external magnetic field [@magneticB], in the Jaynes-Cummings model [@JC], as well in strongly-correlated graph states at thermal equilibrium [@graph1; @graph2].
Bound entanglement can be “activated" (or “unlocked") into free entanglement [@QuantEnt; @act]. For example, in the multipartite case, a distillable ensemble can be obtained from a tensor product or a mixture of some non-distillable (i.e., bound) ensembles [@superactivation; @BI]. Additionally, in multipartite bound-entangled states, if two parties carry out a Bell-type measurement on their particles, the other parties can distill free entanglement by LOCC [@Smolin; @DC]. Alternatively, in the bipartite case, both unitary [@unitary] and nonunitary [@non-unitary] evolutions of bound-entangled states may result in the birth of free entanglement.
Here, we demonstrate that, by attaching an ancillary qubit to a bipartite bound-entangled state and performing two weak measurements between each party of that state and the ancilla, one can transform this bound entanglement to free entanglement with a nonvanishing probability. Measurements we employ here are “weak” in that they do not disturb the initial state strongly; thus, after performing the weak measurements, there is some probability with which strong or projective measurement does not occur [@WM1; @WM2; @Igain]. Note that we do not rule out the existence of similar distillation scenarios with strong or projective measurements; here we only focus on a weak-measurement scenario. Our approach has a lower cost than the protocols suggested in Refs. [@Smolin; @DC; @exp1] because, (i) the measurement operators we use have less nonlocal content than the Bell-type measurements, and (ii) we do not need to share any free-entangled state between parties. Furthermore, unlike existing distillation [@distill1; @distill2; @distill3] and entanglement generation [@entgen] protocols, we do not use any unitary operation either. Some environmentally-induced decoherence scheme may inhibit applying unitary operations, whence our scheme provides a controllable method for distillation of free entanglement from bipartite bound-entangled states.
Bound-entangled states and weak measurement setting
===================================================
We illustrate our scheme through three qutrit-qutrit bound-entangled states shared between an “Alice” and a “Bob”: a state which is complementary to the tiles unextendible product basis [@UPB]; $\chi_1=\frac{1}{4}(\openone-\sum_{i=0}^4 |\psi_i\rangle\langle\psi_i|)$, in which $|\psi_0\rangle=|0\rangle(|0\rangle-|1\rangle)/\sqrt{2}$, $|\psi_1\rangle=(|0\rangle-|1\rangle)|2\rangle/\sqrt{2}$, $|\psi_2\rangle=|2\rangle(|1\rangle-|2\rangle)/\sqrt{2}$, $|\psi_3\rangle=(|1\rangle-|2\rangle)|0\rangle/\sqrt{2}$, and $|\psi_4\rangle=(|0\rangle+|1\rangle+|2\rangle)(|0\rangle+|1\rangle+|2\rangle)/3$; and two Horodeckis’ states [@BEH] $$\begin{aligned}
& \chi_2(a)= \frac{1}{1+8a}\left(
\begin{array}{ccccccccc}
a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\
0 & a & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\
a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{1+a}{2} & 0 & \frac{\sqrt{1-a^2}}{2} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\
a & 0 & 0 & 0 & a & 0 & \frac{\sqrt{1-a^2}}{2} & 0 & \frac{1+a}{2} \\
\end{array}
\right),\nonumber\end{aligned}$$
![(Color online) Degree of weakness of the measurement, $\zeta$, vs parameter $x$ for $\beta=1/10,\;1/5,\;3/10$, and $2/5$ (from bottom to top). $\zeta$ is symmetric around $\beta=1/2$. According to the argument bellow Eq. (\[weakness\]), it is clear that for all $\beta$s, the measurement is strong when $x=1$. Depending on the value of $\beta$, the measurement is significantly weak when $x\in(1/\sqrt{3},1/\sqrt{2})$.[]{data-label="fig1"}](fig1)
and $\chi_3(b)=(2|\Phi^+\rangle\langle\Phi^+|+b\Sigma_+ +(5-b)\Sigma_-)/7$, in which $|\Phi^+\rangle=(|00\rangle+|11\rangle+|22\rangle)/\sqrt{3}$, $\Sigma_{+}=(|01\rangle\langle01|+|12\rangle\langle12|+|20\rangle\langle20|)/3$, and $\Sigma_-$ is the swap of $\Sigma_+$ [@act]. The state $\chi_2$ is a function of parameter $a\in[0,1]$. This state is separable when $a\in\{0,1\}$, otherwise it is bound-entangled [@BEH]. The state $\chi_3$ is a function of a parameter $b\in[2,5]$. For $b\in[2,3]$ this state is separable, for $b\in(3,4]$ it is bound-entangled, and otherwise it is free-entangled [@act].
To quantify the free entanglement created via measurement, we use “negativity" [@Negativity], defined for a bipartite $d\times d'$ system $\varrho_{AB}$ (when $d<d'$) as [@NNegativity] $$\begin{aligned}
\mathcal N= (\Vert \varrho^{T_B}\Vert_1-1)/(d-1).\end{aligned}$$ Here $\varrho^{T_B}$ denotes partial transposition with respect to the second party, which according to the Peres-Horodecki criterion [@PH], is negative when the state $\varrho_{AB}$ is free-entangled. In this case, $\mathcal N$ is positive, otherwise it vanishes. Recall that $\Vert B\Vert_1=\mathrm{Tr}[\sqrt{B^\dagger B}]$, and in a given basis $\varrho^{T_B}_{ij,kl}=\varrho_{il,kj}$.
We note that, since the positivity of the partial transposition of a state does not change with LOCC [@BE], one may need to consume some sort of nonlocality (at least indirectly) in order to transform bound entanglement into free entanglement. We share one of the states, $\chi_1$, $\chi_2(a)$, or $\chi_3(b)$, between Alice and Bob, and attach an ancillary qubit $\varrho^C$ to this system. Now, first Bob (next Alice) performs a joint weak measurement on his (her) own particle and this ancilla as follows: $$\begin{aligned}
M_i=\sum_{j=1}^3 \varepsilon_{j\oplus(i-1)} P_j,\end{aligned}$$ where $i\in\{1,2,3\}$, $\oplus$ denotes modulo $3$ sum, $\varepsilon_l\in[0,1]$ are some real parameters such that $\sum_{l=1}^3\varepsilon_l^2=1$ (hereafter we rename $\varepsilon_0$ as $\varepsilon_3$), and $P_j$s are some orthogonal projectors (specified later) which satisfy $\sum_{j=1}^3 P_j=\openone$. It is thus evident that $\sum_{i=1}^3 M^\dagger_i M_i=\openone$. The result of this weak measurement can be described as $$\begin{aligned}
{\mathcal E}_\zeta(X)=(1-\zeta){\mathcal E}_\mathrm{strong}(X)+\zeta X,\label{weakness}\end{aligned}$$ where $\zeta=\varepsilon_1\varepsilon_2+\varepsilon_2\varepsilon_3+\varepsilon_1\varepsilon_3$, and ${\mathcal E}_\mathrm{strong}(X)=\sum_{j=1}^3 P_j X P_j$ denotes the strong or projective measurement. That is, with probability $(1-\zeta)$ the strong measurement ${\mathcal E}_\mathrm{strong}$ is applied, whereas the state does not undergo any change with probability $\zeta$. The smaller $\zeta$ is, the stronger the measurement is.
The post-measurement state of Alice, Bob, and ancilla becomes $$\begin{aligned}
\varrho^{ABC}_{ij}=\frac{M^{AC}_j M^{BC}_i\left(\chi^{AB}\otimes\varrho^C\right) M^{BC}_i M^{AC}_j}{\mathrm{Tr}\left[M^{AC}_j M^{BC}_i\left(\chi^{AB}\otimes\varrho^C\right) M^{BC}_i M^{AC}_j\right]},\end{aligned}$$ where we choose $$\begin{aligned}
&P_1\equiv\openone_{3\times2}-|\phi\rangle\langle\phi|-|\psi\rangle\langle\psi|,\nonumber\\
&P_2\equiv|\phi\rangle\langle\phi|\;,\;P_3\equiv|\psi\rangle\langle\psi|,\end{aligned}$$ with $|\phi\rangle=\alpha|00'\rangle+\sqrt{1-\alpha^2}|11'\rangle$ and $|\psi\rangle=\sqrt{1-\alpha^2}|00'\rangle-\alpha|11'\rangle$, where $\alpha\in(0,1)$. Here $|0'\rangle$ and $|1'\rangle$ are the basis vectors of the Hilbert space of the ancillary qubit. In addition, we choose $\varepsilon_1=x$, $\varepsilon_2=\sqrt{\beta(1-x^2)}$, and $\varepsilon_3=\sqrt{(1-\beta)(1-x^2)}$, in which $x\in[0,1]$. Note that the case $\beta=1/2$ (i.e., $\varepsilon_2=\varepsilon_3$) is not of interest because in this case no entanglement is induced by measurement between the bound-entangled state and the ancilla when the outcome of measurement is $M_1$. Figure \[fig1\] depicts the behavior of $\zeta$ vs parameter $x$, for some fixed values of $\beta$. We shall demonstrate that our protocol generates free entanglement from bound entanglement when the measurements are generically weak.
Distillation of free entanglement
=================================
For specificity, hereon we fix the value of $\beta$ to $1/10$. Moreover, we take $\alpha=1/\sqrt{2}$, for which the entanglement content of the measurement projectors $P_2$ and $P_3$ becomes maximal. However, we also discuss the $\alpha\neq1/2$ in the sequel as well. Furthermore, we initialize the ancilla qubit in the state $(|0'\rangle+|1'\rangle)/\sqrt{2}$.
Suppose that initially Alice and Bob share the state $\chi_1$. Figure \[fig2\] shows the variation of negativity of their post-measurement state, $\varrho_{ij}^{AB}=\mathrm{Tr}_C[\varrho_{ij}^{ABC}]$, in terms of $x$, for different outcomes of measurements. It is remarkable that in this case, all $x\in(0,1/4]$ yield the free entanglement with certainty.
![(Color online) Genuine tripartite entanglement, $E_{ABC}=\overline{{\mathcal N}}^2_{AB,C}-(\overline{{\mathcal N}}^2_{A,C}+\overline{{\mathcal N}}^2_{B,C})$ [@Monogamy], as a function of $x$, for initial states $\chi_1$ (dashed curve) and $\chi_2(a)$ (for the values of $a$ as in the previous figures) when $\alpha=1/\sqrt{2}$. The lowest curve corresponds to $\chi_2(1/50)$.[]{data-label="fig6"}](fig6)
If Alice and Bob begin with the state $\chi_2(a)$, negativity of the post-measurement state behaves as in Fig. \[fig3\]. This figure demonstrates that: (i) in all outcomes, there exist intervals of $x$ in which transition from bound entanglement to free entanglement occurs. In particular, when $x\in(0,0.05]$, one can assure that the transition takes place with a high probability; that is, seven out of nine total cases show nonvanishing amounts of free entanglement. (ii) For all allowed values of $a\neq0$ this transition occurs (for some values of $x$). (iii) For $a$s not too close to zero, the maximum amount of free entanglement is created when the measurement outcomes $M_1$ and $M_3$ are obtained for Bob and Alice, respectively.
A similar analysis shows that when $\chi_3(b)$ is the initial state, after the measurement, free entanglement is generated between Alice and Bob for all $b\in[2,4]$ with nonzero probability—plots not presented here. The overall behavior of ${\mathcal N}_{A,B}$, for each measurement outcome is also akin to that of $\chi_2(a)$, with the difference that in this case, even when the measurement outcomes are the same ($i=j$), the points at which negativity vanishes depend on both $b$ and $x$.
If $\alpha$ approaches $0$ or $1$, the maximum value of $\mathcal{N}_{A,B}$ may increase or decrease depending on the outcomes and the value of $a$ and $b$ for the initial states $\chi_2(a)$ and $\chi_3(b)$. For example, in the case of the initial state $\chi_1$ and the outcome $M_2^{AC} M_3^{BC}$, $\mathcal{N}_{A,B}$ takes its maximum value at $\alpha\approx 1/\sqrt{4.3}$ (see the dot-dashed curve in the middle panel of Fig. \[fig2\]). The interval of $x$ in which the transition occurs may also increase or decrease for all initial states, but here the point is that for all $\alpha\neq 0,1$ one can distill free entanglement with nonzero probability (for the initial state $\chi_3(b)$, this is correct for $b=4$). For example, we showed the results corresponding to $\alpha=1/100$ in Fig. \[fig4\].
To get further insight on the cost of generating free entanglement between Alice and Bob, we investigate the non-locality content of the applied (weak) measurements. Note that the pre-measurement state of Bob and ancilla is a product state. Thus, the average entanglement of their post-measurement state can be considered as a lower bound on the nonlocality that this measurement contains. A similar argument is also applicable to the measurement performed on the Alice-ancilla state in the next step since this state before measurement is separable—because it can be checked that its negativity is zero and since this state is $3\times2$, positivity of its partial transposed implies its separability. We remark that the method proposed in Ref. [@EntCost] to identify nonlocal cost of a measurement does not seem suitable to apply in our scenario, because in that method the pre-measurement state is maximally mixed, and the method uses a clever trick to remove the effect of the initial state on the nonlocality content of a measurement. Taking all these points into account, an estimate for nonlocality on the measurements in our scheme is as follows: $$\begin{aligned}
\hskip-.5mm M_\mathrm{cost}=\sum_{j=1}^3\left[p_{B,C}(j)\mathcal{N}_{B,C}(j)+\sum_{k=1}^3 p_{A,C}(j,k)\mathcal{N}_{A,C}(j,k)\right]\hskip-1.2mm,\nonumber\end{aligned}$$ where $p_{B,C}(j)$ \[$p_{A,C}(j,k)$\] in the first (second) term denotes the probability of obtaining the result(s) $j$ ($j$ and $k$) after a joint measurement over party $B$ and ancilla $C$ ($A$ and ancilla $C$), and $\mathcal{N}_{B,C}(j)$ \[$\mathcal{N}_{A,C}(j,k)$\] is the negativity of the post-measurement state of the same parties, corresponding to the same result(s). We consider $E_\mathrm{cost}\equiv M_\mathrm{cost}-\overline{{\mathcal N}}_{A,B}$ as the entanglement cost of our distillation scenario (depicted in the right panel of Fig. \[fig5\]). Here, $\overline{{\mathcal N}}_{A,B}$ denotes the average of free entanglement generated between Alice and Bob after the measurements (see the left panel of Fig. \[fig5\]). The curves shown in the right panel of Fig. \[fig5\] present the behavior of $E_\mathrm{cost}$ versus $x$ for different initial states $\chi_1$ and $\chi_2(a)$. It should be remarked that, in our protocol, measurements indeed can generate entanglement in the $(AB)C$, $A(BC)$, and $B(AC)$ bipartitions as well as genuine tripartite entanglement (Fig. \[fig6\]). Nevertheless, since here we are only interested in the generated entanglement between $A$ and $B$, in our analysis we only subtract this entanglement from the measurement cost in order to obtain a lower bound for the entanglement cost of our scenario. One should also note that the existence of all these correlations naturally restricts the amount of free entanglement generated between Alice and Bob after measurements. Figures \[fig1\] and \[fig5\] imply that if Alice and Bob choose the measurement strength from the significantly weak ranges, e.g., $x\in(0.65,0.75)$, they can distill relatively high amounts of free entanglement with relatively low cost. Having said all this, to give a better estimate of how much nonlocality is needed in our scenario, a more detailed analysis in which all sorts of entanglement (bipartite and tripartite) are taken into account is needed. For our purposes, however, the given analysis suffices.
A final remark is in order here. It might be argued that our protocol transfers the initial bound entanglement to the other partitions. However, it is straightforward to show that in the range of parameters where distillation of free entanglement between $A$ and $B$ is successful there does not exist any further bipartite bound entanglement in the total system. Accordingly, our protocol does not distribute bound entanglement within the system.
Summary
=======
Here we have proposed a controllable scheme for distilling free entanglement from bipartite bound-entangled states. Unlike previous entanglement distillation protocols, our protocol employs an ancillary qubit, and is based on weak measurements, obviating the need to share any free-entangled state between the parties. In this sense, our protocol uses a non-distillable entangled state and transforms it into a “useful" type without having to consume other useful states. Rather, the entanglement is unlocked through the very measurement process. Therefore, in order to analyze the cost of our protocol, we have compared how much entanglement needs to be invested in the measurement process, and as a result how much entanglement can be obtained. This argument has implied that our protocol can generate useful entanglement with a set of measurements which do not need much entanglement to be realized.
There may still exist projective or strong measurements which can also transform bound to free entanglement. Our scheme can be a suitable alternative in situations, e.g., where physical realization of such strong measurements is difficult. Although we have illustrated our protocol with specific $3\times 3$ bound-entangled states, generalization to higher dimensional and multipartite systems seems straightforward.\
We appreciate valuable discussions with V. Karimipour. ATR acknowledges partial financial support by Sharif University of Technology’s Office of Vice-President for Research.
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