\section{Introduction}
\label{sec:introduction}
Einstein by formulating his General Relativity \cite{Einstein:1915ca,Einstein:1916vd} showed that there is a relation between geometry of spacetime
and matter contribution. Until now, many
astronomical observations have tested GR giving us that it is the best matching theory for explaining gravitational phenomena which we have in hand.
However, there are still many issues in fundamental physics, astrophysics and cosmology
(for review, see e.g. \cite{Copeland:2006wr,Nojiri:2006ri,Capozziello:2007ec,Carroll:2004de,Sotiriou:2008ve}) indicating
 that GR is not a final theory of gravitational interactions. There are such unsolved
problems as the dark matter puzzle \cite{Cap_Laur, Cap_Far}, inflation \cite{Starobinsky:1980te,Guth:1980zm}, another one is the late-time cosmic acceleration \cite{Huterer:1998qv, Sami} which is explained by the assumption that there
exists an exotic fluid called dark
energy \cite{Copeland:2006wr,Nojiri:2006ri,Capozziello:2007ec}. It is introduced to the dynamics throughout adding the
cosmological constant to the standard Einstein's field equations considered in Friedmann-Robertson-Lemaitre-Walker (FRLW)
spacetime.
Due to the above shortcomings one looks for different approaches in order to find a good theory which will be able to answer the above
problems and to show us directions of future research. The additional argument in a favor of searching generalizations of
gravity is non-renormalization of Einstein's theory. The renormalization problem seems to be solved if extra high curvature
terms are added \cite{stelle}. One would also like to unify gravity theory with the other ones (electromagnetism, weak and strong
interactions has been already unified into the Standard Model) but no satisfactory result has been obtained so far as e.g. string theory,
supersymmetry, that could combine particle physics and gravitation.

One of many ways to deal with the mentioned problems are Extended Theories of Gravity (ETG) \cite{Cap_beyond, cap_invar} which
have been gained a lot of attention. The main arguments which advocate them are a possibility to explain problematic issues without introducing exotic,
immeasurable entities as well as some approaches to quantum gravity \cite{buch} or the geometrization of the Standard Model (plus gravity) as
derived from Chamseddine-Connes spectral action
in noncommutative geometry \cite{nonc, nonc2} which indicate that the effective action for the classical gravity should
be more complicated than the one introduced by Einstein.

The geometric part of the action might be changed in many different ways. One may assume that constant of Nature are not
really constant values \cite{Dabrowski:2012eb, Leszczynska:2014xba, Salzano:2016pny}. A scalar field might be added into Lagrangian
and moreover, it can be minimally or non-minimally coupled to gravity \cite{brans, Bergmann}. One proposes
much more complicated functionals than the simple linear one used in GR, for example $f(R)$
gravity \cite{buchdahl, Starobinsky:1980te}. The latter approach has gained a lot of interest recently as the extra geometric terms could explain not only dark matter
issue \cite{cap_jcap, cap_not} but also dark energy problem because it produces the accelerated late-time effect at low cosmic densities. The field
equations also differ from the Einstein's ones so they could provide different
behavior of the early Universe.

The $f(R)$ gravity is usually treated in two different ways: in the metric approach
\cite{Sotiriou:2008rp,Carroll:2004de,Sotiriou:2008ve,DeFelice:2010aj,Will:1993te}
and Palatini one \cite{Palatini:1919di,Sotiriou:2008rp,Capozziello:2011et,Ferraris:1992dx}. The former arises to the fourth
order differential equations which are more difficult to handle than the second-order field equations of GR
\footnote{ The theory possesses the scalar-tensor representation, that is, it is mathematically equivalent to the Brans-Dicke gravity (with the parameter $\omega=0$)
with a potential of the scalar field. In this representation one deals with the differential equations of the second order on the metric components and a modified Klein-Gordon
equation for the scalar field, see the details e.g. \cite{sotir_pal}.}. Another point is that one believes that physical equations of motion should be of
the second order. In contrast to the metric formalism, the Palatini $f(R)$ gravity provides second order differential equations since
the connection and the metric are treated as independent objects. The Riemann and Ricci tensors are constructed with the connection while
for building the Ricci scalar we also use the physical metric in order to contract the indices. The Palatini approach allows to get
the modified Friedmann equation \cite{alle_bor1, alle_bor2, alle_bor3} in a form that might be compared with the observational
data \cite{bor_kam, BSSW1, BSSW2, BSS, SSB}. It shows the potential of the Palatini formulation when it is applied to the gravitational
problems \cite{lavinia, roshan}.

There also exist disadvantages of such an approach: being in conflict with the Standard Model of particle physics
\cite{sotir_pal, Flanagan:2003rb, Iglesias:2007nv, Olmo:2008ye},
the algebraic dependence of the post-Newtonian
metric on the density \cite{Olmo:2005zr, Sotiriou:2005xe}, and the complications with the initial values problem in the
presence of matter \cite{Ferraris:1992dx, Sotiriou:2006hs}, although
the problem was already solved in \cite{olmo_sanch}. Another one happens at microscopic scales, that is, the theory produces
instabilities in
atoms which disintegrate them. One should also mention limitations which arrive  when one treats the extra terms as fluid-like
\cite{zaeem, zaeem2, zaeem3}. However, it was shown \cite{olmo_tri} that high curvature corrections do not cause such problem.
What is also very promising, some of the Palatini Lagrangians avoid the Big Bang singularity.
Moreover, the effective dynamics of Loop Quantum
Gravity can be reproduced by the Palatini theory which gives the link to one of approaches to Quantum Gravity \cite{olmo_singh}.
High curvature correction of the form  changes the notion of the independent connection: in the simple Palatini $f(R)$ gravity the
connection is auxiliary field while in the more general Palatini theory it is dynamical without making the equations of motion
second order in the fields.

The problem which is our interest in this work concerns astrophysical aspects of Palatini gravity, for example black
holes \cite{diego1,diego2,diego3,diego4}, wormholes \cite{worm} and neutron stars \cite{anab}.
We would like to examine the last objects in the context of Palatini gravity. In
general, the maximal mass of neutron stars is still an open problem: Einstein's gravity gives us, together with the recent observations, the limit
as $2$M$_\odot$. The pulsar PSR J1614-2230 is found to have the limit $1.97$M$_\odot$ \cite{demorest} while another one
is Vela X-1 with the mass $\sim 1.8$M$_\odot$ \cite{rawls}. One also supposes the existence more massive neutron stars as for instance the ones with
masses around $2.1$M$_\odot$ \cite{MassNs} and $2.4$M$_\odot$ (B1957+20) \cite{van}. For the recent neutron stars mass determination see \cite{stellarcollapse}.
There also exists "hyperon puzzle": it turns out that equations of state including hyperons make the maximal mass limit in the case of
neutron stars without magnetic field lower than $2$M$_\odot$ \cite{hyperon1, hyperon2, hyperon3}. Due to that fact it seems that these equations of state could
not be feasible for massive neutron stars in the framework of General Relativity. Some solutions were already proposed, like for example
hyperon-vector coupling, chiral quark-meson coupling and the existence of strong magnetic
fields inside the star. An emissary for the latter idea are works indicating an influence of magnetic fluid which increases masses of the stars
\cite{mag1, mag2, mag3}. This arises to a question if neutron stars without strong magnetic field with mass larger than $2$ M$_{\odot}$ can exist
\cite{astash, astash2, astash3} whose structure can be still described by the equations derived from General Relativity.

Moreover, because of the very recent neutron stars' merger observation \cite{gw} the confrontation of gravitational theories with the observational data is an
appealing task. Neutron stars are perfect objects for tests at high density regimes thus one may perceive possible deviations from GR. It also seems that
using General Relativity as a description of strong gravitational fields \cite{astash3, eksi} can be an extrapolation for fields sourced by neutron
stars since they are many orders of magnitude larger than the one we may probe by solar system tests. Therefore, likely GR should be modified in a strong gravity regime and
in the case of large spacetime curvature \cite{berti}.

Since we are interested in Palatini gravity, one should mention another problem which arose throughout the studies on neutron stars in this formalism.
In \cite{Barausse:2007ys, barau} it was shown that there exist surface singularities of static spherically symmetric objects in the case of polytropic equation
of state. It would seem that all Palatini $f(\hat{R})$ functionals besides Einstein-Hilbert one could be ruled out since they provide unphysical behavior. However in
\cite{olmo_ns} it was demonstrated that the problem appears because of the particular equation of state rather than the dynamics of the theory itself.
Further approach to the problem \cite{fatibene} indicates that polytropic equation of state is not a fundamental issue but only an approximation of the matter
content in the star. Nevertheless, they examined the case of problematic matter description according to the Ehlers-Pirani-Schild (EPS) interpretation
\cite{eps, mauro, fatibene1}, which
we also follow in our work. Considering the conformal metric as the one responsible for the free fall they revealed that in this case the singularities are not
generated in comparison to metric which was used in \cite{Barausse:2007ys}.

Due to the above brief considerations on neutron stars, there are two main problems concerning these objects: how to model dense matter inside the star, which one
would like to describe by an equation of state, and viable equations which depict the macroscopic properties, that is, mass and radius. The alternative
equations, or rather modified Tolman-Oppenheimer-Volkoff equations, describing the macroscopic values of relativistic stars are actively proposed
\cite{AltNS1, AltNS2, AltNS3, AltNS4, AltNS5, AltNS6, AltNS7} and tested with the observational data. In this work we would also like to challenge this problem
and hence in the next section we will present briefly $f(\hat{R})$ gravity in Palatini formalism and then we are going to focus on stars' stability problem. We will also provide
appropriate TOV equations for this generalization of General Relativity. We are using the Weinberg's \cite{weinberg} signature convention, that is, $(-,+,+,+)$.

\section{A brief description of Palatini gravity: frames}
As we have already mentioned, we are interested in Palatini gravity which is a particular case of EPS formalism \cite{eps}. That means that geometry of spacetime
is described by two structures which in Palatini gravity turns out to be the metric $g$ and the connection $\hat{\Gamma}$ which are independent objects.
In addition, the connection is a Levi-Civita connection of a metric conformally related to $g$. Due to this interpretation, one considers motion of a mass particle
appointed by the connection while clocks and distances are ruled by the metric $g$.
Let us briefly, but in a quite detailed way, introduce the basic of the Palatini $f(\hat{R})$ gravity. The action is given by
\begin{equation}
S=S_{\text{g}}+S_{\text{m}}=\frac{1}{2}\int \sqrt{-g}f(\hat{R}) d^4 x+S_{\text{m}},\label{action}
\end{equation}
where $\hat{R}=\hat{R}^{\mu\nu}g_{\mu\nu}$ is the generalized Ricci scalar. The variation of (\ref{action}) with respect to the metric $g_{\mu\nu}$ provides
\begin{equation}
f'(\hat{R})\hat{R}_{\mu\nu}-\frac{1}{2}f(\hat{R})g_{\mu\nu}=T_{\mu\nu},\label{structural}
\end{equation}
where $T_{\mu\nu}=-\frac{2}{\sqrt{-g}} \frac{\delta}{\delta g_{\mu\nu}}S_m  = (\rho+p)u^{\mu}u^{\nu}+pg^{\mu\nu}$ is energy
momentum tensor, $p$ pressure while $\rho$ energy density of the fluid. The vector $u^\alpha$ is a $4$-velocity of the observer co-moving with
the fluid ($g_{\mu\nu}u^\mu u^\nu = -1$).
Taking the trace of the equation (\ref{structural}) with respect to $g_{\mu\nu}$ yields a structural equation, which is given by
\begin{equation}
f'(\hat{R})\hat{R}-2 f(\hat{R})=T.\label{struc}
\end{equation}
Assuming that we are able to solve (\ref{struc}) as $\hat{R}(T)$ we see that $f(\hat{R})$ is also a function
of the trace of the energy momentum tensor, where $T=g^{\mu\nu}T_{\mu\nu}\equiv 3p-\rho$.

The variation with respect to the independent connection gives
\begin{equation}
\hat{\nabla}_\alpha(\sqrt{-g}f'(\hat{R})g^{\mu\nu})=0,\label{con}
\end{equation}
from which we immediately notice that $\hat{\nabla}_\alpha$ is the covariant derivative calculated with respect to $\hat \Gamma$, that is, it is the Levi-Civita connection of the conformal
metric $h_{\mu\nu}=f'(\hat{R})g_{\mu\nu}.$

 It is well known \cite{DeFelice:2010aj} that the action (\ref{action})
is dynamically equivalent to the  constraint system with first order Palatini gravitational Lagrangian:
\begin{align}\label{action1}
 S(g_{\mu\nu}, \Gamma^\lambda_{\rho\sigma}, \chi)=\frac{1}{2\kappa}\int\mathrm{d}^4x\sqrt{-g}\left(f^\prime(\chi)(\hat R-\chi) + f(\chi) \right)\nonumber\\
 + S_m(g_{\mu\nu},\psi),
\end{align}
providing that $f''(\hat R)\neq 0 $.
\footnote{In that case the linear Einstein-Hilbert Lagrangian $f(\hat R)=\hat R -2\Lambda$ is excluded on that level.}
Introducing further a scalar field $\Phi=f'(\chi)$ and taking into account the constraint equation $\chi=\hat R$, one can rewrite the action in dynamically equivalent way as a Palatini action
\begin{equation}\label{actionP}
 S(g_{\mu\nu}, \Gamma^\lambda_{\rho\sigma},\Phi)=\frac{1}{2k}\int\mathrm{d}^4x\sqrt{-g}\left(\Phi \hat R - U(\Phi) \right)+ S_m(g_{\mu\nu},\psi),
\end{equation}
where the potential $U(\Phi)$ "remembers" the form of function $f(\hat R)$. It is defined as
\begin{equation}\label{PotentialP}
U(\Phi)=\chi(\Phi)\Phi-f(\chi(\Phi)
\end{equation}
where $\chi\equiv\hat R  = \frac{d U(\Phi)}{d\Phi}$
and $\Phi = \frac{d f(\chi)}{d\chi}$.

Palatini variation of this action provides
	\begin{align}
	\label{EOM_P
	\Phi\left( \hat R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} \hat R \right)   +{1\over 2} g_{\mu\nu} U(\Phi) - \kappa T_{\mu\nu} = 0\\
	\label{EOM_connectP}
	 \hat{\nabla}_\lambda(\sqrt{-g}\Phi g^{\mu\nu})=0\\
	\label{EOM_scalar_field_P}
	  \hat R   -  U^\prime(\Phi) =0
	\end{align}
The last equation due to the constraint $\hat R =\chi= U^\prime(\phi)$ is automatically satisfied. The middle equation (\ref{EOM_connectP}) implies that
the connection $\hat \Gamma$ is a metric connection for the new metric $h_{\mu\nu}=\Phi g_{\mu\nu}$. The $g$-trace of the first
equation  $-\Phi \hat R+2 U(\Phi)=\kappa T$, can be recast  as
\begin{equation}\label{struc2}
  2U(\Phi)-U'(\Phi)\Phi=\kappa T.
\end{equation}
which provides an analog of the structure equation (\ref{struc}).

Now the equation (\ref{EOM_P}) can be rewritten as a dynamical equation for the
metric $h_{\mu\nu}$ \cite{BSS,SSB}
 \begin{subequations}
	\begin{align}
	\label{EOM_P1}
	 \bar R_{\mu\nu} - \frac{1}{2} h_{\mu\nu} \bar R  &  =\kappa \bar T_{\mu\nu}-{1\over 2} h_{\mu\nu} \bar U(\Phi)

	\end{align}
	\begin{align}
	\label{EOM_scalar_field_P1}
	  \Phi\bar R &  -  \Big(\Phi^2\,\bar U(\Phi)\Big)^\prime =

	\end{align}
\end{subequations}
where we have introduced $\bar U(\phi)=U(\phi)/\Phi^2$ and appropriate energy momentum tensor $\bar T_{\mu\nu}=\Phi^{-1}T_{\mu\nu}$.
Moreover, one notices that $\hat R_{\mu\nu}=\bar R_{\mu\nu}, \bar R= h^{\mu\nu}\bar R_{\mu\nu}=\Phi^{-1} \hat R$ and $h_{\mu\nu}\bar R=\ g_{\mu\nu}\hat R$.
The last equation, together with the trace of (\ref{EOM_P1}), can be replaced by
\begin{equation}\label{EOM_P1c}
 \Phi\,\bar U^\prime(\Phi)  + \kappa \bar T = 0\,.
\end{equation}
Thus the system (\ref{EOM_P1}) - (\ref{EOM_P1c}) corresponds to a scalar-tensor action for the metric $h_{\mu\nu }$ and (non-dynamical) scalar field $\Phi$
\begin{equation}\label{action2}
 S(h_{\mu\nu},\Phi)=\frac{1}{2\kappa}\int\mathrm{d}^4x\sqrt{-h}\bigg(\bar R- \bar U(\Phi) \bigg) + S_m(\Phi^{-1}h_{\mu\nu},\psi),
\end{equation}
where
\begin{equation}\label{em_2}
    \bar T^{\mu\nu} =
-\frac{2}{\sqrt{-h}} \frac{\delta}{\delta h_{\mu\nu}}S_m  = (\bar\rho+\bar p)\bar u^{\mu}\bar u^{\nu}+ \bar ph^{\mu\nu}=\Phi^{-3}T^{\mu\nu}~,
\end{equation}
and $\bar u^\mu=\Phi^{-{1\over 2}}u^\mu$, $\bar\rho=\Phi^{-2}\rho,\ \bar p=\Phi^{-2}p$, $\bar T_{\mu\nu}= \Phi^{-1}T_{\mu\nu}, \ \bar T= \Phi^{-2} T$ (see e.g. \cite{DGB}).
Further, the trace of (\ref{EOM_P1}), provides
\begin{equation}\label{EOM_metric_2}
    \bar R= 2\bar U(\Phi)-\kappa \bar T.
\end{equation}

\section{Stability condition}
Following \cite{weinberg} and \cite{green} we wish to show that Palatini neutron stars whose matter is modeled by perfect fluid are in dynamical equilibrium and are
described by modified TOV equations. Moreover, we are going to demonstrate that this equilibrium is stable which means that a considered configuration is in
thermodynamic equilibrium.
\subsection{TOV equations in ETGs}
It was shown in \cite{aw1} how some modifications of the TOV equations influence a mass and radius of a neutron star. The introduced parameters to the equations not only
allow to see how the mass-radius diagram changes with respect to them but also can be useful to give some clue on a kind of ETG which we should look for in order
to get the desired results which are in agreement with the observations. The details and discussion on gravitational theories which are related to the
parameters can be also found in \cite{aw1, szwab}.

The TOV equations can be generalized for a certain class of theories after the generalization of the energy density and pressure \cite{aw2}:
\begin{align}
 Q(r):=\rho(r)+\frac{\sigma(r)W_{tt}(r)}{\kappa B(r)},\\
\label{def2} \Pi(r):=p(r)+\frac{\sigma(r)W_{rr}(r)}{\kappa A(r)}
\end{align}
where the extra terms above are provided by the modified Einstein field equations of the particular form \cite{mim, mim2, mim3}
\begin{equation}\label{mod1}
 \sigma(\Psi^i)(G_{\mu\nu}-W_{\mu\nu})=\kappa T_{\mu\nu}.
\end{equation}
As usually, the tensor $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor, $\kappa=-8\pi G$, the factor $\sigma(\Psi^i)$ is a coupling to the gravity
while $\Psi^i$ represents for instance curvature invariants or other fields, like
scalar ones. The symmetric tensor $W_{\mu\nu}$ denotes additional geometrical terms which might appear in considered ETG. It should be
noted that (\ref{mod1}) provides a parameterization of gravitational theories at the level of field equations.
The energy-momentum tensor $T_{\mu\nu}$ will be considered as the one of a perfect fluid discussed in the previous section.
Because the tensor $W_{\mu\nu}$ usually includes extra fields like scalar or electromagnetic ones, apart the modified Einstein's
field equations (\ref{mod1}) we also deal with equations for the additional fields.

As shown \cite{aw1}, in the case of the spherical-symmetric metric
\begin{equation}\label{metric}
 ds^2=-B(r)dt^2+A(r)dr^2+r^2d\theta^2+r^2\sin^2{\theta} d\varphi^2,
\end{equation}
the modified Einstain field equations of the form (\ref{mod1}) reproduces the generalized TOV equations
\begin{align}\label{tov}
  \left(\frac{\Pi}{\sigma}\right)'&=-\frac{G\mathcal{M}}{r^2}\left(\frac{Q}{\sigma}+\frac{\Pi}{\sigma}\right)
  \left(1+\frac{4\pi r^3\frac{\Pi}{\sigma}}{\mathcal{M}}\right)\left(1-\frac{2G\mathcal{M}}{r}\right)^{-1}\nonumber\\
	&+\frac{2\sigma}{\kappa r}\left(\frac{W_{\theta\theta}}{r^2}-\frac{W_{rr}}{A}\right)\\
	\mathcal{M}(r)&= \int^r_0 4\pi \tilde{r}^2\frac{Q(\tilde{r})}{\sigma(\tilde{r})} d\tilde{r}.\label{mod_mass}
\end{align}
A specification of a gravitational theory allows us to study stellar configurations. We would like to emphasize the importance of extra field equations which may
appear in the considered theory of gravity, as we have already shown it in the previous work as well as we will always take them into account further. Moreover,
let us notice that the modified mass equation (\ref{mod_mass}) comes directly from the solution of the field equations (\ref{mod1}) with the metric
ansatz (\ref{metric}) so the extra term
proportional to $W_{tt}$ should not be skipped in the examination of the star's configuration. One may treat them as a contribution from additional fields to the
theory, for example electromagnetic or scalar ones, as caused by the influence of modified equation of state \cite{blas1, blas2, blas3} or as just
pure geometric modifications.

\subsection{Stability of Palatini stars}

Let us focus our attention on the discussed stability problem in the case of Palatini gravity. From the structural equation (\ref{struc}) we expect that the crucial
quantity in the Palatini formalism, that is, the conformal factor $\phi=f'(\hat{R})$, will depend on the trace of the energy momentum tensor which in the case of the perfect
fluid has a form $T=3p-\rho$. As we assumed the spherical-symmetric geometry, all physical quantities depend on the radial coordinate $r$ only and hence $\phi=\phi(r)$
which makes the both metric spherical-symmetric.
Another important observation is that the "Palatini-Einstein" tensor $\hat{G}_{\mu\nu}=\hat{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\hat{R}$ built of both, Palatini-Ricci
tensor and the metric $g_{\mu\nu}$ turns out to be equal to the Einstein tensor considered in so-called Einstein
frame $\bar{G}_{\mu\nu}=\bar{R}_{\mu\nu}-\frac{1}{2}h_{\mu\nu}\bar{R}$. Due to that fact the equations in both frames give a solution on the metric $h_{\mu\nu}$ while
the metric $g_{\mu\nu}$ can be obtained from the conformal relation (\ref{con}). Therefore we will refer to the metric (\ref{metric}) as the metric $h_{\mu\nu}$ which
enters the Einstein equations in the Einstein frame (\ref{EOM_P1}). This is a metric responsible for a geodesic motion, that is, it is a Levi-Civita metric of the
connection $\hat{\nabla}_\mu$. According to this interpretation \cite{eps} motion of particles should be considered with respect to the
Palatini connection while the metric $g$ is related to light cones and distances which are measured by a normalized observer $u^\mu$, that is, $u^\mu u^\nu g_{\mu\nu}=-1$.

Writing the equations (\ref{EOM_P1}) as
\begin{equation}
\bar{R}_{\mu\nu}=\kappa\bar{T}_{\mu\nu}-\frac{1}{2}h_{\mu\nu}(\bar{T}+\bar{U}(\phi))
\end{equation}
and using the spherical-symmetric ansatz on the metric $h_{\mu\nu}$ one may write
 \begin{equation}
 \frac{\bar{R}_{rr}}{2A}+\frac{\bar{R}_{00}}{2B}+\frac{\bar{R}_{\theta\theta}}{r^2}=-\frac{A'}{rA^2}-\frac{1}{r^2}+\frac{1}{Ar^2}=\kappa\bar{\rho}+\frac{1}{2}r^2\bar{U}
\end{equation}
from which immediately we get that
\begin{equation}\label{mod_geo}
 A(r)=\left( 1-\frac{2G \mathcal{M}(r)}{r} \right)^{-1},
\end{equation}
where
\begin{equation}\label{nowem}
\mathcal{M}(r)=\int^r_0\left( 4\pi \tilde{r}^2\bar{\rho}(\tilde{r})+\frac{\tilde{r}^2\bar{U}(\tilde{r})}{4G} \right)d\tilde{r}
\end{equation}
or, if one prefers to write the mass distribution for the physical energy density
\begin{equation}\label{nowem2}
 \mathcal{M}(r)=\int^r_0\left( 4\pi \tilde{r}^2\frac{\rho}{\phi(\tilde{r})^2}(\tilde{r})+\frac{\tilde{r}^2U(\tilde{r})}{4G\phi(\tilde{r})^2} \right)d\tilde{r}.
\end{equation}
Let us underline that we took into account the transformations of (\ref{em_2}) together with the conformal transformation of the metric, that is, the diagonal element of the metric $g$
will be denoted by $\tilde{A}(r)=\phi^{-1}A(r)$ and so on, which will be also transparent in the further considerations.

Due to the relations (\ref{def2}) and the equations (\ref{mod1}) one has
\begin{subequations}\label{defq}
 \begin{equation}
   \bar{Q}=\bar{\rho}+\frac{1}{2}\bar{U}=\frac{\rho}{\phi^2}+\frac{U}{2\phi^2}=\phi^{-2}Q,\label{q}
 \end{equation}
\begin{equation}
  \bar{\Pi}=\bar{p}-\frac{1}{2}\bar{U}=\frac{p}{\phi^2}-\frac{U}{2\phi^2}=\phi^{-2}\Pi\label{pi}.
\end{equation}
\end{subequations}
We are dealing with two conserved quantities in Einstein frame
\begin{align}
 \hat{\nabla}_\mu\bar{J}^\mu=\hat{\nabla}_\mu n\bar{u}^\mu&=0,\label{current}\\
 \bar{u}_\nu\hat{\nabla}_\mu\bar{T}_\text{eff}^{\mu\nu}=\bar{u}_\nu\hat{\nabla}_\mu \left(\bar{T}^{\mu\nu}-\frac{1}{2}h^{\mu\nu}\bar{U}\right)&=0\label{bian},
\end{align}
where the first equation is the nucleon number current conservation law while the second one is the Bianchi identity contracted with the vector
field $\bar{u}^\alpha$, which naively could be thought as an observer with respect to the metric $h$ but she is not the one responsible for the measurements. Since the relation
between the physical observer and the vector field $\bar{u}^\alpha$ is as already mentioned
\begin{equation}
      \bar{u}^\alpha=\phi^{-\frac{1}{2}}u^\alpha
\end{equation}
we may apply the vector field $\bar{u}^\alpha$ to the (\ref{bian}) in order to simplify the calculations. The another observation is that the proper nucleon number density $n$
appearing in (\ref{current}) is
frame independent, that is, $n=-u_\alpha J^\alpha_N=-\bar{u}_\alpha \bar{J}^\alpha_N$. From (\ref{bian})
\begin{align}
 \bar{u}^\mu\left( -n\bar{p}\hat{\nabla}_\mu\Big(\frac{1}{n}\Big)-n\hat{\nabla}_\mu\Big(\frac{\bar{\rho}}{n}\Big)-\frac{1}{2\kappa}\Bar{U}'\hat{\nabla}_\mu\phi \right)=0
\end{align}
which we write as
\begin{equation}
 \hat{\nabla}_\mu n=\frac{n}{\bar{\rho}+\bar{p}}\hat{\nabla}_\mu\bar{\rho}+\frac{1}{2\kappa}\frac{n}{\bar{\rho}+\bar{p}}\bar{U}'\hat{\nabla}_\mu\phi,
\end{equation}
where we may also use the relation (\ref{EOM_P1c}) in the last term.
Thus the infinitesimal changes with respect to the infinitesimal chan\-ges of the energy density are
\begin{equation}
 \delta n=\frac{n}{\bar{\rho}+\bar{p}}\delta\bar{\rho}+\frac{1}{2\kappa}\frac{n}{\bar{\rho}+\bar{p}}\bar{U}'\delta\phi,
\end{equation}
which turns out to be
\begin{equation}\label{deltan}
 \delta n=\frac{n}{\bar{Q}+\bar{\Pi}}\delta\bar{Q}
\end{equation}
after applying the definitions (\ref{defq}) and $\delta\bar{\rho}=\delta\bar{Q}-\frac{\bar{U}'}{2\kappa}\delta\phi$.

In order to examine the stability condition in the case of Palatini gravity, we will follow the approach presented, for example, in \cite{weinberg}, that is, the
Lagrange multipliers method.
Following Weinberg, let us formulate the proposition:
\begin{prop}
 A particular stellar configuration in the Palatini gravity with an arbitrary function $f(\hat R)$, with uniform entropy per nucleon and chemical composition,
 will sa\-tisfy the equations ($\phi=f'(\hat{R})$)
 \begin{align*}
\mathcal{M}(r)&=\int^r_0 4\pi \tilde{r}^2\frac{Q(\tilde{r})}{\phi(\tilde{r})^2}d\tilde{r},\\
   \frac{d}{dr}\Big( \frac{\Pi(r)}{\phi^2(r)}\Big)&=-\frac{\tilde{A}G\mathcal{M}}{r^2}\left(\frac{\Pi+Q}{\phi({r})^2}\right)\left(1+4\pi r^3\frac{\Pi}{\phi({r})^2\mathcal{M}}\right)
\end{align*}
for equilibrium, if the quantity $\mathcal{M}$, defined by
$$
\mathcal{M}\equiv\int 4\pi r^2\frac{Q(r)}{\phi({r})^2}dr
$$
is stationary with respect to all variations of $\bar{Q}(r)$ that leave unchanged the quantity (nuclear number)
$$
  N=\int^R_0 4\pi r^2\left(1-\frac{2G\mathcal{M}(r)}{r}\right)^{-\frac{1}{2}} n(r)dr
$$
and that leave the entropy per nucleon and the chemical composition uniform and unchanged. The equilibrium is stable with respect to radial oscillations if
$\mathcal{M}$ is a minimum with respect to all such variations.
\end{prop}
In order to demonstrate the validity of the proposition, let us firstly notice that the nuclear number $N$ also depends on modified geometry
via the relation (\ref{nowem}) or (\ref{nowem2}). Moreover, the extra term appearing in (\ref{tov}) vanishes in our particular theory.
Then we see that $\mathcal{M}$ is stationary with
respect to all variations that leave the nuclear number fixed if there exists a constant $\lambda$ such that $\mathcal{M}-\lambda N$ is stationary with
respect to all variations. It means that we are interested in the case when $\mathcal{M}$ can be an extremum for the unchanged nuclear number, together with uniform
entropy per nucleon mass and the chemical decomposition.
Hence we deal with
\begin{eqnarray}\label{deltaMlambda}
 &0&=\delta \mathcal{M}-\lambda \delta N=\int_0^\infty 4\pi r^2 \delta \bar{Q} dr \nonumber\\
 &-&\lambda \int_0^\infty 4\pi r^2\left(1-\frac{2G\mathcal{M}(r)}{r}\right)^{-\frac{1}{2}}\delta n(r)dr\nonumber\\
 &-&
 \lambda G\int_0^\infty 4\pi r\left(1-\frac{2G\mathcal{M}(r)}{r}\right)^{-\frac{3}{2}}n(r)\delta\mathcal{M}(r)dr.
\end{eqnarray}
Using the relation (\ref{deltan}) and gravitational mass definition (\ref{nowem})
\begin{equation}
 \delta\mathcal{M}(r)=\int^r_04\pi r'^2\delta\bar{Q}(r')dr',
\end{equation}
we interchange the $r$ and $r'$ integrals in the last term after replacing it by the above to get
\begin{align}\label{condit}
 \delta\mathcal{M}&-\lambda\delta N=\int^\infty_0 4\pi r^2\left(
 1-\frac{\lambda n(r)}{\bar{\Pi}(r)+\bar{Q}(r)}\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{1}{2}}\right.\nonumber\\
 &\left.-\lambda G\int^\infty_r 4\pi r'n(r')\left[ 1-\frac{2G\mathcal{M}(r')}{r'} \right]^{-\frac{3}{2}}dr'
 \right)\delta\bar{Q}(r)dr.
\end{align}
From the above expression one sees that $\delta\mathcal{M}-\lambda\delta N$ vanishes for all $\delta\bar{Q}(r)$ if and only if
\begin{align}
 \frac{1}{\lambda}&=\frac{n(r)}{\bar{\Pi}+\bar{Q}}\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{1}{2}}\nonumber\\
 &+G\int^\infty_r 4\pi r' n(r')\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{3}{2}}dr'
\end{align}
for some the Lagrange multiplier $\lambda$. That means that the right-hand side the above result should be independent of $r$ and hence
\begin{align}\label{rach}
 0=&\left( \frac{n'}{\bar{\Pi}+\bar{Q}}-\frac{n(\bar{\Pi}'+\bar{Q}')}{(\bar{\Pi}+\bar{Q})^2} \right)\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{1}{2}}\nonumber\\
 &+\frac{Gn}{\bar{\Pi}+\bar{Q}}\left(4\pi r\bar{Q}-\frac{\mathcal{M}}{r^2}\right)\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{3}{2}}\nonumber\\
 &-4\pi Grn\left[ 1-\frac{2G\mathcal{M}(r)}{r} \right]^{-\frac{3}{2}}.
\end{align}
Since the entropy per nucleon is uniform, thus using
\begin{equation}
 n'(r)=\frac{n(r)\bar{Q}'(r)}{\bar{\Pi}+\bar{Q}}
\end{equation}
we write (\ref{rach}) in the form of the TOV equations in Palatini gravity with an arbitrary Lagrangian function as
\begin{align}\label{tov_kon}
  \left(\frac{\Pi}{\phi({r})^2}\right)'&=-\frac{G\tilde{A}\mathcal{M}}{r^2}\left(\frac{Q+\Pi}{\phi({r})^2}\right)
  \left(1+\frac{4\pi r^3\frac{\Pi}{\phi({r})^2}}{\mathcal{M}}\right)\nonumber\\
	\mathcal{M}(r)&= \int^r_0 4\pi \tilde{r}^2\frac{Q(\tilde{r})}{\phi(\tilde{r})^2} d\tilde{r},
\end{align}
where $\tilde{A}=\phi^{-1}A(r)$.  Thus, $\delta\mathcal{M}$ vanishes for all $\delta Q$ with the constraint $\delta N=0$ if and only if one deals
with (\ref{tov_kon}). We see from (\ref{condit}) that one deals with a stationary point of $\mathcal{M}$ if the
 expression on the right-hand
 side vanishes. It will be so if we are equipped with the generalized TOV equations in the given form since this expression can be transformed into it.

 We have just shown that the mass $\mathcal{M}$ appearing in the TOV equations is stationary, so the necessary condition for the star to pass from stability to
 instability with respect to some radial perturbation is satisfied. In order to have the TOV equations which describe a stable star's configuration, the constrained extremum must be a minimum:
 therefore one deals with $\delta^2\mathcal{M}>0$. We see from the form of (\ref{nowem2}) that the second order in $\delta\bar{Q}$ has to be positive-definite for all
 perturbations, analogously to the $\delta\rho$ in General Relativity. It has to be so since if any of the squared frequencies $\omega_n^2$ from the all perturbation
 modes takes negative values then the frequency is purely imaginary and the time variation of this mode would grow exponentially leading to an unstable system.

\section{Conclusions}
We have examined the stability condition of neutron stars in the case of the Palatini gravity with an arbitrary Lagrangian functional $f(\hat{R})$. We immediately
notice that the obtained result (\ref{tov_kon}) reduces to the General Relativity equations describing equilibrium of the relativistic stars, specifically, when
$f(\hat{R})=\hat{R}$ since $\phi\equiv f'(\hat{R})=1$, where prime denotes here the derivative with respect to the Palatini scalar $\hat{R}$. Then also the term
proportional to $U(\phi)$, which appears in the definitions of generalized energy density and pressure, vanishes.

The performed analysis on the stability criterion for Palatini gravity, which is the simplest representation of EPS formulation, shows that a stability condition of
a neutron star is very similar to the one in General Relativity. Let us notice that in the case of General Relativity
one gives the final statement if the star's system is stable or not after the examination of the stability conditions with respect to an equation of state.
In Palatini gravity one has to do even more: we see that the generalized energy
density ${Q}$ depends on $\rho$ and the form of the $f(\hat{R})$ via the potential $U$ and coupling $\phi$. Thus, besides the equation of state we need to choose a model which
we wish to study in Palatini formulation. We would like to add here that in the case when we are able to solve the master equation to get $\hat{R}=\hat{R}(T)$, the
generalized energy density appears finally to depend on the equation of state only.

With the all discussed points above, and together with the previous result presented in \cite{aw2}, where we considered scalar-tensor gravity with a minimally coupled scalar field and an
arbitrary potential, we conclude that there are other gravitational theories besides General Relativity which allow us to consider possible
stable stars' configurations with respect to radial oscillations. Future and more detailed work along these lines is currently underway.

\section*{Acknowledgements}
AW is grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work.
The work is supported by the NCN grant $DEC-2014/15/B/ST2/00089$.
The author would also like to thank Andrzej Borowiec and Diego Rubiera-Garcia for their comments and helpful discussions.


\section{Introduction}

\section{Associated varieties of vertex algebras}
A \emph{vertex algebra} consists of a vector space $V$ with a distinguished vacuum vector $|0{\rangle}  \in V$ and a vertex operation, which is a linear map $V \otimes V \rightarrow V((z))$, written $u \otimes v \mapsto Y(u,z)v = (\sum_{n \in \mathbb{Z}} u_{(n)} z^{-n-1})v$, such that the following are satisfied:
\begin{itemize}
\item (Unit axioms) $(|0{\rangle})(z) = 1_V$ and $Y(u,z)|0{\rangle}  \in u + zV[[z]]$ for all $u \in V$.

\item (Locality)
$(z-w)^n[Y(u,z),Y(v,w)]=0$ for a sufficiently large $n$
for all $u, v\in V$.
\end{itemize}
The operator $\partial: u \mapsto u_{(-2)}|0{\rangle} $ is called the translation operator and it satisfies $Y(Tu, z) = \partial_z Y(u, z)$. The operators $u_{(n)}$ are called \emph{modes}.

To each vertex algebra $V$
one associates a Poisson algebra
$R_V$,
called the {\em Zhu's $C_2$-algebra},
as follows (\cite{Zhu96}).
 Let $C_2(V)$ be the subspace of $V$ spanned by the elements
 $a_{(-2)}b$, $a,b\in V$,
 and
set
 $R_V=V/C_2(V)$.
Then $R_V$ is a Poisson algebra
by
\begin{align*}
\bar a.\bar b=\overline{a_{(-1)}b},\quad \{ \bar a,\bar b\}=\overline{a_{(0)}b},
\end{align*}
where $\bar a$ denote the image of $a\in V$ in $R_V$.

A vertex algebra is called  {\em strongly finitely generated} if $R_V$ is finitely generated.
In this note we assume that all the vertex algebras are finitely strongly generated.

The
{\em associated variety} $X_V$ of a vertex algebra $V$
is  the affine Poisson variety $X_V$ defined by
\begin{align*}
X_V=\operatorname{Specm}(R_V
\end{align*}
(\cite{Ara12}).

\smallskip

Let $\mf{g}$ be a simple Lie algebra over $\mathbb{C}$,
 $\widehat{\mf{g}}=\mf{g}[t,t^{-1}]\+ \mathbb{C} K$ be the affine Kac-Moody algebra associated with $\mf{g}$
and the normalized invariant inner product $(~|~)$.
Set
$$V^k(\mf{g}):=U(\widehat{\mf{g}})\*_{U(\mf{g}[t]\+ \mathbb{C} K)}\mathbb{C}_k,$$
where $k\in \mathbb{C}$ and $\mathbb{C}_k$ is one-dimensional representation of $\mf{g}[t]\+ \mathbb{C} K$
on which $\mf{g}[t]$ acts trivially and $K$  acts as the multiplication by $k$.
There is a unique vertex algebra structure on
$V^k(\mf{g})$
such that $|0{\rangle} =1\*1$ is the vacuum vector
and $$Y(x,z)=x(z):=\sum_{n\in \mathbb{Z}}(xt^n)z^{-n-1}\quad (x\in \mf{g}),$$
where
we consider $\mf{g}$ as a subspace of $V^k(\mf{g})$ by the embedding
$\mf{g}\hookrightarrow V^k(\mf{g})$, $x\mapsto xt^{-1}|0{\rangle}$.
$V^k(\mf{g})$
 is called the {\em universal affine vertex algebra associated with $\mf{g}$
at level $k$}.

One can regard $V^k(\mf{g})$ as an analogue of the universal enveloping algebra
in the sense that
a $V^k(\mf{g})$-module is the same as
a smooth $\widehat{\mf{g}}$-module of level $k$,
where a $\widehat{\mf{g}}$-module $M$ is called {\em smooth} if
$x(z)m\in M((z))$ for all $m\in M$, $x\in \mf{g}$,
and called  of level $k$ if $K$ acts as the multiplication by $k$.

Any graded quotient $V$ of $V^k(\mf{g})$  as a $\widehat{\mf{g}}$-module has the structure of the quotient vertex algebra.
In particular the unique simple graded quotient $L_k(\mf{g})$ is a vertex algebra
and is called the {\em simple affine vertex algebra associated with $\mf{g}$ at level $k$}.

For any quotient vertex algebra $V$ of $V^k(\mf{g})$,
we have
$R_V=V/\mf{g}[t^{-1}]t^{-2}V$,
and the surjective linear map
\begin{align}
\mathbb{C}[\mf{g}^*]=S(\mf{g})\rightarrow R_V,\quad  x_1\dots x_r\mapsto \overline{(x_1t^{-1}\dots x_rt^{-1}|0{\rangle}}\quad (x_i\in \mf{g})
\label{eq:Poisson-surj}
\end{align}
is a homomorphism of Poisson algebras.
In particular  $X_{L_k(\mf{g})}$ is a subvariety of $\mf{g}^*$,
which is $G$-invariant and conic.
We note that on the contrary to the associated variety of a primitive ideal of $U(\mf{g})$,
$X_{L_k(\mf{g})}$  is not necessarily contained in the nilpotent cone $\mathcal{N}$ of $\mf{g}$.
Indeed,
$L_k(\mf{g})=V^k(\mf{g})$ for a generic $k$ and
$X_{V^k(\mf{g})}=\mf{g}^*$ as \eqref{eq:Poisson-surj} is an isomorphism
for $V=V^k(\mf{g})$ by the PBW theorem.
\smallskip

For a nilpotent element $f$ of $\mf{g}$,
let $\mathscr{W}^k(\mf{g},f)$ be the universal $W$-algebra associated with $(\mf{g},f)$ at level $k$:
\begin{align*}
\mathscr{W}^k(\mf{g},f)=H_{DS,f}^0(V^k(\mf{g})),
\end{align*}
where $H_{DS,f}^\bullet(?)$ is the BRST cohomology functor of the quantized Drinfeld-Sokolov reduction
associated with $(\mf{g},f)$ (\cite{FF90,KacRoaWak03}).
The associated variety
$X_{\mathscr{W}^k(\mf{g},f)}$ is isomorphic to the {\em Slodowy slice} $\mathcal{S}_f=f+\mf{g}^e$,
where $\{e,f,h\}$ is an $\mathfrak{sl}_2$-triple and $\mf{g}^e$ is the centralizer of $e$ in $\mf{g}$ (\cite{De-Kac06}).
For any quotient $V$ of $V^k(\mf{g})$,
$H_{DS,f}^0(V)$ is a quotient vertex algebra of $\mathscr{W}^k(\mf{g},f)$ provided that it is nonzero,
and we have
\begin{align}\label{eq:ass-variety-W}
X_{H_{DS,f}^0(V)}= X_V\cap \mathcal{S}_f,
\end{align}
which is a $\mathbb{C}^*$-invariant subvariety of $\mathcal{S}_f$ (\cite{Ara09b}).

\section{Lisse and quasi-lisse vertex algebras}
A vertex algebra $V$ is called {\em lisse} (or $C_2$-cofinite) if
$\dim X_V=0$,
or equivalently, $R_V$ is finite-dimensional.
For instance,  $L_k(\mf{g})$ is  lisse if and only if $L_k(\mf{g})$ is integrable as a $\widehat{\mf{g}}$-module,
or equivalently,
$k\in \mathbb{Z}_{\geqslant 0}$ (\cite{DonMas06}).
Therefore, the lisse condition generalizes the integrability to an arbitrary vertex algebra.
Indeed, lisse vertex algebras are analogue of finite-dimensional algebras in the following sense.
\begin{Lem}[\cite{Ara12}]
A vertex algebra $V$ is lisse if and only if $\dim \operatorname{Spec}(\gr V)=0$,
where $\gr V$ is the associated graded Poisson vertex algebra with respect to the canonical filtration on $V$ (\cite{Li05}).
\end{Lem}
It is known that lisse vertex algebras have various nice properties
such as
modular invariance of characters of $V$-modules under some mild assumptions (\cite{Zhu96,Miy04}).
However, there are significant vertex algebras that do not satisfy the lisse condition.
For instance, an {\em admissible affine vertex algebra} $L_k(\mf{g})$
(see below)
has a
complete reducibility property (\cite{A12-2})
and the modular invariance property
(\cite{KacWak89}, see also \cite{AEkeren})
in the category $\mathcal{O}$
although it is not lisse
unless it is integrable.
So it is natural to try to relax the lisse condition.

Since
$X_V$ is a Poisson variety
we have a finite partition
$$X_V=\bigsqcup_{k=0}^r X_k,$$
where each $X_k$ is a smooth analytic Poisson variety.
Thus for any point $x\in X_k$ there is a well defined symplectic leaf through it.
A vertex algebra $V$ is called {\em quasi-lisse} (\cite{Arakawam:kq}) if $X_V$ has only finitely many symplectic leaves.
Clearly,
lisse vertex algebras are quasi-lisse.

For example,
consider the simple affine vertex algebra $L_k(\mf{g})$.
Since
symplectic leaves in $X_{L_k(\mf{g})}$ are the coadjoint  $G$-orbits
contained in $X_{L_k(\mf{g})}$,
where $G$ is the adjoint group of $\mf{g}$,
 it follows that
$L_k(\mf{g})$ is quasi-lisse if and only if $X_{L_k(\mf{g})}\subset \mathcal{N}$.
Hence \cite{FeiMal97,Ara09b},
admissible affine vertex algebras  are quasi-lisse.

A theorem of Etingof and Schelder \cite{EtiSch10}
says that if a Poisson variety $\operatorname{Specm}(R)$ has finitely many symplectic leaves then
the zeroth Poisson homology
$R/\{R,R\}$ is finite-dimensional.
It follows \cite{Arakawam:kq} that
a quasi-lisse conformal vertex algebra has only finitely many
simple ordinary  representations.
Here
a $V$-module $M$ is
called ordinary if it is a positive energy representation
on which $L_0$ acts semisimply
and
 each $L_0$-eigenspace is finite-dimensional,
so that
the normalized character
\begin{align*}
\chi_M(\tau)=\operatorname{tr}_M (q^{L_0-\frac{c}{24}})
\end{align*}
is well-defined.

By extending Zhu's argument \cite{Zhu96}
using the theorem of Etingof and Schelder,
we get the following assertion.
\begin{Th}[\cite{Arakawam:kq}]\label{Th:AK}
Let $V$ be a quasi-lisse vertex algebra and $M$ a ordinary $V$-module.
Then $\chi_M$  satisfies a {\em modular linear differential equation}.
\end{Th}

Since the space of solutions of a modular linear differential equation (MLDE) is invariant under the action of $SL_2(\mathbb{Z})$,
this implies that a quasi-lisse vertex algebra possesses a certain  modular invariance property,
although we do not claim that the normalized characters of $V$-modules span the space of the solutions.

 \section{Irreducibility conjecture and Examples of quassi-lisse vertex algebras}
 Let $\hat{\Delta}^{re}$ be the set
 of real roots
 of $\widehat{\mf{g}}$,
 $\hat{\Delta}^{re}_+$ the set of real positive roots.
 For a weight $\lambda$ of $\widehat{\mf{g}}$,
 let $\hat{\Delta}(\lambda)=\{\alpha\in \hat{\Delta}^{re}\mid
 {\langle} \lambda+\rho,\alpha^{\vee}{\rangle}\in \mathbb{Z}\}$,
 the integral roots system of $\lambda$.
 An irreducible highest weight representation
 $L(\lambda)$ of $\widehat{\mf{g}}$ with highest weight $\lambda$ is called admissible if $\lambda$ is regular dominant,
that is,
${\langle} \lambda+\rho,\alpha^{\vee}{\rangle}\not \in \{0,-1,-2,\dots,\}$ for all positive $\alpha\in \Delta_+$,
and $\mathbb{Q}\hat{\Delta}(\lambda)=\mathbb{Q}\hat{\Delta}^{re}$ (\cite{KacWak89}).
The simple affine vertex algebra $L_k(\mf{g})$  is called admissible if
it is admissible as a $\widehat{\mf{g}}$-module.
This condition is equivalent to that
\begin{align*}
k+h^{\vee}=\frac{p}{q},\quad p,q\in \mathbb{N},\  (p,q)=1,\ p\geqslant \begin{cases}h^{\vee}&\text{if }(r^{\vee},q)=1,\\
h&\text{if }(r^{\vee},q)\ne 1,
\end{cases}
\end{align*}
where $h$, $h^{\vee}$,
and $r^{\vee}$ is the Coxeter number, the dual Coxeter number, and the lacing number of $\mf{g}$,
respectively (\cite{KacWak08}).

 As we have already mentioned above
an admissible affine vertex algebra $L_k(\mf{g})$ is quasi-lisse,
that is, $X_{L_k(\mf{g})}\subset \mathcal{N}$.
In fact, the following assertion holds.
\begin{Th}[\cite{Ara09b}]
For an admissible affine vertex algebra
 $L_k(\mf{g})$,
 $X_{L_k(\mf{g})}$ is an irreducible variety contained in $\mathcal{N}$, that is,
 there exits a nilpotent orbit $\mathbb{O}$ such that
 $X_{L_k(\mf{g})}=\overline{\mathbb{O}}$.
\end{Th}
See \cite{Ara09b} for a concrete description of the orbit $\mathbb{O}$ that appears in the above theorem.

 For $\mf{g}=\mathfrak{sl}_2$, it is not difficult to check that  $L_k(\mf{g})$ is quasi-lisse
 if and only if $L_k(\mf{g})$ is admissible
 for a non-critical\footnote{If $k$ is critical, that is, if $k=-h^{\vee}$,
then $X_{L_k(\mf{g})}= \mathcal{N}$ by \cite{FeiFre92,EisFre01,FreGai04} for all simple Lie algebra $\mf{g}$.
}
 $k$, see \cite{Mal90,GorKac07}.
 However, there are non-admissible affine vertex algebras that are quasi-lisse for higher rank $\mf{g}$.

 Recall that the {\em Deligne exceptional series} \cite{De96} is the sequence of simple Lie algebras
 $$A_1\subset A_2\subset G_2\subset D_4\subset F_4\subset E_6\subset E_7\subset E_8.$$
 Let $\mathbb{O}_{min}$ be the unique non-trivial nilpotent orbit of $\mf{g}$.
\begin{Th}[\cite{AM15}]\label{Th:AMJoseph}
Let $\mf{g}$ be a simple Lie algebra that belongs to the Deligne exceptional series,
and let $k$ be a rational number of the form $k=-h^{\vee}/6-1+n$, $n\in \mathbb{Z}_{\geqslant 0}$, such that $k\not \in \mathbb{Z}_{\geqslant 0}$.
Then $$X_{L_k(\mf{g})}=\overline{\mathbb{O}_{min}}.$$
\end{Th}
For types $A_1$, $A_2$, $G_2$, $D_4$, $F_4$,
the simple affine vertex algebra
$L_k(\mf{g})$ appearing Theorem \ref{Th:AMJoseph}
is admissible, and hence, the statement is the special case of \cite{Ara09b}.
However,
this is not the case for for types $D_4$, $F_4$, $E_6$, $E_7$, $E_8$
and
Theorem \ref{Th:AMJoseph}
gives examples of non-admissible quasi-lisse affine vertex algebras

Except for $\mf{g}=\mathfrak{sl}_2$, the classification problem of quasi-lisse affine vertex algebras is  wide open.
(See \cite{AraMor16,AraMor16b} for more for more examples lisse affine vertex algebras.)

All the associated varieties are irreducible in the above examples
of quasi-lisse affine vertex algebras.
We conjecture that this is true in general:
\begin{Conj}[\cite{AraMor16}]\label{Conj1}
The associated variety of an quasi-lisse conical vertex algebra is irreducible.
\end{Conj}

\smallskip

Recall the description of associated variety of $W$-algebras given by \eqref{eq:ass-variety-W}.
This implies that
 if $L_k(\mf{g})$ is quasi-lisse
and $f\in X_{L_k(\mf{g})}$,
then
the $W$-algebra $H_{DS,f}^0(L_k(\mf{g}))$ is quasi-lisse as well,
and so is its simple quotient $\mathscr{W}_k(\mf{g},f)$.
In this way we obtain
a huge number of quasi-lisse $W$-algebras.
(See \cite{{AraMor16b}} for the irreducibility of the corresponding associated varieites.)
Moreover, if  $X_{L_k(\mf{g})}=\overline{G.f}$,
then $X_{H_{DS,f}^0(L_k(\mf{g}))}=\{f\}$ by the transversality of $\mathcal{S}_f$ to $G$-orbits,
so that  $\mathscr{W}_k(\mf{g},f)$ is in fact lisse.
Thus, Conjecture \ref{Conj1} in particular says that
a quasi-lisse affine vertex algebra produces exactly one lisse simple $W$-algebra.

Lisse $W$-algebras thus obtained from admissible affine vertex algebras contain all the {\em exceptional $W$-algebras}
discovered by  Kac and Wakimoto \cite{KacWak08} (\cite{Ara09b}),
in particular, the {\em minimal series principal $W$-algebras} \cite{FKW92},
which are natural generalization of minimal series Virasoro vertex algebras \cite{BPZ84}.
The rationality of
the minimal series principal $W$-algebras  has  been recently
recently proved  by the author (\cite{A2012Dec}).

\section{BL${}^2$PR${}^2$ correspondence and Higgs branch conjecture}
In \cite{BeeLemLie15},
Beem,  Lemos, Liendo, Peelaers, Rastelli,
and van Rees have constructed a remarkable
map
$$\Phi: \{\text{4d $N=2$ SCFTs}\}\rightarrow \{\text{vertex algebras}\},$$
such that,
among other things,
the character of the vertex algebra
$\Phi(\mathcal{T})$ coincides  with the {\em Schur index}
of the corresponding   4d $N=2$ SCFT $\mathcal{T}$,
which is an important invariant.

How do vertex algebras coming from 4d $N=2$ SCFTs look like?
According to \cite{BeeLemLie15}, we have
$$c_{2d}=-12c_{4d},$$
where $c_{4d}$ and $c_{2d}$ are central charges of the 4d $N=2$ SCFT
and the corresponding vertex algebra, respectively.
Since the central charge is positive for a unitary theory,
this implies that the vertex algebras obtained in this way are never unitizable.
In particular integrable affine vertex algebras never appear by this correspondence.

The main examples of vertex algebras
considered in  \cite{BeeLemLie15}
are affine vertex algebras $L_k(\mf{g})$ of types
$D_4$, $F_4$, $E_6$, $E_7$, $E_8$ at level $k=-h^{\vee}/6-1$,
which are non-rational,
non-admissible quasi-lisse affine vertex algebras appeared in Theorem \ref{Th:AMJoseph}.
One can find more examples in the literature, see e.g.\ \cite{Beem:2015yu,BN1,
CorSha16,
BN2,
Dan,SXY,BLN}.

Now, there is another important invariant of a 4d $N=2$ SCFT $\mathcal{T}$,
called the {\em Higgs branch},
which we denote by $Higgs_{\mathcal{T}}$.
The Higgs branch $Higgs_{\mathcal{T}}$
is an affine  algebraic variety
that has the hyperK\"{a}hler structure in its smooth part.
In particular,
$Higgs_{\mathcal{T}}$  is a (possibly singular)
 symplectic  variety.

Let $\mathcal{T}$ be one of the  4d $N=2$ SCFTs studied in  \cite{BeeLemLie15}
such that  that $\Phi(\mathcal{T})=L_k(\mf{g})$ with
$k=h^{\vee}/6-1$
for types
$D_4$, $F_4$, $E_6$, $E_7$, $E_8$ as above. It is known that $Higgs_{\mathcal{T}}=\overline{\mathbb{O}_{min}}$ ,
which equals to $X_{L_k(\mf{g})}$ by Theorem \ref{Th:AMJoseph}.
It is expected that this is not just a coincidence.
\begin{Conj}[Beem and Rastelli \cite{BeeRas}]\label{Conj:Beem and Rastell}
For a 4d $N=2$ SCFT $\mathcal{T}$, we have
\begin{align*}
Higgs_{\mathcal{T}}=X_{\Phi(\mathcal{T})}.
\end{align*}
\end{Conj}
So we are expected to  recover the Higgs branch of a 4d $N=2$ SCFT
from the corresponding vertex algebra, which is
a purely algebraic object!
Note that the associated variety of a vertex algebra is only a Poisson variety in general.
Physical intuition expects that they are all quasi-lisse,
and so
vertex algebras that come from 4d $N=2$ SCFTs via the map $\Phi$ form some special subclass of quasi-lisse vertex algebras.

We note that Conjecture \ref{Conj:Beem and Rastell} is a physical conjecture
since the Higgs branch is not a mathematically defined object at the moment.
 The Schur index is not a mathematically defined object  either.
However, in view of \cite{BeeLemLie15} and Conjecture \ref{Conj:Beem and Rastell},
one can try to define both Higgs branches and Schur indeces
of 4d $N=2$ SCFTs
using vertex algebras.
We note that there is a close relationship between Higgs branches of 4d $N=2$ SCFTs
and {\em Coulomb branches} of three-dimensional  $N=4$ gauge theories whose mathematical definition has been recently  given by
Braverman, Finkelberg and Nakajima \cite{BFN16,BraFinNak17} (see \cite{A17a,A17b}).

In view of
Conjecture \ref{Conj:Beem and Rastell},
Theorem \ref{Th:AK} implies that
the Schur index of a 4d $N=2$ SCFT satisfies a  MLDE,
which is something that has been conjectured in physics (\cite{BeeRas}).

\subsection*{Acknowledgments}
This note is based on the talks given by the author at
 AMS Special Session ^^ ^^ Vertex Algebras and Geometry,"
at the University of Denver, October 2016,
 and at
 ^^ ^^ Exact operator algebras in superconformal field theories",
 at Perimeter Institute for Theoretical Physics, Canada, December 2016.
 He thanks the organizers of these conferences
 and the Simons Collaboration on the Non-perturbative Bootstrap.
 He  benefited greatly from discussion with  Christopher Beem,
 Madalena Lemos,
 Anne Moreau,
 Hiraku Nakajima,
 Takahiro Nishinaka,
 Wolfger Peelaers,
 Leonardo Rastelli,
 Shu-Heng Shao,
 Yuji Tachikawa, and
 Dan Xie.
This research was supported in part
JSPS KAKENHI Grant Numbers 17H01086, 17K18724,
and
by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development \& Innovation.

\newcommand{\etalchar}[1]{$^{#1}$}


\section{Introduction}
\label{sec:intro}
In recent years, research in natural language processing has led to significant advancements on tasks such as sentiment analysis, question-answering, text- and speech-based machine translation, automatic speech recognition, or speech synthesis.  Intelligent language systems such as voice assistants, chat bots, or AR/VR devices start to permeate many aspects of our daily lives.
These novel forms of human-computer interaction require suitable content and crucially depend on the availability of powerful models along with adequate data to train them. This creates inequalities across different languages and particularly poses challenges for low-resource languages and dialects, where data of sufficient quantity and quality is often scarce.

Among the low-resources languages, Swiss German dialects are characterized by being derived from Standard German (High German, DE) with considerable differences in phonetics, vocabulary, morphology and syntax, even lacking clear boundaries and definitions.
Unlike other languages, where dialect usage decreases in favor of standard variants in most social domains, Swiss German dialects are widely used in everyday life (cf.~\cite{liidi2007swiss}). With the expansion of personalized digital media and social platforms, dialect use is growing even more. Despite this increasing demand, the obstacles posed by, for instance, lack of standard orthography, have made data collection of written text (e.g.~transcripts) challenging. These difficulties in collecting consistent and clean data have led to a situation, where Switzerland is lagging behind and struggles to take full advantage of basic NLP tools.

\begin{figure}[t]
\centering
\includegraphics[trim=65 0 350 0, clip, width=0.5\textwidth]{teaser.pdf}
\caption{\textit{A sample sentence from SwissDial. For each sample sentence in High German, we provide the manual translations and the speech samples in 8 Swiss dialects. (The sentence means \textit{"Then he laughs at her and jumps away."} in English)}}
\label{fig:sample}
\end{figure}

This paper presents an annotated parallel corpus of spoken Swiss German dialects, \emph{SwissDial}, for 8 different regions: Aargau (AG), Bern (BE), Basel (BS), Graub\"{u}nden (GR), Luzern (LU), St. Gallen (SG), Wallis (VS) and Z\"{u}rich (ZH). Our corpus consists of web-crawled sentences in High German, manual translations into Swiss German dialects and their audio recordings. Different genres and sources are represented: news stories, Wikipedia articles, weather reports, short stories. The audio recordings are performed by a single speaker for each dialect. Figure \ref{fig:sample} presents a sample from SwissDial. Our primary motivation and focus is to provide a basic dataset that would allow applicability of recent data-driven NLP models, especially for speech synthesis, and step up the ongoing research in Swiss German. To our knowledge, this is the  first publicly available parallel corpus of spoken Swiss German.
Our main contributions are as follows:
\begin{itemize}
    \item We present \textit{SwissDial}, a parallel corpus of text and audio in 8 Swiss German dialects on designated topics. The data collection process is well-documented, allowing for transparency and augmentability. The corpus is available at \url{https://projects.mtc.ethz.ch/swiss-voice-data-collection}.
    \item We validate the suitability of our corpus by running experiments with the recent neural models for (i) single speaker, (ii) multi-speaker-multi-dialect data, and (iii) code-switching (CS) speech synthesis. The samples from these models are available at \url{https://projects.mtc.ethz.ch/projects/swiss-voice/swissdial}.

\end{itemize}

\section{Related Work}
For various high-resource languages, there are parallel text corpora \cite{jawaid2014tagged, koehn2005europarl, buck2014n} and monolingual text corpora such as \cite{kupietz2010german}, Google News, Common Crawl that are used for the tasks of language modeling and machine translation. \cite{panayotov2015librispeech, ljspeech17, park2019css10} present paired audio and text data which are widely used for automatic speech recognition and speech synthesis with various transformations. For these languages, \cite{di2019must} presents a parallel corpus for multilingual speech translation.

However, availability of data is limited for languages containing strong dialectal variations such as Arabic, Chinese, Swiss German which come forward with their special condition where the formal variety of the language differs significantly from varieties that are acquired natively and mostly used for verbal communication.
There are various datasets concerning Swiss German that allow various NLP tasks such as dialect identification \cite{ali2018character, jauhiainen2018heli}, dialect machine translation \cite{honnet2018machine, scherrer2016automatic}, morphology generation \cite{scherrer2011morphology}, automatic speech recognition~\cite{garner2014automatic}.
Archimob \cite{samardzic2016archimob, scherrer2019digitising} presents a general purpose corpus of spoken Swiss German based on oral history interviews with various people from different dialects, which provides pairs of audio and text with annotated normalization and part-of-speech tagging.
Another resource~\cite{swiss_sms}, Swiss SMS corpus, provides text corpus of SMS messages.
NOAH's corpus  \cite{hollenstein2014compilation} presents a collection of text in various genres in Swiss dialects with manually annotated part-of-speech tags.
\cite{honnet2018machine} collects a various parallel written sources (High German and Swiss German) to perform machine translation by normalizing Swiss German.
Recently, \cite{schmidt2020swiss} introduced a dictionary containing forms of common words in various Swiss German dialects normalized into High German.

Most of these existing resources in Swiss German do not provide sufficient data to train end-to-end models for neural machine translation and speech synthesis. They rather provide word-level correspondences which are suboptimal for sentence level NLP due to the lack of full context and syntax. Moreover, existing datasets of spoken language do not fulfill the requirements of today's neural speech synthesizers, which ideally need clean and single speaker audio. SwissDial is thus, to our knowledge, the first available parallel multi-dialectal corpus with transcribed text and clean audio in Swiss German. The text part of our corpus is carefully designed to cover a large set of topics and lexicons providing manual and parallel translations of sentences in dialects. The audio part, paired with the corresponding text, provides clean and single speaker audio for each dialect at sentence level. Moreover, the corpus contains labeled code-mixing samples which would be valuable for code-mixing speech synthesis research. We believe that the parallel structure of our corpus can also be used to improve data efficiency in NLP tasks via transfer learning.

\begin{table*}[t]
\centering
\begin{tabular}{lrrrrrrrrrrr}
\hline \textbf{Topic} &  \textbf{Source} & \textbf{Parallel} & ~\textbf{AG} & \textbf{BE} & \textbf{BS} & \textbf{GR} & \textbf{LU} & \textbf{SG} & \textbf{VS} & \textbf{ZH} & \textbf{Total} \\ \hline
Animals/farming & News & 130 & 137 & 135 & 137 & 137 & 135 & 137 & 137 & 134 & 1'089 \\
Culture & News  & 81 & 85 & 87 & 87 & 86 & 86 & 87 & 87 & 86 & 691 \\
Earth/Space & Wikipedia  & 17 & 18 & 18 & 18 & 18 & 18 & 18 & 18 & 25 & 151 \\
Economics & News  & 60 & 63 & 63 & 62 & 63 & 63 & 63 & 63 & 234 & 674 \\
Int. politics & News  & 140 & 155 & 154 & 153 & 155 & 152 & 155 & 155 & 151 & 1'230 \\
Medicine & News  & 104 & 111 & 110 & 109 & 112 & 110 & 112 & 112 & 140 & 916 \\
Meteorology & News  & 192 & 215 & 201 & 214 & 215 & 202 & 215 & 215 & 211 & 1'688 \\
Random & Wikipedia  & 132 & 149 & 142 & 142 & 148 & 148 & 149 & 147 & 160 & 1'185 \\
Special & Wikipedia  & 1030 & 1'120 & 1'105 & 1'106 & 1'120 & 1'112 & 1'122 & 1'123 & 1'730 & 9'538 \\
Science & News & 189 & 197 & 192 & 195 & 197 & 194 & 197 & 197 & 243 & 1'612 \\
Sports & News  & 88 & 103 & 101 & 101 & 103 & 97 & 102 & 103 & 163 & 873 \\
Story & Internet  & 49 & 50 & 50 & 50 & 49 & 50 & 50 & 50 & 81 & 430 \\
Swiss politics & News  & 228 & 248 & 245 & 245 & 249 & 241 & 249 & 249 & 240 & 1'966 \\
Swiss regional & News  & 88 & 97 & 97 & 94 & 97 & 97 & 96 & 97 & 467 & 1'142 \\
\hline
with Code-Switching & News  & 232 & 247 & 247 & 247 & 249 & 248 & 250 & 248 & 345 & 2'081 \\
\hline
All topics & - & 2528  & 2'748 & 2'700 & 2'713 & 2'749 & 2'715 & 2'752 & 2'753 & 4'065 & 23'195 \\
\hline
\hline
\end{tabular}
\caption{\label{table:topic_distribution} The distribution of sentences among dialects. \emph{Special} indicates the sentences that are manually prepared to catch different vocabulary among the dialects.}
\end{table*}

\section{SwissDial Corpus Content and Creation}

In the following, we present the construction and content of our corpus and its modalities. Acquisition of SwissDial for applications in end-to-end neural models has five main steps: obtaining raw data, selection of dialects and annotators, translation, recording, and post-processing.

\subsection{Raw data}
\label{section:raw_data}
Swiss dialects are spoken colloquially, but only rarely used in written form. Moreover, the available written forms collected from unspecific sources generally contain inconsistencies within the same dialect, since there is no official standard orthography. This poses a challenge for annotators, possibly leading to disagreements with regard to the orthography of the provided text. To increase consistency and reduce ambiguity of the process, we started by collecting High German sentences to be translated by the annotators themselves.

We collected random sentences from news articles to cover a wide range of topics for good generalization. These sentences were labeled with the topic metadata of the article that they belong to. As mentioned earlier, Swiss dialects differ in vocabulary from High German and from each other. We thus crawled the internet for lists of lexical items that show variability among the dialects. We then extracted sentences from Wikipedia articles that contain each word (labeled as \textit{special}). In this way, we aimed to cover vocabulary differences of dialects, while having a wide range of topics for a good generalization in various NLP tasks. Table~\ref{table:topic_distribution} presents the topic distribution of the collected sentences in the corpus.
It is important to note that significant majority of the code-switching (CS) content was the result of the random selection procedure, rather manual addition. There are two main reasons for this relatively large amount of CS content: (i) there is a high linguistic contact between English and German which results in considerable amount of word \textit{borrowing}, (ii) our main sentence source is news articles that contain global topics like politics, science, technology etc, and therefore naturally contain words from English.

\subsection{Selection of dialects}
Swiss dialects are not clearly separated by geological borders, the variation in the regional language is rather continuous. One can separate each dialect into numerous sub-dialects even down to resolution of villages.
Therefore, we had to make discretization where we aimed to represent the differentiation of the dialects as much as possible while addressing the most populated areas. Our selection motivation was to cover a range of dialects while complying with time constraints and availability of annotators.
As a result, our corpus contains data from the following regions: Aargau, Bern, Basel, Graub\"{u}nden, Luzern, St. Gallen, Wallis and Z\"{u}rich.
To represent these areas, we carried out an auditioning where we evaluated the annotator candidates based on their performance on a sample translation and pronunciation task, as well as their vocal quality features inspired by \cite{kempster2009consensus}. We hired one non-professional annotator for each dialect that passed all the tests to perform the translation and audio recording steps.

\subsection{Text translation}
We asked annotators from each dialect to translate the provided High German sentences, described in Section \ref{section:raw_data}, to their native dialect. As mentioned earlier, Swiss dialects have different morphology together with different vocabulary and word choice. In some cases, a High German sentence or expression may not be easily translatable into dialect without sounding artificial. Therefore  annotators had the freedom to skip such sentences. All the  numbers, abbreviations and time expressions were translated as the annotator would read them out loud in the next step. In other words, the translation step aimed to have a one-to-one mapping between the text and the audio to-be-recorded. The foreign words, e.g. code-switching words, in the High German sentences were directly transferred as in the original language, rather than performing translation or phone mapping.

\subsection{Recording}
As the last step of the data collection, the annotators were asked to read out loud their translations in a quiet room, in front of a high-quality recording set-up to obtain clean recordings. We used a Neumann TLM 102 microphone with pop filter and recorded in mono at 48 kHz. The speakers were instructed to speak at a distance of about 15 cm from the microphone. We prepared a user interface that shows the annotator the translated sentences each at a time and allows recording control through keyboard. In this way, the annotators were able to prepare and control their voice and breathing for each sentence, and rest when they needed. The interface also allowed annotators to modify their translations when they notice typos or incorrect wording/expressions before recording each sentence. This can be considered as the last quality control step for the correctness of the translations. The recordings were carried out in multiple sessions, around the same time of the day, that took 1-2 hours each. This scheduling is important to have a consistent voice across all the audio samples without exhausting the annotators and their vocal chords.
Table~\ref{table:audio_statistics} presents the statistics of the collected text data.
\\
\begin{table}
\centering
\begin{tabular}{lccc}
\hline \textbf{Dialect} & \textbf{Total} & \textbf{Average} & \textbf{Average \#word}\\
  & \textbf{Audio (h)} & \textbf{utterance (s)} & \textbf{per utterance}\\ \hline
AG & 2.78 & 3.64 & 11.8\\
BE & 3.41 & 4.55 & 11.9 \\
BS & 3.15 & 4.18 & 12.7 \\
LU & 2.54 & 3.36 & 12.3 \\
GR & 2.93 & 3.83 & 12.5 \\
SG & 3.71 & 4.85 & 12.1 \\
VS & 3.32 & 4.34 & 11.9 \\
ZH & 4.55 & 4.03 & 12.2\\
\hline
\hline
\end{tabular}
\caption{\label{table:audio_statistics} The statistics of the collected audio data.}
\end{table}

\subsection{Post-processing}
After translations and recording, we performed small post-processing on the collected data to make it more convenient for various NLP tasks. For the text data in High German, we post-processed each sentence in the corpus by spelling numeric values into words. For the dialect text, we post-processed each sentence transforming the spelling of the numbers into numeric characters. In this way, we provide two versions for each sentence in all dialects: text with numeric characters and all characters. Providing both versions is helpful to tokenize numbers in various NLP tasks.

For the audio recordings, we performed a cleaning procedure where the aim was to cut off the segments at the beginning and end of the utterances that contain various noises like key pressing, breathing, mouth noises, etc. Some of the utterances contained noise during the sentence reading, rather than beginning/end, which would not be straightforward to clean. We repeated the recording step for these samples whenever it was possible, and eliminated the rest to present clean data which is significant for the speech synthesis task. Then, we down-sampled all recordings to 22050 Hz to be used in our speech synthesis experiments.

\section{Speech Synthesis Experiments with SwissDial}
With the increase in monolingual corpora, the end-to-end architectures~\cite{shen2018natural, oord2016wavenet} have shown high quality speech synthesis results. Using these methods, we performed various experiments to synthesize speech in Swiss dialects using audio-text pairs from Swiss Dial.
All the experiments below are carried out with character level input representations.

\begin{figure*}[t]
\centering
\includegraphics[trim=40 310 50 50, clip, width=0.9\textwidth]{user_study.pdf}
\caption{\textit{MOS results on the single speaker model and code-switching model with standard deviation.}}
\label{fig:user_study}
\end{figure*}

\subsection{Single speaker model}
We implemented the well-established neural speech synthesis model \cite{shen2018natural}, an encoder-decoder architecture with attention, and conducted experiments on each dialect separately.
To increase the data efficiency for our low-resource set-up, we firstly trained on the German part of CSS10~\cite{park2019css10} (16 hours) for a good initialization. The main motivation for pretraining on High German is to transfer the textual and acoustic representations and learned alignment from High German data to Swiss dialects in the tuning stage, since Swiss dialects are derived from High German. Then, we fine-tuned the pretrained model on each dialect separately to obtain a single speaker speech synthesis model. For each dialect, we held out 10\% of the samples as a validation set.

We conducted a 5-scale mean opinion score (MOS) test, with 0.5 increments, for speech naturalness and quality. Fifteen native Swiss speakers from different regions were invited to participate. Randomly chosen ten utterances from a test set are presented to the participants for each dialect. Figure \ref{fig:user_study} shows the mean and deviation of MOS ratings of the participants. One has to keep in mind, that we hired non-professional speakers which naturally have, in comparison to professional speakers, much stronger deviations in quality and consistency of their pronunciation, intonation and voice control. Therefore, we suspect the differences in MOS rating between the dialects to be a result of the varying vocal quality of our non-professional speakers.

\subsection{Multi-speaker-multi-dialect model}
Next, we aimed to encourage sharing of model capacity across different dialects by training a single model on all dialects simultaneously. We implemented the multi-speaker model in~\cite{jia2018transfer}, which is an extension of the previous model, and extended it with learnt dialect embeddings to support multi-dialect training. The speaker and the dialect embeddings are learnt in a similar way following~\cite{wan2018generalized}.
We trained the speaker and dialect verification models by collecting samples from additional speakers from radio shows to enhance the capability of representations, since our dataset would only provide 8 different speakers. We held out 5\% of the samples as a validation set. Different from~\cite{jia2018transfer}, the learnt speaker and dialects embeddings are concatenated at the encoder output and the decoder input inspired by \cite{xue2019building}. An internal subjective comparison between the multi-speaker-multi-dialect model results and the single speaker model results revealed a very similar naturalness and quality for the two models.

\subsection{Code-Switching.}
Lastly, we implemented the model~\textit{SE-DEC}~\cite{xue2019building}, which is again an extension on \cite{shen2018natural}, to explore the performance of our corpus in code-switching set-up with English words mixed in single Swiss dialect, BE, for showcasing. Firstly, we pretrained the model with monolingual recordings in English~\cite{ljspeech17} and High German~\cite{park2019css10}. Then we fine-tuned the model using monolingual English data and the BE part of our dataset which introduces 247 code-switching samples. The MOS result for this experiment is shown in Figure \ref{fig:user_study} with the CS-BE bar. The participants were asked to rate the samples according to (i) Swiss German pronunciation, (ii) English pronunciation, (iii) naturalness and quality of the audio. As one can see in Figure \ref{fig:user_study}, the resulting MOS ratings for BE with and without code-switching are almost identical, which is a very encouraging result for the use of code-switching extensions for Swiss German speech synthesis.

\section{Conclusion}
In this paper, we present SwissDial, a parallel multidialectal corpus of text and audio in 8 Swiss German dialects, by describing the collection procedure in detail. Furthermore, we validate the quality of SwissDial by presenting the results of experiments with the recent neural models for speech synthesis in single-speaker-single-dialect, multi-speaker-multi-dialect, and code-switching set-ups. Last but not least, we believe that applications on SwissDial are not limited only to speech synthesis but it can also empower further NLP research and application in Swiss German in the fields of machine translation and dialect identification.

\bibliographystyle{IEEEtran}