Explicit Entropic Proofs of Irreversibility Theorems for Holographic RG Flows
Evan Deddo James T. Liu Leopoldo A. Pando Zayas Robert J. Saskowski



An equivariant p-adic Artin conjecture
    Ben Forrás
    Version of April 23, 2024
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§ INTRODUCTION  AND SUMMARY

A central organizing principle in quantum field theory is that of renormalization group (RG) flows. In general, we expect the process of coarse-graining along flows to result in a reduction in degrees of freedom, generally measured by a suitable choice of c-function. Establishing the irreversibility of RG flows involves distinct field-theoretic techniques in different spacetime dimensions. For example, the c-theorem, establishing the irreversibility of flows in two dimensions, exploits properties of the correlators of two stress-energy tensors <cit.>, while the four-dimensional a-theorem is proved non-perturbatively with the help of a certain four-point amplitude <cit.>, and progress on the six-dimensional a-theorem has focused on a certain six-point amplitude <cit.>. Entropic approaches first emerged in (re)proving the c-theorem in <cit.> and resulted in the first proof of the three-dimensional F-theorem <cit.>; such approaches have proven to be extremely successful in providing a general framework, and a unified proof for spacetime dimensions two, three, and four was presented in <cit.>. The latter approach is rooted in the strong subadditivity (SSA) property of the relative entanglement entropy  (EE). The authors of <cit.> used SSA to show that for a field theory in d dimensions, a suitably subtracted EE satisfies the inequality


    R ∂^2_R Δ S_EE(R)-(d-3) ∂_R Δ  S_EE(R)   ≤ 0,


where R is the radius of a spherical entangling region. Here,

    Δ S_EE(R)≡ S_EE(R)-S_UV,

where S_UV is the EE of the UV fixed point, the subtraction of which regulates the divergences. The above result, therefore,  establishes a monotonically non-increasing function, 

    c_CTT(R)=R ∂_RΔ S(R) - (d-2) Δ S(R),

along RG flows. In  d≤ 3, this function c_CTT(R) interpolates between the UV and IR central charges or free energies.[As an abuse of language, we will generally refer to both of these as central charges, even in odd dimensions where the Weyl anomaly vanishes.] This is no longer the case in higher dimensions, but in d=4, the inequality is still strong enough to prove the a-theorem. The connection between the amplitude and EE approaches to establishing the irreversibility of RG flows has remained largely open. 


Recently, there has been some work on connecting the correlators that arise in proving c-theorems with the Averaged Null Energy Condition (ANEC), which can be derived both holographically <cit.> and field-theoretically  <cit.>. In CFT, the ANEC provides several constraints on the coupling constants and anomaly coefficients <cit.> and results in the 2d c-theorem and the 4d a-theorem being related to particular three-point correlators involving the averaged null energy <cit.>. Since the averaged null energy condition is related to the modular Hamiltonian <cit.>, this proof may be rephrased in terms of relative entropy, which was also used in <cit.> to prove the c-theorem. A second, related condition is the Quantum Null Energy Condition (QNEC) <cit.>, which has been proved field-theoretically in <cit.> and holographically in <cit.>, and is implied by the ANEC <cit.>. In particular, <cit.> showed that the inequality that arises in proving the 2d c-theorem is precisely the QNEC, and <cit.> used the QNEC to reprove (<ref>) and extend it to defect RG flows.

Much has been accomplished specifically using holographic methods. In particular, one approach is to use the NEC to directly construct a local holographic c-function that monotonically interpolates between the UV and IR central charges (see e.g. <cit.>). This method is powerful and leads to proofs in arbitrary dimensions, but the resulting c-function lacks a clear field-theoretic interpretation away from the fixed points. This is because taking an arbitrary slice of AdS away from the boundary does not necessarily have the interpretation of being dual to a QFT. In particular, the bulk (A)NEC is not holographically dual to the boundary ANEC <cit.>.

A complementary approach is to use holographic EE, and some progress has been made in this direction <cit.>.
As expected of an EE, the holographic EE has been shown to satisfy SSA <cit.>, and so one may use the same proof on the holographic EE as for the field-theoretic EE, although this is quite circuitous. We will mainly be interested in genuinely holographic proofs. In particular, <cit.> considered a strip-shaped entangling surface and used the integral of motion and the NEC to show monotonicity. We expect a spherical entangling region to pick out the A-type central charge, whereas the strip selects some linear combination of central charges. In d=2, the strip is a sphere, and so <cit.> provides a holographic c-function that yields the 2d c-theorem, but for d>2, the strip does not yield the A-type central charge. The two-dimensional c-theorem was further investigated in <cit.>. More generally, the difficulty with spherical entangling regions is that there is no integral of motion for dimensions d>2, which makes them computationally difficult compared with a strip. 

More recently, <cit.> used the Hamilton-Jacobi equations associated to the Ryu-Takayanagi surface <cit.> to derive a sequence of equations for the μ_d-2k coefficients in the IR expansion of the entanglement entropy of the sphere

    S_EE(R)=μ_d-2(ϵ)R^d-2+μ_d-4(ϵ)R^d-4+⋯+
            (-1)^d2+14AlogR/ϵ   d even
    
            (-1)^d-12F   d odd,

where ϵ is a UV cutoff. When compared to the non-flowing theory, this allowed the authors to compute the differences Δμ_d-2k, which resulted in a holographic proof of the c-, F-, and a-theorems as a comparison between the UV and IR central charges, but they did not construct a c-function along the flow. Hence, the first goal of the present work will be to construct such a holographic c-function as in (<ref>) that continuously measures degrees of freedom along the RG flow and to demonstrate its monotonicity.



Additionally, there has been some work on flows across dimensions recently, which may be viewed as flows that start at a D-dimensional UV fixed point, undergo compactification onto a (D-d)-dimensional manifold along the flow, and end at a d-dimensional IR fixed point. This is quite difficult to study directly in field theory but is holographically dual to a gravitational solution interpolating between AdS_D+1 and AdS_d+1. There are many examples of supergravity solutions holographically dual to RG flows interpolating between CFTs of different dimensions <cit.>, and some candidate c-functions for such flows were studied in <cit.>. More recently, explicit c-functions were constructed in <cit.> and further explored in <cit.>. In particular, <cit.> proved the monotonicity of flows to AdS_3, but it is difficult to push the analysis further due to the lack of a first integral for the surface minimization equation. A second goal of this manuscript is to generalize our results for flows within the same dimension to those across dimensions.

In this work, we provide a unified proof, valid in various dimensions, using only the Null Energy Condition (NEC) and the assumption that for field theories with a gravity dual the Ryu-Takayanagi (RT) formula <cit.> determines the entanglement entropy of a region. We start by considering flows within the same dimension, for which we will prove the inequality (<ref>) holographically.  These flows correspond to a domain wall solution

    s^2=1/h(z)^2(- t^2+ z^2+ r^2+r^2Ω_d-2^2),

where Ω_d-2^2 is the line element on the unit sphere S^d-2 and h(z) asymptotes to z/L_UV in the UV and z/L_IR in the IR, where L_UV and L_IR correspond to the UV and IR AdS radii, respectively.  Using the RT prescription, we compute the entanglement entropy as the area of the minimal surface wrapping S^d-2 with radius R on the conformal boundary and use this along with the NEC to obtain the inequality (<ref>).  Due to the spherical symmetry of the geometry, we may assume that the RT surface has a profile r(z) with r(0)=R independent of the angular coordinates.

Motivated by the presence of R derivatives in (<ref>), we introduce a continuous family of such surfaces parameterized by R. Differentiating the equations of motion for r(z) yields a linear equation for r/ R. The key step is then to introduce a more suitable measurement of the separation between infinitesimally close surfaces,

    v(z)≡r(z)/RrR,

which we will show satisfies v'(z)≥ 0. The proof then reduces to the observation that

    R(Δ S'_EE(R)/R^d-3)∼ -v'(z),

so long as the conformal dimension of the perturbing operator satisfies

    Δ≥d-4/2.

Given that Δ>0, this is only an additional condition when d≥ 5. In particular, (<ref>) recovers the results of <cit.> and provides a holographic, monotonic c-function, proving the c-, F-, and a-theorems.

We then consider the case of flows across dimensions, which corresponds to a metric of the form 

    s^2=1/h(z)^2(- t^2+ z^2+ r^2+r^2Ω_d-2^2)+1/j(z)^2 s^2_M_D-d,

subject to the asymptotics

    h(z)z→ 0∼z/L_UV,        h(z)z→∞∼z/L_IR,
    
        j(z)z→ 0∼z/L_UV,        j(z)z→∞∼j_IR.

We again make use of the RT formula to construct a central charge for entangling regions that wrap the internal manifold. We begin by reproving the case of flows from AdS_D+1 to AdS_3, which was originally proved in <cit.>, in our new formalism. We then attempt to extend the proof to general flows to AdS_d+1. We find that additional divergences arise that hamper a proof in general dimensions, but we do prove the monotonicity of flows from AdS_5 to AdS_4.


The rest of this paper is organized as follows. In Section <ref>, we consider flows within the same dimension and prove the c-, F-, and a-theorems using holographic EE and the NEC. In Section <ref>, we extend our results to the case of flows across dimensions. Finally, we conclude in Section <ref> and discuss some future directions.




§ FLOWS WITHIN THE SAME DIMENSION


We start with holographic flows in the same dimension.  In order to prove the inequality, (<ref>), for the subtracted entanglement entropy Δ S_EE(R), we first need to understand the properties of holographic domain wall flow.  This will be constrained by the bulk NEC.  Having understood the background, we then probe its geometry by constructing the RT surface with profile r(z) and computing its corresponding entropy function S(R).  Because of its property as a minimal surface, we can obtain its derivative, S'(R), as a local boundary expression.

In principle, knowledge of S'(R) is all we need to construct the c_CTT(R) function of (<ref>).  However, proof of its monoticity requires a second derivative with respect to R, as in (<ref>), and this requires additional work.  Essentially, what we are looking for is the change in the surface (and its corresponding area) as we vary the boundary radius R.  We find that this is well captured by the separation function v(z) as defined in (<ref>).  By studying the breaking of conformal symmetry as one moves away from the UV boundary, we are able to show that the separation of two nearby minimal surfaces increases as one moves into the interior of AdS_d+1 in the sense that v'(z)≥0.  Finally, we complete the proof of monotonicity of the c-function, ∂_Rc_CTT(R)≤0, by relating this derivative to the behavior of v'(z) near the UV boundary.

In summary, the holographic proof proceeds as follows:

    
  * We first obtain a condition on the domain wall metric warp factor, h”(z)≥ 0 by imposing the NEC in the bulk.  This is simply an application of the usual local holographic c-theorem <cit.>.
    
  * We introduce a profile r(z) for the RT surface, and find an expression for S'(R) via the Hamilton-Jacobi equation. This gives an expression for S' in terms of local boundary values of fields.
    
  * We construct a monotonically non-increasing function 𝒞(z) built from the metric function h(z) and the surface profile r(z) that measures the breaking of conformal symmetry. Moreover, we show that 𝒞≥ 0 everywhere along the surface.
    
  * We define the separation function v(z) according to (<ref>) and prove that the non-negativity of 𝒞(z) and the NEC implies that v'(z)≥ 0.
    
  * Finally, we show that ∂_R c_CTT(R) is proportional to the leading term in -v'(z) near the boundary.  Since v'(z)≥0 everywhere along the flow, this immediately leads to the entropic c-theorem, ∂_R c_CTT(R)≤0.





 §.§ AdS/CFT and domain wall flows

The AdS/CFT correspondence <cit.> posits an equivalence between certain field theories and theories of gravity, leading to a geometrization of various field theory concepts. For example, conformal symmetry in the field theory is mapped to the isometries of the gravity background. We consider a metric of the form

    s^2=1h(z)^2(η_μν x^μ x^ν+ z^2)=1/h(z)^2(- t^2+ z^2+ r^2+r^2Ω_d-2^2),

where η_μν is the Minkowski metric and Ω_d-2^2 is the line element on S^d-2. This spacetime may be viewed as a domain wall solution interpolating between a UV and an IR AdS_d+1, so long as we have appropriate asymptotics of the form

    h(z)z→ 0∼z/L_UV,        h(z)z→∞∼z/L_IR.

Under the AdS/CFT dictionary, we may view this as an RG flow between two d-dimensional CFTs <cit.>. In the rest of this manuscript, we will set L_UV=1 for convenience.

We now constrain the domain wall flow by imposing the null energy condition in the bulk.  Given a future-directed null vector ξ, the NEC requires that

    T_MNξ^Mξ^N≥0,

where T_MN is the stress-energy tensor. Making use of the Einstein equations, the stress-energy tensor may be exchanged for the Ricci tensor

    R̂_MNξ^Mξ^N≥0.

The metric (<ref>) has curvature tensor components

    R̂^μ_νλσ=-h'^2/h^2(δ^μ_λη_νσ-δ^μ_ση_νλ),    R̂^μ_zν z=1/h^2(hh”-h'^2)δ^μ_ν,

and

    R̂_μν=1/h^2(hh”-dh'^2)η_μν,    R̂_zz=d/h^2(hh”-h'^2).

Picking ξ=∂_z±∂_t, the NEC then implies

    h”(z)≥ 0.

Note that this will be the only non-trivial NEC due to the planar symmetry of the domain
wall. The main consequence is that the slope of h must be larger in the IR than in the UV, and so the effective radius 

    L_eff(z)=1/h'(z),

decreases along the flow. Since the AdS radius is a proxy for the number of degrees of freedom, one may naturally write down an (unnormalized) c-function

    c_NEC(z)=L_eff(z)^d-1,

and obtain a holographic c-theorem <cit.>. However, as already emphasized, this c-function only has a rigorous field-theoretic interpretation at the fixed points of the flow.




 §.§ Entanglement entropy and the RT surface

The Ryu-Takayanagi formula <cit.> allows one to compute the entanglement entropy of a region, M, in field theory as the area of the bulk surface homologous to M. For the case of a  minimal surface γ anchored on the maximally spherical region of radius R on the boundary, using the metric (<ref>), we find that

    A(γ) = Vol(S^d-2)∫_ϵ^z_0 z r(z)^d-2/h(z)^d-1√(1+r'(z)^2),

where r(z) is the profile of the minimal surface.  Here ϵ is a UV cutoff and z_0 is the turning point of the minimal surface. The holographic setup is depicted in Figure <ref>.





Since the inequalities derived from SSA in field theory contain derivatives of S, it will be convenient to alternatively parametrize the extremal surface by a profile z(r).  In this case, the entropy is given by

    S(R) = ∫_0^R_c r (z,ż;r),

where

    ℒ̃(z,ż;r)=Vol(S^d-2) r^d-2/h(z(r))^d-1√(1+ż(r)^2).

Here, ż= z/ r  and the surface is anchored such that z(r=R_c)=ϵ. Although S(R) is divergent on its own <cit.>, after suitable subtraction of a reference entropy, S(R) may be identified with the (relative) entropy of the dual field theory. In particular, it is important to emphasize that we have a well-defined holographic interpretation for all values of R, i.e., at all points along the RG flow.

Due to the minimality of the extremal surface, the bulk contributions to S' disappear, leading to a Hamilton-Jacobi equation

    S'(R)    = R_cR|_R_c + ∫_0^R_c r(zzR + żżR) 
       = R_cR|_R_c + żzR|_0^R_c,

where the equation of motion was used to reach the second line. At the cutoff, we have z(R_c) = ϵ. Differentiating this condition gives

    żR_cR|_R_c + zR|_R_c = 0,

and so

    S'(R)    = R_cR(- żż)|_R_c - żzR|_0
       =Vol(S^d-2)(R_cRr^d-2/h^d-1√(1+ż^2)|_R_c - r^d-2ż/h^d-1√(1+ż^2)zR|_0) 
       = -Vol(S^d-2)r'r^d-2/h^d-1√(1+r'^2)rR|_z=ϵ,

where in the second line, the second term evaluates to zero.  In the final line, we have re-expressed S'(R) in terms of the profile function r(z).  Here we are considering a family of surfaces parametrized by R, and both r'≡∂ r/∂ z and ∂ r/∂ R show up in this expression.  It is important to note that S'(R) is a local function that depends only on the fields and their derivatives evaluated at the cutoff z=ϵ. Note also that (<ref>) matches the Hamilton-Jacobi equation obtained in <cit.>, up to a change of coordinates.[However, it is important to note that Ref. <cit.> uses a different anchoring prescription, which is equivalent to z(R)=ϵ, rather than z(R_c)=ϵ.]

Similar results were obtained using the Jacobi equation in <cit.>.





 §.§ Equations of motion for the RT surface


Although S'(R) can be evaluated locally at the boundary, we still need to understand the behavior of the surface at the boundary.  To do so, we turn to the equation of motion for the minimal surface


    r”(z) = (d-2/r(z) + (d-1)h'(z)r'(z)/h(z))(1+r'(z)^2).


For d=2 (i.e., AdS_3), this admits a first integral, r'(z)=h(z)/√(h(z_0)^2-h(z)^2).  However, no general closed-form solution is known for d≥3, except for the semicircle solution, r(z)=√(z_0^2-z^2), in the conformal case where h(z)=z.  Nevertheless, one may construct series expansions near the boundary z=0 and turning point z=z_0. The surface profile has boundary conditions r(0)=R and r(z_0)=0.

Without bulk matter, the canonical way to examine the behavior of the metric in asymptotically AdS spacetimes is via the Fefferman-Graham expansion <cit.>; however, in the presence of a relevant deformation of conformal dimension d-Δ, we will generically have backreaction that leads to a metric boundary expansion of the form[Note that the boundary expansion may be non-analytic.]

    h(z)=z+c z^1+2Δ+o(z^1+2Δ).

This is summarized explicitly for a scalar deformation in Appendix <ref>. Note that we must have Δ>0 to get asymptotic AdS. This is equivalent to the requirement that the deformation is relevant. Note also that the NEC requires c≥ 0.

In the absence of such a perturbation, i.e., in the case of pure AdS, the minimal surface is just the semicircle

    r_0(z)=√(R^2-z^2).

More generally, the geometry near the boundary takes the form

    r(z)=r_0(z)+r_d(R)z^d+ζ_1(R)z^2+2Δ+o(z^2+2Δ)+𝒪(z^d+2Δ),

where

    ζ_1(R)=(d-1)Δ/(d-2-2Δ)(Δ+1)c/R.

Here, r_d(R) is an integration constant that corresponds to global information about the turning point, the z^2+2Δ term is the leading correction due to the backreaction of the metric, and the 𝒪(z^d+2Δ) terms include the cross terms between the r_d(R) term and the metric corrections. Note that in the Fefferman-Graham expansion, r_d precisely corresponds to (r^(d)(0)-r^(d)_0(0))/d!. When the entangling region is small, r_d may be found explicitly (see Appendix <ref>).

Despite the possible non-analyticity near the boundary, the metric expansion around a bulk point is simply a Taylor series, and hence the expansion of z(r) near the r=0 turning point is more straightforward:

    z(r)=z_0 -h'(z_0)/2 h(z_0)r^2 + ((d-1)h'(z_0) h”(z_0)/8(d+1)h(z_0)^2-h'(z_0)^3/8h(z_0)^3)r^4 + 𝒪(r^6).

This will be useful to us later.




 §.§ An approximate constant of motion


In pure AdS, corresponding to unbroken conformal invariance, the RT surface is just a semicircle.  Moreover, there is a constant of motion for the surface associated with scale invariance: z→λ z,   r→λ r. By Noether's theorem, this constant is

    𝒞 = ( - ∂/∂ r'r')z - ∂/∂ r'r = r(z+rr')/z^2√(1+(r')^2) = 0.

For a flow in which conformal symmetry is broken, the above quantity will instead vary along the RT surface as a function of z. However, a generalized form of 𝒞(z), though non-constant, will act as a measure of conformal symmetry breaking and produce useful constraints on the behavior of the surface. 

At any point z=z^*, we may consider the linear approximation for h(z) given by

    h̅(z)=h(z^*)+h'(z^*)(z-z^*),

Since any metric with linear h(z) describes AdS, h̅(z) is effectively an AdS approximation to the flowing theory at z^*. We may then consider the conserved quantity derived from the scale symmetry of this associated AdS metric. In the UV and IR limits, the linear approximation will become exact, and the resulting quantity will be conserved in the flowing theory as well. In the intermediate regime, it will be non-constant.




Since the linear approximation does not in general satisfy h̅(0)=0, we should consider dilatations centered around

    z_c = z^*-h(z^*)/h'(z^*),

where z_c satisfies h̅(z_c)=0. This is depicted schematically in Figure <ref>.  Infinitesimal dilatations with scale factor (1+δλ) then take the form

    r → r + δλ r, 
       z → z + δλ (z-z_c) = z + δλ(z-z^* + h(z^*)/h'(z^*)).

Following Noether's theorem, we define

    𝒞(z) = ( - ∂/∂ r'r')h(z)/h'(z) - ∂/∂ r'r = r^d-2h+h'rr'/h'h^d-1√(1+r'^2),

where for each z we take the generator to be that of the dilatation defined by the linear approximation at that point (by setting z=z^* in (<ref>)). Since h”(z) describes the deviation of the flowing theory from pure AdS, it should also control the variation of 𝒞(z). Indeed, imposing the equations of motion (<ref>) yields

    𝒞z = -r^d-2h”/(h')^2h^d-2√(1+r'^2)≤ 0,

where the inequality results from the NEC (<ref>). Therefore, 𝒞(z) is non-increasing everywhere and zero if and only if the NEC is saturated.
Near the turning point, the expansion (<ref>) gives

    𝒞(z(r)) = h”(z_0)/(d+1)h(z_0)^d r^d+1 + 𝒪(r^d+3),

so 𝒞(z_0) = 0. Thus 𝒞 is always non-negative, and in particular

    h+h'rr' ≥ 0.

This inequality will be used in the proof below.




 §.§ Entropic inequalities


Field theoretically, the authors of <cit.> used strong subadditivity to show that a suitably subtracted EE, Δ S_EE(R), satisfies the inequality (<ref>), namely


    RΔ S”(R)-(d-3)Δ S'(R)≤0,


which may be rewritten as


    R^d-2R(Δ S'/R^d-3)≤0.


To reproduce this result holographically, we must determine the boundary behavior of the minimal surfaces and utilize the expression (<ref>) for S'(R). We now consider the z-evolution of the infinitesimal separation between two minimal surfaces. The differential equation describing this separation is sufficient to derive the desired monotonicity properties.

From (<ref>) we may write

    R(Δ S'/R^d-3)    = -Vol(S^d-2)/h^d-1R(rr'r^d-3/R^d-3√(1+r'^2)rR)|_z=ϵ
           + Vol(S^d-2)/z^d-1R(r_0 r_0'r_0^d-3/R^d-3√(1+r_0'^2)r_0R)|_z=ϵ.

The first term in this difference originates from the entropy of the flowing theory, while the second term is the subtraction of the UV CFT contribution. Here, a factor of rr' has been separated out inside the first set of parentheses, and the leftover part in the parentheses is just 1 to leading order. A similar case holds for the CFT portion. Hence, one may expect that rr' will give the major contribution to the overall expression after subtraction. In fact, we will demonstrate that

    R(Δ S'/R^d-3)∼ -1/h^d-1R(rr')|_z=ϵ,

to leading order in ϵ.

At this point, it is interesting to note that we may rewrite the above derivative as

    R(rr')=r'rR+rr'R=z(rrR).

This then motivates us to consider the quantity

    v(z):=r/RrR.

Intuitively, r/R measures the r-separation between neighboring surfaces, and the r/R factor keeps v(z) finite at the turning point. In pure AdS, v(z) is constant since, in this case, it amounts to a measurement of the perpendicular separation between surfaces with respect to a flat metric in the r-z plane. Thus, one may interpret v(z) as linearizing the deviation of surfaces. Now, we would like to constrain the sign of v'(z), which leads us to the following Lemma.

Lemma: The NEC implies that the separation v(z) never decreases.

    h”(z)≥0       v'(z)≥ 0.

Proof:
We proceed by demonstrating two results:

    
  * v(z) has no local minima.
    
  * Smoothness at the turning point requires v'(z) > 0 for physically reasonable flows, so v'(z) ≥ 0 for all z∈[0,z_0].

If we take the R derivative of the minimal surface equation, (<ref>), and rewrite in terms of v(z), we get a linear second-order ODE

    v” - (d-1)(2 r'/r + h'/ h(1 + 3 r'^2))v' +2(d-1)r'^2/h r^2(h + h'rr')v = 0.
 

Notice that the coefficient of v in (<ref>) is non-negative by (<ref>). By the monotonicity property of 𝒞(z), if h+h'rr' = 0 at any value of z<z_0 it must be zero everywhere and correspond to the case of pure AdS. For nontrivial h(z), the coefficient of the third term of (<ref>) is therefore strictly positive for z<z_0. Separate minimal surfaces in a family cannot intersect, so v(z) should be positive. Therefore, any critical point z_c ∈ (0,z_0) satisfying v'(z_c)=0 has v”(z_c)<0. Local minima are thus forbidden.

Now, consider the behavior at the turning point. One may naïvely expect that smoothness requires v'(z_0) = 0, but the singular behavior of the z parametrization at the turning point leads to a different result. We then substitute the r=0 expansion (<ref>) into (<ref>), and use r'(z(r)) = 1/ż(r). Setting each term of the r expansion of (<ref>) to zero constrains the derivatives of v at z=z_0. The expansion of v(z) is then

    v(z)/v(z_0) = 1 + 2 h”(z_0)/(d+1) h'(z_0)(z-z_0) + 𝒪((z-z_0)^2).


Recall that the NEC implies h”(z_0)≥ 0. For domain wall flows between two fixed points, h”(z) is expected to approach zero only as a limit in the IR. In this case, we have the strict inequality v'(z) > 0 in the neighborhood of z=z_0. Since local minima cannot exist, v'(z) ≥ 0 for all z<z_0. □

Given the above Lemma, the quick proof is that

    R(Δ S'/R^d-3)∼-Vol(S^d-2)/h^d-1v'|_z=ϵ≤ 0

to leading order. Of course, we must ensure that there are no other divergences that ruin this statement, which we check now.

We may evaluate (<ref>) given our metric (<ref>) and the boundary expansion (<ref>) to write

    R(Δ S'/R^d-3)   =-Vol(S^d-2)[(dR(Rr_d(R))+𝒪(ϵ^2Δ))
       +(2d^2(1+Δ)-4d(2+Δ)+4(2+Δ+2Δ^2)/(d-2-2Δ)(2+2Δ)(d-1)c/R^3ϵ^4-d+2Δ+o(ϵ^4-d+2Δ))].

Now, several important points are in order. First, there is a subtle interplay between factors of R and z.  At this level, r_d(R) is a generic function of R and so we can be assured that all corrections are at least 𝒪(ϵ^2Δ); however, the z^2+2Δ term in r(z) has a coefficient which is a non-generic function of R, so extra caution is required. Indeed, if instead of the R dependence following from Equations  (<ref>) and (<ref>), ζ_1(R)∝ R^-1, we had any other R dependence, then we would have to contend with additional, divergent 
 ϵ^2-d+2Δ terms proportional to R(Rζ_1(R)). Thus, to ensure that our power counting is indeed correct, one must additionally check that the subleading terms also have a R^-1 dependence (at least up to order n=⌊d-22Δ⌋-1, past which there is no divergence). Fortunately, this is indeed the case: If one writes the more general expression

    h(z)=z+∑_n=1^∞ c_n z^1+2nΔ+∑_n=2^∞ℓ_n z^1+2nΔlog z,


    r(z)=r_0(z)+(r_d(R)/Rz^d+𝒪(z^d+2Δ))+(∑_n=1^∞ζ_n(R) z^2+2nΔ+∑_n=2^∞ξ_n(R) z^2+2nΔlog z)],

and expands the equations of motion in z, then the leading contribution from the metric corrections to r(z) will be proportional to Rζ_n(R) or Rξ_n(R) (up to the point where the z^2 corrections from the √(R^2-z^2) become important), whereas the leading contribution from the metric itself will always be 𝒪(R^0) and depends on {c_i}_i≤ n and {ℓ_i}_i≤ n. Hence, solving the equations of motion order-by-order in z will give c_n∝ R^-1 and ℓ_n∝ R^-1 up to order nΔ=1 (at which point there become 𝒪(1/R^4) corrections to worry about). Thus, our order counting above is sufficient (up to ϵ^4-d+2Δ), and this is then a second source of ϵ^4-d+2Δ terms.

Now, this naturally leads to an issue for sufficiently large d: If d-4>2Δ, then our expression is divergent. This suggests that there is a lower bound on Δ for the expression to make sense

    Δ≥d-4/2.


To prove monotonicity, we want to connect the quantity (<ref>) to v(z). It is straightforward to expand around z=0 as

    v(z)=1+(1/RR(R r_d)z^d+o(z^d))+(2(d-1)Δ/(d-2-2Δ)(2+2Δ)2c/R^4z^4+2Δ+o(z^4+2Δ)).

Note that the bound 2Δ≥ d-4 automatically implies that z^4+2Δ is, at worst, the same order as z^d (note also that the assumption that Δ>0 means that 4+2Δ > d for d≤ 4, so the possibility of saturating this bound is only an issue for d>4). Thus, from the Lemma, we conclude that (<ref>) is non-positive, and this suffices to prove the monotonicity of (<ref>). In particular, this provides a holographic proof of the c-, F-, and a-theorems, analogous to the field-theoretic version <cit.>. For d≥ 5, the function (<ref>) no longer corresponds to a central charge at the fixed points, but it does still give us some sort of monotonic quantity.





§ FLOWS ACROSS DIMENSIONS

Given the proof of monotonicity in the preceding section for flows within the same dimension, it is natural to attempt to generalize this to the case of flows across dimensions. Hence, we choose a metric

    s^2=1/h(z)^2(- t^2+ z^2+ r^2+r^2Ω_d-2^2)+1/j(z)^2 s^2_M_D-d,

with asymptotics

    h(z)z→ 0∼z,        h(z)z→∞∼z/L_IR,
    
        j(z)z→ 0∼z,        j(z)z→∞∼j_IR,

where j_IR is a constant. This metric then interpolates between a UV AdS_D+1 and an IR AdS_d+1× M_D-d, which is holographically dual to a flow from a CFT_D to a CFT_d, compactified on M_D-d <cit.>. Note that the special case D=d corresponds to the metric (<ref>). The metric (<ref>) has Ricci tensor components

    R̂^μ_ν   =(hh”-d(h')^2-(D-d)hh'j'/j)δ^μ_ν,
    R̂^i_j   =j^2R^i_j+h/j^2(-(D-d-1)h(j')^2+j(hj”-(d-1)h'j'))δ^i_j,
    R̂^z_z   =d(hh”-(h')^2)+(D-d)h/j^2(-2h(j')^2+j(h'j'+hj”)).

We assume that the internal space M_D-d is Einstein with metric g_ij and curvature

    R_ij=(D-d-1)κ/ℓ^2g_ij,

where κ = 1, 0, or -1 for positive, flat, or negative curvature, respectively, and ℓ characterizes the scale of the compact dimensions. Since the D-dimensional isometry is broken by the flow, we end up with two independent NEC inequalities

    NEC1:       h̃”(z) ≥(D-1)(D-d)/(d-1)^2h̃(z)(j'(z)/j(z))^2 ≥ 0,
    NEC2:       d(h'/h)^2+(D-2d+1)h'j'-jh”/hj+j”j-(D-d+1)(j')^2/j^2+(D-d-1)κ/ℓ^2j^2/h^2≥0,

where we have defined the effective warp factor

    h̃(z):=h(z) j(z)^D-d/d-1.

We will consider entanglement regions that wrap M_D-d and are spherical in the remaining d-1 dimensions, with radius R. In particular, the entropy functional

    S(R) = Vol(S^d-2)Vol(M_D-d)∫_ϵ^z_0 z r(z)^d-2/h̃(z)^d-1√(1+r'(z)^2),

and NEC1 appear almost identical to the case of flows within the same dimension. Hence, most of the calculations proceed exactly as before with the replacement h→h̃. In particular, we still have a monotonically decreasing function

    𝒞(z) = r^d-2h̃+h̃'rr'/h̃'h̃^d-1√(1+r'^2),

and the Lemma straightforwardly generalizes to flows across dimensions, i.e., v'(z)≥0. Likewise, we find

    R(Δ S'/R^d-3)    = -Vol(S^d-2)Vol(M_D-d)/h̃^d-1R(rr'r^d-3/R^d-3√(1+r'^2)rR)|_z=ϵ
            + Vol(S^d-2)Vol(M_D-d)/h̃_0(z)^d-1R(r_0 r_0'r_0^d-3/R^d-3√(1+r_0'^2)r_0R)|_z=ϵ,

where h̃_0=z^(D-1)/(d-1) corresponds to the non-flowing geometry with h_0=j_0=z.

The main place where the distinction from flows within the same dimension becomes clear is when considering expansions around the boundary since h̃ has asymptotics

    h̃(z)z→ 0∼z^D-1/d-1.

Not only that, but the internal space M_D-d itself may contribute to the curvature when κ is non-vanishing. This leads to corrections to the metric, even in the absence of any relevant perturbations. Solving the vacuum AdS Einstein equations

    R̂_MN-1/2ĝ_MN+D(D-1)/2=0,

we may find the effective warp factor as a series expansion

    h̃_0(z)=z^D-1/d-1(1+(D-d)(D-d-1)/3(D-2)(d-1)κ/ℓ^2z^2+𝒪(κ/ℓ^2 z^4)),

although the full solution is not known in general. As we will see, this leads to some important differences. 



 §.§ Flows to AdS_3

As a warm-up, we use our new formalism to demonstrate the monotonicity of flows from AdS_D+1 to AdS_3, which was proved originally in <cit.>. We start by finding the unperturbed minimal surface profile r_0(z). Naïvely, we might like to simply solve the equations of motion perturbatively, order by order. Unfortunately, this gives the unsatisfactory solution r_0=R, which is not a closed surface. Fortunately, we may use the integral of motion to find r_0(z).

The entropy functional (<ref>) has no explicit r(z) factors for d=2, which implies the existence of a constant of motion given by

    r'/h̃√(1+(r')^2)=const.

After rearranging and imposing the condition that lim_z→ z_0r'(z)=-∞, we find that

    r'(z)=-h̃(z)/h̃(z_0)/√(1-(h̃(z)/h̃(z_0))^2),

which, for the torus case, i.e., κ=0, becomes simply

    r_0(z)=√(π)Γ(D/2D-2)/Γ(1/2D-2)z_0-z_0/D(z/z_0)^D_2 F_1(1/2,D/2D-2;1+D/2D-2;(z/z_0)^2(D-1)).

Here the turning point z_0 is implicitly a function of R given by the requirement that r_0(0)=R, which may be solved to give

    z_0=Γ(1/2D-2)/Γ(D/2D-2)R/√(π),

and so we get the behavior

    r_0(z)= R-z_0/D(z/z_0)^D+𝒪(z^2D).

The profile r_0 is shown in Figure <ref>.

Note that for D=2, this reduces precisely to the semicircle r_0=√(R^2-z^2) with turning point z_0=R. 

When κ 0, we must use the more general metric (<ref>). This leads to a more general expression for r_0(z):

    r_0(z)   =R-h̃(z_0)/D(z/h̃(z_0))^D_2 F_1(1/2,D/2D-2;3D-2/2D-2;(z/h̃(z_0))^2(D-1))
         -(D-1)(D-d)(D-d-1)/3(D+2)(D-2)(d-1)κ/ℓ^2z^D+2/h̃(z_0)^D-1_2 F_1(1/2,D+2/2D-2;3D/2D-1,-(z/h̃(z_0))^2(D-1))
         +𝒪(z^D+4)
       = R-z_0/D(z/z_0)^D+𝒪(z^D+2).

The main distinction is that in the case κ 0, there is no longer a simple expression for the turning point of r_0. Instead, we will leave it implicit as a function z_0(R).

Hence, we write

    r(z)=r_0(z)+r_D(R) z^D+𝒪(z^D+2Δ).

Importantly, both the metric backreaction and internal curvature pieces are subleading compared to r_D. Hence, we may compute the expansion of v to give

    v(z)=1+(-1/DRR(R/z_0^D-1)+1/RR(R r_D))z^D+𝒪(z^D+2Δ).

For the torus case, this first term is simply

    -1/DRR(R/z_0^D-1)κ→ 0=D-2/Dπ^(D-1)/2/R^D(Γ(D/2D-2)/Γ(1/2D-2))^D-1.


We may then compute the unsubtracted central charge[It should be noted that the original proof of <cit.> did so with the unsubtracted central charge as well. There are no UV divergences so the subtraction (<ref>) is unnecessary for the case d=2.] derivative to find that

    R(RS') = -DVol(S^d-2)Vol(M_D-d)(-1/DRR(R/z_0^D-1)+1/RR(R r_D))+𝒪(ϵ^2Δ).

Hence, given that v'(z)≥0 by the Lemma, we have the monotonicity of flows to d=2, AdS_3.





 §.§ Flows to AdS_d+1

Now we consider the general case of flows with D>d>2. In this case, the equations of motion corresponding to minimizing the entropy functional (<ref>) are

    r”(z) = (d-2/r(z) + (d-1)h̃'(z)r'(z)/h̃(z))(1+r'(z)^2).

Unfortunately, no closed-form solution is known for r_0, even when κ=0. Instead, we must expand perturbatively around z=0, which gives an expansion

    r_0(z)   =R-1/2d-2/D-2z^2/R
       -d-2/6(D-2)^3(D-4)(3(d-2)(D^2-4D-2d+8)/R^2-4(D-d)(D-d-1)(D-2)κ/ℓ^2)z^4/R
       +𝒪(z^6).

Notice that for D=d, this reduces to the series expansion of the semicircle √(R^2-z^2). Since the solution is only known perturbatively, the coefficient of z^D is ill-defined[This can be seen from the fact that the coefficient of z^D would naïvely have a 1/(D-D) prefactor.] and should be absorbed into r_D. As before, we write the general surface profile as

    r(z)=r_0(z)+r_D(R) z^D+𝒪(z^D+2Δ).


Now, we may compute the expansion of v to be

    v(z)=1+(1/RR(R r_D)z^D+o(z^D))+((d-2)^2(D-d)/2(D-2)^3(D-4)z^4/R^4+o(z^4)).

The z^4 term is potentially problematic and notably does not exist for the case D=d, which is due to the fact that

    √(R^2-z^2)/RR√(R^2-z^2)=1.

So there are three cases to consider:

    
  * If D<4, then the r_D term is still the leading order contribution, but since we assumed D>d>2, this is an empty collection of cases.
    
  * If D=4, then we must have d=3. Moreover, the z^4 term arises from the z^4 term in r_0, which, as mentioned before, is ill-defined in D=4. Hence, the leading order contribution is the r_4 term as before, and the Lemma forces it to be positive

    1/RR(R r_4)≥ 0.

Likewise, the derivative of the central charge is given by

    Δ S'R=-4Vol(S^1)Vol(M_1)1/RR(R r_4)+o(ϵ^0).

This proves the monotonicity of flows from AdS_5 to AdS_4.
    
  * If D>4, then the leading term is the z^4 term, and the Lemma merely implies the tautology

    (d-2)^2(D-d)/2(D-2)^3(D-4)1/R^4≥ 0.

However, the central charge derivative is given by

    R(Δ S'/R^d-3)=-DVol(S^d-2)Vol(M_D-d)1/RR(Rr_D)+o(ϵ^0).

Unfortunately, the sign is no longer constrained by the Lemma, and so it becomes difficult to prove with our present approach. 








§ DISCUSSION

Our proof of irreversibility in RG flows contains explicitly several ingredients that could ultimately clarify the two main approaches to establishing such theorems: field-theoretic and entropic. Namely, in the expansion of metric terms for a given background solution, subleading terms in the Fefferman-Graham expansion encode information about the stress-energy tensor and its correlation functions <cit.>. We see how those terms affect the properties of the entanglement entropy, which we computed using the Ryu-Takayanagi prescription. We plan to pursue this lead to bridge the field theory proofs based on properties of correlators and amplitudes and the entropic proof based on strong subadditivity of the relative entropy <cit.>.

One remaining question is how to generalize these results beyond d=4. Some progress has been made in this direction field theoretically in five dimensions <cit.> and six dimensions <cit.>. Of particular relevance to the present context, <cit.> proposed an interpolating c-function

    c_LM(R)=1/(d-2)!!(R∂_R-1)(R∂_R-3)⋯(R∂_R-(d-2))S_EE(R)   d odd
    1/(d-2)!!R∂_R(R∂_R-2)⋯(R∂_R-(d-2))S_EE(R)   d even,

which agrees with (<ref>) for d=2,3 and generalizes to higher dimensions. Unfortunately, one issue with using c_LM is that it is unclear how to generalize v(z), as such a function would have to contain higher-order R derivatives of r(z). A bigger issue, however, is that there is also no guarantee that c_LM is monotonic in general. Non-monotonic examples have been found in d=4, for instance. We hope to return to some of these issues elsewhere.



§ ACKNOWLEDGEMENTS

We thank Jonathan Sorce and Antony Speranza for discussions. This work is partially supported by the U.S. Department of Energy under grant DE-SC0007859. ED and RJS were supported in part by Leinweber Graduate Summer Fellowships. RJS was supported in part by a Rackham One-Term Dissertation Fellowship.



§ DOMAIN WALL FLOWS

Here we discuss the corrections to the metric due to a scalar field. Consider a scalar field coupled to gravity


    e^-1ℒ=R-12(∂ϕ)^2-V(ϕ).


Note that the scalar potential V(ϕ) implicitly includes the cosmological constant and also that the generalization to multiple scalars is straightforward. For a domain wall flow corresponding to the metric (<ref>), the coupled Einstein and Klein-Gordon equations are


    ϕ”-(d-1)h'hϕ'-∂_ϕ Vh^2=0,
       h”h=12(d-1)(ϕ')^2,
       (h'h)^2=12d(d-1)(ϕ')^2-1d(d-1)Vh^2.


One can check that the first and last equations imply the middle one.  For numerical solutions, one would typically solve the first two second-order equations and check that the last “energy” equation is satisfied.

Note that we can obtain a BPS flow by introducing a superpotential


    V=12(∂_ϕ W^2-d2(d-1)W^2).


Then the first-order BPS flow is given by


    h'=12(d-1)W,    ϕ'=1h∂_ϕ W.



To understand the asymptotics, take a potential


    V=-d(d-1)L^2+12m^2ϕ^2,


corresponding to a cosmological constant and minimally coupled massive scalar. We now assume the asymptotics


    ϕ∼α z^Δ_+,     h∼zL(1+h̅(z)).


The Klein-Gordon equation gives


    Δ_+(Δ_+-d)=(mL)^2    ⇒    Δ_+=d2+√((d2)^2+(mL)^2).


The first backreaction correction to the metric is then


    h̅(z)=Δ_+4(d-1)(2Δ_++1)α^2z^2Δ_+.


Note that this expansion assumes Δ_+>0, corresponding to a relevant deformation. This justifies our choice (<ref>).

For a minimally coupled scalar, the series solution can be developed to higher order.  The result is an expansion of ϕ(z) in odd powers of z^Δ_+ and an expansion of h̅(z) in even powers of z^Δ_+. In the main body of the paper, we have dropped the + subscript on Δ.



§ R_D FOR SMALL SURFACES

Since we do not have good control over the whole entangling surface, it is generally quite difficult to write down an expression for r_d. However, when R is small, i.e., when we are close to the UV, one can find an explicit expression, similar to what was done in the small R expansion in <cit.>. Let us take R to be small and

    x=z/R∈[0,z_0/R],

fixed. We will, as before, take the warp factor to have an asymptotic expansion

    h(z)=z+c z^1+2Δ+o(z^1+2Δ).

Then we may expand the equations of motion

    r”(x) = R^2(d-2/r(x) + (d-1)h'(x)r'(x)/R^2h(x))(1+r'(x)^2/R^2),

perturbatively in R and solve order-by-order to get an expansion for the profile

    r(x)   =R√(1-x^2)+R^1+2Δ/√(1-x^2){c_1+c_2x^d _2 F_1(d-1/2,d/2;d+2/2;x^2)
         +2Δ (d-1)cx^2+2Δ/2d(d-2-2Δ[(d-2-2Δ)Γ(Δ+1)/Γ(Δ+2)_3F_2(3/2,1,1+Δ;d+2/2,2+Δ;x^2)
         +2 _2F_1(3-d/2,2+2Δ-d/2;4+2Δ-d/2;x^2)_2F_1(d-1/2,d/2;d+2/2;x^2)]}
         +o(R^1+2Δ).

Fixing the turning point[Note that this allows r(0) to change, but we may always rescale the surface in the end to set r(0)=R. This will not affect r_d to leading order in R.] to be z_0=R and requiring that the surface is well-behaved at x=1 fixes the coefficients c_1 and c_2, which in turn allows us to extract r_d for various dimensions, which we have presented in Table <ref>. Note that these will receive corrections in R as the surface grows.  One can also check that these all satisfy (R r_d)/ R≥ 0 to leading order in R, so long as 2Δ≥ d-4.




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