Topological terms, AdS_2n gravity and renormalized Entanglement Entropy of holographic CFTs

We extend our topological renormalization scheme for Entanglement Entropy to holographic CFTs of arbitrary odd dimensions in the context of the AdS/CFT correspondence. The procedure consists in adding the Chern form as a boundary term to the area functional of the Ryu-Takayanagi minimal surface. The renormalized Entanglement Entropy thus obtained can be rewritten in terms of the Euler characteristic and the AdS curvature of the minimal surface. This prescription considers the use of the Replica Trick to express the renormalized Entanglement Entropy in terms of the renormalized gravitational action evaluated on the conically-singular replica manifold extended to the bulk. This renormalized action is obtained in turn by adding the Chern form as the counterterm at the boundary of the 2n-dimensional asymptotically AdS bulk manifold. We explicitly show that, up to next-to-leading order in the holographic radial coordinate, the addition of this boundary term cancels the divergent part of the Entanglement Entropy. We discuss possible applications of the method for studying CFT parameters like central charges.

For clarity, we begin by reviewing the usual RT proposal [5,6], which considers that the EE of an entangling region A in a holographic CFT is given by the volume of a codimension-2 extremal surface Σ in the AAdS bulk. This surface Σ is such that it has minimal area, its boundary ∂Σ is located at the spacetime boundary B, and ∂Σ is conformal to the entangling surface ∂A which bounds A at the conformal boundary C. We include a pictorial representation of the different submanifolds involved in the RT construction, explaining the relations between them, in FIG. 1.
As it is well known, the RT proposal for the EE is divergent due to the presence of an infinite conformal factor at the boundary B. This is apparent in the Fefferman-Graham (FG) expansion of the AAdS metric [40,41]. There are different methods for renormalizing the EE. For example, the scheme proposed by Taylor and Woodhead [9], which is based on the standard Holographic Renormalization procedure [20][21][22][23][24][25][26]. There is also the alternative topological renormalization scheme proposed in Ref. [1], which is based on the Kounterterms renormalization procedure [27][28][29][30][31].
Both renormalization schemes rely on writing the EE in terms of the Euclidean gravitational action in the bulk, including its corresponding counterterms at the boundary B. As shown by Lewkowycz and Maldacena [8], and by Dong [10], the replica trick can be used to construct a suitable 2n−dimensional bulk replica orbifold M (α) 2n , which is a squashed-cone (conically singular manifold without U(1) isometry) having conical angular parameter α, such that 2π (1 − α) is its angular deficit. Then, using the AdS/CFT correspondence in the semi-classical limit, the EE can be expressed as where I E is the Euclidean gravitational action in the bulk, evaluated on the M (α) 2n orbifold. If I E is chosen as the Euclidean Einstein-Hilbert (EH) action, then Eq.(1) reproduces the RT proposal for the EE. However, it is apparent that if I E is chosen instead as a suitably renormalized gravitational action, then S EE will be renormalized as well.
Here, we consider the renormalized Euclidean action I ren E as given by the Kounterterms scheme for even-dimensional manifolds in Ref. [27]. Then we proceed in the same manner than in Ref. [1], but now considering that the action has to be evaluated on a squashed-cone and ∂A is the entangling surface. On the gravity side, B is the boundary of spacetime, Σ is the minimal surface in the bulk and ∂Σ is its border at the spacetime boundary. Both sides are related such that C is conformal to B and ∂A is conformal to ∂Σ.
of higher even dimension (beyond the particular case of D = 4). After taking the corresponding derivative with respect to the angular parameter, we obtain that the renormalized entanglement entropy S ren EE is given by where is the AdS radius, V ol (Σ) is the volume of the codimension-2 extremal surface Σ and B 2n−3 is the (n − 1) −th Chern form evaluated at ∂Σ, whose explicit form is given in Eq. (36). The derivation of Eq.(2) and its properties are further explained in section III. The first term of Eq.(2) is exactly the RT proposal. Therefore, the counterterm that renormalizes the EE is given by the B 2n−3 part, which depends on the induced metric γ on ∂Σ, and on both the intrinsic curvature of ∂Σ and its extrinsic curvature with respect to the radial foliation.
The expression for S ren EE given in Eq.
(2) can be rewritten in terms of the Euler characteristic and the AdS curvature [32] of the minimal surface Σ, as explained in section III C. This is done by considering the usual Euler theorem for regular manifolds, which relates the Chern form at ∂Σ with the Euler density on Σ. After some algebraic manipulations, S ren EE can be expressed as where is the AdS curvature of Σ, which depends on its intrinsic Riemann curvature tensor, and χ [Σ] is the Euler characteristic of Σ. This topological form of S ren EE is useful for computing the renormalized EE of certain entangling regions, which give rise to constant curvature minimal surfaces in the bulk. For example, in section IV, we discuss the case of ball-shaped entangling regions, where the computation of S ren EE is greatly simplified.
The organization of this paper is as follows: In section II, we discuss a possible generalization of the Euler theorem for squashed cones in even dimensions. In section III, we derive S ren EE by introducing the renormalized gravitational action for AAdS 2n evaluated on the replica orbifold. We verify the cancellation of the divergence in S ren EE up to next-to-leading order in the radial coordinate ρ. We also reinterpret S ren EE in terms of the topological and geometric properties of Σ. In section IV, we rederive S ren EE for spherical entangling surfaces in odd-dimensional CFTs by using the topological renormalization scheme of Eq.(3), recovering the results of Refs. [18,19]. We also comment on the simplicity of the computation using the topological scheme. Finally, in section V, we comment on the application of the renormalization procedure for characterizing odd-dimensional CFTs and future generalizations thereof.

II. EULER DENSITY FOR EVEN-DIMENSIONAL SQUASHED CONES
In Ref. [1], we discussed the Euler theorem for squashed cones in 4D, which was derived by Fursaev, Patrushev and Solodukhin in Ref. [34] considering the form of the quadratic curvature invariants on squashed-cone manifolds, which have non-trivial extrinsic curvature contributions coming from the binormal foliation at the tip of the cone. The theorem states where ε 4 is the Euler density evaluated on a regular manifold M 4 given by the α → 1 limit and ε 2 is the Euler density evaluated on the 2D manifold Σ, which is the codimension-2 surface located at the tip of the cone (defined for integer 1 α as the fixed-point set of the Z ( 1 α ) symmetry). The form of ε 4 , which is also referred to as the Gauss-Bonnet term, is given in terms of the quadratic curvature invariants by and ε 2 = R √ γd 2 x, where the cursive R denotes the Ricci scalar on Σ.
Furthermore in Ref. [1], we derived a relation between the second Chern form at the spacetime boundary and the first Chern form at ∂Σ, given by In order to obtain this, we considered the usual Euler theorem for regular 2n−dimensional manifolds M 2n , which states that where B 2n−1 is the n−th Chern form at the boundary ∂M 2n (whose form in Gauss normal coordinates is given in Eq. (17)) and χ [X] is the Euler characteristic of the manifold X. We also considered the relation satisfied by the Euler characteristic of 4D squashed-cones, given by χ M and derived by FPS in Ref. [34]. Then, Eq.(7) can be obtained by considering n = 1 and n = 2 in Eq. (8), replacing the ε 4 for the B 3 and the ε 2 for the B 1 in Eq.(5), and then using Eq.(9) to eliminate the Euler characteristics.
We now conjecture that Eqs. (5,7,9) have generalizations for squashed-cone manifolds of arbitrary even dimensions. Namely, we propose that and In the above relations, the Euler density ε 2n is given by 2n−1 is the n−th Chern form evaluated at the conically-singular ∂ M 2n−1 is the n−th Chern form evaluated at the regular ∂M 2n boundary (corresponding to the α → 1 limit) and B 2n−3 is the (n − 1) −th Chern form evaluated at ∂Σ. Also, δ [µ 1 ...µ 2n ] is the totally antisymmetric generalization of the Kronecker delta, defined by We mention that Eqs. (10)(11)(12) were proven by Fursaev and Solodukhin in Ref. [33] for the case of conically-singular manifolds with a U(1) rotational isometry about the symmetry axis of the cone. Here we simply assume these equations to hold also for the squashed-cone case, and we show that if they are correct, then the S ren EE for arbitrary odd-dimensional holographic CFTs can be obtained in a manner analogous to the 3D case studied in Ref. [1].
Conversely, as we will explicitly verify (up to next-to-leading order in the holographic radial coordinate ρ) in section III B, the S ren EE obtained in this manner does indeed renormalize the EE, which lends credence to our conjectured generalization of the Euler theorem, although it does not constitute a proof thereof.
The expression given in Eq. (11) is precisely what is used in section III in order to evaluate the Euclidean action in the replica orbifold, as required in the computation of S EE according to Eq.(1). This ultimately gives the expression for S ren EE when considering the renormalized Euclidean action for AAdS 2n , which is discussed in the following section.
In this section, we study the renormalization of EE via the topological scheme. We consider the renormalized Euclidean gravity action for AAdS 2n spacetimes given in Ref. [27], which was obtained by the Kounterterms procedure [27][28][29][30][31]. The preference for this renormalization procedure over the usual holographic renormalization scheme is only due to practical reasons, as both schemes have been shown to give assymptotically equivalent results for the renormalized action [30,31]. Essentially, as explained in Ref. [1], the renormalization of the even-dimensional action in the Kounterterms scheme can be accomplished by the addition of the Chern form (with a specific coupling), which constitutes a single boundary counterterm, whereas in the case of holographic renormalization, the number of required counterterms rapidly grows with the dimension.
The renormalized bulk action, when evaluated in the replica orbifold (as described in the introduction), is given by where and the n−th Chern form B 2n−1 is given by In the expression for the Chern form of Eq. (17), h ij is the metric at the spacetime boundary B, R k ij is the intrinsic Riemann tensor computed with h ij , and K i j is the extrinsic curvature of B with respect to the radial foliation. Also, we note that in Eq.(15), B (α) 2n−1 denotes the Chern form evaluated at the conically singular boundary of theM (α) 2n orbifold, which can be expressed in terms of the Chern form at the boundary of the minimal surface Σ using our conjectured generalization of the Euler theorem to squashed-cones in arbitrary even dimensions, as presented in Eq. (11).
When computing the EE, after taking the α derivative according to Eq.(1), the Einstein-Hilbert part simply gives the usual RT minimal area prescription for the EE, as shown by Lewkowicz and Maldacena [8] and by Dong [10]. The EE counterterm then comes from the term containing the boundary Chern form B (α) 2n−1 . We therefore define the counterterm of the Euclidean action as and we proceed to compute the counterterm of the EE (S ct EE ) as such that S ren EE = S RT EE + S ct EE , where S RT EE is the usual RT prescription for the EE. Finally, using Eq.(11) to evaluate I ct E , we obtain recovering S ren EE as given in Eq.(2).

A. Explicit covariant embedding
Now, we consider the embedding of the minimal surface Σ in the AAdS bulk as given by Hung, Myers and Smolkin [43], and by Schwimmer and Theisen [42]. The embedding is such that the bulk coordinates {ρ, x i } of Σ and its worldvolume coordinates {τ, y a } are related by We also consider the asymptotic expansion of the bulk metric G µν , which is of FG form [40] and is given by as presented in Ref. [41]. Here, ρ is the holographic radial coordinate and h ij is the induced metric at the spacetime boundary B, which is located at ρ = 0.
We now consider the induced metric γ ab at the minimal surface Σ, which is defined as the pullback of G µν in the FG gauge, and is therefore given by Fixing the diffeomorphism gauge as τ = ρ and γ aτ = 0, we obtain a FG-like expansion for γ ab such that In the previous expansions, d is defined as the dimension of the spacetime boundary ij corresponds to the metric of the CFT at the conformal boundary C (conformal to B) and σ (0) ab denotes the induced metric on the entangling surface ∂A (conformal to ∂Σ) which is given by Moreover, g ij , defined as denotes the Schouten tensor of the g (0) ij . Now, considering the definition of γ ab given in Eq. (23) and the embedding of Σ from Eq. (21), we obtain that Here, such thatn i (n) are the orthonormal vectors to ∂A at the conformal boundary C (n = 1, 2), and κ (n) ab are the corresponding extrinsic curvatures of ∂A. In the following subsection, given the explicit embedding of Σ in M (α) 2n and the corresponding FG expansions of G µν and γ ab , we proceed to verify the finiteness of S ren EE , as given in Eq.(2).

B. Proof of finiteness of S ren EE
We now use the previously discussed embedding of Σ in order to verify that S ren EE , as defined in Eq.(2), is free from divergences. In order to do this, we first exhibit the divergence structure of S RT EE , and then we check that the divergences are exactly cancelled by the S ct EE defined in Eq. (20), without modifying the finite universal part. We mention that although the value of S ren EE depends on the particular choice of entangling surface ∂A, the divergence structure of S RT EE and S ct EE do not. We start by computing the RT part of the EE, according to the minimal area prescription [5,6]. We have that where ρ max is the maximum value of the holographic radial coordinate on the Σ surface, which depends on the choice of entangling surface ∂A in the CFT, and ε is a cutoff such that the ε → 0 limit is to be evaluated at the end. Considering the FG-like expansion of γ ab given in Eq. (24), we have that and therefore, by performing the ρ integration, we obtain where C 1 is a finite constant that depends on the value of ρ max . Here we can see that the leading and next-to-leading divergences occur at orders ε − (2n−3) 2 and ε − (2n−5) 2 respectively, as expected.
Now, we consider the form of σ (2) ab , which is the second coefficient in the FG expansion of the induced metric γ ab at ∂Σ and it is given in Eq. (27). We therefore have that where S σ (0) is the trace of the Schouten tensor of the induced metric σ .
Finally, we obtain By replacing Eq.(34) into the radial integral of Eq.(31), and after some simplifications, we can rewrite S RT EE as where C 2 4G is the finite part of the EE, which depends on the choice of ∂A. We now analyze the asymptotic behavior of the EE counterterm, according to Eq. (20).
We have that where R a 1 a 2 b 1 b 2 is the Riemann tensor of the γ metric at ∂Σ, k a b = γ ac k cb and k cb is the extrinsic curvature of ∂Σ with respect to the radial foliation along the holographic coordinate ρ (not to be confused with κ (n) ab for ∂A or with K ij for B). By definition, and therefore, using the FG expansion for γ ab given in Eq. (24) and considering that we have that We also have that γ is given by and that the Riemann tensor of γ ab satisfies where R (0) a 1 a 2 b 1 b 2 is the Riemann tensor of the σ (0) ab metric. We now have everything we need in order to expand B 2n−3 to next-to-leading order in ρ. The explicit step-by-step computation is presented in appendix A.
We therefore obtain that the (n − 1) −th Chern form at ∂Σ, located at ρ = ε, is given by and thus, S ct EE is given by Finally, using Eq.(34), we can rewrite S ct EE as which explicitly cancells the divergences of S RT EE as presented in Eq.(35), up to next-to-leading order in ρ = ε.
By considering the definition of S ren EE , given as the sum of S RT EE and S ct EE , and by using Eq.(35) and Eq.(44), we finally obtain that S ren EE = C 2 4G , where C 2 is O (1). Therefore, we have explicitly verified, up to next-to-leading order, that the definition of renormalized EE as given in Eq.(2) is correct, being finite and equal to the universal part or the EE. Furthermore, we consider the explicitly verified cancellation of divergences as evidence of the validity of the generalization of the Euler theorem to squashed cones of arbitrary even dimensions, which to the best of our knowledge, has no formal proof yet.
Finally, we note that the EE counterterms can be rewritten in purely intrinsic form, in terms of the induced metric γ and its curvature invariants, up to linear order in the curvature. To achieve this, we first invert the FG expansion of γ, starting from Eq. (40), Then, we replace this into Eq.(44), and after some simplifications, which include substituting for tr σ (2) using Eq. (34), and relating the Ricci scalars of γ ab and σ (0) ab using Eq.(41), we have that up to order ε −(2n−5)/2 , S ct EE is given by which is written in terms of purely intrinsic quantities. This intrinsic form of the counterterms allows to make contact with the EE renormalization scheme presented in Ref. [9]. In appendix B, we show how it can be derived starting directly from S ct EE written in terms of the Chern form, as given in Eq. (20).

C. Topological interpretation of renormalized EE for the AdS 2n /CFT 2n−1 case
We now reinterpret S ren EE as given in Eq.(2), in terms of the topological and geometric properties of the minimal surface Σ as an AAdS submanifold. In particular, we rewrite S ren EE in terms of the Euler characteristic of Σ and its AdS curvature [32].
By considering the Euler theorem, as presented in Eq.(8), we have that the Chern form which gives the EE counterterm can be rewritten as where ε 2n−2 is the Euler density of Σ and χ [Σ] is its Euler characteristic. Then, S ren EE can be expressed as We now define the topological constant C χ as and using the definition of ε 2n−2 given in Eq.(13), we can write S ren EE as where R a 1 a 2 b 1 b 2 is the Riemann tensor of the induced metric γ ab on Σ. Finally, we can simplify Eq.(50) by considering the properties of the antisymmetric Kronecker delta. In particular, we have that Now, we express S ren EE in terms of the AdS curvature F AdS , which for a general AAdS manifold is defined as where R µ 1 µ 2 ν 1 ν 2 is the Riemann tensor of the manifold. Then, for the Σ manifold, the product of the Riemann tensors can be reexpressed in terms of the AdS curvature as where we have disregarded the order of the indices due to the presence of the overall Kronecker delta. Now, noting that the k = 0 term cancells the deltas appearing in Eq.(51), we can write Finally, using the properties of the antisymmetric delta, we obtain that where S ren EE has been expressed in terms of contractions of the AdS curvature of Σ (F AdS| Σ ) and its Euler characteristic, as considered in the definition of C χ given in Eq.(49).
We note that Eq.(55) gives an interpretation of renormalized EE in terms of the topological and geometric properties of Σ as an AAdS Riemannian manifold, which generalizes the result obtained in Ref. [1] for the case of AdS 4 /CFT 3 .

IV. EXPLICIT EXAMPLE: BALL-SHAPED ENTANGLING REGION IN
CFT 2n−1 , WITH A GLOBAL ADS 2n BULK In order to exhibit the advantages of the topological renormalization scheme, we now rederive the renormalized EE of a ball-shaped entangling region in the ground state of an holographic (2n − 1) −dimensional CFT, having global AdS 2n as its gravity dual. This computation was firstly done by Kawano, Nakaguchi and Nishioka in Ref. [18], by evaluating the RT area functional. Instead, we make use of the topological form for S ren EE given in Eq.(55).
We consider that the entangling region is delimited by a spherical entangling surface, such that ∂A = S 2n−3 . Also, the metric G µν of global AdS 2n can be written as where the boundary metric has been expressed in spherical coordinates. Now, as shown in appendix C, the minimal surface Σ corresponding to the entangling surface ∂A = S 2n−3 can be parametrized as where R is the radius of the sphere. We proceed to compute the induced metric γ ab on Σ, defined in Eq.(23), considering that its worldvoume coordinates are given by y a = {ρ, θ 1 , ..., θ 2n−3 }, whereas the bulk coordinates are x µ = {ρ, t, r, θ 1 , ..., θ 2n−3 }. In particular, we obtain that Considering the induced metric, we now compute S ren EE using the topological procedure. Given the induced metric of Eq.(58), we compute the AdS curvature on Σ, according to Eq.(52), and we find that it vanishes identically (i.e., F AdS| Σ a 1 a 2 b 1 b 2 = 0). Also, we consider that Σ is topologically equivalent to a (2n − 2) −ball, whose Euler characteristic is χ (Σ) = 1.
Therefore, using S ren EE as given in Eq.(55), we have that which agrees with the standard result as given in Refs. [18,19]. We also conclude from this analysis that for spherical entangling surfaces, the minimal surface Σ is a constant curvature surface, which has R a 1 a 2 . We note that the computation of the renormalized EE using the topological approach of Eq.(55) is performed directly to all orders in ρ = ε. We also mention that, as further explained in section V, the computation of S ren EE for the case with ∂A = S 2n−3 is important because it is related to the a− charge [18,39], such that S ren EE = (−1) n−1 2πa 2n−1 . The a−charge counts the number of degrees of freedom of the CFT and is conjectured to decrease along RG flows between conformal fixed points. Therefore, it can be thought of as a generalization of Zamolodchikov's c-theorem [38].
Using our result for S ren EE , we therefore find that in agreement with the known result as given in Refs. [18,19]. As written, the a−charge is expressed in terms of the bulk gravity quantities (like the AdS radius and Newton's constant G), but it can be rewritten entirely in terms of the CFT quantities by using the standard dictionary.

V. OUTLOOK
We have successfully extended the topological scheme developed in Ref. [1] for computing the renormalized EE to holographic CFTs of arbitrary odd dimensions. This procedure considers adding the Chern form to the usual RT area functional, such that the renormalized EE is written as shown in Eq. (2). Alternatively, S ren EE can be written in terms of the Euler characteristic of the minimal surface Σ and its AdS curvature, as shown in Eq.(55). The latter form greatly simplifies the computation of S ren EE and is of interest because it exhibits the relation between the EE and the topological and geometric properties of Σ. We also make contact with the renormalization procedure developed in Ref. [9], by writing the EE counterterm in terms of the intrinsic quantities on ∂Σ, as shown in Eq.(46).
The renormalized EE is of interest for the study of holographic renormalization group (RG) flows. As firstly mentioned in Ref. [9] and also discussed in Ref. [19], S ren EE for a ballshaped entangling region in the ground state of a CFT is related to its a−charge [39], which encodes information about the number of degrees of freedom of the theory and is conjectured to decrease along RG flows between conformal fixed points. It therefore constitutes a generalization of Zamolodchikov's c-theorem [38]. In particular, for a (2n − 1) −dimensional CFT, S ren EE = (−1) n−1 2πa 2n−1 where (a 2n−1 ) U V ≥ (a 2n−1 ) IR between any two fixed points. Using our topological renormalization approach (see Eq.(55)), the computation of S ren EE for spherical entangling surfaces becomes nearly trivial, and as discussed in section IV, we recover the known result given in Refs. [18,19].
As future work, we will also study how to extend the scheme to AAdS manifolds of arbitrary odd dimensions by using the renormalized Euclidean gravitational action discussed in Ref. [28]. We expect this analysis to be useful for the study of the conformal anomaly of the corresponding holographic CFTs. We also intend to apply our topological renormalization scheme to obtain the renormalized EE for higher-curvature theories of gravity, specially those of the Lovelock class [44,45], and to renormalize other information-theoretic measures of holographic CFTs, like the Entanglement Renyi Entropies (EREs) [10][11][12][13] and the complexity [14][15][16][17].
[a 1 ...a 2n−3 ] [2n −3] . Also, as the final expression is fully contracted, the indices of all tensors are omitted. For example, k a 1 b 1 k a 2 b 2 is denoted by k 2 , and R a 1 a 2 b 1 b 2 , which is the Riemann tensor of the metric γ, is denoted as Rie. To avoid ambiguity, traces of tensors are explicitly indicated.
In the new notation, the definition of the Chern form at ∂Σ, is given by and the FG expansion of the different quantities are given in section III B. We note that, in every FG expansion, the "..." represents O (ρ 2 ) terms.
First, we expand the k 2 terms, were we have that Then, we consider that the term in parenthesis can be expanded as Note that Rie (0) is the Riemann tensor of the metric σ (0) . The product of such terms, to linear order in ρ, becomes Therefore, γk 1 2 Rie − t 2 k 2 n−2 can be expanded as Thus, the t integral of the previous term has a FG expansion given by Now, using that we can finally obtain the contractions of the integrand, such that where R (0) is the Ricci scalar of the σ (0) metric.
Finally, replacing the previous expression into Eq.(A1), we obtain that B 2n−3 is given by where B 2n−3 is given in Eq.(A1). We use the same shorthand notation introduced in appendix A. Now, the Riemann tensor of the γ ab can be written in terms of the Weyl tensor and the Schouten tensor, such that where the antisymmetrization does not have a 1 4 factor. In turn, the Schouten and the Weyl tensors of the metric γ ab at ∂Σ can be expressed in terms of those of the σ (0) ab metric at ∂A, to first order in ρ, as Then, considering the FG expansion of the extrinsic curvature k a b of ∂Σ as given in Eq.(39) and using that d = 2n − 1, we have that Finally, using that as given in Eq.(32), we have that We proceed to simplify the expression for B 2n−3 . Using the shorthand notation, we have that k 2 = k (0) + 2ρk (0) k (2) + ..., where we used the fact that the contractions of the Weyl tensor, after considering the overall Kronecker delta, vanish identically due to its tracelessnes. Now, using that k (0) a b = 1 δ a b , tr[k (2) ] = tr S (0) and we can compute the contractions with the overall Kronecker delta and perform the t integration in order to obtain that which corresponds to Eq.(46) of the main text.
Appendix C: Minimal surface for a ball-shaped entangling region We consider the minimal area condition, as derived in appendix A of Ref. [1], which is given by ∂ a z ,a 4ρ (D−1)/2 m (ρ, x a ) + ∂ ρ z ,ρ 2 ρ (D−3)/2 m (ρ, x a ) = 0, where D is the dimension of the AAdS bulk, z (ρ, x a ) is the embedding function of the surface Σ expressed in Cartesian coordinates and m (ρ, x a ) = 1 + 4 ρ 2 z ,ρ z ,ρ + z ,a z ,a .
We also consider the definition of the minimal surface Σ, which is given in Eq.(57) of the main text, and corresponds to Σ : t = const ; r 2 + 2 ρ = R 2 .
We proceed to verify that this definition of Σ satisfies the minimal area condition of Eq.(C1). Therefore, we first write the corresponding embedding function z (ρ, x) in Cartesian coordinates, considering that r 2 = z 2 + x a x b δ ab , for a, b = 1, ..., (D − 3). Thus we have that z (ρ, x a ) = ± R 2 − 2 ρ − x a x b δ ab . Then, computing the derivatives of the embedding function, we have that z ,a = ∓ which implies that Σ is indeed the minimal surface.