Supersymmetric Renyi Entropy and Weyl Anomalies in Six-Dimensional (2,0) Theories

We propose a closed formula of the universal part of supersymmetric R\'enyi entropy $S_q$ for $(2,0)$ superconformal theories in six-dimensions. We show that $S_q$ across a spherical entangling surface is a cubic polynomial of $\gamma:=1/q$, with all coefficients expressed in terms of the newly discovered Weyl anomalies $a$ and $c$. This is equivalent to a similar statement of the supersymmetric free energy on conic (or squashed) six-sphere. We first obtain the closed formula by promoting the free tensor multiplet result and then provide an independent derivation by assuming that $S_q$ can be written as a linear combination of 't Hooft anomaly coefficients. We discuss a possible lower bound ${a\over c}\geq {3\over 7}$ implied by our result.


Introduction
Exact results in interacting quantum field theories are rare. Even less is known about the six-dimensional (2, 0) theories, although they are the local conformal field theories (CFTs) with maximal supersymmetry in the maximum number of dimensions [1,2], which actually play important roles in understanding lower dimensional supersymmetric physics [3][4][5][6][7]. The main obstacle is that the proper formulation of the interacting theories is still lacking, for instance in the path integral formalism. 1 This also makes it challenging to study the theories in curved spaces. In particular it is unclear how to perform the supersymmetric localization [18][19][20] directly.
Recently alternative approaches to 6d (2, 0) theories, such as effective actions on the moduli space and the superconformal bootstrap, are advocated in [21,22] and in [23,24], respectively. In particular, the Weyl anomaly coefficients a g and c g have been determined for the (2, 0) superconformal field theory (SCFT) characterized by a Lie algebra g, 2ā where r g , d g and h ∨ g are the rank, dimension and dual Coxeter number of the compact simply-laced Lie algebra g, respectively. a and c appear generally as coefficients of the anomalous trace of the stress tensor in a six-dimensional curved background [25,26], where E 6 is the Euler density while I i are Weyl invariants. In the presence of (2, 0) superconformal symmetry, c i=1,2,3 are proportional to a single coefficient c. One interesting fact is that bothā g andc g will be uniquely fixed once we assume that they are linear combinations of the 't Hooft anomaly coefficients, h ∨ g d g and r g . This can be done by combining the large N values ( from holography [27][28][29][30] ) and the free tensor multiplet values [31,32].
As robust observables, the 't Hooft anomalies of the continuous global symmetries in 6d (2, 0) theories have been worked out [33][34][35][36][37][38][39]. They are organized in an 8-form anomaly polynomial, where p 2 (R) is the second Pontryagin class of the field strength of the SO(5) Rsymmetry background and I u(1) is the anomaly polynomial of a free Abelian tensor multiplet.
As in other even dimensions, it is known that a g determines both the universal part 3 of the sphere partition function and the universal entanglement entropy associated with a spherical entangling surface (in flat space) [40]. On the other hand, it was pointed out that c g determines both the 2-point and the 3-point functions of the stress tensor in the vacuum in flat space [23,24]. Due to the intrinsic relations between the flat space stress tensor correlators and the nearly-round sphere partition function, it is therefore attempting to ask whether one can fully determine the partition function on a branched (q-deformed) sphere, 4 which is directly related to the supersymmetric Rényi entropy S q .
The concept supersymmetric Rényi entropy was first introduced in three-dimensions [41][42][43], and later studied in four-dimensions [44,45,47], five-dimensions [48,49] and for free tensor multiplets in six-dimensions [50]. 5 By turning on certain R-symmetry background fields (chemical potentials), one can calculate the partition function Z q on a q-branched sphere S d q , and define the supersymmetric Rényi entropy as which is a supersymmetric refinement of the ordinary Rényi entropy (which is nonsupersymmetric because of the conical singularity). 6 The quantities defined in (1.4) are UV divergent in general but one can extract universal parts free of ambiguities. For instance, for N = 4 SYM in four-dimensions, the log coefficient of S q as a function of q and three chemical potentials µ 1 , µ 2 , µ 3 (corresponding to three independent R-symmetry Cartans) has been shown to be protected from the interactions [44]. It also receives a precise check from the holographic computation on the 5d BPS STU topological black holes [44]. Furthermore, there are universal relations between the Weyl anomaly coefficients a, c and the supersymmetric Rényi entropy in 4d N = 1, 2 SCFTs, which provides a new way to understand the Hofman-Maldacena bounds [45]. 7 The above facts indicate that the supersymmetric Rényi entropy may be used as a new robust observable to understand SCFTs. In this work we show that the supersymmetric Rényi entropy of 6d (2, 0) SCFTs characterized by simply-laced Lie algebra g is given by a cubic polynomial of γ ∶= 1

5)
3 By "universal" we mean scheme-independent. 4 A branched sphere is a sphere with a conical singularity with the deformation parameter q − 1. 5 The supersymmetric Rényi entropy was recently studied in two-dimensional (2, 2) SCFTs [51] in a slightly different way. 6 For CFTs, the Rényi entropy (or supersymmetric one) associated with a spherical entangling surface in flat space can be mapped to that on a sphere. Throughout this work we take the "regularized cone" boundary conditions. 7 Some of a c bounds by Hofman and Maldacena [46] coincide with Rényi entropy inequalities.
with four coefficients where r 1 and r 2 are background parameters denoting the weights of the two U(1) chemical potentials associated to the two R-symmetry Cartans, satisfying the supersymmetry constraint r 1 + r 2 = 1. 8 Since bothā g andc g are linear combinations of the 't Hooft anomaly coefficients, one may rewrite the closed formula S (2,0) γ (1.5) also as a linear combination of h ∨ g d g and r g , with coefficients as cubic polynomials of γ We derive the closed formula (1.5) by promoting the supersymmetric Rényi entropy of a free tensor multiplet. The free tensor multiplet result is nothing but T γ in the alternative expression (1.7), which can be directly computed using heat kernel method. Demanding that entanglement entropy S γ=1 is proportional to a g and both the first and the second γ-derivatives at γ = 1 are proportional to c g , we could fix the coefficients s 0 , s 1 and s 2 in (1.5). We fix the remaining s 3 by demonstrating a precise relation between the large γ (small q) behavior of supersymmetric Rényi entropy and the supersymmetric Casimir energy on extremely squashed sphere. 9 As a nontrivial test of our result (1.5), we show that H γ precisely agrees with the holographic result computed from the BPS topological black hole in 7d gauged supergravity.
The fact that T γ and H γ are precisely the free multiplet result and the holographic result, respectively, actually provides an alternative derivation of the closed formula (1.5). Consider T γ and H γ as independent results from the free field computation and the holographic dual, respectively, one can uniquely determine the closed formula of S (2,0) γ by assuming that S (2,0) γ is a linear combination of the 't Hooft anomaly coefficients. This assumption can be reasonably imposed once we are aware of any two of s n=0,1,2,3 as linear combinations of h ∨ g d g and r g . This paper is organized as follows. We begin with the general study of the relations between the perturbative supersymmetric Rényi entropy around q = 1 and the integrated correlation functions (stress tensor and R-current) in Section 2, which works for general dimensions. We focus on the first and the second derivative at q = 1. Then we review the supersymmetric Rényi entropy of free tensor multiplets in Section 3. Built up on these facts, we propose a way to determine the supersymmetric Rényi entropy for interacting (2, 0) theories in Section 4. In Section 5, we show a general relation between the q → 0 behavior of supersymmetric Rényi entropy and supersymmetric Casimir energy, which is used to determine the remaining unfixed coefficient in the proposed formula in the previous section. Finally we give a precise test of our results by comparing with the holographic results in Section 6.

Near q = 1 expansion
We begin with the perturbative expansion of supersymmetric Rényi entropy (associated with spherical entangling surface) around q = 1. This can be considered as an extension of the previous study of the ordinary Rényi entropy near q = 1. Although our main concern will be 6d (2, 0) SCFTs, we keep the discussions in this section valid for any SCFT with conserved R-symmetries in d-dimensions.
Following the way in [52,53] 10 , we consider the supersymmetric partition function on S 1 q × H d−1 with background gauge fields (R-symmetry chemical potentials), which can be used to compute the supersymmetric Rényi entropy across a spherical entangling surface, see S d−2 , in flat space. We work in grand canonical ensemble. The partition function on S 1 β=2πq × H d−1 can be written as The state variables can be computed as follows where I ∶= − log Z. Therefore we get energy expectation value by (2. In the presence of supersymmetry, both inverse temperature β and chemical potential µ are functions of a single variable q therefore I is considered as The supersymmetric Rényi entropy is defined as Consider the Taylor expansion around q = 1, with δq = q − 1, We will first consider ∂ q I q . The first derivative with respect to q can be written as Using (2.2) and (2.4), we can rewrite it as The q-dependence of the temperature and the chemical potential can be read off from the supersymmetric background (including metric and R-symmetry gauge field), where β(q) is determined by the geometric fact and µ(q) is solved from the Killing spinor equation on the background. α is some number which may be different in various rigid supersymmetric backgrounds. 11 The first q-derivative of I q is simplified by using (2.12) Notice that in general both E and Q are functions of q. Also E and Q here are expectation values rather than operators.

S
From (2.9) we see that (2.14) Let us take one more derivative above on the first derivative (2.13) and take use of (2.5) and (2.6) which can be simplified in the limit q → 1 by using µ = 0 at q = 1 This can be rewritten as connected correlators where we have used [Ê,Q] = 0 to flip the order ofÊ andQ. Given that ⟨ÊQ⟩ c = 0 and ⟨ÊÊ⟩ c has been computed in [52], we get where Now the task is to compute the second term in (2.18). Following the way of computing ⟨T T ⟩ on the hyperbolic space S 1 q=1 ×H d−1 , one can take use of the flat space correlators in the CFT vacuum. The result is 12 where C v is defined in the current correlator in flat space Then our final result of S ′ q=1 becomes which tells us that the first q-derivative of supersymmetric Rényi entropy at q = 1 is given by a linear combination of C T and C v . 13 This is intuitively expected because in the presence of supersymmetry, taking the derivative with respect to q is equivalent to taking the derivative with respect to g τ τ and A τ in the same time. 14 q-deformation can be often equivalent to the squashing b ∶= √ q, therefore this formula also shows the relation between ∂ 2 b=1 of the free energy on squashed sphere and flat space correlators. It is clear from the above derivation that this formula works both for free theories and interacting SCFTs in general d-dimensions. In the particular case of 6d (2, 0) SCFTs, the 2-point function of the stress tensor is determined by the central charge c g in (1.1) [23,24]. Therefore the integrated 2-point function is proportional to c g . Furthermore, S ′ q=1 is also proportional to c g , because the stress tensor and the Rcurrent in the right hand side of (2.23) live in the same supermultiplet. 15 The same thing happens in N = 4 SYM [44].
From (2.9) we see that One may go straightforward to compute I ′′′ q by taking one more derivative above on (2.15) which may be simplified at q = 1 where µ = 0 This can be further written in terms of connected correlation functions,  16 These correlators in flat space can be determined up to some coefficients for general CFTs in d-dimensions by conformal Wald identities [57,58]. In the presence of 6d (2, 0) superconformal symmetry, both the 2-and 3-point functions of the stress tensor supermultiplet are uniquely determined in terms of a single parameter, the central charge c g [23,24]. And the right hand side of (2.27) should be proportional to c g , because the stress tensor and the R-current belong to the same supermultiplet. 17 The same thing can be seen in N = 4 SYM [44].

Abelian tensor multiplet
The six-dimensional (2, 0) superconformal algebra is osp(8 * 4). While it is easy to identify a free Abelian tensor multiplet that realizes the (2, 0) superconformal symmetry, the existence of interacting (2, 0) theories was only inferred from decoupling limits of string constructions [62][63][64]. See for instance [65] for a review of various aspects of 6d (2, 0) theories. Now we review the supersymmetric Rényi entropy of free tensor multiplets [50]. For free fields, the Rényi entropy associated with a spherical entangling surface in flat space can be computed by working on a hyperbolic space S 1 β × H 5 and using heat kernel method. 18 A six-dimensional (2, 0) tensor multiplet includes 5 real scalars, 2 Weyl fermions and a 2-form field with self-dual strength. The 2-form field with self-dual strength can be considered as a chiral 2-form field with half of the degrees of freedom.

Heat kernel
The partition function of free fields on S 1 β=2πq × H 5 can be obtained by heat kernel method, 19 1) 16 We leave the explicit computations of these correlators elsewhere. 17 By representation theory, the stress tensor belongs to a half BPS multiplet. In superspace, the 2-, 3-and 4-point functions of all half BPS multiplets are known to admit a unique structure [59][60][61]. 18 Six-dimensional (2, 0) theories have been studied in AdS 5 × S 1 recently in the viewpoint of rigid holography [66]. 19 For Rényi entropy of free fields in other higher dimensions, see for instance [67][68][69][70].
where K S 1 β ×H 5 (t) is the heat kernel of the associated conformal Laplacian. The kernel can be factorized when the spacetime is a direct product, The kernel on a circle K S 1 β (t) is known to be 20 In the presence of a chemical potential µ, it is twisted to be [55] where f = 0 for scalars and f = 1 for fermions. Finally the kernels on the hyperbolic space K H 5 (t) can be written as follows because H 5 is homogeneous, The regularized volume V 5 = π 2 log(ℓ ǫ). ǫ is the UV cutoff of the theory in the original space 21 and ℓ is the curvature radius of H 5 . Note that the kernels K H 5 (0, t) for free fields with different spins are known. See [50] and references there.

Rényi entropy
The total Rényi entropy of a tensor multiplet can be obtained by summing up the contributions of 5 real scalars, 2 Weyl fermions and a chiral 2-form, where the Rényi entropy for fields with different spins can be computed by using the corresponding heat kernels. For the details of this computation we refer to [50]. We will instead list the results here. The Rényi entropy of a 6d real scalar is and the Rényi entropy of a 6d Weyl fermion is and that of a 6d 2-from field is It is worth to mention that, to get the correct Rényi entropy for the two form field, one has to take into account a q-independent constant shift due to the edge modes [50], like what should be done for the gauge field in 4d [71,72]. Finally the Rényi entropy for a free (2, 0) tensor multiplet is (3.10) It has been checked that ∂ 0 q=1 , ∂ 1 q=1 and ∂ 2 q=1 of S f ree q are consistent [50] with the previous results about the tensor multiplet [31,32,73].

S q and S γ
Before moving on, let us represent S f ree q in terms of The reason why S γ is convenient is that, the series expansion near γ = 1 has finite terms while the expansion of S q near q = 1 has infinite terms. We will use S γ instead of S q to express Rényi entropy and supersymmetric Rényi entropy from now on. It is worth to note the relations between the derivatives with respect to q and the derivatives with respect to γ at q = 1 γ = 1, (3.12)

Supersymmetric Rényi entropy
The supersymmetric Rényi entropy of a free tensor multiplet can be computed by the twisted kernel (3.4) on the supersymmetric background. The R-symmetry group of 6d (2, 0) theories is SO(5), which has two U(1) Cartans. Therefore one can turn on two independent R-symmetry background gauge fields (chemical potentials) to twist the boundary conditions for scalars and fermions along the replica circle S 1 β . A general analysis of the Killing spinor equation on the conic space (S 6 q or S 1 β=2πq × H 5 ) leads to the solution of the R-symmetry chemical potential [50] 22 22 The Killing spinors on round sphere have been explored in [74].
with k 1 and k 2 being the R-charges of the Killing spinor under the two U(1) Cartans, respectively. We choose k 1 = k 2 = 1 2 and the two background fields can be expressed as This is the most general background satisfying (3.13). For each component field in the tensor multiplet, one has to first figure out the Cartan charges k 1 and k 2 and then compute the chemical potential by k 1 A 1 + k 2 A 2 . Then one can compute the free energy on S 1 β × H 5 using the twisted heat kernel and get the supersymmetric Rényi entropy. For details, see [50].
After summing up all the component fields, the final supersymmetric Rényi entropy in terms of γ can be expressed as, 23 It is worth to note that, for a single U(1) background, r 1 = 1, r 2 = 0, the result becomes while for two U(1) backgrounds with equal values, r 1 = r 2 = 1 2 , we have (3.17)

Interacting (2, 0) theories
Having obtained the supersymmetric Rényi entropy (3.15) for a free tensor multiplet, we now try to promote it to a general form which may work for interacting (2, 0) SCFTs, where the coefficients A , B , C , D will depend on the specific theory. 24 The factors carrying r 1 and r 2 should stay the same as that appearing in the free multiplet result (3.15) because they originally come from the α i (α 1 = r 1 , α 2 = r 2 ) in (2.23)(2.27), which are background parameters independent of the specific theory. Later we will see that precisely the same factors appear in the holographic supersymmetric Rényi entropy, which confirms this fact. 23 Although the form of this expression is a series expansion, the result itself is complete. 24 S (2,0) γ should be a cubic polynomial of γ, which is the unique option compatible with both free field result and holographic result (as we will see). The same thing happens in N = 4 SYM. Here we see an essential difference between the ordinary Rényi entropy and the supersymmetric one, because the type of q scaling in the ordinary Rényi entropy is not protected [53,75].

S
(2,0) γ=1 and a g We would like to first determine the coefficient D in (4.1). This can be done by using the fact that, the entanglement entropy associated with a spherical entangling surface, which is nothing but S γ=1 , is proportional to a, where a is the a-type Weyl anomaly. This is true for general CFTs in even dimensions as shown in [40]. Therefore This allows us to fix where we have used the a-type Weyl anomaly result in 6d (2, 0) theories [22].

∂S
γ=1 and c g The coefficients C and B in (4.1) are determined by the first and the second γderivatives of S (2,0) γ at γ = 1, respectively. γ-derivatives can be translated into q-derivatives. Taking q-derivatives can be equivalently considered as taking derivatives with respect to background fields, therefore ∂S Explicitly, the first γ-derivative (which is minus the q-derivative at q = 1 γ = 1) is determined by a linear combination of the integrated stress tensor 2-point function and the integrated R-current 2-point function. The first q-derivative at q = 1 is given by the formula (2.23), which works for general SCFTs with conserved R-symmetries in d-dimensions.
Similarly the second γ-derivative at γ = 1 is related to q-derivatives by (3.12). The second q-derivative at q = 1 is determined by a linear combination of the integrated stress tensor 3-point function, the integrated R-current 3-point function and some mixed 3-point functions. This is given explicitly by (2.27) q=1 ×H d−1 , (4.5) which also works for general SCFTs with conserved R-symmetries in d-dimensions.
In the particular case of 6d (2, 0) SCFTs, all the above two-and three-point functions may be uniquely determined in terms of a single parameter, the central charge c g (1.1), as discussed in Section 2. 25 25 This actually explains the universal ratio 4N 3 between the explicit results on ⟨T T ⟩, ⟨T T T ⟩, ⟨JJ⟩, ⟨JJJ⟩ in holography and those in free tensor multiplets [32,76].
Due to the above facts, the straightforward idea to get ∂S (2,0) γ=1 and ∂ 2 S (2,0) γ=1 for interacting theories is to multiply to the free multiplet values in (3.15). This actually means we can fix The remaining coefficient A will be fixed as in the next section by studying the asymptotic q ∶= 1 γ → 0 behavior of the supersymmetric Rényi entropy. Obviously, the leading contribution in the limit γ → ∞ is controlled only by A.

A closed formula
As a summary, we can completely determine a closed formula of supersymmetric Rényi entropy for (2, 0) SCFTs characterized by simply-laced Lie algebra g where in the last line we have used the normalized Weyl anomalies defined in (1.1). For a single U(1) chemical potential, the result is simplified to be 12 . (4.13) As for two U(1) chemical potentials with equal values, the result is simplified to be

q → 0 asymptotics
In this section we discuss the q → 0 limit (γ → ∞) of supersymmetric Rényi entropy S q . Recall the definition of S q Assuming that in the limit q → 0 the free energy behaves where α ≥ 0, one can easily get in the leading order. This relation does not depend on which geometric background we are working on. The idea is that, S d q can be conformally mapped to H 1 × S d−1 q , therefore the Rényi entropy (or supersymmetric) is invariant [40]. In the case with supersymmetry, one has to make sure that in the limit q → 0, the background field on S d q coincides with that on H 1 × S d−1 q . If that is the case, the asymptotic supersymmetric Rényi entropy S q→0 on S d q will coincide with the minus free energy on H 1 × S d−1 q→0 . The latter is determined by the supersymmetric Casimir energy [77]. We will illustrate the details in the following.

From
We start with the conformal transformation from conic sphere S d q to hyperbolic space H d−p × S p q . Of course S d q can be considered as the special case of p = d. In the particular case p = 1, the transformation is nothing but the Weyl transformation discussed in [40], which offers a convenient way to compute Rényi entropy of CFTs. In this case, the branched d-sphere is described as 26 with domains of coordinates given by and Ω d−2 is a standard d-2-dimensional round sphere. The metric (5.4) can be written as ds 2 = sin 2 θ q 2 dτ 2 + 1 which can be related to the following space by dropping an overall factor sin 2 θ and using a coordinate transformation cot θ = sinh η where η ∈ [0, +∞). This is the space of H d−1 × S 1 q , which indeed fits the case of p = 1. Now we consider the general cases, 1 ≤ p < d. The key observation is that, the branched sphere can be presented in different forms. For instance, we can represent S d q as ds 2 = sin 2 θ(dχ 2 + sin 2 χq 2 dτ 2 ) + dθ 2 + cos 2 θd 2 Ω d−3 , and Ω d−3 is a standard d-3-dimensional round sphere. Again by dropping an overall factor sin 2 θ and using a coordinate transformation cot θ = sinh η for the metric (5.8), one obtains which is the space H d−2 × S 2 q with p = 2. One can follow the same way to eventually figure out the Weyl transformations between S d q and H d−p ×S p q for any integer 1 ≤ p < d. Since the Rényi entropy on S d q can not depend on which particular circle we choose to create the conical singularity, one eventually arrives at the conclusion by employing the same argument in [40]: 27 The universal part of CFT d Rényi entropy is invariant on For later purpose, let us discuss the particular case p = d − 1. In this case we describe the branched sphere S d q as ds 2 = sin 2 θ(dχ 2 + sin 2 χq 2 dτ 2 + cos 2 χd 2 Ω d−3 ) + dθ 2 , (5.11) with domains Again by dropping an overall factor sin 2 θ for the metric (5.11), one obtains where cot θ = sinh η and η ∈ (−∞, +∞). This is the space S d−1 q × H 1 . Here we use H 1 instead of R 1 to emphasize that the volume of H d may be regularized. For free fields, one can compute the CFT Rényi entropy on S d−1 q × H 1 and show explicitly that the result agrees with that computed from S d q or S 1 q × H d−1 . In consideration of supersymmetry, one has to add a background field A τ along the replica τ circle inside S d−1 q , in order to find the agreement. 27 Again by the universal part of Rényi entropy we refer to the scheme independent part.

Coincidence of backgrounds
Our main concern is physical quantities for CFTs. For this purpose we can work on × H 1 because they are related by a scale transformation (5.14) Furthermore, we focus on the limit q → 0. For this purpose, one can instead consider In consideration of supersymmetry, one can use the squashed sphereS d−1 √ q to replace the conic sphere S d−1 √ q in the right hand side of (5.15), because supersymmetric partition functions do not depend on the resolving factor [42,49,[78][79][80][81]. 28 (5.15) is useful in the sense that it offers a way to compute the asymptotic supersymmetric Rényi entropy for interacting SCFTs. To do this, one has to make sure that the background gauge field on S d−1 √ q × S 1 1 √ q agrees with that on the original space S d q . Fortunately we have more knowledge about supersymmetric partition functions on S d−1 × S 1 or its generalized version S d−1 b × S 1 β , where b is the squashing parameter.

Squashed Casimir energy
Now we make a connection between the asymptotic Rényi entropy and Casimir energy. It is known that the partition function Z on S d−1 b × S 1 β is determined by the Casimir energy on S d−1 b in the limit β → ∞ E c ∶= − lim β→∞ ∂ β log Z(β) , (5.16) which is equivalent to say lim In this work, we concern the case with supersymmetry. In the particular case of 6d (2, 0) theories, the supersymmetric Casimir energy has been studied in [82] 29 , where the authors considered a general 5-sphere with squashing parameters ⃗ ω = (ω 1 , ω 2 , ω 3 ). The squashing parameters are defined as parameters appearing in the Killing vector where φ 1 , φ 2 , φ 3 are three circles representing U(1) 3 isometries of S 5 . The supersymmetric Casimir energy of an interacting (2, 0) theory is [82] where σ 1 and σ 2 are chemical potentials for the two Cartans of the SO(5) Rsymmetry and E u(1) is given by For the particular case of S 5 q × S 1 (which is equivalent to for CFTs), we should identify the shape parameters as In the limit q → 0, in order to match our chemical potentials (3.14), we set σ 1 and σ 2 as 30 Therefore the free energy 31 where we have divided a q-independent volume factor Vol [D 4 × S 1 ] = π 3 . Due to (5.15), we have from which we obtain the asymptotic supersymmetric Rényi entropy on S 1 q × H 5 S q→0 = −I q→0 = 1 12 This fixes the undetermined coefficient A in (4.1) as Notice that the fact that the free limit of (5.26) precisely agrees with the leading large γ term of (3.15) by itself is nontrivial, which confirms the validity of (5.25) in the free case.

Large N limit
In the large N limit of the (2, 0) theory with g = A N −1 , the supersymmetric Rényi entropy (4.9) becomes We will demonstrate in this section that the above large N result precisely agrees with the holographic result from the seven-dimensional BPS topological black hole in gauged supergravity.

Gauged supergravity
The seven-dimensional gauged SO(5) supergravity can be obtained by Kaluza-Klein reduction of eleven-dimensional supergravity on S 4 . For our purpose, we consider a truncation where only the metic, two gauge fields associated to two Cartans of SO (5) and two scalars are retained. The seven-dimensional Lagrangian is given by [86] 1 where ⃗ φ = (φ 1 , φ 2 ) are two scalars and The potential is given by Note that for two equal scalars and two equal gauge strengths, the Lagrangian (6.2) can be further truncated. Turn to the CFT side, 6d (2, 0) theories have global SO(5) R-symmetry, which corresponds to the SO(5) gauge group in the bulk supergravity. Also there could be two U(1) background fields used to compensate the singularity on S 6 q , which correspond to A 1 , A 2 in the gauged supergravity.

Topological black hole
The 2-charge 7d AdS black hole solution for (6.2) was found in [86] together with scalars and gauge fields (6.6) dΩ 2 5,k is the metric on a unit S 5 , T 5 or H 5 corresponding to k = 1, 0, −1, respectively. Since our concern is the 6d SCFT on S 1 × H 5 , we are particularly interested in the extremal solution with hyperbolic foliation, where m = 0 and k = −1. We will first proceed in Lorentz signature and assume a well-defined Wick rotation.
The solution (6.5) is a BPS topological black hole with two charges. For convenience, define a rescaled charge where the horizon r H is the largest root of the equation Then the horizon can be expressed in terms of κ i . (6.9) The Hawking temperature of this black hole is . (6.10) When all charges vanish, we get to the temperature of the uncharged black hole . (6.11) The Bekenstein-Hawking entropy is given by the outer horizon area where G 7 is the seven dimensional Newton constant and V 5 is the regularized volume of H 5 . The total charge Q i can be computed by Gauss law (6.13) The chemical potential is (6.14) To match the background gauge fields of the boundary CFT, we set , with r 1 + r 2 = 1 . (6.15) By using these inputs, we can solve κ 1 and κ 2 by (6.14). Then all physical quantities T, S, Q i can be worked out explicitly. One can eventually compute the holographic supersymmetric Rényi entropy using the formula derived in [42] Written in terms of γ ∶= 1 q, the result is given by By identifying the bulk and boundary parameters, This precisely agrees with the field theory result (6.1).

A possible a c bound
As what has been observed in 4d SCFTs [45], the Rényi entropy inequalities indicate the a c bounds in field theories 32 , where H q ∶= S q S 1 . Imposing these conditions to our results (4.10)(4.12)(4.15), one obtains 0 <c a ≤ 7 3 , Note that all the a, c data of the currently known 6d (2, 0) SCFTs, listed in Table 1 in Appendix A, satisfy the inequality (7.5)(7.6). The lowestā c value in the current data, 4 7, supported by the large N limits, is greater than our bound 3 7. Note that the expression of supersymmetric Rényi entropy in terms of a, c anomalies could work for theories beyond the ADE type. It would be interesting to understand whether our bound implies new (2, 0) SCFTs. It would also be interesting to understand similar bounds in SCFTs with less supersymmetry. We leave these questions for future work.