Black hole entropy function for toric theories via Bethe Ansatz

We evaluate the large-N behavior of the superconformal indices of toric quiver gauge theories, and use it to find the entropy functions of the dual electrically charged rotating AdS5 black holes. To this end, we employ the recently proposed Bethe Ansatz method, and find a certain set of solutions to the Bethe Ansatz Equations of toric theories. This, in turn, allows us to compute the large-N behavior of the index for these theories, including the infinite families Ypq , Xpq and Lpqr of quiver gauge theories. Our results are in perfect agreement with the predictions made recently using the Cardy-like limit of the superconformal index. We also explore the index structure in the space of chemical potentials and describe the pattern of Stokes lines arising in the conifold theory case.


Introduction
The pursuit of a theory of quantum gravity is one of the greatest challenges of modern theoretical physics. The description of the microscopic origin of black hole entropy constitutes a highly important step towards such a theory. One of the tremendous successes in this direction was achieved in string theory which provides a microscopic explanation for the entropy of a class of asymptotically flat black holes [1].
Another source that might shed light on this question could be the study of the entropy of asymptotically AdS black holes. This is by virtue of the AdS/CFT correspondence which establishes a connection between a quantum theory of gravity and a conformal field theory (CFT) living on the boundary of the AdS space. This connection provides a natural playground for the study of black hole microstates. In this setting the problem of microstate counting of asymptotically AdS black holes translates into the counting of BPS states in the dual CFT. For a long time, many approaches employed in the attempt to match these dual descriptions had fallen short. However, recent years have seen substantial progress in this direction.
While considerable progress was made in the context of static magnetically charged black holes (in the AdS/CFT setting), the situation as far as rotating electrically charged black holes are concerned had remained unclear. The simplest and canonical example of this kind is that of BPS black holes arising in type IIB supergravity on AdS 5 × S 5 [23][24][25][26]. On the dual side, microstate counting of these black holes should correspond to the counting of 1 16 -BPS states in 4d N = 4 SU(N ) Super Yang-Mills (SYM) theory on S 3 × R. The latter are counted by the superconformal index [27][28][29]. Yet, an attempt to match these two quantities has failed already at the leading order in N , since the number of BPS states grew as N 0 , which is much slower than the growth of N 2 corresponding to the black hole entropy. This slow growth was explained by the huge cancellation between fermionic and bosonic states counted by the index. Later, indices of other theories including the conifold theory [30], the Y pq family [31] and finally toric theories in general [32], were also evaluated at the large-N limit. In each case the situation repeated the one of N = 4 SYM case: the indices appeared to be of order O(1) at large N .
This puzzle has been recently resolved in several steps. The first inspiration came from the extremization principle used to resolve the problem in the case of magnetically charged black holes. In particular, in [33] the entropy function of the rotating AdS 5 black hole with two angular momenta J 1,2 and three charges Q 1,2,3 was proposed. The entropy function is the Legendre transform of the black hole entropy. It is a function of the chemical potentials for the electric charges ∆ 1,2,3 and angular momenta ω 1,2 , correspondingly. The chemical potentials, in turn, appear to be constrained by the relation ∆ 1 + ∆ 2 + ∆ 3 = ω 1 + ω 2 − 1. Later on, similar constructions were proposed for electrically-charged black holes in other dimensions [34,35]. Furthermore, it was also realized that the black hole entropy function may be reproduced by the evaluation of the complexified supergravity on-shell action in the bulk [36,37].
On the dual CFT side, the entropy function has a clear formulation in terms of the corresponding field theory parameters, but a precise relation to the counting of BPS states had remained obscure. An answer to this question appeared in [38] where the authors considered the superconformal index with complex fugacities. The latter allowed them to avoid large cancellations between different states and to obtain an N 2 scaling for the growth in the number of BPS states. More importantly, considering the Cardy-like limit [39], also referred to as the high-temperature limit, the authors were able to identify the superconformal index of N = 4 SYM given by with the corresponding entropy function of the electrically charged AdS black hole. Here ∆ a and ω a are chemical potentials conjugated to three R-charges of SO(6) R-symmetry JHEP04(2020)091 and two angular momenta J 1,2 on S 3 , correspondingly. Just as in the entropy function, these chemical potentials satisfy the constraint mentioned above. This important result was subsequently extended to various 4d N = 1 SCFTs [40][41][42][43][44]. For the purposes of the present paper, the most relevant is the observation made in [44], where the authors considered a Cardy-like limit of the superconformal index for the class of toric quiver gauge theories. The following expression was obtained at the large N limit where ∆ I are chemical potentials dual to d charges under U(1) I trial R-symmetries satisfying the constraint d I=1 ∆ I − ω 1 − ω 2 = 1. The coefficients C IJK are associated with the triangle anomalies of the corresponding U(1) I,J,K symmetries that can be obtained directly from the toric data of the gauge theory quiver. Interestingly, expression (1.2) was already proposed for the entropy function of AdS 5 black holes with d electric charges in appendix A of [34].
Despite the great success in computing the superconformal index at the Cardy-like limit, the importance of that particular limit and the reason why it precisely reporduced the black hole entropy, even at finite temperatures, were unclear. The proper way to treat the problem is to calculate the large N limit of the index with complex fugacities, relying upon no further restrictions or limits. This kind of computation was performed in [45] for the canonical case of N = 4 SYM, for which the authors exploited the Bethe Ansatz technique [46,47]. Its main idea is to reformulate the integral representation of the superconformal index as a sum of residues of the integrand. The positions of the integrand's poles, in turn, are defined by the solutions to the Bethe Ansatz Equations (BAEs) system. Generally, it is very hard to solve this system of equations. However, in [45] it was shown that using a particular class of relatively simple solutions to the BAEs, it is possible to reproduce the entropy function (1.1) at large N , but finite temperature limit, within a certain range of values for the chemical potentials ∆ I .
Another intriguing observation made in [45] is the highly complicated structure of the large N index in the space of complex chemical potentials. In particular, there are many competing exponential contributions to the index. The full space of chemical potentials divides into separate regions with one of the exponential contributions dominating in each of them. These regions are separated by codimension one surfaces, Stokes lines, on top of which different exponents contribute the same order and hence there is no dominant contribution. This also explains why the calculation in [28] resulted in an O(N ) index since it has been performed precisely on such Stokes lines where O(N 2 ) saddles cancel each other.
The BAE approach is considerably more involved than Cardy limit calculations, but it yields a large N answer that is precise in temperature. It further grants access to the Stokes phenomenon and the full large N index structure. It is therefore natural to apply this approach to theories beyond N = 4 SYM. Here, similarly to [44] we consider the superconformal index of various toric theories. In particular, we show that the basic solutions to the BAEs for N = 4 SYM used in [45] also solve the BAEs for all toric quiver theories. This allows us to readily evaluate the leading order contribution of the basic JHEP04(2020)091 solution to the superconformal index of these theories. Note that in principle there might be other solutions contributing at the same order of N which are difficult to find due to the complicated structure of the BAEs. However, just as in the N = 4 SYM case, we find that the Cardy limit prediction of [44] presented in (1.2) is exactly reproduced by the basic solution contribution, and is precise in temperature for a certain range of values for the chemical potentials. We thus provide a more definite support in favor of the conjectured form of the multi-charge black hole entropy function proposed in [34].
Unfortunately, entropy functions (1.2) appear to be quite complicated for performing extremization. In particular, in [44] the only cases considered in detail were those of Y pp and L 0b0 . However, the BAE approach also grants us access to the Stokes phenomenon. Since reproducing the full structure of the large-N index in the space of chemical potentials seems too complicated a task, we consider certain limits. In particular, as was done in [45] we consider the special case of equal chemical potentials ∆ I and also take ω 1 = ω 2 = τ as required by the BAE technique. Consequently, the only parameter on which the large-N index depends is τ , and we can define the positions of Stokes lines and regions of dominance of certain exponents. Interestingly, for the conifold theory we find that the structure of the index in the complex plane of τ exactly reproduces the one presented in [45] for the case of N = 4 SYM.
The paper is organized as follows. In section 2 we briefly review the BAE method for the calculation of the index and present the basic solutions to the BAEs for toric theories. We also present a general formula that can be used to evaluate the contribution of these solutions to the index at the large-N limit. Next, in section 3 we study in detail the simplest possible case of all toric theories, namely the conifold theory. We exploit the relative simplicity that characterizes this case, to clearly illustrate how most of the arguments presented in section 2 work. In section 4, omitting details we directly apply the results of section 2 to many different cases of toric quiver theories, including the infinite families Y pq , X pq , L pqr . Finally, in the last section 5 we consider the large-N index of the conifold theory derived in section 3 and study its structure in the case where all the chemical potentials ∆ I are equal. In particular, we find the positions of Stokes lines and regions of dominance for various contributions to the index.
Note added. When we were finalizing our paper the preprint [48], which has a significant overlap with our work, has been posted on arXiv. The authors' conclusions disagree with ours at various points.

Large N index for toric theories
In this section we will be considering some generalities of index computations that we will later use in order to obtain large-N limits for the indices of toric theories. Let us consider general N = 1 quiver gauge theories with gauge group G, flavor symmetry G F and non-anomalous U(1) R -symmetry. The matter consists of n f chiral multiplets in the representation R a of the gauge group G. Since we are interested in toric theories, we will concentrate only on the cases of adjoint and bifundamental matter leaving the discus-JHEP04(2020)091 sion of matter in the fundamental and other representations for the future. The integral representation of the superconformal index is then given by [29,49] Here p and q are complex fugacities for the angular momentum, v a are flavor fugacities with a = 1, . . . , rk(G F ). The integration variables z i parametrize the maximal torus of G with the integration contour given by rk(G) unit circles. The roots of G are parametrized by vectors α while ρ a and ω a are the weights of the representations R a in which the chiral multiplets transform under gauge group G and flavor symmetry group G F correspondingly. Also, |W G | is the order of the Weyl group. Finally, Γ (z; p, q) and (a; q) ∞ are standard elliptic Gamma function [50] and q-Pochhammer symbol: Throughout the paper, we will also be using the chemical potentials τ, σ, ξ a and complexified holonomies u i as follows p = e 2πiτ , q = e 2πiσ , v a = e 2πiξa , z i = e 2πiu i . (2.4) All the functions in the index (2.1), and hence the index itself, are well-defined for |p| < 1 and |q| < 1. We will be interested in the large N behavior of the index. In order to estimate it, we will employ the technique of Bethe Ansatz Equations (BAE) [46,47], which was recently used to find the large N behavior of the superconformal index of N = 4 SYM [45]. The results obtained in this case perfectly matched the entropy of the corresponding AdS 5 black hole. Below, we briefly describe this method closely following [45,46].
The main observation behind the method is that for τ and σ satisfying we can recast the integral representation (2.1) of the index as the sum over poles located at the solutions to certain transcendental equations called BAEs. Notice that as was shown in [46], the set of τ and σ satisfying (2.5) is dense in the domain {|p| < 1, |q| < 1} so the method is, in principle, applicable for any fugacities p and q. However, even when τ and σ are generic parameters satisfying (2.5), it is hard to perform BAE calculations. Therefore, for simplicity and without loss of generality we will consider the easiest case possible of JHEP04(2020)091 τ = σ. Under this condition, as was shown in [46,47], the poles of the integral (2.1) are located at the solutions to the following equations written in terms of BA operators Q i : where the function P (u) is expressed through the usual θ 0 (u; τ ) ≡ e 2πiu ; e 2πiτ ∞ e 2πi(τ −u) ; e 2πiτ ∞ as follows: After finding all the solutions to the equations (2.6), we can evaluate the residues of the integrand (2.1) and represent the supreconformal index in the form, where we introduce the following notations. Z (û; ξ, τ ) is the integrand of the index (2.1) and can be generally written as where we introduce a newΓ-function, which is just the usual Γ-function defined as the function of chemical potentials, Γ (u; τ, σ) ≡ Γ e 2πiu ; e 2πiτ , e 2πiσ . (2.10) The κ G in (2.8) denotes the constant prefactor appearing integral representation in of the index (2.1) and is defined as The final ingredient to be explained in (2.8) is the Jacobian H (u; ξ, τ ) for the change of variables u i → Q i (u): Notice that as promised we have concentrated on the particular case of p = q for the index. However, all the equations above may be written for the more general case of τ /σ ∈ Q + . All the corresponding expressions can be found in [46]. JHEP04(2020)091

Solutions to the BAE
Let us start by very briefly reviewing the solution to the BAE for N = 4 SU(N ) SYM that was used in [45] in order to match the black hole entropy with the large N behavior of the index. The BAE (2.6) in this case take the following form: where u ij ≡ u i − u j . Notice that due to the double periodicity of Q i , if u i is the solution to the BAE then so are u i + 1 and u i + τ . Therefore, there is in fact an infinite number of solutions to the BAE, but we can group them into equivalence classes. In other words, our holonomies u i live on a torus with modular parameter τ . Following [46] we also introduce the convenient parametrization ∆ for the flavor fugacities defined as follows: where r a are the R-charges of the corresponding multiplets. Then for the case of N = 4 SYM, this parametrization just takes the form ∆ a = ξ a + τ +σ 3 . Finally, λ in (2.13) is the Lagrange multiplier required here since we are considering SU(N ) gauge group instead of U(N ). For the same reason, the holonomies u i satisfy the following constraint: The exact same equations appeared in [9] in the context of the twisted index of N = 4 SU(N ) SYM on T 2 × S 2 with the topological twist on S 2 . The authors also found the following solutions to these equation at the high-temperature limit τ → 0: 1 In [45] it was shown that the solution above is actually not limited to the high-temperature limit and, in fact, solves the BAE (2.13) exactly for both finite N and τ . It was also shown in [9] that a similar solution, namely is valid at the high temperature limit of the BAE for Klebanov-Witten (KW) (or conifold) theory [51] and in general for any toric quiver. The index (a) in the expression above refers to the node of the quiver. We will demonstrate here that, just as in the case of N = 4 JHEP04(2020)091 SYM, this high temperature solution is actually exact in N and τ . Finally, just as in the case of N = 4 SYM, we added the constant u to satisfy the SU(N ) constraint (2.15) at each of the nodes. We will refer to the solution (2.17) as the basic solution.
To understand how this works, notice that in toric theories each node has an equal number of bifundamentals and anti-bifundamentals, i.e. the quiver is non-chiral. The contribution of each such pair to the BAE is given by where ∆ 1,2 are fugacities corresponding to each of the chiral multiplets and u j . Again, all upper indices denote quiver nodes. In particular, the node (a) is the one that we write the equation for, while (b) and (c) are the ones that connect to (a) through bifundamental chirals. The same expression describes the contribution of one adjoint multiplet provided b = c = a. In this case, ∆ aa is just the chemical potential dual to the charges of this multiplet.
Substituting the basic solution (2.17) in the expression above we can use the following chain of manipulations to drastically simplify it Here, between the first line and the second we introduce new summation indices k = i − j in the numerator and k = j − i in the denominator. Between the second line and the third we performed a shift of the index k → k + N in the product l−N k=−1 in the numerator and in the product 1−l k=−1 in the denominator. Now all the terms which do not depend on l, including the ratio of θ-functions product, can be omitted since we can always absorb them into the Lagrange multiplier Λ (a) = e 2πiλ (a) . Combining the l-dependent part with the one coming from the B(u; τ ) parts we get: Q i.e. every dependence on l goes away and we are left with terms independent of l that can be absorbed into the Lagrange multiplier. This proves the validity of the basic solution for the class of non-chiral quivers with only bifundamental or adjoint matter.

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For the case of N = 4 SYM it was shown in [52]  whereĵ = 0, . . . , m−1,k = 0, . . . , n−1 is the parametrization of the index j = 0, . . . , N −1 in case N can be represented as the product N = nm, and 0 ≤ r < n. Since the BAE we get are similar to the SYM case 2 and the basic solution is just identical, it is straightforward that the same general class (2.21) is valid in our case. The family (2.21) is organized into PSL(2, Z) orbits with the following actions: T : {m, n, r} → {m, n, r+m} , S : {m, n, r} → gcd(n, r), mn gcd(n, r) , m(n−r) gcd(n, r) , (2.22) where the last entry of the triplet {m, n, r} is always understood modulo n. We will always be considering only the contribution of the basic solution (2.17) and of the more general family (2.21). In principle, it is not obvious that there are no other solutions to the BAE contributing to the index at the leading N order. This is already true even for the simplest case of N = 4 SYM, let alone for complicated quiver theories, where the situation becomes more involved. Nevertheless, from [45] we know that the class of solutions (2.21) gives the leading contribution to the index which matches the supergravity result. Motivated by this result, we will also proceed to check the contribution of (2.21) against AdS black hole entropy and previous high-temperature index computations [44].

Large-N superconformal index
In this subsection we will be describing the contributions to the index (2.1) at large-N of the basic solution (2.17) and of the more general solutions (2.21) to the BAE. For this, we should just employ eq. (2.8) which is a straightforward yet quite a tedious task. However, as we have shown in the previous section, the solutions we exploit are basically the same solutions that have been previously used in [45] in order to find the large-N behavior of the N = 4 SYM index. Hence, we can exploit the derivations used in the latter case. Below, we summarize briefly the crucial expressions and derivations that are required for the calculation and explain, where relevant, the generalization to quiver theories. For all the details of the derivations we refer the reader to the original paper [45].
Contribution of the integrand. The first and most important thing to estimate is the contribution of eachΓ-function (2.10) in the integrand Z (û; ξ, τ ) given in (2.9) in the most general form. In [45] it was shown that upon substitution of the basic solution (2.17), the leading term of the correspondingΓ-function in the large-N limit is given by:  where we introduced the periodic and discontinuous function This function is endowed with a number of useful properties: i.e. the function [∆] τ shifts ∆ along the real line and brings it to the fundamental domain For convenience we show this strip in figure 1. We also introduce the notation γ for the right boundary of this domain, i.e. the line passing through the points 0 and τ . Notice that the function [∆] τ is discontinuous along the lines γ + Z. The latter are Stokes lines that divide the complex ∆-plane into regions corresponding to different dominating large-N contributions. Along these lines the different contributions compete with each other and a more accurate calculation, including an evaluation of the subleading terms, is required.
As a result, expression (2.23) is not defined in the case ∆ = 0 which sits exactly on the Stokes line. At the same time, this value of ∆ is relevant for the contribution of the vector multiplets in (2.9). In this case, as has been shown in [45], the leading-order contribution of theΓ function is given by These two expressions constitute all the building blocks we need for the evaluation of the integrand contribution to the superconformal index.

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Contribution of the Jacobian. The second possible contribution to the index comes from the Jacobian H of the transformation between the u i and Q i variables given in (2.12).
Since we are considering a quiver theory with SU(N ) groups it is convenient to write the definition of the Jacobian in a clearer form as 1 , . . . , u In the expression above, we wrote down precisely the quiver indices 1, . . . , p, where p is the number of nodes. The resulting expression can be seen as a division of the full N p × N p matrix into p × p blocks of size N × N . Second, notice that due to the SU(N ) gauge groups in the quiver, each set of holonomies u To evaluate expression (2.28), let us write the general form of the BA operators Q (a) i first. As we said, since our quiver is balanced, each node (a) has an equal number of outgoing (bifundamental chiral) arrows (ab) and incoming (antibifundamental chiral) arrows (ca). Then, we can write the corresponding operator Q (a) i using the general expression (2.6) in the following form: where we denote by (a, b) the set of all arrows going from the node (a) to the node (b) of the quiver. Therefore, when we take the product over all b ∈ (a, b) for example, the product is over all the nodes that have incoming arrows originating from the node (a). As discussed before, the number of terms of the products in the numerator and the denominator of (2.29) is the same, since each quiver node has an equal number of incoming and outgoing arrows. Moreover, as before u l . Now, we can easily find the logarithmic derivative of the BA operator (2.29): where for convenience we introduced the following function: Now we can estimate the contributions of each of the logarithmic derivatives (2.30) to the Jacobian (2.28) upon the substitution of the basic solutions (2.17) to the BAE. In particular, we should use

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Notice that the last two terms appear because we consider only N − 1 holonomies u (a) i as independent variables, while due to the SU(N ) constraint at each node u (a) i . Then, we can rewrite the matrix elements of the Jacobian matrix H as follows: where in the first equality we have used the definition of the BAEs Q (a) i = 1. In addition, we introduced δ b∈(a,b) , which is one if node (b) is connected to node (a) with the arrow a → b and zero otherwise. In the same way δ b∈(b,a) is one if node (b) is connected to node (a) with the arrow b → a and zero otherwise.
Finally, we also need the elements of the Jacobian (2.28) containing the derivatives with respect to the Lagrange multipliers λ (a) : ∼ O(1). Indeed, upon substituting the basic solution (2.17) in the function F (u), we see that its argument is τ 1 At the same time, potential divergences can arise only at zeros of the θ 0 (u) function, which take place at u = m + nτ , m, n ∈ Z. As we see from the above, the latter condition can only be satisfied if the chemical potential ∆ ab ∈ m + nτ . In this case, on the boundaries of the Bethe roots distribution the function F u As a result, the contribution of the Jacobian H to the logarithm of the index I given in (2.8) is always subleading with respect to the order N 2 integrand contribution Z on the basic solution (2.17). The remaining term κ G also contributes at order pN log N , and so the only contribution of the basic solution that is relevant to us is the one coming from the integrand.

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Contribution from SL (2, Z) solutions. We should next understand what is the contribution of all the other solutions from the family (2.21). In the case of N = 4 SYM, it was argued that the T -transformed solutions would contribute at the leading order in N [45]. In particular, one can notice that both the integrand Z and the Jacobian H are invariant under τ → τ + r. Hence, once again only the integrand Z contributes to the index at the leading order and the value can be computed from (2.23) and (2.27) by simply substituting τ → τ + r. At the same time, in [45] it was shown that S-transformed basic solutions contribute to the index only at order N and are thus subleading. Based on this, the authors argued that among all the solutions (2.21) labeled by {m, n, r}, only those obtained from the basic one (2.16) by T -transformations contribute at leading order. Since our basic solutions and all the terms in the index (2.8) have exactly the same form as in the case of N = 4 SYM, the arguments above are also valid in our case and we will assume that only the T -transformed solutions (2.17) contribute at leading order to the index.
Summary. Gathering the above derivations, we can write down the leading N 2 term in the superconformal index (2.1). Since we are now convinced that for all toric theories only the basic solution (2.17) and its T -transform contribute at the leading order, we may now directly substitute these solutions into the index (2.8). Hence, each time we have an adjoint or a chiral bifundamental multiplet with corresponding chemical potential ∆, we should add a contribution (2.23) to log I. For each vector multiplet, i.e. for each quiver node, we should add a contribution of (2.27) to the log I. Since, as we saw, the Jacobian H always contributes at the subleading order, these are all the relevant expressions we need in order to calculate the total contribution of the basic solution to the index, namely where p is the total number of the gauge group nodes and in the last term of the Θ({∆}; τ ) function, we summed over all oriented arrows in the quiver graph. This is done to avoid counting twice the contribution of each bifundamental chiral multiplet. In addition, if a = b in the (a, b) pair, then the adjoint multiplet is pictured as one arrow on the graph and should be counted only once.
As discussed above, we should finally include the contributions of the T -transformed solutions (2.36). This is done simply by shifting τ → τ + r in Θ({∆}; τ ). This corresponds JHEP04(2020)091 to the contribution of one of the T -transformed solutions. In the final step we should find a T -transformation (or equivalently such r ∈ Z) that maximizes the real part of −πiΘ: This would give us the final expression for the dominant large-N contribution to the index. Still, this expression is not highly informative on its own. In the following sections of the paper, we will thus be considering various toric theories case by case, and applying (2.39) we will be deriving a compact form for the final answer such that the relation between the U(1) triangular anomalies and large-N superconformal index would be clear. The function I ∞ has a complicated structure. The space of complex parameters ({∆}, τ ) is divided into regions separated by Stokes lines, which are codimension-one surfaces. In each of the regions one solution determined by the value of r that maximizes ImΘ ({∆}, τ + r) dominates over the other solutions. On the Stokes lines themselves, however, two or more values of r 1 , r 2 , · · · ∈ Z yield the same contribution, i.e.
In this case we have two or more equivalent exponents contributing to I ∞ . We should sum these exponents, but in this case their relative sign should be known in order to compute the final answer for I ∞ . That is because the contributions could potentially cancel each other, forcing us to consider the subleading orders. The particular structure of the parameter space and Stokes lines depends on the particular theory considered. We will consider some details pertaining to the Stokes phenomenon for various toric theories in section 5.

The conifold theory
We are now ready to apply the derivations of the previous section to particular examples of toric theories. In principle, as discussed above we can directly apply the final expression (2.39) to any toric theory. However, in this section we will study in detail the BAEs, solutions to it and the large-N index of the simplest possible toric theory beyond N = 4 SYM, namely the conifold theory. This theory was originally proposed in [51] as the worldvolume theory of a stack of N D3-branes probing the tip of the conical singularity xy − zt = 0. In the near-horizon limit this gives the AdS 5 × T 1,1 dual of the theory. Here T 1,1 is the conifold which can be seen as the coset SU(2) × SU(2)/U(1). The conifold is in fact a toric manifold with a toric diagram identified with the vectors: The theory is N = 1 supersymmetric with SU(N ) × SU(N ) gauge group and two pairs of bifundamental chiral multiplets A 1,2 and B 1,2 transforming in the N, N and N, N representations of the gauge groups, correspondingly. The corresponding quiver is shown on the figure 3. The superpotential term is given by The global symmetries of the theory include U(1) R R-symmetry factor, two SU(2) flavor symmetries rotating the A and B doublets, correspondingly, and a U(1) B baryonic symmetry. These symmetries can also be identified with the isometries of T 1,1 . It is further convenient to use instead of SU(2) flavor symmetries Cartans of their combinations, which we denote by U(1) 1 and U(1) 2 . The matter fields' charges under these global symmetries are summarized in the table below.
Also notice that the conifold theory is a particular case of the infinite Y pq class of theories, corresponding to p = 1 and q = 0. We will be considering this class of theories in section 4.1 with less detail. We now have all the ingredients at our disposal to write down the integral representation (2.1) of the superconformal index for the conifold theory.
where we turned on fugacities v F1,2 for the flavor symmetries U(1) 1,2 , and v B for the baryonic symmetry U(1) B . As usually, we introduce the corresponding chemical potentials according to (2.4),

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The expression for κ G is given by (2.11) Throughout this paper we focus on the case where p = q and use the chemical potential τ defined in (2.4). From expression (3.3), it is obvious that the following parameterization of the chemical potentials would be convenient to work with: In particular, with this parametrization the BAEs (2.6), (2.29) for the conifold case may then be written in the following compact form Notice that upon substituting the functions B(u) given in (2.7) into the BAE operators (2.29), we get certain terms depending on the ∆'s but independent of the holonomies u (1,2) i in the exponents. However, we ignored these terms in the expressions written above since they can always be adsorbed into a redefinition of the Lagrange multipliers It is noteworthy that the parametrization (3.6) can be obtained in two natural steps. First, one should perform a mixing of the flavor and baryonic symmetries. This leads to the new basis of U(1) 1,2,3 global symmetries with the corresponding chemical potentials ξ 1,2,3 turned on for each of them. The charges of the conifold theory matter fields in this new basis are summarized in the table below.
Actually, the proper choice of mixed U(1)-symmetries can be seen directly from the geometric data of the toric manifold. To this end, one should apply algorithms defining the matter JHEP04(2020)091 content charges from the toric diagrams as proposed in [53]. The same parametrization of the global symmetries has been used in the calculation of the superconformal indices of toric theories at the high-temperature limit in [44]. In section 4 we will be considering many examples of toric theories, and we will always be using a choice of global symmetries that is determined by the corresponding toric data. The relevant algorithm will be briefly reviewed in that section. Finally, after the flavor symmetry mixing we should perform the shift (2.14) ∆ a =ξ a + r a τ , (3.8) leading exactly to expressions (3.6). It will often prove important to remember in future derivations that the chemical potentials ∆ a are not independent, rather they satisfy the constraint Let us now show how the basic solution (2.17) satisfies the BAEs (3.7). In particular, let us substitute the basic solution into the first BAE and apply a chain of relations similar to the one used in (2.19). We first wish to concentrate on the transformation of θ-functions: where for the notational brevity we introduced Using (3.10) in the first BAE (3.7), and similar relations in the second, it turns out that all l-dependences drop out and we are left simply with (3.12) The equations above merely define the values of the Lagrange multipliers Λ (1,2) ; these are irrelevant to the large-N contribution of the basic solution, as discussed in section 2. However, it is an interesting observation that the values of the Lagrange multipliers obtained from (3.12) precisely reproduce the results of [9] at the high-temperature limit τ → 0. 4

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After showing how the basic solution (2.17) solves the BAEs (3.7) of the conifold theory, we are now ready to directly evaluate the index (3.3) using our BAE formula (2.8). In principle, calculations of all possible contributions were performed in section 2. Nevertheless, to illustrate our estimates for the contribution of the Jacobian (2.12) to the index, let us take a look at the particular explicit form the Jacobian assumes for the conifold theory case, that is Here, each of the diagonal blocks J i has the form (3.14) where the elements A (i) j are all of order N and are given in (2.33). Notice that J i coincides with the Jacobian appearing in the calculations for the case of N = 4 SYM. It can be clearly seen from the expression above that indeed, as expected, the logarithm of the Jacobian log H contributes at order N log N at most. Now using the results of section 2, we can directly obtain expression (2.39) with the function Θ given by where the function [∆] τ is defined in (2.24). However, this form may be drastically simplified if one remembers that our chemical potentials ∆ a satisfy the constraint (3.9), leading to To simplify the function Θ written above, we then need to understand which of the strips shown in figure 1 does the sum [ In particular, there are three possibilities Case III : Upon substituting these expressions back into the chemical potentials constraint (3.16) and into (3.15), we obtain the final expression for the Θ-function: where [∆ 4 ] τ is defined in (3.16) and the function F ([∆] τ ; τ ) is defined as follows: (3.20) and (2.39) completely define the leading term in the superconformal index of the conifold theory. To find the final expression for the index, it is still necessary to perform an r-extremization as prescribed by (2.39). However, it is quite a technically involved task due to the Stokes phenomenon and at the moment we postpone this question to the future. In section 5 we will consider the extremization procedure for the particular case of equal chemical potentials ∆ a . Notice that in case I our large-N index behavior (3.19) is in line with the Cardy-like limit computations (1.2) performed in [44].
Finally, as we have mentioned in the beginning of this section the conifold theory is a particular example of an infinite Y pq class of toric theories with the specification p = 1, q = 0. As we will see further in section 4.1, the general result (4.15) for the large-N index of Y pq theories indeed reduces to the T 1,1 result (3.19) upon this identification.

Other toric models
In this section we extend our analysis to more toric theories and find the large-N asymptotic forms of their indices using the general prescription of section 2, thereby obtaining the entropy functions of the dual black holes. We consider the models discussed in [44], and for most of them (where it is not cumbersome) present the expressions for the entropy function in all the different domains, as in the previous section.
Before going into particular examples of toric quiver theories, let us consider some of their general properties which will be of particular use for us in this section. Toric quiver gauge theories arise as the world-volume theories of D3 branes probing the tip of a toric cone over various five-dimensional Sasaki-Einstein manifolds. In the near-horizon limit, these constructions provide us with the holographic dual description of the corresponding gauge theories. A nice and useful property of the toric models is that all their data can be JHEP04(2020)091 connected to the geometry of the cones [54,55]. In particular, all this data can be encoded in the toric diagrams of the corresponding cones. These diagrams are convex polytopes P built from vectors of the form V i = (·, ·, 1). We summarize below the algorithm of [53] used to extract the assignment of U(1) charges for the chiral fields, as well as other gauge theory data. To illustrate this algorithm, we show all the steps in the example of the conifold theory whose toric diagram is shown in figure 2 and the corresponding vectors V I are given in (3.1).
As a first step, and without any extra constructions, we can extract the total number of gauge group nodes F from the area of the polytope P : Another important fact is that the total number of global symmetries (including R-symmetry) d is equal to the number of external points in the diagram. In the case of the T 1,1 diagram, we can see that the total number of gauge group nodes is 2 × 1 and the total number of global symmetries is 4, as we expect from the gauge theory content. Next, we continue with defining two-component vectors that connect nodes of the toric diagram: v i ≡ V i+1 − V i , and assign numbers for each of them as follows: . In the case of the conifold theory, these vectors are given by To each pair of vectors (ij) = (v i , v j ) such that v j can be rotated into v i clockwise with less than a 180 • turn, we associate a chiral field with a multiplicity of det v T i v T j and charges a i+1 + a i+2 + · · · + a j . In the conifold example, this gives four pairs in total: In the last line of the table we identify a specific bifundamental multiplet of the conifold theory for each of the pairs. The algorithm for this identification requires an introduction for dimer constructions [54] and we omit it here for simplicity. 5 Now, in order to fix the charges of the fields we are free to make any choice of the numbers a i that satisfies the constraints: For example, in the case of the conifold theory we can choose the following set of numbers: (4.5) 5 Notice that for an index computation the precise identification of pairs with particular fields of the quiver theory is not even required.

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Then, a R i will set the assignment of R-charges according to the table of pairs and a I i will assign the charges under the (d − 1) global symmetries U(1) I . The choice presented above precisely reproduces the charges used in section 3.
The final ingredients we will need to extract from the toric data are the triangular anomalies of the U(1) I trial R-symmetries C IJK . In particular we can assign trial Rsymmetry to each vertex of the toric diagram. In this case we can calculate anomalies between any three of these U(1) I,J,K symmetries by just calculating areas of the triangles built from the vertices (V I , V J , V K ) of the toric diagram [56]. Now let us discuss the form of the large-N index we expect to obtain. As we have mentioned in the introduction, the Cardy-like limit computations of [44] has led to the general result (1.2) for the indices of the toric theories. This result, in turn, supported the conjectured form of the entropy function of multi-charged AdS 5 black holes proposed in [33]. However, our computations are slightly different since we do not limit ourselves to any chamber of the chemical potentials. Still, we expect to obtain the same form for this expression, with ∆ a substituted by [∆ a ] τ , at least in some regions of the ∆-space, similarly to the N = 4 SYM case in [45]. In terms of the function Θ, defining the index according to (2.38) we expect the following contribution of the basic solution in one of the chambers: It can be easily checked that in the case of the conifold theory, (3.19) precisely reproduces this expression in case I. As we will see in this section, the same situation takes place for every toric theory we consider. Also in the end of this section we will perform a detailed comparison between results of our computations and Cardy-like limit computations of [44] addressing some subtle questions.

The family Y pq
Let us start our discussion with the infinite family of models Y pq , given by quiver gauge theories with 2p gauge groups and bifundamental chiral fields [57]. In figure 5 we show an example of the Y 32 quiver gauge theory. The notation Y pq denotes a toric manifold with the corresponding diagram shown in figure 4, and is parametrized by the vectors: Using the algorithm summarized above, the charges of the fields under the three global U(1) symmetries and the U(1) R R-symmetry can be identified from the form of the toric diagram shown on figure 4 as follows, As before, we turn on the chemical potentials ξ i for each of the three U(1) i global symmetries and τ for the R symmetry, and perform the shift (2.14): where we have also defined the usual auxiliary chemical potential ∆ 4 . Now we have everything we need in order to evaluate the superconformal index using the expression (2.39), with the function Θ ({∆ i }; τ ) (2.38) given for the case of the Y pq matter content by the following lengthy expression: In this table we fill in the number of the strip in the complex plane to which the particular sum of [∆ i ] τ belongs, with the n-th strip being defined as follows: More explicitly, the green stripe shown in figure 1 corresponds to n = 1, while the strips to its left (right) correspond to n = 2, 3, . . . (n = 0, −1, −2, . . . ). As a simple example that illustrates the notations used in this type of table, which we will also employ below when analyzing other theories, let us consider the first line of the table corresponding to case I.
As we see, all the relevant sums of [∆ i ] τ are in the first strip, hence we get that case I is given by: (4.11) Next, after we distinguished between the various cases, we find the following relations for the bracket functions: Cases IIa, IIb, IIc, IId

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Substituting these relations into (4.9), we get the final form of the function Θ (∆ i ; τ ) in the different cases: Case III (4.15) where, defining for convenience the functions (a = {1, 2, 3, 4}): Θ K (∆ i ; τ ) are given by:  Notice that Θ I (∆ i ; τ ), which can be written more explicitly as has exactly the same structure as the expected result (4.6). In addition to that, it can be shown that the corresponding coefficients are exactly the triangular anomalies which can be obtained from the toric diagram in figure 4. Note also that exactly the same result has been obtained in [44] using the Cardy limit of the superconformal index, and that by specifying to p = 1, q = 0 the expressions in (4.15) reduce to (3.15) obtained in the conifold case.

The family L pqr (p = r)
We next turn to the family of models L pqr , built from quiver gauge theories with p + q gauge groups and bifundamental chiral fields [58][59][60]. The notation L pqr denotes a toric manifold with the corresponding diagram parametrized by the vectors:

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where we have P q = r − ks, s = p + q − r and we assume that p = r. Using our previous discussion, the charges of the fields under the global symmetries can be read from the toric diagram, and are given in the table below.
As in (4.8), we turn on the chemical potentials ξ i and τ for the U(1) i and U(1) R symmetries and perform the shift (2.14): where we have defined ∆ 4 as before. Then, the index is given at large N by (2.39), where

(4.22)
To simplify (4.22) as we did above, we notice that the different cases here are exactly the same as for the Y pq family. Therefore, we can use the same classification along with the relations (4.12), (4.13) and (4.14), and get (4.23) We notice that here as well, Θ I (∆ i ; τ ) (which corresponds to Case I) has the expected form (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index. It is also worth noticing the following interesting fact. If we accurately consider Θ III (∆ i ; τ ) function, we find that it can be obtained directly from and performing an overall shift that does not depend on chemical potentials ∆ i and τ . In particular: where d = 4 in this particular case. Both constraints are consistent with the one used for obtaining dual black hole entropy from the entropy function. Also, the overall constant shift is not relevant in the context of the dual supergravity solution since it does not affect the critical points of the Legendre transformation required to obtain black hole entropy. It would be interesting to better understand the physical meaning of all these observations in the future.

The family X pq
This family is built from quiver gauge theories with 2p + 1 gauge groups and bifundamental chiral fields [61]. The toric diagram of the manifold X pq is parametrized by the vectors: (1, p, 1) , and as discussed above, the corresponding charges of the fields under the global symmetries are as follows, We turn on the chemical potentials ξ i and τ for the U(1) i and U(1) R symmetries and perform the shift (2.14): . . . , 4) , where we have defined the auxiliary ∆ 5 in the usual way. Then, the index is given at large N by (2.39), where (4.28)

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As before, to simplify this expression we need to consider the different domains of the bracket functions. However, for compactness we only present here the result for Case Ia, in which Case Ia : (note that all the sub-sums are also in the same strip). For this case, we have which is of the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index.

The model SPP
We consider the SPP (short for "suspended pinch point") model which was first introduced in [62]. The theory is living on a stack of N D3-branes probing the SPP conical singularity x 2 y = wz. This gauge theory is described by the quiver given in figure 6. The toric manifold in this case is parametrized by the vectors: and the corresponding charges of the fields under the global symmetries are given in the following table: We turn on the chemical potentials ξ i and τ for the U(1) i and U(1) R symmetries and perform the shift (2.14): . . . , 4) , where we have defined the auxiliary ∆ 5 in the usual way. Then, the index is given at large JHEP04(2020)091 X 32 X 23 Figure 6. Quiver diagram corresponding to the SPP model. Each node represents an SU(N ) gauge group and each X ij field transforms in the bifundamental representation, i.e. in the N of the i-th node and in the N of the j-th node. The field φ transforms in the adjoint (N 2 − 1) representation.
As usual, to simplify (4.31) we need to consider the different domains of the bracket functions. Using the notations presented in subsection 4.1, we define the different cases as follows:

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Similarly to subsection 4.1, we find the following relations for the bracket functions: Cases IIa, IIb, IIc, IId Cases IIc, IId, IIIa, IIIb and Cases Ib, IIb, IIc, IIIa Substituting these relations into (4.31), we get the expression for Θ (∆ i ; τ ) in the different cases: (4.32) Here Θ Ia (∆ i ; τ ) (which corresponds to Case Ia) is of the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index. Also it can be checked that there is simple relation between Θ IIIb and Θ Ia , 6 which is similar to our previous observations for T 11 , Y pq and L pqr theories.

The model dP 1
Now we turn to models related to various del Pezzo surfaces [63][64][65][66][67]. We start with the simplest case of the dP 1 model, which is the theory living on a stack of N D3-branes probing the tip of the Calabi-Yau cone whose base is dP 1 (i.e. the first del Pezzo surface). This gauge theory is described by the quiver given in figure 7.
The toric manifold in this case is parametrized by the vectors: and the corresponding charges of the fields under the global symmetries are given in the following table:   Figure 7. Quiver diagram corresponding to the dP 1 model. Each node represents an SU(N ) gauge group and each X ij field transforms in the bifundamental representation, i.e. in the N of the i-th node and in the N of the j-th node.
Carefully examining this matter content, one can notice that it precisely reproduces the one of the Y 21 model. Notice that both the gauge theory quivers and the toric diagrams of these theories are different. However, all that matters for our calculation is the matter content and charges under global U(1) symmetries, which are exactly the same in the two cases. Hence, we can recycle our calculations for Y pq theories and conclude that the final result for the function Θ(∆ i ; τ ) is given by (4.15) and (4.19) upon specifications p = 2 and q = 1. This result of course has the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index.

The model dP 2
Next we consider dP 2 model, the theory living on a stack of N D3-branes probing the tip of the Calabi-Yau cone whose base is dP 2 (i.e. the second del Pezzo surface). This gauge theory is described by the quiver given in figure 8.
The toric manifold in this case is parametrized by the vectors:   This matter content precisely reproduces the one of the X 21 model. Just as in dP 1 case this allows us to recycle our calculations for X pq theories and conclude that the final result for the function Θ(∆ i ; τ ) is given by (4.29) upon specifications p = 2 and q = 1. This result of course has the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index.

The model dP 3
We turn to the dP 3 model, the theory living on a stack of N D3-branes probing the tip of the Calabi-Yau cone whose base is dP 3 (i.e. the third del Pezzo surface). This gauge theory is described by the quiver given in figure 9.
The toric diagram in this case is parametrized by the vectors: JHEP04(2020)091 6 1 2 3 4 5 Figure 9. Quiver diagram corresponding to the dP 3 model. Each node represents an SU(N ) gauge group and each X ij field transforms in the bifundamental representation, i.e. in the N of the i-th node and in the N of the j-th node.
We turn on the chemical potentials ξ i and τ for the U(1) i and U(1) R symmetries and perform the shift (2.14): where we have defined the auxiliary ∆ 6 in the usual way. Then, the index is given at large N by (2.39), where

(4.35)
To simplify this expression, we need to consider the different cases for the bracket functions. However, for compactness we only present here the result for Case Ia, in which Case Ia :

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(note that all the sub-sums are also in the same strip). For this case, we have which is of the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index.

The model (P)dP 4
We consider the (P)dP 4 model, defined using the fourth del Pezzo surface. Unlike all other models considered in this paper (P)dP 4 theory is non-toric [65,66,68,69].The quiver corresponding to this theory is given in figure 10.
The toric manifold in this case is parametrized by the vectors: and the corresponding charges of the fields under the global symmetries are given in the following table: We turn on the chemical potentials ξ i and τ for the U(1) i and U(1) R symmetries and perform the shift (2.14): where we have defined the auxiliary ∆ 7 in the usual way. Then, the index is given at large N by (2.39), where (4.38) To simplify this expression, we need to consider the different cases for the bracket functions. However, for compactness we only present here the result for Case Ia, in which Case Ia : (note that all the sub-sums are also in the same strip). For this case, we have which is of the same form as (4.6) with the coefficients given by the triangular anomalies, and matches the result obtained in [44] using the Cardy limit of the index.

Comparison with the Cardy limit calculations
In all of the cases considered in this section we claimed that our result matches the results of the computations in the Cardy-like limit presented in [44]. Let's now explain what we mean exactly when we say that the match takes place. In both cases the object of study is the superconformal index of a toric theory with d global U(1) charges given by where Q I (I = 1, · · · , d) are the charge operators corresponding to the U(1) I global symmetries. p, q and v i in the expression above are the usual fugacities used by us throughout the paper and are parametrized by the chemical potentials σ, τ and ξ i according to the expressions (2.4). Notice that in (4.40) we choose a basis of the global symmetries such that the (d − 1) charges corresponding to the global non-R symmetries are given by 1 2 (Q i − Q d ) while the R-symmetry charge is given by 1 2d d I=1 Q I . This is precisely the same basis with the symmetric choice of the R-charges that we use throughout our calculations. Now, it has been shown [42,44,45] that the index (4.40) can be written in the following form, where the fugacities y I are given by Notice that the expression (4.41) does not include the (−1) F term under the trace and looks like the BPS partition function. Hence, it is expected to reproduce the entropy function of JHEP04(2020)091 the dual black hole provided we identify Q I /2 with the electric charges on the supergravity side [42,45]. The chemical potentials ∆ i defined in (4.42) coincide with our flavor fugacities ∆ i shifted according to (2.14) for i = 1, · · · , d − 1. However, our definition of ∆ d is different since we choose it such that instead of the constraint (4.42) we have d I=1 ∆ I = 2τ . Consequently, in order to compare the results of our calculations with the Cardy limit calculations we should express the index as a function of ∆ i (i = 1, · · · , d − 1) only. In this case the direct comparison of the two methods will make sense. As the pattern of the expression for the index is similar in each case, let us examine in detail the simplest prototypical example of the conifold theory described in section 3.
We begin by choosing the chemical potentials ∆ 1,2,3 in the chamber corresponding to case I of (3.18). For this we choose the flavor chemical potentials as −1 < ξ 1,2,3 < 0 and −1 < 3 i=1 ξ i < 0 where according to (2.14) ∆ i = ξ i + τ 2 . In this case we have: Substituting these relations into (3.19) we obtain the function Θ (∆ i ; τ ) related to the index through (2.39) in the following form, This is precisely the kind of expression we expect from the computations in the Cardy-like limit performed in [44].
However, notice that the authors of [44] performed the calculations in the chamber of chemical potentials 0 < ξ 1,2,3 < 1/2 and 0 < ξ 1 + ξ 2 + ξ 3 < 1/2 which corresponds to our case III where we have so that the function Θ (∆ i ; τ ) takes the form This result matches the one presented in [44] up to the constant shift −2. However, since this shift does not depend on any chemical potential we can ignore it in the entropy function.
Finally, we can study the chamber of chemical potentials corresponding to case II of (3.18). This case is quite subtle. As an example let us take the part of the chamber corresponding to −1 < ξ 1,2 < 0 , 0 < ξ 3 < 1 and −1 < 3 i=1 ξ i < 0 so that

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Substituting these relations into (3.19) we obtain for the function Θ (∆ i ; τ ) where we have introduced the notations assuming ∆ i should be identified with the chemical potentials on the supergravity side of the duality. There are two problems with this expression. First, there is an extra term −1 + 1/2τ which depends on the chemical potential τ and is not present in the high temperature calculations of [44]. However, this term is subleading in τ and one can argue that the Cardy limit calculations are just insensitive to it. But there is a bigger problem with the fact that the chemical potentials ∆ i satisfy the wrong constraint 4 i=1 ∆ i = 2τ . This contradicts the constraint (4.42) required to eliminate the cancellation of the bosonic and fermionic modes in the superconformal index. It drives us to the conclusion that in the chamber of chemical potentials we consider the index does not count the microstates of the dual black hole.
As we will see in the next section, the analysis of the Stokes phenomena for the case of equal chemical potentials in the conifold theory supports the conclusions of the analysis performed above. In particular, we will see that in the chambers corresponding to Case I and Case III the index is indeed dominated by the single-centered black hole solution. At the same time, Case II corresponds to the Stokes line where presumably the singlecentered black hole solution is unstable and the index counts other states in the dual supergravity picture.
A similar analysis can be performed for the other theories described in this section. The results are qualitatively similar to the ones presented above for the conifold theory. In particular, in Case I where [∆ i ] τ = ∆ i for i = 1, · · · , d − 1 we get the following result for the function Θ, where C IJK are the corresponding anomaly coefficients and the chemical potentials ∆ I satisfy the constraint written above. In the last case for each theory the chemical potentials satisfy [ In this case we obtain the following form for the function Θ,

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which exactly reproduces the result of the Cardy limit computation of [44] up to a constant shift. From our results it looks like this shift equals the number of SU(N ) gauge group nodes. As we have pointed out before, since the constants are independent of any chemical potential we can omit them in the entropy function. Finally, avoiding this complicated analysis we can just summarize our results as follows. If we choose all the chemical potentials to satisfy the conditions of Case I we obtain an index of the form (4.6) with C IJK being the anomaly coefficients and [∆ I ] τ satisfying the constraint d I=1 [∆ I ] τ = 2τ −1. This is consistent with the expected form of the AdS 5 black hole entropy function [34] provided we directly identify [∆ I ] τ with the chemical potentials of supergravity.

Extremization and black hole entropy
As discussed in the introduction, black hole entropy can be obtained from the Legendre transform of the entropy function [33,36]. The latter may be formulated in terms of the dual CFT parameters, and has been shown to coincide with the Cardy-like limit of the index for the case of N = 4 SYM dual to the AdS 5 × S 5 black hole with three electric charges Q 1,2,3 and two angular momenta J 1,2 [23,24]. Its entropy function in particular was shown to be equal to where ω 1,2 are chemical potentials conjugate to two angular momenta and X 1,2,3 are the ones conjugate to three charges. In [45] the large-N behavior of the index was derived using a BAE analysis. In particular, it was shown that the basic contribution to the index is dominant for the case of the dual single-centered black hole. 8 Moreover, the authors showed that (5.1) is reproduced by the large-N index in this case upon the identification where ∆ a are the same chemical potentials (2.14) we use in the present work and d = 3 for the SYM case. Their observations hold under an assumption that the ∆s belong to the region analogous to the one we call "case I" through sections 3 and 4. In this case, [∆ a ] τ satisfy the constraint which is precisely the constraint that should be satisfied by X a . Another argument in favor of the identification (5.2), as shown in [45] for the case of SYM, is that X a should satisfy the same strip inequality as [∆ a ] τ do, namely

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Similar arguments should also work for black holes with d > 3 charges, which on the dual side correspond to theories with d global U(1) symmetries. We thus propose, based on the results of sections 3 and 4, the following entropy functions for black holes dual to toric theories: Here, the coefficients C IJK are defined by the triangular anomalies of the corresponding U(1) global symmetries, and the chemical potentials X a satisfy the constraint (5.3), Notice that this is exactly the proposal of [34] confirmed at the Cardy-like limit in [44].
Let us now briefly revise the procedure of extracting the entropy from the entropy functions S, giving a concrete illustration of the simplest case of N = 4 SYM. The black hole entropy is computed as the Legendre transform of the entropy function S, given by the following function: computed at the critical point Here, as usual Q I are the black hole's d electric charges and J i are its two angular momenta. We additionally introduce a Lagrange multiplier Λ to impose the constraint (5.6). At this point one should, in principle, perform the technically challenging task of solving (5.8) and substituting the solution into (5.7). However, ref. [33] proposed a nice trick to circumvent this complication and obtain the black hole entropy rather easily. Specifically, it was noticed that the entropy function (5.1) is a monomial of X so it can be rewritten in the following form In more general cases of toric theories, the entropy function (5.5) is a homogenous polynomial of degree three so the same argument applies. This relation together with the equations of motion (5.8) allow us to express the black hole entropy (5.7) in terms the Lagrange multiplier Λ alone, as follows: Now, the only thing left to do is to express Λ in terms of the charges Q I and angular momenta J i . For example, in the N = 4 SYM case the equations of motion (5.8) take the JHEP04(2020)091 the shift τ → τ + 2, from which it follows that we may restrict to 0 ≤ Reτ < 2. As we will later see, the choice (5.15) has a nice interpretation in terms of a dual black hole with equal charges Q I . The contribution of the basic solution and its T -transforms to the logarithm of the superconformal index reduces in this case to The definition of the different cases may be found in (3.18). To work with this expression, we must understand what the values of [∆] τ +r are for various r ∈ Z and which cases in (5.16) they correspond to. In particular, it can be checked that we should distinguish between cases of even and odd r as follows: It can also be checked that for even r, one should use "case I" in (5.16), while for odd r "case III" is the suitable one. Now we finally would like to understand the structure of the large-N index in the complex plane of the single parameter we are left with, namely τ . For this we need to understand which values of r maximize ImΘ (∆; τ + r) in the different regions. If in some region there is no r maximizing this function, one then needs to sum over infinitely many exponents contributing at the same order. This would require extra information and probably the inclusion of subleading terms to evaluate the index. That situation precisely corresponds to being on the Stokes line. In our case, the imaginary part of the Θ function (5. This is remarkably similar to the SYM result [45], with only two differences between the two cases. One difference is that in N = 4 SYM there are three possibilities with r = 0, 1, 2 mod 3. For r = 1 mod 3, [∆] τ is not defined since this value is precisely on the Stokes line. The second is that the expressions for r = 0 and r = 2 mod 3 exactly reproduce our expressions for even and odd r, correspondingly, up to an overall factor. However, this is irrelevant since this coefficient only defines the value of the index in the particular region but not the region itself. As illustrated in figure 11, there exists a maximum of ImΘ (∆; τ + r) given in (5.18) with an even r if τ resides in the interior of the semicircle defined by Reτ +r even > 3 |τ +r even | 2 .

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Substituting the above expression into the equations of motion (5.22) and assuming ω 1 = ω 2 = τ , we obtain that all electric charges are equal and that the two angular momenta are equal to each other as well: We are now able to use the usual trick and construct a simple cubic relation between the charges and the angular momenta: where p 0 = Q 3 + cJ 2 , p 1 = 3Q 2 − cJ , p 2 = 3Q + c , c ≡ 27 16 ν . (5.27) It is convenient to measure the charges in units of cν, so we introduce Q ≡ cνQ , J ≡ cνJ , Λ ≡ cνΛ. In this case all our expressions, except for the equations of motion (5.25), appear to be the same as in the case of N = 4 SYM in [45]. In particular, we obtain the following constraint for the charges and entropy 0 = p 1 p 2 − p 0 = c 3 ν 3 8Q 3 + 3Q 2 − 2 3Q + 1 J −J 2 , S = 2π √ p 1 = 2πcν 3Q 2 −J . Finally, we can also solve the equations of motion (5.25) which gives us the parametrization of τ and X: It is interesting to note that the solution for τ is exactly the same as for the case of N = 4 SYM [45]. Moreover, even the parametrization we use in (5.29) is the same as in the SYM case, so the expression appears to be exactly the same. The form of the solution for the chemical potential X is also identical to the one found in the case of N = 4 SYM, up to an overall coefficient only. It is interesting to see how these results look like in figure 11, where we show the structure of the large-N index dual to the entropy of a black hole with equal charges. Using the parametrization (5.29) for the solution (5.30), the line of critical points is shown in orange in figure 11. It is not surprising that the picture we obtain is identical to the one JHEP04(2020)091 obtained for N = 4 SYM in [45]. In particular, we see that for small values of µ below a certain critical value µ < µ * , the solution is in the region with no dominant contribution to the index. Hence, in this region we should take into account other solutions beyond the basic one to evaluate the index, and probably the leading order contribution will no longer be of order N 2 . In [45] it was proposed that such behavior signals the instability of small (i.e. with charge below some critical value, Q < Q * ) single-centered black holes towards other supergravity solutions. On the other hand, when we increase µ at some point it crosses the semicircle (5.19) and the solution we found becomes dominant. Therefore, large singlecentered black holes (with charge Q above the threshold Q * ) appear to be stable solutions of supergravity. The critical value of µ * can be found from the equation Reτ = 3|τ | 2 defining the boundary of the semicircle (5.19). Using the parametrization (5.29) together with the solution (5.30) we can define the following values of critical charges and entropy: In figure 11 we show the line of critical points in green. As can be seen, the form of the curve just mirrors the line of critical points for the case of r = 0. As we have mentioned repeatedly in this section, the picture we obtain for the case of equal black hole charges Q I is exactly identical to the one obtained in [45]. This result could have been expected since we consider a black hole with equal charges and angular momenta. This degenerate case should correspond to the so-called universal black hole [15,22], i.e. the solution of five-dimensional gauged supergravity that can be embedded in any AdS 5 × SE 5 background of type IIB supergravity. This universal black hole is then parametrized only by the central charge of the dual theory which we can clearly see in our derivations above. 10 In principle, the existence of such a universal black hole should suggest that for all the other toric theories the picture should be the same for a choice of the chemical potentials ∆ I corresponding to equal charges of the black hole. However, technically this problem appeared to be very complicated due to the structure of the equations of motion (5.8). It would be interesting to show that the large-N indices of the other toric theories analyzed in this paper also have a similar structure in the case of equal charges.

Discussion and outlook
In this paper we used a BAE analysis to obtain the large-N behavior of the superconformal index for a variety of toric gauge theories. We demonstrated that the basic solution (2.16) JHEP04(2020)091 to the BAEs of N = 4 SYM can be directly generalized to (2.21), which solves the BAEs for any toric quiver theory. Using these solutions, we derived the relatively simple expressions (2.38) and (2.39) for their contribution to the large-N limit of the superconformal index for arbitrary toric quivers. Studying these expressions for a number of theories, including the infinite families Y pq , X pq and L pqr , we found a way to rewrite them in a form that coincides with (1.2) for certain ranges of parameters. The latter is the result of the index computation in the Cardy-like limit [44]. Remarkably, this also reproduces a conjecture made in [34] for the entropy function of a multi-charged rotating AdS 5 black hole.
Finally, we also studied the large-N index structure for the conifold theory in the specific case corresponding to a dual black hole with four equal charges and two equal angular momenta. In this case, we identified Stokes lines and regions in the space of the single complex parameter τ , which is just the chemical potential dual to the angular momentum of the black hole. We further performed an extremization for this case. Curiously, our picture exactly reproduces the one found in the N = 4 case presented in [45].
Many related open questions are still waiting to be answered. First, our results are based only on the basic solutions (2.17) and their SL (2, Z) transforms. However, one should study closely the BAEs for particular theories and try to understand if there are other solutions, at which order of N they contribute to the index, and what the corresponding observables are in the bulk of AdS.
Another open question is the extremization procedure of the large-N index. In this paper, we presented results for the very particular case of T 1,1 with all the chemical potentials being equal. The authors of [44] managed to perform an extremization in the cases of the Y pp and L 0b0 theories. However, for all the other toric cases we have failed to perform a Legendre transform since the algorithms of [33] do not work for them and a straightforward solution is too complicated. It would be interesting to resolve these problems in one way or another.
Finally, it would be interesting to understand if the BAE approach can be used in order to study superconformal indices of larger classes of theories. Of particular interest are theories with matter in the fundamental or anti-fundamental representations. It looks like the basic solution does not work in this case and that one needs to perform a more complicated analysis of the corresponding BAEs.

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