Microstate counting via Bethe Ansätze in the 4d N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 superconformal index

We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe Ansatz approach recently applied by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large N limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of N D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and Yp,q quiver gauge theories. For a suitable choice of the chemical potentials of the theory we find agreement with predictions made for the same theories in the Cardy-like limit. However, for other regions of the domain of chemical potentials the superconformal index is modified and consequently the associated black hole entropy receives corrections. We work out explicitly the simple case of the conifold theory.


Introduction
The understanding of the quantum microstates responsible for the entropy of black holes has long been one of the central questions in the path to a quantum theory of gravity. In the context of the AdS/CFT correspondence it has recently been shown that the entropy of certain asymptotically AdS 4 black holes admits a microscopic explanation in terms of a topologically twisted field theory [1] (see [2,3] for reviews with extensive lists of references).
More recently, the question of microstates for asymptotically AdS 5 black holes dual to N = 4 supersymmetric Yang-Mills (SYM), which was originally tackled in [4], has been revisited providing a microscopic entropy matching using various approaches. A broader interpretation of localization was successfully put forward in [5] while an analysis of the free-field partition function in a particular limit led to the entropy in [6] (see also [7]). Both these groups relied on a particular Cardy-like limit to evaluate the path integral. Another approach, put forward by Benini and Milan in [8], attacked the superconformal index using a Bethe Ansatz approach developed in [9]. Understanding that the superconformal index can be written as a sum over solutions to Bethe Ansatz equations was demonstrated in [10] based on interesting relations between observables on manifolds of different topologies developed in [11]. One key advantage of the Bethe Ansatz approach is that it does not require taking the Cardy limit and thus opens the door for a more in-depth understanding of the superconformal index. In this brief note we simply generalize the large N results obtained for N = 4 SYM using the Bethe Ansatz approach to a large class of N = 1 4d supersymmetric field theories.
Other recent studies demonstrating that the Cardy-like limit of the superconformal index of 4d N = 4 SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographically dual AdS 5 rotating black holes were JHEP03(2020)088 presented in [12,13]. Such analysis has by now been extended to generic N = 1 supersymmetric gauge theories [14,15] including a particular description specialized to arbitrary N = 1 toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data [16]. These powerful results rest on systematic studies of the Cardy limit developed in, for example, [17][18][19][20][21].
In this note we verify that a class of holonomies of the form u i − u j = τ N (i − j), used prominently in [8] for the case of N = 4 SYM, can be generalized to evaluate the superconformal index of generic N = 1 four-dimensional superconformal field theories.
The rest of the note is organized as follows. In section 2 we show that a particular class of holonomies solves the Bethe Ansatz equation for generic 4d N = 1 gauge theories and proceed to evaluate the superconformal index in the large N limit. Section 3 works out explicitly the index for a number of superconformal field theories. We find that there is always a way of redefining the chemical potentials suitably, such that the superconformal index obtained reproduces successfully the entropy of the dual AdS 5 black holes upon extremization of its Legendre transform. We also focus on the conifold theory in which the simplicity of the superconformal index allows us to study it for some region of the domain of chemical potentials that can provide a black hole entropy with corrections purely depending on the angular velocity τ . We conclude in section 5.
Note added. After this manuscript was originally submitted to the arxiv we received [22] with a considerable overlap with this work. The authors of [22] perform a more exhaustive analysis of the behavior of the entropy function for different regions in the domain of complex chemical potentials. The present version of this manuscript contains substantial changes with respect to the first two versions appearing in arxiv. We have essentially found that, selecting a set of chemical potentials that ensures an optimal obstruction of cancellations between bosonic and fermionic contributions to the superconformal index, one can always find a region of chemical potentials where the index accounts for the black hole entropy. This resolves an apparent tension between our conclusions and the ones subsequently reported in [22].

Bethe Ansatz approach to the superconformal index
In this section we generalize the solutions to the Bethe Ansatz type equations proposed in [8][9][10] to evaluate the superconformal index of N = 4 SYM to generic 4d N = 1 supersymmetric gauge theories. For concreteness we will work in the context of toric quiver gauge theories which are naturally decorated with extra global and baryonic symmetries but the results apply more generally to 4d N = 1 supersymmetric gauge theories.
Consider a generic N = 1 theory with semi-simple gauge group G, flavor symmetry G F and non-anomalous U (1) R R-symmetry. The matter content of this theory is taken to be n χ chiral multiplets Φ a in representations R a of G, with flavor weights ω a in some representation R F of G F and superconformal R-charge r a . Let us start by introducing the following quantities which are related to global fugacities and holonomies in the Cartan of JHEP03(2020)088 the gauge group: and the R-charge chemical potential which is fixed by supersymmetry to: With the above data, the integral representation for the superconformal index can be written as [23,24]: 3) The integration variables z i parameterize the maximal torus of the gauge group G and the integration contour is the product of rk (G) unit circles. Following standard notation, ρ a are the weights of the representation R a , α parameterize the roots of G and |W G | is the order of the Weyl group. The notation adopted also denotes z ρa ≡ The other functions involved in the expression for the superconformal index are the Elliptic Gamma function 4) and the q-Pochhamer symbol An interesting result of [9] and [8], based on [10], is to rewrite the above superconformal index in terms of solutions to certain Bethe Ansatz like system of equations taking the generic form of where ω is such that rτ = sσ with r and s coprime integer numbers (in practice we will evaluate the equations for r = s). Furthermore, the "Bethe Ansatz operator" is defined as: where P (u; ω) = e −πi u 2 ω +πiu θ 0 (u; ω) . (2.8) Thus, θ 0 (ρ a (u) + ω a (ξ) + r a ν R ; ω) , (2.9)

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where: θ 0 (u; ω) = e 2πiu ; e 2πiω ∞ e 2πi(ω−u) ; e 2πiω ∞ . (2.10) Now we would like to evaluate the Bethe Ansatz equations for the case of a toric quiver gauge theory. Toric quiver gauge theories describe the low energy dynamics of a stack of N D3 branes probing the tip of a toric Calabi-Yau singularity; there is by now a vast literature detailing how to construct a supersymmetric field theory given toric data (see, for example, [25,26]). Consider a toric quiver gauge theory whose gauge group G has n v simple factors (in all the N = 1 quiver gauge theories we will deal with, the number of simple factors coincides with the number of vector multiplets). We focus, for concreteness, on the case in which all the gauge group factors are SU(N a ), a goes from 1 to n v , with N a = N ∀ a, the same numerical value for all nodes. In these theories the weight vectors ρ are such that for any bi-fundamental field Φ ab (notice that in the more generic notation used in [9], the index a of Φ a would now split into ab): Let us now evaluate the operator P (u; ω) for a generic field Φ ab (when Φ ab transforms in the adjoint representation of G then, in this notation, a = b): where (a, b) run over all the fields Φ ab for a fixed a and r ab are the R-charges of the fields Φ ab . The d − 1 fugacities correspond to the flavor symmetries appearing in the generic toric gauge theories that we will study, d is the number of external points of the toric diagram that are related to the quivers defining the theory [16]. If we denote a, b ≡ (a, b)| ρ (a,b) ij >0 , which implies: Let us now introduce a Lagrange multiplier λ a that accounts for the constraint ensuring the condition i u a i = 0 [8], with its help, equation (2.13) can be written as: (2.14)

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where we have denoted u ia − u j b ≡ u ab ij . Restricting ourselves to the case with τ = σ, we would like to propose a set of u ab ij that makes (2.14) equal to 1, thus solving the Bethe Ansatz equation (2.6). It is natural to make an attempt with a direct generalization of the type of solution encountered in [8], namely: u ab ij = τ N (i a − j b ). These solutions appeared first in [27] while evaluating the topologically twisted of 4d N = 1 theories on T 2 × S 2 in the high temperature limit; it was later shown in [28] that such configuration provides an exact solution to the Bethe Ansatz equations.
Consider one generic factor entering in (2.14) for a fixed value of b: In (2.15) we have used the following properties of the θ 0 function: and for the sake of compactness we have absorbed all the factors independent of i a in the function F (∆ ab , ∆ ba , τ ). Inserting (2.15) back into (2.14) leads to multiplying all the results obtained in (2.15) for all n a values of b connected with a via some field Φ ab :

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Upon a proper choice for the Lagrange multipliers we can ensure that:

Evaluation of the index
The formula for the superconformal index in terms of solutions to the Bethe Ansatz like equations reads [9,10]: We assume that dominant contributions to the index in the large N limit will come from terms analogous to those dominating the expression obtained in [8] for the N = 4 SYM theory. This implies that in order to investigate the large N limit of (2.19), we only need to consider the following term: Note that, the leading contribution coming from the vector multiplets can be obtained from (2.20) by setting ∆ ab = 0. In the large N limit we can write: As a clarifying example, let us now analyze the case of N = 4 SYM theory already studied in [8] and peroform the same calculation using the toric data language of [16]. The corresponding Φ ab are the three chiral fields Φ 1,2,3 appearing in the superpotential:

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with the associated chemical potentials being ∆ 1,2,3 . Accordinng to our definition of the chemical potentials we have that, for the R-charge assignment used in [8]: Using the identity: reduces (2.21) to the following expression: where [∆ ab ] τ is defined such that [∆] τ = ∆ mod 1 [8] and depends on the region withing the domain of complex chemical potentials one is evaluating (for a more detailed description of this function see also [22]). If |∆ ab | < 1, then: which is indeed the necessary structure in order for the superconformal index of N = 4 SYM to account for the entropy of the dual AdS 5 black hole [8].
Before proceeding to generic toric quiver gauge theories, let us comment on the choice of R-charge assignment, since one might expect a more symmetric one based on a-maximization. We notice that, if one chooses a set of chemical potentials and R-charges as the one used in [16], namely where r 1 = r 2 = r 3 = 2 3 , in contrast with the choice r 1 = r 2 = 0, r 3 = 2, then the use of identity (2.24) is not directly possible. This means that, if one starts with the data suggested by a-maximization [16] (r 1 = r 2 = r 3 = 2 3 ), then (2.26) should be understood in terms of shifted chemical potentials that would permit some of the arguments of the Elliptic Gamma functions in (2.21) to have the structure JHEP03(2020)088 ∆ + 2τ as needed in (2.24). Specifically, we have: We can either interpret this as a suitable redefinition of the chemical potentials wich does not affect the physical R-charge obtained via a-maximization or rather as a computation done directly with the more naive R-charge assignment used in [8]. Let us now explore more generically the consequences of shifting ∆ ab in such a way that the arguments of the elliptic Gamma functions in (2.21) look either like ∆ or ∆ + 2τ . Suppose we do such a shift obtaining that a certain number, let us call this number n s , of the total of n χ chiral fields contributions to (2.21) are of the form ∆ + 2τ . Thus, the leading contribution in N to log I takes the form: where [∆ ab ] τ is defined such that [∆] τ = ∆ mod 1 [8], the sum v is carried over the n v vector multiplets and n χ is the number of chiral fields, s ab is 1 if Φ ab effectively has Rcharge 0 and −1 if it has R− charge 2 with a new set of chemical potentials. Conservation of U(1) charges implies Φ ab [∆ ab ] τ = 0, which allows us to eliminate every linear term in [∆ ab ] τ appearing in (2.28), therefore we can write: where we have defined
Let us now analyze the properties of the function we have obtained. Equation (2.29) is very similar to the one obtained in [16] when analyzed in the Cardy-like limit of the index, however, there is an extra contribution of the form iπN 2 3τ 2 (n χ − 2n s − n v ) τ τ − 1 2 (τ − 1) which is still of order O(N 2 ) but sub-leading when τ → 0. Notice that at this point there is no dependence on the holonomies of the gauge groups since we have already evaluated in the solutions of the Bethe Ansatz equations. We still need to determine if we can find a consistent way of redefining the chemical potentials, thus fixing the value of n s and s ab . The shifting has to preserve the R− charge of the superpotential which is ensured by the constrain: where A denotes monomial terms of the superpotential W . Let us call n F the number of elements in A. Using the fact that for these toric quiver gauge theories each chiral field appears only once in exactly two terms in the superpotential, then (2.32) implies that 2n s = n F . To gain a better understanding of the implications that shifting the chemical potentials has on the superconformal index of a toric quiver gauge theory let us consider: where we have used the same basis for the non R-global symmetries used in [16]. Shifting the chemical potentials as allows us to rewrite the index as and identifying e 2πiQ d = (−1) F [16] allows us to express the superconformal index in such a way that bosonic-fermionic cancellations are optimally obstructed [6]:

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The shifting (2.34) which is dictated by the geometry of the toric diagram, in particular by its number of vertices, turns out to be the adequate one in order to reproduce the dual black hole entropy.
Finally, recalling that we are dealing with toric quivers, which can be drawn on a torus providing a polygonalization of the torus [29] , and n χ is the number of edges n E of the graph, n F is associated to the number of faces and n v to the number of vertices then, the last term in (2.30) vanishes due to the Euler relation, n E − n F − n v = 0: Defining ∆ d such that: d I=1 ∆ I −2τ = −1 [8], it can be shown that log I can be writen as: The coefficients C IJK in (2.39) correspond, as pointed out originally in [30] and later in [16], to the Chern-Simons couplings of the holographic dual gravitational description as elucidated in [31]. In the following section we proceed to evaluate the superconformal index for various models, some of them recently discussed in a similar context in [16], and compare our results with (2.39).

The superconformal index of various SCFT's
We will apply our general result (2.39) in various cases in each of which we follow the prescription of charge assignment used in [16]. Indeed, below we will see that in order to obtain (2.39) all the chemical potentials have to be shifted by − 2τ d , exactly like [16]. We will restrict ourselves to the regime of chemical potentials ∆ i of the d − 1 U(1) global symmetries such that: which is inside the fundamental domain: which in our case will be useful to evaluate the function K(∆, τ ) using equation (2.30). The region (3.2) has been highlighted in figure 1 in grey. This regime also coincides with the one in which the existence of a universal saddle point in which all the holonomies vanish according to the analysis carried in [16], can be ensured.

The conifold theory
We would like to study the index in the large N limit and thus investigate it beyond the Cardy-like limit. To do so we start with one of the simplest examples of toric quiver gauge theories -the conifold theory [32] whose quiver diagram is given below. We take the ranks of all the gauge groups equal (N 1 = N 2 = N ) and the sub-index in N i helps describe the representations of the matter fields: The superpotential is The global charges of the conformal field theory are: a U(1) R factor, two SU(2) factors and finally there is a U(1) B baryonic symmetry. A fascinating fact about this theory is that it admits a gravity dual in terms of strings in AdS 5 × T 1,1 . The isometries of T 1,1 realize the mesonic symmetries of the field theory in terms of the isometries of CP 1 × CP 1 ; the U(1) B baryonic symmetry is associated to the unique non-trivial three-cycle of the geometry. It is worth pointing out that the rotating electrically charged black holes dual to the superconformal index have not yet been constructed on the supergravity side, and that remains an outstanding problem.

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We use the basis for the charges suggested by the toric diagram discussed in [16] and we summarize them in the following table: After performing the shifting ∆ 1,2,3 → ∆ 1,2,3 − τ 2 , we are ready to evaluate equation (2.38): After imposing the condition d I=1 ∆ I − 2τ = −1 yields: We see that log I presents the behavior proposed in (2.39).

The suspended pinch point
The suspended pinch point (SPP) gauge theory corresponds to the near horizon limit of a stack of N D3 branes probing the tip of the conical singularity, x 2 y = wz. The SPP gauge theory is described by the following quiver All the ranks are taken to be the same with N 1 = N 2 = N 3 = N and the sub-indices are meant to help understand the representation properties of the matter fields. The superpotential is (3.6) Each X ij transforms in the N representation of the index i-th node and in the N of the j-th node. The field φ transforms in the adjoint representation of the corresponding gauge JHEP03(2020)088 group. The charge assignment for the U(1) R and the extra U(1) i global symmetries can be taken as: We shift now the chemical potentials ∆ 1,2,3,4 → ∆ 1,2,3,4 − 2τ 5 . The next step is to use this information and perform the evaluation (2.38).
]. Now we use: 5 I=1 ∆ I − 2τ = −1 we introduce a fifth fugacity ∆ 5 that permits us to rewrite (3.7) in the following, more symmetric, way: This result is in agreement with equation (2.39) which is what is expected from toric geometry and reinforces the validity of the analysis of [16] which was limited to the Cardy-Like limit.

The dP 1 theory
We consider now the theory arising from a stack of N D3 branes at the tip of the complex Calabi-Yau cone whose base is the first del Pezzo surface. The quiver associated to this theory is:

Y p,q quiver gauge theories
The Y pq model corresponds to quiver gauge theories with 2p gauge groups and a chiral field content of bifundamental fields. The charge assignment and the corresponding multiplicity JHEP03(2020)088 of the fields are shown below: We proceed to perform the shifting of chemical potentials as follows: ∆ 1,2,3 → ∆ 1,2,3 − τ 2 . Now we evaluate the leading, order O(N 2 ), part of the superconformal index (2.38): Finally we eliminate τ from (3.13) using 4 I=1 ∆ I − 2τ = −1, which successfully reproduce the structure of (2.39): (3.14) 4 Corrections to the dual black hole entropy of the conifold theory Thus far we have been focused on a specific region of chemical potentials that permited us to simplify all the computations associated with the function [∆] τ . Ultimately, it is necessary to identify the quantity log I with the corresponding entropy function for the dual black hole gravity solution. The prototypical example is provided by N = 4 SYM, as discussed in [8], where the identification states: provided ∆ I ⇐⇒ X I . More generically, [∆ I ] τ ⇐⇒ X I where τ is the chemical potential associated to the two equal angular momenta J 1 = J 2 = J and X I are the chemical potentials associated to the cherges Q I . As discussed in [8], the identification above is valid provided X I is within the same analyticity domain that [∆] τ . This is the case, since both belong to the domain specified by (3.2), and they satisfy the constraint:

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The next step would be to obtain the black hole entropy by extremizing the Legendre transform of log I with the appropriate constraint: where Λ is a Lagrange multiplier imposing the constraint. The black hole entropy can be thus obtained exactly as done in [5,16]. In particular the result obtained in [22], which appeared after the first version of our manuscript, confirms our results. Going away from the region of chemical potentials we have been restricting ourselves to so far would produce some modifications in the result.The technical reason being that, in different regions of the domain of complex chemical potentials, the functions [ I q I ∆ I ] τ and I q I [∆ I ] τ do not agree. The structure (2.39) can be lost either by spoiling the homogeneity of log I with respect to ∆ I and τ or by failing to completely cancel the term that only depends on τ and comes from the contribution of vector multiplets. Homogeneity in ∆ I and τ is crucial to perform the extremization procedure, whereas the appearance of an extra term exclusively dependent on τ could be easily incorporated in order to explore possible modifications to black the hole entropy. In the particular case of the conifold theory that we studied in section 3.1 we have that, for chemical potentials in the region: therefore, the superconformal index takes the following form: Notice that the appearance of a contribution that only depends on τ in this case is related to the specific details of how the function [· · · ] τ behaves in the different domains of chemical potentials. The case of N = 4 SYM is special because the function log I ∼ ∆ 1 ∆ 2 ∆ 3 is quite simple and consists only of one term. For this reason one could hope to eliminate all contributions depending only on τ by modifying the constraint obeyed by the chemical potentials d I=1 ∆ I − 2τ = −1 → d I=1 ∆ I − 2τ = 1. This is, indeed, the case verified in [8]. However, for more complicated theories where log I has a more than one term dictated by the anomaly coefficients C IJK , the extra pice persists as we see in (4.5).

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Let us now investigate how this extra term modifies the entropy obtained by taking the Legendre transform of log I. Our starting point it to organize the computation as to maximally take advantage of the scaling properties of S E , which implies that now we propose the identification [∆ I ] τ + 1 2 ⇐⇒ X I within the region (4.4). Notice that now the constraint is modified as: Even though the constraint (4.6) has been modified when identifying [∆ I ] τ + 1 2 ⇐⇒ X I we notice that the new constraint corresponds precisely to the other possible choice of relation among the chemical potentials as discussed in [12,14]. If the chemical potentials are in the region (4.4), then: Note that this region does not coincides precisely with the fundamental domain over which the X I are defined (3.2), however there is a non-empty intersection between the two as we illustrate in figure 2. Thus we write: Since S τ is independent of X I we have: The function we need to extremize now is the following: The extremization condition implies: The homogeneity of S E leads to the important relation: Following [16], we insert (4.12) in (4.9) and evaluating on the extremization solutions we find:

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For a particular set of values of X I , the properties of S E allow us to reconstruct (S E ) 2 from suitable combinations of products of its derivatives with respect to X I which generically leads to a cubic equation to determine Λ(Q, J). If we choose X 1 = X 3 , then S E for the conifold theory coincides with S E for the Y p,p theory described in [16], namely: (4.14) Now we can follow the extremization procedure put forward in [16] keeping track of the correction S τ when taking Q 3 → Q 1 for the Y 1,1 quiver gauge theory. It can be shown that S E satisfies: Using equation (4.9), we can obtain a cubic equation for Λ in the same spirit as [16] and also to keep track of the modification produced in the entropy by the presence of S τ in (2.39), hence, we have: Equation (4.16) can be written as: Demanding the condition p 0 = p 1 p 2 , (4.18) the assumption of real charges in [5,16] led to purely imaginary values of Λ and therefore to a real entropy. We need to be more careful since (4.16) is a modified version of the one appearing in [16]. The modifications enter through J and iπN 2 τ (J) (τ (J) − 1) in (4.13). Let us still demand the condition (4.18), which a priory do not ensure real entropy but gives a simplified enough expression that we can work with. Reality of the entropy then would impose that, separately, the correction was a real number: JHEP03(2020)088 of 1/N corrections will naturally translate into interesting aspects in the dual quantum gravity side for AdS 5 black holes. For example, the statistical entropy of certain magnetically charged AdS 4 black holes has recently been given a microscopic explanation in terms of the topologically twisted index [1] (see [2,3] for a reviews with comprehensive lists of references). The investigation of sub-leading (logarithmic in N ) corrections such as those performed recently [35,36] have helped clarify the nature of the degrees of freedom on the gravitational side of the duality. One would hope for similar developments in the context of AdS 5 black holes. There are many other interesting open problems. At the technical level, it would be interesting to generalize the Bethe Ansatz approach to arbitrary fugacities such that a general expression depending on both angular momenta can be achieved. There is little doubt that such generalization will yield the expected results but it will clarify the inner workings of the evaluation of the superconformal index. In this manuscript we have completely avoided the subtle discussion concerning the space of solutions of the Bethe Ansatz equations, we limited ourselves to just one class and showed that it yields a contribution sufficient to extract the dual black hole entropy and its potential corrections in the appropriate domain of chemical potentials. It would be very illuminating to have a better understanding of all the solutions and how one should weight their contributions to the index.
Finally, it is an important open problem to construct explicitly the black holes dual to the field theories discussed in this manuscript. Our computation, as well as those in a number of recent publications [14][15][16], show that it is relatively easy to find the superconformal index in a large class of supersymmetric four-dimensional field theories some of which have known supergravity dual. Moreover, using the entropy formula one can evaluate the entropy and realize that it corresponds to that of large black holes in AdS 5 . However, the explicit black hole construction on the gravity side is still in its infancy, not much is known beyond the AdS 5 black holes dual to N = 4 SYM (and some of its orbifolds). It remains an outstanding challenge for the supergravity community to explicitly construct rotating electrically charged black holes which could be understood as dual of available field theory results. One particular example that comes to mind among the class discussed in this note would be the black holes in asymptotically AdS 5 × T 1,1 and, more generally, AdS 5 × Y p,q .