The SUSY Index Beyond the Cardy Limit

We analyze a set of contributions to the superconformal index of 4d $\mathcal{N} = 4$ $SU(N)$ super Yang-Mills using the Bethe Ansatz approach. These contributions dominate at the large $N$ limit, where their leading order in $N$ reproduces various supersymmetric Euclidean black hole saddles in the dual theory, and they also dominate for finite $N$ in high temperature Cardy-like limits. We compute the $O(N^0)$ terms, including those exponentially suppressed in the Cardy limit, and show that there are no $1/N$ corrections beyond them. Under certain assumptions, it implies that the gravitational perturbative series around these black hole saddles is 1-loop exact.

C Special functions 19 1 Introduction The last few years have seen immense progress in the ability to compute the entropy of black holes and quantum gravity partition functions through their holographic dual conformal field theories, at least in supersymmetric cases.The pioneering works of [1][2][3][4] had succeeded in reproducing the leading contribution to the black hole partition function and reproduced the black hole entropy, while subsequent works had matched both perturbative and non-perturbative effects, in various dimensions and settings .
In this work, we will concentrate on the case of N = 4 SU (N ) super Yang-Mills theory.With a specific choice of chemical potentials, the partition function of this theory essentially becomes a protected quantity, the superconformal index, which is sensitive to 1/16-BPS states in the theory.On the other hand, the gravitational partition function of the dual theory can be computed in a semi-classical manner as a sum over the Euclidean saddles of type IIB supergravity on asymptotically AdS 5 ×S 5 with matching boundary conditions.
Previous works [2,3,37,49] had shown various ways in which information about different Euclidean gravitational saddles, describing black hole and orbifold geometries, can be extracted from the superconformal index -either by showing that at large N specific contributions to the index scale as these geometries, or by taking various "high temperature" Cardy-like limits where on the gravity side one of these geometries becomes dominant, and calculating the asymptotic behavior of the index in that limit.
In the present paper we concentrate on the field theory side, and compute a set of discrete contributions1 to the index using the Bethe Ansatz approach, at finite N and τ , where τ denotes one of the chemical potentials, to be defined below.We then show that in the large N limit the perturbative contribution of our solution in 1/N truncates at O(N 0 ), and explicitly compute these terms.Moreover, we compute the contribution at finite N and in the Cardy limit τ → 0, including O(e −1/τ ) terms up to O(e −N/τ ) and find that it takes a particularly compact form.Since the leading order contributions we compute match those of black hole saddles on the gravity side, we conjecture that the gravitational perturbation theory around these saddles truncates as well.
Given some supercharge Q, the superconformal index of the theory is where J 1,2 + 1 2 R 3 and q 1,2 are conserved charges that commute with Q, for further details see Appendix A. The index can be expressed as an integral over the holonomies {u i } of the SU (N ) gauge field around the thermal cycle.As reviewed in Section 2 and following [50], when τ = σ and after some manipulation this integral can be evaluated using the residue theorem.The residues are solutions to some complicated equations, (2.9), termed the Bethe Ansatz equations.Eventually, the index can be expressed as a sum over configurations of the holonomies, schematically where Z and κ come from the integrand and H is the Jacobian relating the integration variable to the Bethe Ansatz equations.Some residues might have a vanishing Jacobian, and their contribution to the index will be described in an upcoming paper [51].
A general classification of the solutions is still beyond reach, but there is a known family of solutions called the Hong-Liu solutions [52].They are given by symmetric configurations of the N holonomies u i on a lattice of modular parameter τ where N = m • n, r = 0, . . ., n − 1 and ū is chosen such that j u j = 0, as the holonomies are of an SU (N ) matrix.These configurations are sometimes denoted by the triplet {m, n, r}.Their contribution to the index is denoted by I {m,n,r} , and there could be additional contributions as well, The partition function of the theory on S 1 × S3 with a particular choice of chemical potentials is proportional to its superconformal index [53].Since the partition function should match that of the dual gravitational theory, [54] argued2 that in the large N limit each Hong-Liu configuration with n ∼ N and m, r ≪ n reproduces the action of a different Euclidean geometry, in the sense that the "action" of an HL configuration is where c {m,n,r} N 2 is the on-shell action of the corresponding gravitational saddle.These geometries describe (complex) Euclidean black holes and their orbifolds.The argument hinges on evaluating Z and κ in the large N limit, and bounding the contribution of the Jacobian H.Moreover, Z contains O(e −c D 3 N ) terms, corresponding to the on-shell action of additional saddles containing arbitrary number of wrapped supersymmetric D3 branes on top of these geometries.
In this paper we exactly compute the Jacobian H for the Hong-Liu solutions.When n = N → ∞, r ≪ n, m = 1, which is the large N limit corresponding to the various black hole saddles without orbifolds, we find that the series truncates 3 at order N 0 up to exponential corrections log I {1,N,r} We conjecture that this result carries over to the gravity side, meaning that the gravitational perturbative series truncates at order O(G 0 N ) and is one-loop exact around these saddles.Moreover, the new exponentially small corrections we have found are of the same order of magnitude as the exponential correction found before, and therefore can again be considered as arising from saddles with additional branes.
It should be stressed that what we actually compute in this paper is the contribution of a set of Bethe Ansatz solutions to the index, whose leading order behavior matches that of the gravitational saddles.We have not performed any new gravitational computation.However, these are the leading contributions to the index in various Cardy-like limits (see [55,56]).As long as there is no other BA contribution whose action differs from log I {1,N,r} by O(N 0 ), our conjecture should hold.Amongst the Hong-Liu family this is indeed the case.It seems likely that this is the case in general -any other BA configuration would differ by at least one of the eigenvalues u i , and thereby change 4N out of the 4N 2 terms in the relevant log Z, generically changing it by O(N ).
We are also able to analyze the theory in the Cardy limit τ → 0, in which the {1, N, 0} HL configuration dominates.Up to order O(e −1/τ ) our expression agrees with those of [7,8,18,55,56].The only corrections up to O(e −N/τ ) come from the Jacobian.The final expression takes the rather compact form4 where q = e −2πi/τ and ỹa = e 2πi[∆a]τ /τ .Exact details and definition of [∆] τ are given in Section 3. The final 1-loop determinant looks vaguely similar 5 to the multi-graviton partition function of [57], , where y a = e 2πi∆ 1 , q = e 2πiτ , but with the modular transformed fugacities and with a different power of the denominator.It would be interesting to understand if there's a deeper reason for the similarity.
These results raise several questions.The first is, of course, can they be verified by any of the other approaches used to analyze the superconformal index?A direct Cardy limit analysis seems quite hard.One would need to re-sum the exponentially suppressed terms to find the form (1.7).However, computing the first few exponentially suppressed terms in the Cardy limit would test our assumptions about the contribution of any other BA solutions.Another method is the large N saddle point analysis of the N = 4 index matrix model, initiated in [49], in which one of the saddles reproduces the exponential term in (1.7) for the equal angular velocities case.Our results imply that the perturbative expansion around this saddle, combined with terms of lower order in N in the potential used in [49], should also truncate.
A second, more ambitious question would be whether there is an argument from the gravity side for this apparent one loop exactness in G N , maybe through localization on the supergravity side [58][59][60][61][62]. Lastly, we note that a truncation of similar yet different spirit was found numerically in [63][64][65] for the topologically twisted index of the three dimensional ABJM theory and for its superconformal index in the Cardy limit.It would be nice to understand whether this is a generic feature of supersymmetric partition functions, or merely a coincidence.
The paper is organized as follows.Section 2 contains a brief overview of the Bethe Ansatz approach.Section 3 describes the explicit behavior of the Hong-Liu solutions in the large N and Cardy limits, up to terms exponentially suppressed in N .Appendix A includes definitions of the index and our conventions.Appendix B contains the explicit computation of the Hong-Liu contributions.Appendix C contains definitions of the special functions we use throughout the paper.

The Bethe ansatz approach
Here we review the computation of the superconformal index for N = 4 super Yang-Mills (SYM) using the so-called Bethe Ansatz approach.The theory is an SU (N ) superconformal gauge theory, with an SU (4) R R-symmetry, and when put on the cylinder R × S 3 it has an additional SO(4) symmetry due to the isometries of the underlying manifold.It has six real adjoint scalars in the 6 of the R-symmetry group and four adjoint Weyl fermions in its 4. Turning on chemical potentials for two elements of the Cartan which commute with a specific supercharge, and equal chemical potentials for the two angular momenta (see Appendix A for exact details) the superconformal index can be written in the integral form where T N −1 is the torus {∀i : , and u N = − N −1 i=1 u i so the u's can be thought of the eigenvalues of an N × N traceless matrix.We also use the fugacity q = e 2πiτ .The integrand and prefactor are (2. 2) The chemical potentials satisfy 2τ − 3 a=1 ∆ a ∈ Z.The function Γ is the elliptic Gamma function and (q; q) ∞ is the Pochhammer symbol, see Appendix C.
The Bethe Ansatz approach uses the quasi-periodicity of Γ, (C.16).There is an elliptic function 6  Qi ({u i }; ∆, τ ) with periods 1, τ for which where ũ = (u 1 , . . ., u i−1 , u i −τ, u i+1 , . . ., u N −1 , u N +τ ).Now comes the trick.Multiply both the numerator and the denominator of (2.1) by the same factor N −1 i=1 (1 − Qi ) to get 6 The explicit form of this function is where in the last line we've used the property (2.4) to change the contour integral into the new contour , where the sign denotes the direction of the contour.Evaluating the integral by the residue theorem, one finds that poles of the numerator are canceled by poles of the denominator, such that the only poles of the integrand come from zeroes of the denominator.A more thorough analysis will appear in an upcoming paper [51], but a simple sufficient condition is the vanishing of all the terms of the denominator Qi = 1 , ∀i = 1, . . ., N − 1 . (2.6) These are called 7 the Bethe Ansatz equations.If the solutions are isolated first order zeros of 1 − Qi and the Jacobian for the change of variables from {z i } to { Qi } is well defined, then the contribution to the integral is the inverse of the Jacobian times the residue at the pole (which is simply the numerator evaluated at the pole).Finally, the Bethe Ansatz approach to evaluating the index takes the form where the sum is over well-behaved solutions to the Bethe Ansatz equations, and the last term denotes contributions where the Jacobian vanishes.Such contributions will be discussed in [51].Here Z and I U (1) are defined as in (2.2), and H is the Jacobian, we note that I U (1) is the index of the free U (1) N = 4 theory.
In [67] it was argued that the contribution of many solutions to the Bethe Ansatz equations cancel each other, such that effectively only the solutions to the reduced Bethe Ansatz equations contribute to the index.There is a known family of solutions to these equations [52].
In order to find them, note that Q i is invariant to shifting any of the u's by 1 or by τ , meaning that the u's naturally live on a torus.The equations are then automatically solved if one distributes the u's symmetrically on the torus, meaning that for every u i and u j , there exists a u k such that u ij = −u ik on the torus. 8These solutions are termed the Hong-Liu solutions, whose explicit form is (1.3).

The Cardy and large N limits
The contribution of the {m, n, r} Hong-Liu solution to the index is computed in Appendix B, and turns out to be where η a = (1, 1, −1), q = e 2πiτ , the special functions are defined in Appendix C, and We compute log det 1 C by diagonalizing the matrix in Appendix B. The explicit result is given for the cases m = 1, where we find that the eigenvalues of the matrix B −1 C are practically the discrete Fourier transform of G(•, ∆ τ ; − 1 τ ), which is closely related to the logarithmic derivative of θ 0 .We comment on the m ≪ n case in the appendix.
Large n Let us look at the {m, n, r} configuration when n ∼ O(N ) → ∞ and m, r ≪ n.Then using the modular identities for the various special functions log θ 0 m ∆a ; ) where Q is defined in (C.21) and q = e −2πi/τ .
If one shifts the ∆'s by integers to the regime − Im 1 τ > Im m ∆a τ > 0 before performing the modular transformations then the special functions on the right hand side above are exponentially suppressed in n.We denote The special functions are now of order O(q n , ỹn a , (q 2 /ỹ a ) n ) where10 ỹa = e 2πi[m∆a] τ /τ .We now have to split into two cases, as in [2,54]: In order to compute the b N term in (3.1) up to exponential accuracy one needs to shift its second argument using (C.27), which changes where the upper sign is for the first case and the lower for the second.For the first case where The determinant was evaluated in (B.24), and using q = ỹ1 ỹ2 ỹ3 where the corrections are of order O(ỹ n a , (q/ỹ a ) n ).Overall, the large N limit of the {1, N, r} Hong-Liu solution is For the second case, it is convenient to define where We find it convenient to define the fugacities xa = e −2πi[∆a] ′ τ = q/ỹ a which satisfy x1 x2 x3 = q and |q| < |x a | < 1. Picking the bottom sign in (B.24) leaves us with the final expression for the deteminant in the second case where the corrections are of order O(x n a , (q/x a ) n ).Finally, for the second case the contribution of the {1, N, r} Hong-Liu solution to the index is (3.12) The Cardy limit In the Cardy limit τ → 0 the dominant contribution to the index comes from the {m, n, r} = {1, N, 0} solution [2,12,18], where the holonomies approach u i = 0.In order to get a good expansion 12 in 1/τ one should use the modular transformations (3.3) on I U (1) as well as on the rest of (3.1), ) so the index in the Cardy limit is log 3 the expression agrees with [7,8,18,55,56].Specifically, note that the prefactor N 2 changes in this limit to N 2 − 1 and that the log τ term cancels out, both are necessary implications of the EFT argument of [55].But we can do better.Given the exact expression for the determinant term (B.24) and for the special functions involved, we can systematically write down the leading corrections to I {1,N,0} .The rather simple expression for the first case is a , qN , (q/ỹ a ) N (3.15) while for the second case it is ) where ỹa = e 2πi[∆]a/τ , q = e −2πi/τ , x a = q/ỹ a .These formulas receive corrections at order O(x N , ỹN a , qN ).Note that e πi 12 τ I U (1) changes the exponent to N 2 − 1 and adds O(q, ỹa ) terms to the action, so it should also be considered if one wants to expand the order O(e −1/τ ) terms in the Cardy limit order by order.I U (1) is defined in (2.2).
Assuming no other Bethe Ansatz configuration contributes in this limit, these are also the leading asymptotics of the index itself.This assumption applies for all other HL configurations, as they are smaller by O(e −N 2 /τ 2 ).We assume that in the Cardy limit other contributions are also suppressed by at least O(e −N/τ ).
Acknowledgment The author thanks Erez Urbach and Tal Sheaffer for useful discussions, and is grateful to Francesco Benini and Ofer Aharony for many useful conversations and for comments on the manuscript.The work of the author was supported in part by an Israel Science Foundation (ISF) center for excellence grant (grant number 2289/18), by ISF grant no.2159/22, by Simons Foundation grant 994296 (Simons Collaboration on Confinement and QCD Strings), by grant no.2018068 from the United States-Israel Bi-national Science Foundation (BSF), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant "Holography and the Swampland", and by a research grant from Martin Eisenstein.The author would like to thank SISSA for its hospitality, funded by ERC-COG grant NP-QFT No. 864583.The author also thanks the Simons Center for Geometry and Physics, Stony Brook University for organizing the "Supersymmetric Black Holes, Holography and Microstate Counting" workshop which helped facilitate this paper.

A The superconformal index of N = 4 SYM
Using N = 1 language, N = 4 SYM theory consists of a vector multiplet and three chiral multiplets in the adjoint representation.Its R-symmetry is SU (4) R , whose Cartan is U (1) 3 .We pick generators R 1,2,3 , each giving R-charge 2 to a single chiral multiplet and zero to the other two, in a symmetric way.Local operators in the theory (and likewise states on S 3 ) are labeled by two half-integer angular momenta J 1,2 , each rotating an R 2 ⊂ R 4 in which S 3 is embedded.The fermion number is defined as F = 2J 1 .Note that all fields in the theory (and thus all states) have integer charges under R 1,2,3 and obey The superconformal index of N = 4 SYM theory counts, with sign, 1/16-BPS states on S 3 preserving one complex supercharge Q, which we choose to be associated with a specific U (1) R symmetry the generator r = 1 3 (R 1 + R 2 + R 3 ).The index can also keep track of some combinations of the R-charges and the two angular momenta J 1,2 .It is useful to introduce two flavor generators q 1,2 = 1 2 (R 1,2 − R 3 ) that commute with the supercharge and with r.The superconformal index [57,68] is then defined , and thus summation over regular shifts along the two cycles of the (1, τ ) torus is given by 14 The argument of the resulting elliptic Gamma function can be shifted, (B.5) 14 One can check that the series expansions are valid for some domain for the arguments, and then analytically continued to generic argument.For the first identity the domain is 0 < Im( ∆) < 2 Im(τ ), while for the second the domain is 0 Overall, for the {m, n, r} HL solution and when ∆ ̸ = 0 we find where τ = mτ + r.The limit ∆ → 0 can now be taken as well.This is most conveniently done by expressing the elliptic Gamma and theta functions as an infinite product Finally, we can put it all together to compute the total contribution Z with q = e 2πiτ Z u {m,n,r} , ∆1 , ∆2 , τ where we have denoted ∆3 = ∆1 + ∆2 and η a = (1, 1, −1).

B.2 The Jacobian H
The Jacobian can be written as (see appendix B.3 in [54]) where (B.12) We will now compute log det 1 Only the expression for m = 1 will be reproduced here 15 .However, we will restrict to m = 1 only when needed, at (B.21).Let's begin.Using (B.26) and (B.29) in [54] Υ + O ỹN a , (q/ỹ a ) N where the corrections are exponentially small in N .Moreover 17 The Fourier transform of the discrete convolution is where the upper sign is for the first case (3.5) and the lower for the second.Note that all the exponents here are of ∆a τ , like the combination that match the D-branes that wrap an S 3 ⊂ S 5 , but without the corresponding factor of N .We can now evaluate the the determinant and find that there are no perturbative corrections in 1/N .

C Special functions
Throughout we will sometimes use 18q = e 2iπτ , p = e 2πiσ , z = e 2πiu (C.1) q-Pochhammer symbol The q-Pochhamemer symbol is There are also series expansion and Plethystic representation for (z; q) ∞ , The elliptic Gamma function The elliptic Gamma function is defined by This definition gives a meromorphic single value function on |p|, |q| < 1 with simple zeroes at z = p m+1 q n+1 and simple poles at z = p −m q −n for m, n ≥ 0. The infinite product is convergent on the whole domain.We can also give a plethystic definition (C.28)The branch of the logarithm is determined by its series expansion log(1 − z) = − ∞ ℓ=1 z ℓ /ℓ, whereas Li 2 (z) = ∞ ℓ=1 z ℓ /ℓ 2 is the dilogarithm.One can show that the branch cut discontinuities of the logarithm and the dilogarithm cancel in the definition of ψ(t), therefore the latter extends to a meromorphic function on the whole complex plane.Some useful properties of ψ(t) are:

N limits 7 A 11 B
The superconformal index of N = 4 SYM Simplifying the Hong-Liu contribution 13 B.1 Computing Z 14 B.2 The Jacobian H 16