Supersymmetric phases of 4d N=4 SYM at large N

We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N) N=4 super Yang-Mills theory on $S^3 \times S^1$ with one chemical potential $\tau$. The saddle-point configurations are labelled by points $(m,n)$ on the lattice $\Lambda_\tau= \mathbb{Z} \tau +\mathbb{Z}$ with $\text{gcd}(m,n)=1$. The eigenvalues at a given saddle are uniformly distributed along a string winding $(m,n)$ times along the $(A,B)$ cycles of the torus $\mathbb{C}/\Lambda_\tau$. The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and the related Bloch formula allows us to calculate the action at the saddle-points in terms of real-analytic Eisenstein series. The actions of $(0,1)$ and $(1,0)$ agree with that of pure AdS$_5$ and the supersymmetric AdS$_5$ black hole, respectively. The black hole saddle dominates the canonical ensemble when $\tau$ is close to the origin, and there are new saddles that dominate when $\tau$ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit $\tau\to 0$.


Introduction
The subject of this paper is a matrix model defined by an integral over N × N unitary matrices U of the following type, 1 n f (n τ ) tr U n tr (U † ) n . (1.1) Here τ denotes the vector of coupling constants of the model, and DU is the invariant measure. Upon diagonalizing the matrices, one obtains an integral over the N eigenvalues which experience a purely two-particle interaction governed by the single function f (τ ).
The particular model that we analyze arises as the generating function of the superconformal index [1,2] of N = 4 super Yang-Mills (SYM) theory on S 3 × S 1 . In this model U has the interpretation as the holonomy of the gauge field around S 1 , and there are three coupling constants τ , σ, ϕ, which are interpreted, respectively, as the chemical potentials dual to the two angular momenta on the S 3 and a certain combination of the R-charges of the theory. The function f has the interpretation as the single-letter index of the super Yang-Mills theory [1]. The matrix model (1.1) for the supersymmetric index has generated a lot of interest since the work of [2]. The holographically dual gravitational theory with AdS 5 boundary conditions admits supersymmetric black hole solutions which preserve the same supercharges as the superconformal index I captured by the matrix model. This leads to the expectation that the (indexed) degeneracies of states contained in I grow exponentially as a function of the charges when the charges become large enough to form a black hole. The analysis of [2] seemed to suggest that the shape, or functional form, of I does not allow for such a growth of states, thus leading to a long-standing puzzle.
Following the insightful observation of [3], this puzzle has been revisited recently [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] The emergent picture shows that there is actually an exponential growth of states contained in the index, which is captured by the saddle-point estimate if one allows the potentials σ, τ , ϕ to take values in the complex plane away from the pure imaginary values assumed in [2]. 1 There are essentially three strands of analysis contained in the recent progress: 1. The first strand is an analysis of the bulk supergravity [4] which shows that the black hole entropy can be understood as the Legendre transform of the regularized on-shell action of a family of Euclidean solutions lying on the surface σ + τ − ϕ = n 0 , n 0 = −1 . (1. 2) It was first noticed in [3] that this constraint is important to obtain the black hole entropy as an extremization of an entropy function. The result of [4] gives a physical interpretation to this observation-the entropy function is the gravitational action, and the constraint arises from demanding supersymmetry. One is thus led to look at configurations on this surface in the field theory index in order to make accurate saddle-points estimates.
2. The second strand, referred to as the Cardy-like limit [5,10,11,13,14] is the analysis of the index I in the limit when the charges are much larger than all other parameters of the theory including N . In the bulk gravitational dual, this corresponds to a black hole that "fills up all of AdS space". In this limit one finds a configuration of eigenvalues clumped near the origin which has an entropy equal to that of the black hole. This analysis has some subtleties related to convergence of infinite sums and the related puzzle that if one literally puts all the eigenvalues at the origin then the index seems to vanish. The resolution is given by a certain limiting procedure in which the eigenvalues are placed close to zero and one then takes the limit of the entropy as they go to zero. 3. The third strand is the Bethe-ansatz analysis for the subspace τ = σ, initiated by [7,8]. Here the analysis is performed not directly on the matrix model (1.1) but an equivalent representation of the index which one can solve at large N by computing the roots of the Bethe-ansatz equations. These roots are referred to in [8] as saddlepoint-like configurations, suggesting that those configurations could correspond to the saddle-points of the matrix model (1.1). This analysis was shown in [8] to be consistent with that of Point 2 when one further takes the Cardy-like limit.
Put together, these three strands of analysis strongly suggest that there should be a complex saddle-point of the matrix model (1.1) at large N and finite τ , having the same entropy as that of the black hole. In this paper we show that this is indeed the case in the subspace of coupling constants σ = τ . Within this subspace we find an infinite family of saddle-points labelled by points on the lattice Λ τ = Zτ + Z. More precisely, writing a generic lattice point as mτ + n, m, n ∈ Z, the independent saddles are labelled by (m, n) with gcd(m, n) = 1. As we explain below, our solution is best presented in terms of the torus C/Λ τ whose A-cycle is identified with the original circle of unit radius. The eigenvalues at the (m, n) saddle are uniformly distributed along a string winding m and n times along the A and B cycles, respectively, of the torus (see Figure 1). In particular, the saddle (0, 1) is identified with pure AdS 5 and (1, 0) is identified with the supersymmetric AdS 5 black hole.
Our method of analysis is as follows. Firstly we need to extend the action to complex values of the fields. Denoting the eigenvalues of the matrix U as e 2πiu i , i = 1, . . . , N , the variables u i of the original unitary matrix model take values in R/Z, whose representative we choose throughout the paper to be [0, 1). "Complex field configurations" in our context means complex values of u i or, equivalently, moving away from the original circle of unit radius in the variables e 2πiu i . A priori there is no canonical manner to extend the action to the complex plane. A natural first guess may be to use the analytic continuation of the original action away from real values of u i . However, the construction of our solution needs the action to be well-defined on the torus, for which the analytic continuation is not the right choice as it is only quasi-periodic under translations u i → u i + τ (in a manner that generalizes the quasi-periodicity of the Jacobi θ-function). The good choice of continuation of the action to the complex u i plane turns out to be an elliptic function, i.e. one which is strictly periodic under translations of 1 and τ . This choice can be thought of as a real-analytic (in u) deformation of the analytic continuation which cures the "small" non-periodicity at the expense of meromorphy.
The particular elliptic function that underlies our extended action can be identified with the elliptic dilogarithm, a certain single-valued real-analytic extension of the ordinary dilogarithm first studied by D. Wigner and S. Bloch [22]. This identification immediately places us in a very good position. The main technical tool is a formula due to Bloch [22] which has been studied intensively by number theorists [23,24]. Bloch's formula expresses the elliptic dilogarithm in terms of a real analytic lattice sum also known as a Kronecker-Eisenstein series [25], which is not only manifestly elliptic but also modular invariant under the usual SL 2 (Z) action on τ ! 2 The Kronecker-Eisenstein series can be thought of as a Fourier expansion along the lattice directions, and this makes the calculation of the action at the saddle-points very easy as we just have to calculate the zeroth Fourier modes along the direction chosen by a given lattice point.
The resulting actions of (0, 1) and (1, 0) agree precisely with that of pure AdS 5 and the supersymmetric AdS 5 black hole, respectively, which leads us to the above-mentioned identification of these saddles as the duals of the respective gravitational configurations. We also calculate the action and the entropy of the generic (m, n) saddle in Section 4. Since we are working with Euclidean matrix integrals with complex fields, a rigorous analysis of when a given saddle contributes to the partition function should be done using Picard-Lefshetz theory and the ideas of resurgence (a recent comprehensive review can be found in [26]). Here we start a more simple-minded analysis of comparing the various actions at a given value of τ , and calculate the resulting phase diagram for the supersymmetric theory. (The bounding curves of the resulting regions are the anti-Stokes lines.) We find that the dominant phase close to τ = 0 is the AdS 5 black hole. Interestingly there are new dominant phases (m, n) that dominate when τ approaches the rational points −n/m.
The form of the Kronecker-Eisenstein series allows us to easily calculate the asymptotics of the action at a given saddle-point in the Cardy-like limit τ → 0. Applying this procedure to the black hole saddle immediately yields the leading terms in the limit, thus clarifying certain technical issues that arose in the calculations in the Cardy-like limit in [14] by different methods. Our formalism thus unifies the three strands of analysis mentioned above and confirms the following satisfying physical picture of the Cardy-like limit-it is simply the limit when the black hole saddle-point shrinks to zero size, i.e. the configuration of eigenvalues uniformly distributed between 0 and τ all coalesce to the origin.
Finally we also develop a second method to calculate the action of the above saddles, by rewriting the elliptic gamma function as an infinite sum which is convergent along a given direction along the lattice. The second method recovers the results of the first method, and also allows us to make contact with the Bethe-ansatz approach.
The plan of the paper is as follows. In Section 2 we present a simple lemma which allows us to find complex saddle-point configurations for a class of matrix models defined purely by a two-particle potential. We also briefly review the index for N = 4 SYM on S 1 × S 3 and its integral representation in terms of elliptic gamma functions Γ e . In Section 3 we show how to extend the action of the matrix model from the unit circle to a torus using the elliptic dilogarithm. In Section 4 we calculate the action and the entropy of a generic saddle and discuss the phase structure of our supersymmetric matrix model. We also make some comments about the Cardy-like limit. In Section 5 we make some brief comments about interesting aspects of this problem that we do not discuss in this paper. In Appendices A-E we collect the definitions and useful properties of the various functions that enter in our analysis. We then discuss the analytic continuation of the integrand of the index and its relation with our elliptic extension. Using this discussion we recover our previous results and also make contact with the Bethe-ansatz approach of [7,8].
Notation We will use the notation e(x) := e 2πix throughout the paper: We will use τ ∈ H as the modular parameter and z, u ∈ C as elliptic variables. We set q = e(τ ), ζ = e(z). We decompose the modular variable into its real and imaginary parts as τ = τ 1 + iτ 2 , and the elliptic variables into their projections on to the basis (1, τ ), i.e. z 2 = Im(z) τ 2 , z 1 = Re(z) − z 2 τ 1 so that z = z 1 + τ z 2 . N denotes the set of natural numbers and N 0 = N ∪ {0}.
2 Complex saddle-points of N = 4 SYM In this section we present a simple lemma which allows us to find complex saddle-point configurations for matrix integrals of the type (1.1). Upon rewriting these integrals in terms of eigenvalues, the class of models that we consider have no single-particle interactions, and only two-particle interactions which depend only on the distance between the two particles. With these assumptions the basic lemma states that if the potential is periodic in any (complex) direction in field space then the uniform distribution between the origin and the point of periodicity is a saddle-point. We then review some properties of the N = 4 superconformal index and set up the particular problem of interest to us.

Saddle-point configurations for periodic potentials
As is well-known, the matrix integral Z in (1.1) reduces to an integral over the eigenvalues of U which we denote by {e(u i )} i=1,...,N . The product of two traces in (1.1) leads to an overall N 2 in front of the action, which allows us to use large-N methods. In the large-N limit one promotes {u i } to a continuous variable u(x), x ∈ [0, 1], and replaces the sum over i by an integral over x. One can further replace each integral over x in the action by an integral over u with a factor of the eigenvalue density ρ(u) = dx du which obeys the normalization condition du ρ(u) = 1. In this limit the integral (1.1) can be written in terms of an effective action S(u) as where the pairwise potential V is determined by the function f (τ ). The paths u(x) run over the real interval (− 1 2 , 1 2 ] in the unitary matrix model, but here we are interested in complex saddle-points of this model. The main observation is that if the pairwise potential V (z) is periodic with (complex) period T , then the uniform distribution u(x) = xT , x ∈ [0, 1], extremizes the effective action. To see this we first note that the odd part of the potential 1 2 (V (u) − V (−u)) drops out of the integral in (2.1)), and so V (u) can be taken to be an even function. Now, the variational equations arising from the integral (2.1) The partial derivatives here are odd, periodic functions with period T . On the configuration u(x) = xT , the left-hand side of this equation equals 1 where the integral is understood to be along a straight line in the complex plane along 0 to T . Using the periodicity and oddness properties, respectively, we now have (with denoting either the holomorphic or anti-holomorphic derivative), This same argument, with minor changes, can be repeated in the discrete variables. We present this as a lemma which we will use in the following development. Here we set u ≡ {u i } i=1,...,N , and use the measure [Du] = 1 N ! N i=1 du i . We will work with the SU (N ) gauge theory throughout the paper for which we need to include the tracelessness condition N i=1 u i = 0. Consider the N -dimensional integral defined by a pairwise potential between the eigenvalues that only depends on the difference u ij = u i − u j , i.e., The potential V (z) is any smooth complex-valued function, not necessarily meromorphic. We want to solve the variational equations In the above set up, suppose the potential V is periodic with (complex) period T . Then (i) The following configuration solves the variational equations (2.5), where u 0 is fixed in such a way the traceless constraint is obeyed.
(ii) The action S(u) at the saddle (2.6) is given by N 2 times the average value of the potential on the straight line joining 0 and T in the large-N limit.
Remarks The distribution (2.6) becomes the uniform distribution between 0 and T in the large-N limit. Since the potential V is periodic with period T , it admits a Fourier expansion along that direction. The second part of the lemma is equivalent to saying that the saddle-point value of the action is N 2 times the zeroth Fourier coefficient in the direction T .
To prove the first part of the lemma, we assume, without loss of generality as before, that V is an even function. The saddle-point equations (2.5) can be written as On the configuration (2.6) we have u ij = (i − j)T /N . For every point j in this sum the point j = 2i − j (modN ) is equidistant from i and lives on the other side of i, that is i − j = −(i − j ) (modN ). Since the functions ∂ z V , ∂ z V are odd and periodic with period T , these two points cancel against each other and the sums in (2.7) vanish. (For even N the antipodal point i + 1 2 N (modN ) is its own partner, but the argument still applies as The second part of the lemma follows from a simple calculation of the action on the distribution (2.6), which we call the effective action of the saddle: (2.8) In obtaining the second equality we have used the periodicity of the potential, and in the last equality we have used the large-N limit.

The index of N = 4 SYM
The superconformal index of N = 4 SYM theory was calculated in [1,2]. In [4] this index was revisited as a path integral corresponding to the partition function of supersymmetric field theories on S 3 × S 1 with complexified angular chemical potentials σ, τ for rotation in S 3 , and holonomy for the background R-symmetry gauge field A given by S 1 A = iπϕ, with ϕ = i(σ + τ − n 0 ). The introduction of the integer n 0 in [4] corresponds to a shift in the background value of the R-symmetry gauge field. 3 It was shown in [4] that where the function I(σ, τ ; 0) is the familiar Hamiltonian definition of the superconformal index [1,2] I(σ, τ ; 0) = Tr H phys (−1) F e −β{Q,Q}+2πiσ(J 1 + 1 (2.10) The above two equations make two things immediately clear. On one hand they show that the index I(τ, σ; n 0 ) is protected against small deformations of the parameters of the theory exactly as the original index. On the other hand expanding the index as I(σ, τ ; n 0 ) = n 1 ,n 2 d(n 1 , n 2 ; n 0 ) e 2πi(σn 1 +τ n 2 ) (2.11) in terms of the degeneracies d(n 1 , n 2 ), we see from (2.10) that a constant shift of the chemical potentials only changes the phase of the indexed degeneracies, so that |d(n 1 , n 2 ; n 0 )| = |d(n 1 , n 2 ; 0)| . (2.12) Thus we have that the index I, i.e. the canonical partition function, depends on n 0 whose correct value is dictated by the holographic dictionary. For the degeneracies d, i.e. the microcanonical observables, the value of n 0 does not change the absolute value of the degeneracies. It does, however, play a role in the calculation of the saddle-point value of the degeneracies as a function of large charges, since a change of n 0 corresponds to a change in the τ -contour of integration used to define the saddle-point value, and for the purpose of this calculation one should use a value of n 0 that maximizes the answer. In our situation, since the R-charges of the N = 4 SYM theory are quantized in units of 1 3 , the degeneracies only depend on n 0 (mod 3). We can choose the representatives to be n 0 = 0, ±1. As it turns out, the natural contours associated to n 0 = ±1 both have leading entropy at large charges equal to that of the black hole, which dominates the natural contour associated to n 0 = 0. We note that one can map n 0 = +1 → n 0 = −1 by using the transformation (Re σ, Re τ ) → −(Re σ, Re τ ). All this is consistent with the supergravity calculation of the Euclidean black hole [4]. We will use the value n 0 = −1 in the rest of the paper and suppress writing it explicitly.
The superconformal index of SU (N ) N = 4 SYM can be written in the form (1.1) with . (2.13) One can recast the exponential of the infinite sum expression in (1.1) in terms of an infinite product to obtain the following N -dimensional integral, [1,2,4] I(σ, τ ) = e(− σ 24 − τ 24 ) η(σ) η(τ ) N × (2.14) [Du] i =j where the measure factor is and the integral over each u i runs over the real range (− 1 2 , 1 2 ]. Here the elliptic gamma function 4 is defined by the infinite product formula [27], The interpretation of the various pieces in (2.14) in its derivation using supersymmetric localization of the partition function [4] are as follows. The gauge holonomies u i label the localization manifold, and the classical action vanishes at any point on this manifold. The infinite products in the integrand comes from the one-loop determinant of the localization action. The first elliptic gamma function arises from the vector multiplet which has Rcharge 2, and the other three elliptic gamma functions arise from the chiral multiplets which carry R-charge 2 3 . The product over i, j in the integrand of (2.16) reflects the fact that all the fields are in the adjoint representation of the gauge group. For the vector multiplets one needs to remove the zero modes which arise for the Cartan elements i = j for the vector field, and the non-zero modes of the Cartan elements form the pre-factor in front of the integral (2.14).
In the rest of the paper we analyze the model with σ = τ and n 0 = −1 so that the SYM index is The product expression (2.15) makes it clear that Γ e (z; τ, σ) is separately periodic under the translations z → z + 1, σ → σ + 1, and τ → τ + 1. Thus we see from (2.16) that the N = 4 SYM index I(τ ) manifestly has the symmetry τ → τ + 3. It is also clear from (2.16) that by shifting τ by 1 and 2, respectively, we reach the other two independent values of n 0 , thus explicitly showing the relation of n 0 with the contour of integration.
Our goal now is to find complex saddle-points of the integral (2.16). In order to do this we need to extend the integrand to the complex u-plane. One natural guess would be to use analytic continuation of the elliptic gamma function [27]. This function is periodic with period 1, and is almost-but not quite-periodic 5 with period τ : it obeys Γ e (z + τ ; τ, τ ) Γ e (z; τ, τ ) = θ 0 (z; τ ) , (2.17) where the function θ 0 is related to the odd Jacobi theta function ϑ 1 as In order to use the method of the previous subsection we need a strictly periodic function, and so we need to find a periodic extension. Our idea is simple in principle-we deform the integrand away from the real axis such that the resulting function is periodic with period τ . Of course the result will not be meromorphic in u and the best we can hope for a real-analytic function of Re(u), Im(u). (We still demand that the action is mermorphic in τ .) The technical goal is thus to replace the integrand of (2.16) by a function that (a) is periodic under u → u + mτ + n, m, n ∈ Z (an elliptic function), The properties (a)-(c) above do not uniquely fix such a function. We will make a particular choice which has three nice properties. Firstly, our extended action turns out to be closely related to the elliptic dilogarithm function which has, quite remarkably, modular properties under the SL 2 (Z) action on τ . As mentioned in the introduction, these properties are made manifest by a formula of Bloch [22], which allows us to calculate the saddle-point action in one simple step. Secondly, the action of the black hole configuration agrees precisely with the supergravity action of the black hole. The third property looks to be an aesthetic or mathematical one at the moment-the elliptic dilogarithm obeys interesting equations, and its values at special points are interesting from a number-theoretic point of view.
We present the details of the extension of the action to the torus in Section 3, and continue for now by assuming that there is an extension obeying Properties (a)-(c). The periodicity of the extension under translations of the lattice Λ τ = Zτ + Z can be restated as saying that the function is well-defined on the torus C/Λ τ . Now, our lemma about complex saddle-points in the previous subsection means that there is one saddle-point for every lattice point in Zτ + Z. If we think of the uniform distribution of eigenvalues u(x) as a string, then a solution corresponding to the point mτ + n can be thought of as a closed string that winds (m, n) times along the (A, B) cycles of the torus. In terms of the original matrix model, the eigenvalues have moved off the original circle of unit radius into the complex plane, as shown in Figure 1.

Comment about other discrete saddles and gcd condition
In the context of our solution-generating lemma in Section 2.1, suppose V (x) has minimal period T . Then clearly the (discrete) uniform distribution spread between 0 and pT , for any non-zero p ∈ Z also solves the saddle-point equations by the same arguments as those given in the lemma. These solutions all give rise to the same distribution of points on the torus if and only if gcd(p, N ) = 1. When (p, N ) = (dp , dK) with d > 1 and gcd(p , K) = 1, then we have a new solution which is a uniform distribution of K "stacks" of eigenvalues, each stack containing d = N K eigenvalues. Applying these considerations to our problem on the torus, we find a family of saddles that is classified by the following three-label notation: (K| m, n) with gcd(m, n) = 1 and K is a divisor of N . The mτ + n saddles discussed above correspond to K = N , and we will continue to use that notation in that case. If N is prime, saddles are essentially isomorphic to lattice points (m, n) with gcd(m, n) = 1. 6 The one solution which needs to be discussed separately is (1| m, n) ≡ (0, 0). This saddle corresponds to the distribution where all eigenvalues are placed at the origin u = 0. As we discuss at the end of Section E, this distribution of eigenvalues is highly suppressed with respect to the dominant one at finite values of τ . However, one has to be careful to take the limit τ → 0 because other effects start to appear. From now on, we focus on (N | m, n) ≡ (m, n) saddles.
In this section we show how to extend the matrix model to the torus using elliptic functions. We collect the definitions and properties of some standard functions that appear in our presentation in Appendix A.

Elliptic functions, Jacobi products, and the elliptic dilogarithm
In order to define our deformation, and as a warm-up, we begin with a brief discussion of the quasi-elliptic function θ 0 (z) = θ 0 (z; τ ) defined in Equation (2.18). It has the following periodicity properties, and a product representation which follows from the Jacobi product formula, The function θ 0 can be modified slightly in order to obtain a function that is elliptic. Define the related function (recall z = z 1 + τ z 2 ) The elliptic transformations z → z +1, z → z +τ are simply shifts of the real variables z 1 → z 1 + 1, z 2 → z 2 + 1, respectively. It is easy to see that |P | is invariant under z → z + 1. Under z → z + τ , the shift of the theta function in (3.1) is cancelled by the corresponding shift of B 2 (Equation (A.6)), and therefore |P | is an elliptic function. A non-trivial fact is that |P | is also invariant under the modular transformations τ → aτ +b cτ +d , z → z cτ +d . In other words, it is invariant under the full Jacobi group. These properties are immediately demonstrated by the second Kronecker limit formula [25], The right-hand side of (3.5) is a real-analytic Kronecker-Eisenstein series which is manifestly invariant under the Jacobi group.
These properties of θ 0 are classical facts known for over a century (see [25] for a beautiful exposition). Interestingly there is a similar story for the elliptic gamma function, which is much more recent. Following [24,30], we construct the function, for z ∈ C, In order to completely define Q, the equation (3.6) needs to be supplemented by a choice of branch. We will instead work with a modified function Q(z) for which we will give another explicit formula below. For the moment the reader could focus on the absolute value of this function which is unambiguously defined. On the real axis, Q agrees with the Γ e -function, i.e.
The quasi-periodicity relation of the elliptic Gamma-function implies that the function | Q| is periodic under translations of the lattice Zτ + Z. Remarkably, it also satisfies a relation similar to, but more complicated than, the Kronecker limit formula (3.5) for |P |.
To see the relation, one first notices that the function | Q| is very closely related to the Bloch-Wigner dilogarithm function. The Bloch-Wigner dilogarithm function is the single-valued non-holomorphic function (we follow the treatment of [23]), Its elliptic average, which is manifestly invariant under lattice translations, is defined as The function D(q, x) turns out to have a natural imaginary partner which is itself an elliptic average 7 of the function J(x) := log |x| log |1 − x| , (3.12) so that the most elegant formulas are written in terms of the combination The relation of | Q| to F is as follows [24], [30] Now that we have made an identification between Q and the elliptic dilogarithm, the relation to Kronecker-Eisenstein series follows from the following formula [22,23], The function F is manifestly invariant under shifts of u by lattice points. The form of F (τ, z) as a lattice sum also makes it clear that under modular transformations it transforms as a Jacobi form of weight (0, 1). So far we have only considered the real part of the function log Q whose real part is elliptic. As mentioned above, we can build a related periodic function Q whose real part is defined to be, for z ∈ C, τ ∈ H, Re log Q(z; τ ) := Re log Q(z; τ ) . (3.17) The real and imaginary parts of Equation (3.15) give us formulas for log |Q| as well as its modular transform in terms of the elliptic dilogarithm functions The imaginary part of log Q is defined to be the harmonic dual of the real part with respect to τ , namely we demand that log Q is holomorphic in τ . This leaves an ambiguity of a τindependent function, which we parameterize by an elliptic real function Ψ(z) with Fourier coefficients 1 4 K(m, n). These considerations allow us to write a Fourier expansion for log Q as follows The ambiguity of the imaginary part is constrained to ensure that Q and Q agree on the real axis, i.e. Q satisfies Equation (3.7). By comparing this expansion to the expansion (C.3) of Γ e and using Equation (3.7), we obtain that Ψ(z)| z 2 =0 = 0. The function Q should be related to a version of the elliptic dilogarithm holomorphic in τ studied in [31,32], we leave a detailed study of this to the future.

Elliptic extension of the action
Having set up the basic formalism, we can now connect to the action of N = 4 SYM. In order to do so, we define the related function This function is clearly elliptic as all its building blocks are, and it obeys the property The index (2.16) can now be written as Writing this in terms of an effective action as we have In terms of the functions Q and P we have We have thus reached our goal of extending the action to the complex plane in terms of elliptic functions. The action is a real-analytic function on the torus except for a finite number of points where it has singularities, we comment on this in Appendix B.
The fact that the expansions (3.15) and (3.19) are written as a double Fourier series in z 1 , z 2 means that we can read off the average value in any desired direction in the zplane. Secondly this expression in terms of a lattice sum makes the modular properties completely manifest. We will use these properties in the following section to calculate the action at the saddle-points, and to calculate asymptotic formulas in the Cardy-like limit.

Action of the saddles, black holes, and the phase structure
The considerations of the previous two sections have shown that the complex saddles for the supersymmetric index (2.16) of N = 4 SYM on S 3 ×S 1 in the large-N limit are labelled by the lattice points mτ + n ∈ Zτ + Z with gcd(m, n) = 1. 8 We write the index as a sum over saddles as follows, The effective action S eff (m, n; τ ) is the classical action (3.24) evaluated at the saddlepoint configuration (m, n) plus the quantum corrections induced by loop effects around the saddle. In this paper we only calculate the classical part. We show that the action of the (1, 0) saddle agrees with that of the supersymmetric black hole in the dual AdS 5 theory. We then compare the real parts of the actions at the different saddle-points as a function of τ which leads to the phase structure of the theory.

Action of the saddles
In the large-N limit the effective action (3.24) of the SYM theory extended to the torus can be written in terms of a potential as in (2.4), The formula (2.8) says that the value of the action at the saddle-point mτ + n is given by the average value of the action (as a function of z) on the straight line between the origin and that point. In terms of the coefficients of the double Fourier series the average value of V between 0 and mτ + n (gcd(m, n) = 1) is Here in the second equality we have used that the integral of 1 0 e(nx)dx = δ n,0 . We will now use the formula (4.4) to calculate the average value of the action. In preparation for the result we first calculate the average values of the functions log P and log Q. As discussed below Equation (3.18), it is easier to calculate the average value of the real parts log |P | and log |Q|. To reach the complex function we construct an expression holomorphic in τ whose real part agrees with these calculations. (We only present that after having assembled the full action, but it is easy to do this at every step if required.) This still leaves the ambiguity of a τ -independent purely imaginary term, we shall discuss this in the context of the (1, 0) saddle in detail in the following subsection, and more generally in Appendix E.
Average value of log |P |. The double Fourier series given by the Kronecker limit formula (3.5) implies that, for (m, n) = (0, 0), Average value of log |Q|. The double Fourier series given by the Bloch formula (3.15) implies that, for (m, n) = (0, 0), In reaching the second lines in the above formulas we have used the Fourier expansions (A.8), (A.9) for the Bernoulli polynomials.
Average value of log |Q ab |. Putting these two together we can calculate the average value of log Q ab to be Vector multiplet action. The real part of the vector multiplet contribution to the saddle-point action is given by Chiral multiplet action. The real part of the chiral multiplet contribution to the saddle-point action is given by Putting everything together, we reach the following concise formula for the real part of the total action at the (m, n) saddle-point, (4.10) The coefficients of the second and third terms in the above formula are periodic in their arguments and take the values given in Table 1. The symmetry τ → τ + 3 is realized in this formula as (m, n) → (m, n + 3m). Under this operation the combination 1 Using the values of the Bernoulli polynomials in Table 1 we can rewrite the real part of the action as where χ i , i = 0, 1 are the two Dirichlet characters of modulus 3: Upon constructing the holomorphic part of the action as explained above, we obtain S(0, 1; τ ) = 0 and a short calculation shows that, for m = 0, where the τ -independent real phase ϕ is where in terms of the function K in (3.19), we have It is interesting to ask what is the action of a given saddle-point configuration if we use the analytic continuation of the elliptic gamma function instead of the elliptic continuation presented here. We address this question in Appendix E and find that the results coincide, up to a τ -independent imaginary constant. The phenomenon that the action of a given saddle essentially does not change if we use the analytic continuation of the elliptic gamma function instead of the elliptic continuation is interesting. At the practical level, the meromorphic analysis of Appendix E, can be used to extend the current results of the Bethe ansatz approach of [7,8] and, to relate those with the results of the elliptic approach introduced here.

Entropy of supersymmetric AdS 5 black holes
A particularly interesting saddle-point is the black-hole (m, n) = (1, 0), whose action is Up to a τ -independent additive constant that we discuss below, this action is equal to the action of the supersymmetric AdS 5 black hole, calculated in [4] from supergravity after explicitly solving the constraint (1.2) for ϕ and subsequently setting τ = σ. In [4] the black hole entropy was derived by performing a Legendre transform which includes a Lagrange multiplier that enforces the constraint. Here we verify that one can equivalently obtain the result for the black hole entropy by explicitly solving the constraint. As we see below the Legendre transform of the action (4.16) combined with a set of reality conditions on the expectation values of charges and the entropy leads to the Bekenstein-Hawking entropy of the BPS black hole [3,4]. The entropy of the black hole comes from the (1, 0) saddle-point as . There are two τ -independent terms in the exponent of the right-hand side which deserve a comment. One is the last term on the right-hand side which appears from a similar term in the Hamiltonian definition of the index (2.9) with n 0 = −1. The other is the value of the constant ϕ(1, 0) that we fix to zero by the choice K(1, 0) = 12 27 in (4.16), so that In the large-N regime, the integral (4.17) is approximated by its saddle-point value, and as usual we assume that the integration contour goes through the saddle-point. Thus we have to extremize the functional E(τ ). We find that, as a result of this extremization procedure, the black hole entropy obeys P ( i 2π S BH ) = 0, where P is the cubic polynomial This is precisely the polynomial that was found to govern the gravitational entropy of the supersymmetric extremal black holes in [4]. In addition if we impose the reality conditions we find that the charges J and Q obey the following constraint where are the coefficients of P defined in (4.19). With these reality conditions, we obtain the following result for the entropy, which agrees with the Bekenstein-Hawking entropy of the corresponding BPS black holes. Based on the black hole properties, we shall call the SYM saddle-points with real expectation value of charges and entropy as extremal.
The above results can be obtained as follows. Upon extremising E(τ ) one obtains the following relation Plugging this relation into E(τ ) one obtains, at the saddle point The above two equations imply that the quantity s = i 2π S BH obeys, at the saddle point, Putting these two equations together we find that P (s) = 0. Upon imposing the reality condition of J and Q in (4.24) we obtain that the extremal solutions lie on the following real one-dimensional locus in the complex τ -plane (we denote the corresponding quantities by * ), Now imposing the reality of S BH and Q in (4.24), (4.25), we obtain (4.28) One can now check that the expressions (4.28) satisfy the non linear constraint (4.21), and that the right-hand side of (4.25) equals the expression (4.23). One can ask what happens if we choose differently the τ -independent term ϕ(1, 0) in the extremization problem (4.17). This is answered by noticing that without these terms the exponent has the symmetry which shifts J and Q 2 equally in opposite directions keeping 2J + Q invariant. The actual minimization procedure (e.g. in Equation (4.24)) only depends on this combination. However, the reality conditions that we impose break this symmetry. Now it is clear that if we change these terms, e.g. if we change the value of the constant K = K(1, 0) away from 12 27 , then the whole procedure that we described above still goes through, but with the redefinitions Q → Q + N 2 ( K − 12 27 ), J → J − N 2 2 ( K − 12 27 ). It is interesting that the value of K = 12 27 that agrees with the supergravity entropy function is precisely the one given by the analytic continuation of the original action (2.16) (see Appendix E). It will be nice to understand this observation at a deeper level.
We move on to compare these results with the gravitational picture. In [4], a family of complex supersymmetric asymptotically AdS 5 solutions of the dual supergravity theory was studied, using earlier results of [33,34] on general non-supersymmetric black hole solutions. Upon imposing reality conditions on the solutions within this family, one reaches the extremal supersymmetric black hole. The BPS angular velocities of the general complex solutions was defined in [4], using a certain limiting process to the supersymmetric locus, (in the present context of equal angular potentials) as where a ∈ [0, 1) and r + > 0. At extremality, i.e. for r + = √ 2a + a 2 , τ reduces to [4] τ * (a) = (a − 1) −(a + 2)n 0 − 3i a(a + 2) 2(a + 2)(5a + 1) . (4.30) The two values n 0 = ±1 in (4.29) are related in field theory by τ 1 → −τ 1 , and label two complex conjugated solutions in the gravitational theory related by Im ω → −Im ω. At extremality the metric and gauge field configurations of two gravitational solutions are real, and they coincide which means that the two values n 0 = ±1 are simply two descriptions of the very same solution. One can check that for n 0 = ±1 the gravitational extremal curve τ * (a) in (4.30) agrees with the field theory extremal curve τ * , obtained from (4.27) as one expects. 9 We plot these curves in Figure 2. The dashed curve in this figure is the same as the (1, 0) curve in [8]. In Figure 3 we have plotted the regions covered by such solutions as one varies the parameters a and r + . We note that the known non-extremal supersymmetric solutions of discussed in [4] (which have 0 ≤ a < 1 and r + > 0) do not cover the full range of complex 9 We have checked that there are no (m, n) saddles in the n0 = −1 field theory index that matches the n0 = +1 extremal curve. Indeed, once the holographic dictionary is established, it would have been surprising to have two different field-theory saddles corresponding to the same BPS black hole solution. coupling τ ∈ H, and, in particular, the regions with large τ 2 . As we will see in the following subsections, there are other saddles (m, n) which exist (and dominate) other regions.

Entropy of (m, n) solutions
In this subsection we extremize the entropy functional of the ensemble described by (m, n) saddles, and calculate their entropy. The procedure is exactly as in the (1, 0) case, namely that the extremization combined with the imposition of reality of the expectation values of field-theory charges and entropy forces the chemical potential τ to live on a specific one-dimensional curve in the upper half plane.
We start by rewritting (4.13) in terms of the variable τ = mτ + n, S eff (m, n; τ ) = 1 m πi 27 N 2 2 τ + χ 1 (m + n) 3 τ 2 + N 2 πi ϕ(m, n) . (4.31) As before we set ϕ(m, n) = 0 by an appropriate choice of the phases K(m, n). One can reinstate it following the discussion at the end of the previous subsection. To ease presentation we suppress the m and n dependence of χ from now. The entropy of (m, n) saddles is obtained by the following extremization, where the entropy functional E is where we have defined useful auxiliary charges in terms of the physical charges as It should be clear from the above considerations that the entropy of extremal (m, n) solutions has a similar form as that of the (1, 0) solution (4.23) with Q, J replaced by Q, J, respectively. We now spell out some of the details. We have that S (m,n) is a root of the cubic polynomial P ( x 2πi ), where with Next we impose reality conditions. Notice that reality of (J, Q) is equivalent to reality of ( J, Q). Imposing the reality conditions on the charges as well as the entropy S (m,n) , we obtain and the extremal locus Notice that τ 1 = 0 is a cusp of the curve. The corresponding singularity is related to the phenomenon of the solution at this point becoming infinitely large, which can be seen from the fact that the charges (4.39) become infinite. Excluding this point, and demanding that the expression under the square root in (4.40) is positive gives us the following allowed values of τ 1 : Note that the real part of the action S eff is less than (greater than) the limiting value of all saddles as τ 2 → ∞ (which we will discuss in the next section) for the first lines (second lines) in (4.41), (4.42). Relatedly, the first lines reproduce the action and entropy of the known black hole solutions, while the second lines do not correspond to anything that we know of in gravity. 10 For this reason we define the Cardy-like limit (discussed later in more detail) as τ → 0, −χ 1 τ 1 > 0.
In terms of the physical chemical potential τ , (4.40) takes the forms 10 From now on we only refer to the regions in the first lines when we discuss extremal curves. and the first lines of (4.41), (4.42) take the form, along the corresponding extremal curve. For every n m , with m > 0, gcd(m, n) = 1, and such that (m + n) mod 3 = 0, the normalized entropy grows to +∞ when the extremal curve approaches − n m from the left ((m + n) mod 3 = 1), or from the right ((m + n) mod 3 = −1).
From Equation (4.36) it follows that S (m,n) = 0 for solutions such that χ 1 (m + n) = 0. From now on, we focus on solutions such that χ 1 = ±1. The reality conditions on charges and entropy imply that for all m ∈ N and n ∈ Z with (m + n) mod 3 = 0 the following factorisation property holds p 0 = p 1 p 2 . The factorisation (4.45) implies that the algebraic form of the S (m,n) is The constraint (4.45) can be used to eliminate one variable in this expression. An implicit way to do that, is to use (4.39) in the right-hand side of (4.46). In that way one obtains the following parametric expression for the entropy, where the real parameter τ 1 ranges in the domain (4.44). 11 11 The expression (4.47) is independent of the phase ϕ even if we reinstate it in the preceding discussion. From Equations (4.39) and (4.47) it follows that when the extremal (m, n) curve (4.43) approaches the rational number − n m within the range (4.44), the expectation value of charges as well as the entropy (in units of N 2 ) grow to infinity. We refer to such limits as extremal Cardy-like limits (see Figure 4). As the set of rationals Q is dense, it follows that τ 2 = 0 is an accumulation line for such limits. Figure 5 shows a set of extremal curves for m = 1, . . . , 30 and n = −30, . . . , 30. In the J-Q plane, the extremal solution for (1, 0) is the one-dimensional curve (4.21) or, equivalently, (4.28). Now we see that there is a dense set of such extremal curves (4.45) in the J-Q plane labelled by rational numbers.
It would be very interesting to find the gravitational interpretation of the generic (m, n) saddles. These should be asymptotically AdS 5 supersymmetric solutions whose Bekenstein-Hawking entropy should coincide with our predictions (4.46), (4.47). 12 An interesting feature of the (m, n) saddles is that that the labels of the set of saddle-points i.e. (m, n) with gcd(m, n) = 1, can be equivalently thought of as the coset Γ ∞ \Γ, i.e. the set of matrices in Γ = SL(2, Z) (acting as usual via fractional linear transformations on τ ) modulo the subgroup Γ ∞ that fixes the point at infinity. This is the same structure as the AdS 3 black hole Farey Tail [38], and perhaps points to a similar structure of Euclidean saddle-points as in [39].

Phase structure
Now we analyze the relative dominance of the saddles for a given value of τ . Firstly, recall that I(τ ) is invariant under τ → τ + 3, and correspondingly we analyze the region −2 ≤ Re(τ ) < 1. The idea is to compare the real parts of the action at the different saddles at a generic point in the τ -plane. The dominant phase (if a unique one exists) is the saddle with least real action. The phase boundaries where the real parts of the actions of two saddles are equal are called the anti-Stokes lines.
As τ 2 → ∞, the real part of the action (4.12) of all the phases with (m + n) mod 3 = 0 equals − 8 27 πN 2 τ 2 . There are an infinite number of such phases and the imaginary parts and sub-dominant effects will be important to evaluate the path integral. (Curiously, 12 The hairy black holes explored in [35][36][37] could be relevant to this discussion. the entropy of these saddles actually vanish as shown in the previous subsection.) When (m + n) mod 3 = 0, then the third term in (4.12) is always positive, and it dominates over the second term for large enough τ 2 . Therefore these saddles are sub-dominant compared to those with (m + n) mod 3 = 0 in that region. When we start to reduce τ 2 we begin to hit regions where one phase dominates. For example, the action S eff (1, 0; τ ) blows up at τ = 0 and the dominant phase near τ = 0 is (1, 0) (the black hole). The anti-Stokes lines between the black hole and the region at τ 2 → ∞ is given by Similarly, as τ approaches a rational point − n m , −Re S eff (m, n; τ ) becomes very large and that is the corresponding dominant phase near the point.

Cardy-like limit(s) revisited
In this subsection we revisit the Cardy-like limit at finite N of the SU (N ) N = 4 SYM superconformal index (2.9), with n 0 = −1. As mentioned in the introduction, this limit has been analyzed for a class of N = 1 SCFTs including N = 4 SYM. Here the scope is to address the problem from a different angle, by using the results presented so far. First of all, it is clear from the formula (4.12) that the dominant contribution to the large-N expansion of the index (4.1) in the Cardy-like limit, τ → 0 with τ 1 < 0, 15 comes 13 This region lies inside the region in red in the Figure 3 to the right which means that there is a supersymmetric (but not necessarily extremal) black-hole-like solution that exists in supergravity).
14 A partial phase diagram for N = 4 SYM was first presented in [8]. It focused on the subset of (1, r) configurations, with r ∈ Z which is equivalent to our Figure 6 after mapping conventions (see Footnote 24). 15 The second condition is important because the sign of the second term in (4.12), which dominates in absolute value as τ → 0, is controlled by sgn(τ1). from the (1, 0) saddle which we identified with the black hole in the previous section. The large-N action of (1, 0) saddle in this limit is In fact one can do better and go beyond the large-N limit in the Cardy-like limit. We start with the formulas (3.5), (3.15), (3.19) and take the τ → 0 limit. We obtain (4.50) Similarly, for the P function we obtain, from (3.5), with holomorphic part (4.53) The formula (4.53) is precisely the one obtained in the Cardy-like limit studied in [14]. We note, however, that our starting point here is different and that we do not need to go through the subtleties of the Cardy-like limit that were used in [14]. The action (4.53) can be then used for the finite N answer in the Cardy-like limit as in [14], to obtain (4.49). The advantage of our formalism is that we can calculate further corrections in that limit in a systematic way, using the Fourier expansions (in τ 1 ) of the Eisenstein series [24]. The considerations in Section 2 allows to clarify one more aspect of the Cardy-like limit. In [14] we analyzed the different saddles of the theory in the large-N limit along the the real line. Now we can repeat that analysis for finite N . The right-hand side of (4.53) only depends on u 1 , it is differentiable, and periodic with period 1, and thus obeys the conditions of the Lemma in Subsection 2.1. For N finite and prime, its Cardy-like saddles are (0, 0) and (0, 1). Their actions (after properly dealing with the i = j modes in (4.53)) are 16 , (as τ → 0 and exact in N ), These results are consistent with the intuition of a large-N limit followed by Cardy-like limit: in the Cardy-like limit the (1, 0) saddle approaches the (0, 0) and the other large-N saddles approach the (0, 1). Indeed the term proportional to N 2 in (4.55) coincides with (4.49), and that in (4.56) vanishes as does S eff (0, 1; τ ). In Appendix D we expand on the role of the (0, 0) saddle further. At finite N > 2, and in the Cardy-like limit τ → 0 with τ 1 < 0, the saddle (4.55) dominates, as shown in [14] for a generic family of N = 1 SCFTs including N = 4 SYM. For the limit τ → 0 with τ 1 > 0, the solution (4.56) dominates. A more refined statement about this case involves the use of a higher order expansion in τ with N -dependent coefficients. We postpone such an analysis. More generally, there exists a Cardy-like limit for all (m, n) saddles (corresponding to a rational − n m with (m + n) mod 3 = 0), which is τ → 0. From Formula (4.10) it follows that the (m, n) saddle is the leading configuration in that limit. As for the (1, 0) case mentioned above, the result is sensitive to the actual limiting procedure. As mentioned below Equation (4.42) we can define the Cardy-like limit to be τ → 0 with −χ 1 (m+n) τ 1 > 0. In this limit, the real part of the effective action goes to −∞. For χ 1 (m + n) = +1(−1), the left(right) extremal curve drawn in Figure 4 defines a particular example of this limit.
It is interesting to recast the discussion in the above paragraph in terms of the microcanonical ensemble. Naively, the Cardy-like limit is one in which all the charges scale to infinity and, correspondingly, τ → 0. However, as we saw above, there is actually a family of interesting limits in which only τ 2 → 0 and τ 1 approaches a rational number. The microcanonical analog of these new limits has to do with the ambiguity in taking the "largecharge limit" when there are multiple charges. In our situation, Equations (2.11), (2.10) imply that τ couples to 2J +Q, which should scale to infinity. However, the charges J and Q 16 Here we have used the following identity that holds for z ∈ R and any integer n with gcd(N, |n|) = 1 are a priori independent and one needs to specify in what relative manner they scale to infinity. Demanding that the entropy is real specifies a certain relation between the charges. As we discussed in Section 4.3, there is a consistent relation for every (m, n) which defines a one-dimensional sector of BPS states in the charge space, such that the corresponding entropy is S m,n (J, Q). Recalling from [14] that the Cardy limit for (1, 0) is given by Q ∼ Λ 2 , J ∼ Λ 3 , it is easy to see that the Cardy-like limit for a given (m, n) is given simply by scaling the appropriate linear combinations in the same manner, i.e. Q ∼ Λ 2 , J ∼ Λ 3 .

Summary and discussion
In this paper we have found a family of saddle points of the matrix model governing the index of N = 4 SYM on S 3 × S 1 . The solution of this problem is governed by the Bloch-Wigner elliptic dilogarithm function which is a well-defined real-analytic function on the torus C/(Zτ + Z) and transforms like a Jacobi form with weight (0, 1) under SL(2, Z) transformations of the torus. 17 The Bloch formula, which expresses the elliptic dilogarithm as a real-analytic Kronecker-Eisenstein series (a particular type of lattice sum), makes the analysis very elegant. The action at any saddle can be calculated by reading off the zeroth Fourier coefficient of the series along the direction of the saddle. We also develop a second method to calculate the action which uses a series representation of the meromorphic elliptic Gamma functions involved in the original definition of the SYM index. We show that deforming the original action to the Bloch-Wigner type function away from the real axis in this approach agrees with our first approach. Using our formalism we find a family of solutions labelled by (m, n) with gcd(m, n) = 1, and analyze the competition among these phases of the theory at finite τ . The analysis of this paper relates in a natural way to the various approaches to the problem that have been used so far, and shows that they are all consistent with each other. Firstly, the contact with supergravity comes from the fact that the action of the saddle (1, 0) is precisely the supergravity action of the black hole. We also calculate the action and entropy of the generic (m, n) saddle. It is an interesting question to identify the corresponding gravitational configurations. Secondly, the calculations in the Cardy-like limit get clarified and simplified as the limit τ → 0 of the finite-τ answer. The formalism is powerful and can be used to calculate corrections from the Cardy-like limits. Thirdly, there seems to be a relation to the Bethe-ansatz type analysis which is intrinsically a meromorphic analysis. We find that the action of the original mermomorphic elliptic gamma function at the Bethe roots agrees with the action at the saddle-points derived from our approach. This makes it clear that the Bethe roots should indeed be identified with Euclidean saddle points of the matrix model.
The relation between the meromorphic approach and our real-analytic approach remains experimental and it would be nice to understand it at a deeper level. Towards this end it is important to understand the meromorphic function log Q (as opposed to just the 17 Preliminary indications of modular invariance in this system were found in [40]. The calculations of [4], which showed that a generalized supersymmetric Casimir energy of N = 4 SYM on S 1 × S 3 equals the black hole entropy functional, also pointed towards a modular symmetry underlying the system. real part log |Q|) better. It seems to us that this should be related to a related multi-valued function studied in [31,32]. The meromorphic function in question discussed in [31,32] satisfies a certain first-order differential equation in τ (it is the primitive of a holomorphic Kronecker-Eisenstein series), and trying to find such an equation may be one way to completely fix the phase ambiguities that we discussed in this paper. It is interesting to note that there is also a version of the dilogarithm [41][42][43] which seems to be closer to the original elliptic gamma function.
The elliptic dilogarithm (in its real-analytic as well as its meromorphic avatars) has been understood in recent years to play a crucial role in string scattering amplitudes (see e.g. [44,45]). It would be remarkable if there is a direct physical relation between these two appearances. One natural speculation along these lines would be about a holographic relation, with the string amplitudes (perhaps suitably modified) being observables in a bulk theory dual to the SYM theory that we discuss in this paper. A less striking, but nevertheless interesting, possibility would be the following syllogism: the appearance of the same elliptic dilogarithm in string scattering amplitudes and the partition function of AdS 5 black holes could be compared with the appearance of the theta function in string amplitudes and the partition functions of black holes in flat space.
Our analysis also clarifies the role of modular symmetry in this problem. The SYM index itself is not a modular form, instead the Bloch-Wigner function is essentially a period function for the real part of the action of the SYM. More precisely, writing the action as a simple linear combination of the functions log P and log Q (see Equation (3.25)), we have that log |P | is Jacobi invariant while the Bloch-Wigner dilogarithm is a period function of log |Q| (see Equations (3.14), (3.17)). In particular, this explains the fact that the SYM action on S 1 × S 3 as τ → 0 has a τ −2 behavior in contrast with the τ −1 behavior of holomorphic modular forms appearing in theories on S 1 × S 1 .
There are clearly many interesting things to do. For one, our analysis can be enlarged to include 4d N = 1 SCFTs. In another direction, these ideas could lead to progress in similar problems in other dimensions. It would be nice to find the correct mathematical structures to extend the ideas in this paper away from the restriction τ = σ. At a technical level, it would be nice to find a direct physical interpretation of the elliptic deformation in the complex plane of the SYM index, the obvious thing to look for would be a supersymmetryexact operator. Similarly, it would be nice to find a direct relation to the Bethe-ansatz approach, for which there are clues mentioned in this paper.
Finally, it is tempting to speculate that deeper number-theoretical aspects of the functions appearing here play a role in physics. The original motivations of Bloch in uncovering his beautiful formulas arose from algebraic K-theory. Relatedly, the values of the elliptic dilogarithm at particular (algebraic) values of τ are closely related to values of certain L-functions of Hecke characters, and the so-called Mahler measure [46]. It would be really interesting if the physics of SYM and AdS 5 black holes is directly related to these objects, perhaps in a manner that extends the beautiful ideas described in [47].

A Some special functions and their properties
Bernoulli polynomials The Bernoulli polynomials B k (z) for z ∈ C are defined through the following generating function, The first three Bernoulli polynomials are These polynomials obey the following properties Their Fourier series decomposition, for 0 ≤ z < 1 is Clearly this also equals B k ({z}) where {z} := z − z is the fractional part of z. In particular we have for z ∈ R Multiple Bernoulli Polynomials The multiple Bernoulli Polynomials B r,n (z| ω) are defined for z ∈ C, ω = (ω r−1 , . . . , ω 0 ) with ω j ∈ C − {0} through the following generating function [48,49] (we follow the conventions of [49]): t r e z t r−1 j=0 (e ω j t − 1) One has similar formulas which express B n,n (z| ω) with ω 0 = 1 in terms of usual Bernoulli polynomials, to obtain those, one uses (A.15), (A.16), (A.17) and applies the property (A.14) on the corresponding left-hand sides. The above equation will be convenient to analyticallymatch the results of the action in Appendix E and Section 4.

Theta functions
The odd Jacobi theta function is defined as (with q = e(τ ), ζ = e(z)) In the main text we use the following function The function θ(z) is holomorphic and has simple zeros at z = jτ + , j, ∈ Z. The following elliptic transformation properties follows easily from the definition, Modular properties of θ 0 and Γ e In the main text we also introduced the elliptic gamma function in (2.15) which obeys the quasi-periodicity property (3.8). The function Γ e (z, τ, σ) is meromorphic, it has simple zeros at z = (j + 1)τ + (k + 1)σ + and simple poles at z = −jτ − kσ + , j, k ∈ N, ∈ Z. Next, we introduce a set of identities that involve modular transformations of parameters τ , σ and z. Recall that both θ 0 and Γ e are invariant under the transformation z → z − k 0 for k 0 ∈ Z. We will combine this symmetry with the modular transformation properties to obtain new identities. For θ 0 we have, for every k 0 ∈ Z, Using (A.20), (A.21), we can also write this as The analogous identities for elliptic Gamma functions are, for Im( σ τ ) > 0, k 0 ∈ Z, which are minor variations of the ones presented in [27,49],

Factorization properties
By using the representations (C.1), (C.3) and basic trigonometric identities, one obtains the following "factorization" identity of θ 0 . For all m ∈ N we have Upon using this factorization identity, the symmetry of θ 0 under integer shifts of its second argument, and the modular identity (A.22), one obtains, for any k 0 ( ) ∈ Z, = 0, . . . , m−1, Upon using this factorization identity, the symmetry of Γ e under integer shifts of its last two arguments, and the appropriate limit of modular identity (A.24), one obtains, with τ ε = τ + ε, ε → 0, and for any k 0 ( ) ∈ Z, = 1, . . . , m,

B Elliptic extension: zeroes and poles
In this appendix we comment on the singularities of the elliptically extended action that was introduced in Section 3.2. The integrand of the SYM index (3.22) can be rewritten in terms of the function Q a,b which was introduced in (3.20) as, for u ∈ C, Here the Q 1,0 factors arise from the vector multiplet, and the Q − 1 3 , 1 3 factors arise from the chiral multiplets.
The action has singularities whenever the function Q has singularities or zeros. In Figure 8 we present the positions of zeros (blue) and singularities (red and green) of Q in the u-plane. The zeros come from the vector multiplet, the singularities come from the chiral multiplets. As Q − and it has no zero. It thus follows that Q − 1 −1 has two singularities of order 1 3 , located at the green and red points, respectively, in Figure 8. An analogous analysis shows that the only zeroes of (B.1) in the fundamental domain come from the vector multiplet contribution, i.e. from the first factor in right-hand side of (B.1) (the blue points in the figure).
For configurations that pass through the zeroes or singularities of Q, the effective action S(u) is +∞ or −∞, respectively. Indeed, the original contour of integration defining the SYM index crosses the zeroes of Q. In order to deal with this issue, one can turn on a regulator , following the prescription introduced in [4], (and which will be recalled in the following section). This regulator shifts the position of zeroes and singularities in the uplane by an amount proportional to , in a direction parallel to the dashed line, thus making the contour integrals that define the actions of saddles (0, 1) and (1, 0) well-defined 18 . We note that this regulator does not work if the saddle itself is along the direction of the (2, 1) saddle (indicated by the dashed lines in Figure 8). For this saddle a different regularization In the main text we calculated the effective action as integrals of elliptic functions constructed out of log θ 0 (u) and log Γ e (u), along various directions in the complex u-plane, using the double Fourier expansion in u provided by the Bloch formula (see the discussion in Section 4.1). In the first two subsections we develop an alternative method to integrate log θ 0 (u) and log Γ e (u), using certain series expansions that will be called (m, n) representations. We will often use the notion of a (m, n) ray in the u-plane, by which we mean a ray emanating from the origin, in direction of the vector mτ + n with m ∈ N and n ∈ Z In the last subsection we study the definition of the elliptic extensions Q from the perspective of (m, n) representations, thus obtaining a series representation for the even (in u) product of Q's. We end by briefly commenting upon an ambiguous overall phase function that appears in the definition of Q.

Basic series representations
We start by presenting the representations that are building blocks of our forthcoming analysis. For θ 0 that representation is [27] log θ 0 (z; where the series on the right-hand side is absolutely convergent for 0 < Im(z) < Im(τ ) .

Some notation
In the presentation that follows we use the notation for the real components of any complex number z along 1 and τ . This is very similar to the notation z = z 1 + z 2 τ that we use in the main text, the difference being that here in (C.5) we consider τ to be variable, and so in particular we can replace τ by mτ + n. We can write the components explicitly, for any τ and z, as It is clear from definition (C.5) that The (m, n) representations of log θ 0 The series representation (C.1) implies the following formula, which converges in the region Expanding z as in (C.5), we see that this convergence condition is equivalent to where we have used (C.7) to reach the second set of inequalities. For k 0 ∈ Z the inequalities (C.10) are solvable if and only if z ⊥τ / ∈ Z. For fixed k 0 the region in the z-plane given by (C.10) is an infinite ribbon parallel to, and enclosing, the ray τ . The unique value of k 0 that satisfies this condition is, The second equality in (C.11) follows from the fact that z ⊥τ / ∈ Z. Now, combining the formula (C.8) with the modular transformation (A.22) we obtain, in the region (C.10), (C.12) In exactly the same manner we can combine the basic series representations with the factorization formulas (A.27) to obtain, for any given (m, n), which converges when the following condition is satisfied for = 0, . . . , m − 1, (C.14) Representations (C.13) will be called (m, n) representations of log θ 0 (z).
In what follows we will encounter the combination u 2 log θ 0 (z I (u); τ ) − u 2 log θ 0 (z I (−u); τ ) , (C. 15) where the index I stands for either chiral (C) or vector (V ) multiplets, and z I (u) = u + r I 2 (2τ + 1) , (C. 16) with r I the R-charge of multiplet I. Next, we will use representations (C.13) in both terms in (C.15), for which the choice of k 0 is when the argument of the function is not an integer. We note that in the domain |u ⊥mτ +n | < κ(r I , , m, n) , and for the functions k 0 ( , z I (u)) and k 0 ( , z I (− u)) are both equal to and, in particular, they are independent of u.
Each value of R-charge r I defines the width of the ribbon-like domain of absolute convergence (C.18). The middle axis of this ribbon passes through the origin of u-plane, and is parallel to the vector mτ +n. For a theory with various building-block multiplets, we linearly combine the (m, n) representations of (C.15). The domain of absolute convergence of the (m, n) representation of such linear combinations is typically determined by the R-charge of one of the building-block multiplets. For instance, we can ask for the domain of absolute convergence of the (c, 0) representation of a linear combination of terms like (C.15), associated to N = 4 SYM, for which the spectrum of R-charges is r V = 2 and r C = 2 3 . For that case, from equality κ(r V , , m, 0) = 0 and (C. 18), it follows that the domain of absolute convergence of the latter linear combination of (m, 0) representations is empty. To deal with this issue we use the infinitesimal regulator = 0 + that was introduced in [14] and mentioned in Appendix B. For the vector and chiral multiplets, such regulator is introduced as the following infinitesimal deformation of the R-charge spectrum, r V = 2 − and r C = 2+ 3 . In that case, the (c, 0)-representation of the corresponding linear combination of (C.15) is absolutely convergent in |u ⊥cτ | < 2 = 0 + , i.e., in a ribbon of infinitesimal width directed along the (1, 0)-ray.
To summarize, after using the formula (C.13) we obtain the following absolutely convergent series representation in the domain (C.18) and for the choice of k 0 ( ) given in (C.21), Here the pre-factor F (m,n) (u) is and is purely imaginary and is independent of τ . After summing (C.25) over the matter content of N = 4 SYM, and taking the regulator to zero, one obtains where η is a real function. The index I runs over the labels of vector and three chiral multiplets of N = 4 SYM, each of them with their corresponding charge asignment r I . We have checked the following three properties in numerous examples (not proven): (a) that the coefficient η (m, n) is rational, (b) that it depends only on the ratio n/m, and (c) that m 2 η (m, n) ∈ Z (see Table 3).  Table 3: Table of values of η (m, n). Notice that m 2 η is always an integer and that η only depends on the combination n/m. The squared boxes mark the slots corresponding to pairs (m, n) that are proportional to the pair (2, 1) for which the regulator fails (see the last paragraph of Appendix B).
The (m, n) representations of log Γ e (z; τ, τ ) We now develop similar series representations for the function log Γ e (z; τ, τ ) using the basic series representation (C.3) for log Γ e (z; τ, σ) and the modular factorization identities (A.29).
We start with an identity analogous to (C.12) for the Γ e -function by combining (C.3) with the basic modular identity (A.24). The locus σ = τ , however, is not in the range of validity of (A.24), so we need to take a limit σ → τ . After substituting the two elliptic gamma functions in the right-hand side of (A.24) by their respective series representations (C.3) and taking the limit ε → 0 with σ = τ + ε in the resulting expressions, one obtains the following representation, for any k 0 ∈ Z, From domain of convergence of the series (C.3), given in (C.4), it follows that the series on the right-hand side of (C.27) are absolutely convergent in 0 < −z ⊥τ + k 0 < 1. We have numerically checked the veracity of (C.27) in this ribbon, using the defining product representation (2.15) of the Γ e -function. Representations (C.27) are called (m, n) representations of log Γ e (z; τ, τ ). Since our eventual goal is to use these formulas to represent the SYM index, wherein only even combinations of log Γ e (z I (u); τ, τ ) appear, we move directly to a representation of such functions which will be very useful for us in the following. For every (m, n), log Γ e (z I (u); τ, τ ) + log Γ e (z I (−u); τ, τ ) = − 2 F (m,n) (u) + ∞ j=1 2 j c 1 (j; m, n) g(ju; mτ + n) + c 2 (j; c, d) g(j; u, mτ + n) , (C. 28) which follows from the identities (A.29) and the definitions (C.16). Analogously to the (m, n) representations (C.22), the series in the right-hand side of (C.28) are absolutely convergent in domain (C.18) for the k 0 ( ) in (C.21). In Figure 9 we have plotted the real part of a couple of series representations (C.28) and compared them with the representation of the left-hand side of (C.28) that one obtains after using the product definition of log Γ e (z; τ, τ ). The pre-factor F (m,n) (u) is defined as (m−| −m+1|) B 3,3 z I (u)−k 0 ( )+ τ |mτ +n, mτ +n, −1 +(u → −u) .
For u along an (m, n) ray i.e. u = u 2 m (mτ + n), and several values of m, n and Rcharges r I , we have checked that the quantity is purely imaginary. After summing (C.34) over the matter content of N = 4 SYM, and taking the regulator to zero, one obtains  Table 4). The last property will be useful in the next appendix.  is only a function of the quotient n m . The squared boxes mark the slots corresponding to pairs (m, n) that are proportional to the pair (2, 1) for which the regulator fails (see the last paragraph of Appendix B).
The (m, n) representations of the elliptic extension In Section 3 we showed that although Γ e (u) is not elliptic invariant one can dress it with θ 0 (u) u 2 to produce the elliptic function Q −1 (u) that coincides with Γ e (u) at u 2 = 0, and is holomorphic in τ for finite values of u 2 . The function Q was represented by the double Fourier series expansion (3.19) which includes the τ -independent purely imaginary term iΨ which was not fixed. In this subsection, we find an alternate representation of the even-in-u product (from now even product) of Q's that follows from the (m, n) representations of θ 0 and Γ e that we have developed in the previous appendices. As before, we work with the even combination Γ e (z I (u); τ, τ )Γ e (z I (−u); τ, τ ) appearing in the index formula. We show that, along an (m, n) ray, the even product of Q's, coincides with the (m, n) representation of the even product of Γ e 's, once the non-periodic part g of the Γ e 's is projected out 19 . We end this section by using this result to deduce an identity that will be used in Section E to relate the elliptic and meromorphic actions of (m, n) configurations.
We start by proving a useful identity. Upon comparing the coefficient of the third term in (C.28) with the coefficient of the second term in (C.22), we conclude that along an (m, n) ray i.e. for u = u 2 m (mτ + n), This is the (m, n) representation we were looking for. It shows that along an (m, n) ray, the even sum of elliptic functions − log Q in the left-hand side, can be seen, up to a redefinition of Ψ(u) 21 , as the result of projecting out the non periodic part g in (m, n) representation (C.28) of the corresponding even sum of log Γ e . Notice that Ψ(u) in (C.41) is still ambiguous, but it is elliptic, satisfies (C.38), and it does not depend on τ . Equation (C.41) and the definition of the elliptic extension of the action S(u) given in (3.25), implies the following identity S eff (m, n; τ ) = 2 N 2 I F (m,n) (0) + πiN 2 ϕ(m, n) , (C. 42) where ϕ(m,n) 4 ∈ R are the Fourier coefficients of Ψ(u) along the (m, n) ray. They are arbitrary constants that could be matched to ϕ(m, n). We have checked that the right-hand side of (C.42) coincides with the right-hand side of (4.13) for several values of m ∈ N, n ∈ Z and gcd(m, n) = 1. This will be used in Section E to match the elliptic and meromorphic actions (up to a τ -independent imaginary quantity).

D The role of the saddle (0, 0)
We now clarify the role (and the lack of role) of the saddle (0, 0). First let's turn off the regulator so that u i = 0 strictly. In this case the vector multiplet has z V (0) = 2τ + 1 (see Equation (C.16)). From the product representation (2.15) we see that this value of z V is a zero of the elliptic gamma function and therefore the index actually vanishes. Now let's turn on the regulator . From the representation (C.28) we see that, for small enough for fixed τ ∈ H, the quantity −Re S(0, 0) is negative and thus the contribution of saddle (0, 0) is suppressed. These observations are summarized in Figure 10. For the (0, 0) saddle, all values of (m, n) in the representation (C.28) clearly give the same answer. We have used m = 1, n = 0 to generate the figure.
In contrast, for fixed and in the Cardy-like limit (τ → 0 with τ 1 < 0) the quantity −Re S(0, 0) becomes positive. Furthermore, as shown in [14], the full action asymptotes to the action of the black hole. In Section 4.5 we have revisited and confirmed the results of [14], as to how any asymptotic limit τ → 0 of the action of the black hole saddle (1, 0) coincides, in that very same limit, with the -regulated action of the (0, 0) saddle.

E The (m, n) meromorphic action
In this appendix we evaluate the analytic continuation of the integrand of (2.16) on the (m, n) configurations u i = (mτ + n) i N and compare it with the valuation of the elliptic extension of the action calculated in Section 4 on the same configurations. To ease the language, we will call the negative of the logarithm of these two quantities, the meromorphic action and the elliptic action, respectively. We find that, remarkably, although The meromorphic action is defined as the logarithm of the integrand of (2.16) at large N , i.e. (the relevant notation has been summarized near (C.16)), S mero (u) := − I i,j log Γ e u ij + r I 2 (2τ + 1); τ, τ . (E.1) In particular, using the regulator discussed in the previous appendix, we should take r V = 2 − and r C = 2+ 3 , and → 0 + at the end. In order to calculate the meromorphic action we simply evaluate the series representation of the elliptic gamma functions developed in the previous section on the (m, n) configuration, We recall that the (m, n) series representation (C.28) has three parts, the pre-factor F, the periodic part (governed by g(j u; τ )), and the non-periodic part ( g(j u; τ )). We proceed to evaluate them on the configuration (E.2). From the identity, are when k equals integer multiples of N . Noting that the coefficients c 1 (k; m, n) (C.31) are exponentially suppressed as a function of k for large k, we have that the double sum (E.4) behaves like e −N and therefore vanishes in the large-N limit.
The non-periodic function g(u; mτ + n) vanishes as well when evaluated on (E.2). This follows from the identity, k ∈ Z, The only term that contributes to the action (E.7) is the pre-factor F. Using Formula (C. 35), summing over all the Γ e -functions that contribute to the index, and using the fact that on the (m, n) ansatz we obtain that the value of the meromorphic action S mero (u) on the (m, n) configuration is S mero eff (m, n; τ ) = 2 N 2 I F (m,n) (0) − π i 3 N 2 m 2 η(m, n) , (E.7) where 2 F (m,n) (0) is defined by (C.29), and the index I runs over the matter content of N = 4 SYM. The real function η(m, n) is the pure imaginary term discussed in (C.35). Now, recall that the definition of action S is always ambiguous by the addition of an integer multiple of 2πi. Using the property that m 2 η ∈ Z (see the comments below Equation (C.35)), we see that the η-term contributes to the free energy of the saddle only as N 2 m 2 η mod 3. Upon comparing the formulas (E.7) and (C.42), it follows that the elliptic and meromorphic (m, n) effective actions, for relatively prime (m, n), coincide up to a τ -independent imaginary quantity. Thus, for N prime, the meromorphic action (E.7) can be written in the form 23 S mero eff (m, n; τ ) = N 2 π i 27 m 2(mτ + n) + χ 1 (m + n) 3 (mτ + n) 2 + N 2 π i ϕ mero (m, n) , (E.9) where ϕ mero (m, n) for every coprime (m, n). In particular we find that ϕ mero (1, 0) = 0 as mentioned below Equation (4.28). We end this section with a comment about a subtle issue regarding the comparison between the elliptic and meromorphic approaches. Recall from the discussion in Section 2 that, for N prime, the saddles of the elliptic action are labelled by pairs (m, n) such 23 For N not a prime, recall that there are other solutions along (m, n) rays denoted by (K|m, n) with K|N , and for m, n co-prime. We can also compute their large-N action by similar methods to be The gap between the actions of (N |m, n) ≡ (m, n) and (K|m, n) vanishes exponentially fast for large values of K ∼ N because the coefficient c1 is exponentially suppressed for large values of its first argument (theregulator is important to do such calculations). It would be interesting to understand the meaning of the large-N limits of these solutions.
that gcd(m, n) = 1. We could have chosen to reparameterize the saddle (m, n) as (km, kn), where k is any integer-even depending on m, n-and this would not have made any difference to the elliptic action because elliptic symmetry forces the action to be a function of the quotient n/m, and not of m and n independently. In the meromorphic approach, however, we find that the action is not a function of n/m only. In particular, we have checked in several examples that the first summand in the right-hand side of (E.7) only depends on the ratio n/m but the second summand in the right-hand side of (E.7) is not just a function of n/m. If we think of this reparameterization as a "local rescaling" of the space of saddles with a function k(m, n), we are drawn to conclude that the meromorphic action is anomalous under such a local scale-reparametrization. Perhaps the resolution of this puzzle is tied to a better understanding of the imaginary term in the action.

F Relation to the N = 4 SYM Bethe Ansatz formula
In this subsection we make contact with the approach of [7,8]. The Bethe Ansatz formula put forward in these papers has the following form, Here the function Z(u) is the analytic continuation in u of the integrand of the index (2.14)-which is precisely what we discussed in Appendix E, i.e. in our language Z(u) = exp −S mero eff (u) defined in (E.1). The factor H −1 is a Jacobian factor which is subleading in 1 N compared to the effective action. The factor κ N is precisely the pre-factor in the righthand side of our equation (2.16), which is also sub-leading in where f is a certain combination of θ 0 -functions. Solutions to these equations were found in [50], [51], [7] and, in particular, it was shown that the (m, n) configuration i.e. u ij = i−j N (mτ +n) with gcd(m, n) = 1 is a solution to the BAEs. It was also shown that the (1, 0) configuration 24 has the same entropy as the AdS 5 black hole. From our point of view, we notice that the logarithm of the BAEs (F.2) is precisely of the form (2.7) for some V (u) that is even in u and double-periodic on the lattice generated by 1 and τ . Thus, in virtue of our discussion in Section 2, it follows that all the (m, n) 24 The conventions of [8] can be matched to the ones used in this paper under the identifications ∆1,2,3 = ∆ = (2τ there − 1)/3, τ here ↔ τ there − 1, and I here (τ here , τ here ; n0 = −1) ↔ I there (∆1, ∆2, ∆3, τ there , τ there ). This leads to the following map between saddles (m, n) there ↔ (m, n + 1) here . configurations discussed in this paper are Bethe roots. Further, from the discussion in Appendix E, it follows that the action of these roots is given by the formula (E.9). As mentioned below that equation, it agrees with the elliptic action introduced in this paper evaluated at the same saddle, up to a τ -independent imaginary term.