Supersymmetric phases of 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 x S-1 with one chemical potential tau. The saddle-point configurations are labelled by points (rn, n) on the lattice Lambda(tau) = ZT + Z with gcd(rn, 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 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 -> 0.