Supersymmetric phases of N = 4 SYM at large N

We show the existence of an infinite family of complex saddle-points at large N, for the matrix model of the superconformal index of SU(N) N = 4 super Yang-Mills theory on S3 × S1 with one chemical potential τ. The saddle-point configurations are labelled by points (m,n) on the lattice Λτ = Z τ + Z with 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 C/Λτ . The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and its values at (m,n) saddles are determined by Fourier averages of the latter along directions of the torus. The actions of (0,1) and (1,0) agree with that of pure AdS5 and the Gutowski-Reall AdS5 black hole, respectively. The actions of the other saddles take a surprisingly simple form. Generically, they carry non vanishing entropy. The Gutowski-Reall black hole saddle dominates the canonical ensemble when τ is close to the origin, and other saddles dominate when τ approaches rational points.