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 $S^3 \times S^1$withonechemicalpotential$\tau$. 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 $\mathbb{C}\mathrm{/}{\mathrm{\Lambda }}_{\tau }$. 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 $\tau \to 0$.