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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 $τ$. The saddle-point configurations are labelled by points $(m,n)$ on the lattice $Λ_τ= \mathbb{Z} τ+\mathbb{Z}$ 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 $\mathbb{C}/Λ_τ$. 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 $τ$ is close to the origin, and there are new saddles that dominate when $τ$ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit $τ\to 0$.