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

Understand

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.