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

TL;DR

This paper identifies complex saddle-points in the matrix model for the superconformal index of 4d N=4 SYM at large N, relating their actions to elliptic functions and revealing connections to AdS black holes and modular forms.

Contribution

It introduces a family of saddle-points labeled by lattice points, linking their actions to elliptic dilogarithms and Eisenstein series, and explores their dominance in different regimes.

Findings

Saddle-points are labeled by lattice points (m,n) with gcd 1.

Actions at specific points match known AdS and black hole solutions.

New saddle-points dominate near rational points of the modular parameter.

Abstract

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. 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 $\tau\to 0$.