A novel representation of an integrated correlator in $\mathcal{N}=4$ SYM theory

TL;DR

This paper presents a new lattice sum representation of an integrated correlator in $ ext{N}=4$ SYM that is invariant under S-duality and satisfies a novel Laplace equation, connecting different gauge groups and matching known properties.

Contribution

It introduces a novel two-dimensional lattice sum form of the integrated correlator that makes S-duality manifest and relates correlators across different gauge groups via a Laplace equation.

Findings

Reproduces known perturbative and non-perturbative properties for finite N

Extends large-N expansion conjectures

Satisfies a new Laplace equation linking different gauge groups

Abstract

An integrated correlator of four superconformal stress-tensor primaries of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory (SYM), originally obtained by localisation, is re-expressed as a two-dimensional lattice sum that is manifestly invariant under $SL(2,\mathbb{Z})$ S-duality. This expression is shown to satisfy a novel Laplace equation in the complex coupling constant $\tau$ that relates the $SU(N)$ integrated correlator to those of the $SU(N+1)$ and $SU(N-1)$ theories. The lattice sum is shown to precisely reproduce known perturbative and non-perturbative properties of $\mathcal{N}=4$ SYM for any finite $N$, as well as extending previously conjectured properties of the large-$N$ expansion.