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

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.