An integrated correlator of four superconformal stress-tensor primaries of $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory (SYM), originally obtained by localization, is reexpressed 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 nonperturbative properties of $\mathcal{N}=4$ SYM for any finite $N$, as well as extending previously conjectured properties of the large-$N$ expansion.

• Received 19 February 2021
• Accepted 30 March 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.161601

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Published by the American Physical Society