Novel Representation of an Integrated Correlator in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> Supersymmetric Yang-Mills Theory

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