Modular-invariant large-$N$ completion of an integrated correlator in $\mathcal{N}=4$ supersymmetric Yang-Mills theory

The use of supersymmetric localisation has recently led to modular covariant expressions for certain integrated correlators of half-BPS operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory with a general classical gauge group $G_N$. Here we determine generating functions that encode such integrated correlators for any classical gauge group and provide a proof of previous conjectured formulae. This gives a systematic understanding of the relation between properties of these correlators at finite $N$ and their expansions at large $N$. In particular, it determines a duality-invariant non-perturbative completion of the large-$N$ expansion in terms of a sum of novel non-holomorphic modular functions. These functions are exponentially suppressed at large $N$ and have the form of a sum of contributions from coincident $(p,q)$-string world-sheet instantons.