  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.

High Energy Physics - Theory