Modular-invariant large- N completion of an integrated correlator in 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 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 previously 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.