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

TL;DR

This paper derives a systematic, duality-invariant non-perturbative large-$N$ expansion for integrated correlators in $ =4$ supersymmetric Yang-Mills theory, revealing their modular structure and instanton contributions.

Contribution

It provides generating functions for integrated correlators of any classical gauge group and proves conjectured formulas, advancing understanding of finite and large-$N$ properties.

Findings

Established duality-invariant non-perturbative large-$N$ expansion

Derived generating functions for classical gauge groups

Identified instanton contributions from $(p, q)$-strings

Abstract

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.