Large-$N$ integrated correlators in $\mathcal{N}=4$ SYM: when resurgence meets modularity

TL;DR

This paper develops a modular invariant Borel resummation of large-$N$ integrated correlators in $ =4$ SYM, revealing non-perturbative sectors through resurgence and connecting to spectral theory, thus deepening understanding of non-perturbative effects in gauge theories.

Contribution

It introduces a modular invariant Borel resummation method for large-$N$ correlators, extracting exact non-perturbative data via resurgence and linking it to spectral theory.

Findings

Resurgent analysis yields exact non-perturbative sectors.

Modular invariant resummation reduces to known genus expansions.

Spectral theory encodes the same non-perturbative data.

Abstract

Exact expressions for certain integrated correlators of four half-BPS operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $SU(N)$ have been recently obtained thanks to a beautiful interplay between supersymmetric localisation and modular invariance. The large-$N$ expansion at fixed Yang-Mills coupling of such integrated correlators produces an asymptotic series of perturbative terms, holographically related to higher derivative interactions in the low energy expansion of the type IIB effective action, as well as exponentially suppressed corrections at large $N$, interpreted as contributions from coincident $(p,q)$-string world-sheet instantons. In this work we define a manifestly modular invariant Borel resummation of the perturbative large-$N$ expansion of these integrated correlators, from which we extract the exact non-perturbative large-$N$ sectors via resurgence analysis. Furthermore, we show that in the 't Hooft limit such modular invariant non-perturbative completions reduce to known resurgent genus expansions. Finally, we clarify how the same non-perturbative data is encoded in the decomposition of the integrated correlators based on $\rm{SL}(2,\mathbb{Z})$ spectral theory.