An Airy Tale at Large $N$

TL;DR

This paper demonstrates that the large N sphere partition function of certain 3d SCFTs can be expressed as an Airy function, revealing deep connections to topological string theory and M-theory corrections.

Contribution

It proposes and provides evidence that the all-order large N expansion of the partition function resums into an Airy function, linking holographic SCFTs to topological strings and Fermi gas models.

Findings

Partition function expressed as an Airy function at large N

Numerical and saddle point evidence supporting the conjecture

Connections established to topological string theory and M-theory corrections

Abstract

We study the sphere partition function of 3d $\mathcal{N}=2$ holographic SCFTs arising on the worldvolume of $N$ coincident M2-branes in the presence of squashing and real mass deformations. We argue that the all-order large $N$ perturbative expansion of the partition function resums into a compact expression in terms of an Airy function. We provide ample evidence for this conjecture using numerical methods, saddle point analysis in the 't Hooft limit, and the relation of the squashed sphere partition function to the Cardy-like limit of the superconformal index. We also discuss the relation of our results to topological string theory and the statistical mechanics of Fermi gases. Our conjecture has a bearing on the AdS/CFT correspondence and the structure of perturbative M-theory corrections to 11d supergravity which we discuss.