Exact large $N$ expansion of mass deformed ABJM theory on squashed sphere

TL;DR

This paper derives an exact large N expansion of the mass deformed ABJM theory on a squashed sphere, revealing an Airy function structure and providing new insights into its partition function.

Contribution

It presents the first exact large N expansion of the mass deformed ABJM theory on a squashed sphere, including an all-order perturbative series and a new formula for the constant factor.

Findings

Partition function expressed as an Airy function.

Analytic derivation of the 1/N perturbative expansion.

Exact values computed for small N.

Abstract

In this paper we study the partition function of the mass deformed ABJM theory on a squashed three sphere. In particular, we focus on the case with the Chern-Simons levels being $\pm 1$ and apply a duality between this theory and the $\mathcal{N}=4$ $\mathrm{U}\left(N\right)$ super Yang-Mills theory with an adjoint hypermultiplet and a fundamental hypermultiplet. For a special mass parameter depending on the squashing parameter, we find that the partition function can be written as that of an ideal Fermi gas with a non-trivial density matrix. By studying this density matrix, we analytically derive the all order perturbative expansion of the partition function in $1/N$, which turns out to take the form of the Airy function. Our results not only align with previous findings and conjectures but also lead to a new formula for the overall constant factor of the partition function. We also study the exact values of the partition function for small but finite values of $N$.