ABJM theory as a Fermi gas

TL;DR

This paper introduces a novel Fermi gas approach to analyze the partition functions of supersymmetric Chern-Simons-matter theories, providing exact results and insights into non-perturbative effects in M-theory and type IIA string theory.

Contribution

It develops a new method reformulating matrix models as ideal Fermi gases, enabling exact calculations of N^{3/2} behavior and 1/N corrections for ABJM and related theories.

Findings

Derivation of N^{3/2} scaling for ABJM theory.

Full series of 1/N corrections expressed via Airy functions.

Explicit computation of non-perturbative D2-brane instanton effects.

Abstract

The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.