Interacting fermions and N=2 Chern-Simons-matter theories

TL;DR

This paper reformulates N=2 Chern-Simons-matter theories' partition functions as interacting fermion gases, enabling analysis via thermodynamic approximations and revealing connections to known solutions and Airy functions.

Contribution

It introduces an interacting fermion gas framework for N=2 theories, extending the ideal gas approach of N=3 theories, and analyzes large N behavior with novel integral equations.

Findings

Partition function modeled as interacting fermions in 1D.

Large N limit analyzed with Hartree and Thomas-Fermi approximations.

Partition function with no long-range forces approximated by an Airy function.

Abstract

The partition function on the three-sphere of N=3 Chern-Simons-matter theories can be formulated in terms of an ideal Fermi gas. In this paper we show that, in theories with N=2 supersymmetry, the partition function corresponds to a gas of interacting fermions in one dimension. The large N limit is the thermodynamic limit of the gas and it can be analyzed with the Hartree and Thomas-Fermi approximations, which lead to the known large N solutions of these models. We use this interacting fermion picture to analyze in detail N=2 theories with one single node. In the case of theories with no long-range forces we incorporate exchange effects and argue that the partition function is given by an Airy function, as in N=3 theories. For the theory with g adjoint superfields and long-range forces, the Thomas-Fermi approximation leads to an integral equation which determines the large N, strongly coupled R-charge.