Exact Results on the ABJM Fermi Gas

TL;DR

This paper provides exact computations of the ABJM Fermi gas partition function, revealing polynomial structures in 1/pi and establishing methods for both analytical and numerical analysis of the system.

Contribution

It introduces a novel method to compute the ABJM grand partition function exactly and reduces the eigenvalue problem to diagonalizing a Hankel matrix.

Findings

Exact partition functions up to N=9 for k=1

Partition functions expressed as polynomials in 1/pi with rational coefficients

Development of TBA-type integral equations for numerical computation

Abstract

We study the Fermi gas quantum mechanics associated to the ABJM matrix model. We develop the method to compute the grand partition function of the ABJM theory, and compute exactly the partition function Z(N) up to N=9 when the Chern-Simons level k=1. We find that the eigenvalue problem of this quantum mechanical system is reduced to the diagonalization of a certain Hankel matrix. In reducing the number of integrations by commuting coordinates and momenta, we find an exact relation concerning the grand partition function, which is interesting on its own right and very helpful for determining the partition function. We also study the TBA-type integral equations that allow us to compute the grand partition function numerically. Surprisingly, all of our exact results of the partition functions are written in terms of polynomials of 1/pi with rational coefficients.