Quantization conditions and functional equations in ABJ(M) theories

TL;DR

This paper derives an exact spectral determinant and WKB quantization condition for ABJ(M) theories' partition function, revealing new functional equations relating different gauge group ranks.

Contribution

It provides a novel exact expression for the spectral determinant of the ABJ(M) Hamiltonian, extending previous maximally supersymmetric results.

Findings

Exact spectral determinant expression for ABJ(M) theories.

WKB quantization condition matches numerical results.

Functional equations relate spectral determinants across gauge group ranks.

Abstract

The partition function of ABJ(M) theories on the three-sphere can be regarded as the canonical partition function of an ideal Fermi gas with a non-trivial Hamiltonian. We propose an exact expression for the spectral determinant of this Hamiltonian, which generalizes recent results obtained in the maximally supersymmetric case. As a consequence, we find an exact WKB quantization condition determining the spectrum which is in agreement with numerical results. In addition, we investigate the factorization properties and functional equations for our conjectured spectral determinants. These functional equations relate the spectral determinants of ABJ theories with consecutive ranks of gauge groups but the same Chern-Simons coupling.