The Partition Function of ABJ Theory

TL;DR

This paper computes the exact partition function of the ABJ theory, a supersymmetric Chern-Simons-matter model, using lens space matrix models and analytic continuation, confirming dualities and supersymmetry breaking predictions.

Contribution

It provides a novel exact expression for the ABJ partition function through lens space matrix models and analytic continuation, extending the mirror description of ABJM theory.

Findings

Partition function expressed via q-deformed Barnes G-function.

Reproduces perturbative expansions accurately.

Vanishes when |N_1 - N_2| > k, consistent with supersymmetry breaking.

Abstract

We study the partition function of the N=6 supersymmetric U(N_1)_k x U(N_2)_{-k} Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N_1) x U(N_2) lens space matrix model exactly. The result can be expressed as a product of q-deformed Barnes G-function and a generalization of multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N_2 to -N_2. The answer is given by min(N_1,N_2)-dimensional integrals and generalizes the "mirror description" of the partition function of the ABJM theory, i.e. the N=6 supersymmetric U(N)_k x U(N)_{-k} CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for |N_1-N_2|>k in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples.