The partition function of ABJ theory

Abstract

We study the partition function of the N = 6 supersymmetric U(N 1)k × U(N2)-k Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N1) × U(N2) lens space matrix model exactly. The result can be expressed as a product of the q-deformed Barnes G-function and a generalization of the multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N2 to -N2. The answer is given by min(N1, N2)- dimensional integrals and generalizes the "mirror description" of the partition function of the ABJM theory, i.e. the N = 6 supersymmetric U(N) k × U(N)-k CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for |N1 - N 2| > k in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples. © The Author(s) 2013.