The partition function of ABJ theory

We study the partition function of the supersymmetric U(N 1) k × 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) × U(N 2) 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 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 supersymmetric U(N) k × 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.