The Partition Function of ABJ Theory
Abstract
We study the partition function of the N=6 supersymmetric U(N1)k x 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) x U(N2) 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 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 x U(N)-k CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for |N1-N2|>k in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples.
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