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.
Highlights
There has recently been remarkable progress in applications of the localization technique [1] to supersymmetric gauge theories, notably in dimensions D ≥ 3: In D = 4, the Seiberg–Witten prepotential of N = 2 supersymmetric quantum chromodynamics (QCD) [2] was directly evaluated, and the partition functions and BPS Wilson loops of the N = 2 and N = 4 supersymmetric Yang–Mills theories (SYM) were reduced to eigenvalue integrals of the matrix model type [3], providing, in particular, proof of the earlier results on a Wilson loop in the N = 4 SYM [4,5]
In D = 3, similar results were obtained for the partition functions and BPS Wilson loops of N = 2 supersymmetric Chern–Simons-matter (CSM) theories [6,7], including the N = 6 superconformal theories constructed by Aharony, Bergman, Jafferis, and Maldacena (ABJM) [8,9]
In this paper we focus on the partition function of the ABJ theory, i.e., the N = 6 supersymmetric U (N1)k × U (N2)−k CSM theory, which generalizes the equal rank N1 = N2 case of the ABJM theory [9]
Summary
[12,20] in line with the conjecture on (2, 0) 6d superconformal theory compactified on S1 [21,22], despite far a lack of precise agreement with its AdS7 dual It should, be noted that the utility of the localization method, unlike the integrability [23], is limited to a class of supersymmetric observables, such as the partition function and BPS Wilson loops. Mariño and Putrov developed a more elegant approach, the Fermi gas approach, without making any use of matrix model techniques or holomorphic anomaly equations, to compute directly the partition functions of N = 3 and N = 2 CSM theories including the ABJM theory [31,32] They found, in particular, a universal Airy function behavior for the N = 3 theories at large N in the small k M-theory regime.
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