Exact results on ABJ theory and the refined topological string

Abstract

Abstract: We study the partition function of the ABJ theory, which is the (Formula presentef)N = 6 $$ \mathcal{N}=6 $$(/Formula presentef) superconformal Chern-Simons matter theory with gauge group U(N) × U(N + M) and Chern-Simons levels (k, −k). We exactly compute the ABJ partition function on a three sphere for various k, M and N via the Fermi gas approach. By using these exact data, we show that the ABJ partition function is completely determined by the refined topological string on local (Formula presentef)ℙ 1 × ℙ 1 $$ {\mathbb{P}}^1\times {\mathbb{P}}^1 $$(/Formula presentef), including membrane instanton effects in the M-theory dual.