Exact results on ABJ theory and the refined topological string

Abstract

We study the partition function of the ABJ theory, which is the $$ \mathcal{N}=6 $$ 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 $$ {\mathbb{P}}^1\times {\mathbb{P}}^1 $$ , including membrane instanton effects in the M-theory dual.