Exact results on ABJ theory and the refined topological string

We study the partition function of the ABJ theory, which is the N=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=6 $$\end{document} 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 ℙ1×ℙ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{P}}^1\times {\mathbb{P}}^1 $$\end{document}, including membrane instanton effects in the M-theory dual.