Index for three dimensional superconformal field theories with general R-charge assignments

We derive a general formula of an index for three dimensional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} = 2 $\end{document} super-conformal field theories with general R-charge assignments to chiral multiplets by using the localization method in S2 × S1 background. As examples we compute the index for theories in a few mirror pairs, and confirm the agreement of the indices in each mirror pair.