Superconformal index onRP2×S1and mirror symmetry

Abstract

We study $\mathcal{N}=2$ supersymmetric gauge theories on ${\mathbb{R}\mathbb{P}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{1}$ and compute the superconformal index by using the localization technique. We consider not only the round real projective plane ${\mathbb{R}\mathbb{P}}^{2}$ but also the squashed real projective plane ${\mathbb{R}\mathbb{P}}_{b}^{2}$ which turns back to ${\mathbb{R}\mathbb{P}}^{2}$ by taking a squashing parameter $b$ as 1. In addition, we find that the result is independent of the squashing parameter $b$. We apply our new superconformal index to check the simplest case of 3D mirror symmetry, i.e., the equivalence between the $\mathcal{N}=2$ supersymmetric quantum electrodynamics (SQED) and the $XYZ$ model on ${\mathbb{R}\mathbb{P}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{1}$. We prove it by using a mathematical formula called the $q$-binomial theorem. We also comment on the $\mathcal{N}=4$ version of mirror symmetry, mirror symmetry via generalized indices, and possibilities of generalizations from mathematical viewpoints.