Superconformal index on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ and mirror symmetry

Abstract

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

47 pages; v2: discussions and references added; v3: discussions added and published in Physical Review D

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