Single-flavor Abelian mirror symmetry on $\mathbb{RP}^{2} \times \mathbb{S}^{1}$

TL;DR

This paper uses localization to compute the superconformal index on unorientable space $ ext{RP}^2 imes S^1$, providing a rigorous proof of Abelian mirror symmetry in this setting.

Contribution

It introduces two compatible parity conditions on $ ext{RP}^2$ and demonstrates Abelian mirror symmetry through the exact superconformal index calculation.

Findings

Derived superconformal index on $ ext{RP}^2 imes S^1$ using localization.

Established two types of Abelian mirror symmetry on unorientable space.

Validated mirror symmetry with exact index computations.

Abstract

The supercoonformal index on $\mathbb{RP}^{2} \times \mathbb{S}^{1}$ can be derived exactly by the localization technique and applied to the direct proof of Abelian mirror symmetry. We find two sets of parity conditions compatible with the unorientable property of $\mathbb{RP}^{2}$ and then rigorously show two kinds of Abelian mirror symmetry via the index on $\mathbb{RP}^{2} \times \mathbb{S}^{1}$.