Varieties of Abelian mirror symmetry on $\mathbb{RP}^2 \times \mathbb{S}^1$

TL;DR

This paper explores 3d mirror symmetry involving loop operators and multi-flavor cases on the manifold $ ext{RP}^2 imes ext{S}^1$, revealing new structural insights using mod 2 Fourier analysis and SL(2,Z) actions.

Contribution

It introduces a novel analysis of mirror symmetry on $ ext{RP}^2 imes ext{S}^1$ using mod 2 Fourier analysis and clarifies the interchange of parity conditions under mirror symmetry.

Findings

Parity conditions $ ext{P}$-type and $ ext{CP}$-type are exchanged under mirror symmetry.

The mod 2 Fourier analysis is key to understanding the structure of the index formula.

SL(2,Z) actions, including the S-operation, are relevant in the context of 3d SCFTs.

Abstract

We study 3d mirror symmetry with loop operators, Wilson loop and Vortex loop, and multi-flavor mirror symmetry through utilizing the $\mathbb{RP}^2 \times \mathbb{S}^1$ index formula. The key identity which makes the above description work well is the mod 2 version of the Fourier analysis, and we study such structure, the S-operation in the context of a SL$(2,\mathbb{Z})$ action on 3d SCFTs. We observed that two types of the parity conditions basically associated with gauge symmetries which we call $\mathcal{P}$-type and $\mathcal{CP}$-type are interchanged under mirror symmetry. We will also comment on the T-operation.