Abelian 3d mirror symmetry on $\mathbb{RP}^2 \times \mathbb{S}^1$ with $N_f=1$

TL;DR

This paper introduces a new 3d superconformal index on $ ext{RP}^2 imes ext{S}^1$, explores parity conditions, and demonstrates their role in 3d mirror symmetry involving matter merging effects.

Contribution

It provides a generic formula for the superconformal index with multiple U(1) symmetries and investigates mirror symmetry on $ ext{RP}^2 imes ext{S}^1$ with novel parity conditions and matter merging phenomena.

Findings

Derived a general formula for the index with arbitrary U(1) symmetries.

Identified two consistent parity conditions, $ ext{P}$ and $ ext{CP}$.

Discovered matter merging effects crucial for mirror symmetry agreement.

Abstract

We consider a new 3d superconformal index defined as the path integral over $\mathbb{RP}^2 \times \mathbb{S}^1$, and get the generic formula for this index with arbitrary number of U$(1)$ gauge symmetries via the localization technique. We find two consistent parity conditions for the vector multiplet, and name them $\mathcal{P}$ and $\mathcal{CP}$. We find an interesting phenomenon that two matter multiplets coupled to the $\mathcal{CP}$-type vector multiplet merge together. By using this effect, we investigate the simplest version of 3d mirror symmetry on $\mathbb{RP}^2 \times \mathbb{S}^1$ and observe four types of coincidence between the SQED and the XYZ model. We find that merging two matters plays an important role for the agreement.