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

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 $\mathcal{C}\mathcal{P}$. We find an interesting phenomenon that two matter multiplets coupled to the $\mathcal{C}\mathcal{P}$-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.

28 pages, formulas modified and comments added

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• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
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Cited by in corpus (3)

• Pentagon identities arising in supersymmetric gauge theory computations
• Varieties of Abelian mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$
• Single-flavor Abelian mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$