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

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.