Varieties of Abelian mirror symmetry on ℝ ℙ 2 × S 1 $$ mathrm{mathbb{R}}{mathrm{mathbb{P}}}^2times {mathbb{S}}^1 $$

We study 3d mirror symmetry with loop operators, wilson loop and vortex loop, and multi-flavor mirror symmetry through utilizing the $$ mathrm{mathbb{r}}{mathrm{mathbb{p}}}^2times {mathbb{s}}^1 $$ 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 $$ mathrm{s}mathrm{l}left(2,mathrm{mathbb{z}}right) $$ 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 $$ mathcal{c}mathcal{p} $$ -type are interchanged under mirror symmetry. we will also comment on the t-operation.