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

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.