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We study $\mathcal{N} = 2$ supersymmetric gauge theories on $\mathbb{RP}^2\times \mathbb{S}^1$ and compute the superconformal index by using thelocalization technique. We consider not only the round real projective plane$\mathbb{RP}^2$ but also the squashed real projective plane $\mathbb{RP}^2_b$which turns back to $\mathbb{RP}^2$ by taking a squashing parameter $b$ as $1$.In addition, we found that the result is independent of the squashing parameter$b$. We apply our new superconformal index to the check of the simplest 3dmirror symmetry, i.e. the equivalence between the $\mathcal{N}=2$ SQED and theXYZ model on $\mathbb{RP}^2 \times \mathbb{S}^1$. We prove it by using amathematical formula called the $q$-binomial theorem. We comment on the$\mathcal{N}=4$ version of mirror symmetry, mirror symmetry via generalizedindices, and possibilities of generalizations from mathematical viewpoints.

Superconformal index on $\mathbb{RP}^2 \times \mathbb{S}^1$ and mirror symmetry