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

Understand

We study $\mathcal{N} = 2$ supersymmetric gauge theories on $\mathbb{RP}^2 \times \mathbb{S}^1$ and compute the superconformal index by using the localization 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 3d mirror symmetry, i.e.