Abelian 3d mirror symmetry on ℝℙ2×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathbb{R}}{\mathrm{\mathbb{P}}}^2\times {\mathbb{S}}^1 $$\end{document} with Nf = 1

We consider a new 3d superconformal index defined as the path integral over ℝℙ2×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathbb{R}}{\mathrm{\mathbb{P}}}^2\times {\mathbb{S}}^1 $$\end{document}, 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 P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{P} $$\end{document} and CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C}\mathcal{P} $$\end{document}. We find an interesting phenomenon that two matter multiplets coupled to the CP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{C}\mathcal{P} $$\end{document}-type vector multiplet merge together. By using this effect, we investigate the simplest version of 3d mirror symmetry on ℝℙ2×S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathrm{\mathbb{R}}{\mathrm{\mathbb{P}}}^2\times {\mathbb{S}}^1 $$\end{document} 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.