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

Abstract

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 ${\mathbb{RP}}_b^2$ 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. the equivalence between the $\mathcal{N}=2$ SQED and the XYZ model on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$. We prove it by using a mathematical formula called the $q$-binomial theorem. We comment on the $\mathcal{N}=4$ version of mirror symmetry, mirror symmetry via generalized indices, and possibilities of generalizations from mathematical viewpoints.

arXiv comment47 pages; v2: discussions and references added; v3: discussions added and published in Physical Review D

References in corpus (34)

• Localization of gauge theory on a four-sphere and supersymmetric Wilson loops
• An Index for 4 dimensional Super Conformal Theories
• Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
• Mirror Symmetry in Three Dimensional Gauge Theories
• The Exact Superconformal R-Symmetry Extremizes Z
• SUSY Gauge Theories on Squashed Three-Spheres
• Aspects of N=2 Supersymmetric Gauge Theories in Three Dimensions
• The complete superconformal index for N=6 Chern-Simons theory
• On Mirror Symmetry in Three Dimensional Abelian Gauge Theories
• Index for three dimensional superconformal field theories with general R-charge assignments
• Indices for Superconformal Field Theories in 3,5 and 6 Dimensions
• Holomorphic Blocks in Three Dimensions
• N=2 supersymmetric theories on squashed three-sphere
• Superconformal Indices for $\mathcal{N}=6$ Chern Simons Theories
• Factorisation of N = 2 theories on the squashed 3-sphere
• Generalized Superconformal Index for Three Dimensional Field Theories
• Branes and Mirror Symmetry in N=2 Supersymmetric Gauge Theories in Three Dimensions
• Exact Kahler Potential from Gauge Theory and Mirror Symmetry
• Superconformal indices of three-dimensional theories related by mirror symmetry
• Cohomological localization of Chern-Simons theory
• Localization on Three-Manifolds
• Dynamics of N=2 Supersymmetric Chern-Simons Theories
• S^3/Z_n partition function and dualities
• Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold
• Extended Supersymmetry and Super-BF Gauge Theories
• Supersymmetric Boundary Conditions in Three Dimensional N = 2 Theories
• Chern-Simons theory on L(p,q) lens spaces and Localization
• Holomorphic Blocks for 3d Non-abelian Partition Functions
• Exact Partition Functions on RP2 and Orientifolds
• Localization on round sphere revisited
• Superfield theories on $S^3$ and their localization
• Abelian Duality
• Global Aspects of Abelian Duality in Dimension Three
• Abelian duality in three dimensions

Cited by in corpus (5)

• Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1\times D^2$
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• Abelian 3d mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ with $N_f=1$
• Pentagon identities arising in supersymmetric gauge theory computations
• Single-flavor Abelian mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$