Version Of Mirror Symmetry Research Articles

Superconformal index onRP2×S1and mirror symmetry

We study $\mathcal{N}=2$ supersymmetric gauge theories on ${\mathbb{R}\mathbb{P}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{1}$ and compute the superconformal index by using the localization technique. We consider not only the round real projective plane ${\mathbb{R}\mathbb{P}}^{2}$ but also the squashed real projective plane ${\mathbb{R}\mathbb{P}}_{b}^{2}$ which turns back to ${\mathbb{R}\mathbb{P}}^{2}$ by taking a squashing parameter $b$ as 1. In addition, we find that the result is independent of the squashing parameter $b$. We apply our new superconformal index to check the simplest case of 3D mirror symmetry, i.e., the equivalence between the $\mathcal{N}=2$ supersymmetric quantum electrodynamics (SQED) and the $XYZ$ model on ${\mathbb{R}\mathbb{P}}^{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{1}$. We prove it by using a mathematical formula called the $q$-binomial theorem. We also comment on the $\mathcal{N}=4$ version of mirror symmetry, mirror symmetry via generalized indices, and possibilities of generalizations from mathematical viewpoints.

Singularity Theory and Integrable Systems

The workshop brought together three very different areas of mathematics, namely singularity theory, integrable systems and quantum cohomology. They are linked by their applications in topological quantum field theory and by constructions of (often isomorphic) Frobenius manifolds. The first and second are related by a version of mirror symmetry, the link of the second and third has attracted much attention. The connection of the first and third is the least developed and was at the focus of the workshop.