Mirror Symmetry in 2+1 and 1+1 Dimensions

TL;DR

This paper investigates the duality between Coulomb and Higgs branches in 2+1D supersymmetric theories by compactifying to 1+1D, revealing how mirror symmetry emerges through superpotential resummation and supporting the conjecture of all-scale 3D mirror symmetry.

Contribution

It demonstrates the connection between 2+1D Coulomb-Higgs duality and 1+1D mirror symmetry via compactification and superpotential analysis, providing evidence for the all-scale 3D mirror symmetry conjecture.

Findings

Superpotential in the compactified theory matches the Landau-Ginzburg mirror in the R->0 limit.

Resummation of Kaluza-Klein modes reproduces 2D mirror symmetry.

The results support the conjecture of all-scale 3D mirror symmetry.

Abstract

We study the Coulomb-Higgs duality of N=2 supersymmetric Abelian Chern-Simons theories in 2+1 dimensions, by compactifying dual pairs on a circle of radius R and comparing the resulting N=(2,2) theories in 1+1 dimensions. Below the compactification scale, the theory on the Higgs branch reduces to the non-linear sigma model on a toric manifold. In the dual theory on the Coulomb branch, the Kaluza-Klein modes generate an infinite tower of contributions to the superpotential. After resummation, in the limit R->0 the superpotential becomes that of the Landau-Ginzburg model which is the two-dimensional mirror of the toric sigma model. We further examine the conjecture of all-scale three-dimensional mirror symmetry and observe that it is consistent with mirror symmetry in 1+1 dimensions.