Mirror Symmetry in 2+1 and 1+1 Dimensions
Mina Aganagic, Kentaro Hori, Andreas Karch, David Tong
TL;DR
This work investigates how mirror symmetry in 2+1 dimensional N=2 Abelian Chern-Simons theories descends to 1+1 dimensional (2,2) mirror pairs upon compactification on a circle. By tuning parameters and summing the infinite tower of Kaluza-Klein modes on the Coulomb branch, the authors show that the 3d all-scale mirror pair yields, in the $R\to 0$ limit, the Landau-Ginzburg mirror of the toric Higgs branch sigma model, illustrating a precise holomorphic equivalence across dimensions. The analysis provides evidence for an all-scale 3d mirror symmetry, consistent with 2d HV mirror symmetry, and discusses the behavior of the Kahler potential via a squashed toric model, supporting the conjecture KS that 3d duality extends beyond IR. Vortex-electron exchange and the role of regularization (zeta function) are used to connect BPS spectra and superpotential structures across the compactification, highlighting the深 connection between Coulomb and Higgs descriptions under duality.
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.