Mirror Symmetry and Bosonization in 2d and 3d

TL;DR

The paper shows that SUSY-breaking deformations of the 2d Hori-Kapustin cigar⇄Liouville mirror pair renormalize IR dynamics to the classic 2d bosonization duality, with the cigar flowing to Massive Thirring and Liouville to Sine-Gordon; the bosonization map emerges at one loop, matching Coleman’s relation $\beta^2 = \frac{1}{4\pi} + \frac{g}{2\pi^2}$ and confirming one-loop exactness. It then connects this to 3d bosonization by compactifying on $S^1$, finding that the strongly coupled 3d duality reduces to the same 2d bosonization structure via current mappings, where the 2d boson is a compact scalar with $\beta^2 = \frac{1}{4\pi}$ at the free-fermion point. Mass deformations in 2d map cleanly between fermionic and bosonic operators (Dirac mass to $\cos\theta$, Majorana mass to $e^{2i\tilde{\theta}}$), and the circle reduction clarifies how higher-dimensional dualities feed into the canonical 2d bosonization framework. Overall, the work unifies 2d and 3d dualities by showing that bosonization naturally arises as the IR endpoint of higher-dimensional mirror pairs, with a precise one-loop bosonization map and robust current-operator correspondences.

Abstract

We study a supersymmetry breaking deformation of the 2d N=(2,2) cigar=Liouville mirror pair, first introduced by Hori and Kapustin. We show that mirror symmetry flows in the infra-red to 2d bosonization, with the theories reducing to massive Thirring and Sine-Gordon respectively. The exact bosonization map emerges at one-loop. We further compactify non-supersymmetric 3d bosonization dualities on a circle and argue that these too flow to 2d bosonization at long distances.