Bosonization and Mirror Symmetry

TL;DR

This work derives a 2+1D bosonization duality between a free Dirac fermion and scalar QED3 by starting from ${\cal N}=4$ mirror symmetry and performing controlled SUSY-preserving deformations. A chiral SUSY duality between a free ${\cal N}=2$ chiral multiplet and ${\cal N}=2$ SUSY QED3 with a single charged chiral acts as the SUSY completion, which upon breaking SUSY with a background $D$-term yields the non-supersymmetric bosonization duality. The authors explicitly map background deformations $U(1)_A$ and $U(1)_R$, track moduli-space and charge-attachment phenomena via Chern-Simons and BF terms, and show that the two theories share identical phase structures and topological responses near the critical point. The results connect to a broader duality web—including Peskin-Dasgupta-Halperin, Son’s fermion/fermion duality, and modular-transformation-generated dualities—offering a robust 2+1D field-theory framework for fermion-boson equivalences with potential applications to quantum Hall transitions and related condensed-matter systems.

Abstract

We study bosonization in 2+1 dimensions using mirror symmetry, a duality that relates pairs of supersymmetric theories. Upon breaking supersymmetry in a controlled way, we dynamically obtain the bosonization duality that equates the theory of a free Dirac fermion to QED3 with a single scalar boson. This duality may be used to demonstrate the bosonization duality relating an $O(2)$-symmetric Wilson-Fisher fixed point to QED3 with a single Dirac fermion, Peskin-Dasgupta-Halperin duality, and the recently conjectured duality relating the theory of a free Dirac fermion to fermionic QED3 with a single flavor. Chern-Simons and BF couplings for both dynamical and background gauge fields play a central role in our approach. In the course of our study, we describe a chiral mirror pair that may be viewed as the minimal supersymmetric generalization of the two bosonization dualities.

Figures (2)

• Figure 1: Phase diagram of theory A. Phases I-III are separated by second order critical points (indicated by the solid blue line). Setting $\hat{A}_A = 0$, the transition at $\hat{\sigma}_A = \hat{\sigma}_J$ represents the point across which the Chern-Simons level for $\hat{A}_J$ changes by unity. The horizontal axis at $\hat{D}_J = 0$ is described by the SUSY chiral theory A, while the $\hat{D}_J>0$ line is controlled by the free fermion lagrangian in Eq. \ref{['freedirac']}. Phase III is unstable because $m^2_{v_+} < 0$ -- see Eq. \ref{['stabilitycondition']} -- and there are no interactions to stabilize the broken-symmetry vacuum.
• Figure 2: Phase diagram of theory B. Phases I-III are separated by second order critical points (indicated by the solid blue line). Setting $\hat{A}_A = 0$, the transition at $\hat{\sigma}_A = \hat{\sigma}_J$ represents the point across which the Chern-Simons level for $\hat{A}_J$ changes by unity. The horizontal axis at $\hat{D}_J = 0$ is described by the SUSY chiral theory B, while the $\hat{D}_J>0$ line is controlled by the lagrangian in Eq. \ref{['sQED3']}. Phase III cannot be accessed within our framework.