Bosonization and mirror symmetry

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.

Figure 1

Phase diagram of theory A. Phases I-III are separated by second order critical points (indicated by the solid blue line). Setting ${\widehat{A}}_A=0$, the transition at ${\widehat{\sigma }}_A={\widehat{\sigma }}_J$ represents the point across which the Chern-Simons level for ${\widehat{A}}_J$ changes by unity. The horizontal axis at ${\widehat{D}}_J=0$ is described by the SUSY chiral theory A, while the ${\widehat{D}}_J>0$ line is controlled by the free fermion Lagrangian in Eq. (4.2). Phase III is unstable because $m_{v_{+}}^2<0$-see Eq. (3.6)—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 ${\widehat{A}}_A=0$, the transition at ${\widehat{\sigma }}_A={\widehat{\sigma }}_J$ represents the point across which the Chern-Simons level for ${\widehat{A}}_J$ changes by unity. The horizontal axis at ${\widehat{D}}_J=0$ is described by the SUSY chiral theory B, while the ${\widehat{D}}_J>0$ line is controlled by the Lagrangian in Eq. (4.7). Phase III cannot be accessed within our framework.