Duality and bosonization of (2+1)d Majorana fermions

TL;DR

This paper develops a dual bosonized description for massless Majorana fermions in (2+1)d, proposing an $SO(N)_1$ Chern-Simons gauge theory with a critical $SO(N)$ vector bosonic matter field, and explores its implications and generalizations.

Contribution

It introduces a novel duality framework for Majorana fermions in (2+1)d using $SO(N)_1$ Chern-Simons theory, extending bosonization beyond Dirac fermions.

Findings

Identifies the monopole of the $SO(N)$ gauge field with the Majorana fermion.

Establishes correspondence between gapped phases in the dual theories.

Proposes a dual description of an $ ext{N}=1$ supersymmetric fixed point.

Abstract

We construct a dual bosonized description of a massless Majorana fermion in $(2+1)d$. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach. We argue that the dual theory is an $SO(N)_1$ Chern-Simons gauge theory with a critical $SO(N)$ vector bosonic matter field ($N \geq 3$). The monopole of the $SO(N)$ gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to $N_f$ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global $Z_2$ symmetry. Finally, we conjecture a dual description of an $\mathcal{N} = 1$ supersymmetric fixed point in $(2+1)d$, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.