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

TL;DR

This work introduces a nonabelian bosonization framework for a single Majorana fermion in $(2+1)d$, proposing an $SO(N)_1$ Chern-Simons gauge theory coupled to a real vector boson as the dual description and identifying the Majorana mode with the $SO(N)$ monopole. By matching the trivial and $p_x+ip_y$ superconducting phases, employing a parton construction, and establishing a monopole-fermion correspondence, the authors provide evidence for the duality and extend the construction to $N_f$ flavors with a corresponding symmetry-preserving bosonic dual. They also connect the duality to a Gross-Neveu–Yukawa fixed point and discuss a possible $\mathcal{N}=1$ SUSY infrared theory with a specific scaling-dimension relation, suggesting deep links between non-Abelian CS–matter theories and Majorana physics. The paper highlights open questions about time-reversal invariance in the dual description and potential surface topological orders, pointing to future work on deriving explicit time-reversal–invariant duals and exploring broader non-Abelian dualities. Overall, the results extend and enrich the landscape of $(2+1)d$ dualities by providing a concrete nonabelian bosonization framework for Majorana fermions and laying groundwork for further connections to SUSY, SPT phases, and surface order phenomena.

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.

Figures (1)

• Figure 1: The proposed duality and renormalization group flow. The double headed arrow stands for "dual to each other", while the single headed arrow represents the RG flow.