Duality and bosonization of $(2+1)$-dimensional Majorana fermions

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 $\mathrm{SO}{\left(N\right)}_1$ Chern-Simons gauge theory with a critical $\mathrm{SO}\left(N\right)$ vector bosonic matter field ($N\ge 3$). The monopole of the $\mathrm{SO}\left(N\right)$ 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.

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.