Carroll fermions from null reduction: A case of good and bad fermions

TL;DR

This paper derives Carrollian fermionic actions via null reduction from Bargmann spacetimes, revealing how dynamical and constrained fermions relate in electric and magnetic sectors across spacetime dimensions.

Contribution

It introduces a unified approach to construct electric and magnetic Carroll fermions from a single Bargmann-invariant Dirac action, applicable in arbitrary dimensions.

Findings

Carroll fermions decompose into 'good' and 'bad' modes adapted to null frames.

Electric and magnetic sectors are derived from constrained and dynamical modes, respectively.

Two-point functions exhibit expected behavior consistent with Carrollian symmetry.

Abstract

We derive Carrollian fermionic actions using the null reduction method from Bargmann spacetimes. In the Lorentzian light-cone formulation, the Dirac spinor naturally decomposes into dynamical and constrained degrees of freedom $-$ the so-called `good' and `bad' fermions $\Psi_{(\pm)}$. These light-cone projections are intrinsically adapted to the null frame and, unlike the chiral decomposition into left- and right-handed spinors $\Psi_{L(R)}$, are valid in arbitrary spacetime dimensions, both even and odd. As in the case of bosons, the magnetic Carroll sector for fermions is governed by the dynamical modes of the parent theory, while the electric sector arises from the constrained modes. Upon deforming to a Bargmann spacetime, these constraints are removed, promoting the `bad' fermions to dynamical modes that describe the electric Carroll fermions. We construct the Clifford algebra on the Carroll manifold through its embedding in the ambient Bargmann manifold, and obtain both electric and magnetic Carroll fermion actions from a \textit{single} Bargmann-invariant Dirac action. We analyze the canonical structure of both theories, establish their invariance under Carroll transformations, and compute the corresponding two-point functions, which exhibit the expected behavior in both sectors. We conclude with some comments on the quantization of these Carrollian theories.