Global Anomalies and Anyons in 1+1 Dimensions

TL;DR

This paper explores how in 1+1 dimensions, charge conjugation and gauge invariance impose constraints on particle statistics, preventing the existence of anyons unless certain quantization conditions are met.

Contribution

It demonstrates that charge conjugation invariance and abelian instantons in 1+1D restrict particles to be bosons or fermions, revealing a novel anomaly related to anyons.

Findings

Charge conjugation and gauge invariance lead to anomalies unless flux is quantized.

Particles must have spin n+1/2 to avoid anomalies.

Anyons cannot exist without violating these symmetries.

Abstract

We consider the analog in one spatial dimension of the Bose-Fermi transmutation for planar systems. A quantum mechanical system of a spin 1/2 particle coupled to an abelian gauge field, which is classically invariant under gauge transformations and charge conjugation is studied. It is found that unless the flux enclosed by the particle orbits is quantized, and the spin takes a value $n+ 1/2$, at least one of the two symmetries would be anomalous. Thus, charge conjugation invariance and the existence of abelian instantons simultaneously force the particles to be either bosons or fermions, but not anyons.