Making Sense of Singular Gauge Transformations in 1+1 and 2+1 Fermion Models

TL;DR

This paper explores how singular gauge transformations can decouple fermion fields in 1+1 and 2+1 dimensions, revealing their effects on fermionic currents and parity violation through Fujikawa Jacobians.

Contribution

It demonstrates that singular gauge transformations can be used to decouple fermions and derive fermionic current expectations, extending the understanding of gauge transformations in lower-dimensional models.

Findings

Singular gauge transformations lead to chiral transformations in 2D.

Fujikawa Jacobians are crucial for understanding fermion decoupling.

Parity-violating fermionic current expectation values are obtained.

Abstract

We study the problem of decoupling fermion fields in 1+1 and 2+1 dimensions, in interaction with a gauge field, by performing local transformations of the fermions in the functional integral. This could always be done if singular (large) gauge transformations were allowed, since any gauge field configuration may be represented as a singular pure gauge field. However, the effect of a singular gauge transformation of the fermions is equivalent to the one of a regular transformation with a non-trivial action on the spinorial indices. For example, in the two dimensional case, singular gauge transformations lead naturally to chiral transformations, and hence to the usual decoupling mechanism based on Fujikawa Jacobians. In 2+1 dimensions, using the same procedure, different transformations emerge, which also give rise to Fujikawa Jacobians. We apply this idea to obtain the v.e.v of the fermionic current in a background field, in terms of the Jacobian for an infinitesimal decoupling transformation, finding the parity violating result.