Induced Chern-Simons term by dimensional reduction

TL;DR

This paper derives an Abelian Chern-Simons term in 2+1 dimensions through dimensional reduction from a finite-temperature 3+1 dimensional Dirac theory with vector and axial-vector couplings, highlighting the role of the lowest Matsubara mode.

Contribution

It introduces a novel derivation of the induced Chern-Simons term via dimensional reduction, connecting finite-temperature field theory with topological gauge terms.

Findings

The CS term emerges from the lowest Matsubara mode of the vector field.

The coefficient of the CS term is determined by a single parameter of the axial-vector field configuration.

The derivation links finite-temperature effects with topological gauge structures.

Abstract

We derive an induced Abelian Chern-Simons (CS) term in 2+1 dimensions, by dimensional reduction from the finite-temperature theory of a Dirac field with both vector and axial-vector couplings to two Abelian gauge fields, in 3+1 dimensions. In our construction, the CS term emerges for the lowest Matsubara mode of the vector Abelian field, by integrating the fermionic field, under the assumption that the axial vector field is in a "vacuum" configuration. This configuration is characterized by a single number, which in turn determines the coefficient of the induced CS term for the Abelian vector field.