Dimensional reduction, magnetic flux strings, and domain walls

TL;DR

This paper explores how dimensional reduction from 3+1 to 2+1 dimensions transforms magnetic flux strings into domain walls, revealing localized fermionic zero modes and deriving effective actions for planar massless QED.

Contribution

It demonstrates the mapping of flux strings to domain walls and analyzes fermionic zero modes in the reduced theory, providing new insights into lower-dimensional gauge systems.

Findings

Magnetic flux strings become domain walls after dimensional reduction.

Fermionic zero modes localize around domain walls in the reduced theory.

Derived effective action for massless QED in 2+1 dimensions.

Abstract

We study some consequences of dimensionally reducing systems with massless fermions and Abelian gauge fields from 3+1 to 2+1 dimensions. We first consider fermions in the presence of an external Abelian gauge field. In the reduced theory, obtained by compactifying one of the coordinates `a la Kaluza-Klein, magnetic flux strings are mapped into domain wall defects. Fermionic zero modes, localized around the flux strings of the 3+1 dimensional theory, become also zero modes in the reduced theory, via the Callan and Harvey mechanism, and are concentrated around the domain wall defects. We also study a dynamical model: massless $QED_4$, with fermions confined to a plane, deriving the effective action that describes the `planar' system.