Topologically Massive Gauge Theory: Wu-Yang Type Solutions

TL;DR

This paper explores Wu-Yang type solutions in topologically massive gauge theories, revealing their geometric structure, charge configurations, and quantization conditions, with implications for understanding topological effects in gauge fields.

Contribution

It introduces new Wu-Yang solutions in Maxwell-Chern-Simons and Yang-Mills-Chern-Simons theories, utilizing Hopf maps and gauge transformations to analyze their topological and geometric properties.

Findings

Existence of solutions with different Chern numbers.

Identification of a natural length scale related to topological mass.

Connection between topological mass and Hall resistivity.

Abstract

We discuss Wu-Yang type solutions of the Maxwell-Chern-Simons and the Yang-Mills-Chern-Simons theories. There exists a natural scale of length which is determined by the inverse topological mass. We obtain the non-abelian solution by means of a SU(2) gauge transformation of Dirac magnetic monopole type solution. In the abelian case, field strength locally determines the gauge potential up to a closed term via self-duality equation. We introduce a transformation of the gauge potential using dual field strength which can be identified with the gauge transformation in the abelian solution. Then we present Hopf map from S^3 to S^2 including the topological mass. This leads to a reduction of the field equation onto S^2 using local sections of S^3. The local solutions possess a composite structure consisting of both magnetic and electric charges. These naturally lead to topologically massive Wu-Yang solution which is based on patching up the local potentials by means of a gauge transformation. We also discuss solutions with different first Chern numbers. There exist a fundamental scale over which the gauge function is single-valued and periodic for any integer in addition to the fact that it has a smaller period. We also discuss Dirac quantization condition. We present a stereographic view of the fibres in the Hopf map. Meanwhile Archimedes map yields a simple geometric picture for the Wu-Yang solution. We also discuss holonomy of the gauge potential and the dual-field on S^2. Finally we point out a naive identification of the natural length scale introduced by the topological mass with Hall resistivity.