Topologically Massive Gauge Theory: Wu-Yang Type Solutions

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.