Topologically Massive Gauge Theory: A Lorentzian Solution

TL;DR

This paper constructs a Lorentzian solution for topologically massive non-abelian gauge theory on AdS space, utilizing gauge transformations and geometric mappings, revealing a global structure akin to the Hopf map and analyzing the gauge potential's holonomy.

Contribution

It introduces a Lorentzian solution for the non-abelian gauge theory on AdS space through a gauge transformation of the abelian solution and explores the geometric and topological structure of the solution.

Findings

Derived a Lorentzian solution via SU(1,1) transformation

Mapped AdS space to a pseudo-sphere using a Lorentzian Hopf map

Analyzed the gauge potential's holonomy and dual-field strength

Abstract

We obtain a lorentzian solution for the topologically massive non-abelian gauge theory on AdS space by means of a SU(1, 1) gauge transformation of the previously found abelian solution. There exists a natural scale of length which is determined by the inverse topological mass. The topological mass is proportional to the square of the gauge coupling constant. In the topologically massive electrodynamics the field strength locally determines the gauge potential up to a closed 1-form via the (anti-)self-duality equation. We introduce a transformation of the gauge potential using the dual field strength which can be identified with an abelian gauge transformation. Then we present the map from the AdS space to the pseudo-sphere including the topological mass. This is the lorentzian analog of the Hopf map. This map yields a global decomposition of the AdS space as a trivial circle bundle over the upper portion of the pseudo-sphere which is the Hyperboloid model for the Lobachevski geometry. This leads to a reduction of the abelian field equation onto the pseudo-sphere using a global section of the solution on the AdS space. Then we discuss the integration of the field equation using the Archimedes map from the pseudo-sphere to the cylinder over the ideal Poincare circle. We also present a brief discussion of the holonomy of the gauge potential and the dual-field strength on the upper portion of the pseudo-sphere.