Topologically Massive Abelian Gauge Theory

TL;DR

This paper explores mathematical structures in topologically massive abelian gauge theory, including contact structures, self-dual solutions, and a curl transformation, revealing new geometric and quantization insights.

Contribution

It introduces novel contact structures, classifies self-dual solutions via holomorphic functions, and applies curl transformation to analyze solutions and mass quantization.

Findings

Contact structures on various manifolds linked to solutions.

Classification of solutions into self-dual and anti-self-dual.

Representation of solutions as vectors on a sphere in transform space.

Abstract

We discuss three mathematical structures which arise in topologically massive abelian gauge theory. First, the euclidean topologically massive abelian gauge theory defines a contact structure on a manifold. We briefly discuss three solutions and the related contact structures on the flat 3-torus, the AdS space, the 3-sphere which respectively correspond to Bianchi type I, VIII, IX spaces. We also present solutions on Bianchi type II, VI and VII spaces. Secondly, we discuss a family of complex (anti-)self-dual solutions of the euclidean theory in cartesian coordinates on R3 which are given by (anti-)holomorpic functions. The orthogonality relation of contact structures which are determined by the real parts of these complex solutions separates them into two classes: the self-dual and the anti-self-dual solutions. Thirdly, we apply the curl transformation to this theory. An arbitrary solution is given by a vector tangent to a sphere whose radius is determined by the topological mass in transform space. Meanwhile a gauge transformation corresponds to a vector normal to this sphere. We discuss the quantization of topological mass on an example.