Abelian Chern-Simons theories in 2 + 1 dimensions

Abstract

Abelian topological Chern-Simons theories in 2 + 1 dimensions are examined. The Hilbert space is constructed explicitly for all two-dimensional space manifolds and is shown to be finite-dimensional, in the case of compact U(1). It is determined by the genus of the spatial surface, the coefficient of the action k, and a set of vacuum angles, and it corresponds to the space of conformal blocks of a free scalar in two dimensions compactified on a circle of radius squared k. The expectation of an arbitrary knot is then evaluated by relating the theory in the presence of the knot to a theory in a background gauge field. In this framework, the dependence of the result on the regularization and the framing of the knot becomes explicit. Finally, possible extensions of the methods used in this work to nonabelian theories are discussed.