Topics in statistical physics involving braids

Abstract

We review the appearance of the braid group in statistical physics. In particular, we explain its relevance to the anyon model of fractional statistics and conformal field theory. This short contribution reviews two theories in statistical physics: conformal field theory and the anyon model. Our research has concerned studies of the duality equation in conformal theories and the energy spectrum in the anyon model. While our interests have concentrated on physical issues, both of these subjects have intimate connections with the braid group. This article will expand our oral presentation to discuss those mathematical connections. We hope that a review format, where one of the principle goals is to clarify the origin of braid statistics, will aid the present audience and encourage it to address the unanswered mathematical issues presented. The remainder of this contribution is organized as follows. The first section of the article presents some generalities on braid group statistics that are relevant to both theories. The second section describes the anyon model. We have chosen to start with it, because it is inherently simple and does not require discussions of advanced topics in physics. The third section considers correlation functions in conformal field theory. The physical origins of this theory are briefly presented. Then, we develop in some detail the machinery that has led to the construction of correlation functions. The braid group only occurs in the tools of this construction, i.e. the conformal blocks. We end by illustrating the connection with the duality equation. 1991 Mathematics Subject Classification: 82-02, 82Bxx, 81-02, 81Rxx. The paper is in final form and no version of it will be published elsewhere.