Monodromy Representations of the Braid Group

TL;DR

This paper explores the monodromy representations of the braid group arising from chiral conformal blocks in rational conformal field theories, highlighting their mathematical structure and physical implications in quantum phenomena.

Contribution

It provides a comprehensive review of braid group representations in conformal field theory, emphasizing the role of regular solution bases and extensions to indecomposable representations.

Findings

Monodromy representations relate to quantum group symmetries.

Regular bases facilitate analysis of indecomposable representations.

Connections to quantum Hall states and braid statistics.

Abstract

Chiral conformal blocks in a rational conformal field theory are a far going extension of Gauss hypergeometric functions. The associated monodromy representations of Artin's braid group capture the essence of the modern view on the subject, which originates in ideas of Riemann and Schwarz. Physically, such monodromy representations correspond to a new type of braid group statistics, which may manifest itself in two-dimensional critical phenomena, e.g. in some exotic quantum Hall states. The associated primary fields satisfy R-matrix exchange relations. The description of the internal symmetry of such fields requires an extension of the concept of a group, thus giving room to quantum groups and their generalizations. We review the appearance of braid group representations in the space of solutions of the Knizhnik - Zamolodchikov equation, with an emphasis on the role of a regular basis of solutions which allows us to treat the case of indecomposable representations as well.