Introduction to Representations of Braid Groups

Abstract

These are lecture notes prepared for a minicourse given at the Cimpa Research School Algebraic and geometric aspects of representation theory, held in Curitiba, Brazil in March 2013. The purpose of the course is to provide an introduction to the study of representations of braid groups. Three general classes of representations of braid groups are considered: homological representations via mapping class groups, monodromy representations via the Knizhnik-Zamolodchikov connection, and solutions of the Yang-Baxter equation via braided bialgebras. Some of the remarkable relations between these three different constructions are described.

FQHE and tt* geometry

“…What makes the UV approach so nice is its relation to the Kohno connections [48][49][50][51] in the theory of the braid group representation [11,52]. Flatness with respect to a Kohno connection may be seen as a generalization of the Knizhnik-Zamolodchikov equations [9].…”

“…The k-th classical vacuum is to be identified geometrically with the intersection of these two sheets. 52 The absolute number of 2d BPS solitons connecting the k-th and h-vacua is given by the number of sheets they share…”

“…51 An instance of the non-generic situation is described in Example 10. 52 The precise sense of the identification will be clarified momentarily in § §. 4.4.5, 4.4.6.…”