Flat Connections, Irregular Singularities and Quantum Groups

Northeastern University, Boston MA

Investigators

Abstract

This proposal stems from the PI's recent generalisation of the Kohno-Drinfeld theorem according to which the monodromy of the Casimir connection, a flat connection with logarithmic singularities on the root hyperplanes of a simple Lie algebra, is described by the quantum Weyl group operators of the corresponding quantum group. This result had been conjectured independently by the PI and De Concini. The first project, in collaboration with B. Feigin and E. Frenkel, endeavours to establish a precise link between the Casimir connection and Conformal Field Theory by exhibiting new fields in the Wess-Zumino-Witten model whose correlation functions satisfy the corresponding system of differential equations, thus incorporating connections with irregular singularities into this model. This project has a number of important potential applications beyond those to Conformal Field Theory: to the structure and representation theory of simple Lie algebras (quantisation of the shift of argument subalgebra, construction of new basis of irreducible finite dimensional representations) and to Statistical Mechanics (construction of new integrable models generalising the Gaudin model). The second project aims at extending the PI's monodromy theorem at roots of unity. This extension could provide the impetus for the study of generalised braided tensor categories, that is tensor categories whose underlying braid group is a generalised one, and whose definition is implicit in the axioms of a quasi-Coxeter algebra. The last project, in collaboration with R. Rouquier, is concerned with Dynkin diagram cohomology, which was introduced by the PI to control deformations of quasi-Coxeter algebras. Its aim is to arrive at a better combinatorial and topological understanding of this cohomology by computing it for Coxeter groups and the enveloping algebras of semi-simple Lie algebras. Quantum groups are deformations of the most basic symmetry groups of Nature. They were discovered in the mid-eighties as symmetries of certain Quantum mechanical systems and have since appeared in a wide range of fields in Mathematics and Physics such as String Theory, Quantum Statistical Mechanics, the study of finite symmetry groups, Topology and Combinatorics. One of their bewildering aspects is their uncanny ability to address, and solve, long standing problems in these fields which are often formulated without appealling to quantum groups. The PI has been pursuing one such avenue of investigation, by using quantum groups to describe the branching behaviour of solutions of certain systems of differential equations in the complex domain. This proposal aims at further broadening our understanding of this intriguing phenomenon. This project has important potential applications to String Theory and Representation Theory. It is also expected that it will lead to a wide generalisation of Braided Tensor Categories which, in their present guise, have been extensively used in Computer Science, Logic and, more recently, Quantum Computing. The PI has, on several occasions, given lectures to audiences of young researchers and graduate students about his recent results and always found an extremely receptive audience. It is likely that the further developments stemming from them which are presented above will attract such young mathematicians.