Algebraic Structures of Representation Theory of Infinite Dimensional Lie Algebras and Quantum Field Theory

Yale University, New Haven CT

Investigators

Abstract

ABSTRACT The relation between representation theory of certain infinite-dimensional Lie algebras such as affine LIe algebras, Virasoro algebra, etc. and 2-dimensional conformal field theory was actively developed in the past 20 years. It has led to unprecedented synthesis of different areas of mathematics and mathematical physics that include integrable Hamiltonian systems, theory of Riemann surfaces and vector bundles, theory of knots and three dimensional manifolds, representations of the Monster group and other sporadic groups, Brownian motion on compact groups and more general path integration, theory of modular functions, commutative systems of differential and difference equations and theory of generalized hypergeometric functions, integrable models of statistical mechanics, string theory and others. One part of the proposal contains further steps extending this relation. The author proposes an approach to q-deformation of a class of vertex operator algebras and the related problem of elliptic deformation of certain braided tensor categories. There is a strong evidence that the resulting algebraic structures will provide an algebraic foundation of 2-dimensional quantum field theory, which does not possess the conformal invariance. The main part of the proposal is dedicated to the first substantial representation theoretic approach to 4-dimensional quantum field theory. The principal investigator plans to develop the theory of vertex representations, which played the key role in 2-dimensional conformal field theory, using representation theory of families of finite groups called wreath products and associated quadratic algebras. In this algebraic approach, which has also a deep geometric counterpart, various structures of vertex representations studied by the proposer and other mathematicians in the last 20 years can be realized as Grothendieck rings and operations between them. Since the Grothendieck rings are always "tips of icebergs" of the full representation categories underlying these rings, one of the main objectives is to extend this realization to a level of representation categories and functors between these categories. This is the long range goal of the present proposal. It is expected that in the same way as theory of vertex representations is an algebraic foundation of 2-dimensional conformal field theory, the new theory will provide an appropriate algebraic structure for the 4-dimensional generalization, which will lead in the next 20 years (or longer!) to a synthesis of practically all areas of pure mathematics and mathematical physics.