Infinite Dimensional Lie Algebras, Quantum Groups and their Applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This research project lies at the interface of algebra, geometry, and mathematical physics. One of the central problems in modern theoretical physics is to construct a mathematically consistent theory of fundamental interactions. The paradigm for this construction is known as quantum field theory. An underlying theme of the research is the role of the boundary of "space-time." One way to think about this is the usual evolution of time over a finite interval. In this case the boundary of space-time is the union of the space at the initial time and at the target time. Quantum field theory on a space-time with boundary is important not only for high energy physics, but also for low energy physics, for example insulators and other states of matter where the surface (the boundary) has distinguishing physical properties. This project will develop algebraic tools to study two and three dimensional quantum field theories. It will also construct quantum field theories with infinite dimensional symmetries. The proposal consists of three parts. The common theme is the effect of the boundary on local quantum field theory and on local spacial models in statistical mechanics. The first part is focused on representation theory. The main goals are to connect coideal subalgebras to boundary q-Knizhnik-Zamolodchikov equation by studying vertex operators for principal series representations of the corresponding quantum affine algebras; to study integrable systems on symmetric spaces; and to construct invariants of knots with flat connections in the complement using R-matrix intertwining representations of quantum sl(2) at roots of unity and construct the corresponding invariants of 3-manifolds via surgery. In the second part the goals are to construct perturbative topological quantum field theories, which involves constructing perturbative topoplogical quantum field theory for manifolds with boundary; and to construct semiclassical asymptotic for q-6j symbols for simple Lie algebras of rank grater then one. The goal of the third part is the study of limit shapes in statistical mechanics. This part is closely related to the second part and can be regarded as the study of the semiclassical limit of spin models in statistical mechanics. The main concept is that the thermodynamical limit is similar to the semiclassical limit. The models in question involve the stochastic 6-vertex model, ASEP models and others.