Algebraic Topology, Representation Theory, and Theoretical Physics

University Of Texas At Austin, Austin TX

Investigators

Abstract

The PI will continue two main lines of research. The first, joint with Michael Hopkins and Constantin Teleman, builds on our theorem identifying the Verlinde ring in the representation theory of loop groups with a certain twisted equivariant K-theory ring. The Verlinde ring appears as part of a 3-dimensional topological quantum field theory, Chern-Simons theory, and we will attempt to make further constructions in this direction using K-theory. In another direction, the Verlinde ring may be made using correspondence diagrams and integration in K-theory. Consistent orientations of moduli spaces are necessary to carry this out, and the appearance of Madsen-Tillmann spectra in this regard will be further explored. Extensions to families of surfaces and "open strings" potentially connect with other mathematics, for example von Neumann algebras, and may open up new links. The second main line of research, joint with a variety of collaborators, concerns topological ideas in string theory and M-theory. For example, with Greg Moore and Graeme Segal we hope to construct completely the quantum theory of a generalized self-dual field. This will mix theta functions, Heisenberg groups, quadratic forms, and the index theorem in a novel manner. There are also interesting questions involving various aspects of D-branes and the B-field, some of which impact current ideas about the landscape of solutions to string theory. In addition, there are many student projects which relate to these topics as well as to the geometric theory of Dirac operators. This research is part of an interaction between algebraic topology and quantum field theory which began in the mid 1970s. The flow of ideas is bidirectional. Existing mathematics is applied to solve particular problems in physical theories. New ideas emerging from the physics inspire developments and new directions in the mathematics. In particular, over the past 15 years a new topological side to quantum field theory has been developed and has had ramifications not only in topology but other parts of geometry as well. Some of our efforts are devoted to using the integration processes in algebraic topology to shed light on the integration processes in topological quantum field theory--the former are well-defined whereas the latter have as yet to be understood mathematically in the necessary generality. The grant also supports educational activities, including the Saturday Morning Math Group, a highly successful outreach program for middle and high school students.