Algebraic Topology, Representation Theory, and Theoretical Physics

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The PI will continue two main lines of research. The first, joint withMichael Hopkins and Constantin Teleman, builds on our theorem identifying theVerlinde ring in the representation theory of loop groups with a certaintwisted equivariant K-theory ring. The Verlinde ring appears as part of a3-dimensional topological quantum field theory, Chern-Simons theory, and wewill attempt to make further constructions in this direction using K-theory.In another direction, the Verlinde ring may be made using correspondencediagrams and integration in K-theory. Consistent orientations of modulispaces are necessary to carry this out, and the appearance of Madsen-Tillmannspectra in this regard will be further explored. Extensions to families ofsurfaces and "open strings" potentially connect with other mathematics, forexample von Neumann algebras, and may open up new links. The second mainline of research, joint with a variety of collaborators, concerns topologicalideas in string theory and M-theory. For example, with Greg Moore and GraemeSegal we hope to construct completely the quantum theory of a generalizedself-dual field. This will mix theta functions, Heisenberg groups, quadraticforms, and the index theorem in a novel manner. There are also interestingquestions involving various aspects of D-branes and the B-field, some ofwhich impact current ideas about the landscape of solutions to string theory.In addition, there are many student projects which relate to these topics aswell as to the geometric theory of Dirac operators.This research is part of an interaction between algebraic topology andquantum field theory which began in the mid 1970s. The flow of ideas isbidirectional. Existing mathematics is applied to solve particular problemsin physical theories. New ideas emerging from the physics inspiredevelopments and new directions in the mathematics. In particular, over thepast 15 years a new topological side to quantum field theory has beendeveloped and has had ramifications not only in topology but other parts ofgeometry as well. Some of our efforts are devoted to using the integrationprocesses in algebraic topology to shed light on the integration processes intopological quantum field theory--the former are well-defined whereas thelatter have as yet to be understood mathematically in the necessarygenerality. The grant also supports educational activities, including theSaturday Morning Math Group, a highly successful outreach program for middleand high school students.