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.
Topology is the branch of geometry which investigates properties of shapes which do not change under continuous deformation. One often extracts algebraic invariants which measure topological properties and link to questions in geometry. The research in this grant is part of the extremely active contemporary interface between mathematics and theoretical physics. We focus in particular on algebraic topology, but our work touches on other parts of mathematics as well. The projects and papers completed during the term of this grant may be divided into three broad groups. The first concerns "generalized differential cohomology", a relatively new theory on smooth spaces called manifolds. It arose out of considerations in theoretical quantum field theory, but ties in with longstanding ideas in geometry and has inspired new directions. Together with John Lott we proved a refinement of the renowned Atiyah-Singer index theorem to differential K-theory. While this is an important result, there is further work to be done in this direction. With Mike Hopkins we are completing a first paper setting up a general framework in which to investigate these further questions. One application of these results is back to physics, where they are used in certain anomaly cancellations. We expect future applications in geometry as well. A second group of projects is in the new area of topological quantum field theory. While this is part of algebraic topology, as the name suggests it arose out of structures in quantum field theory proper, through work of both physicists and mathematicians close to the physics. Together with Mike Hopkins and Constantin Teleman we construct purely in algebraic topology a 2-dimensional theory which encodes the Verlinde ring, a central object in low-dimensional field theories. Whereas usual quantum mechanics demands complex numbers, this theory is constructed over the integers, so encodes finer invariants. We added Jacob Lurie to the team for a project on some special 3-dimensional field theories. We solved a long-standing problem for these theories, and also proposed ideas and techniques which should find further application. In particular, I am now following up with Constantin Teleman on some of these ideas applied to a general class of 3-dimensional topological quantum theories. The third group of projects are direct applications of algebraic topology and geometry to theoretical physics. There are three main directions, and they are in three parts of theoretical physics: condensed matter theory, quantum field theory, and string theory. Together with Greg Moore we undertake a general investigation of symmetry in quantum mechanical systems. (As part of that I discovered new proofs of a fundamental theorem of Wigner.) Following up on recent work of Kitaev and others, we prove that certain gapped systems of free fermions are classified by topological K-theory, and we extend the classification to account for a bigger symmetry group in many cases. A solo project revisited old work in quantum field theory concerning the effective theory of pions in quantum chromodynamics. We refine the traditional Wess-Zumino-Witten term using modern ideas in generalized differential cohomology, and thereby settle some lingering small difficulties with previous treatments. A longterm project with Jacques Distler and Greg Moore investigates algebro-topological aspects of superstring theory. We use recent ideas about twisted forms of K-theory to prove consistency of certain aspects of orientifold models. This NSF grant included support for graduate students and for an outreach program we run at UT Austin. This Saturday Morning Math Group and Math Circle gathers local middle and high school students for enrichment activities led by faculty in mathematics and the sciences, as well as guest speakers. It is one of the longest running programs of its type in the country.
