Structure in Topological Field Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This research plan integrates several research projects of the PI, unified by the relation to topological field theories which controls the answer to the various questions. One project builds on the PI's structural classification of 2-dimensional (closed string) semi-simple topological field theories, with implications for several standing conjectures on Gromov-Witten invariants. Beyond a careful treatement of the result and its implications, the aim is to extend it to more general theories; this seems to relate to the factorization properties for GW invariants (Ionel-Parker, Li) and to the structural results in the open/closed string case (Kontsevich, Costello). A second project is the study of gauged Gromov-Witten theory, where twisted K-theories and representations of loop groups appear. Loop groups also feature in the third project, pertaining to the geometric Langlands programme, where the cohomological calculations of the PI (with E. Frenkel) offer a way forward form the results of Beilinson-Drinfeld, toward the 'derived' version of the Langlands correspondence. (Recent work of Gukov-Kapustin-Witten suggest a controlling topological field theory.) Current work by the PI (with C.Woodward) extending to Higgs bundles results of coherent cohomology previously known for principal bundles is an important step. Other, and more speculative projects include a study of non-semisimple 3-dimensional TFT's associated to derived categories.     Topological Field theory is a spectacular and unforeseen application of ideas from modern quantum physics to topology: that is the field of mathematics which, broadly speaking, studies the properties of shapes that are stable under continuous deformations. Previous applications of the foundational problems of quantum physics to mathematics had dominated development in mathematical analysis for decades, but their emergence in topology in the 1980's came as a surprise. Crudely put, the new ideas exploit a topological irreversibility of time flow: a topological change in space-time can usually not be 'undone' in the future. This led to the encoding of information in new kinds of algebraic structures (technically, they are monoids rather than groups). The new methods succeeded in unifying existing invariants of knots and links with those of 3-and 4-dimensional structures (manifolds). New invariants could be defined that bear stunning relations to other fields of mathematics which study finer, but less robust structures (algebraic geometry and complex analysis). The PI's research focuses on instances of these structures where, in addition, a continuous group of symmetries is present in the system, and studies the refined structures that emerge.