Groups and Algebraic Structures in Topological Quantum Field Theory

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Abstract Award: DMS-1007255 Principal Investigator: Constantin Teleman The proposed research comprises several, thematically related projects at the interface of topology, 2-dimensional quantum field theory and category theory. The first project, gauged mirror symmetry, combines group actions on projective manifolds, equivariant K-theory and ideas from category theory (open-closed TQFTs and a conjectural Brauer group of equivariant K-theory). One application would be the determination of Gromov-Witten theory of GIT quotients from the gauged Fukaya category of a symplectic manifold. A concrete description of gauged topological quantum field theories in two dimensions is proposed, based on recent progress by Kontsevich, Costello, Hopkins and Lurie on 'extended' topological field theories. The PI has a concrete proposal for coupling a 2-dimensional TQFT to a compact symmetry group and quantizing the gauged theory. This combines ideas from physics (Landau-Ginzburg super-potentials) with earlier work by the PI and collaborators on equivariant (twisted) K-theory and the general index formula on moduli of principal bundles over Riemann surfaces. The second project explores the 'higher algebras' introduced by the PI and collaborators as simplified models of higher categories, and the a toy example of a homotopical TQFT for finite homotopy types. This is hoped to be a good working ground for the interaction of exotic homology theories with ideas from QFT. A third, closely related project is the construction of Chern-Simons gauge theory as a 0-1-2-3 theory by topological methods, along the lines already accomplished by the PI and collaborators for torus groups. To explain the context of this research, one must recall that the fundamental interactions governing energy and matter in the universe are believed to be governed by quantum field theory, a sophisticated mathematical framework that has evolved from the beginnings of quantum mechanics over the last century. Quantum field theory has never been reconciled with general relativity -- another well-supported physical theory -- and much mathematical research over the last six decades has centered around reconciling the two. Topological quantum field theory is a toy attempt to come to grips with the problem while avoiding the analytical difficulties: the notions of distance and magnitude (for instance, mass) are abandoned, and the geometry of space-time is directly related to the algebraic structure of the quantum field theory. Substantial progress in understanding the algebraic structure has been made over the last decade thanks to work by Kontsevich, Hopkins and Lurie. The PI's projects revolve around integrating these recent developments with the idea of symmetry -- in the form of gauge theory -- which is known to be indispensable in realistic physical theories. (One should recall that the so-called 'standard model' of particle physics, comprising the electromagnetic, weak and strong interactions, is a gauge theory.)