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.)
The research project concerned the study of symmetry in a simplidfied model for quantum field theory. The simplifications are * reducing of the number of dimensions (from the physical 4 to 2, one space and one time), * ignoring dynamical, higher-energy effects, which in mathematics is accomplished by the restriction to topological field theories. (Often, this is the zero-energy sector of a genuine quantum field theory.) Symmetries (and groups of symmetries) are prevalent in fundamental particle physics: the currently accepted mathematical model of the universe is based on a theory of symmetries (a so-called gauge theory, with group U(1) x SU(2) x SU(3), covering the electromagnetic, weak and strong interactions). The underlying mathematics is so complex that we have been unable, to date, even to prove that the theory is well-defined. Simplified models of quantum field theory, where such proofs can be carried out, are thus of great theoretical interest. Topological quantum field theories study the zero-energy sector (massless, no dynamics) of physics. While it may sound vacuous, it was understood that in fact such theories capture the algebra and geometry underlying `true' quantum field theories, without the analytical difficulties. The case of two dimensions is especially well understood, because of a wealth of examples coming from complex analytic and algebraic geometry. (The reason is that two-dimensional space-times admit complex structures, turning them into curves described by complex algebraic equations, a classical subject with roots in 19th century and earlier mathematics.) The project initially proposed to set and study the foundations of symmetry in these 2-dimensional topological field theories. It was originally based on the PI's discovery of a computational method computing fied-points of symmetries, but developed beyond the original scope with the discovery of connections with gauge theories in 3 dimensions. Further research revealed connections to four-dimensional physics (not quite the standard model, but a related theory with super-symmetry, more amenable to mathematical calculations) and led to a geometric realization of a duality between these field theories, known as mirror symmetry in 2 dimensions and Langlands duality in 4 dimensions. The consequences of these result lead to many concrete problems and projects which are being currently investigated.
