Geometric Representation Theory, Topology, and Mathematical Physics

I am interested in the connections between mathematical physics, topology, and representation theory, in particular the incarnations of the last two in quantum field theory and string theory. More specifically, I currently study a connection between a certain category of equivariant matrix factorisations, which comes from the LandauGinzburg or B model in string theory, and a certain category of representations of
semisimple Lie groups and their loop groups. The connection is established via a Dirac-type operator that gives rise to twisted equivariant Fredholm bundles, which, when the group is compact, give a relation to twisted K-theory. This equivalence can also be seen as categorification of the Verlinde formula and thus has implications for 3D topological Chern Simons theory and, given the dizzying web of connections and dualities of varies TFT's, is likely related to all kinds of mathematical physics.

The aim of my current project is threefold: Firstly, I am developing the analytic methods to extend the Dirac operator technique to the case of non-compact real semisimple Lie groups and prove a similar equivalence of categories. Secondly, I am a novel proof of the Kirillov character formula for compact groups, loops in compact groups, and tempered representations of real semisimple groups which does not make use of Harish-Chandra's work on orbital integrals - instead it follows directly from the properties of this Dirac operator. Finally, I am using algebraic methods, using semi-infinite cohomology and vertex operator algebras, to develop nice classes of representations of loop groups of noncompact groups which might serve as the correct setting in which to generalize this categorified Verlinde formula, whose existence is hinted at in the topology and physics literature alike.

The new methods used in this project revolve around mathematical physics. The use of matrix factorisations to study representations, inspired by string theory, was introduced by my supervisor and his collaborators a few years ago. As an application, taking chern characters of the Dirac families gives a new proof of the Kirillov character formula for compact groups, which has the advantage of being fit for generalisation to loop groups and real semisimple groups without much change. Other methods include semi-infinte cohomology and vertex operator algebras, which both have physics origins as well, which I use is conjunction with the geometry of the flag variety of loop groups of noncompact semisimple Lie groups to develop analogs of the discrete series.

This project falls within the EPSRC Topology research area