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Group-valued moment maps and moduli spaces of flat G-bundles

He will be speaking at the IGA/AMSI Workshop, Group-valued moment maps with applications to mathematics and physics

5 lectures (2 hours each), September 5-9, 2011. The lectures will be held at the Conference Room 7.15, Level 7, Innova 21 building, University of Adelaide.

Lecture 1: Introduction to G-valued moment maps

The theory of quasi-Hamiltonian G-spaces with G-valued moment maps has its origins in 2-dimensional gauge theory. Its features are similar to the usual Hamiltonian theory, but with interesting modifications. We will give an overview of the theory, with discussions of a convexity theorem and a Kirwan surjecticity theorem.

Lecture 2: Dirac Geometry and Witten's volume formulas

Dirac geometry treats 2-forms and bivector fields within a common framework. We will explain how this leads to a conceptual approach to G-valued moment maps. As an application, we will construct Liouville volume forms, leading to a proof of Witten's volume formulas for moduli spaces of flat G-bundles over surfaces.

Lecture 3: Dixmier-Douady theory and pre-quantization

Dixmier-Douady bundles provide geometric realizations of integral degree three cohomology classes over a space. We will use these bundles to construct distinguished "twisted Spin-c structures" on quasi-Hamiltonian G-spaces, and also define pre-quantizations in similar terms.

Lecture 4: Quantization of group-valued moment maps

We will review Rosenberg's definition of twisted K-homology in terms of Dixmier-Douady bundles and the Freed-Hopkins-Teleman theorem on the twisted K-homology of Lie groups. We then define the quantization of group-valued moment maps as push-forwards in twisted K-homology.

Lecture 5: Application to Verlinde formulas

The quantization of a quasi-Hamiltonian space is computable via localization. In conjunction with a `quantization commutes with reduction' theorem, this leads to the symplectic version of the Verlinde formulas for moduli spaces of flat G-bundles.