For the IGA Lecturers in 2013, click here.
For the IGA Lecturers in 2012, click here.
For the IGA Lecturers in 2010, click here.
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.
Last changed 2011/12/19
Email IGA web person
|BACK TO THE IGA HOME PAGE|