Three talks by Eckhard Meinrenken at the CRCG Workshop - Higher Structures in Topology and Geometry III, June 4-5 2009.
Lecture I: Dirac geometry and moment maps
A Dirac structure on a manifold $M$ is a Lagrangian sub-bundle of the direct sum $TM\oplus T^*M$, satisfying a certain integrability condition. The projection of a Dirac structure onto $TM$ is an integrable generalized distribution, so that one may speak of the `leaves' of the Dirac structure. Basic examples are Poisson structures, with the symplectic leaves, and the Cartan-Dirac structure on a Lie group $G$, with leaves the conjugacy classes in $G$. Following Bursztyn-Crainic, we will use morphisms of Dirac structures to give a nice definition of group-valued moment maps, and develop some of their properties.
Lecture II: Dixmier-Douady theory and twisted K-theory
A classical theorem of Dixmier-Douady identifies the integral degree 3 cohomology group $H^3(X,\Z)$ with Morita isomorphism classes of *-algebra bundles $A\to X$, with typical fiber the compact operators on a Hilbert space. Such bundles (sometimes called `gerbes') are thus a higher degree analogs of line bundles. We will consider the definition of the twisted K-theory of X in terms of such bundles, and discuss some basic examples.
Lecture III: Quantization of group-valued moment maps
In this lecture, we will combine the concepts from lectures I and II to develop a quantization procedure for group-valued moment maps. The `localization' and `quantization commutes with reduction' theorems for our theory give a computation of Verlinde numbers for moduli spaces of flat $G$-bundles over surfaces.
Notes of the talks taken by Christoph Wockel.