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The Moduli Space of Flat Connections on a Surface Poisson Structures and Quantization

by Anders Reiter Skovborg
PhD Dissertations October 2006

The moduli space of flat connections in a principal bundle over a surface is the subject of this dissertation. Poisson structures on this space, parametrized by certain tensors on the structure group, are studied, and it is demonstrated how the Poisson algebra of chord diagrams unifies all these Poisson structures. In the case of a surface with boundary, so-called $*$-products, constructed by Andersen, Mattes and Reshetikhin, quantizing the Poisson structure on chord diagrams are presented, and it is proved that they induce $*$-products on the moduli space if the structure group is either general linear or special linear.

Quantization of the Poisson loop algebras is also addressed; this discussion is interesting in its own right and moreover facilitates an investigation of the relationship between the AMR $*$-products and another $*$-product, due to Bullock, Frohman and Kania-Bartoszynska, on the moduli space of $\mathrm{SL}_2(\mathbf{C})$-connections. The thesis ends with an analysis of the differentiability of the latter $*$-product.

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Thesis advisor: Jørgen Ellegaard Andersen