Kähler Quantization and Hitchin Connections

By Niels Leth Gammelgaard
PhD Dissertations
October 2010
In this thesis, we study geometric quantization as well as deformation quantization of symplectic manifolds endowed with a compatible complex structure. Using Karabegov's classification of star products with separation of variables, we give an explicit, local, combinatorial formula for any such deformation quantization, which uses Feynman graphs to encode the relevant differential operators. In particular, this yields an explicit formula for the Berezin-Toeplitz star product. For geometric quantization, we consider Andersen's generalization of Hitchin's projectively flat connection to a general symplectic manifold, and we extend his construction to geometric quantization with metaplectic correction. We calculate the curvature and prove that the connection is projectively flat if the symplectic manifold does not allow holomorphic vector fields. Furthermore, we prove that the Hitchin connection is asymptotically unitary to any order. Finally, based on ideas by Andersen, we study a formal analog of the Hitchin connection for deformation quantization, and we give explicit formulas for the formal Hitchin connections associated with the ordinary and metaplectic Berezin-Teoplitz star products.
Thesis advisor: Jørgen Ellegaard Andersen
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