Aarhus University Seal / Aarhus Universitets segl

PhD defence by Niels Leth Gammelgaard (2010)

Title: Kähler Quantization and Hitchin Connections

2013.11.04 | Christine Dilling

Date Mon 25 Oct
Time 13:30 16:30
Location Aud. D4

Bedømmelsesudvalg / Assessment Committee:
Martin Schlichenmaier, University of Luxembourg, Grand-Duchy of Luxembourg
Richard A. Wentworth, University of Maryland, USA
Professor Marcel Bökstedt (formand), Aarhus Universitet

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

Format available: PDF (1198.6 kb)

Short URL: math.au.dk/publs

Links til resume og ph.d.-afhandling / Please see abstract and PhD thesis:

PhD defense