seminar: A generalization of Hitchin's connection and Berezin-Toeplitz deformation quantization
speaker: Jorgen Ellegaard Andersen (University of Aarhus)abstract: We establish that Hitchin's connection exist for any rigid holomorphic family of K{\"a}hler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Berezin-Toeplitz quantization we prove that Hitchin's connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin-Toeplitz deformation quantizations.
We shall explain how we see the existence of this connection as a replacement of the paradime "the quantization is independent of the polarization". - In fact it suggest that the quantization {\em should indeed really depend} on the polarization (equivalently on the metric) when this connection is not flat. - A geometric setting thus arrises in which general relativity and quantization can be considered simultaniously, though we only study it in the positive definite compact case.
Finally we will discuss two of the applications of this to the moduli space situation in which Hitchin constructed his connection and showed its projectively flatness: Our proof of Turaev's Asymptotic faithfulness Conjecture and a construction of a mapping class group invariant deformation quantization of the moduli spaces. Such a deformation quantization is unique, so we paraprase this result by stating that: There is a unique way to quantise the moduli spaces mapping class group invariantly.
timetable:
Tue 6 Sep, 9:00 - 10:00, Aula Dini
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