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Hitchin's connection, Toeplitz operators, and symmetry invariant deformation quantization

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  • Jørgen Ellegaard Andersen, Denmark
We introduce the notion of a rigid family of Kähler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin 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. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization.

Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.
Original languageEnglish
JournalQuantum Topology
Volume3
Issue3/4
Pages (from-to)293-325
Number of pages33
ISSN1663-487X
DOIs
Publication statusPublished - 2012

    Research areas

  • Geometric quantization, Hitchin connection, Toeplitz operators, deformation quantization, star products, moduli space of flat connections

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