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

Abstract

We establish that Hitchin's connection exist for any rigid holomorphic family of Kahler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators 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. - 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's 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.

Typo's corrected

References in corpus (4)

• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Abelian Conformal Field theories and Determinant Bundles
• Mapping Class Groups do not have Kazhdan's Property (T)
• Modular functors are determined by their genus zero data

Cited by in corpus (8)

• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Mapping Class Groups do not have Kazhdan's Property (T)
• Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators
• Asymptotics of Toeplitz operators and applications in TQFT
• Hitchin's Connection in Half-Form Quantization
• Degree one cohomology with twisted coefficients of the mapping class group
• Cohomology of mapping class groups and the abelian moduli space
• The Nielsen-Thurston classification of mapping classes is determined by TQFT