Asymptotics of Toeplitz operators and applications in TQFT

Abstract

In this paper we provide a review of asymptotic results of Toeplitz operators and their applications in TQFT. To do this we review the differential geometric construction of the Hitchin connection on a prequantizable compact symplectic manifold. We use asymptotic results relating the Hitchin connec- tion and Toeplitz operators, to, in the special case of the moduli space of flat SU(n)-connections on a surface, prove asymptotic faithfulness of the SU(n) quantum representations of the mapping class group. We then go on to re- view formal Hitchin connections and formal trivializations of these. We discuss how these fit together to produce a Berezin-Toeplitz star product, which is independent of the complex structure. Finally we give explicit examples of all the above objects in the case of the abelian moduli space. We furthermore discuss an approach to curve operators in the TQFT associated to abelian Chern-Simons theory.

28 pages. To appear in Travaux Mathematiques, University of Luxembourg in the volume Lectures of the GEOQUANT school

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• Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to \infty$
• Abelian Conformal Field theories and Determinant Bundles
• Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators
• Hitchin's connection, Toeplitz operators and symmetry invariant deformation quantization
• Modular functors are determined by their genus zero data
• Hitchin's Connection in Half-Form Quantization
• Some asymptotics of TQFT via skein theory
• The mapping class group orbit of a multicurve
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Cited by in corpus (7)

• Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory
• The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
• Mapping class group invariant unitarity of the Hitchin connection over Teichmüller space
• A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications
• Coordinates for the Universal Moduli Space of Holomorphic Vector Bundles
• Felix Alexandrovich Berezin and his work
• Asymptotic aspects of the Teichmüller TQFT