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      Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators

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          Abstract

          In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particular, we recall the differential geometric construction of Hitchin's projectively flat connection in the bundle over Teichmuller space obtained by push-forward of the determinant line bundle over the moduli space of rank n, fixed determinant, semi-stable bundles fibering over Teichmuller space. We recall the relation between the Hitchin connection and Toeplitz operators which was first used by the first named author to prove the asymptotic faithfulness of the SU(n) quantum representations of the mapping class groups. We further review the construction of the formal Hitchin connection, and we discuss its relation to the full asymptotic expansion of the curve operators of Topological Quantum Field Theory. We then go on to identifying the first terms in the formal parallel transport of the Hitchin connection explicitly. This allows us to identify the first terms in the resulting star product on functions on the moduli space. This is seen to agree with the first term in the star product on holonomy functions on these moduli spaces defined by Andersen, Mattes and Reshetikhin.

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          Quantum field theory and the Jones polynomial

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            Invariants of 3-manifolds via link polynomials and quantum groups

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              Classical and quantum conformal field theory

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                Author and article information

                Journal
                24 March 2009
                Article
                0903.4091
                7f1c057e-5a76-4321-8f83-ab7cb1c9238a

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                Grassmannians, moduli spaces and vector bundles, 1-24, Clay Math. Proc., 14, Amer. Math. Soc., Providence, RI, 2011
                24 pages
                math.DG math-ph math.MP

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