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

Andersen, Jørgen Ellegaard; Gammelgaard, Niels Leth

doi:10.48550/arxiv.0903.4091

Cited by 3 publications

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“…G. Faltings [32] has generalized N. Hitchin's approach to the general G-case. For a pure differential geometric approach to the Hitchin connection see also [9], [7] and [10].…”

Section: Geometric Quantization States That the Hilbert Space Which A...mentioning

confidence: 99%

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

Andersen

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We formulate the Asymptotic Expansion Conjecture for the Witten-Reshetikhin-Turaev quantum invariants of closed oriented three manifolds. For finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz-Riemann-Roch Theorem due to Baum, Fulton, MacPherson & Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level k, weighted by a complex phase, which is k times the Chern-Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture and the Growth Rate Conjecture for these finite order mapping tori.

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“…G. Faltings [32] has generalized N. Hitchin's approach to the general G-case. For a pure differential geometric approach to the Hitchin connection see also [9], [7] and [10].…”

Section: Geometric Quantization States That the Hilbert Space Which A...mentioning

confidence: 99%

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

Andersen

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The first named author has obtained other results about this TQFT using the gauge theory approach, such as the asymptotic faithfulness of the quantum representations [3] and the determination of the Nielsen-Thurston classification via these same representations [4] (see also [9]). He has further related these quantum representations to deformation quantization of moduli spaces both in the abelian and in the non-abelian case, please see [2], [7] and [5]. The second named author has answered some open questions by Jeffrey [49] about this TQFT for torus-bundles over S 1 by using cut-and-paste methods to perform spectral flow computations [46].…”

Section: Introductionmentioning

confidence: 99%

The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II

Andersen¹,

Himpel²

2012

Quantum Topol.

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We identify the leading order term of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants for finite order mapping tori with classical invariants for all simple and simplyconnected compact Lie groups. The square root of the Reidemeister torsion is used as a density on the moduli space of flat connections and the leading order term is identified with the integral over this moduli space of this density weighted by a certain phase for each component of the moduli space. We also identify this phase in terms of classical invariants such as Chern-Simons invariants, eta invariants, spectral flow and the rho invariant. As a result, we show agreement with the semiclassical approximation as predicted by the method of stationary phase. * Supported in part by the center of excellence grant "Center for quantum geometry of Moduli Spaces" from the Danish National Research Foundation.

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“…Here further exciting research is going on. In particular, in the realm of TQFT and the construction of modular functors [8], [9,10].…”

Section: Introductionmentioning

confidence: 99%

Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results

Schlichenmaier

2010

Preprint

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This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kähler manifolds. The basic objects, concepts, and results are given. This concerns the correct semi-classical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.

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

Cited by 3 publications

References 0 publications

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

The Witten-Reshetikhin-Turaev invariants of finite order mapping tori II

Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results

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