Formal Connections for Families of Star Products

Andersen, Jørgen Ellegaard; Masulli, Paolo; Schätz, Florian

doi:10.1007/s00220-016-2574-2

Cited by 4 publications

(9 citation statements)

References 30 publications

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“…For references on Toeplitz operator theory and how it is related to deformation quantization please see [56,8] and the following works of Bordemann-Meinrenken, and Schlichenmaier [27,69,70]. For further important work on semi-classical aspects of Toeplitz operator theory related to this paper we refer to the works of Boutet de Monvel and Guillemin and Boutet de Monvel and Sjöstrand [28,29].…”

Section: The Hitchin Connection and Quantum Representationsmentioning

confidence: 99%

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

Andersen

Petersen

2018

Trans. Amer. Math. Soc.

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In this paper we engage in a general study of the asymptotic expansion of the Witten-Reshetikhin-Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten-Reshetikhin-Turaev TQFT via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of Picard-Lefschetz theory for Laplace integrals.

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Section: The Hitchin Connection and Quantum Representationsmentioning

confidence: 99%

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

Andersen

Petersen

2018

Trans. Amer. Math. Soc.

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“…Remark 3.3. Most of the statement already appeared in [2]. Our contribution is the flatness of R ∇ (A, B) and the identification of its first term.…”

Section: A Canonical Compatible Formal Connection On Vmentioning

confidence: 84%

“…We explain, in our context, the work of Andersen-Masulli-Schätz [2] and Masulli [28] who constructed formal connections adapted to families of Fedosov star products. The only difference is that we are dealing with a bundle over the whole infinite dimensional space of symplectic connections and that we are only considering Fedosov star products build with a trivial choice of series of closed 2-form.…”

Section: Formal Connectionsmentioning

confidence: 99%

“…As in [2], to a family of star product { * σ } σ∈T on (M, ω) parametrized by a smooth manifold T , one associate the vector bundle…”

Section: Definitionsmentioning

confidence: 99%

“…We use the notion of formal connections introduced by Andersen-Masulli-Schätz [2] We consider a vector bundle of formal star product algebras on the space of symplectic connections. Topologically, it is simply a product with fiber the space of formal smooth functions on M. Algebraically, on the fiber over the symplectic connection ∇, the star product is * ∇ , the Fedosov star product obtained with symplectic connection ∇ and a trivial formal series of closed 2-form.…”

Section: Introductionmentioning

confidence: 99%

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The formal moment map geometry of the space of symplectic connections

La Fuente-Gravy

2021

Preprint

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We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebra on the space of symplectic connections E. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the * -product trace of the curvature is a formal symplectic form on E. The action of Hamiltonian diffeomorphisms on E preserves the formal symplectic structure and we show the * -product trace can be interpreted as a formal moment map for this action. Finally, we apply this picture to study automorphisms of star products and Hamiltonian diffeomorphisms.

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Asymptotic properties of the Hitchin–Witten connection

Andersen

Malusà

2019

Lett Math Phys

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We explore extensions to SL(n, C)-Chern-Simons theory of some results obtained for SU(n)-Chern-Simons theory via the asymptotic properties of the Hitchin connection and its relation to Toeplitz operators developed previously by the first named author. We define a formal Hitchin-Witten connection for the imaginary part s of the quantum parameter t = k + is and investigate the existence of a formal trivialisation. After reducing the problem to a recursive system of differential equations, we identify a cohomological obstruction to the existence of a solution. We explicitly find one for the first step, in the specific case of an operator of order 0, and show in general the vanishing of a weakened version of the obstruction. We also find a solution of the whole recursion in the case of a surface of genus 1. * Supported in part by the center of excellence grant "Centre for quantum geometry of Moduli Spaces" DNRF95, from the Danish National Research Foundation.

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Formal Connections for Families of Star Products

Cited by 4 publications

References 30 publications

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

The formal moment map geometry of the space of symplectic connections

Asymptotic properties of the Hitchin–Witten connection

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