Identification of Berezin-Toeplitz deformation quantization

Karabegov, Alexander; Schlichenmaier, Martin

doi:10.1515/crll.2001.086

Cited by 82 publications

(160 citation statements)

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“…Also, we can get smoothness of the canonical form (see Proposition 4.7), since it is the Ricci form of the canonical connection. However, to identify the Berezin-Toeplitz product with an E-product in Theorem 4.9, we need the machinery of formal integrals used by Karabegov and Schlichenmaier [20]. Remark 3.17.…”

Section: Corollary 315 Let X Be a Manifold (Possibly With Boundary)mentioning

confidence: 99%

“…See, for instance, [2] [16] and references therein. 3 On the other hand, Karabegov [19] studied special formal deformation quantization adapted to the Kähler structure, and he and Schlichenmaier [20] linked the two approaches.…”

Section: Quantizationmentioning

confidence: 99%

“…Engliš' Berezin-Toeplitz quantization is an example of a deformation quantization with separation of variables. In Theorem 5.9 of [20], Karabegov and Schlichenmaier identified the opposite of the Berezin-Toeplitz star product with a star product with separation of variables whose Karabegov form is equal to −(1/i )ω + ω can and characteristic form is (1/ i )ω − ω can /2i, where ω can is the curvature form of the canonical line bundle. Proof.…”

Section: Strictly Pseudoconvex Boundary Of a Complex Manifoldmentioning

confidence: 99%

“…Proof. According to the definition given in [20], ω can is equal to −i∂∂ν, where i n+1 e ν dzdz is the symplectic volume. (We write dz for dz 1 · · · dz n+1 .)…”

Section: Strictly Pseudoconvex Boundary Of a Complex Manifoldmentioning

confidence: 99%

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Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Leichtnam¹,

Tang²,

Weinstein³

2007

J. Eur. Math. Soc.

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Abstract. Let X be a complex manifold with strongly pseudoconvex boundary M. If ψ is a defining function for M, then − log ψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form σ = i∂∂(− log ψ) is a symplectic structure on the complement of M in a neighborhood of M in X; it blows up along M.The Poisson structure obtained by inverting σ extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M. In addition, when − log ψ is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Engliš for the Berezin-Toeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.

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Section: Corollary 315 Let X Be a Manifold (Possibly With Boundary)mentioning

confidence: 99%

Section: Quantizationmentioning

confidence: 99%

Section: Strictly Pseudoconvex Boundary Of a Complex Manifoldmentioning

confidence: 99%

“…Proof. According to the definition given in [20], ω can is equal to −i∂∂ν, where i n+1 e ν dzdz is the symplectic volume. (We write dz for dz 1 · · · dz n+1 .)…”

Section: Strictly Pseudoconvex Boundary Of a Complex Manifoldmentioning

confidence: 99%

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Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Leichtnam¹,

Tang²,

Weinstein³

2007

J. Eur. Math. Soc.

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“…Karabegov and Schlichenmaier [35] showed that the logarithm of the diagonal value of the weighted Bergman kernel is the Karabegov classifying form of the Berezin quantization. There has been much work on deformation quantization of Kähler manifolds and its applications (see the recent survey [50]).…”

Section: Introductionmentioning

confidence: 99%

Bergman kernel and Kähler tensor calculus

2013

Pure and Applied Mathematics Quarterly

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Fefferman [22] initiated a program of expressing the asymptotic expansion of the boundary singularity of the Bergman kernel for strictly pseudoconvex domains explicitly in terms of boundary invariants. Hirachi, Komatsu and Nakazawa [30] carried out computations of the first few terms of Fefferman's asymptotic expansion building partly on Graham's work on CR invariants and Nakazawa's work on the asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains. In this paper, we prove a formula for coefficients in Nakazawa's asymptotic expansion as explicit summations over strongly connected graphs, and a formula expressing partial derivatives of Kähler metrics (resp. functions) as summations over rooted trees encoding covariant derivatives of curvature tensors (resp. functions). These formulae shall be useful in studying general patterns of Fefferman's asymptotic expansion.

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Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

Roché¹,

Szenes

2002

Advances in Mathematics

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We describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. The resulting algebra is a variant of the quantum moduli algebra introduced by Alekseev, Grosse, and Schomerus and Buffenoir and Roche. We construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of Fedosov and Nest and Tsygan via Verlinde's formula. © 2002 Elsevier Science (USA) Key Words: formal deformations; quantum groups; moduli spaces. Contents.0. Introduction. • the theory of star products or formal deformations of symplectic manifolds, more specifically, the index theorem of Fedosov and Nest and Tsygan (F-NT) [14,19,29].Our first result is a greatly simplified, invariant construction of the quantum moduli algebra. We fix a generic conjugacy class s. The quantum moduli algebra is a canonically defined noncommutative algebra A q , which is finitely generated over a subring D(q) of rational functions in q. The main focus of the present work is shown in the following commuting diagram:We are searching for a natural module K defined over D(q), a trace functional Tr q and a map ev which complete the diagram. In other words, we seek to lift the canonical trace from the infinitesimal world to the global q world.A natural solution to this lifting problem seems to be setting K to be D(q) and ev: D(q) Q CQ(R to be the Laurent expansion at q=1. Surprisingly, cyclic functionals on A Our results are subject to various restrictions and assumption which we have omitted in the discussion so far for the sake of clarity. These will be discussed below and in the main body of the paper.Contents of the paper. In Section 1 we recall the volume formula of Witten and Verlinde's formula. We compute the asymptotics of the q-volume series that we introduce by analogy. In Section 2 we recall the

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Identification of Berezin-Toeplitz deformation quantization

Cited by 82 publications

References 41 publications

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Bergman kernel and Kähler tensor calculus

Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

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