Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Leichtnam, Éric; Tang, Xiang; Weinstein, Alan

doi:10.4171/jems/93

Cited by 11 publications

(10 citation statements)

References 29 publications

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“…Parts of our construction for general contact manifolds reduce to the definitions used in [Raj08] when expressed in terms of a local Darboux chart. Other examples in the literature related to the quantization of contact manifolds include [BdMG81,GS82b,LTW07]; in each case the methods used are related to deformation quantization. A brief sketch of an approach to geometric contact quantization was given by Vaisman in [Vai79]; the first quantization we present for contact manifolds expands upon the suggestion in [Vai79].…”

Section: Introductionmentioning

confidence: 99%

On the geometric quantization of contact manifolds

Fitzpatrick

2011

Journal of Geometry and Physics

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Suppose that $(M,E)$ is a compact contact manifold, and that a compact Lie group $G$ acts on $M$ transverse to the contact distribution $E$. In an earlier paper, we defined a $G$-transversally elliptic Dirac operator $\dirac$, constructed using a Hermitian metric $h$ and connection $\nabla$ on the symplectic vector bundle $E\rightarrow M$, whose equivariant index is well-defined as a generalized function on $G$, and gave a formula for its index. By analogy with the geometric quantization of symplectic manifolds, the $\mathbb{Z}_2$-graded Hilbert space $Q(M)=\ker \dirac \oplus \ker \dirac^{*}$ can be interpreted as the "quantization" of the contact manifold $(M,E)$; the character of the corresponding virtual $G$-representation is then given by the equivariant index of $\dirac$. By defining contact analogues of the algebra of observables, pre-quantum line bundle and polarization, we further extend the analogy by giving a contact version of the Kostant-Souriau approach to quantization, and discussing the extent to which this approach is reproduced by the index-theoretic method.Comment: 25 pages, references added, several corrections and clarifications and some reorganization of conten

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Section: Introductionmentioning

confidence: 99%

On the geometric quantization of contact manifolds

Fitzpatrick

2011

Journal of Geometry and Physics

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“…In local holomorphic coordinates it is written as g lk where the indices k and l are holomorphic and antiholomorphic, respectively. The Jacobi identity for g lk takes the form (13) g lk ∂g qp ∂z k = g qk ∂g lp ∂z k and g lk ∂g qp ∂z l = g lp ∂g qk ∂z l , where we assume summation over repeated indices. If the Kähler-Poisson tensor g lk is nondegenerate on M, then its inverse g kl is the metric tensor of a global pseudo-Kähler structure on M.…”

Section: Deformation Formal Symplectic Groupoid With Separation Of Vamentioning

confidence: 99%

“…Only a few examples of star products with separation of variables on general Kähler-Poisson manifolds are known (see, e.g., [4], [13] , [10]).…”

Section: Deformation Formal Symplectic Groupoid With Separation Of Va...mentioning

confidence: 99%

Infinitesimal Deformations of a Formal Symplectic Groupoid

Karabegov

2011

Lett Math Phys

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Abstract. Given a formal symplectic groupoid G over a Poisson manifold (M, π 0 ), we define a new object, an infinitesimal deformation of G, which can be thought of as a formal symplectic groupoid over the manifold M equipped with an infinitesimal deformation π 0 + επ 1 of the Poisson bivector field π 0 . The source and target mappings of a deformation of G are deformations of the source and target mappings of G. To any pair of natural star products ( * , * ) having the same formal symplectic groupoid G we relate an infinitesimal deformation of G. We call it the deformation groupoid of the pair ( * , * ). We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kähler-Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.

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“…Also, in the category of Banach geometry the study of Lie algebroids was initiated in [1,2] and some significant developments are given in [6]. On the other hand, the study of Riemannian geometry of Lie algebroids is introduced and intensively studied in [4] and a first treatment of (para) Kählerian Lie algebroids can be found in [21]. Other important structures as symplectic, hypersymplectic or Poisson structures on Lie algebroids are studied, see for instance [3,16,18].…”

Section: Introductionmentioning

confidence: 99%

On Almost Complex Lie Algebroids

Ida

Popescu

2015

Mediterr. J. Math.

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The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the E-Chern form of E 1,0 associated to an almost complex connection ∇ on E can be expressed in terms of the matrix JER, where JE is the almost complex structure of E and R is the curvature of ∇. Also, we consider a metric product connection associated to an almost Hermitian Lie algebroid and we prove that the mean curvature section of E 0,1 vanishes and the second fundamental 2-form section of E 0,1 vanishes iff the Lie algebroid is Hermitian.

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Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Cited by 11 publications

References 29 publications

On the geometric quantization of contact manifolds

On the geometric quantization of contact manifolds

Infinitesimal Deformations of a Formal Symplectic Groupoid

On Almost Complex Lie Algebroids

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