A new construction of strict deformation quantization for Lagrangian fiber bundles

Quantization of Poisson algebras associated to Lie algebroids

We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded as a result of a quantization procedure. The C*-algebra of the tangent groupoid of a given Lie groupoid G (with Lie algebra g) is the C*-algebra of a continuous field of C*-algebras over R with fibers A_0=C*(g)=C_0(g*) and A_h=C*(G) for nonzero h. The same is true for the corresponding reduced C*-algebras. Our results have applications to, e.g., transformation group C*-algebras, K-theory, and inde…

Pati-Salam model in curved space-time and sheaf quantization

This note aims to give a self-contain and detail explanation about U(k')times U(k) Pati-Salam model in curved space-time which derived from (1+n)-dimensional square root Lorentz manifold by self-parallel transportation principle. The concepts foundation of manifold from view point of category theory, fiber bundle and sheaf theories are reviewed. There are extra U(k')times U(k)-principal bundle and U(k)-associated bundle than (1+n)-dimensional Lorentz manifold. The conservative currents on square root Lorentz manifold are discussed preliminary. A detail proof of relation from sheaf quantization…

Identification of Berezin-Toeplitz deformation quantization

We give a complete identification of the deformation quantization which was obtained from the Berezin-Toeplitz quantization on an arbitrary compact Kaehler manifold. The deformation quantization with the opposite star-product proves to be a differential deformation quantization with separation of variables whose classifying form is explicitly calculated. Its characteristic class (which classifies star-products up to equivalence) is obtained. The proof is based on the microlocal description of the Szegoe kernel of a strictly pseudoconvex domain given by Boutet de Monvel and Sjoestrand.

辛代数余结构、群体Toeplitz算子与变形量子化

Symplectic algebroids, groupoid Toeplitz operators and deformation quantization

本文利用Toeplitz算子在辛Lie代数余结构诱导的泊松流形上定义星积，推广了Guillemin和Melrose在辛情形下的方法。

Deformation Quantization and Irrational Numbers

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is motivated by the problem of constructing strict deformation quantizations of symplectic manifolds. We show that this type of approximation exists for any real number and also investigate what happens if the number is rational or a quadratic irrational.

Quantization of Planck's Constant

This paper is about the role of Planck's constant, $\hbar$, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a given Poisson manifold, one can use all possible rescalings of the Poisson structure, which can be combined into a single "Heisenberg-Poisson" manifold. The new coordinate on this manifold is identified with $\hbar$. I present an explicit construction for a symplectic groupoid integrating a Heisenberg-Poisson manifold and discuss its geometric quantization. I show that in cases where $\hbar$ canno…

Dynamical Quantization of Contact Structures

We construct a dynamical quantization for contact manifolds in terms of a flat connection acting on a Hilbert tractor bundle. We show that this contact quantization, which is independent of the choice of contact form, can be obtained by quantizing the Reeb dynamics of an ambient strict contact manifold equivariantly with respect to an R+-action. The contact quantization further determines a certain contact tractor connection whose parallel sections determine a distinguished choice of Reeb dynamics and their quantization. This relationship relies on tractor constructions from parabolic geometri…

Toeplitz算子族、族索引与变形量子化

Families of Toeplitz operators, family index and deformation quantization

本文通过构造Szegö投影族和Toeplitz算子族，实现了预量子化辛纤维化的变形量子化，并推导了族索引公式。

A Formal Equivalence of Deformation Quantization and Geometric Quantization (of Higher Groupoids) and Non-Perturbative Sigma Models

Based on work done by Bonechi, Cattaneo, Felder and Zabzine on Poisson sigma models, we formally show that Kontsevich's star product can be obtained from the twisted convolution algebra of the geometric quantization of a Lie 2-groupoid, one which integrates the Poisson structure. We show that there is an analogue of the Poisson sigma model which is valued in Lie 1-groupoids and which can often be defined non-perturbatively; it can be obtained by symplectic reduction using the quantization of the Lie 2-groupoid. We call these groupoid-valued sigma models and we argue that, when they exist, they…

Quantization of the Geodesic flow on Quaternion Projective Spaces

We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle of this complex structure is holomorphically trivial by explicitly constructing a nowhere vanishing holomorphic global section. Then we construct a Hilbert space consisting of a certain class of holomorphic functions on the punctured cotangent bundle by the method of pairing polar…