Rieffel's deformation quantization and isospectral deformations

Flabby strict deformation quantizations and K-groups

We construct examples of flabby strict deformation quantizations not preserving K-groups. This answers a question of Rieffel negatively.

Noncommutative principal torus bundles via parametrised strict deformation quantization

In this paper, we initiate the study of a parametrised version of Rieffel's strict deformation quantization. We apply it to give a classification of noncommutative principal torus bundles, in terms of parametrised strict deformation quantization of ordinary principal torus bundles. The paper also contains a putative definition of noncommutative non-principal torus bundles.

Deformation Quantization: Observable Algebras, States and Representation Theory

In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star product algebra, physical applications require to study representations of this algebra. I review the recent development of a representation theory including techniques like Rieffel induction and Morita equivalence. Applications beyond quantization theory are found in noncommutative field theories.

Strict deformation quantization of locally convex algebras and modules

In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of $\mathbb{R}^n$. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactl…

Connes' Tangent Groupoid and Strict Quantization

We address one of the open problems in quantization theory recently listed by Rieffel. By developping in detail Connes' tangent groupoid principle and using previous work by Landsman, we show how to construct a strict, flabby quantization, which is moreover an asymptotic morphism and satisfies the reality and traciality constraints, on any oriented Riemannian manifold. That construction generalizes the standard Moyal rule. The paper can be considered as an introduction to quantization theory from Connes' point of view.

Parametrised strict deformation quantization of C*-bundles and Hilbert C*-modules

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrised strict deformation quantization. Rieffel's basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrised stric…

Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

$C^*$-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-$*$-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-$*$-algebras instead of $C^*$-algebras. Fourier transformation and representation theory of the mea…

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…

Quantized anti de Sitter spaces and non-formal deformation quantizations of symplectic symmetric spaces

We realize quantized anti de Sitter space black holes, building Connes spectral triples, similar to those used for quantized spheres but based on Universal Deformation Quantization Formulas (UDF) obtained from an oscillatory integral kernel on an appropriate symplectic symmetric space. More precisely we first obtain a UDF for Lie subgroups acting on a symplectic symmetric space M in a locally simply transitive manner. Then, observing that a curvature contraction canonically relates anti de Sitter geometry to the geometry of symplectic symmetric spaces, we use that UDF to define what we call Di…

$\mathbb C^n$ 中多圆盘与球上全纯函数代数的非形式形变

Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$

本文构造多圆盘与球上全纯函数代数的全纯非形式形变，证明其构成 Rieffel 意义下的严格形变量子化，并阐明相关投射性及与形式形变的关系。