Deformation quantization with traces and vanishing of the wheels

Vanishing of the Kontsevich integrals of the wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. We deduce this theorem from the result of \cite{FSh} on the deformation quantization with traces.

PROP profile of deformation quantization and graph complexes with loops and wheels

Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete examples such as wheeled completion of the operad of strongly homotopy Lie algebras and the wheeled completion of the dg prop of Poisson structures. We prove also a deformation quantization theorem of wheeled Poisson structures on arbitrary formal graded manifolds.

On deformation quantization of quadratic Poisson structures

We study the deformation complex of the dg wheeled properad of $\mathbb{Z}$-graded quadratic Poisson structures and prove that it is quasi-isomorphic to the even M. Kontsevich graph complex. As a first application we show that the Grothendieck-Teichmüller group acts on the genus completion of that wheeled properad faithfully and essentially transitively. As a second application we classify all universal quantizations of $\mathbb{Z}$-graded quadratic Poisson structures together with the underlying (so called) homogeneous formality maps. In particular we show that two universal quantizations of …

Deformation quantization of a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface

We give an elementary proof of the result by Leichtnam, Tang, and Weinstein that there exists a deformation quantization with separation of variables on a complex manifold endowed with a Kaehler-Poisson structure vanishing on a Levi nondegenerate hypersurface and nondegenerate on its complement.

论 $C^{\infty}(\mathbb R^d)$ 的量子化

On the quantization of $C^{\infty}(\mathbb R^d)$

本文证明基本万有形变公式，刻画 $C^{\infty}(\mathbf R^d)$ 上初级障碍消失的无穷小形变与 Poisson 结构，并指出三维情形的特殊性质及其宇宙学启示。

Weak associativity and deformation quantization

Non-commutativity and non-associativity are quite natural in string theory. For open strings it appear due to the presence of non-vanishing background two-form in the world volume of Dirichlet brane, while in closed string theory the flux compactifications with non-vanishing three-form also lead to non-geometric backgrounds. In this paper, working in the framework of deformation quantization, we study the violation of associativity imposing the condition that the associator of three elements should vanish whenever each two of them are equal. The corresponding star products are called alternati…