相关论文: Formality and Star Products
Using a variant of the Boardman-Vogt tensor product, we construct an action of the Grothendieck-Teichm\"uller group on the completion of the little n-disks operad $E_n$. This action is used to establish a partial formality theorem for $E_n$…
Based on a closed formula for a star product of Wick type on $\CP^n$, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the…
For compact quantizable K\"ahler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which is explained. As star products the…
We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so…
Formality is a topological property, defined in terms of Sullivan's model for a space. In the simply-connected setting, a space is formal if its rational homotopy type is determined by the rational cohomology ring. In the general setting,…
Given a star product with separation of variables $\star$ on a pseudo-K\"ahler manifold $M$ and a point $x_0 \in M$, we construct an associative algebra of formal distributions supported at $x_0$. We use this algebra to express the formal…
We extend the Kontsevich formality $L_\infty$-morphism $\U\colon T^\ndot_\poly(\R^d)\to\D^\ndot_\poly(\R^d)$ to an $L_\infty$-morphism of an $L_\infty$-modules over $T^\ndot_\poly(\R^d)$, $\hat \U\colon C_\ndot(A,A)\to\Omega^\ndot(\R^d)$,…
We give simple explicit formulas for deformation quantization of Poisson-Lie groups and of similar Poisson manifolds which can be represented as moduli spaces of flat connections on surfaces. The star products depend on a choice of…
The quantizer-dequantizer formalism is developed for mean value and probability representation of qubits and qutrits. We derive the star-product kernels providing the possibility to derive explicit expressions of the associative product of…
An 'isomorphism' between the 'moduli space' of star products on $\R^2$ and the 'moduli space' of all formal Poisson structures on $\R^2$ is established.
We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known…
This is an extended abstract for a talk given at the mini-workshop "Cohomology rings and fundamental groups of hyperplane arrangements, wonderful compactifications, and real toric varieties", held in Oberwolfach, September 30-October 6,…
This is my talk on the Bourbaki seminar, November 1996. It contains an elementary introduction to Borcherds' product formulas.
In the first part of this paper we review several formalisms which give alternative ways for describing the light. They are: the formalism `baroque' and the Majorana-Oppenheimer form of electrodynamics, the Sachs' theory of Elementary…
We study a geometric notion related to formality for Bott-Chern cohomology on complex manifolds.
Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more…
One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $\P$ on $M$ and any torsionfree linear connection $\nabla$ on $M$, a universal deformation quantization associates to this data a…
We conjecture an explicit formula for a cyclic analog of the Formality $L_{\infty}$-morphism [K]. We prove that its first Taylor component, the cyclic Hochschild-Kostant-Rosenberg map, is in fact a morphism (and a quasiisomorphism) of the…
Invited lecture at the XIV-th workshop on geometric methods in physics, Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the…
Starting from formal deformation quantization we use an explicit formula for a star product on the Poincar\'e disk D_n to introduce a Fr\'echet topology making the star product continuous. To this end a general construction of locally…