Notes for an Introduction to Kontsevich’s quantization theorem

نویسندگان

  • B. Keller
  • Freddy Van Oystaeyen
چکیده

In these notes, we present Kontsevich’s theorem on the deformation quantization of Poisson manifolds, his formality theorem and Tamarkin’s algebraic version of the formality theorem. We also introduce the necessary material from deformation theory.

منابع مشابه

Quantizing Poisson Manifolds

This paper extends Kontsevich’s ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds. These notes grew out of the author’s attempt to understand Kontsevich’s ideas [Kon95a] on quantizing Poisson manifolds. We introduce a new di...

متن کامل

Formality and Star Products

These notes, based on the mini-course given at the PQR2003 Euroschool held in Brussels in 2003, aim to review Kontsevich’s formality theorem together with his formula for the star product on a given Poisson manifold. A brief introduction to the employed mathematical tools and physical motivations is also given.

متن کامل

Graph complexes in deformation quantization

Kontsevich’s formality theorem and the consequent star-product formula rely on the construction of an L∞-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential grade...

متن کامل

Bimodules and Branes in Deformation Quantization

We prove a version of Kontsevich’s formality theorem for two subspaces (branes) of a vector space X. The result implies in particular that the Kontsevich deformation quantizations of S(X∗) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet’s recent paper on Koszul duality in deformation quantization.

متن کامل

Quantization of Linear Poisson Structures and Degrees of Maps

1.1. Kontsevich’s quantization. Recently, Kontsevich [3] provided a deformation quantization of the algebra of functions on an arbitrary Poisson manifold. The key ingredient of his construction is the quantization formula for R, which is then extended to the case of general manifolds using formal geometry. The terms of the star product are identified with certain Feynman graphs, and their coeff...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

متن کامل
عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2005