Graph complexes in deformation quantization
Abstract
Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an -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 graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.
Cite
@article{arxiv.math/0505410,
title = {Graph complexes in deformation quantization},
author = {Domenico Fiorenza and Lucian M. Ionescu},
journal= {arXiv preprint arXiv:math/0505410},
year = {2013}
}
Comments
39 pages; 3 eps figures; uses Xy-pic. Final version. Details added, mainly concerning the tree-level approximation. Typos corrected. An abridged version will appear in Lett. Math. Phys