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Graph complexes in deformation quantization

Domenico Fiorenza·Lucian M. Ionescu
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Abstract

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-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.

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BibTeXRIS

Domenico Fiorenza, Lucian M. Ionescu. 2005-09-05. Graph complexes in deformation quantization. https://doi.org/10.1007/s11005-005-0017-7

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