Abstract
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 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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Mathematics Subject Classifications (2000): 53D55, secondary 18G55
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Fiorenza, D., Ionescu, L.M. Graph Complexes in Deformation Quantization. Lett Math Phys 73, 193–208 (2005). https://doi.org/10.1007/s11005-005-0017-7
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DOI: https://doi.org/10.1007/s11005-005-0017-7