A proof of the Tsygan formality conjecture for chains

2000/10/31 by Boris Shoikhet, Shoikhet, Boris
Mathematics · Physics and Astronomy · #13D03 #18G55 #19D55 #81T18 #Commutative Algebra (math.AC) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #K-Theory and Homology (math.KT) #Primary 53D55 #Quantum Algebra (math.QA) #Secondary 57R56 #hep-th #math.AC #math.KT #math.QA #msc:13D03 #msc:18G55 #msc:19D55 #msc:53D55 #msc:57R56 #msc:81T18

paper · pdf · doi:10.48550/arxiv.math/0010321

LaTeX, 24 pages, 5 eps figures

arxiv created 2000/12/05 · arxiv updated 2009/11/30

Abstract

We extend the Kontsevich formality L_∞-morphism \U\colon T^\ndot_\poly(\R^d)→\D^\ndot_\poly(\R^d) to an L_∞-morphism of an L_∞-modules over T^\ndot_\poly(\R^d), \U\colon C_\ndot(A,A)→Ω^\ndot(\R^d), A=C^∞(\R^d). The construction of the map \U is given in Kontsevich-type integrals. The conjecture that such an L_∞-morphism exists is due to Boris Tsygan \citeTs. As an application, we obtain an explicit formula for isomorphism A_*/[A_*,A_*]\simto A/\A,A\ (A_* is the Kontsevich deformation quantization of the algebra A by a Poisson bivector field, and \,\ is the Poisson bracket). We also formulate a conjecture extending the Kontsevich theorem on the cup-products to this context. The conjecture implies a generalization of the Duflo formula, and many other things.

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