Formality theorems for Hochschild chains in the Lie algebroid setting

Abstract

In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary Poisson Lie algebroid. The proofs are based on the use of Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]], Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and Fedosov's resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid.

40 pages, no figures

References in corpus (7)

• Deformation quantization of Poisson manifolds, I
• Deformation quantization of algebraic varieties
• Quantum groupoids
• From local to global deformation quantization of Poisson manifolds
• Deformation quantization with traces
• Characteristic classes of gauge systems
• Formality for Lie algebroids

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• Formality of the homotopy calculus algebra of Hochschild (co)chains
• Caldararu's conjecture and Tsygan's formality
• Weak quantization of Poisson structures
• Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
• Formality and Kontsevich--Duflo type theorems for Lie pairs
• Atiyah classes and dg-Lie algebroids for matched pairs
• Noncommutative calculus and operads
• On global deformation quantization in the algebraic case
• Formality of differential graded algebras and complex Lagrangian submanifolds
• Integration of Cocycles and Lefschetz Number Formulae for Differential Operators
• Lie algebroids as $L_{\infty }$ spaces