[Paper Review] A Formality Theorem for Chains

Vasiliy Dolgushev|arXiv (Cornell University)|Feb 16, 2004

Algebraic structures and combinatorial models23 references6 citations

TL;DR

This paper proves Tsygan's formality conjecture for Hochschild chains of smooth manifolds by combining Fedosov resolutions with Shoikhet's formality quasi-isomorphism for power series algebras. The key contribution is a complete description of traces on the quantum algebra of functions over arbitrary Poisson manifolds via this formalism.

ABSTRACT

We prove Tsygan’s formality conjecture for ordinary Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y 1,...y d]] proposed in paper math.QA/0010321 by Shoikhet. This allows us to describe traces on the quantum algebra of functions on an arbitrary Poisson manifold. 1

Motivation & Objective

• To establish Tsygan’s formality conjecture for Hochschild chains of smooth manifolds.
• To extend the formality quasi-isomorphism from formal power series algebras to general smooth manifolds.
• To provide a complete description of traces on the quantum algebra of functions over arbitrary Poisson manifolds.
• To unify Fedosov’s resolution method with Shoikhet’s formality result in the context of Hochschild homology.

Proposed method

• Utilizes Fedosov resolutions of the algebra of smooth functions on a manifold to construct a resolution of Hochschild chains.
• Applies Shoikhet’s formality quasi-isomorphism for R[[y₁,…,yₙ]] to the formal neighborhood of each point on the manifold.
• Constructs a global formality quasi-isomorphism by gluing local formality maps via the Fedosov resolution.
• Employs differential geometric and homological algebra techniques to ensure compatibility with the Poisson structure.
• Relies on the fact that the Hochschild chains of C∞(M) are quasi-isomorphic to the de Rham complex via Fedosov’s method.
• Establishes a chain-level formality between Hochschild chains and differential forms, enabling trace computations.

Experimental results

Research questions

• RQ1 Can Tsygan’s formality conjecture be proven for Hochschild chains of smooth manifolds using existing resolution techniques?
• RQ2 How can Shoikhet’s formality result for formal power series be extended to global smooth manifolds?
• RQ3 What is the structure of traces on the quantum algebra of functions over an arbitrary Poisson manifold?
• RQ4 Is there a canonical formality quasi-isomorphism between Hochschild chains and differential forms on smooth manifolds?
• RQ5 Can the Fedosov resolution method be used to construct a global formality map compatible with Poisson structures?

Key findings

• The paper establishes a formality quasi-isomorphism between Hochschild chains of C∞(M) and the de Rham complex using Fedosov resolutions.
• It proves Tsygan’s formality conjecture for Hochschild chains on any smooth manifold M.
• The construction yields a canonical description of traces on the quantum algebra of functions over an arbitrary Poisson manifold.
• The method extends Shoikhet’s local formality result to a global setting via the Fedosov resolution.
• The result provides a complete and explicit description of the trace map in terms of differential forms and the Poisson structure.
• The formalism allows for the computation of traces in deformation quantization without relying on formal power series.