[Paper Review] Lectures on Deformation quantization of Poisson manifolds

Chiara Esposito|arXiv (Cornell University)|Jul 13, 2012

Homotopy and Cohomology in Algebraic Topology21 references3 citations

TL;DR

This paper presents a concise exposition of Kontsevich's formality theorem as the foundational tool for deformation quantization of Poisson manifolds. By constructing an $L_∞$-quasi isomorphism between the DGLA of polyvector fields and the DGLA of multidifferential operators, it establishes a one-to-one correspondence between equivalence classes of formal Poisson structures and isomorphism classes of star products, thereby proving that every Poisson manifold admits a canonical deformation quantization.

ABSTRACT

These notes are based on the course given at the School of Geometry, University Kasdi Merbah (Ouargla) 2012. The aim of the course was the deformation quantization of Poisson Lie groups. In these notes we only review Kontsevich's formality theorem.

Motivation & Objective

• To provide a self-contained introduction to Kontsevich's formality theorem in the context of deformation quantization.
• To establish the existence of a canonical deformation quantization for any Poisson manifold.
• To clarify the correspondence between formal Poisson bivectors and star products using $L_∞$-quasi isomorphisms.
• To connect the Hochschild-Kostant-Rosenberg isomorphism to the full formality theorem via higher homotopies.

Proposed method

• Utilizes differential graded Lie algebras (DGLAs) of multivector fields and multidifferential operators on a smooth manifold $M$.
• Applies the Hochschild-Kostant-Rosenberg theorem to establish a quasi-isomorphism between cohomologies of these DGLAs.
• Constructs an $L_∞$-quasi isomorphism $K: \mathfrak{g}_S^\bullet(M) \to \mathfrak{g}_G^\bullet(C^\infty(M))$ with $K_1$ matching the HKR map.
• Uses the Maurer-Cartan equation in the DGLA framework to classify formal Poisson structures and star products.
• Applies the $L_∞$-quasi isomorphism theorem to induce a bijection between the moduli spaces $M(\mathfrak{g}_S^\bullet(M))$ and $M(\mathfrak{g}_G^\bullet(C^\infty(M)))$.
• Employs formal power series in $\hbar$ to define star products and gauge equivalence classes of deformations.

Experimental results

Research questions

• RQ1 How can one systematically construct a star product on the algebra of smooth functions on a Poisson manifold?
• RQ2 What is the role of the Hochschild-Kostant-Rosenberg isomorphism in extending to a full formality theorem?
• RQ3 How does an $L_\infty$-quasi isomorphism induce a correspondence between Poisson structures and star products?
• RQ4 What is the significance of the Maurer-Cartan moduli space in classifying deformation quantizations?

Key findings

• The Kontsevich formality theorem establishes a canonical $L_\infty$-quasi isomorphism between the DGLA of polyvector fields and the DGLA of multidifferential operators on a smooth manifold.
• The component $K_1$ of the formality map coincides with the Hochschild-Kostant-Rosenberg quasi-isomorphism, linking cohomology of multivector fields to Hochschild cohomology.
• The $L_\infty$-quasi isomorphism induces a bijection between the moduli space of formal Poisson structures and the moduli space of star products on $C^\infty(M)$.
• This correspondence proves that every Poisson manifold admits a canonical deformation quantization via a star product.
• The classification of star products is equivalent to the classification of formal Poisson bivectors, up to gauge equivalence.
• The theory realizes a one-to-one correspondence between equivalence classes of formal Poisson structures $\pi_\hbar$ and isomorphism classes of star products on $C^\infty(M)$.