[Paper Review] Polarized deformation quantization

Paul Bressler, J. Donin|ArXiv.org|Jul 30, 2000

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper establishes a polarized deformation quantization framework on symplectic manifolds equipped with a complex polarization, proving the existence of a star-product algebra $ω$-compatible with the polarization and deriving a precise formula linking the Fedosov class $\theta(\mathcal{A})$, the Chern class $c_1(P)$ of the polarization, and the extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ of the Lie algebra of derivations of the polarized subalgebra $\mathcal{O}$. The key result is the identity $\theta(\mathcal{A}) = \frac{1}{t}\mathrm{cl}(\mathcal{A},\mathcal{O}) - \frac{1}{2}c_1(P)$, which generalizes Karabegov's separation-of-variables quantization to polarized settings.

ABSTRACT

Let $A$ be a star product on a symplectic manifold $(M,ω_0)$, $\frac{1}{t}[ω]$ its Fedosov class, where $ω$ is a deformation of $ω_0$. We prove that for a complex polarization of $ω$ there exists a commutative subalgebra, $O$, in $A$ that is isomorphic to the algebra of functions constant along the polarization. Let $F(A)$ consists of elements of $A$ whose commutator with $O$ belongs to $O$. Then, $F(A)$ is a Lie algebra which is an $O$-extension of the Lie algebra of derivations of $O$. We prove a formula which relates the class of this extension, the Fedosov class, and the Chern class of $P$.

Motivation & Objective

• To construct a star-product algebra $\mathcal{A}$ on a symplectic manifold $(M,\omega_0)$ that is compatible with a complex polarization $P$.
• To define a subalgebra $\mathcal{O} \subset \mathcal{A}$ of functions constant along the polarization, preserving commutativity.
• To study the Lie algebra $F(\mathcal{A}) = \{f \in \mathcal{A} \mid [f,\mathcal{O}] \subset \mathcal{O}\}$ and its structure as an $\mathcal{O}$-extension of derivations of $\mathcal{O}$.
• To derive a precise formula relating the Fedosov class $\theta(\mathcal{A})$, the Chern class $c_1(P)$, and the extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ in the context of polarized quantization.

Proposed method

• Adapts the Fedosov method to construct star-products on symplectic manifolds with complex polarization, using a flat connection on the Weyl algebra bundle.
• Introduces a deformation $(M,\omega,\mathcal{P})$ of the triple $(M,\omega_0,P)$, where $\mathcal{P}$ is a deformation of the polarization $P$.
• Defines the polarized star-product $(\mathcal{A},\mathcal{O})$ such that $\mathcal{O}$ consists of functions constant along $\mathcal{P}$, and $f \ast g = fg$ for $f \in \mathcal{O}$, $g \in \mathcal{A}$.
• Constructs the Lie algebra $F(\mathcal{A})$ as the set of elements in $\mathcal{A}$ whose commutator with $\mathcal{O}$ lies in $\mathcal{O}$, forming an $\mathcal{O}$-extension of $\mathrm{Der}(\mathcal{O})$.
• Uses Deligne's obstruction theory and the $t$-derivative of the Fedosov class to relate the extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ to the Fedosov class $\theta(\mathcal{A})$ and the Chern class $c_1(P)$.
• Applies the Fedosov construction to the opposite algebra $\mathcal{A}^{op}$ and the $t$-reversal $\mathcal{A}^\sigma$ to derive the constant term in the relation $\theta(\mathcal{A}) = \frac{1}{t}\mathrm{cl}(\mathcal{A},\mathcal{O}) - c$, ultimately identifying $c = \frac{1}{2}c_1(P)$.

Experimental results

Research questions

• RQ1 Does a star-product $\mathcal{A}$ exist on a symplectic manifold $(M,\omega_0)$ with a complex polarization $P$ such that the subalgebra $\mathcal{O}$ of functions constant along $P$ is preserved under the star-product and remains commutative?
• RQ2 How is the Lie algebra $F(\mathcal{A})$ of elements normalizing $\mathcal{O}$ structured as an $\mathcal{O}$-extension of $\mathrm{Der}(\mathcal{O})$?
• RQ3 What is the precise relationship between the Fedosov class $\theta(\mathcal{A})$, the Chern class $c_1(P)$ of the polarization, and the extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ of the $\mathcal{O}$-extension $F(\mathcal{A})$?
• RQ4 Can the extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ be expressed in terms of the Fedosov class and the Chern class of the polarization?
• RQ5 How does the $t$-dependence of the Fedosov class and the obstruction theory relate to the structure of the extension class in polarized quantization?

Key findings

• A polarized star-product $(\mathcal{A},\mathcal{O})$ exists for any good polarization $P$ and its deformation $(M,\omega,\mathcal{P})$, with $\theta(\mathcal{A}) = \frac{1}{t}[\omega]$, generalizing earlier results for real polarizations.
• The subalgebra $\mathcal{O} \subset \mathcal{A}$ consists of functions constant along the deformed polarization $\mathcal{P}$, and the star-product restricts to the commutative product on $\mathcal{O}$.
• The Lie algebra $F(\mathcal{A})$ of elements normalizing $\mathcal{O}$ forms an $\mathcal{O}$-extension of $\mathrm{Der}(\mathcal{O})$, and this extension is locally split when $P$ is a strong polarization.
• The extension class $\mathrm{cl}(\mathcal{A},\mathcal{O})$ is well-defined in $H^2(M, \frac{1}{t}\mathbb{C}[[t]])$ for strong polarizations.
• The main formula $\theta(\mathcal{A}) = \frac{1}{t}\mathrm{cl}(\mathcal{A},\mathcal{O}) - \frac{1}{2}c_1(P)$ holds, relating the Fedosov class, the extension class, and the Chern class of the polarization.
• The formula generalizes Karabegov's separation-of-variables quantization, with $\mathrm{cl}(\mathcal{A},\mathcal{O})$ coinciding with Karabegov's class in the Kähler case.