Quantization of Whitney functions

TL;DR

This paper extends deformation quantization to Whitney functions on singular sets, demonstrating existence over closed subsets of symplectic manifolds and relating homology to Whitney--de Rham cohomology.

Contribution

It introduces a new framework for deformation quantization of Whitney functions on singular sets and establishes homological properties in the analytic and subanalytic case.

Findings

Existence of deformation quantization for Whitney functions over closed subsets of symplectic manifolds.

Hochschild and cyclic homology of the deformed algebra match Whitney--de Rham cohomology.

Connection to an algebraic index theorem for Whitney functions.

Abstract

We propose to study deformation quantizations of Whitney functions. To this end, we extend the notion of a deformation quantization to algebras of Whitney functions over a singular set, and show the existence of a deformation quantization of Whitney functions over a closed subset of a symplectic manifold. Under the assumption that the underlying symplectic manifold is analytic and the singular subset subanalytic, we determine that the Hochschild and cyclic homology of the deformed algebra of Whitney functions over the subanalytic subset coincide with the Whitney--de Rham cohomology. Finally, we note how an algebraic index theorem for Whitney functions can be derived.