Deformation Quantization in Singular Spaces

TL;DR

This paper introduces a novel method for quantizing singular analytic spaces within smooth manifolds, extending deformation quantization techniques to include spaces with singularities using a super-manifold framework.

Contribution

It develops a super-manifold based modification of Fedosov's construction enabling quantization of singular spaces and provides a systematic way to define quantum algebras for these spaces.

Findings

Quantization method applies to singular spaces.

The approach yields a finite set of PDEs for quantum functions.

Examples demonstrate the method's effectiveness.

Abstract

We present a method of quantizing analytic spaces $X$ immersed in an arbitrary smooth ambient manifold $M$. Remarkably our approach can be applied to singular spaces. We begin by quantizing the cotangent bundle of the manifold $M$. Using a super-manifold framework we modify the Fedosov construction in a way such that the $\star$-product of the functions lifted from the base manifold turns out to be the usual commutative product of smooth functions on $M$. This condition allows us to lift the ideals associated to the analytic spaces on the base manifold to form left (or right) ideals on $(\mc{O}_{\Omega^1 M}[[\hbar]],\starl)$ in a way independent of the choice of generators and leading to a finite set of PDEs defining the functions in the quantum algebra associated to $X$. Some examples are included.