Deformation quantization of submanifolds and reductions via Duflo-Kirillov-Kontsevich map

TL;DR

This paper introduces a method to quantize Poisson submanifolds using the Duflo-Kirillov-Kontsevich map and explores the conjecture that deformation quantization commutes with reduction, verified in the case of the 2-sphere.

Contribution

It proposes a new approach to quantize Poisson submanifolds via factor algebras derived from the Duflo-Kirillov-Kontsevich map and conjectures the equivalence with traditional quantization methods.

Findings

The proposed quantization method matches known results for the 2-sphere.

Conjecture that deformation quantization commutes with reduction is supported.

Explicit star product for the 2-sphere confirms the theoretical constructions.

Abstract

We propose the following receipt to obtain the quantization of the Poisson submanifold $N$ defined by the equations $f_i=0$ (where $f_i$ are Casimirs) from the known quantization of the manifold $M$: one should consider factor algebra of the quantized functions on $M$ by the images of $D(f_i)$, where $D: Fun(M) \to Fun(M)\otimes \CC[\hbar]$ is Duflo-Kirillov-Kontsevich map. We conjecture that this algebra is isomorphic to quantization of $Fun(N)$ with Poisson structure inherited from $M$. Analogous conjecture concerning the Hamiltonian reduction saying that "deformation quantization commutes with reduction" is presented. The conjectures are checked in the case of $S^2$ which can be quantized as a submanifold, as a reduction and using recently found explicit star product. It's shown that all the constructions coincide.