Deformation quantization of the simplest Poisson Orbifold

TL;DR

This paper explores new deformation quantizations of invariant function algebras on Poisson orbifolds, specifically for the reflection symmetry on 2, revealing additional structures beyond standard Kontsevich Formality.

Contribution

It provides explicit formulas for deformed products in Poisson orbifolds with 2 symmetry, extending deformation quantization beyond traditional methods.

Findings

Identified a new deformation related to the 2 symmetry

Connected the deformation to Wigner's work and fuzzy sphere models

Derived explicit formulas using homological perturbation theory

Abstract

Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by Kontsevich Formality. We consider the simplest example of this situation: $\mathbb{R}^2$ with the reflection symmetry $\mathbb{Z}_2$. The usual quantization leads to the Weyl algebra. While Weyl algebra is rigid, the algebra of even or twisted by $\mathbb{Z}_2$ functions has one more deformation, which was identified by Wigner and is related to Feigin's $gl_\lambda$ and to fuzzy sphere. With the help of homological perturbation theory we obtain explicit formula for the deformed product, the first order of which can be extracted from Shoikhet-Tsygan-Kontsevich formality.