Conformally invariant quantization -- towards complete classification

TL;DR

This paper develops explicit formulas for conformally invariant quantization operators on smooth manifolds, extending the understanding of their existence beyond generic weights and identifying critical cases.

Contribution

It provides explicit formulas for conformally invariant quantization for all weights, including critical ones, and describes the set of critical weights with a conjecture on its minimality.

Findings

Explicit formulas for all critical weights

Description of the critical weight set

Conjecture on the minimality of the critical set

Abstract

Let $M$ be a smooth manifold equipped with a conformal structure, $E[w]$ the space of densities with the the conformal weight $w$ and $D_{w,w+\de}$ the space of differential operators from $E[w]$ to $E[w+\delta]$. Conformal quantization $Q$ is a right inverse of the principle symbol map on $D_{w,w+\delta}$ such that $Q$ is conformally invariant and exists for all $w$. This is known to exists for generic values of $\delta$. We give explicit formulae for $Q$ for all $\delta$ out of the set of critical weights. We provide a simple description of this set and conjecture its minimality.