On natural and conformally equivariant quantizations

TL;DR

This paper explores the existence and construction of conformally equivariant quantizations on manifolds, extending previous results to symbols of degree 4 using Cartan connection frameworks.

Contribution

It generalizes the theory of conformally equivariant quantizations to higher degrees, specifically degree 4, building on prior work for degrees up to 3.

Findings

Confirmed existence of conformally equivariant quantizations for symbols of degree 3.

Extended the framework to establish existence for degree 4 symbols.

Reproduced known results for degrees 2 and 3 using Cartan connections.

Abstract

The concept of conformally equivariant quantizations was introduced by Duval, Lecomte and Ovsienko in \cite{DLO} for manifolds endowed with flat conformal structures. They obtained results of existence and uniqueness (up to normalization) of such a quantization procedure. A natural generalization of this concept is to seek for a quantization procedure, over a manifold $M$, that depends on a pseudo-Riemannian metric, is natural and is invariant with respect to a conformal change of the metric. The existence of such a procedure was conjectured by P. Lecomte in \cite{Leconj} and proved by C. Duval and V. Ovsienko in \cite{DO1} for symbols of degree at most 2 and by S. Loubon Djounga in \cite{Loubon} for symbols of degree 3. In two recent papers \cite{MR,MR1}, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we will show how the formalism developed in these works adapts in order to deal with the conformally equivariant quantization for symbols of degree at most 3. This will allow us to easily recover the results of \cite{DO1} and \cite{Loubon}. We will then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4.