Conformally Equivariant Quantization - a Complete Classification

TL;DR

This paper fully classifies when conformally equivariant quantization exists and is unique, linking these cases to the presence of conformally invariant differential operators and providing a comprehensive understanding of the conditions involved.

Contribution

It provides a complete classification of existence and uniqueness of conformally equivariant quantization based on the values of δ and λ, extending previous partial results.

Findings

Unique existence is lost if and only if a nontrivial conformally invariant differential operator exists.

In cases where existence is lost, quantization exists only for finitely many λ values.

The results are proved in a broader context of IFFT (or AHS) equivariant quantization.

Abstract

Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda+\delta$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $\delta$. Later, Silhan has determined the critical values of $\delta$ for which unique existence is lost, and conjectured that for those values of $\delta$ existence is lost for a generic weight $\lambda$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of $\delta$ and $\lambda$. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight $\delta$, and (ii) in that case the conformally equivariant quantization exists only for a finite number of $\lambda$, corresponding to nontrivial conformally invariant differential operators on $\lambda$-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization.