Existence of natural and conformally invariant quantizations of arbitrary symbols

Abstract

A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The quantization map is moreover required to be a linear bijection. It is known that there is in general no natural quantization procedure. However, considering manifolds endowed with additional structures, such as projective or pseudo-conformal structures, one can seek for quantizations that depend on this additional structure and that are natural if the dependence with respect to the structure is taken into account. The question of existence of such a quantization was addressed in a series of papers in the context of projective geometry, using the framework of Thomas-Whitehead connections (see Bordemann, Hansoul and Fox). Recently, we recovered these existence results, using the theory of Cartan projective connections. In the present work, we show that our method can be adapted to pseudo-conformal geometry to yield the so-called natural and conformally invariant quantization for arbitrary symbols, still outside some critical situations. Our method is general enough to analyze the problem of invariant quantizations in the context of manifolds with irreducible parabolic geometries.