Existence of Natural and Conformally Invariant Quantizations of Arbitrary Symbols

A quantization over a manifold can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of principal symbols to the space of differential operators 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 existence of such a quantization was conjectured by P. Lecomte in [19] in the context of projective and conformal geometry. 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 for instance [4, 14, 13, 15]). In [23, 21], we recovered the existence of a quantization that depends on a projective structure and that is natural (provided some critical situations are avoided), 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. Moreover, we give new and more general proofs of some results of [21] and eventually, we notice that the method is general enough to analyze the problem of natural and invariant quantizations in the context of manifolds endowed with irreducible parabolic geometries studied in [9].