Fabian Radoux

Natural and Projectively Equivariant Quantizations by means of Cartan Connections

AbstractThe existence of a natural and projectively equivariant quantization in the sense of Lecomte [20] was proved recently by M. Bordemann [4], using the framework of Thomas–Whitehead connections. We give a new proof of existence using the notion of Cartan projective connections and we obtain an explicit formula in terms of these connections. Our method yields the existence of a projectively equivariant quantization if and only if an

Existence of natural and conformally invariant quantizations of arbitrary symbols

sl(m+1,\mathbb{R})

Second Order Symmetries of the Conformal Laplacian

-equivariant quantization exists in the flat situation in the sense of [18], thus solving one of the problems left open by M. Bordemann.

Explicit Formula for the Natural and Projectively Equivariant Quantization

The concept of conformally equivariant quantization was introduced by C. Duval, P. Lecomte and V. Ovsienko 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 ,t hat 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 and proved by C. Duval and V. Ovsienko for symbols of degree at most 2 and by S. Loubon Djounga for symbols of degree 3. In two recent papers, we investigated the question of existence of projectively equivariant quantizations using the framework of Cartan connections. Here we shall 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 C. Duval, V. Ovsienko and S. Loubon Djounga. We shall then show how it can be modified in order to prove the existence of conformally equivariant quantizations for symbols of degree 4.

Natural and Projectively Invariant Quantizations on Supermanifolds

A quantization can be seen as a way to construct a differential operator with prescribed principal symbol. The map from the space of symbols to the space of differential operators is moreover required to be a linear bijection. In general, there is no natural quantization procedure, that is, spaces of symbols and of differential operators are not equivalent, if the action of local diffeomorphisms is taken into account. However, considering manifolds endowed with additional 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 proved recently in a series of papers in the context of projective geometry. Here, we show that the construction of the quantization based on Cartan connections can be adapted from projective to pseudo-conformal geometry to yield the natural and conformally invariant quantization for arbitrary symbols, outside some critical situations.

On osp(p+1,q+1|2r)-equivariant quantizations

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra

spo(2|2)-equivariant quantizations on the supercircle S^{1|2}

{\mathfrak{pgl}(p+1|q)}