On natural and conformally equivariant quantizations

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.Comment: 19 page

Natural and Projectively Invariant Quantizations on Supermanifolds

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the $\mathfrak{pgl}({n+1|m})$-equivariant quantization on ${\mathbb{R}}^{n|m}$ constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two

Existence of natural and conformally invariant quantizations of arbitrary symbols

peer reviewedA 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

NPK-contents of macrophytes from the Mediterranean Morocco

Cette étude présente une évaluation des teneurs en NPK (azote, phosphore et potassium) des principaux macrophytes. Lemna gibba, les feuilles de Salix purpurea et les parties reproductives de Tamarix africana présentent une teneur en azote élevée. Tandis que Lemna gibba, les parties souterraines de Sparganium erectum et les parties reproductives de Vitex agnus-castus ont une teneur élevée en phosphore.This work presents an assessment of NPK (nitrogen, phosphorus and potassium) -contents of the main macrophytes from the Mediterranean Morocco. Lemna gibba,leaves of Salix purpurea and reproductive parts of Tamarix africana present a high nitrogen content; while Lemna gibba, underground parts of Sparganium erectum and reproductive parts of Vitex agnus-castus have a high phosphorus one

Projectively equivariant quantizations over the superspace Rp∣q\R^{p|q}

We investigate the concept of projectively equivariant quantization in the framework of super projective geometry. When the projective superalgebra pgl(p+1|q) is simple, our result is similar to the classical one in the purely even case: we prove the existence and uniqueness of the quantization except in some critical situations. When the projective superalgebra is not simple (i.e. in the case of pgl(n|n)\not\cong sl(n|n)), we show the existence of a one-parameter family of equivariant quantizations. We also provide explicit formulas in terms of a generalized divergence operator acting on supersymmetric tensor fields.Comment: 19 page