Conformally equivariant quantization: existence and uniqueness

Duval, Christian; Lecomte, Pierre; Ovsienko, Valentin

Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1999-2029.

On établit l’existence et l’unicité d’un calcul symbolique et d’une quantification conformément équivariants sur une variété pseudo-riemannienne conformément plate $(M,g)$, i.e. on met en évidence un isomorphisme canonique entre l’espace des polynômes sur $T^{*}M$ et celui des opérateurs différentiels agissant sur les champs de densités tensorielles, vus tous les deux comme modules sur l’algèbre de Lie $\mathrm{o}(p+1,q+1)$, où $p+q=dim\left(M\right)$. Cette quantification existe pour des valeurs génériques des poids de densités; on calcule les valeurs critiques de ces poids, pour lesquelles un tel isomorphisme n’existe éventuellement pas. Dans le cas des demi-densités, on obtient ainsi un star-produit conformément équivariant.

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold $(M,g)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^{*}M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\mathrm{o}(p+1,q+1)$ where $p+q=dim\left(M\right)$. This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of half-densities, we obtain a conformally invariant star-product.

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