Second-Order Conformally Equivariant Quantization in Dimension 1|2

Conformally equivariant quantization

Let (M, g) be a pseudo-Riemannian manifold and Fλ(M) the space of densities of degree λ on M . We study the space D2 λ,μ(M) of second-order differential operators from Fλ(M) to Fμ(M). If (M, g) is conformally flat with signature p− q, then D2 λ,μ(M) is viewed as a module over the group of conformal transformations of M . We prove that, for almost all values of μ− λ, the O(p+1, q+1)-modules D2 λ...

Conformally Equivariant Quantization – a Complete Classif ication

Conformally equivariant quantization is a peculiar map between symbols of real weight δ and differential operators acting on tensor densities, whose real weights are designed by λ and λ + δ. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight δ. Later, Silhan has determined the critical values of δ for which unique existence is lost, an...

Differential operators on supercircle: conformally equivariant quantization and symbol calculus

We consider the supercircle S equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the correspondin...

Geometric Quantization and Equivariant Cohomology Geometric Quantization and Equivariant Cohomology

A new integrable system on the sphere and conformally equivariant quantization

Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere S, namely the dual Moser system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. M...