Dequantized Differential Operators between Tensor Densities as Modules over the Lie Algebra of Contact Vector Fields

TL;DR

This paper characterizes invariant differential operators between tensor densities as modules over contact vector field subalgebras, revealing their algebraic structure, generators, and spectral properties, with implications for understanding symmetries in geometric analysis.

Contribution

It provides a complete description of invariant tensor fields and operators under contact transformations, and demonstrates the diagonalization of the Casimir operator, linking algebraic invariants to geometric structures.

Findings

Algebras of invariant tensor fields are fully characterized.

Invariant operators are generated by specific geometric lifts and Hamiltonians.

Casimir operator is diagonal, leading to Diophantine conditions on parameters.

Abstract

In recent years, algebras and modules of differential operators have been extensively studied. Equivariant quantization and dequantization establish a tight link between invariant operators connecting modules of differential operators on tensor densities, and module morphisms that connect the corresponding dequantized spaces. In this paper, we investigate dequantized differential operators as modules over a Lie subalgebra of vector fields that preserve an additional structure. More precisely, we take an interest in invariant operators between dequantized spaces, viewed as modules over the Lie subalgebra of infinitesimal contact or projective contact transformations. The principal symbols of these invariant operators are invariant tensor fields. We first provide full description of the algebras of such affine-contact- and contact-invariant tensor fields. These characterizations allow showing that the algebra of projective-contact-invariant operators between dequantized spaces implemented by the same density weight, is generated by the vertical cotangent lift of the contact form and a generalized contact Hamiltonian. As an application, we prove a second key-result, which asserts that the Casimir operator of the Lie algebra of infinitesimal projective contact transformations, is diagonal. Eventually, this upshot entails that invariant operators between spaces induced by different density weights, are made up by a small number of building bricks that force the parameters of the source and target spaces to verify Diophantine-type equations.