Weyl quantization of degree 2 symplectic graded manifolds

Abstract

Let S be a spinor bundle of a pseudo-Euclidean vector bundle (E,g) of even rank. We introduce a new filtration on the algebra D(M,S) of differential operators on S. As a main property, the associated graded algebra grD(M,S) is proved to be isomorphic to the algebra O(M) of smooth functions on M, where M is the degree 2 symplectic graded manifold canonically associated to (E,g). Accordingly, we establish the Weyl quantization of M as a map WQħ:O(M)→D(M,S), and prove that WQħ satisfies all the desired properties of quantizations. As an application, we obtain a bijection between Courant algebroid structures (E,g,ρ,〚⋅,⋅〛) equivalently characterized by Hamiltonian generating functions on M, and skew-symmetric Dirac generating operators D∈D(M,S). The operator D2 gives a new invariant of (E,g,ρ,〚⋅,⋅〛) which generalizes the square norm of the Cartan 3-form of a quadratic Lie algebra. We study this invariant in detail in the particular case of E being the double of a Lie bialgebroid (A,A⁎).