Weyl quantization of degree 2 symplectic graded manifolds

Abstract

Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle $(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra $\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the associated graded algebra $\mathrm{gr}\mathcal{D}(M,S)$ is isomorphic to the algebra $\mathcal{O}(\mathcal{M})$ of functions on $\mathcal{M}$, where $\mathcal{M}$ is the symplectic graded manifold of degree $2$ canonically associated to $(E,\mathrm{g})$. Accordingly, we define the Weyl quantization on $\mathcal{M}$ as a map $\mathcal{WQ}_\hbar:\mathcal{O}(\mathcal{M})\to\mathcal{D}(M,S)$, and prove that $\mathcal{WQ}_\hbar$ satisfies all desired usual properties. As an application, we obtain a bijection between Courant algebroid structures $(E,\mathrm{g},\rho,[\cdot,\cdot])$, that are encoded by Hamiltonian generating functions on $\mathcal{M}$, and skew-symmetric Dirac generating operators $D\in\mathcal{D}(M,S)$. The operator $D^2$ gives a new invariant of $(E,\mathrm{g},\rho,[\cdot,\cdot])$, which generalizes the square norm of the Cartan $3$-form of a quadratic Lie algebra. We study in detail the particular case of $E$ being the double of a Lie bialgebroid $(A,A^*)$.