Higher Covariant Derivative and the Bundle of Dirac Currents

Using the higher covariant derivative on a manifold $M$ equipped with a torsion-free connection, we define a natural surjective bundle map $\Phi$ from $(\otimes(TM))\otimes (\wedge(TM))$ to the vector bundle $\mathcal{U}(M)$ of de Rham currents on $M$ supported in a single (variable) point. The resulting quotient bundle can be thought of as a bundle of generalized Weyl algebras, with the symplectic form replaced with the Riemannian curvature tensor. The fibers of the bundle $\mathcal{U}(M)$ are differential co-algebras, and the boundary, co-product and co-unit stitch together to form bundle maps which lift via $\Phi$ to commuting bundle maps on $(\otimes(TM))\otimes (\wedge(TM))$. Interior product, higher-order covariant differentiation, and their $L^2$ adjoints also form bundle maps on $\mathcal{U}(M)$ which lift via $\Phi$. The higher-order covariant derivative in particular is an $\mathbb{R}$-algebra representation of the space $C^\infty(\otimes(TM))$ equipped with a non-standard, \emph{covariant product}. Its composition with interior product yields a quantization of $\mathcal{U}(M)$ corresponding to a Hopf-algebraic smash product. Finitely supported and locally finitely supported sections functors can be applied to $\mathcal{U}(M)$, yielding the spaces of finitely supported and locally finitely supported currents, respectively. In particular, the finitely supported currents on a smooth manifold are a filtered differential graded co-algebra in duality with differential forms.