Higher Covariant Derivative and the Bundle of Dirac Currents

TL;DR

This paper develops a framework using higher covariant derivatives on manifolds to define a bundle of de Rham currents with algebraic structures, linking differential geometry and algebra.

Contribution

It introduces a natural surjective bundle map from tensor and wedge bundles to currents, forming a bundle of generalized Weyl algebras with rich algebraic and geometric properties.

Findings

Defined a surjective bundle map to de Rham currents supported at points.

Established algebraic structures like co-algebras, Hopf algebra, and quantization on the bundle.

Connected finitely supported currents with filtered differential graded co-algebras.

Abstract

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.