Differential Calculus onN-Graded Manifolds

Abstract

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.