A graded manifold of dimension is defined as a locally ringed space where is an -dimensional smooth manifold and is a -sheaf of Grassmann algebras of rank where is the sheaf of smooth real functions on. The sheaf is called the structure sheaf of the graded manifold, and the manifold is said to be the body of. Sections of the sheaf are called graded functions on a graded manifold. They make up a graded commutative -ring called the structure ring of. The well-known Batchelor theorem and Serre-Swan theorem characterize graded manifolds as follows.
Serre-Swan theorem for graded manifolds
Let be a graded manifold. There exists a vector bundle with an -dimensional typical fiber such that the structure sheaf of is isomorphic to the structure sheaf of sections of the exterior product of, whose typical fibre is the Grassmann algebra. Let be a smooth manifold. A graded commutative -algebra is isomorphic to the structure ring of a graded manifold with a body if and only if it is the exterior algebra of some projective -module of finite rank.
Graded functions
Note that above mentioned Batchelor's isomorphism fails to be canonical, but it often is fixed from the beginning. In this case, every trivialization chart of the vector bundle yields a splitting domain of a graded manifold, where is the fiber basis for. Graded functions on such a chart are -valued functions where are smooth real functions on and are odd generating elements of the Grassmann algebra.
Given a graded manifold, graded derivations of the structure ring of graded functions are called graded vector fields on. They constitute a real Lie superalgebrawith respect to the superbracket where denotes the Grassmann parity of. Graded vector fields locally read They act on graded functions by the rule
Graded exterior forms
The -dual of the module graded vector fields is called the module of graded exterior one-forms. Graded exterior one-forms locally read so that the duality product between and takes the form Provided with the graded exterior product graded one-forms generate the graded exterior algebra of graded exterior forms on a graded manifold. They obey the relation where denotes the form degree of. The graded exterior algebra is a graded differential algebra with respect to the graded exterior differential where the graded derivations, are graded commutative with the graded forms and. There are the familiar relations
In the category of graded manifolds, one considers graded Lie groups, graded bundles and graded principal bundles. One also introduces the notion of jets of graded manifolds, but they differ from jets of graded bundles.
Due to the above-mentioned Serre-Swan theorem, odd classical fields on a smooth manifold are described in terms of graded manifolds. Extended to graded manifolds, the variational bicomplex provides the strict mathematical formulation of Lagrangian classical field theory and Lagrangian BRST theory.
