Graded manifold

In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra.

Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras.

However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces.

is defined as a locally ringed space

-sheaf of Grassmann algebras of rank

is the sheaf of smooth real functions on

is called the structure sheaf of the graded manifold

There exists a vector bundle

is isomorphic to the structure sheaf of sections of the exterior product

, whose typical fibre is the Grassmann algebra

-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

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

yields a splitting domain

are odd generating elements of the Grassmann algebra

are called graded vector fields on

They constitute a real Lie superalgebra

denotes the Grassmann parity of

Graded vector fields locally read They act on graded functions

-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 (interior) product between

takes the form Provided with the graded exterior product graded one-forms generate the graded exterior algebra

denotes the form degree of

One also introduces the notion of jets of graded manifolds, but they differ from jets of graded bundles.

The differential calculus on graded manifolds is formulated as the differential calculus over graded commutative algebras similarly to the differential calculus over commutative algebras.

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.