In algebraic geometry, Graded manifolds are extensions of the manifold concept 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.
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Graded manifolds
A graded manifold of dimension
Serre-Swan theorem for graded manifolds
Let
Let
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
where
Graded vector fields
Given a graded manifold
where
They act on graded functions
Graded exterior forms
The
Provided with the graded exterior product
graded one-forms generate the graded exterior algebra
where
where the graded derivations
Graded differential geometry
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.
Graded differential calculus
The differential calculus on graded manifolds is formulated as the differential calculus over graded commutative algebras similarly to the differential calculus over commutative algebras.
Physical outcome
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.