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.
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 .
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.
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- G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced Classical Field Theory (World Scientific, 2009) ISBN 978-981-283-895-7; arXiv:math-ph/0102016; arXiv:1304.1371.