Classical field theory admits the adequate geometric formulation in the terms of fiber bundles and graded manifolds (Archive). Why? A cornerstone of classical field theory is the generalized Serre – Swan theorem.

Let

*X*be a compact smooth manifold and*C(X)*the ring of smooth functions on*X*. The original Serre – Swan theorem states that a*C(X)*-module is a projective module of finite rank if and only if it is isomorphic to a module of sections of some vector bundle over*X*. This theorem has been extended to an arbitrary smooth manifold due to the fact that any smooth manifold admits a finite manifold atlas.It follows from the Serre – Swan theorem that, if classical fields are assumed to constitute a projective C(X)-module of finite rank, they are represented by sections of a vector bundle.

In a general setting, theory of Grassman-graded even and odd classical fields is considered. There are different models of odd classical fields in the terms of graded manifolds and supermanifolds. Combination of the well-known Batchelor theorem and the above mentioned Serre – Swan theorem results in a generalization of the Serre – Swan theorem to graded manifolds as follows.

Given a smooth manifold

*X*, a graded commutative*C(X)*-algebra is isomorphic to the structure ring of a graded manifold with a body*X*if and only if it is the exterior algebra of some projective*C(X)*-module of finite rank.**References:**

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical Field Theory (WS, 2009)

G.Sardanashvily, Classical field theory. Advanced mathematical formulation, arXiv: 0811.0331