The generalized Serre – Swan theorem is a cornerstone of classical field theory

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