Abstract: Less familiar manipulation with invariant differential forms on a (finite or infinite dimensional) smooth S¹-manifold leads to a mild modification of equivariant cohomology in finite dimensional case of some interest. However, when applied to the free loop space of a manifold, regarded as an infinite dimensional manifold, this leads to an interesting homotopy functor (string cohomology of the manifold) which:

a) unifies Atyah Hirzebruch and Waldhausen algebraic K theory at least for 1-connected space,
b) provides a convenient homological interpretation of expressions ∫ e^ω on the free loop space of interest in string theory.

The functor is computable, and can be actually defined on the category of connected commutative differential graded algebras; however we do not know HOW.