Equivariant cohomology and current algebras

Abstract: Let M be a manifold and g a Lie algebra acting on M. Differential forms Omega(M) carry a natural action of Lie derivatives L(x) and contractions I(x) of fundamental vector fields for x \in g. Contractions (anti-) commute with each other, [I(x), I(y)]=0. Together with the de Rham differential, they satisfy the Cartan's magic formula [d, I(x)]=L(x). In this paper, we define a differential graded Lie algebra Dg, where instead of commuting with each other, contractions form a free Lie superalgebra. It turns out that central extensions of Dg are classified (under certain assumptions) by invariant homogeneous polynomials p on g. This construction gives a natural framework for the theory of twisted equivariant cohomology and a new interpretation of Mickelsson-Faddeev-Shatashvili cocycles of higher dimensional current algebras. As a topological application, we consider principal G-bundles (with G a Lie group integrating g), and for every homogeneous polynomial p on g we define a lifting problem with the only obstruction the corresponding Chern-Weil class cw(p).