EQUIVARIANT COHOMOLOGY AND CURRENT ALGEBRAS

Abstract

This paper touches upon two big themes, equivariant cohomology and current algebras. Our first main result is as follows: we define a pair of current algebra functor which assigns Lie algebras (current algebras) and to a manifold M and a differential graded Lie algebra (DGLA) A. The functors and are contravariant with respect to M and covariant with respect to A. If A = C𝔤, the cone of a Lie algebra 𝔤 spanned by Lie derivatives L(x) and contractions I(x)(x ∈ 𝔤) and satisfying the Cartan's magic formula [d, I(x)] = L(x), the corresponding current algebras coincide, and they are equal to , the space of smooth 𝔤-valued functions on M with the pointwise Lie bracket. Other examples include affine Lie algebras on the circle and Faddeev–Mickelsson–Shatashvili (FMS) extensions of higher-dimensional current algebras. The second set of results is related to the construction of a new DGLA D𝔤 assigned to a Lie algebra 𝔤. It is generated by L(x) and I(x) (similar to C𝔤) and by higher contractions I(x2), I(x3) etc. Similar to C𝔤, D𝔤 can be used in building differential models of equivariant cohomology. In particular, twisted equivariant cohomology (including twists by 3-cocycles and higher odd cocycles) finds a natural place in this new framework. The DGLA D𝔤 admits a family of central extensions Dp𝔤 parametrized by homogeneous invariant polynomials . There is a Lie homomorphism from to the FMS current algebra defined by p. Let G be a Lie group integrating the Lie algebra 𝔤. The current algebras and integrate to groups DG(M) and DpG(M). As a topological application, we consider principal G-bundles, and for every homogeneous polynomial we pose a lifting problem (defined in terms of DG(M) and DpG(M)) with the only obstruction the Chern–Weil class cw(p). When M is a sphere, we study integration of the current algebra . It turns out that the corresponding group is a central extension of DG(M). Under certain conditions on the polynomial p, this is a central extension by a circle.