1 2 M ay 2 00 5 Chiral Equivariant Cohomology I

For a smooth manifold equipped with a compact Lie group action, we construct an equivariant cohomology theory which takes values in a vertex algebra, and contains the classical equivariant cohomology as a subalgebra. The main idea is to synthesize the algebraic approach to the classical equivariant cohomology theory due to H. Cartan, with the chiral de Rham algebra of Malikov-Schechtman-Vaintrob, by using a vertex algebra notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new equivariant cohomology theory that have no classical analogues. Cartan’s theory was further developed by Duflo-Kumar-Vergne [8] and Guillemin-Sternberg [19]. This paper follows closely the latter approach.