J an 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 and Guillemin-Sternberg, with the chiral de Rham algebra of Malikov-SchechtmanVaintrob, 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 derive a spectral sequence, in the abelian case, which is analogous to the well-known spectral sequence for the Cartan model. We give interesting cohomology classes in the new equivariant cohomology theory that have no classical analogue.