Chiral equivariant cohomology II

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of $O({\mathfrak{s}}{\mathfrak{g}})$ algebras. The new cohomology theory can be viewed as a kind of ``chiralization'' of the classical equivariant cohomology, the latter being defined on the category of $G^*$ algebras a là H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call ${\mathfrak{s}}{\mathfrak{g}}[t]$ algebras. In the geometrical setting, our principal example of an $O({\mathfrak{s}}{\mathfrak{g}})$ algebra is the chiral de Rham complex ${\mathcal{Q}}(M)$ of a $G$ manifold $M$. There is an interesting subalgebra of ${\mathcal{Q}}(M)$ which does not admit a full $O({\mathfrak{s}}{\mathfrak{g}})$ algebra structure but retains the structure of an ${\mathfrak{s}}{\mathfrak{g}}[t]$ algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allow us to delineate a number of interesting geometric aspects of the $G$ manifold $M$, sometimes in ways that are quite different from the classical theory.