  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(sg)
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 la 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 sg[t] algebras. In the geometrical setting, our principal example
of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There
is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra
structure but retains the structure of an sg[t] algebra, enough for us to
define its chiral equivariant cohomology. The latter then turns out to have
many surprising features that allows 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.
