  This is the third of a series of papers on a new equivariant cohomology that
takes values in a vertex algebra, and contains and generalizes the classical
equivariant cohomology of a manifold with a Lie group action a la H. Cartan. In
this paper, we compute this cohomology for spheres and show that for any simple
connected group G, there is a sphere with infinitely many actions of G which
have distinct chiral equivariant cohomology, but identical classical
equivariant cohomology. Unlike the classical case, the description of the
chiral equivariant cohomology of spheres requires a substantial amount of new
structural theory, which we fully develop in this paper. This includes a
quasi-conformal structure, equivariant homotopy invariance, and the values of
this cohomology on homogeneous spaces. These results rely on crucial features
of the underlying vertex algebra valued complex that have no classical
analogues.
