The affine-Virasoro Ward identities are a system of non-linear differential
equations which describe the correlators of all affine-Virasoro constructions,
including rational and irrational conformal field theory. We study the Ward
identities in some detail, with several central results. First, we solve for
the correlators of the affine-Sugawara nests, which are associated to the
nested subgroups gβh1βββ¦βhnβ. We also find an
equivalent algebraic formulation which allows us to find global solutions
across the set of all affine-Virasoro constructions. A particular global
solution is discussed which gives the correct nest correlators, exhibits
braiding for all affine-Virasoro correlators, and shows good physical behavior,
at least for four-point correlators at high level on simple g. In rational
and irrational conformal field theory, the high-level fusion rules of the
broken affine modules follow the Clebsch-Gordan coefficients of the
representations.Comment: 45 pages, Latex, UCB-PTH-93/18, LBL-34111, BONN-HE-93/17. We
factorize the biconformal nest correlators of the first version, obtaining
the conformal correlators of the affine-Sugawara nests on g/h_1/.../h_