Abstract
Generalizing the Knizhnik-Zamolodchikov equations, we derive a hierarchy of nonlinear Ward identities for affine-Virasoro correlators. The hierarchy follows from null states of the Knizhnik-Zamolodchikov type and the assumption of factorization, whose consistency we verify at an abstract level. Solution of the equations requires concrete factorization ansätze, which may vary over affine-Virasoro space. As a first example, we solve the nonlinear equations for the coset constructions, using a matrix factorization. The resulting coset correlators satisfy first-order linear partial differential equations whose solutions are the coset blocks defined by Douglas.
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