W-algebras are constructed via quantum Hamiltonian reduction associated with a Lie algebra and an -embedding into . We derive correspondences among correlation functions of theories having different W-algebras as symmetry algebras. These W-algebras are associated to the same but distinct -embeddings.
For this purpose, we first explore different free field realizations of W-algebras and then generalize previous works on the path integral derivation of correspondences of correlation functions. For , there is only one non-standard (non-regular) W-algebra known as the Bershadsky-Polyakov algebra. We examine its free field realizations and derive correlator correspondences involving the WZNW theory of , the Bershadsky-Polyakov algebra and the principal -algebra. There are three non-regular W-algebras associated to . We show that the methods developed for can be applied straightforwardly. We briefly comment on extensions of our techniques to general .