W-Algebras in Conformal Field Theory (original) (raw)

Quantum W-algebras are defined and their relevance for conformal field theories is outlined. We describe direct constructions of W-algebras using associativity requirements. With this approach one explicitly obtains the first members of series of W-algebras, including in particular 'Casimir algebras' (related to simple Lie algebras) and extended symmetry algebras corresponding to special Virasoro-minimal models. We also describe methods for the study of highest weight representations of W-algebras. In some cases consistency of the corresponding quantum field theory already imposes severe restrictions on the admitted representations, i.e. allows to determine the field content. We conclude by reviewing known results on Walgebras and RCFTs and show that most known rational conformal fields theories can be described in terms of Casimir algebras although on the level of W-algebras exotic phenomena occur.