We construct several quantum coset [Formula: see text] algebras, e.g. [Formula: see text] and [Formula: see text] and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying [Formula: see text] algebras of Casimir [Formula: see text] algebras. We show that it is possible to give coset realizations of various types of unifying [Formula: see text] algebras; for example, the diagonal cosets based on the symplectic Lie algebras sp (2n) realize the unifying [Formula: see text] algebras which have previously been introduced as [Formula: see text]. In addition, minimal models of [Formula: see text] are studied. The coset realizations provide a generalization of level-rank duality of dual coset pairs. As further examples of finitely nonfreely generated quantum [Formula: see text] algebras, we discuss orbifolding of [Formula: see text] algebras which on the quantum level has different properties than in the classical case. We demonstrate through some examples that the classical limit — according to Bowcock and Watts — of these finitely nonfreely generated quantum [Formula: see text] algebras probably yields infinitely nonfreely generated classical [Formula: see text] algebras.

No abstract

We show that quantum Casimir W-algebras truncate at degenerate values of the central charge c to a smaller algebra if the rank is high enough: Choosing a suitable parametrization of the central charge in terms of the rank of the underlying simple Lie algebra, the field content does not change with the rank of the Casimir algebra any more. This leads to identifications between the Casimir algebras themselves but also gives rise to new, 'unifying' W-algebras. For example, the kth unitary minimal model of WA n has a unifying W-algebra of type W(2, 3, . . . , k 2 + 3k + 1). These unifying W-algebras are non-freely generated on the quantum level and belong to a recently discovered class of W-algebras with infinitely, non-freely generated classical counterparts. Some of the identifications are indicated by level-rank-duality leading to a coset realization of these unifying W-algebras. Other unifying W-algebras are new, including e.g. algebras of type WD −n . We point out that all unifying quantum W-algebras are finitely, but non-freely generated.

We demonstrate that all rational models of the N = 2 super Virasoro algebra are unitary. Our arguments are based on three different methods: we determine Zhu's algebra A(H 0 ) (for which we give a physically motivated derivation) explicitly for certain theories, we analyse the modular properties of some of the vacuum characters, and we use the coset realisation of the algebra in terms of su(2) and two free fermions. Some of our arguments generalise to the Kazama-Suzuki models indicating that all rational N = 2 supersymmetric models might be unitary.

No abstract