Fermionic and bosonic ghost systems are defined each in terms of a single vertex algebra which admits a one-parameter family of conformal structures. The observation that these structures are related to each other provides a simple way to obtain char
acter formulae for a general twisted module of a ghost system. The U(1) symmetry and its subgroups that underly the twisted modules also define an infinite set of invariant vertex subalgebras. Their structure is studied in detail from a W-algebra point of view with particular emphasis on Z_N-invariant subalgebras of the fermionic ghost system.
For any non-negative integer v we construct explicitly [v/2]+1 independent covariant bilinear differential operators from J_{k,m} x J_{k,m} to J_{k+k+v,m+m}. As an application we construct a covariant bilinear differential operator mapping S_k^{(2)}
x S^{(2)}_{k} to S^{(2)}_{k+k+v}. Here J_{k,m} denotes the space of Jacobi forms of weight k and index m and S^{(2)}_k the space of Siegel modular forms of degree 2 and weight k. The covariant bilinear differential operators constructed are analogous to operators already studied in the elliptic case by R. Rankin and H. Cohen and we call them Rankin-Cohen operators.
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 Zhus algebra (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.
We construct several quantum coset W-algebras, e.g. sl(2,R)/U(1) and sl(2,R)+sl(2,R) / sl(2,R), and argue that they are finitely nonfreely generated. Furthermore, we discuss in detail their role as unifying W-algebras of Casimir W-algebras. We show t
hat it is possible to give coset realizations of various types of unifying W-algebras, e.g. the diagonal cosets based on the symplectic Lie algebras sp(2n) realize the unifying W-algebras which have previously been introduced as `WD_{-n}. In addition, minimal models of WD_{-n} are studied. The coset realizations provide a generalization of level-rank-duality of dual coset pairs. As further examples of finitely nonfreely generated quantum W-algebras we discuss orbifolding of W-algebras which on the quantum level has different properties than in the classical case. We demonstrate in some examples that the classical limit according to Bowcock and Watts of these nonfreely finitely generated quantum W-algebras probably yields infinitely nonfreely generated classical W-algebras.
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 + 3 k + 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.
In 2D conformal quantum field theory, we continue a systematic study of W-algebras with two and three generators and their highest weight representations focussing mainly on rational models. We review the known facts about rational models of W(2,delt
a)-algebras. Our new rational models of W-algebras with two generators all belong to one of the known series. The majority of W-algebras with three generators -including the new ones constructed in this letter- can be explained as subalgebras or truncations of Casimir algebras. Nonetheless, for one solution of W(2,4,6) we reveal some features that do not fit into the pattern of Casimir algebras or orbifolds thereof. This shows that there are more W-algebras than those predicted from Casimir algebras (or Toda field theories). However, most of the known rational conformal field theories belong to the minimal series of some Casimir algebra.
Using the representation theory of the subgroups SL_2(Z_p) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to good fusion algebras. Furthermore, the conformal dimensions
and the central charge of the corresponding rational conformal field theories are calculated. Two series of representations which can be realized by unitary theories are presented. We show that most of the fusion algebras induced by admissible representations are realized in well known rational models.
In this paper we consider the representation theory of N=1 Super-W-algebras with two generators for conformal dimension of the additional superprimary field between two and six. In the superminimal case our results coincide with the expectation from
the ADE-classification. For the parabolic algebras we find a finite number of highest weight representations and an effective central charge $tilde c = 3/2$. Furthermore we show that most of the exceptional algebras lead to new rational models with $tilde c > 3/2$. The remaining exceptional cases show a new `mixed structure. Besides a continuous branch of representations discrete values of the highest weight exist, too.
In this paper we consider extensions of the super Virasoro algebra by one and two super primary fields. Using a non-explicitly covariant approach we compute all SW-algebras with one generator of dimension up to 7 in addition to the super Virasoro fie
ld. In complete analogy to W-algebras with two generators most results can be classified using the representation theory of the super Virasoro algebra. Furthermore, we find that the SW(3/2, 11/2)-algebra can be realized as a subalgebra of SW(3/2, 5/2) at c = 10/7. We also construct some new SW-algebras with three generators, namely SW(3/2, 3/2, 5/2), SW(3/2, 2, 2) and SW(3/2, 2, 5/2).
