Fusion Algebras and Characters of Rational Conformal Field Theories

We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our first main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. Secondly, we show that the conformal characters of various rational models of W-algebras can be determined from the mere knowledge of the central charge and the set of conformal dimensions. We also describe how to actually construct conformal characters by using theta series associated to certain lattices. On our way we develop several tools for studying representations of the modular group on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, are in general useful for the analysis rational models.Comment: 87 pages, AMS TeX, one postscript figure, one exceptional case added to Main theorem 2, some typos correcte

Rankin-Cohen Type Differential Operators for Siegel Modular Forms

Let H_n be the Siegel upper half space and let F and G be automorphic forms on H_n of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on H_n x H_n such that the restriction of D(F(Z_1) G(Z_2)) to Z = Z_1 = Z_2 is again an automorphic form of weight k+l+v on H_n. Since the elliptic case, i.e. n=1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin-Cohen type operators. We also discuss a generalisation of Rankin-Cohen type operators to vector valued differential operators.Comment: 19 pages LaTeX2e using amssym.de

Fusion Algebras Induced by Representations of the Modular Group

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.Comment: (one change informula (4.15), some minor changes) 13 pages (plain TeX), to be published in Int.Jour.Mod.Phys.

Fusion Algebras of Fermionic Rational Conformal Field Theories via a Generalized Verlinde Formula

We prove a generalization of the Verlinde formula to fermionic rational conformal field theories. The fusion coefficients of the fermionic theory are equal to sums of fusion coefficients of its bosonic projection. In particular, fusion coefficients of the fermionic theory connecting two conjugate Ramond fields with the identity are either one or two. Therefore, one is forced to weaken the axioms of fusion algebras for fermionic theories. We show that in the special case of fermionic W(2,d)-algebras these coefficients are given by the dimensions of the irreducible representations of the horizontal subalgebra on the highest weight. As concrete examples we discuss fusion algebras of rational models of fermionic W(2,d)-algebras including minimal models of the $N=1$ super Virasoro algebra as well as $N=1$ super W-algebras SW(3/2,d).Comment: 28 pages (Plain TeX), BONN-HE-93-0

Coset Realization of Unifying W-Algebras

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 that 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.Comment: 60 pages (plain TeX) (final version to appear in Int. J. Mod. Phys. A; several minor improvements and corrections - for details see beginning of file