14 research outputs found
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
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
super Virasoro algebra as well as super W-algebras SW(3/2,d).Comment: 28 pages (Plain TeX), BONN-HE-93-0
Modular Invariance and Uniqueness of Conformal Characters
We show that the conformal characters of various rational models of
W-algebras can be already uniquely determined if one merely knows the central
charge and the conformal dimensions. As a side result we develop several tools
for studying representations of SL(2,Z) on spaces of modular functions. These
methods, applied here only to certain rational conformal field theories, may be
useful for the analysis of many others.Comment: 21 pages (AMS TeX), BONN-TH-94-16, MPI-94-6