59 research outputs found
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.
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
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
Boundary States, Extended Symmetry Algebra and Module Structure for certain Rational Torus Models
The massless bosonic field compactified on the circle of rational is
reexamined in the presense of boundaries. A particular class of models
corresponding to is distinguished by demanding the existence
of a consistent set of Newmann boundary states. The boundary states are
constructed explicitly for these models and the fusion rules are derived from
them. These are the ones prescribed by the Verlinde formula from the S-matrix
of the theory. In addition, the extended symmetry algebra of these theories is
constructed which is responsible for the rationality of these theories.
Finally, the chiral space of these models is shown to split into a direct sum
of irreducible modules of the extended symmetry algebra.Comment: 12 page
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
New N=1 Extended Superconformal Algebras with Two and Three Generators
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 field. 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).Comment: 30 pages (Plain TeX), BONN-HE-92-0
Unifying 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.Comment: 13 pages (plain TeX); BONN-TH-94-01, DFTT-15/9
Galois currents and the projective kernel in Rational Conformal Field Theory
The notion of Galois currents in Rational Conformal Field Theory is
introduced and illustrated on simple examples. This leads to a natural
partition of all theories into two classes, depending on the existence of a
non-trivial Galois current. As an application, the projective kernel of a RCFT,
i.e. the set of all modular transformations represented by scalar multiples of
the identity, is described in terms of a small set of easily computable
invariants
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