43 research outputs found
Conformal Field Theories, Graphs and Quantum Algebras
This article reviews some recent progress in our understanding of the
structure of Rational Conformal Field Theories, based on ideas that originate
for a large part in the work of A. Ocneanu. The consistency conditions that
generalize modular invariance for a given RCFT in the presence of various types
of boundary conditions --open, twisted-- are encoded in a system of integer
multiplicities that form matrix representations of fusion-like algebras. These
multiplicities are also the combinatorial data that enable one to construct an
abstract ``quantum'' algebra, whose - and -symbols contain essential
information on the Operator Product Algebra of the RCFT and are part of a cell
system, subject to pentagonal identities. It looks quite plausible that the
classification of a wide class of RCFT amounts to a classification of ``Weak
- Hopf algebras''.Comment: 23 pages, 12 figures, LateX. To appear in MATHPHYS ODYSSEY 2001
--Integrable Models and Beyond, ed. M. Kashiwara and T. Miwa, Progress in
Math., Birkhauser. References and comments adde
Virasoro Singular Vectors via Quantum DS Reduction
The BRST quantisation of the Drinfeld - Sokolov reduction is exploited to
recover all singular vectors of the Virasoro algebra Verma modules from the
corresponding ones. The two types of singular vectors are shown
to be identical modulo terms trivial in the cohomology. The main
tool is a quantum version of the DS gauge transformation.Comment: 7 pages, plain TeX, SISSA-74/93/E
On Structure Constants of Theories
Structure constants of minimal conformal theories are reconsidered. It is
shown that {\it ratios} of structure constants of spin zero fields of a
non-diagonal theory over the same evaluated in the diagonal theory are given by
a simple expression in terms of the components of the eigenvectors of the
adjacency matrix of the corresponding Dynkin diagram. This is proved by
inspection, which leads us to carefully determine the {\it signs} of the
structure constants that had not all appeared in the former works on the
subject. We also present a proof relying on the consideration of lattice
correlation functions and speculate on the extension of these identities to
more complicated theories.Comment: 32 page
From CFT's to Graphs
In this paper, we pursue the discussion of the connections between rational
conformal field theories (CFT) and graphs. We generalize our recent work on the
relations of operator product algebra (OPA) structure constants of
theories with the Pasquier algebra attached to the graph. We show that in a
variety of CFT built on -- typically conformal embeddings and
orbifolds, similar considerations enable one to write a linear system satisfied
by the matrix elements of the Pasquier algebra in terms of conformal data --
quantum dimensions and fusion coefficients. In some cases, this provides a
sufficient information for the determination of all the eigenvectors of an
adjacency matrix, and hence of a graph.Comment: 44 pages, 6 postscript figures, the whole uuencoded. TEX file, macros
used : harvmac.tex , epsf.tex. Optionally, AMS fonts in amssym.def and
amssym.te
The fusion algebra of bimodule categories
We establish an algebra-isomorphism between the complexified Grothendieck
ring F of certain bimodule categories over a modular tensor category and the
endomorphism algebra of appropriate morphism spaces of those bimodule
categories. This provides a purely categorical proof of a conjecture by Ostrik
concerning the structure of F.
As a by-product we obtain a concrete expression for the structure constants
of the Grothendieck ring of the bimodule category in terms of endomorphisms of
the tensor unit of the underlying modular tensor category.Comment: 16 page
Fusion rules for admissible representations of affine algebras: the case of
We derive the fusion rules for a basic series of admissible representations
of at fractional level . The formulae admit an
interpretation in terms of the affine Weyl group introduced by Kac and
Wakimoto. It replaces the ordinary affine Weyl group in the analogous formula
for the fusion rules multiplicities of integrable representations. Elements of
the representation theory of a hidden finite dimensional graded algebra behind
the admissible representations are briefly discussed.Comment: containing two TEX files: main file using input files harvmac.tex,
amssym.def, amssym.tex, 19p.; file with figures using XY-pic package, 6p.
Correction in the definition of general shifted weight diagra
Non-Rational 2D Quantum Gravity: I. World Sheet CFT
We address the problem of computing the tachyon correlation functions in
Liouville gravity with generic (non-rational) matter central charge c<1. We
consider two variants of the theory. The first is the conventional one in which
the effective matter interaction is given by the two matter screening charges.
In the second variant the interaction is defined by the Liouville dressings of
the non-trivial vertex operator of zero dimension. This particular deformation,
referred to as "diagonal'', is motivated by the comparison with the discrete
approach, which is the subject of a subsequent paper. In both theories we
determine the ground ring of ghost zero physical operators by computing its OPE
action on the tachyons and derive recurrence relations for the tachyon bulk
correlation functions. We find 3- and 4-point solutions to these functional
equations for various matter spectra. In particular, we find a closed
expression for the 4-point function of order operators in the diagonal theory.Comment: TEX-harvmac, revised version to appear in Nuclear Physics
The many faces of Ocneanu cells
We define generalised chiral vertex operators covariant under the Ocneanu
``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a
given rational 2-d conformal field theory. This provides a chiral approach,
which, unlike the conventional one, makes explicit various algebraic structures
encountered previously in the study of these theories and of the associated
critical lattice models, and thus allows their unified treatment. The
triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A},
reappear in several guises. With {\cal A} and its dual algebra {hat A} one
associates a pair of graphs, G and {\tilde G}. While G are known to encode
complete sets of conformal boundary states, the Ocneanu graphs {\tilde G}
classify twisted torus partition functions. The fusion algebra of the twist
operators provides the data determining {\hat A}. The study of bulk field
correlators in the presence of twists reveals that the Ocneanu graph quantum
symmetry gives also an information on the field operator algebra.Comment: 57 pages, 10 figures. Several misprints and the derivation of eq.
(7.38) correcte
Permutation Branes
N-fold tensor products of a rational CFT carry an action of the permutation
group S_N. These automorphisms can be used as gluing conditions in the study of
boundary conditions for tensor product theories. We present an ansatz for such
permutation boundary states and check that it satisfies the cluster condition
and Cardy's constraints. For a particularly simple case, we also investigate
associativity of the boundary OPE, and find an intriguing connection with the
bulk OPE. In the second part of the paper, the constructions are slightly
extended for application to Gepner models. We give permutation branes for the
quintic, together with some formulae for their intersections.Comment: 27 page
Admissible Representations -- Fusion Transformations and Local Correlators
We reconsider the earlier found solutions of the Knizhnik-Zamolodchikov (KZ)
equations describing correlators based on the admissible representations of
. Exploiting a symmetry of the KZ equations we show that the
original infinite sums representing the 4-point chiral correlators can be
effectively summed up. Using these simplified expressions with proper choices
of the contours we determine the duality (braid and fusion) transformations and
show that they are consistent with the fusion rules of Awata and Yamada. The
requirement of locality leads to a 1-parameter family of monodromy (braid)
invariants. These analogs of the ``diagonal'' 2-dimensional local 4-point
functions in the minimal Virasoro theory contain in general non-diagonal terms.
They correspond to pairs of fields of identical monodromy, having one and the
same counterpart in the limit to the Virasoro minimal correlators.Comment: LaTex, 20 pages; misprints corrected, few small addition