513 research outputs found
The action of outer automorphisms on bundles of chiral blocks
On the bundles of WZW chiral blocks over the moduli space of a punctured
rational curve we construct isomorphisms that implement the action of outer
automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms
respect the Knizhnik-Zamolodchikov connection and have finite order. When all
primary fields are fixed points, the isomorphisms are endomorphisms; in this
case, the bundle of chiral blocks is typically a reducible vector bundle. A
conjecture for the trace of such endomorphisms is presented; the proposed
relation generalizes the Verlinde formula. Our results have applications to
conformal field theories based on non-simply connected groups and to the
classification of boundary conditions in such theories.Comment: 46 pages, LaTeX2e. Final version (Commun.Math.Phys., in press). We
have implemented the fact that the group of automorphisms in general acts
only projectively on the chiral blocks and corrected some typo
D-brane conformal field theory
We outline the structure of boundary conditions in conformal field theory. A
boundary condition is specified by a consistent collection of reflection
coefficients for bulk fields on the disk together with a choice of an
automorphism \omega of the fusion rules that preserves conformal weights.
Non-trivial automorphisms \omega correspond to D-brane configurations for
arbitrary conformal field theories.Comment: 7 pages, LaTeX2e. Slightly extended version of a talk given by J.
Fuchs at the 31st International Symposium Ahrenshoop on the Theory of
Elementary Particles, Buckow, Germany, September 199
Solitonic sectors, conformal boundary conditions and three-dimensional topological field theory
The correlation functions of a two-dimensional rational conformal field
theory, for an arbitrary number of bulk and boundary fields and arbitrary world
sheets can be expressed in terms of Wilson graphs in appropriate
three-manifolds. We present a systematic approach to boundary conditions that
break bulk symmetries. It is based on the construction, by `alpha-induction',
of a fusion ring for the boundary fields. Its structure constants are the
annulus coefficients and its 6j-symbols give the OPE of boundary fields.
Symmetry breaking boundary conditions correspond to solitonic sectors.Comment: 9 pages, LaTeX2e. Invited talk by Christoph Schweigert at the TMR
conference ``Non-perturbative quantum effects 2000'', Paris, September 200
Symmetry breaking boundaries II. More structures; examples
Various structural properties of the space of symmetry breaking boundary
conditions that preserve an orbifold subalgebra are established. To each such
boundary condition we associate its automorphism type. It is shown that
correlation functions in the presence of such boundary conditions are
expressible in terms of twisted boundary blocks which obey twisted Ward
identities. The subset of boundary conditions that share the same automorphism
type is controlled by a classifying algebra, whose structure constants are
shown to be traces on spaces of chiral blocks. T-duality on boundary conditions
is not a one-to-one map in general. These structures are illustrated in a
number of examples. Several applications, including the construction of non-BPS
boundary conditions in string theory, are exhibited.Comment: 51 pages, LaTeX2
Completeness of boundary conditions for the critical three-state Potts model
We show that the conformally invariant boundary conditions for the
three-state Potts model are exhausted by the eight known solutions. Their
structure is seen to be similar to the one in a free field theory that leads to
the existence of D-branes in string theory. Specifically, the fixed and mixed
boundary conditions correspond to Neumann conditions, while the free boundary
condition and the new one recently found by Affleck et al [1] have a natural
interpretation as Dirichlet conditions for a higher-spin current. The latter
two conditions are governed by the Lee\hy Yang fusion rules. These results can
be generalized to an infinite series of non-diagonal minimal models, and
beyond.Comment: 9 pages, LaTeX2
A representation theoretic approach to the WZW Verlinde formula
By exploring the description of chiral blocks in terms of co-invariants, a
derivation of the Verlinde formula for WZW models is obtained which is entirely
based on the representation theory of affine Lie algebras. In contrast to
existing proofs of the Verlinde formula, this approach works universally for
all untwisted affine Lie algebras. As a by-product we obtain a homological
interpretation of the Verlinde multiplicities as Euler characteristics of
complexes built from invariant tensors of finite-dimensional simple Lie
algebras. Our results can also be used to compute certain traces of
automorphisms on the spaces of chiral blocks. Our argument is not rigorous; in
its present form this paper will therefore not be submitted for publication.Comment: 37 pages, LaTeX2e. wrong statement in subsection 4.2 corrected and
rest of the paper adapte
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