50 research outputs found
Orders and dimensions for sl(2) or sl(3) module categories and Boundary Conformal Field Theories on a torus
After giving a short description, in terms of action of categories, of some
of the structures associated with sl(2) and sl(3) boundary conformal field
theories on a torus, we provide tables of dimensions describing the semisimple
and co-semisimple blocks of the corresponding weak bialgebras (quantum
groupoids), tables of quantum dimensions and orders, and tables describing
induction - restriction. For reasons of size, the sl(3) tables of induction are
only given for theories with self-fusion (existence of a monoidal structure).Comment: 25 pages, 5 tables, 9 figures. Version 2: updated references. Typos
corrected. Several proofs added. Examples of ADE and generalized ADE
trigonometric identities have been removed to shorten the pape
Semi-regular masas of transfinite length
In 1965 Tauer produced a countably infinite family of semi-regular masas in
the hyperfinite factor, no pair of which are conjugate by an
automorphism. This was achieved by iterating the process of passing to the
algebra generated by the normalisers and, for each , finding
masas for which this procedure terminates at the -th stage. Such masas are
said to have length . In this paper we consider a transfinite version of
this idea, giving rise to a notion of ordinal valued length. We show that all
countable ordinals arise as lengths of semi-regular masas in the hyperfinite
factor. Furthermore, building on work of Jones and Popa, we
obtain all possible combinations of regular inclusions of irreducible
subfactors in the normalising tower.Comment: 14 page
Branching rules of semi-simple Lie algebras using affine extensions
We present a closed formula for the branching coefficients of an embedding p
in g of two finite-dimensional semi-simple Lie algebras. The formula is based
on the untwisted affine extension of p. It leads to an alternative proof of a
simple algorithm for the computation of branching rules which is an analog of
the Racah-Speiser algorithm for tensor products. We present some simple
applications and describe how integral representations for branching
coefficients can be obtained. In the last part we comment on the relation of
our approach to the theory of NIM-reps of the fusion rings of WZW models with
chiral algebra g_k. In fact, it turns out that for these models each embedding
p in g induces a NIM-rep at level k to infinity. In cases where these NIM-reps
can be be extended to finite level, we obtain a Verlinde-like formula for
branching coefficients.Comment: 11 pages, LaTeX, v2: one reference added, v3: Clarified proof of
Theorem 2, completely rewrote and extended Section 5 (relation to CFT), added
various references. Accepted for publication in J. Phys.
From conformal embeddings to quantum symmetries: an exceptional SU(4) example
We briefly discuss several algebraic tools that are used to describe the
quantum symmetries of Boundary Conformal Field Theories on a torus. The
starting point is a fusion category, together with an action on another
category described by a quantum graph. For known examples, the corresponding
modular invariant partition function, which is sometimes associated with a
conformal embedding, provides enough information to recover the whole
structure. We illustrate these notions with the example of the conformal
embedding of SU(4) at level 4 into Spin(15) at level 1, leading to the
exceptional quantum graph E4(SU(4)).Comment: 22 pages, 3 color figures. Version 2: We changed the color of figures
(ps files) in such a way that they are still understood when converted to
gray levels. Version 3: Several references have been adde
From modular invariants to graphs: the modular splitting method
We start with a given modular invariant M of a two dimensional su(n)_k
conformal field theory (CFT) and present a general method for solving the
Ocneanu modular splitting equation and then determine, in a step-by-step
explicit construction, 1) the generalized partition functions corresponding to
the introduction of boundary conditions and defect lines; 2) the quantum
symmetries of the higher ADE graph G associated to the initial modular
invariant M. Notice that one does not suppose here that the graph G is already
known, since it appears as a by-product of the calculations. We analyze several
su(3)_k exceptional cases at levels 5 and 9.Comment: 28 pages, 7 figures. Version 2: updated references. Typos corrected.
su(2) example has been removed to shorten the paper. Dual annular matrices
for the rejected exceptional su(3) diagram are determine
Exceptional quantum subgroups for the rank two Lie algebras B2 and G2
Exceptional modular invariants for the Lie algebras B2 (at levels 2,3,7,12)
and G2 (at levels 3,4) can be obtained from conformal embeddings. We determine
the associated alge bras of quantum symmetries and discover or recover, as a
by-product, the graphs describing exceptional quantum subgroups of type B2 or
G2 which encode their module structure over the associated fusion category.
Global dimensions are given.Comment: 33 pages, 27 color figure
Rigid C^*-tensor categories of bimodules over interpolated free group factors
Given a countably generated rigid C^*-tensor category C, we construct a
planar algebra P whose category of projections Pro is equivalent to C. From P,
we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid
C^*-tensor category Bim whose objects are bifinite bimodules over an
interpolated free group factor, and we show Bim is equivalent to Pro. We use
these constructions to show C is equivalent to a category of bifinite bimodules
over L(F_infty).Comment: 50 pages, many figure
Non-critical string pentagon equations and their solutions
We derive pentagon type relations for the 3-point boundary tachyon
correlation functions in the non-critical open string theory with generic
c_{matter} < 1 and study their solutions in the case of FZZ branes. A new
general formula for the Liouville 3-point factor is derived.Comment: 18 pages, harvmac; misprints corrected, section 3.2 extended, a new
general formula for the Liouville 3-point factor adde
Spectral measures of small index principal graphs
The principal graph of a subfactor with finite Jones index is one of the
important algebraic invariants of the subfactor. If is the adjacency
matrix of we consider the equation . When has square
norm the spectral measure of can be averaged by using the map
, and we get a probability measure on the unit circle
which does not depend on . We find explicit formulae for this measure
for the principal graphs of subfactors with index , the
(extended) Coxeter-Dynkin graphs of type , and . The moment
generating function of is closely related to Jones' -series.Comment: 23 page
Subfactors of index less than 5, part 1: the principal graph odometer
In this series of papers we show that there are exactly ten subfactors, other
than subfactors, of index between 4 and 5. Previously this
classification was known up to index . In the first paper we give
an analogue of Haagerup's initial classification of subfactors of index less
than , showing that any subfactor of index less than 5 must appear
in one of a large list of families. These families will be considered
separately in the three subsequent papers in this series.Comment: 36 pages (updated to reflect that the classification is now complete