80 research outputs found
SU(3)-Goodman-de la Harpe-Jones subfactors and the realisation of SU(3) modular invariants
We complete the realisation by braided subfactors, announced by Ocneanu, of
all SU(3)-modular invariant partition functions previously classified by
Gannon.Comment: 47 pages, minor changes, to appear in Reviews in Mathematical Physic
Modular Invariants from Subfactors: Type I Coupling Matrices and Intermediate Subfactors
A braided subfactor determines a coupling matrix Z which commutes with the S-
and T-matrices arising from the braiding. Such a coupling matrix is not
necessarily of "type I", i.e. in general it does not have a block-diagonal
structure which can be reinterpreted as the diagonal coupling matrix with
respect to a suitable extension. We show that there are always two intermediate
subfactors which correspond to left and right maximal extensions and which
determine "parent" coupling matrices Z^\pm of type I. Moreover it is shown that
if the intermediate subfactors coincide, so that Z^+=Z^-, then Z is related to
Z^+ by an automorphism of the extended fusion rules. The intertwining relations
of chiral branching coefficients between original and extended S- and
T-matrices are also clarified. None of our results depends on non-degeneracy of
the braiding, i.e. the S- and T-matrices need not be modular. Examples from
SO(n) current algebra models illustrate that the parents can be different,
Z^+\neq Z^-, and that Z need not be related to a type I invariant by such an
automorphism.Comment: 25 pages, latex, a new Lemma 6.2 added to complete an argument in the
proof of the following lemma, minor changes otherwis
Modular invariants from subfactors
In these lectures we explain the intimate relationship between modular
invariants in conformal field theory and braided subfactors in operator
algebras. A subfactor with a braiding determines a matrix which is obtained
as a coupling matrix comparing two kinds of braided sector induction
("alpha-induction"). It has non-negative integer entries, is normalized and
commutes with the S- and T-matrices arising from the braiding. Thus it is a
physical modular invariant in the usual sense of rational conformal field
theory. The algebraic treatment of conformal field theory models, e.g.
models, produces subfactors which realize their known modular
invariants. Several properties of modular invariants have so far been noticed
empirically and considered mysterious such as their intimate relationship to
graphs, as for example the A-D-E classification for . In the subfactor
context these properties can be rigorously derived in a very general setting.
Moreover the fusion rule isomorphism for maximally extended chiral algebras due
to Moore-Seiberg, Dijkgraaf-Verlinde finds a clear and very general proof and
interpretation through intermediate subfactors, not even referring to
modularity of and . Finally we give an overview on the current state of
affairs concerning the relations between the classifications of braided
subfactors and two-dimensional conformal field theories. We demonstrate in
particular how to realize twisted (type II) descendant modular invariants of
conformal inclusions from subfactors and illustrate the method by new examples.Comment: Typos corrected and a few minor changes, 37 pages, AMS LaTeX, epic,
eepic, doc-class conm-p-l.cl
Modular invariants and subfactors
In this lecture we explain the intimate relationship between modular
invariants in conformal field theory and braided subfactors in operator
algebras. Our analysis is based on an approach to modular invariants using
braided sector induction ("-induction") arising from the treatment of
conformal field theory in the Doplicher-Haag-Roberts framework. Many properties
of modular invariants which have so far been noticed empirically and considered
mysterious can be rigorously derived in a very general setting in the subfactor
context. For example, the connection between modular invariants and graphs (cf.
the A-D-E classification for ) finds a natural explanation and
interpretation. We try to give an overview on the current state of affairs
concerning the expected equivalence between the classifications of braided
subfactors and modular invariant two-dimensional conformal field theories.Comment: 25 pages, AMS LaTeX, epic, eepic, doc-class fic-1.cl
Modular Invariants, Graphs and -Induction for Nets of Subfactors I
We analyze the induction and restriction of sectors for nets of subfactors
defined by Longo and Rehren. Picking a local subfactor we derive a formula
which specifies the structure of the induced sectors in terms of the original
DHR sectors of the smaller net and canonical endomorphisms. We also obtain a
reciprocity formula for induction and restriction of sectors, and we prove a
certain homomorphism property of the induction mapping.
Developing further some ideas of F. Xu we will apply this theory in a
forthcoming paper to nets of subfactors arising from conformal field theory, in
particular those coming from conformal embeddings or orbifold inclusions of
SU(n) WZW models. This will provide a better understanding of the labeling of
modular invariants by certain graphs, in particular of the A-D-E classification
of SU(2) modular invariants.Comment: 36 pages, latex, several corrections, a strong additivity assumption
had to be adde
Reoptimization of Some Maximum Weight Induced Hereditary Subgraph Problems
The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Î , an optimal solution OPT for Î in I and an instance IâČ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Î in I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph and max split subgraph under vertex insertions and deletion
Spectral Measures and Generating Series for Nimrep Graphs in Subfactor Theory II: SU(3)
We complete the computation of spectral measures for SU(3) nimrep graphs
arising in subfactor theory, namely the SU(3) ADE graphs associated with SU(3)
modular invariants and the McKay graphs of finite subgroups of SU(3). For the
SU(2) graphs the spectral measures distill onto very special subsets of the
semicircle/circle, whilst for the SU(3) graphs the spectral measures distill
onto very special subsets of the discoid/torus. The theory of nimreps allows us
to compute these measures precisely. We have previously determined spectral
measures for some nimrep graphs arising in subfactor theory, particularly those
associated with all SU(2) modular invariants, all subgroups of SU(2), the
torus, SU(3), and some SU(3) graphs.Comment: 38 pages, 21 figure
Topological Quantum Field Theories and Operator Algebras
We review "quantum" invariants of closed oriented 3-dimensional manifolds
arising from operator algebras.Comment: For proceedings of "International Workshop on Quantum Field Theory
and Noncommutative Geometry", Sendai, November 200
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