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

In-flight friction and wear mechanism

A unique mechanism developed for conducting friction and wear experiments in orbit is described. The device is capable of testing twelve material samples simultaneously. Parameters considered critical include: power, weight, volume, mounting, cleanliness, and thermal designs. The device performed flawlessly in orbit over an eighteen month period and demonstrated the usefulness of this design for future unmanned spacecraft or shuttle applications

Integrable Lattice Models for Conjugate An(1)A^{(1)}_n

A new class of $A^{(1)}_n$ integrable lattice models is presented. These are interaction-round-a-face models based on fundamental nimrep graphs associated with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each value of the rank and level. The Boltzmann weights are parameterized by elliptic theta functions and satisfy the Yang-Baxter equation for any fixed value of the elliptic nome q. At q=0, the models provide representations of the Hecke algebra and are expected to lead in the continuum limit to coset conformal field theories related to the $A^{(1)}_n$ conjugate modular invariants.Comment: 18 pages. v2: minor changes, such as page 11 footnot

Spectral Measures for Sp(2)Sp(2)

Spectral measures provide invariants for braided subfactors via fusion modules. In this paper we study joint spectral measures associated to the compact connected rank two Lie group $SO(5)$ and its double cover the compact connected, simply-connected rank two Lie group $Sp(2)$, including the McKay graphs for the irreducible representations of $Sp(2)$ and $SO(5)$ and their maximal tori, and fusion modules associated to the $Sp(2)$ modular invariants.Comment: 41 pages, 45 figures. Title changed and notation corrected. arXiv admin note: substantial text overlap with arXiv:1404.186

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 $Z$ 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. $SU(n)_k$ 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 $SU(2)_k$. 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 $S$ and $T$. 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 ("$\alpha$-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 $SU(2)_k$) 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

Isotopic tiling theory for hyperbolic surfaces

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces

Orbifold subfactors from Hecke algebras II --- Quantum doubles and braiding ---

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M_infinity-M_infinity bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A_2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M_infinity-M_infinity bimodules are described by certain orbifolds (with ghosts) for SU(3)_3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu's orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D_2n, n>2.Comment: 37 pages, Late