Dedicated to John E. Roberts on the occasion of his sixtieth birthday We describe the structure of the inclusions of factors A(E) ⊂A(E ′ ) ′ associated with multi-intervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. In particular, if the net is conformal and the subfactor has finite index, the inclusion associated with two separated intervals is isomorphic to the Longo-Rehren inclusion, which provides a quantum double construction of the tensor category of superselection sectors of A. As a consequence, the index of A(E) ⊂A(E ′ ) ′ coincides with the global index associated with all irreducible sectors, the braiding symmetry associated with all sectors is non-degenerate, namely the representations of A form a modular tensor category, and every sector is a direct sum of sectors with finite dimension. The superselection structure is generated by local data. The same results hold true if conformal invariance is replaced by strong additivity and there exists a modular PCT symmetry.

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞-M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+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∞-M ∞ 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 D2n, n>2. 1

We investigate the structure of the Longo-Rehren inclusion for a finite closed system of endomorphisms of factors, whose categorical structure is known to be the same as the asymptotic inclusion of A. Ocneanu. In particular, we obtain a precise description of the sectors associated with the Longo-Rehren inclusions in terms of half braidings, which do not necessarily satisfy the usual condition of braidings. In doing so, we give new proofs to most of the known statements concerning asymptotic inclusions. We construct a complete system of matrix units of the tube algebra using the half braidings, which will be used in the second part to describe concrete examples of the Longo-Rehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and Evans-Kawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors. 1 Introduction The notion of the asymptotic inclusio...

Abstract not found

We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with µ-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.

Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in two-dimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding two-dimensional theories. 1 Introduction and

Abstract: Various definitions of chiral observables in a given Möbius covariant twodimensional (2D) theory are shown to be equivalent. Their representation theory in the vacuum Hilbert space of the 2D theory is studied. It shares the general characteristics of modular invariant partition functions, although SL(2, Z) transformation properties are not assumed. First steps towards classification are made. 1

We study the recent construction of subfactors by Rehren which generalizes the Longo-Rehren subfactors. We prove that if we apply this construction to a non-degenerately braided subfactor N ⊂ M and α ±-induction, then the resulting subfactor is dual to the Longo-Rehren subfactor M ⊗ M opp ⊂ R arising from the entire system of irreducible endomorphisms of M resulting from α ±-induction. As a corollary, we solve a problem on existence of braiding raised by Rehren negatively. Furthermore, we generalize our previous study with Longo and Müger on multi-interval subfactors arising from a completely rational conformal net of factors on S 1 to a net of subfactors and show that the (generalized) Longo-Rehren subfactors and α-induction naturally appear in this context. 1

Abstract: A theorem is derived which (i) provides a new class of subfactors which may be interpreted as generalized asymptotic subfactors, and which (ii) ensures the existence of twodimensional local quantum field theories associated with certain modular invariant matrices. 1 Introduction and results We consider type III von Neumann factors throughout. Endfin(N) stands for the set of unital endomorphisms λ with finite dimension d(λ) of a factor N. A closed N-system is a set ∆ ⊂ Endfin(N) of mutually inequivalent irreducible endomorphisms such that (i) idN ∈ ∆, (ii) if λ ∈ ∆ then there is a conjugate endomorphism

We survey the current status of Ocneanu’s paragroup theory and its relation to topological quantum field theory. 1