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Multiinterval subfactors and modularity of representations in conformal field theory
 Commun. Math. Phys
"... 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 multiintervals E ⊂ R for a local irreducible net A of von Neumann algebras on the real line satisfying the split property and Haag duality. I ..."
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Cited by 63 (26 self)
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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 multiintervals 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 LongoRehren 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 nondegenerate, 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.
Orbifold subfactors from Hecke algebras
 Comm. Math. Phys
, 1994
"... 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 ..."
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Cited by 39 (23 self)
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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 nondegenerate braiding exists on the even vertices of D2n, n>2. 1
The Structure of Sectors Associated with the LongoRehren Inclusions I. General Theory
 Commun. Math. Phys
, 1999
"... We investigate the structure of the LongoRehren 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 LongoRe ..."
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Cited by 37 (0 self)
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We investigate the structure of the LongoRehren 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 LongoRehren 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 LongoRehren inclusions arising from the Cuntz algebra endomorphisms. We also discuss the case where the original system has a braiding, and generalize Ocneanu and EvansKawahigashi's method for the analysis of the asymptotic inclusions of the Hecke algebra subfactors. 1 Introduction The notion of the asymptotic inclusio...
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Canonical tensor product subfactors
"... Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional 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 exi ..."
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Cited by 21 (6 self)
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Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional 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 twodimensional theories. 1 Introduction and
Classification of twodimensional local conformal nets with c < 1 and 2cohomology vanishing for tensor categories
 Commun. Math. Phys
, 2004
"... We classify twodimensional 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 ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification ..."
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Cited by 17 (9 self)
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We classify twodimensional 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 ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of onedimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of twodimensional local conformal nets. Such nets are also characterized as twodimensional local conformal nets with µindex equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for onedimensional nets, is 2cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.
Chiral observables and modular invariants
 Commun. Math. Phys
, 2000
"... 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, ..."
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Cited by 12 (3 self)
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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
Generalized LongoRehren subfactors and αinduction
 Comm. Math. Phys
, 2002
"... We study the recent construction of subfactors by Rehren which generalizes the LongoRehren subfactors. We prove that if we apply this construction to a nondegenerately braided subfactor N ⊂ M and α ±induction, then the resulting subfactor is dual to the LongoRehren subfactor M ⊗ M opp ⊂ R arisin ..."
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Cited by 5 (2 self)
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We study the recent construction of subfactors by Rehren which generalizes the LongoRehren subfactors. We prove that if we apply this construction to a nondegenerately braided subfactor N ⊂ M and α ±induction, then the resulting subfactor is dual to the LongoRehren 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 multiinterval subfactors arising from a completely rational conformal net of factors on S 1 to a net of subfactors and show that the (generalized) LongoRehren subfactors and αinduction naturally appear in this context. 1
Locality and modular invariance in 2D conformal QFT
 in Mathematical Physics in Mathematics and Physics
"... The relations and differences between various classification problems arising in the context of local twodimensional (2D) conformal quantum field theory, modular invariants, and subfactors, are discussed. The extent to which locality implies modular invariance, is exhibited. AMS Subject classificat ..."
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Cited by 3 (0 self)
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The relations and differences between various classification problems arising in the context of local twodimensional (2D) conformal quantum field theory, modular invariants, and subfactors, are discussed. The extent to which locality implies modular invariance, is exhibited. AMS Subject classification: 81T40, 46L37 (primary); 81T05 (secondary) Modular invariants, 2D conformal QFT, and subfactors One of the great excitements in conformal QFT was the ADE classification of modular invariant coupling matrices for the SU(2) current algebra [5]. At each level k, this algebra has only a finite number of covariant representations with positive energy (superselection