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88
D.E.: Modular Invariants, Graphs and αInduction for Nets of Subfactors II
 In preparation
"... 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 obta ..."
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Cited by 80 (8 self)
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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 ADE classification of SU(2) modular invariants.
Operator algebras and conformal field theory  III. Fusion of positive energy representations of LSU(N) using bounded operators
, 1998
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The Conformal spin and statistics theorem
 Commun. Math. Phys
, 1996
"... During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure provi ..."
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Cited by 64 (23 self)
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During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional spacetime, because of physical motivations such as the desire of a better understanding of twodimensional critical phenomena, and also for its rich mathematical structure providing remarkable connections with different areas such as Hopf algebras, low dimensional topology, knot invariants, subfactors
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.
Operator Algebras and Conformal Field Theory
 COMMUNICATIONS MATHEMATICAL PHYSICS
, 1993
"... We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite typ ..."
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Cited by 55 (2 self)
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We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the TomitaTakesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "backgroundindependent" formulation of conformal field theories.
Nets of subfactors
, 1994
"... A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of th ..."
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Cited by 51 (4 self)
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A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the ‘observables’ in the spacetime region O) of the von Neumann algebras B(O) (the ‘fields ’ localized in O). Every local algebra being a (type III1) factor, the inclusion A(O) ⊂ B(O) is a subfactor. The assignment of these local subfactors to the spacetime regions is called a ‘net of subfactors’. The theory of subfactors is applied to such nets. In order to characterize the ‘relative position ’ of the subtheory, and in particular to control the restriction and induction of superselection sectors, the canonical endomorphism is studied. The crucial observation is this: the canonical endomorphism of a local subfactor extends to an endomorphism of the field net, which in turn restricts to a localized endomorphism of the observable net. The method allows to characterize, and reconstruct, local extensions B of a given theory A in terms of the observables. Various nontrivial examples are given. Several results go beyond the quantum field theoretical application.
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reco ..."
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Cited by 49 (8 self)
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It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
An algebraic spin and statistics theorem
 Commun. Math. Phys
, 1995
"... Dedicated to Hans Borchers on the occasion of his seventieth birthday ..."
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Cited by 42 (11 self)
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Dedicated to Hans Borchers on the occasion of his seventieth birthday
Classification of local conformal nets. Case c < 1
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
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Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by CappelliItzyksonZuber and the method of αinduction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.