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103
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.
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 . . . . . ....
Chiral Structure of Modular Invariants for Subfactors
, 1999
"... In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of MM morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral ” intersection, and we show that the ambichiral braidin ..."
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Cited by 48 (21 self)
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In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of MM morphisms. The relative braiding between the chiral systems restricts to a proper braiding on their “ambichiral ” intersection, and we show that the ambichiral braiding is nondegenerate if the original braiding of the NN morphisms is. Moreover, in this case the dimensions of the irreducible representations of the chiral fusion rule algebras are given by the chiral branching coefficients which describe the ambichiral contribution in the irreducible decomposition of αinduced sectors. We show that modular invariants come along naturally with several nonnegative integer valued matrix representations of the original NN Verlinde fusion rule algebra, and we completely determine their decomposition into its characters. Finally the theory is illustrated by various examples, including the treatment of all SU (2)k modular invariants.
On αinduction, chiral generators and modular invariants for subfactors
 Commun. Math. Phys
, 1999
"... We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle ..."
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Cited by 38 (10 self)
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We consider a type III subfactor N ⊂ M of finite index with a finite system of braided NN morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply αinduction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the αinduced sectors. A matrix Z is defined and shown to commute with the S and Tmatrices arising from the braiding. If the braiding is nondegenerate, then Z is a “modular invariant mass matrix ” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of MM morphisms is generated by the images of both kinds of αinduction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will
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...
Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors
 Commun. Math. Phys
, 2000
"... A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal structure which can be reinterpreted as the diagonal coupling matrix with r ..."
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Cited by 35 (5 self)
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A braided subfactor determines a coupling matrix Z which commutes with the S and Tmatrices arising from the braiding. Such a coupling matrix is not necessarily of “type I”, i.e. in general it does not have a blockdiagonal 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 ± 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 Tmatrices are also clarified. None of our results depends on nondegeneracy of the braiding, i.e. the S and Tmatrices need not be modular. Examples from SO(n) current algebra models illustrate that the parents can be different, Z + ̸ = Z − , and that Z need not be related to a type I invariant by such an automorphism. 1