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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.
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
Category theory for conformal boundary conditions
 FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI
, 2003
"... ... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the d ..."
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Cited by 50 (14 self)
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... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIMrep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 29 (6 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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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.
Topological sectors and a dichotomy in conformal field theory
 Commun. Math. Phys
"... Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index the ..."
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Cited by 27 (14 self)
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Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the nfold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µindex of A is finite then A turns out to be strongly additive. By [31], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is nondegenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.
Quantum automorphism groups of homogeneous graphs
 J. Funct. Anal
"... Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry g ..."
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Cited by 24 (11 self)
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Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry group G(X). For a graph with no edges Tannakian duality makes H(X) correspond to a TemperleyLieb algebra. We study some versions of this correspondence.
Symmetries of a generic coaction
 Math. Ann
, 1999
"... Abstract. If B is C ∗algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂ B), have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂ B) in the case ..."
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Cited by 23 (18 self)
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Abstract. If B is C ∗algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂ B), have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂ B) in the case where B is a matrix algebra (2) if n ≥ 5 then the dual ̂ Gaut ( ̂ B) is not amenable (3) if n ≥ 4 then the fixed point subfactor