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.

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 = F-Vect, where F is a field. An object X ∈ A with two-sided 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 3-manifolds 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

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 consider a type III subfactor N ⊂ M of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction 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 α-induced sectors. A matrix Z is defined and shown to commute with the S- and Tmatrices arising from the braiding. If the braiding is non-degenerate, 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 M-M morphisms is generated by the images of both kinds of α-induction, 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

Abstract not found

Abstract. We study a class of centrally trivial automorphisms for subfactors, and get an upper bound for the order of the group they make (modulo normalizers) in terms of the “dual ” principal graph for AFD type II1 subfactors with trivial relative commutant, finite index and finite depth. We prove that this upper bound is attained for many known subfators. We also introduce χ(M,N) for subfactors N ⊂ M as the relative version of Connes ’ invariant χ(M), and compute this group for many AFD type II1 subfactors with finite index and finite depth including all the cases with index less than 4 and many Hecke algebra subfactors of Wenzl. In these finite depth cases, the group χ(M,N) is always finite and abelian, and we realize all the finite abelian groups as χ(M,N). Analogy between this topic and modular structure of type III factors is also discussed. As an application, we give some classification results for Aut(M,N). For example, for the subfactors of type A2n+1, there are two and only two outer actions of Z2. One is of the “standard” form and the other is given by the “orbifold ” action arising from the paragroup symmetry. As preliminaries, we also prove several statements on central sequence subfactors announced by A. Ocneanu.

Abstract not found

We define “a crossed product by a paragroup action on a subfactor ” as a certain commuting square of type II1 factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu’s paragroup. In addition to the standard axioms of paragroups, we have the intertwining Yang-Baxter equation as the new additional axiom. We will show that Rational Conformal Field Theory in the sense of Moore-Seiberg and orbifold construction in the sense of D. E. Evans, the author, and F. Xu produce paragroup actions on 1 subfactors in the canonical form. As applications, we show that the subfactor N ⊂ M of Goodman-de la Harpe-Jones with index 3 + √ 3 is not conjugate to its dual M ⊂ M1 by showing the fusion algebras of N-N bimodules and M-M bimodules are different, although the principal graph and the dual principal graph are the same. This is the first example of such a subfactor. We also determine the topological quantum field theory of this subfactor. Finally, we make an analogue of the coset construction in RCFT for subfactors in our settings. 1

Abstract. We prove that the (two) connections, or commuting squares, on the Coxeter-Dynkin diagram E7 produce a subfactor with principal graph D10. This was conjectured by J.-B. Zuber in connection with modular invariants in conformal field theory, and solve the last case of computing the flat parts of the connections on Coxeter-Dynkin diagrams with index less than 4. Since V. F. R. Jones initiated a systematic study of subfactors in [Jo], more and more connections of the subfactor theory with topology and quantum field theory have been pointed out. Our aim in this paper is to provide an evidence of a deeper relation between a notion of flatness in subfactor theory and modular invariants in

the relative Jones invariant κ