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 A-D-E classification of SU(2) modular invariants.

Abstract. We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is non-commutative. In the second application we show that the Kac-Wakimoto hypothesis about certain relations between branching rules and S-matrices, which has existed for almost a decade, is not true in at least three examples. Finally we show that the fusion rings of subfactors associated with the conformal inclusions SU(n) (n+2) ⊂ SU(n(n + 1)/2) and SU(n + 2)n ⊂ SU((n + 1)(n + 2)/2) are canonically isomorphic using a version of level-rank duality. Let Gk ⊂ H be a conformal inclusion with both G and H being semisimple compact Lie groups, and k the Dykin index of the inclusion (cf. [GNO] or [KW]). We shall use i (resp. λ) to denote the irreducible projective positive energy representation of loop group LH (resp. LG) at level 1 (resp. k) (cf.[PS]). Denote by biλ

During the recent years Conformal Quantum Field Theory has become a widely studied topic, especially on a low dimensional space-time, because of physical motivations such as the desire of a better understanding of two-dimensional 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

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.

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 . . . . . ....

: A subtheory of a quantum field theory specifies von Neumann subalgebras A(O) (the `observables ' in the space-time region O) of the von Neumann algebras B(O) (the `fields' localized in O). Every local algebra being a (type III 1 ) factor, the inclusion A(O) ae B(O) is a subfactor. The assignment of these local subfactors to the space-time 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 non-trivial examples are given. Several resu...

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

Dedicated to Hans Borchers on the occasion of his seventieth birthday