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 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 construct and analyze modular invariants from SU (n) k loop group subfactors in a forthcoming publication, including the treatment of all SU (2) k modular invariants.

In this paper we further analyze modular invariants for subfactors, in particular the structure of the chiral induced systems of M -M 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 non-degenerate if the original braiding of the N -N 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 α-induced sectors. We show that modular invariants come along naturally with several nonnegative integer valued matrix representations of the original N -N 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, some SU (3) conformal inclusions and the chiral conformal Ising model.

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 A-D 2n -E 6,8 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 Cappelli-ItzyksonZuber and the method of α-induction 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.

We consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. We present a general procedure which associates to every strongly local vertex operator algebra V a conformal net A V acting on the Hilbert space completion of V and prove that the isomorphism class of A V does not depend on the choice of the scalar product on V . We show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V , the map W → A W gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of A V . Many known examples of vertex operator algebras such as the unitary Virasoro vertex operator algebras, the unitary affine Lie algebras vertex operator algebras, the known c = 1 unitary vertex operator algebras, the moonshine vertex operator algebra, together with their coset and orbifold subalgebras, turn out to be strongly local. We give various applications of our results. In particular we show that the even shorter Moonshine vertex operator algebra is strongly local and that the automorphism group of the corresponding conformal net is the Baby Monster group. We prove that a construction of Fredenhagen and Jörß gives back the strongly local vertex operator algebra V from the conformal net A V and give conditions on a conformal net A implying that A = A V for some strongly local vertex operator algebra V .if S ⊂ B(H) is self-adjoint then S ′ is a self-adjoint subalgebra of B(H) which is also unital, i.e. 1 H ∈ S ′ .A self-adjoint subalgebra M ⊂ B(H) is called a von Neumann algebra if M = M ′′ . Accordingly, (S ∪ S * ) ′ is a von Neumann algebra for all subsets S ⊂ B(H) and W * (S) ≡ (S ∪ S * ) ′′ is the smallest von Neumann algebra containing S.A von Neumann algebra M is said to be a factor if M ′ ∩ M = C1 H , i.e. M has a trivial center. B(H) is a factor for any Hilbert space H. Its isomorphism class as an abstract complex * -algebra only depends on the Hilbertian dimension of H. A von Neumann algebra M isomorphic to some B(H) (here H is not necessarily the same Hilbert space on which M acts) is called a type I factor. If H has dimension n ∈ Z >0 then M is called a type I n factor while if H is infinite-dimensional then M is called a type I ∞ factor.There exist factors which are not of type I. They are divided in two families: the type II factors (type II 1 or type II ∞ ) and type III factors (type III λ , λ ∈ [0, 1], cf.[23]).If M and N are von Neumann algebras and N ⊂ M then N is called a von Neumann subalgebra of M. If M is a factor then a von Neumann subalgebra N ⊂ M which is also a factor is called a subfactor. The theory of subfactors is a central topic in the theory of operator algebras and in its applications to quantum field theory. Subfactor theory was initiated in the seminal work [56] where V. Jones introduced and studied an index [M : N] for type II 1 factors. Subfac...