Yasuyuki Kawahigashi

Multi-Interval Subfactors and Modularity of Representations in Conformal Field Theory

Abstract: We describe the structure of the inclusions of factors ?(E)⊂?(E′)′ associated with multi-intervals E⊂ℝ for a local irreducible net ? 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 ?. As a consequence, the index of ?(E)⊂?(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 ? 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.

On α-induction, chiral generators and modular invariants for subfactors

Abstract:We consider a type III subfactor N⊂N 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 T-matrices 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.

Chiral Structure of Modular Invariants for Subfactors

Abstract: 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 non-negative 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.

Orbifold subfactors from Hecke algebras

We apply the notion of orbifold models ofSU(N) solvable lattice models to the Hecke algebra subfactors of Wenzl and get a new series of subfactors. In order to distinguish our subfactors from those of Wenzl, we compute the principal graphs for both series of subfactors. An obstruction for flatness of connections arises in this orbifold procedure in the caseN=2 and this eliminates the possibility of the Dynkin diagramsD2n+1, but we show that no such obstructions arise in the caseN=3. Our tools are the paragroups of Ocneanu and solutions of Jimbo-Miwa-Okado to the Yang-Baxter equation.

Classification of Two-Dimensional Local Conformal Nets with c < 1 and 2-Cohomology Vanishing for Tensor Categories

We classify two-dimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of A-D-E Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of one-dimensional local conformal nets, Dynkin diagrams D2n +1 and E7 do not appear, but now they do appear in this classification of two-dimensional local conformal nets. Such nets are also characterized as two-dimensional local conformal nets with μ-index equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for one-dimensional nets, is 2-cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.

The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions

We complete classification of discrete abelian or finite group actions on injective type III1 factors up to cocycle conjugacy. We also give a proof for Connes’ characterization of the Ker (mod) and Cnt(M) for an injective factor M of type III. §0 Introduction. The purpose of this paper is to give a proof of Connes’ announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective 1980 Mathematics Subject Classification (1985 Revision). 46L40.

Conformal Field Theory, Tensor Categories and Operator Algebras

This is a set of lecture notes on the operator algebraic approach to 2-dimensional conformal field theory. Representation theoretic aspects and connections to vertex operator algebras are emphasized. No knowledge on operator algebras or quantum field theory is assumed.

Local conformal nets arising from framed vertex operator algebras

Abstract We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III 1 ) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel–Lepowsky–Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.

From Vertex Operator Algebras to Conformal Nets and Back

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 Jorss 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.

Centrally trivial automorphisms and an analogue of Connes’s

April, 1992 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.