## Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories (2004)

Venue: | Commun. Math. Phys |

Citations: | 17 - 9 self |

### BibTeX

@ARTICLE{Kawahigashi04classificationof,

author = {Yasuyuki Kawahigashi and Roberto Longo},

title = {Classification of two-dimensional local conformal nets with c < 1 and 2-cohomology vanishing for tensor categories},

journal = {Commun. Math. Phys},

year = {2004},

volume = {244},

pages = {63--97}

}

### OpenURL

### Abstract

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.

### Citations

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Citation Context ... other hand. We first prove that the correspondence A ↦→ Z is surjective. Indeed, by our previous work [31], Z can be realized by α-induction as in [5] for extensions of the 3 (1)sVirasoro nets. (See =-=[38, 52, 3, 6, 7, 4]-=- for more on α-induction.) Then Rehren’s results in [48] imply that θ defined as above (1) is the canonical endomorphism associated with a natural Q-system, and we have a corresponding local extension... |

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Citation Context ... other hand. We first prove that the correspondence A ↦→ Z is surjective. Indeed, by our previous work [31], Z can be realized by α-induction as in [5] for extensions of the 3 (1)sVirasoro nets. (See =-=[38, 52, 3, 6, 7, 4]-=- for more on α-induction.) Then Rehren’s results in [48] imply that θ defined as above (1) is the canonical endomorphism associated with a natural Q-system, and we have a corresponding local extension... |

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Citation Context ...celike separated. Then the Jones index [A(E ′ ) ′ : A(E)] is finite. This index is denoted by µA, the µ-index of A. 9 (5) (6) (7)sThe notion of complete rationality has been introduced and studied in =-=[32]-=- for a local net C on R. If C is conformal, the definition of complete rationality strictly parallels the above one in the two-dimensional case. In general, the above (onedimensional version) of the a... |

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Citation Context ...e following properties. V ∈ Hom(id,θ), (9) W ∈ Hom(θ, θ 2 ), (10) V ∗ V = 1, (11) W ∗ W = 1, (12) V ∗ W = θ(V ∗ )W ∈ R+, (13) W 2 = θ(W )W, (14) θ(W ∗ )W = WW ∗ . (15) Actually, it has been proved in =-=[39]-=- that Condition (15) is redundant. (It has been also proved in [27] that Condition (14) is redundant if (15) is assumed.) In this case, θ is a canonical endomorphism of a certain subfactor of the orig... |

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Citation Context ... R) factors through a representation of G. The above proposition holds as a consequence of spacelike locality, it is a particular case of the conformal spin-statistics theorem and can be proved as in =-=[20]-=-. Because of the above Prop. 2.1, A does extend to a local G-covariant net on the Einstein cylinder E = R × S 1 , the cover of the 2-torus obtained by lifting the time coordinate from S 1 to R. Explic... |

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Citation Context ...ems of irreducible DHR endomorphisms of AL and AR, respectively. The matrix Z =(Zij) is called a coupling matrix. The two nets AL and AR define S- and T -matrices, SL, TL, SR, TR, respectively, as in =-=[46]-=-. We are interested in the case where the S-matrices are invertible. (By the results in [32], this invertibility, which is called non-degeneracy of the braiding, holds if the nets are completely ratio... |

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Citation Context ... bounded intervals; we denote by K the set of double cones. The Möbius group PSL(2, R) acts on R∪{∞} by linear fractional transformations, hence this action restricts to a local action on R (see e.g. =-=[8]-=-), in particular if F ⊂ R has compact closure there exists a connected neighborhood U of the identity in PSL(2, R) such that gF ⊂ R for all g ∈U. It is convenient to regard this as a local action on R... |

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Citation Context ... other hand. We first prove that the correspondence A ↦→ Z is surjective. Indeed, by our previous work [31], Z can be realized by α-induction as in [5] for extensions of the 3 (1)sVirasoro nets. (See =-=[38, 52, 3, 6, 7, 4]-=- for more on α-induction.) Then Rehren’s results in [48] imply that θ defined as above (1) is the canonical endomorphism associated with a natural Q-system, and we have a corresponding local extension... |

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Citation Context ...t verify Condition 3. We label four irreducible objects as in Figure 4. (If n = 3, we set λ1 = id.) Note that the connection with respect to the generator σ has a principal graph as in Figure 5. (See =-=[24]-=-, for example, for the fusion rules of a subfactor with principal graph D2n.) 24s� � � � � ❅ � ❅ � ❅❅� � � � �❅ ❅� � � �❅ � ❅ � ❅❅� � id σ ··· Figure 3: The principal graph for the subfactor D2n � � �... |

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Citation Context ...µm/(ωλ1 ···ωλn). Note that if a 2-cocycle is trivial, then it is scalar-valued, in particular. We now recall the definition of the Longo-Rehren subfactor [38, Proposition 4.10] as follows. (See [40], =-=[43]-=-, [44] for related or more general definitions.) Let ∆ = {λk | k =0, 1,...,n} be a finite system of endomorphisms of a type III factor M where λ0 = id. We choose a system {Vk | k =0, 1,...,n} of isome... |

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34 | On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors
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Citation Context ...ry SU(2)k has a Z/2Zgrading and this generator 1 is odd, Condition 2 (b) also holds. Now the connection values with respect to this σ are the usual connection values of the paragroup Ak+1 as in [42], =-=[30]-=-, [14, Section 11.5], and they are non-zero and Condition 3 holds. The multiplication rule by the generator σ is described with the usual Bratteli diagram for the principal graph Ak+1 as in [28], [14,... |

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Citation Context ... automorphism of G associated with space and time reflection [8]. (iv) � Additivity. Let O be a double cone and {Oi} a family of open sets such that i Oi contains the axis of O. Then A(O) ⊂ � i A(Oi) =-=[15]-=-. (v) Equivalence between rreducibility and uniqueness of the vacuum. A is irreducible on M (that is �� O∈K A(O)� ′′ = B(H)), iff A is irreducible on E, iffΩ is the unique U-invariant vector (up to a ... |

28 | R.: Classification of local conformal nets. Case c < 1
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Citation Context ...1 with c < 1. (b) The classification of irreducible local extensions with finite Jones index of the two-dimensional Virasoro net Virc ⊗ Virc. Point (a) has been completely achieved in our recent work =-=[31]-=-. The Virasoro nets on S 1 with central charge less than one are in bijective correspondence with the pairs of A-D2n-E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to ... |

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Citation Context ...in this direction were obtained, in particular the central charge c>0, an intrinsic quantum label associated with each model, was shown to split in a discrete range c<1 and a continuos one c ≥ 1, see =-=[2, 17, 19]-=- refs in [19]. The main purpose of this paper is to achieve a complete classification of the twodimensional conformal models in the discrete series. In order to formulate such a statement in a precise... |

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Citation Context ...ectively. These two maps are equal in End(Hom(λµ, νσ)). In the above decomposition (20), we set λ = σ = id and µ = ν, then we have τ = ¯µ and π = µ in the summations. With Frobenius reciprocity as in =-=[25]-=- and the above identity of two maps in End(Hom(λµ, νσ)), we obtain the identity C id µ¯µ Cid ¯µµ = 1. (21) We next apply identity (12) to (16) and obtain the following equality ∑ λ,µ∈∆ dλdµK ν λµ = wd... |

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Citation Context ...in [32], that the matrix Z is a modular invariant for the tensor category of representations of the Virasoro net Virc [41], and such modular invariants have been classified by Cappelli-Itzykson-Zuber =-=[9]-=-. We shall show that this map A ↦→ Z sets up a bijective correspondence between the the set of isomorphism classes of two-dimensional maximal local conformal nets with parity and central charge less t... |

22 | Canonical tensor product subfactors
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Citation Context .... Indeed, by our previous work [31], Z can be realized by α-induction as in [5] for extensions of the 3 (1)sVirasoro nets. (See [38, 52, 3, 6, 7, 4] for more on α-induction.) Then Rehren’s results in =-=[48]-=- imply that θ defined as above (1) is the canonical endomorphism associated with a natural Q-system, and we have a corresponding local extension A of Virc⊗Virc and this produces the matrix Z in the ab... |

21 |
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Citation Context ... denote by ΛO the one-parameter subgroup of G defined as follows: ΛO = gΛW g −1 if W is a wedge, ΛW is the boost one-parameter group associated with W , and gO = W with g ∈ PSL(2, R) × PSL(2, R), see =-=[23]-=-. We collect in the next proposition a few basic properties of a local Möbius covariant net. The proof is either in the references or can be immediately obtained from those. All the statements also ho... |

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Citation Context ...1 ···ωµm/(ωλ1 ···ωλn). Note that if a 2-cocycle is trivial, then it is scalar-valued, in particular. We now recall the definition of the Longo-Rehren subfactor [38, Proposition 4.10] as follows. (See =-=[40]-=-, [43], [44] for related or more general definitions.) Let ∆ = {λk | k =0, 1,...,n} be a finite system of endomorphisms of a type III factor M where λ0 = id. We choose a system {Vk | k =0, 1,...,n} of... |

19 | Operatoralgebraic methods in quantum field theory - Baumgärtel - 1995 |

18 |
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Citation Context ...t of CappelliItzykson-Zuber modular invariant on the other hand. We first prove that the correspondence A ↦→ Z is surjective. Indeed, by our previous work [31], Z can be realized by α-induction as in =-=[5]-=- for extensions of the 3 (1)sVirasoro nets. (See [38, 52, 3, 6, 7, 4] for more on α-induction.) Then Rehren’s results in [48] imply that θ defined as above (1) is the canonical endomorphism associated... |

18 | Theory of Operator Algebras, Vols - Takesaki - 2003 |

16 |
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Citation Context ...ion of Q-systems in the sense of [36] having the canonical endomorphism of the form given by eq. (1). This type of classification of Q-systems, up to unitary equivalences, was studied by Izumi-Kosaki =-=[27]-=- as a subfactor analogue of 2-cohomology of (finite) groups. In our setting, we now have a 2-cohomology group of a tensor category, while the 2-cohomology of Izumi-Kosaki does not have a group structu... |

12 |
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(Show Context)
Citation Context ...pectively. These two maps are equal in End(Hom(λµ, νσ)). In the above decomposition (20), we set λ = σ = id and µ = ν, then we have τ =¯µ and π = µ in the summations. With Frobenius reciprocity as in =-=[25]-=- and the above identity of two maps in End(Hom(λµ, νσ)), we obtain the identity C id µ¯µC id ¯µµ =1. (21) We next apply identity (12) to (16) and obtain the following equality � dλdµK λ,µ∈∆ ν λµ = wdν... |

12 | Chiral observables and modular invariants
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(Show Context)
Citation Context ...d we have a corresponding local extension A of Virc⊗Virc and this produces the matrix Z in the above correspondence. To show the injectivity of the correspondence note that, due to the work of Rehren =-=[47]-=-, we have an inclusion Virc(I+) ⊗ Virc(I−) ⊂A+(I+) ⊗A−(I−) ⊂A(O) where A+ ⊗A− is the maximal chiral subnet. By assumption, A+ and A− are isomorphic with central charge c<1, thus they are in the discre... |

11 |
Symmetric enveloping algebras, amenability and AFD properties for subfactors
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(Show Context)
Citation Context ...1 ···ωλn). Note that if a 2-cocycle is trivial, then it is scalar-valued, in particular. We now recall the definition of the Longo-Rehren subfactor [38, Proposition 4.10] as follows. (See [40], [43], =-=[44]-=- for related or more general definitions.) Let ∆ = {λk | k =0, 1,...,n} be a finite system of endomorphisms of a type III factor M where λ0 = id. We choose a system {Vk | k =0, 1,...,n} of isometries ... |

9 |
Classification of local conformal nets, case c<1
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(Show Context)
Citation Context ...S 1 with c<1. (b) The classification of irreducible local extensions with finite Jones index of the two-dimensional Virasoro net Virc ⊗ Virc. Point (a) has been completely achieved in our recent work =-=[31]-=-. The Virasoro nets on S 1 with central charge less than one are in bijective correspondence with the pairs of A-D2n-E6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to ... |

9 |
Locality and modular invariance
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(Show Context)
Citation Context ...iant arising from a decomposition of a two-dimensional net and its µ-index. Müger has then solved the problem affirmatively in [41]. We recall some notions and results necessary for our work here. In =-=[47, 48, 49]-=-, Rehren studied 2-dimensional local conformal quantum field theory B(O) which irreducible extends a given pair of chiral theories A = AL ⊗AR. That is, the mathematical structure studied there is an i... |

8 | Local nature of coset models
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(Show Context)
Citation Context ...her important aspects of this classification, we mention here the occurrence of nets that are not realized as coset models, in contrast to a long standing expectation. (See Remarks after Theorem 7 of =-=[34]-=- on tihs point. Also, Carpi and Xu recently made a progress on classification for the case c = 1 in [10], [54], respectively.) The aim of this paper is to pursue point (b). We shall obtain a complete ... |