## 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 ...onformal Quantum Field Theory is particularly interesting in two spacetime dimensions and has indeed been intensively studied in the last two decades with important motivations from Physics (see e.g. =-=[11]-=-) and Mathematics (see e.g. [14]). Basically the richness of structure is due to the fact that conformal group (with respect to the Minkowskian metric) is infinite dimensional in 1 + 1 dimensions. Alr... |

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Citation Context ...-tensor category and then apply it to the tensor categories related to the Virasoro algebra. We consider a strict C ∗ -tensor category T (with conjugates, subobjects, and direct sums) in the sense of =-=[13, 39]-=- and we assume that we have only finitely many equivalence classes of irreducible objects in T and that each object has a decomposition into a finite direct sum of irreducible objects. Such a tensor c... |

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Citation Context ... particularly interesting in two spacetime dimensions and has indeed been intensively studied in the last two decades with important motivations from Physics (see e.g. [11]) and Mathematics (see e.g. =-=[14]-=-). Basically the richness of structure is due to the fact that conformal group (with respect to the Minkowskian metric) is infinite dimensional in 1 + 1 dimensions. Already at the early stage of inves... |

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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 ... I, the map π is trivial, so the dual canonical endomorphism has the same form of the Longo-Rehren endomorphism [38]. Thus the classification is reduced to classification 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-coh... |

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(Show Context)
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 ... 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... |

64 | Multi-interval subfactors and modularity of representations in conformal field theory
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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 ...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... |

50 | Y.: Chiral structure of modular invariants for subfactors
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(Show Context)
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... |

48 | V.: On a q-analogue of the McKay correspondence and the ADE classification of ŝl2 conformal field theories - Kirillov, Ostrik - 2002 |

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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 � � �... |

42 | Classification of subfactors and of their endomorphisms - Popa - 1995 |

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Quantized symmetry, differential geometry of finite graphs and classification of subfactors’, (Notes by Y. Kawahigashi), University of Tokyo Seminar Notes 45
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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... |

38 | The structure of sectors associated with Longo-Rehren inclusions - Izumi - 2001 |

36 | D.E.: Modular invariants from subfactors: Type I coupling matrices and intermediate subfactors
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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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Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansion, DESY preprint
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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 ... |

27 | 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... |

25 | Conformal subnets and intermediate subfactors - Longo - 2003 |

25 |
Subalgebras of infinite C ∗ -algebras with finite Watatani indices II: Cuntz-Krieger algebras
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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... |

24 | Y.: Longo-Rehren subfactors arising from α-induction
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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... |

21 |
Algebraic and modular structures of von Neumann algebras of physics
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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... |

21 |
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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... |

21 | 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... |

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

18 |
On α-induction, chiral projectors and modular invariants for subfactors
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(Show Context)
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 |

15 |
H.: On the subfactor analogue of the second cohomology
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(Show Context)
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 |
of infinite C∗-algebras with finite Watatani indices
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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
- Rehren
- 2000
(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
- Kawahigashi, Longo
(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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- 2003
(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 ... |