Classification of Modular Invariant Partition Functions in Two Dimensions

Kato, Atsushi

doi:10.1142/s0217732387000732

Cited by 136 publications

(94 citation statements)

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“…CappelliItzykson-Zuber [11] and Kato [36] have made an A-D-E classification of the modular invariant matrices for SU(2) k . That is, for the unitary representation of the group SL(2, Z) arising from SU(2) k as in [14, Subsec.…”

Section: Virasoro Nets and Classification Of The Modular Invariantsmentioning

confidence: 99%

Classification of local conformal nets. Case c < 1

2004

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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.

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Section: Virasoro Nets and Classification Of The Modular Invariantsmentioning

confidence: 99%

Classification of local conformal nets. Case c < 1

2004

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“…The modular invariant partition functions for these models are completely classified [28,29,30] (see also [17]); in this subsection we shall deal with the A series which is sometimes denoted as (A k+1 , A k+2 ).…”

Section: Minimal Models a Seriesmentioning

confidence: 99%

Organizing boundary RG flows

Fredenhagen

2003

Nuclear Physics B

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We show how a large class of boundary RG flows in two-dimensional conformal field theories can be summarized in a single rule. This rule is a generalization of the 'absorption of the boundary spin'-principle of Affleck and Ludwig and applies to all theories which have a description as a coset model. We give a formulation for coset models with arbitrary modular invariant partition function and present evidence for the conjectured rule. The second half of the article contains an illustrated section of examples where the rule is applied to unitary minimal models of the A-and D-series, in particular the 3-state Potts model, and to parafermion theories. We demonstrate how the rule can be used to compute brane charge groups in the example of N = 2 minimal models.

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“…When we continue the expressions from bosonic Liouville theory to c = 1, one might wonder whether it is also possible to continue them to c < 1. This is in fact straightforward for rational central charges of the form c = 1 − 6 (p−q) 2 pq . The corresponding conformal field theories would not be unitary and would have a continuous spectrum.…”

Section: Jhep09(2007)098mentioning

confidence: 97%

“…For unitary theories, the classification has only been achieved for central charge c < 1 resulting in the well known ADE series of minimal models [1,2]. One might hope that the problem still remains tractable for the limiting value c = 1.…”

Section: Introductionmentioning

confidence: 99%

A common limit of super Liouville theory and minimal models

Fredenhagen

Wellig

2007

J. High Energy Phys.

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Abstract:We show that N = 1 supersymmetric Liouville theory can be continued to central charge c = 3/2, and that the limiting non-rational superconformal field theory can also be obtained as a limit of supersymmetric minimal models. This generalises a result known for the non-supersymmetric case. We present explicit expressions for the three-point functions of bulk fields, as well as a set of superconformal boundary states. The main technical ingredient to take the limit of minimal models consists in determining analytic expressions for the structure constants. In the appendix we show in detail how the structure constants of supersymmetric and Virasoro minimal models can be rewritten in terms of Barnes' double gamma functions.

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Classification of Modular Invariant Partition Functions in Two Dimensions

Abstract: The partition functions of c < 1 minimal unitary conformal theories are completely classified. It is proven that the list given by Cappelli, Itzykson and Zuber exhausts the set of physically acceptable solutions.

Cited by 136 publications

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Classification of local conformal nets. Case c < 1

Classification of local conformal nets. Case c < 1

Organizing boundary RG flows

A common limit of super Liouville theory and minimal models

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