# On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

Abstract : The relationship between bulk and boundary properties is one of the founding features of (Rational) Conformal Field Theory. Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of braid translation, which is a natural way to close an open spin chain by adding an interaction between the first and last spins using braiding to bring them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain - the ordinary Temperley-Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic TL algebra. We show that for the Restricted Solid-On-Solid models, which are known to be described by minimal unitary CFTs in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine TL algebra. We then turn to the same kind of analysis for two models whose continuum limits are Logarithmic CFTs - the alternating $gl$(1|1) and $sl$(2|1) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant TL modules does not give rise to the modules known to be present in the periodic chains. In the $gl$(1|1) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit.
Type de document :
Pré-publication, Document de travail
t17/088 62 pp. 2017
Domaine :

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https://hal-cea.archives-ouvertes.fr/cea-01540390
Contributeur : Emmanuelle De Laborderie <>
Soumis le : vendredi 16 juin 2017 - 12:19:50
Dernière modification le : jeudi 19 avril 2018 - 17:46:01
Document(s) archivé(s) le : mercredi 13 décembre 2017 - 13:23:20

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1705.07769.pdf
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• HAL Id : cea-01540390, version 1
• ARXIV : 1705.07769

### Citation

Jonathan Belletête, Azat M. Gainutdinov, Jesper L. Jacobsen, Hubert Saleur, Romain Vasseur. On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories. t17/088 62 pp. 2017. 〈cea-01540390〉

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