Logarithmic minimal models

Pearce, Paul A.; Rasmussen, Jørgen; Zuber, Jean-Bernard

doi:10.1088/1742-5468/2006/11/p11017

Cited by 177 publications

(580 citation statements)

References 78 publications

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“…These transfer matrices are not actually matrices but rather diagrammatic objects (tangles [38,39]) defined in the corresponding planar TL algebra [2,3]. Matrix representations are obtained by acting with these tangles on suitable vector spaces of link states [1,35]. Our results mirror the results obtained in [16] for the corresponding rational RSOS models [11,12], but here they are established directly in the planar algebra.…”

Section: Lm(psupporting

confidence: 69%

“…Since the logarithmic minimal models are su(2) theories, we work throughout with T -and Y -systems of the same form as in [16]. Remarkably, we find that these functional equations hold for arbitrary coprime integers p, p ′ and that the underlying structures are related to the Dynkin diagrams of the affine Lie algebras A (1) p ′ −1 . Indeed, the determinantal structure of the polynomial functional equations of degree p ′ is the same [40] as for the Cyclic Solid-on-Solid (CSOS) models [41][42][43].…”

Section: Lm(pmentioning

confidence: 99%

“…Elementary face operators The dense loop model with crossing parameter λ can be described by the planar TL algebra [1][2][3] generated by the elementary face operators u := s 1 (−u) + s 0 (u) (2.1) This decomposition is in terms of the two possible internal connectivities with the coefficients indicating the corresponding (local Boltzmann) weights. The small quarter arc in the lower-left corner is a marker indicating the orientation of the diagrams in the decomposition.…”

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

“…Because only vacuum boundary conditions are considered, the two relations in (2.23) are simplified versions of the usual BYBE, see for example [1]. In fact, the BYBEs (2.23) are equivalent to the commutation relation (2.20).…”

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

“…In [1], the non-fused lattice loop model LM(p, p ′ ) 1×1 and the corresponding CFT LM(P, P ′ ; 1) were studied and both referred to as the logarithmic minimal model LM(p, p ′ ). To emphasise the strong relation between the lattice model and the corresponding CFT, here we write LM(p, p ′ ) 1×1 ≡ LM(P, P ′ ; 1), P = p, P ′ = p ′ .…”

Section: Introductionmentioning

confidence: 99%

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Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N ∈ N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL N (β) with loop fugacity β = 2 cos λ, λ ∈ R. Similarly on a cylinder, the single-row transfer tangle T (u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models LM(p, p ′ ) comprise a subfamily of the TL loop models for which the crossing parameter λ = (p ′ − p)π/p ′ is a rational multiple of π parameterised by coprime integers 1 ≤ p < p ′ . For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1, 2), with β = 0, D(u) and T (u) satisfy inversion identities that have led to the exact determination of the eigenvalues in any representation and for arbitrary finite system size N . The generalisation for p ′ > 2 takes the form of functional relations for D(u) and T (u) of polynomial degree p ′ . These derive from fusion hierarchies of commuting transfer tangles D m,n (u) and T m,n (u) where D(u) = D 1,1 (u) and T (u) = T 1,1 (u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl-Jones projectors P k on k = m or k = n nodes. Some projectors P k are singular for k ≥ p ′ , but we argue that D m,n (u) and T m,n (u) are nonsingular for every m, n ∈ N in certain cabled link state representations. For generic λ, we derive the fusion hierarchies and the associated T -and Y -systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p ′ translates into functional relations of polynomial degree p ′ for D m,1 (u) and T m,1 (u). We also derive the closure of the Y -systems for the logarithmic theories. The T -and Y -systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

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Section: Lm(psupporting

confidence: 69%

Section: Lm(pmentioning

confidence: 99%

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

Section: Temperley-lieb Loop Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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Some aspects of c = ‐2 theory

Rajabpour

Rouhani

Saberi

2007

Fortschritte der Physik

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We investigate some aspects of the c = ‐2 logarithmic conformal field theory (LCFT). At first, we calculate some correlators of logarithmic conformal fields which have third order singular OPE with the energy‐momentum tensor. Then, we argue about the fields in the c = ‐2 model which are associated with this kind of more general logarithmic primary fields. We go on to find fermionic representations for all the fields in the extended Kac table, in particular the untwisted sector. Moreover, we calculate the various OPEs of the fields, especially for the logarithmic energy‐momentum tensor and by using these OPEs we find the exact finite transformation of this field. We briefly discuss about the important role of the zero modes in the c = ‐2 model. Finally we consider the perturbation of this theory and its relationship with integrable models, and generalization of Zamalodchikov's c‐theorem.

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NGK and HLZ: Fusion for Physicists and Mathematicians

Kanade

Ridout

2019

Springer INdAM Series

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A. In this expository note, we compare the fusion product of conformal field theory, as defined by Gaberdiel and used in the Nahm-Gaberdiel-Kausch (NGK) algorithm, with the P (w )-tensor product of vertex operator algebra modules, as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality of the two "coproducts" derived by NGK is essentially dual to the P (w )-compatibility condition of HLZ and how the algorithm of NGK for computing fusion products may be adapted to the setting of HLZ. We provide explicit calculations and instructive examples to illustrate both approaches. This document does not provide precise descriptions of all statements, it is intended more as a gentle starting point for the appreciation of the depth of the theory on both sides. C 1. Before we begin 1 2. Some notation and conventions 4 3. Fusion: The physicists' approach 5 4. The mathematician's approach I: A universal definition 9 5. The mathematician's approach II: A model for tensor products 13 6. The Huang-Lepowsky-Zhang approach: Double duals 17 7. An explicit example of a fusion calculation: Virasoro at c = −2 22 8. The fusion algorithm of Nahm-Gaberdiel-Kausch 25 9. A (very brief) summary of other approaches to fusion 29References 301.2. How? This note is organised as follows.In Section 2, we first provide a review of the (sometimes wildly different) notation and terminology used by mathematicians and physicists, so as to make it easier for both audiences to follow the rest of the paper. We also take the opportunity to fix some of our own choices in this regard. Our compromises will no doubt outrage many readers, but we take solace in the fact that it is really not possible to please everyone in this respect. We begin our exposition by first presenting Gaberdiel's original approach to defining fusion products in Section 3.Here, we will explain how the fusion product of two modules is morally constructed from their vector space tensor product by imposing a certain equality of two coproducts. These coproducts are derived from locality, so the definition appears natural and uncontroversial. However, we point out concretely that this equality of coproducts does not make sense in general because it involves infinite coefficients when expanded in the usual fashion.We then move on to the HLZ approach. Just like the case of tensor products of modules over commutative rings, they define fusion via a universal property which we cover in Section 4. Here, we introduce and explain the concept of intertwining maps. These maps are a rigorous version of the physicists' rather general notion of a field and are central to the universal property. Of course, this definition of fusion is quite abstract and it is not obvious how to go about actually computing it. One therefore needs a concrete model of this universal gadget to realise the fusion product concretely. We begin to explain this in Section 5, working with a model built out of the vector space tensor product of modules, and we explain its shortcomings in terms of convergence issue...

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Logarithmic minimal models

Cited by 177 publications

References 78 publications

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

Some aspects of c = ‐2 theory

NGK and HLZ: Fusion for Physicists and Mathematicians

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