Logarithmic superconformal minimal models

Pearce, Paul A.; Rasmussen, Jørgen; Tartaglia, Elena

doi:10.1088/1742-5468/2014/05/p05001

Cited by 22 publications

(68 citation statements)

References 103 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%

“…The (2, 2)-fused loop model and its conformal properties are investigated in [35]. acts from below as the identity on the generalised monoid X m,n a , while I n,m acts as the identity from above, that is…”

Section: Fused Face Operatorsmentioning

confidence: 99%

“…For example, the infinite family of higher-level logarithmic minimal models [34] are believed to correspond to m = n and µ = nλ. In the case m = n = 2, in particular, setting µ = 2λ indeed gives rise to logarithmic superconformal minimal models [35]. It is occasionally convenient to indicate explicitly the dependence on the crossing parameter λ by writing…”

Section: Fused Transfer Tangles For Generic λmentioning

confidence: 99%

“…For n > 1, it has been conjectured in [35] that the theories LM(P, P ′ ; n) are realised on the lattice by n × n fusions built from the elementary logarithmic minimal models LM(p, p ′ ), that is LM(p, p ′ ) n×n ≡ LM(P, P ′ ; n) (1.5)…”

Section: Introductionmentioning

confidence: 99%

“…In particular, for n = 2, the theories are logarithmic superconformal field theories [35] LM(p, p ′ ) 2×2 ≡ LM(P, P ′ ; 2), P = |2p − p ′ |, P ′ = p ′ (1.6) also denoted by LSM(p, p ′ ). The first members of the superconformal series include superconformal dense polymers LSM(2, 3) and superconformal percolation LSM (3,4) which are lattice models that are expected to be of independent interest in statistical mechanics.…”

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: Fused Face Operatorsmentioning

confidence: 99%

Section: Fused Transfer Tangles For Generic λmentioning

confidence: 99%

Section: Introductionmentioning

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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The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

Grans-Samuelsson

Jacobsen

Saleur

2021

J. High Energ. Phys.

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We investigate the action of discretized Virasoro generators, built out of generators of the lattice Temperley-Lieb algebra (“Koo-Saleur generators” [1]), in the critical XXZ quantum spin chain. We explore the structure of the continuum-limit Virasoro modules at generic central charge for the XXZ vertex model, paralleling [2] for the loop model. We find again indecomposable modules, but this time not logarithmic ones. The limit of the Temperley-Lieb modules Wj,1 for j ≠ 0 contains pairs of “conjugate states” with conformal weights (hr,s, hr,−s) and (hr,−s, hr,s) that give rise to dual structures: Verma or co-Verma modules. The limit of $$ {W}_{0,{\mathfrak{q}}^{\pm 2}} $$ W 0 , q ± 2 contains diagonal fields (hr,1, hr,1) and gives rise to either only Verma or only co-Verma modules, depending on the sign of the exponent in $$ {\mathfrak{q}}^{\pm 2} $$ q ± 2 . In order to obtain matrix elements of Koo-Saleur generators at large system size N we use Bethe ansatz and Quantum Inverse Scattering methods, computing the form factors for relevant combinations of three neighbouring spin operators. Relations between form factors ensure that the above duality exists already at the lattice level. We also study in which sense Koo-Saleur generators converge to Virasoro generators. We consider convergence in the weak sense, investigating whether the commutator of limits is the same as the limit of the commutator? We find that it coincides only up to the central term. As a side result we compute the ground-state expectation value of two neighbouring Temperley-Lieb generators in the XXZ spin chain.

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Critical $$ \mathcal{N} $$ = (1, 1) general massive supergravity

Deger

Moutsopoulos

Rosseel

2018

J. High Energ. Phys.

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In this paper we study the supermultiplet structure of N = (1, 1) General Massive Supergravity at non-critical and critical points of its parameter space. To do this, we first linearize the theory around its maximally supersymmetric AdS 3 vacuum and obtain the full linearized Lagrangian including fermionic terms. At generic values, linearized modes can be organized as two massless and 2 massive multiplets where supersymmetry relates them in the standard way. At critical points logarithmic modes appear and we find that in three of such points some of the supersymmetry transformations are noninvertible in logarithmic multiplets. However, in the fourth critical point, there is a massive logarithmic multiplet with invertible supersymmetry transformations.

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Logarithmic superconformal minimal models

Cited by 22 publications

References 103 publications

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

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

The action of the Virasoro algebra in quantum spin chains. Part I. The non-rational case

Critical $$ \mathcal{N} $$ = (1, 1) general massive supergravity

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