Logarithmic Minimal Models

Abstract

Working in the dense loop representation, we use the planar Temperley-Lieb algebra to build integrable lattice models called logarithmic minimal models LM(p,p'). Specifically, we construct Yang-Baxter integrable Temperley-Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley-Lieb algebra are inherently non-local and not (time-reversal) symmetric. In the continuum scaling limit, they yield logarithmic conformal field theories with central charges c=1-6(p-p')^2/pp' where p,p'=1,2,... are coprime. The first few members of the principal series LM(m,m+1) are critical dense polymers (m=1, c=-2), critical percolation (m=2, c=0) and logarithmic Ising model (m=3, c=1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights Delta_{r,s}=(((m+1)r-ms)^2-1)/4m(m+1), r,s=1,2,.... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.

40 pages, v3: typos corrected and comments added

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Cited by in corpus (136)

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• Fusion algebra of critical percolation
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• The puzzle of bulk conformal field theories at central charge c=0
• Boundary algebras and Kac modules for logarithmic minimal models
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• W-Extended Logarithmic Minimal Models
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• Geometric Exponents, SLE and Logarithmic Minimal Models
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• Jordan cells in logarithmic limits of conformal field theory
• Integrability and conformal data of the dimer model
• Discrete holomorphicity and integrability in loop models with open boundaries
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra
• Wind on the boundary for the Abelian sandpile model
• A Proof of Selection Rules for Critical Dense Polymers
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• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
• Logarithmic two-point correlators in the Abelian sandpile model
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• On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• The fully packed loop model as a non-rational $W_3$ conformal field theory
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• Off-Critical Logarithmic Minimal Models
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• Fusion and monodromy in the Temperley-Lieb category
• Fusion hierarchies, $T$-systems and $Y$-systems for the dilute $A_2^{(2)}$ loop models on a strip
• Integrable models from singly generated planar algebras
• PT-invariance and representations of the Temperley-Lieb algebra on the unit circle
• One-Dimensional Sums and Finitized Characters of $2\times 2$ Fused RSOS Models
• Contour Integral Representations for the Characters of Logarithmic CFTs