Boundary algebras and Kac modules for logarithmic minimal models

Abstract

Virasoro Kac modules were initially introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley-Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl-Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley-Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley-Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley-Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin-Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.

71 pages. v3: version published in Nucl. Phys. B

References in corpus (21)

• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• From Percolation to Logarithmic Conformal Field Theory
• Virasoro representations and fusion for general augmented minimal models
• From boundary to bulk in logarithmic CFT
• The two-boundary Temperley-Lieb algebra
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Bosonic Ghosts at $c=2$ as a Logarithmic CFT
• Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)
• Radford, Drinfeld, and Cardy boundary states in (1,p) logarithmic conformal field models
• On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories
• W-Extended Logarithmic Minimal Models
• The Verlinde formula in logarithmic CFT
• On the Percolation BCFT and the Crossing Probability of Watts
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• A note on the logarithmic (p,p') fusion
• Kac boundary conditions of the logarithmic minimal models

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• Restriction and induction of indecomposable modules over the Temperley-Lieb algebras
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• On the computation of fusion over the affine Temperley-Lieb algebra
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• Modularity of Bershadsky-Polyakov minimal models
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• Modular Valenced Temperley-Lieb Algebras
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• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Tensor Categories arising from the Virasoro Algebra
• Generators, projectors, and the Jones-Wenzl algebra
• Standard modules, radicals, and the valenced Temperley-Lieb algebra
• Logarithmic Minimal Models with Robin Boundary Conditions
• Free Field Realisations of Staggered Modules in 2D Logarithmic CFTs
• Fused Hecke algebra and one-boundary algebras
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• Staggered and affine Kac modules over $A_1^{(1)}$
• Spin chains as modules over the affine Temperley-Lieb algebra