Lattice fusion rules and logarithmic operator product expansions

Abstract

The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the "lattice" fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator-product expansions of quantum fields.

54pp, many comments added

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Cited by in corpus (45)

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• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• Conformal fields and operator product expansion in critical quantum spin chains
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Emergence of conformal symmetry in quantum spin chains: anti-periodic boundary conditions and supersymmetry
• Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
• Logarithmic ^sl(2) CFT models from Nichols algebras. 1
• Logarithmic operators and logarithmic conformal field theories
• Local logarithmic correlators as limits of Coulomb gas integrals
• Standard Modules, Induction and the Temperley-Lieb Algebra
• Lattice W-algebras and logarithmic CFTs
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Jordan cells of periodic loop models
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• The principal indecomposable modules of the dilute Temperley-Lieb algebra
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• 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 correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• On the reality of spectra of ${\mathbf{U}}_{\mathbf{q}}(\mathbf{s}{\mathbf{l}}_2)$-invariant XXZ Hamiltonians
• On the computation of fusion over the affine Temperley-Lieb algebra
• Dimer representations of the Temperley-Lieb algebra
• Emergence of Kac-Moody Symmetry in Critical Quantum Spin Chains
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• Fusion rules for the Temperley-Lieb algebra and its dilute generalisation
• Representations of ${\bar{U}}_{q}s\ell (2|1)$ at even roots of unity
• Quantum fields for unitary representations of Thompson's groups F and T
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Integrable Floquet systems related to logarithmic conformal field theory
• Infinitely extended Kac table of solvable critical dense polymers
• On the Representation theory of the Infinite Temperley-Lieb algebra
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• Fusion and monodromy in the Temperley-Lieb category
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• The idempotents of the TL_n-modules \otimes^nC^2 in terms of elements of U_qsl_2