Associative-algebraic approach to logarithmic conformal field theories

Abstract

We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non--semi-simple associative algebras appearing in their lattice regularizations (as discussed in a companion paper). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl($n|n$) and gl($n+1|n$), respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge $c=-2$ and $c=0$ respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with $c=0$. Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau-Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields.

References in corpus (33)

• Logarithmic Operators in Conformal Field Theory
• Bits and Pieces in Logarithmic Conformal Field Theory
• A Rational Logarithmic Conformal Field Theory
• Symplectic Fermions
• Indecomposable Fusion Products
• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• Exact exponents for the spin quantum Hall transition
• A Local Logarithmic Conformal Field Theory
• An algebraic approach to logarithmic conformal field theory
• Logarithmic Minimal Models
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• gl(N|N) Super-Current Algebras for Disordered Dirac Fermions in Two Dimensions
• On Modular Invariant Partition Functions of Conformal Field Theories with Logarithmic Operators
• The GL(1|1) WZW model: From Supergeometry to Logarithmic CFT
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Curiosities at c=-2
• On Fusion Rules in Logarithmic Conformal Field Theories
• Conformal Algebras of 2D Disordered Systems
• Solvable Critical Dense Polymers
• Representations of the Virasoro algebra from lattice models
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• Logarithmic Conformal Field Theories via Logarithmic Deformations
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• c-Theorem for Disordered Systems
• Disordered Dirac Fermions: the Marriage of Three Different Approaches
• Multi-Colour Braid-Monoid Algebras
• Fusion in conformal field theory as the tensor product of the symmetry algebra
• The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models
• The O(n) model on the annulus
• Mesoscopic conductance and its fluctuations at non-zero Hall angle
• Critical RSOS and Minimal Models I: Paths, Fermionic Algebras and Virasoro Modules
• Critical RSOS and Minimal Models II: Building Representations of the Virasoro Algebra and Fields

Cited by in corpus (110)

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• From Percolation to Logarithmic Conformal Field Theory
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• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Lattice fusion rules and logarithmic operator product expansions
• Holographic applications of logarithmic conformal field theories
• On Staggered Indecomposable Virasoro Modules
• Fusion rules and boundary conditions in the c=0 triplet model
• The ABC (in any D) of Logarithmic CFT
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula
• sl(2)_{-1/2} and the Triplet Model
• Conformal fields and operator product expansion in critical quantum spin chains
• Branes in the GL(1|1) WZNW-Model
• Fusion algebra of critical percolation
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Conformal two-boundary loop model on the annulus
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Boundary Spectra in Superspace Sigma-Models
• The puzzle of bulk conformal field theories at central charge c=0
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• Boundary algebras and Kac modules for logarithmic minimal models
• Boundary criticality and multifractality at the 2D spin quantum Hall transition
• The Sigma Model on Complex Projective Superspaces
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• W-Extended Logarithmic Minimal Models
• Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
• The Verlinde formula in logarithmic CFT
• On the Percolation BCFT and the Crossing Probability of Watts
• Takiff superalgebras and Conformal Field Theory
• The Nichols algebra of screenings
• Solvable Critical Dense Polymers on the Cylinder
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Bimodule structure in the periodic gl(1|1) spin chain
• A lattice approach to the conformal $\OSp(2S+2|2S)$ supercoset sigma model. Part I: Algebraic structures in the spin chain. The Brauer algebra
• Pair correlations in sandpile model: a check of logarithmic conformal field theory
• Conformal boundary state for the rectangular geometry
• W-Extended Fusion Algebra of Critical Percolation
• A fusion for the periodic Temperley-Lieb algebra and its continuum limit
• 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
• Emergence of conformal symmetry in quantum spin chains: anti-periodic boundary conditions and supersymmetry
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• Logarithmic operators and logarithmic conformal field theories
• Local logarithmic correlators as limits of Coulomb gas integrals
• Standard Modules, Induction and the Temperley-Lieb Algebra
• Logarithmic conformal invariance in the Abelian sandpile model
• Rectangular amplitudes, conformal blocks, and applications to loop models
• Pure scaling operators at the integer quantum Hall plateau transition
• A lattice approach to the conformal $\OSp(2S+2|2S)$ supercoset sigma model. Part II: The boundary spectrum
• Non-scalar operators for the Potts model in arbitrary dimension
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• SLE local martingales in logarithmic representations
• Wind on the boundary for the Abelian sandpile model
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• Logarithmic two-point correlators in the Abelian sandpile model
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• 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
• Restriction and induction of indecomposable modules over the Temperley-Lieb algebras
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Polynomial fusion rings of W-extended logarithmic minimal models
• On the computation of fusion over the affine Temperley-Lieb algebra
• Dimer representations of the Temperley-Lieb algebra
• Accurate expansions of internal energy and specific heat of critical two-dimensional Ising model with free boundaries
• Critical exponents for the homology of Fortuin-Kasteleyn clusters on a torus
• The Generic Critical Behaviour for 2D Polymer Collapse
• Emergence of Kac-Moody Symmetry in Critical Quantum Spin Chains
• Toward logarithmic extensions of ^sl(2)_k conformal field models
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Conformal partition functions of critical percolation from $D_3$ Thermodynamic Bethe Ansatz equations
• Bulk from boundary in finite CFT by means of pivotal module categories
• A note on the identity module in $c=0$ CFTs
• Three-leg correlations in the two component spanning tree on the upper half-plane
• Fusion rules for the Temperley-Lieb algebra and its dilute generalisation
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality
• Three- and four-point connectivities of two-dimensional critical $Q-$ Potts random clusters on the torus
• Quantum Groups as Global Symmetries
• Efficient algortihms for the two dimensional Ising model with a surface field
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• Edge states and conformal boundary conditions in super spin chains and super sigma models
• Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Two-point boundary correlation functions of dense loop models
• Logarithmic Correlations in Quantum Hall Plateau Transitions
• Infinitely extended Kac table of solvable critical dense polymers
• Tensor Categories arising from the Virasoro Algebra
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• Realizing the braided Temperley-Lieb-Jones C*-tensor categories as Hilbert C*-modules
• Staggered modules of $N=2$ superconformal minimal models
• PT-invariance and representations of the Temperley-Lieb algebra on the unit circle
• A non-semisimple non-invertible symmetry
• On correlation functions in models related to the Temperley-Lieb algebra
• Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors
• Fusion and monodromy in the Temperley-Lieb category
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra
• The idempotents of the TL_n-modules \otimes^nC^2 in terms of elements of U_qsl_2
• Junction type representations of the Temperley-Lieb algebra and associated symmetries