Indecomposability parameters in chiral Logarithmic Conformal Field Theory

Abstract

Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the `b-number' introduced initially by Gurarie for the c=0 case. The determination of these indecomposability parameters has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl(n+m|m) and osp(n+2m|2m)-invariant superspin chains and to nonlinear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarie's, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM(1,p) and LM(p,p+1).

32 pages, 2 figures, Published Version

References in corpus (33)

• Logarithmic Operators in Conformal Field Theory
• Conformal Field Theory of AdS Background with Ramond-Ramond Flux
• Symplectic Fermions
• Indecomposable Fusion Products
• Exact exponents for the spin quantum Hall transition
• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• Logarithmic Minimal Models
• Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions
• The Blob Algebra and the Periodic Temperley-Lieb Algebra
• Associative-algebraic approach to logarithmic conformal field theories
• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Curiosities at c=-2
• Conformal Algebras of 2D Disordered Systems
• From Percolation to Logarithmic Conformal Field Theory
• Scaling fields in the two-dimensional abelian sandpile model
• Representations of the Virasoro algebra from lattice models
• Fermionic field theory for trees and forests
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• Conformal Field Theory at central charge c=0 and Two-Dimensional Critical Systems with Quenched Disorder
• Conformal boundary loop models
• On Staggered Indecomposable Virasoro Modules
• Representation theory of sl(2|1)
• c-Theorem for Disordered Systems
• Logarithmic Correlations in Quenched Random Magnets and Polymers
• Fusion Algebras of Logarithmic Minimal Models
• Conformal field theory of the integer quantum Hall plateau transition
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Conformal two-boundary loop model on the annulus
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• W-Extended Fusion Algebra of Critical Percolation
• Fused Potts Models
• Critical behaviour of the dilute O(n), Izergin-Korepin and dilute $A_L$ face models: Bulk properties

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• Logarithmic Conformal Field Theory: Beyond an Introduction
• Logarithmic Conformal Field Theory: a Lattice Approach
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• Logarithmic observables in critical percolation
• Deligne Categories in Lattice Models and Quantum Field Theory, or Making Sense of $O(N)$ Symmetry with Non-integer $N$
• Continuum limit and symmetries of the periodic gl(1|1) spin chain
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• The puzzle of bulk conformal field theories at central charge c=0
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0
• 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
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Local logarithmic correlators as limits of Coulomb gas integrals
• Logarithmic conformal invariance in the Abelian sandpile model
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• A Proof of Selection Rules for Critical Dense Polymers
• Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity
• Critical properties of joint spin and Fortuin-Kasteleyn observables in the two-dimensional Potts model
• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
• 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
• Refined conformal spectra in the dimer model
• Observation of non-scalar and logarithmic correlations in 2D and 3D percolation
• $N$-cluster correlations in four- and five-dimensional percolation
• The action of the Virasoro algebra in quantum spin chains. I. The non-rational case
• Logarithmic operator intervals in the boundary theory of critical percolation
• Finite-size corrections for logarithmic representations in critical dense polymers
• Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
• A note on the identity module in $c=0$ CFTs
• Transfer matrix for spanning trees, webs and colored forests
• Quantum Groups as Global Symmetries
• Interactions as conformal intertwiners in 4D QFT
• From elongated spanning trees to vicious random walks
• Free Field Realisations of Staggered Modules in 2D Logarithmic CFTs
• Staggered modules of $N=2$ superconformal minimal models
• Logarithmic operators in $c=0$ bulk CFTs
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra