Logarithmic Conformal Field Theory: Beyond an Introduction

Abstract

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.

Invited review by J Phys A for a special issue on LCFT; v2 updated references; v3 fixed a few minor typos

References in corpus (124)

• Vertex Algebras and Algebraic Curves
• Logarithmic Operators in Conformal Field Theory
• Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum
• Mock Theta Functions
• Critical Percolation in Finite Geometries
• Strings in AdS_3 and the SL(2,R) WZW Model. Part 2: Euclidean Black Hole
• The abelian sandpile and related models
• Bits and Pieces in Logarithmic Conformal Field Theory
• Strings in AdS_3 and the SL(2,R) WZW Model. Part 3: Correlation Functions
• 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
• 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
• Logarithmic extensions of minimal models: characters and modular transformations
• 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
• Associative-algebraic approach to logarithmic conformal field theories
• The GL(1|1) WZW model: From Supergeometry to Logarithmic CFT
• A crossing probability for critical percolation in two dimensions
• $PSL(n|n)$ Sigma Model as a Conformal Field Theory
• Fusion rules and logarithmic representations of a WZW model at fractional level
• Nonsemisimple Fusion Algebras and the Verlinde Formula
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Equivalence between Chain Categories of Representations of Affine sl(2) and N=2 Superconformal Algebras
• Integrable highest weight modules over affine superalgebras and Appell's function
• Curiosities at c=-2
• Conformal Algebras of 2D Disordered Systems
• From Percolation to Logarithmic Conformal Field Theory
• Modular Data and Verlinde Formulae for Fractional Level WZW Models II
• S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial
• Virasoro representations and fusion for general augmented minimal models
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• Coset Constructions of Logarithmic (1,p)-Models
• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Quasi-rational fusion products
• The su(2)_{-1/2} WZW model and the beta-gamma system
• Conformal Field Theory at central charge c=0 and Two-Dimensional Critical Systems with Quenched Disorder
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• On the SU(2|1) WZW model and its statistical mechanics applications
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• Lattice fusion rules and logarithmic operator product expansions
• On Staggered Indecomposable Virasoro Modules
• Height variables in the Abelian sandpile model: scaling fields and correlations
• From boundary to bulk in logarithmic CFT
• The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity
• Excursion decompositions for $\SLE$ and Watts' crossing formula
• sl^(2)_{-1/2}: A Case Study
• On the structure of Verma module over Virasoro and Neveu-Schwarz algebras
• Fusion rules and boundary conditions in the c=0 triplet model
• c-Theorem for Disordered Systems
• Higher-Level Appell Functions, Modular Transformations, and Characters
• Logarithmic Correlations in Quenched Random Magnets and Polymers
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Non-compact String Backgrounds and Non-rational CFT
• Fusion Algebras of Logarithmic Minimal Models
• Conformal field theory of the integer quantum Hall plateau transition
• Fusion in Fractional Level sl^(2)-Theories with k=-1/2
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Conformal Sigma-Models on Supercoset Targets
• sl(2)_{-1/2} and the Triplet Model
• Radford, Drinfeld, and Cardy boundary states in (1,p) logarithmic conformal field models
• A modular invariant bulk theory for the c=0 triplet model
• Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)
• Operator algebra of the SL(2) conformal field theories
• Fusion algebra of critical percolation
• Branes in the GL(1|1) WZNW-Model
• Logarithmic lift of the su(2)_{-1/2} model
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Fusion of twisted representations
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Boundary Spectra in Superspace Sigma-Models
• The puzzle of bulk conformal field theories at central charge c=0
• Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• Percolation Crossing Formulas and Conformal Field Theory
• The GL(1|1)-symplectic fermion correspondence
• On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories
• Fusion in conformal field theory as the tensor product of the symmetry algebra
• $A_1^{(1)}$ Admissible Representations -- Fusion Transformations and Local Correlators
• On the Percolation BCFT and the Crossing Probability of Watts
• Conformal invariance and quantum integrability of sigma models on symmetric superspaces
• Takiff superalgebras and Conformal Field Theory
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• On the triplet vertex algebra W(p)
• Fusion algebra at a rational level and cohomology of nilpotent subalgebras of sl_2
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• How Complete is the Classification of W-Symmetries ?
• Logarithmic conformal field theory with boundary
• W-Extended Fusion Algebra of Critical Percolation
• Boundary Correlators in Supergroup WZNW Models
• 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
• Fusion rules of chiral algebras
• Representation theory of the affine Lie superalgebra sl(2|1) at fractional level
• Classification of irreducible modules of W_3 algebra with c = -2
• On the extended W-algebra of type sl_2 at positive rational level
• Characters of admissible representations of the affine superalgebra sl(2|1)
• Cohomological Reduction of Sigma Models
• SLE local martingales in logarithmic representations
• Supergroup - extended super Liouville correspondence
• Logarithmic bulk and boundary conformal field theory and the full centre construction
• Embedding Diagrams of N=2 Verma Modules and Relaxed ^sl(2) Verma Modules
• The Extended Algebra of the Minimal Models
• From world-sheet supersymmetry to super target spaces
• The Extended Algebra of the SU(2) Wess-Zumino-Witten Models
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• Branes in the OSP(1|2) WZNW model
• W-Algebras Extending Affine gl(1|1)
• On W-algebras associated to (2,p) minimal models and their representations
• Kazhdan--Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT
• An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}
• Non-Chiral Logarithmic Couplings for the Virasoro Algebra
• A note on the logarithmic (p,p') fusion
• Fusion rules for admissible representations of affine algebras: the case of $A_2^{(1)}$
• New Modular Hopf Algebras related to rational $k$ $\widehat{sl(2)}$
• On modular properties of the AdS3 CFT
• On duality and extended chiral symmetry in the SL(2,R) WZW model
• A braided monoidal category for free super-bosons
• Vertex operator algebras associated to admissible representations of ${\widehat{sl}}_2$
• osp(1|2) Conformal Field Theory
• Integrable highest weight modules over affine superalgebras and number theory

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• Conformal symmetry in quasi-free Markovian open quantum systems
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• Duality structures for module categories of vertex operator algebras and the Feigin Fuchs boson
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• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
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