Bits and Pieces in Logarithmic Conformal Field Theory

Abstract

These are notes of my lectures held at the first School & Workshop on Logarithmic Conformal Field Theory and its Applications, September 2001 in Tehran, Iran. These notes cover only selected parts of the by now quite extensive knowledge on logarithmic conformal field theories. In particular, I discuss the proper generalization of null vectors towards the logarithmic case, and how these can be used to compute correlation functions. My other main topic is modular invariance, where I discuss the problem of the generalization of characters in the case of indecomposable representations, a proposal for a Verlinde formula for fusion rules and identities relating the partition functions of logarithmic conformal field theories to such of well known ordinary conformal field theories. These two main topics are complemented by some remarks on ghost systems, the Haldane-Rezayi fractional quantum Hall state, and the relation of these two to the logarithmic c=-2 theory.

91 pages, notes of lectures delivered at the first School and Workshop on Logarithmic Conformal Field Theory and its Applications, Tehran, September 2001. Amendments in Introduction

References in corpus (9)

• Fusion rules and logarithmic representations of a WZW model at fractional level
• Scaling fields in the two-dimensional abelian sandpile model
• Modular transformation and boundary states in logarithmic conformal field theory
• SU(2)_0 and OSp(2|2)_{-2} WZNW models : Two current algebras, one Logarithmic CFT
• World Sheet Logarithmic CFT in AdS Strings, Ghost-Matter Mixing and M-theory
• Prelogarithmic operators and Jordan blocks in SL(2)_k affine algebra
• A Free Field Representation of the Osp(2|2) current algebra at level k=-2, and Dirac Fermions in a random SU(2) gauge potential
• On the AdS/CFT Correspondence and Logarithmic Operator
• Impulse Action on D-particles in Robertson-Walker Space Times, Higher-Order Logarithmic Conformal Algebras and Cosmological Horizons

Cited by in corpus (187)

• Instability in cosmological topologically massive gravity at the chiral point
• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• Chiral Gravity, Log Gravity and Extremal CFT
• Logarithmic Minimal Models
• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• Logarithmic Conformal Field Theory: Beyond an Introduction
• The GL(1|1) WZW model: From Supergeometry to Logarithmic CFT
• Closed strings from decaying D-branes
• AdS_3/LCFT_2 - Correlators in New Massive Gravity
• Logarithmic Conformal Field Theory: a Lattice Approach
• Magic in the spectra of the XXZ quantum chain with boundaries at Delta=0 and Delta=-1/2
• Product CFTs, gravitational cloning, massive gravitons and the space of gravitational duals
• The WZNW model on PSU(1,1|2)
• Virasoro representations and fusion for general augmented minimal models
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• Graviton 1-loop partition function for 3-dimensional massive gravity
• Temperley-Lieb Stochastic Processes
• The su(2)_{-1/2} WZW model and the beta-gamma system
• AdS_3/LCFT_2 - Correlators in Cosmological Topologically Massive Gravity
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• Height variables in the Abelian sandpile model: scaling fields and correlations
• Holographic applications of logarithmic conformal field theories
• On Staggered Indecomposable Virasoro Modules
• A c=-2 boundary changing operator for the Abelian sandpile
• From boundary to bulk in logarithmic CFT
• Logarithmic torus amplitudes
• sl^(2)_{-1/2}: A Case Study
• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra, Part I
• Higher-Level Appell Functions, Modular Transformations, and Characters
• The ABC (in any D) of Logarithmic CFT
• Decoherence induced CPT violation and entangled neutral mesons
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic CFTs connected with simple Lie algebras
• Reparametrization modes, shadow operators, and quantum chaos in higher-dimensional CFTs
• The logarithmic triplet theory with boundary
• String Field Theory -- A Modern Introduction
• The N=1 triplet vertex operator superalgebras
• Ghosts of Critical Gravity
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• Radford, Drinfeld, and Cardy boundary states in (1,p) logarithmic conformal field models
• Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)
• Free fermion resolution of supergroup WZNW models
• Fusion algebra of critical percolation
• Branes in the GL(1|1) WZNW-Model
• Logarithmic lift of the su(2)_{-1/2} model
• Logarithmic scaling for height variables in the Abelian sandpile model
• Liouville Decoherence in a Model of Flavour Oscillations in the presence of Dark Energy
• A logarithmic generalization of tensor product theory for modules for a vertex operator algebra
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• The puzzle of bulk conformal field theories at central charge c=0
• Abelian Sandpile Model: a Conformal Field Theory Point of View
• Wavefunctions of Symmetry Protected Topological Phases from Conformal Field Theories
• W-Extended Logarithmic Minimal Models
• On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories
• Boundary States in c=-2 Logarithmic Conformal Field Theory
• Pre-logarithmic and logarithmic fields in a sandpile model
• Superspace conformal field theory
• Loop-level gluon OPEs in celestial holography
• Three-Dimensional Tricritical Gravity
• PP-waves and logarithmic conformal field theories
• Short-cut to new anomalies in gravity duals to logarithmic conformal field theories
• Brany Liouville Inflation
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• On $d{S}_4$ extremal surfaces and entanglement entropy in some ghost CFTs
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Logarithmic intertwining operators and vertex operators
• W-Extended Fusion Algebra of Critical Percolation
• Pair correlations in sandpile model: a check of logarithmic conformal field theory
• Unitary Truncations and Critical Gravity: a Toy Model
• Geometry of branes on supergroups
• Boundary Correlators in Supergroup WZNW Models
• Direct Evidence for Conformal Invariance of Avalanche Frontier in Sandpile Models
• The Neveu-Schwarz and Ramond Algebras of Logarithmic Superconformal Field Theory
• Counterterms, critical gravity and holography
• Logarithmic limits of minimal models
• The conformal current algebra on supergroups with applications to the spectrum and integrability
• Fermionic Expressions for the Characters of c(p,1) Logarithmic Conformal Field Theories
• The Jordan Structure of Two Dimensional Loop Models
• Notes on non-trivial and logarithmic CFTs with c=0
• Matrix Product State Description and Gaplessness of the Haldane-Rezayi State
• On logarithmic extensions of local scale-invariance
• No chiral truncation of quantum log gravity?
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Logarithmic Celestial Conformal Field Theory
• Jordan cells in logarithmic limits of conformal field theory
• New boundary conditions for the c=-2 ghost system
• Logarithmic Correlators in Non-relativistic Conformal Field Theory
• Geometric Exponents, SLE and Logarithmic Minimal Models
• $gl(2|2)$ Current Superalgebra and Non-unitary Conformal Field Theory
• Wind on the boundary for the Abelian sandpile model
• Proposal for a CFT interpretation of Watts' differential equation for percolation
• Affine Jordan cells, logarithmic correlators, and hamiltonian reduction
• Boundary conditions and defect lines in the Abelian sandpile model
• Ghost Systems Revisited: Modified Virasoro Generators and Logarithmic Conformal Field Theories
• Ghost-spin chains, entanglement and $bc$-ghost CFTs
• Logarithmic two-point correlators in the Abelian sandpile model
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• Black holes and thermodynamics - The first half century
• On Holographic Realization of Logarithmic GCA
• Polynomial fusion rings of W-extended logarithmic minimal models
• M-Theory Through the Looking Glass: Tachyon Condensation in the E_8 Heterotic String
• Causality from dynamical symmetry: an example from local scale-invariance
• Note on SLE and logarithmic CFT
• Gravity duals for logarithmic conformal field theories
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• Refined conformal spectra in the dimer model
• Boundary States and Symplectic Fermions
• Logarithmic Primary Fields in Conformal and Superconformal Field Theory
• A Non-relativistic Logarithmic Conformal Field Theory from a Holographic Point of View
• On the combinatorics of partition functions in AdS3/LCFT2
• Logarithmic Conformal Field Theories and Strings in Changing Backgrounds
• Conical spaces, modular invariance and $c_{p,1}$ holography
• An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}
• Nonlinear Dynamics of Parity-Even Tricritical Gravity in Three and Four Dimensions
• Logarithmic Exotic Conformal Galilean Algebras
• Logarithmic Conformal Null Vectors and SLE
• Aging Logarithmic Conformal Field Theory : a holographic view
• The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
• Beta-gamma systems and the deformations of the BRST operator
• Notes on Generalised Nullvectors in logarithmic CFT
• On logarithmic solutions to the conformal Ward identities
• Four-Point Functions in Logarithmic Conformal Field Theories
• On unitary subsectors of polycritical gravities
• Quasinormal modes around the BTZ black hole at the tricritical generalized massive gravity
• Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas
• What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT
• Towards the Construction of Local Logarithmic Conformal Field Theories
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• Sandpile models in the large
• The O(n) Model in the $n\to 0$ Limit (self-avoiding-walks) and Logarithmic Conformal Field Theory
• Primary fields and screening currents of gl(2|2) non-unitary conformal field theory
• Graph fusion algebras of WLM(p,p')
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Three-leg correlations in the two component spanning tree on the upper half-plane
• Operator Algebra in Logarithmic Conformal Field Theory
• Moduli space of logarithmic states in critical massive gravities
• On conformal Jordan cells of finite and infinite rank
• Exact partition functions for deformed $\mathcal{N}=2$ theories with $N_f=4$ flavours
• Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
• Kac boundary conditions of the logarithmic minimal models
• Differential operator realizations of superalgebras and free field representations of corresponding current algebras
• Bipartite fidelity of critical dense polymers
• The origin of multiplets of chiral fields in SU(2)_k WZNW at rational level
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• N-level ghost-spins and entanglement
• C-Graded Vertex Algebras and Conformal Flow
• Perturbed Logarithmic CFT and Integrable Models
• BRST quantization of a sixth-order derivative scalar field theory
• Aging Logarithmic Galilean Field Theories
• Free field realization of the $osp(2n|2n)$ current algebra
• On Extensions of Superconformal Algebras
• A differential U-module algebra for U=U_q sl(2) at an even root of unity
• Operator Product Expansion and Zero Mode Structure in Logarithmic CFT
• RG and logarithmic CFT multicritical properties of randomly diluted Ising models
• Logarithmic two-Point Correlation Functions from a z = 2 Lifshitz Model
• Remarks on BMS${}_3$ invariant field theories: correlation functions and nonunitary CFTs
• The exact description of NS5-branes in the Penrose limit
• Two-Point Functions and Logarithmic Boundary Operators in Boundary Logarithmic Conformal Field Theories
• Two-Point Functions and Boundary States in Boundary Logarithmic Conformal Field Theories
• Logarithmic quasinormal modes of a spin-3 field around the BTZ black hole
• Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point
• Loop models for CFTs
• SLE($\kappa ,\rho$)and Boundary Coulomb Gas
• Harmonic Analysis and Free Field Realization of the Takiff Supergroup of GL(1|1)
• On the gl(1|1) Wess-Zumino-Witten Model
• Some Aspects of c=-2 Theory
• A Note on Four-Point Functions in Logarithmic Conformal Field Theory
• Character Vectors of Strongly Regular Vertex Operator Algebras
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• Cosmological singleton gravity theory and dS/LCFT correspondence
• Non-abelian soft radiation data for a celestial theory
• Multi-parametric R-matrix for the sl(2|1) Yangian
• Logarithmic operators in $c=0$ bulk CFTs
• Holographic Lifshitz fluids in 1+1 dimensions
• Massive ghost theories with a line of defects
• Multipoint correlators in the Abelian sandpile model
• Tricritical gravity waves in the four-dimensional generalized massive gravity
• Critical behavior in a non-local interface model
• Pure Gauge SU(2) Seiberg-Witten Theory and Modular Forms
• Correlation Functions in N=3 Superconformal Theory
• On a possible interpretation of fusion in Stochastic Loewner Evolution
• Indecomposable Representations in Z_n Symmetric b,c Ghost Systems via Deformations of the Virasoro Field
• On Factorization Constraints for Branes in the H3+ Model
• Contour Integral Representations for the Characters of Logarithmic CFTs
• Cosmological three-coupled scalar theory for the dS/LCFT correspondence
• Implications of Space-Time foam for Entanglement Correlations of Neutral Kaons