On Fusion Rules in Logarithmic Conformal Field Theories

Abstract

We find the fusion rules for the c_{p,1} series of logarithmic conformal field theories. This completes our attempts to generalize the concept of rationality for conformal field theories to the logarithmic case. A novelty is the appearance of negative fusion coefficients which can be understood in terms of exceptional quantum group representations. The effective fusion rules (i.e. without signs and multiplicities) resemble the BPZ fusion rules for the virtual minimal models with conformal grid given via c = c_{3p,3}. This leads to the conjecture that (almost) all minimal models with c = c_{p,q}, gcd(p,q) > 1, belong to the class of rational logarithmic conformal field theories.

14 pages, LaTeX (or better LaTeX2e), no figures, also available at http://www.sns.ias.edu/~flohr/, some corrections and clarifications

References in corpus (7)

• A Rational Logarithmic Conformal Field Theory
• Indecomposable Fusion Products
• Edge excitations of paired fractional quantum Hall states
• Non-unitary Conformal Field Theory and Logarithmic Operators for Disordered Systems
• World-Sheet Logarithmic Operators and Target Space Symmetries in String Theory
• Curiosities at c=-2
• D Branes from Liouville Strings

Cited by in corpus (73)

• A Rational Logarithmic Conformal Field Theory
• Symplectic Fermions
• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• A Local Logarithmic Conformal Field Theory
• Logarithmic Minimal Models
• Associative-algebraic approach to logarithmic conformal field theories
• A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
• Fusion rules and logarithmic representations of a WZW model at fractional level
• Kazhdan--Lusztig-dual quantum group for logarithmic extensions of Virasoro minimal models
• The Logarithmic Conformal Field Theories
• The Haldane-Rezayi Quantum Hall State and Conformal Field Theory
• Logarithmic Conformal Field Theories via Logarithmic Deformations
• Matrix D-brane Dynamics, Logarithmic Operators and Quantization of Noncommutative Spacetime
• Virasoro representations and fusion for general augmented minimal models
• On Reducible but Indecomposable Representations of the Virasoro Algebra
• Origin of Logarithmic Operators in Conformal Field Theories
• Logarithmic torus amplitudes
• From boundary to bulk in logarithmic CFT
• 2-Dimensional Turbulence: yet another Conformal Field Theory Solution
• Singular Vectors in Logarithmic Conformal Field Theories
• Non-analyticity of the Callan-Symanzik beta-function of two-dimensional O(N) model
• The logarithmic triplet theory with boundary
• Generalized twisted modules associated to general automorphisms of a vertex operator algebra
• Modular transformation and boundary states in logarithmic conformal field theory
• Fusion algebra of critical percolation
• Operator Product Expansion in Logarithmic Conformal Field Theory
• Disordered Dirac Fermions: Multifractality Termination and Logarithmic Conformal Field Theories
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• Towards a Classification of Two-Character Rational Conformal Field Theories
• On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories
• Boundary States in c=-2 Logarithmic Conformal Field Theory
• Ghost Systems: A Vertex Algebra Point of View
• Logarithmic Conformal Field Theory Through Nilpotent Conformal Dimensions
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Boundary states in boundary logarithmic CFT
• Logarithmic conformal field theory approach to topologically massive gravity
• Logarithmic conformal field theory with boundary
• Logarithmic Operators in Conformal Field Theory and The $\W_\infty$-algebra
• The Neveu-Schwarz and Ramond Algebras of Logarithmic Superconformal Field Theory
• Logarithmic limits of minimal models
• Logarithmic operators in the SL(2,R) WZNW model
• Notes on non-trivial and logarithmic CFTs with c=0
• Fermionic Expressions for the Characters of c(p,1) Logarithmic Conformal Field Theories
• Logarithmic conformal field theories with continuous weights
• Jordan cells in logarithmic limits of conformal field theory
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Legendre's Relation and the Quantum Equivalence osp(4|4)_(1) = osp(2|2)_(-2) + su(2)_(0)
• Proposal for a CFT interpretation of Watts' differential equation for percolation
• Vacuum Instability in Chern-Simons Theory, Null Vectors and Two-Dimensional Logarithmic Operators
• Logarithmic Currents in the SU(2)_0 WZNW model
• Minimal Superstrings and Loop Gas Models
• Zamalodchikov's C-Theorem and The Logarithmic Conformal Field Theory
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• Boundary States and Symplectic Fermions
• Logarithmic Conformal Field Theory Solutions of Two Dimensional Magnetohydrodynamics
• Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)
• Extended chiral algebras in the SU(2)_0 WZNW model
• Notes on Generalised Nullvectors in logarithmic CFT
• What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT
• Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas
• The O(n) Model in the $n\to 0$ Limit (self-avoiding-walks) and Logarithmic Conformal Field Theory
• Coset construction of logarithmic minimal models: branching rules and branching functions
• Extended multiplet structure in Logarithmic Conformal Field Theories
• Vacuum Instability in Topologically Massive Gauge Theory
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
• Logarithmic Correlation Functions in Liouville Field Theory
• The origin of multiplets of chiral fields in SU(2)_k WZNW at rational level
• Minimal String Theory is Logarithmic
• Operator Product Expansion and Zero Mode Structure in Logarithmic CFT
• Characters of coinvariants in (1,p) logarithmic models
• A Note on Four-Point Functions in Logarithmic Conformal Field Theory
• Logarithmic operators in SL(2, R) WZNW model, Singletons and AdS_3/(L)CFT_2 correspondence