Coset Graphs in Bulk and Boundary Logarithmic Minimal Models

Abstract

The logarithmic minimal models are not rational but, in the W-extended picture, they resemble rational conformal field theories. We argue that the W-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each W-projective representation arising from fundamental fusion is associated with a boundary condition. Multiplication in the associated Grothendieck ring leads to a Verlinde-like formula involving A-type twisted affine graphs A^{(2)}_{p} and their coset graphs A^{(2)}_{p,p'}=A^{(2)}_{p} x A^{(2)}_{p'}/Z_2. This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in W-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs.

33 pages, v2: comments and references added, v3: minor corrections

References in corpus (75)

• Logarithmic Operators in Conformal Field Theory
• Chiral Gravity in Three Dimensions
• Critical Percolation in Finite Geometries
• The abelian sandpile and related models
• Bits and Pieces in Logarithmic Conformal Field Theory
• Instability in cosmological topologically massive gravity at the chiral point
• A Rational Logarithmic Conformal Field Theory
• Symplectic Fermions
• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• Branes, Rings and Matrix Models in Minimal (Super)string Theory
• Polymers and percolation in two dimensions and twisted N=2 supersymmetry
• Chiral Gravity, Log Gravity and Extremal CFT
• An algebraic approach to logarithmic conformal field theory
• Logarithmic Minimal Models
• Logarithmic extensions of minimal models: characters and modular transformations
• 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
• Fusion rules and logarithmic representations of a WZW model at fractional level
• Non-unitary Conformal Field Theory and Logarithmic Operators for Disordered Systems
• Nonsemisimple Fusion Algebras and the Verlinde Formula
• On the Classification of Bulk and Boundary Conformal Field Theories
• Curiosities at c=-2
• On Fusion Rules in Logarithmic Conformal Field Theories
• From Percolation to Logarithmic Conformal Field Theory
• Solvable Critical Dense Polymers
• Scaling fields in the two-dimensional abelian sandpile model
• Logarithmic Conformal Field Theories via Logarithmic Deformations
• The WZNW model on PSU(1,1|2)
• Virasoro representations and fusion for general augmented minimal models
• 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
• A c=-2 boundary changing operator for the Abelian sandpile
• From boundary to bulk in logarithmic CFT
• The Triplet Vertex Operator Algebra W(p) and the Restricted Quantum Group at Root of Unity
• Logarithmic torus amplitudes
• Logarithmic tensor product theory for generalized modules for a conformal vertex algebra, Part I
• Fusion rules and boundary conditions in the c=0 triplet model
• Fusion Algebras of Logarithmic Minimal Models
• Conformal field theory of the integer quantum Hall plateau transition
• The logarithmic triplet theory with boundary
• sl(2)_{-1/2} and the Triplet Model
• 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)
• Radford, Drinfeld, and Cardy boundary states in (1,p) logarithmic conformal field models
• 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
• Conformal Invariance and Percolation
• Conformal field theory at central charge c=0: a measure of the indecomposability (b) parameters
• Abelian Sandpile Model: a Conformal Field Theory Point of View
• W-Extended Logarithmic Minimal Models
• On Verlinde-Like Formulas in c_{p,1} Logarithmic Conformal Field Theories
• On the Percolation BCFT and the Crossing Probability of Watts
• Boundary States of c=1 and 3/2 Rational Conformal Field Theories
• Solvable Critical Dense Polymers on the Cylinder
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• Pair correlations in sandpile model: a check of logarithmic conformal field theory
• W-Extended Fusion Algebra of Critical Percolation
• Intertwiners and \ade Lattice Models
• Logarithmic limits of minimal models
• Affine Jordan cells, logarithmic correlators, and hamiltonian reduction
• Polynomial fusion rings of W-extended logarithmic minimal models
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• Note on SLE and logarithmic CFT
• Fusion of irreducible modules in WLM(p,p')
• Logarithmic Primary Fields in Conformal and Superconformal Field Theory
• Kazhdan--Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT
• Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)
• Integrals of Motion for Critical Dense Polymers and Symplectic Fermions
• On logarithmic solutions to the conformal Ward identities
• Graph fusion algebras of WLM(p,p')
• Minimal String Theory is Logarithmic

Cited by in corpus (15)

• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Modular invariant partition function of critical dense polymers
• Logarithmic bulk and boundary conformal field theory and the full centre construction
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}
• Conformal partition functions of critical percolation from $D_3$ Thermodynamic Bethe Ansatz equations
• Coset construction of logarithmic minimal models: branching rules and branching functions
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Off-Critical Logarithmic Minimal Models
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Staggered modules of $N=2$ superconformal minimal models
• Logarithmic Minimal Models with Robin Boundary Conditions