Classification of Kac representations in the logarithmic minimal models LM(1,p)

Abstract

For each pair of positive integers r,s, there is a so-called Kac representation (r,s) associated with a Yang-Baxter integrable boundary condition in the lattice approach to the logarithmic minimal model LM(1,p). We propose a classification of these representations as finitely-generated submodules of Feigin-Fuchs modules, and present a conjecture for their fusion algebra which we call the Kac fusion algebra. The proposals are tested using a combination of the lattice approach and applications of the Nahm-Gaberdiel-Kausch algorithm. We also discuss how the fusion algebra may be extended by inclusion of the modules contragredient to the Kac representations, and determine polynomial fusion rings isomorphic to the conjectured Kac fusion algebra and its contragredient extension.

31 pages, v2: comments, subsection and references added

References in corpus (15)

• Logarithmic extensions of minimal models: characters and modular transformations
• Associative-algebraic approach to logarithmic conformal field theories
• From Percolation to Logarithmic Conformal Field Theory
• Virasoro representations and fusion for general augmented minimal models
• Fusion Algebras of Logarithmic Minimal Models
• Logarithmic M(2,p) Minimal Models, their Logarithmic Couplings, and Duality
• 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)
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• W-Extended Logarithmic Minimal Models
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• W-Extended Fusion Algebra of Critical Percolation
• The Jordan Structure of Two Dimensional Loop Models
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)

Cited by in corpus (23)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Logarithmic Conformal Field Theory: a Lattice Approach
• Lattice fusion rules and logarithmic operator product expansions
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Boundary algebras and Kac modules for logarithmic minimal models
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Logarithmic Superconformal Minimal Models
• Logarithmic conformal invariance in the Abelian sandpile model
• Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models
• Integrability and conformal data of the dimer model
• Fusion rules for the logarithmic $N=1$ superconformal minimal models II: including the Ramond sector
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Refined conformal spectra in the dimer model
• Structure of Virasoro tensor categories at central charge $13-6p-6p^{-1}$ for integers $p>1$
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Finite-size corrections for logarithmic representations in critical dense polymers
• 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)
• Yang-Baxter Integrable Dimers on a Strip
• Infinitely extended Kac table of solvable critical dense polymers
• Free Field Realisations of Staggered Modules in 2D Logarithmic CFTs
• Logarithmic Minimal Models with Robin Boundary Conditions
• Staggered and affine Kac modules over $A_1^{(1)}$