Fusion of irreducible modules in WLM(p,p')

Abstract

Based on symmetry principles, we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the W-extended logarithmic minimal model WLM(p,p'). In addition to the irreducible modules themselves, closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules. We conjecture that this fusion algebra is the same as the one obtained by application of the Nahm-Gaberdiel-Kausch algorithm and find that it reproduces the known such results for WLM(1,p') and WLM(2,3). For p>1, this fusion algebra does not contain a unit. Requiring that the spectrum of modules is invariant under a natural notion of conjugation, however, introduces an additional (p-1)(p'-1) reducible yet indecomposable rank-1 modules, among which the identity is found, still yielding a well-defined fusion algebra. In this greater fusion algebra, the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights Delta_{kp-1,1}, k=1,2,3. We also identify polynomial fusion rings associated with our fusion algebras.

25 pages, v3: new section, comments and a reference added

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• A modular invariant bulk theory for the c=0 triplet model
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}
• Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)
• Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ${\mathbb{Z}}_4$ fermions
• Graph fusion algebras of WLM(p,p')
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge