Polynomial fusion rings of W-extended logarithmic minimal models

Abstract

The countably infinite number of Virasoro representations of the logarithmic minimal model LM(p,p') can be reorganized into a finite number of W-representations with respect to the extended Virasoro algebra symmetry W. Using a lattice implementation of fusion, we recently determined the fusion algebra of these representations and found that it closes, albeit without an identity for p>1. Here, we provide a fusion-matrix realization of this fusion algebra and identify a fusion ring isomorphic to it. We also consider various extensions of it and quotients thereof, and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings. One particular extension is reminiscent of the fundamental fusion algebra of LM(p,p') and offers a natural way of introducing the missing identity for p>1. Working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual Moore-Penrose inverses intertwining between the W-fundamental and enlarged fusion algebras.

48 pages

References in corpus (13)

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• W-Extended Logarithmic Minimal Models
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• Polynomial Fusion Rings of Logarithmic Minimal Models

Cited by in corpus (14)

• A modular invariant bulk theory for the c=0 triplet model
• A physical approach to the classification of indecomposable Virasoro representations from the blob algebra
• Fusion Rules of the ${\mathcal{W}}_{p,q}$ Triplet Models
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Classification of Kac representations in the logarithmic minimal models LM(1,p)
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)
• Fusion of irreducible modules in WLM(p,p')
• Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)
• Graph fusion algebras of WLM(p,p')
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge