Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)

Abstract

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $\tau$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $\tau$-dependent S-matrix.

13 pages, v2: example, comments and references added

References in corpus (8)

• Logarithmic extensions of minimal models: characters and modular transformations
• Virasoro representations and fusion for general augmented minimal models
• Logarithmic torus amplitudes
• From boundary to bulk in logarithmic CFT
• Fusion Algebras of Logarithmic Minimal Models
• Integrable Boundary Conditions and W-Extended Fusion in the Logarithmic Minimal Models LM(1,p)
• W-Extended Logarithmic Minimal Models
• Polynomial fusion rings of W-extended logarithmic minimal models

Cited by in corpus (13)

• Logarithmic Conformal Field Theory: Beyond an Introduction
• Modular Data and Verlinde Formulae for Fractional Level WZW Models I
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Boundary algebras and Kac modules for logarithmic minimal models
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Fusion rules for the logarithmic $N=1$ superconformal minimal models I: the Neveu-Schwarz sector
• Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)
• The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre
• Graph fusion algebras of WLM(p,p')
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Staggered modules of $N=2$ superconformal minimal models