Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1,p)

Abstract

The infinite series of logarithmic minimal models LM(1,p) is considered in the W-extended picture where they are denoted by WLM(1,p). As in the rational models, the fusion algebra of WLM(1,p) is described by a simple graph fusion algebra. The corresponding fusion matrices are mutually commuting, but in general not diagonalizable. Nevertheless, they can be simultaneously brought to Jordan form by a similarity transformation. The spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into Jordan canonical form consisting of Jordan blocks of rank 1, 2 or 3. The various similarity transformations and Jordan forms are determined from the modular data. This gives rise to a generalized Verlinde formula for the fusion matrices. Its relation to the partition functions in the model is discussed in a general framework. By application of a particular structure matrix and its Moore-Penrose inverse, this Verlinde formula reduces to the generalized Verlinde formula for the associated Grothendieck ring.

28 pages, v2: section, comments and references added

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Cited by in corpus (9)

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• Boundary algebras and Kac modules for logarithmic minimal models
• Modular Transformations and Verlinde Formulae for Logarithmic $(p_{+},p_{-})$-Models
• Coset Graphs in Bulk and Boundary Logarithmic Minimal Models
• Graph fusion algebras of WLM(p,p')
• W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)
• Hopf algebras and finite tensor categories in conformal field theory
• Staggered modules of $N=2$ superconformal minimal models