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arXiv · 0907.0134

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

Paul A. Pearce·Jorgen Rasmussen·Philippe Ruelle
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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.

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Paul A. Pearce, Jorgen Rasmussen, Philippe Ruelle. 2009-07-01. Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p). https://doi.org/10.1088/1751-8113/43/4/045211

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