Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2009-08-14
J.Phys.A43:105201,2010
Physics
High Energy Physics
High Energy Physics - Theory
28 pages, v2: section, comments and references added
Scientific paper
10.1088/1751-8113/43/10/105201
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.
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