The tensor structure on the representation category of the \(W_p\) triplet algebra

Makes the Nahm–Gaberdiel–Kausch algorithm for computing fusion rigorous, giving a systematic method that works when representations are not completely reducible. Uses it to prove that the representation category of the \(\mathcal{W}_p\) triplet algebras is rigid and to prove the conjectured fusion formulae for all simple and projective modules.

Abstract

We study the braided monoidal structure that the fusion product induces on the abelian category \(\mathcal{W}_p\)-mod, the category of representations of the triplet \(W\)-algebra \(\mathcal{W}_p\). The \(\mathcal{W}_p\)-algebras are a family of vertex operator algebras that form the simplest known examples of symmetry algebras of logarithmic conformal field theories. We formalise the methods for computing fusion products, developed by Nahm, Gaberdiel and Kausch, that are widely used in the physics literature and illustrate a systematic approach to calculating fusion products in non-semi-simple representation categories. We apply these methods to the braided monoidal structure of \(\mathcal{W}_p\)-mod, previously constructed by Huang, Lepowsky and Zhang, to prove that this braided monoidal structure is rigid. The rigidity of \(\mathcal{W}_p\)-mod allows us to prove explicit formulae for the fusion product on the set of all simple and all projective \(\mathcal{W}_p\)-modules, which were first conjectured by Fuchs, Hwang, Semikhatov and Tipunin; and Gaberdiel and Runkel.

Recommended citation: A. Tsuchiya and S. Wood, J. Phys. A, 46 (2013) 445203
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