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The tensor structure on the representation category of the $\mathcal {W}_p$ triplet algebra

Tsuchiya, Akihiro and Wood, Simon ORCID: 2013. The tensor structure on the representation category of the $\mathcal {W}_p$ triplet algebra. Journal of Physics A: Mathematical and Theoretical 46 (44) , 445203. 10.1088/1751-8113/46/44/445203

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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 formalize 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Publisher: IOP
ISSN: 1751-8113
Date of First Compliant Deposit: 6 December 2016
Last Modified: 02 Nov 2022 09:52

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