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

TL;DR

This paper rigorously analyzes the tensor structure of the representation category of the triplet $W$-algebra $\\mathcal{W}_p$, establishing its rigidity and providing explicit fusion product formulas for simple and projective modules.

Contribution

It formalizes fusion product computation methods for non-semi-simple categories and proves the rigidity of the braided monoidal structure of \\mathcal{W}_p-\text{mod}.

Findings

Proved the braided monoidal structure of \\mathcal{W}_p-\text{mod} is rigid.

Derived explicit fusion product formulas for simple and projective modules.

Validated conjectures on fusion products in logarithmic conformal field theories.

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.