Fusion rules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$

TL;DR

This paper analyzes the fusion rules of the triplet W-algebra using vertex tensor category theory, rederives known non-semisimple fusion rules, and demonstrates the self-duality of specific indecomposable modules.

Contribution

It applies vertex tensor category theory to derive and verify fusion rules for the triplet W-algebra, including the self-duality of certain modules, advancing understanding of its representation structure.

Findings

Rederived non-semisimple fusion rules for the triplet W-algebra.

Showed certain indecomposable modules are self-dual.

Enhanced understanding of the algebra's module category.

Abstract

We study the structure of fusion rules for the triplet $W$-algebra $\mathcal{W}_{p_+,p_-}$. By using the vertex tensor category theory developed by Huang, Lepowsky and Zhang, we rederive certain non-semisimple fusion rules given by Gaberdiel-Runkel-Wood and Rasmussen. We further show that certain rank two and three indecomposable modules are self-dual.