Tensor structure on the module category of the triplet superalgebra $\mathcal{{SW}}(m)$

TL;DR

This paper studies the tensor category structure of modules over the $N=1$ triplet superalgebra $ ext{SW}(m)$, establishing fusion rules, rigidity, and explicit solutions to related differential equations using vertex supercategory theory.

Contribution

It determines the fusion product structure and proves the rigidity of the tensor supercategory of $ ext{SW}(m)$-modules, introducing explicit solutions to a key differential equation.

Findings

Fusion rules between simple and projective modules are explicitly determined.

The tensor supercategory on $ ext{SW}(m)$-modules is shown to be rigid.

Explicit solutions to a fourth-order Fuchsian differential equation are constructed.

Abstract

We discuss the tensor structure on the category of modules of the $N=1$ triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced by Adamovi\'{c} and Milas. Based on the theory of vertex tensor supercategories, we determine the structure of fusion products between the simple and projective $\mathcal{SW}(m)$-modules and show that the tensor supercategory on $\mathcal{SW}(m)$-mod is rigid. Technically, explicit solutions of a fourth-order Fuchsian differential equation are important to show the rigidity of $\mathcal{SW}(m)$-modules. We construct solutions of this Fuchsian differential equation using the theory of the Dotsenko-Fateev integrals developed by Sussman.