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

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