On ribbon categories for singlet vertex algebras

TL;DR

This paper constructs and analyzes two braided ribbon tensor categories of modules for singlet vertex algebras, revealing their structure, projective covers, and fusion rules, advancing understanding of non-semisimple tensor categories in vertex algebra theory.

Contribution

It introduces two new non-semisimple braided ribbon tensor categories for singlet vertex algebras, detailing their module structures and fusion properties, and identifies projective covers for irreducible modules.

Findings

Every irreducible module has a projective cover in the second category.

Fusion products involving atypical irreducible modules are explicitly computed.

The categories provide a framework for understanding non-semisimple tensor structures in vertex algebras.

Abstract

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $\mathcal{M}(p)$, $p\geq 2$. The first category consists of all finite-length $\mathcal{M}(p)$-modules with atypical composition factors, while the second is the subcategory of modules that induce to local modules for the triplet vertex operator algebra $\mathcal{W}(p)$. We show that every irreducible module has a projective cover in the second of these categories, although not in the first, and we compute all fusion products involving atypical irreducible modules and their projective covers.