Ribbon tensor structure on the full representation categories of the singlet vertex algebras

TL;DR

This paper establishes a braided tensor category structure on the modules of singlet vertex algebras, computes tensor products, and shows these categories are rigid and ribbon, extending previous work and connecting to quantum groups and orbifold theories.

Contribution

It proves the tensor category structure for generalized modules of singlet vertex algebras, introduces a graded subcategory with projectives, and applies these results to orbifold models.

Findings

Category of modules is a braided tensor category.

Tensor products of modules are explicitly computed.

Categories are shown to be rigid and ribbon.

Abstract

We show that the category of finite-length generalized modules for the singlet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to the category $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite $\mathcal{M}(p)$-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since $\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical $\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of atypical $\mathcal{M}(p)$-modules. We also introduce a tensor subcategory $\mathcal{O}_{\mathcal{M}(p)}^T$, graded by an algebraic torus $T$, which has enough projectives and is conjecturally tensor equivalent to the category of finite-dimensional weight modules for the unrolled restricted quantum group of $\mathfrak{sl}_2$ at a $2p$th root of unity. We compute all tensor products involving simple and projective $\mathcal{M}(p)$-modules, and we prove that both tensor categories $\mathcal{O}_{\mathcal{M}(p)}$ and $\mathcal{O}_{\mathcal{M}(p)}^T$ are rigid and thus also ribbon. As an application, we use vertex operator algebra extension theory to show that the representation categories of all finite cyclic orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ are non-semisimple modular tensor categories, and we confirm a conjecture of Adamovi\'{c}-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.