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

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 Adamović-Lin-Milas on the classification of simple modules for these finite cyclic orbifolds.

65 pages

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Cited by in corpus (6)

• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• On the structure of W-algebras in type A
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• $N=1$ super Virasoro tensor categories
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras