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

Thomas CreutzigRobert McRaeJinwei Yang
摘要
We show that the category of finite-length generalized modules for thesinglet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to thecategory $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite$\mathcal{M}(p)$-modules, and that this category admits the vertex algebraicbraided 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 structureson Heisenberg Fock modules, our results substantially extend our previous workon tensor categories of atypical $\mathcal{M}(p)$-modules. We also introduce atensor subcategory $\mathcal{O}_{\mathcal{M}(p)}^T$, graded by an algebraictorus $T$, which has enough projectives and is conjecturally tensor equivalentto the category of finite-dimensional weight modules for the unrolledrestricted quantum group of $\mathfrak{sl}_2$ at a $2p$th root of unity. Wecompute 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 rigidand thus also ribbon. As an application, we use vertex operator algebraextension theory to show that the representation categories of all finitecyclic orbifolds of the triplet vertex algebras $\mathcal{W}(p)$ arenon-semisimple modular tensor categories, and we confirm a conjecture ofAdamovi\'{c}-Lin-Milas on the classification of simple modules for these finitecyclic orbifolds.
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