On ribbon categories for singlet vertex algebras

Abstract

We construct two non-semisimple braided ribbon tensor categories of modules for each singlet vertex operator algebra $\mathcal{M}(p)$, $p\ge 2$. 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.

66 pages, final version incorporating referee suggestions, to appear in Comm. Math. Phys

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• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Admissible-level ${\mathfrak{sl}}_3$ minimal models
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• 3d Mirror Symmetry and the $\beta \gamma$ VOA
• Rigid tensor structure on big module categories for some $W$-(super)algebras in type $A$
• An ${\mathfrak{sl}}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications
• A general mirror equivalence theorem for coset vertex operator algebras
• Fusion and (non)-rigidity of Virasoro Kac modules in logarithmic minimal models at $(p,q)$-central charge
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• On the structure of W-algebras in type A
• $N=1$ super Virasoro tensor categories
• The non-semisimple Kazhdan-Lusztig category for affine ${\mathfrak{sl}}_2$ at admissible levels
• Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras
• Rigidity of non-negligible objects of moderate growth in braided categories