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.

arXiv comment66 pages, final version incorporating referee suggestions, to appear in Comm. Math. Phys

References in corpus (25)

• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• Logarithmic extensions of minimal models: characters and modular transformations
• On the Superconformal Index of Argyres-Douglas Theories
• Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories
• Braided tensor categories and extensions of vertex operator algebras
• Nonsemisimple Fusion Algebras and the Verlinde Formula
• Schur-Weyl Duality for Heisenberg Cosets
• Modular Data and Verlinde Formulae for Fractional Level WZW Models II
• Nonmeromorphic operator product expansion and C_2-cofiniteness for a family of W-algebras
• Coset Constructions of Logarithmic (1,p)-Models
• The tensor structure on the representation category of the ${\mathcal{W}}_p$ triplet algebra
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• Some remarks on the unrolled quantum group of sl(2)
• Braided tensor categories of admissible modules for affine Lie algebras
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• Modular categories and orbifold models
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• Duality of subregular W-algebras and principal W-superalgebras
• A quasi-Hopf algebra for the triplet vertex operator algebra
• On the tensor structure of modules for compact orbifold vertex operator algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Tensor categories of affine Lie algebras beyond admissible levels
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Non-Semisimple Quantum Invariants and TQFTs from Small and Unrolled Quantum Goups
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions

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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