Simple current extensions beyond semi-simplicity

Abstract

Let V be a simple VOA and consider a representation category of V that is a vertex tensor category in the sense of Huang-Lepowsky. In particular, this category is a braided tensor category. Let J be an object in this category that is a simple current of order two of either integer or half-integer conformal dimension. We prove that $V\oplus J$ is either a VOA or a super VOA. If the representation category of V is in addition ribbon, then the categorical dimension of J decides this parity question. Combining with Carnahan's work, we extend this result to simple currents of arbitrary order. Our next result is a simple sufficient criterion for lifting indecomposable objects that only depends on conformal dimensions. Several examples of simple current extensions that are $C_2$-cofinite and non-rational are then given and induced modules listed.

34 pages

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Cited by in corpus (44)

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• Schur-Weyl Duality for Heisenberg Cosets
• Braided tensor categories of admissible modules for affine Lie algebras
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
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• Duality of subregular W-algebras and principal W-superalgebras
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• Cosets, characters and fusion for admissible-level $\mathfrak{osp}(1|2)$ minimal models
• Twisted modules and $G$-equivariantization in logarithmic conformal field theory
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• Modularity of logarithmic parafermion vertex algebras
• Direct limit completions of vertex tensor categories
• Parity and Spin CFT with boundaries and defects
• Trialities of orthosymplectic $\mathcal{W}$-algebras
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• Unitary and non-unitary $N=2$ minimal models
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
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• Vertex operator superalgebras and 16-fold way
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