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On Ribbon Categories for Singlet Vertex Algebras

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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.

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Acknowledgements

TC acknowledges support from NSERC discovery grant RES0048511. RM thanks the University of Alberta for its hospitality during the visit in which this work was begun.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AL, T6G 2R3, Canada

    Thomas Creutzig & Jinwei Yang

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Robert McRae

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  1. Thomas Creutzig
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  2. Robert McRae
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  3. Jinwei Yang
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Correspondence to Jinwei Yang.

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Creutzig, T., McRae, R. & Yang, J. On Ribbon Categories for Singlet Vertex Algebras. Commun. Math. Phys. 387, 865–925 (2021). https://doi.org/10.1007/s00220-021-04097-9

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