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Classification of Super-Modular Categories by Rank

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Abstract

We pursue a classification of low-rank super-modular categories parallel to that of modular categories. We classify all super-modular categories up to rank = 6, and spin modular categories up to rank = 11. In particular, we show that, up to fusion rules, there is exactly one non-split super-modular category of rank 2, 4 and 6, namely PSU(2)4k+ 2 for k = 0,1 and 2. This classification is facilitated by adapting and extending well-known constraints from modular categories to super-modular categories, such as Verlinde and Frobenius-Schur indicator formulae.

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Authors and Affiliations

  1. Expedia Group, Bellevue, WA, USA

    Paul Bruillard

  2. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    César Galindo

  3. Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA

    Siu-Hung Ng

  4. Department of Mathematics, Indiana University, Bloomington, IN, USA

    Julia Y. Plavnik

  5. Department of Mathematics, Texas A&M University, College Station, TX, USA

    Eric C. Rowell

  6. Microsoft Research Station Q and Department of Mathematics, University of California, Santa Barbara, CA, USA

    Zhenghan Wang

Authors
  1. Paul Bruillard
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  2. César Galindo
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  3. Siu-Hung Ng
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  4. Julia Y. Plavnik
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  5. Eric C. Rowell
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  6. Zhenghan Wang
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Corresponding author

Correspondence to Julia Y. Plavnik.

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Presented by: Jan Stovicek

Publisher’s Note

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The results in this paper were mostly obtained while all six authors were at the American Institute of Mathematics, participating in a SQuaRE. We would like to thank that institution for their hospitality and encouragement. Galindo was partially supported by the Ciencias Básicas funds from vicerrectoria de investigaciones de la Universidad de los Andes, Ng by NSF grant DMS-1501179, Plavnik by CONICET, ANPCyT and Secyt-UNC, Rowell and Plavnik by NSF grant DMS-1410144, and Wang by NSF grant DMS-1411212. The research described in this paper was, in part, conducted under the Laboratory Directed Research and Development Program at PNNL, a multi-program national laboratory operated by Battelle for the U.S. Department of Energy.PNNL Information Release: PNNL-SA-126378.

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Bruillard, P., Galindo, C., Ng, SH. et al. Classification of Super-Modular Categories by Rank. Algebr Represent Theor 23, 795–809 (2020). https://doi.org/10.1007/s10468-019-09873-9

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