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Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

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Abstract

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, 25 ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the pentagon equation.

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Notes

  1. Below \(n_r\) and \(m_r\) are the numbers of fusion rings and complex Grothendieck rings, of multiplicity one and rank r, see also [26, 27].

    $$\begin{aligned} \begin{array}{c|cccccc} r&{}1&{}2&{}3&{}4&{}5&{}6 \\ \hline n_r&{}1&{}2&{}4&{}10&{}16&{}39 \\ \hline m_r&{}1&{}2&{}4&{}9&{}10&{}21 \end{array} \end{aligned}$$

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Acknowledgements

The authors would like to thank Eric C. Rowell for pointing out the new zesting construction [5], Andrew Schopieray for pointing out the PhD thesis of Josiah E. Thornton about generalized near-group categories [34], Ricardo Buring for his help with SageMath [32], Joost Slingerland and Gert Vercleyen for useful discussions and AnyonWiki [30], Arthur Jaffe for his constant encouragement and helpful discussions, and finally the anonymous referee for careful proofreading and relevant comments. The first author would like to thank Harvard University for his hospitality. The first author is supported by Grant 04200100122 from Tsinghua University and 2020YFA0713000 from NKPs. The second author is supported by BIMSA. The third author is supported by Grant TRT 0159, ARO Grants W911NF-19-1-0302 and W911NF-20-1-0082.

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Authors and Affiliations

  1. Department of Mathematics, Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Zhengwei Liu

  2. Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China

    Zhengwei Liu & Sebastien Palcoux

  3. Department of Physics, Harvard University, Cambridge, MA, 02138, USA

    Yunxiang Ren

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  1. Zhengwei Liu
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  2. Sebastien Palcoux
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Correspondence to Sebastien Palcoux.

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Appendix: SageMath code

Appendix: SageMath code

This section provides the SageMath code for the criteria of Theorems 2.3, 2.9, and 2.11 (the only criteria needed to prove Theorem 1.1). They apply in the commutative case only (as needed). Just apply the function Checking below to a fusion ring written as a list M, for example, the following computation shows that the fusion ring N\(^{\underline{\mathrm{o}}}\)5 at rank 4 is non-Drinfeld and non-d-number (as written in Sect. 3.7).

figure b

Here is the code of the function Checking:

figure c

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Liu, Z., Palcoux, S. & Ren, Y. Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Lett Math Phys 112, 54 (2022). https://doi.org/10.1007/s11005-022-01542-1

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