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On the structure of the Witt group of braided fusion categories

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We analyze the structure of the Witt group \({\mathcal{W}}\) of braided fusion categories introduced in Davydov et al. (Journal für die reine und angewandte Mathematik (Crelle’s Journal), eprint arXiv: 1009.2117 [math.QA], 2010). We define a “super” version of the categorical Witt group, namely, the group \({s\mathcal{W}}\) of slightly degenerate braided fusion categories. We prove that \({s\mathcal{W}}\) is a direct sum of the classical part, an elementary Abelian 2-group, and a free Abelian group. Furthermore, we show that the kernel of the canonical homomorphism \({S : \mathcal{W} \to s\mathcal{W}}\) is generated by Ising categories and is isomorphic to \({{\mathbb{Z}}/16\mathbb{Z}}\) . Finally, we give a complete description of étale algebras in tensor products of braided fusion categories.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Hampshire, Durham, NH, 03824, USA

    Alexei Davydov & Dmitri Nikshych

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Victor Ostrik

Authors
  1. Alexei Davydov
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  2. Dmitri Nikshych
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  3. Victor Ostrik
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Correspondence to Dmitri Nikshych.

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Davydov, A., Nikshych, D. & Ostrik, V. On the structure of the Witt group of braided fusion categories. Sel. Math. New Ser. 19, 237–269 (2013). https://doi.org/10.1007/s00029-012-0093-3

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