On the structure of the Witt group of braided fusion categories

Abstract

We analyze the structure of the Witt group $${\mathcal{W}}$$ of braided fusion categories introduced in Davydov et al. (Journal fur 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 etale algebras in tensor products of braided fusion categories.