On the structure of the Witt group of braided fusion categories
Alexei Davydov, Dmitri Nikshych, Victor Ostrik
TL;DR
This work builds on the categorical Witt framework to analyze the structure of braided fusion categories through the introduction of the Witt group sW of slightly degenerate categories and the canonical map $S:\mathcal{W}\to s\mathcal{W}$, whose kernel is generated by Ising categories and is isomorphic to $\mathbb{Z}/16\mathbb{Z}$. It proves a precise decomposition of $s\mathcal{W}$ into a direct sum of the classical part, an elementary Abelian $2$-group, and a free Abelian group, and it provides a complete classification of étale algebras in tensor products, together with a tensor-product decomposition theorem for slightly degenerate categories. The results yield a detailed understanding of the relations in $s\mathcal{W}$ and illuminate the structure of the subgroup generated by C$(\mathfrak{sl}(2),k)$, including explicit Witt-order and factorization data across three families of levels. Overall, the paper establishes a robust, 2-primary picture for the Witt theory of braided fusion categories and supplies concrete tools for computing and decomposing slightly degenerate and non-degenerate centers, with implications for rational conformal field theory and modular data.
Abstract
We analyze the structure of the Witt group W of braided fusion categories introduced in the previous paper arXiv:1009.2117v2. We define a "super" version of the categorical Witt group, namely, the group sW of slightly degenerate braided fusion categories. We prove that sW 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: W --> sW is generated by Ising categories and is isomorphic to Z/16Z. Finally, we give a complete description of etale algebras in tensor products of braided fusion categories.