On the structure of Witt groups and minimal extension conjecture

TL;DR

This paper characterizes the Witt relation for braided fusion categories over Tannakian categories and shows the Witt group decomposes into a direct sum involving classical groups, leading to minimal extensions of certain fusion categories.

Contribution

It provides necessary and sufficient conditions for the Witt relation over Tannakian categories and describes the structure of the Witt group as a direct sum, advancing understanding of fusion categories.

Findings

Witt group decomposes as a direct sum of classical groups.

Characterization of Witt relations over Tannakian categories.

Existence of minimal extensions for certain fusion categories.

Abstract

Let $\mathcal{E}=\text{Rep}(G)$ be a Tannakian fusion category. For a braided fusion category $\mathcal{C}$ over $\mathcal{E}$ we give sufficient and necessary conditions that characterize the Witt relation $[\mathcal{C}]=[\mathcal{E}]$. Then we show the Witt group $\mathcal{W}(\mathcal{E})$ is naturally a direct sum of Witt group $\mathcal{W}:=\mathcal{W}(\text{Vec})$ and the group $\text{H}^4(G,\mathbb{K}^\times)$. Consequently, for any non-degenerate fusion category $\mathcal{C}$ over $\mathcal{E}$, there is a positive integer $n$ (e.g. $n=|G|$) such that $\mathcal{C}^{\boxtimes_\mathcal{E}^n}$ admits a minimal extension.