On the structure of Witt groups and minimal extension conjecture

TL;DR

The paper identifies the structure of the relative Witt group $\mathcal{W}(\mathcal{E})$ for a Tannakian fusion category $\mathcal{E}=\mathrm{Rep}(G)$, proving a canonical isomorphism $\mathcal{W}(\mathcal{E}) \cong \mathcal{W} \oplus \mathrm{H}^4(G,\mathbb{k}^\times)$. It constructs a natural projection $\phi_\mathcal{E}$ to the ordinary Witt group and shows its kernel is isomorphic to $\mathrm{H}^4(G,\mathbb{k}^\times)$ by employing $G$-graded, $G$-quasi-monoidal categories and graded extension theory. The results provide a complete interpretation of the fourth cohomology in terms of the relative Witt group and yield that every non-degenerate fusion category over $\mathcal{E}$ has a minimal extension after a finite base change, with a concrete bound $n$ (e.g., $n=|G|$). Additionally, the paper analyzes the map to the super Witt group and discusses when a global splitting exists, leading to insights and open questions about finiteness and 2-primary torsion in kernels. These findings advance the understanding of how higher cohomology controls Morita-type obstructions in relative Witt theory and minimal extension problems.

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.

Theorems & Definitions (44)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 3.1: OY
• Remark 3.2
• Corollary 3.3
• Lemma 3.4
• proof
• Theorem 3.5