AI 中文总结
该研究分类了实数域上的对称融合范畴，通过伽罗瓦下降将其约化为复域上对称融合范畴的半线性$\text{Z}_2$作用问题，建立了与带$\text{Z}_2 \times \text{BZ}_2$作用的有限群胚的塔纳卡-克赖因对应，得到了德利涅分类结果的实数类似物。
AI 中文摘要
我们证明，每个实数域上的对称融合范畴都等价于一个$\boldsymbol{\text{Z}_2}$分次有限超群的有限维半线性表示范畴。证明利用了复/实数域上张量范畴的伽罗瓦下降，将分类问题约化为复域上对称融合范畴的半线性$\text{Z}_2$作用问题。作为进一步的结构结果，我们建立了实数域上对称融合范畴与带有$\boldsymbol{\text{Z}_2 \times \text{BZ}_2}$作用的有限群胚之间的塔纳卡-克赖因型对应，这给出了德利涅分类结果的完整实数域类似物。
英文摘要
We show that every symmetric fusion category over $\mathbb{R}$ is equivalent to the category of finite-dimensional semi-linear representations of a $\mathbb{Z}_2$-graded finite super group. The proof uses Galois descent for tensor categories over $\mathbb{C}/\mathbb{R}$, reducing the classification to semi-linear $\mathbb{Z}_2$-actions on symmetric fusion categories over $\mathbb{C}$. As a further structural result, we establish a Tannaka-Krein type correspondence between symmetric fusion categories over $\mathbb{R}$ and finite groupoids with a $\mathbb{Z}_2 \times \mathrm{B} \mathbb{Z}_2$-action. This gives a complete real analogue of Deligne's classification result.
Comments33 pages. Comments are welcome
