Classification of symmetric fusion categories over $\mathbb{R}$

Abstract

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.