arXiv:2611.05677v1 Announce Type: new Abstract: This note harmonizes two classical topics: minimal modular extensions of braided fusion categories and the Galois action on modular fusion categories. Their common generalizations are Galois-modular extensions, which always exist when a braided fusion category $\mathcal{B}$ is pseudounitary. Each extension induces a Galois action on the simple objects of $\mathcal{B}$. Up to integral equivalence, the Galois-modular extensions of $\mathcal{B}$ form a torsor over an explicit group of homomorphisms from $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to the invertible objects of order dividing two in the symmetric center $\mathcal{E}$ of $\mathcal{B}$. If $\mathcal{E}$ is Tannakian and $\mathcal{B}$ has a minimal modular extension, all minimal modular extensions induce one canonical Galois action. We prove that many facts about the modular data of modular fusion categories extend to pseudounitary premodular fusion categories as well via the Galois action.
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