Orbifolds and minimal modular extensions

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the category ${\cal E}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category ${\cal E}={\rm Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\cal C}_{V^G}$ is a minimal modular extension of ${\cal E},$ and is equivalent to the Drinfeld center ${\cal Z}({\rm Vec}_G^{\alpha})$ as modular tensor categories for some $\alpha\in H^3(G,S^1)$ with a canonical embedding of ${\cal E}$. Moreover, the collection ${\cal M}_v({\cal E})$ of equivalence classes of the minimal modular extensions ${\cal C}_{V^G}$ of ${\cal E}$ for holomorphic vertex operator algebras $V$ with a $G$-action form a group, which is isomorphic to a subgroup of $H^3(G,S^1).$ Furthermore, any pointed modular category ${\cal Z}({\rm Vec}_G^{\alpha})$ is equivalent to ${\cal C}_{V_L^G}$ for some positive definite even unimodular lattice $L.$ In general, for any rational vertex operator algebra $U$ with a $G$-action, ${\cal C}_{U^G}$ is a minimal modular extension of the braided fusion subcategory ${\cal F}$ generated by the $U^G$-submodules of $U$-modules. Furthermore, the group ${\cal M}_v({\cal E})$ acts freely on the set of equivalence classes ${\cal M}_v({\cal F})$ of the minimal modular extensions ${\cal C}_{W^G}$ of ${\cal F}$ for any rational vertex operators algebra $W$ with a $G$-action.