Orbifolds and minimal modular extensions

Abstract

Let V be a simple vertex operator algebra and G a finite automorphism group of V such that $V^G$ is regular, and the conformal weight of any irreducible g-twisted V-module N for $g\in G$ is nonnegative and is zero if and only if $N=V$. It is established that if V is holomorphic, then the $V^G$-module category $C_{V^G}$ is a minimal modular extension of $E=\mathrm{Rep}\left(G\right)$, and is equivalent to the Drinfeld center $Z\left({\mathrm{Vec}}_G^{\alpha }\right)$ as modular tensor categories for some $\alpha \in H^3(G,S^1)$ with a canonical embedding of $E$. Moreover, the collection $M_v\left(E\right)$ of equivalence classes of the minimal modular extensions $C_{V^G}$ of $E$ for holomorphic vertex operator algebras V with a G-action forms a group, which is isomorphic to a subgroup of $H^3(G,S^1)$. Furthermore, any pointed modular category $Z\left({\mathrm{Vec}}_G^{\alpha }\right)$ is equivalent to $C_{V_L^G}$ for some positive definite even unimodular lattice L. In general, for any rational vertex operator algebra U with a G-action, $C_{U^G}$ is a minimal modular extension of the braided fusion subcategory $F$ generated by the $U^G$-submodules of U-modules. Furthermore, the group $M_v\left(E\right)$ acts freely on the set of equivalence classes $M_v\left(F\right)$ of the minimal modular extensions $C_{W^G}$ of $F$ for any rational vertex operator algebra W with a G-action.