Orbifolds and minimal modular extensions

Abstract

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 ${\mathcal{E}}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category $\mathcal{E}=\mathrm{Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\mathcal{C}}_{V^G}$ is a minimal modular extension of $\mathcal{E},$ and is equivalent to the Drinfeld center $\mathcal{Z}({\mathrm{Vec}}_G^{\alpha })$ as modular tensor categories for some $\alpha \in H^3(G,S^1)$ with a canonical embedding of $\mathcal{E}$. Moreover, the collection ${\mathcal{M}}_v(\mathcal{E})$ of equivalence classes of the minimal modular extensions ${\mathcal{C}}_{V^G}$ of $\mathcal{E}$ 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 $\mathcal{Z}({\mathrm{Vec}}_G^{\alpha })$ is equivalent to ${\mathcal{C}}_{V_L^G}$ for some positive definite even unimodular lattice $L.$ In general, for any rational vertex operator algebra $U$ with a $G$-action, ${\mathcal{C}}_{U^G}$ is a minimal modular extension of the braided fusion subcategory $\mathcal{F}$ generated by the $U^G$-submodules of $U$-modules. Furthermore, the group ${\mathcal{M}}_v(\mathcal{E})$ acts freely on the set of equivalence classes ${\mathcal{M}}_v(\mathcal{F})$ of the minimal modular extensions ${\mathcal{C}}_{W^G}$ of $\mathcal{F}$ for any rational vertex operators algebra $W$ with a $G$-action.

arXiv comment43 pages, correct typos and add more details to the proof of Theorem 7.1

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Cited by in corpus (1)

• One dimensional gapped quantum phases and enriched fusion categories