G-Reconstruction and the Hopf Equivariantization Theorem

Abstract

$G$-structures on fusion categories have been shown to be an important tool to understand orbifolds of vertex operator algebras \cite{Kirillov}\cite{G_crossed_muger}\cite{Orbifold_Paper}. We continue to develop this idea by generalizing Eilenberg-Maclane's notion of an Abelian $3$-cocycle to describe $G$-structures on fusion categories as $G$-(crossed, ribbon) Abelian $3$-cocycles on an algebra $H$. In particular, we show that a $G$-(crossed, ribbon) Abelian $3$-cocycle on $H$ will induce a $G$-(crossed braided, ribbon) tensor structure on its category of modules $\mathrm{Mod}(H)$. We then prove that every $G$-(crossed braided, ribbon) fusion category $\mathcal{C}$ will be equivalent to the category of modules of some finite dimensional algebra $H$ with $G$-structure induced from a $G$-(crossed, ribbon) Abelian $3$-cocycle. We call this $G$-reconstruction. Lastly, we prove that a $G$-ribbon Abelian $3$-cocycle $\Gamma$ on $H$ allows us to describe the equivariantization $(\Mod(H))^G$ as the category of modules of a ribbon (weak) quasi Hopf algebra $H\#_{\Gamma}\mathbb{C}[G]$. We call this the Hopf equivariantization theorem. By $G$-reconstruction this shows that if $\mathcal{V}$ is a strongly rational vertex operator algebra where $G$ acts faithfully on $\mathcal{V}$ such that $\mathcal{V}^G$ is also strongly rational, then there is an equivalence of modular fusion categories: \begin{equation} \mathrm{Mod}\ \mathcal{V}^G \cong \mathrm{Mod}(H\#_{\Gamma}\mathbb{C}[G]) \end{equation} for some finite dimensional algebra $H$ with a $G$-ribbon Abelian $3$-cocycle $\Gamma$. This provides a proof of the Dijkgraaf-Witten conjecture, and generalizes it as far as possible in the semi-simple setting.