Orbifold aspects of the Longo-Rehren subfactors

Abstract

In this article, we will prove that the subsectors of $α$-induced sectors for $M \rtimes \hat{G} \supset M$ forms a modular category, where $M \rtimes \hat{G}$ is the crossed product of $M$ by the group dual $\hat{G}$ of a finite group $G$. In fact, we will prove that it is equivalent to Müger's crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of $Δ$(not necessarily non-degenerate) obtained from a Longo-Rehren inclusion $A \supset B_Δ$ under certain assumptions. We will apply the above description of the quantum double of $Δ$ to the Reshetikhin-Turaev topological invariant of closed 3-manifolds, and we obtain a simpler formula, which is a degenerate version of Turaev's theorem that the Reshetikhin-Turaev invariant for the quantum double of a modular category $\hatΔ$ is the product of Reshetikhin-Turaev invariant of $\hatΔ$ and its complex conjugate.