Orbifold aspects of the Longo-Rehren subfactors

In this article, we will prove that the subsectors of $\alpha$-induced sectors for $M⋊\widehat{G}\supset M$ forms a modular category, where $M \rtimes \hat{G}$isthecrossedproductof$M$bythegroupdual$\hat{G}$ of a finite group $G$. In fact, we will prove that it is equivalent to M"uger’s crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of $\mathrm{\Delta }$(not necessarily non-degenerate) obtained from a Longo-Rehren inclusion $A\supset B_{\mathrm{\Delta }}$ under certain assumptions. We will apply the above description of the quantum double of $\mathrm{\Delta }$ 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 $\widehat{\mathrm{\Delta }}$ is the product of Reshetikhin-Turaev invariant of $\widehat{\mathrm{\Delta }}$ and its complex conjugate.