Orbifold aspects of the Longo-Rehren subfactors

In this article, we will prove that the subsectors of α-induced sectors for M ⋊ Ĝ ⊃ M forms a modular category, where M ⋊ Ĝ is the crossed product of M by the group dual Ĝ 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 ∆(not necessarily non-degenerate) obtained from a Longo-Rehren inclusion A ⊃ B∆ under certain assumptions. We will apply the above description of the quantum double of ∆ 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 ∆̂ is the product of Reshetikhin-Turaev invariant of ∆̂ and its complex conjugate.