2016/07/31 by Marcel Bischoff · 1 citation
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.MP #math.OA #math.QA #math.RT
paper · pdf · doi:10.1142/s0129055x17500027
40 pages, many TikZ figures. Comments are welcome
arxiv created 2016/07/31 · arxiv updated 2017/02/01
Let B be a conformal net. We give the notion of a proper action of a finite hypergroup acting by vacuum preserving unital completely positive (so-called stochastic) maps, which generalizes the proper actions of finite groups. Taking fixed points under such an action gives a finite index subnet BK of B, which generalizes the G-orbifold. Conversely, we show that if A⊂ B is a finite inclusion of conformal nets, then A is a generalized orbifold A=BK of the conformal net B by a unique finite hypergroup K. There is a Galois correspondence between intermediate nets BK⊂ A ⊂ B and subhypergroups L⊂ K given by A=BL. In this case, the fixed point of BK⊂ A is the generalized orbifold by the hypergroup of double cosets L\backslash K/ L. If A⊂ B is an finite index inclusion of completely rational nets, we show that the inclusion A(I)⊂ B(I) is conjugate to a Longo--Rehren inclusion. This implies that if B is a holomorphic net, and K acts properly on B, then there is a unitary fusion category F which is a categorification of K and Rep(BK) is braided equivalent to the Drinfel'd center Z(F). More generally, if B is completely rational conformal net and K acts properly on B, then there is a unitary fusion category F extending Rep(B), such that K is given by the double cosets of the fusion ring of F by the Verlinde fusion ring of B and Rep(BK) is braided equivalent to the Müger centralizer of Rep(B) in Z(F).