Conformal quantum field theory and half-sided modular inclusions of von-Neumann-algebras

Let N, M be von-Neumann-Algebras on a Hilbert space H, Ω a common cyclic and separating vector. Assume Ω to be cyclic and separating also for N intersection M. Denote by JM, JN the modular conjugations to (M, Ω), ΔM and ΔN the associated modular operators. If ΔM-it(N intersection M)ΔMit contains or equal to (N intersection M) for all t≥0, ΔNit(N intersection M)ΔN-it contains or equal to (NcaM) for all t≥0, and JMNJM=N, these data define in a canonical way a conformal quantum field theory on a circle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way. (orig.)