Absence of subsystems for the Haag-Kastler net generated by the energy-momentum tensor in two-dimensional conformal field theory

We show that if A is the Haag-Kastler net generated by the energy-momentum tensor in a chiral quantum field theory, then every subsystem B subset of A which is covariant under the action of SL(2, R) given on A must coincide with A. The result is valid for all the allowed values of the central charge and is obtained using scaling limit techniques.