Classification of subsystems for the Haag-Kastler nets generated by c=1 chiral current algebras

Let F be the Haag-Kastler net generated by the su(2) chiral current algebra at level 1. We classify the SL(2, R)-covariant subsystems B subset F by showing that they are all fixed points nets F^H for some subgroup H of the gauge automorphisms group SO(3) of F. Then using the fact that the net F_1 generated by the u(1) chiral current can be regarded as a subsystem of F we classify the subsystems of F_1. In this case there are two distinct proper subsystems: the one generated by the energy-momentum tensor and the gauge invariant subsystem F_1^{Z_2}.