Gauge invariance in Chern-Simons theory on a torus

In the Chern-Simons gauge theory on a manifold T²×{}R¹ (two-torus×{}time) the unitary operators, which induce large gauge transformations shifting the nonintegrable phases of the two dinstinct Wilson-line integrals on the torus by multiples of 2{π}, do not commute with each other unless the coefficient of the Chern-Simons term is quantized. In U(1) theory this condition gives the statistics phase {θ}={π}/n (n an integer). The condition coincides with the one previously derived on a manifold S³ (three-sphere) for SU(N{≥}3) theory but differs by a factor 2 for SU(2) theory. The requirement of the ZN invariance in pure SU(N) gauge theory imposes a stronger constraint.