Gauge invariance in Chern-Simons theory on a torus

In the Chern-Simons gauge theory on a manifold $T^2$×$R^1$ (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 2π, 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 $S^3$ (three-sphere) for SU(N≥3) theory but differs by a factor 2 for SU(2) theory. The requirement of the $Z_N$ invariance in pure SU(N) gauge theory imposes a stronger constraint.