REMARKS ON LOCAL DISCRETE SYMMETRY

It is Pointed out that discrete symmetries left unbroken after a spontaneous Higgs condensation remain as global symmetries in the low-energy effective theory. The global nature of these discrete symmetries makes the associated charges physical so that they may become "quantum hair" for black holes as has been suggested recently. It is also emphasized that to obtain a theory with a local discrete symmetry G, one must not only introduce sectors in the Hilbert space corresponding to G-twisted boundary conditions, but should also project out G non-invariant states in each sector to preserve the local G gauge invariance. Explicit examples discussed here include a Z(N) model with two scalars and the SO(3) --> O(2) model with Alice strings.