Center symmetry and Abelian projection at finite temperature

At finite temperature, there is an apparent conflict between Abelian projection and critical universality. For example, should the deconfinement transition of an SU(2) gauge theory projected to U(1) lie in the Z(2) universality class of the parent SU(2) theory or in the U(1) universality class? I prove that the projected theory lies in the universality class of the parent gauge theory. The mechanism is shown to be non-local terms in the projected effective action involving Polyakov loops. I connect this to the recent work by Dunne et al. on the deconfinement transition in the 2+1 dimensional Georgi-Glashow model, leading to a discussion of the role of renormalization group fixed points in Abelian projection.