IS THERE A PHASE-TRANSITION IN MAXWELL-CHERN-SIMONS THEORY

We reexamine recent work on the Abelian Higgs model (scalar QED) in 21 dimensions. We compute the effective potential at finite temperature using the Vilkovisky-DeWitt formalism and show that there is spontane ous symmetry breaking at one loop. We use an effective action formalis m to take into account ring corrections and find that classically brok en symmetries can be restored at high temperature. The presence of the Chern-Simons parameter can hinder symmetry breaking for values of the self-couplings very close to the critical value normally required for a phase transition.