Abelian Higgs Model Effective Potential in the Presence of Vortices
  We determine the contribution of nontrivial vacuum (topological) excitations, more specifically vortex--strings of the Abelian Higgs model in 3+1 dimensions, to the functional partition function. By expressing the original action in terms of dual transformed fields we make explicit in the equivalent action the contribution of the vortex--strings excitations of the model. The effective potential of an appropriately defined local vacuum expectation value of the vortex--string field in the dual transformed action is then evaluated both at zero and finite temperatures and its properties discussed in the context of the finite temperature phase transition. 
Rudnei O. Ramos, J. F. Medeiros Neto, Daniel G. Barci and Cesar A.   Linhares