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.