  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.
