  The dual theory describing the 4D Coulomb gas of point-like magnetically
charged objects, which confines closed electric strings, is considered. The
respective generalization of the theory of confining strings to confining
membranes is further constructed. The same is done for the analogous
SU(3)-inspired model. We then consider a combined model which confines both
electric charges and closed strings. Such a model is nothing, but the mixture
of the above-mentioned Coulomb gas with the condensate of the dual Higgs field,
described by the dual Abelian Higgs model. It is demonstrated that in a certain
limit of this dual Abelian Higgs model, the system under study undergoes
naively the dimensional reduction and becomes described by the (completely
integrable) 2D sine-Gordon theory. In particular, owing to this fact, the phase
transition in such a model must be of the Berezinskii-Kosterlitz-Thouless type,
and the respective critical temperature is expressed in terms of the parameters
of the dual Abelian Higgs model. However, it is finally discussed that the
dimensional reduction is rigorously valid only in the strong coupling limit of
the original 4D Coulomb gas. In such a limit, this reduction transforms the
combined model to the 2D free bosonic theory.
