A physical picture for the phase transitions in Z_N symmetric models

We show how the phase transitions in Z_N symmetric spin and gauge theories can be understood as being caused by condensations of topological excitations. For the two-dimensional Z_N periodic gaussian and vector Potts models, there are two phase transitions, the first caused by a condensation of strings (domain boundaries) and the second by an unbinding of vortices. The relationship between our picture and the double Coulomb gas representation of Kadanoff is discussed. Using our representation, we also explain the correspondence between these models and the recent theory of two-dimensional melting of Halperin and Nelson. Finally, we describe generalizations of our picture to higher dimensions and to gauge theories.