Vortices and the low-temperature structure of the x-y model

An exact duality transformation is applied to the partition function Z for the x-y model in two and three dimensions. The fields which appear in the dual representation of Z are integer valued and represent the topological excitations, or vortices of the x-y model. Furthermore, this form of the partition function is particularly simple at low temperatures. In two dimensions, the dual representation of Z at low temperatures describes a two-dimensional Coulomb gas in which point charges are vortices. In three dimensions, the dual form of Z describes a locally invariant gauge theory, analogous to QED, and coupled to integer-valued conserved currents which represent the line vortices of the three-dimensional x-y model. Qualitative comments about the low-temperature behavior of the theories are made. The meaning of vortices on a lattice is also discussed