Vortices and the low-temperature structure of thex−ymodel

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 the 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-value, 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.