Topological Excitations in U(1)-Invariant Theories

A class of U(1)-invariant theories in d dimensions is introduced on a lattice. These theories are labeled by a simplex number s, with $1<~s<d$. The case with $s=1$ is the X-Y model; and $s=2$ gives compact photodynamics. An exact duality transformation is applied to show that the U(1)-invariant theory in d dimensions with simplex number s is the same as a similar theory in d dimensions but which is Z_∞-invariant and has simplex number ̃ \~s=d-s. This dual theory describes the topological excitations of the original theory. These excitations are of dimension ̃ \~s-1.