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.
Robert Savit (Mon,) studied this question.