Dual transformations: An equivalence between the continuum and lattice versions of the two-dimensional Abelian Higgs model

The topological excitations in the two-dimensional Abelian Higgs model are studied in both the continuum and the Villain lattice version, and the results are compared. Using a singular gauge transformation, it is shown that there is an exact equivalence between the dual form of both versions. A fermion form of the theory is useful in studying its confinement properties beyond the dilute-gas limit. Also, the correct measure factor for the continuum version is derived.