Topological excitations in the Abelian Higgs model

A lattice version of the Abelian Higgs model is studied in arbitrary Euclidean dimension. Using an exact duality transformation, the theory is rewritten in terms of its topological excitations. The dual form of the theory specifies in a simple way all the allowed topological excitations as well as their interactions. The combination of the scalar Higgs field and the Abelian gauge field produces excitations found neither in the pure gauge theory nor in the pure scalar theory ($\mathrm{XY}$ model). In three dimensions, for example, we find finite vortex strings terminating on monopoles, as well as closed vortex loops. Implications of these singularities for the critical behavior of the theory are briefly discussed.