String Representation of the Abelian Higgs Theory and Aharonov-Bohm Effect on the Lattice

The partition function of the $4D$ lattice Abelian Higgs theory is represented as the sum over world sheets of Nielsen–Olesen strings. The creation and annihilation operators of the strings are constructed. The topological long–range interaction of the strings and charged particles is shown to exist; it is proportional to the linking number of the string world sheet and particle world trajectory.