Fractals, topological defects, and confinement in lattice gauge theories

Topological defects — monopoles, vortices, and strings — are discussed. It is shown that these objects form clusters with a nonintegral dimension, i.e., they are fractals. The fractal dimension reflects the physical properties of a system. In particular, studies of monopole current clusters in U(1) and SU(2) lattice gauge theories make it possible to identify the confinement mechanism. In the confinement phase the current lines of a magnetic monopole form a percolating cluster and these lines are so dense that their dimension exceeds unity, whereas in the deconfinement phase their dimension is trivial: it is equal to unity. It is also shown that the string tension is proportional to the dimension of extended monopole currents. This is in agreement with a confinement model based on the condensation of magnetic monopoles into a superconducting phase. A string between a quark and an antiquark is then analogous to an Abrikosov vortex in a superconductor. An account is given of the application of the theory of fractals in the problem of gauge fixing in lattice gauge theories. It is also demonstrated that, in SU(2) gluodynamics, domains of the deconfinement phase have a nonintegral dimension near a phase transition point. Apart from monopoles, the review deals also with the properties of vortex and string clusters in three-dimensional and four-dimensional XY models. The corresponding physical objects are vortices in liquid helium and 'global cosmic strings'.