We find, in close analogy to Abelian dominance in the maximal Abelian gauge, the phenomenon of center dominance in the maximal center gauge for SU(2) lattice gauge theory. The maximal center gauge is a gauge-fixing condition that preserves a residual $Z_2$ gauge symmetry; “center projection” is the projection of SU(2) link variables onto $Z_2$ center elements, and “center dominance” is the fact that the center-projected link elements carry most of the information about the string tension of the full theory. We present numerical evidence that the thin $Z_2$ vortices of the projected configurations are associated with “thick” $Z_2$ vortices in the unprojected configurations. The evidence also suggests that the thick $Z_2$ vortices may play a significant role in the confinement process.

We discuss the implementation of the “direct” maximal center gauge (a gauge which maximizes the lattice average of the squared-modulus of the trace of link variables), and its use in identifying $Z_2$ center vortices in Yang-Mills vacuum configurations generated by lattice Monte Carlo calculations. We find that center vortices identified in the vacuum state account for the full asymptotic string tension. Scaling of vortex densities with lattice coupling, change in vortex size with cooling, and sensitivity to Gribov copies are discussed. Preliminary evidence is presented, on small lattices, for center dominance in SU(3) lattice gauge theory.

We argue that the approximate “Casimir scaling” of the string tensions of higher-representation Wilson loops is an effect due to the finite thickness of center vortex configurations. It is shown, in the context of a simple model of the $Z_2$ vortex core, how vortex condensation in Yang-Mills theory can account for both Casimir scaling in intermediate size loops and color-screening in larger loops. An implication of our model is that the deviations from exact Casimir scaling, which tend to grow with loop size, become much more pronounced as the dimensionality of the group representation increases.