We find, in close analogy to Abelian dominance in the maximal Abelian gauge, the phenomenon of center dominance in the maximal center gauge for SU͑2͒ 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 SU͑2͒ 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. ͓S0556-2821͑97͒07104-X͔

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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. 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 is discussed. Preliminary evidence is presented, on small lattices, for center dominance in SU(3) lattice gauge theory.

We show that the confining property of the one-gluon propagator, in Coulomb gauge, is linked to the unbroken realization of a remnant gauge symmetry which exists in this gauge. An order parameter for the remnant gauge symmetry is introduced, and its behavior is investigated in a variety of models via numerical simulations. We find that the color-Coulomb potential, associated with the gluon propagator, grows linearly with distance both in the confined and − surprisingly − in the high-temperature deconfined phase of pure Yang-Mills theory. We also find a remnant symmetry-breaking transition in SU(2) gauge-Higgs theory which completely isolates the Higgs from the (pseudo)confinement region of the phase diagram. This transition exists despite the absence, pointed out long ago by Fradkin and Shenker, of a genuine thermodynamic phase transition separating the two regions.

I review investigations of the quark confinement mechanism that have been carried out in the framework of SU(N) lattice gauge theory. The special role of Z N center symmetry is emphasized.

We estimate the Coulomb energy of static quarks from a Monte Carlo calculation of the correlator of timelike link variables in Coulomb gauge. We find, in agreement with Cucchieri and Zwanziger, that this energy grows linearly with distance at large quark separations. The corresponding string tension, however, is several times greater than the accepted asymptotic string tension, indicating that a state containing only static sources, with no constituent gluons, is not the lowest energy flux tube state. The Coulomb energy is also measured on thermalized lattices with center vortices removed by the de Forcrand-D'Elia procedure. We find that when vortices are removed, the Coulomb string tension vanishes.

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. ͓S0556-2821͑98͒03206-8͔

Local gauge symmetries cannot break spontaneously, according to Elitzur's theorem, but this leaves open the possibility of breaking some global subgroup of the local gauge symmetry, which is typically the gauge symmetry remaining after certain (e.g. Coulomb or Landau) gauge choices. We show that in an SU(2) gauge-Higgs system such symmetries do indeed break spontaneously, but the location of the breaking in the phase diagram depends on the choice of global subgroup. The implication is that there is no unique broken gauge symmetry, but rather many symmetries which break in different places. The problem is to decide which, if any, of these gauge symmetry breakings is associated with a transition between physically different, confining and non-confining phases. Several proposals − Kugo-Ojima, Coulomb, and monopole condensate − are discussed.