Z ( 2 ) vortices and the string tension in SU ( 2 ) gauge theory ∗

We report on recent work on the crucial role of vortices characterized by Z(N) flux for maintaining confinement at weak coupling in SU(N) gauge theory. The present work is a continuation of our on-going project over the last several years [1]. We begin by distinguishing between various types of vortices. Recall that in the continuum formulation there is no local distinction between pure SU(N) and SU(N)/Z(N) gauge theories, but in the lattice formulation there is. Now for continuum SU(N)/Z(N) fields, vortices are topologically classified by π1(SU(N)/Z(N)) = Z(N). A vortex forms a closed 2-dim structures in d = 4. Topologically, it is also possible to have Dirac monopoles, also classified by π1(SU(N)/Z(N)). The Dirac sheet of such a monopole loop (d = 4) may be described as defining a ‘punctured’ vortex. On the lattice the SU(N) and SU(N)/Z(N) theories differ by Z(N) degrees of freedom. Exciting these Z(N) degrees of freedom gives rise to ‘thin’ Z(N) vortices. They are very efficient at disordering at small β; but are directly suppressed by the SU(N) plaquette action and become unimportant at large β. This reflects the fact that the distinction between SU(N) and SU(N)/Z(N) LGT must disappear as the con-