Topological charge carried by nexuses and center vortices

Abstract

In this paper we further explore some questions of topological charge arising from transverse intersections of center vortices and nexuses. Topological charge can also be carried by twist or writhe; we introduce a new nexus carrying twist (or equivalently writhe) which carries Chern-Simons number and in some cases topological charge without reference to an intersection with a separate center vortex. This new nexus can also carry topological charge by linking to vortices. In general, no topological charge in $d=4$ arises from static new nexuses, since the charge is the difference of two (equal) Chern-Simons numbers, but it can arise through dynamic reconnection processes, as the author had suggested earlier. Generally, charge carried by transverse intersections is locally fractionalized in units of $1/N$ for gauge group $\mathrm{SU}(N),$ but globally quantized in integral units. We show explicitly that in $d=4$ such global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy ${\ensuremath{\Pi}}_{3}{(S}^{2})\ensuremath{\simeq}Z;$ this homotopy ensures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy ${\ensuremath{\Pi}}_{2}{(S}^{2})\ensuremath{\simeq}Z,$ a homotopy usually associated with 't Hooft--Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We complete earlier work on vortex-nexus intersections to show explicitly how to express globally integral topological charge as composed of essentially independent units of charge $1/N.$