Center vortices, nexuses, and the Georgi-Glashow model

Abstract

In a gauge theory with no Higgs fields the mechanism for confinement is by center vortices, but in theories with adjoint Higgs fields and generic symmetry breaking, such as the Georgi-Glashow model, Polyakov showed that in $d=3$ confinement arises via a condensate of 't Hooft--Polyakov monopoles. We study the connection in $d=3$ between pure-gauge-theory and the theory with adjoint Higgs fields by varying the Higgs VEV $v.$ As one lowers $v$ from the Polyakov semiclassical regime $v\ensuremath{\gg}g (g$ is the gauge coupling) toward zero, where the unbroken theory lies, one encounters effects associated with the unbroken theory at a finite value $v\ensuremath{\simeq}g,$ where dynamical mass generation of a gauge-symmetric gauge-boson mass $m\ensuremath{\simeq}{g}^{2}$ takes place, in addition to the Higgs-generated non-symmetric mass $M\ensuremath{\simeq}\mathrm{vg}.$ This dynamical mass generation is forced by the infrared instability (in both 3 and 4 dimensions) of the pure-gauge theory. We construct solitonic configurations of the theory with both $m,M\ensuremath{\ne}0$ which are generically closed loops consisting of nexuses (a class of soliton recently studied for the pure-gauge theory), each paired with an antinexus, sitting like beads on a string of center vortices with vortex fields always pointing into (out of) a nexus (antinexus); the vortex magnetic fields extend a transverse distance $1/m.$ An isolated nexus with vortices is continuously deformable from the 't Hooft--Polyakov $(m=0)$ monopole to the pure-gauge--nexus-vortex complex $(M=0).$ In the pure-gauge $M=0$ limit the homotopy ${\ensuremath{\Pi}}_{2}(\mathrm{SU}(2)/U(1))=Z$ [or its analog for $\mathrm{SU}(N)]$ of the 't Hooft--Polyakov monopoles is no longer applicable, and is replaced by the center-vortex homotopy ${\ensuremath{\Pi}}_{1}({\mathrm{SU}(N)/Z}_{N}){=Z}_{N}$ of the center vortices.