YM-Thermodynamics 2024, 36: ANO Vortex Lines and Center-Vortex Loops
Topologically stabilized line-like solitons of an Abelian Higgs model are called Abrisokov-Nielsen-Oelsen vortex lines (abbr. ANO). They are quanta of magnetic flux by their topological charge (winding number), so in YM theories without external sources, neither isolated charges or isolated charge sinks of magnetic flux lines exist, for point-like, condensed magnetic monopoles. Nontrivial elements of the magnetic center groups act locally on a complex background, which creates quasistable loops. In Abelian U(1) Higgs, the topological defects in the deconfining phase of SU(2) or SU(3) YM-theory are screened BPS monopoles, where defects make up screened and closed magnetic flux lines. If the SU(n) is broken to U^n-1(1) by small holonomy (anti)calorons, the nontrivial homotopy group creates stable and unresolvable magnetic (anti)monopoles in the defconfining phase. The preconfining phase magnetic vortex loops are associated with flux lines with cores of magnetic (anti)monopoles, so that monopoles and antimonopoles move in opposite directions within fluxes. This creates a net magnetic flow. At finite g, the massiveness of (anti)monopoles in the core is due to the local restoration of the dual gauge symmetry. The flux penetrating a spatial hyperplane is a line integral along a loop, so that the line integral is a linear Lorentz covariant gauge, the contribution to the integration curve of each moving (anti)monopole is equal to a static (anti)monopole, due to perpendicularity. The magnetic flux of a vortex line then is only determined by the charge of a monopole. ANO take place in the preconfining phase, due to the nontrivial homotopy group Z, so the analytic expression for SU(2) is derived from the action without the external field. The system solves for r > 0 into a magnetic flux of
The negative total pressure exerted by the unresolvable vortex-loop contaminates the monopole-antimonopole condensate, so the vortex-loops start to collapse as soon as they are created under finite g. For an infinite limit of g, P vanishes, so for temperatures below critical, vortex loops exist as particle-like quasistable excitations. A magnetic flux then is equatable to a 2-fold degenerate degree of freedom. Flux creation stems from the local action of nontrivial magnetic center elements of the gauge group (center-vortex loop). If energy is considered to be infinite per vortex length, solutions can be approximated to
