Monopoles in the Abelian Polyakov gauge and projection (in)dependence of the dual superconductor mechanism of confinement

We discuss the effect of the choice of an Abelian projection on the dual superconductor mechanism of confinement in SU(2) gluodynamics. Using qualitative arguments we show that the dual superconductor Lagrangian corresponding to the Abelian Polyakov gauge has a different structure compared to the dual Lagrangian in the maximal Abelian gauge. The difference between these Lagrangians reflects the fact that in the continuum limit the monopoles should be static in the Abelian Polyakov gauge and therefore these monopoles cannot give rise to the confinement of static quarks. Using the Abelian Polyakov gauge as an example, we show that (i) the dual superconductor scenario of confinement may depend on the Abelian projection, and (ii) the condensation of the Abelian monopoles—which may be realized in any Abelian projection—is a necessary but not sufficient condition for confinement. These results do not exclude the existence of a class of Abelian gauges (including the Maximal Abelian gauge) in which the dual superconductor scenario works well.