Monopoles in the Abelian Projection of Gluodynamics

Abstract

We discuss some properties of the abelian monopoles in compact U (1) gauge theory and in the SU (2) gluodynamics both on the lattice and in the continuum. §1. IntroductionAbelian monopoles play a key role in the dual superconductor mechanism of confinement 1) in non-abelian gauge theories. Abelian monopoles appear after the so called abelian projection 2) . According to the dual superconductor mechanism a condensation of abelian monopoles should give rise to the formation of an electric flux tube between the test quark and antiquark. Due to a non-zero string tension the quark and the antiquark are confined by a linear potential. This mechanism has been confirmed by many numerical simulations of the lattice gluodynamics 3), 4) which show that the abelian monopoles in the Maximal Abelian projection are responsible for the confinement. The SU (2) string tension is well described by the contribution of the abelian monopole currents 5) ; these currents satisfy the London equation for a superconductor 6), 7) . In Fig. 1, taken from Ref. 7) , the abelian monopole currents near the center of the flux tube formed by the quark-anti-quark pair are shown. It is seen that the monopoles wind around the center of the flux tube just as the Cooper pairs wind around the center of the Abrikosov string. In Fig. 2 taken from Ref. 8) we show the dependence of the value of the monopole condensate Φ inf c on β is shown. It is clearly seen that Φ inf c vanishes at the phase transition and it plays the role of the order parameters 8), 9) . In Ref. 10) the effective lagrangian for monopoles was reconstructed from numerical data for monopole currents for SU (2) gluodynamics in the Maximal Abelian gauge. It occurs that this lagrangian corresponds to the Abelian Higgs model, the monopole are condensed in the classical string tension of the Abrikosov-Nielsen-Olesen string describes well the quantum string tension of the SU (2) gluodynamics. It means that the description of the gluodynamics at large distances in terms of the monopole variables can be very useful.

The Berry Phase and Monopoles in Non-Abelian Gauge Theories

“…For this reason the Berry phase factor is also called geometrical phase. Indeed, the rich mathematical structure behind (8,9) was discovered by Simon [2] and further elaborated in a number of papers [3,5,6], [13]- [16]. In particular, it was shown in [3] that the adiabaticity requirement is in fact unnecessary.…”

“…For an infinitesimal Wilson loop we obtain the geometrical phase in terms of the continuum gauge potentials and argue that it reflects the non triviality of the Hopf bundle SU(2) → SU(2)/U(1) = S 2 . Our considerations allow to define monopole-like defects in the SU(2) gluodynamics, which are different from the Abelian monopoles considered so far (see [9] for a review). Indeed, the usual definition of Abelian monopoles implies a particular partial gauge fixing which leaves a U(1) subgroup unfixed and the monopoles are defined with respect to this remaining U (1).…”

Monopole content of topological clusters: Have Kraan-van Baal calorons been found?

“…We recall that correlations between Abelian monopoles and topological density have been studied already in the past, both without [25] and with smoothing [26,27,28]. Here we go a step further and use the location and number of Abelian monopoles in order to see the correlation with the Polyakov loop variable inside topological clusters.…”

Gauge problem of monopole dynamics inSU(2)lattice gauge theory

“…In other words, monopole condensation occurs ͓4͔ in the large b region for all gauges. Figures 20,21,22,and 23 show the RG flows projected onto the g 1 -g 2 , g 1 -g 5 , g 1 -g 7 , and g 1 -g 10 coupling planes, respectively. The effective coupling constants for all gauges seem to converge to the identical line for the large b region.…”