Complex Monopoles and Gribov Copies

Complex monopole solutions exist in the three dimensional Georgi-Glashow model with the Chern-Simons term. They dominate the path integral and disorder the Higgs vacuum. Gribov copies of the vacuum and monopole configurations are studied in detail. 1 Monopole-supercurrent duality In the Georgi-Glashow model, SO(3) gauge theory with a triplet Higgs scalar field h, the gauge sysmetry is spontaneously broken to U (1) by the Higgs mechanism. In perturbation theory the U (1) gauge boson remains massless and the Higgs vacuum is ordered with nonvanishing h = 0. In three dimensional spacetime, however, Polyakov 1 showed that due to instantons, or monopoles in the three-dimensional Euclidean space, the Higgs vacuum is disordered h = 0. The U (1) gauge field acquires a finite mass without breaking the U (1) gauge invariance. A pair of positive and negative electric charges is bound by an electric flux tube. Electric charges are linearly confined. Physical mechanism at work behind this becomes clear by the duality transformation. It was shown by one of the authors 2 that the Georgi-Glashow model, or more generally, the compact QED 3 , is dual to the Josephson junction system in the superconductivity. (E 1 , E 2 , B) in the compact QED 3 corresponds to (B 1 , B 2 , E 3) in the barrier region of the Josephson junction. If a magnetic monopole-antimonopole pair is inserted in the barrier region, a magnetic flux of the Abrikosov magnetic vortex would be formed between the poles as su-percurrents flow through the barrier. Instantons (monopoles) in the compact QED 3 are supercurrents flowing through the two-dimensional barrier in the Josephson junction.