SU(N)-gauge theories in Polyakov gauge on the torus

We investigate the Abelian projection with respect to the Polyakov loop ope rator for SU(N) gauge theories on the four torus. The gauge fixed A(0) is t ime-independent and diagonal. We construct fundamental domains for A(0). In sectors with non-vanishing instanton number such gauge fixings are always singular. The singularities define the positions of magnetically charged mo nopoles, strings or walls. These magnetic defects sit on the Gribov horizon and have quantized magnetic charges. We relate their magnetic charges to t he instanton number. (C) 1999 Elsevier Science B.V. All rights reserved.