Abelian projection on the torus for general gauge groups

We consider Yang-Mills theories with general gauge groups G and twists on t he four-torus. We find consistent boundary conditions for gauge fields in a ll instanton sectors. An extended abelian projection with respect to the Po lyakov loop operator is presented, where A(o) is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed A(o) are co nstructed for arbitrary gauge groups. In the sectors with non-vanishing ins tanton number such gauge fixings are necessarily singular. The singularitie s can be restricted to Dirac strings joining magnetically charged defects. The magnetic charges of these monopoles take their values in the co-root la ttice of the gauge group. We relate the magnetic charges of the defects and the windings of suitable Higgs fields about these defects to the instanton number. (C) 1999 Elsevier Science B.V. All rights reserved.