Monopoles , Polyakov - Loops and Gauge Fixing on the Torus

We consider pure Yang Mills theory on the four torus. A set of non-Abelian transition functions is presented which encompass all instanton sectors. It is argued that these transition functions are a convenient starting point for gauge fixing. In particular, we give an extended Abelian projection with respect to the Polyakov loop, where A0 is independent of time and in the Cartan subalgebra. In the nonperturbative sectors such gauge fixings are necessarily singular. These singularities can be restricted to Dirac strings joining monopole and anti-monopole like “defects”. Supported by the Deutsche Forschungsgemeinschaft, DFG-Wi 777/3-1 e–mails: Ford, Mitreuter, Tok and [email protected] e–mail: [email protected]