COMPLETENESS OF WILSON LOOP FUNCTIONALS ON THE MODULI SPACE OF SL(2, C) AND SU(1, 1) CONNECTIONS

The structure of the moduli spaces M := A/G of (all, not just flat) SL (2, C) and SU(1, 1) connections on an n-manifold is analysed. For any topology on the corresponding spaces A of all connections which satisf ies the weak requirement of compatibility with the affine structure of A, the moduli space M is shown to be non-Hausdorff. It is then shown that the Wilson loop functionals-4.e. the traces of holonomies of conn ections around closed loops-are complete in the sense that they suffic e to separate all separable points of M. The methods are general enoug h to allow the underlying n-manifold to be topologically non-trivial a nd for connections to be defined on non-trivial bundles. The results h ave implications for canonical quantum general relativity in four and three dimensions.