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 of the corresponding spaces A of all connections which satisfies 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-i.e. the traces of holonomies of connections around closed loops-are complete in the sense that they suffice to separate all separable points of M. The methods are general enough to allow the underlying n-manifold to be topologically non-trivial and for connections to be defined on non-trivial bundles. The results have implications for canonical quantum general relativity in four and three dimensions.