Completeness of Wilson loop functionals on the moduli space of $SL(2,C)$ and $SU(1,1)$-connections

TL;DR

This paper investigates the structure of moduli spaces of $SL(2,C)$ and $SU(1,1)$ connections, showing Wilson loop functionals are complete for separating points, with implications for quantum gravity.

Contribution

It proves Wilson loop functionals are complete on the moduli space of certain connections, extending understanding of their structure and applications in quantum gravity.

Findings

Moduli spaces are non-Hausdorff under natural topologies.

Wilson loop functionals separate all separable points.

Results apply to non-trivial bundles and topologically complex manifolds.

Abstract

The structure of the moduli spaces $\M := \A/\G$ of (all, not just flat) $SL(2,C)$ and $SU(1,1)$ connections on a n-manifold is analysed. For any topology on 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 4 and 3 dimensions.