On the Holonomy of the Coulomb Connection over 3-manifolds with Boundary

Abstract

Narasimhan and Ramadas showed that the Gribov ambiguity was maximal for the product SU(2) bundle over S^3. Specifically they showed that the holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group. Instead of base manifold S^3, we consider here a base manifold with a boundary. In this with-boundary case we must include boundary conditions on the connection forms. We will use the so-called conductor (a.k.a Dirichlet) boundary conditions on connections. With these boundary conditions, we will first show that the space of connections is a C^\infty Hilbert principal bundle with respect to the associated conductor gauge group. We will consider the holonomy of the Coulomb connection for this bundle. If the base manifold is an open subset of R^3 and we use the product principal bundle, we will show that the holonomy group is again a dense subset of the connected component of the identity of the gauge group.