On the Holonomy of the Coulomb Connection over Manifolds with Boundary

Abstract

Narasimhan and Ramadas showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle $S^3 \times SU(2)\to S^3$. Instead of a base manifold S^3, we consider here a base manifold of dimension $n\ge 2$ with a boundary and use Dirichlet boundary conditions on connections as defined by Marini. A key step in the method of Narasimhan and Ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy Lie algebra with respect to an appropriate Sobolev norm. Our objective is to explore the effect of the presence of a boundary on this construction of the holonomy Lie algebra. Fixing appropriate Sobolev norms, it will be shown that the space spanned, linearly, by the curvature form at any one connection is never dense in the holonomy Lie algebra. In contrast, the linear space spanned by the curvature form and its first commutators at the flat connection is dense and, in the $C^\infty$ category, is in fact the entire holonomy Lie algebra. The former, negative, theorem is proven for a general principle bundle over $M$, while the latter, positive, theorem is proven only for a product bundle over the closure of a bounded open subset of $\mathbb{R}^n$. Our technique for proving absence of density consists in showing that the linear space spanned by the curvature form at one point is contained in the kernel of a linear map consisting of a third order differential operator, followed by a restriction operation at the boundary; this mapping is determined by the mean curvature of the boundary.