On the holonomy of the Coulomb connection over manifolds with boundary

Abstract

Narasimhan and Ramadas showed in [16] 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 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 boundary conditions on connections. With these boundary conditions, we will first show that the space of connections is a C ∞ 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 gauge group.We will be concerned with boundary conditions on p-forms that have been dubbed conductor boundary conditions by Gross in [8]. We say a form ω satisfies conductor boundary conditions if i * (ω) = 0 where i : ∂M → M is the inclusion map. In other words ω(X 1 ∧ . . . ∧ X p ) = 0 if X 1 , . . . , X p are all tangent to the boundary. This is half of the "relative" boundary conditions given by Ray and Singer in [19], and the boundary conditions of the "Dirichlet problem" of Marini in [13].We can extend the notion of conductor boundary conditions to connections. Given an Ehresmann connection ω A on P , we consider the induced Koszul connection ∇ A on the vector bundle E. Given a fixed connection A 0 on P , we say that a connection ∇ A satisfies conductor boundary condtions with respect to ∇ A0 if the 1-form ∇ A − ∇ A0 satisfies conductor boundary conditions. Such a ∇ A equals ∇ A0 on the boundary in tangential directions, giving us a Dirichletlike boundary condition (hence the terminology found in [13]). If we restrict our view to connections satisfying conductor boundary conditions with respect to a fixed A 0 , we must change the gauge group so that it preserves the boundary conditions. A gauge transformation g ∈ G is a section of certain bundle over M with fibers diffeomorphic to K. The conductor gauge group G con consists of those gauge transformations g such that g| ∂M ≡ e, where e is the identity of K. Note that the definition of G con does not depend on the fixed connection ∇ A0 .

The Yang-Mills Heat Semigroup on Three-Manifolds with Boundary

“…Moreover, choosing φ(P ) = e K and U small, one can ensure that φ takes its values in a contractible neighborhood of e K in K and therefore g can be extended to all of M. For a nontrivial bundle over M the boundary conditions B norm = 0 and B tan = 0 are both well defined, as opposed to A norm = 0 and A tan = 0. This has been observed and used by W. Gryc, [20], in his work extending the no-section theorem of Narasimhan and Ramadas, [39], to manifolds with boundary.…”

“…We resume the assumption that M is a compact Riemannian 3-manifold with smooth boundary and assume now that A and ω are in W 1 (M). We will write 20) and, by Sobolev's inequality, there exists a constant κ, depending on the geometry of M but not on A, such that ω 2…”

The Li–Yau–Hamilton estimate and the Yang–Mills heat equation on manifolds with boundary

“…Namely, one may formulate the boundary conditions for the connection ∇(t) directly. The paper [7] takes this particular standpoint; see also [26,15]. It may or may not be more natural to impose the boundary conditions on ∇(t) than to impose ones on R ∇(t) depending on the considered problem and the chosen perspective.…”

“…The results in Section 3 prevail regardless of whether we choose (1.3) or (1.4) to hold on ∂M . Other ways to introduce the boundary conditions in the context of Yang-Mills theory were considered in several works including, for example, [26,36,38,15,7]. We should mention, however, that none of these works except [7] deals with parabolic-type equations like (1.2).…”