Stable Yang–Mills connection


In differential geometry and especially Yang–Mills theory, a stable Yang–Mills connection is a Yang–Mills connection around which the Yang–Mills action functional is positively or even strictly positively curved. Yang–Mills connections are solutions of the Yang–Mills equations following from them being local extrema of the curvature, hence critical points of the Yang–Mills action functional, which are determined by a vanishing first derivative of a variation. stable Yang–Mills connections furthermore have a positive or even strictly positive curved neighborhood and hence are determined by a positive or even strictly positive second derivative of a variation.
stable Yang–Mills connections are named after Yang Chen-Ning and Robert Mills.

Definition

Let be a compact Lie group with Lie algebra and be a principal -bundle with a compact orientable Riemannian manifold having a metric and a volume form. Let be its adjoint bundle., an affine vector space isomorphic to, is the space of connections. These are under the adjoint representation invariant -valued differential forms on and through pullback along smooth sections differ by -valued differential forms on.
The Yang–Mills action functional is given by:
A Yang–Mills connection, hence which fulfills the Yang–Mills equations, is called stable if:
for every smooth family with. It is called weakly stable if only holds. A Yang–Mills connection, which is not weakly stable, is called instable. For comparison, the condition to be a Yang–Mills connection is:
For a stable or instable Yang–Mills connection, its curvature is called a stable or instable Yang–Mills field.

Properties

  • All weakly stable Yang–Mills connections on for are flat. James Simons presented this result without written publication during a symposium on "Minimal Submanifolds and Geodesics" in Tokyo in September 1977.
  • If for a compact -dimensional smooth submanifold in an exists so that:
  • :
  • Every weakly stable Yang–Mills field over with gauge group,, or is either anti self-dual or self-dual.
  • Every weakly stable Yang–Mills field over a compact orientable homogenous Riemannian -manifold with gauge group is either anti self-dual, self-dual or reduces to an abelian field.

Yang–Mills-instable manifolds

A compact Riemannian manifold, for which no principal bundle over it has a stable Yang–Mills connection is called Yang–Mills-instable. For example, the spheres are Yang–Mills-instable for because of the above result from James Simons. A Yang–Mills-instable manifold always has a vanishing second Betti number. Central for the proof is that the infinite complex projective space is the classifying space as well as the Eilenberg–MacLane space. Hence principal -bundles over a Yang–Mills-instable manifold are classified by its second cohomology :
On a non-trivial principal -bundles over, which exists for a non-trivial second cohomology, one could construct a stable Yang–Mills connection.
Open problems about Yang-Mills-instable manifolds include:
  • Is a simply connected compact simple Lie group always Yang-Mills-instable?
  • Is a Yang-Mills-instable simply connected compact Riemannian manifold always harmonically instable? Since for is Yang-Mills-instable, but not harmonically instable, the condition to be simply connected cannot be dropped.

Literature

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