Yang-Mills homogeneous connections on compact simple Lie groups
Let \(M\) be a compact Riemannian manifold and \(P\) a principal \(G\)-bundle, where \(G\) is a compact Lie group with a fixed bi-invariant Riemannian metric. A Yang-Mills connection is a critical point of the Yang-Mills functional; if the second variation is non-negative, then the connection is said to be stable. The compact Riemannian manifold \(M\) is Yang-Mills unstable if for every choice of \(G\) and \(P\), a stable Yang-Mills connection is flat. In an interesting paper [Math. Z. 193, 165-189 (1986; Zbl 0634.53022)], \textit{S. Kobayashi, Y. Ohnita} and \textit{M. Takeuchi} posed the question: Is every simply connected compact simple Lie group Yang-Mills unstable? The author considers an equivariant \(G\)-bundle \(P\) over a compact connected simple Lie group \(L\), obtained by a Lie homomorphism \(R\). (1), (2) and (3) are the following properties with respect to homogeneous connections on \(P\): (1) \(R\) is indecomposable; (2) flat bihomogeneous connections are only (+)-connections and \((-)\)-connections; (3) the (0)-connection is a unique non-flat Yang-Mills homogeneous connection. Then (1) and (2) are equivalent; (3) implies (1). Moreover, if the image of \(R\) contains a regular element, then any non-flat Yang-Mills homogeneous connection is unstable.
scientific article; zbMATH DE number 949859