On the Yang–Mills Functional over Open Four-Manifolds

In this paper we prove that for a large and physically relevant class of smooth complete metrics on open four-manifolds the set of L-norms or energies of (anti-)self-dual finite-energy G-connections is discrete where G is a simple, connected, simply connected and compact Lie group. The proof is based on the Atiyah–Patodi–Singer Index Theorem. To demonstrate our result we consider the Euclidean Schwarzschild manifold. This metric is significantly non-cylindrical hence standard techniques are not applicable for the study of moduli space of instantons in this physically important case. In this case the energy of a (anti-)self-dual connection is an integer.