A Yang–Mills Inequality for Compact Surfaces

We consider the Yang–Mills action functional S_YM on the infinite-dimensional space of connections on a bundle over a compact surface. We find a lower bound for S_YM(ω) in terms of holonomies of the connection ω, the topology of the surface, and the topology of the bundle. An intermediate 'energy inequality' becomes an equality if and only if the connection is a critical point of the Yang–Mills action. Yang–Mills minima can also be understood using these inequalities.