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vanishes at infinity. The name instanton derives from the fact that these fields are localized in space and (Euclidean) time - in other words, at a specific instant.

The Yang-Mills energy is given by

where * is the Hodge dual. If we insist that the solutions to the Yang-Mills equations have finite energy, then the curvature of the solution at infinity (taken as a limit) has to be zero. This means that the Chern-Simons invariant can be defined at the 3-space boundary. This is equivalent, via Stokes' theorem, to taking the integral

.

This is a homotopy invariant and it tells us which homotopy class the instanton belongs to.

Since the integral of a nonnegative integrand is always nonnegative,

for all real θ. So, this means

If this bound is saturated, then the solution is a BPS state. For such states, either *F = F or *F = − F depending on the sign of the homotopy invariant.