Instanton

In mathematical physics, the concept of instanton is more complicated in Minkowski space: in this article, we will focus on instantons in 4D Euclidean space.

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 Stoke's theorem, to taking the integral

.

This is a homotopy invariant and it tells us which homotopy class the instanton belongs to. The Yang-Mills energy is given by where * is the Hodge dual.

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.