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 Stokes' 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.
Referenced By
Bogomol'nyi Prasad Sommerfield bound | List of mathematical topics (G-I) | List of mathematical topics (G-Z)
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