Instanton
In
quantum field theory, an
instanton is a
topologically nontrivial field configuration in four-dimensional
Euclidean space (considered as the
Wick rotation of
Minkowski spacetime). Specifically, it refers to a Yang-Mills
gauge field A which locally approaches pure gauge at spatial
infinity. This means the field strength defined by
A,
-
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.