Local extrema to the classical Lagrangian field equation are minima and are termed instantons. In classical field theory we do not distinguish between global and local extrema of the action. Can we define a classical vacuum? Would the vacuum be (in my naive understanding) a global minima?
In quantum field theory we do distinguish between such solutions and they lead to their own unique phenomena (which I am still trying to learn).
In addition what does the phrase topological charge refer in the following fascinating answer to a similar physics SE question.
