What are the definition and examples of topological excitation?

Question

I read topological excitation in wiki, while it's too brief. What is the precise definition of topological excitation? And can give me some examples and explain why they are topological excitation? Are there some references which give explainations in detail ?

Answer 1

There is no better definition than what Wikipedia offers - in general, a topological excitation is a (field) state, i.e. a localized quantity since fields depend on spacetime, whose integral is a topological invariant.

One prime example are Yang-Mills theories in 4D, where the integral $\int \mathrm{Tr}(F\wedge F)$, as essentially the second Chern class of the underlying principal bundle, is a topological invariant and tells you which instanton is the local vacuum belonging to the $F$ in question, since perturbation (being a small and smooth addition) about any given minimum of the action will not change a discrete (topological) quantity. One would then call the instanton the topological excitation, since its value of the action is only this topological quantity. For more on 4D instantons as vacua/mediation between vacua, see my answer here.