An instanton in $d$ dimensions is often a soliton in $d + 1$ dimensions?

Question

The title of this questions is a "folklore" I've heard from a lot of researchers, but I never understood why this is the case. I know what an instanton and soliton is, respectively in the context they are originally defined (soliton from the KdV/Toda and instanton from the Euclidean YM). Can someone give me a very high level description of how the two are related?

In particularly, I want to know how topology comes into play here. In deriving solitonic equations from KdV, the standard inverse scattering in 1d is used, and I do not see where the "topology" is. In contrast, an instantonic solution of the YM is protected by its Chern class.

Answer 1

The kind of solitons which are meant in this regard are topological solitons which are somewhat different from solitons in non-linear systems (e.g. KdV). Basically, topological solitons are stable time-independent localized solutions of equations of motion of your theory with stability protected by topology (they cannot decay into vacuum through radiation of some particles because soliton and vacuum lie in different topological sectors of the theory). This is a vast subject with classification of solitons occurring in various spacetime dimensions, such as kinks in 1+1, vortices in 2+1, monopoles in 3+1.

Regarding your question, consider some theory in $(d+1)$ dimensions and consider its' spatial part in $d$ dimensions. This spatial part could be viewed as a Euclidean version of another theory, living in $d$ dimensions. Suppose this $d$-dimensional Euclidean theory has instanton solutions, which are localized in $d$ dimensions and are topologically protected, like you mentioned. One could look at these solutions as time-independent solutions of the original $(d+1)$-dimensional theory. Indeed, they solve equations of motion (because time dependence is switched off). Moreover, they are localized in space and stable with stability protected by topology. In other words, they are exactly what people call topological solitons.