topology - Understanding instantons in pure Yang-Mills theory

Yang-Mills Instantons are defined as finite action solutions to the corresponding Euclidean equation of motion. If I understood it correct, then instantons are those classical gauge field configurations $A_\mu(x)$, for which $S_E[A_\mu(x)]<\infty$. Is this statement correct?

There are many such instanton configurations possible, and one configuration cannot be smoothly deformed into another because they belong to distinct topological classes. How many such solutions are possible?

How does a generic instanton solution $A_\mu(x)$ (belonging to a generic class) look like in pure Yang-Mills theory?

Answer

Yes, an instanton is a classical solution to the Euclidean equations of motion with finite action. Its topological charge is given by $k = \frac{1}{8\pi}\int \mathrm{tr}(F\wedge F)$ which is the integral of the divergence of the Chern-Simons current.

There are many different instantons possible. A generic instanton for $\mathrm{SU}(2)$ and topological charge 1 is given by the BPST instanton $$ A_\mu^a(x) = \frac{2}{g}\frac{\eta_{\mu\nu}^a(x-x_0)^\nu}{(x -x_0)^2 - \rho ^2}$$ where $x_0$ is the "center" of the instanton and $\rho$ its scale, also called the radius. The $\eta$ is the 't Hooft symbol.

A large class of instantons of topological charge $k$ may be described as follows: Transforming the BPST instanton by the singular transformation $x^\mu \mapsto \frac{x^\mu}{x^2}$ leads to the expression $$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\frac{\rho^2}{(x-x_0)^2}\right)\right)$$ for the transformed instanton, and one now makes the more general ansatz $$ A_\mu^a(x) = -\eta_{\mu\nu}^a \partial^\nu\left(\ln\left(1+\sum_{l=1}^k \frac{\rho_l^2}{(x-x_{0,l})^2}\right)\right)$$ which leads to an instanton solution of topological charge $k$. This construction can be generalized to other non-Abelian gauge groups.

The generic construction of all instantons on four-dimensional spacetimes of gauge group $\mathrm{SU}(N)$ is given by the ADHM instanton, see also the original paper "Construction of instantons" by Atiyah, Drinfeld, Hitchin and Manin.