Yang-Mills gauge theory is given by the action $$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$ whose Euler-Lagrange equations are the classical equations of motion. The classical solutions are stationary points of this functional under variation. The $M$ is the spacetime base manifold. The $\mathrm{Tr}_\mathfrak{g}$ is the trace over the Lie algebra $\mathfrak{g}$ of the principle compact Lie group $G$-bundle. The $F$ is the field strength of the connection $A$.

Equations of motion are: $$ dF=0, \quad d\star F=0 $$

It is known that one way to satisfy the classical equations of motion are equivalent to $$\star F = \pm F,$$ i.e. the classical solutions of instanton equation to the equation of motion.

It is said that the self-dual $F^+$ and an anti-self-dual $F^-$ are orthogonal to each other, say $$\star F_{\pm} = \pm F_{\pm},$$ w.r.t. the inner product defined by $ (f_1,f_2) = \int f_1\wedge\star f_2$.

Altogether this gives $S_\text{YM}[A] = \int \mathrm{Tr}_\mathfrak{g}(F^+\wedge \star F^+) + \int \mathrm{Tr}_\mathfrak{g}(F^-\wedge \star F^-).$ The second Chern class $$C_2(A) :=\frac{1}{8 \pi^2} \int\mathrm{Tr}_\mathfrak{g}(F\wedge F),$$ (I hope I get the normalization correct.) Notice that $$S_\text{YM}[A] \geq 8 \pi^2 \lvert C_2(A)\rvert ,$$ is locally minimized when the equality holds.

The equality holds exactly when either $F^+ = 0$ or $F^- = 0$, i.e. when the full field strength $F$ is itself either self-dual $F^+$ or anti-self-dual $F^-$, or zero field strength.

Question: At the classical level, it looks that the self-dual instanton configuration ($\star F_{+} = + F_+$) and anti-self-dual instanton configuration ($\star F_{-} = - F_-$) are totally

decoupled. However, at the quantum level, in terms of the Yang-Mills functional $Z= \int [DA]\exp[- S_\text{YM}[A]] = \int [DA]\exp[- \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)]$, do the self-dual instanton and anti-self-dual instanton configurations interact with each other in some nontrivial way? Namely, are there some configurations of $F_{+}$ and $F_{-}$, that they interplay with each other, instead of beingdecoupledwith each other, at

the

*even*classical level (no need for the path integral but only at the classical solutions)or only at the quantum level?

If so, what are the examples? How do they alter the classical theory or quantum theory of path integral?