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: $$ dF=0, \quad d\star F=0 $$

  1. 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.

  2. 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 being decoupled with 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?


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