Consider $SU(2)$ gauge theory. The classical ground state is $F^a_{μν}=0$ . This implies that the vector potential $A^a_μ=U∂_μU^†$. Here $U(x)$ is an element of the gauge group. Now suppose that $U_0(x)$ and $U_1(x)$ are from different homotopy classes. How can we proof that $A^a_{0μ}=U_0(x)∂_μU_0^†$ and $A^a_{1μ}=U_1(x)∂_μU_1^†$ belongs to two different classical vacua that is how can we proof that there is a potential barrier between them.

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    $\begingroup$ This does not make much sense as written: If the $A_i = U_i^{-1}\mathrm{d}U_i$ are globally defined, then they cannot come from different "homotopy classes". If they are singular at some point, then the issue becomes a bit more subtle. In any case, this question is lacking sufficient context to be able to tell what exactly you are talking about - precision is key in this area of gauge theory (come to think of it, it's key everywhere, really). Maybe look at my answers physics.stackexchange.com/a/159200/50583, physics.stackexchange.com/a/317273/50583 to see a bit of the subtlety. $\endgroup$
    – ACuriousMind
    Commented May 17, 2017 at 9:34
  • $\begingroup$ I am trying to understand the $\theta$ vacua. Most books just say that gauge fields with different winding number belongs to different vacua . I am not seeing how $\endgroup$ Commented May 17, 2017 at 9:47


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