The complete generating functional in QCD (starting from the most general renormalizable, Lorentz invariant and gauge invariant Lagrangian) given by $$Z_\theta[J]=\int \mathcal{D}A \exp i\int d^4x~ {\rm Tr}\Big[-\frac{1}{2}F_{\mu\nu}F^{\mu\nu}-\frac{\theta g_s^2}{16\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu}+J_\mu A^\mu\Big]\tag{1}$$ is, in general, a functional integral over all gauge field configurations $A_\mu$ with all possible winding numbers. At the perturbative regime of QCD where the strong coupling $g_s\ll 1$, the tunneling amplitude between two different $n$-vacua with winding numbers $n^\prime$ and $n$, given by $$\sim\exp\big[-\frac{8\pi^2|Q|}{g_s^2}\big],~~~Q=n^\prime-n\tag{2}$$ is heavily suppressed. If the tunneling is completely neglected which amounts to setting $g\to 0$, reduces $(1)$ to $$Z_\theta[J]=\int \mathcal{D}A_{Q=0} \exp i\int d^4x~ {\rm Tr}\Big[-\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+J_\mu A^\mu\Big].\tag{3}$$
Question $1$ In Eq.$(3)$, can one restrict the integral to be over gauge field configurations with zero winding number? If so, is it the (a) correct way to look at this?
Question $2$ If this is true, is it fair to assert that the perturbative vacuum is a topological sector with $n=0$?
