I'm stuck in figuring out why some theories of axions predict the existence of domain walls.
Axions are NG bosons associated with the chiral $U(1)_{PQ}$ symmetry, which was spontaneously broken at some energy scale $f_a$. Incidentally, cosmic strings were formed because \begin{align} \pi_1(G/H) = \pi_1(U(1)) = \mathbb{Z} \neq \{\mathbf{1}\} \end{align} where $G$ is the gauge group before the symmetry breaking ($G=U(1)$) and $H$ is the group of elements in $G$ that doesn't change the vacuum expectation value: \begin{align} H = \{ a \in G\ |\ \left<a\phi_0\right> = \left<\phi_0\right> \} \end{align} In the case above, $H = \{\mathbf{1}\}$.
After the PQ symmetry breaking, the QCD instanton effect affects the potential of the axion field. Some theories of axions have more than one minima in its potential: $$ V(a) \propto 1 - \cos(2\pi N_{DW}\theta) $$ Now, the $U(1)$ symmetry is explicitly broken to $\mathbb{Z}_{N_{DW}}$ ($G = U(1)$ and $H = \mathbb{Z}_{N_{DW}}$). Each causally connected patch in the universe chose one of the degenerated minima, and domain walls are formed on their boundaries. However, \begin{align} \pi_0(G/H) = \pi_0(U(1)/\mathbb{Z}_2) = \pi_0(U(1)) = \{\mathbf{1}\} \end{align} which means the manifold $G/H$ is connected and thus no domain wall is formed.
Could anybody tell me what causes this inconsistency?
