The domain wall number of a UV complete theory of axion is related to the number of PQ-charged heavy quarks that run in the loop. In the case of KSVZ model, $N_{\rm DW}=1$ while in DFSZ, $N_{\rm DW}>1$.
On the other hand, domain walls are topological defects that are formed when discrete symmetry is broken spontaneously.
I find it hard to understand the connection between these two. Is it somehow $Z_{N_{\rm DW}}$ is broken? If so, how is the number of PQ charged quarks related to $Z_{N_{\rm DW}}$ and how is it broken?
I would appreciate any reference or explanation for this.
