The gauge field is associated with the connection, which implies that it lives on the link as for the gauge transformation acting on the sites. In contrast, the domain wall we discussed in the recent paper is a degree of freedom defined on the link. Then the gauge field lives on the site. It is dual due to the Kramers-Wannier. Still, the effects of the twisting, that is, topologically, are not trivial. We discussed the scenario topologically for the extended cluster model and generalized it to the long-range hopping models by pivoting. Look at our article, which appears in the Physical Review Letters, “Topological domain-wall pump with Z2 spontaneous symmetry breaking” by Yoshihito Kuno and Yasuhiro Hatsugai, Accepted 7 May, 2025.  See also arXiv:2411.17219.