We consider a fermion system on a square lattice, where a curved domain-wall is assigned to the mass term. In a similar way to the standard flat domain-wall fermion, chiral edge modes appear at the wall. These edge-localized modes feel gravity, through the induced spin connection or metric due to the Einstein’s equivalence principle.
In the cases of circle $S^1$ and sphere $S^2$ domain-walls embedded into higher dimensional square lattices of one dimension higher, we numerically confirm the existence of the edge-localized modes and the effect of gravity encoded in the spectrum. With $U(1)$ link variables, we also find a good consistency to the anomaly inflow described by the Atiyah-Patodi-Singer index theorem in the continuum theory. This talk is based on a work with Shoto Aoki, https://arxiv.org/abs/2203.03782 and some preliminary results.