  Atiyah-Singer index theorem on a lattice without boundary is well understood
owing to the seminal work by Hasenfratz et al. But its extension to the system
with boundary (the so-called Atiyah- Patodi-Singer index theorem), which plays
a crucial role in T-anomaly cancellation between bulk- and edge-modes in 3+1
dimensional topological matters, is known only in the continuum theory and no
lattice realization has been made so far. In this work, we try to
non-perturbatively define an alternative index from the lattice domain-wall
fermion in 3+1 dimensions. We will show that this new index in the continuum
limit, converges to the Atiyah-Patodi-Singer index defined on a manifold with
boundary, which coincides with the surface of the domain-wall.
