  Gauge anomaly in 4-dimensions can be viewed as a current inflow into an
extra-dimension, where the total phase of the fermion partition function is
given in a gauge invariant way by the Atiyah- Patodi-Singer(APS) eta-invariant
of a 5-dimensional Dirac operator. However, this formalism requires a non-local
boundary condition, with which the physical roles of edge/bulk modes are
unclear and how the causality of the theory is maintained is not obvious. In
this work, we consider a special case where the Dirac operator is in a real
representation and its eta invariant becomes the mod-two type APS index. We
propose a physicist-friendly reformulation of the mod-two index using
domain-wall fermion formalism, which naturally describes how the global anomaly
is canceled between edge and bulk.
