Understanding the index theorems with massive fermions

The index theorems relate the gauge field and metric on a manifold to solution of Dirac equation it. In standard approach, operator must be massless make chirality well defined. physics, however, theorem appears as consequence chiral anomaly, which is an explicit breaking symmetry. It then natural ask if we can understand in massive fermion system does not have this review, discuss how reformulate anomaly with operators, where find nontrivial mathematical relations between fermions. A special focus placed Atiyah–Patodi–Singer index, whose original formulation requires physicist-unfriendly boundary condition, while corresponding domain-wall reformulation not. provides understanding inflow bulk edge particle condensed matter physics.