Atiyah–Patodi–Singer index theorem for domain walls

We consider the index of a Dirac operator on a compact even dimensional manifold with a domain wall. The latter is defined as a co-dimension one submanifold where the connection jumps. We formulate and prove an analog of the Atiyah–Patodi–Singer theorem that relates the index to the bulk integral of Pontryagin density and η-invariants of auxiliary Dirac operators on the domain wall. Thus the index is expressed through the global chiral anomaly in the volume and the parity anomaly on the wall. (paper)