The Atiyah-Patodi-Singer index theorem from the axial anomaly

We give a very simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its small generalization by using the path integral of massless Dirac fermions. It is based on Fujikawa's argument for the relation between the axial anomaly and the Atiyah-Singer index theorem, and only a minor modification of that argument is sufficient to show the APS index theorem. The key ingredient is the identification of the APS boundary condition and its generalization as physical state vectors in the Hilbert space of the massless fermion theory. The APS eta-invariant appears as the axial charge of the physical states.

Automated summary

The paper provides a simple derivation of the Atiyah-Patodi-Singer (APS) index theorem and its generalization using the path integral of massless Dirac fermions. It shows that APS boundary conditions and their generalizations can be identified as physical state vectors in the Hilbert space of the massless fermion theory, with the APS eta-invariant representing the axial charge of the physical states.