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

Shun K. Kobayashi, Kazuya Yonekura

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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 η-invariant appears as the axial charge of the physical states.

Original languageEnglish
Article number073B01
JournalProgress of Theoretical and Experimental Physics
Issue number7
Publication statusPublished - 2021 Jul 1


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