Atiyah-Patodi-Singer index theorem from axial anomaly

Abstract

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 the 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.

16 pages, 3 figures

References in corpus (3)

• Dai-Freed theorem and topological phases of matter
• Atiyah-Patodi-Singer index from the domain-wall fermion Dirac operator
• The algebra of anomaly interplay

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• Modes of the Sakai-Sugimoto soliton