Atiyah-Patodi-Singer index on a lattice

Abstract

We propose a non-perturbative formulation of the Atiyah-Patodi-Singer(APS) index in lattice gauge theory, in which the index is given by the $\eta$ invariant of the domain-wall Dirac operator. Our definition of the index is always an integer with a finite lattice spacing. To verify this proposal, using the eigenmode set of the free domain-wall fermion, we perturbatively show in the continuum limit that the curvature term in the APS theorem appears as the contribution from the massive bulk extended modes, while the boundary $\eta$ invariant comes entirely from the massless edge-localized modes.

arXiv comment14 pages, appendices added, details of key equations added, typos corrected, to appear in PTEP

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Cited by in corpus (4)

• Curved domain-wall fermions
• Anomaly and Superconnection
• A lattice version of the Atiyah-Singer index theorem
• $K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators