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arXiv · 1910.09675

Atiyah-Patodi-Singer index on a lattice

Hidenori Fukaya·Naoki Kawai·Yoshiyuki Matsuki·Makito Mori·Katsumasa Nakayama·Tetsuya Onogi·Satoshi Yamaguchi
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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.

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Hidenori Fukaya, Naoki Kawai, Yoshiyuki Matsuki, Makito Mori, Katsumasa Nakayama, Tetsuya Onogi, Satoshi Yamaguchi. 2019-10-21. Atiyah-Patodi-Singer index on a lattice. https://doi.org/10.1093/ptep%2Fptaa031

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