The Atiyah–Patodi–Singer Index and Domain-Wall Fermion Dirac Operators

Abstract

We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah–Patodi–Singer index. In a previous paper, motivated by the study of lattice gauge theory, the physicist half of the authors derived a formula expressing the Atiyah–Patodi–Singer index in terms of the eta invariant of domain-wall fermion Dirac operators when the base manifold is a flat 4-dimensional torus. In this paper, we generalise this formula to any even dimensional closed Riemannian manifolds, and prove it mathematically rigorously. Our proof uses a Witten localisation argument combined with a devised embedding into a cylinder of one dimension higher. Our viewpoint sheds some new light on the interplay among the Atiyah–Patodi–Singer boundary condition, domain-wall fermions, and edge modes.

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Notes

  1. 1.

    The discrete spectrum of a self-adjoint operator consists of isolated eigenvalues with finite multiplicity.

  2. 2.

    This does not remain true for pseudodifferential operators [16, Theorem 13.12].

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Acknowledgements

The authors wish to express their gratitude to the organisers of the workshop Progress in the Mathematics of Topological States of Matter, which triggered our collaboration. The authors also wish to express their thanks for helpful discussions during the preparation of this paper to S. Aoki, Y. Hamada, M. Hamanaka, K. Hashimoto, S. Hayashi, N. Kawai, Y. Kikukawa, T. Kimura, Y. Kubota, Y. Matsuki, T. Misumi, M. Mori, H. Moriyoshi, K. Nakayama, T. Natsume, H. Suzuki, and K. Yonekura. The authors are also grateful to the anonymous referees for carefully reading the paper and making many useful suggestions. Hidenori Fukaya is supported in part by JSPS KAKENHI Grant Numbers JP18H01216 and JP18H04484. Mikio Furuta is supported in part by JSPS KAKENHI Grant Number JP17H06461. Shinichiroh Matsuo is supported in part by JSPS KAKENHI Grant Number JP17K14186. Tetsuya Onogi is supported in part by JSPS KAKENHI Grant Number JP18K03620. Satoshi Yamaguchi is supported in part by JSPS KAKENHI Grant Number JP15K05054. Mayuko Yamashita is supported in part by JSPS KAKENHI Grant Number 19J22404.

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Fukaya, H., Furuta, M., Matsuo, S. et al. The Atiyah–Patodi–Singer Index and Domain-Wall Fermion Dirac Operators. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-020-03806-0

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