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

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...

Full description

Saved in:
Bibliographic Details
Published in:Communications in mathematical physics 2020-12, Vol.380 (3), p.1295-1311
Main Authors: Fukaya, Hidenori, Furuta, Mikio, Matsuo, Shinichiroh, Onogi, Tetsuya, Yamaguchi, Satoshi, Yamashita, Mayuko
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03806-0