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

Fukaya, Hidenori; Furuta, Mikio; Matsuo, Shinichiroh; Onogi, Tetsuya; Yamaguchi, Satoshi; Yamashita, Mayuko

doi:10.1007/s00220-020-03806-0

Cited by 22 publications

(37 citation statements)

References 23 publications

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“…In [9], three mathematician half of the authors joined the collaboration, and we succeeded in a proof that the above observation of [5] is not a coincidence but generally true. Namely, we proved that for any APS index on a manifold with boundary (we denote X + and its boundary Y ), there exists a domain-wall Dirac operator on a closed manifold without boundary, where its half coincides with X + , and its η invariant is equal to the original APS index.…”

Section: Introductionmentioning

confidence: 77%

“…Recently, the mathematician-half of the authors joined and we succeeded in proving the equivalence of the APS index and the domain-wall η invariant on a general even-dimensional manifold [9]. The main theorem we have proved is Theorem 1.…”

Section: Mathematical Proofmentioning

confidence: 94%

“…(γ 5 (D − M)) 2 = Q, (3.5)which coincides with the AS index. For more mathematical proof, see our paper[9].…”

mentioning

confidence: 99%

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A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

Fukaya¹,

Furuta²,

Matsuo³

et al. 2020

Proceedings of 37th International Symposium on Lattice Field Theory — PoS(LATTICE2019)

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The Atiyah-Patodi-Singer index theorem describes the bulk-edge correspondence of symmetry protected topological insulators. The mathematical setup for this theorem is, however, not directly related to the physical fermion system, as it imposes on the fermion fields a non-local and unnatural boundary condition known as the "APS boundary condition" by hand. In 2017, we showed that the same integer as the APS index can be obtained from the η invariant of the domain-wall Dirac operator. Recently we gave a mathematical proof that the equivalence is not a coincidence but generally true. In this contribution to the proceedings of LATTICE 2019, we try to explain the whole story in a physicist-friendly way.

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Section: Introductionmentioning

confidence: 77%

Section: Mathematical Proofmentioning

confidence: 94%

See 1 more Smart Citation

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

Fukaya¹,

Furuta²,

Matsuo³

et al. 2020

Proceedings of 37th International Symposium on Lattice Field Theory — PoS(LATTICE2019)

Self Cite

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“…To realize this situation, the domain-wall fermion [2,3] was used and it produced the edge-localized mode and bulk Chern-Simons action. The anomaly inflow now covers a wide area of physics such as chiral gauge theory on a lattice [4,5], topological matters [6,7] and index theorem [8,9,10].…”

Section: Introductionmentioning

confidence: 99%

“…In our study, we use the technique based on Ref. [8,9,10] computed both the real and imaginary part of the effective action. Our study reveals the cancellation mechanism for both the real and imaginary part.…”

Section: Introductionmentioning

confidence: 99%

The Anomaly Inflow of the domain-wall fermion in odd dimension

Fukaya¹,

Kawai²,

Matsuki³

et al. 2020

Proceedings of 37th International Symposium on Lattice Field Theory — PoS(LATTICE2019)

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Massless fermions and superconductivity of string-wall composites

Eto,

Suzuki

2024

J. High Energ. Phys.

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An axion cosmic string is known to be a chiral superconductor when the axion couples to an electrically charged fermion. After the QCD phase transition, a QCD axion string is attached by N domain walls. We would like to elucidate the fate of massless fermions on a global string after domain walls attached not only in the axion model but also in general models having string-wall composites. We investigate the Dirac equation under various string-wall composite backgrounds both in the axion(-like) models and in the $$ \mathcal{N} $$ N = 2 supersymmetry inspired Abelian-Higgs models. We give an answer to the elementary question of whether massless fermions exist, and if so, where they are localized. The answer depends on fermion/boson masses in the models, and the massless fermion can be localized either on the string, on one of the domain walls, or in one of the vacua. We find analytic solutions for the fermion zero mode function by which we prove the existence of the massless fermion on the string-wall composites. We also show supercurrents flowing along the string-wall composites and anomalous electric currents flowing in from outside.

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

Cited by 22 publications

References 23 publications

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

The Anomaly Inflow of the domain-wall fermion in odd dimension

Massless fermions and superconductivity of string-wall composites

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