The index of mesoscopic Wilson--Dirac operators: I

Abstract

This is the first of a series of papers which will be devoted to a study of the index of mesoscopic Wilson--Dirac operators on closed Riemannian manifolds. The aim of the series is to discretise the index of the Dirac operator. On flat tori, this is achieved by the Wilson--Dirac operator of lattice gauge theory on a regular lattice. On a general closed Riemannian manifold, however, no regular lattice is available. As an intermediate step between the continuum Dirac operator and its discretisations, we introduce mesoscopic Wilson--Dirac operators on any closed Riemannian manifold, defined by integrating covariant differences over distances comparable to a fixed length scale. In this paper, we define the index of the mesoscopic Wilson--Dirac operators via spectral flow and prove that it agrees with that of the continuum Dirac operator at all sufficiently small scales. The proof rests on an elliptic estimate uniform in the scale and a Rellich-type compactness theorem. In the second paper, we shall pass from mesoscopic operators to discrete Wilson--Dirac operators.

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BibTeX RIS

BibTeX

@misc{furuta2026index,
  title = {{The index of mesoscopic Wilson--Dirac operators: I}},
  author = {Mikio Furuta and Shinichiroh Matsuo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08266},
  eprint = {2611.08266},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08266},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The index of mesoscopic Wilson--Dirac operators: I
AU  - Mikio Furuta
AU  - Shinichiroh Matsuo
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08266
UR  - https://arxiv.org/abs/2611.08266
AB  - This is the first of a series of papers which will be devoted to a study of the index of mesoscopic Wilson--Dirac operators on closed Riemannian manifolds. The aim of the series is to discretise the index of the Dirac operator. On flat tori, this is achieved by the Wilson--Dirac operator of lattice gauge theory on a regular lattice. On a general closed Riemannian manifold, however, no regular lattice is available. As an intermediate step between the continuum Dirac operator and its discretisations, we introduce mesoscopic Wilson--Dirac operators on any closed Riemannian manifold, defined by integrating covariant differences over distances comparable to a fixed length scale. In this paper, we define the index of the mesoscopic Wilson--Dirac operators via spectral flow and prove that it agrees with that of the continuum Dirac operator at all sufficiently small scales. The proof rests on an elliptic estimate uniform in the scale and a Rellich-type compactness theorem. In the second paper, we shall pass from mesoscopic operators to discrete Wilson--Dirac operators.
ER  -

Abstract

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