$η$ invariant of massive Wilson Dirac operator and the index
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arXiv
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| Autori principali: | , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929688740364288 |
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| author | Aoki, Shoto Fukaya, Hidenori Furuta, Mikio Matsuo, Shinichiroh Onogi, Tetsuya Yamaguchi, Satoshi |
| author_facet | Aoki, Shoto Fukaya, Hidenori Furuta, Mikio Matsuo, Shinichiroh Onogi, Tetsuya Yamaguchi, Satoshi |
| contents | We revisit the lattice index theorem in the perspective of $K$-theory. The standard definition given by the overlap Dirac operator equals to the $η$ invariant of the Wilson Dirac operator with a negative mass. This equality is not coincidental but reflects a mathematically profound significance known as the suspension isomorphism of $K$-groups. Specifically, we identify the Wilson Dirac operator as an element of the $K^1$ group, which is characterized by the $η$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $η$-invariant equals to the index of the continuum Dirac operator. Our results indicate that the Ginsparg-Wilson relation and the associated exact chiral symmetry are not essential for understanding gauge field topology in lattice gauge theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $η$ invariant of massive Wilson Dirac operator and the index Aoki, Shoto Fukaya, Hidenori Furuta, Mikio Matsuo, Shinichiroh Onogi, Tetsuya Yamaguchi, Satoshi High Energy Physics - Lattice Strongly Correlated Electrons High Energy Physics - Theory K-Theory and Homology We revisit the lattice index theorem in the perspective of $K$-theory. The standard definition given by the overlap Dirac operator equals to the $η$ invariant of the Wilson Dirac operator with a negative mass. This equality is not coincidental but reflects a mathematically profound significance known as the suspension isomorphism of $K$-groups. Specifically, we identify the Wilson Dirac operator as an element of the $K^1$ group, which is characterized by the $η$-invariant. Furthermore, we prove that, at sufficiently small but finite lattice spacings, this $η$-invariant equals to the index of the continuum Dirac operator. Our results indicate that the Ginsparg-Wilson relation and the associated exact chiral symmetry are not essential for understanding gauge field topology in lattice gauge theory. |
| title | $η$ invariant of massive Wilson Dirac operator and the index |
| topic | High Energy Physics - Lattice Strongly Correlated Electrons High Energy Physics - Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2501.02873 |