$K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915373039747072 |
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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 show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in $K$-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index, our formulation does not require the Ginsparg-Wilson relation and has broader applicability to systems with boundaries and to the mod-two version of the indices in general dimensions. We numerically verify that the $K$ and $KO$ group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah-Singer index on a flat two-dimensional torus and, for the first time, demonstrate that the Atiyah-Patodi-Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23921 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators Aoki, Shoto Fukaya, Hidenori Furuta, Mikio Matsuo, Shinichiroh Onogi, Tetsuya Yamaguchi, Satoshi High Energy Physics - Theory High Energy Physics - Lattice K-Theory and Homology We show that the Wilson Dirac operator in lattice gauge theory can be identified as a mathematical object in $K$-theory and that its associated spectral flow is equal to the index. In comparison to the standard lattice Dirac operator index, our formulation does not require the Ginsparg-Wilson relation and has broader applicability to systems with boundaries and to the mod-two version of the indices in general dimensions. We numerically verify that the $K$ and $KO$ group formulas reproduce the known index theorems in continuum theory. We examine the Atiyah-Singer index on a flat two-dimensional torus and, for the first time, demonstrate that the Atiyah-Patodi-Singer index with nontrivial curved boundaries, as well as the mod-two versions, can be computed on a lattice. |
| title | $K$-theoretic computation of the Atiyah(-Patodi)-Singer index of lattice Dirac operators |
| topic | High Energy Physics - Theory High Energy Physics - Lattice K-Theory and Homology |
| url | https://arxiv.org/abs/2503.23921 |