The Arf-Brown-Kervaire invariant on a lattice
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914219803279360 |
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| author | Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi |
| author_facet | Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi |
| contents | We propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant which takes values in $\mathbb{Z}_8$. Compared to the standard $\mathbb{Z}$-valued index, the ABK invariant is more involved in that it arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-orientable manifolds, and its definition contains an infinite sum over Dirac eigenvalues that requires proper regularization. We employ the massive Wilson Dirac operator, with and without domain-walls, on standard two-dimensional square lattices, and use its Pfaffian for the definition. Twisted boundary conditions and cross-caps, which reverse the orientation, are introduced to realize nontrivial topologies equipped with nontrivial $\mathrm{Pin}^{-}$ structures of Majorana fermions. We verify numerically (and partly analytically) that our formulation on a torus, Klein bottle, real projective plane (as well as its triple connected sum), and two types of Möbius strip reproduces the known values in continuum theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Arf-Brown-Kervaire invariant on a lattice Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi High Energy Physics - Lattice Mesoscale and Nanoscale Physics High Energy Physics - Theory Mathematical Physics We propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant which takes values in $\mathbb{Z}_8$. Compared to the standard $\mathbb{Z}$-valued index, the ABK invariant is more involved in that it arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-orientable manifolds, and its definition contains an infinite sum over Dirac eigenvalues that requires proper regularization. We employ the massive Wilson Dirac operator, with and without domain-walls, on standard two-dimensional square lattices, and use its Pfaffian for the definition. Twisted boundary conditions and cross-caps, which reverse the orientation, are introduced to realize nontrivial topologies equipped with nontrivial $\mathrm{Pin}^{-}$ structures of Majorana fermions. We verify numerically (and partly analytically) that our formulation on a torus, Klein bottle, real projective plane (as well as its triple connected sum), and two types of Möbius strip reproduces the known values in continuum theory. |
| title | The Arf-Brown-Kervaire invariant on a lattice |
| topic | High Energy Physics - Lattice Mesoscale and Nanoscale Physics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2512.11424 |