How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914381029179392 |
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| author | Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi |
| author_facet | Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi |
| contents | Topological invariants and their associated anomalies have played a crucial role in understanding low-energy phenomena in quantum field theories. In lattice gauge theory, the standard $\mathbb{Z}$-valued Atiyah-Singer index is formulated via the overlap Dirac operator through the Ginsparg-Wilson relation, but extensions to more general topological invariants have remained limited. In this work, we propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant, which takes values in $\mathbb{Z}_8$. The ABK invariant arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-oriented manifolds, and its definition involves an infinite sum over Dirac eigenvalues that must be properly regularized. By carefully treating the boundary conditions, with and without a domain-wall mass term, we demonstrate that the ABK invariant can be extracted from Pfaffians of the Wilson Dirac operator. We further provide numerical verification on two-dimensional lattices, showing that the $\mathbb{Z}_8$-valued results on the torus, Klein bottle, real projective plane, and Möbius strip agree with those in the continuum theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice Araki, Sho Fukaya, Hidenori Onogi, Tetsuya Yamaguchi, Satoshi High Energy Physics - Lattice Mesoscale and Nanoscale Physics Topological invariants and their associated anomalies have played a crucial role in understanding low-energy phenomena in quantum field theories. In lattice gauge theory, the standard $\mathbb{Z}$-valued Atiyah-Singer index is formulated via the overlap Dirac operator through the Ginsparg-Wilson relation, but extensions to more general topological invariants have remained limited. In this work, we propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant, which takes values in $\mathbb{Z}_8$. The ABK invariant arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-oriented manifolds, and its definition involves an infinite sum over Dirac eigenvalues that must be properly regularized. By carefully treating the boundary conditions, with and without a domain-wall mass term, we demonstrate that the ABK invariant can be extracted from Pfaffians of the Wilson Dirac operator. We further provide numerical verification on two-dimensional lattices, showing that the $\mathbb{Z}_8$-valued results on the torus, Klein bottle, real projective plane, and Möbius strip agree with those in the continuum theory. |
| title | How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice |
| topic | High Energy Physics - Lattice Mesoscale and Nanoscale Physics |
| url | https://arxiv.org/abs/2603.09354 |