Paper
How to formulate the $\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice
Authors
Sho Araki, Hidenori Fukaya, Tetsuya Onogi, Satoshi Yamaguchi
Abstract
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.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.09354v1</id>\n <title>How to formulate the $\\mathbb{Z}_8$ topological invariant of Majorana fermion on the lattice</title>\n <updated>2026-03-10T08:35:26Z</updated>\n <link href='https://arxiv.org/abs/2603.09354v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.09354v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='hep-lat'/>\n <category scheme='http://arxiv.org/schemas/atom' term='cond-mat.mes-hall'/>\n <published>2026-03-10T08:35:26Z</published>\n <arxiv:comment>10 pages, 1 figure, Talk presented at the 42nd International Symposium on Lattice Field Theory (LATTICE2025), 2-8 November 2025, Tata Institute of Fundamental Research, Mumbai, India</arxiv:comment>\n <arxiv:primary_category term='hep-lat'/>\n <author>\n <name>Sho Araki</name>\n </author>\n <author>\n <name>Hidenori Fukaya</name>\n </author>\n <author>\n <name>Tetsuya Onogi</name>\n </author>\n <author>\n <name>Satoshi Yamaguchi</name>\n </author>\n </entry>"
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