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Arf-Brown-Kervaire invariant on a lattice

Sho Araki, Hidenori Fukaya, Tetsuya Onogi, and Satoshi Yamaguchi

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Phys. Rev. D 114, 014503 – Published 6 July, 2026

DOI: https://doi.org/10.1103/wyp3-bptq

Abstract

We propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant which takes values in Z8. Compared to the standard Z-valued index, the ABK invariant is more involved in that it arises in Majorana fermion partition functions with reflection symmetry on two-dimensional nonorientable 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 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.

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Phys. Rev. D 114, 014503– Published 6 July, 2026
  • Received 3 January 2026
  • Accepted 2 June 2026
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