How to formulate the ${\mathbb{Z}}_8$ topological invariant of Majorana fermion on the lattice

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.

Comments: 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