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

TL;DR

This paper introduces a lattice formulation for the $ ext{ABK}$ invariant, a $ ext{Z}_8$-valued topological invariant relevant to Majorana fermions, verified through numerical simulations on various 2D manifolds.

Contribution

The paper develops the first lattice formulation of the $ ext{ABK}$ invariant, extending topological invariants in lattice gauge theory beyond the Atiyah-Singer index.

Findings

Successfully extracts the $ ext{Z}_8$ invariant from Pfaffians of the Wilson Dirac operator.

Numerical results on different 2D manifolds match continuum theory predictions.

Provides a method to regularize the sum over Dirac eigenvalues for the $ ext{ABK}$ invariant.

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 M\"obius strip agree with those in the continuum theory.