The Arf-Brown-Kervaire invariant on a lattice

TL;DR

This paper develops a lattice-based formulation of the Arf-Brown-Kervaire invariant, enabling numerical computation of this topological invariant in Majorana fermion systems on various non-orientable surfaces.

Contribution

It introduces a novel lattice approach to compute the ABK invariant using the Pfaffian of the Wilson Dirac operator, accommodating non-orientable topologies and Pin^- structures.

Findings

Successfully reproduces known continuum ABK values on various surfaces

Demonstrates the feasibility of lattice computation for complex topological invariants

Provides a framework for future numerical studies of fermionic topological phases

Abstract

We propose a lattice formulation of the Arf-Brown-Kervaire (ABK) invariant which takes values in $\mathbb{Z}_8$. Compared to the standard $\mathbb{Z}$-valued index, the ABK invariant is more involved in that it arises in Majorana fermion partition functions with reflection symmetry on two-dimensional non-orientable 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 $\mathrm{Pin}^{-}$ 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 M\"obius strip reproduces the known values in continuum theory.