Chiral fermions on the lattice

TL;DR

This paper explores the topological challenges of implementing chiral fermions on lattices, establishing a theorem linking chirality to winding numbers and revealing topological differences for odd numbers of Weyl fermions.

Contribution

It generalizes the understanding of topological obstructions in lattice chiral fermions, extending beyond Ginsparg-Wilson fermions with a new theorem relating chirality and winding numbers.

Findings

Proves a theorem connecting total chirality to winding number differences.

Shows particles and anti-particles occupy topologically distinct spaces for odd Weyl fermions.

Highlights topological obstructions in lattice formulations of chiral fermions.

Abstract

We discuss topological obstructions to putting chiral fermions on an even dimensional lattice. The setting includes Ginsparg-Wilson fermions, but is more general. We prove a theorem which relates the total chirality to the difference of generalised winding numbers of chiral projection operators. For an odd number of Weyl fermions this implies that particles and anti-particles live in topologically different spaces.