Left-invariant harmonic spinors on three-dimensional Lie groups

TL;DR

This paper investigates conditions under which three-dimensional Lie groups with left-invariant pseudo-Riemannian metrics admit harmonic spinors, revising formulas and classifying metrics supporting such spinors.

Contribution

It provides new criteria and classifications for the existence of left-invariant harmonic spinors on 3D Lie groups, extending previous formulas to pseudo-Riemannian cases.

Findings

Derived conditions for harmonic spinors on 3D Lie groups.

Classified metrics supporting harmonic spinors up to automorphism.

Revised the Dirac operator formula for left-invariant spinors.

Abstract

We study the existence of left-invariant harmonic spinors on three-dimensional Lie groups equipped with a left-invariant pseudo-Riemannian metric. An existing formula for the spin Dirac operator acting on left-invariant spinors in the Riemannian setting is revised and specialised to our cases, in particular to almost Abelian Lie algebras. Focussing on dimension two and three, we find equivalent conditions for the Lie groups to admit left-invariant harmonic spinors in terms of constraints on the structure equations of the corresponding Lie algebras. We then identify those metrics (up to automorphism) carrying left-invariant harmonic spinors in each case.