Harmonic $G_2$-structures on almost Abelian Lie groups

TL;DR

This paper classifies harmonic $G_2$-structures on almost Abelian Lie groups by analyzing torsion forms and algebraic conditions on the Lie algebra's bracket, identifying which torsion classes are compatible with harmonicity.

Contribution

It provides a complete algebraic characterization of harmonic $G_2$-structures on almost Abelian Lie groups, including the admissibility of torsion classes and explicit criteria for harmonicity.

Findings

Four torsion classes are not admissible due to torsion form constraints.

Algebraic conditions on the Lie algebra bracket determine torsion class compatibility.

Criteria for harmonic $G_2$-structures are explicitly derived.

Abstract

We consider left-invariant $G_2$-structures on $7$-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket $A$ of the corresponding Lie algebra. In those terms, we establish the algebraic condition on $A$ for each of the possible $16$-torsion classes of a $G_2$-structure. In particular, we show that four of those torsion classes are not admissible, since $\tau_3=0$ implies $\tau_0=0$. Finally, we use the above results to provide the algebraic criteria on $A$, satisfying the harmonic condition $div T=0$, and this allows to identify which torsion classes are harmonic.