The $\bar{\nu}$-Invariant of $G_2$-Structures on Aloff-Wallach Spaces

TL;DR

This paper calculates the $ar{ u}$-invariant for specific homogeneous $G_2$-structures on Aloff-Wallach spaces, providing explicit formulas and comparing different structures.

Contribution

It derives explicit formulas for the $ar{ u}$-invariant of homogeneous nearly-parallel $G_2$-structures on Aloff-Wallach spaces and compares different structures.

Findings

$ar{ u}( ext{structure } extstyle ext{various}) = extstyle ext{mp} ext{ for } ext{each structure}$

Explicit expression for $ar{ u}$ in terms of representation-theoretic data

Comparison of $ar{ u}$-invariants from different geometric structures

Abstract

We compute the $\bar{\nu}$-invariant of homogeneous nearly-parallel $G_2$-structures on Aloff--Wallach spaces $N_{k,l} = SU(3)/S^1_{k,l}$. Using Goette's formulas for the $\eta$-invariants of homogeneous spaces, we derive an explicit expression for $\bar{\nu}$ in terms of representation-theoretic data and show that for the two homogeneous nearly-parallel structures $\varphi^\pm$ on $N_{k,l}$ one has \[\bar{\nu}(\varphi^\pm) = \mp 41.\] Additionally, we compare the $\bar{\nu}$-invariants of the nearly-parallel $G_2$-structures arising from the 3-Sasakian structure.