On the nu-invariant of two-step nilmanifolds with closed G2-structure

TL;DR

This paper computes the nu-invariant for specific closed G2-structures on two-step nilmanifolds, revealing conditions for the existence of harmonic spinors and the invariant's vanishing on invariant cases.

Contribution

It provides explicit calculations of the nu-invariant for certain nilmanifolds with closed G2-structures and analyzes harmonic spinors in this context, highlighting new geometric insights.

Findings

Existence of non-invariant harmonic spinors on these manifolds

Determination of the parity of harmonic spinor spaces

Vanishing of the nu-invariant on invariant harmonic spinors

Abstract

For every non-vanishing spinor field on a Riemannian spin seven-manifold, Crowley, Goette, and Nordstr\"om defined the so-called $\nu$-invariant. This is an integer modulo $48$ that detects connected components of the moduli space of $\mathrm G_2$-structures on any seven-dimensional oriented spin manifold. The $\nu$-invariant can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $\eta$-invariants of spin Dirac and odd-signature operator. We compute these data for certain families of left-invariant closed $\mathrm G_2$-structures on compact two-step nilmanifolds with their natural spin structure. Specifically, we establish the existence of non-invariant harmonic spinors and determine the parity of the dimension of the space of harmonic spinors. We deduce the vanishing of $\nu$ on invariant harmonic spinors.