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

For every non-vanishing spinor field on a Riemannian spin seven-manifold, Crowley, Goette, and Nordström 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.