On the nu-invariant of two-step nilmanifolds with closed G2-structure
Anna Fino, Gueo Grantcharov, Giovanni Russo
TL;DR
The paper computes the ν-invariant for specific left-invariant closed G2-structures on compact two-step nilmanifolds by combining spin geometry with Kirillov representation theory. It shows that the space of harmonic spinors is always even-dimensional and contains non-invariant spinors, and that η-invariants vanish via orientation-reversing isometries while the Mathai-Quillen current vanishes for the relevant harmonic spinors. For the h1 and h2 cases, the ν-invariant vanishes on the invariant harmonic spinors for the one- and two-parameter families φ_a and φ_{b1,b2}, respectively. The results provide explicit vanishing of ν in these non-integrable settings and illustrate a workflow to analyze ν on higher-step nilmanifolds using Kirillov theory and invariant geometric data.
Abstract
For every non-vanishing spinor field on a Riemannian spin seven-manifold, Crowley, Goette, and Nordström defined the so-called $ν$-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 $ν$-invariant can be defined in terms of Mathai--Quillen currents, harmonic spinors, and $η$-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 $ν$ on invariant harmonic spinors.