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

Anna Fino; Gueo Grantcharov; Giovanni Russo

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arXiv · 2409.06870

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

Anna Fino·Gueo Grantcharov·Giovanni Russo

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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.

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BibTeX RIS

Anna Fino, Gueo Grantcharov, Giovanni Russo. 2024-09-10. On the nu-invariant of two-step nilmanifolds with closed G2-structure. https://arxiv.org/abs/2409.06870

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