Giovanni Russo: The nu-invariant of two-step nilmanifolds with closed G2-structure

For every non-vanishing spinor field on a Riemannian 7-manifold, Crowley, Goette, and Nordström introduced the so-called nu-invariant. This is an integer modulo 48, and can be defined in terms of Mathai-Quillen currents, harmonic spinors, and eta-invariants of spin Dirac and odd-signature operator. In general, the role of the nu-invariant is to detect connected components of the moduli space of G_2-structures.

We explain how we compute these data on compact two-step nilmanifolds admitting invariant closed G_2-structures, in particular determining harmonic spinors and relevant symmetries of the spectrum of the spin Dirac operator. This is joint work with Anna Fino and Gueo Grantcharov.