On the nu-invariant of two-step nilmanifolds with closed G2-structure
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908816888561664 |
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| author | Fino, Anna Grantcharov, Gueo Russo, Giovanni |
| author_facet | Fino, Anna Grantcharov, Gueo Russo, Giovanni |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the nu-invariant of two-step nilmanifolds with closed G2-structure Fino, Anna Grantcharov, Gueo Russo, Giovanni Differential Geometry 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. |
| title | On the nu-invariant of two-step nilmanifolds with closed G2-structure |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2409.06870 |