Parallel spinors for $\mathrm{G}_2^*$ and isotropic structures
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916661349580800 |
|---|---|
| author | Gil-García, Alejandro Shahbazi, C. S. |
| author_facet | Gil-García, Alejandro Shahbazi, C. S. |
| contents | We obtain a correspondence between irreducible real parallel spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions of an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the Kähler-Atiyah bundle of $(M,g)$. Applying this general framework, we obtain an intrinsic algebraic characterization of $\mathrm{G}_2^*$-structures as well as the first explicit description of isotropic irreducible spinors in signature $(4,3)$ that are parallel under a general connection on the spinor bundle. This description is given in terms of a coherent system of mutually orthogonal and isotropic one forms and follows from the characterization of the stabilizer of an isotropic spinor as the stabilizer of a highly degenerate three-form that we construct explicitly. Using this result, we show that isotropic spinors parallel under a metric connection with torsion exist when the connection preserves the aforementioned coherent system. This allows us to construct a natural class of metrics of signature $(4,3)$ on $\mathbb{R}^7$ that admit spinors parallel under a metric connection with torsion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parallel spinors for $\mathrm{G}_2^*$ and isotropic structures Gil-García, Alejandro Shahbazi, C. S. Differential Geometry Primary 53C27, Secondary 53C10, 53C50, 15A66 We obtain a correspondence between irreducible real parallel spinors on pseudo-Riemannian manifolds $(M,g)$ of signature $(4,3)$ and solutions of an associated differential system for three-forms that satisfy a homogeneous algebraic equation of order two in the Kähler-Atiyah bundle of $(M,g)$. Applying this general framework, we obtain an intrinsic algebraic characterization of $\mathrm{G}_2^*$-structures as well as the first explicit description of isotropic irreducible spinors in signature $(4,3)$ that are parallel under a general connection on the spinor bundle. This description is given in terms of a coherent system of mutually orthogonal and isotropic one forms and follows from the characterization of the stabilizer of an isotropic spinor as the stabilizer of a highly degenerate three-form that we construct explicitly. Using this result, we show that isotropic spinors parallel under a metric connection with torsion exist when the connection preserves the aforementioned coherent system. This allows us to construct a natural class of metrics of signature $(4,3)$ on $\mathbb{R}^7$ that admit spinors parallel under a metric connection with torsion. |
| title | Parallel spinors for $\mathrm{G}_2^*$ and isotropic structures |
| topic | Differential Geometry Primary 53C27, Secondary 53C10, 53C50, 15A66 |
| url | https://arxiv.org/abs/2409.08553 |