Fowdar, U;
(2020)
S¹ -quotient of Spin(7)-structures.
Annals of Global Analysis and Geometry
, 57
(4)
pp. 489-517.
10.1007/s10455-020-09710-z.
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Abstract
If a Spin(7)-manifold N⁸ admits a free S¹ action preserving the fundamental 4-form, then the quotient space M¹ is naturally endowed with a G₂ -structure. We derive equations relating the intrinsic torsion of the Spin(7)-structure to that of the G₂-structure together with the additional data of a Higgs field and the curvature of the S¹-bundle; this can be interpreted as a Gibbons–Hawking-type ansatz for Spin(7)-structures. In particular, we show that if N is a Spin(7)-manifold, then M cannot have holonomy contained in G₂ unless the N is in fact a Calabi–Yau fourfold and M is the product of a Calabi–Yau threefold and an interval. By inverting this construction, we give examples of SU(4) holonomy metrics starting from torsion-free SU(3)-structures. We also derive a new formula for the Ricci curvature of Spin(7)-structures in terms of the torsion forms. We then describe this S¹-quotient construction in detail on the Bryant–Salamon Spin(7) metric on the spinor bundle of S⁴ and on flat R⁸.
| Type: | Article |
|---|---|
| Title: | S¹ -quotient of Spin(7)-structures |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s10455-020-09710-z |
| Publisher version: | https://doi.org/10.1007/s10455-020-09710-z |
| Language: | English |
| Additional information: | © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/). |
| Keywords: | Differential geometrym, Exceptional holonomy, S¹-quotient, G₂-structures, Spin(7)-structures |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10108360 |
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