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S¹ -quotient of Spin(7)-structures

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. Green open access

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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 > 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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