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Geometric Transitions with Spin(7) Holonomy via a Dynamical System

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We clarify the global geometry of two 1-parameter families of cohomogeneity one Spin(7) holonomy metrics with generic orbit the Aloff–Wallach space \(N(1,-1) \cong \text {SU}(3)/\text {U}(1)\) and singular orbits \(S^5\) and \({\mathbb {C}}P^{2}\), which at short distance were shown to exist by Reidegeld. The two families fit into the geography of previously known families of cohomogeneity one metrics with exceptional holonomy and provide a Spin(7) analogue of the well-known conifold transition in the setting of Calabi–Yau 3-folds. Furthermore, we discover that there is another transition to families of Spin(7) holonomy metrics which have a similar asymptotic behaviour on one end, but are singular on the other end. We obtain our results by relating the Spin(7)-equations to a simple dynamical system on a 3-dimensional cube.

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Acknowledgements

This work is a result of the author’s PhD thesis and was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London. I want to thank my PhD supervisors Mark Haskins, Jason Lotay and Lorenzo Foscolo for their support, and my PhD examiners Johannes Nordström and Simon Salamon for their helpful comments. In particular, I want to thank Simon Salamon for generously sharing with me the Mathematica code which I have used to create Figs. 2 and 5.

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  1. Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, USA

    Fabian Lehmann

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Correspondence to Fabian Lehmann.

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Communicated by S. Gukov.

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Lehmann, F. Geometric Transitions with Spin(7) Holonomy via a Dynamical System. Commun. Math. Phys. 394, 309–353 (2022). https://doi.org/10.1007/s00220-022-04400-2

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