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arXiv · 2312.16136

Hyper-K\"ahler instantons, symmetries, and flat spaces

Bernardo Araneda
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Abstract

We find all hyper-K\"ahler 4-manifolds admitting conformal K\"ahler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual dyon) in locally flat spaces. The complex structures of different flat pieces are not compatible however, reflecting that the global geometry is not a linear superposition. For either orientation the space must be Gibbons-Hawking (thus excluding the Atiyah-Hitchin metric), and, if the orientations are opposite, it must also be toric and have an irreducible Killing tensor. We also show that the only hyper-K\"ahler 4-metric with a non-constant Killing-Yano tensor is the half-flat Taub-NUT instanton.

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Bernardo Araneda. 2023-12-26. Hyper-K\"ahler instantons, symmetries, and flat spaces. https://arxiv.org/abs/2312.16136

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