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\(G_2\)-Instantons on the Spinor Bundle of the 3-Sphere

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Abstract

We classify \(G_2\)-instantons admitting \(SU (2)^3\)-symmetries, and construct a new family of examples on the spinor bundle of the 3-sphere, equipped with the asymptotically conical, co-homogeneity one \(G_2\)-metric of Bryant–Salamon. We also show that outside of the \(SU (2)^3\)-invariant examples, any other \(G_2\)-instanton on this metric with the same asymptotic behaviour must have obstructed deformations.

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Notes

  1. See [23] for a discussion of constructing examples near the collapsed limit.

  2. Since \(\lbrace 1 \rbrace \times \lbrace 1 \rbrace \times SU (2)\) acts trivially on the singular orbit, we can identify the singular orbit \(SU (2)^3 / \Delta _{1,2} SU (2) \times SU (2)\) with \(SU (2)^2 / \Delta SU (2)\) in the same way.

  3. These \(SU (2)^2\)-equivariant bundles are referred to, respectively, as \(P_\textrm{Id}\) and \(P_1\) in [16].

  4. See [16, Theorem 2] for an explanation of this limit in terms of the bubbling and removable-singularity phenomenon found in [24].

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Acknowledgements

Special thanks to Simon Salamon, Gonçalo Oliveira, Lorenzo Foscolo, Jason Lotay, and Johannes Nordström for their helpful comments and discussions. The first author was funded by the Royal Society, through a studentship supported by the Research Fellows Enhancement Award 2017 RGF-EA-180171, and the EPSRC through the UCL Research Associates Award EP-W522636-1. The second author was funded by the EPSRC Studentship 2106787 and the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics #488631.

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  1. University College London, London, UK

    Jakob Stein

  2. University of Bath, Bath, UK

    Matt Turner

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Stein, J., Turner, M. \(G_2\)-Instantons on the Spinor Bundle of the 3-Sphere. J Geom Anal 34, 149 (2024). https://doi.org/10.1007/s12220-024-01573-1

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