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arXiv · 2610.41867

The twistor construction of toric gravitational instantons

Maciej Dunajski·Lionel Mason·Paul Tod
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Abstract

We give an explicit construction of all ALF and ALE toric gravitational instantons in terms of certain rational matrix functions of one complex variable on the Riemann sphere; these provide patching matrices for their corresponding twistor holomorphic vector bundles. The reconstruction of the instantons arises from the solution to a Riemann--Hilbert splitting problem and we show how this can be solved via linear algebra, resembling but not equivalent to the Belinski--Zakharov ansatz. This construction gives a complete classification together with an explicit form for all toric gravitational instantons up to the resolutions of conical and orbifold singularities. We illustrate the formalism by computer--assisted numerical construction of two new ALF (but not AF) gravitational instantons with five nodes.

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Maciej Dunajski, Lionel Mason, Paul Tod. 2026-10-30. The twistor construction of toric gravitational instantons. https://arxiv.org/abs/2610.41867

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