The twistor construction of toric gravitational instantons

@misc{dunajski2026twistor,
  title = {{The twistor construction of toric gravitational instantons}},
  author = {Maciej Dunajski and Lionel Mason and Paul Tod},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.41867},
  eprint = {2610.41867},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.41867},
}

TY  - UNPB
TI  - The twistor construction of toric gravitational instantons
AU  - Maciej Dunajski
AU  - Lionel Mason
AU  - Paul Tod
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.41867
UR  - https://arxiv.org/abs/2610.41867
AB  - 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.
ER  -