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Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures

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Abstract

Using twistor methods, we explicitly construct all local forms of four–dimensional real analytic neutral signature anti–self–dual conformal structures (M, [g]) with a null conformal Killing vector. We show that M is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of (M, [g]) by the group action induced by the conformal Killing vector.

We obtain a local classification which branches according to whether or not the conformal Killing vector is hyper-surface orthogonal in (M, [g]). We give examples of conformal classes which contain Ricci–flat metrics on compact complex surfaces and discuss other conformal classes with no Ricci–flat metrics.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK

    Maciej Dunajski & Simon West

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  1. Maciej Dunajski
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  2. Simon West
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Correspondence to Maciej Dunajski.

Additional information

Communicated by G.W. Gibbons

Dedicated to the memory of Jerzy Plebański

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Dunajski, M., West, S. Anti-self-dual Conformal Structures with Null Killing Vectors from Projective Structures. Commun. Math. Phys. 272, 85–118 (2007). https://doi.org/10.1007/s00220-007-0208-4

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