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Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

Published online by Cambridge University Press:  24 April 2014

MACIEJ DUNAJSKI  and
WOJCIECH KRYŃSKI
MACIEJ DUNAJSKI
Affiliation:
Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeWilberforce Road, Cambridge CB3 0WA. e-mail: [email protected]
WOJCIECH KRYŃSKI
Affiliation:
Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeWilberforce Road, Cambridge CB3 0WA, UK, and Institute of Mathematics of the Polish Academy of SciencesŚniadeckich 8, 00-956 Warsaw, Poland. e-mail: [email protected]
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Abstract

We exploit the correspondence between the three–dimensional Lorentzian Einstein–Weyl geometries of the hyper–CR type and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper–CR Einstein–Weyl structures by Kodaira deformations of the flat twistor space $T\mathbb{CP}^1$, and how to recover the pencil of Poisson structures in five dimensions illustrating the method by an example of the Veronese web on the Heisenberg group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 1 , July 2014 , pp. 139 - 150
Copyright
Copyright © Cambridge Philosophical Society 2014 

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