Einstein--Weyl geometry, dispersionless Hirota equation and Veronese webs

TL;DR

This paper establishes a link between Einstein-Weyl geometries of hyper-CR type and Veronese webs, showing they can be described by solutions to the dispersionless Hirota equation, and demonstrates construction methods via twistor space deformations.

Contribution

It reveals the local description of hyper-CR Einstein-Weyl structures through solutions to the dispersionless Hirota equation and introduces a construction approach using Kodaira deformations.

Findings

Einstein-Weyl geometries are characterized by solutions to the dispersionless Hirota equation.

Construction of hyper-CR Einstein-Weyl structures via twistor space deformations.

Illustration of the method with a Veronese web on the Heisenberg group.

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 $T\CP^1$, 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.