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

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.

11 pages. Minor changes. Final version, to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (4)

• Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1
• A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence
• Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy
• Hirota equation and the quantum plane

Cited by in corpus (15)

• The four-dimensional Martinez Alonso--Shabat equation: reductions and nonlocal symmetries
• Symmetry reductions and exact solutions of Lax integrable $3$-dimensional systems
• The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces
• Einstein--Weyl Spaces and Near-Horizon Geometry
• Veronese webs and nonlinear PDEs
• Nonlocal symmetries, conservation laws, and recursion operators of the Veronese web equation
• Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates
• Supersymmetric near-horizon geometry and Einstein-Cartan-Weyl spaces
• On deformations of the dispersionless Hirota equation
• GL(2)-geometry and complex structures
• The Cayley cubic and differential equations
• Webs and the Plebański equation
• A note on the Hyper--CR equation, and gauged $N=2$ supergravity
• Bäcklund transformations as integrable discretization. The geometric approach
• Summary of session A4: Complex and conformal methods in classical and quantum gravity