Einstein–Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein–Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev–Petviashvili (dKP) equation as a special case: If an EW structure admits a constant weighted vector then it is locally given by h = dy2−4dxdt−4udt2, ν = −4uxdt, where u = u(x, y, t) satisfies the dKP equation (ut − uux)x = uyy. Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-selfdual conformal structures with symmetries. All four-dimensional hyper-Kähler metrics in signature (+ + −−) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP-bundle over a Weyl space. It is proved that complex solutions to the dKP equations, modulo a certain coordinate freedom, are in a one-to-one correspondence with minitwistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2)) that admit a section of κ, where κ is the canonical bundle of Z. Real solutions are obtained if the minitwistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ that are invariant under the involution. 1 Three-dimensional Einstein–Weyl spaces The aim of this paper is to study the Einstein–Weyl (EW) equations in relation to integrable systems, and in particular the dispersionless Kadomtsev–Petviashvili equation. We begin by collecting various definitions and formulae concerning three-dimensional Einstein– Weyl spaces (see [26] for a fuller account). In section 2 we construct and characterise a class of new EW structures in 2+1 dimensions out of solutions to the dKP equation. We then show that the dKP solutions give rise to hyper-Kähler metrics in four dimensions. We abuse terminology and call hyper-Kähler (hyper-complex, hyper-Hermitian) metrics which in signature (+ + −−) should be referred to as pseudo-hyper-Kähler (pseudo-hyper-complex, pseudo-hyper-Hermitian). A null vector field (with conformal weight) will play a central role in our discussion so most of our constructions only make sense for Einsetin-Weyl spaces with Lorentzian signature, or complex holomorphic EW spaces (i.e. the complexification of real analytic EW spaces) and for the most part we work with the latter and restrict to a real slice when reality conditions play a role. In section 3 we construct some new examples of EW structures. We obtain all solutions of the dKP equation with the property that the associated EW space admits a family of divergence-free, shear-free geodesic congruences. These solutions give rise to new EW metrics depending on one arbitrary function of one variable. In section 4 a Lax representation of the general EW equations is given, together with a reformulation of the EW equations in terms of a closed and simple two-form on the bundle of spinors. A full twistor characterisation of dKP Einstein–Weyl structures and the corresponding hyper-Kähler metrics will be given in section 5. In section 6 we summarise our present knowledge of conformal reductions of four-dimensional hyper-Kähler metrics in split signature. In the Appendix we show ∗email: [email protected]