Einstein-Weyl geometry, the dKP equation and twistor theory

Abstract

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 $h=d y^2-4d xd t-4ud t^2, \nu=-4u_xd t$, where $u=u(x, y, t)$ satisfies the dKP equation $(u_t-uu_x)_x=u_{yy}$. Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-K\"ahler 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 $\CP^1$-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 ${\cal Z}$ containing a rational curve with normal bundle $\O(2)$) that admit a section of $\kappa^{-1/4}$, where $\kappa$ is the canonical bundle of ${\cal Z}$. 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 $\kappa^{-1/4}$ that are invariant under the involution.