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 $h=d{y}^2-4dxdt-4ud{t}^2,\nu =-4u_{x}dt$, where $u=u(x,y,t)$ satisfies the dKP equation $(u_t-u{u}_x{)}_x=u_{yy}$. 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-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 $\CP^1$-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 $\mathcal{Z}$ containing a rational curve with normal bundle $Ø(2)$) that admit a section of ${\kappa }^{-1/4}$, where $\kappa$ is the canonical bundle of $\mathcal{Z}$. 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.

22 pages, 1 figure

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