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= dy 2−4 dx dt−4u dt 2 , ν=−4u x dt , 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-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 mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2) ) that admit a section of κ −1/4, where κ is the canonical bundle of Z . Real solutions are obtained if the mini-twistor space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of κ −1/4 that are invariant under the involution.