Einstein–Weyl geometry, the dKP equation and twistor theory

By Maciej Dunajski, L. Mason, P. Tod. 2000

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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 −4 d x d t−4u d t 2 , ν=−4u x d t , 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-self-dual conformal structures with symmetries. All four-dimensional hyper-Kahler 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.

Subfields: Astronomy and Astrophysics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2fddd8c40ee19928aa23adae3de8be317579b908