Einstein–Weyl geometry, the dKP equation and twistor theory
<p style="text-align:justify;"> 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...
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Elsevier
2000
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author | Dunajski, M Mason, L Tod, P |
author_facet | Dunajski, M Mason, L Tod, P |
author_sort | Dunajski, M |
collection | OXFORD |
description | <p style="text-align:justify;"> 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<sup>1</sup>-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.</p> |
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format | Journal article |
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institution | University of Oxford |
last_indexed | 2024-03-07T02:36:40Z |
publishDate | 2000 |
publisher | Elsevier |
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spelling | oxford-uuid:a9042f4e-70ea-481c-afa8-2d10cdf4fb2d2022-03-27T03:05:47ZEinstein–Weyl geometry, the dKP equation and twistor theoryJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a9042f4e-70ea-481c-afa8-2d10cdf4fb2dSymplectic Elements at OxfordElsevier2000Dunajski, MMason, LTod, P <p style="text-align:justify;"> 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<sup>1</sup>-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.</p> |
spellingShingle | Dunajski, M Mason, L Tod, P Einstein–Weyl geometry, the dKP equation and twistor theory |
title | Einstein–Weyl geometry, the dKP equation and twistor theory |
title_full | Einstein–Weyl geometry, the dKP equation and twistor theory |
title_fullStr | Einstein–Weyl geometry, the dKP equation and twistor theory |
title_full_unstemmed | Einstein–Weyl geometry, the dKP equation and twistor theory |
title_short | Einstein–Weyl geometry, the dKP equation and twistor theory |
title_sort | einstein weyl geometry the dkp equation and twistor theory |
work_keys_str_mv | AT dunajskim einsteinweylgeometrythedkpequationandtwistortheory AT masonl einsteinweylgeometrythedkpequationandtwistortheory AT todp einsteinweylgeometrythedkpequationandtwistortheory |