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Anti-self-dual four-manifolds with a parallel real spinor | | Maciej Dunajski
; | | Date: |
28 Feb 2001 | | Journal: | Proc.Roy.Soc.Lond. A458 (2002) 1205-1222 | | Subject: | Differential Geometry; Mathematical Physics; Exactly Solvable and Integrable Systems MSC-class: 53C28;53A30 | math.DG gr-qc hep-th math-ph math.MP nlin.SI | | Abstract: | Anti-self-dual metrics in the $(++--)$ signature which admit a covariantly constant real spinor are studied. It is shown that finding such metrics reduces to solving a fourth order integrable PDE, and some examples are given. The corresponding twistor space is characterised by existence of a preferred non-zero real section of $kappa^{-1/4}$, where $kappa$ is the canonical line bundle of the twistor space It is demonstrated that if the parallel spinor is preserved by a Killing vector, then the fourth order PDE reduces to the dispersionless Kadomtsev--Petviashvili equation and its linearisation. Einstein--Weyl structures on the space of trajectories of the symmetry are characterised by the existence of a parallel weighted null vector. | | Source: | arXiv, math.DG/0102225 | | Services: | Forum | Review | PDF | Favorites | |
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