Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 035, 18 pages      math.DG/0606754      https://doi.org/10.3842/SIGMA.2014.035
Contribution to the Special Issue on Progress in Twistor Theory

Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction

David M.J. Calderbank
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received January 21, 2014, in final form March 18, 2014; Published online March 28, 2014

Abstract
I present a construction of real or complex selfdual conformal 4-manifolds (of signature (2,2) in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex 2-manifold. The 4-manifolds obtained are characterized by the existence of a foliation by selfdual null surfaces of a special kind. The classification by Dunajski and West of selfdual conformal 4-manifolds with a null conformal vector field is the special case in which the gauge group reduces to the group of diffeomorphisms commuting with a vector field, and I analyse the presence of compatible scalar-flat Kähler, hypercomplex and hyperkähler structures from a gauge-theoretic point of view. In an appendix, I discuss the twistor theory of projective surfaces, which is used in the body of the paper, but is also of independent interest.

Key words: selfduality; twistor theory; integrable systems; projective geometry.

pdf (438 kb)   tex (32 kb)

References

  1. Bailey T.N., Eastwood M.G., Gover A.R., Mason L.J., The Funk transform as a Penrose transform, Math. Proc. Cambridge Philos. Soc. 125 (1999), 67-81.
  2. Baston R.J., Eastwood M.G., The Penrose transform: its interaction with representation theory, Oxford Mathematical Monographs, Oxford University Press, Oxford, 1989.
  3. Calderbank D.M.J., Selfdual Einstein metrics and conformal submersions, math.DG/0001041.
  4. Calderbank D.M.J., Integrable background geometries, SIGMA 10 (2014), 034, 51 pages, arXiv:1403.3471.
  5. Calderbank D.M.J., Pedersen H., Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics, Ann. Inst. Fourier (Grenoble) 50 (2000), 921-963, math.DG/9911117.
  6. Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172, math.DG/0102097.
  7. Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (2001), 97-113, math.DG/0001164.
  8. Dubois-Violette M., Structures complexes au-dessus des variétés, applications, in Mathématiques et Physiques (Paris, 1979/1982), Progr. Math., Vol. 37, Birkhäuser Boston, Boston, MA, 1983, 1-42.
  9. Dunajski M., Anti-self-dual four-manifolds with a parallel real spinor, Proc. Roy. Soc. London Ser. A 458 (2002), 1205-1222, math.DG/0102225.
  10. Dunajski M., West S., Anti-self-dual conformal structures with null Killing vectors from projective structures, Comm. Math. Phys. 272 (2007), 85-118, math.DG/0601419.
  11. Dunajski M., West S., Anti-self-dual conformal structures in neutral signature, in Recent Developments in Pseudo-Riemannian Geometry, Editors D.V. Alekseevsky, H. Baum, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, 113-148, math.DG/0610280.
  12. Hitchin N.J., Complex manifolds and Einstein's equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., Vol. 970, Editors H.D. Doebner, T.D. Palev, Springer, Berlin, 1982, 73-99.
  13. Hitchin N.J., Geometrical aspects of Schlesinger's equation, J. Geom. Phys. 23 (1997), 287-300.
  14. LeBrun C., Spaces of complex geodesics and related structures, Ph.D. Thesis, University of Oxford, 1980.
  15. LeBrun C., Mason L.J., Zoll manifolds and complex surfaces, J. Differential Geom. 61 (2002), 453-535, math.DG/0211021.
  16. LeBrun C., Mason L.J., Nonlinear gravitons, null geodesics, and holomorphic disks, Duke Math. J. 136 (2007), 205-273, math.DG/0504582.
  17. Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs. New Series, Vol. 15, The Clarendon Press, Oxford University Press, New York, 1996.
  18. Nakata F., Self-dual Zollfrei conformal structures with $\alpha$-surface foliation, J. Geom. Phys. 57 (2007), 2077-2097, math.DG/0701116.
  19. Pedersen H., Tod P., Einstein metrics and hyperbolic monopoles, Classical Quantum Gravity 8 (1991), 751-760.
  20. Penrose R., Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31-52.
  21. Przanowski M., Killing vector fields in self-dual, Euclidean Einstein spaces with $\Lambda\neq 0$, J. Math. Phys. 32 (1991), 1004-1010.
  22. Tafel J., Two-dimensional reductions of the self-dual Yang-Mills equations in self-dual spaces, J. Math. Phys. 34 (1993), 1892-1907.
  23. Tafel J., Wójcik D., Null Killing vectors and reductions of the self-duality equations, Nonlinearity 11 (1998), 835-844.
  24. Tod K.P., The ${\rm SU}(\infty)$-Toda field equation and special four-dimensional metrics, in Geometry and Physics (Aarhus, 1995), Lecture Notes in Pure and Appl. Math., Vol. 184, Editors J.E. Andersen, J. Dupont, H. Pedersen, A. Swann, Dekker, New York, 1997, 307-312.
  25. Ward R.S., On self-dual gauge fields, Phys. Lett. A 61 (1977), 81-82.


Previous article  Next article   Contents of Volume 10 (2014)