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

SIGMA 10 (2014), 034, 51 pages      arXiv:1403.3471      https://doi.org/10.3842/SIGMA.2014.034
Contribution to the Special Issue on Progress in Twistor Theory

### Integrable Background Geometries

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
This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solution of this equation determines a background geometry on which, for any Lie group $G$, an integrable gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual Yang-Mills theory, while the lower-dimensional structures are nondegenerate (i.e., non-null) reductions of this. Any solution of the gauge theory on a $k$-dimensional geometry, such that the gauge group $H$ acts transitively on an $\ell$-manifold, determines a $(k+\ell)$-dimensional geometry ($k+\ell\leqslant4$) fibering over the $k$-dimensional geometry with $H$ as a structure group. In the case of an $\ell$-dimensional group $H$ acting on itself by the regular representation, all $(k+\ell)$-dimensional geometries with symmetry group $H$ are locally obtained in this way. This framework unifies and extends known results about dimensional reductions of selfdual conformal geometry and the selfdual Yang-Mills equation, and provides a rich supply of constructive methods. In one dimension, generalized Nahm equations provide a uniform description of four pole isomonodromic deformation problems, and may be related to the ${\rm SU}(\infty)$ Toda and dKP equations via a hodograph transformation. In two dimensions, the ${\rm Diff}(S^1)$ Hitchin equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while the ${\rm SDiff}(\Sigma^2)$ Hitchin equation leads to a Euclidean analogue of Plebanski's heavenly equations. In three and four dimensions, the constructions of this paper help to organize the huge range of examples of Einstein-Weyl and selfdual spaces in the literature, as well as providing some new ones. The nondegenerate reductions have a long ancestry. More recently, degenerate or null reductions have attracted increased interest. Two of these reductions and their gauge theories (arguably, the two most significant) are also described.

Key words: selfduality; gauge theory; twistor theory; integrable systems.

pdf (731 kb)   tex (91 kb)

References

1. Ablowitz M.J., Clarkson P.A., Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge, 1991.
2. Ashtekar A., Jacobson T., Smolin L., A new characterization of half-flat solutions to Einstein's equation, Comm. Math. Phys. 115 (1988), 631-648.
3. Atiyah M., Hitchin N., The geometry and dynamics of magnetic monopoles, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988.
4. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
5. Atiyah M.F., Ward R.S., Instantons and algebraic geometry, Comm. Math. Phys. 55 (1977), 117-124.
6. Boyer C.P., Finley J.D., Killing vectors in self-dual, Euclidean Einstein spaces, J. Math. Phys. 23 (1982), 1126-1130.
7. Burtsev S.P., Zakharov V.E., Mikhailov A.V., Inverse scattering method with variable spectral parameter, Theoret. and Math. Phys. 70 (1987), 227-240.
8. Calderbank D.M.J., Selfdual Einstein metrics and conformal submersions, math.DG/0001041.
9. Calderbank D.M.J., Mason L.J., Spinor-vortex geometry and microtwistor theory, unpublished, 2002.
10. 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.
11. Calderbank D.M.J., Tod P., Einstein metrics, hypercomplex structures and the Toda field equation, Differential Geom. Appl. 14 (2001), 199-208, math.DG/9911121.
12. Cartan É., Sur une classe d'espaces de Weyl, Ann. Sci. École Norm. Sup. (3) 60 (1943), 1-16.
13. Chakravarty S., Mason L., Newman E.T., Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group, J. Math. Phys. 32 (1991), 1458-1464.
14. Dancer A.S., Scalar-flat Kähler metrics with ${\rm SU}(2)$ symmetry, J. Reine Angew. Math. 479 (1996), 99-120.
15. Dancer A.S., Strachan I.A.B., Cohomogeneity-one Kähler metrics, in Twistor Theory (Plymouth), Lecture Notes in Pure and Appl. Math., Vol. 169, Dekker, New York, 1995, 9-27.
16. Dubrovin B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
17. Dunajski M., The twisted photon associated to hyper-Hermitian four-manifolds, J. Geom. Phys. 30 (1999), 266-281, math.DG/9808137.
18. Dunajski M., Mason L.J., Tod P., Einstein-Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001), 63-93, math.DG/0004031.
19. Dunajski M., Mason L.J., Woodhouse N.M.J., From 2D integrable systems to self-dual gravity, J. Phys. A: Math. Gen. 31 (1998), 6019-6028, solv-int/9809006.
20. Dunajski M., Tod P., Einstein-Weyl spaces and dispersionless Kadomtsev-Petviashvili equation from Painlevé I and II, Phys. Lett. A 303 (2002), 253-264, nlin.SI/0204043.
21. Gauduchon P., Structures de Weyl-Einstein, espaces de twisteurs et variétés de type $S^1\times S^3$, J. Reine Angew. Math. 469 (1995), 1-50.
22. Gauduchon P., Tod K.P., Hyper-Hermitian metrics with symmetry, J. Geom. Phys. 25 (1998), 291-304.
23. Gegenberg J.D., Das A., Stationary Riemannian space-times with self-dual curvature, Gen. Relativity Gravitation 16 (1984), 817-829.
24. Gibbons G.W., Hawking S.W., Gravitational multi-instantons, Phys. Lett. B 78 (1978), 430-432.
25. Glazebrook J.F., Kamber F.W., Pedersen H., Swann A., Foliation reduction and self-duality, in Geometric Study of Foliations (Tokyo, 1993), Editors T. Mizutani, K. Masuda, S. Matsumoto, T. Inaba, T. Tsuboi, Y. Mitsumatsu, World Sci. Publ., River Edge, NJ, 1994, 219-249.
26. Grant J.D.E., Strachan I.A.B., Hypercomplex integrable systems, Nonlinearity 12 (1999), 1247-1261, solv-int/9808019.
27. Gross M., Wilson P.M.H., Large complex structure limits of $K3$ surfaces, J. Differential Geom. 55 (2000), 475-546, math.DG/0008018.
28. Hashimoto Y., Yasui Y., Miyagi S., Ootsuka T., Applications of the Ashtekar gravity to four-dimensional hyper-Kähler geometry and Yang-Mills instantons, J. Math. Phys. 38 (1997), 5833-5839, hep-th/9610069.
29. 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.
30. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
31. Hitchin N.J., Twistor spaces, Einstein metrics and isomonodromic deformations, J. Differential Geom. 42 (1995), 30-112.
32. Hitchin N.J., Geometrical aspects of Schlesinger's equation, J. Geom. Phys. 23 (1997), 287-300.
33. Hitchin N.J., Hypercomplex manifolds and the space of framings, in The Geometric Universe (Oxford, 1996), Editors S.A. Huggett, L.J. Mason, K.P. Tod, S.T. Tsou, N.M.J. Woodhouse, Oxford University Press, Oxford, 1998, 9-30.
34. Husain V., Self-dual gravity as a two-dimensional theory and conservation laws, Classical Quantum Gravity 11 (1994), 927-937, gr-qc/9310003.
35. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306-352.
36. Jones P.E., Tod K.P., Minitwistor spaces and Einstein-Weyl spaces, Classical Quantum Gravity 2 (1985), 565-577.
37. Joyce D.D., Explicit construction of self-dual 4-manifolds, Duke Math. J. 77 (1995), 519-552.
38. LeBrun C., Spaces of complex geodesics and related structures, Ph.D. Thesis, University of Oxford, 1980.
39. LeBrun C., Explicit self-dual metrics on ${\mathbb {CP}}_2\#\cdots\#{\mathbb {CP}}_2$, J. Differential Geom. 34 (1991), 223-253.
40. Mason L.J., Newman E.T., A connection between the Einstein and Yang-Mills equations, Comm. Math. Phys. 121 (1989), 659-668.
41. 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.
42. Maszczyk R., The classification of self-dual Bianchi metrics, Classical Quantum Gravity 13 (1996), 513-527.
43. Maszczyk R., Mason L.J., Woodhouse N.M.J., Self-dual Bianchi metrics and the Painlevé transcendents, Classical Quantum Gravity 11 (1994), 65-71.
44. Nahm W., The construction of all self-dual multimonopoles by the ADHM method, in Monopoles in Quantum Field Theory (Trieste, 1981), World Sci. Publishing, Singapore, 1982, 87-94.
45. Obata M., Affine connections on manifolds with almost complex, quaternion or Hermitian structure, Jpn. J. Math. 26 (1956), 43-77.
46. Park Q.H., Self-dual gravity as a large-$N$ limit of the 2D nonlinear sigma model, Phys. Lett. B 238 (1990), 287-290.
47. Pedersen H., Poon Y.S., Kähler surfaces with zero scalar curvature, Classical Quantum Gravity 7 (1990), 1707-1719.
48. Pedersen H., Tod K.P., Three-dimensional Einstein-Weyl geometry, Adv. Math. 97 (1993), 74-109.
49. Penrose R., Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), 31-52.
50. Plebański J.F., Some solutions of complex Einstein equations, J. Math. Phys. 16 (1975), 2395-2402.
51. Tafel J., Two-dimensional reductions of the self-dual Yang-Mills equations in self-dual spaces, J. Math. Phys. 34 (1993), 1892-1907.
52. Tafel J., Wójcik D., Null Killing vectors and reductions of the self-duality equations, Nonlinearity 11 (1998), 835-844.
53. Tod K.P., Self-dual Einstein metrics from the Painlevé VI equation, Phys. Lett. A 190 (1994), 221-224.
54. Tod K.P., Cohomogeneity-one metrics with self-dual Weyl tensor, in Twistor Theory (Plymouth), Lecture Notes in Pure and Appl. Math., Vol. 169, Editor S.A. Huggett, Dekker, New York, 1995, 171-184.
55. Tod K.P., Scalar-flat Kähler and hyper-Kähler metrics from Painlevé-III, Classical Quantum Gravity 12 (1995), 1535-1547.
56. Tod K.P., `Special' Einstein-Weyl spaces, Twistor Newsletter 42 (1997), 13-15.
57. Todd J.A., Projective and analytical geometry, Pitman Publishing Corporation, New York, 1946.
58. Ueno T., Integrable field theories derived from 4-D self-dual gravity, Modern Phys. Lett. A 11 (1996), 545-552, hep-th/9508012.
59. Ward R.S., Integrable and solvable systems, and relations among them, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), 451-457.
60. Ward R.S., Einstein-Weyl spaces and $\mathrm{SU}(\infty)$ Toda fields, Classical Quantum Gravity 7 (1990), L95-L98.
61. Ward R.S., Linearization of the ${\rm SU}(\infty)$ Nahm equations, Phys. Lett. B 234 (1990), 81-84.
62. Ward R.S., The ${\rm SU}(\infty)$ chiral model and self-dual vacuum spaces, Classical Quantum Gravity 7 (1990), L217-L222.
63. Weyl H., Space, time, matter, Dover, New York, 1952.
64. Yoshida M., Hypergeometric functions, my love (modular interpretations of configuration spaces), Aspects of Mathematics, Vol. E32, Friedr. Vieweg & Sohn, Braunschweig, 1997.
65. Calderbank D.M.J., Selfdual 4-manifolds, projective structures, and the Dunajski-West construction, SIGMA 10 (2014), 035, 18 pages, math.DG/0606754.
66. Donaldson S., Fine J., Toric anti-self-dual 4-manifolds via complex geometry, Math. Ann. 336 (2006), 281-309, math.DG/0602423.
67. Dunajski M., Harmonic functions, central quadrics and twistor theory, Classical Quantum Gravity 20 (2003), 3427-3440, math.DG/0303181.
68. Dunajski M., Solitons, instantons, and twistors, Oxford Graduate Texts in Mathematics, Vol. 19, Oxford University Press, Oxford, 2010.
69. Dunajski M., Grant J.D.E., Strachan I.A.B., Multidimensional integrable systems and deformations of Lie algebra homomorphisms, J. Math. Phys. 48 (2007), 093502, 11 pages, nlin.SI/0702040.
70. Dunajski M., Krynski W., Einstein-Weyl geometry, dispersionless Hirota equation and Veronese webs, Math. Proc. Cambridge Philos. Soc., to appear, arXiv:1301.0621.
71. Dunajski M., Sparling G., A dispersionless integrable system associated to ${\rm Diff}(S^1)$ gauge theory, Phys. Lett. A 343 (2005), 129-132, nlin.SI/0503030.
72. 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.
73. 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.
74. Ferapontov E.V., Huard B., Zhang A., On the central quadric ansatz: integrable models and Painlevé reductions, J. Phys. A: Math. Theor. 45 (2012), 195204, 11 pages, arXiv:1201.5061.
75. Ferapontov E.V., Kruglikov B., Dispersionless integrable systems in 3D and Einstein-Weyl geometry, arXiv:1208.2728.
76. Fine J., Toric anti-self-dual Einstein metrics via complex geometry, Math. Ann. 340 (2008), 143-157, math.DG/0609487.
77. LeBrun C., Mason L.J., Zoll manifolds and complex surfaces, J. Differential Geom. 61 (2002), 453-535, math.DG/0211021.
78. LeBrun C., Mason L.J., Nonlinear gravitons, null geodesics, and holomorphic disks, Duke Math. J. 136 (2007), 205-273, math.DG/0504582.
79. LeBrun C., Mason L.J., The Einstein-Weyl equations, scattering maps, and holomorphic disks, Math. Res. Lett. 16 (2009), 291-301, arXiv:0806.3761.
80. LeBrun C., Mason L.J., Zoll metrics, branched covers, and holomorphic disks, Comm. Anal. Geom. 18 (2010), 475-502, arXiv:1002.2993.
81. Nakata F., Self-dual Zollfrei conformal structures with $\alpha$-surface foliation, J. Geom. Phys. 57 (2007), 2077-2097, math.DG/0701116.
82. Nakata F., Singular self-dual Zollfrei metrics and twistor correspondence, J. Geom. Phys. 57 (2007), 1477-1498, math.DG/0607276.
83. Nakata F., A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence, Comm. Math. Phys. 289 (2009), 663-699, arXiv:0806.2696.