Self-dual Zollfrei conformal structures with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>α</mml:mi></mml:math>-surface foliation

Nakata, Fuminori

doi:10.1016/j.geomphys.2007.05.004

Cited by 6 publications

(10 citation statements)

References 15 publications

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“…The quotient is a surface with a natural projective structure structure, and the gauge field equations are projectively invariant on this background geometry. In [81] the global Mason-LeBrun theories are related by this construction. More recently, similar methods have been applied in Einstein-Weyl geometry [79,83].…”

Section: Addendamentioning

confidence: 99%

Integrable Background Geometries

Calderbank¹

2014

SIGMA

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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., nonnull) reductions of this. Any solution of the gauge theory on a k-dimensional geometry, such that the gauge group H acts transitively on an -manifold, determines a (k + )-dimensional geometry (k + 4) fibering over the k-dimensional geometry with H as a structure group. In the case of an -dimensional group H acting on itself by the regular representation, all (k + )-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 SU(∞) Toda and dKP equations via a hodograph transformation. In two dimensions, the Diff(S 1 ) Hitchin equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while the SDiff(Σ 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.

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Section: Addendamentioning

confidence: 99%

Integrable Background Geometries

Calderbank¹

2014

SIGMA

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“…As in (6.16), each α-surface S q is defined by the equation x = A (q)y for each q ∈ S 3 ⊂ R 4 . In the coordinate (s, t, y) ∈ S 1 × R × S 2 ≃ M, this equation is equivalent to the following system: 20) tanh t = (0, 0, 1)A (q) · y. (6.21)…”

Section: Standard Modelmentioning

confidence: 99%

“…The twistor theory concerning holomorphic disks, developed by C. LeBrun and L. J. Mason, is now progressing steadily (see [14,15,16,17,19,20,21]). In general, LeBrun-Mason type twistor correspondence is characterized in the following way:…”

Section: Introductionmentioning

confidence: 99%

“…The results in this article is also considered as the LeBrun-Mason theory version of the Jones-Tod reduction theory [10]. In contrast, in [19,20], the author studied the LeBrun-Mason theory version of the Dunajski-West reduction theory [2,4]. Particularly in [19], we obtain infinitely many self-dual indefinite Zollfrei conformal structures on S 2 × S 2 with singularity, and their LeBrun-Mason correspondences are established by making use of the Radon transform on R 2 .…”

Section: Introductionmentioning

confidence: 99%

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Wave equations and the LeBrun-Mason correspondence

Nakata

2012

Trans. Amer. Math. Soc.

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The LeBrun-Mason twistor correspondences for S 1 -invariant self-dual Zollfrei metrics are explicitly established. We give explicit formulas for the general solutions of the wave equation and the monopole equation on the de Sitter three-space under the assumption for the tameness at infinity by using Radon-type integral transforms, and the above twistor correspondence is described by using these formulas. We also obtain a critical condition for the LeBrun-Mason twistor spaces, and show that the twistor theory does not work well for twistor spaces which do not satisfy this condition. (2000) : 53C28, 35L05, 53C50, 32G10. Mathematics Subject Classifications

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“…Since this article first appeared on the arXiv, F. Nakata has developed a global approach [18] relating β-surface reduction to the work of C. LeBrun and L. Mason [15,16].…”

Section: Introductionmentioning

confidence: 99%

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

Calderbank¹

2014

SIGMA

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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.

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Self-dual Zollfrei conformal structures withα-surface foliation

Cited by 6 publications

References 15 publications

Integrable Background Geometries

Integrable Background Geometries

Wave equations and the LeBrun-Mason correspondence

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

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