Selfdual Einstein metrics and conformal submersions

Weyl derivatives, Weyl-Lie derivatives and conformal submersions are defined, then used to generalize the Jones-Tod correspondence between selfdual 4-manifolds with symmetry and Einstein-Weyl 3-manifolds with an abelian monopole. In this generalization, the conformal symmetry is replaced by a particular kind of conformal submersion with one dimensional fibres. Special cases are studied in which the conformal submersion is holomorphic, affine, or projective. All scalar-flat Kahler metrics with such a holomorphic conformal submersion, and all four dimensional hypercomplex structures with a compatible Einstein metric, are obtained from solutions of the resulting ``affine monopole equations''. The ``projective monopole equations'' encompass Hitchin's twistorial construction of selfdual Einstein metrics from three dimensional Einstein-Weyl spaces, and lead to an explicit formula for carrying out this construction directly. Examples include new selfdual Einstein metrics depending explicitly on an arbitrary holomorphic function of one variable or an arbitrary axially symmetric harmonic function. The former generically have no continuous symmetries.Comment: 34 page

Integrable Background Geometries

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.Comment: for Progress in Twistor Theory, SIGM

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

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 K\"ahler, hypercomplex and hyperk\"ahler 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.Comment: for Progress in Twistor Theory, SIGM

Integrability via geometry: dispersionless differential equations in three and four dimensions

We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein-Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.Comment: 26 pages, v2 substantially expanded and restructured, with new theorem on projective property of Lax pair in 3

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

We study the Jones and Tod correspondence between selfdual conformal 4-manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3-manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kahler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the Einstein-Weyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new Einstein-Weyl spaces, which we call Einstein-Weyl ``with a geodesic symmetry'', give rise to hypercomplex structures with two commuting triholomorphic vector fields.Comment: 30 pages, 7 figures, to appear in Ann. Inst. Fourier. 50 (2000

Parabolic subalgebras, parabolic buildings and parabolic projection

Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a reductive Lie algebra, we establish a "parabolic projection" process which sends parabolic subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi subquotient. We indicate how these ideas may be used to study geometric configurations and their moduli.Comment: 26 pages, v2 minor clarification