84 research outputs found
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 , 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 -dimensional geometry, such that the gauge group acts
transitively on an -manifold, determines a -dimensional
geometry () fibering over the -dimensional geometry with
as a structure group. In the case of an -dimensional group acting
on itself by the regular representation, all -dimensional geometries
with symmetry group 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 Toda and dKP equations via a
hodograph transformation. In two dimensions, the Hitchin
equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while
the 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
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