Integrable Background Geometries

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 ($k+\ell ⩽4$) 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 $\mathrm{SU}(\infty )$ Toda and dKP equations via a hodograph transformation. In two dimensions, the $\mathrm{Diff}(S^1)$ Hitchin equation is shown to be equivalent to the hyperCR Einstein-Weyl equation, while the $\mathrm{SDiff}({\Sigma }^2)$ Hitchin equation leads to a Euclidean analogue of Plebanski's heavenly equations.

for Progress in Twistor Theory, SIGMA

References in corpus (17)

• Einstein-Weyl geometry, the dKP equation and twistor theory
• Einstein--Weyl spaces and dispersionless Kadomtsev--Petviashvili equation from Painlevé I and II
• Self-dual gravity as a two dimensional theory and conservation laws
• The Twisted Photon Associated to Hyper-Hermitian Four-Manifolds
• On the central quadric ansatz: integrable models and Painleve reductions
• Anti-self-dual conformal structures with null Killing vectors from projective structures
• Hypercomplex Integrable Systems
• From 2D Integrable Systems to Self-Dual Gravity
• Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
• Selfdual Einstein metrics and conformal submersions
• A construction of Einstein-Weyl spaces via LeBrun-Mason type twistor correspondence
• Multidimensional integrable systems and deformations of Lie algebra homomorphisms
• Applications of the Ashtekar gravity to four dimensional hyperkähler geometry and Yang-Mills Instantons
• Self-dual Zollfrei conformal structures with alpha-surface foliation
• A dispersionless integrable system associated to Diff$(S^1)$ gauge theory
• Toric anti-self-dual 4-manifolds via complex geometry
• Singular Self-dual Zollfrei Metrics and Twistor Correspondence

Cited by in corpus (19)

• On the Einstein-Weyl and conformal self-duality equations
• Manton's five vortex equations from self-duality
• Null Kähler geometry and isomonodromic deformations
• The quadric ansatz for the $mn$-dispersionless KP equation, and supersymmetric Einstein-Weyl spaces
• Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• On a class of 2D integrable lattice equations
• An action principle for the Einstein-Weyl equations
• On deformations of the dispersionless Hirota equation
• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• SDYM equations on the self-dual background
• Dispersionless integrable systems and the Bogomolny equations on an Einstein-Weyl geometry background
• GL(2)-geometry and complex structures
• Webs and the Plebański equation
• Differential invariants of Einstein-Weyl structures in 3D
• Self-dual gravity via Hitchin's equations
• Recurrent Lorentzian Weyl spaces
• Einstein Metrics, Projective Structures and the $SU(\infty )$ Toda Equation
• Hydrodynamic integrability via geometry