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 lowerdimensional structures are nondegenerate (i.e., nonnull) reductions of this. Any solution of the gauge theory on a kdimensional geometry, such that the gauge group H acts transitively on an lmanifold, determines a (k+l)dimensional geometry (k+l≤4) fibering over the kdimensional geometry with H as a structure group. In the case of an ldimensional group H acting on itself by the regular representation, all (k +l)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 YangMills 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(S1) Hitchin equation is shown to be equivalent to the hyperCR Einstein{Weyl equation, while the SDiff(σ) 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.
Original language  English 

Article number  34 
Journal  SIGMA: Symmetry, Integrability and Geometry: Methods and Applications 
Volume  10 
DOIs  
Publication status  Published  28 Mar 2014 
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David Calderbank
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching