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 SIGMA, 2014, Volume 10, 034, 51 pp. (Mi sigma899)

Integrable Background Geometries

D. M. J. Calderbank

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

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 ($k+\ell\leqslant4$) 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 $\mathrm{SU}(\infty)$ 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. 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.

Keywords: selfduality; gauge theory; twistor theory; integrable systems.

DOI: https://doi.org/10.3842/SIGMA.2014.034

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ArXiv: 1403.3471
MSC: 53A30; 32L25; 37K25; 37K65; 53B35; 53C25; 58J70; 70S15; 83C20; 83C80
Received: January 21, 2014; in final form March 18, 2014; Published online March 28, 2014
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Citation: D. M. J. Calderbank, “Integrable Background Geometries”, SIGMA, 10 (2014), 034, 51 pp.

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. D. M. J. Calderbank, “Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski–West Construction”, SIGMA, 10 (2014), 035, 18 pp.
2. Kruglikov B., Morozov O., “Integrable Dispersionless Pdes in 4D, Their Symmetry Pseudogroups and Deformations”, Lett. Math. Phys., 105:12 (2015), 1703–1723
3. Dunajski M., Ferapontov E.V., Kruglikov B., “on the Einstein-Weyl and Conformal Self-Duality Equations”, J. Math. Phys., 56:8 (2015), 083501
4. Krynski W., “Webs and the Plebański equation”, Math. Proc. Camb. Philos. Soc., 161:3 (2016), 455–468
5. L. V. Bogdanov, “SDYM equations on the self-dual background”, J. Phys. A-Math. Theor., 50:19 (2017), 19LT02
6. M. Atiyah, M. Dunajski, L. J. Mason, “Twistor theory at fifty: from contour integrals to twistor strings”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 473:2206 (2017), 20170530
7. W. Krynski, “On deformations of the dispersionless Hirota equation”, J. Geom. Phys., 127 (2018), 46–54
8. B. Doubrov, E. V. Ferapontov, B. Kruglikov, V. S. Novikov, “On integrability in Grassmann geometries: integrable systems associated with fourfolds in $\mathbf{Gr}(3,5)$”, Proc. London Math. Soc., 116:5 (2018), 1269–1300
9. B. Kruglikov, E. Schneider, “Differential invariants of Einstein-Weyl structures in 3D”, J. Geom. Phys., 131 (2018), 160–169
10. M. Dunajski, T. Mettler, “Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four”, J. Geom. Anal., 28:3 (2018), 2780–2811
11. L. V. Bogdanov, “Matrix extension of the Manakov–Santini system and an integrable chiral model on an Einstein–Weyl background”, Theoret. and Math. Phys., 201:3 (2019), 1701–1709
12. L. V. Bogdanov, “Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background”, Theoret. and Math. Phys., 205:1 (2020), 1279–1290
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