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 equati...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146817 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Integrable Background Geometries / D.M.J. Calderbank // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 83 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862531925882699776 |
|---|---|
| author | Calderbank, D.M.J. |
| author_facet | Calderbank, D.M.J. |
| citation_txt | Integrable Background Geometries / D.M.J. Calderbank // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 83 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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 ℓ-manifold, determines a (k+ℓ)-dimensional geometry (k+ℓ ≤ 4) fibering over the k-dimensional geometry with H as a structure group. In the case of an ℓ-dimensional 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 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 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(Σ²) 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.
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| first_indexed | 2025-11-24T04:38:42Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-146817 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T04:38:42Z |
| publishDate | 2014 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Calderbank, D.M.J. 2019-02-11T16:21:11Z 2019-02-11T16:21:11Z 2014 Integrable Background Geometries / D.M.J. Calderbank // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 83 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A30; 32L25; 37K25; 37K65; 53B35; 53C25; 58J70; 70S15; 83C20; 83C80 DOI:10.3842/SIGMA.2014.034 https://nasplib.isofts.kiev.ua/handle/123456789/146817 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 ℓ-manifold, determines a (k+ℓ)-dimensional geometry (k+ℓ ≤ 4) fibering over the k-dimensional geometry with H as a structure group. In the case of an ℓ-dimensional 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 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 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(Σ²) 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. This paper is a contribution to the Special Issue on Progress in Twistor Theory. The full collection is available
 at http://www.emis.de/journals/SIGMA/twistors.html.
 I would like to thank Harry Braden, Maciej Dunajski, Evgeny Ferapontov, Paul Gauduchon,
 Nigel Hitchin, Boris Kruglikov, Claude LeBrun, Lionel Mason, Ian Strachan, Paul Tod and Nick
 Woodhouse for helpful discussions. This work was supported by the EPSRC, the Leverhulme
 Trust, and the William Gordon Seggie Brown Trust. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Integrable Background Geometries Article published earlier |
| spellingShingle | Integrable Background Geometries Calderbank, D.M.J. |
| title | Integrable Background Geometries |
| title_full | Integrable Background Geometries |
| title_fullStr | Integrable Background Geometries |
| title_full_unstemmed | Integrable Background Geometries |
| title_short | Integrable Background Geometries |
| title_sort | integrable background geometries |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146817 |
| work_keys_str_mv | AT calderbankdmj integrablebackgroundgeometries |