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SIGMA, 2014, том 10, 034, 51 стр.
(Mi sigma899)
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Эта публикация цитируется в 12 научных статьях (всего в 12 статьях)
Integrable Background Geometries
D. M. J. Calderbank Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Аннотация:
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.
Ключевые слова:
selfduality; gauge theory; twistor theory; integrable systems.
DOI:
https://doi.org/10.3842/SIGMA.2014.034
Полный текст:
PDF файл (731 kB)
Полный текст:
http://emis.mi.ras.ru/journals/SIGMA/2014/034/
Список литературы:
PDF файл
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Реферативные базы данных:
ArXiv:
1403.3471
Тип публикации:
Статья
MSC: 53A30; 32L25; 37K25; 37K65; 53B35; 53C25; 58J70; 70S15; 83C20; 83C80 Поступила: 21 января 2014 г.; в окончательном варианте 18 марта 2014 г.; опубликована 28 марта 2014 г.
Язык публикации: английский
Образец цитирования:
D. M. J. Calderbank, “Integrable Background Geometries”, SIGMA, 10 (2014), 034, 51 pp.
Цитирование в формате AMSBIB
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Образцы ссылок на эту страницу:
http://mi.mathnet.ru/sigma899 http://mi.mathnet.ru/rus/sigma/v10/p34
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
Эта публикация цитируется в следующих статьяx:
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D. M. J. Calderbank, “Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski–West Construction”, SIGMA, 10 (2014), 035, 18 pp.
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Kruglikov B., Morozov O., “Integrable Dispersionless Pdes in 4D, Their Symmetry Pseudogroups and Deformations”, Lett. Math. Phys., 105:12 (2015), 1703–1723
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Dunajski M., Ferapontov E.V., Kruglikov B., “on the Einstein-Weyl and Conformal Self-Duality Equations”, J. Math. Phys., 56:8 (2015), 083501
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Krynski W., “Webs and the Plebański equation”, Math. Proc. Camb. Philos. Soc., 161:3 (2016), 455–468
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L. V. Bogdanov, “SDYM equations on the self-dual background”, J. Phys. A-Math. Theor., 50:19 (2017), 19LT02
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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
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W. Krynski, “On deformations of the dispersionless Hirota equation”, J. Geom. Phys., 127 (2018), 46–54
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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
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B. Kruglikov, E. Schneider, “Differential invariants of Einstein-Weyl structures in 3D”, J. Geom. Phys., 131 (2018), 160–169
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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
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Л. В. Богданов, “Матричное расширение системы Манакова–Сантини и интегрируемая киральная модель на фоне геометрии Эйнштейна–Вейля”, ТМФ, 201:3 (2019), 337–346
; 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 -
Л. В. Богданов, “Бездисперсионные интегрируемые системы и уравнения Богомольного на фоне геометрии Эйнштейна–Вейля”, ТМФ, 205:1 (2020), 41–54
; 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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