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Selecta Mathematica

, Volume 24, Issue 5, pp 4749–4780 | Cite as

A cohomological approach to immersed submanifolds via integrable systems

  • J. de Lucas
  • A. M. Grundland
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Abstract

A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of \({\mathfrak {g}}\)-valued differential forms. We introduce Poincaré-type lemmas for these cohomologies, which appropriately describe the integrability conditions of Lax pairs associated with systems of PDEs. Our methods clarify the structure and properties of the deformations and soliton surfaces for the aforesaid Lax pairs. Our findings allow for the generalization of the theory of soliton surfaces in Lie algebras to general soliton submanifolds. Techniques from the theory of infinite-dimensional jet manifolds and diffieties enable us to justify certain common assumptions of the theory of soliton surfaces. Theoretical results are illustrated through \({\mathbb {C}}P^{N-1}\) sigma models.

Keywords

Cohomology \({\mathbb {C}}P^{N-1}\) sigma model Generalized symmetries \({\mathfrak {g}}\)-valued differential forms \({\mathfrak {g}}\)-valued de Rham cohomology Integrable systems Immersion formulas Soliton surfaces 

Mathematics Subject Classification

Primary 35Q53 Secondary 35Q58 53A05 

Notes

Acknowledgements

A.M. Grundland was partially supported by the research Grant ANR-11LABX-0056-LMHLabEX LMH (France) and from the NSERC (Canada). J. de Lucas and A.M. Grundland acknowledge partial support from Project MAESTRO DEC-2012/06/A/ST1/00256 of the National Science Center (Poland). This work was partially accomplished during the stay of A.M. Grundland and J. de Lucas at the École Normale Superieure de Cachan (CMLA). The authors would also like to thank CMLA for its hospitality and attention during their stay. Finally, we thank an anonymous referee for valuable comments to improve the paper.

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Authors and Affiliations

  1. 1.Department of Mathematical Methods in PhysicsUniversity of WarsawWarsawPoland
  2. 2.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada
  3. 3.Department of Mathematics and Computer ScienceUniversité du Québec à Trois-RivièresTrois-RivièresCanada

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