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Supersymmetric harmonic maps into symmetric spaces. (English) Zbl 1120.53039
The paper studies supersymmetric harmonic maps from the point of view of integrable systems. Similar to the harmonic maps from $$\mathbb{R}^2$$ into symmetric spaces which are known to be solutions of integrable systems, the superharmonic maps from $$\mathbb{R}^{2| 2}$$ into symmetric spaces are shown to be solutions of a first elliptic integrable system in the sense of C.-L. Terng [Geometries and symmetries of soliton equations and integrable elliptic equations, arXiv: math.DG/0212372 (30 Dec. 2002)]. A Weierstrass-type representation is constructed in terms of holomorphic potentials (as well as of meromorphic potentials). Finally, it is shown that the superprimitive maps from $$\mathbb{R}^{2| 2}$$ into a 4-symmetric space provide, by restriction to $$\mathbb{R}^2$$, solutions of the second elliptic system associated with the previous 4-symmetric space.

##### MSC:
 53C43 Differential geometric aspects of harmonic maps 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 58E20 Harmonic maps, etc.
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