Supersymmetric Harmonic Maps into Symmetric Spaces

Abstract

We study supersymmetric harmonic maps from the point of view of integrable system. It is well known that harmonic maps from R^2 into a symmetric space are solutions of a integrable system . We show here that the superharmonic maps from R^{2|2} into a symmetric space are solutions of a integrable system, more precisely of a first elliptic integrable system in the sense of C.L. Terng and that we have a Weierstrass-type representation in terms of holomorphic potentials (as well as of meromorphic potentials). In the end of the paper we show that superprimitive maps from R^{2|2} into a 4-symmetric space give us, by restriction to R^2, solutions of the second elliptic system associated to the previous 4-symmetric space.