Harmonic superconformal maps of surfaces in Hn

Abstract

The class of harmonic superconformal maps from Riemann surfaces into real hyperbolic spaces is considered and harmonic sequences are constructed for these maps. They are used to obtain a rigidity result for such maps and to construct primitive lifts into an auxiliary flag space F m . It is also shown that superconformal harmonic maps into H 2 m and H 2 m−1 are locally described by 2D-affine Toda fields associated to the pair ( so(2m+1,C),σ) , where σ is the involution determined by the non-compact real form so(1,2m) . Applying the Adler–Kostant–Symes integration scheme to appropriate loop algebras we construct finite type primitive maps ψ:H 2→ F m , and harmonic superconformal maps f: H 2→ H 2 m and hence finite type Toda fields.