Supersymmetric extension of Hopf maps: N=4N=4σ -models and the S3→S2S3→S2 fibration

We discuss four off-shell N=4N=4D=1D=1 supersymmetry transformations, their associated one-dimensional σ-models and their mutual relations. They are given by(I)the (4,4)lin(4,4)lin linear “root” supermultiplet (supersymmetric extension of R4R4),(II)the (3,4,1)lin(3,4,1)lin linear supermultiplet (supersymmetric extension of R3R3),(III)the (3,4,1)nl(3,4,1)nl non-linear supermultiplet living on S3S3, and(IV)the (2,4,2)nl(2,4,2)nl non-linear supermultiplet living on S2S2. The I→III→II map is the supersymmetric extension of the R4→R3R4→R3 quadratic map, while the II→IVII→IV map is the supersymmetric extension of the S3→S2S3→S2 first Hopf fibration. The restrictions on the S3S3, S2S2 spheres are expressed in terms of the stereographic projections. The non-linear supermultiplets, whose supertransformations are local differential polynomials, are not equivalent to the linear supermultiplets with the same field content.The σ-models are determined in terms of an unconstrained prepotential of the target coordinates. The Uniformization Problem requires solving an inverse problem for the prepotential.The basic features of the supersymmetric extension of the second and third Hopf maps are briefly sketched.Finally, the Schur's lemma (i.e. the real, complex or quaternionic property) is extended to all minimal linear supermultiplets up to N⩽8N⩽8.