Supersymmetric Extension of Hopf Maps: N=4 sigma-models and the S^3 -> S^2 Fibration

TL;DR

This paper explores supersymmetric extensions of Hopf maps within N=4 one-dimensional sigma-models, analyzing linear and non-linear supermultiplets on spheres, and extending Schur's lemma to minimal supermultiplets up to N=8.

Contribution

It introduces supersymmetric versions of Hopf maps, compares linear and non-linear supermultiplets, and extends Schur's lemma to higher N minimal supermultiplets.

Findings

Supersymmetric extension of the S^3 -> S^2 Hopf fibration is constructed.

Non-linear supermultiplets are not equivalent to linear ones with same field content.

Schur's lemma is extended to all minimal linear supermultiplets up to N=8.

Abstract

We discuss four off-shell N=4 D=1 supersymmetry transformations, their associated one-dimensional sigma-models and their mutual relations. They are given by I) the (4,4)_{lin} linear supermultiplet (supersymmetric extension of R^4), II) the (3,4,1)_{lin} linear supermultiplet (supersymmetric extension of R^3), III) the (3,4,1)_{nl} non-linear supermultiplet living on S^3 and IV) the (2,4,2)_{nl} non-linear supermultiplet living on S^2. The I -> II map is the supersymmetric extension of the R^4 -> R^3 bilinear map, while the II -> IV map is the supersymmetric extension of the S^3 -> S^2 first Hopf fibration. The restrictions on the S^3, S^2 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 sigma-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<=8.