Geometry of theN=4,d=1nonlinear supermultiplet

Abstract

We construct the general action for $N=4$, $d=1$ nonlinear supermultiplet including the most general interaction terms which depend on the arbitrary function $h$ obeying the Laplace equation on ${S}^{3}$. We find the bosonic field $B$ which depends on the components of nonlinear supermultiplet and transforms as a full time derivative under $N=4$ supersymmetry. The most general interaction is generated just by a Fayet-Iliopoulos term built from this auxiliary component. Being transformed through a full time derivative under $N=4$, $d=1$ supersymmetry, this auxiliary component $B$ may be dualized into a fourth scalar field giving rise to a four-dimensional $N=4$, $d=1$ sigma-model. We analyzed the geometry in the bosonic sector and find that it is not a hyper-K\"ahler one. With a particular choice of the target space metric $g$ the geometry in the bosonic sector coincides with the one which appears in heterotic (4, 0) sigma-model in $d=2$.