N=4 mechanics of general (4,4,0) multiplets

We construct the manifestly N=4 supersymmetric off-shell superfield “master” action for any number n of the N=4 supermultiplets (4,4,0) described by harmonic analytic superfields q+a(ζ,u), a=1,…,2n, subjected to the most general harmonic constraints. The action consists of the sigma-model and Wess-Zumino parts. We present the general expressions for the target space metric, torsion and background gauge fields. The generic target space geometry is shown to be weak HKT (hyper-Kähler with torsion), with the strong HKT and HK ones as particular cases. The background gauge fields obey the self-duality condition. Our formulation suggests that the weak HKT geometry is fully specified by the two primary potentials: an unconstrained scalar potential L(q+,q−,u)|θ=0 which is the θ=0 projection of the superfield sigma-model Lagrangian, and a charge 3 harmonic analytic potential L+3a(q+,u)|θ=0 coming from the harmonic constraint on q+a. The reductions to the strong HKT and HK geometries amount to simple restrictions on the underlying potentials. We also show, using the N=2 superfield approach, that the most general bosonic target geometry of the N=4, d=1 sigma models, of which the weak HKT geometry is a particular case, naturally comes out after adding the mirror (4,4,0) multiplets with different transformation laws under N=4 supersymmetry and SO(4)R symmetry. Thus the minimal dimension of the target spaces exhibiting such a “weakest” geometry is 8, which corresponds to a pair of the mutually mirror (4,4,0) multiplets.