Quaternion-K\"ahler N=4 Supersymmetric Mechanics

Using the N=4, 1D harmonic superspace approach, we construct a new type of N=4 supersymmetric mechanics involving 4n-dimensional Quaternion-K\"ahler (QK) 1D sigma models as the bosonic core. The basic ingredients of our construction are {\it local} N=4, 1D supersymmetry realized by the appropriate transformations in 1D harmonic superspace, the general N=4, 1D superfield vielbein and a set of 2(n+1) analytic "matter" superfields representing (n+1) off-shell supermultiplets (4, 4, 0). Both superfield and component actions are given for the simplest QK models with the manifolds \mathbb{H}H^n = Sp(1,n)/[Sp(1) x Sp(n)] and \mathbb{H}P^n = Sp(1+n)/[Sp(1) x Sp(n)] as the bosonic targets. For the general case the relevant superfield action and constraints on the (4, 4, 0) "matter" superfields are presented. Further generalizations are briefly discussed.