2k-dimensional N=8 supersymmetric quantum mechanics

We demonstrate that two-dimensional N=8 supersymmetric quantum mechanics which inherits the most interesting properties of $N=2, d=4$ SYM can be constructed if the reduction to one dimension is performed in terms of the basic object, i.e. the $N=2, d=4$ vector multiplet. In such a reduction only complex scalar fields from the $N=2, d=4$ vector multiplet become physical bosons in $d=1$, while the rest of the bosonic components are reduced to auxiliary fields, thus giving rise to the {\bf (2, 8, 6)} supermultiplet in $d=1$. We construct the most general action for this supermultiplet with all possible Fayet-Iliopoulos terms included and explicitly demonstrate that the action possesses duality symmetry extended to the fermionic sector of theory. In order to deal with the second--class constraints present in the system, we introduce the Dirac brackets for the canonical variables and find the supercharges and Hamiltonian which form a N=8 super Poincar\`{e} algebra with central charges. Finally, we explicitly present the generalization of two-dimensional N=8 supersymmetric quantum mechanics to the $2k$-dimensional case with a special K\"{a}hler geometry in the target space.