Global Symmetries of Quaternion-K$\ddot{\mathbf{a}}$hler $\mathcal{N}=4$ Supersymmetric Mechanics

We analyze the global symmetries of $\mathcal{N}=4$ supersymmetric mechanics involving $4n$-dimensional Quaternion-K"ahler (QK) $1D$ sigma models on projective spaces $\mathbb{H}{\mathrm{H}}^n$ and $\mathbb{H}{\mathrm{P}}^n$ as the bosonic core. All Noether charges associated with global worldline symmetries are shown to vanish as a result of equations of motion, which implies that we deal with a severely constrained hamiltonian system. The complete hamiltonian analysis of the bosonic sector is performed.