Extended supersymmetry in Dirac action with extra dimensions

We investigate a new realization of extended quantum-mechanical supersymmetry. We first show that an $\mathcal{N}=2$ quantum-mechanical supersymmetry is hidden in the four-dimensional (4D) spectrum of the Kaluza-Klein decomposition for the higher dimensional Dirac field, that is, Kaluza-Klein mode functions of 4D right-handed spinors and 4D left-handed ones form $\mathcal{N}=2$ supermultiplets. In addition to $\mathcal{N}=2$ supersymmetry, we discover that an $\mathcal{N}$-extended supersymmetry ($\mathcal{N} = d+2\ (d+1)$ for $d=$ even (odd) extra dimensions) is further hidden in the 4D spectrum. The extended symmetry can explain additional degeneracy of the spectrum. Furthermore, we show that a superpotential can be introduced into the $\mathcal{N}$-extended supercharges and clarify the condition to preserve the supersymmetry. The partial breaking of the supersymmetry is also demonstrated.