A BPS Interpretation of Shape Invariance
  We show that shape invariance appears when a quantum mechanical model is invariant under a centrally extended superalgebra endowed with an additional symmetry generator, which we dub the shift operator. The familiar mathematical and physical results of shape invariance then arise from the BPS structure associated with this shift operator. The shift operator also ensures that there is a one-to-one correspondence between the energy levels of such a model and the energies of the BPS-saturating states. These findings thus provide a more comprehensive algebraic setting for understanding shape invariance. 
Michael Faux and Donald Spector