supersymmetric QM
The number of the supersymmetric ground states is independent of the potential, so we can apply the limit that the strength of the potential approaches infinity, in which case ground states are localized around critical points . Also, the semi-classical ground states sit in the sector with (Morse index) fermions excited, namely, (fermion number/ p-form in Q-cohomology) .
cellular homology
Morse fundamental theorem: Morse index=the dimension of attached cell when going across the critical point.
Morse inequality: can be proved in SQM.
Morse-Bott theory, homotopy
generalization: the Hessian matrix can be degenerated except for the normal direction, so that critical points can form a closed manifold rather than a set of discrete points.
Applied to the proof of Bott periodicity. (generalized Morse fundamental theorem is applied to the loop space and its submanifold , consisting of minimal geodesics/critical points of zero indices)