Is there stringy Morse theory? This question is pretty vague and open. I'm just curious if anyone has considered this.
Morse theory has a nice physical formulation: a Morse function can be thought of as a potential, so the gradient flow is the force experienced by a particle. The equilibria are the critical points of the potential. If we take a supersymmetric extension of this as a quantum theory, its ground state structure computes the homology of the space the particle moves in.
What can strings moving in a manifold subject to a potential tell us about the topology of the manifold?
 A: You have to consider Picard-Lefschetz theory instead Morse theory and find what a stringy analog to $N=2$ SUSY QM is. An analog might be constructed from a new look at the path integral formulation of (complexified) $N=2$ SUSY QM (see Edward Witten's paper and Stephan Zheng's master thesis). You might also consider string topology.
A: Properties of Riemannian, Kähler and hyperKähler manifolds could be seen through dimensional reductions of source spaces (domains) of corresponding quantum SUSY non-linear sigma-models, models whose target spaces (codomains) are the above mentioned manifolds, e.g. you get $N=2$ SUSY QM i.e. quantum superparticle moving in a Riemannian manifold as a dimensional reduction of 2d $N=1+1$ quantum non-linear sigma-model over superMinkowski spacetime (1+1)|(1,1), a quantum non-linear sigma-model which is again a dimensional reduction of 3d $N=2$ quantum non-linear sigma model over superMinkowski spacetime (1+2)|2. $\mathbb{Z}$-grading of forms on a Riemannian manifold is the relic of $R$-symmetry i.e $SO(2)$-group of superPoincare group $P^{3|2}$.
Deformations of the above mentioned SUSY non-linear models are done via superpotentials which are expressed by corresponding real or holomorphic Morse functions. 
Read Dan Freed's book about the above mentioned.
Regarding your question, see also:
http://arxiv.org/abs/hep-th/0311064
http://arxiv.org/abs/hep-th/0401175
http://arxiv.org/abs/hep-th/0407122
http://arxiv.org/abs/hep-th/0509163
and
http://arxiv.org/abs/hep-th/0702137
http://arxiv.org/abs/hep-th/0610149
http://arxiv.org/abs/0803.3302
http://golem.ph.utexas.edu/~distler/blog/archives/001030.html
http://golem.ph.utexas.edu/category/2012/07/notes_from_stringmath_2012.html
