How should I physically understand the slope stability of vector bundles on a manifold X? Very basic question here as I try to come to grips with the geometry associated with string theory.
I can (almost) understand how a manifold $X$ can admit or not admit a particular vector bundle on it. Please correct me if I am wrong but if I take as my example $S^2$ as a real manifold, then I believe it does not admit a globally trivial rank 2 vector bundle.
Does the slope stability imply that some bundles are more "allowed" than others? For instance would a rank 4 bundle that is not slope-stable be seen to "decay" to a rank 2 sub-bundle?
 A: The physical interpretation of slope stability of vector bundles is revealed once one thinks of the vector bundles as being the "Chan-Paton gauge fields" on D-branes. Then then rank of the vector bundle is proportional to the mass density of a bunch of coincident D-branes, while the degree, being the Chern-class, is a measure for the RR-charge carried by the D-branes. 
This reveals that the "slope of a vector bundle" is nothing but the charge density of the corresponding D-brane configuration. 
Now a D-brane stae is supposed to be stable if it is a "BPS-state", which is the higher dimensional generalization of the classical concept of a charged black hole being an extremal black hole in that it carries maximum charge for given mass.
Hence the stable D-branes are those which maximize their charge density, hence the "slope" of their Chan-Paton vector bundles. 
The condition that every sub-bundle have smaller slope hence means that smaller branes can increase their charge density, hence their slope, by forming "bound states" into the larger, stable object. 
Hence slope-stability of vector bundles/coherent sheaves is the BPS stability condition on charged D-branes.
This idea is really what underlies Michael Douglas's discussion of "Pi-stability" of D-branes, which then inspired Tom Bridgeland to his general mathematical definition, now known as Bridgeland stability, which subsumes slope-stability/mu-stability of vector bundles as a special case. But, unfortunately, this simple idea is never quite stated that explicitly in Douglas's many articles on the topic. 
For more along these lines and more pointers see the discussion at 
**nLab: Bridgeland stability -- As stability of BPS D-branes **
