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Okay, I cannot give you a full understanding of what is going on, but I can make the objects we are dealing with more precise: There are two spaces here: The moduli space $M_\text{sh}(r,k)$ of framed torsion-free coherent sheaves of rank $r$ and second Chern class $k$ on the projective scheme $\mathbb{P}^2$ viewed as a complex analytic space with its ...

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This question deserves more than one good answer, so let me offer some comments addressing more directly what's hard about this particular problem. The mass gap is an important challenge because solving it should force mathematical physicists to confront directly the messy question of exactly what the observables of QCD are. It's fairly easy to describe ...

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One less well-known but great reference are the classical field theory notes by Deligne and Freed in the '99 IAS lectures. Some good things about them Very elegant treatment written for mathematicians Begins with a nice discussion of ordinary classical mechanics using principal bundles and connections Useful comments on supersymmetric gauge theories ...

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