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Yang-Mills functional on P3

This is an exercise designed to understand elementary aspects of Yang-Mills theory, especially, the aspects in stable bundles and the Penrose twistor map. Let P3 denote the 3-dimensional complex projective space, and consider a nontrivial rank-2 complex vector bundle E on P3 such that c1(E) = 0 and c2(E) = 1, on which the Yang-Mills functional Y M can be defined as usual. The main purpose is to describe the minimum submanifold MP3 of Y M in various ways: firstly thanks to the complex structure on P3, the description of MP3 can be converted to that of the moduli space of certain stable bundles on P3, the latter being known due to the previous works by Hartshorne and so on. Then through the Penrose map, it is related to the pull-back bundles of instatons over S 4 .