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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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First, the physical thing we care about is $\vec B$. So we can do anything to $\vec A$ we like as long as we get the same $\vec B$. That is, we can do anything that doesn't change the curl of $\vec A$. Now, suppose that $\vec \nabla\cdot\vec A = f$. Here's where Purcell neglects to stress what he means by "analogue of $\vec E$ in electrostatics" - the curl ...

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There is nothing wrong here. Choosing non-linear gauge conditions leads to uncanonical forms of the action. Generically, nothing ensures that for arbitary gauge conditions $F(A) = 0$ we will obtain some sort special form of the gauge fixed action. If one picks "odd" forms for the gauge fixing condition, then the resulting ghost action will contain unusual ...

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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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