Physicists often speak of "Connections modulo gauge transformations" as the natural configuration space of a gauge field. In this sense, the fundamental object of study in gauge theory is the space of pairs,
$$\frac{(E,\nabla)}{\text{gauge transformations}}$$
Where $E$ is a complex vector bundle, and $\nabla$ is a suitable connection. In the case of gauge group $U(1)$ and line bundle $E=L$, there is a natural identification of this configuration space with the set of holomorphic structures on $L$, i.e.,
$$\frac{(L,\nabla)}{\text{gauge transformations}}\cong \text{Pic}^0(X)$$
Where the zero is there because there are no magnetic monopoles in nature.
This is immensely useful, because the space on the RHS is amazingly concrete and visualizable, computable, etc...
The question is, does this correspondence continue for bundles of rank higher than one, and general compact gauge group $G$ (i.e. interpretation as holomorphic structures)? This would allow for a similar analysis of general Yang-Mills theory, and other non-abelian gauge theories of the Standard Model.
Also, does anyone have references for these results?
For the case of $U(n)$, the correspondence continues for higher dimensional complex vector bundles. See page 3 of http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture10.pdf for details.