First, if this post must be broken up in separate questions, please tell me so. I thought it would be better if I simply posed my questions in one thread, as they are directly related to each other.
I am currently trying to understand the construction of the Hitchin moduli space, constructed in this paper (pdf). I'll jump right in and point to the passages in the text where my questions arise, using the same notation as in the text. Below, all principal and vector bundles will be over the same compact riemann surface $M$.
Let's look at the beginning of §2. Here, we consider an $SO(3)$ principal bundle $P$ and make a case distinction based on whether the 2nd Stiefel-Whitney class vanishes or not. There are a few things I do not understand:
- How do we obtain a $SU(2)$ or $U(2)$ principal bundle covering $P$?
- What is $V$, and why does a connection on $P$ induce a connection on $V$ (or am I missing that $V$ is associated not only to a cover of $P$ as above, but to $P$ itself)?
Next is a question that is most likely the result of my ignorance concerning the nature of $V$.
I was under the impression that the result that one wants to show is that the moduli space of solutions to the self-duality equations on $P$ modulo gauge equivalence is a smooth manifold, for some restriction on $M$.
Theorem (5.7) proves this for solutions on a rank 2 vector bundles of odd degree, as long as the genus of $M$ is greater than 1. This is somehow equivalent to looking at $P$, at least if I interpret the introduction correctly. But how are these viewpoints equivalent? Is there a bijection between the moduli spaces of solutions of the self-duality equations on P, respective V?
I am trying to construct this bijection from the results of the paper, but I fear I'm a tad lost at the moment. I would be grateful for answers to any of my questions, or references to literature that might help me progress.
Thank you very much!