ag.algebraic geometry - Gauge theory construction of moduli of vector bundles

Hi Botong, I guess I could say this in person but this is faster.

Given a reductive group acting a finite dimensional Euclidean space, Kempf and Ness proved that the orbits of stable points are precisely the ones on which the norm attains a minimum. This gives a hint of the role of stability in the analytic construction,
where now one would like to minimize a suitable energy on the orbits of an infinite dimensional gauge group. This is spelled out in more detail in
[DK] "The geometry of four manifolds" by Donaldson and Kronheimer, and
[C] "Flat G-bundles with canonical metrics" by Corlette in addition to the references given above.

Regarding your second question, I suspect that the most meticulous comparison of the
algebraic and analytic constructions can be found in Simpson's papers "Moduli
of representations of the fundamental group I & II".

Addendum: Perhaps I should expand the my answer a bit, since it wasn't terribly enlightening. In very rough terms on the algebraic side one proceeds as follows.
One first needs to prove that the set of stable vector bundles with fixed topological type $V$ form a bounded family. This already uses the stability condition in an essential way.
From this one deduces the existence of a scheme $Q$, usually a subscheme of a Quot scheme, such that $Xtimes Q$ carries a vector $E$ such that all the stable bundles in question occur among the fibres $E_q$. The moduli space $M$ would then be quotient $M=Q/G$ for some appropriate reductive $G$ acting by "change of basis". At this point, to apply GIT,
one needs to know that stability in the abstract is related to to stability of the
$G$-action. Note that $E$ typically won't descend i.e. $M$ need not be fine.

On the analytic side, one can form the quotient $N$ of the set of stable
complex structures $S$ on $V$ modulo gauge equivalence. Again stability comes into play
to guarantee that $N$ is reasonable. So then one gets a map of
spaces $Q^{an}to S$ induced by $E$. This should descend to a map $M^{an}to N$. To check that this is an analytic equivalence, one would need a description of the local analytic structures on both sides. Fortunately one has this this, see theorem 4.5.1 of [Huybrechts-Lehn]
and prop 6.4.3 of [DK]. Making this into a proof would take a lot of work of course.