Given a smooth projective complex variety $X$, instead of using Mumford's GIT to construct the moduli of rank $n$ topologically trivial vector bundles, we can also take the gauge theory approach. 


To classify all topologically trivial vector bundles is same as to classify all possible complex structures on the topologically trivial vector bundle $X\times \mathbb{C}^n$ module the ambiguity from the choice of basis. And a complex structure is determined by $\bar\partial=\bar\partial\_0+\eta$, where $\bar\partial\_0$ corresponds to the holomorphic structure as a trivial vector bundle, and $\eta$ is an anti-holomorphic $End(\mathbb{C}^n)$ valued one form, such that $d\eta+\eta\wedge\eta=0$. Let $M$ denote the set of all such $\eta$, and $G$ denote the gauge group with fiber $GL(\mathbb{C}^n)$, then $G$ acts on $M$. Now, set-theoretically the orbits will correspond to the set of isomorphism classes of rank $n$ topologically trivial holomorphic vector bundles, but the quotient topology seems rather bad (non-Hausdorff).


My question is using this approach


(1)Whether stability condition is reflected in this construction?


(2)Can we obtain the same moduli space as using Mumford's GIT?


(3)I guess this is a natural approach. So is there any article or book about this?


The assumption of topologically trivial should not be necessary, just to simplify some notation. We only need to fix the underlying topological vector bundle.



Gauge theory construction of moduli of vector bundles
In the case of a nonsingular algebraic curve, I guess this is the point of Atiyah and Bott's Yang-Mills on Riemann surfaces, or the work of Narasimhan and Seshadri. Choose a Hermitian pairing on your complex bundle and then search for a unitary connection with central curvature. There is a really great book by Kobayashi "The Differential Geometry of Complex Vector Bundles" that explores this approach in more generality.


Once you have reduced to unitary connections with central curvature you can understand them as representations of a central extension of the fundamental group of the manifold, which then puts you in a finite dimensional situation with a compact group acting. The quotient corresponds to the geometric invariant theory quotient.


