A series of results on moduli spaces of connections in the presence of complex structure due to (Donaldson 85, Uhlenbeck-Yau 86), the most famous of which says that on a compact Kähler manifold any semistable holomorphic vector bundle with trivial determinant line bundle admits a Hermite-Einstein connection.
The theorem is recalled for instance as (Scheinost-Schottenloher 96, theorem 1.15). This result is a key step in the construction of the Kähler polarization on a moduli space of flat connections via symplectic reduction (Scheinost-Schottenloher 96, corollary 1.16), a non-abelian version of the Griffiths intermediate Jacobian (see there at Examples – Picard variety).
The correspondence between semi-stable vector bundles and Hermite-Einstein connections holds more generally over complex manifolds, where it is known as the Kobayashi-Hitchin correspondence. The special case of that over Riemann surfaces in turn is essentially the Narasimhan-Seshadri theorem.
The original articles are
Simon Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. LMS 50 (1985) 1-26
Karen Uhlenbeck, Shing-Tung Yau, On the existence of Hermitean Yang-Mills-connections on stable bundles over Kähler manifolds, Comm. Pure Appl. Math. 39 (1986) 257-293
Reviews in the context of discussion of Kähler polarization of moduli spaces of connections is in
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