The fundamental theorem of Narasimhan and Seshadri proved in the mid 1960s established a one-to-one correspondence between stable bundles over a compact Riemann surface and unitary representations of the fundamental group of the surface. Atiyah and Bott introduced at the beginning of the 1980s a new point of view for this correspondence, by means of the theory of Yang–Mills connections on Riemann surfaces.The generalisation of the theorem of Narasimhan and Seshadri to higher dimensions is given by the Hitchin–Kobayashi correspondence between Hermite–Yang-Mills connections and stable bundles over compact Kähler manifolds proved by Donaldson and Uhlenbeck–Yau in the mid 1980s.

A central ingredient for the geometric understanding of these correspondences is provided by the symplectic nature that underlies gauge theory on compact Kähler manifolds. Namely, the fact that the space of connections has a symplectic structure and the Hermite–Yang-Mills equation appears as a moment map equation for the action of the gauge group. This point of view was brought by Donaldson in the late 1990s to Kähler geometry in the study of constant scalar curvature Kähler metrics on a compact complex manifold. In this case the symplectic manifold is the space of complex structures compatible with the symplectic structure on a compact symplectic manifold, and the acting group is the group of Hamiltonian symplectomorphisms. Here the correspondence is given by the Yau–Tian–Donaldson conjecture.

After reviewing some of the basic facts of the theories mentioned above, in this course I will illustrate first how we can exploit the symplectic moment map point of view to produce very interesting and geometrically rich generalizations of the Hitchin–Kobayashi correspondence, involving not only connections but Higgs fields of different sorts. A particularly important case is provided by Hitchin’s theory of Higgs bundles. After that, in a different direction, I will present a programme that applies the moment map perspective to the study of coupled equations for Yang–Mills connections and Kähler metrics. This programme, initiated almost 15 years ago in collaboration with Luis Álvarez-Cónsul and Mario García-Fernández, combines somehow the Donaldson–Uhlenbeck–Yau and the Yau-Tian-Donaldson theories, and is inspired by the coupling in physics of high energy particle gauge theories with gravitation.