In fact the Narasimhan–Seshadri theorem may be obtained as a special case of the nonabelian Hodge correspondence by setting the Higgs field to zero.
It was proven by M. S. Narasimhan and C. S. Seshadri in 1965 that stable vector bundles on a compact Riemann surface correspond to irreducible projective unitary representations of the fundamental group.
[1] This theorem was phrased in a new light in the work of Simon Donaldson in 1983, who showed that stable vector bundles correspond to Yang–Mills connections, whose holonomy gives the representations of the fundamental group of Narasimhan and Seshadri.
[2] The Narasimhan–Seshadri theorem was generalised from the case of compact Riemann surfaces to the setting of compact Kähler manifolds by Donaldson in the case of algebraic surfaces, and in general by Karen Uhlenbeck and Shing-Tung Yau.
Nigel Hitchin introduced a notion of a Higgs bundle as an algebraic object which should correspond to complex representations of the fundamental group (in fact the terminology "Higgs bundle" was introduced by Carlos Simpson after the work of Hitchin).
The first instance of the nonabelian Hodge theorem was proven by Hitchin, who considered the case of rank two Higgs bundles over a compact Riemann surface.
It was shown by Donaldson in this case that solutions to Hitchin's equations are in correspondence with representations of the fundamental group.
[6] The results of Hitchin and Donaldson for Higgs bundles of rank two on a compact Riemann surface were vastly generalised by Carlos Simpson and Kevin Corlette.
The statement that polystable Higgs bundles correspond to solutions of Hitchin's equations was proven by Simpson.
[7][8] The correspondence between solutions of Hitchin's equations and representations of the fundamental group was shown by Corlette.
A Higgs bundle is (semi-)stable if, for every proper, non-zero coherent subsheaf
[9] To each of the three concepts: Higgs bundles, flat connections, and representations of the fundamental group, one can define a moduli space.
The first part was proved by Donaldson in the case of rank two Higgs bundles over a compact Riemann surface, and in general by Corlette.
[6][9] In general the nonabelian Hodge theorem holds for a smooth complex projective variety
The second part of the theorem was proven by Hitchin in the case of rank two Higgs bundles on a compact Riemann surface, and in general by Simpson.
Combined, the correspondence can be phrased as follows: Nonabelian Hodge theorem — A Higgs bundle (which is topologically trivial) arises from a semisimple representation of the fundamental group if and only if it is polystable.
The nonabelian Hodge correspondence not only gives a bijection of sets, but homeomorphisms of moduli spaces.
Indeed, if two Higgs bundles are isomorphic, in the sense that they can be related by a gauge transformation and therefore correspond to the same point in the Dolbeault moduli space, then the associated representations will also be isomorphic, and give the same point in the Betti moduli space.
are naturally complex algebraic varieties, and where it is smooth, the de Rham moduli space
However, even though the Dolbeault and Betti moduli spaces both have natural complex structures, these are not isomorphic.
When the underlying vector bundle is topologically trivial, the holonomy of a Hermitian Yang–Mills connection will give rise to a unitary representation of the fundamental group,
The subset of the Betti moduli space corresponding to the unitary representations, denoted
The special case where the rank of the underlying vector bundle is one gives rise to a simpler correspondence.
In particular, the Higgs field is uncoupled from the holomorphic line bundle, so the moduli space
The gauge group of a line bundle is commutative, and so acts trivially on the Higgs field
In the case of rank one Higgs bundles on compact Riemann surfaces, one obtains a further description of the moduli space.
A version of the nonabelian Hodge theorem holds for these objects, relating principal
[7][8][11] The correspondence between Higgs bundles and representations of the fundamental group can be phrased as a kind of nonabelian Hodge theorem, which is to say, an analogy of the Hodge decomposition of a Kähler manifold, but with coefficients in the nonabelian group
The exposition here follows the discussion by Oscar Garcia-Prada in the appendix to Wells' Differential Analysis on Complex Manifolds.
, but this is not an actual isomorphism of sets, as the moduli space of Higgs bundles is not literally given by the direct sum above, as this is only an analogy.