In previous posts on this blog (for instance briefly in the post Prismatic Cohomology: An Overview) we have made mention of a correspondence between vector bundles with flat (or integrable) connection and local systems of complex vector spaces on complex analytic manifolds. This correspondence is an equivalence of categories, and the functor from the former to the latter can be thought of as assigning solutions to differential equations. This is a primitive version of what is known as the “**Riemann-Hilbert correspondence**“.

It is also worth noting that local systems give rise to representations (called **monodromy representations**) of the fundamental group of the complex analytic manifold they live on; this is a bijection, and is an important part of the formulation of geometric Langlands correspondence (see also The Global Langlands Correspondence for Function Fields over a Finite Field).

In this post we discuss another mathematical object related to vector bundles with flat connection and to local systems of complex vector spaces – **Higgs bundles**. They were first studied by Nigel Hitchin and the theory was then further developed by Carlos Simpson. Hitchin named these objects after the physicist Peter Higgs, apparently because of a physics-inspired motivation which we will not discuss further due to lack of knowledge of this particular history and analogy. However, we will briefly discuss how Higgs bundles also allow us to formulate a non-abelian generalization of the Hodge decomposition. This is just one of many applications these objects have found in mathematics.

Let be a smooth complex projective variety. A rank **Higgs bundle** on is a pair where is a rank holomorphic vector bundle and is a morphism from to satisfying the condition that .

Let us see how these are related to vector bundles with flat connection and local systems. Let us start with an -dimensional representation of the fundamental group. Then as we have stated earlier, there is a local system that gives rise to such a representation. Therefore there is also a corresponding vector bundle with flat connection. Now suppose the representation of the fundamental group is reductive. Then a theorem of Kevin Corlette and Simon Donaldson says that we can equip the vector bundle with flat connection with a harmonic metric.

This harmonic metric splits the flat connection into an skew-Hermitian and a Hermitian part, which in turn gives rise to a holomorphic structure on the vector bundle, and an endomorphism-valued -form; these two objects are precisely what makes up a Higgs bundle, as stated in the previous paragraph. Work of Hitchin and Simpson then established that a Higgs bundle will admit a harmonic metric if it satifies the condition of **polystability**. Putting all of this together, we find a correspondence between reductive representations of the fundamental group and polystable Higgs bundles, which is called the **Corlette-Simpson correspondence**.

The moduli space of Higgs bundles has many fascinating properties and is therefore the object of study of much mathematical research. For instance, in the case that is a curve, the moduli of rank Higgs bundles on admits a map to a space (whose dimension is half the dimension of the moduli space) called the **Hitchin base** , defined as

where is the canonical bundle (i.e. top exterior power of the sheaf of differentials) of . The fibers of this map are abelian varieties, which may be viewed as the Jacobians of what are known as **spectral curves**. The theory of the moduli space of Higgs bundles is relevant to the proof of the fundamental lemma as proven by Ngo Bau Chao, and is also relevant to applications of mirror symmetry (see also An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) to the geometric Langlands correspondence.

Since there is a correspondence between Higgs bundles, vector bundles with flat connections, and local systems, one might expect that their moduli spaces should be the same. While it is true that their moduli spaces are homeomorphic to each other, each of their moduli spaces have extra structure that is not preserved by these homeomorphisms! In fact, these moduli spaces are Kahler manifolds (again see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry), and each of them have different complex structures. These complex structures are related to each other in a way which is reminiscent of Hamilton’s quaternions, where we have elements , , and , satisfying , and . Kahler manifolds which have complex structures with this behavior are called **hyperkahler manifolds**. Once again this is an important aspect that allows one to apply mirror symmetry to the geometric Langlands correspondence.

As an example of the moduli space of Higgs bundles, let us consider the case where is a smooth projective curve (i.e. a Riemann surface) and the vector bundles are rank . Higgs bundles in this case consist of a line bundle on and an -valued -form, but in this case the endomorphisms of are trivial and just corresponds to a -form. The moduli space of line bundles on is the Jacobian and therefore the moduli space of rank Higgs bundles on is given by which is also the cotangent bundle of .

Let us now discuss how the theory of Higgs bundles and the Corlette-Simpson correspondence allows us to formulate a nonabelian generalization of the Hodge decomposition. We recall the classical Hodge decomposition for :

The cohomology group is dual to the homology group ; in turn, the latter is the abelianization of the fundamental group , by the Hurewicz theorem. Therefore, we may also express the Hodge decomposition as

Now we consider Higgs bundles. Consider a Higgs bundle . We may think of the holomorphic vector bundle as an element of the nonabelian cohomology , and we may think of as an element of . Combining this with the Corlette-Simpson correspondence between Higgs bundles and local systems (and hence representations of ), we get

This is the nonabelian generalization of the Hodge decomposition that we have alluded to throughout this post.

There are many other aspects of the theory of Higgs bundles that we have not yet discussed; for instance, the moduli space of Higgs bundles also arises as the space of solutions to the differential equations known as Hitchin’s equations. In another very different direction, since Hodge theory has a p-adic version (see also p-adic Hodge Theory: An Overview) one may also wonder if there is a p-adic version of the theory we have just discussed, and in fact there is! But we will leave all of this hopefully to future posts.

References:

Higgs bundle on Wikipedia

Nonabelian Hodge correspondence on Wikipedia

What is a… Higgs bundle by Steven Bradlow, Oscar Garcia-Prada, and Peter B. Gothen

Nonabelian Hodge Theory by Carlos T. Simpson

Perverse Sheaves and Fundamental Lemmas (lecture notes by Chao Li from a course by Wei Zhang)