An introduction to Higgs bundles via harmonic maps

These notes are an extended version of lecture notes prepared for the 3-hour mini-course"An introduction to cyclic Higgs bundles and complex variation of Hodge structures"that the author gave at University of Illinois at Chicago.

We aim to choose a "best" H. There is a natural Hermitian metric on End(E) induced from the Hermitian metric H on E. Fix any background conformal metric g 0 on Σ, it induces a natural pairing on Ω 1 (Σ, C). Therefore we have the pairing < Ψ H , Ψ H > H,g 0 . Theorem 1.11. (Corlette [6], Donaldson [10]) If D is an irreducible flat connection, then there exists a unique (up to a scalar multiple) harmonic metric H on E.
Why use the name "harmonic"? Let us recall the definition of equivariant harmonic maps. Given a π 1 M -equivariant map f ∶ M 1 → M 2 between two Riemannian manifolds, then df ∈ Ω 0 ( M 1 , T * M 1 ⊗ f −1 T M 2 ) is also π 1 M 1 -equivariant. This implies that e(f ) =< df, df > T * M 1 ⊗f −1 T M 2 ∶ M 1 → R is a π 1 M 1 -invariant function and hence descends to M 1 . Call e(f ) the energy density on M 1 and also on M 1 . The energy E(f ) is the integral of e(f ) with respect to the volume form of M 1 .
Definition 1. 12. The map f is harmonic if it is a critical point of the energy functional E(f ).
In dimension 2, by the definition of energy, it only depends on the conformal class of the metric on the domain. So is the harmonic map.
Step 1: The space of Hermitian metrics on C n is the space of positive definite Hermitian matrices. A Hermitian metric on E =S × ρ C n is an for any two sections s, t ofΣ × C d .
Step 2: The open subset N ⊂ Herm(C n ) has a structure of a Riemannian manifold. If X, Y ∈ This is the unique GL(n, C)-invariant Riemannian metic on N up to scale. Together with Ψ H = − 1 2 f −1 df ( see Lemma 1.13) and use the same background metric (S, g 0 ) (conformal to Σ), we obtain We finally see that H being harmonic (minimizing the functional E(H)) is equivalent to f ∶ (S,g 0 ) → N being harmonic (minimizing the energy of f ).
The following proof is taken from the lecture note of O. Guichard [11].
For s, t two sections ofS × C n , by definition, Let's go one step further.
(2) A Higgs bundle (E, φ) is polystable if it is a direct sum of stable Higgs bundles of the same slope.
In fact, the connection D here is irreducible. (3) The representation variety Rep(π 1 S, SL(n, C)) is the space of conjugation classes of reductive representations from π 1 (S) to SL(n, C).
We obtain a 1-1 correspondence This is called the non-abelian Hodge correspondence.
Remark 1.19. One can generalize the non-abelian Hodge correspondence to general reductive Lie groups G. In later sections, we'll directly mention G-Higgs bundle without more explanation. 4

Hitchin fibration and C * -action
There are two important concepts in the moduli space of SL(n, C)-Higgs bundles.

Hitchin fibration and Hitchin section.
Given a basis of SL(n, C)-invariant homogeneous polynomials p i of degree i over sl(n, C), 2 ≤ i ≤ n. The Hitchin fibration is a map from the moduli space of SL(n, C)-Higgs bundles over Σ to the direct sum of the holomorphic differentials Note that p 2 (φ) is always a constant multiple of tr(φ 2 ). By choosing an appropriate basis of polynomials, the Hitchin section s of the Hitchin fibration can be defined explicitly as follows. Define Hitchin [13] shows that the Higgs bundles in the image of Hitchin section have holonomy in SL(n, R). Moreover, it is a connected component of the moduli space of SL(n, R)-Higgs bundles, called Hitchin component and denoted by Hit n . In later sections, we will use Hitchin component referring to both Higgs bundles and corresponding representations.
Note that the image s(q 2 , 0, ⋯, 0) correspond to the Teichmüller locus. Each representation corresponding to s(q 2 , 0, ⋯, 0) for some q 2 is a Fuchsian representation post composing the unique irreducible representation from SL(2, R) to SL(n, R), called an n-Fuchsian representation.
2.2. C * -action. There is a natural C * -action on the moduli space of SL(n, C)-Higgs bundles:

Cyclic Higgs bundles
Definition 3.1. A cyclic Higgs bundle (E, φ) over Σ is a SL(n, C)-Higgs bundle of the form

cyclic Higgs bundle in the Hitchin component is of the form
When is a Higgs bundle stable? A sufficient condition of (E, φ) being stable is that lim t→0 t⋅[(E, φ)] is stable. Note that it is not a necessary condition.
When is a cyclic Higgs bundle stable?
What is good about cyclic Higgs bundles?
. We can see that H also solves the Hithcin equation of (E, ωφ). By the uniqueness of harmonic metrics, Since the frame e is holomorphic, we have This implies that ∂H = H ⋅ A and A = H −1 ∂H.
Let g 0 = g 0 (z)dz ⊗ dz be the Hermitian metric on K −1 so that △ log g 0 (z) = g 0 (z), meaning that the corresponding metric on Σ is hyperbolic. We then have ∂z∂ z log(hg 0 (z) where u is a smooth function over Σ. We have The operator 1 g 0 (z) ∂z∂ z and q 2 (z) 2 g 0 (z) −2 do not depend on z, denoted by △ g 0 and q 2 2 g 0 respectively. Hence Equation (2) becomes which indeed holds globally. In fact, the function e u here is the holomorphic energy density of the harmonic map f ∶ Σ → (S, ρ(w) dw 2 ). And Equation (3)   Let the Riemann surface Σ be equipped with a background conformal metric g 0 = g 0 (z) dz 2 . Data of harmonic maps are as follows: (a) Energy density: This is also the Morse function on the moduli space of Higgs bundles. (c) Pullback metric g = f * g N is (d) Hopf differential of f is g 2,0 = 4tr(φ 2 ), which is holomorphic. Note that f is conformal if Hopf (f ) = 0. (e) Curvature of f : For the symmetric space N , the sectional curvature of the plane spanned by Y 1 , Y 2 ∈ T I N = Herm(C n ) is given by The curvature of the tangent plane σ of f (Σ), (tr(φφ * H )) 2 .

Selected topics on harmonic maps and minimal surfaces
The left hand side has the same dimension as the right hand side. The map is equivariant with respect to the mapping class group action.
Question 5.1. Is this map a bijection?
• Injective. This is the hard part, called the "Labourie conjecture". What do we know so far? Let's generalize to consider Hitchin representation for real split Lie groups.
The Labourie conjecture holds means that for any Hitchin representation ρ, there exists a unique Riemann surface Σ such that the ρ-equivariant harmonic map f ∶Σ → N is conformal. That is, there exists a unique ρ-equivariant minimal mapping ofS into N .
Remark 5.2. One can also consider the generalized Labourie conjecture for maximal representations. What do we know in this case? (1) G = Sp(4, R), Collier [3]. Take a universal covering π ∶ Y → Σ ∖ D(E, φ), we have the decomposition of the Higgs bundle To talk about asymptotics of harmonic maps, we introduce the vector distance of two metrics. Let V be an n-dimensional complex vector space. For two Hermitian metrics h 1 , h 2 , we can take a base e 1 , e 2 , ⋯, e n of V which is orthogonal with respect to both h 1 and h 2 . We have the real numbers k j (j = 1, 2, ⋯, n) determined by k j = log e j h 2 − log e j h 1 . We impose k 1 ≥ k 2 ≥ ⋯ ≥ k n .
The key estimate is the following theorem on "decoupled Hitchin equation".
Theorem 5.8. (Mochizuki [22]) Under the same assumptions in Theorem 5.8. Then take any neighborhood N 0 of D(E, φ), there exists a constant C 0 > 0 and ǫ 0 > 0 such that the following holds on Σ ∖ D(E, φ), The constants C 0 , ǫ 0 only depend on (Σ, g Σ ), N 0 and (E, φ). For any local section s of End(E) ⊗ Ω p,q (Σ), s Ht,g Σ ∶ Σ → R is the norm of s with respect to H t on End(E) and g Σ on Σ.
(1) For cyclic Higgs bundles in the Hitchin component, Theorem 5.7 and 5.8 were first shown in Collier-Li [4].
(2) For those Higgs bundles which are not generically regular semisimple, Conjecture 5.5 is not necessarily true. But the complete description is still open.

Negative Curvature.
Conjecture 5. 10. Given a Hitchin representation ρ for P SL(n, R) and a Riemann surface Σ, the unique ρ-equivariant harmonic map f ∶Σ → N satisfies that it is never tangential to a flat in N .
For the case f is a minimal mapping, this is the negative curvature conjecture in Dai-Li [7]. Using the expression of curvature form (4), the above conjecture is rephrased as Conjecture 5.11. For (E, φ) in the Hitchin component for P SL(n, R), Remark 5.12.
(1) A direct corollary of Conjecture 5.10 is that the induced metric of minimal surface is strictly negatively curved.
(2) This phenomenon in Equation (6) is opposite to the asymptotical behavior of generically regular semisimple Higgs bundles in which case the Hitchin equation decouples in exponential decay as in Equation (5).
What do we know so far for Conjecture 5.10? (1) for cyclic Higgs bundles in the Hitchin components, Dai-Li [7].
(2) One can also consider the conjecture for maximal representations. For Sp(4, R)-maximal representations in Gothen components, Dai-Li [8]. How about the energy density as a function over Σ along C * -flow?
Conjecture 5.14. Along C * -flow, the energy density e(f t ) of corresponding harmonic maps f t decreases as a function as t decreases.
Dai and Li in [8] showed Conjecture 5.14 holds for stable cyclic Higgs bundles. A weaker conjecture is about the domination comparing with the limit as t → 0. In the following Theorem 5. 16, we see that Conjecture 5.15 holds for each Higgs bundle in the Hitchin section. Renormalize the metric on N such that in the base n-Fuchsian case, the totally geodesic copy of the hyperbolic plane inside N is of curvature −1.
Theorem 5.16. (Li [19]) Let ρ be a Hitchin representation for P SL(n, R), g 0 be a hyperbolic metric on S, and f be the unique ρ-equivariant harmonic map from (S,g 0 ) to the symmetric space N . Then its energy density e(f ) satisfies e(f ) ≥ 1.
Moreover, equality holds at one point only if e(f ) ≡ 1 in which case ρ is the base n-Fuchsian representation of (S, g 0 ).
5.5. Hitchin fibration. We discuss the relation between harmonic maps and the Hitchin fibration.
Conjecture 5.17. (Dai-Li [8]) Inside each Hitchin fiber, the Hitchin section maximizes the energy density of corresponding harmonic maps.
In fact, even the integral version is still open. A weaker conjecture is as follows.
Conjecture 5.18. Inside each Hitchin fiber, the Hitchin section maximizes the energy of corresponding harmonic maps.
The fact that the n-Fuchsian point being maximal gives us a pure topological control on representations arising from such Hitchin fibers as shown in Theorem 5.21.
Definition 5.20. (1) The translation length of γ ∈ π 1 (S) with respect to ρ ∶ π 1 (S) → SL(n, C) is defined by where d(⋅, ⋅) is the distance induced by the Riemannian metric g N on N .
Theorem 5.21. (Li [20]) Any representation arises from a polystable SL(n, C)-Higgs bundle in the Hitchin fiber at (q 2 , 0, ⋯, 0) is strictly dominated by a n-Fuchsian representation unless it is a n-Fuchsian representation.