Studying Deformations of Fuchsian Representations with Higgs Bundles

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichm\"uller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747].

These notes are based on a three hour minicourse given by the author at University of Illinois at Chicago in June 2018. The main goal is to explain how many components of the character variety of a closed surface are either deformations spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family, is of particular interest and is related to the field of Higher Teichmüller theory. Our main tool is the theory of Higgs bundles. In these notes we try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how The author is funded by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship, NSF MSPRF no. 1604263. they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the notes of Q. Li who gave a concurrent minicourse.
1. An introduction to the character variety Let S be a closed surface of genus g ≥ 2. Denote the fundamental group of S by Γ, and recall that Γ has the standard presentation Γ = a 1 , . . . , a g , b 1 , . . . , b g | g j=1 [a j , b j ] = 1 .
Fix also a real reductive Lie group G. For example G could be one of the following groups GL(n, R) , GL(n, C) , SL(n, C) , Sp(2n, R), PSL(n, R) = SL(n, R)/ ± Id , but G cannot be a group like P = {( a b 0 c ) ∈ GL(2, C)} . One main property of a reductive Lie group G is that, up to conjugation, there is a unique maximal compact subgroup H < G. We will heavily use this property. In fact there is a homotopy equivalence H ≃ G .
The quotient space Hom(Γ, G)/G consists of conjugacy classes of representations. Unless G is compact, Hom(Γ, G)/G is not Hausdorff.
To get a Hausdorff quotient we restrict to the subset of Hom(Γ, G). A representation ρ : Γ → G is called reductive if the composition with the adjoint representation Γ ρ PSL(2, C)-character variety, then it is a closed connected subset which is no longer open.
1.2. Connected components. One fundamental problem is to determine how many connected components the character variety has. Surprisingly, this question has not been answered in full generality. There is a topological invariant which helps distinguish connected components.
Denote the set of isomorphisms classes of topological principal G-bundles on S by B G (S), this is the set of homotopy classes of maps from S to the classifying space of G. For connected groups we have B G (S) = H 2 (S, π 1 G). Every representation ρ : Γ → G defines a principal G-bundle E ρ → S E ρ = (S × G)/Γ , where Γ acts on its universal coverS by deck transformations and by multiplication by ρ(Γ) on G. Thus, we have a map Hom(Γ, G) → B G (S). Moreover, this map is continuous and descends to a map (1.1) τ : π 0 (X (Γ, G)) / / B G (S) .
If X ω (Γ, G) = τ −1 (ω), then X (Γ, G) decomposes as ω∈B G (S) X ω (Γ, G) . Remark 1.5. Note that when τ is injective, the question of connected components counts is not very interesting. We will mainly be interested in when τ is not injective and understanding the special features of these components.
This question has been answered for many groups, but is open in general. For compact groups the map τ is injective, this was proven by Narasimhan-Seshadri [22] for G = U(n) and Ramanathan [23] in general. Theorem 1.6. If G is compact (i.e. G = H), then τ is injective. Furthermore, if G is also semisimple, then τ is a bijection.
Since H and G are homotopic, we have B G (S) = B H (S). Moreover, for each ω ∈ B H (S) X ω (Γ, H) ⊂ X ω (Γ, G) . For complex groups, the map τ is also injective. This was proven for by J. Li [19] for semisimple groups and Garcia-Prada and Oliveira [10] in general. Theorem 1.7. If G is complex (i.e. G = H C ), then τ is injective.
We will prove these results using Higgs bundles in Section 6.5. For G a semisimple complex Lie group, the following corollary follows immediately from the two above theorems. It holds in general. Corollary 1.8. For G a complex reductive Lie group, every representation ρ : Γ → G can be continuously deformed to a compact representation Γ → H ֒→ G.
The above corollary says that the connected components of the character variety are not interesting. For real groups, the situation more subtle. Example 1.9. For G = PSL(2, R), the maximal compact subgroup is H ∼ = SO (2). Since a circle bundle on a closed surface is determined by its degree, we have B H (S) ∼ = Z. Thus, However, the space X d (Γ, PSL(2, R)) is empty when |d| > 2g − 2 [21]. Moreover, when |d| ≤ 2g − 2, the space X d (Γ, PSL(2, R)) is nonempty and connected [12]. We will prove these statements using Higgs bundles in Section 4.
For a PSL(2, R) representation ρ, the integer invariant can be defined as follows. Pick any ρ-equivariant map f ρ : S → PSL(2, R)/H ∼ = H 2 . Such maps always exits since H 2 is contractible. We have a principal H-bundle PSL(2, R) → H 2 , thus define τ to be minus the degree of the pullback bundle: If ρ is a Fuchsian representation, then we may choose f ρ to be the equivariant diffeomorphism uniformizing the Riemann surface H 2 /ρ(Γ). In this case, τ is given by the degree of the cotangent bundle. Namely, τ = 2g − 2. Thus, we have In fact, the above inclusion is an equality, thus Teich(S) ∼ = X 2g−2 (Γ, PSL(2, R)) . Example 1.10. For G = PSL(n, R) the maximal compact subgroup is SO(n) if n is odd and SO(n)/ ± Id when n is even. In this case, In the case n = 2k + 1, the invariant ω ∈ H 2 (S, Z 2 ) is the second Stiefel-Whitney class of the SO(n) bundle.
Example 1.11. For G = SO(p, q), the maximal compact subgroup is S(O(p) × O(q)). We have In the above cases, the element of H 1 (S, Z 2 ) is the first Stiefel-Whitney class of an orthogonal bundle and each element of H 2 (S, Z 2 ) is the second Stiefel-Whitney class of an orthogonal bundle. The case of p = 2 or q = 2 is slightly more complicated.

Deforming Fuchsian representations
We have seen that the Teichmüller space of the surface S is identified with the connected component X 2g−2 (Γ, PSL(2, R)) of the PSL(2, R)-character variety. As a result, the representations in this component have special geometric significance. Given an embedding ι : PSL(2, R) → G, we have ι(Fuch(Γ)) ⊂ X (Γ, G) .
There are many examples of interesting embeddings of PSL(2, R) into other Lie groups, below we discuss some particular interesting ones.
it is discrete and faithful and ρ(Γ) acts cocompactly on a convex domain in H n .
In fact, we have the following: Proposition 2.4. For n > 2, any representation ρ ∈ ι 1,n (Fuch(Γ)) can be continuously deformed to a compact representation.
Proof. Note that it suffices to prove the statement for SO 0 (1, 3). Recall that there is an isomorphism of Lie groups SO 0 (1, 3) ∼ = PSL(2, C). The result now follows from Corollary 1.8.
Remark 2.5. Another interesting embedding is given by the isomorphism PSL(2, R) ∼ = PU (1,1) and the embedding PU(1, 1) → PU(1, n) ∼ = Isom(CH n ) into the isometry group of the complex hyperbolic space. Deformations of Fuch(Γ) ⊂ X (Γ, PU(1, n)) under this embedding satisfy a rigidity phenomenon [11]. This is a special case of the more general situation of maximal representations into a Hermitian Lie group of non-tube type (see for example [5]). We will not discuss this situation further.

Principal embedding.
Recall that for each dimension n, there is a unique irreducible representation which is given by the (n − 1) st -symmetric product of the standard representation. Moreover, it is straight forward to check that this induces an embedding ι pr : PSL(2, R) → PSL(n, R) .
We will call this embedding the principal embedding.
More generally, if G is an split real Lie group of adjoint type, there is a unique preferred (principal) embedding (2.2) ι pr : PSL(2, R) → G .
We will not go into the Lie theory necessary to define the principal embedding in general, see [17] for more details on the general setup. We will explicitly describe ι pr for the classical groups.
The deformation space of ι pr (Fuch(Γ)) ⊂ X (Γ, G) is called the Hitchin component or Hitchin components. Definition 2.7. Let G be a split real simple adjoint Lie group, a Hitchin component is a connected component containing a component of ι pr (Fuch(Γ)).
Unlike the embedding ι 1,n : PSL(2, R) → SO 0 (1, n), representations in Hit(G) cannot be deformed to compact representations. Theorem 2.8. (Hitchin [15]) If ρ ∈ Hit(G) then ρ cannot be deformed to a compact representation. In particular, |π 0 (X (Γ, G))| ≥ 1 + |π 0 (X (Γ, H))| . Remark 2.9. Labourie showed that all representations in the Hitchin component satisfy a certain dynamical property called the Anosov property [18] which generalize the notion of convex cocompactness to higher rank Lie groups. As a consequence, every representation in a Hitchin component is discrete and faithful. Moreover, like Fuch(Γ) the representations in the Hitchin component are all holonomies of certain geometric structures on compact manifolds [13]. Since Hit(G) shares many features with the Teichmüller space of S, it has been called a higher Teichmüller component (see for example [6] and [26]). We will not discuss this perspective anymore, however the components discussed in these notes which are deformation spaces of Fuchsian representations are intimately related with the field of higher Teichmüller theory.
For the group PSL(n, R), Hitchin also proved that there are no other components.
Introductory words about the below corollary Corollary 2.12. If ρ ∈ X (Γ, PSL(n, R)), then there is a dichotomy: either ρ can be deformed to compact representation or ρ can be deformed to a Fuchsian representation in ι pr (Fuch(Γ)).
Remark 2.13. A generalization of the embedding (2.1) is given by The embedding will play an important role in Theorem 7.13. In fact, when q = p the principal embedding ι pr : PSL(2, R) → SO(p, p) is given by principal into SO(p, p−1) followed by ι p,p

Higgs bundles
We now shift our focus to a moduli space of holomorphic objects on a Riemann surface called Higgs bundles. This theory was developed Hitchin [14,16] and Simpson [24,25]. At first look, Higgs bundles and surface group representations seem to have little to do with each other. However, a remarkable theorem, known as the Nonabelian Hodge Correspondence, gives a homeomorphism between the two moduli spaces. Higgs bundles thus provide a powerful tool for addressing certain questions about the topology of the character variety.
Theorem 3.1 (Nonabelian Hodge Correspondence). Let S be a closed orientable surface of genus at least two. For each Riemann surface structure X on S, the moduli space of G-Higgs bundles on X is homeomorphic to character variety X (π 1 (S), G).
One direction of the nonabelian Hodge correspondence asserts that to each polystable G-Higgs bundle, there is a special metric from which one builds a flat G-connection. For principal bundles a metric is by definition a reduction of structure group to the maximal compact subgroup. The other direction asserts that for each reductive representations and each choice of Riemann surface structure on S, there is an equivariant harmonic map X → G/H from the universal cover to the Riemannian symmetric space. From such a map one constructs a polystable G-Higgs bundle. For details on this case we refer the reader to Q. Li's lectures in this volume.
3.1. Definitions. As before, let G be a reductive Lie group with maximal compact H and Cartan decomposition g = h ⊕ m. Complexifying gives an Ad H C -invariant decomposition g C = h C ⊕ m C . Fix a compact Riemann surface X with genus g ≥ 2 and let K denote its holomorphic cotangent bundle.
Definition 3.2. A G-Higgs bundle on X is a pair (P, ϕ) where • P → X is a holomorphic principal H C -bundle and • ϕ is a holomorphic section of the associated bundle P[m C ] ⊗ K. The holomorphic section ϕ is called the Higgs field. Example 3.3. If G is compact, then g = h and m = {0}. In this case, a G-Higgs bundle is just a holomorphic principal G C -bundle.
In this case, a G-Higgs bundles is a pair (P, ϕ) where P is a holomorphic G-bundle and ϕ is a holomorphic section of the adjoint bundle P[g] twisted by K.
Rather than work with principal bundles, we will usually pick a faithful linear representation of G C and work with vector bundles. A faithful representation G C → GL(V ) defines a representation β : H C → GL(V ) and an embedding m C ֒→ End(V ). With this data fixed, a G-Higgs bundle (P, ϕ) gives rise to a pair ( Example 3.5. When G = SL(n, C) we take G → GL(C n ) to be the standard representation. An SL(n, C)-Higgs bundle is thus defines a pair (E, Φ) where E → X is a holomorphic rank n vector bundle and Φ ∈ H 0 (End(E) ⊗ K) satisfies tr(Φ) = 0. Moreover, the standard volume form on C n is preserved by the standard representation of SL(n, C), and a holomorphic principal SL(n, C)-bundle is equivalent to a holomorphic vector bundle E equipped with a holomorphic volume form ω ∈ H 0 (Λ n E). Thus, an SL(n, C)-Higgs bundle is equivalent to a triple (E, ω, Φ). Note that the holomorphic volume form ω is equivalent to a holomorphic trivialization of the determinant line bundle Λ n E. We will usually suppress ω from the notation. Example 3.6. For G = SL(n, R) we have H = SO(n) and the Cartan decomposition is given by is the vector space of traceless symmetric matrices. Again using the standard representation of SL(n, C) we see that an SL(n, R)-Higgs bundle gives rise to a triple (E, ω, Φ) as in the previous example.
Since H C = SO(n, C), the restriction of the standard representation of SO(n, C) preserves a nondegenerate symmetric complex bilinear form on C n , a holomor- An SL(n, R)-Higgs bundle is thus equivalent to a tuple (E, ω, Q E , Φ).
. The Lie algebra of SO(p, q) is given by This implies that A ∈ so(p), D ∈ so(q) and B = −C T , thus the Cartan decomposition is given by Similar to the previous examples, we use the standard representation. Since H C = S(O(p, C) × O(q, C)), the restriction of the standard representation of SO(p + q, C) preserves an orthogonal splitting C p+q = C p ⊕ C q . As in the previous example, a holomorphic principal , where V and W respectively have rank p and q and quadratic forms Q V and Q W . Using the Cartan decomposition and the description of the Lie algebra, the Higgs Since SO(p, q)-Higgs bundles will be a main object of study we record this in a proposition.
and the associated SL(p + q, C)-Higgs bundle is given by forgetting Q E .
We will often suppress Q V , Q W and ω from the notation and just refer to an SO(p, q)-Higgs bundle as a triple (V, W, η). We will also denote the associated Higgs bundle schematically by where we have suppressed the twisting by K from the notation.

3.2.
Stability and the moduli space. The moduli space of Higgs bundle parameterizes isomorphism classes of Higgs bundles. The isomorphism group for Higgs bundles is called the gauge group. Just as with the character variety, to get a nice moduli we restrict to a special class of Higgs bundles whose gauge orbits are closed. Given a smooth principal H C -bundle P → X, the G-gauge group G H C is group of bundle automorphisms f : P → P. The elements of G H C are given by sections of an associated bundle of groups P [H C ] = P × Ad H C H C : Recall that for a holomorphic structure on a vector bundle E is equivalent to a Dolbeault operator. That is a differential operator for all functions f ∈ Ω 0 (C) and sections s ∈ Ω 0 (E). Note that a Dolbeault operator is equivalent to the (0, 1)-part of a connection on E. In particular, the space of holomorphic structures on E is an infinite dimensional affine space with underlying vector space Ω 0,1 (End(E)).
For principal bundles, an analogous theory holds. Namely a holomorphic structure on a principal H C -bundle P → X is equivalent to a section∂ P ∈ Ω 0,1 (P, h C ) which defines the (0, 1)-part of a connection. Thus, the space of holomorphic structures on P an infinite dimensional affine space with the space of basic (0, 1)-forms as underlying vector space. Equivalently, this vector space is given by sections Ω 0,1 (X, P [h C ]) of the adjoint bundle of P.
If we fix a smooth H C -bundle P → X, the set of all Higgs bundles with underlying bundle P is given by Fixing a holomorphic structure on P defines H(G) as a quadratic subspace of a vector space: .
The tangent space is thus given by sections (α, ψ) satisfying this equation to first order: Remark 3.12. The space of Higgs bundles H(G) has a natural complex structure given by I(α, ψ) = (iα, iψ). It also has a natural symplectic structure given by The orbits of the gauge group are not closed, and to form a nice moduli space we need a notion of (poly)stability. The moduli space M(G) of G-bundles is then defined to be the set G H C -orbits of polystable G-Higgs bundles. In general, the notion of the stability involves considering how all holomorphic structure group reductions of an H C -bundle to a parabolic subgroup interact with the Higgs field. Instead of developing this theory in general, we will develop the appropriate stability conditions in the vector bundle situation. For the general theory see [9].
Recall that a GL(n, C)-Higgs bundle is equivalent to a rank n holomorphic vector bundle E a section Φ ∈ H 0 (End(E) ⊗ K). For SL(n, C) the bundle E is equipped with a trivialization of Λ n E, thus, deg(E) = 0.
For general groups G the notion of stability is functorial in the sense that if G is a real form of a reductive subgroup of SL(n, C), then a G-Higgs bundle is polystable if and only if the associated SL(n, C)-Higgs bundle is polystable. Let H ps (G) ⊂ H(G) denote the set of polystable Higgs bundles. The gauge group G H C -preserves H ps (G), and the gauge orbits in H ps (G) are closed. We define the moduli space M(G) to be the quotient space We note that the complex structure I and the symplectic form ω I from Remark 3.12 are preserved by the gauge group action and thus descend to the moduli space.
For the general notion of stability, it is not the case that a G-Higgs bundle is stable if and only if the associated SL(n, C)-Higgs bundle is stable. However, can detect stable G-Higgs bundles inside of the set of polystable Higgs bundles with the following proposition.
Proposition 3.14. Let G be a real form of a complex semisimple subgroup of SL(n, C). A G-Higgs bundle (P, ϕ) is stable if it is polystable and has finite automorphism group. Moreover, the set of stable G-Higgs bundles is open. Remark 3.15. Note that if the SL(n, C)-Higgs bundle associated to a G-Higgs bundle is stable, then it is stable as a G-Higgs bundle.
Let H s (G) ⊂ H ps (G) be the stable locus, the quotient is an orbifold. At a stable Higgs bundle one can show that the real dimension of the tangent space to 4. SO(1, q)-Higgs bundles especially when q = 2 In this section we will describe the moduli space of SO(1, q)-Higgs bundles and SO 0 (1, q)-Higgs bundles. When q = 2 we have SO 0 (1, 2) ∼ = PSL(2, R). In this case we will recall Hitchin's parameterization of most of the components of M(SO 0 (1, 2)) especially the Teichmüller component.
Recall from Proposition 3.9 that an SO(1, n)-Higgs bundle consists of a tuple (V, Q V , W, Q W , ω, η) where rk(V ) = 1 and rk(W ) = q. We can take (V, Using the notation from (3.1), the associated SL(1 + q, C)-Higgs bundle is given by When q = 1, we have η ∈ H 0 (K) and the first Stiefel-Whitney class sw 1 (W ) ∈ H 1 (X, Z 2 ) of W labels the components of M(SO(1, 1)). Namely, For q > 1, the first and second Steifel-Whitney classes (sw 1 , sw 2 ) ∈ H 1 (X, Z 2 ) × H 2 (X, Z 2 ) of (W, Q W ) give a decomposition of the moduli space The first Steifel-Whitney class of W vanishes if and only if the O(q, C)-bundle reduces to SO(q, C). Thus, for q = 2 and sw 1 = 0 the bundle W reduces to an SO(2, C)-bundle. Since C * = SO(2, C), in this case the degree of the C *bundle provides a refinement of the second Stiefel-Whitney class. More precisely, if sw 1 (W, Q W ) = 0, then there is a line bundle L ∈ Pic(X) such that This gives a decomposition of the moduli space as 1, 2)) .
For Higgs bundles in M d (SO(1, 2)) the splitting W = L⊕L −1 gives a decomposition of the Higgs field η : where β ∈ H 0 (LK) and γ ∈ H 0 (L −1 K). Using Q W = 0 1 1 0 , we can write the associated SL(3, C)-Higgs bundle schematically as where we recall that we suppress the twisting by K from the notation. The stability condition limits the objects we are considering. For d = | deg(L)| > 0, we can parameterize the moduli space M d (SO(1, 2)), this was done by Hitchin in [14] for the group PSL(2, R). [14]] For d > 0, the moduli space M d (SO(1, 2)) is smooth and diffeomorphic to the total space of a rank (d + g − 1)-complex vector bundle over the (2g − 2 − d)-symmetric product Sym 2g−2−d (X) of the Riemann surface X.

Theorem 4.2. [Hitchin
Proof. By the above discussion a point in M d (SO(1, 2)) is determined by a triple (L, γ, β) where L ∈ Pic d (X), γ ∈ H 0 (L −1 K)\{0} and β ∈ H 0 (LK). The S(O(1, C)× O(2, C))-bundle is given by For two triples (L, β, γ) and (L ′ , β ′ , γ ′ ) to define isomorphic SO(1, 2)-Higgs bundles it is necessary that | deg(L)| = | deg(L ′ )|. Thus we may assume L = L ′ as elements Pic d (X). The remaining holomorphic gauge transformation of the S(O(1, C)× O(2, C)) bundle is given by for λ ∈ C * . This gauge transformation acts on the Higgs field by In particular, we note that the automorphism group of such an SO(1, 2)-Higgs bundle is trivial since γ = 0. Thus, the moduli space M d (SO(1, 2)) is smooth and Recall that the space of effective divisors on X of degree n is given by the n thsymmetric product Sym d (X). Taking the projective class of γ ∈ H 0 (L −1 K) \ {0} defines a surjective map to the space of effective degree 2g − 2 − d divisors on X : .
We claim that the fiber of this map is a vector space of rank (d + g − We now collect many corollaries of the above theorem. The cohomology ring of a symmetric product of a Riemann surface was computed in [20], as a result this computes the cohomology ring of M d (SO(1, 2)). When d = 2g − 2, the space is contractible and we have the following. Proof. An SO(1, 2)-Higgs bundle in M 2g−2 (SO(1, 2)) is determined by a triple (L, β, γ) where deg(L) = 2g − 2, β ∈ H 0 (LK and γ ∈ H 0 (L −1 K) \ {0}. The condition on γ implies that L = K and thus β ∈ H 0 (K 2 ). If we normalize γ to by γ = 1 ∈ H 0 (O), then there is no more gauge freedom, and so Consider the SO(3, C)-Hitchin fibration .
Translating these statements to the character variety X (Γ, SO(1, 2)) via the nonabelian Hodge correspondence gives the following.  (1, 2)) has a connected component X d (SO(1, 2)) which is smooth and diffeomorphic to a real rank 2d + 2g − 2 vector bundle over the symmetric product Sym 2g−2−d (S).

The Hitchin fibration and Hitchin section
So far we have seen that the character variety X (Γ, G) is homeomorphic to the moduli space of G-Higgs bundles. The upshot of this correspondence is that the Higgs bundle moduli space has a lot of useful structures which the character variety is lacking. In this section we define the Hitchin component and see use this additional structure to construct the Hitchin component from Definition 2.7.
5.1. The Hitchin fibration. Suppose G is a complex simple Lie group. Similar to Chern-Weyl theory, we can apply an invariant polynomial to the Higgs field and obtain a holomorphic differential. Fixing a homogeneous basis p 1 , · · · , p rk(G) of the Ad G -invariant polynomials C[g] with deg(p j ) = m j defines a map called the Hitchin fibration. For example, when G = SL(n, C) we have m j = j + 1 for 1 ≤ j ≤ n − 1, and when G = SO(2n + 1, C) we have m j = 2j for 1 ≤ j ≤ n.
In general, a computation using the Riemann-Roch theorem shows that the base is half the dimension of the moduli space: .

In [16], Hitchin showed that for any choice of basis the Hitchin fibration is proper.
In fact, the generic fibers of the Hitchin fibration are half dimensional tori and makes M(G) into a algebraic completely integrable system, we will not make use of this additional structure.
Remark 5.1. Notice that the dimension of the base of the Hitchin fibration is the same as the dimension of the moduli space of G r -Higgs bundles for G r < G any real form. For example, the Hitchin base of SO(2n + 1, C) has the same dimension as M(SO(p, q)) for all p and q satisfying p + q = 2n + 1.

The Hitchin section.
Let g be a semisimple complex Lie algebra. For s ⊂ g a subalgebra isomorphic to sl(2, C), consider the decomposition of g into irreducible sl(2, C)-representations For any such s ⊂ g we have N ≥ rk(g), and when N = rk(g) the the three dimensional subalgebra s is called principal. Up to the conjugation, there is a unique principal three dimensional subalgebra [17]. In this case we have dim(V j ) = 2m j +1 where 1 = m 1 ≤ m 2 ≤ · · · ≤ m rk(g) are the exponents of g. Moreover, when we restrict a principal embedding sl(2, C) → g to the real subalgebra sl(2, R), the image lies in the a split real subalgebra of g. This defines an embedding We will prove the above theorem for G = SO(2p+ 1, C), namely we will construct the Hitchin section and prove that it maps onto a component for the group SO(p, p+ 1). For SO(2p + 1, C) the Hitchin fibration is given by Consider the rank p holomorphic orthogonal bundle Note that K p has a natural orthogonal structure Q p = 1 . . . 1 We claim that the image of Ψ is contained in the stable Higgs bundles H s (SO(p, p+ 1)) and that the induced map Ψ : Proof. Consider the stable SL(2, C)-Higgs bundle Since the unique irreducible (2p + 1)-dimensional representation of SL(2, C) is given by the 2p-symmetric product, the SL(2p + 1, C)-Higgs bundle given by is also stable. Moreover this is gauge equivalent to After rearranging the summands of K p ⊕ K p+1 , the SL(2p + 1, C)-Higgs bundle associated to Ψ(0, · · · , 0) is given by (5.5). Thus, Ψ(0, · · · , 0) is a stable SO(p, p+1)-Higgs bundle. Since stability is an open condition, for q 2 , · · · , q 2p sufficiently close to zero, the Higgs bundle Ψ(q 2 , · · · , q 2p ) is also stable.
Proof. In the general setting of a complex semisimple Lie group the existence of such a basis was proven by Kostant in [17]. For SO(2p + 1, C) we construct such a basis by direct computation. We explain how this works for p = 2 and leave the general case to the reader.
After rearranging the summands, the SO(5, C)-Higgs bundle (E, Q, Φ) associated to the SO(2, 3)-Higgs bundle Ψ(q 2 , q 4 ) is given by We have tr(Φ 2 ) = 8q 2 and tr(Φ 4 ) = 14q 2 2 + 5q 4 , thus we choose the basis By the previous two propositions, the map Ψ gives rise to a well defined map Ψ : which is a section of the Hitchin fibration for M(SO(2p + 1, C)) → Proof. For openness, not that the spaces have the same dimension. Moreover, by Proposition 5.4, no two Higgs bundles in the image of Ψ are gauge equivalent. For closedness suppose (q j 2 , · · · , q j 2p ) is a divergent sequence of points in p j=1 H 0 (K 2j ).
By Proposition 5.4 and properness of the Hitchin fibration we conclude that the sequence Ψ(q j 2 , · · · , q j 2p ) also diverges in M(SO(p, p + 1)). To complete the proof we need to show that under the nonabelian Hodge correspondence, the component defined by Ψ( p j=1 H 0 (K 2j )) is the Hitchin component Hit(SO(p, p + 1)) from Definition 2.7. It suffices to show that the representation associated to Ψ(0, · · · , 0) is in Hit (SO(p, p+1)). By Remark 4.7, the Higgs bundle (5.4) defines an SL(2, R)-Higgs bundle whose corresponding representation in in Fuch(Γ). From Example 2.6, the principal embedding ι pr : PSL(2, R) → SO(p, p + 1) is given by taking the 2p-symmetric product of the standard representation of SL(2, R). Thus, the representation associated to Ψ(0, · · · , 0) is contained in Hit(SO(p, p + 1)). 6. Structure of the moduli space 6.1. Tangent space and deformation complex. In this section we will assumer for simplicity that G is the real form of a complex semisimple Lie group. Under this assumption, the automorphism group of a stable G-Higgs bundle is discrete (see Proposition 3.14). Recall that H(G) is the set of pairs (∂ P , ϕ) where∂ P is a Dolbeault operator on smooth H C -bundle P → X and ϕ ∈ Ω 1,0 (P [m C ]) such that ∂ P ϕ = 0.
Since the space of Dolbeault operators is an affine space with underlying vector space isomorphic Ω 0,1 (P [h C ]) the tangent space of H ps (G) at (∂ P , ϕ) is given by the set of (α, ψ) ∈ Ω 0,1 (P [h C ]) ⊕ Ω 1,0 (P [m C ]) so that ϕ + ψ is holomorphic with respect to the Dolbeault operator (∂ P + α) to first order. That is, The moduli space of G-Higgs bundles is a set of gauge equivalence classes: where H ps (G) denotes the set of polystable pairs. At stable points of the moduli space, the tangent space can be interpreted as a quotient of the tangent space to the gauge orbit G H C · (∂ P , ϕ): This is because, under our assumption on G, the automorphism group of a stable G-Higgs bundle is discrete, and so the gauge group action is locally free. The tangent space to the gauge orbit of a stable G-Higgs bundle can thus be identified with tangent space at the identity of the gauge group The identification of Ω 0 (P [h C ]) with the tangent space T (∂P ,ϕ) G H C · (∂ P , ϕ) is given by the map .
Proof. First, the elements in the kernel of the map ad ϕ correspond to tangent vectors of one parameter families of automorphisms of (∂ P , ϕ). Thus stability implies that ker(ad ϕ ) = 0. Next, note that the kernel of the map i : is given by the set of (0, ψ) = (∂ P x, [ϕ, x]). Thus, the kernel of i equals the image of the map ad ϕ : since α +∂ P x defines the same cohomology class as α. Any representative of an element of the kernel of π is a pair (∂ P x, ψ) such that∂ P ψ + [∂ P x, ϕ] = 0. Any such pair is equivalent to (0, ψ − ad ϕ x). Thus, the kernel of π is the image of i. Finally, the condition∂ P ψ + [ϕ, α] = 0 implies that ad ϕ (α) is zero in the cohomology group H 1 (P [m C ] ⊗ K). Thus, the image of π is the kernel of ad ϕ . Remark 6.3. For strictly polystable Higgs bundles we have an analogous sequence which fails to be exact on the left and may or may not also fail to be exact on the right. On way to describe this is with a deformation complex (see [3]). Namely, the sheaf map ad ϕ : P [h C ] → P [m C ] ⊗ K defines a long exact sequence in hypercohomology In general, H 0 (∂ P , ϕ) is the space of infinitesimal automorphisms of (∂ P , ϕ), and for stable Higgs bundles, the tangent space T [∂P ,ϕ] M(G) is identified with H 1 (∂ P , ϕ).

6.2.
The C * -action. There is a natural action of C * -action on the G-Higgs bundle moduli space given by scaling the Higgs field .
Note that Hitchin fibration (5.1) is equivariant with respect to a weighted C * -action: Thus, the fixed points of the C * -action are contained in the nilpotent cone h −1 (0). Moreover, the properness of h implies that lim λ→0 [∂ P , λϕ] always exists and is a C *fixed point.
Since we are dealing with isomorphism classes, being a C * -fixed point does not imply ϕ = 0. Rather, it implies that there is a holomorphic gauge transformation g λ such that Ad g −1 λ ϕ = λϕ for all λ ∈ C * . For SL(n, C), the C * -fixed points are classified by the following proposition. Proposition 6.4. Let (E, Φ) be a polystable SL(n, C)-Higgs bundle. Then (E, Φ) is gauge equivalent to (E, λΦ) for all λ ∈ C * if and only if there is a holomorphic splitting E = E 1 ⊕ · · · ⊕ E ℓ in which the Higgs field is given by where ϕ j : E j → E j+1 ⊗ K is a holomorphic bundle map.
Remark 6.5. We will usually represent such a fixed point schematically as where we suppress the twisting by K from the notation. The moduli space of such fixed points are a special case of the moduli of holomorphic chains.
For SL(n, C) we have H C = SL(n, C) and m C = sl(n, C). For SL(n, C)-Higgs bundles fixed by the C * -action the H C -bundle has a holomorphic reduction E = E 1 ⊕ · · · ⊕ E k to a subgroup of block diagonal matrices. Such a reduction gives a Z-grading on the bundle Moreover, with respect to this Z-grading we have Φ ∈ H 0 (End(E) 1 ⊗ K). The characterization of G-Higgs bundles fixed by the C * -action is given by the following proposition.
Proposition 6.6. A polystable G-Higgs bundle (P, ϕ) defines a fixed point of the C * -action on M(G) if and only if (1) There is a Z-grading . Remark 6.7. In terms of vector bundles, the G-Higgs bundle which are C * -fixed points are given by holomorphic chains with extra symmetries which reflect the symmetries of a G-Higgs bundle. For example, the SL(p + q, C)-Higgs bundle associated to an SO(p, q)-Higgs bundle (V, W, η) is given by (V ⊕ W, 0 η † η 0 ), so the associated fixed points are direct sums of holomorphic chains of the form Here r and s are half integers and the additional symmetry on the grading comes from the orthogonal structure. Namely, the quadratic forms give isomorphisms W −j ∼ = W * j and V −j ∼ = V * j . 6.3. Critical points of a Morse-Bott function. So far we have not used the full power of the nonabelian Hodge correspondence. Since we have a special metric associated to each polystable Higgs bundle, we can take the L 2 -norm of the Higgs field. Namely, consider the nonnegative function f : M(G) → R defined by where the norm |ϕ| is taken with respect to the Hermitian metric (∂ P , ϕ) from the nonabelian Hodge correspondence. The C * -action does not in preserve the metric from the nonabelian Hodge correspondence, however, the the metric is preserved by the restriction of the action to U(1) ⊂ C * . Thus, the function f is U(1)-invariant. Moreover, in [15,Section 8], Hitchin showed that the U(1)-action is Hamiltonian with respect to the symplectic structure ω I from Remark 3.12, and that the function f is a moment map for this action. That is, grad(f ) = IX , where X is the vector field generating the U(1)-action. This implies f is a Morse-Bott function on the smooth locus of M(SL(n, C)) and critical submanifolds of f are exactly the components of the fixed point set of the U(1)-action. In fact, Hitchin's arguments also hold for the moduli space M(G).
Since the moduli space M(G) is usually not smooth, we cannot use the full power of Morse theory to do things like compute the cohomology ring. However, using Uhlenbeck compactness Hitchin showed that the function f proper [14] even on the singular locus. Hence, f attains a minima on every closed subset. This implies that on each connected component, f has a local minima. Thus, There are three manifolds which intersect at a C * -fixed point [∂ P , ϕ] 6.4. Local minima criterion. We first describe how for fixed points of the C *action we get a decomposition of the tangent space into weight spaces. Recall from Proposition 6.6 that associated to a polystable G-Higgs bundle (P, ϕ) fixed by the C * -action there is a Z-grading g C = j h C j ⊕ m C j and a holomorphic structure group For such a fixed point, the map ad ϕ : For stable fixed points this gives a decomposition of the exact sequence (6.2), that is, for all j we have (6.5) 0 where, similar to (6.1), T j [∂P ,ϕ] M(G) is defined by This following result was proven for SL(n, C) by Hitchin in [15], the general case follows from arguments analogous to the Morse-Bott function's index computation of Hitchin in [15,Section 8].
Theorem 6.10 (Hitchin [15,14]). Let f : M(G) → R be the Morse-Bott function from (6.4). For a stable G-Higgs bundle we have the following: • [∂ P , ϕ] is a fixed point of the C * -action if and only if it is a critical point of the function f , Theorem 6.13 (Garcia-Prada and Oliveira [10]). Let G be a complex reductive Lie group with maximal compact subgroup H. Then there is a bijection between the components of the moduli space of polystable G-Higgs bundles and the moduli space of polystable G-bundles: π 0 (M(G)) = π 0 (M(H)) .
Proof. By the above discussion and Remark 6.9, it suffices to show that a polystable G-Higgs bundle [P, ϕ] is a local minima of the Morse-Bott function f from (6.4) if and only if ϕ = 0. Let [P, ϕ] be a local minima of f. Since [P, ϕ] is a C * -fixed point, by Proposition 6.6 there is a Z-grading . First suppose [∂ P , ϕ] is a stable local minima of f. Then by Proposition 6.12 we have ad ϕ : is an isomorphism for all j > 0. But, since g is complex, we have h C ∼ = m C , and the only way In this case we have ϕ = 0. To rule out strictly polystable minima with nonzero Higgs field, we note that a G-Higgs bundle which is strictly polystable has a holomorphic reduction to a Levi factor L of a parabolic subgroup of G which is stable as a L-Higgs bundle. Now repeat the above argument for the moduli space M(L).
As a immediate corollary we have the following. Corollary 6.14. If G is a complex reductive Lie group, then every polystable G-Higgs bundle [∂, ϕ] can be continuously deformed to a polystable G-bundle, i.e., a polystable G-Higgs bundle with zero Higgs field.
Using the nonabelian Hodge correspondence, Theorem 1.7 now follows as a corollary direct corollary of the above theorem.
Corollary 6.15. If G is a complex reductive Lie group with maximal compact H, then the map τ : π 0 (X (Γ, G)) → B G (S) from (1.1) is injective. In particular, every representation ρ : Γ → G can be deformed to a compact representation Γ → H ֒→ G.
Using the methods described above, Hitchin gave a complete component count of M (PSL(n, R)). The proof idea is to first classify the stable local minima using Proposition 6.6, then construct explicit deformations of strictly polystable fixed points with nonzero Higgs field which decreases the value of f. We thus have the following corollary.   1, 2)). For n > 2 and odd we have B PSL(n,R) = Z 2 and there is only one Hitchin component. This gives three components. For n > 2 and n-even we have B PSL(n,R) is isomorphic to Z 2 × Z 2 or Z 4 depending on the parity of n 2 (see (1.2)). Moreover, there are two Hitchin components by Remark 2.11, this gives six components. By Theorem 6.16 there are no other components.
For the character variety X (Γ, PSL(n, R)) we of course have the same count.  n-odd, 6 n > 2 and even Theorem 6.16 also gives a dichotomy for deformations of representations into PSL(n, R), namely for n > 2 the components of X (Γ, PSL(n, R)) are either deformations spaces of compact representations or deformation spaces of special Fuchsian representations.
Corollary 6.19. For each n > 2 and each ρ ∈ X (Γ, PSL(n, R)), exactly one of the following holds • ρ can be deformed to a compact representation Γ → PSO(n) ֒→ PSL(n, R) • ρ can be deformed to a representation where ρ F uch ∈ Fuch(Γ) is a Fuchsian representation and ι pr is the principal embedding from (2.2).

SO(p, q)-Higgs bundles
We now apply the techniques of understand the components of the SO(p, q)character variety X (Γ, SO(p, q)). In her thesis [2], Aparicio-Arroyo discovered that Higgs bundle moduli space M(SO(p, q)) had stable local minima of the Morse-Bott function (6.4) with nonzero Higgs field and which did not arise from the Hitchin section. This was done by classifying stable SO(p, q)-Higgs bundles which where fixed points of the C * -action and satisfied Proposition 6.12. Due to the potential singularities, these results were not strong enough to classify the components of the moduli space M(SO(p, q)). We start by recalling the classification of stable minima.
Remark 7.1. The case SO(2, q) is rather special since SO(2, q) is a group of Hermitian type. This special type of group has its own very interesting connected component results. Since we have not said much about this situation, we will only discuss the non-Hermitian case of, that is, for 2 < p ≤ q. For the case of SO(2, q) we refer the reader to [8] and [1].
Remark 7.3. Note that in case three of the above theorem when deg(M ) = p(2g − 2) the existence of a nonzero section of M −1 K p implies M = K p . In this case, the minima is the minima in the SO(p, p+ 1)-Hitchin component defined by Ψ(0, · · · , 0) from (5.3).
In [1] all of the local minima are classified. The result basically says that the only minima not accounted for in Theorem 7.2 arise from polystable Higgs bundles with zero Higgs field and from by the orthogonal bundle W 0 to by strictly polystable. (1) η = 0 (2) there is a polystable rank q − p + 1 orthogonal bundle W 0 with determinant bundle I = det(W 0 ) such that (3) q = p + 1 and there is a line bundle M ∈ Pic d (X) with d ∈ (0, p(2g − 2)] and µ ∈ H 0 (M −1 K p ) \ {0} so that where η µ = 0 · · · 0 µ : K p → M −1 K.
To show that each of the above local minima define a connected component of moduli space M(SO(p, q)) we define a map Θ p,q from a parameter space into M(SO(p, q)) so that (1) Θ p,q is a homeomorphism onto its image, When q = p + 1, we have M(O(q − p + 1)) = M(O(2)) and the number of connected components is 2 2g+1 − 1. Combining these with the p(2g − 2) components of local minima of type 3 in the above theorem give 2 2g+1 − 1 + p(2g − 2) connected components of local minima in M(SO(p, p + 1)) with η = 0. Finally, when q = p we have q − p + 1 = 1 and thus there are 2 2g components of local minima with η = 0. In this case all such minima define Hitchin components.
Combined with the 2 2g+1 -components of M(S(O(p) × O(q)), for 2 < p ≤ q the following theorem of [1] establishes the component count of M (SO(p, q)).
Theorem 7.5. For 2 < p ≤ q, we have To sketch the idea of the above theorem, we need to slightly generalize our notion of Higgs bundles. Definition 7.6. A K p -twisted G-Higgs bundle is a pair (P, ϕ) where • P → X is a holomorphic H C -bundle • ϕ is a holomorphic section of the associated bundle P[m C ] ⊗ K p .
The notions of stability for K p -twisted Higgs bundles are defined completely analogous to the notions of stability for regular Higgs bundles, i.e. for K 1 -twisted Higgs bundles. We will denote the space of polystable K p -twisted G-Higgs bundles and the resulting moduli space by H ps K p (G) and M K p (G) = H ps K p (G)/G H C . Recall from Section 4 that an SO(1, n)-Higgs bundle is given by a triple (V, W, η) = (Λ n W 0 , W 0 , η), where η ∈ H 0 (Λ q W 0 ⊗ W 0 ⊗ K). The map η can be interpreted as a holomorphic bundle map η : Λ q W 0 → W 0 ⊗ K. Similarly, a K p -twisted SO(1, n)-Higgs bundle is given by a triple (Λ n W 0 , W 0 , η p ) where η p ∈ H 0 (Λ q W 0 ⊗ W 0 ⊗ K), which we may interpret at as holomorphic bundle map η p : K 1−p → W 0 ⊗ K .
Remark 7.7. Note that Λ p (I ⊗ K p ) = I p and Λ q ((I ⊗ K p−1 ) ⊕ W 0 ) = I p , so this indeed defines an SO(p, q)-Higgs bundle.
In particular, the SO(p, q)-Higgs bundle is a direct sum of an SO(p, p − 1)-Higgs bundle in the Hitchin component (twisted by an O(1, C)-bundle) with an polystable O(q − p + 1)-Higgs bundle. Remark 7.11. The proof of the n = 1 and n > 2 are by direct computation, namely we show that every fixed point of the C * -action can be deformed to on with zero Higgs field. The additional p(2g − 2) components in the n = 2 case are analogous to the components in Theorem 4.2. In particular, the Higgs field in these components is never zero and these components are parameterized by certain vector bundles over an appropriate symmetric product of the surface.
Combining Remarks 7.8 and 7.11 it follows that if 2 < p < q − 1, then every Higgs bundle in the image of Θ p,q can be deformed to the direct sum of a polystable orthogonal bundle W 0 and a Higgs bundle in the SO(p, p − 1)-Hitchin component twisted by the determinant of W 0 . Apply the nonabelian Hodge correspondence to this statement gives a dichotomy for the character variety X (Γ, SO(p, q)) when q > p + 1 which is analogous to Corollary 2.12.
Theorem 7.13. Suppose 2 < p < q − 1. If ρ ∈ X (Γ, SO(p, q)), then there is a dichotomy: either ρ can be deformed to compact representation or ρ can be deformed to a Fuchsian representation of the form • ρ F uch : Γ → PSL(2, R) is a Fuchsian representation • ι pr : PSL(2, R) → SO(p, p − 1) is the principal embedding • ι p,q : SO(p, p − 1) → SO(p, q) is the embedding given by (2.3) • α : Γ → O(q − p + 1) is a compact representation. In particular, every component of X (Γ, SO(p, q)) is either the deformation space of compact representations or the deformation space of certain Fuchsian representations.
Remark 7.14. For the case SO(p, p + 1) there is a trichotomy, since their are p(2g − 2) − 1 components which cannot be deformed to compact representations and cannot be deformed to Fuchsian representations. In [7], the SO(p, p + 1)-case is studied in great detail, and it is conjectured that every representation in these p(2g − 2) − 1 components is Zariski dense.