Studying Deformations of Fuchsian Representations with Higgs Bundles

Studying Deformations of Fuchsian Representations
with Higgs Bundles††This paper is a contribution to the Special Issue on
Geometry and Physics of Hitchin Systems. The full collection is available at
https://www.emis.de/journals/SIGMA/hitchin-systems.html

Brian COLLIER

B. Collier

Department of Mathematics, University of Maryland, College Park, MD 20742, USA
<EMAIL_ADDRESS>https://www.math.umd.edu/~bcollie2/

Received October 16, 2018, in final form February 02, 2019; Published online
February 12, 2019

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 Teichmüller 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].

Higgs bundles; character varieties; higher Teichmüller theory

14D20; 14F45; 14H60

##  1 An introduction to the character variety

Let $S$ be a closed oriented surface of genus $g\geq 2$. Denote the
fundamental group of $S$ by $\Gamma$, and recall that $\Gamma$ has the
standard presentation

$\displaystyle\Gamma=\left\langle
a_{1},\dots,a_{g},b_{1},\dots,b_{g}\,|\,\prod\limits_{j=1}^{g}[a_{j},b_{j}]=1\right\rangle.$

Fix also a real reductive Lie group ${\mathsf{G}}$. For example ${\mathsf{G}}$
could be one of the following groups

$\textstyle{{\mathsf{GL}}(n,{\mathbb{R}}),}$$\textstyle{{\mathsf{GL}}(n,{\mathbb{C}}),}$$\textstyle{{\mathsf{SL}}(n,{\mathbb{C}}),}$$\textstyle{{\mathsf{Sp}}(2n,{\mathbb{R}}),}$$\textstyle{{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})={\mathsf{SL}}(n,{\mathbb{R}})/\pm{\mathrm{Id}},}$$\textstyle{{\mathsf{U}}(n),}$$\textstyle{{\mathsf{SO}}(n),}$$\textstyle{{\mathsf{SU}}(p,q),}$$\textstyle{{\mathsf{SO}}(p,q),}$$\textstyle{{\mathsf{S}}({\mathsf{O}}(p)\times{\mathsf{O}}(q)),}$

but ${\mathsf{G}}$ _cannot_ be a group like

$\displaystyle{\mathsf{P}}=\big{\\{}\left(\begin{smallmatrix}a&b\\\
0&c\end{smallmatrix}\right)\in{\mathsf{GL}}(2,{\mathbb{C}})\big{\\}}.$

One main property of a reductive Lie group ${\mathsf{G}}$ is that, up to
conjugation, there is a unique maximal compact subgroup
${\mathsf{H}}<{\mathsf{G}}$. We will heavily use this property. In fact there
is a homotopy equivalence

$\displaystyle{\mathsf{H}}\simeq{\mathsf{G}}.$

A group homomorphism $\rho\colon\Gamma\to{\mathsf{G}}$ will be referred to as
a _representation_. Consider the set ${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$ of
all such representations. This set has a topological structure induced by the
inclusion

$\textstyle{{\mathrm{Hom}}(\Gamma,{\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathsf{G}}^{2g},}$$\textstyle{\rho\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\rho(a_{1}),\dots,\rho(b_{g})).}$

Since a representation $\rho$ must satisfy
$\prod\limits_{j=1}^{g}[\rho(a_{j}),\rho(b_{j})]={\mathrm{Id}}$, the image
consists of $2g$-tuples of elements of ${\mathsf{G}}$ which satisfy this
relation. For semisimple groups, the dimension of
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$ was computed in [13] to be

$\displaystyle\dim({\mathrm{Hom}}(\Gamma,{\mathsf{G}}))=2g\cdot\dim{\mathsf{G}}-\dim{\mathsf{G}}.$

The group ${\mathsf{G}}$ acts on ${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$ by
conjugation: for $\gamma\in\Gamma$ and $g\in{\mathsf{G}}$,

$\displaystyle(\rho\cdot g)(\gamma)=g^{-1}\rho(\gamma)g.$

The quotient space ${\mathrm{Hom}}(\Gamma,{\mathsf{G}})/{\mathsf{G}}$ consists
of conjugacy classes of representations. Unless ${\mathsf{G}}$ is compact,
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})/{\mathsf{G}}$ is not Hausdorff.

######  Example 1.1.

Let $\rho_{1}(\gamma)={\mathrm{Id}}$ for all $\gamma\in\Gamma$ and define
$\rho_{2}$ by $\rho_{2}(a_{j})={\mathrm{Id}}$ for all $j$ and

$\displaystyle\rho_{2}(b_{j})=\begin{cases}{\mathrm{Id}},&j\neq g,\\\
\left(\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\right),&j=g.\end{cases}$

If $g_{t}=\left(\begin{smallmatrix}t&0\\\ 0&t^{-1}\end{smallmatrix}\right)$,
then $g_{t}^{-1}\rho_{2}(a_{j})g_{t}={\mathrm{Id}}$ for all $j$ and

$\displaystyle
g_{t}^{-1}\rho_{2}(b_{j})g_{t}=\begin{cases}{\mathrm{Id}},&j\neq g,\\\
\left(\begin{smallmatrix}1&t^{-2}\\\
0&1\end{smallmatrix}\right),&j=g.\end{cases}$

The ${\mathsf{G}}$-orbits through $\rho_{1}$ and $\rho_{2}$ are disjoint in
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$, but

$\displaystyle\rho_{1}\in\overline{\rho_{2}\cdot G}.$

Thus, $[\rho_{1}]$ and $[\rho_{2}]$ are not separable in
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})/{\mathsf{G}}$.

To get a Hausdorff quotient we restrict to a subset of
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$. A representation
$\rho\colon\Gamma\to{\mathsf{G}}$ is called _reductive_ if the composition
with the adjoint representation

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{{\mathsf{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{Ad}}}$$\textstyle{{\mathsf{GL}}({\mathfrak{g}})}$

decomposes as a direct sum of irreducible representations. This is equivalent
to the ${\mathsf{G}}$-orbit through $\rho$ being closed in
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})$. When ${\mathsf{G}}$ is a complex
algebraic group, this is also equivalent to the Zariski closure of
$\rho(\Gamma)$ being a reductive subgroup.

Denote the set of reductive representations by
${\mathrm{Hom}}^{+}(\Gamma,{\mathsf{G}})$. The _${\mathsf{G}}$ -character
variety of $\Gamma$_ is defined to be the Hausdorff space

$\displaystyle{\mathcal{X}}(\Gamma,{\mathsf{G}})={\mathrm{Hom}}^{+}(\Gamma,{\mathsf{G}})/{\mathsf{G}}.$

For semisimple groups, the dimension of ${\mathcal{X}}(\Gamma,{\mathsf{G}})$
is $(2g-2)\dim({\mathsf{G}})$ [13].

######  Remark 1.2.

When ${\mathsf{G}}$ is complex algebraic, one gets the same spaces as the GIT-
quotient.

###  1.1 Fuchsian representations

Important examples of points in a character variety come from Fuchsian
representations and are related to hyperbolic metrics on $S$. These examples
will play a fundamental role throughout the article.

Let $X$ be a Riemann surface structure on $S$. By the uniformization theorem,

$\displaystyle X={\mathbb{H}}^{2}/\rho_{X}(\Gamma),$

where ${\mathbb{H}}^{2}$ is the upper half-plane and
$\rho_{X}\colon\Gamma\hookrightarrow{\mathsf{Bihol}}({\mathbb{H}}^{2})\cong{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$.
By classical results of Riemann, there are $3g-3$ complex parameters for the
deformations of the Riemann surface $X$. Such deformations give rise to a
$(6g-6)$-real dimensional subset of
${\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$. Moreover,
such deformations define an open neighborhood of
$\rho_{X}\in{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$
since $6g-6$ is the dimension of
${\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$. This leads
us to define the set of _Fuchsian representations_ :

$\displaystyle{\mathsf{Fuch}}(\Gamma)=\\{\rho\colon\Gamma\to{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\,|\,\text{discrete
and faithful}\\}/{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}).$

Since being discrete and faithful is a closed condition in
${\mathcal{X}}(\Gamma,{\mathsf{G}})$, ${\mathsf{Fuch}}(\Gamma)$ is a closed
subset. By the above argument, ${\mathsf{Fuch}}(\Gamma)$ is also open. Thus,
${\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$
is an open and closed subset which is identified with the Teichmüller space of
the oriented surface $S$ and the surface $\bar{S}$ with the opposite
orientation

$\displaystyle{\mathsf{Fuch}}(\Gamma)\cong{\mathsf{Teich}}(S)\sqcup{\mathsf{Teich}}\big{(}\bar{S}\big{)}.$

The Teichmüller space of an oriented surface $S$ is defined to be the space of
isotopy classes of (marked) Riemann surface structures on $S$. This space is
famously an open cell of real dimension $6g-6$.

The group ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$ is also the orientation
preserving isometry group of the hyperbolic plane. Thus,
${\mathsf{Fuch}}(\Gamma)$ also parameterizes the set of isotopy classes of
hyperbolic metrics on the surface $S$ and $\bar{S}$.

######  Remark 1.3.

The two connected components of
${\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$
arise because elements of ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$ preserve
the orientation of ${\mathbb{H}}^{2}$. The group
${\mathsf{P}}{\mathsf{GL}}(2,{\mathbb{R}})$ is the full isometry group of the
hyperbolic plane. Thus, if we consider ${\mathsf{Fuch}}(\Gamma)$ as a subset
of the ${\mathsf{P}}{\mathsf{GL}}(2,{\mathbb{R}})$-character variety, then
there is only one component. Similarly, if we consider
${\mathsf{Fuch}}(\Gamma)$ as a subset of
${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{C}})$-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 a
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 isomorphism classes of topological principal
${\mathsf{G}}$-bundles on $S$ by ${\mathcal{B}}_{{\mathsf{G}}}(S)$, this is
the set of homotopy classes of maps from $S$ to the classifying space of
${\mathsf{G}}$. For connected groups we have
${\mathcal{B}}_{\mathsf{G}}(S)\cong H^{2}(S,\pi_{1}{\mathsf{G}})$. Every
representation $\rho\colon\Gamma\to{\mathsf{G}}$ defines a principal
${\mathsf{G}}$-bundle $E_{\rho}\to S$

$\displaystyle E_{\rho}=\big{(}\tilde{S}\times{\mathsf{G}}\big{)}/\Gamma,$

where $\Gamma$ acts on its universal cover $\tilde{S}$ by deck transformations
and by multiplication by $\rho(\Gamma)$ on ${\mathsf{G}}$. Thus, we have a map
${\mathrm{Hom}}(\Gamma,{\mathsf{G}})\to{\mathcal{B}}_{{\mathsf{G}}}(S)$.
Moreover, this map is continuous and descends to a map

$\displaystyle\tau\colon\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
26.6599pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{G}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
50.6599pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.6599pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{B}}_{\mathsf{G}}(S).}$}}}}}}}\ignorespaces}}}}\ignorespaces$
(1.3)

If ${\mathcal{X}}^{\omega}(\Gamma,{\mathsf{G}})=\tau^{-1}(\omega)$, then
${\mathcal{X}}(\Gamma,{\mathsf{G}})$ decomposes as

$\displaystyle\coprod\limits_{\omega\in{\mathcal{B}}_{\mathsf{G}}(S)}{\mathcal{X}}^{\omega}(\Gamma,{\mathsf{G}}).$

######  Question 1.4.

When is the map $\tau$ from (1.3) injective? In other words, when does the
topological invariant distinguish the connected components of the character
variety?

######  Remark 1.5.

Note that when $\tau$ is injective, the question of the connected component
count is not very interesting. We will mainly be interested in when $\tau$ is
_not_ injective and understanding the special features of these components.

Question 1.4 has been answered for many groups, but is open in general. For
compact groups the map $\tau$ is injective, this was proven by
Narasimhan–Seshadri [28] for ${\mathsf{G}}={\mathsf{U}}(n)$ and Ramanathan
[30] in general.

######  Theorem 1.6.

If ${\mathsf{G}}$ is compact $($i.e., ${\mathsf{G}}={\mathsf{H}})$, then
$\tau$ is injective. Furthermore, if ${\mathsf{G}}$ is also semisimple, then
$\tau$ is a bijection.

Since ${\mathsf{H}}$ and ${\mathsf{G}}$ are homotopic, we have
${\mathcal{B}}_{\mathsf{G}}(S)={\mathcal{B}}_{\mathsf{H}}(S)$. Moreover, for
each $\omega\in{\mathcal{B}}_{\mathsf{H}}(S)$

$\displaystyle{\mathcal{X}}^{\omega}(\Gamma,{\mathsf{H}})\subset{\mathcal{X}}^{\omega}(\Gamma,{\mathsf{G}}).$

For complex groups, the map $\tau$ is also injective. This was proven for by
J. Li [24] for semisimple groups and García-Prada and Oliveira [12] for
reductive groups.

######  Theorem 1.7.

If ${\mathsf{G}}$ is complex $($i.e.,
${\mathsf{G}}={\mathsf{H}}^{\mathbb{C}})$, then $\tau$ is injective.

We will prove these results using Higgs bundles in Section 6.5. For
${\mathsf{G}}$ a semisimple complex Lie group, the following corollary follows
immediately from the two above theorems. It holds in general for complex
reductive groups.

######  Corollary 1.8.

For ${\mathsf{G}}$ a complex reductive Lie group, every representation
$\rho\colon\Gamma\to{\mathsf{G}}$ can be continuously deformed to a compact
representation $\Gamma\to{\mathsf{H}}\hookrightarrow{\mathsf{G}}$.

The above corollary says that, for complex groups, the connected components of
the character variety are not interesting. For real groups, the situation is
more subtle.

######  Example 1.9.

For ${\mathsf{G}}={\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$, the maximal
compact subgroup is ${\mathsf{H}}\cong{\mathsf{SO}}(2)$. Since a circle bundle
on a closed surface is determined by its degree, we have
${\mathcal{B}}_{\mathsf{H}}(S)\cong{\mathbb{Z}}$. Thus,

$\displaystyle{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))=\coprod\limits_{d\in{\mathbb{Z}}}{\mathcal{X}}^{d}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})).$

However, the space
${\mathcal{X}}^{d}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$ is empty
when $|d|>2g-2$ [27]. Moreover, when $|d|\leq 2g-2$, the space
${\mathcal{X}}^{d}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$ is
nonempty and connected [14]. We will prove these statements using Higgs
bundles in Section 4.

For a ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$ representation $\rho$, the
integer invariant can be defined as follows. Pick any $\rho$-equivariant map
$f_{\rho}\colon\widetilde{S}\to{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})/{\mathsf{H}}\cong{\mathbb{H}}^{2}$.
Such maps always exist since ${\mathbb{H}}^{2}$ is contractible. We have a
principal ${\mathsf{H}}$-bundle
${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathbb{H}}^{2}$. By
equivariance, the pullback bundle
$f_{\rho}^{*}{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$ descends to a circle
bundle on $S$. Define $\tau$ to be minus the degree of this bundle:

$\displaystyle\tau(\rho)=-\deg((f_{\rho}^{*}{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))/\rho(\Gamma)).$

The bundle ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathbb{H}}^{2}$ is
identified with the unit tangent bundle of ${\mathbb{H}}^{2}$. If $\rho$ is a
Fuchsian representation, then we may choose $f_{\rho}$ to be the equivariant
diffeomorphism uniformizing the Riemann surface
${\mathbb{H}}^{2}/\rho(\Gamma)$. In this case, $\tau$ is given by the degree
of the cotangent bundle. Namely, $\tau=2g-2$. Thus, we have

$\displaystyle{\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}^{2g-2}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))\sqcup{\mathcal{X}}^{2-2g}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})).$

Since ${\mathsf{Fuch}}(\Gamma)$ is open and closed and
${\mathcal{X}}^{2g-2}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$ is
connected, the above inclusion is an equality.

######  Example 1.10.

For ${\mathsf{G}}={\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})$ the maximal
compact subgroup is ${\mathsf{SO}}(n)$ if $n$ is odd and
${\mathsf{SO}}(n)/\pm{\mathrm{Id}}$ when $n$ is even. In this case,

$\displaystyle{\mathcal{B}}_{{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})}(S)\cong
H^{2}(S,\pi_{1}{\mathsf{H}})\cong\begin{cases}{\mathbb{Z}}&\text{if $n=2$,}\\\
{\mathbb{Z}}_{2}&\text{if $n=2k+1$,}\\\
{\mathbb{Z}}_{2}\oplus{\mathbb{Z}}_{2}&\text{if $n=4k$,}\\\
{\mathbb{Z}}_{4}&\text{if $n=4k+2$.}\end{cases}$ (1.4)

For $n=2k+1$, the invariant $\omega\in H^{2}(S,{\mathbb{Z}}_{2})$ is the
second Stiefel–Whitney class of the ${\mathsf{SO}}(n)$ bundle.

######  Example 1.11.

For ${\mathsf{G}}={\mathsf{SO}}(p,q)$, the maximal compact subgroup is
${\mathsf{S}}({\mathsf{O}}(p)\times{\mathsf{O}}(q))$. We have

$\displaystyle{\mathcal{B}}_{{\mathsf{SO}}(p,q)}(S)\cong\begin{cases}H^{1}(S,{\mathbb{Z}}_{2})\cong{\mathbb{Z}}_{2}^{2g}&\text{if
$p=q=1$},\\\ H^{1}(S,{\mathbb{Z}}_{2})\times
H^{2}(S,{\mathbb{Z}}_{2})\cong{\mathbb{Z}}_{2}^{2g+1}&\text{if $p=1$ and
$2<q$},\\\ H^{1}(S,{\mathbb{Z}}_{2})\times H^{2}(S,{\mathbb{Z}}_{2})\times
H^{2}(S,{\mathbb{Z}}_{2})\cong{\mathbb{Z}}_{2}^{2g+2}&\text{if $2<p\leq
q$}.\end{cases}$

In the above cases, the element of $H^{1}(S,{\mathbb{Z}}_{2})$ is the first
Stiefel–Whitney class of an orthogonal bundle and each element of
$H^{2}(S,{\mathbb{Z}}_{2})$ is the second Stiefel–Whitney class of an
orthogonal bundle. The case of $p=2$ or $q=2$ is slightly more complicated.

##  2 Deforming Fuchsian representations

We have seen that the Teichmüller space of the surface $S$ is identified with
the connected components ${\mathsf{Fuch}}(\Gamma)$ of the
${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$-character variety. As a result,
the representations in this component have special geometric significance.
Given an embedding
$\iota\colon{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{G}}$, we have

$\displaystyle\iota({\mathsf{Fuch}}(\Gamma))\subset{\mathcal{X}}(\Gamma,{\mathsf{G}}).$

######  Question 2.1.

What is the deformation space of
$\iota({\mathsf{Fuch}}(\Gamma))\subset{\mathcal{X}}(\Gamma,{\mathsf{G}})$
like?

Here we use the word deformation space of $\iota({\mathsf{Fuch}}(\Gamma))$ to
mean the connected component of ${\mathcal{X}}(\Gamma,{\mathsf{G}})$ which
contains $\iota({\mathsf{Fuch}}(\Gamma))$. There are many examples of
interesting embeddings of ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$ into
other Lie groups, below we discuss some particular interesting ones.

###  2.1 Isometry groups of hyperbolic spaces

There is a natural embedding into
${\mathsf{Isom}}^{+}({\mathbb{H}}^{n})\cong{\mathsf{SO}}_{0}(1,n)$ given by

$\textstyle{{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\cong{\mathsf{Isom}}^{+}({\mathbb{H}}^{2})\cong{\mathsf{SO}}_{0}(1,2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\ \ \ \ \
\iota_{1,n}}$$\textstyle{{\mathsf{SO}}_{0}(1,n)\cong{\mathsf{Isom}}^{+}({\mathbb{H}}^{n}).}$
(2.3)

This gives
$\iota_{1,n}({\mathsf{Fuch}}(\Gamma))\subset{\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,n))$,
and small deformations of these representations are holonomies of complete
hyperbolic $n$-manifolds called quasi-Fuchsian manifolds.

######  Definition 2.2.

A representation $\rho\colon\Gamma\to{\mathsf{Isom}}({\mathbb{H}}^{n})$ is
called _convex cocompact_ if it is discrete and faithful and $\rho(\Gamma)$
acts cocompactly (properly discontinuously with compact quotient) on a convex
domain in ${\mathbb{H}}^{n}$. The set of quasi-Fuchsian representations
${\mathsf{QFuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,n))$
is defined to be

$\displaystyle{\mathsf{QFuch}}(\Gamma)=\\{[\rho]\in{\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,n))\,|\,\text{$\rho$
is convex cocompact}\\}.$

######  Remark 2.3.

If $[\rho]\in{\mathsf{QFuch}}(\Gamma)$, then can be deformed to
$\iota_{1,n}({\mathsf{Fuch}}(\Gamma))$. Moreover, the set of convex cocompact
representations is open in ${\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,n))$.
Thus, any sufficiently small deformation of a Fuchsian representation in
$\iota_{1,n}({\mathsf{Fuch}}(\Gamma))$ is quasi-Fuchsian. However, unlike
${\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,2))$,
the set
${\mathsf{QFuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,n))$
is _not closed_. Namely, there families of convex cocompact representations
whose limit is discrete and faithful, but not convex cocompact.

In fact, we have the following:

######  Proposition 2.4.

For $n>2$, any representation $\rho\in\iota_{1,n}({\mathsf{Fuch}}(\Gamma))$
can be continuously deformed to a compact representation.

###### Proof.

Note that it suffices to prove the statement for ${\mathsf{SO}}_{0}(1,3)$.
Recall that there is an isomorphism of Lie groups
${\mathsf{SO}}_{0}(1,3)\cong{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{C}})$. The
result now follows from Corollary 1.8. ∎

######  Remark 2.5.

Another interesting embedding is given by the isomorphism

$\displaystyle{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\cong{\mathsf{P}}{\mathsf{U}}(1,1)$

and the embedding
${\mathsf{P}}{\mathsf{U}}(1,1)\to{\mathsf{P}}{\mathsf{U}}(1,n)\cong{\mathsf{Isom}}({\mathbb{C}}{\mathbb{H}}^{n})$
into the isometry group of the complex hyperbolic space. Deformations of
${\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{U}}(1,n))$
under this embedding satisfy a rigidity phenomenon [15, 33]. 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 [7]). We will not discuss this
situation further.

###  2.2 Principal embedding

Recall that for each dimension $n$, there is a unique irreducible
representation

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SL}}\big{(}{\mathbb{R}}^{n}\big{)},$

which is given by the $(n-1)^{\rm st}$-symmetric product of the standard
representation. Moreover, it is straight forward to check that this induces an
embedding

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}).$

We will call this embedding the _principal embedding_.

More generally, if ${\mathsf{G}}$ is a split real Lie group of adjoint type,
there is a unique preferred (principal) embedding

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{G}}.$ (2.4)

We will not go into the Lie theory necessary to define the principal embedding
in general, see [22] for more details on the general setup. We will explicitly
describe $\iota_{\rm pr}$ for the classical groups.

######  Example 2.6.

When $n=2p+1$ the principal embedding $\iota_{\rm
pr}\colon{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SL}}\big{(}{\mathbb{R}}^{2p+1}\big{)}$
preserves a symmetric nondegenerate quadratic form of signature $(p,p+1)$.
Thus, we have an embedding

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SO}}_{0}(p,p+1)\subset{\mathsf{P}}{\mathsf{SL}}(2p+1,{\mathbb{R}}).$

This is the principal embedding (2.4) for the split group
${\mathsf{G}}={\mathsf{SO}}_{0}(p,p+1)$.

Similarly, when $n=2p$ the principal embedding $\iota_{\rm
pr}\colon{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SL}}({\mathbb{R}}^{2p})$
preserves a nondegenerate symplectic form. Thus, we have an embedding

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{P}}{\mathsf{Sp}}(2p,{\mathbb{R}})\subset{\mathsf{P}}{\mathsf{SL}}(2p,{\mathbb{R}}).$

This is the principal embedding (2.4) for the split group adjoint
${\mathsf{G}}={\mathsf{P}}{\mathsf{Sp}}(2p,{\mathbb{R}})$.

The deformation space of $\iota_{\rm
pr}({\mathsf{Fuch}}(\Gamma))\subset{\mathcal{X}}(\Gamma,{\mathsf{G}})$ is
called the _Hitchin component_ or _Hitchin components_.

######  Definition 2.7.

Let ${\mathsf{G}}$ be a simple split real Lie group of adjoint type, a Hitchin
component

$\displaystyle{\mathsf{Hit}}({\mathsf{G}})\subset{\mathcal{X}}(\Gamma,{\mathsf{G}})$

is a connected component containing a component of $\iota_{\rm
pr}({\mathsf{Fuch}}(\Gamma))$.

Unlike the embedding
$\iota_{1,n}\colon{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SO}}_{0}(1,n)$,
representations in ${\mathsf{Hit}}({\mathsf{G}})$ cannot be deformed to
compact representations.

######  Theorem 2.8 (Hitchin [21]).

If $\rho\in{\mathsf{Hit}}({\mathsf{G}})$, then $\rho$ cannot be deformed to a
compact representation. In particular,

$\displaystyle|\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{G}}))|\geq
1+|\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{H}}))|.$

######  Remark 2.9.

In [23], Labourie showed that all representations in a Hitchin component
satisfy a certain dynamical property called the Anosov property which
generalizes 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 ${\mathsf{Fuch}}(\Gamma)$, representations in a
Hitchin component are holonomies of certain geometric structures on compact
manifolds [17]. Since ${\mathsf{Hit}}({\mathsf{G}})$ shares many features with
the Teichmüller space of $S$, it has been called a _higher Teichmüller
component_ (see for example [8] and [34]). We will not discuss this
perspective any more, however the components discussed in this article which
are deformation spaces of Fuchsian representations are intimately related with
the field of higher Teichmüller theory.

For the group ${\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})$, Hitchin also proved
that there are no other components.

######  Theorem 2.10 (Hitchin [21]).

For $n>2$ we have

$\displaystyle|\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))|=\begin{cases}3&\text{if
$n$ is odd},\\\ 6&\text{if $n$ is even}.\end{cases}$

######  Remark 2.11.

Recall that
${\mathsf{Fuch}}(\Gamma)\subset{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))$
has two connected components, however, these components are isomorphic via an
outer automorphism of ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$. The number
of Hitchin components
${\mathsf{Hit}}({\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$ depends on the
parity of $n$. Namely, the map $\iota_{\rm
pr}\colon{\mathsf{Fuch}}(\Gamma)\to{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$
is $2:1$ when $n$ is odd and injective when $n$ is even. There are thus two
Hitchin components for
${\mathsf{G}}={\mathsf{P}}{\mathsf{SL}}(2n,{\mathbb{R}})$ and one Hitchin
component for ${\mathsf{G}}={\mathsf{P}}{\mathsf{SL}}(2n+1,{\mathbb{R}})$.

######  Corollary 2.12.

If $\rho\in{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$,
then there is a dichotomy: either $\rho$ can be deformed to compact
representation or $\rho$ can be deformed to a Fuchsian representation in
$\iota_{\rm pr}({\mathsf{Fuch}}(\Gamma))$.

######  Remark 2.13.

A generalization of the embedding (2.3) is given by

$\displaystyle\iota_{p,q}\colon\ {\mathsf{SO}}(p,p-1)\to{\mathsf{SO}}(p,q).$
(2.5)

The embedding

$\textstyle{{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\rm
pr}}$$\textstyle{{\mathsf{SO}}(p,p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{p,q}}$$\textstyle{{\mathsf{SO}}(p,q)}$

will play an important role in Theorem 7.13. In fact, when $q=p$ the principal
embedding $\iota_{\rm
pr}\colon{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SO}}(p,p)$ is
given by the principal embedding into ${\mathsf{SO}}(p,p-1)$ followed by
$\iota_{p,p}$

$\textstyle{{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\rm
pr}}$$\scriptstyle{\iota_{\rm
pr}}$$\textstyle{{\mathsf{SO}}(p,p-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{p,p}}$$\textstyle{{\mathsf{SO}}(p,p).}$

##  3 Higgs bundles

We now shift our focus to a moduli space of holomorphic objects on a Riemann
surface called Higgs bundles. Roughly, a Higgs bundle is a holomorphic bundle
with some extra data, and the moduli space parameterizes isomorphism classes
of Higgs bundles which are called polystable. This theory was developed by
Hitchin [19, 20] and Simpson [31, 32]. At first glance, 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 ${\mathsf{G}}$-Higgs bundles
on $X$ is homeomorphic to the character variety
${\mathcal{X}}(\pi_{1}(S),{\mathsf{G}})$. Moreover, the smooth loci of each
space are diffeomorphic.

######  Remark 3.2.

One direction of the nonabelian Hodge correspondence asserts that, for each
polystable ${\mathsf{G}}$-Higgs bundle, there is a special metric which can be
used to construct a flat ${\mathsf{G}}$-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 representation
and each choice of Riemann surface structure $X$ on $S$, there is an
equivariant harmonic map $\widetilde{X}\to{\mathsf{G}}/{\mathsf{H}}$ from the
universal cover to the Riemannian symmetric space. From such a map one
constructs a polystable ${\mathsf{G}}$-Higgs bundle. For more details on the
correspondence, we refer the reader to Q. Li’s survey article [25].

###  3.1 Definitions

As before, let ${\mathsf{G}}$ be a reductive Lie group with maximal compact
${\mathsf{H}}$ and Cartan decomposition
${\mathfrak{g}}={\mathfrak{h}}\oplus{\mathfrak{m}}$. Complexifying gives an
${\operatorname{Ad}}_{{\mathsf{H}}^{\mathbb{C}}}$-invariant decomposition
${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{h}}^{\mathbb{C}}\oplus{\mathfrak{m}}^{\mathbb{C}}$.
Fix a compact Riemann surface $X$ with genus $g\geq 2$ and let $K$ denote its
holomorphic cotangent bundle. Given a principal
${\mathsf{H}}^{\mathbb{C}}$-bundle $P$, let
$P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}$ denote the associated bundle with
fiber ${\mathfrak{m}}^{\mathbb{C}}$:

$\displaystyle
P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}=\big{(}P\times{\mathfrak{m}}^{\mathbb{C}}\big{)}/{\sim},$

where $(p\cdot h,v)\sim(p,{\operatorname{Ad}}_{h^{-1}}v)$ for
$h\in{\mathsf{H}}^{\mathbb{C}}$.

######  Definition 3.3.

A ${\mathsf{G}}$-Higgs bundle on $X$ is a pair $({\mathcal{P}},\varphi)$ where

  * •

${\mathcal{P}}\to X$ is a holomorphic principal
${\mathsf{H}}^{\mathbb{C}}$-bundle and

  * •

$\varphi$ is a holomorphic section of the associated bundle
${\mathcal{P}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K$.

The holomorphic section $\varphi$ is called the _Higgs field_.

######  Example 3.4.

If ${\mathsf{G}}$ is compact, then ${\mathsf{G}}={\mathsf{H}}$ and
${\mathfrak{m}}=\\{0\\}$. In this case, a ${\mathsf{G}}$-Higgs bundle is just
a holomorphic principal ${\mathsf{H}}^{\mathbb{C}}$-bundle. So, for compact
groups, the moduli space ${\mathcal{M}}({\mathsf{H}})$ of Higgs bundles is
identical to the moduli space of holomorphic
${\mathsf{H}}^{\mathbb{C}}$-bundles.

######  Example 3.5.

If ${\mathsf{G}}$ is complex, then
${\mathfrak{g}}={\mathfrak{h}}^{\mathbb{C}}$ and
${\mathfrak{g}}^{\mathbb{C}}={\mathfrak{h}}^{\mathbb{C}}\oplus{\mathfrak{m}}^{\mathbb{C}}\cong{\mathfrak{g}}\oplus{\mathfrak{g}}$.
In this case, a ${\mathsf{G}}$-Higgs bundles is a pair
$({\mathcal{P}},\varphi)$, where ${\mathcal{P}}$ is a holomorphic
${\mathsf{G}}$-bundle and $\varphi$ is a holomorphic section of the adjoint
bundle ${\mathcal{P}}[{\mathfrak{g}}]$ twisted by $K$.

Rather than working with principal bundles, we will usually pick a faithful
linear representation of ${\mathsf{G}}^{\mathbb{C}}$ and work with vector
bundles. A faithful representation
${\mathsf{G}}^{\mathbb{C}}\to{\mathsf{GL}}(V)$ defines a representation
$\beta\colon{\mathsf{H}}^{\mathbb{C}}\to{\mathsf{GL}}(V)$ and an embedding
${\mathfrak{m}}^{\mathbb{C}}\hookrightarrow{\mathrm{End}}(V)$. With this data
fixed, a ${\mathsf{G}}$-Higgs bundle $({\mathcal{P}},\varphi)$ gives rise to a
pair $(E,\Phi)$, where $E\to X$ is the holomorphic vector bundle
${\mathcal{P}}[V]$ and $\Phi\in H^{0}({\mathrm{End}}(E)\otimes K)$ is given by
$\varphi$ under the inclusion
${\mathcal{P}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\hookrightarrow{\mathrm{End}}({\mathcal{P}}[V])\otimes K$.

######  Example 3.6.

When ${\mathsf{G}}={\mathsf{SL}}(n,{\mathbb{C}})$ we take
${\mathsf{G}}\to{\mathsf{GL}}({\mathbb{C}}^{n})$ to be the standard
representation. An ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle thus defines a
pair $(E,\Phi)$, where $E\to X$ is a holomorphic rank $n$ vector bundle and
$\Phi\in H^{0}({\mathrm{End}}(E)\otimes K)$ satisfies
${\operatorname{tr}}(\Phi)=0$. Moreover, the standard volume form on
${\mathbb{C}}^{n}$ is preserved by the standard representation of
${\mathsf{SL}}(n,{\mathbb{C}})$, and a holomorphic principal
${\mathsf{SL}}(n,{\mathbb{C}})$-bundle is equivalent to a holomorphic vector
bundle $E$ equipped with a holomorphic volume form $\omega\in
H^{0}(\Lambda^{n}E)$. Thus, an ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle is
equivalent to a triple $(E,\omega,\Phi)$. Note that the holomorphic volume
form $\omega$ is equivalent to a holomorphic trivialization of the determinant
line bundle $\Lambda^{n}E$. We will usually suppress $\omega$ from the
notation.

######  Example 3.7.

For ${\mathsf{G}}={\mathsf{SL}}(n,{\mathbb{R}})$ we have
${\mathsf{H}}={\mathsf{SO}}(n)$ and the Cartan decomposition is given by

$\displaystyle{\mathfrak{sl}}(n,{\mathbb{R}})\cong{\mathfrak{so}}(n)\oplus\mathrm{Sym}_{0}\big{(}{\mathbb{R}}^{n}\big{)},$

where $\mathrm{Sym}_{0}\big{(}{\mathbb{R}}^{n}\big{)}$ is the vector space of
traceless symmetric matrices. Again, using the standard representation of
${\mathsf{SL}}(n,{\mathbb{C}})$, we see that an
${\mathsf{SL}}(n,{\mathbb{R}})$-Higgs bundle gives rise to a triple
$(E,\omega,\Phi)$ as in the previous example.

Since ${\mathsf{H}}^{\mathbb{C}}={\mathsf{SO}}(n,{\mathbb{C}})$, the
restriction of the standard representation of ${\mathsf{SO}}(n,{\mathbb{C}})$
preserves a nondegenerate symmetric complex bilinear form on
${\mathbb{C}}^{n}$, a holomorphic principal
${\mathsf{SO}}(n,{\mathbb{C}})$-bundle111For ${\mathsf{O}}(n,{\mathbb{C}})$ a
holomorphic principal bundle is equivalent to a pair $(E,Q_{E})$. is
equivalent to a triple $(E,\omega,Q_{E})$ where $Q_{E}\in
H^{0}(S^{2}(E)\otimes K)$ is everywhere nondegenerate. Equivalently, $Q_{E}$
defines a symmetric holomorphic isomorphism $Q_{E}\colon E\to E^{*}$. Since
${\mathfrak{m}}^{\mathbb{C}}=\mathrm{Sym}_{0}({\mathbb{C}}^{n})$, the Higgs
field $\Phi$ is symmetric with respect to the quadratic form $Q_{E}$, i.e.,

$\displaystyle\Phi^{T}Q_{E}=Q_{E}\Phi.$

An ${\mathsf{SL}}(n,{\mathbb{R}})$-Higgs bundle is thus equivalent to a tuple
$(E,\omega,Q_{E},\Phi)$.

######  Example 3.8.

For ${\mathsf{G}}={\mathsf{SO}}(p,q)$, we have
${\mathsf{H}}={\mathsf{S}}({\mathsf{O}}(p)\times{\mathsf{O}}(q))$ and
${\mathfrak{h}}={\mathfrak{so}}(p)\oplus{\mathfrak{so}}(q)$. With respect to a
splitting ${\mathbb{R}}^{p+q}={\mathbb{R}}^{p}\oplus{\mathbb{R}}^{q}$ we may
decompose a matrix $X\in{\mathrm{End}}\big{(}{\mathbb{R}}^{p+q}\big{)}$ as
$X=\left(\begin{smallmatrix}A&B\\\ C&D\end{smallmatrix}\right)$.The Lie
algebra of ${\mathsf{SO}}(p,q)$ is given by

$\displaystyle{\mathfrak{so}}(p,q)=\left\\{\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)\Big{|}\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)^{T}\left(\begin{matrix}{\mathrm{Id}}&0\\\
0&-{\mathrm{Id}}\end{matrix}\right)+\left(\begin{matrix}{\mathrm{Id}}&0\\\
0&-{\mathrm{Id}}\end{matrix}\right)\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)=0\right\\}.$

This implies that $A\in{\mathfrak{so}}(p)$, $D\in{\mathfrak{so}}(q)$ and
$B=-C^{T}$, thus the Cartan decomposition is given by

$\displaystyle{\mathfrak{so}}(p,q)=({\mathfrak{so}}(p)\oplus{\mathfrak{so}}(q))\oplus{\mathrm{Hom}}\big{(}{\mathbb{R}}^{p},{\mathbb{R}}^{q}\big{)}.$

Similar to the previous examples, we use the standard representation. Since
${\mathsf{H}}^{\mathbb{C}}={\mathsf{S}}({\mathsf{O}}(p,{\mathbb{C}})\times{\mathsf{O}}(q,{\mathbb{C}}))$,
the restriction of the standard representation of
${\mathsf{SO}}(p+q,{\mathbb{C}})$ preserves an orthogonal splitting
${\mathbb{C}}^{p+q}={\mathbb{C}}^{p}\oplus{\mathbb{C}}^{q}$. As in the
previous example, a holomorphic principal
${\mathsf{SO}}(p+q,{\mathbb{C}})$-bundle is equivalent to a triple
$(E,\omega,Q_{E})$. A holomorphic principal
${\mathsf{S}}({\mathsf{O}}(p,{\mathbb{C}})\times{\mathsf{O}}(q,{\mathbb{C}}))$-bundle
is thus equivalent to a triple $(E,\omega,Q_{E})$ which decomposes as

$\displaystyle(E,\omega,Q_{E})=\left(V\oplus
W,\omega,\left(\begin{smallmatrix}Q_{V}&\\\
&-Q_{W}\end{smallmatrix}\right)\right),$

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 field $\Phi\in H^{0}({\mathrm{End}}(V\oplus W)\otimes
K)$ is given by

$\displaystyle\Phi=\left(\begin{matrix}0&\eta^{\dagger}\\\
\eta&0\end{matrix}\right),$

where $\eta\in H^{0}({\mathrm{Hom}}(V,W)\otimes K)$ and, regarding $Q_{V}$ and
$Q_{W}$ as isomorphisms $V\to V^{*}$ and $W\to W^{*}$ respectively,
$\eta^{\dagger}=-Q_{V}^{-1}\eta^{T}Q_{W}$.

######  Remark 3.9.

Taking the determinant of the isomorphisms $Q_{V}\colon V\to V^{*}$ defines an
isomorphism of determinant line bundles $\det(V)\cong\det(V^{*})$, or
equivalently $\det(V)^{2}\cong{\mathcal{O}}$. Thus, the determinant line
bundle of an orthogonal bundle $(V,Q_{V})$ on $X$ is one of the $2^{2g}$ order
two points in the Jacobian of $X$. The above volume form $\omega$ defines an
isomorphism $\Lambda^{p+q}(V\oplus
W)=\Lambda^{p}V\otimes\Lambda^{q}W\to{\mathcal{O}}$. Using the orthogonal
structures, this implies that
$\Lambda^{p}V\cong\Lambda^{q}W^{*}\cong\Lambda^{q}W$.

Since ${\mathsf{SO}}(p,q)$-Higgs bundles will be a main object of study we
record this in a proposition.

######  Proposition 3.10.

An ${\mathsf{SO}}(p,q)$-Higgs bundle on $X$ is equivalent to the data

  * •

a holomorphic rank $p$ vector bundle $V\to X$,

  * •

a holomorphic symmetric isomorphism $Q_{V}\colon V\to V^{*}$,

  * •

a holomorphic rank $q$ vector bundle $W\to X$,

  * •

a holomorphic symmetric isomorphism $Q_{W}\colon W\to W^{*}$,

  * •

a holomorphic isomorphism $\omega\colon\Lambda^{p}V\to\Lambda^{q}W^{*}$,

  * •

a holomorphic section $\eta\in H^{0}({\mathrm{Hom}}(V,W)\otimes K)$.

The ${\mathsf{SO}}(p+q,{\mathbb{C}})$-Higgs bundle associated to a tuple
$(V,Q_{V},W,Q_{W},\omega,\eta)$ is

$\displaystyle(E,\omega,Q_{E},\Phi)=\left(V\oplus
W,~{}\omega,\left(\begin{matrix}Q_{V}&0\\\
0&-Q_{W}\end{matrix}\right),\left(\begin{matrix}0&\eta^{\dagger}\\\
\eta&0\end{matrix}\right)\right),$

and the associated ${\mathsf{SL}}(p+q,{\mathbb{C}})$-Higgs bundle is given by
forgetting $Q_{E}$.

We will often suppress $Q_{V}$, $Q_{W}$ and $\omega$ from the notation and
just refer to an ${\mathsf{SO}}(p,q)$-Higgs bundle as a triple $(V,W,\eta)$.
We will also denote the associated Higgs bundle schematically by

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{W,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\dagger}}$
(3.3)

where we have suppressed the twisting by $K$ from the notation.

######  Remark 3.11.

Recall that the group ${\mathsf{SO}}(p,q)$ has two connected components,
${\mathsf{SO}}_{0}(p,q)<{\mathsf{SO}}(p,q)$ denotes the connected component of
the identity. The maximal compact subgroup of ${\mathsf{SO}}_{0}(p,q)$ is
${\mathsf{SO}}(p)\times{\mathsf{SO}}(q)$. Thus, an ${\mathsf{SO}}(p,q)$-Higgs
bundle $(V,W,\eta)$ reduces to an ${\mathsf{SO}}_{0}(p,q)$-Higgs bundle if and
only if both $V$ and $W$ have trivial determinant.

###  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 ${\mathsf{H}}^{\mathbb{C}}$-bundle $P\to X$, the
${\mathsf{H}}^{\mathbb{C}}$-gauge group
${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$ is the group of bundle
automorphisms $f\colon P\to P$. The elements of
${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$ are given by sections of an
associated bundle of groups
$P[{\mathsf{H}}^{\mathbb{C}}]=P\times_{{\operatorname{Ad}}_{{\mathsf{H}}^{\mathbb{C}}}}{\mathsf{H}}^{\mathbb{C}}$:

$\displaystyle{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}=\Omega^{0}\big{(}X,P\big{[}{\mathsf{H}}^{\mathbb{C}}\big{]}\big{)}.$

Recall that a holomorphic structure on a vector bundle $E$ is equivalent to a
Dolbeault operator. That is, a differential operator

$\displaystyle\bar{\partial}_{E}\colon\ \Omega^{0}(E)\to\Omega^{0,1}(E)$

so that222Dolbeault operators must also satisfy the integrability condition
$\bar{\partial}_{E}^{2}=0$, but this is automatic on a Riemann surface.
$\bar{\partial}_{E}(fs)=\bar{\partial}f\otimes s+f\bar{\partial}_{E}s$ for all
functions $f\in\Omega^{0}({\mathbb{C}})$ and sections $s\in\Omega^{0}(E)$.
Note that the $(0,1)$-part of a connection on $E$ defines a Dolbeault
operator. In particular, the space of holomorphic structures on $E$ is an
infinite-dimensional affine space with underlying vector space
$\Omega^{0,1}({\mathrm{End}}(E))$.

For principal bundles, an analogous theory holds. Namely a holomorphic
structure on a principal ${\mathsf{H}}^{\mathbb{C}}$-bundle $P\to X$ is
equivalent to a section
$\bar{\partial}_{P}\in\Omega^{0,1}\big{(}P,{\mathfrak{h}}^{\mathbb{C}}\big{)}$
which defines the $(0,1)$-part of a connection. In particular, a holomorphic
structure on $P$ defines a Dolbeault operator on any associated vector bundle.
The space of holomorphic structures on $P$ is an infinite-dimensional affine
space with the space of basic ${\mathfrak{h}}^{\mathbb{C}}$-valued
$(0,1)$-forms as underlying vector space. Equivalently, this vector space is
given by sections
$\Omega^{0,1}\big{(}X,P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}$ of
the adjoint bundle of $P$.

If we fix a smooth ${\mathsf{H}}^{\mathbb{C}}$-bundle $P\to X$, the set of all
Higgs bundles with underlying bundle $P$ is given by

$\displaystyle{\mathcal{H}}({\mathsf{G}})=\big{\\{}\big{(}\bar{\partial}_{P},\varphi\big{)}\,|\,\bar{\partial}_{P}\varphi=0\big{\\}}.$

Fixing a holomorphic structure on $P$ defines ${\mathcal{H}}({\mathsf{G}})$ as
a _quadratic_ subspace of a vector space:

$\displaystyle{\mathcal{H}}({\mathsf{G}})\hookrightarrow\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}.$

######  Remark 3.12.

Note that when ${\mathsf{G}}$ is complex
$\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}\cong\Omega^{1}(P[{\mathfrak{g}}])$
since
${\mathfrak{h}}^{\mathbb{C}}\oplus{\mathfrak{m}}^{\mathbb{C}}\cong{\mathfrak{g}}\oplus{\mathfrak{g}}$.

For
$(\alpha,\psi)\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}$,
we have
$\big{(}\bar{\partial}_{P}+\alpha,\varphi+\psi\big{)}\in{\mathcal{H}}({\mathsf{G}})$
if

$\displaystyle\bar{\partial}_{P}\varphi+\bar{\partial}_{P}\psi+[\alpha,\varphi]+[\alpha,\psi]=0.$

The tangent space is thus given by sections $(\alpha,\psi)$ satisfying this
equation to first order:

$\displaystyle
T_{\bar{\partial}_{P},\varphi}{\mathcal{H}}({\mathsf{G}})=\big{\\{}(\alpha,\psi)\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}\,|\,\bar{\partial}_{P}\psi+[\alpha,\varphi]=0\big{\\}}.$

######  Remark 3.13.

The space of Higgs bundles ${\mathcal{H}}({\mathsf{G}})$ has a natural complex
structure given by

$\displaystyle I(\alpha,\psi)=({\rm i}\alpha,{\rm i}\psi).$

When ${\mathsf{G}}$ is complex, ${\mathcal{H}}({\mathsf{G}})$ also has a
natural complex symplectic form given by

$\displaystyle\Omega_{I}^{\mathbb{C}}((\alpha_{1},\psi_{1}),(\alpha_{2},\psi_{2}))={\rm
i}\int_{X}{\operatorname{tr}}(\psi_{2}\wedge\alpha_{1}-\psi_{1}\wedge\alpha_{2}).$

For real groups ${\mathsf{G}}$, it can be shown that
${\mathcal{H}}({\mathsf{G}})\subset{\mathcal{H}}\big{(}{\mathsf{G}}^{\mathbb{C}}\big{)}$
is a Lagrangian subspace.

The gauge group ${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$ acts on
${\mathcal{H}}({\mathsf{G}})$ by pullback, namely for
$g\in{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$

$\displaystyle\big{(}\bar{\partial}_{P},\varphi\big{)}\cdot
g=\big{(}{\operatorname{Ad}}_{g^{-1}}\bar{\partial}_{P},{\operatorname{Ad}}_{g^{-1}}\varphi\big{)}.$

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
${\mathcal{M}}({\mathsf{G}})$ of ${\mathsf{G}}$-Higgs bundles is then defined
to be the set ${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$-orbits of polystable
${\mathsf{G}}$-Higgs bundles. The orbits of the
${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$-action on
${\mathcal{H}}({\mathsf{G}})^{\rm ps}$ are closed (in an appropriate space)
and the moduli space $\mathcal{M}({\mathsf{G}})$ becomes a Hausdorff
topological space.333For technical reasons, one needs to work with suitable
Sobolev completions to give the moduli space a topology; see [3], and also
[18, Section 8] where the straightforward adaptation to Higgs bundles is
discussed in the case ${\mathsf{G}}={\mathsf{GL}}(n,{\mathbb{C}})$.

######  Remark 3.14.

Using the harmonic metric from Remark 3.2, one can also define a symplectic
structure $\omega_{I}$ on ${\mathcal{M}}({\mathsf{G}})$ which is a Kähler form
for the complex structure $I$. In fact, when ${\mathsf{G}}$ is complex, the
moduli space ${\mathcal{M}}({\mathsf{G}})$ is hyper-Kähler. We will not focus
on this structure in this article.

In general, the notion of stability involves considering how all holomorphic
structure group reductions of an ${\mathsf{H}}^{\mathbb{C}}$-bundle to a
parabolic subgroup interact with the Higgs field (see [11]). Instead of
developing this theory in general, we will develop the appropriate stability
conditions in the vector bundle situation.

Recall that a ${\mathsf{GL}}(n,{\mathbb{C}})$-Higgs bundle is equivalent to a
rank $n$ holomorphic vector bundle $E$ and a section $\Phi\in
H^{0}({\mathrm{End}}(E)\otimes K)$. For ${\mathsf{SL}}(n,{\mathbb{C}})$ the
bundle $E$ is equipped with a trivialization of $\Lambda^{n}E$, thus,
$\deg(E)=0$.

######  Definition 3.15.

An ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle $(E,\Phi)$ is

  * •

semistable if for all proper holomorphic subbundles $F\subset E$ such that
$\Phi(F)\subset F\otimes K$ we have $\deg(F)\leq 0$,

  * •

stable if for all proper holomorphic subbundles $F\subset E$ such that
$\Phi(F)\subset F\otimes K$ we have $\deg(F)<0$, and

  * •

polystable if $(E,\Phi)=\bigoplus_{j}(E_{j},\Phi_{j})$ with $(E_{j},\Phi_{j})$
stable and $\deg(E_{j})=0$ for all $j$.

For general groups ${\mathsf{G}}$ the notion of semistability and
polystability is functorial in the sense that if ${\mathsf{G}}$ is a real form
of a reductive subgroup of ${\mathsf{SL}}(n,{\mathbb{C}})$, then a
${\mathsf{G}}$-Higgs bundle is semistable (respectively polystable) if and
only if the associated ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle is
semistable (respectively polystable). Moreover, the set of semistable
${\mathsf{G}}$-Higgs bundles is open in ${\mathcal{H}}({\mathsf{G}})$.

Let ${\mathcal{H}}^{\rm ps}({\mathsf{G}})\subset{\mathcal{H}}({\mathsf{G}})$
denote the set of polystable Higgs bundles. The gauge group
${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}$-preserves ${\mathcal{H}}^{\rm
ps}({\mathsf{G}})$, and the gauge orbits in ${\mathcal{H}}^{\rm
ps}({\mathsf{G}})$ are closed in the open set of semistable
${\mathsf{G}}$-Higgs bundles. We define the moduli space
${\mathcal{M}}({\mathsf{G}})$ to be the quotient space

$\displaystyle{\mathcal{M}}({\mathsf{G}})={\mathcal{H}}^{\rm
ps}({\mathsf{G}})/{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}.$

We note that the complex structure $I$ (and the complex symplectic form
$\Omega_{I}^{\mathbb{C}}$ when ${\mathsf{G}}$ is complex) from Remark 3.13 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
${\mathsf{G}}$-Higgs bundle is stable if and only if the associated
${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle is stable. However, one can
detect stable ${\mathsf{G}}$-Higgs bundles inside of the set of polystable
Higgs bundles with the following proposition.

######  Proposition 3.16.

Let ${\mathsf{G}}$ be a real form of a complex semisimple subgroup of
${\mathsf{SL}}(n,{\mathbb{C}})$. A ${\mathsf{G}}$-Higgs bundle
$({\mathcal{P}},\varphi)$ is stable if it is polystable and has finite
automorphism group. Moreover, the set of stable ${\mathsf{G}}$-Higgs bundles
is open in ${\mathcal{H}}({\mathsf{G}})$.

######  Remark 3.17.

Note that if $({\mathcal{P}},\varphi)$ is a ${\mathsf{G}}$-Higgs bundle whose
associated ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle is stable as an
${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle, then $({\mathcal{P}},\varphi)$
is stable as a ${\mathsf{G}}$-Higgs bundle.

Let ${\mathcal{H}}^{s}({\mathsf{G}})\subset{\mathcal{H}}^{\rm
ps}({\mathsf{G}})$ be the stable locus, the quotient

$\displaystyle{\mathcal{H}}^{s}/{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}\subset{\mathcal{M}}({\mathsf{G}})$

is an orbifold. At a stable Higgs bundle one can show that the real dimension
of the tangent space to
$T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$ is
$\dim_{\mathbb{R}}({\mathsf{G}})(2g-2)$ (see Remark 6.2). Thus the real
dimension of ${\mathcal{M}}({\mathsf{G}})$ is given by
$\dim_{\mathbb{R}}({\mathsf{G}})(2g-2)$.

##  4 $\boldsymbol{{\mathsf{SO}}(1,q)}$-Higgs bundles especially when
$\boldsymbol{q=2}$

In this section we will describe the moduli space of
${\mathsf{SO}}(1,q)$-Higgs bundles and ${\mathsf{SO}}_{0}(1,q)$-Higgs bundles.
When $q=2$ we have
${\mathsf{SO}}_{0}(1,2)\cong{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$. In
this case we will recall Hitchin’s parameterization of all but one of the
components of ${\mathcal{M}}({\mathsf{SO}}_{0}(1,2))$. In particular, we
recall the Higgs bundle parameterization of Teichmüller space.

Recall from Proposition 3.10 that an ${\mathsf{SO}}(1,n)$-Higgs bundle
consists of a tuple $(V,Q_{V},W,Q_{W}$, $\omega,\eta)$, where
$\operatorname{rk}(V)=1$ and $\operatorname{rk}(W)=q$. We can take
$(V,Q_{V})=(\Lambda^{n}W,\det(Q_{W}))$ and $\omega=\det(Q_{W})\colon
V\to\Lambda^{q}W^{*}$. Thus, such a tuple is determined by the triple
$(W,Q_{W},\eta)$, where

$\displaystyle\eta\in H^{0}\big{(}W\otimes(\Lambda^{q}W)^{-1}\otimes
K\big{)}.$

Using the notation from (3.3), the associated
${\mathsf{SL}}(1+q,{\mathbb{C}})$-Higgs bundle is given by

$\textstyle{\Lambda^{n}W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{W.\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta^{\dagger}}$

When $q=1$, we have $\eta\in H^{0}(K)$ and the first Stiefel–Whitney class
$sw_{1}(W)\in H^{1}(X,{\mathbb{Z}}_{2})$ of $W$ labels the components of
${\mathcal{M}}({\mathsf{SO}}(1,1))$. Namely,

$\displaystyle{\mathcal{M}}({\mathsf{SO}}(1,1))=\coprod\limits_{sw_{1}\in
H^{1}(X,{\mathbb{Z}}_{2})}{\mathcal{M}}_{sw_{1}}({\mathsf{SO}}(1,1)),$

and each space ${\mathcal{M}}_{sw_{1}}({\mathsf{SO}}(1,1))$ is parameterized
by $H^{0}(K)$.

For $q>1$, the first and second Stiefel–Whitney classes $(sw_{1},sw_{2})\in
H^{1}(X,{\mathbb{Z}}_{2})\times H^{2}(X,{\mathbb{Z}}_{2})$ of $(W,Q_{W})$ give
a decomposition of the moduli space

$\displaystyle{\mathcal{M}}({\mathsf{SO}}(1,n))=\coprod\limits_{sw_{1},sw_{2}}{\mathcal{M}}_{sw_{1}}^{sw_{2}}({\mathsf{SO}}(1,n)).$

The first Stiefel–Whitney class of $W$ vanishes if and only if the
${\mathsf{O}}(q,{\mathbb{C}})$-bundle reduces to
${\mathsf{SO}}(q,{\mathbb{C}})$. Thus, for $q=2$ and $sw_{1}=0$, the bundle
$W$ reduces to an ${\mathsf{SO}}(2,{\mathbb{C}})$-bundle. Since
${\mathbb{C}}^{*}\cong{\mathsf{SO}}(2,{\mathbb{C}})$, in this case the degree
of the ${\mathbb{C}}^{*}$-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 $L\in{\mathsf{Pic}}(X)$ such that

$\displaystyle(W,Q_{W})\cong\left(L\oplus L^{-1},\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right)\right).$

The integer $\deg(L)$ satisfies $sw_{2}(W,Q_{W})=\deg(L)\mod 2$, and the
isomorphism switching $L$ with $L^{-1}$ preserves the
${\mathsf{O}}(2,{\mathbb{C}})$-structure. Thus, $|\deg(L)|\in{\mathbb{N}}$
which is a well defined invariant of ${\mathsf{O}}(2,{\mathbb{C}})$-bundles
with vanishing $sw_{1}$.

This gives a decomposition of the moduli space as

$\displaystyle\coprod\limits_{sw_{1}\neq
0,sw_{2}}{\mathcal{M}}^{sw_{2}}_{sw_{1}}({\mathsf{SO}}(1,2))~{}\amalg~{}\coprod\limits_{d\in{\mathbb{N}}}{\mathcal{M}}_{d}({\mathsf{SO}}(1,2)).$

For Higgs bundles in ${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ the splitting
$W=L\oplus L^{-1}$ gives a decomposition of the Higgs field
$\eta\colon{\mathcal{O}}\to W\otimes K$ as

$\displaystyle\eta=\left(\begin{matrix}\beta\\\
\gamma\end{matrix}\right)\colon\ {\mathcal{O}}\to LK\oplus L^{-1}K,$

where $\beta\in H^{0}(LK)$ and $\gamma\in H^{0}\big{(}L^{-1}K\big{)}$. Using
$Q_{W}=\left(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right)$, we can
write the associated ${\mathsf{SL}}(3,{\mathbb{C}})$-Higgs bundle
schematically as

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{\mathcal{O}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\beta}$$\textstyle{L^{-1},\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$
(4.3)

where we recall that we suppress the twisting by $K$ from the notation.

The stability condition limits the objects we are considering.

######  Proposition 4.1.

If $\big{(}{\mathcal{O}},L\oplus L^{-1},\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}\beta\\\
\gamma\end{smallmatrix}\right)\big{)}$ is a polystable
${\mathsf{SO}}(1,2)$-Higgs bundle with vanishing first Stiefel–Whitney class,
then $|\deg(L)|\leq 2g-2$. Moreover, if $\deg(L)\in(0,2g-2]$, then $\gamma\neq
0$ and if $\deg(L)\in[2-2g,0)$, then $\beta\neq 0$.

###### Proof.

Consider the associated ${\mathsf{SL}}(3,{\mathbb{C}})$-Higgs bundle (4.3). By
stability, if $\deg(L)>0$ then $\gamma\neq 0$ since otherwise $L$ would define
a positive degree invariant subbundle. But, $\gamma\in
H^{0}\big{(}L^{-1}K\big{)}$ so if $\deg(L)>2g-2$ then $\gamma=0$,
contradicting stability. Similarly, if $\deg(L)<0$, then stability forces
$\beta\neq 0$ and we conclude $\deg(L)>2-2g$. ∎

For $d=|\deg(L)|>0$, we can parameterize the moduli space
${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$, this was done by Hitchin in [19] for
the group ${\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})$.

######  Theorem 4.2 (Hitchin [19]).

For $d>0$, the moduli space ${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ is smooth
and diffeomorphic to the total space of a rank $(d+g-1)$-complex vector bundle
over the $(2g-2-d)$-symmetric product ${\mathrm{Sym}}^{2g-2-d}(X)$ of the
Riemann surface $X$.

###### Proof.

By the above discussion, a point in ${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ is
determined by a triple $(L,\gamma,\beta)$ where $L\in{\mathsf{Pic}}^{d}(X)$,
$\gamma\in H^{0}\big{(}L^{-1}K\big{)}\setminus\\{0\\}$ and $\beta\in
H^{0}(LK)$. The
${\mathsf{S}}({\mathsf{O}}(1,{\mathbb{C}})\times{\mathsf{O}}(2,{\mathbb{C}}))$-bundle
is given by

$\displaystyle(V,Q_{V},W,Q_{W})=\left({\mathcal{O}},\left(\begin{matrix}1\end{matrix}\right),L\oplus
L^{-1},\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)\right),$

and the Higgs field is $\eta=\left(\begin{smallmatrix}\beta\\\
\gamma\end{smallmatrix}\right)\colon V\to W\otimes K$.

For two triples $(L,\beta,\gamma)$ and
$(L^{\prime},\beta^{\prime},\gamma^{\prime})$ to define isomorphic
${\mathsf{SO}}(1,2)$-Higgs bundles it is necessary that
$|\deg(L)|=|\deg(L^{\prime})|$. Thus we may assume $L=L^{\prime}$ as elements
${\mathsf{Pic}}^{d}(X)$. The remaining holomorphic gauge transformation of the
${\mathsf{S}}({\mathsf{O}}(1,{\mathbb{C}})\times{\mathsf{O}}(2,{\mathbb{C}}))$
bundle is given by

$\displaystyle(g_{V},g_{W})=\left(1,\left(\begin{matrix}\lambda&0\\\
0&\lambda^{-1}\end{matrix}\right)\right),$

for $\lambda\in{\mathbb{C}}^{*}$. This gauge transformation acts on the Higgs
field by

$\displaystyle g_{W}^{-1}\eta g_{V}=\left(\begin{matrix}\lambda^{-1}&0\\\
0&\lambda\end{matrix}\right)\left(\begin{matrix}\beta\\\
\gamma\end{matrix}\right)\left(\begin{matrix}1\end{matrix}\right)=\left(\begin{matrix}\lambda^{-1}\beta\\\
\lambda\gamma\end{matrix}\right).$

In particular, we note that the automorphism group of such an
${\mathsf{SO}}(1,2)$-Higgs bundle is trivial since $\gamma\neq 0$. Thus, the
moduli space ${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ is smooth and given by
${\mathbb{C}}^{*}$-equivalence classes $[L,\beta,\gamma]$ where
$(L,\beta,\gamma)\sim(L^{\prime},\beta^{\prime},\gamma^{\prime})$ if and only
if $L=L^{\prime}\in{\mathsf{Pic}}^{d}(X)$, $\beta=\lambda\beta^{\prime}$ and
$\gamma=\lambda^{-1}\gamma^{\prime}$ for $\lambda\in{\mathbb{C}}^{*}$.

Recall that the space of effective divisors on $X$ of degree $n$ is given by
the $n^{\rm th}$-symmetric product ${\mathrm{Sym}}^{n}(X)$. Taking the
projective class of $\gamma\in H^{0}\big{(}L^{-1}K\big{)}\setminus\\{0\\}$
defines a surjective map to the space of effective degree $2g-2-d$ divisors on
$X$:

$\textstyle{{\mathcal{M}}_{d}({\mathsf{SO}}(1,2))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathrm{Sym}}^{2g-2-d}(X),}$$\textstyle{[L,\gamma,\beta]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{[\gamma].}$

We claim that the fiber of this map is a vector space of rank $(d+g-1)$.
Denote by ${\mathcal{O}}([\gamma])$ the line bundle associated to the divisor
$[\gamma]$. The line bundle $L$ is given by $L={\mathcal{O}}([\gamma])^{-1}K$
and $\beta\in H^{0}\big{(}{\mathcal{O}}([\gamma])^{-1}K^{2}\big{)}$. Thus $L$
is determined by $[\gamma]$ and $\beta$ can be any element of the
$(d+g-1)$-dimensional vector space
$H^{0}\big{(}{\mathcal{O}}([\gamma])^{-1}K^{2}\big{)}$. ∎

We now collect many corollaries of the above theorem.

######  Corollary 4.3.

For $d>0$ the moduli space${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ is connected
and homotopy equivalent to the symmetric product ${\mathrm{Sym}}^{2g-2-d}(X)$.

The cohomology ring of a symmetric product of a Riemann surface was computed
in [26], as a result this computes the cohomology ring of
${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$. When $d=2g-2$, the space is
contractible.

Consider the following map

$\textstyle{{\mathcal{M}}({\mathsf{SO}}(3,{\mathbb{C}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}\big{(}K^{2}\big{)},}$$\textstyle{[E,Q_{E},\Phi]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\frac{1}{4}{\operatorname{tr}}\big{(}\Phi^{2}\big{)}.}$

This is the Hitchin fibration for ${\mathsf{SO}}(3,{\mathbb{C}})$, we will
discuss the Hitchin fibration in more generality in subsequent sections.

######  Corollary 4.4.

The moduli space ${\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$ is parameterized
by the $(3g-3)$-dimensional complex vector space $H^{0}\big{(}K^{2}\big{)}$ of
holomorphic differentials. Moreover,
${\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$ is the image of a section of the
${\mathsf{SO}}(3,{\mathbb{C}})$-Hitchin fibration.

###### Proof.

An ${\mathsf{SO}}(1,2)$-Higgs bundle in
${\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$ is determined by a triple
$(L,\beta,\gamma)$ where $\deg(L)=2g-2$, $\beta\in H^{0}(LK)$ and $\gamma\in
H^{0}\big{(}L^{-1}K\big{)}\setminus\\{0\\}$. The condition on $\gamma$ implies
that $L=K$ and thus $\beta\in H^{0}\big{(}K^{2}\big{)}$. If we normalize
$\gamma$ to by $\gamma=1\in H^{0}({\mathcal{O}})$, then there is no more gauge
freedom, and so

$\displaystyle{\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))\cong
H^{0}\big{(}K^{2}\big{)}.$

Using the above parameterization of ${\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$
by $H^{0}\big{(}K^{2}\big{)}$, the ${\mathsf{SO}}(3,{\mathbb{C}})$-Higgs
bundle associated to $q_{2}\in H^{0}\big{(}K^{2}\big{)}$ is given by

$\displaystyle[E,Q_{E},\Phi]=\left({\mathcal{O}}\oplus K\oplus
K^{-1},\left(\begin{matrix}-1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right),\left(\begin{matrix}0&1&q_{2}\\\ q_{2}&0&0\\\
1&0&0\end{matrix}\right)\right).$

For this Higgs bundle
$\frac{1}{4}{\operatorname{tr}}\big{(}\Phi^{2}\big{)}=q_{2}$. ∎

Translating these statements to the character variety
${\mathcal{X}}(\Gamma,{\mathsf{SO}}(1,2))$ via the nonabelian Hodge
correspondence gives the following.

######  Corollary 4.5.

For each $0<d\leq 2g-2$, the character variety
${\mathcal{X}}(\Gamma,{\mathsf{SO}}(1,2))$ has a connected component
${\mathcal{X}}_{d}({\mathsf{SO}}(1,2))$ which is smooth and diffeomorphic to a
real rank $2d+2g-2$ vector bundle over the symmetric product
${\mathrm{Sym}}^{2g-2-d}(S)$.

######  Corollary 4.6.

Every representation $\rho\in{\mathcal{X}}_{d}(\Gamma,{\mathsf{SO}}(1,2))$
factors through the connected component of the identity
${\mathsf{SO}}_{0}(1,2)$ and the Fuchsian representations are given by

$\displaystyle{\mathcal{X}}_{2g-2}(\Gamma,{\mathsf{SO}}(1,2))\cong{\mathsf{Fuch}}(\Gamma).$

###### Proof.

The space ${\mathsf{Fuch}}(\Gamma)$ consists of two connected component of the
character variety
${\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))={\mathcal{X}}(\Gamma,{\mathsf{SO}}_{0}(1,2))$
which are identified with the Teichmüller space of $S$. Since the
representations in these components are conjugate by an element of
${\mathsf{SO}}(1,2)$ which is not in ${\mathsf{SO}}_{0}(1,2)$, the two
components of ${\mathsf{Fuch}}(\Gamma)$ are identified in
${\mathcal{X}}(\Gamma,{\mathsf{SO}}(1,2))$. Since ${\mathsf{Fuch}}(\Gamma)$ is
contractible and the only $d$ for which
${\mathcal{X}}_{d}(\Gamma,{\mathsf{SO}}(1,2))$ is contractible is $d=2g-2$ we
are done. ∎

######  Remark 4.7.

Since the second Stiefel–Whitney class invariant of the Higgs bundles in
${\mathcal{M}}_{d}({\mathsf{SO}}(1,2))$ is given by $d\mod{2}$, the associated
${\mathsf{SO}}(3,{\mathbb{C}})$-Higgs bundles lift to
${\mathsf{Spin}}(3,{\mathbb{C}})$ if and only if $d$ is even. Recall that the
isomorphism ${\mathsf{Spin}}(3,{\mathbb{C}})={\mathsf{SL}}(2,{\mathbb{C}})$,
is given by the $2$ to $1$ map
${\mathsf{SL}}(2,{\mathbb{C}})\to{\mathsf{SO}}(3,{\mathbb{C}})$ which is
induced by the action on the second symmetric product
$S^{2}\big{(}{\mathbb{C}}^{2}\big{)}$. Here, the volume form on
${\mathbb{C}}^{2}$ induces a nondegenerate symmetric form on the second
symmetric product.

The ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundles which give rise to the Higgs
bundles in ${\mathcal{M}}_{2d}({\mathsf{SO}}(1,2))$ are thus given by

$\displaystyle(E,\omega,Q_{E},\Phi)\cong\left(N\oplus
N^{-1},\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right),\left(\begin{matrix}0&\beta\\\
\gamma&0\end{matrix}\right)\right),$

where we view the volume form $\omega\in H^{0}\big{(}\Lambda^{2}E\big{)}$ is
viewed as a skew symmetric homomorphism $\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)\colon E^{*}\to E$. Indeed, taking the second
symmetric product gives

$\displaystyle\big{(}S^{2}E,S^{2}\omega,S^{2}\Phi\big{)}\cong\left({\mathcal{O}}\oplus
N^{2}\oplus N^{-2},\left(\begin{matrix}-1&0&0\\\ 0&0&1\\\
0&1&0\end{matrix}\right),\left(\begin{matrix}0&\gamma&\beta\\\ \beta&0&0\\\
\gamma&0&0\end{matrix}\right)\right).$

In particular, the ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundles which define
points in the component ${\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$ are given
by

$\displaystyle\left(K^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}},\left(\begin{matrix}0&-1\\\
1&0\end{matrix}\right),\left(\begin{matrix}0&1\\\
1&0\end{matrix}\right),\left(\begin{matrix}0&q_{2}\\\
1&0\end{matrix}\right)\right)$

for one of the $2^{2g}$ choices of square root $K^{\frac{1}{2}}$ of $K$. In
particular, there are $2^{2g}$-connected components of
${\mathcal{M}}({\mathsf{SL}}(2,{\mathbb{R}}))$ which project to
${\mathcal{M}}_{2g-2}({\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}}))\cong{\mathcal{M}}_{2g-2}({\mathsf{SO}}(1,2))$.

##  5 The Hitchin fibration and Hitchin section

So far we have seen that the character variety
${\mathcal{X}}(\Gamma,{\mathsf{G}})$ is homeomorphic to the moduli space of
${\mathsf{G}}$-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 use
this additional structure to construct the Hitchin component from Definition
2.7.

###  5.1 The Hitchin fibration

Suppose ${\mathsf{G}}$ is a complex simple Lie group. Similar to Chern–Weil
theory, we can apply an invariant polynomial to the Higgs field and obtain a
holomorphic differential. Fixing a homogeneous basis
$p_{1},\dots,p_{\operatorname{rk}({\mathsf{G}})}$ of the
${\operatorname{Ad}}_{\mathsf{G}}$-invariant polynomials
${\mathbb{C}}[{\mathfrak{g}}]$ with $\deg(p_{j})=m_{j}+1$ defines a map

$\displaystyle\begin{split}&h\colon\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.51292pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{M}}({\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.41307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.41307pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\displaystyle\bigoplus\limits_{j=1}^{\operatorname{rk}({\mathsf{G}})}H^{0}\big{(}K^{m_{j}+1}\big{)},}$}}}}}}}{\hbox{\kern-16.51292pt\raise-18.3pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\big{[}\bar{\partial}_{P},\varphi\big{]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.51292pt\raise-18.3pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
40.51292pt\raise-18.3pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
40.51292pt\raise-18.3pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(p_{1}(\varphi),\dots,p_{\operatorname{rk}({\mathsf{G}})}(\varphi))}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(5.1)

called the Hitchin fibration. For example, when
${\mathsf{G}}={\mathsf{SL}}(n,{\mathbb{C}})$ we have $m_{j}=j$ for $1\leq
j\leq n-1$, and when ${\mathsf{G}}={\mathsf{SO}}(2n+1,{\mathbb{C}})$ we have
$m_{j}=2j-1$ for $1\leq j\leq n$.

In general, a computation using the Riemann–Roch theorem shows that the base
is half the dimension of the moduli space:

$\displaystyle\dim_{\mathbb{C}}\left(\bigoplus\limits_{j=1}^{\operatorname{rk}({\mathsf{G}})}H^{0}\big{(}K^{m_{j}+1}\big{)}\right)=\frac{1}{2}\dim_{\mathbb{C}}\big{(}{\mathcal{M}}({\mathsf{G}})\big{)}=\dim_{\mathbb{C}}({\mathsf{G}})(g-1).$

Moreover, the Hitchin fibration is a proper map [29]. In fact, the generic
fibers of the Hitchin fibration are half-dimensional tori and this makes
${\mathcal{M}}({\mathsf{G}})$ into a algebraic completely integrable system
[20], 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 ${\mathsf{G}}^{r}$-Higgs bundles for
${\mathsf{G}}^{r}<{\mathsf{G}}$ any real form. For example, the Hitchin base
of ${\mathsf{SO}}(2n+1,{\mathbb{C}})$ has the same dimension as
${\mathcal{M}}({\mathsf{SO}}(p,q))$ for all $p$ and $q$ satisfying $p+q=2n+1$.

###  5.2 The Hitchin section

Let ${\mathfrak{g}}$ be a semisimple complex Lie algebra. For
${\mathfrak{s}}\subset{\mathfrak{g}}$ a subalgebra isomorphic to
${\mathfrak{sl}}(2,{\mathbb{C}})$, consider the decomposition of
${\mathfrak{g}}$ into irreducible
${\mathfrak{sl}}(2,{\mathbb{C}})$-representations

$\displaystyle{\mathfrak{g}}=\bigoplus\limits_{j=1}^{N}V_{j}.$

For any such ${\mathfrak{s}}\subset{\mathfrak{g}}$ we have
$N\geq\operatorname{rk}({\mathfrak{g}})$, and when
$N=\operatorname{rk}({\mathfrak{g}})$ the three-dimensional subalgebra
${\mathfrak{s}}$ is called _principal_. Up to conjugation, there is a unique
principal three-dimensional subalgebra [22]. In this case we have
$\dim(V_{j})=2m_{j}+1$ where $1=m_{1}\leq m_{2}\leq\cdots\leq
m_{\operatorname{rk}({\mathfrak{g}})}$ are the exponents of ${\mathfrak{g}}$.
Moreover, when we restrict a principal embedding
${\mathfrak{sl}}(2,{\mathbb{C}})\to{\mathfrak{g}}$ to the real subalgebra
${\mathfrak{sl}}(2,{\mathbb{R}})$, the image lies in a split real subalgebra
of ${\mathfrak{g}}$. This defines an embedding

$\displaystyle\iota_{\rm pr}\colon\
{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{G}}^{\rm split}.$

######  Theorem 5.2 (Hitchin [21]).

Let ${\mathsf{G}}$ be a complex simple Lie group, then the Hitchin fibration
(5.1) has a section

$\displaystyle s_{h}\colon\
\bigoplus\limits_{j=1}^{\operatorname{rk}({\mathsf{G}})}H^{0}\big{(}K^{m_{j}+1}\big{)}\longrightarrow{\mathcal{M}}({\mathsf{G}}),$

which maps onto a component of the moduli space for the split real form
${\mathcal{M}}\big{(}{\mathsf{G}}^{\rm split}\big{)}$. Under the nonabelian
Hodge correspondence $($Theorem 3.1), the image of this section defines the
Hitchin component ${\mathsf{Hit}}\big{(}{\mathsf{G}}^{\rm
split}\big{)}\subset{\mathcal{X}}\big{(}{\mathsf{G}}^{\rm split}\big{)}$ from
Definition 2.7.

We will prove the above theorem for
${\mathsf{G}}={\mathsf{SO}}(2p+1,{\mathbb{C}})$, namely we will construct the
Hitchin section and prove that it maps onto a component for the group
${\mathsf{SO}}(p,p+1)$. For ${\mathsf{SO}}(2p+1,{\mathbb{C}})$ the Hitchin
fibration is given by

$\displaystyle{\mathcal{M}}({\mathsf{SO}}(2p+1,{\mathbb{C}}))\to\bigoplus\limits_{j=1}^{2p}H^{0}\big{(}K^{2j}\big{)}.$

Consider the rank $p$ holomorphic orthogonal bundle

$\displaystyle{\mathcal{K}}_{p}=K^{p-1}\oplus K^{p-3}\oplus\cdots\oplus
K^{3-p}\oplus K^{1-p}.$ (5.2)

Note that ${\mathcal{K}}_{p}$ has a natural orthogonal structure
$Q_{p}=\left(\begin{smallmatrix}&&1\\\ &\iddots&\\\
1\end{smallmatrix}\right)\colon{\mathcal{K}}_{p}\to{\mathcal{K}}_{p}^{*}$.

Consider the map

$\displaystyle\widehat{\Psi}\colon\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
29.58777pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\bigoplus\limits_{j=1}^{2p}H^{0}\big{(}K^{2j}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
53.58777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.58777pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{H}}({\mathsf{SO}}(p,p+1))}$}}}}}}}\ignorespaces}}}}\ignorespaces$

defined by

$\displaystyle\widehat{\Psi}(q_{2},\dots,q_{2p})=\left({\mathcal{K}}_{p},Q_{p},{\mathcal{K}}_{p+1},Q_{p+1},\left(\begin{matrix}q_{2}&q_{4}&\dots&q_{2p}\\\
1&q_{2}&\dots&q_{2p-2}\\\ &\ddots&\ddots&\vdots\\\ &&1&q_{2}\\\
&&&1\end{matrix}\right)\colon{\mathcal{K}}_{p}\to{\mathcal{K}}_{p+1}\otimes
K\right).$ (5.3)

We claim that the image of $\widehat{\Psi}$ is contained in the stable Higgs
bundles ${\mathcal{H}}^{s}({\mathsf{SO}}(p,p+1))$ and that the induced map
$\Psi\colon\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}\to{\mathcal{M}}({\mathsf{SO}}(p,p+1))$
is injective and has an open and closed image.

######  Proposition 5.3.

The image of $\Psi$ consists of stable ${\mathsf{SO}}(p,p+1)$-Higgs bundles.

###### Proof.

Consider the stable ${\mathsf{SL}}(2,{\mathbb{C}})$-Higgs bundle

$\displaystyle(E,\Phi)=\left(K^{\frac{1}{2}}\oplus
K^{-\frac{1}{2}},\left(\begin{matrix}0&0\\\ 1&0\end{matrix}\right)\right).$
(5.4)

Since the unique irreducible $(2p+1)$-dimensional representation of
${\mathsf{SL}}(2,{\mathbb{C}})$ is given by the $2p$-symmetric product, the
${\mathsf{SL}}(2p+1,{\mathbb{C}})$-Higgs bundle given by

$\displaystyle\big{(}S^{2p+1}E,S^{2p+1}\Phi\big{)}=\left(\\!K^{p}\oplus
K^{p-1}\oplus\cdots\oplus K^{1-p}\oplus K^{-p},\left(\begin{matrix}0&\\\
p-1&0\\\ &p-2&0\\\ &&\ddots&\ddots\\\ &&&p-1&0\end{matrix}\right)\\!\right)$

is also stable. Moreover this is gauge equivalent to

$\displaystyle\left(K^{p}\oplus K^{p-1}\oplus\cdots\oplus K^{1-p}\oplus
K^{-p},\left(\begin{matrix}0&\\\ 1&0\\\ &\ddots&\ddots\\\
&&1&0\end{matrix}\right)\right).$ (5.5)

After rearranging the summands of
${\mathcal{K}}_{p}\oplus{\mathcal{K}}_{p+1}$, the
${\mathsf{SL}}(2p+1,{\mathbb{C}})$-Higgs bundle associated to
$\widehat{\Psi}(0,\dots,0)$ is given by (5.5). Thus,
$\widehat{\Psi}(0,\dots,0)$ is a stable ${\mathsf{SO}}(p,p+1)$-Higgs bundle.
Since stability is an open condition, for $q_{2},\dots,q_{2p}$ sufficiently
close to zero, the Higgs bundle $\widehat{\Psi}(q_{2},\dots,q_{2p})$ is also
stable.

Scaling the Higgs field does not preserve the image of $\widehat{\Psi}$.
However, for each $\lambda\in{\mathbb{C}}^{*}$, the Higgs bundle obtained by
scaling the Higgs field of $\widehat{\Psi}(q_{2},q_{4},\dots,q_{2p})$ by
$\lambda$ is gauge equivalent to
$\widehat{\Psi}\big{(}\lambda^{2}q_{2},\lambda^{4}q_{4},\dots,\lambda^{2p}q_{2p}\big{)}$.
Since stability is preserved by scaling the Higgs field, all Higgs bundles in
the image of $\widehat{\Psi}$ are stable. ∎

######  Proposition 5.4.

Let $\Phi(q_{2},\dots,q_{2p})$ be the Higgs field of the
${\mathsf{SO}}(2p+1,{\mathbb{C}})$-Higgs bundle associated to the
${\mathsf{SO}}(p,p+1)$-Higgs bundle $\widehat{\Psi}(q_{2},\dots,q_{2p})$.
There is a basis $(p_{1},\dots,p_{p})$ of the invariant polynomials
${\mathbb{C}}[{\mathfrak{so}}(2p+1,{\mathbb{C}})]^{{\mathsf{SO}}(2p+1,{\mathbb{C}})}$
so that for all $j$

$\displaystyle p_{j}(\Phi(q_{2},\dots,q_{2p}))=q_{2j}.$

###### Proof.

In the general setting of a complex semisimple Lie group the existence of such
a basis was proven by Kostant in [22]. For ${\mathsf{SO}}(2p+1,{\mathbb{C}})$
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 ${\mathsf{SO}}(5,{\mathbb{C}})$-Higgs
bundle $(E,Q,\Phi)$ associated to the ${\mathsf{SO}}(2,3)$-Higgs bundle
$\widehat{\Psi}(q_{2},q_{4})$ is given by

$\displaystyle\left(K^{2}\oplus K\oplus{\mathcal{O}}\oplus K^{-1}\oplus
K^{-2},\left(\begin{matrix}&&&&-1\\\ &&&1\\\ &&-1\\\ &1\\\
-1\end{matrix}\right),\left(\begin{matrix}0&q_{2}&0&q_{4}&0\\\
1&0&q_{2}&0&q_{4}\\\ 0&1&0&q_{2}&0\\\ 0&0&1&0&q_{2}\\\
0&0&0&1&0\end{matrix}\right)\right).$

We have ${\operatorname{tr}}\big{(}\Phi^{2}\big{)}=8q_{2}$ and
${\operatorname{tr}}\big{(}\Phi^{4}\big{)}=14q_{2}^{2}+5q_{4}$, thus we choose
the basis

$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
35.02397pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{p_{1}(\Phi)=\frac{1}{8}{\operatorname{tr}}\big{(}\Phi^{2}\big{)}}$}}}}}}}{\hbox{\kern
59.02397pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\text{and}}$}}}}}}}{\hbox{\kern
105.13512pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{p_{2}(\Phi)=\frac{1}{5}{\operatorname{tr}}\big{(}\Phi^{4}\big{)}-\frac{1}{14}{\operatorname{tr}}\big{(}\Phi^{2}\big{)}^{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

∎

By the previous two propositions, the map $\widehat{\Psi}$ gives rise to a
well defined map

$\displaystyle\Psi\colon\
\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}\to{\mathcal{M}}({\mathsf{SO}}(p,p+1)),$

which is a section of the Hitchin fibration for
${\mathcal{M}}({\mathsf{SO}}(2p+1,{\mathbb{C}}))\to\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}$.
We now show that the image of $\Psi$ is open and closed.

######  Proposition 5.5.

The image of the map
$\Psi\colon\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}\to{\mathcal{M}}({\mathsf{SO}}(p,p+1))$
is open and closed.

###### Proof.

For openness, we use invariance of domains. Namely, the spaces have the same
dimension, and, by Proposition 5.4, no two Higgs bundles in the image of
$\widehat{\Psi}$ are gauge equivalent. Thus, $\widehat{\Psi}$ is an injective
map between manifolds of the same dimension, and is therefore open.

For closedness suppose $\big{(}q_{2}^{j},\dots,q_{2p}^{j}\big{)}$ is a
divergent sequence of points in
$\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}$. By Proposition 5.4 and
properness of the Hitchin fibration we conclude that the sequence
$\Psi\big{(}q_{2}^{j},\dots,q_{2p}^{j}\big{)}$ also diverges in
${\mathcal{M}}({\mathsf{SO}}(p,p+1))$. ∎

To complete the proof we need to show that under the nonabelian Hodge
correspondence, the component defined by
$\Psi\Big{(}\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}\Big{)}$ is the
Hitchin component ${\mathsf{Hit}}({\mathsf{SO}}(p,p+1))$ from Definition 2.7.
It suffices to show that the representation associated to $\Psi(0,\dots,0)$ is
in ${\mathsf{Hit}}({\mathsf{SO}}(p,p+1))$. By Remark 4.7, the Higgs bundle
(5.4) defines an ${\mathsf{SL}}(2,{\mathbb{R}})$-Higgs bundle whose
corresponding representation is in ${\mathsf{Fuch}}(\Gamma)$. From Example
2.6, the principal embedding $\iota_{\rm
pr}\colon{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\to{\mathsf{SO}}(p,p+1)$ is
given by taking the $2p$-symmetric product of the standard representation of
${\mathsf{SL}}(2,{\mathbb{R}})$. Thus, the representation associated to
$\Psi(0,\dots,0)$ is contained in ${\mathsf{Hit}}({\mathsf{SO}}(p,p+1))$.

##  6 Structure of the moduli space

###  6.1 Tangent space and deformation complex

In this section we will assume for simplicity that ${\mathsf{G}}$ is a real
form of a complex semisimple Lie group. Under this assumption, the
automorphism group of a stable ${\mathsf{G}}$-Higgs bundle is discrete (see
Proposition 3.16). Recall that ${\mathcal{H}}({\mathsf{G}})$ is the set of
pairs $\big{(}\bar{\partial}_{P},\varphi\big{)}$ where $\bar{\partial}_{P}$ is
a Dolbeault operator on a smooth ${\mathsf{H}}^{\mathbb{C}}$-bundle $P\to X$
and
$\varphi\in\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}$
such that $\bar{\partial}_{P}\varphi=0$.

Since the space of Dolbeault operators is an affine space with underlying
vector space isomorphic
$\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}$, the
tangent space of ${\mathcal{H}}^{\rm ps}({\mathsf{G}})$ at
$\big{(}\bar{\partial}_{P},\varphi\big{)}$ is given by the set of
$(\alpha,\psi)\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}$
so that $\varphi+\psi$ is holomorphic with respect to the Dolbeault operator
$(\bar{\partial}_{P}+\alpha)$ to first order. That is,

$\displaystyle
T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}})=\big{\\{}(\alpha,\psi)\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}\,|\,\bar{\partial}_{P}\psi+[\alpha,\varphi]=0\in\Omega^{1,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\big{\\}}.$

The moduli space of ${\mathsf{G}}$-Higgs bundles is a set of gauge equivalence
classes:

$\displaystyle{\mathcal{M}}({\mathsf{G}})={\mathcal{H}}^{\rm
ps}({\mathsf{G}})/{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}},$

where ${\mathcal{H}}^{\rm ps}({\mathsf{G}})$ 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
${\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}\cdot\big{(}\bar{\partial}_{P},\varphi\big{)}$:

$\displaystyle
T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})=T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}})/T_{(\bar{\partial}_{P},\varphi)}{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}\cdot\big{(}\bar{\partial}_{P},\varphi\big{)}.$

This is because, under our assumption on ${\mathsf{G}}$, the automorphism
group of a stable ${\mathsf{G}}$-Higgs bundle is discrete, and so the gauge
group action is locally free. The tangent space to the gauge orbit of a stable
${\mathsf{G}}$-Higgs bundle can thus be identified with the tangent space at
the identity of the gauge group

$\displaystyle
T_{e}{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}\cong\Omega^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}.$

The identification of
$\Omega^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}$ with the
tangent space
$T_{(\bar{\partial}_{P},\varphi)}{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}\cdot\big{(}\bar{\partial}_{P},\varphi\big{)}$
is given by the map

$\textstyle{\Omega^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)},}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{(}\bar{\partial}_{P}x,[\varphi,x]\big{)}.}$

Note that $(\bar{\partial}_{P}x,[\varphi,x])\in
T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}})$ since
$\bar{\partial}_{P}([\varphi,x])+[\bar{\partial}_{P}x,\varphi]=0$.

The tangent space to ${\mathcal{M}}({\mathsf{G}})$ at a stable Higgs bundle
$\big{[}\bar{\partial}_{P},\varphi\big{]}$ is thus identified with

$\displaystyle
T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})\cong\dfrac{\big{\\{}(\alpha,\psi)\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}\,|\,\bar{\partial}_{P}\psi+[\alpha,\varphi]=0)\big{\\}}}{\big{\\{}(\bar{\partial}_{P}x,[\varphi,x])\in\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\big{)}\,|\,x\in\Omega^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\big{\\}}}.$
(6.1)

The tangent space fits into a very useful exact sequence.

######  Proposition 6.1.

For a stable ${\mathsf{G}}$-Higgs bundle
$\big{(}\bar{\partial}_{P},\varphi\big{)}$, we have an exact sequence

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
25.50003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
25.50003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
79.77228pt\raise 6.25pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.61111pt\hbox{$\scriptstyle{{\operatorname{ad}}_{\varphi}\ \ \
}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 98.223pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
98.223pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
169.41508pt\raise 5.30833pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30833pt\hbox{$\scriptstyle{\ \ i}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 192.93207pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
192.93207pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces{\hbox{\kern
242.7854pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
228.7854pt\raise-3.00008pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-22.99316pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.00127pt\raise-25.33557pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\cirbuild@}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
6.33907pt\raise-32.84251pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\pi\ \ \ \ \ \ \ \ \ }$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
25.50003pt\raise-28.33557pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
25.50003pt\raise-28.33557pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
79.77228pt\raise-34.58556pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.61111pt\hbox{$\scriptstyle{{\operatorname{ad}}_{\varphi}\ \ \
}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern
96.8341pt\raise-28.33557pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
96.8341pt\raise-28.33557pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{H^{1}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\big{)},}$}}}}}}}{\hbox{\kern 216.35869pt\raise-28.33557pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(6.2)

where the map $i$ is induced by the inclusion

$\textstyle{H^{0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}}),}$$\textstyle{\psi\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(0,\psi),}$

and the map $\pi$ is induced by the projection

$\textstyle{T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)},}$$\textstyle{(\psi,\alpha)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\alpha.}$

######  Remark 6.2.

In fact, the sequence (6.2) is exact on the right, however we have not
developed the techniques to prove this. Using exactness of this sequence, a
Riemann–Roch calculation implies that the real dimension of the tangent space
at a stable Higgs bundle is given by $\dim_{\mathbb{R}}({\mathsf{G}})(2g-2)$.

###### Proof.

First, the elements in the kernel of the map ${\operatorname{ad}}_{\varphi}$
correspond to tangent vectors of one parameter families of automorphisms of
$\big{(}\bar{\partial}_{P},\varphi\big{)}$. Thus stability implies that
$\ker({\operatorname{ad}}_{\varphi})=0$.

Next, note that the kernel of the map $i\colon
H^{0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K\big{)}\to
T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$ is given by the
set of $(0,\psi)=\big{(}\bar{\partial}_{P}x,[\varphi,x]\big{)}$. Thus, the
kernel of $i$ equals the image of the map ${\operatorname{ad}}_{\varphi}\colon
H^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\to
H^{0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K\big{)}$.

The projection
$T_{(\bar{\partial}_{P},\varphi)}{\mathcal{H}}({\mathsf{G}})\to\Omega^{0,1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}$
descends to a map

$\displaystyle\pi\colon\
T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})\to
H^{1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}$

since $\alpha+\bar{\partial}_{P}x$ defines the same cohomology class as
$\alpha$. Any representative of an element of the kernel of $\pi$ is a pair
$\big{(}\bar{\partial}_{P}x,\psi\big{)}$ such that
$\bar{\partial}_{P}\psi+\big{[}\bar{\partial}_{P}x,\varphi\big{]}=0$. Any such
pair is equivalent to $(0,\psi-{\operatorname{ad}}_{\varphi}x)$. Thus, the
kernel of $\pi$ is the image of $i$.

Finally, the condition $\bar{\partial}_{P}\psi+[\varphi,\alpha]=0$ implies
that ${\operatorname{ad}}_{\varphi}(\alpha)$ is zero in the cohomology group
$H^{1}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K\big{)}$.
Thus, the image of $\pi$ is the kernel of ${\operatorname{ad}}_{\varphi}$. ∎

######  Remark 6.3.

For any Higgs bundle 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. One way to
describe this is with a deformation complex (see [5]). Namely, the sheaf map
${\operatorname{ad}}_{\varphi}\colon
P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\to
P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K$ defines a long exact
sequence in hypercohomology

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}}^{0}\big{(}\bar{\partial}_{P},\varphi\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{ad}}_{\varphi}\
\ \
}$$\textstyle{H^{0}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}}^{1}\big{(}\bar{\partial}_{P},\varphi\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{1}\big{(}P\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{ad}}_{\varphi}\
\ \
}$$\textstyle{H^{1}\big{(}P\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{H}}^{2}\big{(}\bar{\partial}_{P},\varphi\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

In general, ${\mathbb{H}}^{0}\big{(}\bar{\partial}_{P},\varphi\big{)}$ is the
space of infinitesimal automorphisms of
$\big{(}\bar{\partial}_{P},\varphi\big{)}$, and for stable Higgs bundles, the
tangent space $T_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$ is
identified with ${\mathbb{H}}^{1}\big{(}\bar{\partial}_{P},\varphi\big{)}$.

###  6.2 The $\boldsymbol{{\mathbb{C}}^{*}}$-action

There is a natural action of ${\mathbb{C}}^{*}$ on the ${\mathsf{G}}$-Higgs
bundle moduli space given by scaling the Higgs field

$\textstyle{{\mathbb{C}}^{*}\times{\mathcal{M}}({\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{M}}({\mathsf{G}}),}$$\textstyle{\big{(}\lambda,\big{[}\bar{\partial}_{P},\varphi\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\big{[}\bar{\partial}_{P},\lambda\varphi\big{]}.}$

Note that the Hitchin fibration (5.1) is equivariant with respect to a
weighted ${\mathbb{C}}^{*}$-action:

$\displaystyle
h\big{(}\big{[}\bar{\partial}_{P},\lambda\cdot\varphi\big{]}\big{)}=\big{(}\lambda^{m_{1}+1}p_{1}(\varphi),\dots,\lambda^{m_{\operatorname{rk}({\mathsf{G}})}+1}p_{\operatorname{rk}({\mathsf{G}})}(\varphi)\big{)}.$

Thus, the fixed points of the ${\mathbb{C}}^{*}$-action are contained in the
_nilpotent cone_ $h^{-1}(0)$. Moreover, the properness of $h$ implies that
$\lim\limits_{\lambda\to 0}\big{[}\bar{\partial}_{P},\lambda\varphi\big{]}$
always exists and is a ${\mathbb{C}}^{*}$-fixed point.

Since we are dealing with isomorphism classes, being a
${\mathbb{C}}^{*}$-fixed point does not imply $\varphi=0$. Rather, it implies
that there is a holomorphic gauge transformation $g_{\lambda}$ such that
${\operatorname{Ad}}_{g_{\lambda}^{-1}}\varphi=\lambda\varphi$ for all
$\lambda\in{\mathbb{C}}^{*}$. For ${\mathsf{SL}}(n,{\mathbb{C}})$, the
${\mathbb{C}}^{*}$-fixed points are classified by the following proposition.

######  Proposition 6.4.

Let $(E,\Phi)$ be a polystable ${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundle.
Then $(E,\Phi)$ is gauge equivalent to $(E,\lambda\Phi)$ for all
$\lambda\in{\mathbb{C}}^{*}$ if and only if there is a holomorphic splitting
$E=E_{1}\oplus\cdots\oplus E_{\ell}$ in which the Higgs field is given by

$\displaystyle\Phi=\left(\begin{matrix}0&\\\ \varphi_{1}&0\\\
&\ddots&\ddots\\\ &&\varphi_{\ell-1}&0\end{matrix}\right),$

where $\varphi_{j}\colon E_{j}\to E_{j+1}\otimes K$ is a holomorphic bundle
map.

######  Remark 6.5.

We will usually represent such a fixed point schematically as

$\textstyle{E_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\textstyle{E_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{2}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\ell-2}}$$\textstyle{E_{\ell-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{\ell-1}}$$\textstyle{E_{\ell},}$

where we suppress the twisting by $K$ from the notation. The moduli space of
such fixed points is a special case of the moduli of holomorphic chains.

For ${\mathsf{SL}}(n,{\mathbb{C}})$ we have
${\mathsf{H}}^{\mathbb{C}}={\mathsf{SL}}(n,{\mathbb{C}})$ and
${\mathfrak{m}}^{\mathbb{C}}={\mathfrak{sl}}(n,{\mathbb{C}})$. For
${\mathsf{SL}}(n,{\mathbb{C}})$-Higgs bundles fixed by the
${\mathbb{C}}^{*}$-action, the ${\mathsf{H}}^{\mathbb{C}}$-bundle has a
holomorphic reduction $E=E_{1}\oplus\cdots\oplus E_{k}$ to a subgroup of block
diagonal matrices. Such a reduction gives a ${\mathbb{Z}}$-grading on the
bundle
${\mathcal{E}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}={\mathrm{End}}(E)=\bigoplus_{j}{\mathrm{End}}(E)_{h}$,
where

$\displaystyle{\mathrm{End}}(E)_{j}=\bigoplus\limits_{b-a=j}{\mathrm{Hom}}(E_{a},E_{b}).$

Moreover, with respect to this ${\mathbb{Z}}$-grading we have $\Phi\in
H^{0}({\mathrm{End}}(E)_{1}\otimes K)$. The characterization of
${\mathsf{G}}$-Higgs bundles fixed by the ${\mathbb{C}}^{*}$-action is given
by the following proposition.

######  Proposition 6.6.

A polystable ${\mathsf{G}}$-Higgs bundle $({\mathcal{P}},\varphi)$ defines a
fixed point of the ${\mathbb{C}}^{*}$-action on ${\mathcal{M}}({\mathsf{G}})$
if and only if

  1. $1.$

There is a ${\mathbb{Z}}$-grading
${\mathfrak{g}}^{\mathbb{C}}=\bigoplus_{j}{\mathfrak{g}}^{\mathbb{C}}_{j}=\bigoplus_{j}{\mathfrak{h}}_{j}^{\mathbb{C}}\oplus{\mathfrak{m}}^{\mathbb{C}}_{j}$
so that $[{\mathfrak{g}}_{j},{\mathfrak{g}}_{i}]\subset{\mathfrak{g}}_{i+j}$.

  2. $2.$

There is a holomorphic reduction
${\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\subset{\mathcal{P}}$ to an
${\mathsf{H}}^{\mathbb{C}}_{0}$-bundle, where
${\mathsf{H}}^{\mathbb{C}}_{0}<{\mathsf{H}}^{\mathbb{C}}$ is the Lie group
with Lie algebra ${\mathfrak{h}}_{0}^{\mathbb{C}}$.

  3. $3.$

With respect to the decomposition
${\mathcal{P}}_{{\mathsf{H}}_{0}^{\mathbb{C}}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}=\bigoplus_{j}{\mathcal{P}}_{{\mathsf{H}}_{0}^{\mathbb{C}}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j}\big{]}$,
we have

$\displaystyle\varphi\in
H^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}_{1}^{\mathbb{C}}\big{]}\otimes
K\big{)}.$

######  Remark 6.7.

In terms of vector bundles, the ${\mathsf{G}}$-Higgs bundles which are
${\mathbb{C}}^{*}$-fixed points are given by holomorphic chains with extra
symmetries which reflect the symmetries of a ${\mathsf{G}}$-Higgs bundle. For
example, the ${\mathsf{SL}}(p+q,{\mathbb{C}})$-Higgs bundle associated to an
${\mathsf{SO}}(p,q)$-Higgs bundle $(V,W,\eta)$ is given by $\Big{(}V\oplus
W,\left(\begin{smallmatrix}0&\eta^{\dagger}\\\
\eta&0\end{smallmatrix}\right)\Big{)}$, so the associated fixed points are
direct sums of holomorphic chains of the form

$\textstyle{V_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{r}}$$\textstyle{W_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{2-r}^{\dagger}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{r-2}}$$\textstyle{W_{1-r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{r}^{\dagger}}$$\textstyle{V_{-r}}$

and

$\textstyle{W_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{1-s}^{\dagger}}$$\textstyle{V_{s-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{s-1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{s-1}^{\dagger}}$$\textstyle{V_{1-s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta_{1-s}}$$\textstyle{W_{-s}.}$

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 $W_{-j}\cong W_{j}^{*}$ and $V_{-j}\cong V_{j}^{*}$.

###  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 $\mathrm{L}^{2}$-norm of the Higgs field. Namely, consider the
nonnegative function $f\colon{\mathcal{M}}({\mathsf{G}})\to{\mathbb{R}}$
defined by

$\displaystyle
f\big{(}\big{[}\bar{\partial}_{P},\varphi\big{]}\big{)}=\int_{X}|\varphi|^{2},$
(6.3)

where the norm $|\varphi|$ is taken with respect to the Hermitian metric
associated to $\big{(}\bar{\partial}_{P},\varphi\big{)}$ from the nonabelian
Hodge correspondence (see Remark 3.2).

######  Remark 6.8.

Note that the function $f\big{(}\bar{\partial}_{P},\varphi\big{)}=0$ if and
only if $\varphi=0$. Equivalently, the global minima of $f$ are given by
points in the moduli of polystable ${\mathsf{H}}^{\mathbb{C}}$-bundles
${\mathcal{M}}({\mathsf{H}})\subset{\mathcal{M}}({\mathsf{G}})$.

For $\lambda\in{\mathsf{U}}(1)$, the metrics from the nonabelian Hodge
correspondence associated $\big{(}\bar{\partial}_{P},\varphi\big{)}$ and
$\big{(}\bar{\partial}_{P},\lambda\varphi\big{)}$ are the same.444This is not
necessarily true when $\lambda\in{\mathbb{C}}^{*}$. Thus, the function $f$ is
${\mathsf{U}}(1)$-invariant. Moreover, in [21, Section 8], Hitchin showed that
the ${\mathsf{U}}(1)$-action is Hamiltonian with respect to the symplectic
structure $\omega_{I}$ from Remark 3.14, and that the function $f$ is a moment
map for this action. That is,

$\displaystyle\mathrm{grad}(f)=IX,$

where $X$ is the vector field generating the ${\mathsf{U}}(1)$-action. This
implies that $f$ is a Morse–Bott function on the smooth locus of
${\mathcal{M}}({\mathsf{SL}}(n,{\mathbb{C}}))$ and critical submanifolds of
$f$ are exactly the components of the fixed point set of the
${\mathsf{U}}(1)$-action. In fact, Hitchin’s arguments also hold for the
moduli space ${\mathcal{M}}({\mathsf{G}})$.

Since the moduli space ${\mathcal{M}}({\mathsf{G}})$ 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 [19] even on the singular locus. Hence, $f$ attains a
minimum on every closed subset. In particular, we have the following upper
bound on the number of components

$\displaystyle|\pi_{0}({\mathcal{M}}({\mathsf{G}}))|\leq|\pi_{0}(\\{\text{local
minima of $f$}\\})|.$

There are three subvarieties which intersect at a ${\mathbb{C}}^{*}$-fixed
point $\big{[}\bar{\partial}_{P},\varphi\big{]}$

  1. 1)

$W^{s}\big{(}{\big{[}\bar{\partial}_{P},\varphi\big{]}}\big{)}=\big{\\{}\big{[}\bar{\partial}_{P}^{\prime},\vartheta\big{]}\,|\,\lim\limits_{\lambda\to
0}\big{[}\bar{\partial}_{P}^{\prime},\lambda\vartheta\big{]}=\big{[}\bar{\partial}_{P},\varphi\big{]}\big{\\}}$,

  2. 2)

$W^{u}\big{(}{\big{[}\bar{\partial}_{P},\varphi\big{]}}\big{)}=\big{\\{}\big{[}\bar{\partial}_{P}^{\prime},\vartheta\big{]}\,|\,\lim\limits_{\lambda\to\infty}\big{[}\bar{\partial}_{P}^{\prime},\lambda\vartheta\big{]}=\big{[}\bar{\partial}_{P},\varphi\big{]}\big{\\}}$,

  3. 3)

$W^{0}\big{(}{\big{[}\bar{\partial}_{P},\varphi\big{]}}\big{)}$ the connected
component of the fixed point locus containing
$\big{[}\bar{\partial}_{P},\varphi\big{]}$.

For a stable fixed point $\big{[}\bar{\partial}_{P},\varphi\big{]}$, these are
exactly the stable, unstable and critical submanifolds of the Morse–Bott
function $f$ at the critical point $\big{[}\bar{\partial}_{P},\varphi\big{]}$.
A fixed point $\big{[}\bar{\partial}_{P},\varphi\big{]}$ is thus a local
minima of $f$ if and only if
$W^{u}\big{(}{\big{[}\bar{\partial}_{P},\varphi\big{]}}\big{)}=\big{\\{}\big{[}\bar{\partial}_{P},\varphi\big{]}\big{\\}}$.

######  Remark 6.9.

Recall from before that we have a lower bound on
$\pi_{0}({\mathcal{M}}({\mathsf{G}}))$ given by
$\pi_{0}({\mathcal{M}}({\mathsf{H}}))$. Moreover, when ${\mathsf{H}}$ is
semisimple we have $\pi_{0}({\mathcal{M}}({\mathsf{H}}))$ is in bijective
correspondence with topological ${\mathsf{H}}$-bundles ${\mathcal{B}}_{H}(X)$.
This gives the following inequality

$\displaystyle|\pi_{0}({\mathcal{M}}({\mathsf{H}}))|\leq|\pi_{0}({\mathcal{M}}({\mathsf{G}}))|\leq|\pi_{0}(\\{\text{local
minima of $f$}\\})|.$

We now have a strategy for counting the components of the character variety
${\mathcal{X}}({\mathsf{G}})$. Namely we should classify local minima of $f$
and show that every component of local minima defines a component of the
moduli space. In particular, if the only local minima have $\varphi=0$, then
every Higgs bundle can be reduced to ${\mathsf{H}}$, and if a component of
${\mathcal{M}}({\mathsf{G}})$ has the property that the Higgs field can never
be deformed to zero, then no representation in the associated component of the
character variety can be deformed to a compact representation.

###  6.4 Local minima criterion

We first describe how for fixed points of the ${\mathbb{C}}^{*}$-action we get
a decomposition of the tangent space into weight spaces. Recall from
Proposition 6.6 that associated to a polystable ${\mathsf{G}}$-Higgs bundle
$({\mathcal{P}},\varphi)$ fixed by the ${\mathbb{C}}^{*}$-action there is a
${\mathbb{Z}}$-grading
${\mathfrak{g}}^{\mathbb{C}}=\bigoplus_{j}{\mathfrak{h}}^{\mathbb{C}}_{j}\oplus{\mathfrak{m}}^{\mathbb{C}}_{j}$
and a holomorphic structure group reduction
${\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}$ so that $\varphi\in
H^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}_{1}^{\mathbb{C}}\big{]}\otimes
K\big{)}$.

For such a fixed point, the map
${\operatorname{ad}}_{\varphi}\colon{\mathcal{P}}\big{[}{\mathfrak{h}}^{\mathbb{C}}\big{]}\to{\mathcal{P}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes
K$ defines a map

$\displaystyle{\operatorname{ad}}_{\varphi}\colon\
{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\to{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j+1}\big{]}\otimes
K.$

For stable fixed points this gives a decomposition of the exact sequence
(6.2), that is, for all $j$ we have

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}_{j}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{ad}}_{\varphi}\
\ \
}$$\textstyle{H^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}_{j+1}^{\mathbb{C}}\big{]}\otimes
K\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\
\
i}$$\textstyle{T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\
\ \ \ \ \ \ \ \
}$$\textstyle{H^{1}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}_{j}^{\mathbb{C}}\big{]}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\operatorname{ad}}_{\varphi}\
\ \
}$$\textstyle{H^{1}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}_{j+1}^{\mathbb{C}}\big{]}\otimes
K\big{)},}$

where, similarly to (6.1),
$T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$ is defined
by

$\displaystyle
T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})=\dfrac{\big{\\{}(\alpha,\psi)\in\Omega^{0,1}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j+1}\big{]}\big{)}\,|\,\bar{\partial}_{P}\psi+[\alpha,\varphi]=0\big{\\}}}{\big{\\{}\big{(}\bar{\partial}_{P}x,[\varphi
x]\big{)}\in\Omega^{0,1}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\big{)}\oplus\Omega^{1,0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j+1}\big{]}\big{)}\,|\,x\in\Omega^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}_{j}^{\mathbb{C}}\big{]}\big{)}\big{\\}}}.$

The following result was proven for ${\mathsf{SL}}(n,{\mathbb{C}})$ by Hitchin
in [21], the general case follows from arguments analogous to the Morse–Bott
function’s index computation of Hitchin in [21, Section 8].

######  Theorem 6.10 (Hitchin [19, 21]).

Let $f\colon{\mathcal{M}}({\mathsf{G}})\to{\mathbb{R}}$ be the Morse–Bott
function from (6.3). For a stable ${\mathsf{G}}$-Higgs bundle we have the
following:

  * •

$\big{[}\bar{\partial}_{P},\varphi\big{]}$ is a fixed point of the
${\mathbb{C}}^{*}$-action if and only if it is a critical point of the
function $f$,

  * •

$T_{[\bar{\partial}_{P},\varphi]}W^{u}\big{(}\big{[}\bar{\partial}_{P},\varphi\big{]}\big{)}=\bigoplus\limits_{j>0}T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$,

  * •

$T_{[\bar{\partial}_{P},\varphi]}W^{s}\big{(}\big{[}\bar{\partial}_{P},\varphi\big{]}\big{)}=\bigoplus\limits_{j<0}T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$,

  * •

$T_{[\bar{\partial}_{P},\varphi]}W^{0}\big{(}{\big{[}\bar{\partial}_{P},\varphi\big{]}}\big{)}=T^{0}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})$.

######  Corollary 6.11.

A stable fixed point
$\big{[}\bar{\partial}_{P},\varphi\big{]}\in{\mathcal{M}}({\mathsf{G}})$ is a
local minimum of $f$ if and only if
$\bigoplus\limits_{j>0}T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})=0$.

Using the sequence (6.2), if
${\operatorname{ad}}_{\varphi}\colon{\mathcal{P}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\to{\mathcal{P}}[{\mathfrak{m}}_{j+1}]\otimes
K$ is an isomorphism for all $j>0$, then
$T^{j}_{[\bar{\partial}_{P},\varphi]}{\mathcal{M}}({\mathsf{G}})=0$ for all
$j>0$ and we are at a local minimum of the $f$. In fact the converse holds as
well (see [6, Section 3.4]), and we have a classification of stable local
minima of the Morse–Bott function $f$.

######  Proposition 6.12.

A stable ${\mathsf{G}}$-Higgs bundle
$\big{(}\bar{\partial}_{P},\varphi\big{)}$ which is a ${\mathbb{C}}^{*}$-fixed
point is a local minima of the function $f$ from (6.3) if and only if

$\displaystyle{\operatorname{ad}}_{\varphi}\colon\
{\mathcal{P}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\to{\mathcal{P}}[{\mathfrak{m}}_{j+1}]\otimes
K$

is an isomorphism for all $j>0$.

###  6.5 Some component results

We have now developed necessary tools to show the map
$\tau\colon{\mathcal{X}}(\Gamma,{\mathsf{G}})\to{\mathcal{B}}_{\mathsf{G}}(S)$
from (1.3) is injective. In fact, the proof is very simple with the above set
up.

######  Theorem 6.13 (García-Prada and Oliveira [12]).

Let ${\mathsf{G}}$ be a complex reductive Lie group with maximal compact
subgroup ${\mathsf{H}}$. Then there is a bijection between the components of
the moduli space of polystable ${\mathsf{G}}$-Higgs bundles and the moduli
space of polystable ${\mathsf{G}}$-bundles:

$\displaystyle\pi_{0}({\mathcal{M}}({\mathsf{G}}))=\pi_{0}({\mathcal{M}}({\mathsf{H}})).$

###### Proof.

By the above discussion and Remark 6.9, it suffices to show that a polystable
${\mathsf{G}}$-Higgs bundle $[{\mathcal{P}},\varphi]$ is a local minima of the
Morse–Bott function $f$ from (6.3) if and only if $\varphi=0$. Let
$[{\mathcal{P}},\varphi]$ be a local minima of $f$. Since
$[{\mathcal{P}},\varphi]$ is a ${\mathbb{C}}^{*}$-fixed point, by Proposition
6.6 there is a ${\mathbb{Z}}$-grading
${\mathfrak{g}}^{\mathbb{C}}=\bigoplus_{j}{\mathfrak{h}}^{\mathbb{C}}_{j}\oplus{\mathfrak{m}}^{\mathbb{C}}_{j}$
so that $({\mathcal{P}},\varphi)$ is isomorphic to
$({\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{o}},\varphi)$ with $\varphi\in
H^{0}\big{(}{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{1}\big{]}\otimes
K\big{)}$.

First suppose $\big{[}\bar{\partial}_{P},\varphi\big{]}$ is a stable local
minima of $f$. Then by Proposition 6.12 we have

$\displaystyle{\operatorname{ad}}_{\varphi}\colon\
{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}\to{\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j+1}\big{]}\otimes
K$

is an isomorphism for all $j>0$. But, since ${\mathfrak{g}}$ is complex, we
have ${\mathfrak{h}}^{\mathbb{C}}\cong{\mathfrak{m}}^{\mathbb{C}}$, and the
only way
${\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}$
can be isomorphic to
${\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{m}}^{\mathbb{C}}_{j+1}\big{]}\otimes
K$ for all $j>0$ is for
${\mathcal{P}}_{{\mathsf{H}}^{\mathbb{C}}_{0}}\big{[}{\mathfrak{h}}^{\mathbb{C}}_{j}\big{]}=0$
for all $j>0$. In this case we have $\varphi=0$.

To rule out strictly polystable minima with nonzero Higgs field, we note that
a ${\mathsf{G}}$-Higgs bundle which is strictly polystable has a holomorphic
reduction to a Levi factor ${\mathsf{L}}$ of a parabolic subgroup of
${\mathsf{G}}$ which is stable as a ${\mathsf{L}}$-Higgs bundle. Now repeat
the above argument for the moduli space ${\mathcal{M}}({\mathsf{L}})$. ∎

As a immediate corollary we have the following.

######  Corollary 6.14.

If ${\mathsf{G}}$ is a complex reductive Lie group, then every polystable
${\mathsf{G}}$-Higgs bundle $\big{[}\bar{\partial}_{P},\varphi\big{]}$ can be
continuously deformed to a polystable ${\mathsf{G}}$-bundle, i.e., a
polystable ${\mathsf{G}}$-Higgs bundle with zero Higgs field.

Using the nonabelian Hodge correspondence, Theorem 1.7 now follows as a
corollary of the above theorem.

######  Corollary 6.15.

If ${\mathsf{G}}$ is a complex reductive Lie group with maximal compact
${\mathsf{H}}$, then the map
$\tau\colon\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{G}}))\to{\mathcal{B}}_{\mathsf{G}}(S)$
from (1.3) is injective. In particular, every representation
$\rho\colon\Gamma\to{\mathsf{G}}$ can be deformed to a compact representation
$\Gamma\to{\mathsf{H}}\hookrightarrow{\mathsf{G}}$.

Using the methods described above, Hitchin gave a complete component count of
${\mathcal{M}}({\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$. 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 $f$.

######  Theorem 6.16 (Hitchin [21]).

For $n>2$, the only local minima of the Morse–Bott function (6.3) on
${\mathcal{M}}({\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$ are $\varphi=0$ and
the image of 0 in the Hitchin section.

We thus have the following corollary.

######  Corollary 6.17.

$\displaystyle|\pi_{0}({\mathcal{M}}({\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))|=\begin{cases}4g-3&n=2,\\\
3&\text{$n$-odd,}\\\ 6&\text{$n>2$ and even}.\end{cases}$

###### Proof.

For $n=2$ the component count is the same as for
${\mathcal{M}}({\mathsf{SO}}_{0}(1,2))$. For $n>2$ and odd we have
${\mathcal{B}}_{{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})}={\mathbb{Z}}_{2}$
and there is only one Hitchin component. This gives three components. For
$n>2$ and $n$-even we have
${\mathcal{B}}_{{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})}$ is isomorphic to
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ or ${\mathbb{Z}}_{4}$ depending on
the parity of $\frac{n}{2}$ (see (1.4)). 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
${\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$ we of course
have the same count.

######  Corollary 6.18.

$\displaystyle|\pi_{0}({\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))|=\begin{cases}4g-3&n=2,\\\
3&\text{$n$-odd,}\\\ 6&\text{$n>2$ and even}.\end{cases}$

Theorem 6.16 also gives a dichotomy for deformations of representations into
${\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}})$, namely for $n>2$ the components
of ${\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$ are
either deformations spaces of compact representations or deformation spaces of
special Fuchsian representations.

######  Corollary 6.19.

For each $n>2$ and each
$\rho\in{\mathcal{X}}(\Gamma,{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}))$,
exactly one of the following holds

  * •

$\rho$ can be deformed to a compact representation

$\displaystyle\Gamma\to{\mathsf{P}}{\mathsf{SO}}(n)\hookrightarrow{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}),$

  * •

$\rho$ can be deformed to a representation

$\textstyle{\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho_{\rm
Fuch}\ \ \ \
}$$\textstyle{{\mathsf{P}}{\mathsf{SL}}(2,{\mathbb{R}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{\rm
pr}}$$\textstyle{{\mathsf{P}}{\mathsf{SL}}(n,{\mathbb{R}}),}$

where $\rho_{\rm Fuch}\in{\mathsf{Fuch}}(\Gamma)$ is a Fuchsian representation
and $\iota_{\rm pr}$ is the principal embedding from (2.4).

##  7 $\boldsymbol{{\mathsf{SO}}(p,q)}$-Higgs bundles

We now apply the techniques of the previous section to understand the
components of the ${\mathsf{SO}}(p,q)$-character variety
${\mathcal{X}}(\Gamma,{\mathsf{SO}}(p,q))$. In her thesis [1], Aparicio-Arroyo
discovered that the Higgs bundle moduli space
${\mathcal{M}}({\mathsf{SO}}(p,q))$ has stable local minima of the Morse–Bott
function (6.3) with nonzero Higgs field and which do not arise from the
Hitchin section. This was done by classifying stable
${\mathsf{SO}}(p,q)$-Higgs bundles which are fixed points of the
${\mathbb{C}}^{*}$-action and satisfied Proposition 6.12. Due to the potential
singularities, these results are not strong enough to classify the components
of the moduli space ${\mathcal{M}}({\mathsf{SO}}(p,q))$.

It should be noted that the methods outlined in the previous section are not
the only approach to studying the question of components. For example, by
examining spectral data on generic fibers of the Hitchin fibration for
${\mathcal{M}}({\mathsf{SO}}(p+q,{\mathbb{C}}))$, Schaposnik and Baraglia [4]
have given evidence for the existence of additional components of the moduli
space ${\mathcal{M}}({\mathsf{SO}}(p,q))$. Although these methods do not
currently distinguish connected components, they provide an intriguing
alternative perspective.

We start by recalling the classification of stable minima.

######  Remark 7.1.

The case ${\mathsf{SO}}(2,q)$ is rather special since ${\mathsf{SO}}(2,q)$ 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, that is, for
$2<p\leq q$. For the case of ${\mathsf{SO}}(2,q)$ we refer the reader to [2,
10].

Recall the notation of ${\mathcal{K}}_{p}=K^{p-1}\oplus
K^{p-3}\oplus\cdots\oplus K^{3-p}\oplus K^{1-p}$ from (5.2) and the map
$\widehat{\Psi}(0,\dots,0)\colon{\mathcal{K}}_{p}\to{\mathcal{K}}_{p+1}\otimes
K$ from (5.3). Denote by $\eta_{0}$ the following transpose

$\displaystyle\eta_{0}=\widehat{\Psi}(0,\dots,0)^{T}\colon\
{\mathcal{K}}_{p}\to{\mathcal{K}}_{p-1}\otimes K.$

######  Theorem 7.2 (Aparicio-Arroyo [1]).

Suppose $2<p\leq q$. If $[V,W,\eta]$ is a stable ${\mathsf{SO}}(p,q)$-Higgs
bundle, then $(V,W,\eta)$ defines a local minima of the Morse–Bott function
(6.3) if and only if one of the following holds

  1. $1)$

$\eta=0$,

  2. $2)$

there is a stable rank $q-p+1$ orthogonal bundle $W_{0}$ with determinant
bundle $I=\det(W_{0})$ such that

$\displaystyle(V,W,\eta)=\left({\mathcal{K}}_{p}\otimes
I,({\mathcal{K}}_{p-1}\otimes I)\oplus W_{0},\left(\begin{matrix}\eta_{0}\\\
0\end{matrix}\right)\colon V\to W\otimes K\right),$

  3. $3)$

$q=p+1$ and there is a line bundle $M\in{\mathsf{Pic}}^{d}(X)$ with
$d\in(0,p(2g-2)]$ and $\mu\in H^{0}\big{(}M^{-1}K^{p}\big{)}\setminus\\{0\\}$
so that

$\displaystyle(V,W,\eta)=\left({\mathcal{K}}_{p},M\oplus{\mathcal{K}}_{p-1}\oplus
M^{-1},\left(\begin{matrix}0\\\ \eta_{0}\\\
\eta_{\mu}\end{matrix}\right)\colon V\to W\otimes K\right),$

where
$\eta_{\mu}=\left(\begin{matrix}0&\cdots&0&\mu\end{matrix}\right)\colon{\mathcal{K}}_{p}\to
M^{-1}K$.

######  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 ${\mathsf{SO}}(p,p+1)$-Hitchin component
defined by $\widehat{\Psi}(0,\dots,0)$ from (5.3).

In [2] 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 Higgs bundles similar to the second
case of Theorem 7.2, but where the bundle $W_{0}$ is allowed to be strictly
polystable.

######  Theorem 7.4.

Assume $2<p\leq q$ and $(V,W,\eta)$ is a polystable ${\mathsf{SO}}(p,q)$-Higgs
bundle. Then $(V,W,\eta)$ defines a local minima of the Morse–Bott function
(6.3) if and only if

  1. $1)$

$\eta=0$,

  2. $2)$

there is a polystable rank $q-p+1$ orthogonal bundle $W_{0}$ with determinant
bundle $I=\det(W_{0})$ such that

$\displaystyle(V,W,\eta)=\left({\mathcal{K}}_{p}\otimes
I,({\mathcal{K}}_{p-1}\otimes I)\oplus W_{0},\left(\begin{matrix}\eta_{0}\\\
0\end{matrix}\right)\colon V\to W\otimes K\right),$

  3. $3)$

$q=p+1$ and there is a line bundle $M\in{\mathsf{Pic}}^{d}(X)$ with
$d\in(0,p(2g-2)]$ and $\mu\in H^{0}\big{(}M^{-1}K^{p}\big{)}\setminus\\{0\\}$
so that

$\displaystyle(V,W,\eta)=\left({\mathcal{K}}_{p},M\oplus{\mathcal{K}}_{p-1}\oplus
M^{-1},\left(\begin{matrix}0\\\ \eta_{0}\\\
\eta_{\mu}\end{matrix}\right)\colon V\to W\otimes K\right),$

where
$\eta_{\mu}=\left(\begin{matrix}0&\cdots&0&\mu\end{matrix}\right)\colon{\mathcal{K}}_{p}\to
M^{-1}K$.

To show that each of the above local minima types defines a connected
component of moduli space ${\mathcal{M}}({\mathsf{SO}}(p,q))$, we define a map
$\Theta_{p,q}$ from a parameter space into ${\mathcal{M}}({\mathsf{SO}}(p,q))$
so that

  1. 1)

$\Theta_{p,q}$ is a homeomorphism onto its image,

  2. 2)

the image of $\Theta_{p,q}$ is open and closed,

  3. 3)

each component of the image of $\Theta_{p,q}$ contains exactly one connected
component of the local minima with $\eta\neq 0$ from Theorem 7.4.

The connected components of the local minima of type 2 in the above theorem
are determined by the number of components of polystable
${\mathsf{O}}(q-p+1,{\mathbb{C}})$-bundles, that is, the components of
${\mathcal{M}}({\mathsf{O}}(q-p+1))$. For $q>p+1$ the group
${\mathsf{O}}(q-p+1)$ is simple thus, by Theorem 1.6

$\displaystyle|\pi_{0}({\mathcal{M}}({\mathsf{O}}(q-p+1))|=|{\mathcal{B}}_{{\mathsf{O}}(q-p+1)}(S)|=2^{2g+1}.$

When $q=p+1$, we have
${\mathcal{M}}({\mathsf{O}}(q-p+1))={\mathcal{M}}({\mathsf{O}}(2))$ 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 gives
$2^{2g+1}-1+p(2g-2)$ connected components of local minima in
${\mathcal{M}}({\mathsf{SO}}(p,p+1))$ with $\eta\neq 0$. Finally, when $q=p$
we have $q-p+1=1$ and thus there are $2^{2g+1}$ components of local minima
with $\eta\neq 0$. In this case all such minima define Hitchin components.

Combined with the $2^{2g+1}$-components of
${\mathcal{M}}({\mathsf{S}}({\mathsf{O}}(p)\times{\mathsf{O}}(q))$, for
$2<p\leq q$ the following theorem of [2] establishes the component count of
${\mathcal{M}}({\mathsf{SO}}(p,q))$.

######  Theorem 7.5.

For $2<p\leq q$, we have

$\displaystyle|\pi_{0}({\mathcal{M}}({\mathsf{SO}}(p,q)))|=2^{2g+1}+\begin{cases}2^{2g+1}&q=p,\\\
2^{2g+1}-1+p(2g-2)&q=p+1,\\\ 2^{2g+1}&\text{else}.\end{cases}$

To sketch the idea of the proof of the above theorem, we need to slightly
generalize our notion of Higgs bundles.

######  Definition 7.6.

A $K^{p}$-twisted ${\mathsf{G}}$-Higgs bundle is a pair
$({\mathcal{P}},\varphi)$ where

  * •

${\mathcal{P}}\to X$ is a holomorphic ${\mathsf{H}}^{\mathbb{C}}$-bundle,

  * •

$\varphi$ is a holomorphic section of the associated bundle
${\mathcal{P}}\big{[}{\mathfrak{m}}^{\mathbb{C}}\big{]}\otimes K^{p}$.

The notions of stability for $K^{p}$-twisted Higgs bundles are defined
similarly 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 ${\mathsf{G}}$-Higgs bundles and the resulting moduli space by

$\displaystyle{\mathcal{H}}^{\rm
ps}_{K^{p}}({\mathsf{G}})\qquad\text{and}\qquad{\mathcal{M}}_{K^{p}}({\mathsf{G}})={\mathcal{H}}^{\rm
ps}_{K^{p}}({\mathsf{G}})/{\mathcal{G}}_{{\mathsf{H}}^{\mathbb{C}}}.$

Recall from Section 4 that an ${\mathsf{SO}}(1,n)$-Higgs bundle is given by a
triple $(V,W,\eta)=(\Lambda^{n}W_{0}$, $W_{0},\eta)$, where $\eta\in
H^{0}(\Lambda^{q}W_{0}\otimes W_{0}\otimes K)$. The map $\eta$ can be
interpreted as a holomorphic bundle map $\eta\colon\Lambda^{q}W_{0}\to
W_{0}\otimes K$. Similarly, a $K^{p}$-twisted ${\mathsf{SO}}(1,n)$-Higgs
bundle is given by a triple $(\Lambda^{n}W_{0},W_{0},\eta_{p})$ where
$\eta_{p}\in H^{0}(\Lambda^{q}W_{0}\otimes W_{0}\otimes K^{p})$, which we may
interpret at as holomorphic bundle map $\eta_{p}\colon K^{1-p}\to W_{0}\otimes
K$.

Recall the definition of the map
$\widehat{\Psi}\colon\bigoplus\limits_{j=1}^{p}H^{0}\big{(}K^{2j}\big{)}\to{\mathcal{H}}({\mathsf{SO}}(p,p+1))$
from (5.3). Taking an transpose of this map defines the
${\mathsf{SO}}(p+1,p)$-Hitchin component. For our applications we need the map
for defining the ${\mathsf{SO}}(p,p-1)$-Hitchin component. We call this map
$\widehat{\Psi}\colon\bigoplus\limits_{j=1}^{p-1}H^{0}\big{(}K^{2j}\big{)}\to{\mathcal{H}}({\mathsf{SO}}(p,p-1))$
as well, explicitly it is given by

$\displaystyle\widehat{\Psi}(q_{2},\dots,q_{2p-2})=\left({\mathcal{K}}_{p},Q_{p},{\mathcal{K}}_{p-1},Q_{p-1},\left(\begin{matrix}1&q_{2}&\dots&q_{2p-2}\\\
&\ddots&\ddots&\vdots\\\
&&1&q_{2}\end{matrix}\right)\colon{\mathcal{K}}_{p}\to{\mathcal{K}}_{p-1}\otimes
K\right).$

Consider the map

$\displaystyle\widehat{\Theta}_{p,q}\colon\ {\mathcal{H}}^{\rm
ps}_{K^{p}}({\mathsf{SO}}(1,q-p+q))\times\bigoplus\limits_{j=1}^{p-1}H^{0}\big{(}K^{2j}\big{)}\longrightarrow{\mathcal{H}}({\mathsf{SO}}(p,q))$
(7.1)

defined by

$\displaystyle\widehat{\Theta}_{p,q}(W_{0},\eta_{p},q_{2},\dots,q_{2p-2})=\big{(}I\otimes{\mathcal{K}}_{p},(I\otimes{\mathcal{K}}_{p-1})\oplus
W_{0},\big{(}\begin{matrix}\widehat{\Psi}(q_{2},\dots,q_{2p-2})&\hat{\eta}_{p}\end{matrix}\big{)}\big{)},$

where $I=\Lambda^{q-p+1}W_{0}$ and

$\displaystyle\hat{\eta}_{p}=\left(\begin{matrix}0&\cdots&0&\eta_{p}\end{matrix}\right)\colon\
I\otimes\big{(}K^{p-1}\oplus K^{p-3}\oplus\cdots\oplus K^{3-p}\oplus
K^{1-p}\big{)}\to W_{0}\otimes K.$

######  Remark 7.7.

Note that $\Lambda^{p}(I\otimes{\mathcal{K}}_{p})=I^{p}$ and
$\Lambda^{q}\big{(}\big{(}I\otimes K^{p-1}\big{)}\oplus W_{0}\big{)}=I^{p}$,
so this indeed defines an ${\mathsf{SO}}(p,q)$-Higgs bundle.

######  Remark 7.8.

Note also that for $W_{0}$ a polystable orthogonal bundle of rank $q-p+1$, we
can take $\eta_{p}=0$. In this case the image of
$\widehat{\Theta}_{p,q}(W_{0},0,0,\dots,0)$ is given by

$\displaystyle\big{(}\Lambda^{q-p+1}(W_{0})\otimes{\mathcal{K}}_{p},\big{(}\Lambda^{q-p+1}(W_{0})\otimes{\mathcal{K}}_{p-1}\big{)}\oplus