# A Gentle Introduction to the Non-Abelian Hodge Correspondence

###### Abstract

We aim at giving a pedagogical introduction to the non-abelian Hodge
correspondence, a bridge between algebra, geometric structures and complex
geometry. The correspondence links representations of a fundamental group, the
character variety, to the theory of holomorphic bundles.

We focus on motivations, key ideas, links between the concepts and
applications. Among others we discuss the Riemann–Hilbert correspondence,
Goldman’s symplectic structure via the Atiyah–Bott reduction, the
Narasimhan–Seshadri theorem, Higgs bundles, harmonic bundles and hyperkähler
manifolds.

Alexander Thomas111Max-Planck Institute Bonn<EMAIL_ADDRESS>Basic
Research Community for Physics BRCP

###### Contents

  1. 1 Introduction
  2. 2 Starting point: Character varieties
  3. 3 Bundles and connections
  4. 4 A crash course in symplectic geometry
    1. 4.1 Symplectic structures
    2. 4.2 Hamiltonian reduction
  5. 5 Atiyah–Bott reduction
  6. 6 GIT quotients and stability conditions
  7. 7 Stable bundles and the Narasimhan–Seshadri theorem
  8. 8 Higgs bundles and the non-abelian Hodge correspondence
  9. 9 The proof strategy: Harmonic bundles
  10. 10 Hyperkähler geometry
  11. 11 Application: Hitchin components
  12. 12 Generalizations and research directions

##  1 Introduction

This paper is an introduction to the non-abelian Hodge correspondence,
focusing on the key principles, motivations and links to other areas.
Technical details and computations are mostly referred to references, while
the ideas and concepts are presented. No new results are presented, but the
way of presentation is original.

We base our exposition on geometric and sometimes on physical intuition, with
some emphasis on symplectic geometry. The paper should be useful to get a
first glimpse on the topic or to step back from technical details to see the
clear conceptual picture. It should be accessible to a broad audience, in
particular to master students in mathematics with interest in mathematical
physics. Some knowledge of differential geometry, Riemann surfaces and Lie
groups are welcome, but not a must have.

The non-abelian Hodge correspondence is the huge achievement due to many
mathematicians, above all Nigel Hitchin [Hit87], Carlos Simpson [Sim88], Kevin
Corlette [Cor88] and Simon Donaldson [Don87]. It links three worlds together:
the topological and algebraic world of representations of fundamental groups,
the differential geometry world of connections and the complex geometry world
of holomorphic bundles. It can be interpreted as a diffeomorphism between
moduli spaces, which play an important role in theoretical physics. The
correspondence is an incarnation of a very strong structure on these moduli
spaces, a hyperkähler structure.

The paper is structured as follows:

  * •

_Introduction and motivations:_ Section 2 presents the multiple facets of the
character variety. In Section 3 we review bundles and connections and describe
the Riemann–Hilbert correspondence between flat connections and the character
variety.

  * •

_The toolbox:_ an important topic is how to define quotients. One way is the
hamiltonian reduction, which is presented together with a crash course in
symplectic geometry in Section 4. As an application, Section 5 constructs the
Goldman symplectic structure on the character variety. Another way uses
stability conditions exposed in Section 6 and applied to holomorphic bundles
in Section 7.

  * •

_The core:_ Section 8 introduces the notion of a Higgs bundle and states the
non-abelian Hodge correspondence. The main ideas for the proof are given in
Section 9 via harmonic map theory and the Hitchin–Simpson theorem. Section 10
gives a deeper understanding coming from hyperkähler geometry.

  * •

_Applications and generalizations:_ In Section 11 we construct Hitchin
components as an application of the non-abelian Hodge correspondence. The
final section 12 exposes generalizations and research directions.

Some paragraphs are marked with an asterisk. They are more advanced and not
necessary for the other (non-advanced) topics. I used the material of this
paper for a master lecture at the University of Bonn in the summer semester
2022.

#### Acknowledgments.

I warmly thank Vladimir Fock, from whom I learned most of the ideas and
concepts. I’m also grateful to Florent Schaffhauser and Georgios Kydonakis for
helpful discussions. I gratefully acknowledge support from the Max-Planck
Institute for Mathematics in Bonn.

##  2 Starting point: Character varieties

Character varieties are at the crossroad between many fields: representation
theory, geometry, theoretical physics, dynamics, number theory,… They form a
playground where techniques from various fields can be applied, and at the
same time they have deep connections to multiple research streams.

#### Motivation.

Consider a group $\Gamma$ which we want to understand. The natural way to
understand a group is to let it act. On the other hand, the theory we
understand the best in mathematics is probably Linear Algebra. So we can try
to let $\Gamma$ act on a vector space $V$, say of dimension $n$ defined over
$\mathbb{C}$. In other words: we try to find matrices which mimic $\Gamma$.
This is the basic idea of representation theory. Hence we consider
$\operatorname{Hom}(\Gamma,\operatorname{GL}_{n}(\mathbb{C}))$.

As usual, we are not really interested in the set of all representations, but
only in the isomorphism classes. For a linear action, any two representations
into $\operatorname{GL}_{n}(\mathbb{C})$ are isomorphic whenever we can obtain
one from the other by a simple base change in $V$. Hence, the space of
isomorphism classes is

$\operatorname{Hom}(\Gamma,\operatorname{GL}_{n}(\mathbb{C}))/\operatorname{GL}_{n}(\mathbb{C}):=\operatorname{Rep}(\Gamma,\operatorname{GL}_{n}(\mathbb{C}))$

where we quotient by the conjugation action. Note that in order to get a nice
topological space (separable), we restrict to representations
$\Gamma\to\operatorname{GL}_{n}(\mathbb{C})$ which are completely reducible.
We come to this in Section 6.

Now, we can consider vector spaces with more structure, for example equipped
with a hermitian product, a symplectic structure… Then we ask for
representations preserving this structure. Thus, we analyze the space
$\operatorname{Rep}(\Gamma,G)$ for some Lie group $G$, typically a subgroup of
$\operatorname{GL}_{n}(\mathbb{C})$.

In the specific case when $\Gamma$ is the fundamental group of a manifold $M$,
we call $\operatorname{Rep}(\pi_{1}M,G)$ the character variety of $M$ and $G$.

The case if a fundamental group of a manifold is interesting because the
character variety has several geometric meanings: it describes
$(G,X)$-structures (geometric structures where the manifold $M$ is locally
modeled on some space $X$ with transition functions in $G$) and flat
$G$-connections. The link to flat connections is explained in Section 3 via
the Riemann–Hilbert correspondence.

For $M=\Sigma$ a surface, the character variety appears in physics, especially
in string theory. It describes geometric structures on the world sheet, the
surface traced out by a string in time. In mathematics, representations of
surface groups into real groups ($G=\operatorname{SL}_{n}(\mathbb{R})$ for
example) have interesting dynamical properties. This goes under the name of
_higher Teichmüller theory_ (see Section 11 for more details).

#### Many viewpoints.

Since the character variety sits at the intersection of many mathematical
areas, it allows many equivalent descriptions. The main goal of the paper is
to understand all these incarnations.

Topological interpretation: The character variety by its very definition is a
space of representations of $\pi_{1}\Sigma$ which does only depend on the
topology of $\Sigma$.

Smooth interpretation: The character variety is also described as the space of
all flat connections (on a trivial bundle over $\Sigma$) modulo gauge
equivalence. This description is called the _Riemann–Hilbert correspondence_ ,
which we will see at the end of Section 3.

Holomorphic interpretation: The character variety can be described by
holomorphic objects on a Riemann surface $S$ whose underlying smooth surface
is $\Sigma$. These objects are _stable Higgs bundles_ , which we introduce in
Section 8. This description is the content of the _non-abelian Hodge
correspondence_.

##  3 Bundles and connections

To study the character variety, its link to the space of flat connections, the
“smooth interpretation”, is fundamental. We motivate and recall the basic
concepts of differential geometry and show the link to character varieties. To
deepen the subject, I highly recommend the outstanding book of Baez and
Muniain [BM94].

#### Dictionary.

Most of modern physical theories, like Maxwell’s theory of electromagnetism or
the standard model of elementary particles, are using concepts of differential
geometry such as bundles and connections. Why are these concepts so
fundamental?

To me, the reason comes from what I like to call the _Global-to-Local
heuristics_ , the idea that our observations describe only local properties of
our universe. When we try to describe natural phenomena which surround us, we
get the impression that we live in a Euclidean space. Indeed, the forces
applied to a same point add up in a vectorial way, and even on a larger scale
like our solar system, nature seems to be well described by Newtonian
mechanics in which the universe is an affine space modeled on
$\mathbb{R}^{3}$. But our perceptions and observations are always limited in
space and time. Nothing prevents Nature to behave only locally like an affine
space, but to bend and twist on a global scale.

A good illustration is the surface of our Earth, which locally is well
described by a flat part of the plane $\mathbb{R}^{2}$ (although there are
mountains and valleys, on average it seems to be flat). Globally of course,
the Earth is a ball, since it bends far away from the observation scales of
our daily life.

The “Global-to-Local heuristics” can be summarized in the following
dictionary:

Linear algebra / Classical mechanics | Riemannian geometry / General relativity
---|---
Absolute space $\mathbb{R}^{n}$ | Manifold
Functions | Sections of a bundle
Differential equations | Connections

To the global concept of absolute space corresponds a mathematical concept
which looks only locally like an open subset of $\mathbb{R}^{n}$: the concept
of a manifold.

In Newtonian mechanics, there are ingredients other than space: physical
quantities are described by functions (temperature, speed, electromagnetic
field, …) whose evolution is described by differential equations. _What is the
local concept of a function?_ I.e. what is the mathematical concept which
looks locally like a function, but which might bend on large scales?

The reader who has never thought about that question should take a second to
think about it. The answer is a bit tricky: a “generalized function” is a
section of a fiber bundle! Consider a function $f:M\to F$ where $M$ is a
manifold and $F$ is some target space ($F=\mathbb{R}$ for the temperature for
example). An equivalent way to describe $f$ is through its graph
$\mathrm{gr}(f)$ which is the subset of $M\times F$ given by all points of the
form $(x,f(x))$. We can also say that a function is the choice of an inverse
to the projection $M\times F\to M$.

To generalize the notion of a function, we first need a space $E$ which
locally looks like $U\times F$ where $U\subset M$ is a small open subset of
$M$. This is precisely the notion of a fiber bundle on $M$ with fiber $F$.
When $F$ is a vector space, we speak of a vector bundle. To be precise, a
vector bundle with fiber $V$ is a manifold $E$ together with a surjective map
$p:E\to M$ such that there exists an atlas on $M$ with charts $U_{i}$ such
that $p^{-1}(U_{i})=U_{i}\times V$ and transition maps which are linear and
linear, i.e. in $\mathrm{GL}(V)$. You can imagine to built up $E$ by taking
$U_{i}\times V$ and identifying them on $(U_{i}\cap U_{j})\times V$ using the
transition functions of $M$ and a fiberwise element of $\mathrm{GL}(V)$.

A “generalized function” is a section of $p$, i.e. a map $s:M\to E$ such that
$p\circ s=\mathrm{id}_{M}$. The space of all sections is denoted by
$\Gamma(E)$. For the trivial bundle $E=M\times F$, a section is nothing but a
function on $M$ with values in $F$. For $E=TM$ the tangent bundle, a section
is a vector field.

$D$$s$$M$ Figure 3.1: Illustration of a bundle and a connection

######  Remark.

The “Global-to-Local heuristics” should have a counterpart to the microscopic
level: what we observe in our daily life is only at a _mesoscopic_ scale.
Nothing prevents Nature to be different on an atomic scale. The search for a
mathematical concept which gives a Euclidean structure at mesoscopic scales
(and a manifold on the global scale) is still open. Candidates exist, for
example discrete models or non-commutative geometry. This concept should play
a paramount role in a hypothetical future theory of quantum gravity.

#### Connections.

What is the local concept of a linear differential equation? How to define the
derivative of a section of a vector bundle? The answer is given by the notion
of a connection.

In a first approximation, we can say that a connection is a generalization of
a directional derivative, i.e. it allows to derive a section $s$ in a
direction given by a vector field $X$. We write the result as $\nabla_{X}(s)$,
which is again a section.

In a second approximation, we can say that a connection is a _matrix-valued
1-form_. Locally it can be written $d+A$ where
$A\in\Omega^{1}(M,\mathfrak{gl}_{n}(\mathbb{C}))$. To see this, fix basis
vectors $e^{i}(x)$ in each fiber, varying smoothly in $x$. Then a section $s$
can be written $s=\sum_{i}s_{i}e^{i}(x)$. Locally, we can differentiate $s$ in
the direction $X=\sum_{j}X_{j}\frac{\partial}{\partial x^{j}}$ in the usual
way:

$d_{X}(s)=\sum_{i}d_{X}(s_{i})e^{i}(x)+\sum_{i,j}s_{i}X_{j}d_{j}(e^{i}(x))$

where $d_{j}$ denotes the derivative in direction $x^{j}$. Since $(e^{i}(x))$
is a basis, there are matrices $A_{j}(x)$ defined by

$d_{j}(e^{i}(x))=A_{j}(x)e^{i}(x).$

Putting $A=\sum_{j}A_{j}dx^{j}$, we can write
$d_{X}(s)=(d+\sum_{i}A_{i}dx^{i})_{X}(s)$, so the connection is $d+A$. Note
that the matrix-valued 1-form $A$ appears from the fact that the basis
$(e^{i}(x))$ depends on $x$.

To be precise, a connection (or covariant derivative) is a map
$D:\Gamma(E)\times\Gamma(TM)\to\Gamma(E)$, where $E$ is a vector bundle, which
satisfies $\forall s,t\in\Gamma(E),f,g\in\mathcal{C}^{\infty}(M)$ and
$X,Y\in\Gamma(TM)$:

$\displaystyle(i)$ Linearity for sections: $\displaystyle
D_{X}(s+t)=D_{X}(s)+D_{X}(t),$ $\displaystyle(ii)$ Linearity for vector
fields: $\displaystyle D_{fX+gY}(s)=fD_{X}(s)+gD_{Y}(s),$ $\displaystyle(iii)$
Leibniz’ rule: $\displaystyle D_{X}(fs)=df(X)\,s+fD_{X}(s).$

The name “connection” comes from the fact that a connection allows to connect
different fibers of $p:E\to M$. A section $s$ is said to be flat if
$D_{X}(s)=0$ for all $X\in\Gamma(TM)$.

To see the link to differential equations, consider first the case of a
1-dimensional manifold, for example $M=\mathbb{R}$. A linear differential
equation is given by

$(d^{n}+t_{1}(x)d^{n-1}+t_{2}(x)d^{n-2}+...t_{n}(x))\psi(x)=0$

where $d=\frac{d}{dx}$.

This is equivalent to a matrix-valued differential equation of order 1:
$(d+A)s=0$ where

$A=\begin{pmatrix}&-1&&\\\ &&\ddots&\\\ &&&-1\\\
t_{n}&t_{n-1}&\cdots&t_{1}\end{pmatrix}\;\text{ and
}\;s=\begin{pmatrix}\psi\\\ d\psi\\\ \vdots\\\ d^{n-1}\psi\end{pmatrix}.$

In dimension 2, you get a system of two differential equations. You can write
it as

$\left\\{\begin{array}[]{cl}D_{1}\psi(x,y)&=0\\\
D_{2}\psi(x,y)&=0\end{array}\right.$

As in dimension 1, there is a standard form:

$\left\\{\begin{array}[]{cl}\partial_{x}\Psi(x,y)&=A_{x}(x,y)\Psi(x,y)\\\
\partial_{y}\Psi(x,y)&=A_{y}(x,y)\Psi(x,y)\end{array}\right.$

where $\Psi$ is a vector whose entries are suitable derivatives of $\psi$.

A natural question then arises: under which conditions there is a full set of
solutions?

A necessary condition, which turns out to be sufficient, is that
$\partial_{x}\partial_{y}\Psi=\partial_{y}\partial_{x}\Psi$ which gives

$\partial_{x}A_{y}-\partial_{y}A_{x}+[A_{x},A_{y}]=0.$ (3.1)

This expression is the _curvature of the connection $d+A$_, as defined below.
In terms of the differential operators $D_{1}$ and $D_{2}$, if the curvature
does not vanish, then you can reduce the system to a smaller one222Explicitly
to $D_{1}\psi=0$ and $([D_{1},D_{2}]\mod\langle D_{1},D_{2}\rangle)\psi=0$.

######  Exercise 3.1.

Work through an explicit example. For instance
$D_{1}=\partial_{x}^{2}-y\partial_{x}$ and $D_{2}=\partial_{y}^{2}-x$.

The curvature of a connection $D$ measures the failure of the covariant
derivatives $D_{i}$ to commute. It is a 2-tensor $F_{D}$ given by

$F_{D}(X,Y)=[D_{X},D_{Y}]-D_{[X,Y]}\;\;\text{ for }X,Y\in\Gamma(TM).$

The fact that it is a tensor means that $F_{D}(X,Y)(fs)=fF_{D}(X,Y)(s)$ for
$f\in\mathcal{C}^{\infty}(M)$. If the curvature vanishes, we call the
connection flat.

Locally we can write $D=d+A$ and take for $X$ and $Y$ coordinate vector
fields. Then we get

$F_{D}(\partial_{i},\partial_{j})=[D_{i},D_{j}]-D_{[\partial_{i},\partial_{j}]}=[D_{i},D_{j}]=[\partial_{i}+A_{i},\partial_{j}+A_{j}]=\partial_{i}A_{j}-\partial_{j}A_{i}+[A_{i},A_{j}].$

We get Equation (3.1). This can be written concisely as

$F(A)=dA+A\wedge A$ (3.2)

which is a matrix-valued 2-form.

#### Gauge transformations and structure group.

The natural symmetry group acting on bundles are bundle automorphisms, also
called gauge transformations.

When working with a vector bundle locally, we often fix a basis
$(e_{1}(x),...,e_{r}(x))$ in each fiber varying smoothly with $x$. Changing
this basis by a matrix $A(x)$ is a gauge transformation, or just gauge in
short. It is a smooth map from $M$ to $\mathrm{GL}(V)$ where $V$ is the fiber
of the vector bundle $E$. Mathematically a gauge transformation is nothing but
a _bundle automorphism_ , i.e. an invertible map $E\to E$ which preserves the
fibers where it acts linearly.

How do connections and the curvature behave under gauge transformations? Since
a gauge transformation acts fiberwise, we can work locally and write $D=d+A$.

######  Proposition 3.2.

The gauge action on connections is given by:

$g.A=gAg^{-1}+gd\left(g^{-1}\right).$

On the curvature tensor we get:

$g.F(A)=gF(A)g^{-1}.$

The first point simply comes from the conjugation action of a gauge on a
connection:

$g(d+A)g^{-1}=d+gAg^{-1}+gd\left(g^{-1}\right).$

Hence the action of $g$ on $A$ is given by
$g.A=gAg^{-1}+gd\left(g^{-1}\right)$ which is an affine transformation.
Indeed, the space of all connections $\mathcal{A}(E)$ is an affine space,
whose underlying vector space is $\Omega^{1}(M,\mathfrak{gl}(V))$. This means
that the difference of two connections $D-D_{0}$ is a matrix-valued 1-form
$A$. Hence, while we can locally write $D=d+A$, we can always globally write

$D=D_{0}+A.$

######  Exercise 3.3.

Check the gauge transformation of the curvature tensor using $F(A)=dA+A\wedge
A$.

In general, the transition functions of a bundle are any elements of
$\mathrm{GL}(V)$, but it might happen that they all are in some subgroup
$G\subset\mathrm{GL}(V)$. In that case, we say that $V$ is a bundle with
structure group $G$. For example, if our vector space $V$ is equipped with a
scalar product, we might require the transition functions to respect this
structure, i.e. $G=\mathrm{O}(V)$. Then, we have a well-defined scalar product
in each fiber of $E$. The structure group allows to put more structure on the
bundle, hence its name. There is also a more abstract notion, that of a
_principal bundle_ , which we will not treat in these lectures.

For a more detailed treatment of bundles and connections and their links to
physics, we warmly recommend the book of John Baez and Javier Munian [BM94],
in particular Chapter 2.

#### Parallel transport.

We have already said that a connection allows to connect different fibers. We
make this precise.

Consider two points $x,y\in M$ and a path $\gamma:[0,1]\to M$ connecting them,
i.e. $\gamma(0)=x,\gamma(1)=y$. Along $\gamma$ there is a unique flat section
$s_{p}$ with given starting point $p\in\pi^{-1}(x)$, i.e. $s_{p}(x)=p$.

The map $p\mapsto s_{p}(y)$ is called parallel transport along $\gamma$. It is
a linear map (element in the structure group $G$). In general, the parallel
transport depends on the path. _For a flat connection, the parallel transport
only depends on the homotopy class of the path_ (i.e. you can change $\gamma$
by isotopies).

For flat connections, the parallel transport along loops is called the
monodromy. The monodromy is an element of the character variety
$\operatorname{Rep}(\pi_{1}M,G)$. Indeed, for loops based at $x$, we get a
point in $\operatorname{Hom}(\pi_{1}(M,x),G)$ and changing $x$ conjugates the
monodromy.

#### Riemann–Hilbert correspondence.

The first fundamental result about character varieties is that they describe
the moduli space of flat connections, i.e. the space of flat connections
modulo gauge equivalence. This gives a link between the topological and the
smooth interpretation of character varieties.

We have just seen how to associate a point in the character variety to a flat
connection, simply by considering its monodromy. This map is a diffeomorphism:

######  Theorem 3.4 (Riemann–Hilbert correspondence).

The character variety is the space of flat $G$-connections on the trivial
bundle on $M$ modulo gauge:

$\boxed{\operatorname{Rep}(\pi_{1}M,G)\cong\\{\text{flat
}G-\text{connections}\\}/\text{gauge}.}$

The idea of the proof is to construct explicitly the inverse map: given a
representation $\rho:\pi_{1}M\to G$, we can consider the diagonal action of
$\pi_{1}M$ on $\widetilde{M}\times G$, where $\widetilde{M}$ denotes the
universal cover of $M$ on which $\pi_{1}M$ acts by deck transformations. The
quotient

$E_{\rho}=(\widetilde{M}\times G)/\pi_{1}M$

is a (principal) $G$-bundle over $M$ and the trivial connection $d$ on
$\widetilde{M}\times G$ descends to a flat $G$-connection. Finally, changing
$\rho$ by conjugation corresponds to changing the connection by a gauge
transformation.

##  4 A crash course in symplectic geometry

Grown out of modern treatments of classical mechanics in the early 19th
century, symplectic geometry is a very active mathematical domain today. We
give motivations and introduce basic concepts, in particular the symplectic
quotient called Hamiltonian reduction. This will be used for the study of
character varieties in the next section.

To deepen the subject, I warmly recommend the book of McDuff and Salamon
[MS17] (especially Sections 1, 3 and 5), and the book of Kirillov [Kir04] (in
particular Chapter 1, Section 4 and Appendix II, Section 3). For the physical
interpretation, I advocate Arnold’s classical book [Arn13].

###  4.1 Symplectic structures

The physical motivations for symplectic structures are sometimes a bit
obscure. We will describe how they naturally arise.

Consider a physical system, for example the motion of a particle which is
restricted to stay on some surface $S$. The first important idea is to
consider the space of all possible states of our system, which is called the
phase space $M$. In our example the system is uniquely given by knowing the
position and the momentum of the particle, so the phase space is the cotangent
bundle333It turns out that while the velocity lives in the tangent bundle, the
momentum lives in the cotangent bundle. $M=T^{*}S$. The picture you might have
in mind is the following: we replace a complicated system by one point in a
complicated space, which describes all possible states. Then the evolution of
the system is nothing but a path in the phase space (see left of Figure 4.1).

Figure 4.1: Phase space with volume preserving evolution

Now, in practice we never precisely know at which point of $M$ our system is,
since for all quantities, we have some uncertainty in the measurement. What we
know is that our system is somewhere in an open set $U\subset M$. An important
observation is that when the system is isolated (i.e. does not exchange energy
or information with the environment), then the evolution of $U$ through the
time flow is _volume-preserving_. This translates the fact that _we will not
gain (or lose) information about the system by simply waiting some time!_

It turns out that it is even better: the physical quantities come in pairs, to
each quantity we can associate a so-called _conjugated_ quantity. For example,
the conjugate of the position in a direction is the momentum in that direction
and vice versa. The time evolution does not only preserve the total volume of
$U$ in the phase space, but also the (induced) volume of all 2-dimensional
slices given by a pair of conjugated quantities. This means that we may gain
information on the position, but we will lose information on the momentum.

This property of “preserving volumes of special 2-dimensional slices” is
captured by the fact that _the phase space has a symplectic structure and the
time evolution preserves that structure_.

######  Definition 4.1.

A symplectic structure on a manifold $M$ is a two-form $\omega$ which is
closed and non-degenerate.

The notion of conjugated variables arise by some linear algebra of two-forms:

######  Exercise 4.2.

Let $V$ be a vector space equipped with a antisymmetric bilinear form $\omega$
which is non-degenerate (i.e. $\omega(x,y)=0\;\forall\,y\Rightarrow x=0$).
Show that the dimension of $V$ is even and that $V$ admits a basis
$(e_{1},...,e_{2n})$ such that
$\omega(e_{2i-1},e_{2i})=1=-\omega(e_{2i},e_{2i-1})$ for all $i=1,...,n$ and
$\omega(e_{i},e_{j})=0$ for all other $i,j$.

The standard examples arise as the simplest phase spaces of physical systems.

  * •

A free particle moving on $\mathbb{R}$ gives $M=T^{*}\mathbb{R}$. The
symplectic form is nothing but the area form, given by $\omega_{0}=dp\wedge
dx$ (where $(p,x)$ are coordinates on $T^{*}\mathbb{R}\cong\mathbb{R}^{2}$).

  * •

More generally, for a free particle moving in $\mathbb{R}^{n}$, its phase
space is $\mathbb{R}^{2n}=T^{*}\mathbb{R}^{n}$ (position and momentum) with
$\omega_{0}=\sum_{i}dp_{i}\wedge dx_{i}$, called the _standard symplectic
structure_.

  * •

If the particle is constraint to stay on some manifold $N$, the phase space
becomes the cotangent bundle $T^{*}N$. Again we can write

$\omega=\sum_{i}dp_{i}\wedge dx_{i}.$ (4.1)

This needs some explanation. We have

$T_{(p,x)}T^{*}N\cong T^{*}_{x}N\oplus T_{x}N.$

In a coordinate independent manner, we have
$\omega((\varphi,X),(\varphi^{\prime},X^{\prime}))=\varphi(X^{\prime})-\varphi^{\prime}(X)$
which can be identified with Equation (4.1). Note that $\omega=d\lambda$ is
exact where $\lambda=\sum_{i}p_{i}dx_{i}$ is called the _Liouville form_.

An important result in the local theory of symplectic manifolds is the
following:

######  Theorem 4.3 (Darboux theorem).

On any symplectic manifold $(M,\omega)$ there exists local coordinates
$(p_{i},x_{i})$ such that $\omega=\sum_{i}dp_{i}\wedge dx_{i}$.

This means that there is no local invariant in symplectic geometry. Note that
this is not the case for Riemannian geometry where the curvature is a local
invariant. Coordinates with the property of the theorem are called _Darboux
coordinates_.

######  Exercise 4.4.

Show that a non-degenerate 2-form on a compact manifold without boundary
cannot be exact. Deduce that there is no symplectic embedding of a symplectic
manifold into $(\mathbb{R}^{2n},\omega_{0})$. Which spheres admit symplectic
structures?

#### Symplectic gradient.

The definition of a symplectic structure is very similar to that of a
Riemannian structure, the main difference being that the 2-tensor is symmetric
for the Riemannian structure, and anti-symmetric for a symplectic structure.
One of the key notions in Riemannian geometry, the gradient of a function, is
still available in symplectic geometry: the symplectic gradient of a function
$f$ is a vector field $\mathrm{sgrad}(f)$ such that

$\omega(\mathrm{sgrad}(f),X)=-df(X)$

for all $X\in\Gamma(TM)$. This is exactly the same definition as for the
gradient (we need only non-degeneracy of $\omega$ to get a well-defined
notion).

So why is the geometry of a phase space symplectic and not Riemannian? One
reason is that a symplectic structure gives naturally the notion of conjugated
variables. Another reason is that it allows to treat the important issue of
_conserved quantities_. While the usual gradient of a function $f$ points in
the direction of biggest change of $f$, the symplectic gradient points in the
direction where $f$ stays constant (the level set). This allows to refine our
picture from above: the physical evolution of a system is described by a
vector field on the phase space which is the symplectic gradient of a special
function, called the Hamiltonian of the system, which is conserved in time.
For isolated systems, _the Hamiltonian is the total energy_.

######  Example 4.5.

Consider $M=\mathbb{S}^{2}\subset\mathbb{R}^{3}$ the sphere with symplectic
form being the area form, and the height function
$h:\mathbb{S}^{2}\to\mathbb{R}$ given by the $z$-coordinate. Then the
hamiltonian flow is the rotation around the $z$-axis. This of course preserves
the level-sets.

Figure 4.2: Height function generates rotation for a sphere

######  Proposition 4.6.

The 1-parameter group of diffeomorphisms of $M$, obtained by integrating the
symplectic gradient of a function, preserve the symplectic structure of $M$.

The proof uses Cartan’s magical formula and illustrates the importance of
$\omega$ being closed.

###### Proof.

We want to show that $f_{t}^{*}\omega=\omega$ for all $t\in\mathbb{R}_{+}$,
where $f_{t}$ denotes the flow associated to a Hamiltonian $H$. For that, it
is sufficient to show that the derivative of $f_{t}^{*}\omega$ is zero. We
restrict attention to $t=0$, the other values are similar.

By definition of the Lie derivative $\mathcal{L}$, we have

$\frac{d}{dt}\bigg{\rvert}_{t=0}f_{t}^{*}\omega=\mathcal{L}_{X_{H}}\omega$

where $X_{H}$ denotes the symplectic gradient of $H$. Using Cartan’s magical
formula and writing $\iota_{X}$ for the inner product, we get

$\mathcal{L}_{X_{H}}\omega=(d\circ\iota_{X_{H}}+\iota_{X_{H}}\circ
d)\omega=d\left(\omega(X_{H},.)\right)+\iota_{X_{H}}d\omega.$

Since $\omega$ is closed, the second term vanishes. The first term vanishes as
well since by definition $\omega(X_{H},.)=-dH$ is exact. ∎

A diffeomorphism $f$ of $M$ preserving its symplectic structure (i.e.
$f^{*}\omega=\omega$) is called a symplectomorphism. The special case coming
from the time 1 flow of a symplectic gradient is called a Hamiltonian
diffeomorphism. Not all symplectomorphisms are Hamiltonian. This distinction
can be seen on the “infinitesimal” level: the Lie algebra of the
diffeomorphism group is the space of vector fields. A vector field $X$ is
called

  * •

symplectic if $\omega(X,.)$ is a closed 1-form,

  * •

Hamiltonian if $\omega(X,.)$ is exact.

A vector field associated to a symplectomorphism (resp. hamiltonian
diffeomorphism) is symplectic (resp. hamiltonian).

Note that there is no constraint on the Hamiltonian of a physical system: any
function in $\mathcal{C}^{\infty}(M)$ can in principle be used to generate
time evolution, and it is the physicists job to find out which function it is
for a given system. Jürgen Jost [Jos09] puts it in these words:

“The aim of physics is to write down the Hamiltonian of the universe.
The rest is mathematics.”

Hence, we can give two physical meanings to a function: as _observable
quantity_ or as _infinitesimal generator of a transformation_. This shows the
importance of the space of functions.

#### Poisson bracket.

The space of smooth functions $\mathcal{C}^{\infty}(M)$ comes with an extra
structure: a Poisson bracket. Concretely, the symplectic form $\omega$ allows
to associate to two functions $f,g\in\mathcal{C}^{\infty}(M)$ a function
denoted by $\\{f,g\\}$ defined by

$\\{f,g\\}=\omega(\mathrm{sgrad}(f),\mathrm{sgrad}(g)).$

Abstractly, a Poisson bracket is a bilinear map
$\mathcal{C}^{\infty}(M)\times\mathcal{C}^{\infty}(M)\to\mathcal{C}^{\infty}(M)$
which is symmetric, satisfies Leibniz’ rule $\\{fg,h\\}=f\\{g,h\\}+\\{f,h\\}g$
and the _Jacobi identity_

$\\{f,\\{g,h\\}\\}+\\{h,\\{f,g\\}\\}+\\{g,\\{h,f\\}\\}=0.$

Roughly speaking, a Poisson bracket is a Lie bracket and a derivation.

######  Exercise 4.7.

Show that the Jacobi identity of the Poisson bracket is equivalent to $\omega$
being closed.

In Darboux coordinates $(x_{i},p_{i})$, the Poisson bracket has a simple
expression:

$\\{.,.\\}=\sum_{i}\frac{\partial}{\partial
p_{i}}\wedge\frac{\partial}{\partial x_{i}}.$ (4.2)

This means that

$\\{f,g\\}=\sum_{i}\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial
x_{i}}-\frac{\partial f}{\partial x_{i}}\frac{\partial g}{\partial p_{i}}.$

The important point is to know the interpretation of the Poisson bracket:
_$\\{f,g\\}(x)$ is the change of $g$ at $x$ along the flow line generated by
the symplectic gradient of $f$_. In physical words: if $H$ is the Hamiltonian
of a physical system and $g$ some observable, then

$\boxed{\frac{dg}{dt}(x)=\\{H,g\\}}$ (4.3)

where $t$ is the parameter of the family of Hamiltonian diffeomorphisms
generated by $H$, and $\frac{dg}{dt}=dg(X_{H})$. In particular, the
Hamiltonian is preserved in time since for $g=H$ we get zero in Equation
(4.3). This is the well-known fact that the total energy of an isolated system
is conserved.

The strength of Equation (4.3) is that we can recover, in a uniform way, all
differential equations from classical systems. All exercises from physics
class in High School about determination of the equation of motion become easy
(at least systematic)!

######  Example 4.8.

Consider a point particle with mass $m$ moving on $\mathbb{R}$ in a potential
$V$ with gravity $g$. The phase space is $M=T^{*}\mathbb{R}$ with standard
symplectic structure. The total energy is $H=\frac{p^{2}}{2m}+mgV(x)$ (kinetic
+ potential energy). Therefore we get

$\dot{x}=\\{H,x\\}=\frac{\partial H}{\partial p}=p/m,$

which is nothing new, since $p=mv=m\dot{x}$. Then

$\ddot{x}=\\{H,\dot{x}\\}=\\{H,p/m\\}=-\frac{1}{m}\frac{\partial H}{\partial
x}=-gV^{\prime}(x)$

which is nothing but Newton’s law.

######  Example 4.9.

Consider a pendulum: a mass $m$ turning around a fixed point at distance
$\ell$ (see Figure 4.3). The phase space is $M=T^{*}\mathbb{S}^{1}$ with
Darboux coordinates $(p,\ell\theta)$. The total energy is
$H=\frac{p^{2}}{2m}-mg\ell\cos\theta$. Hence

$\dot{\theta}=\\{H,\theta\\}=\frac{1}{\ell}\\{H,x\\}=\frac{1}{\ell}\frac{\partial
H}{\partial p}=\frac{p}{m\ell},$

which is nothing new, since $p=mv=m\dot{x}=m\ell\dot{\theta}$. Then

$\ddot{\theta}=\\{H,\dot{\theta}\\}=\frac{1}{m\ell}\\{H,p\\}=-\frac{1}{m\ell^{2}}\frac{\partial
H}{\partial\theta}=-\frac{g}{\ell}\sin\theta$

which is the usual law for a pendulum.

Figure 4.3: Pendulum

A manifold $M$ with a Poisson bracket on its function space
$\mathcal{C}^{\infty}(M)$ is called a Poisson manifold. They are more general
than symplectic manifolds:

######  Example 4.10.

Using the cross product, we can construct a Poisson structure on
$\mathbb{R}^{3}$, without being symplectic (since the dimension is odd). The
Poisson bracket for linear functions (which can be represented by vectors) is
given by the cross product. This means that $\\{x,y\\}=z,\\{y,z\\}=x$ and
$\\{z,x\\}=y$.

Then we extend the bracket to all functions using the Leibniz rule. As a
bivector, we can write

$\\{.,.\\}=z\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial
y}+x\frac{\partial}{\partial y}\wedge\frac{\partial}{\partial
z}+y\frac{\partial}{\partial z}\wedge\frac{\partial}{\partial x}.$

The main thing to know about a Poisson manifold is that _it is the disjoint
union of symplectic manifolds_ (in a unique way), called symplectic leaves.
For $(\mathbb{R}^{3},\times)$ from above, the symplectic leaves are all
spheres with radius $r\geq 0$. Note in particular that the various symplectic
leaves might have different dimensions.

The following is a very important example. In some sense, it is the universal
$G$-Poisson manifold.

######  Example 4.11.

A dual Lie algebra $\mathfrak{g}^{*}$ is always Poisson444The dual is taken in
the sense of linear algebra: $\mathfrak{g}^{*}$ is the vector space of all
linear forms on $\mathfrak{g}$.. For linear functions, which can be identified
with $\mathfrak{g}^{**}\cong\mathfrak{g}$, the Poisson bracket is given by the
Lie bracket in $\mathfrak{g}$. Then we extend to all functions by the Leibniz
rule. Concretely for two functions
$f,g\in\mathcal{C}^{\infty}(\mathfrak{g}^{*})$ and $\xi\in\mathfrak{g}^{*}$ we
have

$\\{f,g\\}(\xi)=\langle\xi,[d_{\xi}f,d_{\xi}g]\rangle$

where $\langle.,.\rangle$ denotes the canonical pairing between
$\mathfrak{g}^{*}$ and $\mathfrak{g}$.

The symplectic leaves of $\mathfrak{g}^{*}$ are the _coadjoint orbits_ , i.e.
the orbits under the action of the Lie group $G$ acting on
$\xi\in\mathfrak{g}^{*}$ by

$\langle
g.\xi,x\rangle=\langle\xi,\mathrm{Ad}_{g^{-1}}(x)\rangle\;\forall\,x\in\mathfrak{g}$

where $\mathrm{Ad}$ denotes the adjoint action of $G$ on $\mathfrak{g}$.

######  Exercise 4.12.

For $\mathfrak{g}=\mathfrak{so}(3)$, show that $\mathfrak{g}^{*}$ is
$(\mathbb{R}^{3},\times)$ from Example 4.10 above, and deduce the symplectic
leaves.

###  4.2 Hamiltonian reduction

One of the most important operations on spaces is the quotient under a group
action. For symplectic manifolds, this quotient is known as the Hamiltonian
reduction (sometimes called Marsden–Weinstein quotient). From the physics
perspective, given a phase space with some symmetry, this allows to define a
reduced phase space.

Let $(M,\omega)$ be a symplectic manifold and $G$ be a Lie group acting on
$M$. The action is called symplectic if all $g\in G$ act by
symplectomorphisms, i.e. $g^{*}\omega=\omega$ for all $g$.

The infinitesimal action of an element $\xi\in\mathfrak{g}$ gives a vector
field $X_{\xi}$. Explicitly it is given by

$X_{\xi}(x)=\frac{d}{dt}\bigg{\rvert}_{t=0}\exp(t\xi).x$

where we write $g.x$ for the action of $g\in G$ on $x\in M$. Note that for any
group action, we have $[X_{\xi},X_{\nu}]=X_{[\xi,\nu]}$ for all
$\xi,\nu\in\mathfrak{g}$. In the case of a symplectic action, the vector
fields $X_{\xi}$ are symplectic.

In order to define the quotient of $M$ by $G$ and to ensure to obtain a
symplectic manifold, the action has to satisfy some extra conditions.

######  Definition 4.13.

The action of $G$ on $M$ is called weakly Hamiltonian if each $g\in G$ acts by
a Hamiltonian diffeomorphism.

For a weakly Hamiltonian action, $X_{\xi}$ is a hamiltonian vector field, i.e.
the symplectic gradient of some function $H_{\xi}$. The function $H_{\xi}$ is
not uniquely defined from the vector field, but only up to addition of an
overall constant.

######  Definition 4.14.

A weakly Hamiltonian action of $G$ on $M$ is called Hamiltonian if there is a
Lie algebra homomorphism $H:\mathfrak{g}\to\mathcal{C}^{\infty}(M)$ such that
$H_{\xi}$ generates the Hamiltonian vector field associated to
$\xi\in\mathfrak{g}$. In particular, we have

$H_{[\xi,\nu]}=\\{H_{\xi},H_{\nu}\\}.$ (4.4)

In more abstract terms, we can say that the action is

  * •

_weakly Hamiltonian_ if there is a Lie algebra homomorphism
$\mathfrak{g}\to\Gamma_{H}(TM)$, where $\Gamma_{H}(TM)$ denotes the space of
Hamiltonian vector fields,

  * •

_Hamiltonian_ if there is a Lie algebra homomorphism
$\mathfrak{g}\to\mathcal{C}^{\infty}(M)$.

As we will see below, the difference between a weakly Hamiltonian action and a
Hamiltonian action is some 2-cocycle in $\mathrm{H}^{2}(\mathfrak{g})$. To see
a bit clearer in this zoo of symplectic, weakly hamiltonian and hamiltonian
actions, there are some useful facts from topology:

  * •

If $H^{1}(M)=0$ or $H^{1}(\mathfrak{g})=0$, then symplectic implies weakly
hamiltonian.

  * •

If $H^{2}(\mathfrak{g})=0$, then weakly hamiltonian implies hamiltonian.

From a general fact in Lie algebra cohomology, the _Whitehead lemma_ , we know
that $H^{1}(\mathfrak{g})=0=H^{2}(\mathfrak{g})$ for finite-dimensional
semisimple $\mathfrak{g}$. Therefore:

######  Proposition 4.15.

If $G$ is semisimple and finite-dimensional, then a symplectic action is
automatically Hamiltonian.

In the presence of a Hamiltonian action, we can define the moment map
$\mu:M\to\mathfrak{g}^{*}$ defined by

$\mu(m).\xi=H_{\xi}(m).$

The name “moment map” comes from some basic examples where we recover momenta:

######  Example 4.16.

Consider $M=T^{*}\mathbb{R}^{2}$ with translation symmetry by $G=\mathbb{R}$
acting by $r.(p,x)=(p,x+r)$. One checks that the action is Hamiltonian with
moment map $\mu:T^{*}\mathbb{R}^{2}\to\mathbb{R}^{*}\cong\mathbb{R}$ given by
the momentum $p$, i.e. $\mu(p,x).t=pt$.

We recommend to be careful when dealing with moment maps since they are not
very intuitive at the beginning (in particular since the target is the dual
Lie algebra). Here is a general procedure to compute the moment map:

  * •

Step 1: Determine the vector field $X_{\xi}$ for $\xi\in\mathfrak{g}$ by
computing $g.x$ for $g=1+\varepsilon\xi$ to first order in $\varepsilon$. The
result is of the form $(1-\varepsilon\xi).x=x+\varepsilon X_{\xi}(x)$.

  * •

Step 2: Compute $\omega(X_{\xi},\delta x)$ and put it into the form
$\delta(\text{something})$, where $\delta$ denotes the variation. The
expression for “something” is the moment map.

Some explanations for the procedure: First, the mysterious $\delta x$ is a
“variation around $x\in M$”. To be precise, given a path $x:[0,1]\to M$ with
$x(0)=x$, the variation is defined to be

$\delta x=\frac{d}{dt}\bigg{\rvert}_{t=0}x(t).$

It is a tangent vector in $T_{x}M$. We can generalize and define the variation
of a function $f$ around $x$:

$\delta f(x)=\frac{d}{dt}\bigg{\rvert}_{t=0}f(x(t)).$

The variation $\delta$ acts like a derivation, i.e.
$\delta(fg)=\delta(f)g+f\delta(g)$.

Second, let us see why the procedure works. Since $\mu:M\to\mathfrak{g}^{*}$,
we have $d_{x}\mu:T_{x}M\to T_{\mu(x)}\mathfrak{g}^{*}\cong\mathfrak{g}^{*}$.
Hence $d_{x}\mu(\xi,X)=d_{x}H_{\xi}(X)=\omega_{x}(X_{\xi},X)$. Let $x:[0,1]\to
M$ be a path with $x(0)=x$. Then

$\delta\mu(x,\xi)=\frac{d}{dt}\bigg{\rvert}_{t=0}\mu(x(t),\xi)=d_{x}\mu(\xi,x^{\prime}(0))=\omega(X_{\xi},\delta
x).$

This justification seems complicated, but in practice, the procedure is
extremely efficient. Let us see an example:

######  Example 4.17.

Take the phase space of a particle moving in three-space,
$M=T^{*}\mathbb{R}^{3}\cong\mathbb{R}^{3}\times\mathbb{R}^{3}$. Consider the
diagonal action of $G=\mathrm{SO}(3)$. Since the group is simple, we only have
to show that the action is weakly Hamiltonian. For that, we directly compute
the moment map.

Step 1: For $\xi\in\mathfrak{so}(3)$, the vector field is simply given by
$X_{\xi}(x,p)=(\xi(x),\xi(p))$.

Step 2: We compute for $X=(x,p)$:

$\displaystyle\omega((\xi(x),\xi(p)),(\delta x,\delta p))$
$\displaystyle=\sum_{i}(\xi(x)_{i}\delta p_{i}-\xi(p)_{i}\delta x_{i})$
$\displaystyle=\sum_{i,j}(\xi_{ij}x_{j}\delta p_{i}-\xi_{ij}p_{j}\delta
x_{i})$ $\displaystyle=\sum_{i,j}(\xi_{ij}x_{j}\delta
p_{i}+\xi_{ji}p_{j}\delta x_{i})\;\;\text{ using antisymmetry of }\xi$
$\displaystyle=\delta\left(\textstyle\sum_{i,j}\xi_{ij}x_{i}p_{j}\right)$
$\displaystyle=\delta\langle p,\xi(x)\rangle.$

Hence the moment map is given by $\mu(x,p).\xi=\langle p,\xi(x)\rangle$. Under
some identification of $\mathbb{R}^{3}$ with $\mathfrak{so}(3)$ (see [MS17,
Example 5.3.1]), we can write $\mu(x,p)=x\times p$. Hence the moment map is
nothing but the _angular momentum_.

######  Exercise 4.18.

Show that the natural action of the linear symplectic group
$\mathrm{Sp}_{2n}(\mathbb{R})$ on $(\mathbb{R}^{2n},\omega_{0})$ is
hamiltonian with moment map $\mu(x).A=\frac{1}{2}\omega(x,Ax)$.

In the two examples, we have seen that to a continuous symmetry, we can
associate a conserved quantity, which is described by the moment map. For
translations, we get the momenta, and for rotations, we get the angular
momenta.

In general, the moment map behaves nicely under the action of $G$:

$\mu(g.m)=\mathrm{Ad}^{*}_{g}.\mu(m)$ (4.5)

It shows that for an abelian group $G$ (a torus action for example), the
moment map is constant along the $G$-orbits. Thus, we can think of it as the
_collection of all preserved quantities associated to the symmetry_. For $G$
not abelian, this is not exactly true, but we precisely know how the moment
map changes on the $G$-orbit.

Equation (4.5) can be seen as one way to formulate the famous Noether theorem,
stating that _to each continuous symmetry in a physical system, there is an
associated conserved quantity_.

######  Exercise 4.19.

Show Equation (4.5). Notice the importance of the requirement
$H_{[\xi,\nu]}=\\{H_{\xi},H_{\nu}\\}$.555Hint: For the infinitesimal action,
check that
$\mu((1+\xi).m).\nu-\mu(m).\nu=\frac{dH_{\nu}}{dt}(m)=\\{H_{\xi},H_{\nu}\\}(m)=H_{[\xi,\nu]}(m)=\mathrm{ad}^{*}_{\xi}\mu(m).\nu$.

Now, we can define the Hamiltonian reduction:

######  Definition 4.20.

For a Hamiltonian action of $G$ on $(M,\omega)$, suppose that
$0\in\mathfrak{g}^{*}$ is a regular value of the moment map $\mu$. Then the
Hamiltonian reduction of $M$ by $G$ (over 0) is defined to be

$M\sslash_{0}G:=\mu^{-1}(\\{0\\})/G.$

More generally, for a coadjoint orbit $\mathcal{O}\subset\mathfrak{g}^{*}$, we
define

$M\sslash_{\mathcal{O}}G:=\mu^{-1}(\mathcal{O})/G.$

Note that by Equation (4.5), the level surface $\mu^{-1}(\mathcal{O})$ is
invariant under $G$.

######  Theorem 4.21.

The hamiltonian reduction $M\sslash_{\mathcal{O}}G$ inherits a symplectic
structure from $M$. To be precise: if $M\sslash_{\mathcal{O}}G$ is a manifold,
i.e. if the action of $G$ on the level set $\mu^{-1}(\\{\mathcal{O}\\})$ is
free and proper, then it symplectic.

The situation is even more exciting: if we take the simple quotient $M/G$, we
get a Poisson manifold (which is not symplectic). Indeed, a function on $M/G$
is nothing but a $G$-invariant function on $M$ and the Poisson bracket of
$G$-invariant functions stays $G$-invariant. The natural question you should
always ask when you have a Poisson manifold is that of its symplectic leaves.
Well:

######  Theorem 4.22.

For a Hamiltonian action of $G$ on $M$, the quotient $M/G$ is a Poisson
manifold (potentially singular) whose symplectic leaves are given by
$M\sslash_{\mathcal{O}}G$ where $\mathcal{O}$ describes all coadjoint orbits
of $\mathfrak{g}^{*}$.

######  Example 4.23.

Consider $M=T^{*}G$ with the natural $G$-action. Since
$T^{*}G\cong\mathfrak{g}^{*}\times G$, we have $M/G=\mathfrak{g}^{*}$ which is
Poisson. Further, $\mu$ is simply the projection map. Hence the reduced space
$M\sslash_{\mathcal{O}}G$ is nothing but $\mathcal{O}$ itself. We recover that
the symplectic leaves of $\mathfrak{g}^{*}$ are its coadjoint orbits.

######  Example 4.24.

Consider the action of $\mathbb{S}^{1}=U(1)$ acting on $M=\mathbb{C}^{n}$ via
$\lambda.(z_{j})=(\lambda z_{j})$. Since $\mathbb{C}^{n}\cong
T^{*}\mathbb{R}^{n}$, we have the standard symplectic structure which in
complex coordinates is given by $\omega_{0}=\frac{i}{2}\sum_{j}dz_{j}\wedge
d\bar{z}_{j}$. Note that the action is symplectic since
$\lambda\bar{\lambda}=1$.

Identifying $\mathfrak{u}(1)\cong i\mathbb{R}$, step 1 gives the vector field
$Z_{ir}(z_{j})=(irz_{j})$ for $r\in\mathbb{R}$. Then step 2 gives

$\omega(Z_{ir},\delta
z)=\frac{i}{2}\sum_{j}r(iz_{j}\delta\bar{z}_{j}+i\bar{z}_{j}\delta
z_{j})=\delta\left(-\frac{1}{2}r\left\|z\right\|^{2}\right).$

Since $\mathbb{S}^{1}$ is abelian, every point of its dual Lie algebra is a
coadjoint orbit. The hamiltonian reduction over -2 gives

$\mathbb{C}^{n}\sslash_{\\{-2\\}}\mathbb{S}^{1}=\mu^{-1}(\\{-2\\})/\mathbb{S}^{1}\cong\mathbb{C}P^{n-1}.$

Hence, we have shown that _complex projective spaces carry a natural
symplectic structure_ (called the Fubini–Study structure). Note that the
coadjoint orbit of 0 is not a regular value for $\mu$ in this example.

An important fact which we will use quite often when computing the cotangent
bundle of moduli spaces is left as an exercise:

######  Exercise 4.25.

For $G$ acting on a manifold $X$, show that (under mild conditions)

$\boxed{T^{*}(X/G)\cong T^{*}X\sslash G.}$

######  Exercise 4.26.

Consider the diagonal action of $G$ on $T^{*}\mathfrak{g}$. Show that the
action is hamiltonian and compute the moment map666Using the Killing form to
identify $\mathfrak{g}^{*}$ with $\mathfrak{g}$, you should find
$\mu(X,Y)=[X,Y]$..

#### Weakly hamiltonian reduction*.

Finally, without entering into too much detail, let us analyze the case of a
weakly Hamiltonian action. For more details, we refer to [Kir04, Chapter 1,
Section 4]. The slogan is:

_“You can always transform a weakly Hamiltonian action into a Hamiltonian one
by replacing the group $G$ by a central extension.”_

Consider a weakly Hamiltonian action of $G$ on $(M,\omega)$. Since every $g\in
G$ acts by Hamiltonian diffeomorphisms, each vector field associated to
$\xi\in\mathfrak{g}$ is the symplectic gradient of some function $H_{\xi}$.
Define constants $c(\xi,\nu)$ by

$\\{H_{\xi},H_{\nu}\\}=H_{[\xi,\nu]}+c(\xi,\nu).$

######  Exercise 4.27.

Prove that $c(\xi,\nu)$ is a constant function777Hint: Show that the
symplectic gradient of $\\{H_{\xi},H_{\nu}\\}$ and $H_{[\xi,\nu]}$ coincide..

One checks that the Jacobi identity implies that $c$ form a 2-cocycle in the
cohomology of $\mathfrak{g}$. This means that

$c([X,Y],Z)+c([Y,Z],X)+c([Z,X],Y)=0\;\forall\,X,Y,Z\in\mathfrak{g}.$

Changing the functions $H_{\xi}$ by constants changes $c$ by a coboundary
term. Hence, the cohomology class $[c]\in H^{2}(\mathfrak{g})$ is well-
defined.

Since elements in $H^{2}(\mathfrak{g})$ describe central extensions, there is
an associated central extension
$\widehat{\mathfrak{g}}=\mathfrak{g}\oplus\mathbb{C}c$. The bracket is defined
by

$[(\xi_{1},a_{1}),(\xi_{2},a_{2})]=([\xi_{1},\xi_{2}],c(\xi_{1},\xi_{2})).$

Consider the group $\widehat{G}$ associated to $\widehat{\mathfrak{g}}$. It is
an extension of $G$ by a one-dimensional subgroup $Z$ consisting in central
elements:

$1\to Z\to\widehat{G}\to G\to 1.$

Define the action of $\widehat{G}$ on $M$ to be the one of $G$, with the
subgroup $Z$ acting trivially. By construction, this action is Hamiltonian.
Indeed we can put $H_{(\xi,a)}=H_{\xi}+a$ and then we get

$\\{H_{(\xi_{1},a_{1})},H_{(\xi_{2},a_{2})}\\}=\\{H_{\xi_{1}},H_{\xi_{2}}\\}=H_{[\xi_{1},\xi_{2}]}+c(\xi_{1},\xi_{2})=H_{([\xi_{1},\xi_{2}],c(\xi_{1},\xi_{2}))}=H_{[(\xi_{1},a_{1}),(\xi_{2},a_{2})]}.$

The only thing which changes is the structure of the dual Lie algebra, which
is now $\widehat{\mathfrak{g}}^{*}$. So the moment map remembers one more
extra information.

We will see a nice application of this construction in the Atiyah–Bott
reduction for surfaces with boundary in the next section.

##  5 Atiyah–Bott reduction

We have seen in the Riemann–Hilbert correspondence that the character variety
is the moduli space of flat connections. In the special case where the
manifold $M=\Sigma$ is a closed surface, the character variety gets a
symplectic structure, called the Goldman symplectic structure. More precisely,
the character variety is the Hamiltonian reduction of a very simple, but
infinite-dimensional space, the space of all connections. For a surface with
boundary, the character variety has a Poisson structure.

Let $\Sigma$ be a surface and $G$ be a subgroup of
$\operatorname{GL}_{n}(\mathbb{C})$ with Lie algebra $\mathfrak{g}$ (it can
easily be adapted to general semisimple Lie groups). Let $E$ be a trivial
$G$-bundle over $\Sigma$. Denote by $\mathcal{A}$ the space of all
$\mathfrak{g}$-connections on $E$. We have seen in Section 3 that
$\mathcal{A}$ is an affine space modeled over the vector space of
$\mathfrak{g}$-valued 1-forms $\Omega^{1}(\Sigma,\mathfrak{g})$. Further,
denote by $\mathcal{G}$ the space of all gauge transforms, i.e. bundle
automorphisms. We can identify the gauge group with $G$-valued functions:
$\mathcal{G}=\Omega^{0}(\Sigma,G)$.

On the space of all connections $\mathcal{A}$, there is a natural symplectic
structure given by

$\hat{\omega}=\int_{\Sigma}\operatorname{tr}\;\delta A\wedge\delta A$ (5.1)

where $\operatorname{tr}$ denotes an $\mathrm{Ad}$-invariant non-degenerate
form on the Lie algebra, which we call the _trace form_. For matrix Lie
algebras $\mathfrak{g}\subset\mathfrak{gl}_{n}$, we can simply choose the
trace (for semisimple Lie algebras, we can take the Killing form).

Let us explain this simple looking expression for $\hat{\omega}$. Since
$\mathcal{A}$ is an affine space, its tangent space at every point is
canonically isomorphic to $\Omega^{1}(\Sigma,\mathfrak{g})$. So given
$A\in\mathcal{A}$ and $B,C\in
T_{A}\mathcal{A}\cong\Omega^{1}(\Sigma,\mathfrak{g})$, we can write
$B=B_{x}dx+B_{y}dy$ with $B_{x},B_{y}\in\mathfrak{g}$ and similar for $C$.
Then we have

$\hat{\omega}_{A}(B,C)=\int_{\Sigma}\operatorname{tr}\;B\wedge
C=\int_{\Sigma}\operatorname{tr}(B_{x}C_{y}-B_{y}C_{x})dx\wedge dy.$

Note that $\hat{\omega}$ is independent of $A$ so $d\hat{\omega}=0$. Further,
the 2-form $\hat{\omega}$ is clearly antisymmetric and non-degenerate (since
the trace form is). Note that this construction only works on a surface.
Finally, if $G$ is a complex Lie group, $\hat{\omega}$ is a complex symplectic
structure.

######  Remark.

When using the wedge product and commutator of $\mathfrak{g}$-valued 1-forms,
some intuitions from usual exterior calculus are not valid anymore. In
particular we have

$[A,B]=[B,A]\;\;\;\text{ and }\;\;\;A\wedge A=\frac{1}{2}[A,A]$ (5.2)

which is non-zero in general. Check these equalities by plugging in
$A=A_{x}\,dx+A_{y}\,dy$ and similar for $B$.

On the space of connections, we have the natural action by the gauge
transforms. The surprising observation of Atiyah and Bott (see end of chapter
9 in [AB83] for unitary case, see section 1.8 in Goldman’s paper [Gol84] for
the general case) is the following:

######  Theorem 5.1 (Atiyah, Bott 1983).

The action of gauge transforms on the space of connections is Hamiltonian with
moment map the curvature.

Let us explain the moment map with more detail: it is a map
$m:\mathcal{A}\to\mathrm{Lie}(\mathcal{G})^{*}$. The Lie algebra
$\mathrm{Lie}(\mathcal{G})$ is equal to $\Omega^{0}(\Sigma,\mathfrak{g})$, so
its dual is isomorphic to $\Omega^{2}(\Sigma,\mathfrak{g})$ via the pairing
$\langle X,f\rangle=\int_{\Sigma}\operatorname{tr}fX$ for
$f\in\Omega^{0}(\Sigma,\mathfrak{g})$ and
$X\in\Omega^{2}(\Sigma,\mathfrak{g})$. On the other hand, given a connection
$A$, its curvature $F(A)$ is a $\mathfrak{g}$-valued 2-form, i.e. an element
of $\Omega^{2}(\Sigma,\mathfrak{g})$. Hence, the theorem asserts that the
moment map $\mu$ evaluated at $A\in\mathcal{A}$ and
$f\in\Omega^{0}(\Sigma,\mathfrak{g})$ is given by

$\mu(A).f=\int_{\Sigma}\operatorname{tr}fF(A).$ (5.3)

We give a sketch of the proof, a computation of the moment map which is the
curvature, neglecting all issues about infinite-dimensional spaces.

###### Sketch of proof.

The action of a gauge transform $g$ on a connection $A$ is given by
$g.A=gAg^{-1}+gdg^{-1}$ coming from expanding $g(d+A)g^{-1}$. So the action on
a tangent vector $\delta A$ is given by $g.\delta A=g\delta Ag^{-1}$.

Step 1: Let us compute the infinitesimal action by an element
$g=1+\varepsilon\xi$ to show that the action is weakly Hamiltonian. We get
$(1+\varepsilon\xi).A=A+\varepsilon([\xi,A]-d\xi)$. So we have a vector field
$A_{\xi}=[\xi,A]-d\xi$ on $\mathcal{A}$.

Step 2: Now we compute

$\displaystyle\hat{\omega}(A_{\xi},\delta A)$
$\displaystyle=\int_{\Sigma}\operatorname{tr}\;A_{\xi}\wedge\delta
A=\int_{\Sigma}\operatorname{tr}\;([\xi,A]-d\xi)\wedge\delta A$
$\displaystyle=\int_{\Sigma}\operatorname{tr}[\delta
A,A]\xi+\int_{\Sigma}\operatorname{tr}\;\xi\;d\delta A$
$\displaystyle=\delta\left(\int_{\Sigma}\operatorname{tr}(\xi\;(dA+A\wedge
A))\right)$

where we used integration by parts and the following facts:

  * •

the so-called cyclicity property of the trace:
$\operatorname{tr}[A,B]C=\operatorname{tr}[B,C]A$,

  * •

by Equation (5.2), we have $\delta(A\wedge
A)=\frac{1}{2}\delta([A,A])=\frac{1}{2}([\delta A,A]+[A,\delta A])=[\delta
A,A].$

Therefore using the identification by Equation (5.3), we get
$\mu(A)=dA+A\wedge A=F(A)$ which is the curvature.

Step 3: to show that the action is Hamiltonian, we compute

$\displaystyle\\{H_{\xi_{1}},H_{\xi_{2}}\\}$
$\displaystyle=\hat{\omega}(d\xi_{1}+[A,\xi_{1}],d\xi_{2}+[A,\xi_{2}])$
$\displaystyle=\int_{\Sigma}\operatorname{tr}\left(d\xi_{1}\wedge
d\xi_{2}+d\xi_{1}\wedge[A,\xi_{2}]+[A,\xi_{1}]\wedge
d\xi_{2}+[A,\xi_{1}]\wedge[A,\xi_{2}]\right)$
$\displaystyle=\int_{\Sigma}\operatorname{tr}\left(d(\xi_{1}d\xi_{2})+[\xi_{1},\xi_{2}](dA+A\wedge
A)\right)$ $\displaystyle=H_{[\xi_{1},\xi_{2}]}$

where we used integration by parts and cyclic properties of the trace. ∎

The important consequence of the Atiyah–Bott theorem is:

######  Corollary 5.2.

We have

$\boxed{\operatorname{Rep}(\pi_{1}\Sigma,G)\cong\\{\text{flat
connections}\\}/\mathcal{G}=\mathcal{A}\sslash\mathcal{G}.}$

In particular for a closed surface $\Sigma$, the character variety
$\operatorname{Rep}(\pi_{1}\Sigma,G)$ is a symplectic manifold.

###### Proof.

Carrying out the hamiltonian reduction of the gauge action on the space of
connections $\mathcal{A}$, we get

$\mathcal{A}\sslash\mathcal{G}=\\{\text{flat connections}\\}/\mathcal{G}.$

By the Riemann–Hilbert correspondence, the moduli space of flat connections is
diffeomorphic to the character variety. ∎

######  Remark.

Goldman [Gol84] gives an explicit formula for this symplectic structure. This
is why it is called the _Goldman symplectic structure_. He computes the
tangent space to $\operatorname{Rep}(\pi_{1}\Sigma,G)$ at a point
$\varphi\in\operatorname{Rep}(\pi_{1}\Sigma,G)$ in terms of cohomology (group
cohomology with coefficients in a twisted module):
$T_{\varphi}\operatorname{Rep}(\pi_{1}\Sigma,G)=\mathrm{H}^{1}(\pi_{1}\Sigma,\mathfrak{g}_{\mathrm{Ad}(\varphi)}).$
Combining the cup-product with the trace form, he gets a map

$\mathrm{H}^{1}(\pi_{1}\Sigma,\mathfrak{g}_{\mathrm{Ad}(\varphi)})\otimes\mathrm{H}^{1}(\pi_{1}\Sigma,\mathfrak{g}_{\mathrm{Ad}(\varphi)})\to
H^{2}(\pi_{1}S,\mathbb{R})\cong\mathbb{R}$

which is nothing but the symplectic form.

#### Surface with boundary*.

Consider now the case of a surface with boundary. In step 2 above, we get an
extra boundary term from the integration by parts:

$H_{\xi}(A)=\int_{\Sigma}\operatorname{tr}\xi(dA+A\wedge
A)+\int_{\partial\Sigma}\operatorname{tr}\xi A.$

Since we can find these functions, the action is weakly Hamiltonian. In step
3, we also get an extra term due to integration by parts:

$\\{H_{\xi_{1}},H_{\xi_{2}}\\}=H_{[\xi_{1},\xi_{2}]}+\int_{\partial\Sigma}\xi_{1}d\xi_{2}.$

Thus we have a non-trivial cocycle
$c(\xi_{1},\xi_{2})=\int_{\partial\Sigma}\xi_{1}d\xi_{2}$.

######  Theorem 5.3.

For a surface with boundary, the character variety
$\operatorname{Rep}(\pi_{1}\Sigma,G)$ is a Poisson manifold whose symplectic
leaves are flat connections with prescribed conjugacy class for the monodromy
around each boundary component.

The idea of the proof is to reduce the problem to the boundary
$\partial\Sigma$ which is a disjoint union of circles. The remaining gauge
group on the circle is the _loop group_. Since the action is only weakly
Hamiltonian, we have to consider a central extension of the loop group, the
famous _affine Lie group_. A good reference for loop groups is the
Pressley–Segal book [PS86], especially Chapter 4.

###### Proof.

Consider $\mathcal{G}_{0}\subset\mathcal{G}$ the subgroup of those gauge
transformations which are the identity on $\partial\Sigma$. For
$\mathcal{G}_{0}$, the cocycle vanishes, so we can define the symplectic
quotient $\mathcal{A}\sslash\mathcal{G}_{0}=\\{\text{flat
connections}\\}/\mathcal{G}_{0}$. By some gymnastics, we get

$\operatorname{Rep}(\pi_{1}\Sigma,G)=\\{\text{flat
connections}\\}/\mathcal{G}=(\\{\text{flat
connections}\\}/\mathcal{G}_{0})\big{/}(\mathcal{G}/\mathcal{G}_{0}).$

Hence we obtain the character variety as the quotient of a symplectic
manifold. By theorem 4.22, it is a Poisson manifold whose symplectic leaves
are given by all possible Hamiltonian reductions.

To determine the symplectic leaves, note that the remaining gauge group is
$\mathcal{G}/\mathcal{G}_{0}\cong\mathcal{L}G^{k}$ where
$\mathcal{L}G=\mathcal{C}^{\infty}(\mathbb{S}^{1},G)$ denotes the _loop group_
and $k$ the number of boundary components of $\Sigma$. Without loss of
generality we consider $k=1$ in the sequel.

The action of $\mathcal{L}G$ on $\mathcal{A}\sslash\mathcal{G}_{0}$ is only
weakly Hamiltonian, since we have the cocycle $c$. Hence we have to consider
the central extension
$\widehat{\mathfrak{g}}=\mathcal{L}\mathfrak{g}\oplus\mathbb{C}$ with Lie
bracket given by

$[(A(z),a),(B(z),b)]=([A(z),B(z)],\textstyle\oint\operatorname{tr}AdB)$

where $z\in\mathbb{S}^{1}\subset\mathbb{C}$ and
$\oint\operatorname{tr}AdB=\int_{\mathbb{S}^{1}}\operatorname{tr}AdB$
represents the cocycle $c$. This is the famous _affine Kac–Moody algebra_ of
type $\mathfrak{g}$.

Since the action of $\widehat{G}$ is only on the boundary circle, we can
restrict our connection to $\partial\Sigma$. Surprisingly, the moment map for
the action of $\widehat{G}$ is nearly the identity (it is an inclusion):

######  Lemma 5.4.

The dual affine Lie algebra $\widehat{g}^{*}$ can be identified with the space
of all $k$-connections of type $G$ on the circle. The coadjoint action of
$\widehat{G}$ is the gauge action.

A _$k$ -connection_ is a generalization of a connection where the Leibniz rule
is replaces by $D(fs)=k\,dfs+fD(s)$. Locally, a $k$-connection is of the form
$kd+A$. For $k=1$ we get usual connections and for $k=0$ we get
$\mathfrak{g}$-valued 1-forms. In particular, the moment map of the action of
$\widehat{G}$ on the space of connections on $\mathbb{S}^{1}$ is the inclusion
(with $k=1$).

###### Proof of lemma.

Elements of $\widehat{\mathfrak{g}}^{*}$ are pairs $(X(z),k)$ where $X$ is a
$\mathfrak{g}$-valued 1-form on $\mathbb{S}^{1}$ and $k\in\mathbb{C}$. The
pairing with $\widehat{\mathfrak{g}}$ is given by

$\langle(X(z),k),(A(z),a)\rangle=\textstyle\oint\operatorname{tr}A(z)X(z)+ka.$

We first show the infinitesimal version of the lemma by computing the
$\widehat{\mathfrak{g}}$-coadjoint action:

$\displaystyle\langle\mathrm{ad}^{*}_{(A,a)}(X,k),(B,b)\rangle$
$\displaystyle=\langle(X,k),[(-A,-a),(B,b)]\rangle$
$\displaystyle=-\textstyle\oint\operatorname{tr}X[A,B]-k\textstyle\oint\operatorname{tr}AdB$
$\displaystyle=\textstyle\oint\operatorname{tr}B([A,X]+k\,dA)$
$\displaystyle=\langle([A,X]+k\,dA,0),(B,b)\rangle.$

Hence $\mathrm{ad}^{*}_{(A,a)}(X,k)=[A,X]+k\,dA$ which can be identified with
the action of an infinitesimal gauge $A$ on a $k$-connection $-kd+X$. After
integration, the $\widehat{G}$-coadjoint action is given by the gauge action.
∎

Finally, we can conclude on the symplectic leaves. The coadjoint orbits are
the gauge-equivalence classes of connections on the circle. Now cut of the
circle at a point. Using the gauge, we can trivialize the connection. The only
information we can not change is how to glue the two ends together. This is
precisely the conjugacy class of the monodromy. Therefore, the condition from
the second moment map prescribes the conjugacy class of the monodromy around
$\partial\Sigma$. ∎

##  6 GIT quotients and stability conditions

The geometric invariant theory (GIT) allows to define in a quite general
setting a quotient of a manifold $M$ by some group action $G$. It introduces
the notion of stable and unstable points which are treated differently to
define a well-behaved quotient $M/G$. These stability conditions play a
crucial role for defining moduli spaces, in particular for flat connections.
We recommend [Tho05] for more details.

#### Introduction.

Let $G$ be a group acting on some manifold $M$. We wish to define a nice space
“$M/G$”. It should satisfy a universal property: whenever there is a
$G$-invariant map $M\to X$, it should factor through $M/G$.

If we take the set-theoretic quotient, i.e. the space of orbits, we often get
a space which is not Hausdorff. This happens for example whenever two orbits
$\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ get arbitrarily close, i.e. if
$\bar{\mathcal{O}}_{1}\cap\bar{\mathcal{O}}_{2}\neq\emptyset$. Indeed any open
sets around the points representing $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ in
$M/G$ intersect.

######  Example 6.1.

Consider the action of $G=\mathbb{C}^{*}$ on $M=\mathbb{C}^{2}$ given by
$\lambda.(x,y)=(\lambda x,\lambda^{-1}y)$. The orbits are shown in Figure 6.1.
Note that the only non-closed orbits are $(x=0,y\neq 0)$ and $(x\neq 0,y=0)$
whose closures intersect. The set-theoretic quotient gives $\mathbb{C}$ with
three points at the origin.

Figure 6.1: Example of non-Hausdorff quotient

We would like to define a quotient such that in the example, the result is
$\mathbb{C}$. For that, two options seem to be reasonable:

  1. 1.

Only keep the close orbits.

  2. 2.

Identify orbits whenever their closure intersect.

In the first case, we would “throw away” the two non-closed orbits, and in the
second case we would identify all three points above the origin to a single
point. The GIT quotient construction takes the second case, but as we will see
is equivalent to the first one in the affine case.

#### Affine GIT quotient.

We will use the very basic idea of algebraic geometry: the dictionary between
geometry and algebra. Consider a smooth affine variety $M$, i.e. the zero-set
of a set of polynomials. Replacing these polynomials by the ideal $I$ they
generate (which does not change $M$, but gives more structure) we can write

$M=\\{x\in\mathbb{C}^{n}\mid P(x)=0\forall x\in I\\}.$

To $M$, we associate the space of all polynomial functions on $M$. Such a
function is a restriction of any element of $\mathbb{C}[x_{1},...,x_{n}]$ to
$M$. Since any element of $I$ is zero on $M$ (by definition), we get

$\operatorname{Fun}(M)=\mathbb{C}[x_{1},...,x_{n}]/I.$

Finally from the quotient
$\operatorname{Fun}(M)=\mathbb{C}[x_{1},...,x_{n}]/I$, we can directly get
$M$: every point of $M$ corresponds to a unique maximal ideal in
$\operatorname{Fun}(M)$. The algebraic procedure of taking all maximal ideals
of a ring $A$ is called the spectrum $\mathrm{Spec}(A)$.

The idea of the GIT quotient is now simple to explain: the functions on $M/G$
have to be the $G$-invariant functions on $M$. By the correspondence between
geometry and algebra, we can define the quotient this way:

$M/_{GIT}\,G:=\mathrm{Spec}(\operatorname{Fun}(M)^{G}).$ (6.1)

In Example 6.1 above, where $\mathbb{C}^{*}$ acts on $\mathbb{C}^{2}$, we get

$\mathrm{Spec}(\operatorname{Fun}(\mathbb{C}^{2})^{\mathbb{C}^{*}})=\mathrm{Spec}(\mathbb{C}[xy])=\mathbb{C}.$

Note that the function $xy$ does not distinguish between the three orbits
$(0,0),(x=0,y\neq 0)$ and $(x\neq 0,y=0)$.

We see that two orbits $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ get identified
if $\overline{\mathcal{O}}_{1}\cap\overline{\mathcal{O}}_{2}\neq\emptyset$:
indeed any smooth $G$-invariant function is constant on
$\overline{\mathcal{O}}_{1}$ and $\overline{\mathcal{O}}_{2}$. Since their
closures intersect, the function takes the same value on them. Since no
$G$-invariant function can distinguish the two orbits, they get identified in
the GIT quotient.

Finally, notice that in the closure of any orbit $\overline{\mathcal{O}}$,
there is a unique closed orbit. That’s why we can also say that the affine GIT
quotient keeps only the closed orbits.

######  Exercise 6.2.

a) Consider the action of $\mathrm{GL}_{n}(\mathbb{C})$ on
$\mathfrak{gl}_{n}(\mathbb{C})$ by conjugation. Show that the orbit of a
matrix $M$ is closed iff $M$ is diagonalizable and that its orbit is of
maximal dimension if the eigenvalues are pairwise distinct.

b) Compute the GIT quotient888Hint: Use that the diagonalization is unique up
to permutation, and that the ring of symmetric polynomial functions is free..

c) Show that the invariant functions are generated by the coefficients of the
characteristic polynomial999Hint: Use the Frobenius form of the matrix..

######  Remark.

The exercise generalizes to any semisimple complex Lie group $G$, the notion
of diagonalizable being replaces by regular elements. The Chevalley
restriction theorem gives $\mathfrak{g}^{reg}/G=\mathfrak{h}/W$ where
$\mathfrak{h}$ denotes the Cartan subalgebra (the diagonal matrices above) and
$W$ the Weyl group (the permutation group above). In addition
$\operatorname{Fun}(\mathfrak{h}/W)=\mathbb{C}[p_{1},...,p_{r}]$ is a free
algebra generated by invariant polynomials, which play an important role in
Lie theory.

Let us now turn to the character variety. The space
$\operatorname{Hom}(\pi_{1}S,G)$ is an affine variety if $G$ is an affine
algebraic group. This can be seen from an explicit presentation of $\pi_{1}S$:

$\pi_{1}S=\langle(a_{i},b_{i})_{1\leq i\leq
g}\mid\textstyle\prod_{i}[a_{i},b_{i}]=1\rangle.$

This presentation comes from the fact that a surface of genus $g$ can be
obtained by a special gluing of a $4g$-gon. Since the only relation is
algebraic, the space $\operatorname{Hom}(\pi_{1}S,G)$ is an algebraic subset
of $G^{2g}$.

To go to the character variety, we have to quotient by the conjugation action
by $G$. To get a nice quotient, we have to take the GIT quotient. In the case
$G=\operatorname{GL}_{n}(\mathbb{C})$, we have

######  Theorem 6.3.

A point
$\rho\in\operatorname{Hom}(\pi_{1}S,\operatorname{GL}_{n}(\mathbb{C}))$ is
polystable iff $\rho$ is completely reducible, i.e. all $\rho$-invariant
subspaces of $\mathbb{C}^{n}$ admit an invariant supplement. If $\rho$ is
irreducible (there are no non-trivial $\rho$-invariant subspaces of
$\mathbb{C}^{n}$), then it is a stable point.

For the proof, which is quite delicate, we refer to [Sik12, Section 7]. From
the algebraic geometric perspective, a point
$\rho\in\operatorname{Hom}(\pi_{1}S,G)$ is regular iff $\rho$ is irreducible,
see [Gun21, Theorem 26].

#### Projective GIT quotient.

The construction of an affine GIT quotient does not always give a satisfactory
answer. Consider the following example:

######  Example 6.4.

Let $\mathbb{C}^{*}$ act on $\mathbb{C}^{2}$ via $\lambda.(x,y)=(\lambda
x,\lambda y)$. Then all orbit closures intersect and the GIT quotient would
give just one point. Of course we would like to get $\mathbb{C}P^{1}$ as
quotient.

The idea is to slightly modify the functors $\operatorname{Fun}$ and
$\mathrm{Spec}$ in the defining Equation (6.1) above. For that, we consider a
_projective_ variety $X\subset\mathbb{C}P^{n}$ and we suppose that $G$ acts
via $G\to\operatorname{SL}_{n+1}(\mathbb{C})$. Then the action lifts to the
cone $\tilde{X}\subset\mathbb{C}^{n+1}-\\{0\\}$.

Figure 6.2: Projective cone over $X$

The main difference with the affine case is that on $\tilde{X}$ we have a
$\mathbb{C}^{*}$-action, which induces a grading of its function space into
homogeneous parts. We define

$M/G=\mathrm{Proj}\left(\mathrm{Projfun}(\tilde{X})^{G}\right)$ (6.2)

where $\mathrm{Projfun}$ is the direct sum of all homogeneous polynomial
function of degree $k\geq 1$ on $\tilde{X}$, and $\mathrm{Proj}$ associates a
projective variety to a ring (by taking its maximal essential ideals).

In Example 6.4 above, the action extends to $\mathbb{C}^{3}$ via
$\lambda.(x,y,z)=(\lambda x,\lambda y,\lambda^{-2}z)$ (since the action takes
values in $\operatorname{SL}_{3}(\mathbb{C})$). The invariant homogeneous
polynomials are generated by monomials $x^{a}y^{b}z^{(a+b)/2}$ with $a+b$
even. Taking $z=1$ gives
$\mathrm{Projfun}(\tilde{X})^{\mathbb{C}^{*}}=\oplus_{k\geq
1}\mathbb{C}_{2k}[x,y]$, whose associated projective variety is
$\mathbb{C}P^{1}$ (with structure bundle $\mathcal{O}(2)$).

The good news is that you don’t have to know much about the functors
$\mathrm{Projfun}$ and $\mathrm{Proj}$ since there is a _simple geometric
recipe to compute the projective GIT quotient_. For that, we define types of
points and treat each type differently: a point $x\in X$ is

  * •

Unstable, if for all non-constant homogeneous $G$-invariant polynomial $f$, we
have $f(x)=0$.

  * •

Semistable, if it is not unstable.

  * •

Polystable, if it is semistable and has a closed orbit in $\tilde{X}$.

  * •

Stable, if it is semistable and its orbit in $\tilde{X}$ is closed and of
maximal dimension.

The recipe of the projective GIT quotient can then be given as follows: _Throw
away all unstable points and identify semistable points if their orbit
closures in $\tilde{X}$ intersect._

The explanation is that unstable points are not seen by $G$-invariant
homogeneous functions. The same argument as for the affine GIT quotient
explains why to identify orbits whose closure intersect.

In addition to the simple recipe, there is a nice geometric characterization
of unstable points:

######  Proposition 6.5.

A point $x$ is unstable iff 0 is in the orbit closure of $\tilde{x}$ (any
preimage of $x$ of $\tilde{X}\to X$).

The so-called Hilbert–Mumford criterion states that it is sufficient to check
that property for all 1-parameter subgroups of $G$.

In Example 6.4, the point $(0,0)$ is unstable since
$\lambda.(0,0,1)=(0,0,\lambda^{-2})\to 0$ for $\lambda\to\infty$. Any other
point $(x,y)$ is semistable since $x^{2}z$ or $y^{2}z$ is an invariant
homogenous function not vanishing on the point. They are even stable since the
orbit in $\mathbb{C}^{3}$ is closed and the stabilizer is trivial. Hence, we
throw away the origin and take the set-theoretic quotient of the rest: we get
precisely $\mathbb{C}P^{1}$.

The projective GIT quotient is summarized in the following table:

Type | Algebraic | Geometric | GIT quotient
---|---|---|---
Unstable | $f(x)=0\,\forall f\in\mathrm{Fun}(\tilde{X})^{G}$ | $0\in\overline{G.\tilde{x}}$ | throw away
Semistable | not unstable | $0\notin\overline{G.\tilde{x}}$ |  $\mathcal{O}_{1}\sim\mathcal{O}_{2}$ if $\overline{\mathcal{O}}_{1}\cap\overline{\mathcal{O}}_{2}\neq\emptyset$
Polystable | semistable, closed orbit |  $G.\tilde{x}$ is closed | keep
Stable | maximal polystable |  $+\;\mathrm{Stab}(\tilde{x})$ finite | keep

Note that in the affine case, all orbits are semistable (since we do not
exclude the constant functions) and in every orbit closure there is a unique
polystable orbit. This is not always true in the projective case.

#### Link to symplectic quotient.

The link between the GIT quotient and the symplectic quotient is given by the
Kempf–Ness theorem. Roughly, it tells that

$\boxed{X\sslash K\cong X/_{GIT}\,K^{\mathbb{C}}=X^{ps}/K^{\mathbb{C}}.}$
(6.3)

To be more precise:

######  Theorem 6.6 (Kempf–Ness).

Let $V$ be a complex vector space with hermitian inner product,
$K\subset\mathrm{U}(V)$ a closed subgroup and put $G=K^{\mathbb{C}}$. Let
$X\subset V$ be a $G$-invariant affine variety. Then the action of $K$ on $X$
is Hamiltonian and the Hamiltonian reduction $X\sslash K$ equals the GIT
quotient $X/G$.

Denoting by $\mu$ the moment map, one can show that $X^{ps}=G.\mu^{-1}(0)$,
i.e. that a $G$-orbit intersects $\mu^{-1}(0)$ iff it is closed. We refer to
the original article [KN79] for the proof.

To give an example, let us reconsider the case of projective space, already
seen in Example 4.24:

######  Example 6.7.

Consider $G=\mathrm{U}(1)\subset\mathbb{C}$ acting on $\mathbb{C}^{n}$ by
scaling all coordinates by some factor (of module 1). We have seen in Example
4.24 that the symplectic reduction over the coadjoint orbit $\\{-2\\}$ gives
$\mathbb{C}P^{n-1}$. This coincides with the projective GIT quotient (see
Example 6.4).

##  7 Stable bundles and the Narasimhan–Seshadri theorem

The goal of this section is to characterize flat bundles with unitary
monodromy, i.e. understand $\operatorname{Rep}(\pi_{1}\Sigma,\mathrm{U}(n))$.
We will see that the corresponding bundles carry a holomorphic structure and
satisfy some stability condition (in the GIT sense). Before, we will see some
basic theory about bundles.

#### Different types of bundles.

Up to now, we were a bit sloppy when speaking about bundles. In fact, there
are three different types of bundles: real (or topological), complex and
holomorphic bundles.

Before, we always treated _real bundles_ , where the transition functions are
smooth. If the fiber is a complex vector space, we speak about a _complex
vector bundle_. If in addition the base manifold $X$ is complex, we can define
a _holomorphic bundle_ to be a complex bundle with holomorphic transition
functions. To summarize:

Real | Complex | Holomorphic
---|---|---
smooth transitions | complex structure on fiber | holomorphic transitions
$\check{\mathrm{H}}^{1}(X,\mathcal{C}^{\infty}(\operatorname{GL}_{n}(\mathbb{R})))$ | $\check{\mathrm{H}}^{1}(X,\mathcal{C}^{\infty}(\operatorname{GL}_{n}(\mathbb{C})))$ | $\check{\mathrm{H}}^{1}(X,\mathrm{Hol}(\operatorname{GL}_{n}(\mathbb{C})))$

If you know about sheaf cohomology, the last line indicates how to
characterize the different types of bundles in topological terms. The set of
all transition functions form a 1-cocycle in sheaf cohomology, and the
isomorphism class of the bundle is described by the associated cohomology
class.

The case of line bundles is particularly interesting since they form a group
under tensor product. The transition functions simply get multiplied when you
tensor line bundles. Since
$\operatorname{GL}_{1}(\mathbb{C})\cong\mathbb{C}^{*}$, the transition
function vanish nowhere, so are invertible.

######  Remark.

Again for the reader familiar with algebraic topology, some remarks on the
classification of line bundles:

  * •

Using
$0\to\mathbb{Z}/2\mathbb{Z}\to\mathcal{C}^{\infty}(\mathbb{R}^{*})\to\mathcal{C}^{\infty}(\mathbb{R}_{>0})\to
0$ given by $f\mapsto f^{2}$, one can show that
$\check{\mathrm{H}}^{1}(X,\mathcal{C}^{\infty}(\mathbb{R}^{*}))\cong\mathrm{H}^{1}(X,\mathbb{Z}/2\mathbb{Z})$,
so real line bundles are classified by the _first Stiefel–Whitney class_.

  * •

Using
$0\to\mathbb{Z}\to\mathcal{C}^{\infty}(\mathbb{C})\to\mathcal{C}^{\infty}(\mathbb{C}^{*})\to
0$ given by $f\mapsto\exp(f)$, one can show that
$\check{\mathrm{H}}^{1}(X,\mathcal{C}^{\infty}(\mathbb{C}^{*}))\cong\mathrm{H}^{2}(X,\mathbb{Z})$,
so complex line bundles are classified by the first Chern class. For
$X=\Sigma$ a surface, we have $H^{2}(X,\mathbb{Z})\cong\mathbb{Z}$, so the
first Chern class equals the degree of the bundle.

  * •

Holomorphic line bundles are much more abundant. On a Riemann surface $S$ of
genus at least 2, there are smooth families of them. The space of all line
bundles is called the _Picard variety_ $\mathrm{Pic}(S)$ and is given by
$\mathrm{Pic}(S)\cong\mathbb{Z}\times\mathrm{Jac}(S)$ where the first factor
gives the degree and the second factor is the _Jacobian variety_.

Let us give some examples of line bundles over $X=\mathbb{C}P^{1}$.

######  Example 7.1.

The Riemann sphere $\mathbb{C}P^{1}$ is given by two charts $U_{0}=\mathbb{C}$
and $U_{1}=\mathbb{C}$ with transition map $U_{0}\cap
U_{1}=\mathbb{C}^{*}\to\mathbb{C}^{*}$ given by $z\mapsto 1/z$. Define the
line bundle $\mathcal{O}(k)$ in the following way: it is made of two pieces,
$U_{0}\times\mathbb{C}$ and $U_{1}\times\mathbb{C}$ which are glued together
via

$\left\\{\begin{array}[]{cl}(U_{0}\cap U_{1})\times\mathbb{C}&\to(U_{0}\cap
U_{1})\times\mathbb{C}\\\ (z,v)&\mapsto(1/z,z^{k}v)\end{array}\right.$

It turns out that the $\mathcal{O}(k)$ describe all holomorphic line bundles:
$\mathrm{Pic}(\mathbb{C}P^{1})\cong\mathbb{Z}$.

To any vector bundle $E\to M$, we can associate a line bundle in the following
way. There is an open covering of $M$ by open sets $U_{\alpha}$ where $E$ is
trivial and with transition maps $f_{\alpha\beta}$. Define the determinant
line bundle $\det(E)$ to be the line bundle which is trivial over the
$U_{\alpha}$ with transition functions $\det(f_{\alpha\beta})$. The first
Chern class is preserved: $c_{1}(E)=c_{1}(\det E)$.

To understand a bundle, we can try to decompose it into simpler pieces, to
write it as a direct sum. On real or complex line bundles, we can put a scalar
or hermitian product on the fibers (varying smoothly) which allows to
construct canonically a complement to any subbundle. This means that whenever
we have a subbundle $F\subset E$, we can write $E=F\oplus F^{\prime}$. We say
that a bundle is irreducible if it has no non-trivial subbundles.

This is not true for holomorphic bundles since there might be no hermitian
structure varying holomorphically with the point. We speak about an
indecomposable holomorphic bundle whenever it cannot be written as a direct
sum of two other bundles.

Since line subbundles are nothing but non-vanishing sections, we can look for
these to decompose the bundle:

######  Proposition 7.2.

Let $E\to M$ be a vector bundle with $M$ a real $m$-dimensional manifold.

  * •

If $E$ is real of rank more than $m$, then there is a non-vanishing section.

  * •

If $E$ is complex of (complex) rank more than $m/2$, then there is a non-
vanishing section.

The proof is simple: a generic perturbation of any section works.

######  Corollary 7.3.

A complex bundle $E\to\Sigma$ of rank $k$ over a surface is isomorphic to
$\det(V)\oplus\mathbb{C}^{k-1}$ (as complex bundle).

###### Proof.

By the previous proposition we find $k-1$ non-vanishing independent sections.
So $V\cong L\oplus\mathbb{C}^{k-1}$ where $L$ is some line bundle. Then
$\det(V)=\det(L)=L$. ∎

Now we can give an example illustrating the difference between the three kinds
of bundles:

######  Example 7.4.

On $\mathbb{C}P^{1}$, consider the holomorphic bundle
$\mathcal{O}(1)\oplus\mathcal{O}(1)$. Its underlying complex vector bundle is
$\mathbb{C}\oplus T\mathbb{C}P^{1}$, since it can be easily checked that
$\det(\oplus\mathcal{O}(a_{i}))=\mathcal{O}(\sum a_{i})$ and $\mathcal{O}(2)$
is the tangent bundle of $\mathbb{C}P^{1}$. Note that $\mathbb{C}\oplus
T\mathbb{C}P^{1}$ is not trivial, since there is no non-vanishing section of
$T\mathbb{C}P^{1}$ (the “hairy ball theorem”).

Finally, its underlying real bundle is _trivial_. You can check that
$T(\mathbb{S}^{2})\oplus\mathbb{R}$ is a trivial bundle. You just take the
unit sphere $\mathbb{S}^{2}\subset\mathbb{R}^{3}$ and consider its normal
bundle, which is trivial (there is a constant section). Hence its direct sum
with the tangent bundle gives the tangent bundle of $\mathbb{R}^{3}$, which is
trivial, restricted to the sphere. Therefore
$T(\mathbb{S}^{2})\oplus\mathbb{R}^{2}\cong\mathbb{R}^{4}$.

######  Remark.

A theorem of Grothendieck asserts that the set of all holomorphic bundles on
$\mathbb{C}P^{1}$ are the $\oplus\mathcal{O}(a_{i})$. This can be proven using
methods from loop groups.

#### Flat bundles.

Now that we have seen the different kind of bundles, we may ask which bundles
arise in the Riemann–Hilbert correspondence.

Recall that to a representation
$\rho:\pi_{1}M\to\operatorname{GL}_{n}(\mathbb{C})$, we associate the bundle
$E_{\rho}=(\widetilde{M}\times\mathbb{C}^{n})/\pi_{1}M$. This is a complex
vector bundle which is trivial, since $\pi_{1}M$ acts faithfully on
$\widetilde{M}$. On a surface, complex bundles are classified by their degree.
We have:

######  Proposition 7.5.

A complex vector bundle $E$ over a surface $\Sigma$ admits a flat connection
iff $\deg(E)=0$ (iff $E$ is trivial).

Over a general manifold, a complex vector bundle is trivial iff all its Chern
classes vanish.

######  Remark.

More generally, there is a beautiful link between characteristic classes and
the curvature of a connection, described by _Chern–Weil theory_. For the
degree of a bundle $E$ over $M$, the link is

$\deg(E)=c_{1}(E)=\left[\frac{i}{2\pi}\operatorname{tr}F(A)\right]\in
H^{2}(M,\mathbb{Z}).$

Since trivial complex bundles can have non-trivial holomorphic structures, we
can ask which _holomorphic_ bundles can arise through the Riemann–Hilbert
correspondence? For this to make sense, we have to equip the smooth surface
$\Sigma$ with a complex structure. The resulting Riemann surface is denoted by
$S$.

We have seen the notion of indecomposable holomorphic bundles (not the direct
sum of others). By definition, they form the building blocks for all
holomorphic bundles:

######  Proposition 7.6.

Every holomorphic bundle over a compact Riemann surface is the direct sum of
indecomposable bundle in a unique way (up to permutation of the factors).

For holomorphic bundles, we have the following theorem due to Weyl:

######  Theorem 7.7 (Weyl).

A holomorphic bundle $E=\oplus E_{i}$ which is a direct sum of indecomposable
bundles $E_{i}$ admits a flat connection if and only if
$c_{1}(E_{i})=0\,\forall\,i$.

A proof can be found in [Gun21, Theorem 16].

#### Cauchy–Riemann operator.

On a holomorphic bundle $E$, there is a natural connection
$\bar{\partial}_{E}$ generalizing the Cauchy–Riemann operator (which defines
what a holomorphic function is). It turns out that holomorphic structures are
in bijection with these operators.

A Cauchy–Riemann operator $\bar{\partial}_{E}$ (also called _Dolbeault
operator_) on a complex bundle $E$ over a complex manifold $M$ is a
connection, i.e. a map $\Gamma(E)\otimes\Gamma(TM)\to\Gamma(E)$, such that
$\bar{\partial}_{E}^{2}=0$ and

$\bar{\partial}_{E}(fs)=(\bar{\partial}f)s+f\bar{\partial}_{E}(s)$ (7.1)

for all sections $s\in\Gamma(E)$ and functions $f\in\mathcal{C}^{\infty}(M)$.
Note that $\bar{\partial}$ is the usual Cauchy–Riemann operator, which is
well-defined since $M$ is a complex manifold.

######  Proposition 7.8.

A holomorphic structure on a complex vector bundle $E$ is equivalent to the
existence of a Cauchy–Riemann operator $\bar{\partial}_{E}$.

We refer to the thesis of McCarthy [McC18, Theorem 3.3.3] and [AB83, p.555]
for a proof. The delicate point is to construct holomorphic transition
functions from $\bar{\partial}_{E}$ which leads to an elliptic system to
solve.

On a Riemann surface $S$, a complex bundle admits plenty of holomorphic
structures: any connection gives one. Indeed we can decompose a connection
$\nabla=\nabla^{1,0}+\nabla^{0,1}$ into the $dz$ and $d\bar{z}$-part. Then
$\nabla^{0,1}$ is a Cauchy–Riemann operator (since
$d=\partial+\bar{\partial}$).

Consider now a trivial complex bundle $E$ over $S$. Using the differential $d$
as canonical base point, we can identify the space of $G$-connections with
$\Omega^{1}(S,\mathfrak{g})$. Hence, we can describe holomorphic structures by
$\Omega^{0,1}(S,\mathfrak{g})$. Then, two holomorphic structures are
equivalent (under some bundle automorphism) iff the corresponding operators
are gauge-equivalent. Note that this action is given by

$g.B=gBg^{-1}+g\bar{\partial}\left(g^{-1}\right).$

Denote by $\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))$ the moduli
space of holomorphic structures on $E=S\times\mathbb{C}^{n}$ and by
$\mathcal{G}^{\mathbb{C}}=\mathcal{G}(\operatorname{GL}_{n}(\mathbb{C}))$ the
gauge group. Since the trivial complex bundle has degree 0, we have

$\operatorname{Hol}_{\deg=0}(S,\operatorname{GL}_{n}(\mathbb{C}))\cong\Omega^{0,1}(S,\mathfrak{gl}_{n}(\mathbb{C}))/\mathcal{G}^{\mathbb{C}}.$
(7.2)

#### Stability for bundles.

The space of holomorphic structures is given by the quotient (7.2). To get a
nice space, we have to interpret this quotient in the GIT sense. The
associated stability condition is described now.

For a holomorphic bundle $E\to M$, we define its slope $\mu(E)$ by

$\mu(E)=\frac{\deg(E)}{\operatorname{rk}(E)}.$ (7.3)

To memorize: the degree can be zero, so cannot be in the denominator.

######  Definition 7.9 (Mumford).

A holomorphic bundle $E$ is stable if for all holomorphic subbundles $F\subset
E$ we have $\mu(F)<\mu(E)$. The bundle is semistable if the inequality is not
strict.

Let us see two properties of stable bundles:

######  Proposition 7.10.

Let $E$ be a stable holomorphic bundle. Then

  1. 1.

$E$ cannot be a direct sum $E_{1}\oplus E_{2}$.

  2. 2.

$E$ has only trivial holomorphic automorphisms (of the form
$\lambda\operatorname{id}$ for some constant $\lambda$).

###### Proof.

Part (1) is simply proved by contradiction: if $E=E_{1}\oplus E_{2}$, then
$\deg(E)=\deg(E_{1})+\deg(E_{2})$ and
$\operatorname{rk}(E)=\operatorname{rk}(E_{1})+\operatorname{rk}(E_{2})$.
Since $E$ is stable, we have $\mu(E_{1})<\mu(E)$ and $\mu(E_{2})<\mu(E)$ which
leads to a contradiction:

$\deg(E)=\deg(E_{1})+\deg(E_{2})<\deg(E)\left(\frac{\operatorname{rk}(E_{1})}{\operatorname{rk}(E)}+\frac{\operatorname{rk}(E_{2})}{\operatorname{rk}(E)}\right)=\deg(E).$

For part (2), consider a holomorphic automorphism $\varphi:E\to E$. Then its
characteristic polynomial is constant, since its coefficients are holomorphic
functions on a compact manifold. If there are at least two different
eigenvalues, we can decompose $E$ into a direct sum which is impossible by
(1). So there is a constant $\lambda\in\mathbb{C}^{*}$ such that
$\varphi=\lambda\operatorname{id}+\psi$ where $\psi$ is nilpotent. Suppose
$\psi\neq 0$. Since $\psi$ is still a holomorphic automorphism of $E$, we have
$\mu(E)>\mu(\mathrm{Im}(\psi))$ and $\mu(E)>\mu(\mathrm{ker}(\psi))$ by
stability. Since $\mathrm{Im}(\psi)=E/\mathrm{ker}(\psi)$, we also get
$\mu(E)<\mu(\mathrm{Im}(\psi))$, a contradiction. Hence
$\varphi=\lambda\operatorname{id}$. ∎

The second property gives a hint why Mumford’s stability is the appropriate
notion, since stable objects have usually a small automorphism group.

#### Unitary character varieties.

We are now ready to characterize the holomorphic bundles arising through
unitary representations of the fundamental group:

######  Theorem 7.11 (Narasimhan–Seshadri [NS65]).

The character variety for the unitary group is in bijection with semistable
holomorphic bundles of degree 0:

$\boxed{\operatorname{Rep}(\pi_{1}\Sigma,\mathrm{U}(n))\cong\operatorname{Hol}^{ss}_{\deg=0}(S,\operatorname{GL}_{n}(\mathbb{C})).}$

Moreover, the stable points correspond to irreducible representations.

In other words: a holomorphic bundle $E=\oplus E_{i}$ with $E_{i}$
indecomposable, comes from a unitary representation iff all $E_{i}$ are stable
and of degree 0. Note that we can generalize Narasimhan–Seshadri theorem to
any group $K\subset\mathrm{U}(n)$:

$\operatorname{Rep}(\pi_{1}\Sigma,K)\cong\operatorname{Hol}^{ss}_{\deg=0}(S,K^{\mathbb{C}})$

where the holomorphic bundles have structure group $K^{\mathbb{C}}$.

Let us try to understand the profound meaning of the theorem. One direction is
less surprising: given a representation coming from a flat connection
$\nabla$, its $(0,1)$-part defines a holomorphic structure. The theorem
asserts that this holomorphic bundle is semistable.

The other direction is much more surprising: given a stable holomorphic
bundle, there is preferred _flat_ unitary connection on it! This is surprising
since a holomorphic structure is given by a $(0,1)$-part of a connection.
Looking for unitary connections, we can complete this $(0,1)$-part into a full
connection, but there is no reason for that connection to be flat! The point
is that we work on the level of moduli spaces, so within gauge equivalence
classes. The theorem asserts that in the gauge-orbit of a stable holomorphic
structure, there is a representative, given by a Cauchy–Riemann operator
$\bar{\partial}_{E}$, whose associated unitary connection is flat.

We present the idea of the proof in layers, like peeling an onion. The proof
strategy presented here is due to Donaldson [Don83] which uses the ideas of
the Kempf–Ness theorem.

###### Idea of proof.

We start by the Atiyah–Bott reduction which gives

$\operatorname{Rep}(\pi_{1}\Sigma,\mathrm{U}(n))\cong\mathcal{A}(\mathfrak{u}(n))\sslash\mathcal{G}$

where $\mathcal{A}(\mathfrak{u}(n))$ denotes the space of all unitary
connections on a trivial complex bundle over $S$ and
$\mathcal{G}=\mathcal{G}(\mathrm{U}(n))$ denotes the unitary gauge group.

By the principle of the Kempf–Ness theorem, we have

$\mathcal{A}(\mathfrak{u}(n))\sslash\mathcal{G}\cong\mathcal{A}^{ss}(\mathfrak{u}(n))/\mathcal{G}^{\mathbb{C}}$

where $\mathcal{G}^{\mathbb{C}}$ denotes the complex gauge group. Note that
the difficult part of the proof is to make this isomorphism work since it is
an infinite-dimensional analog of the Kempf–Ness theorem.

Since a unitary connection is uniquely determined by its $(0,1)$-part:

$\mathcal{A}^{ss}(\mathfrak{u}(n))/\mathcal{G}^{\mathbb{C}}\cong\Omega^{ss,(0,1)}(\mathfrak{gl}_{n}(\mathbb{C}))/\mathcal{G}^{\mathbb{C}}.$

Finally, since a Cauchy–Riemann operator determines a holomorphic structure,
we get from Equation (7.2):

$\Omega^{ss,(0,1)}(\mathfrak{gl}_{n}(\mathbb{C}))/\mathcal{G}^{\mathbb{C}}\cong\operatorname{Hol}^{ss}_{\deg=0}(S,\operatorname{GL}_{n}(\mathbb{C})).$

∎

This elegant proof, combining nicely all the material we have seen before, is
only the first layer, lacking lots of important details. In a second layer,
one has to prove two things: the stability condition appearing in the proof
idea is identical with the Mumford slope-stability, and the Kempf–Ness theorem
can be adapted to the infinite-dimensional setting.

To carry out the latter, Donaldson imitates the proof strategy of the
Kempf–Ness theorem. The rough idea is to show that the complex gauge orbit of
a unitary connection intersects the zero-set of the moment map (the flatness
condition) iff the connection is semistable. To achieve this, one uses a
gradient descent method. The function we consider for that is simply the norm
of the moment map (for some adapted $L^{2}$-norm):

$A\mapsto\left\|F(A)\right\|^{2}.$

This is called the _Yang–Mills functional_. The absolute minima of this
function are obviously given by flat connections.

So you start with a point, apply the gradient flow to get a sequence of
connections in the same complex gauge orbit. What you have to show is that you
converge to an absolute minimum iff your starting point is semistable.

This can be done in a third layer using the Uhlenbeck–Yau compactness theorem.
See Donaldson’s paper [Don83] for details.

We might ask, what is so special about bundles coming from unitary
representations? One aspect is the following: since the transition functions
are unitary, they are in particular bounded. Hence any holomorphic section of
$E$ is constant, by the maximum principle ($S$ is compact and transitions are
bounded).

To understand character varieties for non-unitary groups, especially non-
compact groups, we need the notion of Higgs bundles.

##  8 Higgs bundles and the non-abelian Hodge correspondence

In this section we will see how the notion of a Higgs bundle naturally arises.
We then state the main theorem of our lecture, the non-abelian Hodge
correspondence.

#### Cotangent space to
$\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))$.

The Narasimhan–Seshadri theorem has told us that stable holomorphic bundles
correspond to irreducible unitary representations of $\pi_{1}\Sigma$. In some
sense, the holomorphic structure of a bundle is encoded in a $(0,1)$-form
which can be uniquely completed to a unitary connection.

To describe representations
$\pi_{1}\Sigma\to\operatorname{GL}_{n}(\mathbb{C})$, we need more than
holomorphic bundles: something which is encoded both in a $(0,1)$-form and a
$(1,0)$-form. Put $\mathfrak{g}=\mathfrak{gl}_{n}(\mathbb{C})$. We remark that

$T^{*}\Omega^{0,1}(S,\mathfrak{g})\cong\Omega^{0,1}(S,\mathfrak{g})\oplus\Omega^{1,0}(S,\mathfrak{g}).$
(8.1)

Indeed, the tangent space at any point is given by
$\Omega^{0,1}(S,\mathfrak{g})$ itself (since it is a vector space). For
$\alpha\in\Omega^{1,0}(S,\mathfrak{g})$ and
$\beta\in\Omega^{0,1}(S,\mathfrak{g})$ the map

$\langle\alpha,\beta\rangle=\int_{S}\operatorname{tr}\alpha\wedge\beta$

is a non-degenerate pairing, which explains Equation (8.1).

This leads to the idea to consider the cotangent bundle of
$\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))$.

######  Proposition 8.1.

$T^{*}\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))\cong\\{(\bar{A},\Phi)\in\Omega^{0,1}\times\Omega^{1,0}\;\big{|}\;\bar{\partial}\Phi+[\bar{A},\Phi]=0\\}/\mathcal{G}$

where the gauge group $\mathcal{G}$ acts by
$g.\bar{A}=gAg^{-1}+gd\left(g^{-1}\right)$ and $g.\Phi=g\Phi g^{-1}$.

The gauge action is explained by the idea that $\Phi$ is a cotangent vector,
so gives a small deformation $\bar{\partial}+\varepsilon\Phi+\bar{A}$. Since
the gauge parameter does not involve $\varepsilon$, it acts on $\Phi$ simply
by conjugation.

The main ingredient to proof the proposition is the formula
$T^{*}(X/G)=T^{*}X\sslash G$ seen in Exercise 4.25.

###### Proof.

Since
$\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))=\Omega^{0,1}(S,\mathfrak{g})/\mathcal{G}$,
we have

$T^{*}\operatorname{Hol}(S,\operatorname{GL}_{n}(\mathbb{C}))=T^{*}\Omega^{0,1}(S,\mathfrak{g})\sslash\mathcal{G}=(\Omega^{0,1}(S,\mathfrak{g})\oplus\Omega^{1,0}(S,\mathfrak{g}))\sslash\mathcal{G}$

by Equation (8.1). The symplectic form is given by

$\omega=\int_{S}\operatorname{tr}\delta\bar{A}\wedge\delta\Phi.$

To compute the moment map, we first compute the infinitesimal gauge action by
$g=1+\varepsilon$. We easily get the vector fields representing an
infinitesimal change

$\bar{A}_{\varepsilon}=-\bar{\partial}\varepsilon+[\varepsilon,\bar{A}]$

and

$\Phi_{\varepsilon}=[\varepsilon,\Phi].$

By the usual procedure, we compute

$\displaystyle\iota_{(\bar{A}_{\varepsilon},\Phi_{\varepsilon})}\omega(\delta\bar{A},\delta\Phi)$
$\displaystyle=\int_{S}\operatorname{tr}(\bar{A}_{\varepsilon}\delta\Phi-\delta\bar{A}\Phi_{\varepsilon})$
$\displaystyle=\int_{S}\operatorname{tr}((-\bar{\partial}\varepsilon+[\varepsilon,\bar{A}])\delta\Phi-\delta\bar{A}[\varepsilon,\Phi])$
$\displaystyle=\int_{S}\operatorname{tr}\varepsilon(\bar{\partial}\delta\Phi+[\bar{A},\delta\Phi]+[\delta\bar{A},\Phi])$
$\displaystyle=\delta\left(\int_{S}\operatorname{tr}\varepsilon(\bar{\partial}\Phi+[\bar{A},\Phi])\right).$

Hence, the moment map is given by
$\mu(\bar{A},\Phi)=\bar{\partial}\Phi+[\bar{A},\Phi]$, which gives the
Proposition. ∎

We can understand the Proposition in more conceptual terms: $\bar{A}$ defines
a Cauchy–Riemann operator on $E$, which induces a holomorphic structure on
$\operatorname{End}(E)$ given by a Cauchy–Riemann operator
$\bar{\partial}_{\operatorname{End}(E)}$. On a section $\Phi$ of
$\operatorname{End}(E)$ it acts via

$\bar{\partial}_{\operatorname{End}(E)}\Phi=\bar{\partial}{\Phi}+[\bar{A},\Phi].$

If $\Phi$ is the $\operatorname{End}(E)$-valued $(1,0)$-form from above, the
Proposition tells us that $\bar{\partial}_{\operatorname{End}(E)}\Phi=0$,
hence $\Phi$ is holomorphic. Such an object is called a Higgs field. In
technical terms, we have

$\Phi\in\mathrm{H}^{0}(S,\operatorname{End}(E)\otimes K)$

where $K$ denotes the canonical bundle (holomorphic $(1,0)$-forms), but you
really should think of a Higgs field as a cotangent vector to the moduli space
of holomophic structures.

######  Definition 8.2.

A Higgs bundle is a holomorphic bundle $E$ equipped with a Higgs field
$\Phi\in\mathrm{H}^{0}(S,\operatorname{End}(E)\otimes K)$.

#### Moduli space.

We want to define the moduli space of Higgs bundles by

$\mathcal{M}_{H}(S,\operatorname{GL}_{n}(\mathbb{C}))=\\{\text{Higgs
bundles}\\}/\mathcal{G}.$

As usual, we have to take the GIT quotient to get a nice topological space (in
fact we get a manifold).

The appropriate stability condition is the following:

######  Definition 8.3.

A Higgs bundle $(E,\Phi)$ is stable, if for all $\Phi$-invariant holomorphic
subbundles $F\subset E$ we have $\mu(F)<\mu(E)$. It is called semistable if
the inequality is not necessarily strict.

Note that $\Phi$-invariant means that for all vector fields $X\in\Gamma(TS)$,
we have $\Phi(X).F\subset F$.

######  Example 8.4.

If $E$ is a stable holomorphic bundle, then $(E,\Phi)$ is a stable Higgs
bundle for all Higgs fields $\Phi$.

######  Example 8.5.

Fix a so-called spin structure on $S$, i.e. a line bundle denoted by $K^{1/2}$
whose square is the canonical bundle $K$. Then, consider

$\left(E=K^{1/2}\oplus K^{-1/2},\Phi=\begin{pmatrix}0&0\\\
1&0\end{pmatrix}\right).$

Note that the non-zero entry in $\Phi$ makes sense since it is an element of

$\operatorname{Hom}(K^{1/2},K^{-1/2})\otimes K\cong K^{-1}\otimes
K\cong\mathcal{O}$

which is the trivial line bundle. It is easy to check that the only non-
trivial $\Phi$-invariant subbundle is $K^{-1/2}$. Since
$\deg(K^{-1/2})<0=\deg(E)$, we get that $(E,\Phi)$ is a stable Higgs bundle.

Note that in the last example, $E$ is not stable as holomorphic bundle since
it has $K^{1/2}$ as holomorphic subbundle.

From the two examples, we see that

$T^{*}\mathrm{Hol}^{s}(S,\mathrm{SL}_{n}(\mathbb{C}))\subset\mathcal{M}_{H}\subset
T^{*}\mathrm{Hol}(S,\mathrm{SL}_{n}(\mathbb{C})),$ (8.2)

i.e. the moduli space of Higgs bundle sits between two cotangent bundles of
holomorphic structures.

######  Proposition 8.6.

Stability is an open condition, i.e. if $(E,\Phi)$ is a stable Higgs bundle,
then any Higgs bundle sufficiently close to it is also stable.

#### Non-abelian Hodge correspondence

We are now ready to state the main theorem of our lectures:

######  Theorem 8.7 (non-abelian Hodge correspondence).

The moduli space of polystable Higgs bundles of degree 0 is diffeomorphic to
the $\operatorname{GL}_{n}(\mathbb{C})$-character variety:

$\boxed{\mathcal{M}^{ps}_{H,\deg=0}(S,\operatorname{GL}_{n}(\mathbb{C}))\cong\operatorname{Rep}^{c.r.}(\pi_{1}\Sigma,\operatorname{GL}_{n}(\mathbb{C})).}$

The non-abelian Hodge correspondence gives a remarkable link between purely
topological objects, completely reducible representations of the fundamental
group, and holomorphic objects, polystable Higgs bundles. Irreducible
representations correspond to stable Higgs bundles.

It generalizes the Narasimhan–Seshadri theorem: we will see that an
irreducible unitary representation corresponds to a stable Higgs bundle
$(E,\Phi)$ with vanishing Higgs field $\Phi=0$. So necessarily $E$ is stable.

Why is it called “non-abelian Hodge correspondence”? The answer here is
inspired by the introduction of Simpson’s paper [Sim88].

A basic motto in algebraic topology is that _all cohomology theories are (more
or less) equivalent_. This is why there is an axiomatization of cohomology (by
Eilenberg–Steenrod). One instance of this motto is the _Hodge correspondence_
, which gives a link between de Rham cohomology (defined using differential
geometry) and Dolbeault cohomology (defined using holomorphic objects):

$H^{k}_{dR}(X,\mathbb{C})\cong\bigoplus_{p+q=k}H^{p,q}_{Dol}(X).$

For $k=1$ we get

$H^{1}(S,\mathbb{C})\cong H^{0,1}(S)\oplus H^{1,0}(S)\cong
H^{1}(S,\mathcal{O}_{S})\oplus H^{0}(S,\Omega^{1}(S)),$

where we used sheaf cohomology in the last term.

Replacing $\mathbb{C}$ by $\operatorname{GL}_{n}(\mathbb{C})$, which is non-
abelian for $n>1$, we can interpret
$\operatorname{Rep}(\pi_{1}S,\operatorname{GL}_{n}(\mathbb{C}))$ as
$H^{1}(S,\operatorname{GL}_{n}(\mathbb{C}))$ since for $n=1$:

$H^{1}(S,\operatorname{GL}_{1}(\mathbb{C}))=H^{1}(S,\mathbb{C}^{*})\cong\operatorname{Hom}(\pi_{1}S,\mathbb{C}^{*})=\operatorname{Rep}(\pi_{1}S,\mathbb{C}^{*})$

where we used in the last equality that $\mathbb{C}^{*}$ is abelian.

The generalization of $H^{1}(S,\mathcal{O}_{S})$ is
$\check{H}^{1}(S,\operatorname{Hol}(\operatorname{GL}_{n}(\mathbb{C})))$ which
describes holomorphic bundles of rank $n$. Finally $H^{0}(S,\Omega^{1}(S))$
becomes $H^{0}(S,\operatorname{GL}_{n}(\mathbb{C})\otimes K)$ giving the Higgs
field.

The analogy can even be enlarged: the most basic cohomology theories
(simplicial, singular or cellular) are purely topological. They are sometimes
calles Betti cohomology. By the basic motto, they all coincide with the de
Rham or Dolbeault cohomology.

The non-abelian analogs are (names were given by Simpson):

  * •

Betti moduli space: space of representations of the fundamental group, i.e.
the character variety
$\operatorname{Rep}(\pi_{1}\Sigma,\operatorname{GL}_{n}(\mathbb{C}))$.

  * •

de Rham moduli space: space of flat connections.

  * •

Dolbeault moduli space: space of Higgs bundles.

The Betti and de Rham moduli space are equivalent by the Riemann–Hilbert
correspondence. They are equivalent to the Dolbeault moduli space by the non-
abelian Hodge correspondence.

We will see even more analogies: in the presence of a metric there is a
preferred representative in each de Rham cohomology class, a harmonic form.
The analog leads to the notion of harmonic bundles which are the key
ingredient to prove the non-abelian Hodge correspondence.

##  9 The proof strategy: Harmonic bundles

In this section we will see two notions of harmonic bundles, giving
representatives for flat bundles and Higgs bundles respectively. The existence
of harmonic representatives are described by the theorems of
Corlette–Donaldson and Hitchin–Simpson, which together give the non-abelian
Hodge correspondence.

#### Hermitian bundles.

The basic ingredient to harmonic theory is the notion of a _hermitian bundle_
, which is a complex bundle with a hermitian product $(.,.)$ in each fiber,
varying in a smooth manner.

In a holomorphic bundle $E$, a hermitian structure determines a preferred
connection, similar to the Levi–Civita connection for a Riemannian manifold:

######  Proposition 9.1.

In a hermitian holomorphic bundle $E$, there is a unique connection $\nabla$,
called the Chern connection, which is compatible with

  1. 1.

the holomorphic structure: $\nabla^{0,1}=\bar{\partial}_{E}$,

  2. 2.

the hermitian structure: $d(s_{1},s_{2})=(\nabla s_{1},s_{2})+(s_{1},\nabla
s_{2})$ for all sections $s_{1},s_{2}$.

Let us analyze the space of all hermitian structures on a given flat bundle
$E=E_{\rho}$, where $\rho:\pi_{1}\Sigma\to
G=\operatorname{GL}_{n}(\mathbb{C})$ is the monodromy. At one point it is
described by

$\mathrm{Herm}^{++}=\\{H\in\mathfrak{gl}_{n}\mid H^{\dagger}=H,\text{ positive
definite}\\}.$

Indeed, a matrix $H\in\mathrm{Herm}^{++}$ determines a Hermitian product by
$(x,y)_{H}=x^{\dagger}Hy$. On $\mathrm{Herm}^{++}$, there is an action of
$G=\operatorname{GL}_{n}(\mathbb{C})$ by $(x,y)_{g.H}=(gx,gy)_{H}$ (where
$g\in\operatorname{GL}_{n}(\mathbb{C})$). Hence it is given by
$g.H=g^{\dagger}Hg$. The action is transitive and the stabilizer of
$\operatorname{id}\in\mathrm{Herm}^{++}$ is $K=\mathrm{U}(n)$. Therefore

$\mathrm{Herm}^{++}\cong\operatorname{GL}_{n}(\mathbb{C})/\mathrm{U}(n)=G/K.$

Note that $\mathrm{U}(n)$ is not a normal subgroup, so the quotient is merely
a set.

Locally, in a given trivialization of $E$, a hermitian structure is given by a
map $U\subset\Sigma\to G/K$. Globally over $\Sigma$, the map is not well-
defined since taking a non-trivial loop $\gamma\in\pi_{1}\Sigma$ results in a
conjugated $G/g^{\dagger}Kg$ where $g=\rho(\gamma)$.

To get a well-defined map, we have to consider the universal cover
$\widetilde{\Sigma}$ and maps $\widetilde{\Sigma}\to G/K$ which are
equivariant with respect to $\pi_{1}\Sigma$, which acts on the universal cover
by deck transformations and on $G/K$ via $\rho$.

######  Proposition 9.2.

The space of hermitian structures on $E=E_{\rho}$ can be identified with the
space of $\pi_{1}\Sigma$-equivariant functions $u:\widetilde{\Sigma}\to G/K$.

For more details, we refer to [Tou15, Prop. 2].

#### Harmonic flat bundles and Corlette–Donaldson theorem.

For a point in the moduli space of flat connections (a flat connection modulo
gauge equivalence), we wish to define a nice representative, a harmonic flat
bundle. This representative should exist whenever the point in the de Rham
moduli space is semistable. We have seen in Section 6 that this is the case
whenever the monodromy is completely reducible.

Thus, to find the appropriate notion of a harmonic flat bundle, we have to
translate the property of being completely reducible from the Betti moduli
space to the de Rham moduli space, i.e. in terms of flat bundles.

Recall the Riemann–Hilbert correspondence:

$\rho\in\operatorname{Rep}(\pi_{1}\Sigma,\operatorname{GL}(V))\mapsto(E_{\rho},\nabla)\text{
flat bundle given by }E_{\rho}=(\widetilde{\Sigma}\times V)/\pi_{1}\Sigma.$

Clearly, a subrepresentation corresponds to a $\nabla$-invariant subbundle.
Thus, completely reducible representations correspond to completely reducible
flat bundles.

Consider for example a hermitian structure $h$ on the vector space $V$ such
that $\mathrm{Im}(\rho)\subset U(V,h)$. If $F$ is a $\nabla$-invariant
subbundle, then $F^{\perp_{h}}$ as well and
$\nabla=\nabla_{F}+\nabla_{F^{\perp}}$ since $\nabla$ is unitary.

######  Goal.

Find a condition on $\nabla$, where $(E,\nabla)$ is a flat bundle, which
ensures complete reducibility of $E$.

For a fixed hermitian structure $h$, we can decompose $\nabla=d_{A}+\Psi$
where $d_{A}$ is a unitary connection and $\Psi$ is the hermitian part of
$\nabla$. Locally, this is nothing but writing a matrix as a sum of a
hermitian and an anti-hermitian matrix.

Consider $F\subset E$ a $\nabla$-invariant subbundle. As complex bundles, we
have $E=F\oplus F^{\perp}$. So we can write

$\nabla=\begin{pmatrix}\nabla_{1}&\eta\\\ 0&\nabla_{2}\end{pmatrix}\;\text{
with }\;\eta\in\Omega^{1}(\Sigma,\operatorname{Hom}(F^{\perp},F)).$

Being reducible means that $\eta=0$, so we look for a condition which forces
$\eta$ to vanish. For that, the idea due to Corlette is to consider
$\operatorname{End}(E)$ with induced connection $d_{A}$ and the section
$s=-\operatorname{id}_{F}\oplus\operatorname{id}_{F^{\perp}}$. Decompose
$\nabla_{i}=d_{A_{i}}+\Psi_{i}$ into unitary and hermitian parts, hence

$\nabla=\begin{pmatrix}\nabla_{1}&\eta\\\
0&\nabla_{2}\end{pmatrix}=\begin{pmatrix}d_{A_{1}}&\eta/2\\\
-\eta^{*}/2&d_{A_{2}}\end{pmatrix}+\begin{pmatrix}\Psi_{1}&\eta/2\\\
\eta^{*}/2&\Psi_{2}\end{pmatrix}.$

Then compute $d_{A}s$:

$d_{A}s=\begin{pmatrix}d_{A_{1}}&0\\\
0&d_{A_{2}}\end{pmatrix}\begin{pmatrix}-\operatorname{id}_{F}&0\\\
0&\operatorname{id}_{F^{\perp}}\end{pmatrix}+\left[\begin{pmatrix}0&\eta/2\\\
-\eta^{*}/2&0\end{pmatrix},\begin{pmatrix}-\operatorname{id}_{F}&0\\\
0&\operatorname{id}_{F^{\perp}}\end{pmatrix}\right]=\begin{pmatrix}0&\eta\\\
\eta^{*}&0\end{pmatrix}.$

Hence

$\langle\Psi,d_{A}s\rangle_{L^{2}}=\int_{\Sigma}(\Psi,d_{A}s)_{h}=\int_{\Sigma}\operatorname{tr}\begin{pmatrix}\Psi_{1}&\eta/2\\\
\eta^{*}/2&\Psi_{2}\end{pmatrix}\begin{pmatrix}0&\eta\\\
\eta^{*}&0\end{pmatrix}=\langle\eta,\eta\rangle_{L^{2}}.$

We also have $\langle\Psi,d_{A}s\rangle=\langle d_{A}^{*}\Psi,s\rangle$ using
the adjoint. So to get $\eta=0$, it is sufficient to require

$d_{A}^{*}\Psi=0.$

######  Definition 9.3.

A metric $h$ on $(E,\nabla)$ is harmonic if $d_{A}^{*}\Psi=0$ where
$\nabla=d_{A}+\Psi$.

A flat bundle equipped with a harmonic hermitian metric is called _harmonic
flat bundle_.

######  Proposition 9.4.

A bundle $(E,\nabla=d_{A}+\Psi,h)$ is a harmonic flat bundle iff
$F(A)+\Psi\wedge\Psi=0$ and $d_{A}\Psi=0=d_{A}^{*}\Psi$.

###### Proof.

The onyl thing to check is that $\nabla$ is flat. The curvature of $\nabla$ is
$F(A)+\Psi\wedge\Psi+d_{A}\Psi$. The unitary part $F(A)+\Psi\wedge\Psi$ and
the hermitian part $d_{A}\Psi$ have to vanish both. ∎

The main theorem which will give half of the non-abelian Hodge correspondence
is the Corlette–Donaldson theorem [Cor88, Don87]:

######  Theorem 9.5 (Corlette–Donaldson).

A flat bundle $(E,\nabla)$ admits a harmonic metric iff it is completely
reducible, i.e. iff its monodromy is completely reducible. In addition, this
harmonic metric is unique up to an overall positive constant factor.

In other words

$\boxed{\operatorname{Rep}^{c.r.}(\pi_{1}\Sigma,G)\cong\\{\text{flat
}G\text{-bundles}\\}/\text{gauge}\cong\\{\text{harmonic
}G\text{-bundles}\\}/\text{constants}.}$

This is the analog of the harmonic representative in each de Rham cohomology
class.

Let us see how to prove one direction of the non-abelian Hodge correspondence
from the Corlette–Donaldson theorem: we can associate a Higgs bundle to a
given flat bundle $(E,\nabla)$. Use a harmonic metric to decompose
$\nabla=d_{A}+\Psi$.

Using a complex structure on $\Sigma$, we can further decompose
$\Psi=\Phi+\Phi^{*_{h}}$ where $\Phi$ is the $(1,0)$-part of $\Psi$. The
$(0,1)$-part of the unitary connection $d_{A}$ gives a holomorphic structure
on $E$. Since $d_{A}\Psi=0$, we also have $d_{A}^{0,1}\Phi=0$. Hence $\Phi$ is
a Higgs field. So to $(E,\nabla,h)$, we can associate
$(E,d_{A}^{0,1},\Psi^{1,0})$ which is a Higgs bundle. The fact that it is
semistable comes from the second half, the Hitchin–Simpson theorem.

#### Harmonic map theory.

To get an idea of the proof of the Corlette–Donaldson theorem, we give a brief
introduction to harmonic maps.

The general setting is as follows: consider two Riemannian manifolds $(M,g)$
and $(N,G)$. To a smooth map $f:M\to N$, we associate the so-called Dirichlet
energy

$E(f)=\int_{M}\left\|df^{2}\right\|d\mathrm{vol}(g)=\int_{M}\frac{\partial
f^{i}}{\partial x^{\alpha}}\frac{\partial f^{j}}{\partial
x^{\beta}}G_{ij}g^{\alpha\beta}\sqrt{\lvert\mathrm{det}(g)\rvert}dx$ (9.1)

where we use the Einstein sum convention, $x^{\alpha}$ are coordinates on $M$
and $(g^{\alpha\beta})$ is the inverse of the matrix $(g_{\alpha\beta})$.

A map $f$ is called harmonic if it is a critical point of $E$. This is the
case iff $\Delta_{g,G}f=0$, where $\Delta_{g,G}$ is some generalization of the
Laplacian, which explains the name “harmonic”.

Some examples:

  * •

For $\dim M=1$, a harmonic map $f:M\to N$ is the same as a geodesic in $N$
parametrized by $M$.

  * •

For $\dim N=1$, being harmonic is equivalent to $\Delta_{M}f=0$ where
$\Delta_{M}$ is the Laplace–Beltrami operator.

  * •

For $\dim M=2$, the energy only depends on a conformal class of $g$, i.e. the
metrics $g$ and $e^{\varphi}g$ for $\varphi$ a function on $M$ gives the same.
In dimension 2, a conformal class of a metric is the same as a complex
structure, so we can do harmonic map theory with $M$ being a Riemann surface.

The main result of harmonic map theory is the following theorem due to Eells
and Sampson [ES64]:

######  Theorem 9.6 (Eells–Sampson).

If $(M,g)$ and $(N,G)$ are compact Riemannian manifolds where $N$ has non-
positive sectional curvature, then there is a unique harmonic map in each
homotopy class of functions $[M,N]$.

The main idea for the proof is to start with any function in a given class and
to apply a heat flow (some kind of steepest descent flow). One has to show
that the flow exists for a short time, then for all times and that when time
goes to infinity, we get a well-defined limit which is harmonic.

Now, we are ready to see the proof idea of the Corlette–Donaldson theorem. It
reduces nearly to the Eells–Sampson theorem, in an equivariant setting.

Remember that we wish to show that in the gauge-orbit of a flat connection
(with completely reducible monodromy) there is a harmonic representative. So
we have a flat bundle $(E,\nabla)$ with fixed hermitian metric $h$ and we vary
$\nabla$ in its gauge-orbit.

The first important idea is to notice that we can fix $\nabla$ and vary $h$
instead. To determine the action of a gauge transformation $g$ on $h$, just
note that $\Psi$ is hermitian with respect to $h$, i.e. $\Psi^{*_{h}}=\Psi$.
So $g.\Psi$ has to be hermitian with respect to $g.h$. Locally, we can write
$\Psi^{*_{h}}=h\Psi^{\dagger}h^{-1}$. Hence

$g\Psi g^{-1}=g.\Psi=(g.\Psi)^{*_{g.h}}=(g.h)(g\Psi
g^{-1})^{\dagger}(g.h)^{-1}.$

Using $\Psi=\Psi^{*h}=h\Psi^{\dagger}h^{-1}$, we deduce

$g.h=ghg^{\dagger},$

which is the usual action on hermitian structures.

We have now a flat bundle $(E,\nabla)$ with varying hermitian metric $h$. We
have seen in Proposition 9.2 that a hermitian metric is a
$\pi_{1}\Sigma$-equivariant map $u:\widetilde{\Sigma}\to G/K$.

The second important observation is that $h$ is harmonic iff $u$ is harmonic
in the sense of harmonic map theory. Note that $\widetilde{\Sigma}$ is the
hyperbolic plane and $G/K$ is a symmetric space, so both carry a natural
Riemannian structure. One can check that changing $u$ by a homotopy is
equivalent to changing $h$ by a gauge transform.

Since $K$ is the maximal compact subgroup of $G$, the symmetric space $G/K$
has non-positive sectional curvature. Thus, we are almost in the setting of
the Eells–Sampson theorem. The only problem is that $\widetilde{\Sigma}$ and
$G/K$ are not compact. But our map $u$ is equivariant and the fundamental
domain is $\Sigma$ which is compact. In his paper [Cor88], Corlette imitates
the proof strategy of the Eells–Sampson theorem in the given setting. I
recommend his paper to find more details.

Finally, note that the Dirichlet energy for a flat connection
$\nabla=d_{A_{h}}+\Psi_{h}$ decomposed using a hermitian metric $h$, is given
by

$E_{\nabla}(h)=\int_{\Sigma}\left\|\Psi_{h}\right\|^{2}_{L^{2}}d\mathrm{vol}_{\Sigma}.$

#### Harmonic Higgs bundles and the Hitchin–Simpson theorem.

We present the notion of a harmonic Higgs bundle, giving a link between Higgs
bundles and flat connections. The existence of harmonic representatives, the
Hitchin–Simpson theorem, completes the proof of the non-abelian Hodge
correspondence.

To a stable Higgs bundle $(E,\Phi)$, we wish to associate a flat connection.
The idea is to fix a hermitian structure $h$ on $E$. Since $E$ is holomorphic,
we get the Chern connection $\nabla_{A}$. Then we consider

$\mathcal{A}=\Phi+\nabla_{A}+\Phi^{*_{h}}.$

The appearance of $\Phi+\Phi^{*_{h}}$ is not surprising since we wish to get
an equivalence with harmonic flat bundles $(E,\nabla=d_{A}+\Psi,h)$. Since
$\Phi=\Psi^{1,0}$, we have $\Psi=\Phi+\Phi^{*_{h}}$.

The strategy is to find a point in the gauge-orbit of $(E,\Phi)$, i.e. a point
in the moduli space $\mathcal{M}_{H}$, such that $\mathcal{A}$ is flat. There
is one important observation to be made: a $\mathbb{C}^{*}$-action on
$\mathcal{M}_{H}$ simplifying the flatness condition.

The flatness of $\mathcal{A}$ is a priori one complicated equation. The trick
is to split this into five much simpler equations. The fact which allows this
decomposition is a $\mathbb{C}^{*}$-action on $\mathcal{M}_{H}$ given by

$\lambda.[(E,\Phi)]=[(E,\lambda\Phi)],$

i.e. we simply scale the Higgs field. This is well defined since the scaling
commutes with the gauge action (where $\Phi$ simply gets conjugated) and one
easily checks that $(E,\lambda\Phi)$ stays stable.

Using this action, we are looking actually at a whole family of connections

$\mathcal{A}(\lambda)=\lambda\Phi+\nabla_{A}+\lambda^{-1}\Phi^{*_{h}}.$ (9.2)

The reason why to consider $\Phi^{*_{h}}$ with weight $\lambda^{-1}$ comes
from twistor theory, explained in Section 10.

The curvature of $\mathcal{A}(\lambda)$, which is a Laurent polynomial in
$\lambda$, is flat for all $\lambda$ iff all its coefficients are zero. In a
local chart where
$\mathcal{A}(\lambda)=d+\lambda\Phi+A+\lambda^{-1}\Phi^{*_{h}}$, we get

  1. 1.

$\Phi\wedge\Phi=0$ and $\Phi^{*}\wedge\Phi^{*}=0$ (coefficients of
$\lambda^{2}$ and $\lambda^{-2}$),

  2. 2.

$\bar{\partial}\Phi+[A^{0,1},\Phi]=0$ and
$\partial\Phi^{*}+[A^{1,0},\Phi^{*}]=0$ (coefficients of $\lambda$ and
$\lambda^{-1}$),

  3. 3.

$F(A)+[\Phi,\Phi^{*}]=0$ (coefficient for constant term).

Note that the couples of equations on the same line are equivalent (by taking
the hermitian conjugate). Equation 1. is automatic since $\Phi$ is of type
$(1,0)$ and we are on a surface101010For the notion of a Higgs bundle on a
higher-dimensional manifold, one requires $\Phi\wedge\Phi=0$ in the
definition. We see here why.. Equation 2. is also automatic since $\Phi$ is a
Higgs field, so holomorphic (see Proposition 8.1).

The only remaining equation is the so-called Hitchin equation:

$\boxed{F(A)+[\Phi,\Phi^{*}]=0.}$ (9.3)

######  Remark.

One can obtain this equation as a dimensional reduction of the Yang–Mills
equation in dimension 4.

######  Definition 9.7.

A harmonic Higgs bundle is a Higgs bundle $(E,\Phi)$ equipped with a hermitian
metric $h$, such that Hitchin’s equation (9.3) holds.

The main result is the theorem of Hitchin [Hit87] and Simpson [Sim88]:

######  Theorem 9.8 (Hitchin–Simpson).

In the complex gauge orbit of a stable Higgs bundle $(E,\Phi)$ with
$\deg(E)=0$, there is a unique (up to unitary gauge) harmonic representative
iff $(E,\Phi)$ is polystable.

Note that for $\Phi=0$, the Hitchin–Simpson theorem reduce to the
Narasimhan–Seshadri theorem, since the Hitchin equation becomes $F(A)=0$,
giving a flat unitary connection and $(E,0)$ is polystable iff $E$ is.

The proof is similar in spirit to the one of the Narasimhan–Seshadri theorem.
If there is a harmonic representative, one shows polystability by a direct
argument, see for example [WGP08, Section 3.2].

For the converse, one defines a gradient flow using the functional

$f(\Phi,h)=\int_{S}\left\|F(A)+[\Phi,\Phi^{*_{h}}]\right\|^{2}_{L^{2}}.$

Note that the integrand is nothing but some $L^{2}$-norm of the term from the
Hitchin equation. One has to show that the flow stays inside the gauge-orbit,
that a minimizing sequence $(A_{n},\Phi_{n})$ converges (for stable
$(E,\Phi)$) and that the limit solves the Hitchin equation. Again details can
be found in [WGP08, Section 3.2] and the original papers by Hitchin [Hit87]
and Simpson [Sim88].

We will see another proof sketch, similar to our proof sketch of the
Narasimhan–Seshadri theorem, by interpreting the Hitchin equation as a moment
map. To do so, we will introduce the notion of hyperkähler geometry and the
hyperkähler quotient in Section 10.

#### Non-abelian Hodge correspondence.

Now that we have the notions of harmonic representatives for both, flat
bundles and Higgs bundles, the non-abelian Hodge correspondence reduces to a
simple observation, the equivalence of harmonic flat bundles and harmonic
Higgs bundles.

To a harmonic flat bundle $(E,\nabla=d_{A}+\Psi,h)$, we associate the Higgs
bundle $(E,\bar{\partial}_{E}=d_{A}^{0,1},\Phi=\Psi^{1,0})$. Together with the
hermitian metric $h$, we actually get a harmonic Higgs bundle since the
flatness

$F(A)+\Psi\wedge\Psi=0$

is equivalent to Hitchin’s equation

$F(A)+[\Phi,\Phi^{*_{h}}]=0$

since $\Psi=\Phi+\Phi^{*_{h}}$.

In the reverse direction, to a harmonic Higgs bundle
$(E,\bar{\partial}_{E},\Phi,h)$, we associate $(E,d_{A}+\Phi+\Phi^{*_{h}},h)$,
where $d_{A}$ is the Chern connection. This is a harmonic flat bundle. Both
constructions are inverse to each other.

This finishes the proof sketch of the non-abelian Hodge correspondence, which
we summarized in Figure 9.1.

Figure 9.1: Proof scheme of non-abelian Hodge correspondence via harmonic
representatives

Taking a step back, we can say that from a flat connection $\nabla=d_{A}+\Psi$
decomposed into a unitary and hermitian part, we can easily get a holomorphic
structure $d_{A}^{0,1}$ and a Higgs field $\Psi^{1,0}$ (by forgetting half of
the information). The difficulty lies in finding a preferred decomposition,
which is achieved by using a harmonic hermitian structure.

The more surprising part of the non-abelian Hodge correspondence is that from
half of the data, a holomorphic structure and a Higgs field, we can recover
the flat connection by choosing an appropriate hermitian structure. The
flatness condition reduces to solving Hitchin’s equation.

To get a better and deeper understanding of the correspondence, we introduce
hyperkähler geometry. The hyperkähler structure is the strongest possible
geometric structure on a manifold (in a certain sense). This viewpoint will
unify the character variety and the moduli space of Higgs bundles into one big
picture.

##  10 Hyperkähler geometry

We give an overview on hyperkähler manifolds, in particular the quotient and
twistor construction. This allows to understand the non-abelian Hodge
correspondence as a natural diffeomorphism in the twistor space of the moduli
space of Higgs bundles. A nice reference is Hitchin’s paper [Hit92a].

#### Kähler trilogy.

A Kähler structure on a manifold is a Riemannian, symplectic and complex
structure which interact nicely such that any two structures determine the
third.

To start, let us see what happens at one point, i.e. we reduce to linear
algebra. Consider $\mathbb{R}^{2n}$. We have the following correpondences
between geometric structures and their symmetry groups:

  * •

Riemannian structure $\longleftrightarrow$ $\mathrm{O}_{2n}(\mathbb{R})$

  * •

Symplectic structure $\longleftrightarrow$ $\mathrm{Sp}_{2n}(\mathbb{R})$

  * •

Complex structure $\longleftrightarrow$ $\mathrm{GL}_{n}(\mathbb{C})$

######  Proposition 10.1.

The intersection of any two of these three groups is $\mathrm{U}(n)$.

Figure 10.1: Kähler trilogy for linear groups

The group $\mathrm{U}(n)$ corresponds to hermitian structures on
$\mathbb{R}^{2n}\cong\mathbb{C}^{n}$. Our convention is that a hermitian
product $h(X,Y)$ is $\mathbb{C}$-linear in $Y$ and anti-$\mathbb{C}$-linear in
$X$.

######  Proposition 10.2.

If $h\in\mathrm{Herm}^{++}$, then $h=g+i\omega$ where $g$ is a Riemannian and
$\omega$ a symplectic structure.

This realizes the Kähler trilogy in one equation. In addition, we have

$\omega(X,iY)=\mathrm{Im}(h(X,iY))=\mathrm{Im}(ih(X,Y))=\mathrm{Re}(h(X,Y))=g(X,Y).$

On a manifold, we gather these three structures together to define:

######  Definition 10.3.

A Kähler manifold is $(M,g,\omega,I)$ where $g$ is a Riemannian, $\omega$ a
symplectic and $I$ a complex structure such that

$g(X,Y)=\omega(X,IY)\;\;\forall\;X,Y\in\Gamma(TM).$

Note that in this definition, we consider the endomorphism
$I\in\mathrm{End}(TM)$ satisfying $I^{2}=-\mathrm{id}$, which mimics the
multiplication by $i$. Such a structure is called an _almost-complex
structure_. Any complex structure induces an almost-complex structure, but the
converse is not true. An almost-complex structure which comes from a complex
structure is called integrable. The symplectic form $\omega$ is usually called
the _Kähler form_.

Using the fact that any two structures determine the third, we can
equivalently define a Kähler manifold in the following ways:

  * •

Riemannian viewpoint: Riemannian manifold with almost-complex structure $J$
which is orthogonal ($g(X,Y)=g(IX,IY)$) and with vanishing covariant
derivative (using the Levi–Civita connection).

  * •

Complex viewpoint: Complex manifold with hermitian structure $h$ such that
$\mathrm{Im}(h)$ is closed.

######  Example 10.4.

Consider the simplest case $M=\mathbb{C}$ with hermitian metric $h=dz\otimes
d\bar{z}$. Using real coordinates $z=x+iy$, we see that

$h=(dx+idy)\otimes(dx-idy)=(dx^{2}+dy^{2})-i(dx\wedge dy).$

Hence the real part of $h$ is the standard Riemannian structure, and the
negative imaginary part the standard symplectic structure on $\mathbb{R}^{2}$.

Other examples:

  * •

$\mathbb{C}P^{n}$ is Kähler since the hamiltonian reduction of
$\mathbb{C}^{n+1}$ by $\mathbb{S}^{1}$ gives not only a symplectic, but also a
complex structure111111There is the notion of Kähler reduction as for
symplectic manifolds..

  * •

Complex submanifolds of Kähler manifolds are again Kähler, so in particular
any complex projective variety. In particular all Riemann surfaces are Kähler.

######  Proposition 10.5.