FACULTY OF SCIENCE

Moduli Spacesof Flat Connections

Daan MICHIELS

Promotors: Prof. dr. F. DillenKU LeuvenProf. dr. J. Van der VekenKU Leuven

Copromotors: Prof. dr. M. CrainicUtrecht UniversityDr. F. SchätzUtrecht University

Thesis presented infulfillment of the requirements

for the degree of Master of Sciencein Mathematics

Academic year 2012-2013

c©Copyright by KU LeuvenWithout written permission of the promotors and the authors it is forbidden to repro-

duce or adapt in any form or by any means any part of this publication. Requests forobtaining the right to reproduce or utilize parts of this publication should be addressedto KU Leuven, Faculteit Wetenschappen, Geel Huis, Kasteelpark Arenberg 11 bus 2100,3001 Leuven (Heverlee), Telephone +32 16 32 14 01.

A written permission of the promotor is also required to use the methods, products,schematics and programs described in this work for industrial or commercial use, and forsubmitting this publication in scientific contests.

Preface

The goal of this thesis is to construct and study moduli spaces of flat connections. Thesemoduli spaces are the spaces of all possible flat principal bundles over a given manifoldwith a given structure group, up to isomorphism. This defines the space as a mere set, andour main goal is to endow the moduli space with extra structure (a topology, a smoothstructure, a symplectic structure).

This thesis was written in the 2012-2013 academic year at two different universities.From September 2012 until January 2013 I was at Utrecht University as an exchange stu-dent, where I started learning about moduli spaces of flat connections under the supervisionof professor Marius Crainic and the guidance of doctor Florian Schätz. From February 2012until June 2013, my thesis research continued at the KU Leuven. Supervision was providedby professor Franki Dillen, whose sad passing away in April 2013 meant a great loss tothe department of mathematics. Later supervision was provided by professor Joeri Vander Veken. I am grateful to Marius, Florian, Franki and Joeri for their patience, help andguidance. I would also like to thank professor Johan Quaegebeur for his help in practicalmatters.

The text assumes only that the reader has a working knowledge of differential geometry,is familiar with the basic ideas behind fundamental groups and covering spaces and knowswhat morphisms in a category are. In particular, we will make use of manifolds, smoothmaps and their derivatives, Lie brackets of vector fields, differential forms, fundamentalgroups of surfaces and lifts of curves to covering spaces without explanation. In contrast,the reader need not know anything about Lie groups, Lie algebras, bundles and vector-valued differential forms as their basics are explained in the first few chapters.

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Summary (samenvatting)

(Dutch version on next page – Nederlandstalige versie op volgende pagina)

The aim of this work is to understand moduli spaces of flat connections. To a givenmanifold Σ and Lie group G we can associate the moduli spaceM(Σ, G) of flat principal G-bundles overM . This moduli space can be equipped with a topology. We also equip it witha smooth structure in a certain sense, although the moduli space typically has singularities.Inspired by the procedure of symplectic reduction, we describe how the moduli space mayin some cases be equipped with a symplectic structure.

Chapters 1, 2, 3 and 4 contain material that is necessary to understand what the modulispaces M(Σ, G) are. The first chapter discusses Lie groups and Lie algebras on a basiclevel. The second chapter introduces principal bundles and their connections starting fromthe more general concept of a fibre bundle. Vector bundles are also treated separately. Thethird chapter contains some of the theory of vector-valued differential forms. These arenecessary in later chapters to understand the symplectic structure on the moduli space.The fourth chapter talks about symplectic and Poisson geometry. The most importantconcept here is symplectic reduction, a procedure we will later use to equip the modulispace with a symplectic structure.

Chapters 5, 6 and 7 study the moduli spaces themselves. The fifth chapter is the heartof the text, as it explains how the moduli space can be identified with Hom(π1(Σ), G)/G,the space of all morphism from the fundamental group of Σ to G up to conjugation. Thisleads to a topology and a smooth structure onM(Σ, G). In this chapter we also explainhow the moduli space may be regarded as a symplectic quotient of an infinite-dimensionalsymplectic manifold under a Hamiltonian action if Σ is a compact orientable surface. Thesixth chapter contains a topological characterization of the reducible part of the modulispace for compact orientable surfaces. The case G = SU(2) is treated in greater detail,where it is shown that a large part of the moduli space is smooth. The last chapter derivesan explicit formula for the symplectic structure on the moduli space, which expresses thesymplectic form as a integral over a 1-chain instead of a 2-chain. Finally, the abelian caseis discussed, in which the new formula simplifies considerably and yields an integral-freeexpression for the symplectic form.

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Het doel van dit werk is het begrijpen van moduliruimten van vlakke connecties. Aan eengegeven differentiaalvariëteit Σ en Lie-groep G kunnen we de moduliruimte M(Σ, G) vanvlakke G-hoofdvezelbundels overM associëren. Deze moduliruimte kan voorzien worden vaneen topologie. In zekere zin voorzien we ze daarenboven van een gladde structuur, hoewel demoduliruimte singulariteiten heeft. Naar analogie met symplectische reductie beschrijvenwe hoe de moduliruimte soms uitgerust kan worden met een symplectische structuur.

Hoofdstukken 1, 2, 3 en 4 bevatten theorie die nodig is om te begrijpen wat de mo-duliruimten M(Σ, G) zijn. Het eerste hoofdstuk behandelt de basis van Lie-groepen enLie-algebra’s. Het tweede hoofdstuk introduceert hoofdvezelbundels en connecties daaropvanuit het algemenere concept van vezelbundels. Vectorbundels worden apart behandeld.Het derde hoofdstuk bevat een deel van de theorie van vectorwaardige differentiaalvormen.Deze zijn nodig om in latere hoofdstukken de symplectische structuur op de moduliruimte tebegrijpen. Het vierde hoofdstuk gaat over symplectische meetkunde en Poisson-meetkunde.Het belangrijkste concept hier is symplectische reductie, een procedure die we later zullengebruiken om de moduliruimte uit te rusten met een symplectische strucuur.

Hoofstukken 5, 6 en 7 behandelen de eigenlijke moduliruimten. Het vijfde hoofdstukis de spil van deze tekst en legt uit hoe de moduliruimte geïdentificeerd kan worden metHom(π1(Σ), G)/G, de ruimte van alle morfismen van de fundamentaalgroep van Σ naarG modulo conjugatie. Dit resulteert in een topologie en een gladde structuur opM(Σ, G).In dit hoofdstuk leggen we ook uit hoe de moduliruimte geïnterpreteerd kan worden alseen symplectisch quotiënt van een oneindigdimensionale symplectische ruimte onder eenHamiltoniaanse actie indien Σ een compact orienteerbaar oppervlak is. Het zesde hoofdstukbevat een topologische karakterisatie van het reducibele gedeelte van de moduliruimte voorcompacte oriënteerbare oppervlakken. Het geval G = SU(2) wordt in meer detail behandeld,en er wordt getoond dat een groot deel van de moduliruimte glad is. Het laatste hoofdstukleidt een expliciete formule af voor de symplectische structuur op de moduliruimte, die desymplectische vorm uitdrukt als een integraal over een 1-keten in plaats van een 2-keten.Tenslotte wordt het abelse geval besproken, waarin de nieuwe formule vereenvoudigt en eenintegraal-vrije uitdrukking geeft voor de symplectische vorm.

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List of symbols and conventions

A(P ) Set of connections on PAd, ad, Adg, adg Adjoint action (of g)Ad∗, ad∗, Ad∗g, ad∗g Coadjoint action (of g)[A1, . . . , An] Equivalence class of the tuple (A1, . . . , An)[α ∧ β] Wedge product followed by Lie bracket〈α ∧ β〉 Wedge product followed by bilinear pairingAp Connection A at p regarded as 1-formdA Covariant exterior derivativeFA Curvature of connection Af ∗E Pullback of E along fF(M) Set of smooth real-valued functions on MΓ(E) Set of sections of the bundle EG(P ) Gauge group of PgP Adjoint bundle of PHolq Extended holonomy for base point qHolp(A, γ) Holonomy of A along γ for base point pHp, Vp Horizontal space at p, vertical space at pIdA Identity map A→ A

λg Left translation by gΛk(E) k-th exterior power of vector bundle ELie(G), g Lie algebra of GLX Lie derivative along XM(Σ, G),M Moduli space of flat connections on principal G-bundles over ΣMP (Σ, G) P -block ofM(Σ, G)µξ ξ-component of moment map µωAB Atiyah-Bott 2-formΩk

Ad,H(P, g) Space of horizontal equivariant g-valued k-forms on P

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Ωk(M), Ωk(M,E) Space of differential k-forms on M (with values in E)Ptγ, Ptγ,t Parallel transport along γ (for time t)Tf , Txf Tangent map of the map f (at x)θ Maurer-Cartan formW (T ) Weyl group of torus TXp Vector field X at point p[·, ·] Lie bracket of vector fields, Lie bracket on Lie algebra·, · Poisson bracket∇v(s) Covariant derivative of s along v

By smooth we mean of class C∞. Group units are denoted by e. However, sometimes amore specific notation is adopted, such as writing 1 for the group (C×, ·) and 0 for (R,+),or Id for matrix groups. Manifolds are required to be Hausdorff and second countable.They need not be connected. By a submanifold we mean an embedded submanifold (notmerely an immersed one).

We will make extensive use of group actions. To let the group element g act on a fromthe left, we will write g · a. To let it act from the right, we write a · g. Suppose G is agroup acting on sets A and B. A map f : A → B will be called equivariant (with respectto the G-actions) iff

• both actions are left actions and f(g · a) = g · f(a) for all a ∈ A and g ∈ G.

• both actions are right actions and f(a · g) = f(a) · g for all a ∈ A and g ∈ G.

• the action on A is a left action and the action on B is a right action and f(g · a) =f(a) · g−1 for all a ∈ A and g ∈ G.

• the action on A is a right action and the action on A is a left action and f(a · g) =g−1 · f(a) for all a ∈ A and g ∈ G.

Note that to any right action A × G → A : (a, g) 7→ a · g we can associate a left actionG× A→ A : (g, a) 7→ a · g−1 and vice versa. Under this association, all four of the aboveconditions boil down to the same requirement.

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Contents

Preface ii

Summary (samenvatting) iii

List of symbols and conventions v

Contents vii

1 Lie groups and Lie algebras 11.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The functor Lie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 The (co)adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 The adjoint representation . . . . . . . . . . . . . . . . . . . . . . . 91.3.2 The coadjoint representation . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Compact Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Bundles and connections 142.1 Fibre bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 Structure groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Pullback of a bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.4 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.5 Parallel transport and holonomy . . . . . . . . . . . . . . . . . . . . 20

2.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2.2 Linear connections . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.3 The covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Principal bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Principal homogeneous spaces . . . . . . . . . . . . . . . . . . . . . 242.3.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 The gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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2.3.4 Principal connections . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.5 Holonomy in principal bundles . . . . . . . . . . . . . . . . . . . . . 29

3 Vector-valued differential forms 303.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1.1 Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.2 Wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.3 Exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Associated bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3 The covariant exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . 343.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Symplectic and Poisson structures 384.1 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2.1 Moment maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2.2 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.1 Poisson structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3.2 Symplectic manifolds are Poisson . . . . . . . . . . . . . . . . . . . 434.3.3 The splitting theorem and symplectic leaves . . . . . . . . . . . . . 444.3.4 Example: coadjoint orbits . . . . . . . . . . . . . . . . . . . . . . . 47

5 Constructing the moduli space 515.1 The moduli space as a space of representations . . . . . . . . . . . . . . . . 51

5.1.1 The correspondence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.2 . . . is a bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.3 A topology on the moduli space . . . . . . . . . . . . . . . . . . . . 55

5.2 A smooth structure on the moduli space . . . . . . . . . . . . . . . . . . . 565.2.1 Singular points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2 Application to the moduli space . . . . . . . . . . . . . . . . . . . . 58

5.3 The moduli space as a symplectic reduction . . . . . . . . . . . . . . . . . 595.3.1 Picking Σ and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3.2 The plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.3 The Atiyah-Bott 2-form . . . . . . . . . . . . . . . . . . . . . . . . 615.3.4 The curvature as a moment map . . . . . . . . . . . . . . . . . . . 63

6 A topological observation 676.1 The reducible part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.2 The case G = SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

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7 The symplectic structure 747.1 Flat connections on trivial bundles over simply connected bases . . . . . . 747.2 Application to compact orientable surfaces . . . . . . . . . . . . . . . . . . 76

7.2.1 Pulling back to the universal cover . . . . . . . . . . . . . . . . . . 777.2.2 Integrating over a horizontal domain . . . . . . . . . . . . . . . . . 787.2.3 Bringing in the holonomy . . . . . . . . . . . . . . . . . . . . . . . 797.2.4 The abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Conclusions and further research 85

Bibliography 86

Vulgariserende samenvatting 88

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Chapter 1

Lie groups and Lie algebras

In this first chapter, we discuss Lie groups and Lie algebras. Lie groups are in some sense“smooth” groups, and they are often appropriate to capture symmetries in differentialgeometry. Lie algebras are the infinitesimal, algebraic counterparts to Lie groups, as theydescribe what a Lie group looks like locally, but fail to describe global topological features.Our main reference for the first two sections of this chapter is the book by John Lee [Lee03].In the last section, we base the definitions of (co)adjoint actions and representations onthe book by William Fulton and Joe Harris [FH91].

1.1 Lie groupsDefinition 1. A Lie group is a group G that is also a smooth manifold, in such a waythat the multiplication G×G→ G and the inversion G→ G are smooth maps.

Definition 2. A morphism of Lie groups is a smooth map between Lie groups that is alsoa group morphism.

These two definitions define a category, the category of Lie groups. This means wealso have a notion of isomorphisms between Lie groups (a Lie group morphism that hasan inverse that is also a Lie group morphism), and of automorphisms of a Lie group (anisomorphism of a Lie group with itself). Let us consider a few examples of Lie groups.

• The group of real numbers R under addition.

• The group of positive real numbers R+ under multiplication. This Lie group isisomorphic to the previous one, and an isomorphism is given by exp : R→ R+.

• The group of complex numbers of modulus 1 under multiplication, the so-called circlegroup. As a manifold, this is the circle S1. This is a subgroup of the Lie group ofnon-zero complex numbers.

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• GL(V ), the group of automorphisms of a real vector space V of finite dimension n.Fixing a basis of V , we can identify this group with the group of invertible (n× n)-matrices. It is common practice to denote this group by GL(n).Note that the group operations are indeed smooth: the entries of a matrix product arepolynomial functions of the entries of the factors, and the entries of the inverse of amatrix are rational functions of the entries of this matrix with non-zero denominator(the denominator is the determinant of the matrix).

• Many subgroups of GL(n) are important Lie groups. Examples are SL(n), the groupof (n×n)-matrices of determinant 1, O(n), the group of orthogonal (n×n)-matrices,and SO(n) = O(n) ∩ SL(n).

• The group of unitary (n× n)-matrices (with complex entries) is the Lie group U(n).The subgroup of U(n) consisting of those matrices that have determinant 1 is calledSU(n), the special unitary group. Note that U(1) and SO(2) are isomorphic, and areisomorphic to the circle group.

Definition 3. Let G be a Lie group. A morphism R → G is called a one-parametersubgroup of G.

Often, the image of the morphism rather than the morphism itself is called one-parameter subgroup. As an example, consider G = C×, the Lie group of non-zero complexnumbers. The positive real numbers form a one-parameter subgroup of G by the morphism

R→ C : t 7→ exp(t).

Also the circle group U(1) is a one-parameter subgroup of C×, by the morphism

R→ C : t 7→ exp(2πit).

In the next section, we shall see that all one-parameter subgroups of C× are exactly themorphisms of the form

R→ C : t 7→ exp(tξ)

for some ξ ∈ C (the images of these one-parameter subgroups are logarithmic spiralspassing through 1, with the cases ξ ∈ R and ξ ∈ iR leading to a degenerate spiral). Anexample is shown in figure 1.

1.2 Lie algebrasWe now discuss Lie algebras, the “infinitesimal versions” of Lie groups.

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1

ξ

Figure 1: A one-parameter subgroup of C, given by t 7→ exp(tξ). The image is a logarithmicspiral passing through 1. In this case ξ = 0.2 + i, which is the velocity of the curve whenpassing through 1.

1.2.1 The functor LieGiven a Lie group G and an element h ∈ G, we can consider left translation by h

λh : G→ G : g 7→ hg.

All of the maps λh are diffeomorphisms of the manifold G.A vector field on a Lie group G is called left-invariant if it equals its own pushforward

along any left multiplication map. In symbols, a vector field X on G is left-invariant iff

(Teλh)(Xe) = Xh for all h ∈ G.

A left-invariant vector field is completely determined by its value at the identity (which isan element of TeG). Indeed, given a tangent vector v ∈ TeG, we define the vector field Xby

Xg = (Teλg)(v).This does indeed define a smooth vector field on G, because in local coordinates theJacobian of the map λg varies smoothly with g. It is also clear that this equality isnecessary for X to be left-invariant.

The set of left-invariant vector fields on a Lie group G is a vector space (of the samedimension as G, because it is essentially TeG as shown above). We denote it by Lie(G),but very often it is also denoted by the same letter as the group, in lowercase fraktur (forexample Lie(G) = g). Note that the Lie bracket of two left-invariant vector fields is againleft-invariant (indeed, the Lie bracket commutes with diffeomorphisms). This turns Lie(G)into a so-called Lie algebra.

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Definition 4. A Lie bracket on a vector space L is a bilinear operation

[·, ·] : L× L→ L

that is antisymmetric and satisfies the Jacobi identity

[ξ, [η, ν]] + [η, [ν, ξ]] + [ν, [ξ, η]] = 0

for all ξ, η, ν ∈ L.

To clarify: by antisymmetry we mean that [ξ, ξ] = 0 for all ξ ∈ L, which is equivalentto [ξ, η] = −[η, ξ] for all ξ, η ∈ L if the characteristic of the field is not 2 (as always in thistext).

Definition 5. A Lie algebra is a finite-dimensional real vector space L equipped with aLie bracket.

Definition 6. A morphism of Lie algebras is a linear map φ : L→ L′ such that

φ([ξ, η]) = [φ(ξ), φ(η)] for all ξ, η ∈ L.

Note that in this equality, the bracket on the left hand side is the one of L, while the bracketon the right hand side is the bracket of L′.

We have thus defined the category of Lie algebras. As an example, let us discuss the Liealgebra of GL(n). Because GL(n) is an open subset of the vector space of (n×n)-matrices,the tangent space to each point can be identified with this vector space. In particular, thetangent space at e can be identified with the space of (n× n)-matrices, and hence the Liealgebra gl(n) of GL(n) is just the n2-dimensional vector space of (n× n) square matrices.This characterizes gl(n) as a vector space. To characterize it as a Lie algebra, we also needto know what the bracket operation is.

Theorem 7. The bracket operation [·, ·] on the Lie algebra gl(n) of GL(n) is the commu-tator of matrices, meaning that

[ξ, η] = ξη − ηξ for all A,B ∈ gl(n)

where the multiplication on the right is ordinary matrix multiplication after the identifica-tion of gl(n) with the space of (n× n)-matrices.

Proof. Let ν ∈ Te GL(n) be an element of the Lie algebra. If p ∈ GL(n) is an element of theLie group, the tangent space Tp GL(n) can be identified with the space of (n×n)-matricesin the usual way, and under this identification, the vector at p of the left-invariant vectorfield associated to ν is pν (an ordinary matrix product).

Let us call an R-valued function on GL(n) linear if it is the restriction to GL(n) ofa linear function on the space of (n × n)-matrices. Note that any covector at e is thedifferential of some linear function on GL(n). To check equality of two vector fields on

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GL(n) at e, it therefore suffices to show that they act the same way on linear functions onGL(n).

Now pick a linear function A on GL(n). If ν ∈ gl(n) is a left-invariant vector field onGL(n) (considered as a matrix by identifying with Te GL(n)), we can let ν act on A andwe get

(ν(A))(p) = d

dt |t=0(A(p+ tpν)) = A(pν).

Notice that ν(A) is again a linear function on GL(n). Applying this to ν ∈ ξ, η, we find

[ξ, η](A)(p) = ξ(η(A))(p)− η(ξ(A))(p)= η(A)(pξ)− ν(A)(pη)= A(pξη)− A(pηξ)= A(p(ξη − ηξ))= (ξη − ηξ)(A)(p).

This proves that [ξ, η] and (ξη− ηξ) act the same way on linear functions, establishing theresult.

Let us give a few more examples of Lie algebras.

• The Lie algebra of R is R with the zero bracket.

• The circle group has as its Lie algebra the tangent space to the circle at 1. As aLie algebra, this is just R (indeed, there is only one 1-dimensional Lie algebra up toisomorphism; its bracket is identically zero).

• The Lie algebra of SL(n) is written sl(n) and consists of all (n × n)-matrices withzero trace. To see this, note that the differential of the determinant at the identityis the trace.

• The Lie algebra of O(n) consists of all (n × n) skew-symmetric matrices. BecauseSO(n) is the component of the identity of O(n) and only this component plays a rolein determining the Lie algebra of a Lie group, SO(n) has the same Lie algebra asO(n).

• The Lie algebra of SU(n) consists of the traceless anti-hermitian (n × n) complexmatrices.

Note that in all these cases, the bracket is just the commutator.Lie algebras are algebraic objects, and we have seen that to any Lie group G we can

associate a Lie algebra Lie(G). In fact, to any morphism f : G→ H of Lie groups, we canassociate a morphism of Lie algebras Lie(f), turning Lie into a functor from the categoryof Lie groups to the category of Lie algebras. The map Lie(f) is just defined by

Lie(f) = Te(f),

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the differential of the map f at the identity (which is a map from TeG ∼= Lie(G) toTeH ∼= Lie(H)).Theorem 8. If f : G→ H is a morphism of Lie groups, the map Lie(f) : Lie(G)→ Lie(H)is a morphism of Lie algebras.

The proof of this theorem rests on a simple observation.Lemma 9. Let ν ∈ Lie(G) be a left-invariant vector field on G. Take ν ′ to be the left-invariant vector field on H associated to (Tef)(ν) ∈ TeH. In other words, ν ′ is the imageof ν under Lie(f). Then ν and ν ′ are f -related.Proof. We have to check that for any g ∈ G it holds that

(Tgf)(ν(g)) = ν ′(f(g)).

By left-invariance, the left hand side is

(Tgf)(ν(g)) = (Tgf)((Teλg)(ν(e))) = (Te(f λg))(ν(e)).

Because f is a morphism of Lie groups, we have f λg = λf(g) f . This implies

(Tgf)(ν(g)) = (Te(λf(g) f))(ν(e))= (Teλf(g))(Te(f)(ν(e))= (Teλf(g))(ν ′(e))= ν ′(f(g))

where in the last step we used left-invariance of ν ′. This proves the lemma.

Proof of theorem 8. Let ξ, η ∈ Lie(G). We have to show that

Lie(f)[ξ, η] = [Lie(f)(ξ),Lie(f)(η)].

By the lemma, Lie(f)(ξ) is f -related to ξ and Lie(f)(η) is f -related to η. This implies that[Lie(f)(ξ),Lie(f)(η)] is f -related to [ξ, η]. In other words,

Tgf([ξ, η](g)) = [Lie(f)(ξ),Lie(f)(η)](f(g)).

In particular, for g = e we get

Lie(f)([ξ, η]) = [Lie(f)(ξ),Lie(f)(η)].

This proves the result.

An important facet of Lie theory is describing the extent to which Lie is an equivalenceof categories. We will not discuss this issue in detail. Let us just mention one result. Aproof can be found in [Lee03] at theorem 15.32 (page 396) and theorem 15.35 (page 398),keeping in mind that we have already established theorem 8.Theorem 10. The functor Lie from the category of simply connected Lie groups (a fullsubcategory of the category of Lie groups) to the category of Lie algebras is an equivalenceof categories.

6

1.2.2 The exponential mapWe now introduce a crucial ingredient to understanding Lie groups and Lie algebras: theexponential map.

Suppose γ : R→ G is a one-parameter subgroup of the Lie group G. By differentiationwe can associate to this one-parameter subgroup the element γ′(0) ∈ TeG. This map

one-parameter subgroups of G → g = TeG

is in fact a bijection.

Theorem 11. The map

ϕ : one-parameter subgroups γ of G → TeG : γ 7→ γ′(0)

is a bijection.

Proof. Let ξ ∈ TeG be any element of the Lie algebra of G. Consider this element as aleft-invariant vector field on G. This is the vector field X with Xg = (Teλg)(ξ).

If γ is to be a one-parameter subgroup that is mapped to ξ, then it has to satisfy theconditions

γ′(0) = ξ

γ(t+ s) = γ(t)γ(s) = λγ(t)(γ(s)) for all t, s ∈ R.

By differentiating the second condition with respect to s at s = 0, we find that γ has tosatisfy

γ′(t) = (λγ(t))∗(ξ). (∗)This is exactly the condition that γ is an integral curve of the left-invariant vector fieldassociated to ξ. By uniqueness of integral curves, the map ϕ is injective.

To show surjectivity, let γ : (−ε, ε) → G be the local solution to (∗) with γ(0) = e.For a fixed t ∈ (−ε, ε), the curve s 7→ γ(t + s) is the integral curve of the vector field Xthrough γ(t) for s = 0. The curve s 7→ γ(t)γ(s) is an integral curve of the vector field(λγ(t))∗X = X through γ(t) for s = 0. By uniqueness of integral curves, we conclude that

γ(t+ s) = γ(t)γ(s) for all t, s sufficiently small.

So far γ is only defined in a neighbourhood of 0, but by demanding that γ(t + s) =γ(t)γ(s) for all s, t ∈ R, we can extend γ in a unique way to a one-parameter subgroupof G. This gives us a one-parameter subgroup that is mapped to ξ under the given map,showing surjectivity of ϕ.

Note that we have implicitly shown that any left-invariant vector field is complete(meaning its flow can be extended to all of the real line).

We now define the exponential map for the Lie group G as

exp : g→ G : ξ 7→ (ϕ−1(ξ))(1).

7

In other words, given a left-invariant vector field ξ on G, the point exp(ξ) is where you endup after flowing along ξ for one unit of time (starting at e).

Using standard results in ordinary differential equations, one proves

Theorem 12. The exponential map exp : g→ G is smooth.

Proof. Consider the manifold G × g and the vector field X on it, such that at the point(g, ξ) the vector is ((λg)∗(ξ), 0). Then exp(ξ) is just a component of the value at 1 of theintegral curve of X under the initial conditions (e, ξ). By smooth dependence on initialconditions of the solutions to differential equations, exp is smooth.

If we denote by γξ : R→ G the integral curve of the left-invariant vector field ξ startingat e, then it is clear that

exp(tξ) = γξ(t).

Differentiating this with respect to t at t = 0 gives us

Theorem 13. The differential of the exponential map at zero is the identity.

As a basic example, let us consider the exponential map of the Lie group C×, thegroup of non-zero complex numbers. Note that the tangent space to C× at any point canbe identified with C. The Lie algebra of C× is thus T1C× ∼= C. This means that theexponential map is a map

exp : C→ C×.

Note that by exp we mean the exponential map in the Lie group sense, not the usualexponential map on the complex numbers (although they will turn out to be identical).Let z ∈ C be an element of the Lie algebra. The left-invariant vector field on C× associatedto z is

C× → C : w 7→ wz.

The differential equation for the integral curve γ : R→ C of this left-invariant vector fieldis then

γ′(t) = γ(t)z.

Solving this differential equation with initial condition γ(0) = 1 gives

γ(t) = etz

where e··· is the usual exponential map on the complex numbers. Evaluating this at t = 1gives

exp(z) = ez,

proving that the exponential map (in the Lie sense) on C is the same as the usual ex-ponential map. By theorem (11), the one-parameter subgroups of C× are exactly of theform

t 7→ exp(tz)

8

for z ∈ C.For matrix groups (Lie groups of invertible matrices with multiplication as the group

operation, for example SU(n)), the exponential map turns out to be the classical exponen-tial map of matrices as given by the usual power series. A proof can be found in [Lee03]on page 383 (proposition 15.18).

Theorem 14. Let G be a matrix group. Then

exp(A) =∑k≥0

Ak

k!

where the right-hand side is an absolutely convergent power series.

1.3 The (co)adjoint representationLet us now study some important representations of Lie groups and Lie algebras.

Definition 15. A representation of a Lie group G on a finite-dimensional real vector spaceV is a morphism of Lie groups

G→ GL(V ).

Definition 16. A representation of a Lie algebra g on a finite-dimensional real vectorspace V is a morphism of Lie algebras

g→ gl(V ).

Notice that a representation of a Lie group induces a representation of the correspondingLie algebra by application of the functor Lie.

1.3.1 The adjoint representationLet G be a Lie group. Given an element h ∈ G, we can consider the map

G→ G : g 7→ hgh−1.

It is an automorphism of G, and therefore the differential of this map is an automorphismof Lie(G) = g. Associating to h this differential gives a representation of G on its own Liealgebra called the adjoint representation. We write Ad for the adjoint representation, so

Ad : G→ GL(g) : h 7→ Adh

whereAdh : g→ g : ξ 7→ d

dt |t=0

(h exp(tξ)h−1

).

This adjoint representation is our first example of a representation of a Lie group.

9

Let us now apply the functor Lie (essentially, we will differentiate) to this representationto get a representation of the Lie algebra. The resulting representation of g will also becalled adjoint representation but will be denoted by ad. We calculate:

adξ(η) = d

dt |t=0

(Adexp(tξ)(η)

)= d

dt |t=0

d

ds |s=0(exp(tξ) exp(sη) exp(−tξ))

= [ξ, η]

where in the last step we recognized the geometric definition of the Lie bracket (using flowsof vector fields). This gives us a Lie algebra representation

ad : g→ gl(g) : ξ 7→ adξ .

By theorem (8), we know that this is a Lie algebra morphism. In this case, however,explicit verification is easy. Indeed, the equality

ad([ξ, η]) = [ad(ξ), ad(η)]

is equivalent to[[ξ, η], α] = [ξ, [η, α]]− [η, [ξ, α]] for any α ∈ g,

which is just the Jacobi identity for the Lie bracket.Let us state one easy result that will be useful later on.

Lemma 17. Suppose G is a matrix group. Then the Lie algebra g is a vector space ofmatrices. The adjoint action of G on g is given by conjugation.

Proof. Let M ∈ G be a matrix and ξ ∈ g a matrix in the Lie algebra. Then

AdM(ξ) = d

dt |t=0

(M exp(tξ)M−1

)= M

(d

dt |t=0exp(tξ)

)M−1

= MξM−1.

1.3.2 The coadjoint representationLet G be a Lie group and h ∈ G. By dualizing the map

Adh−1 : g→ g

we get a mapAd∗h : g∗ → g∗.

10

(Note that by this notation we mean (Ad∗)h rather than (Adh)∗. In cases where we meanthe latter, we will always write the brackets explicitly.) This gives us a representation ofG on the dual of its Lie algebra,

Ad∗ : G→ GL(g∗) : h 7→ Ad∗h = (Adh−1)∗

where the last star means dualization of linear maps. This definition turns Ad∗ into arepresentation of the Lie group, because

Ad∗gh = (Ad(gh)−1)∗ = (Adh−1 Adg−1)∗ = (Adg−1)∗ (Adh−1)∗ = Ad∗g Ad∗h .

This representation Ad∗ of G on g∗ is called the coadjoint representation of G.We now want to give an appropriate definition of the coadjoint representation of g.

There are two reasonable approaches:

• we can dualize the adjoint representation of g;

• or we can differentiate the coadjoint representation of G.

Fortunately, both approaches give the same result. Let us define the coadjoint representa-tion of g using the first approach. In other words, we set

ad∗ξ = (ad−ξ)∗

and callad∗ : g→ gl(g∗) : ξ 7→ ad∗ξ

the coadjoint representation of g. This yields a representation of g as is easily verified(using the fact that ad is a representation, or in other words using the Jacobi identity).Let us now differentiate the coadjoint representation of G to get a representation of gand check that this gives ad∗ (this renders the verification that ad∗ is a representationunnecessary because representations of G automatically differentiate to representations ofg).

We want to establish that

d

dt |t=0

(Ad∗exp(tξ)

)= ad∗ξ .

To check their equality, we evaluate both of them on α ∈ g∗. We are done if the results(which are elements of g∗) are the same. To check equality of the results, we evaluate themboth on an element η ∈ g. In other words, what we will check is

〈ad∗ξ(α), η〉 = 〈 ddt |t=0

(Ad∗exp(tξ)

)(α), η〉

where 〈·, ·〉 is the pairing between g and g∗. The left hand side is (by definition of thecoadjoint representation) equal to 〈α, ad−ξ(η)〉 = −〈α, [ξ, η]〉. The right hand side is equal

11

to

〈 ddt |t=0

(Ad∗exp(tξ)

)(α), η〉 = 〈 d

dt |t=0

(Ad∗exp(tξ)(α)

), η〉

= d

dt |t=0〈Ad∗exp(tξ)(α), η〉

= d

dt |t=0〈α,Adexp(−tξ)(η)〉

= 〈α, ddt |t=0

Adexp(−tξ)(η)〉

= 〈α, ad−ξ(η)〉= 〈α, [−ξ, η]〉.

This shows that the coadjoint representation of g is indeed obtained by differentiating thecoadjoint representation of G.

1.4 Compact Lie groupsIn this section we mention several results about compact Lie groups without proofs. Theyintroduce some concepts that we will make use of later, notably maximal tori and the Weylgroup. These results can be found in [DK99] (corollary 1.12.4 on page 61 and theorem 3.7.1on page 152).

Theorem 18. Every connected abelian Lie group is isomorphic to Rm × U(1)n for somenatural numbers m,n. In particular, every compact connected abelian Lie group is isomor-phic to U(1)n for some n.

Definition 19. If G is a Lie group, then a torus in G is a compact connected abelian Liesubgroup of G. A maximal torus in G is a torus in G that is not strictly contained in anyother torus in G.

Theorem 20. Every maximal torus in a compact connected Lie group is a maximal abeliansubgroup (an abelian subgroup not strictly contained in any other abelian subgroup). Everyelement of a compact connected Lie group G is contained in some maximal torus, and allmaximal tori are conjugate (so if T and T ′ are maximal tori, there is some g ∈ G suchthat T ′ = gTg−1).

Definition 21. If G is a compact connected Lie group and T is a maximal torus in it,then we define its Weyl group W (T ) to be

W (T ) = N(T )T

,

where N(T ) is the normalizer of T , meaning that N(T ) = g ∈ G | gTg−1 = T.

12

Theorem 22. If G is a compact connected Lie group and T a maximal torus in it, thenthe Weyl group W (T ) is finite.

Note that the Weyl group W (T ) acts on T via group automorphisms (by conjugation)and that this action is faithful because T is a maximal abelian subgroup of G. The followinglemma is lemma 4.33 in [Ada69] (page 96).

Lemma 23. Suppose G is a compact connected Lie group and T a maximal torus in it. Ift, t′ ∈ T are conjugate (so that t′ = gtg−1 for some g ∈ G), then there is an element of theWeyl group W (T ) sending t to t′.

For the case G = SU(2), a maximal torus is given by

T =

ω 0

0 ω

| ω ∈ S1

.Its Weyl group is the group of two elements, with the non-trivial element acting byω 0

0 ω

7→ω 0

0 ω

.

13

Chapter 2

Bundles and connections

In this chapter we study fibre bundles and connections on them. Starting from the generalconcept of a fibre bundle with a connection, we later focus on two important special cases:vector bundles, where the fibre is a vector space, and principal bundles, where the fibre isa principal homogeneous space. Our references for this chapter are [Hec12] (for definitionsof the bundles) and [KN63] and [Mic88] (for connections).

2.1 Fibre bundlesWe start by studying fibre bundles in general. We also treat connections and paralleltransport, both of which will later be specialized to the cases of vector bundles and principalbundles.

2.1.1 DefinitionA fibre bundle is a space that locally looks like a product of two manifolds, but may bedifferent from an ordinary product globally. Let us give the formal definition.

Definition 24. A (smooth) fibre bundle is a tuple (E,Σ, π, F ) where E, Σ and F aresmooth manifolds and π : E → Σ is a smooth surjection. Furthermore, we require thatevery x ∈ Σ has an open neighbourhood U ⊂ Σ such that π−1(U) is diffeomorphic to U ×Fvia a map ϕ : π−1(U)→ U × F in such a way that the following diagram commutes (here,proj1 is projection onto the first factor):

π−1(U) U × F

U

ϕ

πproj1

14

We call E the total space, Σ the base space, π the projection and F the fibre. The set ofthe (Ui, ϕi) is called a local trivialization of the bundle. If x ∈ Σ, the preimage π−1(x)is called the fibre over x, denoted by Ex.

Very often, we will refer to “the fibre bundle E” instead of mentioning the whole tuple.We also speak of a bundle over Σ to indicate that Σ is the base space.

Let us look at a few examples to gain some intuition.• An obvious example is the trivial fibre bundle. Let Σ and F be two smooth manifolds,

and consider E = Σ×F with π the projection onto the first factor. Then (E,Σ, π, F )is clearly a fibre bundle (indeed, in the definition we can take U = Σ for all x andϕ = Id).

• A familiar example of a fibre bundle is the tangent bundle of a manifold. If M is asmooth m-dimensional manifold, then the tangent bundle TM is a fibre bundle overM (with the usual projection TM →M).

• An interesting example is (S1, S1, z 7→ z2, 0, 1). This is a fibre bundle that is nottrivial, because the trivial fibre bundle would be S1×0, 1, which is not connected,whereas S1 is.

• Another example of a fibre bundle is the Möbius strip, pictured in figure 2. It isa fibre bundle over the circle with fibre (0, 1) (or you could include the boundary,resulting in a fibre bundle with fibre [0, 1], which would require a slight generalizationof our definition of fibre bundle to allow manifolds with boundary).

The last two examples are in fact intimately related. To see this, note that we couldconsider the boundary of the Möbius strip, which is diffeomorphic to S1, and look at theprojection of this boundary circle onto the red one. This gives exactly the fibre bundle(S1, S1, z 7→ z2, 0, 1).Definition 25. Let (E,Σ, π, F ) be a fibre bundle. A section of this fibre bundle is a smoothmap s : Σ→ E such that π s = IdΣ. The set of all sections of the fibre bundle is writtenΓ(E).

A fibre bundle is a way of associating a copy of the space F to every point of Σ and asection is just a choice of a point in every such copy (in a smooth way). Let us also definemorphisms of fibre bundles.Definition 26. Let (E,Σ, π, F ) and (E ′,Σ′, π′, F ′) be two fibre bundles. A morphism offibre bundles, also called a bundle map is smooth map φ : E → E ′ such that there exists asmooth map ϕ : Σ→ Σ′ such that following diagram commutes.

E E ′

Σ Σ′

φ

π π′

ϕ

15

Figure 2: The Möbius strip, obtained by gluing two ends of a stroke of paper togetherwith a half-turn twist. It is a fibre bundle over the circle S1 with fibre (0, 1). The basespace can be identified with the circle marked in red. Projection onto the base space thencorresponds to mapping any point of the Möbius strip to the nearest red point.

In other words, a morphism of fibre bundles is a smooth map that preserves fibres.Note that φ determines ϕ uniquely by surjectivity of π. We say that φ is a bundle mapcovering ϕ.

2.1.2 Structure groupsVery often, a fibre bundle will carry some extra structure called a structure group. Let(E,Σ, π, F ) be a fibre bundle and G a Lie group, and suppose that G acts on the fibre Ffrom the left. A local trivialization of the fibre bundle is called a G-atlas if for any twooverlapping charts (Ui, ϕi) and (Uj, ϕj) the transition function

ϕj ϕ−1i : (Ui ∩ Uj)× F → (Ui ∩ Uj)× F

is given by(x, ξ) 7→ (x, tij(x)ξ)

for some smooth function tij : (Ui ∩ Uj)→ G. Two G-atlases are called equivalent if theirunion is also a G-atlas.

Definition 27. A fibre bundle with structure group G (a G-bundle for short) is a fibrebundle with an equivalence class of G-atlases.

Non-rigorously stated, equipping a fibre bundle with a structure group G is a way ofdemanding that the different locally trivial pieces are glued together only in ways given bythe action of G on F , not by arbitrary diffeomorphisms of F . It is easy to check that theMöbius strip admits a (Z/2Z)-atlas. Note that a fibre bundle is trivial exactly if it admitsthe trivial group as structure group.

16

2.1.3 Pullback of a bundleSuppose M and N are two smooth manifolds, and f : M → N is a smooth map. Let(E,N, π, F ) be a fibre bundle over N . We can then pull back this bundle via f to get afibre bundle f ∗E over M . This pullback bundle is defined as

f ∗E = (x, e) ∈M × E | f(x) = π(e) ⊂M × E.

It is a submanifold ofM×E by the implicit function theorem (by noting that the differentialof π has full rank). The projection τ for the pullback bundle is given by projection ontothe first factor. In symbols:

τ : f ∗E →M : (x, e) 7→ x.

The fibre of f ∗E over x ∈M is essentially the fibre of E over f(x). Using the full notation,the pullback bundle is (f ∗E,M, τ, F ). It is not hard to check that this is indeed a fibrebundle. Moreover, pullback of bundles respects structure groups: if E is a fibre bundlewith structure group G, then so is f ∗E. Projection onto E induces a map f : f ∗E → E :(x, e) 7→ e. This is a bundle map covering f . Sections can be pulled back too. Given asection s ∈ Γ(E), we have f ∗s = s f ∈ Γ(f ∗E).

Let us look at an example. Consider the map f : S1 → S1 : z 7→ z2. We have seen thatthe Möbius strip E is a fibre bundle over S1, so that we can pull it back via f to obtain afibre bundle f ∗E over S1. The situation is illustrated in figure 3. Note that the pullbackf ∗E is trivial. If one were to make a paper model of the bundle as shown in the figure, themodel would be a stroke of paper with both ends glued together will a full twist (the twohalf-turns don’t cancel out). Nevertheless, the bundle is trivial. And indeed, the bundle isdiffeomorphic (as manifolds) to the untwisted version, even though they are not isotopicas embedded submanifolds of Euclidean space.

2.1.4 ConnectionsIf (E,Σ, π, F ) is a fibre bundle, then all the fibres are disjoint copies of F that have nothingto do with each other except for being glued together in a smooth fashion. We now wantto identify nearby fibres in a stronger way. To this end, we will consider connections on E.

First observe that if p ∈ E, then there is a distinguished subspace Vp of TpE, called thevertical space at p, that is the tangent space to the fibre p is contained in. In symbols, thiscan be expressed as

Vp = ker(Tpπ) ⊆ TpE.

The dimension of this vertical space is dim(F ). So given a fibre bundle, we can tell whethera vector is “vertical” by checking whether it is mapped to zero by the differential of π. Ona mere fibre bundle, however, we cannot tell whether a tangent vector is “horizontal”. Thisis precisely what a connection provides.

17

Figure 3: Pulling back the Möbius strip via the map S1 → S1 : z 7→ z2. On the rightthe Möbius strip in blue, with the base space in red. On the left the pullback bundle. Wevisualize the map z 7→ z2 as “squashing down vertically” the base space on the left to getthe circle on the right.

Definition 28. A connection on a fibre bundle (E,Σ, π, F ) is a choice of linear complementto the vertical space at every point of E, in a smooth way. In other words, a connection isa smooth choice of subspaces Hp of TpE such that

TpE = Vp ⊕Hp for every p ∈ E.

We call Hp the horizontal space at p.If a connection on E is given, the map Tpπ is an isomorphism between Hp and Tπ(p)Σ.

If a smooth vector field X on Σ is then given, we can lift this to a smooth horizontal vectorfield on E in a unique way. This lift is the unique vector field X on E such that Xp ∈ Hp

and Tpπ(Xp) = Xπ(p) for every p ∈ E.Note that a connection can also be encoded by specifying at every point the projection

Ap : TpE → Vp with kernel Hp. Conversely, if at every point we are given a linear mapAp : TpE → Vp that restricts to the identity on Vp, then we can recover a connection fromthis by defining

Hp = ker(Ap).A connection also allows us to locally lift curves horizontally. Suppose that γ : I → Σ

is a smooth curve in Σ (or a piecewise smooth one by a straightforward extension of whatwe explain here). We want to lift γ horizontally, which means we are looking for a curveγ in E such that π γ = γ and γ′(t) is horizontal for all t. Let t0 ∈ (0, 1) and pick a pointp ∈ π−1(γ(t0)). In a neighbourhood U of γ(t0), the bundle is the trivial bundle U × F .Take local coordinates on E around p by picking local coordinates xi on U and yi on F .The horizontal lift of the tangent vector (v1, . . . , vdim(Σ)) at the point q ∈ E is given by

(v1, . . . , vdim(Σ), f1(q, v), . . . , fdim(F )(q, v)),

18

where the fi are smooth functions of q and v = (v1, . . . , vdim(Σ)) (they are in fact linear inv). The differential equations x′(t) = γ′(t)

y′i(t) = fi(γ′(t))

together with the initial condition (x(t0), y(t0)) = p then specify a local horizontal lift ofγ (and any two such lift coincide on their common domain of definition).

We are mainly interested in connections where any curve can in fact be lifted globally,meaning that the lift is defined on the whole domain of definition of γ.

Definition 29. A connection on a fibre bundle (E,Σ, π, F ) is called complete if any curveγ : I → Σ defined on some interval I has a horizontal lift γ : I → E. In this case, theconnection is also called an Ehresmann connection.

The point in this definition is that γ is defined on all of I.Let us give an example of a connection that is not complete. Consider the trivial bundle

over R with fibre R, which is R2 with projection onto the first coordinate. Write

A = (x, y) ∈ R2 | x ≥ 0 or xy ≤ 1

andB = (x, y) ∈ R2 | x < 0 and xy ≥ 2.

These are two disjoint closed subsets of R2. Pick a smooth function R2 → R that is 0on A and 1 on B (it is a standard result that any two closed disjoint sets in a manifoldcan be separated in this way; it follows immediately from proposition A.8 on page 224 of[MrT97]). Now consider the vector field

R2 → R2 : (x, y) 7→

(1, 0) if x ≥ 0,(1, f(x, y) · −3

x2

)otherwise.

It is smooth and defines a connection on the bundle by demanding that it is horizontal.However, the curve R<0 → R2 : t 7→ (t, 3/t) is an integral curve of this vector field. Thisshows that the curve R → R : t 7→ t cannot be lifted for this connection if we start at(−3,−1), because lifting it on R<0 first results in the map R<0 → R2 : t 7→ (t, 3/t), whichcannot even be continuously extended to all of R.

Just as vector fields on a compact manifold are always complete, we have the followingnice result. A proof can be found in [Mic88] on page 41, under paragraph 9.10. We willnot need this result.

Theorem 30. If a fibre bundle has compact fibre, any connection on it is complete.

19

A

B

Figure 4: The third quadrant in the plane. In dark blue the region A, in light blue theregion B. To construct an incomplete connection, we start with a curve in E which wewant to be the lift of some curve in the base space and that runs off to infinity in finitetime. Here, we pick a branch of a hyperbola, shown on the picture as a thick line andcompletely contained in B. Locally around the curve, we take a nowhere-vertical vectorfield tangent to it. We then extend this vector field to all of the plane in a nowhere-verticalway, thus defining an incomplete connection.

2.1.5 Parallel transport and holonomySuppose the fibre bundle (E,Σ, π, F ) is equipped with a complete connection. Pick a curveγ : I → Σ and assume 0 ∈ I. Using completeness of the connection, we get a map

Ptγ : Eγ(0) × I → E : (p, t) 7→ (lift of γ at p)(t).

Moving along a horizontal curve in a fibre bundle with a connection is called paralleltransport. We say that Ptγ(p, t) is the result of parallel transport of p along γ for a time t.Often, we will leave out t from the notation and just assume that t = 1 (then we must ofcourse have 1 ∈ I). Note that Ptγ is a smooth map since the solutions of smooth differentialequations depend smoothly on the initial conditions. Also note that reparametrization ofγ does not change parallel transport along it.

If we fix t we get a mapPtγ,t : Eγ(0) → Eγ(t).

This map is a diffeomorphism, since an inverse is given by parallel transport along γ in thereverse direction. Suppose now that γ(0) = γ(t), so that we are doing parallel transportalong a loop in the base space. Then the resulting Ptγ,t is an automorphism of Eγ(0) calledthe holonomy of the connection around the loop γ.

20

2.2 Vector bundlesWe now treat a very important type of fibre bundle: vector bundles. These are roughlyspeaking fibre bundles where the fibre is a vector space.

2.2.1 DefinitionDefinition 31. A vector bundle is a fibre bundle (E,Σ, π, F ) where the fibre F is a finite-dimensional real vector space with structure group GL(F ) acting on F in the usual way.The dimension of F is called the rank of the vector bundle.

Note that every fibre of a vector bundle is a vector space. Indeed, the operations ofaddition and scalar multiplication can be carried out in any chart of the GL(F )-atlas, andthe result will not depend on the chosen chart because the transition maps are fibrewiselinear. Also note that any vector bundle has a section, the zero section mapping the pointx ∈ Σ to the origin of the vector space Ex. Locally, the vector bundle is just U×F for someopen U ⊆ Σ. This means that we can pick local coordinates on E by picking coordinateson U and identifying the vector space F with Rn. The resulting local coordinates on Ewill be called linear coordinates.

We want morphisms between vector bundles to preserve the extra structure.

Definition 32. A vector bundle morphism is a bundle map that is linear on each fibre.

A well-known example of a vector bundle is the tangent bundle to a smooth manifold,with the projection given by mapping a vector to its base point. If f : M → N is a smoothmap between manifolds, then Tf : TM → TN is a morphism of vector bundles coveringf . The space of sections of a vector bundle is an F(Σ)-module (and a real vector space).

If E and F are two vector bundles over Σ and Ex is a linear subspace of Fx for everyx ∈ Σ, then we say that E is a subbundle of F . As an example, on any fibre bundle witha connection the horizontal spaces at each point together form a subbundle of the tangentbundle to the given fibre bundle.

Many constructions with vector spaces can be generalized to vector bundles. For exam-ple, if E and F are vector bundles defined over some manifold Σ, then we can form theirdirect sum E ⊕ F , which as a set is just

E ⊕ F =⋃x∈Σ

Ex ⊕ Fx.

It carries the structure of a vector bundle where the projection maps the elements ofEx ⊕ Fx to x. To show this, observe that over some neighbourhood U of x both E and Fare trivial. Take local coordinates (x1, . . . , xdim(Σ)) on E and extend these to local linearcoordinates (x1, . . . , xdim(Σ), e1, . . . , erank(F )) on E and (x1, . . . , xdim(Σ), f1, . . . , frank(F )) onF . Then (x1, . . . , xdim(Σ), e1, . . . , erank(E), f1, . . . , frank(F )) form local linear coordinates onE ⊕ F .

21

In an analogous fashion one defines the tensor product of vector bundles and the dualand the exterior powers of a vector bundle. We leave the details to the reader. The readercan also check the following: if E and F are vector bundles over a common base space, thena bundle map E → F covering the identity is essentially a section of E∗⊗F . The proof is aslight adaptation of the one used to show that Hom(V,W ) ∼= V ∗⊗W for finite-dimensionalreal vector spaces V and W .

2.2.2 Linear connectionsOn vector bundles, we are interested in complete connections that preserve the linearstructure under parallel transport. In other words, we want Ptγ,t : Eγ(0) → Eγ(t) to belinear for all γ and t. A moment’s thought shows that this is equivalent to the followingdefinition.Definition 33. Suppose (E,Σ, π, F ) is a vector bundle equipped with a complete connec-tion. If γ is a curve in Σ and p is a point of the fibre Eγ(0), then write γp for the horizontallift of γ starting at p. The connection on E is called linear if

γt·p = t · γp for all curves γ, points p ∈ Eγ(0) and t ∈ R

andγp+q = γp + γq for all curves γ, points p, q ∈ Eγ(0).

Recall that a connection can also be specified by giving projections Ap : TpE → Vp, thelink between these projections and the horizontal spaces being

Hp = ker(Ap).

Note that we can identify the vertical space at p with the fibre through p because thetangent space to a vector space can be identified with this vector space. We can thusregard Ap as a map from TpE to Eπ(p).

2.2.3 The covariant derivativeWe now introduce the covariant derivative, which measures how much a section deviatesfrom being “constant” (where constancy is measured by horizontalness in a sense we willmake precise). To motivate the definition of the covariant derivative, we first try to quantifyto what extent curves fail to be horizontal.

Take a curve γ : I → E with 0 ∈ I. We will try to quantify how much γ deviates fromhorizontalness at time t = 0. If γ were horizontal, we would have

Pt−1πγ,t(γ(t)) = γ(0) for all t ∈ I.

This suggests a way to quantify the deviation of γ from horizontalness at time t = 0. Ifwe write this deviation as D0γ, then a reasonable definition would be

D0γ = d

dt |t=0

(Pt−1

πγ,t(γ(t))− γ(0)). (∗)

22

This definition is not very easy to work with, so let us reformulate it. Write p = γ(0)and x = π(p). Note that (π γ)∗E is a submanifold of R× E. We can consider the map

ϕ : R× Ex → (π γ)∗E : (t, q) 7→ (t,Ptπγ,t(q)).

Its differential at (0, p) is an isomorphism, implying that ϕ is a diffeomorphism betweensome neighbourhood U ⊆ R × Ex of (0, p) and a neighbourhood V ⊆ (π γ)∗E of (0, p).Composing its inverse with projection onto Ex results in a map

$ : V → Ex.

Now for |t| small enough we can consider t 7→ $(t, γ(t)), which by construction equalsPt−1

πγ,t(γ(t)). Differentiating with respect to t at t = 0 gives

D0γ = d

dt |t=0$(t, γ(t))

= (T(0,p)$)(1, γ′(0))

Now the map TpE → Ex : v 7→ (T(0,p)$)(1, v) is a linear map that vanishes on horizontalvectors and is the identity on Vp by (∗). This uniquely determines the linear map, whichtherefore equals the projection Ap. We find that

D0γ = Ap(γ′(0)).

This is the reformulation of (∗) we were looking for, and it is this definition we use fordifferentiating sections along vector fields. If we are given a tangent vector v ∈ TxΣ and asection s, then we want to define ∇v(s) to quantify how much the section s deviates fromhorizontalness in the direction of v. To this end, we can pick a curve γ in Σ such thatγ′(0) = v and then calculate

D0(s γ) = As(x)((s γ)′(0))= As(x)((Txs)(v)).

Definition 34. Suppose (E,Σ, π, F ) is a vector bundle equipped with a linear connection.Let s be a section of the bundle and let v be a tangent vector to Σ at x. Then we definethe covariant derivative of s along v to be

∇v(s) = As(x)((Txs)(v)) ∈ Ex.

If X is a vector field on Σ, we define the covariant derivative of s along X to be the section∇X(s) given by

∇X(s)(x) = ∇Xx(s).

The covariant derivative satisfies the following properties for X and Y vector fields onΣ, f and g smooth functions on Σ and s and s′ sections of the bundle:

23

• ∇fX+gY (s) = f∇X(s) + g∇Y (s),

• ∇X(s+ s′) = ∇X(s) +∇X(s′),

• ∇X(fs) = f∇X(s) + df(X)s.The first of these is immediate from the definition. The second one follows from linearityof D0 as a map from the vector space of curves lifting π γ to Ex (and this linearity isclear from (∗) on page 22). The third property follows from the definition by a calculationusing local coordinates. Every covariant derivative with these properties defines a uniquelinear connection on E.

2.3 Principal bundlesWe now come to the most important type of fibre bundle for our purposes: principalbundles. To get some intuition for these, let us first discuss the notion of a principalhomogeneous space.

2.3.1 Principal homogeneous spacesDefinition 35. Let G be a Lie group. A principal homogeneous space for G is a manifoldon which G acts smoothly (say from the right), in such a way that the action is both freeand transitive.

This is a technical definition to formalize what is intuitively a group that has forgottenwhere its identity is. Indeed, given a point x of the principal homogeneous space X, wecan identify X with G by identifying x · g with g. However, this identification depends ona choice of this point x and there is no canonical way of turning X into a group.

An example is enlightening. Consider G = U(1), the circle group. Pick any circleX = Γ in the plane, which is a smooth manifold. There is no canonical way of turning Xinto a group. However, there is nice action of U(1) on Γ: the element eiθ rotates the circleover an angle θ. This action is free and transitive. What is the difference between U(1)and Γ? Not much: the essential difference is that G has a “special point”, 1. If we pickany point γ ∈ Γ and declare this to be special, Γ is converted into a group by the rule

(eiθγ) ∗ (eiθ′γ) = eiθeiθ′γ.

In a principal homogeneous space, there is no special point (hence homogeneous) and only“differences” (or “quotients”) between elements are group elements. A principal homoge-neous space is to a group as an affine space is to a vector space.

Note that G is a principal homogeneous space for itself, with the action given by rightmultiplication.Definition 36. Let G be a Lie group and X and Y be two principal homogeneous spacesfor G. A morphism of principal homogeneous spaces is a smooth map φ : X → Y suchthat φ(x · g) = φ(x) · g for all x ∈ X and g ∈ G.

24

Any morphism of principal homogeneous spaces is an isomorphism. This means thatthe category of principal homogeneous spaces of G is a groupoid. Considering G as aprincipal homogeneous space for itself with G acting on G by right multiplication, theautomorphisms of G as a principal homogeneous space are exactly given by left multipli-cation by some group element. In the next subsection, we will define a principal bundleroughly to be a bundle with principal homogeneous spaces as fibres. It is the observationthat automorphisms of G as a principal homogeneous space are given by left multiplicationthat motivates our demand for G to be the structure group by left multiplication.

2.3.2 DefinitionDefinition 37. Let G be a Lie group. A principal bundle is a fibre bundle (P,Σ, π,G)with structure group G which acts on the fibre G via left multiplication.

To emphasize the structure group, one often speaks of a principal G-bundle. Given aprincipal bundle (P,Σ, π,G), the group G acts on this bundle from the right in a naturalway, by right multiplication in any trivialization (which is well-defined). This right G-action turns the fibres into principal homogeneous spaces for G.

Definition 38. A morphism of principal bundles is a bundle map that is a morphismof principal homogeneous spaces on each fibre. In other words, if both (P,Σ, π,G) and(P ′,Σ′, π′, G) are principal bundles (with equal structure groups), then a bundle map φ :P → P ′ is a morphism of principal bundles iff

φ(p · g) = φ(p) · g for all p ∈ P and g ∈ G.

Let us consider some examples of principal bundles.

• The trivial principal bundle Σ×G (with the equivalence class of G-atlases given bythe obvious trivialization).

• Let (E,Σ, π, F ) be a vector bundle. Let n denote the dimension of the vector spaceF . Then we associate to this vector bundle E a principal GL(n)-bundle called theframe bundle. The fibre over x ∈ Σ of this principal bundle is the set of all orderedbases of Ex, on which GL(n) acts from the right by change of basis. We will notdiscuss this in detail. If we take E to be the tangent bundle to Σ, then the manifoldΣ is called parallelizable iff this associated frame bundle is trivial. In general, this isa strong condition on manifolds, but all Lie groups are parallelizable.

• Consider Σ = S2 and take P to be the set of all unit tangent vectors to Σ, with πthe projection P → Σ. Letting S1 act on P by rotation (so that eiθ acts on v ∈ Pby rotating it over an angle θ around the outward normal), we have a principalbundle (P,Σ, π, S1). Note that this principal bundle has no sections: a section wouldcorrespond to a smooth choice of unit tangent vector at every point of the sphere,which does not exist by the famous Hairy Ball Theorem. By the following theorem,

25

this shows that we have a non-trivial principal bundle. This bundle is called the Hopffibration.This example can be generalized by replacing S2 by a closed orientable surface ofgenus g. The bundle will be trivial precisely if g = 1 ([Hir76], theorem 2.2 on page133 and theorem 2.10 on page 137).

In strong contrast with the situation for vector bundles (which always have sections),we have the following result.

Theorem 39. A principal bundle is trivial iff it admits a section.

Proof. It is clear that the trivial bundle admits a section. Let (P,Σ, π, F ) be a principalbundle admitting a section s : Σ→ P . Consider the map

Σ×G→ P : (x, g) 7→ (s(x) · g).

This is clearly a bundle map, and it is surjective by transitivity of the group action on thefibres of P . It is injective by freeness of the group action on the fibres of P . It is thereforean isomorphism of bundles. This proves the theorem.

Let us introduce some more notation that will be useful later on. If ξ ∈ g, then wewrite

p · ξ = d

dt |t=0(p · exp(tξ)) .

Notice thatp · ξ | ξ ∈ g = ker(Tpπ) = Vp

If v ∈ TpP and g ∈ G, then we write v · g for the pushforward of v under the action of g.An interesting result is the following theorem. Its proof requires the use of homotopy

theory and classifying spaces, for which we refer to [Hor13] (and to [Hus66] section 4.13for background on universal bundles).

Theorem 40. If G is a compact simply connected Lie group and Σ is a compact orientablesurface, then every principal G-bundle over Σ is trivial.

2.3.3 The gauge groupIn the study of moduli spaces of flat connections, gauge transformations play an importantrole.

Definition 41. Let (P,Σ, π,G) be a principal bundle. A gauge transformation of thisbundle is an automorphism of P covering the identity IdΣ.

26

Very often gauge transformations are encoded as maps P → G instead of P → P .Indeed, if u : P → P is a gauge transformation, then we can associate to it a uniquesmooth map u : P → G such that

u(p) = p · u(p) for all p ∈ P

because u is fibre-preserving.On the other hand, if we are given a map u : P → G, we can associate to it the map

u : P → P : p 7→ p · u(p). This associated map is a bundle map covering the identity, butit need not be a morphism of principal bundles. The condition for u to be a morphism ofprincipal bundles is

u(p · g) = u(p) · g for all p ∈ P , g ∈ G,

which is equivalent to demanding that

u(p · g) = g−1u(p)g for all p ∈ P , g ∈ G.

This shows that the group of automorphisms of P covering the identity is essentially

G(P ) =u : P → G | u(p · g) = g−1u(p)g for all p ∈ P and g ∈ G

.

This set is called the gauge group of the principal bundle. Note that we have not spec-ified the operation turning G(P ) into a group. We want this operation to correspond tocomposition in terms of gauge transformations, so that uv = u v. This requires

p · (uv)(p) = u(p · v(p)) = p · v(p) · u(p · v(p)) = p · u(p) · v(p)

so that the group operation on G(P ) is just pointwise multiplication.

2.3.4 Principal connectionsJust like vector bundles, principal bundles carry a class of connections that are compatiblewith their extra structure. For vector bundles, we demanded that parallel transport inducemorphisms of vector spaces between fibres. For principal bundles, we demand that paralleltransport induces morphisms of principal homogeneous spaces. In other words, we wantPtγ,t : Pγ(0) → Pγ(t) to be a morphism of principal homogeneous spaces for all γ and t. Amoment’s thought shows that this is equivalent to the following definition.

Definition 42. Suppose (P,Σ, π, F ) is a principal bundle equipped with a complete con-nection. If for any curve γ in Σ with horizontal lift γ starting at p, the curve

γ · g : t 7→ γ(t) · g

is the horizontal lift of γ starting at p · g, the connection on P is called a principal connec-tion.

27

Lemma 43. A complete connection on a principal bundle is principal iff its horizontalsubspaces are invariant under the right G-action on the bundle, by which we mean that

Hp · g = Hp·g for all p ∈ P and g ∈ G.

Here, Hp · g is the set v · g | v ∈ Hp, recalling that v · g is the pushforward of v under theaction of g.

Proof. If the horizontal subspaces are invariant under the G-action, then clearly the con-nection is principal. To prove the converse, suppose that the connection is principal andpick p ∈ P and g ∈ G. Taking any v ∈ Hp, it suffices to show that v · g ∈ Hp·g, becausethis shows that Hp · g ⊆ Hp·g and repeating the argument for p · g and g−1 instead of p andg will then yield the other inclusion.

It thus suffices to show that v · g ∈ Hp·g. Pick any horizontal curve through p withinitial velocity v (for example, lift a curve in Σ with initial velocity (Tpπ)(v)). Applyingg to this curve yields a curve through p · g with initial velocity v · g, and it is horizontalbecause the connection is principal.

Note that in the definition of principal connection, we demand that the connectionbe complete. In fact, any connection on a principal bundle whose horizontal subspacesare invariant under the right G-action is automatically complete and thus a principalconnection ([KN63], proposition 3.1 on page 69).

If p ∈ P is a point in a principal bundle, the vertical space Vp can be identified with gby identifying ξ ∈ g with p · ξ. This allows us to regards the projections Ap : TpP → Vpassociated to any connection as linear maps

Ap : TpP → g.

We shall do this from now on. Specifying a (not necessarily principal) connection on aprincipal bundle can then be done by specifying a linear map Ap : TpP → g for every pin a smooth way, such that Ap(p · ξ) = ξ for all ξ ∈ g (the latter condition amounts todemanding that Ap : TpP → Vp restricts to the identity on Vp).

When is a connection specified by such Ap principal? We have the following result.

Lemma 44. Let (P,Σ, π,G) be a principal bundle and suppose that the kernels of the linearmaps Ap : TpP → g specify a connection on it, meaning that Ap depends smoothly on pand Ap(p · ξ) = ξ for all p ∈ P and ξ ∈ g. Then the connection specified by these kernelsis principal iff

Ap·g(v · g) = Adg−1(Ap(v)) (∗)for all p ∈ P , g ∈ G, v ∈ TpP .

Proof. The connection is principal precisely if for any p ∈ P and g ∈ G, the map Ap·g :Tp·gP → g is the unique linear map that maps (p · g) · ξ to ξ and has kernel Hp · g. Thisunique linear map is in fact the map

Tp·gP → g : v 7→ Adg−1(Ap(v · g−1))

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as one verifies by evaluating on horizontal and vertical vectors. We conclude that theconnection is principal iff for all p ∈ P , g ∈ G, v ∈ Tp·gP we have

Ap·g(v) = Adg−1(Ap(v · g−1)),

which is equivalent to (∗) by replacing v by v · g.

For principal connections, we will use this specification of a connection by its associatedprojections Ap frequently. In fact, we will use the name A for the principal connection withprojections Ap, and the set of all principal connections on a given principal bundle P willbe denoted by A(P ).

If P and Q are two principal bundles over Σ, equipped with connections A and Brespectively, then these bundles are called gauge equivalent if there is an isomorphism ofprincipal bundles f : P → Q such that (Tpf)(HA

p ) = HBf(p) for all p ∈ P (here, HA

p

denotes the horizontal subspace of TpP as given by the connection A and HBf(p) denotes the

horizontal subspace of Tf(p)Q as given by B). Gauge equivalences are the isomorphisms ofprincipal bundles that are equipped with a connection.

2.3.5 Holonomy in principal bundlesSuppose (P,Σ, π,G) is a principal bundle equipped with a principal connection A. If γ isa loop in Σ, then the holonomy of A around γ is a morphism of principal homogeneousspaces

Ptγ : Pγ(0) → Pγ(0).

For every point p ∈ Pγ(0) there is then a unique g ∈ G such that Ptγ(p) = p · g. Thiselement is written Holp(A, γ). In symbols,

Ptγ(p) = p · Holp(A, γ) for all p ∈ Pγ(0).

It follows from the fact that Ptγ is a morphism of principal homogeneous spaces that

Holp·g(A, γ) = g−1 Holp(A, γ)g. (†)

Now suppose that x, y ∈ Σ are in the same component of Σ. Taking a curve δ from x to yand lifting it horizontally at a point p ∈ Px, we get a curve in P that ends in q ∈ Py. Forany loop γ in Σ at y we then have

Holp(δ ? γ ? δ) = Holq(γ).

In the above equation, the star means concatenation of the curves from left to right. Notethat reparametrizing a curve does not influence the holonomy of the connection around it.We conclude that

Hol(A) = Holp(A, γ) | γ is a piecewise smooth curve in Σ

only depends on the base point p up to conjugation. We can therefore speak of the holonomyof the connection A as the conjugacy class of the subgroup Hol(A) ⊆ G.

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Chapter 3

Vector-valued differential forms

An ordinary differential k-form on a manifold M is the specification of a linear mapΛk(TpM) → R at every point p ∈ M depending smoothly on p. In this chapter weextend this notion to differential forms taking values in some vector bundle over M . Wethen reformulate some of the material in this new terminology, and we discuss the co-variant exterior derivative on principal bundles that are equipped with a connection. Ourreferences for this chapter are [Jan05] and [KN63] (section II.5).

3.1 Definitions and propertiesLet M be a smooth manifold. The space of differential forms of degree k on M is denotedΩk(M). A differential k-form onM consists of a smooth choice of linear maps Λk(TpM)→R at every point. We can state this as follows: a differential k-form on M is a morphism ofvector bundles from Λk(TM) to M ×R covering the identity. This motivates the followingdefinition.

Definition 45. Let M be a smooth manifold and E a vector bundle over M . An E-valueddifferential k-form on M is a morphism of vector bundles from Λk(TM) to E covering theidentity IdM .

The space of E-valued k-forms over M will be denoted by Ωk(M,E). If V is a fixedfinite-dimensional vector space, we can speak of V -valued differential forms onM , in whichcase we mean differential forms with values in the trivial bundle M × V . For example, ifA is a principal connection on a principal bundle, the projections Ap : TpP → g specify ag-valued 1-form on P , the so-called connection 1-form.

3.1.1 PullbackLet M and N be smooth manifolds and f : M → N a smooth map. Suppose E is a vectorbundle over N and α is an E-valued k-form on N . Then we can pull back α to M via f

30

to get f ∗α. It is an (f ∗E)-valued form on M . The pullback form is defined by

(f ∗α)x(v1, . . . , vk) = (x, α(Txf(v1), . . . , Txf(vk))) ∈ (f ∗E)x.

The gauge group of a principal bundle acts on a connection on this bundle by pullback ofthe connection 1-form.

3.1.2 Wedge productLet M be a smooth manifold and D,E vector bundles over M . Assume that α is a D-valued k-form over M and β is an E-valued l-form over M . We then define their wedgeproduct in a similar way as the wedge product of ordinary differential forms, but we usethe tensor product as the bilinear operation. The wedge product α ∧ β is a D ⊗E-valued(k + l)-form on M given by

(α ∧ β)x(v1, . . . , vk, vk+1, . . . , vk+l)

= 1k!l!

∑σ∈Sk+l

sgn(σ)αx(vσ(1), . . . , vσ(k))⊗ βx(vσ(k+1), . . . , vσ(k+l)).

for all x ∈ M and vi ∈ TxM . Calculation shows that this product is associative (just likethe ordinary wedge product). The wedge product is compatible with pullbacks in the sensethat

(f ∗α) ∧ (f ∗β) = f ∗(α ∧ β).We will often be concerned with differential forms on a principal G-bundle that take

values in the Lie algebra g. Suppose α, β are g-valued forms on a smooth manifold, ofdegree k and l respectively. We can then form a new g-valued form [α ∧ β] given by

[α ∧ β]x(v1, . . . , vk, vk+1, . . . , vk+l)

= 1k!l!

∑σ∈Sk+l

sgn(σ)[αx(vσ(1), . . . , vσ(k)), βx(vσ(k+1), . . . , vσ(k+l))].

Note that [α∧ β] is essentially the composition of α∧ β with the bracket g⊗ g→ g. Fromthe definition one checks that

[α ∧ β] = (−1)kl+1[β ∧ α].

In particular, if k is even, we have [α ∧ α] = 0. There is also an analogue of the Jacobiidentity for this product. If γ is a g-valued j-form, then

(−1)kj[α ∧ [β ∧ γ]] + (−1)kl[β ∧ [γ ∧ α]] + (−1)lj[γ ∧ [α ∧ β]] = 0.

An analogous construction works for a symmetric bilinear form on any vector space.Suppose 〈−,−〉 is some fixed symmetric bilinear form on some finite-dimensional realvector space V , and let α and β be V -valued forms on a smooth manifold of degree k and

31

l respectively. We can then form an ordinary (k + l)-form 〈α ∧ β〉 on this manifold bydefining

〈α ∧ β〉x(v1, . . . , vk, vk+1, . . . , vk+l)

= 1k!l!

∑σ∈Sk+l

sgn(σ)〈αx(vσ(1), . . . , vσ(k)), βx(vσ(k+1), vσ(k+l))〉.

We have the identity〈α ∧ β〉 = (−1)kl〈β ∧ α〉.

3.1.3 Exterior derivativeWe want to define an exterior derivative for vector-valued forms. Here, we shall only do sofor forms taking values in a trivial vector bundle. Suppose V is a finite-dimensional realvector space and M is a smooth manifold. Let α be a V -valued k-form on M . Then wedefine the exterior derivative dα by the usual formula

(dα)(X0, . . . , Xk) =k∑i=0

(−1)iXi(α(X0, . . . , Xi−1, Xi, Xi+1, . . . , Xk))

+∑

0≤i<j≤k(−1)i+jα([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , Xk).

The usual properties

d(dα) = 0 d(α ∧ β) = (dα) ∧ β + (−1)kα ∧ (dβ)

still hold for any k-form α and any l-form β. This exterior derivative commutes withpullbacks. In section 3.3 we will discuss exterior derivatives in greater detail.

If G is a Lie group, there is an important g-valued 1-form on G that we now brieflydiscuss.

Definition 46. If G is a Lie group, the g-valued 1-form on G given by

θ : TG→ g : v ∈ TgG 7→ (Tg(λg−1))(v) ∈ TeG = g

is called the Maurer-Cartan form on G.

In words, the Maurer-Cartan form identifies every tangent space TgG with TeG = gthrough left translation. To illustrate the exterior derivative of a differential form withvalues in a fixed vector space (and because we will need it later), we prove the Maurer-Cartan structure equation.

Lemma 47 (Maurer-Cartan structure equation). If G is a Lie group with Maurer-Cartanform θ, then for any two vectors v, w ∈ TgG we have

(dθ)g(v, w) = −[θg(v), θg(w)].

This equation is often written as dθ + 12 [θ, θ] = 0.

32

Proof. Extend v and w to left-invariant vector fields V and W on G (these are the left-invariant vector fields associated to θ(v) and θ(w)). Then θ(V ) and θ(W ) are constant onG. We find

(dθ)g(v, w) = (dθ)g(V,W )= V (θ(W ))−W (θ(V ))− θg([V,W ])= −θg([V,W ]).

By the definition of the Lie bracket on g, the latter equals −[θg(V ), θg(W )] and the lemmais proved.

The following result is useful for differentiation on Lie groups. We leave the proof tothe reader.

Lemma 48. Let G be a Lie group and α : (−ε, ε)→ G and β : (−ε, ε)→ G two curves inG. Then

• θ(ddt |t=0(α(t)β(t))

)= Adβ(0)−1

(θ(ddt |t=0α(t)

))+ θ

(ddt |t=0β(t)

),

• θ(ddt |t=0α(t)−1

)= −Adα(0)

(θ(ddt |t=0α(t)

)).

3.2 Associated bundlesSuppose (P,Σ, π,G) is a principal bundle and V is a finite-dimensional vector space. As-sume f : G → GL(V ) is a representation of G. We can then construct the associatedbundle P ×f V , which is a vector bundle over Σ with fibre V . This associated bundle is

P ×f V = (P × V )/ ∼

where ∼ is the equivalence relation (p · g, v) ∼ (p, f(g)(v)). The corresponding projectionis

τ : P ×f V → Σ : [p, v] 7→ π(p).

We now consider an important special case.

Definition 49. Given a principal bundle (P,Σ, π,G), we define the adjoint bundle gP tobe gP = P ×Ad g.

Note that as a set, this bundle is the quotient of P ×g under the equivalence (p ·g, ξ) ∼(p,Adg(ξ)).

Suppose now that we have given a connection on the principal bundle. We will beinterested in g-valued differential forms on this bundle P that are both horizontal andequivariant.

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Definition 50. Let (P,Σ, π,G) be a principal bundle. A g-valued k-form α on P is calledhorizontal if at every point p ∈ P the map αp returns zero whenever one of its argumentsis vertical. It is called equivariant if for all g ∈ G, p ∈ P and vectors v1, . . . , vk ∈ TpP wehave

αp·g(v1 · g, . . . , vk · g) = Adg−1(αp(v1, . . . , vk)).

As an example, connection 1-forms are equivariant. The set of g-valued horizontal,equivariant k-forms on P will be denoted by Ωk

Ad,H(P, g). The space ΩkAd,H(P, g) can be

identified with the space Ωk(Σ, gP ) of gP -valued forms on Σ as we now explain. Supposeα ∈ Ωk

Ad,H(P, g). Then we can create β ∈ Ωk(Σ, gP ) as follows: given x ∈ Σ and vectorsv1, . . . , vk ∈ TxΣ, we define

βx(v1, . . . , vk) = [p, αp(v1, . . . , vk)] ∈ (gP )x (∗)

where p is any point in the fibre over x and vi is the horizontal lift of vi. This is well-definedby equivariance of α. Conversely, if β ∈ Ωk(Σ, gP ), then we can construct α ∈ Ωk

Ad,H(P, g)by demanding (∗). From now on, we will identify Ωk(Σ, gP ) and Ωk

Ad,H(P, g).

3.3 The covariant exterior derivativeIn this section we discuss the covariant exterior derivative, also known as the exteriorcovariant derivative. Our goal is to find an appropriate way of differentiating an elementof Ωk(Σ, gP ) to obtain an element of Ωk+1(Σ, gP ).

Let us pick α ∈ Ωk(Σ, gP ), regarded as a horizontal and equivariant g-valued k-formon P . The obvious candidate for a derivative of α is dα, which is indeed an equivariantg-valued (k + 1)-form on P . It need not be horizontal, however. We will try to define thecovariant exterior derivative dAα as the “horizontal version” of dα.

Given a principal connection A on a principal bundle P , there is a bundle map

h : TP → TP

covering the identity P → P that sends a vector to its horizontal part (in fact, this is abundle map from TP to its subbundle H of horizontal subspaces). It induces a bundlemap

hk : Λk(TP )→ Λk(TP )

for every k by setting h(α∧β) = h(α)∧h(β). These are bundle maps covering the identity,and by abuse of notation we will just write h = hk.

If E is a vector bundle over P , then an E-valued k-form over P is a bundle mapΛk(TP )→ E covering the identity. We can compose such a form with h (applying h first)to obtain a new differential form over P . This new form is clearly horizontal. We are thusled to define dA as follows.

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Definition 51. Given a principal bundle (P,Σ, π,G) equipped with a connection A, wedefine the covariant exterior derivative dA as

dA : Ωk(Σ, gP )→ Ωk+1(Σ, gP ) : α 7→ dα h.

In words, to apply dAα to a set of vectors v1, . . . , vk+1 ∈ TpP , we first take the horizontalparts h(vi) of these vectors and then apply dα to the resulting horizontal vectors. Thecovariant exterior derivative can also be defined by a more hands-on formula.Theorem 52. For α ∈ Ωk(Σ, gP ) we have

dAα = dα + [A ∧ α]. (∗)Here, we consider the connection A as a g-valued 1-form on P and α as an equivarianthorizontal g-valued k-form on P .Proof. We will check the equality (∗) by applying both sides to vectors. By linearity,it suffices to check this on a set of vectors v0, . . . , vk all of which are either horizontal orvertical. We may assume that the firstm of these are vertical and the others are horizontal.

The case m = 0. In this case all the vectors are horizontal and(dAα)(v0, . . . , vk) = (dα)(v0, . . . , vk).

The right hand side of (∗) yields the same result because[A ∧ α](v0, . . . , vk) =

∑j

(−1)j[A(vj), α(v0, . . . , vj, . . . , vk)]

and all the factors A(vj) are zero. This means the term [A ∧ α] will not contribute in (∗).The case m ≥ 1. We want to apply the right hand side of (∗) to

p · ξ0 = v0, . . . , p · ξm−1 = vm−1, vm, . . . , vk

where the vectors vm, . . . , vk are horizontal (and all of the above are elements op TpP ).Since m ≥ 1 we expect to find zero. We may assume that the vm, . . . , vk are linearlyindependent (and by horizontalness then so are their projections onto Σ).

Because we want to use the coordinate-free definition of the exterior derivative, we willextend the vectors vi to local vector fields around p. We extend the vectors p · ξi by takingq · ξi at every point q ∈ P (these are in fact global extensions). We extend the vectors vi(i ≥ m) by extending their projection onto Σ locally in a mutually commuting way andlifting these vector fields to horizontal vector fields around p.

We can now calculate:(dα)p(p · ξ0, . . . , p · ξm−1, vm, . . . , vk) + [A ∧ α](p · ξ0, . . . , p · ξm−1, vm, . . . , vk)

=∑j

(−1)jvj(α(v0, . . . , vj, . . . , vk))

+∑i<j

(−1)i+jα([vi, vj], v0, . . . , vi, . . . , vj, . . . , vk)

+ 1k!

∑σ∈Sk+1

sgn(σ)[A(vσ(0)), α(vσ(1), . . . , vσ(k))].

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If m ≥ 2, then every term above contains a factor of α with at least one vertical vectoramong its arguments. Because α is horizontal, the result is zero in this case. All thatremains is the case m = 1.

We inspect the three terms above for the case m = 1. The second term will notcontribute: either v0 (the only vertical vector) appears in the Lie brackets, in which casethe bracket is zero because v0 commutes with all the other fields (as is easily verified,keeping in mind that the horizontal spaces are G-invariant), or v0 appears as an argumentto α which results in zero because α is horizontal.

The first term has only one contribution: the case where the vertical vector is notsupplied to α (and is hence upfront). The third term only has a contribution in the casewhere the vertical vector is not supplied to α. After noticing that the k! compensates forthe repeated terms in the third term, what remains is

(dAα)p(p · ξ0, v1, . . . , vk) = (p · ξ0)(α(v1, . . . , vk)) + [A(p · ξ0), α(v1, . . . , vk)]= (p · ξ0)(α(v1, . . . , vk)) + [ξ0, α(v1, . . . , vk)].

Now notice that vi(p · g) = vi(p) · g by the way we extended the vi. We get

(dAα)p(p · ξ0, v1, . . . , vk) = d

dt |t=0(α(v1(p) · exp(tξ0), . . . , vk(p) · exp(tξ0)))

+ [ξ0, α(v1, . . . , vk)]

= d

dt |t=0Adexp(−tξ0)(α(v1, . . . , vk)) + [ξ0, α(v1, . . . , vk)]

= ad−ξ0(α(v1, . . . , vk)) + [ξ0, α(v1, . . . , vk)]= 0.

This shows that the formula (∗) is correct.

3.4 CurvatureWe now have a suitable notion of exterior derivative for Ωk(Σ, gP ). One may wonderwhether the important relation d2 = 0 still holds for this new exterior derivative dA, sowhether d2

A = 0. Suppose that α ∈ Ωk(Σ, gP ). Then

dAdAα = dA(dα + [A ∧ α])= d2α + d[A ∧ α] + [A ∧ dα] + [A ∧ [A ∧ α]]

= [dA ∧ α] + 12[[A ∧ A] ∧ α]

= [(dA+ 12[A ∧ A]) ∧ α].

This motivates the following definition.

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Definition 53. Let A be a connection on a principal bundle. The two-form

FA = dA+ 12[A ∧ A]

is called the curvature of the connection A. A connection is called flat if its curvaturevanishes everywhere on the principal bundle. Given a principal bundle, the set of all flatconnections on it is denoted by A0. A flat principal bundle is a tuple (P,A) where P is aprincipal bundle and A a flat connection on it.

We will often refer to the flat principal bundle (P,A) as P , keeping in mind that it isequipped with a certain flat connection. It is easy to verify that

FA = dA h

where h is the projection onto the horizontal subbundle, which is analogous to the definitionof the covariant exterior derivative for Ωk(Σ, gP ).

There is another very important motivation for considering the curvature of a connec-tion. Let us describe it briefly. We first state the import Frobenius theorem. For a proof,we refer to [Lee03], page 359 (theorem 14.5).

Theorem 54 (Frobenius). Let H be a rank k subbundle of the vector bundle TP . Thenthe following conditions are equivalent:

• for every two local sections X and Y of H (local vector fields on P with Xp, Yp ∈ Hp

for every p), the Lie bracket is also a local section of H;

• around every point p in P there exists a chart such that H is spanned by the first kcoordinate vector fields ∂

∂x1, . . . , ∂

∂xk.

If either (and hence both) of these conditions is satisfied, the bundle H is called integrable.

Now let X and Y be local horizontal vector fields around a point p ∈ P . Consider thefollowing calculation:

FA(Xp, Yp) = (dA)(Xp, Yp) + 12[A ∧ A](Xp, Yp)

= (dA)(Xp, Yp) + [A(Xp), A(Yp)]= (dA)(Xp, Yp)= Xp(A(Y ))− Yp(A(X))− A([X, Y ]p)= −A([X, Y ]p).

This calculation shows that flatness of the connection is equivalent to the first conditionmentioned in the Frobenius theorem, and hence to the second. In other words, flatnessof the connection is precisely integrability of the horizontal subbundle. In other words, aconnection is flat if its horizontal subspaces line up to be tangent to the leaves of a foliation.

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Chapter 4

Symplectic and Poisson structures

Symplectic geometry is the branch of geometry concerned with manifolds that are equippedwith a closed and non-degenerate two-form. Poisson geometry is concerned with the studyof manifolds that are equipped with a Poisson structure, and is in a sense a generalizationof symplectic geometry. We discuss some of the basic ideas. Our main references are[Hec12] (for symplectic manifolds) and [Wat07] (for Poisson manifolds).

4.1 Symplectic manifoldsDefinition 55. A symplectic structure on a smooth manifold M is a closed and non-degenerate two-form on M .

Definition 56. A symplectic manifold is a smooth manifold equipped with a symplecticstructure.

Definition 57. A map φ : M → N between symplectic manifolds (M,ω) and (N, ν) is amorphism of symplectic manifolds (also called a symplectic map or a symplectomorphism)if φ∗ν = ω.

A non-degenerate two-form on a manifold M gives an isomorphism between T ∗pM andTpM at each point p ∈M . We will denote this isomorphism by

] : T ∗pM → TpM : v 7→ v]

and its inverse by[ : TpM → T ∗pM : v 7→ v[.

These isomorphisms are completely determined by the condition

v[(w) = ω(v, w) for all v, w ∈ TpM .

A different way to write this isv[ = ivω

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where i denotes the interior product.To any smooth function f ∈ F(M) we can associate a vector field Xf by

Xf = (df)].

Definition 58. A vector field X on M is called Hamiltonian if it equals Xf for somesmooth function f . The flow of this vector field is then called the (Hamiltonian) flow of f .

Definition 59. A vector field on M is called symplectic if its flow preserves the symplecticstructure. In other words, a vector field X on M is symplectic iff

LX ω = 0.

Lemma 60. Every Hamiltonian vector field is symplectic.

Proof. Let f ∈ F(M). Using Cartan’s formula, we have

LXf (ω) = (diXf + iXfd)(ω)= d(iXfω)= d(X[

f )= d(df)= 0.

We can also find an expression for the extent to which the flow of f preserves g ∈ F(M):

Xf (g) = (dg)(Xf )= (dg)((df)])= ω((dg)], (df)])= −ω(Xf , Xg).

In particular, the function f is constant along its own flow. In classical mechanics, this isa very important observation: the evolution of a system will be given by the Hamiltonianflow of a certain function H (called the Hamiltonian), which is in many contexts the totalenergy of the system. The observation that H is constant along its own flow is just thelaw of conservation of energy.

4.2 Symplectic reductionIn classical mechanics, many physical systems can be described as a tuple (M,ω,H), calleda Hamiltonian system, where (M,ω) is a symplectic manifold and H ∈ F(M) is a functioncalled the Hamiltonian. The manifold M is the phase space of the system, and the dy-namics of the system (how the system evolves through time) are given by the flow of theHamiltonian vector field XH . In many cases, the physical system will exhibit some kind

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of symmetry, in the sense that a Lie group acts on the symplectic manifold by symplecto-morphisms which leave H invariant. Under certain technical assumptions, the symmetryof the physical system will lead to conserved quantities and to symplectic reduction, inwhich the dynamics of the original system are reduced to those of a Hamiltonian systemof lower dimension. Let us discuss symplectic reduction.

4.2.1 Moment mapsSuppose (M,ω) is a symplectic manifold and G is a Lie group that acts smoothly on Mfrom the left (in the sense that the map G×M → M is smooth). We can then associateto any ξ ∈ g a vector field Xξ on M by setting

Xξ(p) = d

dt |t=0(exp(tξ) · p) .

Definition 61. A smooth action G×M → M of a Lie group on the symplectic manifoldM is called Hamiltonian if the vector fields Xξ are Hamiltonian for all ξ ∈ g.

Suppose now that G acts on (M,ω) in a Hamiltonian fashion. To any ξ ∈ g we canthen associate the smooth function f such that Xξ = Xf . This choice of f is not unique,but it is unique up to a locally constant function. We get a map

ν : g→ F(M) : ξ 7→ (f such that Xf = Xξ)that can be made linear by appropriate choices for the functions f . Indeed, pick a basisξ1, . . . , ξn for g and pick arbitrary smooth functions f1, . . . , fn such that Xξi = Xfi fori ∈ 1, . . . , n. Define ν by ν(ξi) = fi and extending linearly.

Note that we can viewM as a subset of the dual of F(M) by identifying a point p ∈Mwith the map

evalp : F(M)→ R : f 7→ f(p).Let us now take the dual of the linear map ν and restrict it to M ⊂ F(M)∗. The result isa map µ : M → g∗, and it is an example of a so-called moment map.Definition 62. Let G act on (M,ω) in a Hamiltonian fashion. A linear map

µ : M → g∗

is called a moment map for this action if for any ξ ∈ g the Hamiltonian vector field of thefunction

µξ : M → R : p 7→ µ(p)(ξ)is equal to Xξ.

The map µξ is called the ξ-component of µ.Definition 63. Let G act on (M,ω) in a Hamiltonian fashion, with moment map µ : M →g∗. The moment map is called equivariant if

µ(g · p) = Ad∗g(µ(p)) for all g ∈ G, p ∈M .In this case, (M,ω, µ) is called a Hamiltonian G-space.

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4.2.2 Symplectic quotientsIf (M,ω, µ) is a Hamiltonian G-space, then the inverse image N = µ−1(0) is invariantunder G. Assume that 0 is a regular value of µ. Then N is a submanifold of M . Writei : N →M for the inclusion of N into M . We can restrict ω to a two-form i∗ω on N . Thistwo-form will be closed (recall that pullback and exterior derivative commute). However,in general it need not be non-degenerate (and indeed, N could even be odd-dimensional,forcing i∗ω to be degenerate everywhere).

Let us try to understand what prevents i∗ω from being a symplectic form. To makethis more precise, at every point p ∈ N the two-form i∗ω gives a map TpN → T ∗pN .Non-degeneracy of the two-form is just injectivity of this map, so we shall try to find itskernel.

Let us first describe TpN . Note that TpN = ker(Tpµ). Take ξ ∈ g. By the definition ofa moment map, we have that

Xξ = (dµξ)].This implies

ωp(v, (Xξ)p) = ωp(v, ((dµξ)])p) = −(dµξ)p(v) = −〈(Tpµ)(v), ξ〉

for any vector v ∈ TpM . This shows that the kernel of Tpµ is exactly (Xξ)p | ξ ∈ gωwhere the superscript ω indicates the orthogonal complement with respect to ω, so

TpN = (Xξ)p | ξ ∈ gω.

We are now in a position to find the kernel of the map TpN → T ∗pN : it is just

TpN ∩ (TpN)ω = (Xξ)p | ξ ∈ gω ∩ (Xξ)p | ξ ∈ g.

Take ξ, η ∈ g and note that

ωp(Xξ, Xη) = −〈(Tpµ)(Xη), ξ〉.

Now (Tpµ)(Xη) = 0 because it is just ddt |t=0 (µ(exp(tη · p)) and µ is identically zero on N .

We conclude thatker(TpN → T ∗pN) = (Xξ)p | ξ ∈ g.

Suppose now that G is compact and acts freely. We can then apply the following resultabout group actions. A proof can be found in [Lee03] on page 153 (theorem 7.10).Lemma 64. If a compact Lie group G acts freely on a manifold N , then the quotient spaceN/G has a natural manifold structure such that π : N → N/G is a smooth submersion.

In this case, we see that the kernel of TpN → T ∗pN is exactly the kernel of dpπ withπ the projection N → N/G as in the lemma. This means that N/G carries a symplecticstructure ω0 defined by the equation

π∗ω0 = i∗ω.

The space (N/G, ω0) is called the symplectic quotient of M by the action of G. It is asymplectic manifold of dimension dim(M)− 2 dim(G).

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4.3 Poisson manifoldsWe now introduce Poisson manifolds. We explain why every symplectic manifold can beseen as a Poisson manifold, and we show that a general Poisson manifold is a foliation-likecollection of symplectic leaves (though different leaves can have different dimensions).

4.3.1 Poisson structuresDefinition 65. Let A be an algebra. A Poisson structure on A is a bilinear map

·, · : A× A→ A : (f, g) 7→ f, g

that is a Lie bracket on the vector space A and that satisfies the Leibniz rule

f, gh = gf, h+ f, gh

for all f, g, h ∈ A. If the algebra A is equipped with a Poisson structure, it is called aPoisson algebra.

Definition 66. A Poisson structure (or Poisson bracket) on a smooth manifold M is aPoisson structure on the algebra F(M) of smooth functions on M .

Definition 67. A Poisson manifold is a smooth manifold equipped with a Poisson structure.

Definition 68. Let M and N be two Poisson manifolds. A smooth map φ : M → N is amorphism of Poisson manifolds (also called a Poisson map) if for all f, g ∈ F(N) we have

f φ, g φ = f, g φ.

Suppose M is a Poisson manifold. By the Leibniz rule, for any fixed f ∈ F(M) themap g 7→ −f, g is a derivation of F(M). It is known from differential geometry thatevery derivation of F(M) is in fact a vector field. This means that we can associate to fa vector field Xf corresponding to the derivation, i.e. the vector field satisfying

f, g = −Xf (g) for all g ∈ F(M).

This vector field Xf is called the Hamiltonian vector field associated to f .Note that the calculation

f, g(p) = −(Xf )p(g) = −(dpg)(Xf ) = (Xg)p(f) = (dpf)(Xg)

shows that the Poisson bracket of f and g at p only depends on their differentials dpf anddpg. This means that a Poisson structure gives rise to a linear mapping

T ∗pM → TpM : dpf 7→ Xf

at every point of the manifold. The rank of this map will be called the rank of the Poissonmanifold at the point p.

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4.3.2 Symplectic manifolds are PoissonLet (M,ω) be a symplectic manifold. We will show that it is also a Poisson manifold.Consider

·, · : F(M)×F(M)→ F(M) : (f, g) 7→ f, g = ω(Xf , Xg) = −Xf (g).

We claim that this map is a Poisson structure on M . Bilinearity and antisymmetry of thebracket follow from bilinearity and antisymmetry of ω.

The Jacobi identity for this bracket follows immediately from the following lemma. Itsproof is based on page 67 of [Hec12].

Lemma 69. Let M be a smooth manifold and ω a non-degenerate two-form on it. Usethis two-form to associate to any smooth function f a vector field Xf = (df)]. Define abracket on M by setting f, g = ω(Xf , Xg). Then this bracket satisfies the Jacobi identityiff ω is closed.

Proof. First note that the Jacobi identity for the bracket is equivalent to

[Xf , Xg] = −Xf,g for all f, g ∈ F(M).

Indeed, taking f and g in F(M) we have [Xf , Xg] = −Xf,g exactly if [Xf , Xg](h) =−Xf,g(h) for all h ∈ F(M). The latter is equivalent to Xf (Xg(h)) − Xg(Xf (h)) =f, g, h, or in other words f, g, h − g, f, h = f, g, h, which is just theJacobi identity.

Note that f, g = ω(Xf , Xg) = −Xf (g) = −LXf (g). Consider now the calculation

df, g = −d(LXf (g)) = −LXf (dg) = −LXf (iXgω) = −i[Xf ,Xg ]ω − iXg(LXf ω)

where we have used that LX iY − iY LX = i[X,Y ]. This shows that [Xf , Xg] = −Xf,g forall f, g if and only if LXf ω = 0 for all f . This is equivalent to dω = 0 since LXf ω =iXf (dω).

We now know that every symplectic manifold is also Poisson. If f is a smooth functionon a symplectic manifold M , we have two notions of Hamiltonian vector field associatedto f : one for M as a symplectic manifold, one for M as a Poisson manifold. By theirdefinitions, these coincide. Note that the linear map T ∗pM → TpM induced by the Poissonstructure on a symplectic manifold is just the identification between T ∗pM and TpM usingthe symplectic structure (and in the case of a symplectic manifold, this is an isomorphismbecause the symplectic structure is non-degenerate).

The converse is also true: if the map T ∗pM → TpM is an isomorphism everywhere, thenthe manifold is symplectic.

Theorem 70. Suppose M is a Poisson manifold that has full rank everywhere. Then Mis a symplectic manifold.

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Proof. The symplectic structure on M can be reconstructed by inverting the map T ∗pM →TpM to get a map TpM → T ∗pM (which we then view as a bilinear map from TpM × TpMto R). This bilinear map is then a symplectic structure: antisymmetry follows easilyfrom antisymmetry of the Poisson bracket, non-degeneracy follows immediately from theassumption of full rank and closedness follows from the Jacobi identity for the Poissonbracket according to the previous lemma.

4.3.3 The splitting theorem and symplectic leavesSome Poisson manifolds are not symplectic, and we have seen that the obstruction tobeing symplectic is degeneracy of the map T ∗pM → TpM . We now prove a result originallyby Weinstein about the local structure of Poisson manifolds: near a point p, a Poissonmanifold is the product of a symplectic manifold and a Poisson manifold of rank zero atp. The original reference for this result is [Wei83].

Definition 71. Given two Poisson manifolds M and N , we define their product to bethe smooth manifold M × N equipped with the bracket such that the two projections πM :M ×N →M and πN : M ×N → N are Poisson maps, and such that

f πM , g πN = 0 for all f ∈ F(M) and g ∈ F(N).

This product Poisson manifold admits a more conceptual description if we look at themaps

π∗M : F(M)→ F(M ×N) : f 7→ f πMand

π∗N : F(N)→ F(M ×N) : g 7→ g πN .

It is just the smooth manifold M × N equipped with the bracket such that π∗M and π∗Nare bijective Lie algebra homomorphisms onto commuting subalgebras of F(M ×N). Oneverifies that this determines the bracket on M ×N completely.

If we take local coordinates xi on M and yi on N , then the product manifold is definedby

xi πM , yj πN = 0

andxi πM , xj πM = xi, xj πM yi πN , yj πN = yi, yj πN .

We now come to the splitting theorem, which clarifies what Poisson manifolds look likelocally. The proof given here is based on [Wat07] and [Vai94] (theorem 2.16 on page 27).

Theorem 72 (Splitting theorem). Let M be a Poisson manifold and take p ∈ M . Thenthere exists an open neighbourhood U of p with coordinates q1, . . . , qr, p1, . . . , pr, andn1, . . . , ndim(M)−2r such that

qi, qj = pi, pj = qi, nj = pi, nj = 0

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for all i, j andqi, pj = δij

andni, nj(p) = 0.

These relations are called the canonical commutation relations and the coordinates arecalled canonical coordinates around the point p. Taking S to be the 2r-dimensional localsubmanifold of M determined by the qi and pi and N to be the 2r-codimensional localsubmanifold of M determined by the ni, we have that

M = S ×N locally around p

as Poisson manifolds. Note that S is a symplectic manifold and that N is a Poissonmanifold of rank zero at p.

If M is symplectic, then the canonical coordinates are called Darboux coordinates, andthe splitting theorem is known as the Darboux theorem. Note that this theorem highlightsan important fundamental difference between Riemannian and symplectic geometry: anytwo points in symplectic manifolds have neighbourhoods that are isomorphic as symplecticmanifolds. This is far from true in the Riemannian case, where curvature is a local invariant.This makes symplectic geometry inherently global, since there is nothing interesting to saylocally (except for the linear algebra of symplectic forms).

Proof of the splitting theorem. We prove this by induction on dim(M). The theorem isclear for dim(M) ∈ 0, 1. Suppose therefore that dim(M) ≥ 2 and that the result is truefor all Poisson manifolds of lower dimension.

If M has rank zero at p, we can take local coordinates ni and we are done. If M haspositive rank at p, there is a function p1 ∈ F(M) such that Xp1(p) 6= 0. By a standardresult in differential geometry, there exist local coordinates qi around p such that Xp1 = ∂

∂q1in a neighbourhood of p.

Considering q1 as a smooth function around p, we have q1, p1 = Xp1(q1) = 1 locally.Note that Xp1 and Xq1 are independent at p, for otherwise we would have

1 = q1, p1 = −Xq1(p1) = −λXp1(p1) = λp1, p1 = 0.

Note also that the vector fields Xp1 and Xq1 commute, because

[Xp1 , Xq1 ] = −Xp1,q1 = −Xconstant function 1 = 0.

We can therefore find new coordinates α1, . . . , αdim(M) in a neighbourhood of p such that

Xp1 = ∂

∂α1and Xq1 = ∂

∂α2.

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Because ∂p1∂α1

= Xp1(p1) = 0, ∂p1∂α2

= Xq1(p1) = −1, ∂q1∂α1

= Xp1(q1) = 1 and ∂q1∂α2

= Xq1(q1) =0, the Jacobian of (p1, q1, α3, . . . , αdim(M)) in this new coordinate system is given by

0 −1∗

1 00 Iddim(M)−2

which is invertible (here, the star denotes an appropriately sized block with unknownentries, the zero in the lower left is a block of zeroes and Idn denotes the n × n identitymatrix). By the inverse function theorem, this means that p1, q1, α3, . . . , αdim(M) form localcoordinates in some neighbourhood of p. We have the bracket relations

p1, αi = q1, αi = 0 (i ≥ 3).

Using the Jacobi identity, we get

∂αi, αj∂α1

= p1, αi, αj = 0 for i, j ≥ 3,

with an analogous result for α2 and p2, implying the bracket relations

αi, αj = ϕij(α3, . . . , αdim(M)) for i, j ≥ 3

for some smooth functions ϕij.Write S1 for the two-dimensional local submanifold through p determined by p1 and

q1, and N1 for the local submanifold through p of codimension two determined by αi(i ≥ 3). By the bracket relations established above, M is locally the product of thePoisson manifolds S1 and N1. Note that S1 is in fact symplectic.

By induction, N1 has local coordinates q2, . . . , qr, p2, . . . , pr, n1, . . . , ndim(M)−2r satisfy-ing the canonical commutation relations. Taking the coordinates q1, . . . , qr, p1, . . . , pr andn1, . . . , ndim(M)−2r around p then proves the result.

Let M be a Poisson manifold. Define an equivalence relation on M as follows: twopoints p and q are equivalent iff there is a piecewise smooth curve from p to q such thatevery segment of this curve is a integral curve of a Hamiltonian vector field. The equivalenceclasses of this relation are called the symplectic leaves of M . These symplectic leaves arein fact injectively immersed submanifolds of M , as we now show.

Let L be a symplectic leaf of M . Take a point p ∈ M where the rank of M is r. Wecan find an open neighbourhood that is of the form S × N with S a symplectic manifoldand N a Poisson manifold of rank zero at p. Without loss of generality, we shall assumethat p has all its canonical coordinates zero. Using canonical coordinates, one sees thatS × 0 ⊂ L. If we use the coordinates qi, pi on S, this makes L into a smooth manifold.The inclusion L →M is an immersion by the very definition of the coordinates on L. Thisshows that M is the disjoint union of injectively immersed symplectic manifolds. Note

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that the symplectic leaves are by definition path connected. Also note that the symplecticleaves can have varying dimension (indeed, the dimension of the symplectic leaf throughpoint p is just the rank of the Poisson manifold at p). The symplectic leaves of a symplecticmanifold are its components. Because the local submanifold S in the splitting theoremcarries a symplectic structure, so do the symplectic leaves.

4.3.4 Example: coadjoint orbitsLet G be a Lie group. The dual of its Lie algebra is naturally a Poisson manifold for thebracket

f, g(α) = 〈α, [(df)α, (dg)α]〉 for all f, g ∈ F(g∗) and α ∈ g∗.

Here, Tαf and Tαg are seen as elements of the Lie algebra g: they are both linear mapsTαg

∗ ∼= g∗ → R. The bracket 〈·, ·〉 is the pairing between g and g∗. Antisymmetry and theJacobi identity follow from the corresponding properties of the Lie bracket. The Leibnizrule follows immediately from the usual Leibniz rule d(gh) = g dh+ h dg.

Lemma 73. Suppose G is a connected Lie group. The symplectic leaves of g∗ are the orbitsof the coadjoint action Ad∗.

Proof. Pick a point α ∈ g∗ and write L for the symplectic leaf through α. In canonicalcoordinates, one sees that the tangent space TαL consists precisely of the vectors of theform Xf (α) for some function f ∈ F(g∗) (that is, the tangent space to the symplectic leafis precisely the image of the map T ∗αg∗ → Tαg

∗ induced by the Poisson structure). Becauseany covector at α is in fact the differential at α of some linear function f ∈ (g∗)∗ ∼= g, thetangent space consists of the vectors at α of Hamiltonian vector fields of linear functionson g∗. Take two linear functions f, g ∈ g. Then

f, g(α) = 〈α, [(df)α, (dg)α]〉 = 〈α, adf (g)〉 = −〈ad∗f (α), g〉.

This shows that Xf (α) = ad∗f (α) and so the tangent space to the symplectic leaf at α is

TαL = ad∗f (α) | f ∈ g.

The coadjoint orbit of α is the image of the map ϕ : G→ g∗ : g 7→ Ad∗g(α). For a pointg ∈ G we have Tgϕ : f 7→ Ad∗g(ad∗f (α)). This demonstrates that

Im(Tgϕ) = Tϕ(g)L.

From this, we conclude two things:

• The inverse image ϕ−1(L) is open. Then of course also the inverse images of all otherleaves are open. However, because ϕ−1(L) is the complement of the union of theinverse images of the other leaves, it is also closed. By connectedness of G. we haveϕ−1(L) = G. In words: orbits are subsets of symplectic leaves.

47

• The image of ϕ (this is the orbit of α) is open in L. Then of course also any otherorbits that lie in L are open subsets of it. But again, the orbit of α is the complementof the union of the other orbits in L, and hence closed. By connectedness of symplecticleaves, the orbit of α is exactly L.

As an example, let us consider the case G = SU(2). It follows from the definition ofSU(2) that

SU(2) =

α −ββ α

| α, β ∈ C, |α|2 + |β|2 = 1

.Note that the map (α, β) 7→

(α −ββ α

)is an embedding of the four-dimensional real vector

space into the eight-dimensional real vector space. Its restriction to the 3-sphere is anembedding of the 3-sphere whose image is SU(2). We conclude that SU(2) is a Lie groupdiffeomorphic to the 3-sphere.

The Lie algebra of SU(2) is three-dimensional, with basis

u =0 i

i 0

, v =0 −1

1 0

, w =i 0

0 −i

and Lie brackets

[u, v] = 2w [v, w] = 2u [w, u] = 2v.

Lemma 74 ((Co)adjoint action of SU(2)). Relative to the basis (u, v, w) of su(2), theadjoint action of M =

(α −ββ α

)is given by

N =

a2 − b2 − c2 + d2 −2ab+ 2cd 2ac+ 2bd

2ab+ 2cd a2 − b2 + c2 − d2 −2ad+ 2bc−2ac+ 2bd 2ad+ 2bc a2 + b2 − c2 − d2

where α = a + bi and β = c + di (implying that a2 + b2 + c2 + d2 = 1). Relative to thedual basis of (u, v, w), the coadjoint action of M is given by the same matrix. This isan orthogonal matrix, so that we have a map χ : SU(2) → SO(3). This map is a doublecovering map.

Proof. Let us write M =(α −ββ α

). We investigate the adjoint action of M on the basis

(u, v, w) of su(2). Recalling lemma 17, we see that

AdM(u) = MuM−1 =−iαβ − iαβ iα2 − iβ2

−iβ2 + iα2 iαβ + iαβ

,

AdM(v) = MvM−1 =αβ − αβ −α2 − β2

β2 + α2 −αβ + αβ

,

48

and

AdM(w) = MwM−1 =iαα− iββ 2iαβ

2iβα −iαα + iββ

.Note that it is easy to write these right hand sides as linear combinations of the u, v andw: the coefficient for u is just the imaginary part of the lower left entry, the coefficient forv is the real part of the lower left entry, and the coefficient for w is the upper left entrydivided by i. We can thus write down the matrix for the adjoint action of M relative tothe basis (u, v, w), resulting in N . To verify that this matrix is orthogonal, one checks that

NTN =

(a2 + b2 + c2 + d2)2 0 0

0 (a2 + b2 + c2 + d2)2 00 0 (a2 + b2 + c2 + d2)2

and the right hand side is the identity matrix by the condition a2 + b2 + c2 + d2 = 1. Bythe definition of coadjoint action, Ad∗M = (AdM−1)∗ = (Ad−1

M )∗, which shows that relativeto the dual basis, the matrix of Ad∗M is given by the transpose of the inverse of N , whichis just N .

It only remains to prove that the map χ : SU(2) → SO(3) is a double covering map.We first prove that the map is surjective. Observe that we can pick b = c = 0, a = cos(θ)and d = sin(θ) to get

N =

1 0 00 cos(2θ) − sin(2θ)0 sin(2θ) cos(2θ)

so that all rotations about the first coordinate axis are in the image of χ. An analogousargument shows that all rotations about the other two coordinate axes are also containedin the image of χ. Because rotations about the coordinate axes generate SO(3), the mapχ is surjective.

Straightforward calculation shows that

TIdχ(u) =

0 0 00 0 −20 2 0

TIdχ(v) =

0 0 20 0 0−2 0 0

TIdχ(w) =

0 −2 02 0 00 0 0

,proving that χ has full rank at Id. Now let M ∈ SU(2). Write λM : SU(2) → SU(2) forleft multiplication by M and write λχ(M) : SO(3)→ SO(3) for left multiplication by χ(M).Then consider the following commutative diagram:

TId SU(2) TId SO(3)

TM SU(2) Tχ(M) SO(3)

TIdχ

TIdλM

TMχ

TIdλχ(M)

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Because TIdχ, TIdλM and TIdλχ(M) are isomorphisms, so is TMχ. This proves that χ hasfull rank everywhere and is therefore a local diffeomorphism.

Let us now find the kernel of χ. For χ(M) to equal the identity, we need

a2 − b2 − c2 + d2 = a2 − b2 + c2 − d2 = a2 + b2 − c2 − d2 = a2 + b2 + c2 + d2 = 1,

which is equivalent tob = c = d = 0 and a2 = 1.

This shows that the kernel of χ is exactly ± Id.We can finally prove that χ is a double covering. Take any point N ∈ SO(3). Its

inverse image under χ consists of two points ±M . Take disjoint neighbourhoods of M and−M that are mapped diffeomorphically to neighbourhoods of N . The intersection of theseneighbourhoods of N is clearly evenly covered by χ, proving that χ is a covering map.Because the kernel of χ has cardinality 2, it is a double cover.

50

Chapter 5

Constructing the moduli space

In this chapter we consider the moduli space of flat connections for a fixed structure groupG and base space Σ. This moduli spaceM(Σ, G) is the set of all flat principal G-bundlesover Σ up to gauge equivalence. Slightly more formally, we shall regard M(Σ, G) as aquotient of the set of all flat principal G-bundles (P,A) on Σ, the equivalence relationbeing gauge equivalence. We shall denote the gauge equivalence class of (P,A) by [P,A].Throughout the chapter, we assume that Σ is connected.

5.1 The moduli space as a space of representationsOur first approach to studying the moduli space will be built on the idea that a flat principalG-bundle over Σ up to gauge equivalence is essentially a group morphism π1(Σ) → G upto conjugation. Let us sketch the idea.

The holonomy of a flat connection around a loop only depends on the homotopy classof this loop. Picking a base point p in a flat principal bundle, we can construct a groupmorphism π1(Σ) → G by mapping any loop to the holonomy around it (up to inversion).Changing base points conjugates the resulting morphism, so that we can associate to anyflat principal bundle a group morphism π1(Σ) → G up to conjugation. It turns out thatthis group morphism actually determines the flat bundle up to gauge equivalence.

At the end of this section we will be able to identify

M(Σ, G) ∼=Hom(π1(Σ), G)

G,

where G acts on Hom(π1(Σ), G) by conjugation. This allows us to considerM(Σ, G) notjust as a set, but as a topological space (and in many cases, the moduli space will inheriteven more structure). The space Hom(π1(Σ), G) is often called a representation variety, asfor example in [Lab08].

51

5.1.1 The correspondence. . .We will formalize the correspondence outlined above. Our goal is a bijective map

Ψ :M(Σ, G)→ Hom(π1(Σ), G)G

.

We would like to define Ψ([P,A]) as the equivalence class of the group morphism

π1(Σ)→ G : [γ] 7→ Holp(A, γ)−1

where p is some fixed point of P . Many things need to be checked to construct Ψ:

• the above map is well-defined, meaning that Holp(A, γ) only depends on the homo-topy class of γ,

• the above map is a group morphism,

• the above map depends on p only up to conjugation,

• the above map only depends on the gauge equivalence class of (P,A).

The first of these uses flatness of the connection and follows from the lemma below. Thesecond is a relatively easy verification using (†) on page 29 and we leave it to the reader.The third is a direct consequence of that same equation. The fourth should be clear fromthe definition of gauge equivalence.

Lemma 75. The holonomy of a flat principal connection around a contractible loop istrivial.

Proof. Let (P,Σ, π,G) be a principal bundle with a flat connection A, and fix a point pof P . Write Q for the subset of P of all points that are the endpoint of piecewise smoothhorizontal curves in P that start at p. Around any point q ∈ Q, there is a chart (of P ) suchthat the horizontal space at every point in this chart is spanned by the first k coordinatevector fields. This shows that Q is an immersed submanifold of P , and the tangent spaceto Q consists of vectors that are horizontal in P . Because horizontal vectors are neververtical, the projection τ : Q → Σ (the restriction of π to Q) has bijective differential atevery point. By the inverse function theorem, this shows that τ is locally a diffeomorphismat every point. We see that τ is a surjective local diffeomorphism. In fact, it is a coveringmap: take a point q ∈ Q. Then there is a neighbourhood U of q and a neighbourhood Vof τ(q) such that τ : U → V is a diffeomorphism. This neighbourhood V of τ(q) is evenlycovered by τ because for any other preimage q · g of τ(q) the map τ gives a diffeomorphismbetween U · g and V .

We can now use the theory of covering maps. Taking a null-homotopic loop γ in Σstarting at x, it lifts to a loop in Q, meaning that the horizontal lift of γ starting at p alsoends at p. This shows that

Holp(A, γ) = e,proving the lemma.

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5.1.2 . . . is a bijectionLet us now prove that the correspondence Ψ is indeed a bijection. We first prove injectivity.

Theorem 76. The correspondence Ψ is injective.

Proof. Suppose (P,A) and (Q,B) are two flat principal G-bundles on Σ and Ψ([P,A]) =Ψ([Q,B]). We shall find a gauge equivalence between the two flat bundles. Pick pointsp ∈ P and q ∈ Q lying above the same point x ∈ Σ. By assumption, there is some g ∈ Gsuch that

Holp(A, γ) = gHolq(B, γ)g−1

for every loop γ in Σ starting at x. By replacing q by q · g, we may assume that in fact

Holp(A, γ) = Holq(B, γ).

We will construct a gauge equivalence f : P → Q. Pick a point y ∈ Σ. Take a curve δ inΣ from x to y. Lift δ to an A-horizontal curve δA in P and to a B-horizontal curve δB inQ. The idea is to demand that

f(δA(1)) = δB(1).(Indeed, this is the only possibility for f(δA(1)) because f δA should be horizontal ac-cording to B, so that we want f δA = δB.) This completely determines f on the fibreabove y because f is to be a bundle map. In fact, this determines f on the fibre above yin a way that is independent of the chosen curve δ as we now verify.

Suppose γ is another curve from x to y. Lift γ to curves γA and γB. We have to showthat f as constructed above (using δ) satisfies f(γA(1)) = γB(1). This follows from thefollowing calculation:

f(γA(1)) = f(δA(1) · HolδA(1)(A, δ ? γ))= f(δA(1)) · HolδA(1)(A, δ ? γ)= δB(1) · Holp(A, γ ? δ)= δB(1) · Holq(B, γ ? δ)= δB(1) · HolδB(1)(B, δ ? γ)= γB(1).

This shows that we can indeed define f using the procedure above without worryingabout which curve to pick for every y ∈ Σ. From considerations in local coordinates, it iseasy to see that f is smooth. It is thus a bundle map covering the identity (and thereforean isomorphism of principal bundles). By construction, Tf maps horizontal vectors tohorizontal vectors. We conclude that f is a gauge equivalence.

We have thus shown that a flat principal bundle is essentially determined by its holon-omy. We now prove that every group morphism π1(Σ)→ G actually occurs as the holonomyof some flat bundle.

53

Theorem 77. The correspondence Ψ is surjective.

Proof. Suppose ψ : π1(Σ) → G is a group morphism. We will construct a flat principalG-bundle P over Σ and a point p ∈ P such that

Holp(A, γ)−1 = ψ([γ])

for every loop γ in Σ starting at π(p). Here, A is the flat connection on P and π is theprojection P → Σ, as always.

Let Σ be the universal cover of Σ with projection τ : Σ→ Σ. Pick a point x ∈ Σ. Notethat π1(Σ, τ(x)) acts on Σ via monodromy (from the left). Note also that we can let π1(Σ)act on G from the left via

π1(Σ)×G→ G : ([γ], g) 7→ ψ([γ])g.

We can then consider the quotient

P = Σ×Gπ1(Σ)

where [γ] · (x, g) = ([γ] · x, [γ] · g) for (x, g) ∈ Σ × G. Because π1(Σ) acts properly andfreely on Σ×G, this quotient is a smooth manifold such that

Σ×G→ P

is a smooth submersion (even a local diffeomorphism). In fact, P is a principal G-bundlefor the projection [x, g] 7→ τ(x) (which is well-defined), and the right G-action on P isgiven by [x, g] · h = [x, gh] (which is also well-defined).

We will now construct a suitable connection on P . Call a tangent vector to Σ × Ghorizontal if it only has a component in the Σ-direction, not in the G-direction (in otherwords, if it is of the form (v, 0) where v ∈ T Σ). The action of π1(Σ) on Σ × G will maphorizontal vectors to horizontal vectors, so that the notion of horizontalness descends to thequotient P . We thus define horizontal spaces at each point of P , determining a connectionA on P . This connection is easily seen to be flat by integrability of the horizontal subbundle.

Write p = [x, e]. We claim that ψ(A, p) = ψ, proving the result. Let γ : I → Σ be anyloop starting at τ(x). Lift γ to a curve γ on Σ starting at x. The curve

t 7→ [γ(t), e]

is then the horizontal lift of γ to P starting at p. It ends at

[[γ] · x, e] = [x, ψ([γ])−1] = p · ψ([γ])−1,

showing that Holp(A, γ) = Holp(A, γ)−1 = ψ([γ]). The result is proved.

54

5.1.3 A topology on the moduli spaceBy the results of the previous subsection, we can identify the moduli space with a set ofequivalence classes of group morphisms:

M(Σ, G) ∼=Hom(π1(Σ), G)

G.

This identification allows us to endowM(Σ, G) with more structure than that of a mereset. The first extra structure we will study is a topology. To topologize the moduli space,we will ask that π1(Σ, G) be finitely generated, a mild assumption.

If the group π1(Σ) is finitely generated, we can pick generators a1, . . . , an of π1(Σ). Agroup morphism from π1(Σ) to G is then completely determined by the images of the ai,so that we can consider Hom(π1(Σ), G) to be a subset of Gn. This turns Hom(π1(Σ), G)into a topological space, andM(Σ, G) then inherits the quotient topology. The followingresult is straightforward and ensures that our way of topologizing the moduli space makessense.

Lemma 78. The topology on Hom(π1(Σ), G) as given above is independent of the choiceof generators.

Proof. Suppose a1, . . . , an and b1, . . . , bm are two sets of generators for π1(Σ). Write Ha

for Hom(π1(Σ), G) equipped with the topology inherited from Gn using the ai, and writeHb for Hom(π1(Σ), G) equipped with the topology inherited from Gm using the bi. Notethat Ha = Hb as sets. We will prove that the identity map from Ha to Hb is continuous,implying that the topology on Ha is finer (stronger, has more opens) than the topology onHb. Reversing the roles of the ai and bi then shows that the topologies are in fact equal,establishing the lemma.

The situation can be illustrated in a diagram.

Gn Gm

Ha Hb

f

Since the ai generate π1(Σ), there is a way to express bj as

bj = fj(a1, . . . , an),

where fj(g1, . . . , gn) is a product of (inverses of) the gi. There will in general be many waysto do so, but pick one for each bj. We then get a map

f : Gn → Gm

whose components are the fj. This map is smooth because G is a Lie group, so it ismost certainly continuous. Its restriction to Ha is exactly the identity Ha → Hb, which istherefore continuous.

55

Write $ for the quotient map Hom(π1(Σ), G) →M(Σ, G). Suppose S ⊆ M(Σ, G) isa subset of the moduli space. Then S inherits a topology from M(Σ, G). However, onecould propose a second way to topologize S, namely by topologizing $−1(S) as a subset ofHom(π1(Σ), G) and then putting the corresponding quotient topology on S. So there aretwo ways to construct a topology on S:

• it inherits a subspace topology fromM(Σ, G) (the “subspace-of-quotient topology”),

• it inherits a quotient topology from $−1(S) ⊆ M(Σ, G) (the “quotient-of-subspacetopology”).

It is known from general topology that for arbitrary topological spaces the subspace-of-quotient topology and the quotient-of-subspace topology need not be equal (the latter isalways at least as fine as the former). Fortunately, we need not worry about this in oursituation, as shown below. This rather technical result will be useful later.

Lemma 79. Suppose S is a subset of M(Σ, G). It inherits a subspace topology fromM(Σ, G). This topology is equal to the topology of S as a quotient of $−1(S).

Proof. We will prove that every open of the quotient-of-subspace topology is also openin the subspace-of-quotient topology. The converse statement is well-known for generaltopological spaces and is verified easily.

Suppose therefore that U ⊆ S is open for the quotient-of-subspace topology. This meansthat $−1(U) is an open subset of $−1(S), so that there is an open V ⊆ Hom(π1(Σ), G)such that

$−1(U) = $−1(S) ∩ V.We have to find an open W ⊆ M(Σ, G) such that U = S ∩W . We claim that choosingW = $(V ) does the trick. Note that $(V ) is open inM(Σ, G) because quotient maps ofgroup actions are always open. It only remains to verify that

U = S ∩$(V ).

To show that U ⊆ S ∩ $(V ) we pick an element x ∈ U . This is obviously in S, andpicking y ∈ $−1(x) we observe that y ∈ V so that x ∈ $(V ). The first inclusion isproved.

To show that U ⊇ S ∩$(V ) we pick an element x ∈ S ∩$(V ). There is an y ∈ V suchthat $(y) = x, and clearly y ∈ $−1(S). This shows that y ∈ $−1(U) so that x ∈ U . Thelemma is proved.

5.2 A smooth structure on the moduli spaceIn this section, we will explain how the moduli space can be equipped with more structurethan a mere topology. In particular, we will identify some points as smooth and others assingular.

56

5.2.1 Singular pointsWe start this chapter with a fairly general and straightforward discussion of the notion of asingularity. This subsection is not specific to the study of moduli spaces of flat connections.

Suppose f : M → R is a smooth function. Then its zero set S = f−1(0) is a closedsubset of M . If f is submersive at every point of S, then S is a submanifold of M . Ingeneral, however, f need not be submersive at every point of S. The points of S wheref is not submersive are called singular points of f . As an example, consider the mapf : R2 → R : (x, y) 7→ x3− y2. Its zero set is the so-called cusp. There is exactly one pointon the cusp that is singular for f : the origin. The cusp in shown in figure 5, and it showsthat the cusp is “not smooth” in the origin.

x

y

Figure 5: The cusp. It is the zero set of the polynomial x3 − y2 and has a singularity atthe origin.

Now consider the map g : R2 → R : (x, y) 7→ x2. Its zero set is the y-axis, which is asmooth submanifold of the plane. However, every point on the y-axis is singular for g. Ifwe look at the map h : R2 → R : (x, y) 7→ x, we also have the y-axis as zero set, but thistime the map is a submersion.

This leads us to the following question: how can we distinguish “bad” points of a zeroset from “good” ones in a way that is geometrically sensible? So given a closed subset Sof a manifold M , which criterion can tell us whether a point in S is a “singularity”? Ageometrically satisfying definition may be the following.

Definition 80. If S is a closed subset of a manifoldM , then a point p ∈ S is called smoothof dimension n if there is a neighbourhood U of p in M such that S ∩ U is a submanifoldof U of dimension n. If p is not smooth of any dimension, p is called a singularity of S.

57

In other words, a point is smooth if around the point the inclusion S → M looks likean inclusion Rn → Rm. The reader may verify that according to this definition the originis the only singularity of the cusp.

Definition 81. If S is a closed subset of a manifold M , N is a manifold and U is an opensubset of S, then a map f : U → N is called smooth if there is an open neighbourhood Vof U in M and a map f : V → N such that f|U = f .

Lemma 82. Let S be a closed subset of a manifold M and p ∈ S. Then p is smoothof dimension n iff there is a neighbourhood U of p in S, an open subset V of Rn and ahomeomorphism f : V → U such that a map g : U → R is smooth precisely if g f issmooth.

Proof. Suppose p is smooth of dimension n. Then locally around p the inclusion S → Mlooks like Rn → Rm, and a function Rn → R is smooth precisely if it can be extended toa smooth function Rm → R. This proves one of the two implications.

Now suppose there exist U , V and f as in the lemma. Then f : V → U is smooth anda homeomorphism. We claim that f is an embedding, and the only thing that is left toprove is that f is an immersion. Now f−1 : U → V is smooth: indeed, its i’th componentf−1i : U → R is smooth because g f−1

i is just projection Rn → R onto the i’th component.This means that there is a map f−1 : U → V (with U some open neighbourhood of U inM) that restricts to f−1 on U . By construction f−1 f is the identity on V . Its differentialis therefore injective, which means that f must have injective differential.

The lemma says that smoothness of points of S can be detected just by looking at thesmooth functions on (open subsets of) S. It is this lemma that will motivate our definitionof singularities of the moduli space. We can also reformulate the lemma using languagemore commonly used in algebraic geometry: if we turn S into a ringed space using its sheafof smooth functions, then a point of S is smooth of dimension n iff it has a neighbourhoodthat is isomorphic to an open part of Rn as ringed spaces. This point of view using sheavesis standard in algebraic geometry, but is not widely used in differential geometry. For atreatment of differential geometry in the language of sheaves, we recommend [GdS03].

5.2.2 Application to the moduli spaceInspired by the previous subsection, we make the following definition.

Definition 83. Pick a set of n generators of π1(Σ) to consider Hom(π1(Σ), G) ⊆ Gn.Let U be an open subset of Hom(π1(Σ), G). A map U → R is called smooth if it is therestriction to U of a smooth function on some open subset V of Gn.

The reader is encouraged to check that this is independent of the choice of generators byimitating the proof of lemma 78. We can now say when a real-valued function onM(Σ, G)is smooth.

58

Definition 84. Let U be an open subset of M(Σ, G). Write $ : Hom(π1(Σ), G) →M(Σ, G) for the quotient map. A map f : U → R is called smooth if f $ : $−1(U) issmooth.

Definition 85. A point p ∈M(Σ, G) is called smooth of dimension n if there is a neigh-bourhood U of p, an open subset of Rn and a homeomorphism φ : U → V such that afunction f : V → R is smooth exactly if f φ is smooth.

A point that is not smooth is called singular.

5.3 The moduli space as a symplectic reductionIf Σ is a compact orientable surface and g admits an Ad-invariant non-degenerate symmetricbilinear form, then the moduli space of flat connections can in a sense be obtained as asymplectic reduction of some infinite-dimensional symplectic space (or in fact a disjointunion of several such reductions). The goal of this section is to explain this point of view.It will allow us to find a symplectic structure on (parts of) the moduli space later on, whenwe are studying concrete examples of moduli spaces. The material in this section is basedon chapter 25 of [dS01] and on [Jan05].

It should be noted that this section is far from mathematically rigorous. We will beworking with an infinite-dimensional symplectic manifold without bothering to define whatthat actually is. The role of Lie group acting on the space will be played by the gauge group,which is also infinite-dimensional. This means that many assumptions in the symplecticreduction procedure are violated: even if one builds up a well-founded theory of infinite-dimensional manifolds, for example, we demanded in the original recipe for symplecticreduction that the group acting on the manifold be compact. There is little hope for thisin our infinite-dimensional context. We will ignore these issues entirely and regard thissection as a source of inspiration for finding a symplectic structure on the moduli space inconcrete cases, without claiming mathematical correctness. This symplectic structure canbe placed on firm footing using algebraic techniques from group cohomology, as done in[Kar92]. We do not discuss this. Let us get started.

5.3.1 Picking Σ and G

To constructM(Σ, G) as a symplectic reduction, we need several restrictions on Σ and G.For Σ, the restriction is easy: since we will later want to integrate 2-forms over Σ, we willrequire that Σ be a compact orientable surface. It is well-known that compact orientablesurfaces are classified by a single natural number, the genus g of the surface. For g = 0the surface is the sphere S2 and for g ≥ 1 the surface is the connected sum of g tori. Thefirst few of these surfaces are shown in figure 6.

For us, the case g = 0 is not interesting, since the moduli spaceM(S2, G) is a singleton.We will henceforth assume that Σ is a compact orientable surface of genus g ≥ 1. The

59

Figure 6: From left to right the compact orientable surfaces of genus 0, 1 and 2. Thecompact orientable surface of genus g is the connected sum of g tori, or the sphere if g = 0.To constructM(Σ, G) as a symplectic reduction, we will assume Σ is a compact orientablesurface.

standard presentation of the fundamental group of this surface is

π1(Σ) = 〈a1, b1, . . . , ag, bg |g∏i=1

aibia−1i b−1

i = e〉.

In other words, the fundamental group of Σ is generated by 2g generators ai, bi, subject tothe single relation that ∏i aibia

−1i b−1

i be the group unit.We also restrict our choice of structure group G somewhat, but not as severely as our

choice of base space. The Lie algebra g should admit an Ad-invariant non-degeneratesymmetric bilinear form. This means there should be some symmetric bilinear map

〈−,−〉 : g× g→ R

that is non-degenerate and satisfies

〈Adg(ξ),Adg(η)〉 = 〈ξ, η〉.

This will allow us to produce an ordinary (R-valued) differential form on Σ given twogP -valued forms: if α ∈ Ωk(Σ, gP ) and β ∈ Ωl(Σ, gP ), then we can form their wedgeproduct

α ∧ β ∈ Ωk+l(Σ, gP ⊗ gP ).In a local trivialization, α∧β assumes values in g⊗g, so that we can apply the bilinear formto obtain an ordinary (R-valued) differential form 〈α ∧ β〉 ∈ Ωk+l(Σ). The Ad-invarianceof the pairing ensures that the result is independent of the chosen trivialization, so thatwe have a non-degenerate bilinear pairing on every fibre of the bundle gP given by

〈[p, ξ], [p, η]〉 = 〈ξ, η〉.

If we restrict ourselves to Lie groups whose Lie algebra admits an Ad-invariant non-degenerate symmetric bilinear form, we should ask ourselves how severe this restrictionis. Are there many such Lie groups? Fortunately, Lie groups admitting an Ad-invariantnon-degenerate symmetric bilinear form on their Lie algebra are not rare. All compactLie groups admit an appropriate form, because we can average an arbitrary inner product

60

on g over the group (resulting in a form that is even positive definite). Moreover, thereis another large class of Lie groups admitting the kind of form we want: the so-calledsemisimple Lie groups. These semisimple Lie groups have a non-degenerate Killing form,which satisfies all our requirements. We will not study the theory of semisimple Lie groupsany further.

The following lemma will be useful to us in the future. The proof is a long calculationinvolving lots of manipulation of permutations, and is analogous to the usual proof thatthe exterior derivative is a derivation of degree 1.

Lemma 86. If α ∈ Ωk(Σ, gP ) and β ∈ Ωl(Σ, gP ), then

d〈α ∧ β〉 = 〈dAα ∧ β〉+ (−1)k〈α ∧ dAβ〉,

analogous to the situation for the ordinary exterior derivative.

5.3.2 The planLet us explain in more detail how we plan to equipM(Σ, G) with a symplectic structure.Pick Σ and G as explained in the previous subsection. The moduli spaceM(Σ, G) containsequivalence classes [P,A] of flat bundles. If two flat bundles (P,A) and (Q,B) are gaugeequivalent, then in particular the bundles P and Q are isomorphic (as mere bundles withoutconnections). This shows that we can split up M(Σ, G) into disjoint parts, one for eachisomorphism class of principal G-bundles over Σ. Some of these parts may be empty,because not all principal G-bundles over Σ need to admit a flat connection.

We will use the notation

MP (Σ, G) = [Q,B] ∈M(Σ, G) | Q is isomorphic to P as a principal bundle

for these disjoint parts. Every such part will be called a block of the moduli space. Inparticular, we shall callMP (Σ, G) the P -block of the moduli space. We have the followingresult. For a proof we refer to [Bel10].

Lemma 87. Every blockMP (Σ, G) is a union of path components ofM(Σ, G).

The plan is to construct MP (Σ, G) as a symplectic reduction of A, the space of allconnections on P . In this way MP (Σ, G) will inherit a symplectic structure, and thusM(Σ, G) will also have a symplectic structure. We start by equipping A with a symplecticstructure. Later we show that MP (Σ, G) is a symplectic reduction of A for the momentmap A 7→ FA.

5.3.3 The Atiyah-Bott 2-formLet (P,Σ, π,G) be a principal bundle, as always. Suppose A and B are two principalconnections on P . Recall that these principal connections both specify an equivariant g-valued 1-form on P through their projections Ap : TpP → g. The difference α between

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these is again equivariant, and because the projections Ap and Bp have equal restrictionsto Vp, the 1-form α is also horizontal. This means that α ∈ Ω1

Ad,H(P, g) = Ω1(Σ, gP ).Conversely, given a connection A on P and an α ∈ Ω1(Σ, gP ), the sum of α and the

connection 1-form of A will be the connection 1-form of some principal connection B. Weconclude that the set A of all principal connections on P is an affine space in the sensethat

A = A+ Ω1(Σ, gP ).

We will equip A with a symplectic structure and show thatMP (Σ, G) can be obtainedas a symplectic reduction of A. Of course, it is not entirely clear what is meant by asymplectic structure on A, but since the space is an affine space as explained above, wecan say that

TAA = Ω1(Σ, gP )

for every A ∈ A. The symplectic structure we will construct on A is called the Atiyah-Bott2-form after Sir Michael Atiyah and Raoul Bott, who constructed this form in their paper[AB83]. It is given by

ωAB(α, β) =∫

Σ〈α ∧ β〉.

This is an antisymmetric pairing TAA×TAA → R. It is non-degenerate as shown in thefollowing lemma. It is closed because it has constant coefficients (although this is just ananalogy from the finite-dimensional case). This is why we consider ωAB to be a symplecticstructure on A.

Lemma 88. The Atiyah-Bott 2-form

ωAB : Ω1(Σ, gP )× Ω1(Σ, gP )→ R

is non-degenerate.

Proof. Suppose α ∈ Ω1(Σ, gP ) is not zero. We will show that there is some β ∈ Ω1(Σ, gP )such that ωAB(α, β) 6= 0.

Because α 6= 0, there is some p ∈ Σ and some v ∈ TpΣ such that α(v) 6= 0. In someneighbourhood of p the bundle gP is trivial. Working in this neighbourhood we can find achart around p such that coordinates on gP are given by (x, y, ξ1, . . . , ξdim(G)) where (x, y)are coordinates on Σ and the ξi are linear coordinates on g. Changing coordinates on Σwe may assume that v = ∂

∂xat p.

Using these coordinates we can consider α(v) as an element of g, and because thepairing 〈−,−〉 is non-degenerate there is some ξ ∈ g such that 〈α(v), ξ〉 6= 0. In fact,working in an even smaller neighbourhood U of p and replacing ξ by −ξ if necessary, wemay assume that 〈α( ∂

∂x), ξ〉 > 0 in U . Now take β to be the 1-form that vanishes outside

U and is given in U byβ = f(x, y)ξ ⊗ dy

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where f is a smooth non-negative function with f(p) = 1 and support contained in U .Then by construction we have

ωAB(α, β) =∫

Σ〈α ∧ β〉

=∫Uf(x, y)〈α

(∂

∂x

), ξ〉 dxdy > 0.

This proves the result.

5.3.4 The curvature as a moment mapWe have seen in the previous subsection that A is a symplectic space. Our goal is toconstructMP (Σ, G) as a symplectic reduction of this space under the action of the gaugegroup.

Recall that the gauge group is

G(P ) = u : P → G | u(p · g) = g−1u(p)g for all p ∈ P and g ∈ G

and that it acts on A by pullback. We can identify the Lie algebra of the gauge group with

Lie(G(P )) = h : P → g | h(p · g) = Adg−1(h(p)) for all p ∈ P and g ∈ G.

In other words,Lie(G(P )) = Ω0(Σ, gP ).

Integration provides a non-degenerate bilinear pairing

Ω2(Σ, gP )× Ω0(Σ, gP )→ R : (F, h) 7→∫

Σ〈F ∧ h〉,

and we will use it to identify Lie(G(P ))∗ = Ω2(Σ, gP ) (the proof of non-degeneracy issimilar to and slightly easier than the proof of lemma 88). We claim that the action ofG(P ) on A is Hamiltonian with equivariant moment map

µ : A → Lie(G(P ))∗ = Ω2(Σ, gP ) : A 7→ −FA.

For this claim to hold, we have to verify that

• the map µ is equivariant for the action of G(P ),

• the Hamiltonian vector field of the function µh is the vector field Xh on A (whereXh is the vector field associated to h ∈ Ω0(Σ, gP ) as in subsection 4.2.1).

We formulate the verifications as lemmas.

Lemma 89. The curvature moment map µ is equivariant.

63

Proof. To check that µ is equivariant, we have to check that

Fu∗A = Ad∗u(FA) for all A ∈ A and u ∈ G(P ).

We check this by pairing with an arbitrary h ∈ Ω0(Σ, gP ). We then have to check that

〈Fu∗A ∧ h〉 = 〈FA ∧ Adu(h)〉.

This is an equality of ordinary 1-forms on Σ, which we check by evaluating on the vectorsv, w ∈ TxΣ. To evaluate the left hand side, we pick some fixed point p ∈ Px. Then welift v and w to u∗A-horizontal elements v and w of Tp·u(p)−1P and to A-horizontal elementsv, w of TpP . Then we have

〈Fu∗A ∧ h〉(v, w) = 〈Fu∗A(v, w), h(x)〉= 〈[p · u(p)−1, Fu∗A(v, w)], [p, h(p)]〉= 〈[p · u(p)−1, FA(Tp·u(p)−1u(v), Tp·u(p)−1u(w))], [p, h(p)]〉= 〈[p · u(p)−1, FA(v, w)], [p, h(p)]〉= 〈[p · u(p)−1, FA(v, w)], [p · u(p)−1,Adu(p)(h(p))]〉= 〈FA(v, w),Adu(p)(h(p))〉

where in the fourth step we observed that FA is horizontal and the differences T u(v) − vand T u(w)− w are vertical. Evaluating the right hand side on (v, w) is easier:

〈FA ∧ Adu(h)〉(v, w) = 〈[p, FA(v, w], [p,Adu(h)(p)]〉= 〈[p, FA(v, w)], [p,Adu(p)(h(p))]〉= 〈FA(v, w),Adu(p)(h(p))〉.

This proves equivariance of µ.

Lemma 90. The curvature moment map µ is a moment map for the action of G(P ) onA(P ).

Proof. This will boil down to Stokes’ theorem. Suppose h ∈ Lie(G(P )) = Ω0(Σ, gP ). Thenµh = 〈µ, h〉 is a component of µ (the brackets denote the pairing of Lie(G(P )) with itsdual). We have to check that dµh equals iXhωAB, where Xh is the Hamiltonian vector fieldon A associated to h. We will verify this by applying both to an arbitrary tangent vectorα ∈ TAA = Ω1(Σ, gP ). We will therefore check that

(dµh)A(α) = ωAB(Xh, α). (‡)

First of all, note that µ(A) = −dA h so that µ(A+ tα) = −d(A+ tα) h = −FA− tdAα.We have thus found the left hand side of (‡):

(dµh)A(α) = −∫

Σ〈dAα ∧ h〉.

64

We now investigate the right hand side of (‡). We claim that

Xh = dAh.

It suffices to check this on horizontal tangent vectors to P , so pick v ∈ Hp. Then

(Xh)(v) = d

dt |t=0((exp(th)∗A)p(v)− Ap(v))

= d

dt |t=0(exp(th)∗A)p(v)

= d

dt |t=0Ap·exp(th(p))((Tput)(v)).

Here, ut is the gauge transformation associated to the element ut = exp(th) of the gaugegroup. We now calculate (Tput)(v). To this end, we take γ to be a curve in P with initialvelocity v. Then

(Tput)(v) = d

ds |s=0(γ(s) · ut(γ(s)))

= v · ut(γ(0)) + d

ds |s=0(γ(0) · ut(γ(s))).

The first of these two terms is horizontal by invariance of the horizontal subbundle. Wethus find

(Xh)(v) = d

dt |t=0Ap·exp(th(p))

(d

ds |s=0[γ(0) · ut(γ(s))]

)

= d

dt |t=0Ap·exp(th(p))

(d

ds |s=0[(p · exp(th(p))) · (ut(γ(0))−1ut(γ(s)))]

)

= d

dt |t=0θ

(d

ds |s=0ut(γ(0))−1ut(γ(s))

).

Write f for the map (−ε, ε)2 → G : (s, t) 7→ ut(γ(0))−1ut(γ(s)). Then the above expressionis exactly

d

dt |t=0(f ∗θ)

(d

ds |s=0

),

which by the Maurer-Cartan structure equation and the fact that ddt

and dds

commute,equals

d

ds |s=0(f ∗θ)

(d

dt |t=0

)+[θ

(T(0,0)f

(d

ds

)), θ

(T(0,0)f

(d

dt

))].

The first vector in the Lie brackets is zero. We continue our calculation noting that

65

u0(p) = e for all p ∈ P .

(Xh)(v) = d

ds |s=0θ

(d

dt |t=0ut(γ(0))−1ut(γ(s))

)

= d

ds |s=0

(Adu0(γ(s))u0(γ(0))(h(γ(0))) + h(γ(s))

)= (dh)(v).

This proves that Xh = dAh. To prove the lemma, it thus suffices to show that

−∫

Σ〈dAα ∧ h〉 =

∫Σ〈dAh ∧ α〉.

By Stokes’ theorem and the fact that Σ has no boundary, it suffices to show that

d〈α ∧ h〉 = 〈dAα ∧ h〉 − 〈α ∧ dAh〉.

This follows from lemma 86.

If Σ is a compact orientable surface with boundary, the moduli space M(Σ, G) has aPoisson structure whose symplectic leaves are obtained by fixing the conjugacy class of theholonomy around every boundary component (see [Wei95], page 9).

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Chapter 6

A topological observation

In this chapter we will split the moduli space into two parts. We then completely describethe topology of one of these, the so-called reducible part (which is the smaller and worse-behaved part, in a sense). Throughout the chapter, G is a compact connected Lie groupand Σ is a compact orientable surface of genus g ≥ 1. We writeM =M(Σ, G).

6.1 The reducible partThe fundamental group of Σ has the well-known presentation

π1(Σ) = 〈a1, b1, . . . , ag, bg |g∏i=1

aibia−1i b−1

i = e〉,

so it is generated by 2g generators satisfying a single relation. Consider the map

µ : G2g → G : (A1, B1, . . . , Ag, Bg) 7→g∏i=1

AiBiA−1i B−1

i .

Then Hom(π1(Σ), G) is the subset µ−1(e) of G2g (and this is how we equipped the setHom(π1(Σ), G) with a topology). The moduli spaceM is the quotient of Hom(π1(Σ), G)under conjugation and therefore contains equivalence classes of 2g-tuples of elements ofG. We will write [A1, B1, . . . , Ag, Bg] for the equivalence class represented by the tuple(A1, B1, . . . , Ag, Bg) ∈ Hom(π1(Σ), G) ⊆ G2g.

Definition 91. A tuple (A1, . . . , Bg) ∈ G2g is called reducible if all of its 2g entriescommute pairwise. Otherwise the tuple is called irreducible.

Note that conjugation does not change irreducibility of a tuple, so that we can makeanalogous definitions for the moduli space.

Definition 92. A point [A1, . . . , Bg] ∈ M is called reducible if the tuple (A1, . . . , Bg) is.Otherwise the point is called irreducible.

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Write R for the subset ofM containing all reducible points. Write S for the subset ofM containing all irreducible points. Then

M = Rt S.

Recall the notions of maximal torus and Weyl group from section 1.4 on page 12. Themain result of this section is the following.

Theorem 93. Suppose G is a compact Lie group and Σ is a compact orientable surface ofgenus g. Then the reducible part R of the moduli space is homeomorphic to

T 2g

W (T )

where T is a maximal torus in G and W (T ) its Weyl group, acting on all copies of Tsimultaneously.

Proof. WriteW = W (T ). We will construct a homeomorphism Π : R → T 2g/W . Pick anypoint [A1, . . . , Bg] ∈ R. The fact that (A1, . . . , Bg) is irreducible is exactly the fact thatall the Ai and Bi lie in a common torus in G, and hence in some maximal torus. Becauseall maximal tori are conjugate, we can conjugate to obtain a tuple in T 2g. We take thistuple to be a representative of Π([A1, . . . , Bg]).

Of course, we still have to check that this map Π is well-defined, because there may bemultiple ways to conjugate (A1, . . . , Bg) to get a tuple in T 2g, and we have to verify thatall possible resulting tuples in fact represent the same equivalence class in T 2g/W . This islemma 23 on page 13.

Recall that a continuous bijection from a compact space to a Hausdorff space is auto-matically a homeomorphism. We will prove that R is compact and that Π is continuous,surjective and injective. Because T 2g/W is Hausdorff (recall that W is finite), the resultwill follow.

Let us first show that R is compact. Let R denote $−1(R) where $ : µ−1(e)→M isthe projection. Because R is a quotient of R (by lemma 79), it suffices to prove that R iscompact. Now R is a closed subset of G2g, which itself is compact. This proves that R iscompact. We have showed that R is compact.

It is easy to see that Π is surjective: the equivalence class of (A1, . . . , Bg) is the imageof [A1, . . . , Bg] under Π. It is also easy to see that Π is injective: if Π([A1, . . . , Bg]) =Π([A′1, . . . , B′g]) then by definition of W the tuples (A1, . . . , Bg) and (A′1, . . . , B′g) are con-jugate, proving that [A1, . . . , Bg] = [A′1, . . . , B′g].

It remains to prove that Π : R → T 2g/W is continuous. Because R is a quotient of R,it suffices to show that the composition

R projection−−−−−→ R Π−→ T 2g/W

is continuous. We will call this composition Π, and we will prove its continuity usingsequences. Let (A1, B1, . . . , Ag, Bg) be a point of R. Take a sequence ((An1 , . . . , Bn

g ))n in

68

R converging to this point, indexed by an upper index for notational reasons. For everyn, take an element Cn ∈ G that conjugates the tuple (An1 , . . . , Bn

g ) to lie in T 2g, i.e. suchthat

(CnAn1 (Cn)−1, . . . , CnBng (Cn)−1) ∈ T 2g.

This means that

Π(An1 , . . . , Bng ) = [CnAn1 (Cn)−1, . . . , CnBn

g (Cn)−1].

By compactness of G, we can take a subsequence (Cm(n))n that converges to C ∈ G. Thenbecause Am(n)

i → Ai, Bm(n)i → Bi and Cm(n) → C we find that

limn→+∞

Cm(n)Am(n)i (Cm(n))−1 = CAiC

−1

andlim

n→+∞Cm(n)B

m(n)i (Cm(n))−1 = CBiC

−1.

As limits of sequences in T , all CAiC−1 and CBiC−1 lie in T , which means that

Π(A1, . . . , Bg) = [CA1C−1, . . . , CBgC

−1].

We conclude thatlim

n→+∞Π(Am(n)

1 , . . . , Bm(n)g ) = Π(A1, . . . , Bg). (∗)

We have now proved the following statement: any sequence ((An1 , . . . , Bng ))n in R has a

subsequence such that (∗) holds. This suffices to establish continuity of Π at (A1, . . . , Bg):suppose Π(An1 , . . . , Bn

g ) does not converge to Π(A1, . . . , Bg). Then there is a subsequencewhose image points under Π stay out of a certain neighbourhood of Π(A1, . . . , Bg). Thissubsequence then contradicts the statement. We have shown that Π (and hence Π) iscontinuous, finishing the proof.

The above theorem completely determines the topology of the moduli space in the caseg = 1. Indeed, in the case Σ = S1 × S1 we immediately have

R =M and S = ∅

from the definition of reducibility (because π1(Σ) is abelian).

Corollary 94. Suppose G is a compact connected Lie group. Then M(S1 × S1, G) ishomeomorphic to

T 2

W (T )where T is a maximal torus in G and W (T ) its Weyl group, acting on both copies of T atthe same time.

69

As an example, consider G = SU(2). The maximal torus is then diffeomorphic to S1,with W ∼= Z2 acting by complex conjugation. The moduli space of flat connections on theprincipal SU(2)-bundle over S1 × S1 is then homeomorphic to

S1 × S1

Z2

where Z2 acts by simultaneous complex conjugation of both circles. After cutting andpasting appropriately, one sees that this moduli space is homeomorphic to the sphere S2.In the next section, we discuss the example G = SU(2) in greater detail, because in thiscase we can also say something about the irreducible part.

6.2 The case G = SU(2)Let us study the moduli space of flat connections on the trivial (the only) SU(2)-bundleover a compact orientable surface of genus g. The topology of the case g = 1 was alreadydiscussed in the previous section: the moduli space is homeomorphic to S2.

In the last section, we split the moduli space into a reducible part R and an irreduciblepart S. We already know the reducible part topologically by theorem 93. Let us state theresult for the sake of concreteness.

Theorem 95. If Σ is a compact orientable surface of genus g, then the reducible part ofM(Σ, SU(2)) is homeomorphic to

(S1 × S1 × · · · × S1︸ ︷︷ ︸2g copies

)/Z2

where Z2 acts on the circles by complex conjugation.

Our goal for this section is the following result.

Theorem 96. If Σ is a compact orientable surface of genus g, then the irreducible part ofM(Σ, SU(2)) is a smooth manifold of dimension 6g − 6.

Note that 6g − 6 is the dimension one expects the moduli space to have. Indeed, wehave

M = µ−1(e)/ SU(2),(where µ : G2g → G is the map defined at the start of this chapter) and since µ is a mapfrom SU(2)2g to SU(2), we expect µ−1(e) to have dimension

2g · dim(SU(2))− dim(SU(2)) = 6g − 3,

leading to a quotient of dimension

6g − 3− dim(SU(2)) = 6g − 6.

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This dimension count also serves as a rough sketch of our proof strategy: we first showthat µ is submersive at all irreducible tuples (so that µ−1(e) is a smooth manifold), andthen we prove thatM is a quotient of this manifold under a free action of a compact groupof the same dimension as SU(2). Our first task is to investigate the rank of µ.

The following is a slight generalization of lemma 18 in [Jan05].

Lemma 97. Suppose G = SU(2) and let µ : G2g → G be as before. Then the rank of µ is3 at all irreducible points, 0 at the points (Ai, Bi) with all Ai and Bi equal to ± Id and 2at all other points. In particular, µ has full rank exactly at the irreducible points.

In fact, [Gol84] mentions the following generalization (section 3.7). A proof of thisgeneralization can also be found there.

Theorem 98. Let G be a Lie group and

µ : G2g → G : (A1, B1, . . . , Ag, Bg) 7→g∏i=1

AiBiA−1i B−1

i .

Then the rank of µ at (A1, . . . , Bg) equals the codimension of the centralizer of A1, . . . , Bgin G.

To prove lemma 97, we will use the following result.

Lemma 99. Two elements A and B of SU(2) commute iff their (co)adjoint actions arerotations about the same axis or either of them is ± Id.

Proof. First suppose that A and B are commuting elements of SU(2). After conjugation,we may assume that they are both diagonal. From lemma 74 we conclude that theircoadjoint action are both rotations about the third coordinate axis.

Now suppose that A and B act on su(2) (or su(2)∗) by rotation about a commonaxis. After conjugation, we may assume that this axis is the third coordinate axis. Fromlemma 74 we deduce that A and B are both diagonal and therefore commute.

We now prove lemma 97. The proof is a rewritten and expanded version of the onegiven by Janner in [Jan05].

Proof of lemma 97. We shall differentiate µ at the point (A1, . . . , Bg) in the direction(a1, . . . , bg) ∈ g2g for a general Lie group G (so for now, we don’t use the fact thatG = SU(2)). More precisely, consider a curve (A1(t), . . . , Bg(t)) in G2g with Ai(0) = Aiand Bi(0) = Bi such that

θ

(d

dt |t=0Ai(t)

)= ai and θ

(d

dt |t=0Bi(t)

)= bi

where θ is the Maurer-Cartan form. Applying µ gives the curve

t 7→ γ(t) =g∏i=1

Ai(t)Bi(t)Ai(t)−1Bi(t)−1.

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Using lemma 48 we can differentiate this curve. This results in

θ

(d

dt |t=0γ(t)

)=

g∑i=1

Ad(∏j>i

AjBjA−1j B−1

j

)−1

(AdBiAiB−1

i(ai)

+ AdBiAi(bi)− AdBiAi(ai)

− AdBi(bi))

=g∑i=1

Ad(∏j>i

AjBjA−1j B−1

j

)−1BiAi

((AdB−1

i−1)ai + (1− AdA−1

i)bi) .

The rank of µ at (A1, . . . , Bg) is therefore the rank of the linear map g2g → g given by

(a1, . . . , bg) 7→g∑i=1

Ad(∏j>i

AjBjA−1j B−1

j

)−1BiAi

((AdB−1

i−1)ai + (1− AdA−1

i)bi) .

Suppose now that µ does not have full rank at (A1, . . . , Bg). Then in particular for anyfixed i the map

g2 → g : (a, b) 7→ (AdB−1i−1)a+ (1− AdA−1

i)b

is not surjective, for otherwise take aj = bj = 0 for j 6= i and we obtain all of g as theimage of T(A1,...,Bg)µ already by just varying ai and bi.

Let us now specialize to the case G = SU(2). We claim that the non-surjectivity ofthe above map g2 → g implies that Ai and Bi commute. Indeed, if either is ± Id, we aredone. Otherwise, (AdB−1

i−1) has as image the plane perpendicular to the rotation axis of

AdBi , and (1−AdA−1i

) has as image the plane perpendicular to the rotation axis of AdAi .The sum of these planes (as linear subspace of three-dimensional space su(2)) must havedimension less than three, implying that they coincide. We conclude that AdAi and AdBihave the same axes of rotation, so that Ai and Bi commute. Because this argument canbe made for any i, we conclude that if µ does not have full rank at (A1, . . . , Bg), then

θ

(d

dt |t=0γ(t)

)=

g∑i=1

AdBiAi((AdB−1

i−1)ai + (1− AdA−1

i)bi).

The j’th term in this sum takes as values all elements of the plane perpendicular to therotation axis of Aj (and of Bj, because these are the same axes), unless Aj and Bj are both± Id (then it is always zero). For µ not to have full rank, these planes must coincide forall possible j’s, so that all the Ai and Bj that are not ± Id have the same axis of rotation.In particular, all the Ai and Bj commute, proving that µ has full rank at all irreduciblepoints. We also conclude that in the reducible case, µ does not have full rank, and thatthis rank is 0 if all Ai and Bi are ± Id and exactly 2 otherwise. This proves the lemma.

72

Proof of theorem 96. Notice that the action of SU(2) on µ−1(e) is not free, because − Id ∈SU(2) acts trivially. We can, however, let the quotient SU(2)/± Id ∼= SO(3) act onµ−1(Id). We claim that this action is free, implying that S is the quotient of a manifoldunder a free action of a compact group, hence a manifold of dimension

dim(µ−1(e))− dim(SO(3)) = dim(SU(2)2g)− dim(SU(2))− dim(SO(3)) = 6g − 6

(recall lemma 64). Suppose that [C] ∈ SU(2)/± Id acts trivially on (Ai, Bi) ∈ µ−1(e).This means that C commutes with all Ai and Bi. Because (Ai, Bi) is irreducible, there aretwo matrices in A1, . . . , Bg that do not commute. Call these matrices X and Y .

The situation is as follows: C commutes with both X and Y , but X and Y do notcommute. The rotation axes of AdX and AdY are different. The rotation axis of AdCcannot equal both of these axes, so AdC must be the identity, proving that C = ± Id. Thisproves that the action of SU(2)/± Id on µ−1(e) is free, establishing the result.

73

Chapter 7

The symplectic structure

In this chapter we study in greater detail the symplectic structure on the block of themoduli space corresponding to the trivial bundle. We first build up some theory aboutthe moduli space for simply connected base spaces. In this case, a flat connection hasno holonomy, which simplifies the theory a lot. Recall theorem 40, which shows thatrestricting ourselves to the trivial bundle is not necessarily as drastic as it may seem.

7.1 Flat connections on trivial bundles over simplyconnected bases

We discuss flat connections on trivial principal bundles over simply connected base spaces.Suppose Σ is a simply connected manifold and G a Lie group. If we are given a flatconnection A on the trivial bundle Σ × G, then we can encode this connection as a mapE : P → G as follows.Definition 100. Suppose A is a flat connection on the trivial principal bundle Σ×G, withthe base space Σ simply connected and G any Lie group. Then we define the elevation withrespect to p of this connection as the map E : P → G such that

q = (end point of horizontal lift of curve from π(p) to π(q) starting at p) · E(q)for all q ∈ Σ×G.

Note that the elevation E is well-defined and smooth by flatness of the connection andsimply connectedness of Σ. The elevation has a nice interpretation: it tells you how mucha point differs from p if you identify all the fibres by parallel transport. In particular, acurve is horizontal precisely if it is contained in a level set of E.

We have the following obvious result.Lemma 101. If E is the elevation of a flat connection on a trivial bundle over a simplyconnected base space, then

E(p · g) = E(p) · gfor all p ∈ Σ×G and g ∈ G.

74

The elevation of a connection determines the connection completely, as shown by thefollowing lemma.

Lemma 102. If A is a connection with elevation E, then we can recover A as

Ap(v) = θ(TpE(v)) for all v ∈ Tp(Σ×G).

Here, θ is the Maurer-Cartan form as before (see page 32).

Proof. Both sides of the equality are linear in v, so it suffices to check it on horizontaland vertical vectors. On horizontal vectors, both sides are zero (the left one by definition,the right one because horizontal curves have constant elevation). On vertical vectors, theresult follows from the previous lemma.

We will be interested in tangent vectors to the space of all flat connections. What wethus want to do is to consider a family of flat connections At parametrized by a parametert. We want to consider this as a curve in A0, the space of all flat connections on our trivialbundle.

We can define the tangent vector to this curve in A0 to be A′ : TP → g given by

A′(v) = d

dt |t=0At(v) for all v ∈ TP .

Earlier we saw that A′ ∈ Ω1(Σ, gP ). Our goal for now is to recover A′ from the elevationsEt of the connections At. To this end, define E ′ : P → g as

E ′(p) = θ

(d

dt |t=0Et(p)

).

We would like to find a link between A′ and E ′. Notice that E ′ is equivariant, in the sensethat

E ′(p · g) = Adg−1(E ′(p)) for all p ∈ Σ×G and g ∈ G.

This means we can consider E ′ ∈ Ω0(Σ, gP ), and we already have A′ ∈ Ω1(Σ, gP ). The linkbetween A′ and E ′ is now as nice as one could hope for.

Lemma 103. We haveA′ = dAE

′

where A = A0, the flat connection at time t = 0.

Proof. We will show that

A′(v)− dE ′(v) = [Ap(v), E ′(p)] for all v ∈ Tp(Σ×G),

where we consider A′ and E ′ as g-valued forms on Σ×G.

75

Let v be a tangent vector to Σ×G at p. Take a curve γ in Σ×G with γ′(0) = v. Weevaluate the left hand side of the above equality. We find

(A′)p(v)− (dE ′)p(v) = d

dt |t=0

(Atp(v)

)− d

ds |s=0(E ′(γ(s)))

= d

dt |t=0

(θ

(d

ds |s=0Et(γ(s))

))− d

ds |s=0

(θ

(d

dt |t=0Et(γ(s))

)).

Consider the map f : (−ε, ε)2 → G given by f(s, t) = Et(γ(s)). Now note that

θ

(d

ds |s=0Et(γ(s))

)= (f ∗θ)(0,t)(

d

ds),

where we have pulled back the g-valued Maurer-Cartan form via f . An analogous resultholds for the t-variable. We have found

(A′)p(v)− (dE ′)p(v) = d

dt |t=0

((f ∗θ)(0,t)

(d

ds

))− d

ds |s=0

((f ∗θ)(s,0)

(d

dt

)).

Since dds

and ddt

commute, this is the same thing as

(A′)p(v)− (dE ′)p(v) = (d(f ∗θ))(0,0)

(d

dt,d

ds

).

This leads to

(A′)p(v)− (dE ′)p(v) = f ∗(dθ)(0,0)

(d

dt,d

ds

)

= (dθ)(T(0,0)f

(d

dt

), T(0,0)f

(d

ds

)).

Using the Maurer-Cartan structure equation (lemma 47 on page 32) this reduces to

(A′)p(v)− (dE ′)p(v) = −[θ

(d

dt |t=0Et(γ(0))

), θ

(d

ds |s=0E0(γ(s))

)]= − [E ′(p), Ap(v)] .

This is precisely what we wanted, and the lemma is proved.

7.2 Application to compact orientable surfacesSuppose Σ is a compact orientable surface of genus g ≥ 1 and G is a Lie group admittingan Ad-invariant non-degenerate symmetric bilinear form on its Lie algebra (and a fixedsuch form is chosen). Take two families At1 and At2 of flat connections on the trivial

76

S

Figure 7: The fundamental domain S we will integrate over, here for genus 2. It is a polygonwith 4g sides. To recover the surface Σ topologically, we can appropriately identify the sidespairwise. Integrating a differential 2-form over the compact orientable surface Σ can beaccomplished by integrating the pullback of this form to Σ over this fundamental domain.As opposed to Σ, the fundamental domain S has a boundary (but trivial fundamentalgroup).

bundle P = Σ × G parametrized by a real parameter t in a neighbourhood of 0, in sucha way that A0

1 = A02 = A. In other words, take two curves of flat connections on Σ × G

which pass through A at time t = 0. Associate to these curves their initial velocitiesA′1, A

′2 ∈ Ω1(Σ, gP ). Explicitly, define

A′1(v) = d

dt |t=0

(At1(v)

)for all v ∈ T (Σ×G),

with an analogous definition for A′2. We shall try to find an expression for

ωAB(A′1, A′2).

The idea is to pull back the bundle to the universal cover of Σ to get rid of all holonomy.

7.2.1 Pulling back to the universal coverThe universal cover of a compact orientable surface of genus g ≥ 1 is diffeomorphic toan open disk. This universal cover can be tiled with 4g-gons in such a way that τ mapsthe interior of each polygon injectively to Σ and the edges of the polygon get mapped tocurves on Σ (the standard generators of the fundamental group). All the vertices of sucha polygon get mapped to the same point. Restricting τ to one of these polygons results ina topological quotient map from the polygon to Σ. This kind of polygon in Σ is called afundamental domain, and integrating a 2-form over Σ can be accomplished by integratingits pullback to Σ over a fundamental domain. This is the strategy we will use to study theAtiyah-Bott 2-form. The case g = 2 is illustrated in figure 7.

77

Let τ : Σ→ Σ be the universal cover of Σ. Then we have a map

τ × IdG : Σ×G→ Σ×G.

This map is a local diffeomorphism and allows us to pull back the connections At1 and At2to flat connections At1 and At2 on P = Σ × G. Moreover, we can pull back A′1 and A′2 toget A′1, A′2 ∈ Ω1(Σ, gP ). Note that A′1 and A′2 are the initial velocities of the curves At1 andAt2. The 2-form 〈A′1 ∧ A′2〉 is the pullback of 〈A′1 ∧ A′2〉 via the map τ .

We have

ωAB(A′1, A′2) =∫

Σ〈A′1 ∧ A′2〉

=∫S〈A′1 ∧ A′2〉.

Now associate to At1 and At2 the elevations Et1 and Et

2 for a fixed base point p ∈ Σ × G.Write E0

1 = E02 = E, so that E is the elevation of A = A0

1 = A02. Then we have

ωAB(A′1, A′2) =∫S〈dAE ′1 ∧ dAE ′2〉.

By lemma 86 and Stokes’ theorem, this means

ωAB(A′1, A′2) =∫∂S〈E ′1 ∧ dAE ′2〉.

7.2.2 Integrating over a horizontal domainWe will try to simplify the above expression for ωAB(A′1, A′2) further. Our next step is to getrid of the covariant exterior derivative and replace it with an ordinary exterior derivative.The idea is to consider 〈E ′1 ∧ dAE ′2〉 as a g-valued 1-form on P and integrate this over ahorizontally embedded version of S in P .

Recall that we fixed a base point p when we constructed the elevations Et1 and Et

2. Theset of all points in P = Σ × G that can be reached from p using an A-horizontal curveforms an embedded submanifold of P . Indeed, it is the graph of the map

Σ→ G : x 7→ E(x, e)−1

since the points in this set are exactly the points for which the elevation is zero (there isexactly one such point in P above every point of x ∈ Σ, and it is the point E(x, e)−1).

Write f : Σ → P : x 7→ (x,E(x, e)−1). We can consider E ′1 and dAE′2 as g-valued

forms on P of degrees 0 and 1 respectively. Their pullbacks f ∗(E ′1) and f ∗(dAE ′2) areg-valued forms on Σ. Note that 〈f ∗(E ′1)∧ f ∗(dAE ′2)〉 is exactly the 1-form 〈E ′1 ∧ dAE ′2〉 onΣ. However, because Tf maps every vector in T Σ to a horizontal vector in P , we havef ∗(dAE ′2) = d(f ∗E ′2). This shows that in fact

ωAB(A′1, A′2) =∫∂S〈f ∗E ′1 ∧ d(f ∗E ′2)〉.

78

Writing Fi = f ∗(E ′i) for i ∈ 1, 2 results in

ωAB(A′1, A′2) =∫∂S〈F1 ∧ dF2〉.

Note that Fi is a smooth g-valued function on Σ given by

Fi(x) = E ′i(x,E(x, e)−1) = AdE(x,e)(E ′(x, e)).

7.2.3 Bringing in the holonomyPick q = (x, g) ∈ P . For any y ∈ Σ, we can pick a curve γy from x to y in Σ and lift it toan A-horizontal curve γy,q in Σ starting at q. Then (τ × Idg) γy,q is the A-horizontal liftof τ γ (which is a curve in Σ from τ(x) to τ(y)). It starts at the point (τ(x), g) and endsat some point (τ(y), gy).

Definition 104. Suppose τ : Σ→ Σ is the universal cover of the manifold Σ and supposethat G is a Lie group. Given a flat connection on the principal bundle Σ × G, we defineits extended holonomy with respect to q ∈ Σ×G to be the map

Holq : Σ→ G : y 7→ g−1gy. (∗)

If y happens to lie above τ(x), then Holq(y) is exactly the holonomy of the connectionA around the curve τ γy for the base point (τ(x), g). Note that Holq(y) does not dependon the chosen curve γy: all possible choices of such a curve are homotopic by simplyconnectedness of Σ, and horizontal lifts of homotopic curves have equal end points byflatness of the connection.

Lemma 105. The extended holonomy as defined above is given by

Holq(y) = E(q)−1E(x, e)E(y, e)−1E(q).

Note that this is in fact equal to E(q)−1E(x, g)E(y, g)−1E(q) for any g ∈ G.

Proof. If we lift γy to a horizontal curve γy,(x,E(x,e)−1) in P starting at (x,E(x, e)−1) (wherethe elevation is e), then it will end in (y, E(y, e)−1) (where the elevation is also e). Byinvariance of the horizontal subbundle, the curve γy,(x,E(x,e)−1) · E(x, e)g is a horizontallift of γy starting at q, so it is γy,q. In particular, γy,q ends in (y, E(y, e)−1E(x, e)g), sothat gy = E(y, e)−1E(x, e)g. The result now follows from the definition of Holq and theobservation that E(q) = E(x, e)g.

Recall the situation we were in: we have connections Ati on P and Ati on P (for i ∈1, 2). The connection Ati has elevation Et

i with respect to some fixed base point p ∈ P .We fix some point q ∈ P , which we will later take to equal p, but for now we will do ourcalculations for general q. Associate extended holonomies Holti,q to the connections Ati.

79

Write Holq for the extended holonomy map of A (so that Holq = Hol01,q = Hol02,q). We wantto find a link between the quantities E ′i and the holonomies. We do this by differentiating

Holti,q(y) = Eti (q)−1Et

i (x, e)Eti (y, e)−1Et

i (q)

with respect to time at t = 0. Using lemma 48 gives

θ

(d

dt |t=0Holti,q(y)

)= −AdE(q)−1E(y,e)E(x,e)−1E(q)(E ′i(q))

+ AdE(q)−1E(y,e)(E ′i(x, e))− AdE(q)−1E(y,e)(E ′i(y, e))+ E ′i(q).

Let us now take q = p = (x, g) to simplify matters. Because all our elevations have p as abase point, Et

i (p) = e and therefore Fi(x) = E ′i(p) = 0 and also E ′i(x, e) = 0. We get

θ

(d

dt |t=0Holti,p(y)

)= −AdE(y,e)(E ′i(y, e))

= −Fi(y).

We conclude the following.

Theorem 106. Given

• a Lie group G with a fixed Ad-invariant symmetric non-degenerate bilinear form onits Lie algebra,

• a compact orientable surface Σ of genus g ≥ 1,

• two families At1 and At2 of flat connections on Σ×G smoothly parametrized by t suchthat A0

1 = A02 = A,

write A′i ∈ Ω1(Σ, gP ) for the initial velocity of the curve Ati, and pick a fundamental domainS in the universal cover τ : Σ→ Σ as before. Then

ωAB(A′1, A′2) =∫∂S〈F1 ∧ dF2〉

whereFi : Σ→ G : y 7→ −θ

(d

dt |t=0Holti,p(y)

)

and Holti,p is the extended holonomy of Ati.

80

7.2.4 The abelian caseIn this subsection we will apply theorem 106 to the case where G is abelian. We write thegroup operation additively.

We start with a lemma. It is a direct consequence of the definitions and the fact thatchanging base points within a fibre does not change the holonomy.

Lemma 107. Suppose G is an abelian Lie group and Σ a manifold. Let τ : Σ → Σbe the universal cover of Σ, and let A be a flat connection on the trivial bundle Σ × G.Let Holp : Σ → G be the extended holonomy of A for some base point p ∈ Σ × G. Ifγ : [0, 1]→ Σ is a curve in Σ such that τ γ is a loop, then

Holp(γ(1))− Holp(γ(0))

is the holonomy of A around τ γ (for any base point lying above τ(γ(0)) = τ(γ(1)); notethat changing base points within a fibre of Σ × G does not change the holonomy in theabelian case).

Proof. Write p = (x, g). Take a curve δ in Σ from x to γ(0). Lift δ to an A-horizontal curveδp in Σ×G. Then δp ends at (γ(0), g+Holp(γ(0))) by definition of the extended holonomy.Lift γ to an A-horizontal curve γδp(1) starting at δp(1). Then the concatenation δp ? γδp(1)is a horizontal lift of a curve in Σ from x to γ(1), so it ends at (γ(1), g + Holp(γ(1))).

This shows that γδp(1) is a horizontal lift of γ starting at (γ(0), g + Holp(γ(0))) andending at (γ(1), g + Holp(γ(1))). Then (τ × Id) γδp(1) is a horizontal lift of τ γ startingat (τ(γ(0)), g+ Holp(γ(0))) and ending at (τ(γ(1)), g+ Holp(γ(1))). The holonomy of thiscurve is therefore

Holp(γ(1))− Holp(γ(0))as was to be proved.

We are now ready to apply theorem 106 to the case of an abelian structure group G.As in the theorem, we have

ωAB(A′1, A′2) =∫∂S〈F1 ∧ dF2〉.

Let us integrate over the edges of ∂S separately. The polygon S has 4g edges, which wewill call a1, b1, a1, b1, . . . , ag, bg, ag, bg in counterclockwise order. Under the map τ : Σ→ Σ,the edge ai gets identified with ai (in the opposite direction), and the edge bi gets identifiedwith bi (also in the opposite direction). The case g = 2 was shown before in figure 7 onpage 77.

Parametrize the edges ai and ai as curves γia : [0, 1] → Σ and γia : [0, 1] → Σ in such away that τ(γia(λ)) = τ(γia(1 − λ)) for all λ ∈ [0, 1]. Take analogous parametrizations forthe bi and bi. The situation is illustrated in figure 8.

For any λ ∈ [0, 1], we can consider the curve γ as indicated in the right hand sidepicture in figure 8. This is a curve in ∂S running from γia(λ) to γia(1− λ). Split this curveinto three parts:

81

ai

ai

bi

bi

γa(λ)

γa(1− λ)

Figure 8: A group of four edges ai, bi, ai, bi on the boundary of S (the others left out).On the left: the naming and identification conventions for the edges. On the right: theparametrizations γia and γia illustrated. The two marked points coincide under τ : Σ→ Σ.If γia and γia are parametrized by arc-length on the drawing, then λ ≈ 1

3 . Also indicated isthe curve γ in red, running from γia(λ) to γia(1− λ) along ∂S.

• γ1 running from γia(λ) to γia(1) = γib(0),

• γ2, which is just γib,

• γ3 running from γib(1) = γia(0) to γia(1− λ).

Then by lemma 107 we have (for j ∈ 1, 2)

Holtp,j(γia(1− λ))− Holtp,j(γia(λ)) = Hol(Atj, τ γ)

where we have conveniently forgotten a base point on the right hand side because thechoice of base point does not matter. Using γ = γ1 ? γ2 ? γ3 we get

Holtp,j(γia(1− λ))− Holtp,j(γia(λ)) = Hol(Atj, τ γ1) + Hol(Atj, τ γ2) + Hol(Atj, τ γ3).

However, τ γ1 and τ γ3 are the same curve traversed in the opposite direction, so thattheir contributions cancel. Moreover, τ γ2 is just the generator bi of π1(Σ). Writing bti,jfor the holonomy of Atj around this generator, we find

Holtp,j(γia(1− λ))− Holtp,j(γia(λ)) = bti,j.

An analogous relation holds after switching the roles of a and b (with an additional minussign because ai is enclosed between bi and bi, not ai):

Holtp,j(γib(1− λ))− Holtp,j(γib(λ)) = −ati,j.

82

We now differentiate these relations with respect to t at t = 0. Writing θ(ddt |t=0b

ti,j

)=

b′i,j and θ(ddt |t=0a

ti,j

)= a′i,j this gives

Fj(γia(1− λ))− Fj(γia(λ)) = b′i,j (†)

andFj(γib(1− λ))− Fj(γib(λ)) = −a′i,j. (‡)

Let us now further evaluate the integral in theorem 106. We use the parametrizationsto integrate over all edges separately. This leads to

ωAB(A′1, A′2) =g∑i=1

(∫ 1

0〈F1(γia(λ)), (dF2)(γi′a (λ))〉 dλ

)

+g∑i=1

(∫ 1

0〈F1(γia(λ)), (dF2)(γi′a (λ))〉 dλ

)

+g∑i=1

(∫ 1

0〈F1(γib(λ)), (dF2)(γi′b (λ))〉 dλ

)

+g∑i=1

(∫ 1

0〈F1(γib(λ)), (dF2)(γi′b (λ))〉 dλ

).

In the second and fourth of these terms, we substitute λ→ (1− λ). By differentiating (†)and (‡) with respect to λ, we see that

dFj(γi′a (1− λ)) = −dFj(γi′a (λ))

and analogously for b. Also by (†) and (‡), the first two terms can then be combined, ascan the last two terms. We get

ωAB(A′1, A′2) =g∑i=1

(∫ 1

0〈−b′i,1, (dF2)(γi′a (λ))〉 dλ

)

+g∑i=1

(∫ 1

0〈a′i,1, (dF2)(γi′b (λ))〉 dλ

).

This can be transformed to

ωAB(A′1, A′2) =g∑i=1〈−b′i,1, F2(γia(1))− F2(γia(0))〉

+g∑i=1〈a′i,1, F2(γib(1))− F2(γib(0))〉.

Now by lemma 107 this is equivalent to

ωAB(A′1, A′2) =g∑i=1

(〈−b′i,1,−a′i,2〉+ 〈a′i,1,−b′i,2〉

)

=g∑i=1〈b′i,1, a′i,2〉 − 〈b′i,2, a′i,1〉.

83

This explicit formula for the Atiyah-Bott 2-form is as nice as we could have hoped for.Notice that the antisymmetry of ωAB is clear from this formula, whereas it wasn’t apparentin theorem 106. All that is left for us to do in this chapter is to reformulate the result alittle bit.

Theorem 108 (Symplectic structure in the abelian case, trivial bundle). Suppose Σ is acompact orientable surface of genus g ≥ 1, and suppose G is an abelian Lie group with afixed symmetric non-degenerate bilinear form on its Lie algebra. Then the moduli spaceM(Σ, G) is diffeomorphic to G2g as shown before.

Write a1, b1, . . . , ag, bg for the 2g projections to the copies of G composed with the inver-sion map G→ G so that by abuse of notation ai ∈ G is the holonomy around the generatorai of the fundamental group (and analogously for b). Then θ(Tai) : T Hom(π1(Σ), G)→ gand θ (Tbi) : T Hom(π1(Σ), G) → g are g-valued 1-forms on Hom(π1(Σ), G). Call these1-form dai and dbi. Then the symplectic structure on the blockMΣ×G(Σ, G) of the modulispace is given by

ωAB =g∑i=1〈dbi ∧ dai〉.

Note that this formula reminds us of Darboux coordinates, and in the case of a 1-dimen-sional abelian group (there are two of those: the reals and the circle group), the bi and aiare in fact Darboux coordinates.

84

Conclusions and further research

This thesis has given the reader an introduction to moduli spaces of flat connections.After introducing the basic tools necessary for their study, we have constructed the modulispacesM(Σ, G) as sets. We were able to equip the spaces with a topology and a smoothstructure (in the sense that we have a notion of smooth function on the moduli spaces) bycharacterizing flat connections using their holonomy. The topology of the reducible partof the moduli space was found explicitly as a quotient of the product of maximal tori. Inthe special case G = SU(2) the irreducible part is smooth.

Based on the procedure of symplectic reduction, we have introduced a symplectic formon the moduli space. We have derived a formula for this symplectic structure as integralover a 1-chain instead of a 2-chain. In the case of an abelian structure group, this yieldsan explicit integral-free formula for the symplectic form.

The work done in this thesis can be extended in various interesting ways for futureresearch. First of all, there are various directions in which the moduli spaces can beinvestigated without regard for their symplectic structure: it may be possible to extendthe result obtained for SU(2) (theorem 96) to other structure groups such as SO(3), whichis very similar so SU(2). Another interesting question is whether theorem 93 holds in thesmooth sense, so whether the reducible part is actually diffeomorphic to the given quotientinstead of merely homeomorphic. Besides these generalizations that seem to be close to thework already done, it may be worthwhile investigating the topology of the moduli spaceusing the tools of cohomology theory and homotopy theory (as is done in part in [Jan05]).

Also the symplectic structure allows for many routes into further research. The mostobvious one is to rigorously define the symplectic structure on the moduli space, for exampleusing group cohomology (as in [Gol84]). As a second option, one could try to make theformula for the symplectic structure as given in theorem 106 even more explicit as wasdone in the abelian case. Indeed, all of chapter 7 was originally written by the author onlyfor the abelian case, and most of it was later generalized to the non-abelian situation, sothe same may be possible for the explicit formula. Finally, it may be interesting to allowfor Σ a compact orientable surface with boundary, in which case the symplectic structuredegenerates into a Poisson structure ([Wei95]).

85

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[Ada69] J. Frank Adams, Lectures on lie groups, Midway Reprints, University of ChicagoPress, Chicago, 1969.

[Bel10] Igor Belegradek (http://mathoverflow.net/users/1573), When do two holon-omy maps determine flat bundles that are isomorphic as just bundles (w/o re-gard to the flat connections)?, MathOverflow, 2010, http://mathoverflow.net/questions/17093 (version: 2010-03-08).

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Vulgariserende samenvatting

In dit werk behandelen we zogenaamdemoduliruimten van vlakke connecties. Deze hogerdi-mensionale ruimten duiken op bij de studie van hoofdvezelbundels, objecten die belangrijketoepassingen hebben in de topologie en de differentiaalmeetkunde.

De tekst veronderstelt een zekere achtergrond in de differentiaalmeetkunde, maar bouwtook veel van de benodigde voorkennis zelf op. Ongeveer de helft van de tekst bestaat uitvereiste voorkennis uit de differentiaalmeetkunde en aanverwante gebieden. Lie-groepen,bundels, vectorwaardige differentiaalvormen, symplectische meetkunde en Poissonmeet-kunde komen aan bod.

De tweede helft van de tekst is gewijd aan de studie van de moduliruimten van vlakkeconnecties zelf. Als eerste wordt uitgelegd wat deze ruimten juist zijn, en hoe we ze precieskunnen beschouwen als (topologische) ruimte. Ook wordt er uiteengezet in welke zin demoduliruimten singulariteiten bevatten. Verder wordt de topologie van deze moduliruimtenbestudeerd. Ten slotte wordt uitgelegd hoe de moduliruimten van vlakke connecties insommige gevallen uitgerust kunnen worden met een symplectische structuur, en wordt ereen expliciete formule opgesteld voor deze symplectische structuur in speciale gevallen.

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