Infinite Dimensional Moment Map Theory in Differential Geometry

Infinite Dimensional Moment Map Theory in Differential Geometry

Master Thesis Daniel Robert-Nicoud Tuesday 8th September, 2015

Advisor: Prof. Dr. D. A. Salamon Department of Mathematics, ETH Z¨ urich

Abstract Following Donaldson, we study the moment map picture for some actions of infinite dimensional analogues of Lie groups on reasonable infinite dimensional spaces. We present the derivation of Teichm¨ uller space and the Weil-Petersson metric in this setting as the main application.

i

Aknowledgements

First and foremost, I would like to thank my parents for their unwavering support during these five years at ETH, and my supervisor Prof. Dr. Dietmar A. Salamon for taking me as a student, his patient and helpful supervision all along this work, the long and instructive discussions and for suggesting such an interesting topic to study. This was a very formative experience. I would also like to extend my gratitude to the professors Anton Alekseev, Damien Calaque and Giovanni Felder, and to the fellow students Adrian Clough, Andrea Ferrari, Mario Schulz and Claudio Sibilia for the useful discussions during the past months.

iii

Contents

Contents

v

1 Introduction 1.1 This work and its goals . . . . . . . . . . . . . . . . . . . . . . 1.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2

2 Moment maps, classical mechanics and symplectic quotients 3 2.1 Moment maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Symplectic quotients . . . . . . . . . . . . . . . . . . . . 9 2.3.2 Marsden-Weinstein quotients on K¨ahler manifolds . . . 13 2.3.3 A short cochain complex in moment map theory . . . . 14 3 Gauge theory and flat connections 3.1 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Principal bundles and associated bundles . . . . . . . 3.1.2 Connections on principal bundles and their curvature 3.1.3 Connections in differential geometry . . . . . . . . . . 3.1.4 Parallel transport and holonomy groups . . . . . . . . 3.1.5 The (global) gauge group and its action . . . . . . . . 3.1.6 The stabilizer of a connection . . . . . . . . . . . . . . 3.1.7 The moduli space of flat connections . . . . . . . . . . 3.2 The moment map picture . . . . . . . . . . . . . . . . . . . .

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17 17 17 20 27 28 31 35 37 39

4 Actions of groups of diffeomorphisms 4.1 Maps and volume-preserving diffeomorphisms . . . . . . . . 4.2 SL(n)-frames and exact volume-preserving diffeomorphisms 4.2.1 A 2-form on general associated bundles . . . . . . . 4.2.2 The bundle of SL(n)-frames . . . . . . . . . . . . .

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47 47 52 52 55

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v

Contents 4.2.3

4.3

The space of sections and the action of exact volume preserving diffeomorphisms . . . . . . . . . . . . . . . . 4.2.4 The infinitesimal action . . . . . . . . . . . . . . . . . . 4.2.5 The dual of the Lie algebra Lie G ex . . . . . . . . . . . . 4.2.6 The moment map . . . . . . . . . . . . . . . . . . . . . Teichm¨ uller space and the Weil-Petersson metric . . . . . . . . 4.3.1 The Siegel upper half space . . . . . . . . . . . . . . . . 4.3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 The infinitesimal action . . . . . . . . . . . . . . . . . . 4.3.4 The moment map . . . . . . . . . . . . . . . . . . . . . 4.3.5 The Marsden-Weinstein quotient . . . . . . . . . . . . . 4.3.6 How to get to Teichm¨ uller space . . . . . . . . . . . . . 4.3.7 The Weil-Petersson metric . . . . . . . . . . . . . . . . . 4.3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

56 58 60 60 71 71 75 78 79 82 83 88 89

A Basic concepts of symplectic geometry 91 A.1 Symplectic vector spaces . . . . . . . . . . . . . . . . . . . . . . 91 A.1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . 91 A.1.2 Special subspaces . . . . . . . . . . . . . . . . . . . . . . 92 A.2 Symplectic manifolds and symplectomorphisms . . . . . . . . . 93 A.3 Hamiltonian vector fields and Hamiltonian flows . . . . . . . . 94 A.4 Complex structures and K¨ahler manifolds . . . . . . . . . . . . 96 A.4.1 Almost complex and complex structures . . . . . . . . . 96 A.4.2 Compatible almost complex structures and K¨ahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 A.4.3 Riemann surfaces and the uniformization theorem . . . 98 B Hodge theory B.1 Differential operators on vector bundles . . . B.2 L2 -inner product and adjoints . . . . . . . . . B.3 Sobolev spaces of sections . . . . . . . . . . . B.4 The finiteness theorem . . . . . . . . . . . . . B.5 Hodge’s theorem . . . . . . . . . . . . . . . . B.5.1 The Hodge star operator . . . . . . . . B.5.2 The Laplace-Beltrami operator . . . . B.5.3 Harmonic forms and Hodge’s theorem

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99 99 101 102 103 107 107 108 108

C Teichm¨ uller space and the Weil-Petersson metric

111

Bibliography

119

Chapter 1

Introduction

1.1

This work and its goals

The main objective of this thesis is to study and understand the general theory used in Simon K. Donaldson’s paper [Donaldson, 2003] and its application in the derivation of Teichm¨ uller space. It consists mainly in the application of standard methods (namely, the Marsden-Weinstein quotient) of symplectic geometry to infinite dimensional objects. This kind of procedure can sometimes give new insights on classical problems and produce interesting objects to study (such as some moduli spaces of solutions of geometrical PDEs). Similar methods were applied for example by Donaldson to give an alternative proof of the Narashiman-Seshadri theorem in [Donaldson, 1983], and to the study of anti self-dual instantons on 4-manifolds (and thus relate to the DonaldsonThomas invariants of 4-manifolds; see [Donaldson and Kronheimer, 1990]). We begin with a brief recapitulation of classical moment map theory in chapter 2. After defining what a moment map is and giving the basic results on this object, we explain its application in classical mechanics, where it gives the conserved quantities associated to a symmetry of a physical system in a reformulation of Noether’s theorem. Then we move to symplectic quotients, with a special focus on Marsden-Weinstein quotients of symplectic and K¨ahler manifolds. This special kind of quotient in the K¨ahler case has interesting relations to geometric invariant theory (GIT): let (M , ω ) be a K¨ahler manifold, and G act with a Hamiltonian action on M , and assume that the action extends to an action of the complexified group GC . Then (at least in a suitable open set of “stable points”) we have M G = M /GC .

We will only give an overview of these results without touching the vast subject of GIT. For a more in-depth treatment, the interested reader can consult the first part of [Grundel, 2005] and [Georgoulas et al., 2013]. 1

1. Introduction In chapter 3 we study the first and most important classical application of the techniques of Marsden-Weinstein quotients in an infinite dimensional setting: Yang-Mills theory over Riemann surfaces. We start with an overview of gauge theory, where we define all the relevant objects, to pass to discuss the moment map picture for the moduli space of flat connections up to gauge equivalence. It was in fact remarked by Atiyah and Bott in their paper [Atiyah and Bott, 1983] that the moduli space of flat connections over a Riemann surface can be expressed as the Marsden-Weinstein quotient of the space of all principal connections over the global gauge group. We give the details of how this happens, and a short study of the resulting space. In particular, we show that it is always finite dimensional (but not always regular). Finally, in chapter 4 we pass to the more recent applications of moment map theory found in the papers [Donaldson, 2000] and [Donaldson, 2003], where actions of groups of diffeomorphisms are treated. We focus mostly on the general theory, and only treat the example of Teichm¨ uller space, where the Weil-Petersson metric arises naturally as the metric induced by the L2 -metric on the space of complex structures over the Riemann surface of interest. Besides the main material contained in the chapters 2 to 4, this thesis contains three appendices. Appendix A contains the basic notions of symplectic geometry underlying the whole work, plus some background on complex and K¨ ahler geometry. The main references used are [McDuff and Salamon, 2015] and [Ballmann, 2006]. Appendix B illustrates Hodge theory on vector bundles following [Bertin et al., 2002]. Appendix C gives an overview of Teichm¨ uller space and the Weil-Petersson metric using the methods of [Tromba, 1992].

1.2

Conventions

We will not try to axiomatize what the “appropriate infinite dimensional objects” are, and work on a purely formal level. The spaces considered are spaces of smooth maps, and thus we have well defined notions of smooth paths, and therefore also a good notion of (formal) tangent space by differentiating such paths. When defining the dual (Lie G )∗ of a Lie algebra Lie G in the infinite dimensional setting, we usually take only a subspace of the actual algebraic dual. What we require of (Lie G )∗ is that there exists a bilinear pairing h·, ·i : (Lie G ∗ ) × Lie G −→ R such that if ξ ∈ Lie G is such that hη ∗ , ξi = 0 for all η ∗ ∈ (Lie G )∗ , then ξ = 0. In particular, this is enough to make it so that lemma 2.23 holds. Throughout the text, all manifolds are assumed to be closed (i.e. compact and without boundary) and connected, unless stated otherwise. 2

Chapter 2

Moment maps, classical mechanics and symplectic quotients

In this chapter we give an overview of the basic notions about moment maps in the classical, finite dimensional case, and we explain the origin of the name via a small detour in physics. We will base ourselves mainly on [McDuff and Salamon, 2015, ch. 5].

2.1

Moment maps

Let (M , ω ) be a symplectic manifold, and let G be a Lie group acting on M via symplectomorphisms. In other words, we have a smooth group homomorphism G −→ Symp(M , ω ) sending g ∈ G to a symplectormorphism ψg . Denote by g the Lie algebra of G. The action of G induces a Lie algebra homomorphism g −→ Γ(T M , ω ) ξ 7−→ Xξ (m) :=



d dt t=0 ψexp(tξ ) (m)

where Xξ is in fact a symplectic vector field because of proposition A.13. It satisfies the following properties: Xadg−1 ξ = ψg∗ Xξ , X[ξ,η ] = [Xξ , Xη ]. This map is called the infinitesimal action of G on M . We will often use the notation gm = ψg (m), ξm = Xξ (m). Remark 2.1 If the group acts on the right, we adopt the convention that

Xξ (m) = −

d exp(tξ )m. dt t=0 3

2. Moment maps, classical mechanics and symplectic quotients This is equivalent to considering instead the left action defined by g • m = mg −1 . Definition 2.2 The action of G on M is weakly Hamiltonian if Xξ is a Hamiltonian vector field for every ξ ∈ g. Thus, if the action is weakly Hamiltonian, the 1-form ιXξ ω is exact for every ξ, and is given by dHξ for some Hamiltonian Hξ . Those Hamiltonians are determined only up to a constant, and by setting those constants correctly we can make it so that the map g −→ C ∞ (M ) ξ 7−→ Hξ is linear. Definition 2.3 A moment map for the action of G is a smooth map µ : M −→ g∗ such that for every m ∈ M and ξ ∈ g we have Hξ (m) = hµ(m), ξi. Equivalently, the moment map condition can be expressed as dhµ, ξi = ιXξ ω.

(2.1)

Notice that, when they exist, moment maps are unique up to constants. Indeed if µ, µ0 are two moment maps, for all ξ ∈ g we have dhµ − µ0 , ξi = 0 ⇒hµ − µ0 , ξi ≡ const. ⇒µ − µ0 ≡ const. ∈ g∗ . The following fact will be used in various places. Lemma 2.4 Let µ be a moment map. Then µ is G-equivariant, i.e. µ(gm) = ad∗g µ(m), where ad∗ denotes the coadjoint action of G on g∗ . In particular, the difference between two possible choices of moment map must be an element of g∗ which is fixed by the coadjoint action. 4

2.2. Classical mechanics Definition 2.5 The action of G on M is (strongly) Hamiltonian if the map above is a Lie algebra homomorphism from g with its usual Lie bracket and C ∞ (M ) with the Poisson bracket given by {F , G} = ω (XF , XG ) = dG(XF ). Lemma 2.6 Assume that the action is weakly Hamiltonian. Then the action is Hamiltonian if, and only if there exists a moment map. Proof Assume that the action is Hamiltonian. Fix a basis ξi ∈ g and let ξ i ∈ g∗ be the dual basis. Then the map µ(m) =

X

Hξi (m)ξ i

i

defines a moment map. Conversely, assume that we have a moment map µ : M → g∗ . We show that the map ξ 7−→ Hξ is a Lie algebra homomorphism. We have Hξ +η = hµ, ξ + ηi = hµ, ξi + hµ, ηi = Hξ + Hη , showing that the map is linear. To conclude, notice that H[ξ,η ] (m) =hµ(m), [ξ, η ]i d = hµ(m), adexp(tξ ) ηi dt t=0 d = had∗exp(−tξ ) µ(m), ηi dt t=0 d = hµ(exp(tξ )m), ηi dt t=0 =hdµm (Xξ (m)), ηi

=ω (Xξ , Xη ) ={Hξ , Hη }.

2.2



Classical mechanics

We wish to give a motivation of the name moment map. In order to do so, we must leave our path for a little while and delve a bit into classical mechanics. The following material is based principally on [Abraham et al., 1978, chapter 4]. A good place to learn the physical side of the topic is the book [Goldstein et al., 2014]. 5

2. Moment maps, classical mechanics and symplectic quotients We start by recalling the Hamiltonian formalism of classical mechanics by means of an example. Consider a particle moving in Euclidean space. Its state is completely described by its position and momentum, thus we can ∼ R6 with coordinates (x, p) (representing see it as a point in M = T ∗ R3 = position and momentum). The dynamics of the particle are encoded in the Hamiltonian function H : R × M −→ R (t, p, x) 7−→ Ht (p, x) =

p2 2m

+ V (x, t)

2

p where m is the mass of the particle, the term 2m is the kinetic energy of the particle, and the function V (x, t) is the potential energy. Thus H is in fact the total energy of the system. The equations of motion are given by the system of ordinary differential equations

(

t −p˙i (t) = ∂H ∂qi (x, p) ∂Ht x˙i (t) = ∂pi (x, p)

(2.2)

called Hamilton’s equations. We can express this in terms of the standard symplectic form on T ∗ R3 , given by ω0 = dp1 ∧ dx1 + dp2 ∧ dx2 + dp3 ∧ dx3 by noticing that 



∂Ht t −dHt = − 3i=1 ∂H ∂pi dpi + ∂xi dqi P3 = − i=1 (x˙i dpi − p˙i dxi )   P3 ∂ ∂ = ω0 i=1 p˙i ∂pi + x˙i ∂xi , ?

P

where the equality between the first and second line follows from equations (2.2). In other words, the dynamics of the particle is given exactly by the Hamiltonian flow of Ht , that is −dHt = ιXHt ω q˙ (t) = XHt (q (t)) where q : R → M is the trajectory of the physical system. It can be shown that the above is equivalent to Newton’s laws. This generalizes to arbitrary sympletic manifolds to describe a vast variety of physical systems. In the general case, we take some symplectic manifold (M , ω ) (what physicists call the phase space) whose points describe the position of the physical system, together with a Hamiltonian H : R × M → R (usually given by the total energy of the system), and require that the dynamics of the system are given by the Hamiltonian flow. 6

2.2. Classical mechanics Example 2.7 As a simple but not trivial example, take the ideal pendulum of length 1 and mass m. Its position is described by point on the circle S 1 , ∼ S 1 × R. We will denote points thus the phase space of the system is T ∗ S 1 = ∗ 1 on T S by (θ, p) (for position and momentum), where we identify S 1 with R/2πZ. The symplectic form is given by ω = dp ∧ dθ and the Hamiltonian is H (θ, p) = Thus we compute dH =

p2 + mg cos(θ ). 2m

p dp − mg sin(θ )dθ m

and, writing XH = Xθ dθ + Xp dp, ιXH ω = −Xθ dp + Xp dθ. This gives p θ˙ = Xθ = , m p˙ = Xp = mg sin(θ ), which are exactly the equations of motion for the pendulum. Now G by a Lie group acting by symplectomorphisms on M and assume that it leaves Ht invariant, that is Ht (ψg (x)) = Ht (x) for every g ∈ G and every x ∈ M . Assume that the action is Hamiltonian. (When doing computations try not to confuse the Hamiltonian H with the Hamiltonian of the action!) Then it admits a moment map µ : M → g∗ , and the following theorem, due to Emmy Noether, identifies the moment map with some conserved quantities of the system (the special cases of momentum and angular momentum motivating the name). Theorem 2.8 (Noether) Let the situation be as described above. Then µ is conserved along flow trajectories, that is µ(φtH (x)) = µ(x) for every x ∈ M and t ∈ R, where φH denotes the Hamiltonian flow of H. 7

2. Moment maps, classical mechanics and symplectic quotients Proof For simplicity we assume that H is independent of time. The same argument carries over to the time-dependent case. Let x ∈ M , ξ ∈ G and t ∈ R. Then we have: d d hµ, ξi(φtH ) = (φtH )∗ Hξ = (φtH )∗ LXH Hξ dt dt =(φtH )∗ dHξ (XH ) = (φtH )∗ ω (Xξ , XH )

= − (φtH )∗ dH (Xξ ) = 0 where XH is the Hamiltonian vector field associated to H and Xξ is the infinitesimal action of ξ on G. The first equality is the definition of Hξ , and the last one follows from G-invariance of H.  To conclude this section, we give a standard example of application of this theorem. Example 2.9 Consider the first case we discussed, where M = T ∗ R3 3 (x, p) with Hamiltonian p2 Ht (x, p) = + V (x, t). 2m Let G = R3 act on M by translation on the first factor, that is: v · (x, p) = (x + v, p) for v ∈ R3 . Assume H is invariant under the action. We compute the moment map. Let e1 , e2 , e3 ∈ G be the standard basis. For simplicity we abuse of notation and identify them with the corresponding elements of the Lie algebra. It is enough to do the computations for these three elements. We have: Xei (x, p) =



d dt t=0

(x + exp(tei ), p) =

d hµ, ei i = ιXei ω = ι

∂ ∂xi

∂ ∂xi

ω = −dpi

Thus, up to a constant hµ, ei i (x, p) = pi and so the associated conserved quantity is the momentum p. Another standard example, which we will not treat here, is the fact that rotational symmetry (i.e. G = SO (3) acting on both factors of M = T ∗ R3 by matrix multiplication) leads to conservation of angular momentum.

2.3

Symplectic quotients

We will start this section by showing that if we have a symplectic manifold and a coisotropic submanifold (recall definition A.10), then this submanifold is 8

2.3. Symplectic quotients foliated by isotropic leaves, and under some regularity assumptions the moduli space of the leaves is again a symplectic manifold in a natural way. We will then proceed by showing that if µ is the moment map for a Hamiltonian action, then µ−1 (0) is always a coisotropic submanifold. Taking the quotient, we obtain the so-called symplectic quotient of the manifold by the action of the Lie group. This will be the main object of interest in the rest of this thesis.

2.3.1

Symplectic quotients

Recall that in differential geometry, a distribution on a manifold M is a subset ∆ ⊆ T M of the tangent space T M that locally admits a smooth basis of vector fields. The dimension of ∆ is the dimension of ∆ ∩ Tm M for any point m ∈ M (as M is assumed to be connected, this is independent of m). A distribution is integrable if the Lie bracket of any two vector fields in ∆ is again in ∆. Recall also the definition of a foliation of M . Definition 2.10 A p-foliation of M is a covering of M by charts ϕi : Ui → Rn , Ui ⊂ M , such that the transition maps ψij = ϕj ϕ−1 are of the form i 1 2 ψij (x, y ) = (ψij (x), ψij (x, y ))

where (x, y ) are coordinates of Rn = Rn−p × Rp . In particular, we can glue together the “stripes” with x ≡ const. to obtain submanifolds of M of dimension p, called the leaves of the foliation. We will use the following classical result. Theorem 2.11 (Frobenius) Let M be a manifold. We have a bijective correspondence between integrable p-distribution and p-foliations of M . The distributions arise as the tangent bundle to the leaves of the foliations. We will use the fact that the leaves the foliation are constructed using the flow of the vector fields tangent to the distribution. For a proof, see for example [Robbin and Salamon, 2013]. Definition 2.12 Let M be a foliated manifold, and let m1 ∼ m2 if m1 and m2 are on the same leaf. Denote by Fm the leaf through a point m. The foliation is called regular if the following condition holds: The quotient M / ∼ is Hausdorff, and for every m ∈ M there exists a submanifold m ∈ S ⊂ M which intersects every leaf at most once and which is such that Tm0 M = Tm0 S ⊕ Tm0 Fm0 for every m0 ∈ S. Such a submanifold S is called a local slice. 9

2. Moment maps, classical mechanics and symplectic quotients It can be shown that the foliation is regular if, and only if the quotient M / ∼ is again a manifold and the quotient map M → M / ∼ is smooth. To show that the quotient is a manifold, we take the slices as charts. Back to our specific setting, let (M , ω ) be a symplectic manifold. Take Q ⊆ M a coisotropic submanifold. Then since for every p ∈ M we have that the subspaces Tp Qω ⊆ Tp Q are isotropic, the set T Qω =

[

{p} × Tp Qω

p∈M

gives rise to an isotropic distribution in T Q. Lemma 2.13 The distribution T Qω is integrable. Proof Let X, Y ∈ Ω0 (Q; T Qω ) be vector fields on Q with value in T Qω . Choose any vector field Z ∈ Ω0 (Q; T Q). Then we compute ω ([X, Y ], Z ) =LX ω (Z, Y ) + LY ω (X, Z ) + LZ ω (Y , X )

+ ω ([Y , Z ], X ) + ω ([Z, X ], Y ) + ω ([X, Y ], Z ) =dω (X, Y , Z ) = 0 where the last equality comes from the fact that ω is closed. For the first equality, notice that, since ω (X, Z ) = ω (Z, Y ) = 0 by assumption, we can immediately eliminate two terms in the first line. Since Q is coisotropic, T Qω ⊆ T Q and thus also ω (X, Y ) = 0. Since T Q is integrable, we cancel the last two superfluous terms by the same reasoning, since [Y , Z ], [Z, X ] ∈ Γ(T Q → Q). Thus, by Frobenius’ theorem, T Qω gives a foliation of Q by isotropic leaves, and it makes sense to consider the moduli space Q = Q/ ∼ of this foliation. We denote the equivalence class of a point p ∈ Q by [p] ∈ Q. Definition 2.14 The coisotropic submanifold Q is called regular if the associated foliation by isotropic leaves is regular. Proposition 2.15 Let Q ⊆ M be a regular coisotropic submanifold. Then the moduli space Q is a symplectic manifold with symplectic form given by the unique 2-form ω ∈ Ω2 (Q) such that π ∗ ω = i∗ ω, where π : Q → Q is the quotient map, and i : Q → M is the inclusion. The details can be found in [McDuff and Salamon, 2015, Prop. 5.4.4], we only give a sketch of the proof. Proof Let S0 be a local slice. Then for q ∈ S0 we have by the regularity assumption ∼ Tq0 Q , Tq0 S0 = Tq0 Qω 10

2.3. Symplectic quotients and thus a natural symplectic form on it by lemma A.7. Given two local slices S0 , S1 intersecting some common leaf, we obtain the transition map between them by sending q ∈ S0 to the unique point (if any) in S1 ∩ Fq . It can be shown (by choosing foliation charts) that the transition maps are in fact symplectomorphisms with respect to the symplectic forms obtained as above, and the result follows.  Assume now that we have a Hamiltonian group action of a Lie group G on our symplectic manifold M with a moment map µ : M → g∗ . We will show that under some assumptions the level set µ−1 (0) is a coisotropic submanifold which is isotropically foliated by the orbits of G. Therefore, we will be able to apply the result we have just shown to obtain a new symplectic manifold. Theorem 2.16 (Marsden-Weinstein) Assume that: i. 0 ∈ g∗ is a regular value of µ, and ii. G acts freely and properly on µ−1 (0). Then µ−1 (0) is a coisotropic submanifold and the corresponding isotropic foliation is given by the orbits of G. We can then apply proposition 2.15 to obtain a new symplectic manifold M G = µ−1 (0)/G of dimension dim M − 2 dim G. It is called the Marsden-Weinstein (or symplectic) quotient of M . Proof Assumption i. guarantees that µ−1 (0) is a submanifold of M , while assumption ii. that the quotient µ−1 (0)/G is again a smooth manifold. Let Gp = {ψg (p)|g ∈ G} be the orbit of p ∈ µ−1 (0) under G. Then Tp (Gp) = {ξ · p|ξ ∈ g}. We have to show that this space is the symplectic complement of Tp µ−1 (0). Notice that it has the correct dimension, as µ−1 (0) has codimension dim G in M . Moreover, since ω (ξp, v ) = hdµp v, ξi = 0 for any v ∈ Tp µ−1 (0), it is contained in the symplectic complement, and thus we have equality. In particular, µ−1 (0) is coisotropic since 0 is fixed under the coadjoint orbit. The rest follows from proposition 2.15.  Remark 2.17 The assumptions i. and ii. in the statement of the theorem can be weakened a bit. What we really need is that µ−1 (0) is a submanifold, and that the quotient µ−1 (0)/G is again a smooth manifold. In some cases, we could have for example that the action of G is not free, but is such that the isotropy subgroup of every point in the orbit stays always the same H ⊆ G (so 11

2. Moment maps, classical mechanics and symplectic quotients that it is a normal subgroup). Then we can reduce to the case of a free action by simply considering the action of G/H. If instead we only assume that 0 is a regular value of µ, then the space that we obtain is in general only an orbifold. Remark 2.18 In fact, the following more general version of the above theorem is true: Let O ⊂ g∗ be an orbit of the coadjoint action of G on g∗ , assume that every element of O is a regular value of µ and that G acts freely on O. Then the manifold MO = µ−1 (O )/G has dimension dim MO = dim M + 2 dim O − 2 dim G and it is symplectic with symplectic form given by ωMO ([v ], [w ]) = ω (v, w ) − hµ(p), [ξ, ξ 0 ]i, where v, w ∈ Tp µ(O ) and ξ, ξ 0 ∈ g are such that ad∗ξ µ(p) + dµp v = 0,

ad∗ξ0 µ(p) + dµp v 0 = 0.

For a proof, see [McDuff and Salamon, 2015]. Example 2.19 Consider M = R2n with coordinates

(x, y ) = (x1 , y1 , . . . , xn , yn ). We endow it with the standard symplectic structure given by ω=

n X

dxi ∧ dy i .

i=1

We have the action of S 1 = R/2πZ on R2n by rotations φ · (x, y ) = (cos(φ)x − sin(φ)y, sin(φ)x + cos(φ)y ), which can easily be checked to preserve the symplectic form. The infinitesimal action of ξ ∈ Lie S 1 = R is ξ · (x, y ) = (−ξy, ξx), and thus *

ιξ·(x,y ) ω = −ξ (xdx + ydy ) = 12

!

x2 + y 2 + c d − ,ξ 2

+

2.3. Symplectic quotients for any c ∈ R = (Lie S 1 )∗ (where the pairing between the Lie algebra and its dual is given by multiplication). Therefore, µ(x, y ) = −

x2 + y 2 + c 2

Under the natural identification of R2n with Cn , z = x + iy, we have µ(z ) = −

kzk2 + c . 2

Notice that c = 0 is not a regular value of µ, and for c > 0 we have µ−1 (0) = ∅. ∼ S 2n−1 , and the resulting Marsden-Weinstein For c < 0, we have µ−1 (c) = quotient is isomorphic to CPn−1 . Compare this with subsection 2.3.2: the complexification of S 1 is C∗ , the set of stable points in Cn is given by Cn \{0}, and thus ∼ CPn−1 = ∼ M G. M s /GC = This method can be applied in classical mechanics to greatly simplify problems. Indeed, if G acts on M leaving the Hamiltonian H invariant, then we have an Hamiltonian action and an associated conserved quantity along the physical evolution of the system given by the value of the moment map. So we can restrict ourselves to the preimage of the (coadjoint orbit of the) value of the moment map. Passing to the symplectic quotient allows us to study the evolution of the system up to motion internal to the orbits. The problem is simplified as we find ourselves working in a symplectic manifold of lower dimension.

2.3.2

Marsden-Weinstein quotients on K¨ ahler manifolds

Assume now that (M , ω, J ) is K¨ahler, and that a connected, compact Lie group G acts on M with a Hamiltonian action which preserves J, and a moment map µ : M → g∗ . We give an overview of how in this case the Marsden-Weinstein quotient has a naturally induced K¨ahler structure (where the induced symplectic form is given by the one of theorem 2.16). The reader is invited to consult the first part of [Grundel, 2005] and [McDuff and Salamon, 2015, sect. 5.7] for details. We look at G as a real manifold, and therefore at its Lie algebra as an algebra over R. The first important result we need is the existence of a complexification of G. This is given by the following theorem by Guillemin and Sternberg. Theorem 2.20 (Guillemin-Sternberg) Up to isomorphisms, there exist a unique connected complex Lie group GC with Lie algebra g ⊗ C and containing G as a maximal compact subgroup. 13

2. Moment maps, classical mechanics and symplectic quotients Example 2.21 The easiest example of this is given by G = S 1 ⊂ C∗ = GC . More generally, for G = U (n) we have GC = GL(n, C). As the action of G respects the complex structure, it can be canonically extended to an action of GC . We call the following formula the gradient flow equation: 1 m ˙ = grad(kµ(m)k2 ). (2.3) 2 The solutions of this equation can be shown to be always completely contained in a single GC -orbit. We define a point m ∈ M to be stable if the solution m(t) of (2.3) satisfies limt→∞ m(t) ∈ µ−1 (0). Denote by M s the set of stable points. Theorem 2.22 We have an isomorphism ∼ M s /GC . M G = The desired complex structure on the Marsden-Weinstein quotient is induced by the natural complex structure on M s /GC .

2.3.3

A short cochain complex in moment map theory

Let (M , ω ) be a symplectic manifold with a Hamiltonian (left) G-action, and µ : X → g∗ a moment map for the action. Then we obtain a sequence of maps dµx

L

x Tx X −−−→ g∗ −→ 0, 0 −→ g −−−→

where the first map is given by the infinitesimal action of G on X at x, that is ξ 7−→ Lx ξ = ξx, and the second map is dµx , i.e. the differential of the moment map at x. The composition of the two maps is given by

d µ(exp(tξ ) · x) dt t=0 d = ad∗exp(−tξ ) µ(x) dt t=0 = − Ad∗ξ µ(x).

dµx (Lx ξ ) =

Thus the sequence of maps above is a cochain complex if (and only if) µ(x) is fixed by the coadjoint action. Lemma 2.23 Let x ∈ X be such that c = µ(x) is fixed by the coadjoint action. Let M = µ−1 (c)/G be the Marsden-Weinstein quotient. Then the following statements about the cohomology of the above cochain complex are true: 14

2.3. Symplectic quotients 1. H 0 = Lie(Stabx ) is the Lie algebra of the isotropy subgroup Stabx = {g ∈ G|g · x = x} of x. 2. If H 0 = 0, then H 2 = 0 and in particular x is a regular point for µ, the Marsden-Weinstein is a regular orbifold around [x] ∈ M G, and ∼ T M. If moreover Stabx = {e}, then the quotient is a manifold H1 = [x] around [x]. 3. If H 0 = 0 for all x ∈ µ−1 (c), then c is a regular value of µ, and therefore M G is an orbifold. If moreover the action of G is free on µ−1 (c), then the Marsden-Weinstein quotient is a manifold. Proof For the first statement we have H 0 = ker(dLx ) = {ξ ∈ g dLx ξ = 0}   d exp ( tξ ) · x = 0 = ξ∈g dt t=0   d = ξ∈g exp ( sξ ) exp ( tξ ) · x = 0 for all s ∈ R dt t=0   d = ξ∈g exp(tξ ) · x = 0 for all s ∈ R dt t=s = {ξ ∈ g exp(tξ ) · x = x for all t ∈ R}

=Te Stabx = Lie Stabx . For the second statement, notice that saying that H 2 = 0 is equivalent to say that dµx is surjective, that is for all ξ ∈ g\{0} there exist some xˆ ∈ Tx X such that hdµx x, ˆ ξi 6= 0. Assume this were not true for some ξ ∈ g\{0}. Then for all xˆ we would have 0 = hdµx x, ˆ ξi = ω (Lx ξ, xˆ ). But as ω is non-degenerate, this implies that Lx ξ = 0, and therefore ξ = 0 as by assumption ker Lx = H 0 = 0. Therefore the stabilizer of x is a discrete group. It follows that M is a smooth orbifold around [x], and for H 1 we have Tx µ−1 (c) {ξx ξ ∈ g} ker(dµx ) = im(Lx )

T[x] M =

=H 1 . If the stabilizer is trivial, then we automatically get a manifold instead of an orbifold. The third statement follows directly from the second one.

 15

2. Moment maps, classical mechanics and symplectic quotients We will use this short complex in the next chapters, to show that some spaces are actually finite dimensional. The remaining chapters of this thesis will be devoted to the application of these methods to actions of infinite-dimensional analogues of Lie groups on infinite-dimensional analogues of manifolds.

16

Chapter 3

Gauge theory and flat connections

The first important example of an application of moment map techniques to infinite dimensional objects was given in 1983 by M. Atiyah and R. Bott in their paper [Atiyah and Bott, 1983]. There, they noticed that it is possible to introduce a natural symplectic (in fact K¨ahler) structure on the space of principal connections of a principal bundle over a Riemann surface, that the action of the gauge group is Hamiltonian and admits a moment map, and they showed that the moment map is in fact given by the curvature of the connection. The Marsden-Weinstein quotient is then given by the moduli space of flat connections up to gauge equivalence. In this chapter, we present this fundamental example starting with a review of gauge theory, then going on to the moment map picture, and finally a short analysis of the obtained moduli spaces from the moment map point of view.

3.1

Gauge theory

For the rest of this section, let M be a smooth manifold (the base space), G a compact Lie group (the structure group, or local gauge group), and let g = Lie(G) be the Lie algebra of G.

3.1.1

Principal bundles and associated bundles

Definition 3.1 A principal G-bundle over M is a locally trivial fibre bundle π

P −→ M together with a right action of G on P such that: 1. The action is free, i.e. p · g = p implies g = 1. 2. The action is transitive on the fibres, i.e. if π (p) = π (p0 ), then there exists g ∈ G with p0 = p · g. 17

3. Gauge theory and flat connections Notice that this implies that the fibres of P are isomorphic to G. Example 3.2 The trivial bundle P = M × G together with the obvious projection and right action by (right) multiplication on the group element is a principal G-bundle. The Hopf bundle (see for example [Naber, 2011, sect. 0.3]) exhibits S 3 as a principal S 1 -bundle over S 2 . Fix a principal G bundle P for the rest of this section. Let V be a vector space, and let ρ : G → Aut(V ) be a representation of G on V . We get a right G-action on P × V by

(p, v ) · g = (p · g, ρ(g −1 )v ), and thus an associated bundle ρ(P ) over M by quotienting by this action, i.e. ρ(P ) = P ×ρ V :=

P ×V −→ M , ∼

where the equivalence relation is given by

(p · g, v ) ∼ (p, ρ(g )v ). It is a vector bundle: let [p, v ], [p0 v 0 ] ∈ ρ(P ) be in the same fibre. In particular p and p0 are in the same fibre of P , so there is g ∈ G so that p0 = p · g. Thus we define a natural additive structure via

[p, v ] + [p0 , v 0 ] =[p, v ] + [p · g, v 0 ] =[p, v ] + [p, ρ(g )v 0 ] =[p, v + ρ(g )v 0 ]. Example 3.3 Let M be an n-manifold. The frame bundle Fr(M ) of M is the principal GL(n)-bundle given by Fr(M ) = {(m, θ ) m ∈ M , θ : Rn → Tm M linear isomorphism} with the right action given by g (m, θ ) = (m, θ ◦ g ) for g ∈ GL(n), i.e. by change of basis of Rn . This bundle is of fundamental importance in differential geometry, as we can define all of the natural vector bundles associated to a manifold as associated bundle of Fr(M ). For example, the tangent bundle is the vector bundle induced by the representation of GL(n) on Rn by the usual matrix multiplication. Given additional structure on M , it is often possible to use it to restrict to a subbundle of Fr(M ) encoding the further information. Some evident examples are: • Given an orientation on M , we obtain a principal GL(n)+ -bundle of frames compatible with the orientation. 18

3.1. Gauge theory • Given a volume form on M , we get a principal SL(n)-bundle. This will be treated in detail in subsection 4.2.2. • A Riemannian metric on M induces a principal SO (n)-bundle. And so on. Definition 3.4 Let E → M be a vector bundle. The space of E-valued kforms on M is the space of sections of the bundle Λk T ∗ M ⊗ E −→ M , where the tensor product is taken fibrewise. We write Ωk (M ; E ) for this space. If E = M × V is a trivial bundle, we also write it as Ωk (M ) ⊗ V . Lemma 3.5 We have the following identification: ∼ {β ∈ Ωk (P ) ⊗ V |ιpξ β = 0 and (ψ ∗ β )p = ρ(g −1 ) ◦ βp }, Ωk (M ; ρ(P )) = g where g ∈ G and ξ ∈ g are arbitrary. Proof The identification is given by αm (m ˆ 1, . . . , m ˆ k ) = [p, βp (pˆ1 , . . . , pˆk )], where α ∈ Ωk (M ; ad P ), β ∈ Ωk (P ) ⊗ V , π (p) = m and dπp (pˆi ) = m ˆ i for i = 1, . . . , k. Assume we are given β as in the statement of the lemma. Then we define α as above. We only have to check that it is well defined. First notice that α is independent from the choice of the lift of the vectors m ˆ i . Indeed, two different lifts only differ by a vertical vector of the form pξ for some ξ ∈ g. Therefore, if pˆi lifts m ˆ i at p, we can choose pˆi g as lift at pg. We have

[pg, βpg (pˆ1 , . . . , pˆk )] = [pg, ρ(g −1 )βp (pˆ1 , . . . , pˆk )] = [p, βp (pˆ1 , . . . , pˆk )], so that α is indeed well defined. Conversely, given α ∈ Ωk (M ; ad P ) there exists a unique β ∈ Ωk (P ) ⊗ V satisfying the relation above. By choosing two different lifts of a vector at a single point we obtain that ιpξ β = 0 for all ξ ∈ g. The following equality must hold !

[p, βp (pˆ1 , . . . , pˆk )] = [pg, βpg (pˆ1 g, . . . , pˆk g )] = [p, ρ(g )βpg (pˆ1 g, . . . , pˆk g )]. It follows that (ψg∗ β )p = ρ(g −1 ) ◦ βp .



From now on, we will often implicitly identify elements of Ωk (M ; ρ(P )) with elements of Ωk (P ) ⊗ V according to the lemma. Example 3.6 An important case of interest is when we take V = g, ρ = ad the adjoint action given by d −1 adg ξ = gξg = g exp(tξ )g −1 dt t=0 for g ∈ G and ξ ∈ g. The resulting associated bundle is denoted by ad P . 19

3. Gauge theory and flat connections

3.1.2

Connections on principal bundles and their curvature

A (principal) connection on a principal bundle is, as we will see shortly, an object giving us a way to differentiate sections of the associated bundles. We start by giving two equivalent definitions of what a connection is. Later on, we will speak about the curvature of a connection, and we will see how the concept of a connection on a principal bundle generalizes the concept of a connection on a manifold. Definition 3.7 A connection on P → M is one of the following: 1. A g-valued 1-form A ∈ Ω1 (P ) ⊗ g satisfying • A(Xξ ) = ξ for all ξ ∈ g. • ψg∗ A = adg−1 ◦A for all g ∈ G. 2. A distribution of horizontal planes H ⊂ T P , that is a distribution H satisfying • Transversality to the fibres: Tp P = Hp ⊕ Tp (π −1 (m)) for any p ∈ P , where m = π (p). • Invariance under the action of G: dψg (Hp ) = Hp·g for any g ∈ G. Lemma 3.8 The two definitions above are equivalent. Proof Given a connection A as in the first definition, define Hp = ker Ap . It is a distribution, and we easily check that it satisfies the two necessary properties. We have the short exact sequence Ap

0 −→ ker Ap ,−→ Tp P −→ g −→ 0 which splits by the map g → Tp P given by ξ 7→ pξ. The image of this last map is exactly Tp (π −1 (m)), and thus we get the first property. For the second property, let v ∈ Hp = ker Ap , then we have Ap·g (dψg v ) = (ψg∗ A)p (v ) = ρ(g −1 ) Ap (v ) = 0. | {z } =0

Conversely, given a distribution H as in the second definition, let hp : Tp P → Hp be the projection onto Hp for every point ∈ P . Then every element v ∈ Tp P splits into v = hp v + v v , where v v = v − hp v is the vertical part of v and is tangent to the Tp (π −1 (m)), and thus of the form pξ for some ξ = ξ (v ). We define Ap ( v ) = ξ ( v ) . 20

3.1. Gauge theory The first condition on A is trivially satisfied. For the second we notice dψg v =dψg (hp v + v v )

=dψg (hp v + pξ (v ) d =dψg (hp v ) + p · exp(tξ (v ))g dt t=0 d =dψg (hp v ) + p · g adg−1 exp(tξ (v )) dt t=0 = dψg (hp v ) +(pg )(adg−1 ξ (v )). |

{z

∈Hp·g

}

Applying A on both sides gives us the result.



Definition 3.9 A vector pˆ ∈ Tp P is called vertical if there exist a ξ ∈ g such that pˆ = pξ. Given a connection A, we call pˆ horizontal if Ap (pˆ) = 0. Given a vector m ˆ ∈ Tm M and a point p ∈ π −1 (m), a horizontal lift of m ˆ at p is a horizontal vector pˆ ∈ Tp P such that dπp (pˆ) = m. ˆ Horizontal lifts always exist and are unique. To see this, notice that dπp |Hp : Hp −→ Tπ (p) M is a linear isomorphism for all p ∈ P . The lifts are given by taking the inverse image of vectors on M . We denote by A ⊂ Ω1 (P ) ⊗ g the space of connections on P . The following very important theorem can be found in [Kobayashi and Nomitzu, 1963, p. 67]. Theorem 3.10 Let P be a principal bundle over a manifold M . Then A is non-empty. The space of connections A has some very nice properties, as we will now proceed to show. Lemma 3.11 A is an affine space modelled on Ω1 (M ; ad P ). In particular, its (formal) tangent space at a connection A ∈ A is given by TA A = Ω1 (M ; ad P ). Proof Let A, B ∈ A, define a = A − B ∈ Ω1 (P ) ⊗ g. It satisfies a(Xξ ) = 0 ψg∗ a = adg−1 ◦a and thus is an element of Ω1 (M ; ad P ) by lemma 3.5. Conversely, if A ∈ A and a ∈ Ω1 (M ; ad P ), it is straightforward to verify that A + a ∈ A.  21

3. Gauge theory and flat connections Now fix a connection A ∈ A, let H ⊂ T P be the correspondent distribution of horizontal planes and as before set h : T P → H to be the projection onto H. Since both H and the tangent space of the fibres are invariant under G, we have h ◦ dψg = dψg ◦ h. Let ρ : G → Aut(V ) be a representation of G on V . Let h∗ : T ∗ P → T ∗ P be the map dual to h, then h∗ acts on forms by

(h∗ β )(v1 , . . . , vk ) = β (hv1 , . . . , hvk ). Let α ∈ Ωk (P ) ⊗ V , then, borrowing the notation found in the book [Kobayashi and Nomitzu, 1963], we call it • horizontal if h∗ α = α. • equivariant if ψg∗ α = ρ(g −1 ) ◦ α. • basic if it is both horizontal and equivariant. We can restate lemma 3.5 by saying that Ωk (M ; ρ(P )) can be identified with the set of basic forms in Ωk (P ) ⊗ V . The usual exterior derivative induces a new exterior derivative d ⊗ idV : Ωk (P ) ⊗ V −→ Ωk+1 (P ) ⊗ V , which by a slight abuse of notation we denote again by d. Definition 3.12 The exterior covariant derivative associated to A is dA : Ωk (P ) ⊗ V −→ Ωk+1 (P ) ⊗ V given by dA β = h∗ dβ. Notice that as h∗ is not given by the pullback by a smooth map, we have in general that h∗ d 6= dh∗ , and thus we don’t necessarily always have d2A = 0. Lemma 3.13 The exterior covariant derivative restricts to a map dA : Ωk (M ; ρ(V )) −→ Ωk+1 (M ; ρ(V )). Proof Let β ∈ Ωk (M ; ρ(P )). We have h∗ dA β = h∗ h∗ dβ = h∗ dβ = dA β, thus the form dA β is horizontal. As we have already seen, ψg∗ commutes with h∗ . It also commutes with d, since it is the pullback by a smooth map. Thus ψg∗ dA β = dA ψg∗ β = dA (ρ(g −1 ) ◦ β ) = ρ(g −1 ) ◦ dA β, so that the form is also equivariant, and thus basic. 22



3.1. Gauge theory In order to go on, we need to elaborate a bit on what it means to take a wedge product of forms with values in some vector bundle. The usual wedge product ∧ : Ωp (P ) ⊗ Ωq (P ) −→ Ωp+q (P ) induces in a natural way a wedge product on vector valued forms ∧ : (Ωp (P ) ⊗ V ) ⊗ (Ωq (P ) ⊗ W ) −→ Ωp+q (P ) ⊗ (V ⊗ W ) whose action on 1-forms, for example, is given by

(α ∧ β )(u, v ) = α(u) ⊗ β (v ) − α(v ) ⊗ β (u). We can now compose this with some linear map V ⊗ W → Z to another vector space Z to obtain a wedge product

(Ωp (P ) ⊗ V ) ⊗ (Ωq (P ) ⊗ W ) −→ Ωp+q (P ) ⊗ Z. There are four naturally arising cases of this kind of induced wedge product that we will use. 1. V = End(W ), then we have the natural pairing End(W ) ⊗ W 7→ R and we get a wedge product ∧ : (Ωp (P ) ⊗ End(W )) ⊗ (Ωq (P ) ⊗ W ) −→ Ωp+q (P ) ⊗ W . We will denote it simply by α ∧ β. 2. V = W has an associative algebra structure, then we can use multiplication as our bilinear map. We obtain ∧ : (Ωp (P ) ⊗ V ) ⊗ (Ωq (P ) ⊗ V ) −→ Ωp+q (P ) ⊗ V , again simply denoted by α ∧ β. It is antisymmetric and it is the easiest natural generalization of the usual wedge product. 3. V = W = g is a Lie algebra, then the Lie bracket gives us the desired map [·, ·] : g ⊗ g → g, and we get an associated wedge product

[· ∧ ·] : (Ωp (P ) ⊗ g) ⊗ (Ωq (P ) ⊗ g) −→ Ωp+q (P ) ⊗ g. Its action on 1-forms is given by

[α ∧ β ](u, v ) = [α(u), β (v )] − [α(v ), β (u)]. Notice that, since the Lie bracket is antisymmetric, this wedge product is symmetric. 4. V = W has an inner product h·, ·i : V ⊗ V → R, then we have the wedge product h· ∧ ·i(Ωp (P ) ⊗ V ) ⊗ (Ωq (P ) ⊗ V ) −→ Ωp+q (P ). 23

3. Gauge theory and flat connections We can now give an explicit formula for the exterior covariant derivative. Lemma 3.14 The exterior covariant derivative of β ∈ Ωk (M ; ρ(P )) is given by dA β = dβ + ρ˙ (A) ∧ β, where ρ˙ : g → End(V ) is the tangent map of ρ at the identity of G. Proof We prove the formula for sections and 1-forms. It is then an easy exercise to apply induction to prove it for general k. Let s ∈ Ω0 (M ; ρ(P )), u ∈ Tp P . We have dA s(u) =ds(hu)

=ds(u) − ds(uv ) =ds(u) − ds(pA(u)) d =ds(u) − s(p · exp(tA(u))) dt t=0 d =ds(u) − ρ(exp(−tA(u)))s(p) dt t=0 =sd(u) + ρ˙ (A(u))s(p), where to go from the second to the third line we used the fact that uv is vertical and so has the form pξ for some ξ ∈ g, and thus ξ = A(uv ) = A(uv ) + A(hu) = A(u) since h∗ A = 0. Now let β ∈ Ω1 (M ; ρ(P )), and u, v ∈ Tp P . We have dA β (u, v ) = dβ (hu, hv ) = dβ (u, v ) − dβ (uv , hv ) − dβ (hu, v v ) − dβ (uv , v v ). We want to keep the first term. For the second term, let γ : R2 −→ P be such that γ (0, 0) = p, γ (t, s) = γ (0, s) · exp(tA(u)) and ∂s γ (0) = hv, d . Write γ0 (s) = γ (0, s), then where ∂s = ds s=0

dβ (uv , hv ) =∂t β (∂s γ ) − ∂s β (∂t γ )

=∂t β ((∂s γ0 ) · exp(tA(u))) =∂t ρ(exp(−tA(u)))β (∂s γ0 ) = − ρ˙ (A(u))β (hv ) = − ρ˙ (A(u))β (v ) 24

3.1. Gauge theory where the second term in the first line vanished as β (∂t γ ) = 0 for all s, since ∂t γ is always vertical. Similar arguments for the remaining two pieces (noticing that the last piece vanishes by verticality) give us dA β (u, v ) =dβ (u, v ) + ρ˙ (A(u))β (v ) − ρ˙ (A(v ))β (u) 



=dβ (u, v ) + ρ˙ (A) ∧ β (u, v ) 

as we wanted.

As already stated, in general d2A 6= 0. We have the following result giving an explicit formula to compute d2A . Lemma 3.15 For β ∈ Ωk (M ; ρ(P )) we have 1 d2A β = ρ˙ dA + [A ∧ A] ∧ β. 2 



Proof We have d2A β =dA (dβ + ρ˙ (A) ∧ β )

= d2 β +ρ˙ (A) ∧ dβ + |{z}

d(ρ˙ (A) ∧ β ) |

=0

{z

+ρ˙ (A) ∧ ρ˙ (A) ∧ β

}

=dρ˙ (A)∧β−ρ˙ (A)∧dβ

=ρ˙ (dA) ∧ β + ρ˙ (A) ∧ ρ˙ (A) ∧ β. Now notice that End(V ) is at the same time an associative algebra (multiplication given by composition) and a Lie algebra with Lie bracket given by the commutator. Thus we have 1 ρ˙ (A) ∧ ρ˙ (A) = [ρ˙ (A) ∧ ρ˙ (A)] 2 1 = ρ˙ ([A ∧ A]) 2 as ρ˙ is a Lie algebra homomorphism. Inserting this at the end of the computation above concludes the proof.  Our main case of interest is when we take V = g and ρ = ad the adjoint representation. Then ρ˙ : g → End(g) is given by Adξ η = [ξ, η ], and therefore we have d2A β = AddA+ 1 [A∧A] β 2



=

1 dA + [A ∧ A] ∧ β . 2 



25

3. Gauge theory and flat connections Definition 3.16 We call the 2-form 1 FA = dA + [A ∧ A] ∈ Ω2 (M ; ad P ) 2 the curvature of the connection A. A connection A is called flat if FA = 0. Notice that for A seen as a form in Ω2 (P ) ⊗ g we have FA = d2A 1 = dA A. The form FA measures the amount for which the complex d

d

d

A A A Ω0 (M ; ad P ) −→ Ω1 (M ; ad P ) −→ Ω2 (M ; ad P ) −→ ···

fails to be a honest chain complex. Properties of the curvature Lemma 3.17 (Bianchi identity) The curvature satisfies the Bianchi identity: dA FA = 0. Proof We have dA FA =h∗ d(dA + 21 [A ∧ A]) 1 = h∗ ([dA ∧ A] + [A ∧ dA]) 2 =0 as A is zero on horizontal vectors.



Lemma 3.18 Let X, Y ∈ Ω0 (P ; H ) be horizontal vector fields on P , then

[X, Y ]v (p) = −pFA (X, Y ). In particular, A is flat if, and only if the corresponding distribution H is integrable. Proof By the definitions, we have

[X, Y ]v (p) = pA([X, Y ]), and A([X, Y ]) = − dA(X, Y ) + d(A(Y ))(X ) − d(A(X ))(Y )

= − dA(X, Y ), 26

3.1. Gauge theory where we used the fact that A is zero on horizontal vector fields. At the same time 1 FA (X, Y ) =dA(X, Y ) + [A ∧ A](X, Y ) 2 =dA(X, Y ),



giving what we wanted.

Later in lemma 3.33 we will prove that the curvature is also equivariant with respect to the action of the global gauge group.

3.1.3

Connections in differential geometry

As we have seen in example 3.3, the classical bundles of differential geometry arise as bundles associated to the frame bundle Fr(M ) of the n-manifold M . It is natural to hope to have some similar relation between connections on the manifold and principal connections on Fr(M ), and indeed we have the following correspondence between the two. Identify T M with E = Fr(M ) ×GL(n) Rn via

[(m, θ ), v ] ∈ E 7−→ θ (v ) ∈ Tm M . This gives a bijective correspondence between sections X of T M and sections X of E by X (m) = [m, θ, θ−1 X (m)] for any choice θ of frame at m. Let A ∈ Ω1 (Fr(M )) ⊗ gl(n) be a connection, then A gives us a way to differentiate sections of E, namely dA : Ω0 (M ; E ) −→ Ω1 (M ; E ). So given A we can define a covariant derivative by X 7−→ X 7−→ dA X ∈ Ω1 (M ; E ) and then sending the resulting object back in Ω1 (M ; T M ). Conversely, starting with a section X ∈ Ω0 (M ; E ), we can send it to X ∈ Ω0 (M ; T M ), take the covariant derivative ∇X and send it to ∇X. This is the required association. Proving that these two operations are inverses one to the other is simply a matter of unraveling definitions. The connection A relates with the Christoffel symbols of the associated connection on M as follows. Lemma 3.19 Let (m, θ ) ∈ FrM , and take a local basis for T M given by {θei }i at m. Then for a fixed i we have for the matrix of Christoffel symbols Γkij (m) = θA(θei , 0)θ−1 . 27

3. Gauge theory and flat connections Proof Extend θ to a local section θ : M → Fr(M ), and let Vi (m) = θ (m)ei . Then on one side we have k ∇A Vi Vj = Γij Vk . On the other side, we can compute this covariant derivative in terms of A. We use θ to locally trivialize Fr(M ), i.e. we write (m, θg ) for elements of π −1 (m), with g ∈ SL(n). The section V j : FrM → Rn associated to Vj is given by V j (m, θg ) = g −1 ej . Without loss of generality, we can assume θ to be given by parallel transport along the flow ϕti of Vi . Let (Vi (m), θξ ) ∈ T(m,θ ) Fr(M ) for ξ ∈ sl(n), then we have d (Vi (m), θξ ) = (ϕti (m), θ (ϕti (m)) exp(tξ )), dt t=0 and thus d dVj (m,θ ) (Vi (m), θξ ) = Vj (ϕti (m), θ (ϕti (m)) exp(tξ )) dt t=0 d = exp(−tξ )ej = −ξej . dt t=0 Moreover, A(m,θ ) (Vi (m), θξ ) = A(m,θ ) (Vi (m), 0) + ξ, and thus

(dA V i )(m,θ) V j (m, θ ) = A(m,θ) (Vi (m), 0)Vj (m), which implies what we wanted to prove.

3.1.4



Parallel transport and holonomy groups

This subsection is partly based on the book [Kobayashi and Nomitzu, 1963, ch. 3]. The exposition we give here, however, is more focused on the case of flat connections. Fix a principal connection A on P . Let γ : [0, 1] → M be a path in M and denote by m0 = γ (0). Take p ∈ π −1 (m), then a horizontal lift of γ at p is a path γ˜ : [0, 1] −→ P with γ˜ (0) = p and such that its velocity vector field is always horizontal, that is   d γ˜ (t) = 0. Aγ˜ (t) dt It is a general fact that horizontal lifts always exist and are unique (in fact, this is true also for paths that are only piecewise C 1 ). We will not prove this in full generality, but instead restrict to the case where A is flat. If A is flat, then by lemma 3.18, the associated distribution of horizontal planes in integrable, and thus by Frobenius’ theorem gives a foliation. Let Fp ⊂ P be the leaf through p. 28

3.1. Gauge theory Lemma 3.20 The natural projection π|Fp : Fp −→ M is a covering map. Proof Let p0 ∈ Fp . As dπp0 : Tp0 P → Tπ (p0 ) M restricted to Hp0 = Tp0 Fp is an isomorphism, π|Fp is a local isomorphism. We show that π|Fp is surjective. Let m ∈ M be in the image of π, then by what we have just said, there is a neighborhood of m also contained in the image. Assume instead that m is not in the image. Take any point q ∈ π −1 (m), and let U be a small neighborhood of q in Fq . Then π −1 (π (U )) = U · G. Notice F that this set is given by g∈G U · g, and that U · G ⊂ Fqg , and thus it cannot contain any point in Fp . Therefore, π (U ) is an open neighborhood of m in the complement of the image of π|Fp . Since M is connected, it follows that π|Fp is surjective. To see that it is a covering map, let m ∈ M and take any q ∈ π −1 (m) ∩ Fp . Let U ⊂ Fp be an open subset of the leaf that is mapped isomorphically by π|Fp onto its image V = π (U ). Using G-invariance of the horizontal distribution H and the fact that the leaves of the foliation are obtained by taking flows of vector fields tangent to H (see the proof of Frobenius’ theorem), we obtain that G U g ⊂ Fp (π|Fp )−1 (V ) = g∈G s.t. pg∈Fp

and this set is evenly covered by π|Fp .



Existence and uniqueness of horizontal lifts follow as a direct corollary of this lemma by general results on covering spaces (e.g. [Munkres, 2000, p. 342-343]). In fact, in this case we have something stronger than in the case of an arbitrary principal connection. Corollary 3.21 Given m ∈ M and p ∈ π −1 (m), every path and homotopy of paths in M starting at m admit a unique horizontal lift to P . In particular, we obtain two maps: • Given a path γ : [0, 1] → M from m0 to m1 , we obtain a smooth map, called the parallel transport along γ, τγ : π −1 (m0 ) −→ π −1 (m1 ), given by sending p ∈ π −1 (m0 ) to γ˜ (1), where γ˜ is the horizontal lift of γ starting at p. If two paths γ1 , γ2 are homotopic keeping the endpoints fixed, then τγ1 = τγ2 . • We get a group homomorphism holA p : π1 (M , m) −→ G, 29

3. Gauge theory and flat connections called the holonomy at p. It is given as follows. Given an element [γ ] ∈ π1 (M , m), lift γ to P with basepoint p. Then γ˜ (1) is in the same fibre as p, and thus can be written as pg for some g ∈ G. Define holA p [γ ] = g. Proof As we have said, the existence of lifts follows from the general theory, and the statements about the parallel transport map are obvious. The holonomy map is well-defined as we can lift homotopies. To see that it is a group homomorphism, let [γ1 ], [γ2 ] ∈ π1 (M , m) and let γ˜ 1 , γ˜ 2 be the lifts at p. Denote by gi = holA ˜ 1 (1) = pg1 is p [γi ], then the lift of γ2 with basepoint γ given by γ˜ 2 (t)g1 . Therefore, we get the lift of γ1 ∗ γ2 (concatenation of paths) by concatenating γ˜ 1 and γ˜ 2 g1 . Then the endpoint of this path is given by γ˜ 2 (1)g1 = (pg2 )g1 = p(g1 g2 ),



completing the proof.

Remark 3.22 In the general case, we can still lift paths, but not homotopies, and thus the holonomy is defined as a map with the space Ωm M of loops at m as domain. Lemma 3.23 We have the following properties of hol: A 1. holA pg = adg ◦ holp , and

2. if δ : [0, 1] → M is a path from m to n in M , p ∈ π −1 (m), δ˜ is the horizontal lift of δ, q = δ˜ (1) and Φ : π1 (M , m) −→ π1 (M , n) is the isomorphism induced by δ, namely Φ[γ ] = [δ −1 ∗ γ ∗ δ ], then we have holA q ( Φ [γ ]) = holp [γ ]. Proof If γ˜ (t) lifts γ at p, then γ˜ (t)g lifts γ at pg. So the first assertion follows from A γ˜ (1)g = (p holA p [γ ])g = (pg )(adg holp [γ ]). The proof for the second assertion works on similar lines.



This shows that the holonomy is completely determined by holA p for any point p. A In particular, we have that the images holp (π1 (M , π (p))) are all isomorphics for any choice of p ∈ P , therefore, we call this group the holonomy group of A, and denote it by Hol(A). Holonomy can be used to study, classify and construct principal bundles. An example of how this can be done is found in [Kobayashi, 2014, sect. 1.2], where representations of the fundamental group are used to classify flat vector bundles. An easy consequence of what we just did is the following. 30

3.1. Gauge theory Lemma 3.24 A principal bundle P admits a flat connection A with trivial holonomy group Hol(A) if, and only if it is trivial. Proof If such an A exists, then any leaf of the associated foliation is a 1-sheet covering of the base manifold M . Fix one such leaf F , then we can use F to trivialize P globally by P −→ M × G p 7−→ (π (p), g ) where g is the unique element of G such that pg −1 ∈ F . Conversely, if P = M × G is trivial, then the connection A(m,g ) (m, ˆ gξ ) = ξ



satisfies the assumptions.

In particular, if the base is simply connected, then P admits flat connections if, and only if it is trivial. Example 3.25 The Hopf bundle S 3 → S 2 is a principal S 1 -bundle that doesn’t admit any flat connection.

3.1.5

The (global) gauge group and its action

We define now the good notion of “group of global symmetries of P .” Definition 3.26 The (global) gauge group G (P ) of P is the group of vertical automorphisms of P , that is G (P ) = {u¯ : P → P diffeomorphism|π ◦ u¯ = π and u¯ (p · g ) = u¯ (p) · g}. It acts on A by pullback. Group multiplication is given by composition. Let A ∈ A and u¯ ∈ G (P ). We check that u¯ ∗ A ∈ A. Let ξ ∈ g, then we have

d u¯ (p · exp(tξ )) dt t=0 d = u¯ (p) exp(tξ ) dt t=0 =u(p)ξ

du¯ p (pξ ) =

and thus u¯ ∗ Ap (pξ ) = Au(p) (du¯ p (pξ )) = ξ. Since ψg commutes with u¯ by definition, u¯ ∗ A is equivariant under the action of G, and thus we are done. 31

3. Gauge theory and flat connections Lemma 3.27 We have the following identification: G (P ) = {u : P → G|u(pg ) = ad g −1 u(p)}. Group multiplication is given by pointwise multiplication in G. From now on, we will write u ∈ G (P ) when we mean maps u : P → P , and u ∈ G (P ) to mean maps P → G. Proof Given u : P → G as in the statement, define u(p) = pu(p). Then u¯ is an isomorphism and it is easy to check that its inverse is given by the map u−1 (p) = pu(p)−1 . Since the fibres of P are invariant under the G-action, we have π ◦ u = π. Finally u(pg ) =(pg )u(pg ))

=p(gu(pg )) =p(u(p)g ) =(pu(p))g =u(p) · g Thus u ∈ G (P ). Conversely, let u ∈ G (P ). Take p ∈ P , since u¯ (p) is in the same fibre of p, there exist an unique g ∈ G so that p · g = u¯ (p). Define u(p) = g. Then we have

!

pu(p)g = u(p)g = u(pg ) = pgu(pg ), and therefore, u(pg ) = adg−1 u(p).



Remark 3.28 If the center Z (G) is non-trivial, then the constant functions u(p) ≡ g ∈ Z (G) define elements of G (P ). These elements are somewhat uninteresting. For example, they always leave connections fixed. For ease of notation, we will often implicitly mean G (P )/Z (G) when speaking of G (P ). We can now derive an explicit expression for the action of G (P ) on A in terms of u : P → G. Lemma 3.29 Let A ∈ A, u ∈ G (P ). Then we have u∗ A = u−1 du + u−1 Au. 32

3.1. Gauge theory Proof The associated map u¯ : P → P is given by the composition (1,u)

R

u¯ = P −→ P × G −→ P where R(p, g ) = p · g. Thus (1,du)

dR

du¯ = Tp P −−−→ Tp P × Tu(p) G −→ Tu¯ (p) P . We have dR(p,g ) (p, ˆ gξ ) = (pg )ξ + pg ˆ for pˆ ∈ Tp P and ξ ∈ g. Thus dup pˆ = (pu(p))(u(p)−1 dup pˆ) + pu ˆ (p), and

(u∗ A)p (pˆ) =Ap·u(p) ((pu(p))(u(p)−1 dup pˆ)) + pu ˆ (p)) =u(p)−1 dup pˆ + (ψu∗ (p) A)p (pˆ) =u(p)−1 dup pˆ + adu(p)−1 Ap (pˆ) or, written more succinctly u∗ A = u−1 du + u−1 Au



as we wanted. Lemma 3.30 The action of G (P ) on T A is given by u∗ a = adu−1 ◦a = u−1 au, where u ∈ G (P ) and a ∈ TA A is seen as a form in Ω1 (P ) ⊗ g.

Proof This is just a simple computation. Since A is affine, we can consider just the path A + ta.

d u a = u∗ (A + ta) dt t=0 d = du + u−1 (A + ta)u dt t=0 =u−1 au. ∗



Finally, we find the Lie algebra of G (P ). ∼ Ω0 (M ; ad P ). Lemma 3.31 We have Lie G (P ) = 33

3. Gauge theory and flat connections Proof Let ut : R → G (P ) with u0 (p) = e for every p ∈ P . Then we obviously have d ut (p) ∈ g, dt t=0 and we compute



d d ut (p · g ) = adg−1 ut (p) dt t=0 dt t=0 d = adg−1 ut (p). dt t=0 Thus Lie G (P ) ⊆ Ω0 (M ; ad P ). The other inclusion is given by considering ut (p) = exp(ts(p)), where s : P → g is any element of Ω0 (M ; ad P ).



We identify the dual of Lie G (P ) with the space Ωn (M ; ad P ), where n = dim M , via the pairing Z hα ∧ βi

hα, βi =

M

for α ∈ Ωn (M ; ad P ) and β ∈ Ω0 (M ; ad P ). This is not the algebraic dual of Lie G (P ) (which would contain distributions), but it is a good notion of dual space as for every β ∈ Ω0 (M ; ad P ) non-zero it is easy to construct α ∈ Ωn (M ; ad P ) such that hα, βi 6= 0. Lemma 3.32 Let ξ ∈ Lie G (P ) = Ω0 (M ; ad P ). Then the infinitesimal action of ξ is the vector field Xξ : A → T A given by Xξ (A) = dA ξ. Proof Looking at ξ as a map ξ : P → g, we have

d Xξ (A) = − exp(tξ )∗ A dt t=0  d  =− exp(−tξ )d(exp(tξ )) + adexp(−tξ ) ◦A dt 0  t=  d =− exp ( −tξ ) t ( d exp ) dξ − Ad ◦A ( ) tξ ξ dt t=0 = − (d exp)0 dξ − AdA ◦ξ

= − dξ − AdA ◦ξ = − dA ξ where we used lemma 3.29 in the second line.



The following result is of fundamental importance for what we will do later, as it shows that our moment map will be equivariant. 34

3.1. Gauge theory Lemma 3.33 The action of the gauge group on the curvature is given by Fu∗ A = u∗ FA . Proof This is proven by the very simple computation 1 Fu∗ A =d(u∗ A) + [(u∗ A) ∧ (u∗ A)] 2   1 =u∗ dA + [A ∧ A] 2 ∗ =u FA .

3.1.6



The stabilizer of a connection

Let A ∈ A be flat, we would like to understand the gauge transformations u ∈ G (P ) stabilizing A, i.e. such that u∗ A = A. Lemma 3.34 Let u ∈ StabA . Then u is invariant under parallel transport with respect to A, that is u ◦ τγ = u for any path γ in M . In particular, u is completely determined by its value at a single point. Proof As we have A = u−1 du + u−1 Au, it follows that du = uA − Au, and thus if γ˜ is the horizontal lift of γ we have du = 0 along γ. ˜ The statement follows.  Lemma 3.35 If u ∈ StabA , then u(p) commutes with all the elements in holA p (π1 ( Σ, π (p))). Proof Again, we have the condition A = u∗ A = u−1 du + u−1 Au, which is satisfied if, and only if du + Au − uA = 0. Let p ∈ P , and let γ : [0, 1] → P be a horizontal loop. Then we have 0 = du(γ˙ ) + A(γ˙ )u − uA(γ˙ ) = du(γ˙ ), 35

3. Gauge theory and flat connections as A(γ˙ ) = 0. This implies that u(γ (1)) = u(γ (0)), but γ (1) = γ (0)g for some g in the holonomy. Therefore, !

u(p) = u(pg ) = g −1 u(p)g,



proving the result.

The following infinitesimal version of the statement above is a direct consequence. Corollary 3.36 Let A ∈ A be flat, and let ξ ∈ Ω0 (Σ; ad P ) lie in Lie StabA . Then adg ξ = ξ for all g ∈ holA p (π1 ( Σ, π (p))). A partial converse of last lemma is the following. Lemma 3.37 Let p ∈ P , assume g ∈ G commutes with all the elements of holA p (π1 ( Σ, π (p))). Then the function u(p0 ) = adh−1 g for h ∈ G any element of G such that p0 ∈ Fp , is an element of the gauge group that stabilizes A. Proof We show u is well defined. If h ∈ G is a different choice of element such that p0 h ∈ Fp , then h0 = zh for some z ∈ holA p (π1 (M , m)), and thus adh0−1 g = h−1 z −1 gzh = h−1 gh = adh−1 g. It is obviously an element of G (P ). To show that it stabilizes A, it suffices to show that u∗ A is zero on horizontal vectors. But this is equivalent to showing that du = 0 on horizontal vectors. As u is constant on the leaves of the foliation, we are done.  Putting the last three results together, we obtain a complete characterization of the stabilizer of a connection in terms of its holonomy. Proposition 3.38 We have ∼ CG (holA (π1 (M , m))) StabA = p by sending u ∈ StabA to its value at p. Here, CG (H ) denotes the centralizer of H in G, i.e. the subgroup of G consisting of all elements commuting with all the elements in H. 36

3.1. Gauge theory

3.1.7

The moduli space of flat connections

The space that will be of interest to us in the moment map picture is the moduli space of flat connections over P M = A[ /G (P ), where A[ is the space of connections A ∈ A such that FA = 0. In this subsection we will give a short overview of why this space is interesting, and a couple of basic results. We give two motivations for the importance of this moduli space in modern mathematics. The first one is the Narasimhan–Seshadri theorem (proven originally in [Narasimhan and Seshadri, 1965]), which links stable holomorphic vector bundles over a Riemann surface Σ of genus gΣ ≥ 2 with the moduli spaces of flat connections of principal bundles over Σ. The second reason of the importance of this space comes from gauge theory, and most importantly Chern-Simons theory, which is a topological quantum field theory that can be used to define invariants on 3-manifolds, where M can appear as the phase space. See for example [Himpel, 2015] for an overview. Let P → M be a principal G-bundle, fix a point p ∈ P and let m = π (p). As we have seen, a flat connection A ∈ A[ defines a homomorphism holA p : π1 (M , m) −→ G. Lemma 3.39 The map A[ −→ hom(π1 (M , m), G) given by A 7−→ holA p descends to a well defined map hol : M −→ hom(π1 (M , m), G)/G, where G acts on hom(π1 (M , m), G) on the right by ρ · g = adg−1 ◦ρ. This basically follows from Lemma 3.23. For a full proof, the interested reader is invited to refer to [Himpel, 2015, Sect. 3.10]. It is possible to invert this map. Denote by M(M , G) the space of couples (P , [A]), where P is a principal Gbundle over M and [A] is the equivalence class of a flat connection on P up to gauge equivalence. Theorem 3.40 There is a well defined map hom(π1 (M , m), G)/G −→ M(M , G) associating to an element [ρ] ∈ hom(π1 (M , m), G)/G a couple (P , [A]) such that hol[A] = [ρ]. It is bijective, and its inverse is given by the map hol of the previous lemma. 37

3. Gauge theory and flat connections Notice that from this theorem we immediately recover lemma 3.24 as a corollary. Again, we refer to [Himpel, 2015] for a formal proof and only give a sketch ˜ →M here. Let ρ : π1 (M , m0 ) → G. We construct (P , A) as follows: let q : M ˜ with be the universal cover of M . Fixing a point m0 ∈ M , we can identify M the space of homotopy classes of paths in M starting at m0 (where we only consider homotopies with fixed endpoints). Then we define ˜ × G)/ ∼, P = (M where we identify (m, ˜ g ) ∼ (m ˜ ∗ γ, ρ([γ ])−1 g ) for [γ ] ∈ π1 (M , m0 ). This can easily be checked to be a principal G-bundle over M . We define A as the connection given by the distribution associated to the image of the distributions ˜ × {g}, g ∈ G in the quotient. To be more precise, let M ˜ × G −→ P q:M be the quotient map, then we define 



H[Am,g ˜ ) T(m,g ˜ ) (M × {g}) . ˜ ] = dq(m,g From this description, it is easy to see that we obtain the correct holonomy map. This association then descends to the map we wanted. Remark 3.41 The connection A can also be described by taking the pullback ˜ × G → G of the Maurer-Cartan form θ ∈ Ω1 (G, g) to along the projection M ˜ ×G → M ˜ . This connection obtain a connection A˜ on the trivial bundle M then descends to a well defined connection A on P . Assume that G acts on some vector space V . Then hom(π1 (M , m), G) is a space of representations on V , and therefore it is called the representation variety. As G acts on the representation variety by adjunction, it leaves the trace of the representations invariant, and thus we can use the trace to parametrize hom(π1 (M , m), G)/G. This is exactly the same as taking the character of a representation, and therefore, hom(π1 (M , m), G)/G is called the character variety. Definition 3.42 A representation ρ ∈ hom(π1 (M , m), G) is irreducible if its stabilizer Stabρ = {g ∈ G|ρ · g = ρ} with respect to the action of G coincides with the center Z (G) of G. It is clear from 3.38 that irreducible representations are the ones corresponding to connections with the smallest possible stabilizer. It will become evident from the moment map picture that these points correspond exactly to the regular points of the moduli space. 38

3.2. The moment map picture Example 3.43 Let M be a manifold with H1 (M ; Z) = 0, and P a non-trivial principal SU (2)-bundle over M . Take A ∈ A[ a flat connection on P , then the image of its holonomy

holA p : π1 (M , m) −→ G

cannot be abelian. If it were, then by Hurewicz’s theorem holA p would factor through H1 (M ; Z), so that A would have trivial holonomy, in contradiction with the fact that P is non-trivial. Therefore, the image of the holonomy is non-abelian, and as the centralizer of any non-abelian subgroup of SU (2) is the center Z (SU (2)) = {±1}, it follows by proposition 3.38 and what we have just said that [A] ∈ M is a regular point. But this is true for any A, and thus M is globally regular.

3.2

The moment map picture

We get now to the heart of this first example. Let (M = Σ, J ) be a compact Riemann surface. We will define a good notion of “symplectic form” on A, and then find the moment map associated to the action. It is necessary that we have a Riemann surface, as we need an orientation in order to integrate, and the fact that it is 2-dimensional to define the symplectic form. The complex structure induces a complex structure on the space of connections via the action of the Hodge ∗-operator on 1-forms, which is given by

∗α = −α ◦ J

for α ∈ Ω1 (Σ; ad P ). Fix an ad-invariant inner product h·, ·i on g (it exists by compactness of G, see [Knapp, 2002, p. 249]). As explained in section 3.1.2, this gives rise to a wedge product

h· ∧ ·i : TA A ⊗ TA A −→ Ω2 (Σ) after identifying TA A = Ω1 (Σ; ad P ) ⊂ Ω1 (P ) ⊗ g. The result of such a wedge product is actually an element of Ω2 (P ), but it descends to an honest element of Ω2 (Σ), as shown in the following diagram. 39

3. Gauge theory and flat connections

Ω1 (P )⊗2

T A⊗2

⊗ g⊗2

(ψg∗ )⊗2 ⊗ 1 = 1 ⊗ (adg−1

)⊗2

induced by ψg∗ −invariance

∧⊗1

h· ∧ ·i

Ω2 (P ) ⊗ g⊗2

Ω1 (P )⊗2 ⊗ g⊗2

∧⊗1 ψg∗ ⊗ 1

= 1 ⊗ (adg−1

)⊗2

Ω2 (P ) ⊗ g⊗2

1 ⊗ h·, ·i

Ω2 ( Σ )

1 ⊗ h·, ·i ψg∗

Ω2 (P )

ψg∗ −invariant forms

=1

Ω2 (P )

where the statements in red in the rightmost part of the diagram hold for elements in T A⊗2 . Thus it makes sense to define a bilinear pairing Ω : T A⊗2 −→ R by Ω(a, b) =

Z Σ

ha ∧ bi.

It is obviously antisymmetric, and it is also non-degenerate, as we have Ω(a, ∗a) 6= 0. It is also easy to see that it is closed: for A ∈ A, b1 , b2 , b3 ∈ TA A, let γ (t1 , t2 , t3 ) = A + t1 b1 + t2 b2 + t3 b3 , then we have dΩ(b1 , b2 , b3 ) =∂1 ΩA+t1 b1 (∂2 γ, ∂3 γ ) + cyclic

=∂1 ΩA+t1 b1 (b2 , b3 ) + cyclic =0,

where ∂i = dtdi , and for the last step we noticed that ΩA+t1 b1 (b2 , b3 ) is t=0 independent of t1 (and similarly for the cyclic permutations). Therefore, Ω satisfies all the properties defining a symplectic form. Lemma 3.44 The action of G (P ) on T A preserves Ω. 40

3.2. The moment map picture Proof Since for a ∈ T A we have u∗ a = adu−1 ◦a, this is a direct consequence of ad-invariance of h·, ·i.  Consider now the curvature map : A −→ Ω2 (P ) ⊗ g A 7−→ FA = dA + 21 [A ∧ A]

F

assigning to a connection A its curvature FA . Notice that FA can be seen as an element of (Lie G (P ))∗ via the pairing FA (ξ ) =

Z Σ

hξ ∧ FA i.

as we said before. The 2-form hξ ∧ FA i ∈ Ω2 (P ) is a well-defined form on Σ, following the general principle that h· ∧ ·i : Ωi (Σ; ad P ) × Ωj (Σ; ad P ) −→ Ωi+j (Σ) (which can be proved in a way analogous to the one used to prove that the symplectic form Ω is well defined). Theorem 3.45 An equivariant moment map for the action of G (P ) on A is given by µ(A) = FA . ∼ TA A. First Proof Let ξ ∈ Ω0 (Σ; ad P ) = Lie G (P ) and b ∈ Ω1 (Σ; ad P ) = notice that dhξ ∧ bi = hdξ ∧ bi + hξ ∧ dbi. At the same time, writing ρ for the adjoint action of G on g, we have hdA ξ ∧ bi =hdξ ∧ bi + hρ˙ (A)ξ ∧ bi hξ ∧ dA bi =hξ ∧ dbi + hξ ∧ (ρ˙ (A) ∧ b)i

=hξ ∧ dbi − hρ˙ (A)ξ ∧ bi and thus dhξ ∧ bi = hdA ξ ∧ bi + hξ ∧ dA bi.

(3.1)

Now we compute

(ιXξ Ω)(b) =Ω(−dA ξ, b) = = =

Z ZΣ ZΣ Σ

h−dA ξ ∧ bi hξ ∧ dA bi − dhξ ∧ bi hξ ∧ dA bi

d = hξ ∧ FA+tb i dt t=0 Σ d = hµ(A + tb), ξi = dt t=0 Z

hdµA (b), ξi, 41

3. Gauge theory and flat connections where the third equality is given by formula (30), the fourth by Stokes’ theorem, and the fifth by the easy computation

d FA+tb = dA b, dt t=0



which we leave as an exercise.

To summarize what we have done until now, we have shown that the space A of connections on a Riemann surface is symplectic, meaning that there is something that looks very much as a symplectic form on it, even though it is infinite-dimensional. We have proved that the action of the Gauge group on this space is Hamiltonian, and that it admits a moment map, which is given by the curvature. Thanks to all this, we are now ready to take the MarsdenWeinstein quotient of A, which gives us the structure of a symplectic manifold on the moduli space M = A G (P ) = µ−1 (0)/G (P ) of flat connections up to gauge equivalence. We will now use the general theory of subsection 2.3.3 to show that this moduli space nice. In fact, it turns out to always be a finite dimensional object. The short cochain complex of subsection 2.3.3 is given by −d

d

A A 0 −→ Ω0 (Σ; ad P ) −→ Ω1 (Σ; ad P ) −→ Ω2 (Σ; ad P ) −→ 0

|

{z

=Lie G (P )

}

|

{z

=TA A

}

|

{z

=(Lie G (P ))∗

}

where the first map is given by the infinitesimal action of Lie G (P ), and the second one is dµA . Notice that as Σ is 2-dimensional and A is flat (and thus d2A = 0), this is exactly the generalized deRham complex (Ω• (Σ; ad P ), dA ) (up to an irrelevant minus sign). The fact that the resulting space is finite dimensional is obtained by applying the standard Hodge-theoretic methods of appendix B. To find out the exact dimension of the space we proceed as follows. Lemma 3.46 Let P → Σ be a principal bundle, A ∈ A[ , and define the operator LA = dA + d∗A : Ω0 (Σ; ad P ) ⊕ Ω2 (Σ; ad P ) −→ Ω1 (Σ; ad P ) where dA is seen as an operator acting only on Ω0 (Σ; ad P ), and d∗A only acts on Ω2 (Σ; ad P ). Then: 1. The operator LA is Fredholm and its index is index(LA ) =

2 X

(−1)k dim H k (Σ; ad P , dA ),

k =0

that is the Euler characteristic of the complex (Ω• (Σ; ad P ), dA ). 42

3.2. The moment map picture 2. Let A, A0 ∈ A[ , and a = A0 − A ∈ Ω1 (Σ; ad P ). Denote by K = LA0 − LA . Then K is a compact operator. In particular, index(LA ) is independent of the choice of flat connection. 3. We have index(LA ) = 2(1 − gΣ ) dim G. We will only prove the third point for P a trivial bundle. The statement holds in general, but we are not aware of an easy proof. One possible way to proceed is to use an excision argument on Σ to pass from the trivial bundle to more and more twisted bundles (using the so-called excision property, see for example [Donaldson and Kronheimer, 1990, Prop. 7.1.2]). Proof For the proof of the first point, we rely heavily on lemma B.22 and the computations in its proof. We have Ω1 (Σ; ad P ) im LA im dA ⊕ im d∗A ⊕ H1 (Σ; ad P , dA ) = im dA ⊕ im d∗A

coker LA =

=H1 (Σ; ad P , dA ) = H 1 (Σ; ad P , dA ), and therefore, dim coker LA = dim H 1 (Σ; ad P , dA ) < ∞ by theorem B.23. At the same time ker LA = ker dA ⊕ ker d∗A as they live in different spaces. We notice that ker dA = H 0 (Σ; ad P , dA ), and furthermore we have dim ker d∗A = dim coker(dA : Ω1 → Ω2 ) = dim H 2 (Σ; ad P , dA ). The statement follows. For the second point, consider now LA and LA0 as operators from the space of H s -forms (with s ≥ 1) to the space of H s−1 -forms. Then dA0 − dA is given by multiplication by a, and thus takes value again in the space of H s -forms. Therefore, by Rellich’s theorem (theorem B.13), it is compact. A similar argument holds for d∗A0 − d∗A , and as the sum of two compact operators is compact, it follows that K is compact. For the third and last point, notice that if we take P the trivial bundle and A = 0, then we have H k (Σ; ad P , dA ) = H k (Σ) ⊗ g, which gives the correct result. By point 2. this then holds for any connection of P .  43

3. Gauge theory and flat connections Assume now that Z (G) is trivial, so that we have connections with trivial stabilizer. By the last result, it follows that the dimension of M at its regular points is given by dim M = 2(gΣ − 1) dim G. Remark 3.47 If Z (G) 6= {e}, then at regular points dim H 0 (Σ; ad P ) = dim Lie Z (G). We can identify Lie G (P ) with its dual using the Hodge ∗operator, which respects the natural dual pairing. This gives us a splitting Lie G (P ) = ker(LA ) ⊕ im(dµA ), ∼ H 2 (Σ; ad P ). It follows that in the and thus an isomorphism H 0 (Σ; ad P ) = general case dim M = 2(gΣ − 1) dim G + 2 dim Lie Z (G). If we can compute the holonomy of the connections, we can then study the regular points of M using proposition 3.38. For example, let P be the trivial principal SU (2)-bundle over our Riemann surface Σ (it can be proven that it is the only principal SU (2)-bundle in this case). Then for a flat connection A ∈ A[ we have three possibilities: 1. If Hol(A) = {e}, then StabA = SU (2) and the point is as singular as it can get. As the bundle is trivial, by choosing a trivialization and taking horizontal planes as horizontal distribution we get such a connection. Therefore, the moduli space is always singular. 2. If Hol(A) 6= {e} but is abelian, then it is contained in a single circle S 1 ⊂ SU (2). It follows that StabA = S 1 and the point is singular again, but less so. 3. If Hol(A) contains two non-commuting elements, then StabA = {±1}, and the point is regular. It can be shown that the moduli space has a lot of regular points. Heuristically, this can be seen by considering equivalence classes of flat connections as equivalence classes of representations of π1 (Σ) into SU (2). The standard presentation of the fundamental group is *

π1 ( Σ ) =

a1 , b1 , . . . , agΣ , bgΣ

gΣ Y

+

[ak , bk ] ,

k =1

and thus hom(π1 (Σ), SU (2)) is the inverse image of the identity under the map SU (2)2gΣ −→ SU (2) given by

(a1 , b1 , . . . , agΣ , bgΣ ) 7−→

gΣ Y k =1

44

[ak , bk ].

3.2. The moment map picture The identity is not a regular value, however a point in the preimage is regular if, and only if the centralizer of the subgroup of SU (2) generated by a1 , b1 , . . . , agΣ , bgΣ is Z (SU (2)) = {±1} (this precisely what we have already stated above, formulated in a slightly different language). But this condition is not satisfied if, and only if all of a1 , b1 , . . . , agΣ , bgΣ lie on the same circle inside SU (2), so that any slight perturbation of a singular point gives us a regular point.

45

Chapter 4

Actions of groups of diffeomorphisms

In this last chapter we treat actions of groups of special kinds of diffeomorphisms to some spaces of smooth maps following the papers of Donaldson. In the last section we work out the details of the new derivation of Teichm¨ uller space as (the quotient of) a Marsden-Weinstein quotient found in [Donaldson, 2003].

4.1

Maps and volume-preserving diffeomorphisms

We start this chapter by giving an overview of the first part of Donaldson’s paper [Donaldson, 2000], where a first example of moment map theory applied to actions of groups of diffeomorphisms was given. Similar methods to those applied here will appear in the next section. Let (M , ω ) be a symplectic manifold, S a k-manifold and ρ ∈ Ωk (S ) a fixed volume form. We consider the infinite dimensional space X = {f : S → M in a fixed homotopy class}. Then the (formal) tangent space of X at f ∈ X can be identified with the sections of f ∗ (T M ), that is Tf X = {fˆ : S → T M |fˆ(s) ∈ Tf (s) M }. There is a natural symplectic form on X given by Ωf (fˆ1 , fˆ2 ) =

Z S

ωf (fˆ1 , fˆ2 )ρ

for f ∈ X and fˆ1 , fˆ2 ∈ Tf X . It is obviously antisymmetric. Lemma 4.1 The 2-form Ω is non-degenerate and closed. 47

4. Actions of groups of diffeomorphisms Proof Let fˆ ∈ Tf X be non-zero, and let s ∈ S be such that fˆ(s) 6= 0. Take charts φS : Rk → U ⊂ S and φM : Rn → V ⊂ M around s and f (s). Then we define fˆ0 ∈ Tf X as follows: let v ∈ Rn be such that ωf (s)(fˆ(s), dφM v ) > 0 (existence is guaranteed by non-degeneracy of ω). By continuity, in a small neighborhood U˜ ⊂ U of s we have ωf (s˜)(fˆ(s˜), dφM v ) > 0 for s˜ ∈ U˜ . Let χ : S → R be a bump function with support contained in U˜ and satisfying χ ≡ 1 near s, and define fˆ0 (s) = χ(s)dφM v. Then Ωf (fˆ, fˆ0 ) > 0, proving non-degeneracy. To show that Ω is closed, let u : R3 → X be a map such that u(0, 0, 0) = f , then using d ∂i = dti ti =0 we have dΩf (∂1 u, ∂2 u, ∂3 u) =∂1 Ωu (∂2 u, ∂3 u) + cyclic

= =

Z  ZS S



∂1 ωu (∂2 u, ∂3 u) + cyclic ρ

dωf (∂1 u, ∂2 u, ∂3 u)ρ

=0 as dω = 0, concluding the proof.



We will now define the infinite dimensional Lie group of interest. Definition 4.2 A vector field V ∈ Ω0 (S; T S ) on S is called volume preserving (or divergence free) if LV ρ = 0 ⇔ dιV ρ = 0. It is called exact volume preserving if ιV ρ = dαV for some αV ∈ Ωk−1 (S ). Lemma 4.3 The vector spaces of volume preserving and exact volume preserving vector fields are both naturally Lie algebras with Lie bracket given by the usual Lie bracket of vector fields. Proof Let V , W ∈ Ω0 (S; T S ) be volume preserving, then dι[V ,W ] ω = L[V ,W ] ω = [LV , LW ]ω = 0, so that [V , W ] is also volume preserving. If V , W are additionally exact, we have ι[V ,W ] ω = [LV , ιW ]ω = dLV αW , showing that [V , W ] as again exact. 48



4.1. Maps and volume-preserving diffeomorphisms We denote the Lie algebra of volume preserving vector fields by Lie G. We can integrate it to give rise to our (infinite dimensional) Lie group. Definition 4.4 The group G of interest is given by n

o

G = φ : M → M φ = ϕtVt flow of path Vt ∈ Lie G , that is we take the solution to d t ϕ = Vt (ϕtVt ) dt Vt at some time t ∈ R. It is the path connected component of the group of volume preserving diffeomorphisms containing the identity. The (right) action of G on X is given by pullback, that is φ∗ f = f ◦ φ, and thus the action on Tf X is also by pullback: φ∗ fˆ = fˆ ◦ φ. The infinitesimal action is

d −LV f = − ( ϕt ) ∗ f dt t=0 V = − df (V ). The action of G is easily seen to preserve the symplectic form Ω. The dual of the Lie algebra is given by the space Ω1 (S ) dΩ0 (S )

(Lie G )∗ = via the pairing h[a], V i =

Z

a(V )ρ.

S

The symplectic form is preserved by the action of G, indeed if fˆ1 , fˆ2 ∈ Tf X and φ ∈ G we have Ωφ∗ f (fˆ1 , fˆ2 ) =

= =

Z ZS ZS S

ωf ◦φ (fˆ1 ◦ φ, fˆ2 ◦ φ)ρ 



ωf (fˆ1 , fˆ2 ) ◦ φ ρ

ωf (fˆ1 , fˆ2 )ρ

=Ωf (fˆ1 , fˆ2 ), where in the third line we made a change of variables and used the fact that φ is volume preserving, and thus (φ−1 )∗ ρ = ρ. Thus, it makes sense to look for a moment map for this action. In order to find one, we have to make some assumptions. Namely, we assume that 49

4. Actions of groups of diffeomorphisms 1. ∀f ∈ X we have f ∗ [ω ] = 0 ∈ H 2 (S ), and 2. H 1 (S ) = 0. Notice that as the maps in X are all in the same homotopy class, for the first condition to be satisfied it is enough that there exists one f ∈ X such that f ∗ [ω ] = 0. Now, the first assumption implies that for any f ∈ X we can write f ∗ ω = daf for some af ∈ ω 1 (S ). Let af and a0f be two different choices of primitive for f ∗ ω, then af − a0f is necessarily closed, and thus by the second assumption it is exact, af − a0f = dβ for some β ∈ Ω0 (S ). It follows that we have a well defined assignment µ : X −→ (Lie G )∗ given by µ(f ) = [af ]. Lemma 4.5 Let fˆ ∈ Tf X . Then we have dµf (fˆ) = [ιfˆω ] ∈

Ω1 (S ) , dΩ0 (S )

where ιfˆω denotes the 1-form on S given by

(ιfˆω )s (sˆ) = ωf (s) (fˆ(s), dfs sˆ) for sˆ ∈ Ts S. Proof Let



d fˆ = ft , dt t=0

then we have 

d



d af dt t=0 t





d = daft dt t=0 d = ft∗ ω. dt t=0

At this point, we would like to use Cartan’s magic formula to get our result. However, the maps ft are not necessarily generated by a vector field on M (for example, consider the case where f0 = f is the constant map at a point, 50

4.1. Maps and volume-preserving diffeomorphisms or a map with a similarly singular image). Therefore, we start with extending the map f to its graph f˜ : S → S × M , given by f˜(s) = (s, f (s)). We obviously have the identities prS f˜ = idS ,

prM f˜ = f .

Similarly, fˆ can be made into a map W : S → T (S × M ) by W (s) = (0, fˆ(s)) ∈ Ts S × Tf (s) M . As f˜ is injective, we can produce a vector field W ∈ Γ(T (S × M )) such that W ◦ f˜ = W . This can be done because W is defined along the image of f˜, which is injective. Let ϕt be the flow of W , and define ft ∈ X by ft = prM ◦ϕt ◦ f˜. Denote by ω = pr∗M ω ∈ Ω2 (S × M ), then



d d ft∗ ω = f˜∗ (ϕt )∗ pr∗M ω dt t=0 dt t=0 ∗ d ˜ =f ( ϕt ) ∗ ω dt t=0 =f˜dιW ω =df˜∗ (ι pr∗M ω ) W

=df ∗ (ιprM ◦W ω ) =ιfˆω, where in the third line we were finally able to apply Cartan’s formula. The statement follows.  Proposition 4.6 The map µ is an equivariant moment map. Proof Let φ ∈ G, then

(f ◦ φ)∗ ω = φ∗ f ∗ ω = φ∗ daf = dφ∗ af , showing equivariance. For the moment map equation, let V ∈ Lie G. Then we have hdµf (fˆ), V i =

Z S

ωf (fˆ, df (V ))ρ

=Ωf (fˆ, −LV f ), and we are done.

 51

4. Actions of groups of diffeomorphisms Remark 4.7 It is possible to get rid of the two topological assumptions we have made. However by doing so we might either lose part of the group action (by having to restrict to a subgroup of G), or the equivariance of the moment map. This is discussed in [Donaldson, 2000]. This general theory can be applied to study Lagrangian submanifolds of M (taking S to have half the dimension of M , and considering only the subset of X consisting of embeddings; in fact we get a moduli space of Lagrangian submanifolds together with a volume form) and special Lagrangian submanifolds (“special” meaning that we are considering (M , ω, J ) to be K¨ahler, fixing a holomorphic n-form θ ∈ Ωn,0 (M ) and asking that the restriction of θ to the submanifold be a real n-form; this is done by considering a different subspace of X ). See [Donaldson, 2000] for these applications, or [Grundel, 2005] for a more detailed treatment.

4.2

SL(n)-frames and exact volume-preserving diffeomorphisms

We arrive finally at the main case of interest treated in Donaldson’s article [Donaldson, 2003]. For the rest of this section except subsection 4.2.1, let M be any closed, oriented n-manifold with a fixed volume form ρ ∈ Ωn (M ), and let (X, ω ) be a fixed symplectic manifold with a (left) SL(n)-action admitting a moment map µ : X → sl(n)∗ .

4.2.1

A 2-form on general associated bundles

Let M be any smooth manifold, G a Lie group, P a principal G-bundle over M and (X, ω ) a symplectic manifold with a Hamiltonian left G-action and a moment map µ : X → g∗ . We have the associated bundle X = P ×G X =

P ×X ∼

with (pg, x) ∼ (p, gx) for p ∈ P , x ∈ X and g ∈ G. Lemma 4.8 The tangent space of X is given by TX =

TP × TX , ∼

where the equivalence relation ∼ is generated by the two relations i. (pg, ˆ xˆ ) ∼ (p, ˆ g xˆ ) for pˆ ∈ Tp P , xˆ ∈ Tx X and g ∈ G. ii. (p · ξ, 0) ∼ (0, ξ · x) for p ∈ P , x ∈ X and ξ ∈ g. 52

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms Notice that relation i. gives us a way to change basepoint from (pg, x) to (p, gx), while relation ii. tells us that the vertical part of the tangent bundle is fibrewise isomorphic to T X. Proof Take three paths

(p(t), x(t), g (t)) : R −→ P × X × G

d (p(t), x(t), g (t)) = (p, ˆ x, ˆ gˆ = gξ ). with (p(0), x(0), g (0)) = (p, x, g ) and dt t=0 Differentiating the defining relation of X,

(p(t)g (t), x(t)) ∼ (p(t), g (t)x(t)), we obtain for the left hand side



 d  d −1 g ( t ) , x ( t ) ( p ( t ) g ( t ) , x ( t )) = p ( t ) gg dt t=0 dt t=0 



= pg ˆ + (pg )(g −1 gˆ ), xˆ 



= pg ˆ + (pg )ξ, xˆ . For the right hand side we have



 d d  −1 ( p ( t ) , g ( t ) x ( t )) = p, ˆ ( g ( t ) g )( gx ( t ) dt t=0 dt t=0 



= p, ˆ (adg ξ )(gx) + g xˆ . Therefore the equivalence relation on the tangent is given by 







pg ˆ + (pg )ξ, xˆ ∼ p, ˆ (adg ξ )(gx) + g xˆ ,

which is easily shown to be equivalent to the two relations given above.



In subsection 4.2.6, the following general construction will greatly help us to study a moment map. Fix a principal connection A on P and define the 2-form ω (p,x) ((pˆ1 , xˆ 1 ), (pˆ2 , xˆ 2 )) =

= ωx (xˆ 1 + Lx Ap (pˆ1 ), xˆ 2 + Lx Ap (pˆ2 )) − hµ(x), FA (pˆ1 , pˆ2 )i as an element of Ω2 (P × X ). Lemma 4.9 We have ω = pr∗X ω − dαA , where αA = hpr∗X µ, pr∗P Ai, and prP , prX denote the projections from P × X to P and X respectively. 53

4. Actions of groups of diffeomorphisms Proof Let (pˆi , xˆ i ) ∈ Tp P × Tx X, i = 1, 2. Then

(pr∗X ω )(p,x) ((pˆi , xˆ i )i=1,2 ) = ωx (xˆ 1 , xˆ 2 ). For the second piece, let u : R2 → P × X be such that u(0, 0) = (p, x) and ∂i u(0) = (pˆi , xˆ i ), and write u(t1 , t2 ) = (p(t1 , t2 ), x(t1 , t2 )). Then dαA (∂1 u, ∂2 u) =∂1 hµ(x), Ap (∂2 p)i − (1 ↔ 2) 



= hdµx xˆ 1 , Ap (pˆ2 )i + hµ(x), ∂1 Ap (∂2 p)i − (1 ↔ 2) 



= ωx (Lx Ap (pˆ2 ), xˆ1 ) − (1 ↔ 2) + hµ(x), dAp (pˆ1 , pˆ2 )i 

= − ωx (xˆ 1 , Lx Ap (pˆ2 )) + ωx (Lx Ap (pˆ1 ), xˆ 2 ) + hµ(x), FA (pˆ1 , pˆ2 ) − [A(pˆ1 ), A(pˆ2 )].i Using the fact that hµ(x), [A(pˆ1 ), A(pˆ2 )].i = ωx (Lx Ap (pˆ1 ), Lx Ap (pˆ2 )) and putting all together, the statement follows.



Proposition 4.10 The following assertions on ω ∈ Ω2 (F (M ) × X ) hold: 1. ω is closed. 2. ω descends to a closed 2-form on X. Proof For point 1 we use the form of ω given in lemma 4.9. We have dω =d(pr∗X ω − dαA )

= pr∗X dω − d2 αA =0, so that ω is closed. We pass now to point 2. Thanks to lemma 4.8, to show that ω is a well defined form on X it is sufficient to show that the following two relations hold: • ω (p,x) ((p · ξ, 0), (p, ˆ xˆ )) = ω (p,x) ((0, ξ · x), (p, ˆ xˆ )) and • ω (pg,x) ((pˆ1 g, xˆ 1 ), (pˆ2 g, xˆ 2 )) = ω (p,gx) ((pˆ1 , g xˆ 1 ), (pˆ2 , g xˆ 2 )). Both are really easy to check, and thus we leave them as exercises to the reader. The fact that ω ∈ Ω2 (X ) is closed follows from point 1 by taking equivalence classes.  54

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms

4.2.2

The bundle of SL(n)-frames

In example 3.3 we have seen that to any smooth manifold we can associate canonically a principal GL(n)-bundle of frames. The further structure given by the volume form ρ permits us to restrict instead to the following principal SL(n)-bundle: F (M ) = {(m, θ ) ∈ Fr(M )|ρ(θe1 , . . . , θen ) = 1} , where {ei }ni=1 is the standard basis of Rn . It is a fibre bundle over M through the obvious projection map π (m, θ ) = m. In what follows, we will call this bundle simply the frame bundle of M . The last condition is equivalent to ρ(θv1 , . . . , θvn ) = det(v1 | · · · |vn ). The action of SL(n) on F (M ) is given by composition on the right, that is for g ∈ SL(n) and (m, θ ) ∈ F (M )

(m, θ ) · g = (m, θg ). The resulting element is still in the frame bundle, indeed ρ(θgv1 , . . . , θgvn ) = det(gv1 | · · · |gvn )

= det(g ) det(v1 | · · · |vn ) = det(v1 | · · · |vn ). The action is obviously free and transitive on the fibres. In what follows, we will often need to speak about the tangent bundle of F (M ). This object is closely linked to the double-tangent bundle T (T M ). To describe it, fix a principal connection A ∈ Ω1 (F (M )) ⊗ sl(n). Lemma 4.11 The connection A induces an isomorphism Tp F (M ) −→ Tm M × sl(n) by pˆ 7−→ (dπp p, ˆ Ap (pˆ)). Proof Linearity and surjectivity are obvious. For injectivity, notice that dπp pˆ = 0 if, and only if pˆ = pξ for some ξ ∈ sl(n), but then Ap (pξ ) = 0 implies ξ = 0.  55

4. Actions of groups of diffeomorphisms As seen at the end of subsection 3.1.2, fixing such a principal connection is equivalent to choosing a connection ∇ on M whose Christoffel symbols satisfy Γjij = 0 (which is equivalent to the condition that A takes values in sl(n), by lemma 3.19), then the isomorphism is given as follows: let pˆ be the equivalence class of the path (m(t), θ (t)) ∈ T M with (m(0), θ (0)) = p. We map 

pˆ 7−→



d m ( t ) , ∇t θ , dt t=0 

where ∇t θ ∈ L(Rn , Tm M ) is given by

( ∇t θ ) v = ∇t ( θ ( t ) v ) and the covariant derivative is taken along m(t). The previous picture is obtained by noticing that Ap (pˆ) = θ−1 ∇t θ. From now on, we will mainly use this second point of view, and we will furthermore restrict to torsion free connections. Remark 4.12 Let s : F (M ) → X be any map (usually, an SL(n)-equivariant one). In what follows we will often need to speak about the tangent map ds : T F (M ) −→ T X associated to it. Given a connection A on F (M ) as above, we will write ds(m,θ ) (m, ˆ Φθ ) with (m, θ ) ∈ F (M ), m ˆ ∈ Tm M , and Φ : Tm M → Tm M a traceless en∼ domorphism for the composition of ds with the identification of T F (M ) = −1 Tm M × sl(n) applied to the element (m, ˆ θ Φθ ) ∈ Tm M × sl(n). It is important to always know what connection one is using, as the same map applied to what is at a first glance the same argument under two different identifications can give very different results.

4.2.3

The space of sections and the action of exact volume preserving diffeomorphisms

We have the following bundle over M associated with M and X: X = F (M ) ×SL(n) X =

F (M ) × X , ∼

where the equivalence relation is given by (m, θg, x) ∼ (m, θ, gx). Our main space of interest will be the space X = Γ(X → M ) of sections of this bundle. Similarly to what we have seen in the case of gauge theory, we can identify 56

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms X with the space of SL(n)-equivariant maps from F (M ) to X, i.e. maps s : F (M ) → X satisfying s(m, θg ) = g −1 s(m, θ ). The identification is given by s(m) = [(m, θ ), s(m, θ )] for s ∈ X , s : F (M ) → X equivariant, m ∈ M and (m, θ ) ∈ π −1 (m) any point in the fibre. Again as in the previous section, the tangent space of X is given by Ts X = {sˆ : F (M ) → T X sˆ(m, θ ) ∈ Ts(m,θ ) X, sˆ(m, θg ) = g −1 sˆ(m, θ )}, i.e. equivariant sections of s∗ T X → F (M ) (equivalently, sections of s∗ T X → M ), and we have a symplectic form on X given by Ωs (sˆ1 , sˆ2 ) =

Z

ωs (sˆ1 , sˆ2 )ρ,

M

where s ∈ X and sˆ1 , sˆ2 ∈ Ts X . Lemma 4.13 Ω is well-defined, antisymmetric, non-degenerate and closed. Proof To show that Ω is well-defined, we have to show that ωs (sˆ1 , sˆ2 ) is constant on fibres, and thus descends to a function on M . So consider two points (m, θ ), (m, θ0 ) ∈ π −1 (m). Write θ0 = θg for some g ∈ SL(n), then we have ωs(m,θg ) (sˆ1 (m, θg ), sˆ2 (m, θg )) =ωg−1 s(m,θ ) (g −1 sˆ1 (m, θ ), g −1 sˆ2 (m, θ ))

=ωs(m,θ) (sˆ1 (m, θ ), sˆ2 (m, θ )) where in the last line we used the invariance of ω under the action of SL(n). The rest of the proof goes through exactly as for lemma 4.1.  Let now Lie G ex be the space of exact volume preserving vector fields. The Lie group of interest is given exactly as in definition 4.4, but this time integrating only exact divergence free vector fields. It is called the group of exact volume preserving diffeomorphisms, and we denote it by G ex . The group G ex acts (on the right) on X as follows. Let s : F (M ) → X be a section of X, φ ∈ G ex . We define

(φ∗ s)(m, θ ) = s(φ(m), dφm ◦ θ ) for (m, θ ) ∈ F (M ). Notice that this action is well-defined because φ is volume preserving, and thus dφm ◦ θ is still an SL(n)-frame. The same action also works for non-exact volume preserving diffeomorphisms. 57

4. Actions of groups of diffeomorphisms The action of G ex on T X is easily seen to be given by φ∗ sˆ(m, θ ) = sˆ(φ(m), dφm ◦ θ ). We show that the action preserves the symplectic form. Lemma 4.14 The symplectic form Ω is invariant under the action of G. Proof The computations at the beginning of the proof of lemma 4.13 show that ωs (sˆ1 , sˆ2 )(m, θ ) is in fact independent of θ. Thus for φ ∈ G ex we have ωφ∗ s (φ∗ sˆ1 , φ∗ sˆ2 )(m) =ωs (sˆ1 , sˆ2 )(φ(m))

=φ∗ (ωs (sˆ1 , sˆ2 ))(m), therefore, Ωφ∗ s (φ∗ sˆ1 , φ∗ sˆ2 ) =

=

Z ZM

φ∗ (ωs (sˆ1 , sˆ2 ))ρ ωs (sˆ1 , sˆ2 )ρ

M

=Ωs (sˆ1 , sˆ2 ).

4.2.4



The infinitesimal action

We study now the infinitesimal action of the Lie algebra Lie G ex on X . Let V ∈ Lie G ex be an exact volume preserving vector field, ϕtV its flow. Take s ∈ X . We have

d  t ∗ s(m, θ ) ϕ dt t=0 V d = − s(ϕtV (m), (dϕtV )m ◦ θ ) dt t=0   d t t = − ds(m,θ) (ϕ (m), (dϕV )m ◦ θ ) dt t=0 V

(−LV s)(m, θ ) = −





= − ds(m,θ) V (m), ∇t (dϕtV )m ◦ θ )



= − ds(m,θ) (V (m), (∇V )(m) ◦ θ )) where in the last line we exchanged the two derivatives ∇t and d, and used the defining property of the flow. The term (∇V )(m) ◦ θ is to be interpreted as the element of L(Rn , Tm M ) given by w 7−→ ∇θw V (m). Remark 4.15 We use the Lie derivative as notation for the infinitesimal action, as it is similar the infinitesimal variation of the pullback by the flow. 58

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms We state now two important properties of ds that will be useful in the future. Lemma 4.16 The map ds : T F (M ) → T X satisfies the following two properties: ˆ ) = g −1 ds(m,θ ) (m, 1. Equivariance: ds(m,θg ) (m, ˆ θg ˆ θˆ) for any g ∈ SL(n). 2. Verticality: ds(m,θ ) (0, θξ ) + Ls(m,θ ) ξ = 0 for any ξ ∈ sl(n). Proof Equivariance is obtained by differentiating the relation s(m(t), θ (t)g ) = g −1 s(m(t), θ (t)), where (m(t), θ (t)) : R → F (M ) is a smooth path. Similarly, we get verticality by differentiating s(m, θg (t)) = g (t)−1 s(m, θ ), where g (t) : R → SL(n) with g (0) = e.



A second point of view We also have the following equivalent way to proceed. We can lift any element φ ∈ G ex ⊂ Diff (M ) to φ˜ ∈ Diff (F (M )) by φ˜ (m, θ ) = (φ(m), dφm ◦ θ ). ˜ Then, if We have obviously that φ ◦ π = π ◦ φ.

d V = φt ∈ Lie G, dt t=0 we automatically obtain an associated vector field

d V˜ = φ˜t ∈ Γ(T F (M )) dt t=0 with dπ ◦ V˜ = V ◦ π. Now, we have that for s ∈ X , the action φ∗ s is given by the actual pullback φ˜ ∗ s, and

d ∗ −LV s = − φ˜t s = −ds(V˜ ). dt t=0 Remark 4.17 As we stated above, this is perfectly equivalent to what we did before, but the explicit formulas of above are now hidden in the notation. This formulation will be handy later on. 59

4. Actions of groups of diffeomorphisms

4.2.5

The dual of the Lie algebra Lie G ex

The moment map for the action of G ex on X must take values in (Lie G ex )∗ . Thus, we must identify the dual of this Lie algebra. We start by noticing that we have n−2 (M ) ∼ = Ω Lie G ex −→ n−2 Ωcl (M ) by sending V ∈ Lie G ex to the equivalence class of the associated (n − 2)-form αV (equivalently, we can identify Lie G ex with dΩn−2 (M ) by sending V to ιV ρ). Therefore, we have

(Lie G ex )∗ = dΩ1 (M ) by identifying dβ with the element of (Lie G ex )∗ acting by hdβ, [αV ]i =

Z

dβ ∧ αV =

M

Z

β ∧ ιV ρ =

Z

M

(ιV β )ρ.

M

Remark 4.18 Notice that this space is not the correct function analytical dual of Lie G ex (which would contain distributions). It is however already “big enough” for what we need, namely it satisfies what we asked when setting the conventions in section 1.2, and it will contain the image of the moment map.

4.2.6

The moment map

In his paper, Donaldson expresses the moment map as µ(s) = ωs (∇s, ∇s) − hµs , Ri + d(c(∇µs )) ∈ Ω2 (M ).

(4.1)

In this subsection we will first of all interpret and explain the various terms in this formula, show that it satisfies all the necessary properties, and finally explain how it can be interpreted as a moment map for our action. Remark 4.19 The map (4.1) is not an honest moment map. Indeed, we will see that it takes value in the closed 2-forms on M , but in general not in d(Ω1 (M )) as we would like it to. See remark 4.25 for further details. For the first term, we start by defining a map ∇s eating a point (m, θ ) ∈ F (M ) and a vector m ˆ ∈ Tm M and giving back ∇s(m,θ ) m ˆ = ds(m,θ ) (m, ˆ 0) ∈ Ts(m,θ ) X. Namely, let m(t) : R → M be a path with m(0) = m and ·m(0) = m. ˆ Then we can parallel transport θ with respect to ∇ along m(t), i.e. we define θ (t)ei as the parallel transport of θei along the path. This gives a well defined path (m(t), θ (t)) ∈ F (M ), and thus an element of the tangent space T(m,θ) F (M ) by taking its equivalence class, and we define ∇sm,θ ) m ˆ = ds(m,θ ) [m(t), θ (t)]. 60

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms Notice that ∇s depends heavily on the choice of ∇. Indeed, for different choices of connection we obtain different notions of parallel transport. This dependence will be compensated by the other terms of the moment map. Remark 4.20 An alternative description of ∇s is given by ∇sp m ˆ = dsp (pˆ − Lp Ap (pˆ)), where p = (m, θ ) ∈ F (M ) and pˆ ∈ Tp P is any vector such that dπp pˆ = m. ˆ In view of the verticality property of ds of lemma 4.16 we can immediately rewrite this as ∇sp m ˆ = dsp pˆ + Ls(p) Ap (pˆ). Using ∇s, we define the first term of the moment map as the 2-form on M given by ωs (∇s, ∇s)m (m ˆ 1, m ˆ 2 ) = ωs(m,θ ) (∇s(m,θ ) m ˆ 1 , ∇s(m,θ ) m ˆ 2 ). It is easily seen to be independent from the choice of frame θ. For the second term, consider the map µ ◦ s : F (M ) −→ sl(n)∗ . It can be seen as a map µs taking values in the dual of the space of traceless endomorphisms End0 (T M ) of T M by hµs , Φi = hµ ◦ s, θ−1 Φθi for Φ ∈ End0 (T M ). This pairing gives a well defined map on M . Indeed for g ∈ SL(n) hµ ◦ s(m, θg ), (θg )−1 Φ(θg )i =hµ(g −1 s(m, θ )), g −1 θ−1 Φθgi

=hg −1 (µ ◦ s(m, θ ))g, g −1 θ−1 Φθgi =hµ ◦ s(m, θ ), θ−1 Φθi. Let R ∈ Ω2 (M ; End(T M )) be the curvature tensor of ∇. It takes values in End0 (T M ), so that hµs , Ri gives a 2-form on M by hµs , Rm (m ˆ 1, m ˆ 2 )i. Remark 4.21 Another way to go is to notice that this 2-form is in fact given by hµs , FA i, where A is the principal connection associated to ∇, and thus FA takes values in sl(n). 61

4. Actions of groups of diffeomorphisms Finally, for the last term, c(∇µs ) is the following contraction of ∇µs : T ∗ M ⊗2 ⊗ T M → R. Fix a local basis {ei }ni=1 for T M , then c(∇µs )(m ˆ)=

n D X

E

∇ei µs , (me ˆ ∗i )0 ,

(4.2)

i=1

where {e∗i }ni=1 is the dual basis, me ˆ ∗i ∈ End(T M ) is the map given by

(me ˆ ∗i )(ej ) = δij m, ˆ and (−)0 denotes taking the traceless part of an endomorphism, that is

( Φ )0 = Φ −

1 tr(Φ) idT M . n

It is easy to check that it is independent of the choice of basis. Remark 4.22 Another possible way to describe this map would be to use the (indefinite, non-degenerate) trace pairing

(A, B ) ∈ sl(n) × sl(n) 7−→ tr(AB ) ∈ R to identify sl(n) and sl(n)∗ . Then we could look at µ as a map µ˜ : X −→ sl(n), and write hµs , Φi = tr((µ ◦ s)θ−1 Φθ ) without need to take the traceless part (as multiples of the identity would be sent to zero anyway). Now what we did in subsection 4.2.1 becomes handy. In our specific case we have the 2-form ω (m,θ,x) ((m ˆ 1 , θξ1 , xˆ 1 ), (m ˆ 2 , θξ2 , xˆ 2 )) = D

E

= ωx (xˆ 1 + Lx ξ1 , xˆ 2 + Lx ξ2 ) − µ(x), θ−1 Rm (m ˆ 1, m ˆ 2 )θ . It can be used to encode the first two terms of the moment map. Lemma 4.23 Let s ∈ X . Then s∗ ω = ωs (∇s, ∇s) − hµs , Ri. 62

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms Proof Writing p = (m, θ ) ∈ F (M ) and pˆi = (m ˆ i , θξi ) for i = 1, 2, we have

(s∗ ω )p (pˆ1 , pˆ2 ) =ω [p,s(p)] ([pˆ1 , dsp pˆ1 ], [pˆ2 , dsp pˆ2 ]) =ω (p,s(p)) ((pˆ1 , dsp pˆ1 ), (pˆ2 , dsp pˆ2 )) for s ∈ X . Notice that this is exactly the pullback of ω along the associated section s : M → X. Then

(s∗ ω )p (pˆ1 , pˆ2 ) = = ω (p,s(p)) ((pˆ1 , dsp pˆ1 ), (pˆ2 , dsp pˆ2 )) D

= ωs(p) (dsp pˆ1 + Ls(p) ξ1 , dsp pˆ2 + Ls(p) ξ2 ) − µ(s(p)), θ−1 Rm (m ˆ 1, m ˆ 2 )θ D

= ωs(p) (∇sp m ˆ 1 , ∇sp m ˆ 2 ) − µ(s(p)), θ−1 Rm (m ˆ 1, m ˆ 2 )θ

E

E

where in the last line m ˆ i = dπp p. ˆ Notice that this 2-form in fact depends only from m = π (p) and m ˆ i ∈ Tm M , and not from the choice of frame.  We are finally set up to prove the main result of this section. Theorem 4.24 The map µ : X −→ Ω2 (M ) defined by the expression (4.1) is independent of the choice of the connection ∇, equivariant under the action of G ex , takes values in the space of closed 2-forms Ω2cl (M ), and satisfies the moment map equation. Remark 4.25 Notice that we do not claim that µ is a moment map for the action of G ex on X , as in general we cannot expect it to land in (Lie G ex )∗ = d(Ω1 (M )). This will not be a problem in the application that we will consider, as we will see. In general, the problem can be solved by either restricting to the path components of X where µ(s) is exact, or by considering the whole X but substituting µ by µ(s) − α[s] , where [α[s] ] is an element in the deRham cohomology class of µ(s) (kept fixed over components [s] ∈ π0 (X )). In this last case, however, the resulting moment map will not be equivariant. As the proof of the theorem is very long, we subdivide it into four lemmas. Lemma 4.26 The map µ is independent of the choice of connection. Proof To show independence from ∇, we will need to work in coordinates. Fix a local basis {ei }i=1,2 of T Σ and let {ei }i=1,2 be the dual basis. Then we can write every vector field V : Σ → T Σ as V (m) = V i (m)ei for some smooth functions V i : Σ → R (where we sum implicitly over repeated indices), and similarly if α ∈ Ω1 (Σ) and Φ ∈ End(T Σ) we have αm = αi (m)ei ,

Φm ei = Φji (m)ej , 63

4. Actions of groups of diffeomorphisms and so on. In what follows, we will omit the point of evaluation and write simply V i , αi , Φji , . . . for V i (m), αi (m), Φji (m), . . . We also write V;li , αi;l , Φji;l , . . . for the components of ∇el V , ∇el α, ∇el Φ, . . . Let ∇0 be another connection. Then ∇0 = ∇ + γ, where γ ∈ Ω1 (End(T Σ)). Identifying End(T Σ) with T Σ ⊗ T ∗ Σ, we write i γ = γjk (ei ⊗ ej ⊗ ek ),

meaning that i γ (ej )ek = γjk ei . i = γi . As ∇ and ∇0 are torsion free, we have γjk kj The variation in the Riemann curvature tensor is given by

R 0 ( ei , ej ) ek =

= ∇0ei ∇0ej ek − ∇0ej ∇0ei ek − ∇0[ei ,ej ] ek 



= R(ei , ej )ek + ∇ei (γ (ej )) − ∇ej (γ (ei )) − γ ([ei , ej ]) + [γ (ei ), γ (ej )] ek 



= R(ei , ej )ek + (∇ei γ )(ej ) − (∇ej γ )(ei ) + [γ (ei ), γ (ej )] ek 1 = R(ei , ej )ek + dγ (ei , ej ) + [γ ∧ γ ](ei , ej ) ek , 2 



where in the second line we inserted our expression for ∇0 and regrouped some terms, and in the third line we used the fact that ∇ei (γ (ej )) = (∇ei γ )(ej ) + γ (∇ei ej ), together with the fact that the connection is torsion free. Thus in coordinates l l l t l t (R0 )lijk = Rijk + γjk;i − γik;j + γitl γjk − γjt γik .

Next, we look at the map µs as a section µs : Σ −→ End(T Σ)∗ , acting on the traceless part of the endomorphisms. Using our basis, we identify End(T Σ)∗ with T ∗ Σ ⊗ T Σ, where the action is given by contracting the corresponding slots (basically, this is the trace pairing of remark 4.22), ∗ that i j ∗ k l is, if Φ = Φj (ei ⊗ e ) ∈ End(T Σ) and Ψ = Ψl (e ⊗ ek ) ∈ End(T Σ) , then hΨ∗ , Φi = Ψji Φij . 64

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms This way, we write µs in components as µji (ei ⊗ ej ). Now it is easy to find the variation of the term hµs , Ri. It is given by 1 hµs , R0 i =hµs , Ri + hµs , dγ + [γ ∧ γ ]i 2   k l l t l t =hµs , Ri + µl γjk;i − γik;j + γitl γjk − γjt γik (ei ⊗ ej ). To find the variation in c(∇µs ), notice first that

(∇ei ej )(ek ) + ej (∇ei ek ) =∇ei (ej (ek )) =∇0ei (ej (ek )) =(∇0ei ej )(ek ) + ej (∇0ei ek ) =(∇0ei ej )(ek ) + ej (∇ei ek ) + ej (γ (ei )ek ) where we used the fact that all connections act the same way on functions, and thus j k ∇0ei ej − ∇ei ej = −γik e . Now ∇0ei (µkj (ej ⊗ ek )) =(∇0ei µkj )(ej ⊗ ek ) + µkj ((∇0ei ej ) ⊗ ek ) + µkj (ej ⊗ ∇0ei ek ) k =∇ei (µkj (ej ⊗ ek )) + (µjl γij − µkj γilj )(el ⊗ ek ),

so that k − µkj γilj )(el ⊗ ek ⊗ ei ) ∇0 µs = ∇µs + (µjl γij

and after contraction we obtain k l c(∇0 µs ) = c(∇µs ) + (µli γkl − µkl γki )ei . k = γ k = 0 (where we used torsion freedom for the first equality But since γkl lk and the fact that both ∇ and ∇0 come from a principal connection in the second), we get l i c(∇0 µs ) = c(∇µs ) − µkl γki e.

Applying the exterior derivative to a 1-form αi ei gives dα(ei , ej ) =∇ei (α(ej )) − ∇ej (α(ei ))

=αj;i − αi;j , so that dα = (αj;i − αi;j )(ei ⊗ ej ). Applying this to our case, we obtain the total variation of the third term of the moment map: 



l l l l ) ( ei ⊗ ej ) . dc(∇0 µs ) − dc(∇µs ) = − µkl;j γki − µkl;i γkj + µkl (γki;j − γkj;i

65

4. Actions of groups of diffeomorphisms Putting this together with what we did before, we notice that there is already some cancellation, and the total variation for hµs , Ri + d(c(∇µs )) is given by 



t l t l l − µkl (γitl γjk − γjt γik ) + µkl;j γki − µkl;i γkj (ei ⊗ ej ).

We are left with the task of finding the variation of ωs (∇s, ∇s). Let m ˆ = [δ ], where [δ ] is an equivalence class of paths with δ (0) = m, take (m, θ ) ∈ π −1 (m). Take two lifts (δ (t), θ (t)), (δ (t), θ˜(t)) of δ in F (M ) with θ (0) = θ˜(0) = θ and such that ∇t θ = 0, ∇0t θ˜ = 0. Then 0 =∇0t θ˜ =∇t θ˜ + γ (m ˆ )θ, and thus

(∇0 s − ∇s)(m ˆ ) =ds[δ (t), θ (t)] − ds[δ (t), θ˜(t)] =ds(m, ˆ ∇t θ˜) − ds(m, ˆ ∇t θ ) 0˜ =ds(m, ˆ ∇t θ −γ (m ˆ )θ ) − ds(m, ˆ 0) |{z} =0

=ds(m, ˆ 0) − ds(0, γ (m ˆ )θ ) − ds(m, ˆ 0) = − ds(0, γ (m ˆ )θ ) where we used the identification of the tangent space given by ∇. It follows that for mˆ 1 , mˆ 2 ∈ Tm M we have ωs (∇0 s, ∇0 s)(m ˆ 1, m ˆ 2 ) − ωs (∇s, ∇s)m (m ˆ 1, m ˆ 2) =

= −ωs (ds(0, γ (m ˆ 1 )θ ), ds(m ˆ 2 , 0)) + ωs (ds(0, γ (m ˆ 2 )θ ), ds(m ˆ 1 , 0))+ ˜ ˜ + ωs (ds(0, δ (m ˆ 1 )θ ), ds(0, δ (m ˆ 2 )θ )) where we left away all the basepoints to ease notation. Using ds(0, γ (m ˆ )θ ) = −Ls (θγ (m ˆ )θ ) and the defining property of the moment map µ : X → sl(n)∗ , we have that the above equals the expression D

E

D

E

D

E

dµs (m ˆ 2 ), γ (m ˆ 1 ) − dµs (m ˆ 1 ), γ (m ˆ 2 ) + µs , [ γ ( m ˆ 1 ), γ (m ˆ 2 )] .

In coordinates, this reads l l t l t − µkl;i γjk + µkl (γitl γjk − γjt γik ))(ei ⊗ ej ). (µkl;j γik

This is exactly what we needed to eliminate the contribution of the other two terms, showing independence from the choice of ∇.  66

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms Lemma 4.27 For any s ∈ X , µ(s) is a closed 2-form. Proof The fact that the image of any s ∈ X is closed follows easily from proposition 4.10. Indeed, d(µ(s)) =d (s∗ ω + dc(∇µs ))

=s∗ (dω ) + d2 c(∇µs ) =0 since dω = 0 and d2 = 0.



Lemma 4.28 The map µ is equivariant under the action of G ex . Proof Let φ ∈ G ex , then µ(φ∗ s) = φ˜ ∗ s∗ ω + dc(∇µφ˜ ∗ s ). For the second term, let E ∈ End0 (T M ), then we have hµφ∗ s , Ei(m) =hµφ∗ s (m, θ ), θ−1 Eθi

=hµs (φ(m), dφm ◦ θ ), (dφm ◦ θ )−1 dφm E (dφm )−1 (dφm ◦ θ )i =hµs , φ∗ Ei(φ(m)). But we have φ∗ (me ˆ ∗i )0 = [(dφm m ˆ )(e∗i (dφm )−1 )]0 , so that taking φ∗ e∗i = e∗i (dφm )−1 as a basis for Tφ∗(m) M the above gives c(∇µφ∗ s ) = φ∗ c(∇µs ), and thus µ ( φ∗ s ) = φ∗ µ ( s ) ,



proving equivariance. Lemma 4.29 The map µ satisfies the moment map equation.

Proof We write the equation explicitly. Let s ∈ X , sˆ ∈ Ts X be the derivative at t = 0 of a path st ∈ T X , and V ∈ Lie G ex The moment map equation reads D

d(µ(s))s, ˆ V

E

=

Z M





d ! µ(st ) ∧ αV = Ωs (LV s, sˆ), dt t=0 

(4.3)

where dαV = ιV ρ. We start from the right hand side. We have Ωs (LV s, sˆ) =

Z

ωs (s, ˆ ds ◦ V˜ )ρ.

M

67

4. Actions of groups of diffeomorphisms Now, if the 1-form ζs,sˆ = ωs (s, ˆ ds ◦ −) ∈ Ω1 (F (M )) had the property that for any vector field Y ∈ Γ(T F (M )) the map ζs,sˆ (Y ) ∈ C ∞ (F (M )) descended to a map in C ∞ (M ), then we would be almost done, as what we will do below will show. However, it is in general not the case, so we correct it by taking away the vertical, non-invariant part: ωs (s, ˆ ds ◦ −) = !

ˆ A(−)i, = ωs (s, ˆ ds ◦ −) − hdµ(s)s, ˆ A(−)i +hdµ(s)s, |

{z

}

=:βs,sˆ

where A ∈ Ω1 (F (M ); sl(n)) is the principal connection associated to ∇. The 1-form βs,sˆ has the property we stated above. We have Z M

(1) βs,sˆ (V˜ )ρ =

Z

(2)

ZM

(3)

Z

=

M

=

M

βs,sˆ ∧ dαV dβs,sˆ ∧ αV 



d s˜t ∗ ω ∧ αV , dt t=0 

where we explain the various steps, as well as the definition of s˜t , below: 1. Consider the (n + 1)-form βs,sˆ ∧ π ∗ ρ ∈ Ωn+1 (F (M )). A priori, it eats n + 1 vector fields on F (M ) and spits out an element of C ∞ (F (M )), but in fact, by the property of βs,sˆ , it descends to an element Ωn+1 (M ), so that by a dimension argument we have βs,sˆ ∧ π ∗ ρ = 0 and 0 =ιV˜ (βs,sˆ ∧ π ∗ ρ) =βs,sˆ (V˜ )π ∗ ρ − βs,sˆ ∧ π ∗ ιV ρ

=βs,sˆ (V˜ )π ∗ ρ − βs,sˆ ∧ π ∗ dαV , which proves the first step (after descending to M ). 2. We have d(βs,sˆ ∧ αV ) = dβs,sˆ ∧ αV − βs,sˆ ∧ dαV , and the step follows by applying Stokes’ theorem. 3. We proceed in a way similar to what we did in lemma 4.5. To any map f : F (M ) → X we can associate the graph f˜ : F (M ) → F (M ) × X given by f˜(m, θ ) = (m, θ, f (m, θ )). To our sˆ ∈ Ts X we associate its flow st : F (M ) → X, and define

W = 68

d s˜t : F (M ) → T (F (M ) × X ). dt t=0

4.2. SL(n)-frames and exact volume-preserving diffeomorphisms It is a vector field along s, ˜ that is W (m, θ ) ∈ Ts˜(m,θ ) (F (M ) × X ), and we have d prX ◦W =s, ˆ d prF (M ) ◦W =0. As W is defined along s, ˜ which is injective, we can extend W to a vector field W ∈ Γ(T (F (M ) × X )) such that W ◦ s˜ = W . It follows that the flow ϕtW extends s˜t : t ϕW ◦ s˜ = s˜t . We are now set to prove the third step. We have



d d s˜t ∗ ω = s˜t ∗ pr∗X ω − d dt t=0 dt t=0





d s˜t αA , dt t=0 

where we have used the form of ω given in lemma 4.9. The second term gives



E d d D A ∗ ∗ s ˜ α = µ ◦ pr ◦ s ˜ , s ˜ pr A t t t X F (M ) dt t=0 dt t=0 D E d = µ ◦ st , A dt t=0 =hdµ(s)s, ˆ A(−)i,

i.e. the second piece of βs,sˆ , while the first term is



d d s˜t ∗ pr∗X ω = (ϕt ◦ s˜)∗ pr∗X ω dt t=0 dt t=0 W d = s˜∗ (ϕtW )∗ pr∗X ω dt t=0 =s˜∗ LW (pr∗X ω )

=s˜∗ dιW pr∗X ω =d(s˜∗ ιW pr∗X ω ) where in the fourth line we used Cartan’s magic formula together with the fact that ω is closed. To conclude, we get 

(s˜∗ ιW pr∗X ω )(V ) =ωprX s˜ (d prX W ) ◦ s, d prX dsV ˜



=ωs (s, ˆ ds · V ) where we noticed that (d prX W ) ◦ s = d prX W = s. ˆ 69

4. Actions of groups of diffeomorphisms This gives the first half of the moment map equation. We are left with the correction term hdµ(s)s, ˆ A(−)i to treat. We claim that Z

hdµ(s)s, ˆ A(V˜ )iρ = −

Z

M

d(c(∇µs )) ∧ αV .

(4.4)

M

First of all, notice that A(V˜ ) corresponds to the section of End(T M ) given by ∇V . In order to proceed we have to fix a local basis {ei }ni=1 of T M (as c(∇µs ) is defined in coordinates). Choose it so that we have i. dιei ρ = 0, and ii.

Pn

∗ i = 1 ∇ei e i

= 0, where {e∗i }ni=1 is the dual basis.

We define the function fV =hµs , ∇V i + c(∇µs )(V ) n D X

µs , ((∇ei V

=

i=1 n X

=

i=1 n X

=

D

Lei µs , (V

)e∗i )0

e∗i )0

D

Lei µs , (V e∗i )0

E

E

D

+ ∇ei µs , (V

e∗i )0

! E

!

D

− µs , ( V

(∇ei e∗i ))0

E

i=1

where in the last step we brought the sum inside the second term and used property ii. of the local basis we chose. We denote D

E

fi = µs , (V e∗i )0 . As fV is linear in V , we can assume without loss of generality that V is supported in the chart where the local basis is defined (take a partition of unity subordinated to a finite atlas...) Thus Z

fV ρ =

Z

M

M

=

Z

n X i=1 n X

M i=1

!

Lei fi ρ 

ιei (df ∧ ρ) + dfi ∧ ιei ρ

=0 where we used Cartan’s formula in the second line, the fact that ρ is topdimensional, and property ii. of the local basis in the last step. 70

4.3. Teichm¨ uller space and the Weil-Petersson metric Thus we have Z M

hµst , ∇V iρ = −

= =

Z

Z ZM M

M

c(∇µst )(V )ρ

c(∇µst ) ∧ dαV dc(∇µst ) ∧ αV .

Now differentiate with respect to t. Putting together what we did on βs,sˆ and equation (4.4) we obtain exactly the moment map equation (4.3). 

4.3

Teichm¨ uller space and the Weil-Petersson metric

We will now apply the general theory above to a special case, rederiving the Teichm¨ uller space of complex structures on a fixed Riemannian surface and the Weil-Petersson metric on it, as Donaldson does in [Donaldson, 2003]. For a remainder of what the Teichm¨ uller space is, refer to appendix C.

4.3.1

The Siegel upper half space

The following space was introduced by Siegel in his paper [Siegel, 1939]. It generalizes the upper half space, and we can define on it an action reducing to the M¨ obius transformations in the case n = 1. Definition 4.30 The Siegel upper half space Sn is the space Sn = {X + iY ∈ L(Cn , Cn ) X, Y ∈ L(Rn , Rn ) symm., Y pos. def.} . ∼ H the usual upper We will only need the case n = 1, where we have Sn = half plane, so we will state the results for the general case, but prove them only in this special case. For a more complete treatment, refer to Siegel’s original paper and to the book [McDuff and Salamon, 2015, p. 71]. The paper [Ohsawa, 2015] also offers a complete covering of the topic, furthermore presenting the Siegel upper half space as the Marsden-Weinstein quotient of 2 T ∗ R2n by the action of O (2n) by multiplication on the right. The group Sp(n) of symplectic (2n × 2n)-matrices with real entries acts on Sn as follows: let Z ∈ Sn and Ψ=

A B C D

!

∈ Sp(n),

then Ψ∗ Z = (AZ + B )(CZ + D )−1 . This action obviously corresponds to the usual M¨obius transformations in the special case n = 1. 71

4. Actions of groups of diffeomorphisms Lemma 4.31 This action is well-defined, holomorphic, transitive and the isotropy subgroup of i1 ∈ Sn is U (n) ⊂ Sp(n). In particular, we have an isomorphism ∼ Sp(n)/U (n). Sn = Proof The fact that the action is well-defined (in the sense that it sends elements of S1 to other elements of S1 ) is a straightforward calculation. It is holomorphic as it is given by rational functions, while transitivity is given by the easily checked formula 

1



x

y2

1 y2

Ψ=

1

y− 2

0

,

Ψ∗ i = x + iy.

Now consider the action of Sp(1) on i. It is given by ai + b ci + d ac + bd 1 = 2 +i 2 2 c +d c + d2

Ψ∗ i =

where we used that Sp(1) = SL(2). It is easy to see that Ψ∗ i = i if, and only if Ψ ∈ SO (2) = U (1).  Thus we have the identification H = SL(2)/SO (2). Notice that the inverse of the isomorphism of lemma 4.31 is given by Φ :

H

−→ " SL(2)/SO (!# 2) 1 1 − y 2 xy 2 x + iy 7−→ . 1 0 y− 2

Furthermore, it can be shown that Sn is isomorphic to the space of ω0 compatible complex structures on R2n . Lemma 4.32 The map j : Sn → J (R2n , ω0 ) given by Z 7−→ j (Z ) =

XY −1 −Y − XY −1 X Y −1 −Y −1 X

!

is the unique diffeomorphism such that j (i1) = J0 , and j (Ψ∗ Z ) = Ψj (Z )Ψ−1 . 72

4.3. Teichm¨ uller space and the Weil-Petersson metric Proof In the case n = 1 the formula is given by 1 j (x + iy ) = y

!

x − ( x2 + y 2 ) . 1 −x

It is an easy computation to show that j (z )2 = −1, j (i) = J0 and that it is equivariant with respect to the action of Sp(2). The proof that this map is the unique bijection with those properties can be found in [McDuff and Salamon, 2015, Sect. 2.5], and the fact that it is a diffeomorphism follows from the formula.  The induced SL(2)-equivariant isomorphism SL(2)/SO (2) → J (R2 , ω0 ) is given as follows: Let C ∈ SL(2)/SO (2), take one of the two representatives A ∈ C that are of the form ! ∗ ∗ A= 0 ∗ (there are exactly two, given by A and −A). Then we map C 7−→ AJ0 A−1 . The K¨ ahler structure on the Siegel upper half space The tangent space of the Siegel upper half space is easily seen to be given by TX +iY Sn = {B ∈ L(Cn , Cn )|B symmetric}. We have the obvious complex structure on Sn given by i1 ∈ L(Cn , Cn ). Moreover, it is possible to define a Hermitian metric on Sn that in our case of interest n = 1 reduces to dz ⊗ dz hx+iy = . y2 This gives a Riemannian metric and a symplectic structure by taking the real and imaginary parts respectively, that is  1 dx ⊗ dx + dy ⊗ dy , 2 y dx ∧ dy =− . y2

gH = ωH

This metric is exactly the hyperbolic metric on the upper half plane. For the general case, see for example [Siegel, 1943] or [Ohsawa, 2015]. We will now express these objects explicitly in the identification of ∼ SL(2)/SO (2). H= 73

4. Actions of groups of diffeomorphisms We have the identification of TA (SL(2)/SO (2)) for A ∈ SL(2)/SO (2) with sl(2)/so(2) by left multiplication by A−1 . Let !

σd =

1 0 , 0 −1

!

0 0 , 1 0

σl =

σr =

0 1 0 0

!

span sl(2), then in sl(2)/so(2) we have σl = σr , and thus we can see it as the Lie algebra spanned by σd and σr . d Let ∂x = dt (x + t) + iy ∈ Tx+iy H. It is mapped to t=0

Φ(x + iy )

−1



1 d Φ((x + t) + iy ) = σr ∈ sl(2)/so(2). dt t=0 y

Similarly, ∂y 7−→

1 σd . 2y

It follows that we have dx 7−→ yσr∗ , dy 7−→ 2yσd∗ , and thus g H =σr∗ ⊗ σr∗ + 4σd ⊗ σd , ω H = − 2σr∗ ∧ σd∗ . The action of SL(2) and the moment map The group SL(2) acts on ∼ SL(2)/SO (2) by left multiplication. Notice that the identification is H = equivariant under this action, that is AΦ(x + iy ) = Φ(A∗ (x + iy )) for any A ∈ SL(2). The action is Hamiltonian. To find the moment map, we compute the infinitesimal action. Let ξ=

a b c −a

!

∈ sl(2).

Then

d Xξ (x + iy ) =Φ(x + iy ) exp(tξ )Φ(x + iy ) dt t=0 =Φ(x + iy )−1 ξΦ(x + iy ) −1

!

=(a − cx)σd +

74

b + 2ax − cx2 + cy σr ∈ sl(2)/so(2) y

4.3. Teichm¨ uller space and the Weil-Petersson metric and "

ιXξ ω = − 2

!

b + 2ax − cx2 + cy σd∗ − (a − cx)σr∗ y

#

#

"

a − cx 2ax − cx2 b =2 dy dx − 2 + c + y y y2 2ax − cx2 + b =d − cy y *

=d

2x ∗ 1 ∗ σ + σ − y d y r

!

!

+

x2 + y σl∗ , ξ . y

Therefore, the moment map µ : H → sl(2)∗ is 2x ∗ 1 ∗ µ(x + iy ) = σ + σ − y d y r

!

x2 + y σl∗ . y

An important observation is the fact that hµ(z ), ξi = tr(j (z )ξ ).

4.3.2

(4.5)

Preliminaries

From now on, we will use the same notation as in section 4.2 with the following objects: • M = Σ is a closed, oriented surface of genus gΣ ≥ 2 and a fixed volume form ρ ∈ Ω2 (Σ). • X = H is the hyperbolic plane with its K¨ahler structure defined as in the last subsection. • X = F (Σ) ×SL(2) H and X is the space of sections of X seen as a bundle ∼ J (Σ, ρ) in a canonical way. over Σ. We’ll shortly see that X = • G ex is the group of exact volume-preserving diffeomorphisms of Σ, and its Lie algebra is given by the exact divergence-free vector fields. By the identification we stated in the previous point, these vector fields correspond exactly to the Hamiltonian vector fields, and thus G ex = Ham(Σ, ρ), where we look at ρ as a symplectic form on Σ. As X encodes the space of complex structures on R2 , the elements of X can be identified with the complex structures on Σ compatible with the orientation (and thus with ρ) as follows: let s : F (Σ) → X be SL(2)-equivariant. Then the associated complex structure is given by Js (m) = θj (s(m, θ ))θ−1 ∈ Aut(T M ) 75

4. Actions of groups of diffeomorphisms for any (m, θ ) ∈ π −1 (m), where s(m, θ ) : R2 → R2 is such that s(m, θ )2 = − idR2 . By equivariance, this is well-defined, and we have Js (m)2 = θj (s(m, θ ))2 θ−1 = − idTm M . Since we are working on a surface, Js is an honest complex structure, and not just an almost complex structure (see lemma A.20). From now on, we will ∼ J (Σ, ρ). usually implicitly make the identification X = Since we are in a special case of what we have treated in the last section, we have the symplectic form Ωs (sˆ1 , sˆ2 ) =

Z Σ

ωs (sˆ1 , sˆ2 )ρ

and the map µ(s) = ωs (∇s, ∇s) − hµs , Ri + dc(∇µs ) defined as in (4.1), satisfying the moment map equation. Moreover, we have an obvious almost complex structure on X , given by J sˆ = isˆ for sˆ ∈ T X , and a K¨ ahler metric Gs (sˆ1 , sˆ2 ) =

Z Σ

gsH (sˆ1 , sˆ2 )ρ.

Notice that the function we’re integrating is well defined on Σ (read: independent of θ since te action of SL(2) on H is Hamiltonian and the metric g H is K¨ ahler). We will now express this K¨ahler structure in the picture X = J (Σ, ρ). Lemma 4.33 Let s ∈ X and φ ∈ G ex . Then Jφ∗ s = φ∗ Js . Proof Let (m, θ ) ∈ F (Σ). Then Js (m) = θj (s(m, θ ))θ−1 . Thus, Jφ∗ s (m) =θj ((φ∗ s)(m, θ ))θ−1

=θj (s(φ(m), dφm ◦ θ ))θ−1 =(dφm )−1 (dφm ◦ θ )j (s(φ(m), dφm ◦ θ ))(dφm ◦ θ )−1 dφm =(dφ−1 )φ(m) Js (φ(m))dφm =(φ∗ Js )(m). 76



4.3. Teichm¨ uller space and the Weil-Petersson metric The expressions for the complex structure, the K¨ahler metric and the symplectic form are as follows. Lemma 4.34 The complex structure acts on J (Σ, ρ) by J (Jˆs ) = −Js ◦ Jˆs = Jˆs ◦ Js for Jˆs ∈ TJs J (Σ, ρ). Proof We have J (Jˆs ) = θ (djs (isˆ))θ−1 . The differential of the map j : F (Σ) → J (Σ, ρ) can be easily computed and is given by 1 y

djz ∂x =

− 2x y , 0 − y1

djz ∂y =

− yx2 − y12

!

x2 y2

!

−1 x y2

,

where z = x + iy ∈ H. Using the fact that i∂x = ∂y and i∂y = −∂x and checking the relations between djz (i∂x ) and djz (i∂x ) and the same for ∂y , we see that djz (izˆ ) = −j (z )djz (zˆ ).



The result follows. The second equality in the proof is given by

0=

d J 2 = Jˆs Js + Js Jˆs . dt t=0 st

Lemma 4.35 The complex structure respects the action, i.e. J (φ∗ Jˆs ) = φ∗ (J (Jˆs )) for Jˆs ∈ TJs J (Σ, ρ). Proof Using the last lemma twice, we have J (φ∗ Jˆs ) =(φ∗ Jˆs ) ◦ Jφ∗ s =(φ∗ Jˆs ) ◦ (φ∗ Js )

=φ∗ (Jˆs ◦ Js ) =φ∗ (J (Jˆs )).

 77

4. Actions of groups of diffeomorphisms To write the metric in terms of J (Σ, ρ), first notice that tr((djz (∂k ))(djz (∂l ))) =

2 δkl y2

for k, l ∈ {x, y} and z = x + iy. Thus, we have gsH(m,θ ) (sˆ1 (m, θ ), sˆ2 (m, θ )) =

1 tr(Jˆ1 (m)Jˆ2 (m)) 2

where Jˆk is the element of TJs J (Σ, ρ) associated to sˆk . Therefore, 1 GJ (Jˆ1 , Jˆ2 ) = 2

Z Σ

tr(Jˆ1 Jˆ2 )ρ.

These objects given a K¨ ahler structure on X , indeed it is easy to see that Ω(−, J−) = G(−, −). In particular, we have 1 ΩJs (Jˆ1 , Jˆ2 ) = − 2

4.3.3

Z Σ

tr(Jˆ1 Js Jˆ2 ).

The infinitesimal action

∼ J (Σ, ρ), we have the natural identification As X = ∼ TJ J (Σ, ρ) = Ω0,1 (Σ; T Σ) Ts X = s for s ∈ X . The isomorphism is given by

d θ ((j ◦ st )(m, θ ))θ−1 = θ (djs (sˆ)(m, θ ))θ−1 , Jˆs (m) = dt t=0 and the second equality comes from the fact that ˆ + J Jˆ = 0}, TJ J (Σ, ρ) = {Jˆ ∈ End(T M )|JJ and therefore is composed exactly by anti-holomorphic 1-forms with values in T Σ. Lemma 4.36 The infinitesimal action of V ∈ Lie G ex on J ∈ J (Σ, ρ) is given by −LV J = −2J∂J V , where ∇ is the Levi-Civita connection of the Riemannian metric associated to J, and ∂J V = 12 (∇V + J (∇J V )) is the Cauchy-Riemann operator. The minus sign is, as always, conventional for contravariant actions. 78

4.3. Teichm¨ uller space and the Weil-Petersson metric Proof Let W : Σ → T Σ. We have −(LV J )(W ) = − (LV (JW ) − JLV W )

= − ([V , JW ] + J [V , W ]) = − (∇V (JW ) − ∇JW V − J∇V W + J∇W V ) = − J (J∇JW V + ∇W V ) = − 2J (∂J V )(W ), where in the third line we used torsion freedom of ∇, and in the fourth line we used lemma A.24 to get ∇V (JW ) − J∇V W = 0.  Lemma 4.37 We have J (−LV J ) = −L−JV J. Proof We compute J (LV J ) = − J (−2J∂J V )

= − 2J∂J (−JV ) which is −L−JV J. In the last line we used the fact that ∂J commutes with J. 

4.3.4

The moment map

Since we fixed a volume form ρ ∈ Ω2 (Σ), any s ∈ X induces a Riemannian metric g s on Σ by s gm (m ˆ 1, m ˆ 2 ) = ρm (m ˆ 1 , Js (m)m ˆ 2)

and thus a (Levi-Civita) connection ∇s . Lemma 4.38 We have ωs (∇s s, ∇s s) = 0 and ∇s µs = 0. Proof Let (m(t), θ (t)) ∈ F (Σ) be any path, denote by (m, θ ) = (m(0), θ (0)) and (m, ˆ θˆ) = (m ˙ (0), ∇st θ (0)). Let s(t) = s(m(t), θ (t)) and write Js(t) = Js (m(t)) = θ (t)j (s(t))θ (t)−1 . Then we have Js(t) θ (t) = θ (t)j (s(t)). Differentiating both sides with respect to t and evaluating at t = 0 we get 



Js (m)θˆ = Js(t) ∇st θ (t)

=∇st

 t=0 Js(t) θ (t) t=0  



=∇st θ (t)j (s(t))

ˆ (s(m, θ )) + θdjs(m,θ ) s˙ (0) =θj 79

4. Actions of groups of diffeomorphisms where in the second line we used the fact that Js commutes with the LeviCivita connection ∇s of g s . Therefore, we have ˆ (s(m, θ )). djs(m,θ ) ds(m, ˆ θˆ) = j (s(m, θ ))θ−1 θˆ − θ−1 θj

(4.6)

As j : H → J (R2 , ω0 ) is an isomorphism, djs(m,θ ) also is an isomorphism, and it follows that if we take θˆ = 0 we have ∇s s(m,θ ) m ˆ = ds(m,θ ) (m, ˆ 0) = 0, proving the first statement. For the second statement, let Φ ∈ Ω0 (Σ; End0 (T Σ)) be any section of the traceless endomorphisms of T Σ. Then h∇s µs , Φi = dhµs , Φi − hµs , ∇s Φi. Let (m(t), θ (t)) ∈ F (Σ) be as before. We get dhµs , Φim ˆ = d = hµs (m(t), θ (t)), θ (t)−1 Φ(m(t))θ (t)i dt t=0   d = tr j (s(m(t), θ (t)))θ (t)−1 Φ(m(t))θ (t) dt t=0   d −1 −1 = tr (djs(m,θ) s˙ (0))θ Φ(m)θ + j (s(m, θ )) (θ (t) Φ(m(t))θ (t)) , dt t=0 where we used the description of the moment map µ : H → sl(s)∗ given in equation 4.5, and the fact that the trace is linear and thus commutes with the derivative. Notice that s˙ (0) = ∇s s(m, ˆ θˆ). Now we have

d 0 = (θ (t)−1 θ (t)) dt t=0   d −1 = θ (t) θ + θ−1 θˆ dt t=0 so that



d ˆ −1 , θ (t)−1 = −θ−1 θθ dt t=0

and

d ˆ −1 Φ(m)θ + θ−1 (∇mˆ Φ)θ + θ−1 Φ(m)θ. ˆ (θ (t)−1 Φ(m(t))θ (t)) = −θ−1 θθ dt t=0 Inserting this expression and equation (4.6) into the what we have found for dhµs , Φim, ˆ and choosing a path such that θˆ = 0 (which we can do, since hµs , Φi is independent of θ) we obtain dhµs , Φim ˆ = hµs , ∇smˆ Φi, showing that h∇s µs , Φi = 0. As Φ was arbitrary, we are done. 80



4.3. Teichm¨ uller space and the Weil-Petersson metric It follows (by independence of the moment map from the auxiliary connection) that µ(s) = hµs , Ri. Lemma 4.39 On Riemann surfaces we have R = KJ ⊗ ρ, where K is the Gaussian curvature. Proof Fix a local basis {e1 , e2 = Je1 } of T Σ. Then we have 



g ρ(e1 , e2 )KJe1 , e2 =Kρ(e1 , e2 )g (e1 , e2 )

= − Kρ(e1 , e2 )ρ(e1 , e2 ) g (R (e1 , e2 )e1 , e2 ) =− ρ ( e1 , e2 ) 2 det g where in the second line we used that g (Je1 , e2 ) = ρ(Je1 , Je2 ) = −ρ(e1 , e2 ) and in the third line we inserted the definition of K. But we have g (e1 , e1 ) g (e1 , e2 ) g (e2 , e1 ) g (e2 , e2 )

det g =

!

= − ρ(e1 , e2 )2 and thus the last line becomes simply g (R (e1 , e2 )e1 , e2 ). This concludes the proof.  This, together with the expression of equation (4.5), allows us to compute µ(s) =hµs , KJs iρ

=K tr(Js2 )ρ = − 2Kρ. As already said in remark 4.25, this map is not a moment map for the action. However, we obtain one by Ξ(s) = µ(s) − cρ,

(4.7)

where we fix c by asking that Ξ(s) vanishes on every 2-cycle (thus implying exactness of Ξ(s)). But the only 2-cycle in this case is given by the whole Σ itself, and we have !

0=

Z Σ

=−

Ξ (s)

Z Σ

(c + 2K )ρ

= − c volρ (Σ) − 8π (1 − gΣ ) 81

4. Actions of groups of diffeomorphisms where we used Gauss-Bonnet’s theorem in the third line. Therefore, c=

8π (gΣ − 1) >0 volρ (Σ)

is independent of s. As cρ is invariant under the action of G, it follows that Ξ is equivariant. We summarize what we have done in a lemma. Lemma 4.40 The map Ξ : X → (Lie G )∗ is an equivariant moment map for the action of G on X .

4.3.5

The Marsden-Weinstein quotient

We want now to take the Marsden-Weinstein quotient of X . The set Ξ−1 (0) = µ−1 (cρ) is given by the complex structures J ∈ J (Σ, ρ) such that the induced Riemannian metric g J (−, −) = ρ(−, J−) has constant Gaussian curvature K ≡ − 12 c. Without any loss of generality, we can rescale ρ in such a way that c = 2. Then we have Ξ−1 (0) = J−1 , where J−1 is the subset of J (Σ, ρ) consisting of complex structures inducing Riemannian metrics with constant curvature −1. We omit Σ and ρ in the notation, as they are fixed. Looking at ρ as a symplectic form on Σ, we obtain that X G ex = J−1 / Ham(Σ, ρ).

We will now study a bit this quotient. By the general theory of subsection 2.3.2, it has a naturally induced K¨ahler structure. We show directly what we proved in general in subsection 2.3.3, i.e. that the sequence L

dΞ

J J 0 −→ Lie G ex −→ TJ X −→ (Lie G ex )∗ −→ 0

is a cochain complex. Indeed, we have

dΞJ (LJ V ) =

d Ξ((ϕtV )∗ J ) dt t=0

and as

(ϕtV )∗ J = (d(ϕtV )−1 )ϕtV (J ◦ ϕtV )dϕtV we have ρ(−, ((ϕtV )∗ J )−) =ρ(d(ϕtV )−1 )ϕt dϕtV −, (d(ϕtV )−1 )ϕt (J ◦ ϕtV )dϕtV −) V

=ρ(dϕtV −, (J ◦ ϕtV )dϕtV −) =(ϕtV )∗ (ρ(−, J−)). 82

V

4.3. Teichm¨ uller space and the Weil-Petersson metric It follows that for the curvature we have t

K ( ϕV )

∗J

= K J ◦ ϕtV ≡ −1

as we are on the level set Ξ(s) = 0. Thus dΞJ (LJ V ) = 0 as we wanted. Lemma 4.41 The first map LJ = −2J∂J is injective. Proof Assume V : Σ → T Σ is an exact divergence free vector field such that LJ V = 0. This implies ∂J V = 0, so that V is holomorphic. In particular, either V vanishes everywhere, or it has only isolated zeros (by the identity theorem). Moreover, it is orientation-preserving, so that the index of every zero of V is strictly positive. But as the Euler characteristic of Σ is given by χ(Σ) = 2(1 − g ) < 0, this gives a contradiction to the Poincar´e-Hopf theorem.  Therefore, the Lie algebra of the stabilizer of J is always trivial, and thus the obtained moduli space is an orbifold. In fact, the action is free, so that the moduli space is really a manifold. A proof of the stronger statement that the action of D0 on J (Σ, ρ) is free can be found in [Tromba, 1992, Sect. 2.2].

4.3.6

How to get to Teichm¨ uller space

Our quotient X G ex is not yet Teichm¨ uller space. However, we have the following result. Lemma 4.42 The Teichm¨ uller space T (Σ) = J (Σ, ρ)/D0 is isomorphic to J−1 / Symp(Σ, ρ)0 via the natural map ζ : J−1 / Symp0 (Σ, ρ) −→ J (Σ, ρ)/D0 given by ζ [J ]T 7−→ [J ]T , where we use abuse of notation and use [−]T to denote the equivalence classes in both of those spaces. 83

4. Actions of groups of diffeomorphisms Proof The map is obviously well defined. We show that ζ is injective. Let J1 , J2 ∈ J−1 be such that ζ [J1 ]T = ζ [J2 ]T . This means that there exists a diffeomorphism f ∈ D0 such that f ∗ J2 = J1 . Now by the uniformization theorem (theorem A.27), for every complex structure J ∈ J (Σ) there exists a unique 2-form β ∈ Ω2 (Σ) such that g J (−, −) = β (−, J−) is a Riemannian metric with constant curvature −1. We have f ∗ g J (−, −) =ρf (df −, Jdf −)

=(f ∗ ρ)(−, (f ∗ J )−). Noticing that for both J1 and J2 the associated 2-form is given by ρ (as we are in J−1 ), we get that f ∗ ρ = ρ, and thus f must be a symplectomorphism. We are left to show that f is symplectically isotopic to the identity. We do this using Moser’s trick (the general theory is explained in [McDuff and Salamon, 2015, sect. 3.2]). Let ft be a path linking idΣ to f in D. We define ρt,s = sft∗ ρ + (1 − s)ρ. We would like to have Ft,s such that Ft,s ρt,s = ρ and Ft,0 = idΣ . Assume Ft,s is given as the flow of some vector fields ξt,s , that is d Ft,s = ξt,s ◦ Ft,s . ds Notice that

d ρt,s = ρ − ft∗ ρ ds is exact, as ft are isotopic to the identity (and therefore ρ and ft∗ ρ are in the same cohomology class). Denote its primitive by dλt . Now we have 0=

d (F ∗ ρt,s ) ds t,s

d Lξt,s ρt,s + ρt,s ds

∗ =Ft,s







∗ =Ft,s dιξt,s ρt,s + dλt .

Notice that ρt,s is non-degenerate, as the space of volume forms defining the same orientation is convex, and thus the convex combination ρt,s is again a volume form. Therefore, we can solve the equation ιξt,s ρt,s + λt = 0. this gives us the vector fields ξt,s , and thus Ft,s by taking the flow, with in particular ξ0,s = ξ1,s = 0. Let Ht,s = ft ◦ Ft,s , then the path Ht,1 gives us the desired symplectic isotopy. For surjectivity, let J ∈ J (Σ, ρ). To have J ∈ J−1 we would need that 84

4.3. Teichm¨ uller space and the Weil-Petersson metric curvature of g J = ρ(−, J−) to be −1. Let β ∈ Ω2 (Σ) be the unique 2-form such that β (−, J−) is a Riemannian metric. We use Moser’s trick again. Let ρt = tβ + (1 − t)ρ. Then d ρt = β − ρ. dt As the metric g s has constant curvature −1 (and β is its associated volume form), Gauss-Bonnet’s theorem gives us volβ (Σ) = 4π (g − Σ − 1) = volρ (Σ), and thus the above 2-form is exact (as two top-dimensional forms are in the same cohomology class if, and only if the integrate to the same value). Applying the same methods as before, we obtain diffeomorphisms ψt of Σ such that ψ0 = idΣ and ψt∗ ρt = ρ. In particular, ψ1∗ β = ρ. Then ψ1∗ J is in the image of ζ, completing the proof.  Thus, we have a natural map ∼ T (Σ) X G ex = J−1 / Ham(Σ, ρ) −→ J−1 / Symp0 (Σ, ρ) = that makes X G ex into a Symp0 (Σ, ρ)/ Ham(Σ, ρ)-principal bundle. But we have the standard result that ∼ H 1 (Σ)/ΓΣ , Symp0 (Σ, ρ)/ Ham(Σ, ρ) = where ΓΣ is the flux group of Σ (see [McDuff and Salamon, 2015, Sect. 10.2]). Moreover, since gΣ ≥ 2 we have ΓΣ = 0 (see [Kedra, 1999]). We will now proceed to show that we can quotient X G by the action of this group to obtain a K¨ ahler structure on Teichm¨ uller space. To be more precise, we will show that the orbits of J−1 / Ham(Σ, ρ) under the action of Symp0 (Σ, ρ)/ Ham(Σ, ρ) are complex submanifolds. The action is obviously free. It will follow that the quotient by the action of the group is again a K¨ahler manifold. Thus we get a K¨ ahler structure on Teichm¨ uller space. Lemma 4.43 The orbits of J−1 / Ham(Σ, ρ) under the action of Symp0 (Σ, ρ)/ Ham(Σ, ρ) are complex submanifolds. Proof The action of Ham(Σ, ρ) on X extends to a well defined action of Symp0 (Σ, ρ) with exactly the same formulas. In particular for V ∈ Lie(Symp(Σ, ρ)) 85

4. Actions of groups of diffeomorphisms (i.e. a divercence-free vector field, not necessarily exact) we have LJ V = −2J∂J V ∈ Ω0,1 (Σ, T Σ), where J ∈ X , the induced infinitesimal action on the quotient is given by L[J ] [V ] = [LJ V ] for [J ] ∈ X G ex and V ∈ Lie(Symp0 (Σ, ρ))/ Lie(Ham(Σ, ρ)) (which is independent from the choice of representatives s and V ), and J [LJ V ] = [LJ (−JV )] as before. Since a vector is tangent to the orbits if, and only if it is of the form L[J ] [V ], we are done.  Now we have the following general results. Lemma 4.44 Let (X, ω ) be a symplectic manifold, and let G be a Lie group acting symplectically on X on the left. Assume that the action of G is free and proper, and that the orbits of X under the action are symplectic submanifolds. Then the quotient X/G is again a symplectic manifold in a natural way. Proof By the assumptions, the quotient X/G is a manifold. Consider the distribution H ⊆ T X given by 

Hx = Tx (Gx)

ω

.

Then Tx X = Hx ⊕ Tx (Gx). Indeed, if (V , ω ) is a symplectic vector space and W ⊂ V is a symplectic subspace, we have by definition W ω ∩ W = {0}. Consider the map φ : V → ∼ im(φ) via ω. W ∗ given by φ(v ) = ω (v, −)|W . Then W ω = ker(φ) and W = Thus dim(V ) = dim(ker(φ)) + dim(im(φ)) = dim(W ω ) + dim(W ) as we wanted. Therefore, we have an isomorphism ψx

0 T[x] (X/G) −→ H x0

for any x0 ∈ [x] = Gx, satisfying ψgx0 = gψx0 . We define the symplectic form on X/G by ω[x] ([xˆ 1 ], [xˆ 2 ]) = ωx0 (ψm0 [xˆ 1 ], ψx0 [xˆ 2 ]). It is independent of the choice of x0 ∈ [x] by equivariance of ψ and the fact that the action is symplectic. Since ω is closed, it follows that it is closed, and it is non-degenerate since ω|H is non-degenerate (if W ⊆ V is symplectic, then W ω is symplectic).  86

4.3. Teichm¨ uller space and the Weil-Petersson metric Corollary 4.45 Let (X, ω, J ) be a K¨ ahler manifold, and let G be a Lie group acting symplectically on X on the left. Assume that the action of G is free and proper, and that the orbits of X under the action are complex submanifolds. Then the quotient X/G is again a K¨ ahler manifold in a natural way. Proof As complex submanifolds of a K¨ahler manifold are symplectic, it follows by the lemma that X/G is symplectic. Then (using the same notation as in the last proof) J [xˆ ] = [J xˆ ] gives a compatible complex structure, finishing the proof.



Therefore, it follows that J−1 / Ham(Σ, ρ) ∼ J−1 / Symp (Σ, ρ) = ∼ T (Σ) = 0 Symp0 (Σ, ρ)/ Ham(Σ, ρ) is a K¨ahler manifold. As it will be the main object of interest in the next subsection, we give a more explicit description of the K¨ahler metric. Let [Jˆ]T ∈ T[J ]T T (Σ). By the proof of lemma 4.44, for any [J ] ∈ J−1 / Ham(Σ, ρ) there exist a canonical horizontal lift [Jˆ] of [Jˆ]T in T[J ] J−1 / Ham(Σ, ρ), given by the unique element satisfying 0 = Ω[J ] ([Jˆ], [Jˆ0 ]) = G[J ] ([Jˆ], J [Jˆ0 ]) for all [Jˆ0 ] tangent to the orbit of the group H 1 (Σ) acting on our space. But as every such element can be written as [LX J ] for some symplectic vector field X, and as furthermore the orbits are complex submanifolds, the condition above is equivalent to G[J ] ([Jˆ], [LX J ]) = 0 for all symplectic vector fields X on Σ. The induced metric on Teichm¨ uller space (again denoted by G) is then given by applying the metric of our original space J−1 / Ham(Σ, ρ) to the horizontal lifts of the tangent vectors. We can also compute the dimension of J−1 / Ham(Σ, ρ). Lemma 4.46 We have J−1 dim Symp0 (Σ, ρ) 



J (Σ, ρ) = dim D0 



= 6 (gΣ − 1 ).

It follows also that J−1 dim Ham(Σ, ρ) 



= 8gΣ − 6.

Proof By lemma 4.42 we have an isomorphism 

T[J ]T

J−1 Symp0 (Σ, ρ)



∼T = [J ]T



J (Σ, ρ) D0



=

Ω0,1 (Σ; T Σ) , −2J∂J (Ω0 (Σ; T Σ)) 87

4. Actions of groups of diffeomorphisms as TJ J (Σ, ρ) = Ω0,1 (Σ; T Σ), and the infinitesimal action of Lie D0 = Ω0 (Σ; T Σ) is as always given by the map −2J∂J . Therefore, dim

Ω0,1 (Σ; T Σ) −2J∂J (Ω0 (Σ; T Σ))

!

= dim coker(−2J∂J ).

But as ∂J is injective (lemma 4.41), this equals to − indexR (∂J ), and by Riemann-Roch’s theorem we have indexR (∂J ) = 2 indexC (∂J ) = 2(1 − gΣ + d), where d = deg(T Σ) = 2 − 2gΣ . The first statement follows. The second statement is a consequence of the fact that J−1 / Ham(Σ, ρ) is a principal H 1 (Σ)-bundle over J−1 / Symp0 (Σ, ρ), and that dim H 1 (Σ) = 2gΣ . 

4.3.7

The Weil-Petersson metric

We can finally prove that the metric we obtained on Teichm¨ uller space agrees with the Weil-Petersson metric. Theorem 4.47 Let J ∈ J−1 and [Jˆ1 ]T , [Jˆ2 ]T ∈ T[J ]T J−1 / Symp0 (Σ, ρ). Under the natural identification with Teichm¨ uller space, we have G[J ]T ([Jˆ1 ]T , [Jˆ2 ]T ) = h[Jˆ1 ]T , [Jˆ2 ]T iW P . Proof We have the commutative square J (Σ, ρ) = A

J−1

J−1 / Symp0 (Σ, ρ)

∼ =

T (Σ) = A/D0

where the top arrow is the inclusions, and the bottom isomorphism is given by lemma 4.42. So let [J ]T ∈ T (Σ), and [Jˆ]T ∈ T[J ]T T (Σ). Then the horizontal lift (as in appendix C) Jˆ ∈ TJ A of [Jˆ]T at J ∈ J−1 is an element of TJ J−1 . ˆ LX J ) = 0 for all vector fields Indeed, Jˆ is horizontal if, and only if GJ (J, 0 X ∈ Ω (Σ; T Σ). So let V ∈ Lie G be any element of the Lie algebra. We have hdΞJ (Jˆ), V i =ΩJ (−LV J, Jˆ) ˆ −JLV J ) =GJ (J, ˆ LJV J ) = 0, =GJ (J, 88

4.3. Teichm¨ uller space and the Weil-Petersson metric where in the first line we used the moment map equation, in the second one the K¨ahler structure and in the last one lemma 4.37. It follows that dΞJ (Jˆ) = 0, so that Jˆ is tangent to the level set J−1 . Therefore, we compute G[J ]T ([Jˆ1 ]T , [Jˆ2 ]T ) =GJ (Jˆ1 , Jˆ2 ) Z 1 = tr(Jˆ1 Jˆ2 )ρ 2 Σ Z 1 = tr(Jˆ1 Jˆ2 )dµg 2 Σ =h[Jˆ1 ]T , [Jˆ2 ]T iW P where in the third line we noticed that g (J ) = ρ(−, J−) and dµg = ρ, and the fourth line is given by theorem C.4. 

4.3.8

Summary

To complete the chapter and this thesis with it, we give a summary of what we have done in the sections 4.2 and 4.3, forming the gist of this work. In section 4.2 we treated the general theory. We defined the principal bundle F (M ) of SL(n)-frames on an n-dimensional manifold M with a fixed volume form ρ, and were able to study its tangent space with the help of an auxiliary connection (subsection 4.2.2). Taking then a symplectic manifold (X, ω ) with a Hamiltonian (left) SL(n)-action we formed an associated bundle X −→ M and defined our main space of interest X to be the space of sections of this bundle. We endowed X with a natural symplectic form and a symplectic action of the group G ex of exact volume-preserving diffeomorphisms of M (subsection 4.2.3). We went on studying the infinitesimal action of G ex on X and the dual of the Lie algebra Lie G ex (subsections 4.2.4 and 4.2.5) and were finally able to define a function very close to a moment map µ for the action via the formula (4.1). This expression was a priori dependent on the choice of an auxiliary connection ∇ on M , but in theorem 4.24 we proved that it is in fact invariant under changes of connection, G ex -equivariant and that it satisfies the moment map equation. This was the content of subsection 4.2.6. In section 4.3 we applied this theory to a special case giving an alternative derivation of Teichm¨ uller space, with the Weil-Petersson metric arising naturally. We took M to be a closed, oriented surface Σ of genus at least 2 and X to be the hyperbolic plane H with the K¨ahler structure given in subsection 4.3.1. This gives ∼ J (Σ, ρ), X =

G ex = Ham(Σ, ρ) 89

4. Actions of groups of diffeomorphisms in a natural way (see page 75, and in general subsection 4.3.2 for the details). After a short study of the K¨ahler structure on X and of the action of G ex , in subsection 4.3.4 we derived the expression µ(J ) = −2K J ρ for the map µ in this special case. Here K J denotes the curvature of the Riemannian metric on M naturally associated to J ∈ J (Σ, ρ). Thanks to this, and to the fact that we were working in dimension 2, we were able to define an honest moment map for the action by expression (4.7). The MarsdenWeinstein quotient we obtained thanks to this was not yet Teichm¨ uller space, but a space slightly bigger. In fact, we showed that J (Σ, ρ) Ham(Σ, ρ) −→ T (Σ) is a principal bundle with fibre given by the first deRham cohomology group H 1 (Σ) of Σ. Therefore, we were able to obtain the desired space by further quotienting by the action this group, and then we transferred the K¨ahler structure of the Marsden-Weinstein quotient thanks to a general lemma about K¨ ahler manifolds. This was done in subsection 4.3.6. Finally, in subsection 4.3.7 we showed how the K¨ahler metric we obtained on Teichm¨ uller space corresponds exactly to the Weil-Petersson metric.

90

Appendix A

Basic concepts of symplectic geometry

We present here a brief review of the relevant notions of symplectic geometry underlying all of this work, following closely the first chapters of [McDuff and Salamon, 2015].

A.1

Symplectic vector spaces

A.1.1

Basic concepts

Definition A.1 A symplectic vector space (V , ω ) is a finite dimensional vector space V over R together with a skew-symmetric, nondegenerate bilinear form ω : V × V → R, called the symplectic form. Lemma A.2 A symplectic vector space is necessarily even-dimensional. Proof It is enough to show that any real, skew-symmetric matrix ω of odd dimension has non-zero kernel. So let dim V be odd, ω a skew-symmetric matrix on V . We have det ω = det ω T = det(−ω ) = − det ω, where in the last equality we used the fact that the dimension is odd. Therefore, det ω = 0 and thus the kernel is non-trivial.  ∼ Example A.3 The typical example of symplectic vector space is V = R2n = n n R × R with coordinates (x, y ) and symplectic form ω0 =

n X

dxi ∧ dy i .

i=0

91

A. Basic concepts of symplectic geometry In fact it can be shown that every finite dimensional symplectic vector space is isomorphic (that is, linearly symplectomorphic) to a space of this kind. The following important fact holds. Lemma A.4 Let V be a 2n-dimensional real vector space and ω a skewsymmetric bilinear form on V . Then ω is non-degenerate if, and only if ω n := ω ∧ . . . ∧ ω 6= 0. The relevant notion of morphism of symplectic vector spaces is as follows. Definition A.5 A morphism of symplectic vector spaces from a symplectic vector space (V , ωV ) to another one (W , ωW ) is a linear map φ : V → W preserving symplectic structures, that is ωV (x, y ) = (φ∗ ωW )(x, y ) = ωW (φ(x), φ(y )). A linear symplectomorphism of (V , ω ) is an isomorphism of symplectic vector spaces from V to itself. We denote the group of all such maps by Sp(V , ω ) and call it the symplectic group of (V , ω ).

A.1.2

Special subspaces

Let (V , ω ) be a symplectic vector space. We define the symplectic complement of W by W ω = {v ∈ V |∀w ∈ W : ω (v, w ) = 0}. Definition A.6 Let V be a symplectic vector space, W ⊆ V a linear subspace. Then W is: • isotropic if W ⊆ W ω . • coisotropic if W ω ⊆ W . • symplectic if W ∩ W ω = {0}. • Lagrangian if W = W ω . In particular, if W is Lagrangian, then it is both isotropic and coisotropic. In order to be able to do symplectic quotients, we need the following results showing how every coisotropic subspace of a symplectic vector space gives rise to a new symplectic vector space by taking its quotient by its symplectic complement. Lemma A.7 (Symplectic reduction) Let W ⊆ V be a coisotropic subspace of a symplectic vector space (V , ω ). Then the quotient W /W ω is again a symplectic vector space where the symplectic form (abusing notation) is given by ω (v + W ω , w + W ω ) = ω (v, w ). 92

A.2. Symplectic manifolds and symplectomorphisms Proof We only have to check that the induced bilinear form is well-defined and non-degenerate (skew-symmetry is obvious). Let u1 , u2 ∈ W ω ⊆ W , then ω (v + u1 , w + u2 ) =ω (v, w ) + ω (u1 , w ) + ω (v, u2 ) + ω (u1 , u2 )

=ω (v, w ), and thus it is well defined. For non-degeneracy, let v + W ω and assume ω (v + W ω , w + W ω ) = 0 for all w + W ω . This implies that ω (v, w ) = 0 for all w ∈ W , and thus that v ∈ W ω , so that v + W ω = 0 in the quotient. 

A.2

Symplectic manifolds and symplectomorphisms

Definition A.8 A symplectic manifold (M , ω ) is a smooth manifold without boundary M of even dimension dim(M ) = 2n, together with a closed 2-form ω ∈ Ω2 (M ) whose restriction to Tm M is a symplectic form for every m ∈ M . By lemma A.4, the n-fold wedge product ω ∧ . . . ∧ ω can never be zero, and thus it defines a volume form. In particular, M must be orientable. Remark A.9 Not all even-dimensional manifolds admit a symplectic structure. This can be shown with an easy cohomological argument. Let (M , ω ) be a symplectic manifold. Since ω is closed, it defines a deRham cohomology class [ω ] ∈ H 2 (M ). If M is closed, then ω cannot be exact, else ω n would also be exact and thus Z ω n = 0, M

which is a contradiction to the fact that ω n is a volume form. Therefore 0 6= [ω ] and 0 6= [ω n ] = [ω ] ∪ . . . ∪ [ω ]. In some manifolds, such as spheres S 2n for n ≥ 2, these conditions cannot be satisfied (either singularly or simultaneously). The notions of isotropic, coisotropic, symplectic and Lagrangian vector spaces carry over to the tangent spaces of submanifolds, giving rise to the following definition. Definition A.10 Let (M , ω ) be a symplectic manifold, and let Q ⊆ M be a submanifold. Take a point p ∈ Q, then we look at Tp Q as a subspace of Tp M . We say that Q is: • a (co)isotropic submanifold of M if Tp Q is a (co)isotropic subspace of Tp M for every p ∈ Q. 93

A. Basic concepts of symplectic geometry • a symplectic submanifold of M if Tp Q is a symplectic subspace of Tp M for every p ∈ Q. Equivalently, ω restricts to a symplectic form on Q, thus, Q is a symplectic manifold itself. • a Lagrangian submanifold of M if Tp Q is a Lagrangian subspace of Tp M for every p ∈ Q. Equivalently, Q is an isotropic submanifold of maximal dimension (i.e. half the dimension of M ). The correct notion of a map between symplectic manifolds is obviously one that preserves both the structure of smooth manifold and the symplectic structure. Definition A.11 Let (N , ωN ) and (M , ωM ) be symplectic manifolds. A morphism of symplectic manifolds between N and M is a smooth map φ : N → M such that φ ∗ ωM = ωN . An isomorphism of symplectic manifolds is an invertible morphism of symplectic manifolds such that its inverse is also a morphism of symplectic manifolds. A symplectomorphism of (M , ω ) is an isomorphism of symplectic manifolds from M to itself. We denote the group of symplectomorphisms of M by Symp(M , ω ) = {φ ∈ Diff (M )|φ∗ ω = ω}. Since ω is non-degenerate, it induces a natural module isomorphism between the C ∞ (M )-modules of vector fields of 1-forms on M : Γ(M ) −→ Ω1 (M ) X 7−→ ιX ω Definition A.12 A symplectic vector field X is a vector field X ∈ Γ(M ) such that the 1-form ιX ω is closed. We denote the space of symplectic vector fields by Γ(T M , ω ) = {X ∈ Γ(T M ) : dιX ω = LX ω = 0} Proposition A.13 Let (M , ω ) be a closed symplectic manifold. Then the Lie algebra of Symp(M , ω ) is given by the symplectic vector fields Γ(T M , ω ) together with the usual Lie bracket of vector fields. Moreover, if X, Y ∈ Γ(T M , ω ), then we have ι[X,Y ] ω = df , where f = ω (X, Y ). The proof of this fact is found in [McDuff and Salamon, 2015, Sect. 3.1].

A.3

Hamiltonian vector fields and Hamiltonian flows

Fix a smooth function H : M → R, called the Hamiltonian. To such a function we can associate a symplectic vector field XH ∈ Γ(T M , ω ) by requiring that 94

A.3. Hamiltonian vector fields and Hamiltonian flows it satisfies the identity ιXH ω = dH. We call XH the Hamiltonian vector field associated to H. Definition A.14 A (necessarily symplectic) vector field X ∈ Γ(T M , ω ) is called a Hamiltonian vector field if there exists a smooth function H : M → R satisfying the relation above for X. A Hamiltonian flow is the flow of a (possibly time-varying) Hamiltonian vector field. We denote by Ham(M , ω ) the group of Hamiltonian flows. Every symplectic manifold is also a Poisson manifold, with the Poisson bracket on C ∞ (M ) is given by {F , G} = ω (XF , XG ) = dF (XG ). Lemma A.15 Let (M , ω ) be a symplectic manifold with a fixed Hamiltonian H : M → R. Denote by φtH the Hamiltonian flow. Then: 1. A function F ∈ C ∞ (M ) is constant along φtH if, and only if {F , H} = 0. In particular, H is constant along the Hamiltonian flow. 2. Let ψ ∈ Symp(M , ω ), then ψ ∗ XH = XH◦ψ . 3. For any two functions F , G ∈ C ∞ (M ) we have [XF , XG ] = X{F ,G} . Proof For point 1, F is constant along the flow of H if, and only if d t ∗ (φ ) F dt H =(φtH )∗ LXH F !

0=

=(φtH )∗ dF (XH ) =(φtH )∗ {F , H}. For point 2, d(H ◦ ψ ) =ψ ∗ dH

=ψ ∗ (ιXH ω ) =ιψ∗ XH ψ ∗ ω =ιψ∗ XH ω. Finally, for point 3 we have

[XF , XG ] =LXF XG d = − (φtF )∗ XG dt t=0 d = − XG◦φtF dt t=0 95

A. Basic concepts of symplectic geometry where in the last line we used the fact that φtF is a symplectomorphism, together with point 2. Therefore,

d ι[XF ,XG ] ω = − ιXG◦φt ω dt t=0 F d = − dG ◦ φtF dt t=0 d = − d G ◦ φtF dt t=0 = − d(dG(XF ))

=d{F , G}.

A.4

Complex structures and K¨ ahler manifolds

A.4.1

Almost complex and complex structures



Let V be a real vector space with dim(V ) = n. Definition A.16 An almost complex structure on V is a linear map J : V −→ V such that J 2 = − idV . Notice that an almost complex structure is necessarily an automorphism of V , and as (−1)n = det(− idV ) = det(J 2 ) = det(J )2 ≥ 0, the dimension of V must be even. A classification of almost complex structures on Rn is given by the Siegel upper half space (see subsection 4.3.1). Definition A.17 An almost-complex manifold is a manifold M together with a section J : M → End(T M ) of the endomorphism bundle such that J (m) ∈ End(Tm M ) is an almost complex structure for all m ∈ M . Then J is called an almost complex structure on M . We denote the space of all almost complex structures on M by J (M ). By what we have seen above, the dimension of a complex manifold is necessarily even. Definition A.18 A complex manifold is a manifold with a holomorphic atlas, that is an atlas of charts φ : U → M , U ⊂ Cn , such that the transition maps are holomorphic functions. A complex manifold admits an almost complex structure in a natural way, by taking the section of the endomorphism bundle induced by the multiplication by i in the charts. The relation between complex and almost complex manifolds is given by the following celebrated theorem. 96

A.4. Complex structures and K¨ahler manifolds Theorem A.19 (Newlander-Nirenberg) If (M , J ) is an almost complex manifold, then there exists a holomorphic atlas on M inducing J as almost complex structure if, and only if the Nijenhuis tensor NJ (X, Y ) = [X, Y ] + J ([JX, Y ] + [X, JY ]) − [JX, JY ] vanishes identically. An almost complex structure J on M such that NJ = 0 is called a complex structure. Lemma A.20 Assume dim(M ) = 2. Then every almost complex structure on M is a complex structure. Proof Let J be an almost complex structure on M . Take any vector field X : M → T M , then an easy computation shows that NJ (X, X ) = 0

and

N (X, JX ) = 0

everywhere. But as M is 2-dimensional we can choose a local basis of T M by fixing a local, non-vanishing vector field e1 and taking e2 = Je1 . It follows that NJ = 0, and thus J is integrable. 

A.4.2

Compatible almost complex structures and K¨ ahler manifolds

Now let (M , ω ) be a symplectic manifold. Definition A.21 An almost complex structure J ∈ J (M ) on M is called ω-compatible if the symmetric (0, 2)-tensor ω (−, J−) defines a Riemannian metric on M . The space of ω-compatible almost complex structures is denoted by J (M , ω ). A particularly nice situation is when we have a compatible almost complex structure which is also integrable. Definition A.22 A K¨ ahler manifold is a triple (M , ω, J ) with M a manifold, ω a symplectic form on M and J complex structure compatible with ω. Alternatively, we can define K¨ ahler manifolds as complex manifolds (M , J ) endowed with either • a Riemannian metric g such that g (−, J−) is a symplectic form, or • a Hermitian form h. In this case we recover g and ω as the real and imaginary part of h respectively. 97

A. Basic concepts of symplectic geometry Example A.23 The following K¨ ahler structure on the complex projective space CPn is called the Fubini-Study metric. Endow CPn with the standard complex structure, and let [Z0 , . . . , Zn ] denote homogeneous coordinates. Then the symplectic form is given by ωF S =

i ∂∂ log kZk2 . 2π

Notice that this expression is well defined, as two different choices of Zi change log kZk2 by adding a constant scalar. Products and complex submanifolds of K¨ahler manifolds are again K¨ahler. The following basic results are found in [Ballmann, 2006, ch. 4]. Lemma A.24 Let (M , J, ω ) be a K¨ ahler manifold, let g be its K¨ ahler metric and denote by ∇ the associated Levi-Civita connection. Then ∇J = 0. Lemma A.25 Let M be as above, and let dimR M = 2n. Then ω n = n!dµg . Theorem A.26 Let M be as above, and assume furthermore that M is closed. Then for all 0 ≤ k ≤ n we have [ω k ] 6= 0 in H 2k (M ). In particular, H 2k (M ) 6= 0. This gives an easy cohomological obstruction to the existence of a K¨ahler structure on a manifold.

A.4.3

Riemann surfaces and the uniformization theorem

A Riemann surface is a complex manifold of (real) dimension 2. We have the usual (topological) classification of (closed, connected, oriented) surfaces by genus, and it can be shown that for every g ≥ 0, the genus g surface Σg admits a complex structure. In fact, there is more. Theorem A.27 (Uniformization theorem) Let (Σ, J ) be a Riemann surface of genus g ≥ 2, then there exists a unique 2-form β ∈ Ω2 (Σ) such that β (−, J−) is a Riemannian metric of constant Gaussian curvature K ≡ −1.

98

Appendix B

Hodge theory

For a more detailed treatment of this topic, see the book [Bertin et al., 2002], which we followed closely. For the basic concepts of functional analysis and PDEs, a place where to start is Evan’s book [Evans, 2010]. Let M be a closed manifold of dimension n, and let E be a real or complex smooth vector bundle over M . Definition B.1 A connection D on E is a linear differential operator of order one D : Ωk (M ; E ) −→ Ωk+1 (M ; E ), where Ωk (M ; E ) = C ∞ (M ; Λk T ∗ M ⊗ E ), satisfying the Leibniz rule D (f ∧ α) = df ∧ α + (−1)p f ∧ Dα for all f ∈ Ωp (M ) and α ∈ Ωk (M ; E ). Equip E with an Euclidean (or Hermitian, if E is complex), smooth inner product h·, ·i. It induces a canonical bilinear (respectively sesquilinear) pairing {·, ·} : Ωp (M ; E ) × Ωq (M ; E ) −→ Ωp+q (M ). A connection D is Hermitian if it satisfies d{α, β} = {Dα, β} + (−1)|α| {α, Dβ}.

B.1

Differential operators on vector bundles

Let now E and F be vector bundles on M , or rank rE and rF respectively. Definition B.2 A (linear) differential operator of degree δ from E to F is a linear operator L : Γ(E ) −→ Γ(F ) 99

B. Hodge theory which has the following form in local coordinates X

Lu(x) =

Aα (x)Dα u(x),

|α|≤δ

where α ∈ Nn are multi-indices, Dα = Ex → Fx are linear maps (matrices).



∂ ∂x1

α1

...



∂ ∂xn

 αn

and Aα (x) :

Example B.3 Let Ω ⊂⊂ Rn , E = F = Ω × R. Then sections u ∈ Γ(Ω × R) are given by smooth functions u : Ω → R, and differential operators from E to F correspond in the obvious way to the usual differential operators on (smooth) maps. We can generalize the concept of “leading order term” as follows: given a differential operator L : C ∞ (Ω) −→ C ∞ (Ω) of the form X

Lu(x) =

Aα ( x ) D α u ( x ) ,

|α|≤δ

we have e−tf (x) L(etf (x) u(x)) = tδ

X

Aα (x)Dα f (x) + l.o.t. in t.

|α|=δ

We see that the leading term of L is encoded by |α|=δ Aα (x)Dα f (x). For general differential operators on vector bundles we have similarly P

e−tf (x) L(etf (x) u(x)) = tδ σL (dfx )u(x) + l.o.t. in t, where σL : T ∗ M → hom(E, F ) is a polynomial bundle map. Definition B.4 The map σL of above is called the principal symbol of L. Definition B.5 A differential operator L is elliptic if σL (vx∗ ) ∈ hom(Ex , Fx ) is injective for all x ∈ M and vx∗ ∈ Tx∗ M \{0}. Example B.6 Let L : C ∞ (Ω) → C ∞ (Ω) be a second order operator given by Lu(x) =

n X

aij (x)uxi xj (x) +

i,j =1

n X

bk (x)uxk + cu(x).

k =1

Its principal symbol is given by σL (vx∗ ) =

n X i,j =1

100

aij (x)vi∗ vj∗ : R −→ R.

B.2. L2 -inner product and adjoints Thus it is simply multiplication by a scalar, and injectivity is equivalent to n X

aij (x)vi∗ vj∗ 6= 0.

i,j =1

L is elliptic if, and only if this is always non-zero, and thus either positive on all of T ∗ M , or negative on all of T ∗ M , and thus ellipticity is equivalent to asking for either L or −L to be elliptic in the “usual” sense.

B.2

L2 -inner product and adjoints

Assume now that M is also oriented, and fix a volume form ρ. We define the space L2 (M ; E ) of square integrable sections of the bundle E as the set of sections of E with measurable coefficients such that kukL2 (M ;E ) =

Z

ku(x)kE ρ < ∞.

M

It is a Hilbert space with inner product hhu, vii =

Z

hu(x), v (x)iρ.

M

Definition B.7 Let L : Γ(E ) → Γ(F ) be a differential operator. A formal adjoint of L is a differential operator L∗ : Γ(F ) → Γ(E ) satisfying hhLu, vii = hhu, L∗ vii for all u ∈ Γ(E ) and all v ∈ Γ(F ). Lemma B.8 Every differential operator L : Γ(E ) → Γ(F ) between Euclidean (or Hermitian) bundles admits a unique formal adjoint L∗ : Γ(F ) → Γ(E ). Remark B.9 In the case of M non-compact, in the definition of adjoint we require that hhLu, vii = hhu, Lvii whenever supp(u) ∩ supp(v ) ⊂⊂ M . The result above then holds, and the only adaptation we have to do in the proof is to consider only compactly supported smooth sections. Proof First assume a formal adjoint exists, we show it is then necessarily unique. We have that Γ(E ) = C ∞ (M ; E ) ⊂ L2 (M ; E ). Assume L∗1 , L∗2 are two formal adjoints, and take u ∈ L2 (M ; E ). We can then choose a sequence {un } ⊂ Γ(E ) converging to u in the L2 -norm. Then we have hhu, L∗1 vii = lim hhun , L∗1 vii n→∞

= lim hhLun , vii n→∞

= lim hhun , L∗2 vii n→∞

= hhu, L∗2 vii . 101

B. Hodge theory As this holds for all u ∈ L2 (M ; E ), we obtain L∗1 v = L∗2 v, and thus L∗1 = L∗2 . To prove existence, choose an atlas for M such that F can be trivialized over the charts, and let {ηα } be a partition of unity subordinated to the atlas. Then we have **

hhLu, vii =

++

Lu,

X

=

ηα v

α

X

hhLu, ηα vii ,

α

and thus it is enough to prove that L∗ exists locally, that is over trivializing charts. Let Ω ⊂ Rn be such a chart On it we can write Lu(x) =

X

Aα (x)Dα u(x),

|α|≤δ

ρ(x) =γ (x)dx1 . . . dxn . Then integration by parts gives (with implicit summation over i, j) hhLu, vii =

= =

Z

X Ω

Z Ω

Z Ω

α 1 n Aij α (x) (D ui (x)) vj (x)γ (x)dx . . . dx

|α|≤δ







|α|≤δ

ui (x)

X

1 n (−1)|α| γ −1 (x)Dα Aij α (x)vj (x)γ (x) γ (x)dx . . . dx

|α|≤δ

**

=



1 n (−1)|α| γ −1 (x)Dα Aij α (x)vj (x)γ (x) γ (x)dx . . . dx

X

ui (x)

u,

X

|α| −1

(−1) γ

(x)D

α



Aij α ( x ) vj ( x ) γ ( x )

++ 

|α|≤δ

|

{z

=:L∗ v

}

where we used the fact that γ (x) ∈ R, and thus γ (x) = γ (x).



From the explicit formula above, the principal symbol of L∗ can be easily found to be given by σL∗ = (−1)δ (σL )† , where † is the conjugate-transpose. In particular, if E and F have the same rank, then L is elliptic if, and only if L∗ is.

B.3

Sobolev spaces of sections

Definition B.10 For integers s ≥ 0 we define the Sobolev space H s (M ; E ) as the space of sections of E with coefficients locally in H s (Rn ) on all open charts. 102

B.4. The finiteness theorem Notice in particular that H 0 (M ; E ) = L2 (M ; E ). The following two results are generalizations of the standard theorems for bounded subsets of Euclidean space. Theorem B.11 (Sobolev) For k ∈ N and s ≥ k + embedding H s (M ; E ) ,−→ C k (M ; E ).

n 2

we have a continuous

Corollary B.12 We have Γ (E ) = C ∞ (M ; E ) =

\

H s (M ; E ).

s≥0

Theorem B.13 (Rellich) If t > s, then the embedding H t (M ; E ) ,−→ H s (M ; E ) is a compact operator. Finally, we have the following important classical result. Theorem B.14 (G˚ arding) Let L : Γ(E ) → Γ(F ) be an elliptic differential operator of degree δ, assume that E and F have the same rank and let L˜ be an extension of L to the space of distributional sections D0 (M ; E ). Then for ˜ ∈ H s (M ; F ) we have u ∈ H s+δ (M ; F ) and any u ∈ L2 (M ; E ) such that Lu the estimate 



˜ H s (M ;F ) + kukL2 (M ;E ) , kukH s+δ (M ;E ) ≤ cs kLuk where cs > 0 is a constant depending only on s and M . Notice that such an extension of L always exists by Hahn-Banach’s theorem since L is bounded.

B.4

The finiteness theorem

First of all we recall the following standard result in Fredholm theory (see for example [McDuff and Salamon, 2012, Appendix A.]). Theorem B.15 Let X, Y , Z be Banach spaces, L : X → Y a bounded linear operator and K : X → Z a compact operator. Assume that there exist a constant c > 0 such that the bound kukX ≤ c(kLukY + kKukZ ) is satisfied. Then: 1. ker L is finite dimensional. 103

B. Hodge theory 2. L(X ) is closed in Y . Proof To show that ker L is finite dimensional, it is enough to prove that its unit ball is compact. Take any sequence {un } ⊂ ker L ∩ BX (1). Then the sequence {Kun } ⊂ Z admits a converging subsequence by compactness of K, and thus we can assume that Kun → z ∈ Z with respect to k · kZ . Then we have n,m→∞ k un − um kX ≤ ckKun − Kum kZ −−−→ 0. |

{z

∈ker L

}

Thus the sequence is Cauchy and converges. To show that the image of L is closed, assume without loss of generality that L is injective (if it is not, restrict to a complement of ker L, which exists by Hahn-Banach’s theorem). Let y ∈ L(X ) ⊂ Y . Take a sequence {un } ⊂ X such that Lun → y in Y . We show that {un } is bounded. Assume it were not. Then by taking a subsequence we can assume without loss of generality that kun kX → ∞. Consider now the sequence u˜ n =

un . kun kX

We have Lu˜ n → 0 and ku˜ n kX = 1. Again by compactness of K, we can assume without loss of generality that K u˜ n converges, and thus as before n,m→∞

ku˜ n − u˜ m kX ≤ c(kLu˜ n − Lu˜ m kY + kK u˜ n − K u˜ m kZ ) −−−→ 0. Thus {u˜ n } converges to some limit u˜ ∈ X, which satisfies kuk ˜ X = 1 and Lu˜ = 0, a contradiction to injectivity of L. Thus {un } is a bounded sequence. As before, we can assume that {Kun } converges, and by the same estimates we obtain that {un } converges to some limit u ∈ X. Then by continuitiy of L we have Lu = y, proving that the image of L is closed.  To prove the next important result, we first need an auxiliary bound on the L2 -norm of elements on H k (M ; E ). Lemma B.16 Let k ≥ 1. For all ε > 0 there exists an integer N and a finite k set of elements V = {vi }N i=1 ⊂ H (M ; E ) so that kukL2 (M ;E ) ≤ εkukH k (M ;E ) +

N X

|hhu, vi ii|

i=1

for all u ∈ H k (M ; E ). Proof For any finite subset V ⊂ H k (M ; E ), define (

KV =

k

u ∈ H (M ; E ) εkukH k (M ;E ) +

N X i=1

104

)

|hhu, vi ii| ≤ 1 .

B.4. The finiteness theorem Then the closure KV with respect to the L2 -norm is compact, since by Rellich’s theorem (theorem B.13) the embedding H k (M ; E ) ,→ L2 (M ; E ) is compact. We also have that \ KV = {0}, V ⊂H k (M ;E ) finite

because if it weren’t so, then we could take V = {au} for u in the intersection and a ∈ R big enough to obtain εkukH k (M ;E ) + |hhu, auii| ≥ (ε + a)kukL2 (M ;E ) ≥ 1, which would be a contradiction. It follows that there exists a finite set V so that KV ⊆ BL2 (M ;E ) (1) (the closed unit ball in L2 (M ; E )). This can be shown as follows: First notice that KV ∪V 0 ⊆ KV ∩ KV 0 . Now for any finite set V define CV = KV \BL2 (M ;E ) (1). If there is a V such that CV = ∅ we are done. Otherwise, notice that CV ⊂ KV , which is a compact subset of L2 (M ; E ) as we showed before. If there existed T a finite family V1 , . . . , Vr such that ri=1 CVi = ∅, then we would have CV1 ∩...∩Vr = ∅, a contradiction. Else, if any finite family of such CV ’s has non-empty intersection, by compactness it follows that \ V

⊂H k (M ;E )

CV 6= ∅, finite

in contradiction to what we have proved above. Finally take such a V with KV ⊆ BL2 (M ;E ) (1). Then for u ∈ H k (M ; E ) define u˜ =

u P ∈ H k (M ; E ). εkukH k (M ;E ) + v∈V |hhu, vii|

It follows that u˜ ∈ KV , and thus kuk ˜ L2 (M ;E ) ≤ 1, or equivalently kukL2 (M ;E ) ≤ εkukH k (M ;E ) +

N X

|hhu, vi ii| ,

i=1

which is exactly what we wanted.

 105

B. Hodge theory Theorem B.17 (Finiteness theorem) Let E and F be Euclidean (or Hermitian) vector bundles of the same rank on a compact manifold M , and let L : Γ(E ) → Γ(F ) be an elliptic differential operator of degree δ ≥ 1. Then we have: 1. ker L ⊆ Γ(E ) is finite dimensional. 2. L(Γ(E )) is closed and has finite codimension in Γ(F ). moreover, we have the decomposition Γ(F ) = L(Γ(E )) ⊕ ker L∗ as orthogonal direct sum in L2 (M ; E ). Proof Let k ≥ 0, extend L to a bounded linear operator H k+δ (M ; E ) −→ H k (M ; F ). This is well defined by G˚ arding’s theorem (theorem B.14). Take V as in lemma 1 B.16 for ε = 2ck , and define an operator K : H k+δ (M ; E ) −→ L2 (M ; E ) by X

Ku =

v∈V

hhu, vii v. kvkL2 (M ;E )

Notice that K is compact, and we have 

kukH k+δ (M ;E ) ≤ck kLukH k (M ;E ) + kukL2 (M ;E )

 !

≤ck kLukH k (M ;E ) + εkukH k+δ (M ;E ) +

X

|hhu, vi ii|

v∈V



=ck kLukH k (M ;E ) + εkukH k+δ (M ;E ) + kKukL2 (M ;E )



where we used G˚ arding for the first inequality, and lemma B.16 for the second one. Rearranging we obtain 



kukH k+δ (M ;E ) ≤ 2ck kLukH k (M ;E ) + kKukL2 (M ;E ) . Therefore, by theorem B.15 we obtain that ker L is finite dimensional and L(H k+δ (M ; E )) is closed in H k (M ; E ). By taking the intersection over all k ≥ 0, we have that ker L is finite dimensional also for our original operator on smooth sections, and that L(Γ(E )) is closed in Γ(F ). Now take k = 0. Since Γ(E ) ⊂ H δ (M ; E ) is dense, we have L(H δ (M ; E ))⊥ = L(Γ(E ))⊥ = ker L∗ 106

B.5. Hodge’s theorem and thus it follows that L2 (M ; E ) = L(H δ (M ; E )) ⊕ ker L∗ . Since L∗ is also elliptic, by the first point of the theorem we have that ker L∗ is finite dimensional and contained in Γ(F ). Then by G˚ arding’s theorem we have that H k (M ; E ) = L(H k+δ (M ; E )) ⊕ ker L∗ and thus, taking the intersection over all k ≥ 0, that Γ(F ) = L(Γ(E )) ⊕ ker L∗ ,



proving the theorem.

B.5

Hodge’s theorem

Let (M , g ) be a closed Riemannian manifold with volume form ρ, E → M an Euclidean or Hermitian vector bundle of rank r with an Euclidean (respectively Hermitian) inner product h·, ·i.

B.5.1

The Hodge star operator

Let ∂x∂ 1 , . . . , ∂x∂n be a local basis of T M , and e1 , . . . , er a local basis of E. Denote by dx1 , . . . , dxn and e1 , . . . , er the corresponding dual bases (with respect to g and the inner product h·, ·i respectively). There is a natural inner product

(·, ·) : Λk T ∗ M × Λk T ∗ M −→ C ∞ (M ) given by

(a1 ∧ . . . ∧ ak , b1 ∧ . . . ∧ bk ) = det(g (ai , bj ))i,j , where ai , bj ∈ T ∗ M . This, together with the inner product on E induces an inner product on Λ• T ∗ M ⊗ E (where elements of different degrees are taken to be orthogonal), which we denote again by (·, ·). Definition B.18 The bilinear (respectively sesquilinear) pairing {·, ·} induces an isometry, called Hodge ∗-operator ∗ : Ωk (M ; E ) −→ Ωn−k (M ; E ) by requiring {α, ∗β} = (α, β )ρ. It can be easily verified that the Hodge ∗-operator is in fact a well defined isometry. In local coordinates, it is given by 



c

∗ ajI dxI ⊗ ej = sgn(I, I c )ajI dxI ⊗ ej , 107

B. Hodge theory where I are multi-indices and we adopt implicit summation over repeated indices. Here I c is the complement of I in {1, . . . , n}, and sgn(I, I c ) is the sign of the permutation sending the concatenation of I and I c to the ordered list. We have ∗ ∗ α = (−1)|α|(n−1) α.

B.5.2

The Laplace-Beltrami operator

Let dE be a (Hermitian) connection on the (Hermitian) vector bundle E. Lemma B.19 The formal adjoint of dE : Ωk (M ; E ) → Ωk+1 (M ; E ) is given by d∗E = (−1)nk+1 ∗ dE ∗ . Proof Let α ∈ Ωk (M ; E ) and β ∈ Ωk+1 (M ; E ). Then hhdE α, βii =

=

Z

hdE α, βiρ

ZM

{dE α, ∗β}

M

=

Z



d{α, ∗β} − (−1)k {α, dE ∗ β}



M

=(−1)

k +1

Z

{α, dE ∗ β}

M

=(−1)k+1 (−1)k(n−1)

Z

{α, ∗ ∗ dE ∗ β}

M

=(−1)kn+1 hhα, ∗dE ∗ βii , where in the third equality we used the fact that the connection is Hermitian, and in the fourth one Stokes’ theorem.  Definition B.20 The Laplace-Beltrami operator associated to the connection dE is the second order differential operator ∆E = dE d∗E + d∗E dE . This operator is clearly self adjoint. A short computation using the Leibniz rule shows that σ∆E (v ∗ )u = −|v ∗ |2 u, and thus that ∆E is always an elliptic operator.

B.5.3

Harmonic forms and Hodge’s theorem

Now assume that the connection is flat, i.e. d2E = 0. We obtain a generalized deRham complex d

d

d

E E E Γ(E ) = Ω0 (M ; E ) −→ Ω1 (M ; E ) −→ Ω2 (M ; E ) −→ ···

108

B.5. Hodge’s theorem We denote by H • (M ; E, dE ) its cohomology groups. The main theorem we will prove shortly is the relation of these groups with harmonic forms, and in particular the fact that they are always finite dimensional. Definition B.21 A k-form α ∈ Ωk (M ; E ) is harmonic with respect to dE if ∆E α = 0. We denote by Hk (M ; E, dE ) the space of harmonic k-forms. Lemma B.22 For all k we have the orthogonal decomposition Ωk (M ; E ) = Hk (M ; E, dE ) ⊕ dE (Ωk−1 (M ; E )) ⊕ d∗E (Ωk+1 (M ; E )). Proof We have hh∆E α, αii = kdE αk2 + kd∗E αk2 , so that α ∈ Hk (M ; E, dE ) if, and only if both dE α = 0 and d∗E α = 0. In particular Hk (M ; E, dE ) is orthogonal to the image of dE and d∗E . Moreover the image of dE and the image of d∗E are orthogonal, since hhdE α, d∗E βii = hhd2E α, βii = 0. |{z} =0

Applying theorem B.17 to ∆E = ∆∗E we obtain the decomposition Ωk (M ; E ) = ∆E (Ωk (M ; E )) ⊕ Hk (M ; E, dE ).

(B.1)

But since ∆E = dE d∗E + d∗E dE , the image of ∆E must be contained in the (actually direct) sum of the images of dE and d∗E , and since these images are orthogonal to Hk (M ; E, dE ), they must be contained in the image of ∆E by the decomposition (B.1). This proves the lemma.  Theorem B.23 (Hodge’s isomorphism theorem) We have ∼ Hk (M ; E, dE ). H k (M ; E, dE ) = In particular, these groups are all finite dimensional. Proof The group of boundaries in Ωk (M ; E ) is given by B k = dE (Ωk−1 (M ; E )), and the group of cycles by Z k = ker dE 

= d∗E (Ωk+1 (M ; E ))

⊥

=Hk (M ; E, dE ) ⊕ dE (Ωk−1 (M ; E ), 109

B. Hodge theory where we used the decomposition of lemma B.22. Thus we get the isomorphism. The fact that the groups are finite dimensional is then given by noticing that Hk (M ; E, dE ) = ker ∆E , which is finite dimensional by theorem B.17.



A special case of this is given by taking E = M × R and dE = d the usual exterior derivative. Then our theorem above shows that the deRham cohomology groups of a closed manifold are always finite dimensional.

110

Appendix C

Teichm¨ uller space and the Weil-Petersson metric

In this appendix, we give a short presentation of Teichm¨ uller space and the Weil-Petersson metric following closely Tromba’s book [Tromba, 1992, ch. 1-3], as we find that its geometrical approach is better suited to the present work than the classical treatment via quasi-conformal maps. The motivation for the definition of Teichm¨ uller space is the study of complex structures on a fixed Riemann surface up to some equivalence. More precisely, let Σ be a fixed Riemann surface and C ={holomorphic atlases for Σ}, D ={orientation preserving diffeomorphisms of Σ}. We would like to understand the space R(Σ) = C/D, where the action of D is given by pullback of the atlas charts. This is called the Riemann moduli space, and represents the space of “holomorphically equivalent” Riemann surfaces. However, the direct study of this space has revealed problematic. A breakthrough was made by studying instead the quotient space T (Σ) = C/D0 by the group of diffeomorphisms that are isotopic to the identity (first by Fricke and Klein, and then by Teichm¨ uller in his paper [Teichm¨ uller, 1939]). This new quotient space is called the Teichm¨ uller space. When the genus gΣ of Σ is at least 2, it turns out that we have a diffeomorphism ∼ R6gΣ −6 . T (Σ) = We will now proceed by introducing the important notions and giving a sketch of how Teichm¨ uller space is studied. 111

¨ ller space and the Weil-Petersson metric C. Teichmu Additionally to the spaces already described above, we define: A ={J : T Σ → T Σ|oriented almost complex structure}, Met ={g|Riemannian metric on Σ}, Met−1 ={g ∈ M|scalar curvature is constant R = −1}, P ={p ∈ C ∞ (Σ)|p > 0}. Remark C.1 In the main body of the present work, the space A was denoted by J (Σ, ρ), which is exactly the space of complex structures that agree with the orientation given by the volume form ρ. We use this alternative notation as we don’t want to refer to a fixed volume form. We have the sequence of isomorphisms T (Σ) :=C/D0 (1)

∼ A/D0 =

(2)

∼ (Met /P )/D0 =

(3)

∼ Met−1 /D0 . =

Then, using the last description, we have (4)

(5)

∼ S T T (g ) = ∼ {holomorphic quadratic differentials}, T[g ] T ( Σ ) = 2 where S2T T (g ) = {h symm. (0, 2)−tensor|∇∗g h = 0, trg h = 0} with ∇∗g denoting the adjoint operator to the gradient ∇g : C ∞ ( Σ ) → Γ ( T Σ ) . It follows by the Riemann-Roch theorem that dim T[g ] T (Σ) = 6gΣ − 6. Finally, we have (6) T (Σ) is a smooth manifold, and (7) T (Σ) is diffeomorphic to Euclidean space. We sketch now how the points (1)–(7) are proved. [Tromba, 1992].

For details, refer to

For the isomorphism (1), we denote by As , D0s the spaces of almost complex structures and diffeomorphisms that are of class H s . Then f ∈ D0 acts on both A and As by (f ∗ J )x = (dfx )−1 Jf (x) dfx . 112

Similarly, D0s+1 acts on As , but D0s does not (as the action involves a derivative). We have a bijective map C → A given as follows: if c is a holomorphic atlas of Σ, then for a point x ∈ Σ choose a chart ϕ ∈ c with domain containing x. We define an almost complex structure by Jx = (dϕx )−1 J0 dϕx , where J0 is the standard complex structure on C. This map is D0 -equivariant, and thus gives the desired isomorphism. For isomorphism (2), let Mets be the space of H s -Riemannian metrics. It is an open subset of the space S2s of symmetric (0, 2)-tensors, and thus its tangent space is given by ∼ Ss, Tg Mets = 2 showing that it has the structure of a Hilbert manifold. D0 acts on Met and Mets by the usual pullback. Moreover, we have an L2 -metric on Mets : if ∼ S s , then we have the scalar product induced by g which is ξ, η ∈ Tg Mets = 2 given in coordinates by hξ, ηig = g ik g jl ξij ηkl ∈ C ∞ (M ). We also have the standard volume form dµg = ∗g 1, and this gives the inner product Z hhξ, ηiig =

Σ

hξ, ηig dµg .

It can be shown that we have an L2 -orthogonal splitting 

Tg Ms = S2s (g )

c



⊕ S2s (g )

T

,

where the two summands are the “conformal” and “traceless” parts of a tangent vector, that is:  

S2s (g )

S2s (g )

c

T

={pg|p ∈ P}, ={h ∈ S2s | trg h = 0}.

The same is true for Met (leaving the s away everywhere). As P s is a Hilbert Lie group (modeled on H s (Σ)), and it acts in the obvious way on Mets with a free and proper action, we have that the quotient Mets /P s is again a Hilbert manifold, and its quotient is given by (by the L2 -decomposition above) 





∼ S s (g ) T[g ] Mets /P s = 2

T

.

The action of D0 descends to the quotient. We have a smooth diffeomorphism (of Hilbert manifolds) Φs : Mets /P s −→ As 113

¨ ller space and the Weil-Petersson metric C. Teichmu which is given by 



Φs ([g ]) = counterclockwise roteation by a g−right angle , where we notice that a metric induces a notion of “angle” which is independent of the conformal class. This map is D0s+1 -equivariant. Similarly, we have a diffeomorphism Φ : Met /P −→ A which is D0 -equivariant. For point (3), we need the following result. Theorem C.2 (Poincar´ e) Let Σ be a closed, oriented surface of genus at least 2. Then for any g ∈ Mets there exists a unique λ ∈ P s such that R(λg ) = −1. Here R : Σ → R denotes the scalar curvature. As it depends on the metric, we can write it as a map R : Mets −→ H s−2 (Σ, R) for s ≥ 2 (the loss in regularity is due to the fact that R depends on the second derivative of the metric tensor). We can apply one of the infinite dimensional versions of the implicit function theorem to obtain that Mets−1 is a smooth (Hilbert) submanifold of Mets . Let π : Mets −→ Ms /P s be the quotient map. Then the map π−1 = π|Mets−1 is bjective by theorem C.2, and thus a diffeomorphism. The group D0s+1 acts on Mets−1 by pullback (noticing that R(f ∗ g ) = f ∗ R(g ) = R(g ) ◦ f ), and π−1 is equivariant. This action is free and proper if gΣ ≥ 2, and therefore the quotient Mets−1 /D0s+1 is again a Hilbert smooth manifold. We pass to the smooth setting as follows. Let g0 ∈ Met−1 ⊂ Mets−1 , then the orbit of g0 under D0s+1 is a smooth submanifold of Mets−1 as the action is smooth, and its tangent space at g0 is given by Tg0 (D0s+1 g0 ) = {LX g0 X an H s −vector field} . We have an L2 -orthogonal splitting of h ∈ Tg0 Mets−1 by h = h0 + LX g0 with h0 ∈ S T T (g0 ). Therefore, S T T (g0 ) is a good candidate for Tg0 T (Σ). We use the following result. 114

Theorem C.3 (Gauss, existence of conformal coordinates) For every g ∈ Mets and every x ∈ Σ there exists a local coordinate chart ϕ : U → R2 around x such that (ϕ−1 )∗ g = λgR2 for some λ ∈ P s , where gR2 denotes the standard Euclidean metric on R2 . Thanks to this theorem, we can go to conformal coordinates, where an easy computation shows that for every h ∈ S2T T (g ) locally there exists a unique ξ : Σ → C holomorphic such that 



h = Re ξ (z )dz 2 . In particular, S T T (g0 ) is composed only by smooth tensors (and thus is independent from the choice of s), and by the Riemann-Roch theorem we have dim S T T (g0 ) = 6gΣ − 6. This proves point (5). To conclude the proof of (4), it can be shown that for any g0 ∈ Met−1 there exists a local smooth manifold g0 ∈ S ⊂ Mets−1 of dimension 6gΣ − 6, containing only smooth metrics, with Tg0 S = S T T (g0 ) ˜ ⊂ Mets−1 , g0 ∈ W ⊂ S and and such that there exist neighborhoods g0 ∈ W s+1 idΣ ∈ V ⊂ D0 such that the map Θ : S × D0s+1 −→ Mets−1 (g, f ) 7−→ f ∗g ∼ W ˜ . Then by restricting S if necessary, it gives a diffeomorphism W × V = follows from the fact that the action is proper and free that every point of S corresponds to exactly one orbit of D0 (that is, every orbit intersects S at most once). By taking these slices S as charts, we obtain a smooth, finite dimensional atlas for T (Σ), proving at the same time points (6) and (4). The main idea for the proof of point (7) is to construct a smooth, proper map E : T (Σ) −→ R (given in fact by the Dirichlet energy) with a unique (and therefore, nondegenerate) critical point. Then standard arguments of Morse theory show ∼ Rk for some k. that T (Σ) = The Weil-Petersson metric There are many metrics on the Teichm¨ uller space (not all of them Riemannian). The metric of interest for us is the so-called Weil-Petersson metric introduced by Weil in his paper [Weil, 1958] (in French; English version: [Weil, 1979]). It is a Riemannian (in fact K¨ahler) metric on T (Σ). 115

¨ ller space and the Weil-Petersson metric C. Teichmu Let ξdz 2 , ηdz 2 ∈ T[g ] T (Σ) be two holomorphic quadratic differentials. We use theorem C.3 to locally express the metric as g = λgR2 for some λ ∈ P. Then we have gij dxi dxj = λ|dz 2 | and it follows that

(ξdz 2 )(ηdz 2 ) ξη = 2 (λ|dz 2 |)2 λ

is a well defined function that can be integrated with respect to λdxdy to give the Weil-Petersson metric Z

hξ, ηiW P = Re

ξη dxdy . λ 

Σ

We also have the following alternative description in terms of the metric induced by the L2 -metric on Met. The obvious map π : Met−1 → T (Σ) makes it into what is called a weak principal bundle with fibre D0 , that is, π is smooth with respect to the H s -topology on Met−1 , and it is locally trivializable (via the map Θ), but the objects in questions are not manifolds and Lie groups in the traditional sense. We have the following L2 -metrics: hhh, kiig =

Z Σ

hξ, ηig dµg

on Met−1 as before (g ∈ Met−1 , h, k ∈ Tg Met−1 ), and hhH, KiiJ =

Z Σ

tr(HK )dµg

on A (J ∈ A, H, K ∈ TJ A), where g = g (J ) is the metric of constant curvature −1 associated to J. Contrarily to what could be expected, the identification between Met−1 and A is not an isometry with respect to these metrics. However, they allow us to define a notion of vertical and horizontal vectors in Met−1 and A respectively via canonical L2 -splittings, and they agree on horizontal vectors. In particular, they give a notion of horizontal lift of vectors from T (Σ) to Met−1 and A respectively, and thus a metric on Teichm¨ uller space by

 ˜ K˜ hH, Ki[J ] = H, J, ˜ K˜ are the unique horizontal lifts of H and K at J. where J ∈ [J ] and H, The identification of T (Σ) with Met−1 /D0 gives exactly the same metric of Teichm¨ uller space. Finally, we have the following result, which can be proven via a short calculation in conformal coordinates. 116

Theorem C.4 The Weil-Petersson metric and the one induced by the L2 metrics on Met−1 and A agree up to a factor 21 . Namely, we have h−, −iW P =

1 hh−, −ii . 2

117

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