Integrable Systems on the Moduli Space of Flat Connections

Integrable systems on the moduli space of ﬂat connections

Adam Artymowicz

Master of Science

Department of Mathematics and Statistics

McGill University

Montreal,Quebec 2018-12-13

A thesis submitted to McGill University in partial fulﬁllment of the requirements of the degree of Masters of Science. c Adam Artymowicz

ACKNOWLEDGEMENTS

My sincere thanks go out to my advisor, professor Jacques Hurtubise, for his support and direction throughout the writing of this thesis, and to professor Niky Kamran for his faith and encouragement during my Masters. Lastly, I am grateful to all of my friends at Oˆ Claf for supporting me throughout my time at McGill.

ii ABSTRACT

In this thesis we treat the moduli space of ﬂat SU(n) connections over a surface, as well as the various related spaces that appear when the surface is allowed to have boundary. The properties of these spaces and the relationships between them are used to construct an integrable system on the moduli space over a closed surface which generalizes the toric moment map of Jeﬀrey and Weitsman [10].

iii ABREG´ E´

Dans cette thèse, on s’intéresse àl’espace de modules des connexions plates sur une surface avec groupe de structure SU(n), ainsi qu’aux espaces associés qui aparaissent quand on permet àla surface d’avoir une frontière. Les propriétés de ces espaces et la relation entre eux nous sert a construire un système intégrable sur l’espace de modules sur une surface fermée qui généralise l’application moment torique de Jeffrey et Weitsman [10].

iv TABLE OF CONTENTS

ACKNOWLEDGEMENTS ...... ii

ABSTRACT ...... iii ABREG´ E...... ´ iv

1 Introduction...... 1 2 Symplecticgeometry ...... 4

2.1 LagrangianandHamiltonianformalism ...... 4 2.2 Symplecticandhamiltoniangroupactions ...... 8 2.3 Coadjointorbits...... 10 2.4 Symplecticreduction ...... 15 2.5 Integrablesystems...... 17 2.6 TheGelfand-Cetlinintegrablesystem ...... 18 3 Themodulispaceofﬂatconnections ...... 25

3.1 Gauge-theoreticorigins ...... 25 3.2 Flatconnections...... 29 3.3 Themodulispaceoveraclosedsurface ...... 31 3.4 Themodulispaceoverasurfacewithboundary ...... 35 3.5 Extendedandframedmoduli...... 37 3.6 Gluing ...... 45 3.7 TheJeﬀrey-Weitsmansystem ...... 46 3.8 The integrable system on symplectic leaves of S ...... 50

References...... 52

v CHAPTER 1 Introduction

There are many reasons to study moduli spaces of bundles over a surface. The ﬁrst, coming from algebraic geometry, is perhaps the most direct: it is to understand the parameter space of all holomorphic bundles of a given rank on a surface – one might describe this as the theory of higher-rank analogues of the Picard variety of the surface. Another motivation comes from the theory, again classical, of systems of holomorphic ODEs on a Riemann surface. On a disc such systems admit unique solutions, but these solutions become multi-valued when the domain is not simply connected, and this multi-valued behaviour manifests itself as a representation of the fundamental group of the surface (the monodromy representation) [5]. Surfaces with boundary enter naturally when the ODEs are permitted to have singularities – if these singularities are suﬃciently well-behaved, the classical Riemann-Hilbert correspondence equates such a system of ODEs with its monodromy representation, up to a suitable notion of isomorphism.

Relaxing the assumption that these objects be holomorphic, we enter the world of principal bundles with connection, for which we have a similar correspondence with fundamental group representations. Here we make substantial contact with physics, as principal bundles with connection are the fundamental objects of study in gauge theory [4].

1 The examples given above by no means constitute a full description of the zoo that exists of moduli spaces of bundles with some extra structure. They are only intended to indicate the richness of the theory, and the moral that when we study one space, we are indirectly studying them all at once. Each carries a different flavour, but a recurring theme is that of correspondences which arise when one views the same space from different viewpoints. Such bridges between algebra, geometry, and analysis are part of what make these moduli spaces interesting objects of study.

The subject of this thesis is the moduli space of flat connections on a surface with structure group SU(n), which is is a smooth symplectic manifold on an open dense set. The main result is the construction of a densely-defined integrable Hamiltonian system on this space, which generalizes the toric integrable system on the SU(2) moduli space due to Jeffrey and Weitsman [10] [9]. The Jeffrey-Weitsman system on SU(2) has allowed for a computation of certain invariants of the space, in particular its symplectic volume and the dimension of its geometric quantization. Both were calculated using the image of the toric moment map, which is a polytope: the first by computing the volume of the polytope with respect to the pushforward of the

Liouville measure, and the second by counting the points in the polytope belonging to a particular lattice (the Bohr-Sommerfeld points). The motivation behind con- structing the integrable system in this thesis is to eventually carry out analogous calculations for the SU(n) moduli space.

In the ﬁrst chapter we introduce some of the notions used in the thesis. There is nothing original in these sections, save for some aspects of the presentation of the

Gelfand-Cetlin integrable system in section 2.6. Speciﬁcally, we introduce a slight

2 generalization to the usual system which will be needed in the following sections.

The second chapter is devoted to describing several versions of the moduli space of ﬂat connections on a surface with boundary. The extended moduli spaces described in section 3.5 are due to Lisa Jeﬀrey [8]. The framed spaces of section 3.5 are also well-known, but to the author’s knowledge the presentation of the local calculations on these spaces, and their descriptions via representations of relative fundemantal groups is novel. Finally, in sections 3.6, 3.7, and 3.8 the technology presented in the preceding sections is used to construct the SU(n) integrable system.

3 CHAPTER 2 Symplectic geometry

In this chapter we will recall a few of the standard notions and theorems from symplectic geometry which will be used throughout this thesis. The aim of the chapter is not to act as an introduction to the subject; instead the goal is to present the point of view where homogeneous symplectic manifolds play a central role. We begin by motivating their study from a physical point of view.

2.1 Lagrangian and Hamiltonian formalism

Symplectic geometry has its roots in the study of classical mechanics by the likes of Lagrange, Hamilton, and Poisson. The classical-mechanical point of view of symplectic geometry will be indispensable in what follows, so the basic tenets are outlined in this section. The references for this section are [1] and [15].

Classical mechanics deals with the motion of a system whose configuration is given as a point on an n-dimensional manifold X, the configuration space. For instance, a system of N noninteracting particles in 3-dimensional space has configuration space R3N , a pendulum has configuration space the circle S1, a double pendulum S1 S1, × and so on. To solve a classical mechanical system is to solve the equations that govern the motion of the system in its configuration space.

4 It is a basic fact of classical mechanics that the equations of motion are second order ordinary differential equations in the coordinates x1,...,xn of configuration space X1 . The usual trick from ODE theory is to turn a system of n coupled second order equations into 2n coupled first-order equations by introducing auxiliary

dxi variables y1,...,yn, with the extra relations dt = yi. Geometrically, this amounts to rewriting our equation as a first-order ODE on the tangent bundle TX – the variable yi is just the tangent coordinate corresponding to xi. A first order ODE is a precisely a vector field v, and the flow of this vector field gives the dynamics of our system. With this point of view, one may understand a classical mechanical system as a manifold X of configurations, together with a particular vector field v on TX.

The solution to the system is the one-parameter group of diﬀeomorphisms generated by v.

Mathematics alone can only take us this far – to give this description any physical meaning, we must appeal to some experimentally verified physical principles to tell us what the vector field on TX is. There is no unique way to formulate such a ‘fundamental principle’ or set of principles, and different but (more or less) equivalent ways of phrasing the same principle give rise to different but (more or less) equivalent perspectives or ‘formalisms’ in mechanics.

One such principle is the principle of least action, which states that there is a function

on TX (possibly depending on time), which for a point particle is the diﬀerence L

1 Consider the prototypical example F = mx¨

5 between the kinetic and potential energies of the particle, such that for any interval of time [t0,t1] the trajectories of the system in TX minimize the action integral

t1 S := (x, x,t˙ )dt, L Zt0 where the minimization is taken over all paths with ﬁxed beginning and end points

(x, x˙)(t0) and (x, x˙)(t1). The minimizer for this variational problem is given by the

d ∂ ∂ Euler-Lagrange equation L = L , which, if is convex on ﬁbres, can be solved dt ∂x˙ ∂x˙ L to give the vector ﬁeld on TX .

There is a geometric interpretation of the least action principle. A trajectory of the system between times t and t can be thought of in terms of its graph γ :[t ,t ] 0 1 0 1 → TX [t ,t ]. An inﬁnitesmal variation of the trajectory is a vector ﬁeld δγ on the × 0 1 image of γ. For any such δγ, the least-action principle says that

dt = dt mod ǫ2. L L Zγ Zγ+ǫδγ Writing η := dt, the above can be rewritten as L d 0= dt dt dǫ L − L Zγ Zγ+ǫX 1 = dη(γ′(t),δγ) Z0

= ιδγdη. Zγ

Since this is true for all variations δγ, it follows that γ′(t) ker dη for all t, in other ∈ words the submanifold im γ is tangent to the (involutive) distribution ker dη. It turns

6 out that this distribution is 1-dimensional, so the solutions are uniquely determined by this condition.

The transition from the Lagrangian to the Hamiltonian approach is achieved by working on the cotangent bundle T ∗X instead of the tangent bundle TX. These bundles are isomorphic, but not canonically so. However, if the Lagrangian is L convex on ﬁbres, it provides an isomorphism between TX and T ∗X given by

v T X d T ∗X. x ∈ x 7→ L|TxX ∈ x

Remark. It is worth noting that if we require the above isomorphism to be linear on ﬁbres, then the Lagrangian is forced to be quadratic on ﬁbres, with no linear terms.

In this case we may separate into a constant U C∞(X) and a family of quadratic L ∈ ∂ forms Ex x X on ﬁbres: = π∗U + E. In coordinates x on X, the quantity L { } ∈ L ∂x˙ is linear in x˙, so we may write it as mx˙, where m is a vector of constants. The

Euler-Lagrange equation then becomes

mx¨ = U. ∇

So U is identiﬁed with the negative potential, and E with the kinetic energy.

1 Vis a vis the isomorphism T ∗X = TX described above, the form η := dt Ω (TX) ∼ L ∈ 1 appearing in the least action principle becomes α Hdt Ω (T ∗X), where α is the − ∈ tautological 1-form on T ∗X and H is the Legendre transform of . L So in the Hamiltonian approach, classical mechanics lives on the cotangent bundle

T ∗X, and a starring role is played by the symplectic form ω = dα. One may now

7 forget the Lagrangian origins of the theory, and simply state that any function H ∈ ♭ C∞(T ∗X) generates dynamics on T ∗X via the hamiltonian vector ﬁeld dH . Notice that the convexity requirement in the Lagrangian theory is gone – in particular, some Hamiltonians have no corresponding Lagrangians. In this sense Hamiltonian mechanics generalizes its Lagrangian counterpart.

Symplectic geometry, then, is another generalization still – instead of living on a cotangent bundle, it lives on any space locally isomorphic to a cotangent bundle, where the notion of isomorphism must preserve the symplectic form. This is precisely the deﬁnition of a symplectic manifold as a manifold with a symplectic atlas, which in turn is equivalent to the more common deﬁnition as a manifold with a closed nondegenerate 2-form.

2.2 Symplectic and hamiltonian group actions

In physics one is interested not just in classical mechanical systems, but particularly in ones with symmetry. To formalize this:

Deﬁnition 2.2.1. A smooth action of a Lie group G on a symplectic manifold (M, ω) is symplectic if it preserves the symplectic form, ie. if g∗ω = ω for every g G. ∈ The next deﬁnition is motivated by Noether’s theorem, which establishes a rela- tionship between symmetries of a classical mechanical system and its conserved quantities. The most basic example of this phenomenon can be seen as follows: if f C∞(M) is an observable that commutes with the Hamiltonian H, then the ∈ 1-parameter group of transformations of M generated by the Hamiltonian vector

8 ﬁeld of f is a 1-dimensional symmetry which corresponds to the conserved quantity f. Conversely, if ρ is a symplectic R-action on M, then (ρ )♯ is closed so (ρ )♯ = df ∗ ∗ for some multi-valued function f – this is the conserved quantity corresponding to the symmetry ρ.

Deﬁnition 2.2.2. A symplectic action ρ of a Lie group G on a symplectic manifold

(M, ω) is said to be Hamiltonian if there is a map µ : M g∗, termed the moment → map, such that for any X g we have ∈

ρ (X)♯ = dµX , ∗

X X where µ C∞(M) is defined by µ (p) := µ(p),X . ∈ h i There are two obstructions to making a symplectic action Hamiltonian. The first obstruction arises if the fundamental vector fields ρ (X) of the action aren’t Hamil- ∗ tonian. To put it precisely, the composite map

ρ∗ g X(M) H1(M) −→ → taking an element X g to the cohomology class of ρ (X)♯ must be zero. If this is ∈ ∗ satisﬁed, we are guaranteed that each fundamental vector ﬁeld ρ (X) is generated ∗ by a Hamiltonian HX .

However, the Hamiltonians HX are defined only up to an additive constant, and the second obstruction arises in fixing this ambiguity in a consistent way. Let Ham(M) ⊂ X(M) denote the Hamiltonian vector fields – in other words the set of symplectic vector fields v that satisfy [v♯]=0 H1(M). This is itself a Lie algebra; in fact, ∈ 9 Ham(M) is the Lie algebra of the symplectomorphism group of M. It fits into the exact sequence

0 R C∞(M) Ham(M) 0, (2.1) → → → → in other words C∞(M) (endowed with the Poisson bracket) is a central extension of Ham(M) by R. The central extensions of a Lie algebra are in one-to-one correspondence with its second Lie algebra cohomology (see [16]). The extension class corresponding to (2.1) belongs to H2(Ham(M), R), the second Lie algebra cohomology of Ham(M) with values in the trivial Ham(M)-module R. The pullback of this universal class by the map ρ : g Ham(M) lies in H2(g, R), and represents the ∗ → second obstruction to promoting a symplectic action to a hamiltonian one. Thus we have shown:

Proposition 2.2.1. Every symplectic action of a Lie group G on a symplectic manifold M corresponds to a Hamiltonian action of some central extension of G by R on some cover of M.

(for a detailed proof see section 15.2 of [11]). In the case that G is compact, the second obstruction vanishes because one may introduce a left-invariant volume form R on M and ﬁx the -ambiguity in the Hamiltonians by requiring M HX = 0. R 2.3 Coadjoint orbits

In this section we summarize relevant facts about coadjoint orbits, which form an important class of examples of compact symplectic manifolds carrying Hamiltonian group actions. Let G be a Lie group and g ρ Diﬀ(G) be the conjugation action 7→ g ∈

10 deﬁned by

1 ρ : h ghg− . g 7→ Since each ρ ﬁxes the identity element e G, we may diﬀerentiate every ρ at e to g ∈ g obtain a map G Aut(g): → Ad : g d ρ . 7→ e g This is the adjoint representation of G. The derivative of the adjoint action is precisely the Lie bracket:

d Ad Y = Y dt exp tX LX :=[X,Y ].

If we dualize the adjoint representation we obtain the coadjoint representation of G:

Ad∗ : g (d ρ −1 )∗ Aut(g∗). 7→ e g ∈

In other words, for X g and ξ g∗: ∈ ∈

Adg∗ ξ,X = ξ, Ad g X . h i h − i where , denotes the pairing between g∗ and g (we write Ad for Ad(g), etc.). h i g

The adjoint representation preserves the Lie bracket on g, ie. Adg([X,Y ]) = [Adg X, Adg Y ] (this is immediate if G is a matrix group; see Example 2.3.1).

11 Example 2.3.1. If G is a matrix group then the adjoint action is given by matrix conjugation:

1 AdA(X)= AXA− .

The orbit of ξ g∗ under the coadjoint action of G is called the coadjoint orbit of ∈ ξ, denoted O . The stabilizer Stab(ξ) G is a closed subgroup of G, so O is a ξ ≤ ξ 2 manifold via the identiﬁcation Oξ ∼= G/ Stab(ξ) . This gives us an algebraic char- acterization of coadjoint orbits as quotients of G. A corresponding “inﬁnitesmal” statement characterizes the tangent spaces of coadjoint orbits as quotients of g:

Proposition 2.3.1. For ξ g∗, let stab(ξ) g denote the Lie algebra of Stab(ξ) ∈ ⊂ ⊂ G. Then Tξ(Oξ) ∼= g / stab(ξ).

The coadjoint orbit Oξ through ξ admits a natural symplectic form (a closed nondegenerate 2-form), known as the Kirillov-Kostant-Souriau symplectic form. We begin by deﬁning a skew-symmetric bilinear form ωξ on g by:

ω (X,Y )= ξ, [X,Y ] . ξ −h i

The following proposition allows ωξ to descend to a nondegenerate skew-symmetric bilinear form on g / stab(ξ) ∼= Tξ(Oξ):

2 For details on homogeneous manifolds, see chapter 21 of [14].

12 Proposition 2.3.2. The kernel of ω is stab(ξ) g. ξ ⊂

We call the reduced 2-formω ˜ξ; it is a nondegenerate 2-form on TξOξ. Explicitly:

ω˜ (X,Y )= ξ, [X,Y ] , ξ −h i where X denotes the fundamental vector ﬁeld on O generated by X g, ie. X := ξ ∈ d (Ad∗ )∗. Since G acts on O , we may ask ifω ˜ is compatible with this action. dt |t=0 exp tX ξ The answer is in the aﬃrmative:

(Ad∗)∗ω˜(X ,Y )= Ad∗ ξ, [Ad X, Ad Y ] g ξ ξ −h g g g i

= Ad∗ ξ, Ad [X,Y ] −h g g i = ξ, [X,Y ] . −h i

In other wordsω ˜ is G-invariant. Knowing now thatω ˜ is a (G-invariant) nondegenerate 2-form on Oξ, we must now only check that it is closed. This follows from a direct computation:

dω˜(X ,Y ,Z )= X(˜ω (Y ,Z)) Y (˜ω (X,Z)) + Zω˜ (X,Y ) ξ ξ ξ ξ − ξ ξ ω˜ ([X,Y ],Z)+˜ω ([X,Z],Y ) ω˜ ([Y ,Z],X). − ξ ξ − ξ

The ﬁrst three terms all vanish becauseω ˜ξ is G-invariant, so

d Xω˜(Y ,Z)= (Ad∗ )∗ω˜(Y ,Z)=0, dt exp(tX) t=0

and so on. The last three terms can be rewritten as the expression

ξ([[X,Y ],Z] [[X,Z],Y ]+[[Y,Z],X]) −

13 which vanishes due to the Jacobi identity. We may identify g with g∗ using an inner product on g. If this inner product is Ad-invariant, it identiﬁes adjoint orbits with coadjoint orbits. An Ad-invariant inner product always exists, for instance, if G is compact [12], or if G is a linear algebraic group. In the latter case it is given by

(X,Y ) = tr(XY ). So far we’ve established that the coadjoint orbit Oξ has a symplectic formω ˜, and that G acts on Oξ by symplectomorphisms (ie.ω ˜ is G-invariant). In fact, it is Hamiltonian:

. Theorem 2.3.1. The inclusion O ֒ g∗ is a moment map for the G-action on O ξ → ξ

One case which will be important in what follows is of coadjoint orbits of U(n) and SU(n). The Lie algebra u(n) of U(n) can be identiﬁed with iMsa, where Msa is the set of n n self-adjoint complex matrices. Coadjoint orbits of U(n) are then × conjugacy classes of self-adjoint matrices under conjugation by U(n). By the spectral theorem, these are untarily diagonalizable to a matrix of the form diag(iλ1,...,iλn) with λ ... λ , so conjugacy classes in u(n) are indexed by these increasing 1 ≥ ≥ n sequences of real numbers. A similar correspondence exists when one replaces U(n) with U(n) and requires matrices in Msa to be trace-free. Both of these are special cases of a more general correspondence, valid for compact Lie groups, between the set of coadjoint orbits and points in a positive Weyl chamber in the Lie algebra of a maximal torus.

14 The coadjoint orbits of U(n) and SU(n) also admit descriptions as ﬂag varieties:

Proposition 2.3.3. The coadjoint orbit of U(n) (resp. SU(n)) corresponding to

n eigenvalues λ1 >...>λn is isomorphic to the space Fl(n) of full ﬂags in C .

Proof. Given an element iX u(n)= iM , we have a decomposition Cn = n W ∈ sa j=1 λj into 1-dimensional eigenspaces. The spaces Vk := j k Wj then give the ﬂagL corre- ≤ sponding to iX. L

Conversely, given a flag W ... W , one can choose an orthonormal basis 1 ≤ ≤ n h ,...,h of Cn such that W = span h ,...,k . This is ambiguous up to mul- 1 n k { 1 k} iθj tiplication of each hj by some phase e , so H := (h1,...,hn) is defined up to right-multiplication by T = diag(eiθ1 ,...,eiθn ) . The element of corre- { } Oiλ1,...,iλn 1 sponding to the flag W1,...,Wn is then H diag(λ1,...,λn)H− . The proof of the SU(n) case is essentially identical.

There is a similar description for non-generic orbits (ie. where some of the λi’s coincide) in terms of partial ﬂag varieties, but this won’t be necessary for our pur- poses.

2.4 Symplectic reduction

A hamiltonian action of a Lie group G on a symplectic manifold M allows us to simplify the situation by quotienting or ‘reducing’ by the symmetry. This technique is central to classical mechanics – if a system exhibits some symmetry, one can remove the symmetry to reduce the system to a lower-dimensional one.

15 A classical example is a particle of unit mass in the plane with radial potential U: let r R2 be the position, written in polar coordinates as r = (r, ϕ). In this this ∈ problem the angular momentum M =ϕr ˙ 2 is conserved, and the problem can be rewritten as a system depending only on r, given by the reduced hamiltonian

r˙ M 2 H (r)= + U(r)+ . red 2 2r2

Once the reduced system (involving only r) is solved, the angular component ϕ can be determined from r(t) (for more details see [1], chapter 8A).

For the general case, one reduces at a coadjoint orbit:

Theorem 2.4.1. Let (M, ω) be a symplectic manifold, G a Lie group, and ρ a hamiltonian action with moment map µ, and suppose that M/G is a manifold. Then there is a Poisson structure on M/G that is related to the Poisson structure on M by the projection map p : M M/G. Furthermore, each symplectic leaf of M/G is → 1 1 equal to p(µ− ( )) = µ− ( )/G for some coadjoint orbit . O O O A proof can be found in [3].

Deﬁnition 2.4.1. Consider hamiltonian action of a Lie group G on a symplectic manifold M with moment map µ, and let g∗ be a coadjoint orbit. The space O⊂

1 µ− ( )/G := M// G, O O

(if it is a manifold) is called the symplectic reduction of M by G at . O

16 Example 2.4.1. A Lie group G acts on its cotangent bundle T ∗G by left-multiplication on the base and the coadjoint action on ﬁbres, with moment map the projection

T ∗G = G g∗ g∗. The quotient T ∗G/G = g∗ carries the Lie-Poisson structure ∼ × → ∼ and the symplectic leaves are coadjoint orbits. Thus coadjoint orbits naturally arise as symplectic quotients.

Example 2.4.2. Recall the situation of a particle in R2 with radial potential U(r).

The rotational invariance of the problem manifests itself as a Hamiltonian action of

2 2 U(1) on T ∗ R that preserves H. The associated moment map T ∗ R u(1)∗ = R is → ∼ the angular momentum M =ϕr ˙ 2. The equivariance of the moment map shows that

♭ 2 the hamiltonian vector ﬁeld dH preserves the ﬁbration µ : T ∗ R R, in other words → the dynamics lives on level sets of µ. Since the Hamiltonian is U(1)-invariant, it is

1 obtained from a function Hred on µ− (M)/U(1), which, as it turns out, is isomorphic to T ∗ R, the phase space of the reduced system.

2.5 Integrable systems

An integrable system is, roughly speaking, a system with as many symmetries as one could hope for. The classical deﬁnition, found for instance in [1], states that an integrable system is a symplectic manifold of dimension 2n with functions F =

(f ,...,f ) C∞(M) that Poisson-commute and are functionally independent, in 1 n ∈ the sense that df1,...,dfn are independent everywhere (equivalently one may say that F is a submersion). The Hamiltonian ﬂows of the functions f1 ...fn deﬁne an

n 1 action of R on M that preserves the level sets F − (x) which, if they are compact, are tori. For more on integrable systems, see [1].

17 2.6 The Gelfand-Cetlin integrable system

Let T U(n) be a ﬁxed maximal torus and R+ t a choice of positive roots. The ⊂ ⊂ eigenspaces of a generic element X of the positive Weyl chamber in t, in decreasing

n order of eigenvalue, give a ﬂag W1,...,Wn on C . Explicitly:

Wi := Vλ,

λ λi M≥ where λ1,...,λi are the eigenvalues of X in decreasing order, and Vλ is the λ- eigenspace of X. Writing Pi for the orthogonal projection onto Wi, we may consider, for any element iX of a coadjoint orbit , the eigenvalues µ i of P XP in de- O { ij}j=1 i i creasing order.

Proposition 2.6.1. The eigenvalues µ satisfy µ µ µ . i,j i,j ≥ i+1,j ≥ i,j+1

Proof. The statement follows from the expression, for any self-adjoint matrix A, of the kth largest eigenvalue of A as

x, Ax sup inf h i. Cn x V 2 V ∈ x dim≤V =k k k

18 This property can be conveniently visualized as

µ1,1 µ1,2 µ1,3 ... µ1,n

µ2,1 µ2,2 ... µ2,n 1 −

µ3,1 ... µ3,n 2 −

...

µn,1 where arrows indicate inequalities. On a coadjoint orbit , the top row is Oiλ1,...,iλn constant and consists of the eigenvalues λ1 through λn. Fixing the top row this way and letting the rest of the values vary, we obtain a polytope:

n(n 1) Deﬁnition 2.6.1. For λ ... λ R, the set of − -tuples of numbers 1 ≥ ≥ n ∈ 2

(νij)i=2,...n satisfying j=1...i

ν λ ν , 1 j n 1 2,j ≥ j ≥ 2,j+1 ≤ ≤ − ν ν ν , 3 i n, 1 j n i +1 i,j ≥ i+1,j ≥ i,j+1 ≤ ≤ ≤ ≤ − is called the Gelfand-Cetlin polytope ∆ of the coadjoint orbit . λ1,...,λn Oiλ1,...,iλn Whenever a coadjoint orbit is mentioned without explicit reference to its cor- O responding eigenvalues iλ1,...,iλn, we will suppress notation and write ∆ for O

∆λ1,...,λn .

19 Proposition 2.6.2. For a coadjoint orbit , the map (µij): ∆ is an integrable O O → O system.

The proof can be found in [6].

Definition 2.6.2. The relative Gelfand-Cetlin functions F T : R with respect ij O → + T to (T,R ) are defined to be the components of the map F := (µij) : ∆ O → O M Stab M above. For a generic M u(n)∗, we define F := F , with the positive roots ∈ determined by decreasing order of the eigenvalues of M.

Taking T to be the standard (diagonal) maximal torus of U(n) recovers the usual deﬁnition of the Gelfand-Cetlin maps, as in [6]. In fact, if the torus T is considered

ﬁxed, this ‘relative’ version is equivalent to the standard one by the following equivariance property:

Proposition 2.6.3. For a maximal torus T , the Gelfand-Cetlin map enjoys the property:

g−1Tg 1 T F (g− Mg)= F (M).

Equivalently, for a generic element N u(n)∗, ∈

g−1Ng 1 N F (g− Mg)= F (M).

The Hamiltonian ﬂows induced by the Gelfand-Cetlin functions can be computed explicitly for generic coadjoint orbits as follows.

20 Lemma 2.6.1. Let A be a smooth family of n n matrices with A self-adjoint. t × 0

Suppose λ0 is an eigenvalue of A0 of multiplicity one. Let λt be the eigenvalue of At corresponding to λ0 (this makes sense for small values of t). Then, writing

dAt Y := dt (0), we have dλ t (0) = tr(Y P ), dt λ0 where Pλ0 is the projection onto the λ0-eigenspace of A0.

Proof. Let v be a λ0-eigenvector of A0. We wish to solve the equation

2 (A0 + ǫY )(v + ǫw)=(λ0 + αǫ)(v + ǫw) mod ǫ , for the unknowns w Cn and α = dλt C. Expanding and equating the ǫ- ∈ dt ∈ coeﬃcients yields

(Y αI)v =(λ I A )w, − 0 − 0 in other words,

Yv αv im(λ I A ) = ker(λ I A )⊥ − ∈ 0 − 0 0 − 0

= v ⊥, h i so Yv,v α = h i = tr(Y P 0 ). v 2 λ k k

+ Now ﬁx a maximal torus T and R a set of positive roots, let W1,...,Wn be the associated ﬂag, and write P for the orthogonal projection onto W . For1 i n 1, i i ≤ ≤ −

21 write M for the map P M : W W . If T is the standard maximal torus, the |Wi i i → i matrix PiMPi is simply

Mi 0 PiMPi =   , 0 0   where M = M is the ith principal minor (submatrix of size i, top-left justiﬁed) i |Wi of M. The above lemma allows us to calculate the components of dF T .

Proposition 2.6.4. Let M and suppose that F T (M) lies in the inte- ∈ Oiλ1,...,iλn T rior of ∆ . Write Qij for the orthogonal projection onto the eigenspace of M Wi O | corresponding to the eigenvalue F T (M). Then for √ 1X u(n): ij − ∈

T T dFij (XM ) = tr(M[Qij,X]).

√ 1tX √ 1tX d Proof. Let Mt = exp − M exp− − . Since dt PiMtPi = Pi[X,M]Pi, lemma 2.6.1 gives

T T dFij (XM ) = tr(QijPi[X,M]Pi)

T = tr(Qij[X,M])

T = tr(M[Qij,X]).

So M QT (M) is the Hamiltonian vector ﬁeld of F T . More precisely, we may 7→ ij ij regard M √ 1QT (M) as a map u(n), which becomes the Hamiltonian 7→ − ij O →

22 vector ﬁeld of F T after composing with the projection u(n) u(n)/ stab M = T ij → ∼ M O at each M . ∈O

T The flows of the Hamiltonian vector fields Qij are given by conjugation by tori which vary over . The bundle of tori implementing these flows is given as follows: for O each M , define ∈O

g leaves Wi invariant,   Si(M)= g U(n): gM = Mg on Wi, and  .  ∈    g =id on Wi⊥     We may conjugate the situation so that T is the standard maximal torus – in this case, the subgroup S (M) U(n) is the the following group of block-diagonal ma- i ⊂ trices:

Stab(Mi) 0 Si(M)=   , 0 I     (recall we write Mi for the ith principal minor of M). Conjugating by elements of S gives a well-deﬁned action on , thanks to the following lemma: i O

1 Lemma 2.6.2. Let M and t S (M), then S (t− Mt)= S (M). ∈O ∈ i i i

Proof. Let s Stab(M ). Then ∈ i

1 1 s− 0 M11 M12 s 0 M11 s− M12       =   , 0 I M21 M22 0 I M21s M22                 and this operation leaves the ith principal minor unchanged.

23 Thus separates into strata, each of which is the orbit under some S (M), and dif- O i ferent strata correspond to diﬀerent subgroups S U(n). One would like to patch i ⊂ these up into a single action on . However, over non-generic strata, the groups O

Si(M) may not even be tori, so one must restrict to a suitable open dense subset of by removing the problematic strata. O

Deﬁnition 2.6.3. Let be a coadjoint orbit in general position. The generic locus O ˜ (taken with respect to a maximal torus T with associated ﬂag W ) is the open O { i} dense subset of consisting of those M such that the eigenvalues of M are O ∈O |Wi distinct.

Over ˜, the subgroups S (M) are i-dimensional tori, although they still vary ac- O i cording to stratum. To integrate these into a single torus action we require a global trivialization of the torus bundle S (M) M, which is given by the following: i 7→

Proposition 2.6.5. For each M ˜, the map Ri u(n) given by e √ 1QT ∈ O → j 7→ − ij descends to an isomorphism φ : Ri /2π Zi S (M). i,M → i We now have, for each 1 i n 1 an action of the torus Ri /2π Zi on ˜, deﬁned ≤ ≤ − O as

1 t.M = φi,M (t)− Mφi,M (t).

Proposition 2.6.6. For 1 i

n(n 1) 1 So, all in all, we have a torus of dimension 1 + ... +(n 1) = − = dim ˜ − 2 2 O acting on ˜. O

24 CHAPTER 3 The moduli space of ﬂat connections

The moduli space of ﬂat connections is a symplectic manifold that one associates to a principal G-bundle over a manifold M. Rougly speaking, it indexes the zero-energy vacuum states of the spacetime M. In the proceeding section we outline the way in which it arises.

3.1 Gauge-theoretic origins

In classical electrodynamics the spacetime is M = R4 with the ﬂat Minkowski metric

(dx)2 +(dy)2 +(dz)2 (dt)2. The electric field E is a time-varying 1-form on R3, and − the magnetic field B a time-varying 2-form. These two quantities can be encoded as the coefficients of a single 2-form F (the electromagnetic tensor) by setting F = B + E dt. In terms of F , Maxwell’s equations in vacuum are ∧

dF =0, d⋆F =0. (3.1)

The ﬁrst equation says that F is closed, so since H1(R4) = 0 we may write write F = dA, where A is a 1-form on R4 termed the electromagnetic four-potential. Notice that A is ambiguous up to the addition of a global scalar potential dχ and therefore should be taken to live in Ω1(M)/dΩ0(M). The two equations (3.1) imply that F is

25 harmonic, in other words F extremizes the functional

S(F )= F ⋆F ∧ ZM 2 over ZdR(M), the set of closed 2-forms on M. Equivalently, in a vacuum the electromagnetic potential A extremizes the functional

S(A)= dA ⋆dA. (3.2) ∧ ZM This is a variational principle for classical electrodynamics – the ‘path space’ is Ω1(M)/dΩ0(M) and the action is (3.2). Although the physical meaning of the space

Ω1(M)/dΩ0(M) is not yet clear, one at least gets the feeling that the Lagrangian is natural: if we accept the intuition from quantum mechanics that the gradient of a

ﬁeld is related to its momentum, the functional (3.2) bears some resemblance to the Lagrangian for a free classical-mechanical particle.

It turns out that the space Ω1(M) has a meaningful interpretation as the space of connections on a line bundle over M which arises very naturally in quantum mechanics. Consider a normalized wave function ψ : M C of an electron. As we know, a → global change of phase ψ eiθψ does not change any predictions of the theory, in other words the space of wave functions admits a global U(1) symmetry.

The key is now to insist on local U(1) symmetry – instead of multiplying wave functions by a global phase factor eiθ, we multiply by a factor eiβ(x) that is a function

M U(1): such a transformation is called a (local) change of gauge. What we ﬁnd → is that the wave equations that govern the dynamics of ψ (one may consider the Schrodinger equation as a prototypical example) are not covariant with respect to

26 local change of gauge. The problem arises from the operation of diﬀerentiation. We may express the faliure of the partial diﬀerentiation operator ∂µ to be covariant with respect to a gauge transformation g : M U(1) as follows: →

1 ∂µ(gψ)= g(∂µψ + g− ∂µgψ). (3.3)

1 In other words, in the new gauge the operator ∂µ has become ∂µ + g− ∂µg. The physicist’s remedy is to introduce a correction (or ‘additional degrees of freedom’) to the ∂µ operator that will counteract the above discrepancy, by deﬁning:

Dµ = ∂µ + Aµ,

where in the original gauge Aµ = 0, and a gauge transformation g acts on Aµ by

1 g.A = A g− ∂ g. Replacing the ordinary derivative ∂ with the covariant deriva- µ µ − µ µ tive Dµ makes the equations governing ψ gauge-covariant. What this corresponds to on the mathematical side is viewing wave functions as sections of the hermitian line bundle M C. Sections of the associated principal U(1)-bundle M U(1) are changes × × in trivialization which preserve the hermitian structure, and under such a change of trivialization, the standard ﬂat connection d transforms according to (3.3).

The correction term A (or mathematically, the connection matrix A Ω1(M) µ ∈ ⊗ u(1) = iΩ1(M)) is precisely the four-potential in electrodynamics, and the dΩ0(M)- ambiguity in A is gauge freedom. Indeed, with respect to a gauge transformation

27 g(x)= eiβ(x), the connection matrix A transforms as

iβ iβ g.A = A e− de − = A idβ. −

The electromagnetic field F = dA is then simply interpreted as the curvature of the connection d + A. In this sense, the electromagnetic field may be seen as an outgrowth of the local U(1) ambiguity present in the electron wave function, and the interaction between an electromagnetic field F = dA and an electron wave function ψ is encoded by the covariant derivative D.

Other gauge theories exist, where the spacetime M may be topologically nontrivial, the role of the phase bundle M U(1) is played by a diﬀerent principal bundle P × (which need not be trivial) with structure group some compact Lie group G (which need not be Abelian). Connections on P are called gauge potentials, and the curvature of a gauge potential is a gauge ﬁeld – these are responsible for forces in nature. The generalization of (3.2) is the Yang-Mills action on connections:

S (A)= (F F ), YM B A ∧ A ZM where is an Ad-invariant inner product and F = dA +[A A] is the curvature of B A ∧ the connection A. Minima of the Yang-Mills functional describe the possible gauge

ﬁelds in a vacuum. For a general principal bundle P , the Chern-Weil theory prevents the curvature of a connection on P from vanishing identically – this phenomenon is known as a topological charge. The minimizers of the Yang-Mills functional are

28 instantons, and aside from their obvious relevance in physics they have been studied extensively by mathematicians.

When the bundle is trivial the minima of the Yang-Mills functional are precisely those connections for which the curvature vanishes identically. Such connections are called flat, and the quotient of the space of flat connections by the gauge group is the moduli space of flat conections.

3.2 Flat connections

We’ll begin by recalling a few notions regarding connections on principal bundles and ﬁxing some notation.

Let P π M be a principal bundle over a manifold M with structure group G.A −→ connection on P is a G-invariant splitting of the exact sequence 0 ker dπ T P dπ → → −→ TM 0 of vector bundles over M. The image of the splitting is called the horizontal → bundle and is denoted H T P . At each p P , the projection T P ker(dπ) = g ⊂ ∈ p → p ∼ gives a G-invariant g-valued 1-form ω (Ω1(P ) g)G called the connection form. ∈ ⊗ Pullback of dω by the projection h : T P H is a G-invariant 2-form F on P called → the curvature of ω.

The automorphism group of the bundle P (in the category of G-bundles on M) is called the gauge group . It can be realized as the space of G-equivariant maps G P M G, where M G is equipped the G-action by conjugation on the G factor. → × × An element of the gauge group is called a gauge transformation and describes local changes of trivialization.

29 If the bundle P is trivial the gauge group with smooth maps M G under pointwise → multiplication, and G-invariant forms on P correspond to forms on M. In particular the connection and curvature forms descend to g-valued forms on M, and in fact the space of connections can be identified with Ω1(M) g. These identifications are not ⊗ canonical – they depend on the choice of isomorphism P = M G. From now on, ∼ × we’ll work with the trivial G-bundle over M with a fixed trivialization, ie. we fix P = M G. × The Lie algebra of the gauge group is the space Ω0(M) g of smooth maps G ⊗ M g. An infinitesmal gauge transformation a Ω0(M) g acts on a connection → ∈ ⊗ A Ω1(M) g as follows ∈ ⊗

d ta d ta ta (e .A)= Ad ta .A + e− de dt dt e =[A, a]+ da.

Deﬁning d (a) := da +[A, a], we have a map d : Ω0(M) g Ω1(M) g that one A A ⊗ → ⊗ may check is a (graded) derivation with respect to both the wedge product and the

Lie bracket, and therefore has a unique extension d : Ωk(M) g Ωk+1(M) g A ⊗ → ⊗ for every k 0, which takes the form ≥

d (η)= dη +[A η]. A ∧

It is a fact that Stokes’ theorem remains valid for dA:

30 Proposition 3.2.1. For a Ω∗(M) g and A a connection, we have ∈ ⊗

dAa = a ZM Z∂M

We also have:

Proposition 3.2.2 (Structure equation of E. Cartan [13]). The curvature of a connection A is d (A) Ω2(M) g. A ∈ ⊗

2 A consequence of the structure equation is that (dA) (a) is the derivative of the curvature of A with respect to the inﬁnitesmal gauge transformation a. In particular:

Corollary 3.2.1. If the connection A is ﬂat, (ie. its curvature vanishes identically)

2 then (dA) =0.

3.3 The moduli space over a closed surface

For the rest of the document let G be a connected compact Lie group with discrete center, and denote by g its Lie algebra. Let Σ a surface of genus g, and consider the trivial principal G-bundle G Σ, with gauge group = Map(Σ,G). The space × G ∼ A of connections on Σ is aﬃne, modeled on the space Ω1(Σ) g. Let denote the ⊗ Afl subspace consisting of connections whose curvature vanishes. The moduli space of

ﬂat connections on Σ is deﬁned to be the quotient / . Mg Afl G

31 The moduli space is generically a smooth manifold. The local properties of Mg Mg are encoded in the complex

d d 0 Ω0(Σ) g A Ω1(Σ) g A Ω2(Σ) g 0 (3.4) → ⊗ −→ ⊗ −→ ⊗ →

(Corollary 3.4 ensures that this is indeed a complex).

Proposition 3.3.1. k Denoting by HdA the kth cohomology of this complex, we have

0 Hd = LieStab ([A]) A G H1 = T dA [A]Mg 2 0 HdA =(HdA )∗

Proof. The first two are essentially by definition, and the third follows from Poincaré duality with g-coefficients (using the inner product on g).

Thanks to the above proposition we can compute the index of the complex (3.4) as dim T[A] g 2dimStab ([A]). But the index is also dim G χ(Σ), so we have M − G ·

dim T[A] g = (2g 2)dim G + 2dimStab ([A]). M − G

Thus dim T[A] g is minimal at those points where dimStab ([A]) is minimal – one M G can show that this happens when Stab ([A]) = Z(G) and that this forms an open G dense subset of . At these points, is a smooth manifold of dimension (2g Mg Mg − 2)dim G + 2dim Z(G).

32 The beneﬁt of working over a surface Σ is that the wedge product of two ﬂat connections is a g g-valued 2-form and can be integrated over Σ using an Ad-invariant ⊗ inner product, giving a symplectic form on . Mg

Deﬁnition 3.3.1. The Atiyah-Bott symplectic form on takes the form Mg

ω([α], [β]) = (α β), B ∧ ZΣ for [α], [β] T . ∈ [A]Mg Notice that this is well-deﬁned on gauge-equivalence classes of tangent vectors: if α = d a for some a Ω0(M) g, then α β is a d -coboundary and the integral van- A ∈ ⊗ ∧ A ishes by proposition 3.2.1. One may check directly that ω is closed and nondegenerate, but it is more satisfying to appeal to the following argument. The space of con- A nections on M G carries a natural symplectic form given by ω(α, β)= (α β). × M B ∧ The action of the gauge group on is hamiltonian, and the moment mapR takes a A connection A to its curvature. We then have

Proposition 3.3.2. The moduli space of ﬂat connections is the symplectic quo- Mg tient // of the space of all connections by the hamiltonian action of the gauge A 0G group.

The moduli space can also be deﬁned in terms of representations of the fundamental group of Σ in G. For a loop γ based at some x Σ, the holonomy HolA : P P 0 ∈ γ x0 → x0 with respect to a connection A is given by multiplication by an element g G. In ∈ A general the holonomy Holγ is not homotopy-invariant, and this dependence is coded

33 for by the curvature of the connection. Indeed, by proposition 3.2.1, if H : [0, 1]2 Σ → is a homotopy between two paths γ and γ′, then

A A Holγ Holγ′ = H∗FA. − 2 Z[0,1] A In particular if A is flat then Holγ is homotopy-invariant and descends to a homo- morphism π (Σ,x ) G. This is the monodromy representation associated to the 1 0 → flat connection A, and the map taking a flat connection to its monodromy representation is the monodromy map.

Proposition 3.3.3. The monodromy map descends to an isomorphism Hom(π (Σ),G)/G, Mg → 1 the quotient of Hom(π1(Σ),G) by the conjugation action.

Remark. 1 C This can be thought of as a nonabelian version of the isomorphism HdR(Σ, ) ∼=

(H1(M))∗ = Hom(π1(M), C).

Proof. First one checks that the gauge group acts on the monodromy representation by conjugation:

A 1 A g. Holγ = g(x0)− Holγ g(x0), and that change of basepoint conjugates the monodromy representation by a constant element of G, and so the monodromy map is well-defined. The inverse to the monodromy map is constructed as follows. Given a representation ρ : π (Σ) G, 1 → one constructs the bundle P := Σ˜ G over Σ, where Σ˜ is the universal cover of Σ ×ρ with the usual π -action. The canonical flat connection on Σ˜ G then descends to 1 × a flat connection on P .

34 In terms of the standard presentation for π1(Σ), the space Hom(π1(Σ),G) is given as g A ,...,A ,B ,...B G : [A ,B ]=1 G. 1 g 1 g ∈ i i ( i=1 ) Y 

3.4 The moduli space over a surface with boundary

When the surface Σ is allowed to have boundary, the moduli space ceases to be symplectic. The problem is that the Atiyah-Bott symplectic form on the space of connections no longer descends to a well-deﬁned form on the quotient / A Afl G because ω(d a, β)= d ([a β]) = [a β] A A ∧ ∧ ZΣ Z∂Σ no longer vanishes. The space instead has a regular Poisson structure which can be described as follows:

Proposition 3.4.1. The moduli space of a genus g surface with N bound- Mg,N ary components is Poisson manifold whose symplectic leaves are obtained by ﬁxing conjugacy classes of the monodromy around the punctures.

A full proof will be given after the extended moduli space is introduced in the next section; however in the remainder of this section we will sketch argument, following

[2], which gives a better idea of the big picture.

Recall that in the case where Σ had no boundary, the moduli space was the symplectic quotient of by the hamiltonian action of the gauge group. When ∂Σ = , A 6 ∅

35 the action of the gauge group is still symplectic but fails to be Hamiltonian.

Proposition 3.4.2 ([2] Proposition 2.2.2). For a Ω0(Σ) g = Lie , the funda- ∈ ⊗ G mental vector ﬁeld ρ (a) has a hamiltonian given by ∗

µa(A)= (a F )+ (a A). B ∧ A B ∧ ZΣ Z∂Σ These functions satisfy

µa,µb = µ[a,b] + (a d b). { } B ∧ A Z∂Σ

One can check that the map (a,b) (a d b) is a cocycle and that its class 7→ ∂Σ B ∧ A in H2(Lie , R) is the obstruction mentionedR in section 2.2. It follows that the G corresponding central extension ˆ of has a hamiltonian action on . The moment G G A map takes the form

A (F ,A ) Ω2(Σ, g) Ω1(∂Σ, g). 7→ A |∂Σ ∈ ⊕

2 1 Here Ω (Σ, g) is identiﬁed with (Lie )∗ as before, and Ω (∂Σ, g) is isomorphic to N G copies of the dual of the Lie algebra of the loop group Map(S1,G), with acting by G ﬁrst restricting a gauge transformation to a boundary component and then applying the coadjoint action. Proposition 3.4.1 then follows from theorem 2.4.1, combined with the fact that the coadjoint orbits of the loop group Map(S1,G) are in one-to-one correspondence with the conjugacy classes in G [3].

36 3.5 Extended and framed moduli

In this section we examine some ways to obtain new moduli spaces from old ones. From now on, ﬁx G to be a compact Lie group.

First, one may replace the gauge group with a normal subgroup ′ to obtain a G G larger moduli space. We will be using subgroups of the following form: let V be a closed set in Σ and deﬁne

V := g : g(x)=id for x V . G0 { ∈G ∈ }

Deﬁnition 3.5.1. The quotient / V is called the (generalized) extended moduli Afl G0 space . Mext Because the restricted gauge group V leaves a ﬂat connection unconjugated over V , G0 we obtain a map Λ : (V ). Now, the quotient group / V = (V ) has a Mext →Afl G G0 G residual action on (Σ), and the quotient by this action brings us to the ordinary Afl moduli space. To summarize, we have a commuting square

(Σ) (int V ) Mext Afl

(Σ) (int V ). M M (arrows to the right denote restriction, and arrows down denote a quotient by (V )). G Another operation that one can apply to a moduli space is to restrict the space of ﬂat connections. Again this restriction will be local – let V Σ be closed and let ⊂ ′ (V ) (V ) be a subset such that every equivalence class in (V ) under Afl ⊂ Afl Afl

37 Map(V,G) has a representative in ′ (V ). Write ′ (Σ) for the space of ﬂat connec- Afl Afl tions that restrict on V to elements of ′ (V ). Let ′ Stab( ′ (Σ)) be a subgroup Afl G ⊂ Afl of gauge transformations on Σ that preserves the space ′ (Σ). Afl

Deﬁnition 3.5.2. The framed moduli space is

= ′ / ′. Mfr Afl G

Notice that according to these deﬁnitions, extended moduli spaces are particular examples of framed moduli spaces. Now let Σ be a surface (not necessarily connected) with N boundary components, and G a simply connected reductive Lie group. Around each boundary component Ci, choose a collar neighbourhood Vi with coordinates s R / Z,r [ ǫ, 0]. Let g Ω1(Σ) g denote the space ∈ ∈ − Afl ⊂ ⊗

g := A : X ,...,X g such that A = X dθ on some neighbourhood of C , Afl { ∈Afl ∃ 1 N ∈ i i} and let g be the subset of gauge transformations that are constant on some G ⊂ G neighbourhood of each Ci. These deﬁne the g-framed moduli space

S g := g / g. Afl G

Replacing g with G

:= g : g = 0 on some neighbourhood of each C , G0 { ∈G i}

38 we obtain the g-extended moduli space which we denote

g := g / . M Afl G0

There is a map Λ : (Σ) gN taking a ﬂat connection to (X ,...,X ), which Afl → 1 n descends to a map S g(Σ) (g / Ad G)N , which will also be denoted Λ. As above, → we have a commuting square

g Λ gN M

S g Λ (g / Ad G)N .

The vertical arrows represent a quotient by g/ = GN . The local properties of G G0 ∼ g are encoded in the complex M

d d 0 Ω0(Σ; g) A Ωg,1(Σ; g) A Ω2(Σ; g) 0, (3.5) → c −→ −→ c →

k g,1 where Ωc is the space of k-forms with compact support and Ω (Σ; g) is the space of g-valued 1-forms on Σ that take the form Xdθ on some neighbourhood of each boundary component. We’ll denote the complex (3.5) by Ω∗ g (Σ). The analog of M proposition 3.3.1 for this complex states

Proposition 3.5.1. The cohomology of Ω∗ g (Σ) is M

H0 =0

H1 = T g, [A]M 2 0 (H )∗ = γ H (Σ) : (γ,X)=0 for all X g and 1 i N . ∼ ∈ dA B ∈ ≤ ≤ ZCi 

39 Proof. The vanishing of H0 follows from Lemma 2.2 in [8], and the equality H1 =

T g is true by deﬁnition. To obtain the expression for H2 consider the following [A]M exact sequence of chain complexes:

0 Ωc∗(Σ, g; dA) Ω∗ g (Σ) g˜1 0, → → M → → where g˜ is the complex with g in the 1st entry and 0 elsewhere. The LES on cohomology gives the desired expression. The full proof can be found in ([8]).

This again lets us compute the dimension of T g (and consequently the smooth [A]M locus of g) thanks to the following fact. The space Ωg,1(Σ; g) admits a decompo- M sition via

0 Ω1(Σ; g) Ωg,1(Σ; g) gN 0, → c → → → which admits a splitting by choosing, for each 1 i N, a 1-form ψ with ψ = dθ ≤ ≤ i i on some neighbourhood of C . The splitting then takes the form (X ,...,X ) i 1 N 7→ N i=1 ψiXi. Knowing this, one computes the index of Ω∗ g (Σ) as dim(G)χ(Σ) + M PN dim(G). Since this is independent of [A] g, one ﬁnds that the smooth locus of ∈M g consists of those connections for which dim H2 is minimized, and the dimension M of the moduli space can be computed. In [8] the smooth locus in g is computed M this way.

0 g,0 If in Ω∗ g (Σ) one replaces Ω (Σ; g) with Ω (Σ; g), the space of functions Σ g M → that are constant on a neighbourhood of each Ci, one obtains the chain complex

40 computing the tangent space of S g(Σ):

d d 0 Ωg,0(Σ; g) A Ωg,1(Σ; g) A Ω2(Σ; g) 0, → −→ −→ c →

We’ll call this complex ΩS∗ g (Σ).

Proposition 3.5.2. The cohomology of the complex ΩS∗ g (Σ) is

N H0 = Lie Stab A G |Ci i=1 M 1 g H = T[A]S ,

2 2 H (ΩS∗ g (Σ)) = H (Ω∗ g (Σ)) M

Proof. Using the exact sequence of chain complexes

0 Ω∗ g (Σ) ΩS∗ g (Σ) g˜0 0 → M → → →

(where g˜0 is the chain complex with g in the 0th entry) we obtain via the LES on cohomology:

0 0 1 H (Ω∗ g (Σ)) = ker δ : H (g˜0) H (Ω∗ g (Σ)) . M → M The map δ is computed to take the form

δ(X ,...,X )=([A ,X ],..., [A ,X ]), 1 N |C1 1 |CN N from which the expression for H0 follows. The expression for H1 is, as usual, by definition. Finally, the equality on H2 is because the two complexes differ only in the first entry.

41 This allows for a computation of the smooth locus and dimension of S g(Σ) in the same way as for g(Σ). M It is worthwhile to pause and outline the differences between the extended and framed moduli spaces and the ordinary moduli space (the quotient of all flat connec- Mg,N tions by all gauge transformations). We will denote the ordinary moduli space of a surface Σ by S (Σ). This way, if Σ is a genus g surface with N boundary components, we have = S (Σ). For the remainder of this section, let Σ be a fixed Mg,N surface, g its genus, and N the number of its boundary components.

S For an equivalence class [A] in the usual moduli space (Σ), the holonomy HolCi (A) around a puncture is a conjugacy class in G. Using the notation established above, we say that the map Λ on takes values in (G/ Ad G)N . Mg,N For the framed moduli space S g the codomain of Λ is (g / Ad G)N , and for the extended moduli space g it is gN . This gives us the following tower of spaces M g Λ gN M

S g Λ (g / Ad G)N

S Λ (G/ Ad G)N .

The map S g S is deﬁned via the inclusion of g into , the space of all ﬂat → Afl Afl connections on Σ.

The Atiyah-Bott symplectic form is well-deﬁned at generic points of g, making M it symplectic on an open dense set [8]. The quotient g/ = GN acts on by G G0 ∼ M

42 conjugating the Xi’s – this action is hamiltonian with moment map Λ. The g-framed moduli space S g is the (ordinary) quotient of g by this action. This is the situation M of theorem 2.4.1, so S g inherits a Poisson structure from g whose symplectic leaves M are the symplectic reductions // G at various GN -orbits in gN . In other words, M O for any choice of X ,...,X g, if one restricts S g to those ﬂat connections whose 1 N ∈ holonomy around the ith puncture is conjugate to Xi, one gets a symplectic leaf,

g 1 which is precisely the symplectic reduction of at Λ− ( 1 ... ). M OX × ×OXN

The representation-theoretic description of the moduli space as Hom(π1,G)/G can be carried over to the setting of extended and framed moduli spaces by writing

Rep(π (Σ,A),G) S (Σ) = 1 , Map(A, G) where A is a closed subset of Σ and π1(Σ,A) is the relative fundamental groupoid of Σ with respect to A. This description of the moduli space generalizes the two deﬁnitions given so far – indeed, taking A = x recovers Hom(π ,G)/G, while setting A = Σ { 0} 1 we obtain = / thanks to the Riemann-Hilbert correspondence between local M Afl G systems and ﬂat connections. If we take A = ∂Σ we get

Rep(π (Σ,∂Σ),G) S (Σ) = 1 . N 1 i=1 Map(S ,G)

The space Rep(π1(Σ,∂Σ),G) is a representation-theoreticQ version of the extended moduli space. To conclude this section, we introduce an analogue of the g-extended and g-framed moduli spaces, where the ﬂat connection is restrcted to take values in a ﬁxed maximal torus on the boundary of Σ.

43 Pick a maximal torus T G and let t be its Lie algebra. In analogy with the ⊂ g-framed moduli space, we set

t := A : X ,...,X t such that A = X dθ on some neighbourhood of C . Afl { ∈Afl ∃ 1 N ∈ i i}

A quotient by (the space of gauge transformations that equal the identity on a G0 neighbourhood of ∂Σ) gives us the t-extended moduli space:

t := t/ . M A G0

The same analysis is possible for t as was performed for g to compute the M M dimension and smooth locus (this is done in detail in [8]). If one deﬁnes t to be the G space of gauge transformations that restrict on a neighbourhood of each boundary component Ci to constant maps with values in the normalizer N(T ) of the torus, we obtain the t-framed moduli space

S t := t/ t. A G

We have a similar ‘tower’ of spaces

t Λ tN M

S t Λ (t/W )N

S Λ (G/ Ad G)N .

One advantage of working with torus framing is that the map S t S can be → realized more explicitly. Let Λ = ker(exp : t T ) and deﬁne ˜t to be the set of → G

44 gauge transformations that, on a neighbourhood of each boundary component Ci, restrict to

g(ri,θi)= M exp(λiθi) for some M N(T ) and λ Λ. A quick calculation shows that t is a normal ∈ i ∈ G subgroup of ˜t and their quotient is ΛN . Thus the projection G

S t = t / t t / ˜t Afl G →Afl G is a covering with elements of Λ acting as deck transformations. As it turns out, this is precisely the cover S t . →Mg,N

Proposition 3.5.3. The map t / ˜t S Afl G → which is induced by the inclusion t is an isomorphism. Afl 3.6 Gluing

Now, with Σ as above, pick two boundary components C1 and C2. By gluing these two together, we may form another surface Σˆ with two fewer boundary components.

One has a map α : S (Σ)ˆ S (Σ) taking a ﬂat connection to its pullback by the → quotient map Σ Σ.ˆ The image of α is the set of ﬂat connections [A] where A is → |C1 conjugate to A . The corresponding extended moduli spaces (Σ) and (Σ)ˆ are |C2 M M N 1 related in the following way: consider the map G G taking g (g,g− ,e,e,...,e). → 7→ Through this map, G acts on (Σ) with moment map µ(A) = X X . Then M 1 − 2

45 g g (Σ)ˆ = (Σ)// −1 G. Thus we have the following diagram M M µ (0)

g(Σ)ˆ g(Σ) M M

S (Σ)ˆ α S (Σ), where the dotted arrow indicates symplectic reduction. Using this diagram one proves:

Lemma 3.6.1. Let f,g be two functions on S (Σ). If f,g =0, then α∗f,α∗g = { } { } 0.

3.7 The Jeﬀrey-Weitsman system

Let Σ denote the disjoint union of 2g 2 pairs of pants. One can obtain the genus-g 0 − closed surface Σ by a sequence of 3g 3 gluings −

Σ0 Σ1 ... Σ3g 3 = Σ. → → → −

This gives a sequence of maps on moduli spaces

α3g−3 α3g−4 α1 2g 2 g = S (Σ3g 3) S (Σ3g 4) ... S (Σ0)= S (P ) − M − −−−→ − −−−→ −→

(here P denotes the three-holed sphere or ‘pair of pants’). Repeated applications of lemma 3.6.1 show that functions on obtained by pulling back Poisson-commuting Mg 2g 2 functions on S (P ) − continue to Poisson-commute.

Recall that the ﬁbers of Λ : S (P ) (G/ Ad G)3 are the symplectic leaves of S (P ). → It follows that any function on S (P ) obtained via pullback by Λ is a Casimir. The

46 ﬁrst few functions of the integrable system on S (Σ) are obtained this way.

Deﬁnition 3.7.1. For 1 i 3 and 1 j 2g 2, the Goldman function ≤ ≤ ≤ ≤ − F˜ij : S (Σ ) G/ Ad G is deﬁned as k k →

F˜ij := Λ π α ... α , k i ◦ j ◦ 1 ◦ ◦ k

2g 2 where π : S (P ) − S (P ) is the jth projection. j → There are 3(2g 2) Goldman functions on Σ , and at each gluing, the eﬀective − 0 number of Goldman functions drops by one because the holonomies at the two glued boundary components are forced to coincide. Thus S (Σ ) has 3(2g 2) k Goldman k − − functions, and in particular S3g 3 = g has 3g 3 of them (the Goldman functions − M − on are the conjugacy classes of the holonomies at the 3g 3 loops in Σ induced Mg − ˜ij by its pants decomposition). Since each Fk has dim(G/ Ad G)=rk G components, there are (3(2g 2) k)rk G independent Goldman functions on S (Σ ), giving rise − − k (3(2g 2) k)rk G to a map F : S (Σ ) R − − . k k → To deﬁne the remaining components of the integrable system we return to S (P ).

Since the Goldman functions are constant on ﬁbres of Λ : S (P ) (G/ Ad G)3, it → is reasonable to look now for functions that, in an appropriate sense, live on the ﬁbres of Λ. More precisely, suppose that for each (X ,X ,X ) (G/ Ad G)3 we have 1 2 3 ∈ 1 an integrable system h(X1,X2,X3) on Λ− (X1,X2,X3), and that these functions vary smoothly, in other words that

h :[A] h ([A]) 7→ Λ([A])

47 is a smooth function from S (P ) to Rd/2, where d := dim G 3rk G +2dim Z(G) is − the dimension of the ﬁbre of Λ. Then the components of h will Poisson-commute on S (P ) by the following:

Lemma 3.7.1. Let f,g be functions on a Poisson manifold M. If, for each symplectic leaf S of M, the restrictions f and g Poisson-commute on S, then f,g |S |S Poisson-commute on M.

Proof. By the Weinstein splitting theorem, M is Poisson-isomorphic near any x M ∈ to S N, where S is a neighbourhood of x in the symplectic leaf of M going through × x, and N is a Poisson manifold whose Poisson tensor vanishes at x. Then for any f,g C∞(M) ∈

πM (dxf,dxg)= πS(dxf,dxg)+ πN (dxf,dxg)=0.

The functional independence of these functions and Λ is given by

Proposition 3.7.1. (Λ,h): S (P ) R3rk G+d/2 is a submersion. →

Supposing now that such a family of integrable systems h(X1,X2,X3) on the symplectic leaves of S (P ) has been found, the remaining components of our integrable system are deﬁned by pulling these back to S (Σk):

48 Deﬁnition 3.7.2. The function Hj : S (Σ ) Rd/2 is deﬁned as k k →

Hj = h π α ... α . k ◦ j ◦ 1 ◦ ◦ k

j The components of Hk Poisson-commute with each other, and they will automati- cally Poisson-commute with the Goldman functions because these are Casimirs. To conclude that (Fk,Hk) forms an integrable system on S (Σk), we need only to check that the components are functionally independent, or equivalently that (Fk,Hk)isa submersion. This is done inductively with the aid of

Lemma 3.7.2. Consider the diagram

αk S (Σk) S (Σk 1) −

Fk Fk−1

(3(2g 2) k)rk G (3(2g 2) (k 1)) rk G R − − R − − − .

On each ﬁbre of Fk, the map αk is a submersion onto the corresponding ﬁbre of Fk 1. −

Proof. The proof proceeds via the diagram

g g (Σk) (Σk 1) M M − p

α S (Σk) S (Σk 1) −

Let A be a flat connection on Σk, A′ be corresponding flat connection on Σk 1, and v a − tangent vector in the fibre of Fk 1 containing α([A])=[A′]. Lift v to a tangent vector − g 1 g v˜ T[A′] (Σk 1) such that dΛ(˜v)=0. Thenv ˜ is tangent to µ− (0) (Σk 1), ∈ M − ⊂ M −

49 and so descends to a tangent vectorw ˜ of g(Σ ) at [A]. Then w := dp(˜w) satisﬁes M k dα(w)= v.

This lemma shows that that (Fk,Hk) is a submersion if (Fk 1,Hk 1) is. The base − − case is given by proposition 3.7.1. Thus we conclude

Theorem 3.7.1. Let h(X1,X2,X3) be a family of integrable systems on the ﬁbres of Λ : S (P ) (G/ Ad G)3 that varies smoothly with (X ,X ,X ). Then the function → 1 2 3

(F3g 3,H3g 3) is an integrable system on g. − − M 3.8 The integrable system on symplectic leaves of S

We now define a family of integrable systems on symplectic leaves of S (P ), for the case G = SU(n). It is easiest to begin by defining the system first on symplectic leaves of the framed moduli space S g(P ). The symplectic leaves of S g(P ) cover the symplectic leaves of S (P ) and this can be used to give an integrable system there.

Recall that the symplectic leaves of S g(P ) are the ﬁbres of Λ : S g (g / Ad G)3, → and that points of g / Ad G are (co)adjoint orbits of G. We have

1 Λ− ( , , )= (X,Y,Z) : X + Y + Z =0 /G, O1 O2 O3 { ∈O1 ×O2 ×O3 } where G acts simultaneously on by conjugation. This action is hamil- O1 ×O2 ×O3 1 tonian with moment map X + Y + Z, so we ﬁnd that Λ− ( , , ) is nothing O1 O2 O3 but the symplectic quotient //G at zero. This space is called the triple O1 ×O2 ×O3 reduced product and is denoted P( , , ). O1 O2 O3

50 There is a densely-deﬁned integrable system on the triple reduced product P( , , ) O1 O2 O3 given by (X,Y,Z) GX (Y ). 7→ The equivariance property of the relative Gelfand-Cetlin map (proposition 2.6.3) ensures that this deﬁnition is coherent. One sees that this is an integrable system by picking a maximal torus T and an element µ T and writing ∈ ∩O2

P( , , )= X ,Y : X + µ + Y =0 /T O1 O2 O3 { ∈O1 ∈O3 }

(see [7]).

From the diagram S g(P )Λ (g / Ad G)3

p exp

S (P )Λ (G/ Ad G)3

g we ﬁnd that p take a symplectic leaf S of S (P ) to a symplectic leaf S′ of S (P ).

One checks easily that the Atiyah-Bott form on S and S′ agree, so p is a symplectomorphism and in particular a local diﬀeomorphism and so (since S (P ) is compact) it is a covering map. Identifying an open dense set of S′ with a fundamental domain in S, we obtain a densely-deﬁned integrable system on S, as desired.

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