Introduction to Symplectic Geometry

Candidate Number: 36954

Abstract

This is an M-level Dissertation submitted as a whole-unit of the Part C course, Honour Schoolof Mathematics, University of Oxford in the year 2006-2007. The dissertation is supervised byDr. Andrew Dancer (Mathematical Institute, University of Oxford).

The development of symplectic geometry originally came from the studies of classical mechanics.Mathematicians and physicists first assign a mechanical system a symplectic structure, and bystudying the symplectic structure one can get information on the behaviour of the mechanicalsystems.

However, the subject has been discovered of great use in many other fields of mathematicsand physics in the current research, for example quantum physics, representation theory, stringtheory etc.

This dissertation first gives an introduction of some of the basic concepts in symplectic ge-ometry and its properties. Some important examples of symplectic manifolds, such as phasespace, coadjoint orbit are introduced in Chapters 1 and 3. Chapters 2 and 4 cover the localand global theory of symplectic manifolds respectively. For the local theory (Chapter 2), twoversions of Mosers Theorem and Darbouxs Theorem are given. For the global theory (Chapter4), properties of some special vector fields and group actions on symplectic manifolds are inves-tigated. A detailed account of the moment map, including its definition, existence, uniquenessand examples are given at the end of Chapter 4.

A more modern approach of the subject is taken afterwards. Chapter 5 focuses on the Marsden-Weinstein-Meyer Theorem, which serves as a tool for reducing the dimension of a symplecticspace into another. Chapter 6 gives a proof of a special case of convexity theorem, which statesthat the image of the moment map of a Hamiltonian abelian group action is a convex polytope,which is called a moment polytope. This finally leads to the final chapter of identifying somespecial kinds of symplectic manifolds with their moment polytopes. Some examples of symplec-tic manifolds with their moment polytopes are given, which is very useful in the current researchof representation theory.

The dissertation keeps track on the classical mechanical origin of some notions whenever possible,yet it is not the main focus of the dissertation.

Contents

1 Introduction 21.1 Symplectic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Local Theory of Symplectic Manifolds 52.1 Isotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Moser Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Darbouxs Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Coadjoint Orbit 113.1 Adjoint and Coadjoint Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Symplectic Structure of Coadjoint Orbits . . . . . . . . . . . . . . . . . . . . . . 12

4 Group Action on Symplectic Manifolds 154.1 Symplectic and Hamiltonian Vector Fields . . . . . . . . . . . . . . . . . . . . . . 154.2 Symplectic and Hamiltonian Actions . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 Moment Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4 Existence and Uniqueness of Moment Maps . . . . . . . . . . . . . . . . . . . . . 204.5 Examples of Moment Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5 Reduced Spaces 245.1 Marsden-Weinstein-Meyer Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 245.2 Examples of Reduced spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Convexity Theorems 296.1 Morse-Bott Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.3 The Theorems of Schur and Horn and its Generalisations . . . . . . . . . . . . . 33

7 Toric Actions on Symplectic Manifolds 357.1 Delzants Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357.2 Examples of Moment Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1

Chapter 1

Introduction

In this chapter, we define what a symplectic manifold is, and give some basic definitions andproperties of it. Finally some simple examples of symplectic manifolds are given, which we willuse quite a lot later.

1.1 Symplectic Forms

Definition 1.1. A symplectic form on a vector space V is a bilinear map F : V V Rsuch that

(a) F is skew-symmetric, i.e. F (v, w) = F (w, v) w, v V(b) F is non-degenerate, i.e. if F (v, w) = 0 w V then v = 0

And (V,F) is called a symplectic vector space.

Remark. Suppose F is a symplectic form on a finite dimensional vector space V , it is easy toshow that there exists a basis e1, . . . , en, f1, . . . , fn satisfying:

F (ei, ej) = F (fi, fj) = 0

F (ei, fj) = ij

i,j = 1,2,. . . ,n

The basis {e1, . . . , en, f1, . . . , fn} obtained above is called a symplectic basis of (V, F ). Also,note that dimV = 2n in the above case, hence the dimension of V is always even.

Definition 1.2. Let (V, F ) be a symplectic vector space, then for any subspace W of V

(a) W is isotropic if F (w1, w2) = 0 w1, w2 W . Denote WF by WF := {v V : F (w, v) =0 w W}, it is easy to see W is isotropic W WF .

(b) W is Lagrangian if W is an isotropic space of maximal dimension.

2

Introduction 3

Definition 1.3. Let (V, F ) and (V , F ) be symplectic vector spaces. A symplectomorphismbetween them is a linear isomorphism : V V such that F ((u), (v)) = F (u, v) u, v V .In other words, F = F . If a symplectomorphism between (V, F ) and (V , F ) exists, we say(V, F ) and (V , F ) are symplectomorphic.

1.2 Symplectic Manifolds

Definition 1.4. A symplectic form on a manifold M is a differential 2-form on M suchthat

(a) p : TpM TpM R is symplectic p M(b) is closed, i.e. d = 0

If M has a symplectic form , then (M,) is a symplectic manifold.

Remark. By definition of a differential 2-form p is skew-symmetric. What we need is thereforeits non-degeneracy and closedness. Also, as in the Remark after Definition 1.1, (M,) is asymplectic manifold implies dimM = dimTpM is even.

Examples 1.5. (a) The easiest example is M = R2n with coordinates x1, . . . , xn, y1, . . . , yn.The 2-form

:=ni=1

dxi dyi

is symplectic, with symplectic basis(

x1

)p

, . . . ,

(

xn

)p

,

(

y1

)p

, . . . ,

(

yn

)p

for each p in M .

(b) This example comes from classical mechanics. It may look artificial when in its first ap-pearance, but its importance will become apparent in Chapter 4 (See Example 4.3). Considera particle (x1, x2, x3) moving in R3 with momentum (p1, p2, p3). Suppose the energy functionH(x, p) = 12m |p|2 + V (x), where V (x) is the potential energy function satisfying

md2x

dt2= V (x)

Then we have the following Hamilton equations

dxidt

=1mpi =

H

pidpidt

= md2xidt2

= Vxi

= Hxi

Introduction 4

In classical mechanics, solving Hamilton equations is equivalent to solving the Euler-Lagrangeequation of the Lagrangian L (see Cannas da Silva [2](pp.120-122) for details). Let oursymplectic manifold be R6 with coordinates (x1, x2, x3, p1, p2, p3), and the symplectic form

=3i=1

dxi dpi

as before. We will see there is a strong interplay between the Hamilton equations and thesymplectic form.

Chapter 2

Local Theory of SymplecticManifolds

We begin our studies of symplectic manifolds by discovering their local properties. The maingoal of this chapter is to prove Darbouxs theorem, which says all symplectic manifolds arelocally symplectomorphic to the canonical form in R2n. So our main interest on symplecticmanifolds will be on its global properties. This chapter follows the approach of Cannas da Silva[2] and McDuff [9].

2.1 Isotopy

Definition 2.1. Let M be a manifold and suppose : M R M . If t : M M satisfies0 = idM and t(m) := (m, t) is a diffeomorphism for every t R, then is an isotopy.

For every isotopy , we can construct a vector field Xt on M by letting

d

dtt = Xt t (2.1)

or, in other words,

(

t|(m,t)) = Xt|t(m)

where : T(m,t)(M R) Tt(m)M is the push-forward of . Hence there is a family ofvector fields {Xt} in M . On the other hand, given a R-family of compactly-supported vectorfields {Xt}, we can solve the differential equation (2.1) to get back the isotopy .

Definition 2.2. A one parameter group of diffeomorphisms of a manifold M is an iso-topy with the extra property that

s+t = s t

Proposition 2.3. For the one-parameter group of diffeomorphisms , Xt = X for all t R. Conversely, if X is a complete vector field, then it generates a one-parameter group ofdiffeomorphisms.

5

Introduction 6

Proof. Suppose s(m) = n M , considerd

dts+t(m) = Xs+t s+t(m)d

dtt(n) = Xs+t t(n) = Xt t(n)

So Xs+t t(n) = Xt t(n). Taking t = 0, Xs|n = X0|n. But n and s can be chosen arbitrarily,hence Xs = X0 for all s R.

Conversely, suppose a complete vector field X in M is given. By definition of completeness,we can integrate (2.1) to get t uniquely for all t R and m M . It is indeed a one-parametergroup of diffeomorphisms - consider s(t(x)), by fixing t and vary s, s is the unique integralcurve of X through t(x). On the other hand, s+t(x) is an integral curve passing through t(x)at s = 0. Hence we must have s t(x) = s+t(x).

Example 2.4. The exponential map, or the flow, of a vector field X on a manifold M isdefined as t :M M with

0(m) = m for every m M, andd

dt(t(m)) = X(t(m))

From Proposition 2.3, t is a one-parameter group, i.e. s+t = s t for all s, t R. Hencewe often denote t by exp(tX).

Definition 2.5. The Lie derivative of a differential form along a vector field X is given by

LX := ddtt|t=0

where t is the one-parameter group generated by the vector field X by Proposition 2.3.

Remark. The above definition can actually be extended to any tensor forms. Also we haveCartans formula for the Lie derivative of differential forms:

LX = dX + Xd (2.2)

Proposition 2.6. For a family of 2-forms t,

d

dttt =

t (LXtt +

dtdt

) (2.3)

where Xt is the family of vector fields generated by t.

Introduction 7

Proof. We claim

d

dtt = tLX (2.4)

Then the result easily follows since

d

dttt = (

d

dss)|s=tt + t (

d

dss)|s=t

= tLXtt + tdtdt

= t (LXtt +dtdt

)

To prove our claim, first note that both sides commutes with the exterior derivative d (useCartans formula for the commutativity on the right hand side). And for f 0(M),

d

dt(t f) =

d

dt(f t) := (Xt t)f

t (LXf) = t (Xt(f)) = Xt(f) t = (Xt t)f

Now for differential forms , so that both agree our claim, i.e.

d

dtt =

tLX ,

d

dtt =

tLX

Then

d

dt(t ( )) =

d

dt(t t )

= (d

dt(t)) t + t (

d

dtt )

= tLX t + t tLX= t (LX + LX )= t (LX( ))

For every differential form , it can be written locally as =fa1a2...ak(x)dxa1 dxak

for some k N. Now (2.4) agree on 0-forms. And by commutativity of d, it agrees for exact1-forms. Also we have shown that (2.4) agree on wedge product. Hence the formula holds forany k-forms (by induction).

Remark. The technique involved in proving equation (2.4) can be used to prove Cartansformula as well. We will come across this again in Lemma 4.5.

2.2 Moser Theorems

First of all, an exposition of Mosers Theorem in a special case.

Introduction 8

Theorem 2.7. Let M be a compact manifold, and 0, 1 2(M) be in the same de Rhamcohomology group. Suppose

t = (1 t)0 + t1be symplectic for all t [0, 1], then there is an isotopy such that tt = 0 for all t [0, 1].Proof. With the assumption given in the theorem,

dtdt

= 1 0

Also, since 0, 1 are in the same cohomology class, there exists 1(M) such that 10 =d. Hence

dtdt

= d

From Cartans formula,LXtt = Xtdt + dXtt = dXtt

since t is closed. Recall Proposition 2.6:

d

dttt =

t (LXtt +

dtdt

)

If there exists Xt such that

LXtt +dtdt

= dXtt + d = 0

then we are done by finding its corresponding isotopy. The above equation can be furtherreduced to the Mosers equation:

Xtt + = 0 (2.5)

But t is non-degenerate, so we can solve Xt for each t [0, 1] smoothly by the uniquenesstheorem of differential equations. Given such Xt we can find its isotopy by compactness ofM .

For the general case of any submanifolds Q of M , we need the following technical result whichthe proof is omitted. A sketch proof of it can be found in Canna da Silva (pp.36-37).

Theorem 2.8 (Tubular Neighbourhood Theorem). Suppose Q is an submanifold of amanifold M, the normal bundle of Q is defined by

NQ = {(q, n) : q X;n NxQ := TxMTxQ

}

Then there exist a convex neighbourhood U of the zero section of NQ, a neighbourhood U of Q,and a diffeomorphism : U U such that (q, 0) = q for all q Q.

Introduction 9

Lemma 2.9. Let U be a tubular neighbourhood of a submanifold Q i M . If a closed k-form on U is such that i = 0 on Q, then is exact, and k1(U) can be chosen such that = d with |Q = 0.Proof. By Theorem 2.8, we consider U on NQ instead. For t [0, 1], let t : U U by(q, n) 7 (q, tn).If hk : k(U) k1(U) is defined for each k by:

hk() := 10t (Xt)dt

where Xt is the family of vector field generated by t. Then

(hk+1 d+ d hk) = 10t (Xtd)dt+ d

10t (Xt)dt

= 10t (Xtd+ dXt)dt

= 10tLXtdt

= 10

d

dttdt

= 1 0

But is closed, d = 0. Also 1 = id|U and 0 : (q, n)pi x i0 (q, 0). So 0 = (i0 pi) =

pi(i0) = 0 by our assumption, hence we end up having

d(hk) =

Taking := hk = 10

t (Xt)dt, = d. Also note that t is constant on {(x, 0) : x X} for

all t, hence Xt = 0 on Q and = 0 on Q.

Theorem 2.10. Let M be a manifold and Q be a submanifold, and 0, 1 be symplectic formssuch that they agree on Q. Then there exists open neighbourhoods U0, U1 of Q and a diffeomor-phism : U0 U1 such that

|Q = idQ1 = 0

Proof. Take a tubular neighbourhood U0 of Q, then in U0, 1 0 is closed and is zero on Q.Hence by Lemma 2.9 there is a 1-form on U0 such that 1 0 = d and |Q = 0.Now consider the family of closed forms on U0

t = 0 + t(1 0) = 0 + td

Introduction 10

Note that t|Q = 0|Q, so t is non-degenerate on Q for every t. Since non-degeneracy is anopen property, by shrinking U0, t can be made non-degenerate for every t [0, 1]. Now thecriteria of Theorem 2.7 is fulfilled (note that [t] = [0] for all t [0, 1]), so we can now solveMosers formula (2.5):

Xtt + = 0 (2.6)

to get our family of vector fields Xt.

We now try to solve (2.1) with the Xt above. Since = 0 on Q, (2.6) says Xt = 0 on Qfor all t. On integrating, t|Q = id|Q for all t [0, 1]. For each q Q there is an open neigh-bourhood Uq such that is uniquely determined on Uq [0, 1]. So, take our new neighbourhoodU0 to be U0

qQ Uq to ensure there is an isotopy

: U0 [0, 1]Msuch that tt = 0 for all t [0, 1]. Finally let := 1, U1 = (U0). Then 1 = 0 and|Q = idQ.

2.3 Darbouxs Theorem

The theorem above is actually a stronger result than Darbouxs Theorem.

Theorem 2.11 (Darbouxs Theorem). Every symplectic form of a 2n-dimensional sym-plectic manifold M is locally diffeomorphic to the standard form

ni=1

dxi dyi

on R2n.

Proof. For any m M , take the symplectic basis of for some neighbourhood U of m, so

|m =ni=1

dpi dqi|m

Now, andn

i=1 dpidqi are symplectic forms in U which agrees at m. By Theorem 2.10 (withX = {m}) there is a neighbourhood U0, U1 of m and a diffeomorphism : U0 U1 such thaton U0:

(ni=1

dpi dqi) = ni=1

dpi dqi = ni=1

d(pi ) d(qi ) =

So take xi := pi , yi := qi and we are done.

Chapter 3

Coadjoint Orbit

Coadjoint orbit is a very special example of symplectic manifolds, which deserves us a separatechapter to investigate its properties. It asserts that for any Lie Group G with Lie Algebra g, thedual g is composed of some disjoint symplectic manifolds. It is useful among different aspectsof mathematics, for example in representation theory, as we will see in the last two chapters.The main references are Meinrenken [10](p.67) for coadjoint orbits, and Hall [6] for results inLie Groups in this chapter.

3.1 Adjoint and Coadjoint Maps

Let G be a Lie Group, and g := LieG be the Lie Algebra of G, i.e. the set of all left invariantvector fields of G. We see how G acts on the the dual space g by coadjoint maps, which isto be defined below.

For any Lie Algebra g of a Lie Group G, there is an isomorphism g = TeG by sending anyvector field X to its image at identity Xe. Therefore there is a corresponding isomorphismbetween g and T eG.

Consider A : G G by g 7 g0gg10 . Its derivative at the identity is therefore given by

Adg0 : TeG TeGor, with our identification above

Adg0 : g gHence it induces a map

Ad : G Autgg0 7 Adg0

And the coadjoint map, Ad, is now given by

Ad : G Autg

g 7 Ad(g) = (Ad(g1))

11

Introduction 12

We are now ready to define the coadjoint orbit:

Definition 3.1. The coadjoint orbit of g is given byO := {Ad(g) : g G} g (3.1)

However, the definition in (3.1) does not give us a very precise picture of how O is like, forexample there may be some g G such that (Adg) = . To solve this problem, consider Oas the image of the map g G 7 Ad(g). By the Orbit-Stabiliser Theorem,

O = G/G (3.2)where G := {g G | (Adg) = }.Now, suppose O related with by = Ad(h) for some h G, it can be easily checkedthat the map G/G G/G given by [g] 7 [hgh1] is a well-defined isomorphism.

3.2 Symplectic Structure of Coadjoint Orbits

Given the submanifold structure of O g, we now try to find a non-degenerate 2-form of O.But first of all we see how the tangent space of O looks like. Taking the infinitesimal version of(3.1),

TO = {ad(X) : X g} (3.3)where d(Ad) = ad and ad(X) : g g is given by

(ad(X))(Y ) = (ad(X)(Y )) = ([X,Y ])By taking (3.2) into account as well, there is an identification of TO by g/g , where g :={X g | ad(X) = 0}. For any vector field X g,

X = [X] g/g = TOX = [Ad(h)X] g/g = TO

with our different identifications of O by G/G and G/G.Proposition 3.2. For each g, define : g g R by

(X,Y ) := ([X,Y ])

Then ker = g, and for any h G, Ad(h)()(Ad(h)X,Ad(h)Y ) = (X,Y )Proof.

ker = {X g | (X,Y ) = 0 Y g}= {X g | [X,Y ] = 0 Y g}= {X g | (ad(X)(Y )) = 0 Y g}= {X g | (ad(X))(Y ) = 0 Y g}= {X g | ad(X) = 0} = g

Introduction 13

Suppose g G , then

Ad(h)()(Ad(h)X,Ad(h)Y ) = (Ad(h))([Ad(h)X,Ad(h)Y ])

= (Ad(h))(Ad(h)[X,Y ])

= Ad(h1)Ad(h)([X,Y ])= ([X,Y ])= (X,Y )

Theorem 3.3. Let be such that

([X], [Y ]) := (X,Y )

then it is a well-defined G-invariant symplectic form of O.Proof. First of all, check the definition of is well-defined, i.e. ifX, X be such that [X] = [X]then (X,Y ) = (X, Y ). But it is just straight from our first assertion in Proposition 3.2.Also by the setting of it is obviously skew-symmetric and linear. Hence the map is a well-defined 2-form for O:

: TO TO Rfor each O.

To show the 2-form is G-invariant (hence smooth), consider

(X, Y) = ([Ad(h)X], [Ad(h)Y ])= (Ad(h)X,Ad(h)Y )= (X,Y ) by Proposition 3.2= ([X] , [Y ])= (X, Y)

Again, by Proposition 3.2, the 2-form is obviously non-degenerate, and we are left to show it isclosed. It can be seen easily after introducing the notion of moment maps in the next chapter,which gives us the equation below (one can refer it to (6.2), where : O g is the inclusionmap)

dX = X

where X is the vector field generated by the one-parameter group exp(tX) as defined in Example2.4. Recall Cartans formula,

Xd = LX dX = LX 0

Using Definition 2.5 of Lie derivatives, the invariance of along O implies LX = 0. HenceXd = 0 for all X and hence d = 0.

Introduction 14

Example 3.4. Let G = SO(3). There is a natural identification of so(3) with R3 by thefollowing: 10

0

0 0 00 0 10 1 0

,010

0 0 10 0 01 0 0

,001

0 1 01 0 0

0 0 0

Under these identifications, one can easily see the following nice properties: For S, T so(3)and its corresponding s, t R3,

12tr(ST ) = s, tAdST = S.t

Now identify so(3) with its dual so(3) by the inner product given as above, i.e. for any so(3)the identification is A , where (B) = 12 tr(AB). For any Y so(3), the coadjoint actionbecomes

AdZY = Z.ywhere y (R3) = R3 is the corresponding element of Y . Hence the coadjoint orbit containingY is isomorphic O = {Z.y : Z SO(3)} = {w : |w| = |y|}, which is a sphere of radius |y|.

The final example of coadjoint orbits in this chapter seems to be an artificial one, but it will beof good use in Chapter 6 (See Theorem 6.10).

Example 3.5. Let G = U(n), and Tn U(n) be the subgroup of all diagonal matrices inU(n). Let u(n) = {M Mn(C) : M + M = 0} identify with u(n) by the inner productA,B = tr(AB) as in last example. Under such identification,

Ad(S)(T ), U = T,Ad(S1)(U)=T, S1US

= tr(T S1US)

= tr(T SU(S1))

= tr((S1)T SU)

= tr((STS1)U)

=STS1, U

So Ad(S)T = STS1, which means the coadjoint action on g corresponds to the usual adjointaction of matrices with our identification of u(n) with u(n). Also notice that for any S U(n),STS1 still lies in u(n) by direct verification or Lie Group theory.

Now, let T tn g = g, i.e. T u(n) is diagonal. The coadjoint orbit containing T , O,is isomorphic to

U(n)/Stab(T ) = U(n)/{S U(n) : (AdS)(T ) = T}= U(n)/{S U(n) : STS1 = T}= U(n)/Tn

since T is diagonal. Also, there is a symplectic form on O given by T (A,B) := T, [A,B],where A, B u(n) (or u(n)/tn to be precise).

Chapter 4

Group Action on SymplecticManifolds

This chapter investigates some global properties of symplectic manifolds. We first introduce cer-tain kinds of vector fields on a symplectic manifold, of which originate from classical mechan-ics. Afterwards, a generalisation of this notion to the group actions of symplectic manifoldsis given, which leads to the definition of the moment map, the most important tool in thestudies of symplectic manifolds.

4.1 Symplectic and Hamiltonian Vector Fields

Definition 4.1. Let (M,) be a symplectic manifold. A vector field X on M is symplectic ifLX = 0.Remarks. Recall the definition of the Lie derivative. X is symplectic if and only if

t =

for all t, where t is the one-parameter group generated by X. Also, by Cartans formula (2.2),it is easy to see d(X) = 0 X is a symplectic vector field.By the non-degeneracy of the symplectic form , we can find a unique vector field Xf suchthat Xf = df for any given smooth function f :M R. We haveDefinition 4.2. Let (M,) be a symplectic manifold and X be a vector field on M , then X iscalled the Hamiltonian vector field with Hamiltonian function f if X satisfies

X = df (4.1)

Obviously, a Hamiltonian vector field is a symplectic vector field.

Example 4.3. Continuing our classical mechanics example in 1.5(b), let X be the vector fieldgenerated by the motion of the particle, i.e. (t) = (x1(t), x2(t), x3(t), p1(t), p2(t), p3(t)), t R.From the Hamilton equations, we obtain

X =ni=1

(H

pi

xi Hxi

pi)

15

Introduction 16

It is a Hamiltonian vector field with Hamiltonian function exactly equal to H(x, p) = 12m |p|2 +V (x). To see this, consider

X =3i=1

X(dxi dpi) =3i=1

(H

pidpi +

H

xidxi) = dH

In conclusion, the motion of the particle in R3 is indeed characterised by the Hamiltonian vectorfield on its corresponding symplectic manifold R6, which is called the phase space of theparticle.

Definition 4.4. The Poisson bracket of smooth functions f, g :M R is

{f, g} := (Xf , Xg) (4.2)

Lemma 4.5. For any vector fields X,Y of M and any differential forms ,

[X,Y ] = LXY Y LX (4.3)

Proof. We use the same technique as in Proposition 2.6. The formula is obviously true for0-forms, i.e. smooth functions on M . If df is exact in 1(M), then

(LXY Y LX)(df) = LX(Y (f)) Y (X(ddf) + d(X(df)))= X(Y (f)) Y (d(X(f)))= X(Y (f)) Y (X(f))= [X,Y ](f)= [X,Y ](df)

So the equation holds for exact 1-forms. Now for any differential forms , of degree p, qrespectively,

[X,Y ]( ) = [X,Y ] + (1)p [X,Y ](LXY Y LX)( ) = (LXY Y LX)() + (1)p (LXY Y LX)()

Our desired result now follows.

Proposition 4.6. If X,Y are symplectic(or Hamiltonian) vector fields on symplectic manifold(M,) then [X,Y ] is Hamiltonian with Hamiltonian function (Y,X). Hence,

X{f,g} = [Xf , Xg]

Introduction 17

Proof. Consider [X,Y ]. By Lemma 4.5,

[X,Y ] = LXY Y LX= (Xd+ dX)Y Y (Xd+ dX)= 0 + d(XY ) + Y X(d) + 0= d((Y,X))

where the second last line comes from the fact that X, Y are closed(or exact for Hamiltonianvector spaces), and the last line comes from the fact that is closed.

From above, [Xg ,Xf ] = d((Xf , Xg)). So X{f,g} := X(Xf ,Xg) = [Xg, Xf ] = [Xf , Xg]

4.2 Symplectic and Hamiltonian Actions

Every one-parameter group of diffeomorphisms t :M M for each t R has a correspondingvector field X on M . In the last section, a vector field X on a symplectic manifold (M,) is aHamiltonian vector field if X satisfies

X = df

for some smooth function f :M R.

This section deals with the analog g : M M of each g in any Lie group G. We try toextend the notion of Hamiltonian vector field to every G-parameter group of diffeomorphisms.

Definition 4.7. Let M be any manifold. An action of a Lie group G on M is a group homo-morphism

: G D(M)g 7 g

where D(M) is the set of all diffeomorphisms M M . The action is said to be smooth if themap (m, g) 7 g(m) is smooth.

Example 4.8. The one-parameter group of diffeomorphisms is an R-action on M . The map isgiven by

: t 7 t

Definition 4.9. Let (M,) be a symplectic manifold. A smooth action : G D(M) is calleda symplectic action if g = for every g G.

Introduction 18

Remark. We are now extending the notion of symplectic vector field to symplectic action.For the case in one-parameter group of diffeomorphisms, if X is a complete vector field on Mwe have our usual smooth actions of R, t, on M . But from Remark after Definition 4.1,

X is a symplectic vector field t = for all t is a symplectic actionHence these two notions hold in one-parameter group of diffeomorphisms.

Example 4.10. Let M = S2 and = ddh, where can be treated as the angle and h as theheight of S2. Let X = , then the one-parameter group generated by X is {(+ t, h) : t R}.Since + 2pi = , the map

t : S2 S2(, h) 7 ( + t, h)

defines an action on S2. Note that it is an symplectic action on S2, i.e.

t (d dh) = d( + t) dh = d dhfor every t R.

4.3 Moment Map

The name moment map actually comes from the generalisation of linear momentum and angularmomentum in classical mechanics. One can refer to McDuff [9] (p.165) for more details.

Definition 4.11. Let G be a Lie Group with Lie algebra g, which acts on a symplectic manifold(M,), i.e.

g :M M for every g GFor any X g, let X be a vector field on M generated by the one-parameter group of diffeomor-phisms:

t := exp(tX)(e) :M Mwhere exp(tX) is the exponential map defined in Example 2.4, e is the identity element in G.Then is weakly Hamiltonian if every X is exact.

A Hamiltonian action is a weakly Hamiltonian action with a map

:M g (4.4)called the moment map, such that

(a) for every X g, X : M R given by X(m) := (m)(X) is the Hamiltonian functionof X defined above, i.e.

X = dX (4.5)

Introduction 19

(b) is equivariant with respect to the action on M and the coadjoint action Ad on g, i.e.the following diagram commutes:

Mg M

y yg

Ad(g) g

Remarks. We have the following inclusions

Hamiltonian actions weakly Hamiltonian actions symplectic actionsIn some other texts, the moment map is defined as

C(M) g

such that

(a) , X := X is the Hamiltonian function for X as above, and(b) [X,Y ] = {X , Y }

The first condition above meets the first one in Definition 4.11, and the second condition meetsas well after the next proposition.

Proposition 4.12. The equivariance condition is equivalent to

[X,Y ] = {X , Y } (4.6)Proof. We prove (4.6) is the infinitesimal version of the equivariance condition. For momentmap , X,Y g,

X g = Adg XX g =

,Adg1(X)

by definition of Ad

X exp(tY )(e) =,Ad(exp(tY )(e))1(X)

dX(Y ) = ,[Y,X] by taking the infinitesimal versionX(Y ) = , [X,Y ](X, Y ) = [X,Y ]

{X , Y } = [X,Y ]

Now, the preceding lines can be reversed, since every Y g is complete (See Hall [6](p.311)), itgenerates a unique one-parameter group of diffeomorphisms back. Hence the two conditions areequivalent.

Introduction 20

4.4 Existence and Uniqueness of Moment Maps

By definition, every Hamiltonian action is a symplectic action. In this section, we try to findout the circumstances when these two are equal. In order to achieve this, we need to study thecohomology on g.

Consider the set of all alternating k-linear maps g g R is denoted by kg. Theelements of kg are called the k-cochains on g.

Definition 4.13. The coboundary map on g :=

k kg is defined as

s(X0, . . . , Xn) =i

Introduction 21

Theorem 4.16 (Uniqueness). Let G be a compact Lie Group acting on (M,). The momentmaps for the Hamiltonian actions are unique if H1(g) = 0.

Proof. Let 1, 2 be two moment maps of a given action . Then for every X g, X1 , X2 areboth Hamiltonian functions of X. So d(X1 X2 ) = 0 X1 X2 = f(X) = constant for f g.

Now, for any X, Y g,

f([X,Y ]) = [X,Y ]1 [X,Y ]2= {X1 , Y1 } {X2 , Y2 }= {X2 + f(X), Y2 + f(Y )} {X2 , Y2 }= 0

So f ann[g, g] = {0} since H1(g) = 0. Hence X1 = X2 for all X g, and 1 = 2.Remark. From the proof of Theorem 4.16, if is a moment map then + k is another one ifk ann[g, g]. One particular case is g being commutative, which means ann[g, g] = g. Then itis legitimate to add every value of g to any moment map to get another one.

4.5 Examples of Moment Maps

Example 4.17. Recall Example 4.10. With the same setting as in the example, G = S1 actson (S2, d dh) by:

t : S2 S2(, h) 7 ( + t, h)

It is a Hamiltonian action, with X = and : S2 g = R is given by the height function

h : S2 R (Recall X = dh).

Example 4.18. Let (M,) = (Cn,n

i=1 ridri di) (polar coordinates) acted by G = Tnthrough

: GM M(ei1 , . . . , ein).(1, . . . , n) 7 (ei11, . . . , einn)

To find its corresponding moment map, consider the moment maps of each coordinate first. Forthe jth one,

Xj =

j

Hence

Xj = Xj (n

k=1

rkdrk dk) = rjdrj = d(12r2j + cj) = d

j

Introduction 22

where ck is a constant. Hence there is :M Rn by

(1, . . . , n) 7 (c1, . . . , cn) 12(|1|2, . . . , |n|2)

For equivariance of , consider the commutative diagram in Definition 4.11. Ad(g) = id sinceg = Rn is commutative. So we just need to check is invariant under action of G, which isobvious in this case.

Example 4.19. ([3]) Let (M,) = (Ck, i2

i dzi dzi) and G = U(k), the group of unitarymatrices {V Mkk : V V = Ik}. The action is just the multiplication map : (V, z) 7 V z.

It is a Hamiltonian action: identify u(k) = {A Mkk : A + A = 0}, the Lie Algebra ofU(k), with its dual by u(k) A u(k), where

(B) = trace(AB) = trace(AB)

(Check that A,B = trace(AB) is an inner product). Also, for w Ck,

Aw = Aw

by taking the infinitesimal action of . Now, by writing w as a k 1 matrix, the usual innerproduct (w, v) in Ck is equal to wv.

Consider vw, wv Mkk,

(i

2(vw + wv)) =

i2(wv + vw) = i

2(vw + wv)

Hence i2(vw + wv) u(k).

Now, we claim the following equation holds:

(Av,w) =A,

i

2(vw + wv)

(4.8)

Then, supposing moment map exists,

dv(w), A = (Av, w)= (Av,w)

=i

2A, vw + wv

for every A u(k). So dv(w) = i2(vw + wv), and we can take

: Ck u(k)v 7 i

2vv

Introduction 23

To see equivariance, i.e. S = Ad(S) , recall Example 3.5 that Ad(S)T = STS1. So

(S(v)) = i2(Sv)(Sv) = S( i

2vv)S1 = Ad(S)((v))

We are now left to prove (4.8). Since = i2dzi dzi, we have (, ) = =

(, ) (, ). So

(Av,w) =i

2((Av,w) (Av,w))

=i

2((Av,w) (w,Av))

=i

2((Av,w) (Aw, v))

=i

2((Av,w) + (Aw, v))

But (Av,w) =

i,j Aijviwj , and viwj = (wv)ji. Hence (Av,w) =

i,j Aij(wv

)ji = trace(A(wv)).Working similarly with (Aw, v),

i

2((Av,w) + (Aw, v)) =

i

2(trace(A(wv)) + trace(A(vw))

(Av,w) = trace(A[i

2(vw + wv)])

(Av,w) =A,

i

2(vw + wv)

Example 4.20. Continuing from Example 4.19, consider the action of U(k) on Ckn by matrixmultiplication. We can treat this as n copies of the above action acting on each k 1 column:Let A Ckn and Ar = (A1r, . . . , Akr)T , r = 1, . . . , n. Each r gives a moment map r satisfying

r(Ar) = i2ArAr u(k)

So the corresponding moment map : Ckn u(k) should be

((A))ij =r

(r(Ar))ij = i2r

AirAjr = i2r

AirArj =

i

2(AA)ij

So (A) = i2AA in this case.

Chapter 5

Reduced Spaces

The idea of symplectic reduction arose from classical mechanics. Continuing from Example4.3, consider a mechanical system of n particles in R3. It can be modelled by a symplecticmanifold - the phase space of dimension 6n. It was discovered that when there is a symmetryof dimension k in the system, the degree of freedom of the particles will be reduced by 2k in thephase space. We give the mathematical approach to this feature, and will see its importance indifferent fields of Mathematics the last two chapters.

The main reference in this chapter are Cannas da Silva [2], Cieliebak [3] and the original paperof Marsden and Weinstein [7].

5.1 Marsden-Weinstein-Meyer Reduction

Here is the central theorem of the chapter:

Theorem 5.1. Let (M,) be a symplectic manifold, G be a Hamiltonian action on M withmoment map . Suppose g satisfies the following properties:(a) is a regular value of :M g.(b) G := {g G : Ad(g) = } is compact in G.(c) G acts freely on 1().

Then M := 1()/G (the orbit space of ) has a manifold structure, and the projectionpi : 1() M is of C. Also there is a symplectic form on M satisfying i = pi,where i : 1()M is the inclusion map. This can be shown in the diagram below:

1() i Mpi

yM

Definition 5.2. With the same notation as above, (M, ) is called the reduced space at of (M,) with respect to G,.

24

Introduction 25

Remarks. Condition (a) in the theorem simply implies that 1() is a submanifold of M ofcodimension dimG, by the inverse function theorem.

In writing 1()/G, we need 1() is preserved by G. That is, for any p 1()and h G, h.p 1(). This can be seen by the equivariance property of the moment map:

h.p 1() (h.p) = Adh((p)) = (p) = p 1()

Lemma 5.3. Let M be any manifold and G be a Lie group acting on M by : G M M .Suppose G acts freely on M, and the map (g,m) 7 (m,g(m)) is proper. Then the orbit spaceM/G is a manifold with a C projection map pi :M M/G. The coordinates of M/G is givenby slice of each action.

Proof. It involves the use of principal bundles. One can refer to Bourbaki [1] for a wholeproof, or Cannas da Silva (pp.141-143) for a sketch proof. Note that if G is compact then(g,m) 7 (m,g(m)) is always proper.

Lemma 5.4. For all p M , the kernel and image of the tangent map dp : TpM g are(a) ker dp = (TpOp)p

(b) im dp = ann(gp), the annihilator of gp

where Op = {g.p : g G}, gp is the Lie Algebra of the stabiliser of p.Proof. Recall from our definition of moment map that

X = dX

so, for any p M , X g and v TpM ,p(Xp, v) = dp(v)(X) (5.1)

Hence, ker dp = {v : p(Xp, v) = 0 for all X g} = {Xp : X g}p = (TpOp)p

Now, for any Y gp, Yp = 0 by the infinitesimal action of the stabiliser of p G. Hence,with the same equation (5.1),

0 = p(Yp, v) = dp(v)(Y )

for any v TpM , so im dp ann(gp). But by the non-degeneracy of ,dim(im dp) = dim(M) dim(ker dp)

= dim(M) dim(TpOp)p= dim(TpOp)= dim(G) dim(Gp)= dim(g) dim(gp)= dim(ann(gp))

So they have the same dimension, and hence they are equal.

Introduction 26

Corollary 5.5. Suppose p 1(), and let Op := {h.p : h G} Op. Then

{Xp : X g} =: TpOp = TpOp Tp(1()) (5.2)

Also, TpOp is an isotropic subspace of TpM .Proof. We have already seen that 1() is indeed a submanifold of M . Now, let X g andXp TpOp. Then (5.2) is a consequence of equivariance property of moment map: for anyh G , (h.p) = Adh((p)) = Adh() = . By taking its infinitesimal version, X g dp(Xp) = 0 and hence

Xp TpOp X g dp(Xp) = 0 Xp ker dp = Tp(1())

(note that for every p 1(), Tp(1()) = ker dp). Hence the first result follows. For thesecond result, recall Lemma 5.4 that Tp(1()) = (TpOp)p . But Op 1(), so

TpOp Tp(1()) = (TpOp)p (TpOp)p (5.3)

where the last inclusion comes from the fact that U V V U. Hence Op is isotropic.

We are finally ready to prove the Theorem:

Proof of Theorem 5.1. 1() is a submanifold of codimension dimG. Now, by taking G = Gand M = 1() in Lemma 5.3(and its proof), M is a manifold with a smooth projectionmap pi : 1()M by p 7 [p]. We want to find out its symplectic structure.

For p 1(), considerT[p]M = Tp(1())/TpOp

So every v Tp(1()) corresponds to [v] T[p]M by taking modulo TpOp. Let by

[p]([v], [w]) := p(v, w) (5.4)

The definition guarantees i = pi, and it is obviously skew-symmetric. Also, [p] is well-defined: for any , TpOp,

p(v + ,w + ) = p(v, w) + p(,w) + p(v, ) + p(, )= p(v, w) + 0 + 0 + p(, )= p(v, w) + 0 + 0 + 0 = p(v, w)

where the second last equality comes from the fact that (and similarly ) TpOp TpOp, andw(and similarly v) Tp(1()) = (TpOp)p (Check (5.3)), and the last equality comes fromthe fact that TpOp is isotropic.

It is also true that is smooth on M , since it has a smooth pull-back to M by pi. Hence

Introduction 27

it is a smooth 2-form on M . To see is indeed a symplectic form, we need to show fur-ther it is non-degenerate and closed. For non-degeneracy, suppose [p]([v], [w]) = 0 for allw Tp(1()) = (TpOp)p . Then

p(v, w) = 0 w (TpOp)p v ((TpOp)p)p v TpOp v TpOp Tp(1()) v TpOp by(5.2) [v] = 0

Finally, the formulai = pi

implies pid = id = 0. But pi is injective implies d = 0 and hence it is closed.

5.2 Examples of Reduced spaces

Proposition 5.6. Take = 0 in Theorem 5.1. Then G0 = G and the conditions in Theorem5.1 can be reduced as

(b) G is a compact Lie group.

(c) G acts freely on 1(0).

Proof. G0 = G is obvious from our definition of G in Theorem 5.1. So conditions (b),(c)correspond to conditions (b),(c) respectively. To see (c) implies (a), let p 1(0). Then (c)implies g.p = p g = id. Hence gp = {0}. By Lemma 5.4(b), im dp will be the whole spaceg, which implies dp is surjective. So p is a regular point of : M g for every p 1(0).So 0 is a regular value of , which is precisely (a).

Remark. The space M0 = 1(0)/G is called the reduced space of M at zero level. It isused more frequently than other levels since it involves fewer conditions.

Example 5.7. Let M = R2n+2 = Cn+1 with the usual symplectic form , and consider theT1-action on M by multiplication of eit on each coordinate. As in Example 4.18, the action isHamiltonian with moment map

: Cn+1 R(= (t1))z 7 1

2 12|z|2

Introduction 28

We can see M0 = S2n+1/T1 = CPn here, and its symplectic form is the Fubini-Study form,which is defined by the following:

nFS = i [i

2log(|z|2 + 1)]

where i : {[z0, . . . , zn] : zi 6= 0} Cn by [z0, . . . , zn] 7 (z0/zi, . . . , zi/zi, . . . , zn/zi), and , maps differential forms to differential forms by:

(

I=(I1,I2)

I dzI1 dzI2) :=j

I=(I1,I2)

Izj

dzj dzI1 dzI2

(

I=(I1,I2)

I dzI1 dzI2) :=j

I=(I1,I2)

Izj

dzj dzI1 dzI2

Also, we have the equation pinFS = i from our setting of reduced space.

Example 5.8. Recall Example 4.20. The action of U(k) on Ckn = Mkn by matrix multi-plication is Hamiltonian with moment map (A) = i2AA. We can actually add the identitymatrix to the moment map, making its zero level be 1(0) = {A Ckn : AA = Idk}.

We claim the corresponding reduced space 1(0)/U(k) is the Grassmanian Grk(Cn), where

Grk(Cn) := {W Cn : dimC(W ) = k}

To see this, note that each k-dimensional subspace W is spanned by vectors v1, . . . , vk Cn.

Arranging them into V =

v1 v2 . . .

vk

, a kn matrix. By choosing suitable linear combinationsof vis, we can pick a suitable V satisfying V V = Idk while Span{v1, . . . , vk} is still equal toW . Hence there is a surjection

{V Mkn : V V = Idk} Grk(Cn)

Now, two subspaces of dimension k, {v1, . . . , vk}, {w1, . . . , wk} are equal iff there exists A GLk(C) such that W = AV . If V V =WW = Idk,

Idk =WW = (AV )(AV ) = AV V A = A(Idk)A = AA

Hence in this case A U(k), and finally conclude that

1(0)/U(k) = {V Mkn : V V = Idk}/U(k) = Grk(Cn)

Chapter 6

Convexity Theorems

In this chapter, we find out some properties of the image of moment maps, namely the convexitytheorems. Here is a particular version of it, which is to be proved in this chapter:

Theorem 6.1 (Atiyah-Guillemin-Sternberg). Let (M,) be a compact connected symplecticmanifold with a Hamiltonian action by Tn. Then the moment map :M Rn has the propertythat the image is the convex polyhedron spanned by the image of the fixed points of the action.

The approach of this chapter follows from Cieliebak [3], Guillemin [5] and McDuff [9].

6.1 Morse-Bott Functions

To prove Theorem 6.1, we first need to show for any t, 1() is connected by using theproperties of Morse-Bott functions.

Definition 6.2. Let M be a manifold and f : M R be a function of M . Then f is aMorseBott function if the set of critical points of f , Crit(f), is a submanifold of M , andthe Hessian 2f(p) : TpM TpM has the property that Tp(Crit(f)) = ker 2f(p).Remarks. We can decompose Crit(f) into finitely many connected critical submanifolds C,with the decomposition

TpM = TpC Pp Npwhere Pp, Np are the positive and negative eigenspaces of 2f(p) respectively.

Denote the index of a component of Crit(f) be the number of negative eigenvalues of 2f(p),i.e. ind(C) = dimNp. And the coindex coind(C) = dimPp.

Proposition 6.3. Let M be a compact and connected manifold and f :M R be a Morse-Bottfunction which has no critical components of index or coindex 1. Then every level set of f isconnected, and in particular it has a unique local maximum and minimum.

Proof. See Cieliebak(Part B, p.31) or McDuff(pp.184-185).

29

Introduction 30

Lemma 6.4. Let (M,) be a symplectic manifold, x M be a fixed point of a Hamiltonianaction by Tn. Then there is a diffeomorphism between neighbourhood of x and neighbourhood of0 in Cn which corresponds to the standard symplectic form 0 in Cn. Also, the action aroundx corresponds to a diagonal linear action

(ei1 , . . . , ein).(z1, . . . , zn) 7 (eik1(1,...,n)z1, . . . , eikn(1,...,n)zn)

(when the neighbourhood of x is identified with Cn) where ki Zn Rn = (Rn), with momentmap

(z) = (x) +12

nj=1

kj |zj |2

Proof. See Cieliebak(Part B, pp.26-28).

Corollary 6.5. For any H G, the set {p M : Gp = H} is a symplectic submanifold. Inparticular, Fix(G) := {p M : g.p = p g G} is a symplectic manifold.Proof. Consider the Hamiltonian action onM by H G. By Lemma 6.4, each point in Fix(H)corresponds to a diagonal linear action, which is symplectic (it is indeed Hamiltonian withmoment map). So Fix(H) is a symplectic manifold.

Theorem 6.6. Let (M,) be a symplectic manifold with Hamiltonian action by Tn. Suppose :M Rn is the moment map. Then for any X tn,(a) Crit(X) is a symplectic submanifold of M.

(b) X :M R is a Morse-Bott function.(c) Every critical component of X has even index and coindex. Hence (from Proposition 6.3)

every level set of X is connected and has a unique local maximum and minimum.

Proof. In the proof, we just consider X tn = Rn such that all their components are indepen-dent rationally. Then the result follows by taking suitable linear combinations of them.

(a) Let p Crit(X), i.e. dXp = 0 = Xp. Then Xp = 0 and hence exp(tX).p = 0 for all t.But {exp(tX) : t R} is dense in Tn, p Fix(Tn) and hence Fix(Tn) = Crit(X). SoCorollary 6.5 implies Crit(X) is a symplectic submanifold.

(b) On the neighbourhood of a critical point p Crit(X) = Fix(Tn), there exists a localdiffeomorphism of p to 0 Cn with a moment map

(z) = (p) +12

nj=1

kj |zj |2

Introduction 31

with

X(z) = X(p) + 12n

j=1(kj , X)|zj |2 (6.1)according to Lemma 6.4. Rearranging kis, we can get ki 6= 0 for all i r and 0 otherwise.Then from (6.1), Crit(X) = {z Cn : z1 = = zr = 0}.

Now, for any z such that zj = 0 for all j = r + 1, . . . , n, 2X(z) = 0 z = 0. Itis because X is rationally independent, yielding (kj , X) 6= 0 for all j (recall kj Zn). SoX is non-degenerate and is a Morse-Bott function.

(c) Consider (6.1) again. Taking standard coordinates (x1, y1, . . . , xn, yn) of Cn = R2n, theHessian is

(k1, X) 0 0 0 0

0 (k1, X)...

. . .

(kr, X) 0... . . .

...0 (kr, X)

0 . . . 0...

0 . . . 0 0 0

So ind(x) = 2#{j : (kr, X) < 0} which is even. Since dimCn = 2n the coindex is evenas well.

Corollary 6.7. Let M be a compact connected symplectic manifold with a Hamiltonian toricaction, and let be the moment map. Then every level set 1() is connected.

Proof. Here is just a sketch proof of it. Let Tn = S1 Sn, where each Si is a circle. Then = (1, . . . , n) in this case. Suppose = (1, . . . , n) is a regular value of each i, which isdense in im, then by Corollary 6.6 11 (1) is connected. So the reduced space

11 (1)/S1 is

also connected, and is acted by S2 Sn with moment map (2, . . . , k).

By Theorem 6.6, (1, 2)1(1, 2)/S1 is connected. However, S1 is connected and hence thetotal space (1, 2)1(1, 2) of the bundle map

(1, 2)1(1, 2) (1, 2)1(1, 2)/S1is also connected. Repeating the process gives us the desired result.

For other values im, we argue by contradiction. If there exists disjoint U1, U2 suchthat 1() U1 U2, we can, by Slice Theorem, get open neighbourhoods V1, V2 of (U1),(U2) respectively. Now V1, V2 both contains , there exists some regular near such that V1 V2 and 1() U1 U2. But then U1 U2, which is a contradiction.

Introduction 32

6.2 Proof of the Theorem

Proposition 6.8. With the same setting as in Theorem 6.1, im is convex for any Tn-action.

Proof. Consider the action on M by a closed subgroup H = Tn1 of G = Tn. Since H is closed,its corresponding Lie algebra h is a hyperplane in Rn = tn with rational normal vector.

Let pi : g h be the dual of the inclusion map h g, the moment map correspondingto the action by H is

H = pi :M h

by Corollary 6.7, 1H () is connected for each h, hence (1H ()) is connected in Rn. Butnote that (1H ()) is just the intersection of im with the 1-dimensional line pi

1() g,which has slope equal to the normal vector of the hyperplane, i.e. it has rational slope. Hencefor any two rational points in im, fix one point and find suitable H G such that the straightline (1H ()) im contains these two points.

It is obvious that the set of rational points in im is dense in im, hence every two pointsin im can be joined by a straight line which lies totally in im such that its end points areexactly these two points, i.e. im is convex.

Proof of Theorem 6.1. We have already proved that the image of the moment map is convex.We now try to show the images of the fix points of the action forms the vertices of the polygon.

Corollary 6.5 shows that Fix(Tn) is a symplectic submanifold. Let U1, . . . , Uk be its connectedcomponents. Then (Uj) = j for each j = 1, . . . , k. By Proposition 6.8, the convex polyhedronspanned by the is is contained in the image of .

Now suppose Rn such that it does not lie inside the convex polyhedron spanned by theis. Then pick X Rn = tn such that all their components are rationally independent and(,X) > (j , X) for all j = 1, . . . , k. Recall the proof of Theorem 6.6(a), Crit(X) = Fix(Tn).So the maximum value of X on M is in one of the Uj s.

But on Uj , X = (j , X) and hence (,X) > (j , X) for all j implies (,X) > suppM ((p), X).In particular, cannot be in the convex polyhedron spanned by the is. This completes theproof.

Here is a simple example illustrating the theorem is true:

Example 6.9. Recall Example 4.17 (S1 = T1 act on S2 by rotation). The moment map is theheight function h : S2 R. So the image is just the interval [1, 1], and the vertices 1 and1 of the image corresponds to the South and North Pole of the 2-sphere, which is fixed underrotation.

Introduction 33

6.3 The Theorems of Schur and Horn and its Generalisations

This example comes from McDuff(p.182), we will see how to use symplectic geometry on groupand representation theory. Recall Example 3.5, G = U(n), M = O = U(n)/Tn the coad-joint orbit of a diagonal matrix T u(n). Then the symplectic form T (A,B) = T, [A,B],T u(n) = u(n), A, B u(n).

First of all, we claim the coadjoint action is Hamiltonian with moment map : O g issimply the inclusion map i (when O is seen as an orbit in g). To see this, the coadjoint actionT 7 Ad(S)T of T O induces an infinitesimal action AT = ad(A)T . Consider

BT(AT ) = T (BT , AT ) = T, [B,A] = T,ad(A)(B) = ad(A)T,B =AT , B

diB(AT ) =

di.AT , B

=AT , B

for every AT . So

BT = diB (6.2)

For equivariance, the commutative diagram in Definition 4.11 is easily satisfied since it is aninclusion map. Hence it is a moment map.

Under the identification O by U(n)/Tn, Example 3.5 tells us Ad(S)T = STS1 u(n). So : U(n)/Tn u(n) = u(n) is given by

(S) = STS1

However, it is not a toric action on O, so restrict our action G = U(n) into the subgroup of alldiagonal matrices Tn. Now

G = Tn,M = O = {Ad(S)(T ) : S U(n)}

In this case the corresponding moment map is just the restriction of onto the diagonalentries:

(S) = diag(STS1)

(note that by definition of u(n), the diagonal entries of STS1 have purely imaginary values,hence im has real dimension n only) The fixed points of the action are

{V O :MVM1 = V M Tn} = {V O : V is diagonal}

Also, W O W = STS1 for some S U(n), hence W , T must have the same eigenvalues.If V is a fixed point, V must be diagonal and has the same eigenvalues as T , which is onlypossible by permuting the diagonal entries of T . Finally, we use the convexity theorem:

If T u(n) is a diagonal matrix with entries i1, . . . , in, i R. Then (O) = ({STS1 : S U(n)}) = im(diag{STS1 : S U(n)}) is a convex polytope with vertices diag{(1), . . . , (n)},where Sn.

Introduction 34

To see the theorem in work, suppose W u(n) with eigenvalues i1, . . . , in, i R, thenW = STS1 for some S U(n), where T is the diagonal matrix

T =

i1 0 . . . 00 i2 . . . 0

. . .0 . . . 0 in

In conclusion:

Theorem 6.10. For any matrix W u(n) with eigenvalues (which are purely imaginary num-bers) i1, . . . , in, the diagonal entries (W11, . . . ,Wnn) must lie inside the convex polytope in Rnwith vertices {(1), . . . , (n)}, where Sn. Conversely, for any elements (a1, . . . , an) insidea polytope with vertices (x(1), . . . , x(n)) there exists a matrix V u(n) such that Vjj = aj forj = 1, . . . , n and has eigenvalues ix1, . . . , ixn.

Remarks. By replacing W = iH, where H is any Hermitian matrix H = H, we get the sameresult in Theorem 6.10, simply by replacing the imaginary eigenvalues with the real eigenvaluesin H.

The analogue of Theorem 6.10 in the Hermitian case has a historical background. Schur provedthe first part of the theorem in 1923, while Horn proved the converse more than 30 years later(1954). Both of them got their results without using convexity theorems. Atiyah merged thesetwo results using symplectic geometry in 1982, which is precisely the proof above.

Chapter 7

Toric Actions on SymplecticManifolds

From last chapter, all toric actions on compact symplectic manifolds give convex moment poly-topes. We will try to find out the properties and the shapes of these polytopes, and will see howthey link to different aspects of Mathematics.

Definition 7.1. Let (M,) be a symplectic manifold acted by a Lie group G. The action iseffective if every non-identity element g G moves at least a point p M . In other words,

pMGp = {1}

Proposition 7.2. Let (M,) be a compact connected symplectic manifold with an effectiveHamiltonian action by Tn. Then the there are at least n+1 fixed points, and dimM 2n.Proof. Refer to Cannas da Silva [2](p.170).

Definition 7.3. A toric manifold is a connected compact symplectic manifold (M,) with aneffective Hamiltonian action of Tn such that dimM = 2n and a corresponding moment map :M Rn.

7.1 Delzants Theorem

This is a very powerful classification theorem given by Delzant in late 1980s. It states that thereis a one-one correspondence between toric manifolds M and the images of the moment maps.We will only give a full proof on the half of the theorem as in Cannas da Silva [2], and give asketch proof of the other half.

35

Introduction 36

Theorem-Definition 7.4. For each n N, there is a one-one correspondence between 2n-dimensional toric manifolds and n-dimensional polygons of the properties below:

(a) There are exactly n edges meeting at each vertex of the polygon.

(b) Each edge meeting at a vertex p is rational, i.e. each edge has a parameterisation {p+twi :t 0} with wi (Zn) = Zn.

(c) The vectors w1, . . . , wn forms an integer basis of Zn.

with one side of the correspondence is given by the moment map (M,) 7 (M).

The polygons with the above properties are called Delzant polytopes.

Proposition 7.5. Under the assumptions in Theorem 7.4, the map (M,) 7 (M) is well-defined.

Proof. By the definition of toric manifolds, (M) is a convex polytope of dimension n. We justneed to show (M) is indeed a Delzant polytope. Recall Lemma 6.5, since x M is a fixed point,(x) is a vertex of the polytope and the statement in the lemma is exactly condition (b) above.

We prove (a), (c) together by claiming that the matrix formed by the kis, A =

k1 ... kn

,defines an isomorphism Zn Zn.

Suppose not, then there exists such that A Zn while / Zn, i.e. ki() Z while / Zn.But then the action exp() fixes every element at the neighbourhood of x, hence contradictingthe fact the the action is effective.

A. Construction of the Inverse Map

We now try to find the inverse of the correspondence in Theorem 7.4, i.e. given a Delzantpolytope Rn, find the toric manifold (M,) such that (M) = .

Definition 7.6. Let be a Delzant polytope in Rn. The facets of are the (n-1)-dimensionalfaces of it. Suppose there are totally d facets, then can be represented as the intersection ofthe half-spaces divided by the facets:

= {x (tn) = (Rn) : x, vi i, i = 1, . . . , d}

where vi is outward the normal vector of the ith facet. Note that vi can be represented by integralentries since every facet is spanned by vectors in Zn. Take vi Zn such that its entries have nocommon factor in N. Then vi is said to be a primitive vector in Zn.

Introduction 37

Consider a larger space Rd first with canonical basis {ei = (0, . . . , 1ithentry

, . . . , 0) : i = 1, . . . , d}.

With our representation of above, define the map

pi : Rd Rn

by ei 7 vi. We claim the map restricts to pi : Zd Zn surjectively. Then we will have theinduced map

pi : Td Tn

To prove the claim, check the set {v1, . . . , vd} spans Zn. For each vertex in , the edges arespanned by w1, . . . , wn which forms a basis for Zn. So by letting vik = {w1, . . . , wk, . . . , wn},{vi1 , . . . , vin} forms a basis of Zn as well.

So there is a short exact sequence

0 G i Td pi Tn 0

where G = kerpi. And it induces another two exact sequences

0 g i td = Rd pi tn = Rn 0

0 (Rn) pi (Rd) i g 0Consider the Td-action on (Cd, ), where is the usual symplectic form, by

(ei1 , . . . , eid).(z1, . . . , zd) = (ei1z1, . . . , eidzd)

Then from Example 4.18 the moment map is given by : Cd (td) = (Rd) by

(z1, . . . , zd) = (c1, . . . , cd) 12(|z1|2, . . . , |zd|2)

Take ci = i, and consider the inclusion map

i : G Td

where G is a closed subgroup of Td. Then the restricted action of G to (Cd, ) is also Hamiltonianwith moment map

i : Cd g

(to see this, check that for every X g, exp(tX) g for all t R. But this is true since G is aclosed subgroup).

We now claim the inverse map in Theorem 7.4 to be

7 ((i )1(0)/G, ) := (N/G, )

where (N/G, ) is the reduced space of (Cd, ) with the G-action as above at zero level. How-ever, from Proposition 5.6, the theorem below has to be true:

Introduction 38

Theorem 7.7. Under the above construction, N = (i )1(0) is compact and G acts freelyon it.

Proof. Here is a sketch proof of it. Let = pi() (Rd). We can actually show(N) = = pi() (7.1)

by tracing the last exact sequence above, and since is compact and is a compact map bydefinition, N is compact.

Now, from (7.1), for any x N , w = (x) (N) w = pi(q) for some q . If q lieson a k-dimensional face of , we can deduce that xj 6= 0 for exactly d (nk) coordinates of x,which can be reordered to be the last d (n k) coordinates. Suppose h = (eih1 , . . . , eihd) Tdstabilises x, then eihj = 1 for all j > n k. So

S := {h Td : h.x = x} = {(eih1 , . . . , eihnk , 1, . . . , 1) : hj R}Taking {v1, . . . , vn} = {pi(e1), . . . , pi(en)} as a basis of Zn, the map pi|S : S Tn is injective.Suppose g G such that g.x = x. Then g S G and pi(g) = 0 since g G = kerpi. On theother hand pi(g) = 0 g = 0 since pi|S is injective. Hence the action of G on N is free.

B. Sketch Proof of the Theorem

We are now ready to show one direction of the inverse map is true:

Theorem 7.8. There exists moment map N/G on N/G such that (N/G) =

Proof. Recall the last part of Theorem 7.7, take a basis {v1, . . . , vn} on Tn and letf : Tn Td, vi ei

Then pi f = id, and hence the exact sequence 0 G Td Tn 0 splits, i.e. Td = GTn.

Consider the Tn-action on Cd by (u, z) 7 f(u).z. This is a Hamiltonian action with mo-ment map f . Now, the sequence above splits implies the Tn-action is commutative withthe G-action. So we can get a induced Tn-action on N/G = (i )1(0)/G with moment mapN/G satisfying the following equation

N/G pr = (f ) inwhere pr, in are the projection and inclusion maps

N = (i )1(0) in Cdpr

yN/G

So imN/G = (f )(N) = f((N)) = f(pi()) = , where the second last equality signcomes from (7.1), and the last equality comes from our setting of f such that pi f = id.

Introduction 39

On the contrary, the other side of inverse requires more technicality. The theorem we need is:

Theorem 7.9. Any two toric manifolds with the same moment polytope are isomorphic in thesense that there exists :M1 M2, :M2 M1 such that there are correspondences:

(M1, 1,Tn, 1) (M2, 2,Tn, 2)

(M2, 2,Tn, 2) (M1, 1,Tn, 1)

In later context, the word isomorphic is used in the above sense.

Let M be a toric manifold, the other way of inverse is given by M 7 (M) 7 N/G. Theorem7.8 asserts that M , N/G has the same moment polytope, hence Theorem 7.9 gives us thedesired result - M = N/G. Here is the description of the proof, or one can refer to Cieliebak[3](Part B, pp.40-46) for a full proof.

Sketch proof of Theorem 7.9. For any two toric manifolds (M0, 0,Tn, 0) and (M1, 1,Tn, 1)which have the same moment polytope , we can construct a semi-isomorphism betweenthem, i.e. an isomorphism between Mi and i but not necessarily i. This can be done byconstructing isomorphisms between the preimages of 0 and 1 for each k-dimensional faces of.

We can show further that t = t1 + (1 t)0 is a symplectic form for all t [0, 1], andby Duistermaat-Heckman Theorem, every t is indeed in the same cohomology class. Sowe can use Mosers Theorem (Theorem 2.7) here to get a a symplectomorphism from 0 to 1,which is precisely the statement of the Theorem.

7.2 Examples of Moment Polytopes

In order to use the convexity theorem, we must get a compact symplectic manifold. However,many of our previous examples are (Cn, ) which are non-compact. Therefore, we will try toconstruct some simple examples of moment polytope using (CPn, nFS) and their product in-stead. Such constructions are useful in many aspects of Mathematics, and we will give a tasteof it in the last example.

Examples 7.10. Consider (CP1, 1FS) with T1-action given by

ei.[z0, z1] = [z0, eiz1]

Recall Example 5.7, pi1FS = i. So we can just consider the T1-action on (S3, i) byei.(z0, z1) = (z0, eiz1), which has a moment map (z0, z1) = 12 |z1|2. Hence the correspondingmoment map on CP1 is

[z0, z1] = 12|z1|2

|z0|2 + |z1|2

Introduction 40

Note that im = [12 , 0], and [z0, z1] = 12 , 0 when z0 or z1 = 0 respectively, i.e. [z0, z1] =[1, 0], [0, 1]. It is easy to see the action fixes these 2 points.

With the same technique, we can generalise the example to (CPn, nFS) with Tn-action givenby

(ei1 , . . . , ein).[z0, . . . , zn] = [z0, ei1z1, . . . , einzn]

It is a Hamiltonian action with moment map : CPn Rn given by

[z0, . . . , zn] = 12(|z1|2

|z0|2 + + |zn|2 , . . . ,|zn|2

|z0|2 + + |zn|2 ) (7.2)

The image of the moment map is the (n 1) simplex with interior for each n, with vertices

(0, . . . , 0), (12, 0, . . . , 0), . . . , (0, . . . , 0,1

2)

corresponding to fixed points

[1, 0, . . . , 0], [0, 1, 0, . . . , 0], . . . , [0, . . . , 0, 1]

The figure below shows the moment polytopes of of above action for n = 1, 2, 3 respectively.

Example 7.11. Let T3 act on (CP2 CP1, 2FS 1FS) by(ei1 , ei2 , ei3).([z0, z1, z2], [w0, w1]) = ([z0, ei1z1, ei2z2], [w0, ei3w1])

Then : CP2 CP1 R3 is given by

([z0, z1, z2], [w0, w1]) = 12(|z1|2

|z0|2 + |z1|2 + |z2|2 ,|z2|2

|z0|2 + |z1|2 + |z2|2 ,|w1|2

|w0|2 + |w1|2 )

The image of the moment map is just I, where is a right-angled triangle and I is aninterval. The figure of the moment polytope is:

Introduction 41

Example 7.12. Let T2 be the kernel of the map : T3 T1, (ei1 , ei2 , ei3) = ei(2123).Then we have the exact sequence

0 ker f T3 T1 0

where ker = T2 and f is given by f(ei1 , ei2) = (ei(1+2), e2i1 , e2i2). Then, as a mapbetween Lie Groups, it is f(x, y) = (x+ y, 2x, 2y). The dual map f is therefore given by

f(x, y, z) = (x+ 2y, x+ 2z)

Consider the case when n = 3 in Example 7.10. The new action of T2 on (CP3, 3FS) givesthe moment map f . So the new moment polytope is just the projection of the 2-simplexgiven by f above. The vertices (0,0,0), (12 , 0, 0), (0,12 , 0), (0, 0,12) maps to (0, 0), (12 ,12),(-1,0), (0,-1) respectively. Hence its moment polytope is:

Now consider the case in Example 7.11. The moment polytope is:

Example 7.13. Our final example refers back Section 6.3. Let M be the SU(3)-coadjoint orbit

of the matrix D =

i 0 00 i 00 0 2i

su(3) = su(3), i.e.

M = {Ui 0 00 i 00 0 2i

U1 : U SU(3)}

Introduction 42

The stabiliser of the action is SU(2)SU(1), since the stabiliser of D is simply(A 00 z

), where

AA = Id22 and |z| = 1, and hence the other element in M will only be its adjoint matrix. So,by the Orbit-Stabiliser Theorem,

M = SU(3)/(SU(2) SU(1)) = Gr1(C3) = CP2 (7.3)

Also, dimM = dim(SU(3)) dim(SU(2) SU(1)) = 8 4 = 4

As in Section 6.3, let G = T2 SU(3) act on M by the coadjoint action (note the actionis T2 instead of T3 in this case since the determinant constraint of SU(3) reduces one dimensionof freedom of G). So the action given by

S.T = STS1

for S T2, T M is obviously effective. Hence dimM = 2dim(T2) = 4 and M is a toricmanifold.

From Theorem 6.10, the moment polytope ofM is the convex polytope with vertices (, ,2),(,2, ), (2, , ), i.e. a triangle. Recall the image of the moment map of (CP2, 2FS) isalso a triangle. Hence by Delzants Theorem M = CP2, which agrees with our calculation in(7.3).

We can actually extend the above result into n dimensions, i.e.

M = {U

i 0 . . . 00 i . . . 0

. . .0 . . . 0 ni

U1 : U SU(n)} = CPn1

One can refer to Guillemin [4] for more examples on the orbits of SU(3) and SU(4) (pp.146-152),and the implications of these results in representation theory (pp.xi-xiv).

Bibliography

[1] Bourbaki, N., Varietes differentielles et analytiques, Fascicule de resultats, FasciculeXXXIII, Hermann, Paris, 1971, Topologie Generale, Livre III, ch.3, Fasc. III, Hermann,Paris, 1960

[2] Cannas da Silva, A., Lectures on Symplectic Geometry, Lecture Notes in Mathematics,Springer-Verlag, 2001

[3] Cieliebak, K., Symplectic Geometry, lecture notes available at http://www.mathematik.uni-muenchen.de/kai/classes/257spr01/index.html, 2004

[4] Guillemin, V. and Lerman, E. and Sternberg, S., Symplectic Fibrations and MultiplicityDiagrams, Cambridge University Press, 1996

[5] Guillemin, V. and Sjamaar, R., Convexity Properties of Hamiltonian Group Actions, CRMMonograph Series, American Mathematical Society, 2005

[6] Hall, B., Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Grad-uate Texts in Mathematics 222, Springer-Verlag, 2003

[7] Marsden, J. and Weinstein, A., Reduction of Symplectic Manifolds with Symmetry, Reportson Mathematical Physics 5, 121-130, 1974

[8] Marsden, J. and Weinstein, A., Some Comments on the History, Theory and Applicationsof Symplectic Reduction, Quantization of singular symplectic quotients. N. Landsman, M.Pflaum, M. Schlichenmanier eds., Birkhauser, Boston, 1-20, 2001

[9] McDuff, D. and Salamon, D., Introduction to Symplectic Topology, Oxford University Press,1998

[10] Meinrenken, E., Symplectic Geometry, lecture notes available at http://www.math.utoronto.ca/mein/teaching/sympl.ps

43