Symplectic Geometry

SYMPLECTIC GEOMETRY

Eckhard Meinrenken

Lecture Notes UniversityofToronto

These are lecture notes for two courses taughtattheUniversityofToronto in Spring

and in Fall Our main sources have b een the b o oks Symplectic Techniques

by GuilleminSternb erg and Intro duction to Symplectic Top ology by McDuSalamon

and the pap er Stratied symplectic spaces and reduction Ann of Math

by SjamaarLerman

Contents

Chapter Linear symplectic algebra

Symplectic vector spaces

Subspaces of a symplectic vector space

Symplectic bases

Compatible complex structures

The group SpE of linear symplectomorphisms

Polar decomp osition of symplectomorphisms

Maslov indices and the Lagrangian Grassmannian

The index of a Lagrangian triple

Linear Reduction

Chapter Review of Dierential Geometry

Vector elds

Dierential forms

Chapter Foundations of symplectic geometry

Denition of symplectic manifolds

Examples

Basic prop erties of symplectic manifolds

Chapter Normal Form Theorems

Mosers trick

Homotopy op erators

Darb ouxWeinstein theorems

Chapter Lagrangian brations and actionangle variables

Lagrangian brations

Actionangle co ordinates

Integrable systems

The spherical p endulum

Chapter Symplectic group actions and moment maps

Background on Lie groups

Generating vector elds for group actions

Hamiltonian group actions

Examples of Hamiltonian Gspaces

CONTENTS

Symplectic Reduction

Normal forms and the DuistermaatHeckman theorem

The symplectic slice theorem

Chapter Hamiltonian torus actions

The AtiyahGuilleminSternb erg convexitytheorem

Some basic MorseBott theory

Lo calization formulas

Frankels theorem

Delzant spaces

CHAPTER

Linear symplectic algebra

Symplectic vector spaces

Let E b e a nitedimensional real vector space and E its dual The space E

can b e identied with the space of skewsymmetric bilinear forms E E R

v w w v

Definition The pair E is called a symplectic vector space if E is

nondegenerate that is if the kernel

ker fv E j v w for all w E g

is trivial Two symplectic vector spaces E andE are called symplectomorphic

if there is an isomorphism A E E with A The group of symplectomor

phisms of E is denoted Sp E

Since SpE is a closed subgroup of GlE it is by a standard theorem of Lie group

theory a Lie subgroup

Example Let E R with basis fe e f f gThen

n n

e e f f e f

i j i j i j ij

denes a symplectic structure on E Examples of symplectomorphisms are Ae

f Af e or Ae e f Af f Also

j j j j j j j j

X X

Ae B e Af B f

j jk k j kj k

k k

for anyinvertible n nmatrix B is a symplectomorphism

Example Let V bearealvector space of dimension nandV its dual space

Then E V V has a natural symplectic structure v v v v

If B V V is any isomorphism and B V V the dual map B B E E

is a symplectomorphism

Example Let E b e a complex vector space of complex dimension n with com

plex p ositive denite inner pro duct Hermitian metric h E E C Then E

viewed as a real vector space with bilinear form the imaginary part Imhisa

symplectic vector space Every unitary map E E preserves h hence also and is

therefore symplectic

LINEAR SYMPLECTIC ALGEBRA

Exercise Show that these three examples of symplectic vector spaces are in

fact symplectomorphic

Subspaces of a symplectic vector space

Definition Let E b e a symplectic vector space For any subspace F E

we dene the p erp endicular space F by

F fv E v w for all w F g

With our assumption that E is nite dimensional is nondegenerate if and only if

the map

E E h v wi v w

is an isomorphism F is the preimage of the annihilator ann F E under From

this it follows immediately that

dim F dim E dim F

and

F F

Definition A subspace F E of a symplectic vector space is called

a isotropic if F F

b coisotropic if F F

c Lagrangian if F F

d symplectic if F F fg

The set of Lagrangian subspaces of E is called the Lagrangian Grassmannian and denoted

Lag E

Notice that F is isotropic if and only if F is coisotropic For example every

dimensional subspace is isotropic and every co dimension subspace is coisotropic

Example In the ab ove example E R letL spanfg g g where for

all i g e or g f ThenL is a Lagrangian subspace

i i i i

Lemma For any symplectic vector space E there exists a Lagrangian sub

space L Lag E

Proof Let L b e an isotropic subspace of E which is maximal in the sense that

it is not contained in any isotropic subspace of strictly larger dimension Then L is

Lagrangian For if L L then cho osing any v L nL would pro duce a larger isotropic

subspace L span v

An immediate consequence is that any symplectic vector space E has even dimension

For if L is a Lagrangian subspace dim E dimL dimL dim L

Lemma can b e strengthened as follows

Lemma Given any nite col lection of Lagrangian subspaces M M one

can nd a Lagrangian subspace L with L M fg for al l j j

COMPATIBLE COMPLEX STRUCTURES

Proof Let L b e an isotropic subspace with L M fg and not prop erly contained

in a larger isotropic subspace with this prop ertyWe claim that L is Lagrangian If not

L is a coisotropic subspace prop erly containing LLet L L L b e the quotient

map Cho ose any dimensional subspace F L L suchthatboth F is transversal to

all M L This is p ossible since M L is isotropic and therefore has p ositive

j j

co dimension Then L F is an isotropic subspace with L L and L M fg

This contradiction shows L L

Symplectic bases

Theorem Every symplectic vector space E of dimension n is symplecto

morphic to R with the standard symplectic form from Example

Proof Picktwo transversal Lagrangian subspaces L M Lag E The pairing

L M R v w v w

is nondegenerate In other words the comp osition

E L M E

where the last map is dual to the inclusion L E is an isomorphism Let e e

M By denition of the pairing b e a basis for L and f f the dual basis for L

is given in this basis by

Definition Abasis fe e f f g of E for which has the stan

n n

dard form is called a symplectic basis

Our pro of of Theorem has actually shown a little more

Corollary Let E i be two symplectic vector spaces of equal di

i i

mension and L M Lag E such that L M fg Then there exists a symplecto

i i i i i

morphism A E E such that AL L and AM M

Compare with isometries of inner pro duct spaces which are much more rigid In the

following section wegive an alternative pro of of Theorem using complex structures

Compatible complex structures

Recall that a complex structure on a vector space V is an automorphism J V V

such that J Id

Definition A complex structure J on a symplectic vector space E is called

compatible if

g v w v J w

denes a p ositive denite inner pro duct This means in particular that J is a symplec

tomorphism

J v w JvJw g Jvw g w J v w J v w v v w

We denote by J E the space of compatible complex structures

LINEAR SYMPLECTIC ALGEBRA

We equip J E with the subset top ology induced from EndE Later we will see

that it is in fact a smo oth submanifold

Example In Example a compatible almost complex structure J is given by

n n

Je f Jf e This identies R Jwith C

i i i i

A compatible complex complex structure makes E into a Hermitian vector space

complex inner pro duct space with Hermitian metric

hv w g v w v w

That is h is complexlinear with resp ect to the second entry and complexantilinear with

resp ect to the rst entry

p p

hv whJvw hv w hv J w

and hv v for v We will show b elowthatJ E Assuming this for a

moment let J JE and pick an orthonormal complex basis e e Letf Je

n i i

Then e e f f is a symplectic basis

n n

e f Imhe Je Im he e e e Imhe e

i j i j i j ij i j i j

and similarly f f This is the promised alternative pro of of Theorem

i j

The next Theorem gives a convenient metho d for constructing compatible complex

structures For anyvector space V let RiemV denote the convex op en subset of the

space S V of symmetric bilinear forms consisting of p ositive denite inner pro ducts

Theorem Let E be a symplectic vector space Thereisacanonical contin

uous surjective map

F RiemE JE

The map G J E RiemE J g associating to each compatible complex struc

ture the corresponding Riemannian structureisasection ie F GJ J

Proof Given k RiemE letA GlE b e dened by

k v w v Aw

Since is skewsymmetric A is skewadjoint with resp ect to k A A It follows

that in the p olar decomp osition A J jAj with jAj A A A J and jAj

commute Therefore J Id The equation

v J w v AjAj w k v jAj w k jAj v jAj w

shows that g v w v J w denes a p ositive denite inner pro duct Wethus obtain

acontinuous map F RiemE JE By construction it satises F G id in

particular it is surjective

Corollary The space J E is contractible In particular any two com

patible complex structures can be deformed into each other

THE GROUP Sp E OF LINEAR SYMPLECTOMORPHISMS

Proof Let X RiemE and Y J E The space X is contractible since

itisaconvex subset of a vector space Cho ose a contraction I X X where

Id and is the map onto some p ointinX Then F Id Gisthe

required retraction of Y

Given a Lagrangian subspace L of E any orthonormal basis e e of L is a an

orthornormal basis for E viewed as a complex Hermitian vector space The map taking

this to an orthonormal basis e e of L Lag E is unitary Hence UE acts

transitively on the set of Lagrangian subspaces The stabilizer in UE ofL Lag E

is canonically identied with the orthogonal group OL This shows

Corollary Any choiceofL Lag E and J JE identies the set of

Lagrangian subspaces of E with the homogeneous space

Lag E UE OL

nn nn

Hence Lag E is a manifold of dimension n

The group SpE of linear symplectomorphisms

Let E b e a symplectic vector space of dimension dim E n and SpE GlE

the Lie group of symplectomorphisms A E E A Its dimension can b e

found as follows Since anytwo symplectic vector spaces of the same dimension are

symplectomorphic the general linear group GlE acts transitively on the op en subset

of E consisting of nondegenerate forms The stabilizer at is SpE It follows

that

n n dim SpE dim GlE dim E n

The Lie algebra spE of SpE consists of all glE suchthat v w v w

The following example really a rep etition of example shows in particular that

SpE is not compact

Example Let L M Lag E b e transversal Lagrangian subspaces and identify

M L so that E L L Given B GlLletB GlL the dual map Then

A B B is a symplectomorphismThus for any splitting E L M there is a

canonical emb edding

GlL SpE

as a closed subgroup Note that any A Sp E preserving L M is of this typ e

Example Another natural subgroup of SpE isthegroup U E SpE of

automorphisms preserving the Hermitian structure for a given compatible complex struc

ture J JE

Let us now x a compatible complex structure J JE Let UE SpE

denote the unitary group and g the inner pro duct dened by J If J is another

LINEAR SYMPLECTIC ALGEBRA

compatible complex structure the map A E E taking an orthonormal basis with

resp ect to Jginto one for J g is symplectic and satises A J J This shows

Corollary The action of the symplectic group SpE on the space J E of

compatible complex structures is transitive with stabilizer at J JE equal to the

unitary group U E That is J E may be viewed as a homogeneous space

J E SpE U E

This shows in particular that SpE isconnected We see that J E isanon

compact smo oth manifold of dimension n n n n nWe will show b elow

that the choice of J actually identies J E withavector space Let denote the

transp ose of an endomorphism with resp ect to g

Lemma An automorphism A GlE is in SpE if and only if

A JA J

where A is the transpose of A with respect to g An endomorphism glE is in

spE if and only if

Proof A SpE if and only if for all v w E Av Aw v w or equiv

alently g JAvAw g Jvw ie A JA J The other identityischecked simi

larly

Exercise For E R with the standard symplectic basis and the standard

symplectic structure J is given by a matrix in blo ck form J Writing

a b

T T

A verify that A Sp E is symplectic if and only if a c b d are symmetric

c d

T T

and a d b c I In particular for n wehaveSpR Sl R

Theorem Symplectic eigenvalue Theorem Let A SpE Then detA

and al l eigenvalues of A other than come in either pairs

or quadruples

The members of each multiplet al l appear with the same multiplicity The multiplicities

of eigenvalues and are even

Proof Since detJ the Lemma shows that detA Hence detA

since Sp E is connected For any A GlE the eigenvalues app ear in complex

conjugate pairs of equal multiplicityFor A SpA the eigenvalues have equal

multiplicity since by the Lemma the matrices A and A are similar A JA J

The multiplicities of eigenvalues and havetobeeven since the pro duct of all

eigenvalues equals det A

POLAR DECOMPOSITION OF SYMPLECTOMORPHISMS

Lemma Suppose A SpE is symmetric A A so that A is diagonalizable

and al l eigenvalues arereal Let E kerA denote the eigenspace Then

E E

In particular al l E for eigenvalues f g are isotropic while the eigenspaces for

is symplectic f g are symplectic Moreover E E

Proof For v E w E wehave

v w Av Aw v w

This together with a check of dimensions proves the Lemma

Polar decomp osition of symplectomorphisms

We will use Lemma to obtain the p olar decomp osition of symplectomorphisms

Recall that for any A GlE the p olar decomp osition is the unique decomp osition

A CB into an orthogonal matrix C and a p ositive denite symmetric matrix B

Explicitly B jAj A A and C AjAj Since the exp onential map denes a

dieomorphism from the space of symmetric matrices onto p ositive denite symmetric

matrices this shows that GlE is dieomorphic to a pro duct of OE and a vector

space Wewant to showthatifA SpE then b oth factors in the p olar decomp osition

are in Sp E Thus let

T T

p f spE j g P fA SpE j A A Ag

b e the intersection of Sp E with the set of p ositive denite automorphism and of spE

with the space of symmetric endomorphisms

Lemma The exponential map restricts to a dieomorphism exp p P

Proof Since clearly expp P itsucestoshow that exp p P is onto

Given A P let glE the unique symmetric endomorphism with exp AWe

havetoshow spE or equivalently that A exps Sp E for all s RThe

s s

power A acts on v E as IdLetv E w E If then v w and

s s s s s s

also A v A w v wIf then A v A w v w

v w This shows A SpE

Theorem Polar decomp osition The map UE P SpE C B A

CB is a dieomorphism

Proof Wehavetoshow thatthemapisonto Let A SpE By Lemma

T T

A JAJ SpE Therefore using Lemma jAj A A SpE and

AjAj SpE OE U E

Since J E SpE UE wehavethus shown

Corollary Any xed J JE denes a canonical dieomorphism between

J E and the vector space p

LINEAR SYMPLECTIC ALGEBRA

In particular we see once again that J E iscontractible

Corollary SpE is homotopical ly equivalent to UE In particular it is

connected and has fundamental group SpE Z

Remark Let g spE the Lie algebra of the symplectic group and k uE

spE oE the Lie algebra of the orthogonal group Then g k p as vector spaces

and

k k k k p p p p k

The Killing form on g is p ositive denite on k and negative denite on pFrom these

facts it follows that g k p is a Cartan decomp osition of the semisimple Lie algebra

spE In particular K UE is a maximal compact subgroup of G SpE

Remark The symplectic group SpE should not b e confused with the symplec

tic group Sp n from the theory of compact Lie groups They are however two dierent

real forms of the same complex Lie group

Maslov indices and the Lagrangian Grassmannian

Let E Jg as b efore dim E n The determinant map det UE S induces

an isomorphism of fundamental groups UE S Z Comp osing with the

identication UE SpE we obtain an isomorphism

SpE Z

called the Maslov index of a lo op of symplectomorphisms It is indep endentofthe

choice of J since anytwochoices are homotopic If A B are twoloopsand AB there

pointwise pro duct AB AB Duallywe can pullback the generator

Z by det to nd a canonical class H SpE Z called the Maslov H S Z

class

There are other Maslov indices related to the geometry of the Lagrangian Grass

mannian

Lag E UE OL

Since det U E S takes values onO L its square descends to a welldened

function det Lag E S

Theorem Arnold The map det LagE S denes an isomorphism of

fundamental groups Lag E S Z independent of the choiceofL or

J It is cal ledtheMaslov index of a lo op of Lagrangian subspaces

n n n

Proof Cho ose an orthonormal basis for L to identify L R C and E C so

that Lag E UnO n For t let A t Un b e the diagonal matrix with

entries exp kt Since A On we obtain a lo op L t A tR

k k k

This lo op has Maslov index k which shows that is surjective

To show that is injective it is enough to showthatanyloop Ltwith L

can b e deformed into one of the lo ops L To see this lift Lt to a path L R k

MASLOV INDICES AND THE LAGRANGIAN GRASSMANNIAN

At Ut not necessarily closed with endp oints A and A equal to a diagonal

matrix with entries Wehave to showthatAt can b e deformed into one

of the paths A t while keeping the endp oints xed If the endp oint is the identity

matrix diag so that At is actually a lo op this is clear b ecause U n

S Z is an isomorphism in this case k must b e even If A diag

the path B t At A tpointwise pro duct is a lo op ie can b e deformed into A

for some l Thus At B tA t can b e deformed into A A A

l l

Pulling back the generator H S Zbydet we nd an integral cohomology

class H Lag E Z called the Maslov class

Proposition Let A S SpE and L S Lag E beloops of symplec

tomorphisms resp of Lagrangian subspaces Then

AL LA

Proof Using the notation from the previous pro of wemay assume that A takes

values in Un since SpE U E pAnysuch A is homotopic to a lo op A where

l A The Prop osition follows since A A A

l k k l

Given M Lag E consider the subset

Lag E M fL Lag E j L M fgg

Lemma For any xed L Lag E M one has a canonical dieomorphism

S L N S Lag E M

with the space of symmetric bilinear forms on LInparticular it is contractible The

kernel kerS L is the intersection N L

Proof Recall that the nondegenerate pairing L M R dened by identies

M L Any ndimensional subspace N transversal to M is of the form N fv

S v j v Lg for some linear map S L M One has N L kerS Since

N N N

L M wemayviewS as a bilinear form S L L The condition that N is

Lagrangian is equivalentto

v S v w S w v S w w S v

N N N N

for all v w L In terms of S this says precisely that S S L is symmetric

N N

We note in passing that this Lemma denes co ordinate charts on the Lagrangian

Grassmannian

Remark The co ordinate version of this Lemma is as follows Let L Lag R

suchthat L is transversal to the span of f f ThenL has a unique basis of the

form g e S f The condition L L translates into S b eing a symmetric

i i ij j

are two such Lagrangian subspaces and g g the corresp onding bases matrix If L L

R is given by the pairing L L

g g S S

i ij

j ij

LINEAR SYMPLECTIC ALGEBRA

That is the dimension of the intersection L L L L equals the nullityof S S

The fact that Lag E M iscontractible can b e used to generalize the Maslovindex

to paths L Lag E which are not lo ops but satisfy the b oundary conditions

LL Lag E M Indeed we can complete L toaloop L S RZ Lag E

with Lt Lt for t andLt Lag E M for t suchthat

L L The Maslov intersection index is dened as

L M L Z

which is indep endentofthechoice of L

Remark Maslovs index can b e interpreted as a signed intersection number

of L with the singular cycle Lag E n Lag E M It was in this form that Maslov

originally intro duced his index The diculty of this approach is that the singular cycle

is not a smo oth submanifold of Lag E Given a path in Lag E one p erturbs this path

untilitintersects only the smo oth part of the singular cycle and all intersections are

transversal It is then necessary to prove that the index is indep endentofthechoice of

p erturbation

Maslovinvented his index in the context of geometrical optics high frequency

asymptotics and quantum mechanics semiclassical approximation It app ears phys

ically as a phase shift when a lightray passes through a fo cal p oint a phenomenon dis

covered in the th century Mathematically Maslovs theory gaverisetoHormanders

theory of Fourier integral op erators in PDE

Maslovs index can b e generalized to paths L that are not necessarily transversal to

M at the end p oints This was rst done by Dazord in a pap er and rediscovered

several times since then We will describ e one such construction in the following section

The index of a Lagrangian triple

In this section we describ e a dierent approachtowards Maslov indices using the

HormanderKashiwara index of a Lagrangian triple As a motivation consider the action

of SpE on Lag E Wehave seen that this action is transitive Moreover anytwo

ordered pairs of transversal Lagrangian subspaces can b e carried into each other by some

symplectomorphism An analogous statement is true if one xes the dimension of the

intersection dimL L

Exercise Show that for any L L Lag E there exists a symplectic basis

in which L is spanned by the e e and L by e e f f where k

n k k n

dimL L It follows that the action of SpE on Lag E Lag E has n orbits

lab eled by the dimension of intersections

Is this true also for triples of Lagrangian subspaces

THE INDEX OF A LAGRANGIAN TRIPLE

Exercise Let E R with symplectic basis e f Let L spanfeg L

span ff g What is the form of a matrix of the most general symplectomorphism pre

serving L L Let L spanfe f gandL L L a second triple of Lagrangian

subspaces with L L L L Showby direct computation that there exists a sym

plectic transformation A Sp E with L AL forallj if and only if

L span fe f g with

Thus sp ecifying the dimensions of intersections is insucient for describing the orbit

of a Lagrangian triple L L L There is another invariant called the Hormander

Kashiwara index of a Lagrangian triple

Before we dene the index let us recall that the signature SigB Zof a symmetric

matrix B is dened to b e the number of its positive eigenvalues minus the number of

its negative eigenvalues More abstractly letting sign R R denote the sign function

if t

signt

if t

and dening signB by functional calculus wehaveSigB trsignB The signature

has the prop erty Sig AB A SigB foranyinvertible matrix AIfk S V isa

symmetric bilinear form equivalently a quadratic form on a vector space V one denes

Sigk SigB

where B is the matrix of k in a given basis of V The signature and the nullity are the

only invariants of a symmetric bilinear form That is the action of GlV on S V has

a nite numb er of orbits lab eled bydimkerV and Sig k

Given three Lagrangian subspaces not necessarily transversal consider the symmet

ric bilinear form QL L L on their direct sum L L L given by

QL L L v v v v v v v v v v v v

The index of the the Lagrangian triple L L L is the signature

sL L L Sig QL L L Z

It is due to Hormander in his famous pap er on Fourier integral op erators and

in greater generality Kashiwara according to the b o ok by LionVergne Clearly s is

invariant under the action of SpE onLagE

Cho osing bases for L L L the denition gives QL L L as a symmetric

n nmatrix One can reduce to signatures of n nmatrices as follows Cho ose

a symplectic basis e e f f of E such that L L L are transversal to the

n n

span of f f Let S denote the symmetric bilinear forms on the span of e e

n j n

corresp onding to S In terms of the basis S is just a matrix and QL L L isgiven

j j

LINEAR SYMPLECTIC ALGEBRA

by a symmetric matrix

S S S S

QL L L

S S S S

Lemma sL L L SigS S SigS S SigS S

Proof Brian Feldstein Let

An elementary calculation shows that detT andthatTQL L L T is the

symmetric matrix

S S

From this the Lemma is immediate

Theorem The signature s LagE Z of a Lagrangian triple has the fol

lowing properties

a s is antisymmetric under permutations of L L L

b Cocycle Identity For al l quadruples L L L L Lag E

sL L L sL L L sL L L sL L L

c If M t is a continuous path of Lagrangian subspaces such that M t is always

transversal to L L Lag E then sL L Mt is constant as a function of t

d Any ordered triple of Lagrangian subspaces is determined up to symplectomorphism

by the ve numbers

dimL L dimL L dimL L dimL L L sL L L

Proof The rst prop ertyisimmediate from the denition while the second and

third prop erty follow from the Lemma The fourth prop erty is left as a nontrivial

exercise Perhaps try it rst for the case that the L are pairwise transversal

Lemma Suppose L tL t Lag E are two paths of Lagrangian subspaces

a t b Suppose there exists M Lag E transversal to L t and L t for al l

t a b then the dierence

L L sL aL aM sL bL bM

is independent of the choiceofsuchM

THE INDEX OF A LAGRANGIAN TRIPLE

Proof Let M M be twochoices By the co cycle identity the rst term changes

sL aL aM sL aL aM sL aMM sL aMM

Wehavetoshow that this equals the change of the second term

sL bL bM sL bL bM sL bMM sL bMM

But sL tMM and sL tMM are indep endentof t since L stay transversal

to M M

We dene the Maslovintersection index for two arbitrary paths L L a b

Lag E as follows Cho ose a sub division a t t t b such that for all j

k there is a Lagrangian subspace M transversal to L tL t for t t t

j j

Then put

j j

L L sL t L t M sL t L t M

j j j j

Clearly this is indep endentofthechoice of sub division and of the choice of the

M Note that this denition do es not require transversality at the endp oints The

intersection is additive under concatenation of paths It is antisymmetric L L

L L

Exercise Show that for any path of symplectomorphisms A a b SpE

AL AL L L

Exercise Let E R and let L L a b Lag E bedenedby L t

span f teand L t span f Find L L Howdoesitdependona b

Exercise Let L L L a b Lag E b e three paths of Lagrangian sub

spaces Show that

L L L L L L sL aL aL a sL bL bL b

The approach can also b e used to dene Maslov indices of paths not necessarily

lo ops of symplectomorphisms Let E denote E with minus the symplectic form and let

E E b e equipp ed with the symplectic form pr pr where pr are the pro jections

to the rst and second factor

Proposition For any symplectomorphism A SpE the graph

fAv v j v E gE E

is a Lagrangian subspace

Proof Let pr pr denote the pro jections from E E to the resp ective factor

For all v v E wehave

Av v Av v v v Av Av pr pr

LINEAR SYMPLECTIC ALGEBRA

In this sense Lagrangian subspaces of E E maybeviewed as generalized symplec

tomorphisms If A a b Sp E is a path of symplectomorphisms one can dene

a Maslov index where E E is the diagonal For lo ops based at the

identity this reduces up to a factor of to the index Aintro duced earlier

Linear Reduction

Supp ose F E is a subspace Then the kernel of the restriction of to F is just

F F bythevery denition of F It follows that the quotient space E FF F

inherits a natural symplectic form Letting F FF F b e the quotient

map wehave

v w v w

for all v w F The space E is called the reducedspace or symplectic quotient

Proposition Suppose F E is coisotropic and L Lag E Lagrangian Let

L be the image of L F under the reduction map F FF E Then

F F

L Lag E

F F

Proof Since L F is isotropic it is immediate that L is isotropic Toverify that

for any F L is Lagrangian wejusthavetocount dimensions Using F F F

F F E we compute

dimL F dim E dimL F

dim E dimL F

dim E dim L dim F dimL F

dimL F dim F dimL

This shows that

dim L dimL F dimL F dimF dim L

on the other hand

dim E dimF dim F dimF dim E dim F dimL

For any symplectic vector space E letE denote E with symplectic form

Supp ose E E E are symplectic vector spaces and let

E E E E E

Then the diagonal E consisting of vectors v v v v is coisotropic Given

the direct pro duct and Lag E E Lagrangian subspaces Lag E E

is a Lagrangian subspace of E Thecomposition of is dened as

Lag E E

LINEAR REDUCTION

This is really the comp osition of relations

fv v jv E with v v v v g

If A A are symplectomorphisms

A A A A

Similarlyfor L Lag E wehave L AL

Exercise Let F E b e coisotropic Showthat

fv w E E jw F v w g

F F

where F E is the pro jection is Lagrangian It satises L L for all

F F F

L Lag E

LINEAR SYMPLECTIC ALGEBRA

CHAPTER

Review of Dierential Geometry

Vector elds

We assume familiarity with the denition of a manifold charts smo oth maps etc

We will always take paracompactness as part of the denition this condition ensures

that every op en cover U of M admits a sub ordinate partition of unity f C M

That is f is a nonnegative function supp orted in U near every p oint only a nite

number of f s are nonzero and f

Let XM denote the vector space of derivations X C M C M That is

X XM if and only is

X fgX f g fXg

for all f g C M From this denition it follows easily that the value of X f at

m M dep ends only on the b ehavior of f in an arbitrarily small neighb orho o d of m

ie on the germ of f at m One can showthatinanychart U M with lo cal

co ordinates x x every X XM has the form

a X f

where a are smo oth functions Elements of XM are called vector elds The space

XM isa C M mo dule that is for all f C M X XM one has fX XM

and it is a Lie algebra with bracket

X Y X Y Y X

P P

a In lo cal co ordinates if X b and Y then

i i

x x

i i

X X

b a

i i

X Y a b

j j

x x x

j j i

i j

A tangentvector at m M is a linear map v C M R such that

v fgv f g mf m v g

for all smo oth f g The space of tangentvectors at m is denoted T M and the union

TM T M

REVIEW OF DIFFERENTIAL GEOMETRY

is the tangent bundle In lo cal co ordinated every v T M is of the form

v f v m

One can use this to dene a manifold structure on TM with lo cal co ordinates

x x v v The natural pro jection map TM M is smo oth

n n

Clearlyevery vector eld X denes a tangentvector v X by X f X f m

m m

Converselyevery smo oth map X M TM with X id denes a vector eld

Thus vector elds can b e viewed as sections of the tangentbundleTM

For any smo oth function F M N one has a linear pul lback map

F C N C M F g g F

This denes a smo oth pushforward map

F TM TN v F v

where F v g v F g This map is b erwise linear but do es not in general carry

vector elds to vector elds There are two problems If F is not surjectivewehaveno

candidate for the section N TN away from the image of F IfF is not injective we

mayhave more than one candidate over some p oints in the image of F Vector elds

X XM and Y XN are called F related write X Y ifforallg C N

X F g F Y g

This is equivalentto F X Y for all m M From the denition one sees easily

F m

the imp ortant fact

X Y X Y X X Y Y

F F F

If F is a dieomorphism we denote by F X the unique vector eld that is F related to

A global ow on M is a smo oth map

R M M t m m

with

M t s ts

Every global owonM denes a vector eld X XM by

f X f

t t

For compact manifold M every vector eld arises in this wayFor noncompact M one

has to allow for incomplete ows that is one has to restrict the denition of to some

op en neighborhood of fgM in R M and require m m whenever

t s ts

DIFFERENTIAL FORMS

these terms are dened Supp ose is the ow dened by X XM We dene the

Lie derivative of vector elds Y XM by

L Y Y XM

X t

t t

It is a very imp ortant fact that for all vector elds X Y

L Y X Y

Dierential forms

Let E be a vector space of dimension n over RAk form on E isamultilinear map

E E R

k times

k k

antisymmetric in all entries The space of k forms is denoted E Onehas E fg

for kn E R and

dim E

E is an algebra with pro duct given by for k n The space E

antisymmetrization of

X X X X X X

k l

k k k l

k l

for E and E Given v E there is a natural contraction map

k k

E E given by

This op erator is a graded derivation

v v v

A over a commuta Remark In general given a Zgraded algebra A

k Z

tive ring Raderivation of degree r of A is a linear map D AAtaking A into

k r

A and satisfying the graded Leibniz rule

D abD ab aDb

k l

Der A denote the graded space of graded for aA bA Let DerA

r Z

A derivations of r Then DerA is a graded Lie algebra over R That is for D Der

j onehas

r r

D D Der A D D D D

REVIEW OF DIFFERENTIAL GEOMETRY

Supp ose nowthatM is a manifold Then we can dene vector bundles

k k

T M T M

Smo oth sections of T M are called k forms and the space of k forms is denoted

k k

M is a graded algebra over C M Equivalently M The space M

k forms can b e viewed as C M multilinear antisymmetric maps

XM XM C M

k times

Note M C M For anyvector eld X XM there is a contraction op erator

k k

M M which is a graded derivation

For any function f C M denote bydf M the form suchthatdf X

X f

Theorem The map d M M extends uniquely to a graded deriva

tion d M M of degree in such a way that d

k k k

The quotientspace H M kerdj imdj is called the k th de Rham co

homology group of M Wedge pro duct gives H M the structure of a graded algebra

For M compact the de Rham cohomology groups are always nitedimensional vector

spaces

Supp ose now that F M N is smo oth Then F denes a unique pullback map

k k

F N M by

F v v F v F v

k k

for all v T M The exterior dierential resp ects F that is

j m

dF F d

Supp ose is the owof X X The Lie derivativeof M with resp ect to X is

dened as follows

t t

Since pullbacks commute with the exterior dierential L dd L The op erator

X X

L is a graded derivation of degree

L L L

X X X

One has the following relations b etween the derivations dL

X X

d d d dL L L L

X X Y XY

L d L

X Y X Y X X

The last formula

d dL

X X X

DIFFERENTIAL FORMS

is known as Cartans formula These relations show that the linear subspace of

DerM spanned by L d is in fact a subalgebra of the graded Lie algebra

X X

DerM

Let us reexpress some of the ab ove in lo cal co ordinates x x forachart U M

Atevery p oint m M the forms d x dx are a basis of T M dual to the basis

of T M Every form is given on U by an expression

x x

i i

wehave X where are smo oth functions For X

i i

X X

i i

More generally for every ordered tuple I i i of cardinality jI j k put

dx dx dx

I i i

Then every M has the co ordinate expression

I I

where are smo oth functions and the sum is over all ordered tuples i i One

I k

has

X X

d dx dx

j I

indeed this formula is forced on us by the derivation prop erty of d and the conditions

Exercise Verify that this formula indeed gives d using the equalityof

mixed partials

Exercise Work out the co ordinate expressions for Lie derivatives L and con

tractions

Avolume form on a manifold M of dimension n is an n form M with

for all mIfM admits a volume form it is called orientable The choice of an

equivalence class of volume forms where are equivalentif f with f

everywhere is called an orientation

Exercise For anyvolume form and X XM the divergence of X with

resp ect to is the function div X suchthat L div X Find a formula for

L in lo cal co ordinates X

REVIEW OF DIFFERENTIAL GEOMETRY

CHAPTER

Foundations of symplectic geometry

Denition of symplectic manifolds

Definition A symplectic manifold is a pair M consisting of a manifold

M together with a closed nondegenerate form M Given two symplectic

manifolds M a symplectomorphism is a dieomorphism FM M suchthat

i i

F The group of symplectomorphism of M onto itself is denoted SympM

The space of vector elds X with L is denoted XM

Nondegeneracy means that for each m M the form is a symplectic form on

T M in particular dim M n is even Equivalently the top exterior p ower is

nonzero

Definition The volume form exp is called the Liouville

dim M

form

Definition For any H C M R the corresp onding Hamiltonian vector

eld X is the unique vector eld satisfying

The space of vector elds X of the form X X is denoted X M

H Ham

Proposition Every Hamiltonian vector eld is a symplectic vector eld That

X M XM

Ham

Proof Supp ose X X that is dH Then

H X

L d ddH

X X

Examples

Example op en subsets of R

Example The basic example are op en subsets U R Let

q q p p be co ordinates with resp ect to a symplectic basis

n n

FOUNDATIONS OF SYMPLECTIC GEOMETRY

e e f f for R This identies e and f In terms of

n n j j

q p

j j

the dual forms dq dp the symplectic form is given by

dq dp

j j

and the Liouville form reads

dq dp d q dp

n n

Given a smo oth function H on U wehave

H H

p q q p

j j j j

Hence the ordinary dierential equation dened by X is

H H

q p

j j

p q

j j

Note that Hamiltonians H q pp generate translation in q direction while H q p

j j

q generate translation in minus p direction

j j

Definition Let M b e a symplectic manifold We denote by X M

Ham

the space of Hamiltonian vector elds on M and by XM the space of vector elds

X on M preserving ie L

Example Surfaces Let b e an orientable manifold and a

volume form Then is nondegenerate since everywhere and closed

since every top degree form is obviously closed A symplectomorphism is just a volume

preserving dieomorphism in this case By a result of Moser anytwovolume forms on a

compact manifold M dening the same orientation and having the same total volume

are related by some dieomorphism of M In particular every closed symplectic

manifold is determined up to symplectomorphism by its genus and total volume

Example Cotangent bundles Avery imp ortant example for symplectic

manifolds are cotangent bundles Let Q b e a manifold M T Q its cotangent bundle

There is a canonical form T Qgiven as follows Let M T Q Q the

bundle pro jection d TM TQ its tangent map Then for vectors X T M

m m

h X i hm d X i

m m m m

Q is a covector at m we can pair it with the pro jection of X to since m T

the base An alternativecharacterization of the form is as follows

Proposition is the unique form T Q with the property that for

Q on the base any form

EXAMPLES

where on the right hand side is viewedasasection Q T Q M

Proof Tocheck the prop ertyletw T Q Then d Y pro jects to w Wehave

q q

therefore

h wi h d w i h wi

q q q

Uniqueness follows since every tangentvector v T M except the vertical ones ie

those in the kernel of d can b e written in the form d w where Q with

m q

q m and w T Q Nonvertical vectors span all of T M

q m

It is useful to work out the form in lo cal co ordinates

For anyvector bundle E Q of rank k one obtains lo cal co ordinates over a

bundle chart W Q bycho osing lo cal co ordinates q q on Q and a basis

n k

Q E for the C W mo dule of sections of E j Any p oint m E is then given by

the co ordinates q of its base p oint q m and the co ordinates p p suchthat

i k

P P

m p q Thus if W E j is any section over W the pullbacks

i i i i W

j i

of q p viewed as functions on E j are wehave p and q q

i i W i i i i

In our case E T Q k n and a natural basis for the space of sections is given by

the forms dq The corresp onding co ordinates q q p p on T Qj

i i n n W

are called cotangent coordinates

Lemma In local cotangent coordinates q q p p on T Q the

n n

canonical form is given by

jT W p dq

j j

dq b e a form on W Then p q q thus Proof Let

j j j j j j

X X

p dq dq

j j j j

j j

Theorem Let M T Q beacotangent bund le and its canonical form

Then d is a symplectic structureon M

Proof In lo cal cotangent co ordinates dq dp

j j

We will now describ e some natural symplectomorphisms and Hamiltonian vector

elds on M T Q

Let f Q Q b e a dieomorphism Then the tangentmapdf is a dieomorphism

TQ TQ and dually there is a dieomorphism

F df T Q T Q

FOUNDATIONS OF SYMPLECTIC GEOMETRY

called cotangentliftoff covering f For all Q one has a commutative

diagram

T Q T Q

Q Q

Proposition Naturality of the canonical form Let F T Q T Q be

the cotangent lift of f Then F preserves the canonical form F hence F is a

symplectomorphism F

Proof This is clear since our denition of the canonical form was co ordinatefree

For the sceptic check the prop erty Wehave

F F f f f f f f

This gives a natural group homomorphism

Di Q SympT Q f df

Another subgroup of SympT Q is obtained from the space of closed forms Z Q

Q For any Q let G T Q T Q b e the dieomorphism obtained by

adding

Proposition For al l Q

Thus G is a symplectomorphism if and only if d that is Z Q

Proof Let Q Then

G G

By the characterizing prop ertyof this proves the Prop osition

Wethus nd a group homomorphism

Z Q SympT Q

Recall that for any representation of a group G on a vector space V one denes G n V

to b e the group whose underlying set is G V and with pro duct structure

g v g v g g v g v

In this case we can let Di Q act on Z Qby f f Itiseasytocheckthat

the homomorphisms and combine into a group homomorphism

Di Q n Z Q SympT Q

The semidirect pro duct Di Q n Z Qmaybeviewed as an innitedimensional gen

eralization of the Euclidean group of motions On nR

EXAMPLES

We will now show that the generators of the action of Di Q are Hamiltonian vector

elds Let Y be a vector eld on Q Then there is a unique vector eld X LiftQon

T Q with the prop erty that the owofX is the cotangent lift of the owofY We call

the map

Lift XQ XT Q

the cotangent lift of a vector eld Note that X pro jects onto Y thatisX Y Let

H C T Q R b e dened as the contraction H

Lemma The cotangent lift X of Y is a Hamiltonian vector eld with Hamil

tonian H

Proof Since the owof X preserves L Therefore

dH d d L d

X X X X X

WhatareX and H in these Supp ose Y is given in lo cal co ordinates by Y Y

Y co ordinatesSince X pro jects to Y under we knowthatX is a vertical

vector eld ie a linear combination of Thevertical part do es not contribute to

since p dq is a horizontal form Hence

X j j

X X

Y q p H q p Y q

j j X j

j j

From this werecover

n n

X X

X Y p

j j

q q p

j k k

From these equations we see that the Hamiltonians corresp onding to cotangent lifts are

those whicharelinear along the bers of T Q Other interesting ows are generated by

Hamiltonians that are constant along the bers of T Q ie of the form H f with

f C Q The ow generated by suchanH is given in terms of the notation G

intro duced ab ove by G The Hamiltonian vector eld corresp onding to H is

t tdf

in lo cal cotangent co ordinates

q p

j j

Exercise Verify these claims

On the total space of anyvector bundle E Q there is a canonical vector eld

EXE called the Euler vector eld its ow is b erwise multiplication by e In

our case E T Qwehave in lo cal cotangent co ordinates

E p

j j

FOUNDATIONS OF SYMPLECTIC GEOMETRY

Proposition The Euler vector eld satises

E E

In particular E is the vector eld corresponding to under the isomorphism TQ

T Q

Proof The form is homogeneous of degree along the b ers That is un

t t t

der b erwise multiplication by e it transforms according to e e Taking the

derivative this shows

Applying d gives the rst formula in the Prop osition The second formula is obtained

using the Cartan formula for the Lie derivative together with

Remark A symplectic manifold M together with a free Raction whose gen

erating vector eld E satises such that L is called a symplectic cone Thus

cotangent bundles minus their zero section are examples of symplectic cones Another

example is R fg

Proposition For any closed form Q the sum is a sym

plectic form on T Q The Liouvil le form of equals that for

Proof Since vanishes on tangentvectors to b ers of itskernel at m T Q

contains a Lagrangian subspace ie its kernel is coisotropic The claim now follows

from the following Lemma

Lemma Let E be a symplectic vector space and E a form such

n n

that ker is coisotropic Then Inparticular is nondegenerate

Proof Since ker is coisotropic it contains a Lagrangian subspace LLete e

k nk k nk

a basis for LFor knwehave e e Therefore and

n n

it follows that In particular is symplectic since its top p ower is a

volume form

This has the following somewhat silly corollary For any manifold Q withaclosedform

there exists a symplectic manifold M and an emb edding Q M suchthat

Pro of Take M T Q with symplectic form d

Example Kahler manifolds An almost complex manifold is a manifold

Q together with a smo othly varying complex structure on each tangent space ie a

smo oth section J Q GlTQ satisfying J id A complex manifold is a man

ifold together with an atlas consisting of op en subsets of C insuchaway that the

transition functions are holomorphic maps Every complex manifold is almost complex

the automorphism J given bymultiplication by in complex co ordinate charts

The NewlanderNirenb erg theorem see eg the b o ok by KobayashiNomizu gives a

necessary and sucient criterion vanishing of the Nijenhuis tensor for when an almost

complex structures is integrable ie comes from a complex manifold

EXAMPLES

An almost complex structure J on a symplectic manifold M is called

compatible if it is compatible on every tangent space T M We denote by J M

the space of compatible almost complex structures on M The constructions from lin

ear symplectic algebra can b e carried out b erwise Letting RiemM denote the space

of Riemannian metrics the space of sections g M S T M suchthateach g is an

inner pro duct on T M wehave a canonical surjectivemap

RiemM JM

which is a left inverse to the map J M RiemM asso ciating to J the corre

sp onding inner pro ducts on T M In particular J M is nonempty Similar to the

linear case one nds that anytwo J J JM can b e smo othly deformed within

J M More precisely there exists a smo oth map J M GlTM suchthat

J t m JT M and J J J J

m m

For any J JM the triple M J is called an almost Kahler manifold some

times also almost Hermitian manifold If J comes from an honest complex structure

then M J is called a Kahler manifold An example of a Kahler manifold is M C

Proposition Let M J beaKahler manifold Let N J beancomplex

manifold and N M an complex immersion That is J d d J Then

N J is an Kahler manifold Similar assertions hold for the almost Kahler cate

gory and almost complex immersions

Proof Obviouslyevery complex subspace of a Hermitian vector space is Hermitian

Applying this to eachd T N T M we see that the closed form is non

n n

degenerate and J JN

This shows in particular that every complex submanifold of C is a symplectic mani

fold Notice that if N C is the zero lo cus of a collection of homogeneous p olynomials

such that N is smo oth away from fg then N nfg is a symplectic cone

We next consider complex pro jective space

n n

C P nC nfgC nfgS S

n n n

Let S C the emb edding and S C P n the pro jection Atevery

point z S wehave a canonical splitting of tangent spaces

T C T C P n span fz g

as complex vector spaces Since T C P n is a complex subspace it is also symplectic

This induces a nondegenerate form on C P n whichby construction is compatible

with the complex structure Letting b e the symplectic structure of C wehave

showing that is closed This shows that Therefore d d

C P nisaKahler manifold The form is called FubiniStudy form Later we will

see this construction of more systematically as a symplectic reduction

By the ab ove prop osition every nonsingular pro jectivevarietyisaKahler manifold

FOUNDATIONS OF SYMPLECTIC GEOMETRY

Wehave seen that every symplectic vector space E admits a compatible almost

complex structure It is natural to ask whether it also admits a compatible complex

structure ie whether every symplectic manifold is Kahler The answer is negative A

rst counterexample was found by Ko daira and later rediscovered byThurston By now

there are large families of counterexamples see eg work of McDu and Gompf For

a discussion of the Ko dairaThurston counterexample see the b o ok McDuSalamon

Intro duction to Symplectic Top ology

Basic prop erties of symplectic manifolds

Hamiltonian and symplectic vector elds Wewillnow study the Lie

algebras of Hamiltonian and symplectic vector elds in more detail Let M bea

symplectic manifold By denition a vector eld X is Hamiltonian if dH for

some smo oth function H This means that the isomorphism b etween vector elds and

forms XM M dened by restricts to an isomorphism

X M B M

Ham

with the space B M M imd of exact forms Similarly a vector eld is

symplectic if and only if L whichby Cartans identity means d Thus

X X

wehave an isomorphism

XM Z M

with the space Z M M kerd of closed forms Thus the quotient

XM X M is just the rst deRham cohomology group cohomology H M

Ham

Z M B M and wehave an exact sequence of vector spaces

X M XM H M

Ham

We conclude that if H M fg eg for simply connected spaces suchasM C or

M C P nevery symplectic vector eld is Hamiltonian

Proposition For al l Y Y XM one has

Y Y X

Y Y

Proof Let Y Y XM Then

d Y Y d

Y Y

L d

Y Y Y Y

Y Y

L Y Y Y

Prop osition shows that XM XM X M In particular

Ham

X M is an ideal in the Lie algebra XM and the quotient Lie algebra

Ham

XM X M is ab elian It follows that is an exact sequence of Lie alge

Ham

M carries the trivial Lie algebra structure bras where H

BASIC PROPERTIES OF SYMPLECTIC MANIFOLDS

Poisson brackets Consider next the surjective map C M

X M H X Its kernel is the space Z M H M of lo cally con

Ham H

stant functions Wethus have an exact sequence of vector spaces

Z M C M X M

Ham

We will now dene a Lie algebra structure on C M tomakethisinto an exact sequence

of Lie algebras Prop osition indicates what the right denition of the Lie bracket

should b e

Definition Let M b e a symplectic manifold The Poisson bracket of two

functions F G C M R is dened as

fF Gg X X

F G

From the denition it is immediate that the Poisson bracket is antisymmetric Using

that X dG by denition together with Cartans identity one has the alternative

formulas

fF Gg L G L F

X X

F G

These formulas show for example that if F Poisson commutes with a given Hamiltonian

G then F is an integral of motion for X That is F is constant along solution curves

of X

Proposition The Poisson bracket denes a Lie algebra structure on

C M R That is it is antisymmetric and satises the Jacobi identity

fF fG H gg fG fH F gg fH fF Ggg

for al l F G H The map C M XM F X is a Lie algebra homomorphism

X X X

F G

fFGg

Proof Equation is just a sp ecial case of Prop osition The rst statement

follows from the calculation

fF fG H gg L fG H g

L X X

X G H

X X X X X X

F G H G F H

X X X X

H G

fFGg fFH g

fH fF Ggg fG fF H gg

An immediate consequence of is

Corollary If F G C M Poissoncommute the ows of their Hamilton

ian vector elds X X commute

F G

FOUNDATIONS OF SYMPLECTIC GEOMETRY

Definition An algebra A together with a Lie structure is called a Poisson

algebra if

FGH F G H F H G

For any algebra A the canonical Lie bracket F G FG GF satises this prop erty

Proposition The algebra C M R f g is a Poisson algebra

Proof

fFGHg L FGL F G F L GfF H g G F fG H g

X X X

H H H

Proposition For any compact connected symplectic manifold Lie algebraex

tension hasacanonical splitting That is there exists a canonical Lie algebra homo

morphism X M C M R that is a right inverse to the map F X

Ham F

Proof The required map asso ciates to every X X M the unique H such

Ham

that X X and H where is the Liouville form The equality

Z Z Z

fF Gg L G L G

X X

F F

M M M

shows that this is indeed a Lie homomorphism

Let us give the expression for the Poisson bracket for op en subsets U R with

symplectic co ordinates q q p p Wehave

n n

F F

p q q p

j j j j

hence fF Gg X Gisgiven by

F G F G

fF Gg

p q q p

j j j j

Exercise Verify directly in lo cal co ordinates that the right hand side of this

formula denes a Lie bracket

Review of some dierential geometry As a preparation for the following

section we briey review the notions of submersions brations and foliations Let Q be

an ndimensional manifold

Submersions A submersion is a smo oth map f Q B to a manifold B

such that for all q Q the tangentmapd f f q T Q T B is surjective For

q q

f q

any submersion the b ers f a are smo oth emb edded submanifolds of Q of dimension

k n dim B InfactQ is foliated by such submanifolds in the following sense Every

point q Q has a co ordinate neighborhood U with co ordinates x x with q cor

resp onding to x and f q a co ordinate neighb orho o d with co ordinates y y

BASIC PROPERTIES OF SYMPLECTIC MANIFOLDS

with f q corresp onding to y suchthatthemapf is given by pro jection to the

rst n k co ordinates

Let Q B b e a submersion Let the space of vertical vector elds

X QfX XQj X g

vert

b e the space of all vector elds taking values in ker ie tangent to the b ers Let

Q f Qj X X Qg

hor X vert

the space of horizontal forms and

Q f Qj L X X Qg

basic X X vert

the space of basic forms Notice that Q preserved by the exterior dierential d

basic

that is it is a sub complex of Q d

Fibrations A submersion is called a bration if it is surjective and has the

lo cal triviality condition There exists a manifold F called standard b er suchthat

every p ointinM there is a neighborhood U and a dieomorphism U f U F

in sucha way that f is just pro jection to the rst factor One can showthatevery

surjective submersion with compact b ers is a bration In particular if Q is compact

every submersion from Q is brating

For any bration Q B pullback denes an isomorphism

k k

B Q

basic

This is easily veried in bundle charts U F U

Distributions and foliations Avector subbundle E TQ of rank k is called

a distribution For example if f Q B is a submersion or more generally if the

tangent map f has constant rank then E kerf TQ is a distribution Also

if X XQisnowhere vanishing vector eld span X is a distribution of rank A

submanifold S Q is called an integral submanifold if TS E j For example the

integral submanifolds of spanX are just integral curves of X A distribution E of rank

k is called integrable if through every p oint q Q there passes a k dimensional integral

submanifold As one can show this is is equivalent to the condition that every p ointhas

such that E j kerf For this a neighb orho o d U and a submersion f U R

reason integrable distributions are also called foliations

Example Let S C the unit sphere For each z S let E T S denote

z z

the orthogonal complement of the complex line through z The distribution E is not

integrable

A necessary condition for integrability of a distribution is that for every twovector

elds X X taking values in E their Lie bracket X X takes values in E This

follows b ecause the Lie bracket of twovector elds tangent to a submanifold S Q is

also tangenttoS Frob enius theorem states that this condition is also sucient

Theorem Frob enius criterion A distribution E TQ is integrable if and

only if the spaceofsections of E is closed under the Lie bracket operation

FOUNDATIONS OF SYMPLECTIC GEOMETRY

Example Let Q R with co ordinates x y z and let E TQ the vector

subbundle spanned bythetwovector elds

X X x

x y z

is not in E the distribution E is not integrable Since X X

Example A smo oth map f Y Y is called a constant rank map if the

image of the tangent map f TY TY has constant dimension Examples are

submersions f surjective or immersions f injective The kernel of every constant

rank map denes an integrable distribution Indeed twovector elds X X are in the

kernel if and only if X Hence also X X

j f f

Here wehave used that for anyvector bundle map F E E of constant rank

the kernel and image of F are smo oth vector subbundles In particular kerf isa

smo oth vector subbundle

Lagrangian submanifolds Let M b e a symplectic manifold

Definition A submanifold or more generallyanimmersion N M is

called coisotropic resp isotropic Lagrangian symplectic if at anypoint n N T N

is a coisotropic resp isotropic Lagrangian symplectic subspace of T M

For example RP n C P n is a Lagrangian submanifold Also the b ers of a

cotangent bundle T Q are Lagrangian Another imp ortant example is

Proposition The graph T Q of a form Q T Q is Lagrangian

if and only if is closed

Proof

d d d

P P

In lo cal co ordinates If dq the pullbackof dq dp to the graph

j j j j

j j

of is given by

X X

dq d dq dq d

j j j k

Hence is closed if and only if the pullbackof to the graph vanishes

Let N Q b e a submanifold of Q Dual to the inclusion d TN TQj

there is a surjectivevector bundle map

d T Qj T N

Its kernel kerd ie the preimage of the zero section N T Qj consists of

covectors that vanish on all tangentvectors to N One calls AnnTNkerd the

annihilator bundle of TN or also the conormal bund le

BASIC PROPERTIES OF SYMPLECTIC MANIFOLDS

Proposition The conormal bund le to any submanifold N Q is a Lagrangian

submanifold of T Q More general ly let N be a form on N Then the pre

image of the graph T N under the map d T Qj T N is a Lagrangian

submanifold of T Q

Proof Lo cally near any p ointof N wecancho ose co ordinates q q on Q such

that N is given by equations q q In the corresp onding cotangentcoor

k n

dinates q p on T Q AnnTN is given by equations q q p p

j j k n k

dq dp vanishes on this submanifold More Clearly each summand in

j j

generally the preimage d isgiven by equations

q q p p

k n k k

Hence the pullbackof to this submanifold is given by

k k

X X

dq d d q dq

i j i j

i ij

so that vanishes on this submanifold if and only if is closed

Proposition Let M be symplectic manifolds and let M denote M

j j

with symplectic form A dieomorphism F M M is a symplectomorphism if

and only if its graph

fF mmj m M gM M

is Lagrangian

Proof Similar to the linear case

Here is a nice application of these considerations

Theorem Tulczyjew Let E B beavector bund le E B its dual bund le

T E Thereisacanonical symplectomorphism T E

OutlineofProof Consider the vector space direct sum N E E as a smo oth

submanifold of Q E E The natural pairing b etween E and E denes a function

f N E E Rlet df Bytheabove denes a Lagrangian submanifold

L of T Q T E T E Onechecks that the pro jections from L onto b oth factors T

and T E are dieomorphisms hence L is the graph of a symplectomorphism

Tulczyjew only considered the case E TQ It was p ointed out byDRoyten

b erg in his thesis that the argumentworks for anyvector bundles In fact he uses a

generalization to sup ervector bundles

FOUNDATIONS OF SYMPLECTIC GEOMETRY

Constant rank submanifolds A submanifold N M is called a con

stant rank submanifold if the dimension of the kernel of is indep endentof n N

Proposition Let N be a manifold together with a closed form of constant

rank Then the subbund le ker is integrable ie denes a foliation

Proof We use Frob enius criterion Supp ose X X XN with Since

is closed this implies L d Hence

X X

j j

L L

X X X X X X

If this socalled nullfoliation of N is brating ie if the leaves of the foliation are

the b ers of a submersion N B where B is the space of leaves of the foliation

The form is basic for this bration since and L for all vertical vector

X X

elds It follows that B inherits a unique form such that

Definition The symplectic manifold B is called the symplectic reduc

tion of N

Remark Note that the ab ove discussion carries over for any closed dierential

form of constant rank on N

In typical applications N is a constant rank submanifold of a symplectic manifold

with Of course for a random constant rank submanifold it rarely happ ens that

the null foliation is brating unless additional symmetries are at work

Coisotropic submanifolds An imp ortant sp ecial case of constant rank sub

manifolds of a symplectic manifold M are coisotropic submanifolds ie submani

folds N M with TN TNFor any submanifold N M let

C M fF C M j F j g

N N

denote its vanishing ideal The tangentbundleTN TMj and its annihilator have

the following algebraic characterizations

T N fv T M j v F for all F C M g

n n N

AnnT N f T M j dF j for some F C M g

n n N

The map T M T M identies the annihilator bundle with TN TMj and

n N

dF with X Thus

T N fX nj F C M g

n F N

Theorem The fol lowing three statements areequivalent

a For al l F C M the Hamiltonian vector eld X is tangent to N

N F

b The space C M is a Poisson subalgebraofC M

c N is a coisotropic submanifold of M

BASIC PROPERTIES OF SYMPLECTIC MANIFOLDS

Proof X is tangenttoN if and only if X Gforall G C M Since

F F N

X GfF Gg this shows that a and b are equivalent Wehaveseenabovethat

TN is spanned by restrictions of Hamiltonian vector elds X j with F C M

F N N

Hence N is coisotropic TN TN if and only if every suchvector eld is tangentto

N This shows that a and c are equivalent

Lemma Suppose F M R is a submersion Suppose that the components

of F Poissoncommute fF F g Then the bers of F arecoisotropic submanifolds

i j

of M of codimension k

Proof Let N F a The vector elds X are tangenttoN since X F

F F i

j j

fF F g Since dF dF span AnnTNateach p ointofN the Hamiltonian

j i k

vector elds X span TN This shows TN TN

Remark a In particular given a submersion F M R where dim M

nwithPoissoncommuting comp onents the b ers of F dene a Lagrangian foli

ation of M This is the setting for completely integrable systems We will discuss

this case later in much more detail

b The assumptions of the Prop osition can b e made more geometric bysaying that

F should b e a Poisson map for the trivial Poisson structure on R

The following Prop osition shows that lo callyany coisotropic submanifold is obtained

in this way

Proposition Let N M beacodimension k coisotropic submanifold

For any m N there exists a neighborhood U M containing mandasmooth sub

mersion F U R such that the components of F Poissoncommute fF F g for

i j

al l i j andN U F

Proof The pro of is by induction Supp ose that wehave constructed Poisson

commuting functions F F U RlksuchthatF F U R is

l l

a submersion and F F U N By the previous Lemma the b ers of

F F F dene a foliation of U by coisotropic submanifolds Let X X

l i F

The fact that the F vanish on N means that X is tangentto N ie N is invari

i i

ant under the ow and is contained in a unique b er N of F Cho osing U smaller

if necessarywe can pick a co dimension l submanifold S U transverse to N and a

submersion F S R such that N S F Cho osing U and S even smaller

we can extend F to a function on U invariant under the ows of the commuting vec

tor elds X X This means that F F all Poisson commute and the map

l l

F F is a submersion near N

Later wewillobtaina much b etter version of this result lo cal normal form theorem

showing that one can actually take U to b e a tubular neighb orho o d of N

FOUNDATIONS OF SYMPLECTIC GEOMETRY

CHAPTER

Normal Form Theorems

Mosers trick

Mosers trickwas used by Moser in a very short pap er to showthatonany

compact oriented manifold anytwo normalized volume forms are dieomorphism equiv

alent A volume form is a top degree nowhere vanishing dierential form In order to

describ e his pro of we recall the following fact from dierential geometry

Lemma Let X VectQ t R a timedependent vector eld on a manifold

Q with ow For every dierential form Q

t t

on the region U Q where U Q is dened

Note that if is a form this is just the denition of the ow The general case

follows b ecause b oth sides are derivations of M commuting with d

Theorem Moser Let Q beacompact oriented manifold and two vol

R R

ume forms such that Then there exists a smooth isotopy Di Q

M M

such that

Proof Mosers argument is as follows First note that every t t

t t

is a volume form Second since and have the same integral they dene the same

cohomology class d for some n form Thus

t d

Since each is a volume form the map X from vector elds to n forms

t X t

n dim Q is an isomorphism It follows that there is a unique time dep endentvector

eld X solving

X t

Let denote the owof X Then

t t

d L

X t t

t t

t X

This shows Nowput

NORMAL FORM THEOREMS

Mosers theorem shows that volume forms on a given compact oriented manifold Q

are classied up to dieomorphism by their integral The idea from Mosers pro of applies

to many similar problems A typical application in symplectic geometry is as follows

Theorem Let be a family of symplectic forms on a compact manifold M

depending smoothly on t Suppose that

t t

for some smooth family of forms M Then there exists a family of dieomor

phisms ie a smooth isotopy such that

for al l t

Proof Dene a timedep endentvector eld X by

X t t

Let denote its ow Then is indep endentof t

t t

L d

t X t t X t t

t t

t t t

Alan Weinstein used Mosers argumenttogive a simple pro of of Darb ouxs theorem

saying that symplectic manifold havenolocalinvariants and some generalizations We

will presentWeinsteins pro of b elow after some review of homotopy op erators in de

Rham theory

Homotopy op erators

Let Q Q b e smo oth manifolds and f f Q Q two smo oth maps Supp ose

f f are homotopic ie that they are the b oundary values of a continuous map

f Q Q

As one can show f can always taken to b e smo oth Dene the homotopy op erator

k k

H Q Q

as a comp osition

H f

k k

Here Q Q isberintegration ie integrating out the

s variable The integral of a form not containing ds is dened to b e

DARBOUXWEINSTEIN THEOREMS

Exercise Verify that for any form on Q

Z Z

d d

where Q Q fj g are the two inclusions Hint fundamental theorem of calculus

As a consequence the map H has the prop erty

k k

d H Hdf f Q Q

Thus if is a closed form on Q then H solves

f f d

induce the same map in cohomology and f Hence f

Example Poincare lemma Let U R b e an op en ball around Let

fgU b e the inclusion and U fg the pro jection Then induces an

k k k

isomorphism H U H pt with inverse That is every closed form U

with k is a cob oundary d

Proof Since it is obvious that is the identitymapwe only need

to showthat is the identity map in cohomology Let f U U

be multiplication by t The de Rham homotopy op erator H shows that f f

induce the same map in cohomology The claim follows since f idand f

Darb ouxWeinstein theorems

Theorem Darb oux Let M be a symplectic manifold of dimension

dim M n and m M Then there exist open neighborhoods U of m and V of

fgR and a dieomorphism V U such that m and dq d p

j j

Co ordinate charts of this typ e are called Darboux charts

Proof Using any co ordinate chart centered at mwemay assume that M is an

op en ball U around m R with some p ossibly nonstandard symplectic form

Let and the standard symplectic form Since anytwo symplectic forms on

the vector space T R are related by a linear transformation wemay assume that

agrees with at the origin Using the homotopy op erator from Example let

H U

As in the original Moser trickput

For all t agrees with at Hence taking U smaller if necessarywemay

assume is nondegenerate on U for all t Dene a timedep endentvector eld

X on U by

X t t

NORMAL FORM THEOREMS

The owofthisvector eld will not b e complete in general Since vanishes at

the form and therefore the vector eld X also vanish at Hence we can nd a

smaller neighborhood U of such that the ow U U is dened for all t

The ow satises

L d d d

t X t X t

t t

t t t

hence byintegration Darb ouxs theorem follows by setting

Again the pro of has shown a bit more In a suciently small neighborhood of R

anytwo symplectic forms are isotopic

Darb ouxs theorem says that symplectic manifolds have no local invariants in sharp

contrast to Riemannian geometry where there are manylocalinvariants curvature in

variants Darb ouxs theorem can b e strengthened to the statement that the symplectic

form near any submanifold N of a symplectic manifold M is determined by the restriction

of to TMj

Theorem Let M j be two symplectic manifolds and N

j j j j

M given submanifolds Suppose there exists a dieomorphism N N coveredby

a symplectic vector bund le isomorphism

TM j TM j

N N

such that restricts to the tangent map TN TN Then extends to a

symplectomorphism from a neighborhoodof N in M to a neighborhood N in M

Proof Any submanifold N of a manifold M has a tubular neighb orho o d dieo

morphic to the total space of the normal bundle TMj T N Since induces an

N N

isomorphism wemay assume that M M M is the total spaces of a

N N

vector bundle M N over a given manifold N N N with two given sym

k k

plectic forms that agree along N LetH M M b e the standard

homotopy op erator for the vector bundle M N and put H and

td Since agrees with along N it is in particular symplectic on a

t t

neighb orho o d of N in M On that neighborhood we can dene a timedep endentvector

eld X with Let b e its ow dened on an even small neighborhood

t X t t

for all t and put By Mosers argument

Theorem Let M be a symplectic manifold L M aLagrangian sub

manifold There exists a neighborhood U of L in M a neighborhood U of L in T L

and a symplectomorphism from U to U xing L

Proof By theorem it is enough nd a symplectic bundle isomorphism TMj

T T Lj Cho ose a compatible almost complex structure J on M ThenJ TL TMj

L L

is a Lagrangian subbundle complementary to TL and is therefore isomorphic by means

of the symplectic form to the dual bundle It follows that

TMj TL T L

DARBOUXWEINSTEIN THEOREMS

as a symplectic vector bundle The same argument applies to M replaced with T L

Thus

TMj TL T L T T Lj

L L

This typ e of result generalizes to constant rank submanifolds as follows First we

need a denition

Definition For any constant rank submanifold N M of a symplectic

manifold M the symplectic normal bund le is the symplectic vector bundle

TN T N TN

Note that for coisotropic submanifolds the symplectic normal bundle is just For

an isotropic submanifold of dimension k it has rank n k where n dimM The

following theorem is due to Marle extending earlier results of Weinstein for the cases

N Lagrangian or symplectic and Gotay for the case N coisotropic

Theorem Constant rank emb edding theorem Let N M j

j j j

two constant rank submanifolds of symplectic manifolds M Let

j j

TN TN F TN

j j

be their symplectic normal bund les Suppose there exists a symplectic bund le isomorphism

F F

covering a dieomorphism N N such that

Then extends to a symplectomorphism of neighborhoods of N in M such that

j j

induces

Thus a neighb orho o d of a constant rank submanifold N M is characterized up

to symplectomorphism by together with the symplectic normal bundle In particular

if N is coisotropic a neighb orho o d is completely determined by

Proof Supp ose N M is a compact constant rank submanifold of a symplectic

manifold M There are three natural symplectic vector bundles over N

E TNTN TN

F TN TN TN

G TN TN TN TN

Identifying E with a complementary subbundle to TN TN in TN wehave

TN E TN TN

isomorphism of bundles with forms likewise

TN F TN TN

NORMAL FORM THEOREMS

Therefore TN TN E F TN TN Let J be an compatible complex

structure on TMj preserving the two subbundles E F Then the isotropic subbundle

J TN TN TMj is a complementto TN TN whichby means of is identied

with TN TN This shows

TMj E F G

as symplectic vector bundles To prove the constantrankemb edding theorem cho ose

isomorphisms of this typ e for b oth TM j ThenE E and G G are symplectomor

i N

F by assumption of the theorem phic since N N preserves twoforms and F

Now apply Theorem

CHAPTER

Lagrangian brations and actionangle variables

Lagrangian brations

We had seen that for any submersion F M R from a symplectic manifold

M such that the the comp onents F Poissoncommute the b ers of F are coisotropic

submanifolds of co dimension k In particular if k n dim M the b ers are

Lagrangian submanifolds

Definition Let M b e a symplectic manifold A Lagrangian bration is a

bration M B such that every b er is a Lagrangian submanifold of M

This implies in particular dim B n dim M

Examples a The b ers of a cotangentbundle M T Q Q are a

Lagrangian bration

n n n

b If Q RZ T is an ntorus wehave a natural trivialization T T

n n n n n

T R and the map T T R denes a Lagrangian bration of T T

Is it p ossible to generalize the second example to compact manifolds Q other than a

torusThat is is it p ossible to nd a Lagrangian bration of T Q such that the zero

section Q T Q is one of the leavesW e will show in this section that the answer is

no The leaves of a Lagrangian bration are always dieomorphic to an op en subsets

of pro ducts of vector spaces with tori

We will need some terminology group actions on manifold If G is a Lie group and

Q a manifold a group action is a smo oth map

A G Q Q g q Ag q gq

suchthat eq q and g g q g g q For q Q the set Gq AG q is called

the orbit the subgroup G fg Gj gqg the stabilizer Note that Gq GG IfG

q q q

is compact then Gq is a manifold

Lemma Let G beaconnected Lie group and A G Q Q agroup action on

aconnected manifold Q Then the fol lowing areequivalent

a The action is transitive For some henceall q Q Gq Q

b The action is local ly transitive That is for al l q Q there is a neighborhood U

such that Gq U U

c There exists q Q such that the tangent map to G Q g gq is surjective

LAGRANGIAN FIBRATIONS AND ACTIONANGLE VARIABLES

Proof a just means that the orbits are op en and closed Both b and c imply

that condition The converse is obvious

Of particular interest is the case dim Q dimGWewillsaythatQ is an principal

homogeneous Gspace if Q comes equipp ed with a free transitive action of GAnychoice

of a base p oint q Q identies Q Gq GAnytwosuchidentications dier bya

translation in GIfG is a torus wecallQ an ane torusifG is a vector space we call

Q an ane vector space

Supp ose V is an ndimensional vector space acting transitively on an ndimensional

manifold Q The stabilizer V is a discrete subgroup of V indep endentofq Thus Q

carries the structure of a principal homogeneous H VV space Note that since H is

compact connected and ab elian it is a pro duct of a vector space and a torus That is

Q is a pro duct of an ane torus and an ane vector space

The ab ove considerations also make sense for b er bundles If GB is a group

bundle ie a b er bundle where the b ers carry group structures and admitting bundle

U F that are b erwise group isomorphisms with a xed group F charts U

one can dene actions on b er bundles E B to b e smo oth maps

G E E

that are b erwise group actions For example if E is a vector bundle and GlE isthe

bundle of general linear groups one has a natural action of GlE on E In particular an

ane torus bundle M B is a b er bundle with a b erwise free transitive action

of a torus bundle TB Thus if M B has compact b ers then every b erwise

transitive action of a vector bundle E B with dim E dim M gives M the structure

of an ane torus bundle The torus bundle TB is the quotient bundle E where

B is the bundle of stabilizer groups Any section B M makes M into a torus

bundle however in general there are obstructions to the existence of such a section On

the other hand if T is trivial then M B becomes a T principal bundle

Let us now consider a Lagrangian bration M B The following discussion

is based mainly on the pap er J J Duistermaat On global actionangle variables

Comm Pure Appl Math For simplicitywe will usually assume

that the b ers of are compact and connected

Theorem Let M be a symplectic manifold and M B beaLagrangian

bration with compact connectedbers Then thereisacanonical berwise transitive

vector bund le action T B M M Thus M B has canonical ly the structureof

an ane torus bund le

Proof We denote by VM TM the vertical tangent bundle V M kerd

m m

By assumption VM is a Lagrangian subbundle of the symplectic vector bundle TMFor

any form B let X XM bethevector eld dened by

LAGRANGIAN FIBRATIONS

For all vertical vector elds Y XM

X Y

Y X Y

Since VM is a Lagrangian subbundle it follows that X is a vertical vector eld Note

that the value of X at m M dep ends only on Thus wehave constructed a

linear map V M T B Clearly this map is an isomorphism Taking all of these

isomorphisms together wehave constructed a canonical bundle isomorphism

VM T B

Let F denote the owofthevector eld X and F F M M the time one ow

Since the X are vertical the ow preserves the b ers and again F m dep ends only

on Thus we obtain a b er bundle map

T B M M m F m

B b

Toshow that this is a vector bundle action we need to show that the vector elds X

all commuteThus let be two forms and X X the vector elds they dene

Wehave

L L

X X X X

X X

L d

X X X

L d

X X

since and d are basic forms on M B Since is nondegenerate this

j j

B V is an isomorphism it follows that veries X X Since eachmapT

the action is b erwise transitive

Lemma Let B be a form and X F the corresponding vector eld

and its ow Then

F t d

In particular F is a symplectomorphism if and only if is closed

Proof The Lemma follows byintegrating

t t t

F F L F d d

from to t

Let T B b e bundle of stabilizers for the T B action and

T T B B

the torus bundle

B Proposition is a Lagrangian submanifold of T

LAGRANGIAN FIBRATIONS AND ACTIONANGLE VARIABLES

U Z Proof Supp ose U B is an op en subset such that is trivial over U j

Any sheet of j U is given by the graph of a form B T B suchthat

F id By the previous Lemma this means d Thus d showing

that the sheet corresp onding to is a Lagrangian submanifold

Consider the Lagrangian bration p T B B with standard symplectic form d

on T B Any B denes a vertical vector eld X on T B and a ow F We

had discussed this ow in the section on cotangent bundles where it was denoted G

We had proved that F is the dieomorphism of T B given byadding t Let us

briey recall the argument For all forms the canonical form on T B transforms

according to

F tp

This follows byintegrating

t t t t

F F L F X d F p p

from to t By the prop erty of the canonical form we nd

t t

F F tp t

showing that F adds on t

Proposition The symplectic form on T B descends to T T B The

projection TB isaLagrangian bration

Proof Wehave seen that lo cal sections of are given by closed forms but adding

a closed form in T B is a symplectic transformation

To summarize the discussion up to this p oint Given any symplectic bration

M B with compact connected b ers we constructed a torus bundle TB with

a transitive bundle action of

T M M

This shows that M is an ane torus bundle Moreover T itself was found to carry a

symplectic structure such that TB is a Lagrangian bration Nonetheless in

general T is dierent from M The ane torus bundle M b ecomes a torus bundle only

after wecho ose a global section B M but in general there may b e obstructions

to the existence of such a section

Let us assume that these obstructions vanish and let B M b e a global section

Any other section diers by the action of a section of T The choice of sets up a

unique T equivariant b er bundle isomorphism

A T T B M

sending the identitysection of T to Equivariance means that

A F F A

for all forms B

ACTIONANGLE COORDINATES

The cohomology class H B is indep endentofthechoice of Indeed if

F is another choice of then

F d d

Let us assume that of course this is automatic if is exact eg if M is

an op en subset of a cotangent bundle T Q Then d for some form and

replacing with F we can assume ie the graph of is a Lagrangian

submanifold Note that b oth the existence of and the condition are automatic

if B is contractible ie always lo cally

Proposition The choice of any section B M with denes a

M symplectomorphism A T T B

Proof Let A T B M b e the map covering A It suces to show that A is

a lo cal symplectomorphismie A d The map A is uniquely dened byits

prop erty

A F

for every form B T B Wehave

A A

Since this is true for any we conclude A d

Actionangle co ordinates

Wearenow ready to dene actionangle coordinates Recall that up to this p oint we

havemadetwo assumptions on the bration M B First we assume that it admits

a section B M second we assume that the cohomology class H M

dened indep endentofthechoice of is zero Then wecho ose to b e a Lagrangian

section and this gives a symplectomorphism TM

Let us now also assume that the base B is simply connected This implies that the

group bundle B is trivial Cho ose an isomorphism Z for some b and identify

Actually with a little bit of cheating A kform on a b er bundle over B is determined by its

pullbacks under sections of the b er bundle if and only if dim B k Hence we need dim B

However the case dim B isobvious anyway

LAGRANGIAN FIBRATIONS AND ACTIONANGLE VARIABLES

bycho osing a path from b to b This is indep endentofthechoice of path if B

b b

is simply connected Thus wehave an isomorphism

B Z

and anytwo such isomorphisms dier by an action of AutZ Gln Z the group of

invertible matrices A such that b oth A and A haveinteger co ecients this implies

det A Let B b e the closed forms corresp onding to this

trivialization Atany p oint b B they dene a basis for hence also for T B Thus

wehave also trivialized

B R T B

and

n n n

M T B R Z B T

n n

The map s M RZ T given by pro jection to the second factor are the angle

coordinates Since the forms takevalues in they are closed They dene symplectic

vector elds X Vect U whose ows F in terms of the angle co ordinates are

given by

s s if j i

j j

s s t

i i

Since B is by assumption simply connected the are in fact exact

i i

The I s or their pullbacks to U are called action coordinates The choice of

I denes an emb edding B R That is B is dieomorphic to an op en subset of

R It is clear that I s I s lift to the standard cotangent co ordinates on

n n

T B T R Therefore takes on the form in actionangle co ordinates

d I d s

i i

Notice that the choice of actionangle co ordinates is very rigid The s are determined

up to the action of a matrix C Gln Z and the choice of I is determined up to a

constant That is any other set of actionangle co ordinates is of the form

X X

C I d C s c I s

ji j i ij j i

i i

j j

where the c are functions on B and d are constants The I can b e viewed as functions

i i j

on B note that this induces in particular an anelinear structure on B

If we drop the assumption that B is simply connected there are obstructions to the

existence of global actionangle co ordinates Even if the Lagrangian section B M

exists The rst most serious obstruction is the monodromy obstructionTointro duce

angle co ordinates wehave to assume that the bundle is trivial this however will only

b e true if the mono dromymap

B Aut Gln Z

INTEGRABLE SYSTEMS

is trivial If the mono dromy obstruction vanishes there can still b e an obstruction to

the existence of action co ordinates The forms dened by the angle co ordinates are

closed but they need not b e exact in general

Of course all obstructions vanish lo callyegover contractible op en subsets U B

On the other hand Duistermaat shows that for a very standard integrable system the

spherical p endulum the mono dromy obstruction is nonzero

Exercise Supp ose M is a symplectic manifold suchthat is exact d

for some form Let M B b e a Lagrangian bration with compact connected

b ers with B simply connected Given b B let

A b A b RZ b

b e smo oth lo ops in b generating the fundamental group of the b er Supp ose that

the A b dene continuous functions A B RZ M Show that the formula

i i

I m

A m

denes a set of action variables

For an uptodate discussion of Lagrangian foliations including a review of recent

developments see the preprintNguyen Tien Zung Symplectic Topology of Integrable

Hamiltonian Systems II Characteristic Classes p osted as mathDG

Integrable systems

After this lengthy general discussion let us nally make the connection with the the

ory of integrable systems Let M b e a compact symplectic manifold H C M R

a Hamiltonian and X its vector eld In general the owofX can b e very compli

H H

cated unless there are manyintegrals of motion An integral of motion is a function

G C M Rsuch that X G or equivalently fH Gg Anintegral of mo

tion denes itself a Hamiltonian ow X which commutes with the owofX since

G H

X X X

H G fHGg

Definition The dynamical system M H is called integrable over if there

exists n integrals of motion G G C M R fG Hg such that

n j

a The G are in involution ie they Poissoncommute fG G g

i i j

b The map G M R is a submersion almost everywhere ie d G dG

on an op en dense subset

Theorem LiouvilleArnold Let M H bea completely integrable Hamil

tonian dynamical system with integrals of motion G Suppose G isaproper map Let

M M be the subset on which G is a submersion let B be the set of connectedcom

ponents of bers of Gj B the induced map Then M B is a and M

Lagrangian bration with compact connectedbers hence it is an ane torus bund le

LAGRANGIAN FIBRATIONS AND ACTIONANGLE VARIABLES

The ow F of X is vertical and preserves the ane structure in local actionangle

coordinates it is given by

I t I

j j

s ts t

j j

Proof It remains to prove the description of the owofX Since fG Hg

H j

is constant along the b ers of G ie is the pullback of a function on B for all j H j

In lo cal actionangle co ordinates I s for the bration this means that H is a function

of the I s and the Hamiltonian vector eld b ecomes

I s

j j

The spherical p endulum

As one of the simplest nontrivial examples of an integrable system let us briey

discuss the spherical p endulum We rst give the general description of the motion of a

particle on a Riemannian manifold Q in a p otential V Q R

One denes the kinetic energy T C TQby T v jjv jj Using the identica

tion g TQ T Q given by the metric view T as a function on T Q The Hamiltonian

is the total energy H T V C T Q

In lo cal co ordinates q on Q

T v g q q q

ij i j

where g is the metric tensor The relation b etween velo cities and momenta in lo cal

co ordinates is p g q Thus

i ij j

T q p hq p p

ij i j

where hq is the inverse matrix to g q and

ij ij

hq p p V q H q p

ij i j

The conguration space Q of the spherical p endulum is the sphere whichbyan

appropriate normalization we can take to b e the unit sphere in R Let

b e p olar co ordinates on S thatis

x sin cos x sin sin x cos

The p otential energy is

V cos

and the kinetic energy is

T x sin

THE SPHERICAL PENDULUM

Thus

H cos p p

sin

The apparent singularityat comes only from the choice of co ordinates An

integral of motion for this system is given by G p Indeed

fH Gg

b ecause H do es not dep end on ie b ecause the problem has rotational symmetry

around the x axis Since dim T S itfollows that the spherical p endulum is a

completely integrable system

The image of the map G H has the form H f G where f G is a symmetric

function shap ed roughly like a parab ola The minimum of f is the p ointG H

corresp onding to the stable equilibrium The set of singular valuesofG H consists of

the b oundary of the region ie the set of p oints G F G together with the unstable

equilibrium

Removing these singular p oints from the image of G H we obtain a nonsimply

connected region B and one can raise the question ab out existence of global actionangle

variables Duistermaat shows that they do not exist in this system The lattice bundle

B is nontrivial ie the mono dromy obstruction do es not vanish

LAGRANGIAN FIBRATIONS AND ACTIONANGLE VARIABLES

CHAPTER

Symplectic group actions and momentmaps

Background on Lie groups

A Lie group is a group G with a manifold structure on G such that group multiplica

tion is a smo oth map This implies that inversion is a smo oth map also A Lie subgroup

H G is a subgroup which is also a submanifold By a theorem of Cartan every closed

subgroup of a Lie group is an emb edded Lie subgroup ie smo othness is automatic

In this case the homogeneous space GH inherits a unique manifold structure suchthat

the quotient map is smo oth

Let L G G a ga denote left translation and R G G a ag right

g g

translation A vector eld X on G is called leftinvariantifitis L related to itself for all

g GAny leftinvariantvector eld is determined byitsvalue X e at the identity

g T GBut element Thus evaluation at e gives a vector space isomorphism X G

X G is closed under the Lie bracket op eration of vector elds The space g T G

with Lie bracket induced in this way is called the Lie algebra of GThatiswedene

the Lie bracket on g by the formula

L L L

L L

where is the leftinvariantvector eld with e For matrix Lie groups ie

closed subgroups of GL n R the Lie bracket coincides with the commutator of ma

trices Working with the space of rightinvariantvector elds would have pro duced the

opposite bracket Wehave

R R R

t L

Let F G G b e the owof One denes the exp onential map

exp g G

L R

by exp F e In terms of exp the owof is g g expt while the owof

is g expt g For matrix Lie groups exp is the usual exp onential of matrices

Let Ad L R ie Ad a gAg Clearly Ad xes e so it induces a linear

g g g g

transformation still denoted Ad ofg T G One denes

g e

ad Ad

expt

t t

It turns out that ad whichgives an alternativeway of dening the Lie

bracket on g

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Generating vector elds for group actions

Definition Let G b e a Lie group An action of G on a manifold Q is a smo oth

map

A G Q Q g q A q g q

such that the map G Di Q g A is a group homomorphism

A manifold Q together with a Gaction is called a Gmanifold A map F Q Q

between two Gmanifolds is called equivariantifitintertwines the Gactions that is

gF q F gq

Example a There are three natural actions of any Lie group G on itself

The leftaction ga ga the right action ga ag and the adjoint action

ga gag

b Any nite dimensional linear representation of G on a vector space V is a Gaction

on V

Definition Let g b e a Lie algebra A Lie algebra action of g on Q is a smo oth

vector bundle map

Q g TQ q q

such that the map g XQ is a Lie algebra homomorphism

Supp ose Q is a Gmanifold For g let the generating vector eld b e the unique

vector eld with ow

q expt q

If F Q Q is an equivariant map then

Q F Q

Example For any g let denote the unique leftinvariantvector eld

L R R

with e Similarly let b e the rightinvariantvector eld with e The

generating vector eld for the leftaction is rightinvariant since the leftaction commutes

with the rightaction and its value at e is Hence the leftaction is generated by

R L L R

Similarly the rightaction is generated by and the adjointactionby

Proposition Let G be a Lie group with Lie algebra gFor any action of a Lie

group G on a manifold Q the map

g XQ

is a Lie algebra action of g on QFor g G one has

g Ad

Q g Q

Proof The idea of pro of is to reduce to the case of the action R of G on itself

Let Q G Q with Gaction ga q ag q The generating vector elds for this

L L L L

action are Since by denition of the Lie bracket

Q Q Q

HAMILTONIAN GROUP ACTIONS

The map F G Q Q a q a q is a Gequivariant bration In particular

and implies For the second equation we note that for

F Q Q Q Q

the action of G on itself by left multiplication

L L L

R Ad Ad

g g

Hence g Ad Again since F is Gequivariant this implies g Ad

g Q g Q

Q Q

If G is simply connected and Q is compact the converse is true every gaction on

Q integrates to a Gaction

Any action of G on Q gives rise to an action on TQ and T QIfq Q is xed under

the action of G then these actions induce linear Gactions ie representations on T Q

Q In particular the conjugation action of G on itself induces an action on the and T

Lie algebra g T G called the adjoint action and on g called the coadjoint action

The two actions are related by

hg i h g i g g

Exercise Using the identications T g g g and T g g g determine

the generating vector elds for the adjoint and coadjoint actions

Hamiltonian group actions

A Gaction g A on a symplectic manifold M is called symplectic if A

g g

SympM for all g Similarly a gaction is called symplectic if XM

M M

for all Clearlythe gaction dened by a symplectic Gaction is symplectic The G

action or gaction is called weakly Hamiltonian if all are Hamiltonian vector elds

That is in the Hamiltonian case there exists a function C M for all such

that X One can always cho ose to dep end linearly on dene rst for

a basis of g and then extend by linearity The map can then b e viewed as a

function C M g

Definition A symplectic Gaction on a symplectic manifold M is called

weakly Hamiltonian if there exists a map called moment map

C M g

such that for all the function hiC M is a Hamiltonian for

dhi

It is called Hamiltonian if is equivariant with resp ect to the Gaction on M and the

coadjoint action of G on g

Similarly one denes moment maps for Hamiltonian gactions one requires to b e

gequivariant in this case

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

It is obvious that if a group G acts in a Hamiltonian wayandifH G is a

homomorphism eg inclusion of a subgroup then the action of H is Hamiltonian the

moment map is the comp osition of the Gmoment map with the dual map g h

We sometimes write hi for the comp onent of The equivariance condi

tion means that g g Ad or in detail

i h Ad i hg i hAd

for all m M gWriting g expt and taking the derivativeat t we nd the

innitesimal version of this condition reads

For an ab elian group the conjugation action is trivial so that equivariance simply means

invariance

One wayofinterpreting the equivariance condition is as follows For a weakly

Hamiltonian Gaction the fundamental vector elds dene a Lie homomorphism ho

momorphism g X M XM This can always b e lifted to a linear map

Ham

g C M R The action is Hamiltonian if it can b e lifted to an equivariant map

which for a connected group is equivalent to the following

Lemma For a Hamiltonian Gaction the map

g C M hi

is a Lie homomorphism

Proof

f g X

Notice that we dened the the Poisson bracket in suchawaythat C M XM

is a Lie homomorphism

From nowonwe will always assume that the moment map is equivariantunless

stated otherwise

Lemma Any weakly Hamiltonian action of a Lie group G on M is Hamil

tonian if

a G is compact or

b M is compact

Proof Supp ose is any moment map not necessarily equivariant Dene

g g Ad

EXAMPLES OF HAMILTONIAN GSPACES

so that is equivariant if and only if g for all g We claim that g isalsoa

moment map for the Gaction Indeed

dhg i g dh Ad i

g g

If G is compact we obtain a Gequivariantmoment map byaveraging over the group

Thenis If M is compact and connected normalize by the condition

invariant since g

We remark that the ab ove Lemma also holds for actions of connected simply con

nected semisimple Lie groups ie g gg or more generally for any connected

simply connected group G for which the rst and second Lie algebra cohomology vanish

See Theorem in GuilleminSternberg Symplectic techniques in PhysicsCambridge

University Press for details

Examples of Hamiltonian Gspaces

Linear momentum and angular momentum Recall that for anyvector

eld Y on a manifold Q the cotangent lift Y XT Q is a Hamiltonian vector eld

Y X with H Weleave it as an exercise to check that linear map

XQ C T QY

is actually a Lie algebra homomorphism using the Poisson bracket on T Q Hence if

Q is a Gmanifold we obtain a moment map for the cotangent lift of the Gaction to

T Q by comp osing this map with the generating vector elds g XQ Recall that

in lo cal cotangent co ordinates if Y Y q then the corresp onding Hamiltonian

is H q p Y q p

j j

n n

For example let M T R R with standard symplectic co ordinates q p Let

j j

n n

G R act on itself by translation The Lie algebra is g R withexponential map

exp g G the identitymapof R

The generating vector elds b for b R are obtained from the calculation

exptb f q f q tb b b f q

j R

t t t t q

b and the moment map for the cotangentliftis Thus b

j R

hbi b p

j j

n n n

Using the standard inner pro duct b b b b on R to identify R R we nd

that

q p p

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

That is the moment map is just linear momentum up to an irrelevant sign that just

n n

comes from the chosen identication R R

Consider next the cotangent lift of the action of G Gln Ron R The Lie algebra

g gln R is the space of all real matrices with exp onential map the exp onential map

for the action of A gln R for matrices We compute the generating vector eld A

as follows

f q A exptA f q

t t

f exptAq

t t

A q

jk k

that is A A q A moment map for the cotangent lift of the action is

R jk k

hAi A p q

jk j k

Hence using the nondegenerate bilinear form A A trA A ongln R to identify

gln R gln R wehave

q p p q

ij i j

Note that the pairing on gln Risnotinvariant under the adjoint action ie it do es

not identify the adjoint and coadjoint action Instead the coadjoint action b ecomes the

contragredient action gA Adg A

The pairing do es however restrict to an invariant pairing on the subalgebra h on

of skewsymmetric matrices A A The corresp onding connected Lie subgroup of

Gln R is the sp ecial orthogonal group H SOn The moment map T R

on for the action of SOn reads

q p p q p q

ij i j j i

For n we can further identify so R with the standard rotation action of

SO and just b ecomes just angular momentum q p p q up to an irrelevant

factor which again just dep ends on the chosen identication so so

Exact symplectic manifolds The previous examples generalize to cotangent

bundles T Q of Gmanifolds Q or more generally to what is called exact symplectic

manifolds A symplectic manifold M is called exact if d for a form

which is sometimes called a symplectic p otential Note that a compact symplectic

manifold is never exact unless it is dimensional Indeed if d is exact then

EXAMPLES OF HAMILTONIAN GSPACES

n n

also the Liouville form d is exact Hence if M were compact Stokes

theorem would show that M has zero volume

Proposition Suppose M is an exact symplectic manifold d Then

any Gaction on M preserving is Hamiltonian with moment map

Note that if G is compact one can construct an invariant byaveraging If

H M R then is indep endent of the choice of invariant

Proof We calculate d d L The resulting mo

M M M M

ment map is equivariant

g hi g g g g Ad h Ad i

M M M g M g

The examples considered ab ovewere of the form M T QwithG acting bythe

cotangent lift of a Gaction on Q Another example is provided by the dening action

n n

of UnonC R

Unitary representations Intro duce complex co ordinates z q ip on

j j j

n n

C R and write the symplectic form as

X X

dq dp z dz d hdz dz

j j j j

j j

where hw w w w is the Hermitian inner pro duct Then d with

Imhz dz z d z z dz

j j j j

This choice of is preserved under the unitary group The Lie algebra unofUn

consists of skewadjoint matrices The generating vector elds are

z z

jk k C kj k

z z

j j

where unisaskewHermitian matrix Hence hi is given by

i i

z z hz z hz i

jk j k

where h is the Hermitian metric denes a moment map Using the inner pro duct on

un Tr to identify the Lie algebra and its dual

z z z z z

kj j k k j

This example also shows that any nite dimensional unitary representation G Un

denes a Hamiltonian action of G on C the moment map is the comp osition of the

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Un moment map with the pro jection un g dual to g un For example the

moment map for the scalar S action is given by jjz jj

Pro jective Representations The action of Un on C induces an

action on C P nwhich also turns out to b e Hamiltonian In homogeneous co ordinates

z z the moment map is

z z

i j k jk

jz j

We will verify this fact later in the context of symplectic reduction

Symplectic representations Generalizing the case of unitary representa

tions consider any symplectic representation of G on a symplectic vector space E

That is G acts by a homomorphisms G SpE into the symplectic group A moment

map for such an action is given by the formula

hv i v v

Indeed if we identify T E E the generating vector eld for g is just v v

v E

Thus for w E wehave

v w w v

On the other hand the map dened ab ove satises

d w v tw v tw

w v v w

w v

verifying that is a momentmap

Coadjoint Orbits As a preparation let us note that for any Hamiltonian G

space M the moment map determines the pullbackof to any Gorbit Indeed

since is the Hamiltonian vector eld for

f g

M M

In particular if M is a homogeneous Hamiltonian Gmanifold ie if M is a single

Gorbit is completely determined by

Theorem KirillovKostantSouriau Let Og be an orbit for the coadjoint

action of G on g There exists a unique symplectic structureon O for which the action

is Hamiltonian and the moment map is the inclusion O g

Proof For any O consider the skewsymmetric bilinear form on g

B h i

EXAMPLES OF HAMILTONIAN GSPACES

Writing B h ad i had i we see that the kernel of B consists of all

with ad ie It follows that

h i

O O

is a welldened symplectic form on T O The calculation

B g g hg g gi hg g i h i B

shows that the resulting form on O is Ginvariant and therefore smo oth and

equation gives the moment map condition dhi for the inclusion map

O g

dhi L hi had i h i

O O O O

Tocheckd we compute

d L d

O O O

As remarked ab ove the moment map uniquely determines the symplectic form

Example Let G SO Identify the Lie algebra so with R by identifying

the standard basis vectors of so as follows

A A A

e e e

This identication takes the adjoint action of SO to the standard rotation action on

trAB on so to the standard R and takes the invariant inner pro duct A B

R The coadjoint orbits inner pro duct on R The inner pro duct also identies so

for SO are the spheres around together with the orgin fg

Theorem KostantSouriau Let G be a Lie group and M is a Hamil

tonian Gspace on which G acts transitively Then M is a covering spaceofacoadjoint

orbit with form obtained by pul lback of the KKS form on O

Proof Let O M It is clear that the map M O is a submersion We

had already seen that the form on M is determined by the moment map condition and

the formula shows that the map M O preserves the symplectic form Hence its

tangent map is a bijection everywhere and so M Ois a lo cal dieomorphism

Note that while M is a homogeneous space GG its image under the moment map

is a homogeneous space GG and the covering map is a bration with discrete

b er G G Hence nontrivial coverings can b e obtained only if the stabilizer G

m m

is disconnected for compact connected Lie group do not have prop er subgroups of the

same dimension

If G is a compact connected Lie group then it is known that all stabilizer groups G

for the coadjoint action are connected so one has

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Theorem If G is compact connectedandM is a homogeneous Hamil

tonian Gspace the moment map induces a symplectomorphism of M with the coadjoint

orbit O M

The connectedness of G can b e shown as follows Using the bration G GG

and the fact that G and GG are connected it suces to show that the base GG is

simply connected Given p p G cho ose a path G connecting them

pro jects to a closed path in GG andcanbecontracted to a constant path By the

homotopy lifting prop erty this homotopy can b e lifted to a homotopy with xed end

points of This will then b e a path in G connecting p p The simplyconnectedness

of GG in turn follows from Morse theory this will b e explained later

Poisson manifolds Moment maps t very nicely into the more general cate

gory of Poisson manifolds

Definition APoisson manifold is a manifold M together with a bilinear map

f g C M C M C M such that

a f g is a Lie algebra structure on C M and

b for all H C M the map C M C M F fH F g is a derivation

A smo oth map M M between Poisson manifolds is called a Poisson map if

fF F g f F F g

Avector eld X XM is called Poisson if

X fF F g fX F F g fF XF g

Since any derivation of C M isgiven bya vector eld any H denes a socalled

Hamiltonian vector eld X by X F fH F g

H H

Exercise ShowthatX isaPoisson vector eld Show that the owofany

complete Poisson vector eld is Poisson

Examples of Poisson manifolds are of course symplectic manifolds with the Poisson

bracket asso ciated to the symplectic structure Another imp ortant example due to

of a Lie algebra gFor any g and any function F C g Kirillov is the dual g

identify

g g dF T g

Then dene

fF Ggh d F dG i

Exercise Verify that this is a Poisson structure on g Show that the inclusion

O g of a coadjoint orbit is a Poisson map

The denition of a moment map carries over to Poisson manifolds A Gaction is

Poisson if it preserves Poisson brackets and anysuch action is Hamiltonian if there

exists an equivariant smo oth map M g suchthat

h i

EXAMPLES OF HAMILTONIAN GSPACES

for all k

Exercise Show that the coadjoint action of G on g is Hamiltonian with mo

ment map the identitymap

Exercise Let M b e a symplectic manifold and G a connected Lie group

acting symplectically on M Supp ose there exists a momentmap M g in the

weak sense Show that is equivariantifandonlyif isa Poisson map for the Kirillov

Poisson structure Conversely show that for every Poisson map M g the

equation

dhi

denes a symplectic Lie algebra action of g on M More generally show that for any

Poisson manifold M anyPoisson map M g denes a Poisson gaction on M

d Gauge Theory Let us now sp end some time discussing somewhat infor

mallyaninteresting dimensional example We b egin with a rapid intro duction to

what we call with some exaggeration gauge theory

Let b e a compact oriented manifold G a compact Lie group and gthe

kforms on with values in g The space A g will b e viewed as an innite

dimensional manifold called the space of connections For any A A there is a

corresp onding covariant derivative

k k

d g g

dened

d d A

using the wedge pro duct The gauge action of G C GonA is dened

g A Ad A dgg

where the rst term is the p ointwise adjoint action The second term is written for

matrixgroups so that dgg makes sense as a form on with values in g More

invariantly it is the pullback under g G of the rightinvariant MaurerCartan

form G g ie is the unique rightinvariantformsuch that for any right

R R

invariantvector eld The gauge action is dened in suchaway that the

following is true

Lemma For al l g

d Ad Ad d

g A g g A

Proof Using the denition and using that for all g

dAd dgg

dgg Ad d gd g

dgg Ad Ad dgg Ad d

g g g

dgg Ad Ad d

g g

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

wehave

d Ad dAd Ad A dgg Ad

g A g g g g

Ad d dgg Ad Ad A dgg Ad

g g g g

Ad d

g A

Due to the presence of the gauge term the square of d is usually not zero

A A g be the curvatureofA Then Lemma Let curvA dA

for al l g

curvA d

The curvaturetransforms equivariantly

curvg A Ad curvA

Proof

d d A A dA A

dA A A

where the last term is obtained by the Jacobi identity Equivariance follows bythe

transformation prop ertyford

By the lemma d is precisely if the curvature is zero

Wenow view A as a G space What are the generating vector eldsThe Lie

algebra of the gauge group is identied with g

Lemma The generating vector elds for g are

Proof

A Ad A dexpt expt

A expt

t t

A d d

Wenow sp ecialize to two dimensions dim Let us x an invariant inner pro duct

on g unique up to scalar if G is simple Then A gisan dimensional

symplectic manifold The form is

ab a b

g for all a b T g using the inner pro duct

EXAMPLES OF HAMILTONIAN GSPACES

Theorem The gauge action of G on A is Hamiltonian with moment

map minus the curvature A curvA ie

hAi curv A

Proof We calculate For all a g viewed as a constantvector eld

hd hi ai hA tai

A t t

curvA ta

t t

A A t a a dA tda tA a

t t

d a

a d

It is interesting to extend this calculation to manifolds with b oundary Every

thing carries over but the partial integration pro duces an extra b oundary term so that

the momentmapis

Z Z

hAi curvA A

That is the informally dual space to the Lie algebra g of the gauge group is

identied with g g with the natural pairing and the moment map is

A g g g A curvA

Notice however that this moment map is no longer equivariant in the usual sense for

the action on the second summand is still the gauge action This leads one to dene a

central extension of the gauge group Dene

c G G Ucg g expi g d g dg g

and let G G U with pro duct

g z g z g g z z cg g

One can show that this do es indeed dene a group structure ie c is a co cycle The

R R

d d d Lie algebra of this new group is g R with dening co cycle

g g R with action and its dual is

g z Ad Ad dgg

g g

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

It follows that the moment map for the action of the extended gauge group where

the extra circle acts trivially is equivariant the image of the original moment map is

identied with the hyp erplane

Symplectic Reduction

The MeyerMarsdenWeinstein Theorem Let M b e a Hamilton

ian Gspace As usual we assume that is an equivariant moment map One of the

basic prop erties of the moment map is the following

Lemma For al l m M the kernel and image of the tangent map to are given

kerd T G m

m m

imd anng

m m

Proof By the dening condition of the moment map

mX X d hi hd X i

m M m

Therefore kerd f mj gg T G m

m M m

By the same equation g implies that hd X i for all X hence

m m

imd anng Equality follows by dimension count using nondegeneracy of

m m

dimimd dim M dimkerd

m m

dim M dimT G m

dimG mdimG dim G dim anng

m m

Theorem Apoint g is a regular value of if and only if for al l m

the stabilizer group G is discrete In this case is a constant rank

submanifold The leaf of the nul l foliation through m is the orbit G m

Proof Since is a regular value d is surjective for all m M By the Lemma

this means that ann g g or equivalently g fg This shows that G G is

m m m

discrete Nowlet M b e the inclusion Using T kerd the

m m

kernel of is

T T ker

m m

T T G m

m m

T G m

SYMPLECTIC REDUCTION

Theorem MarsdenWeinstein Meyer Let M be a Hamiltonian G

space Suppose is a regular value of and that the foliation of by G orbits

is a bration This assumption is satisedif G is compact and the G action is free

Let

G M

be the quotient map onto the orbit space There exists a unique symplectic form on

the reduced space M such that

Proof Since the nullfoliation is given by the G orbits this is a sp ecial case of the

theorem on reduction of constant rank submanifolds

The reduced space at is often denoted M MG this notation is useful if several

groups are involved The symplectic quotients M dep end only on the coadjointorbit

O G Let O b e the same Gspace but with minus the KKS form as a symplectic

form and minus the inclusion as a moment map The moment map for the diagonal

action on M O is

M O g m m

Proposition Shiftingtrick is a regular value of if and only if is a

regular value of

M O g m m

Moreover the G action on is free if and only if the Gaction on is free

Thereisacanonical symplectomorphism

M O G M

Proof Consider the map O by m m m Clearly this map

is a Gequivariant bijection Since O G the Gaction on O has

discrete resp trivial stabilizers if and only if the G action on has discrete

resp trivial stabilizers The map M M O m m preserves forms hence

so do es the map m m It follows that the maps

M G M fgG G

are all symplectomorphisms

Example Let M C with the standard symplectic form

dz dz

j j

and scalar S RZaction multiplication byexpit We recall that a moment

map for this action is given byz jjz jj The reduced space at level is C P N

The reduced form is known as the FubiniStudy form Reducing at a dierentvalue

amounts to rescaling the symplectic form by

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Example Let M b e an exact symplectic manifold d and supp ose

is invariant under some Gaction where G is a compact Lie group Let b e the

corresp onding momentmap hi Then if is a regular value of and the

Gaction on is free the pullback is invariant and horizontal hence descends

to a form on M such that and one has

It follows that the symplectic quotient of an exact Hamiltonian Gspace at an exact

symplectic manifold

Example As a subexample consider the case M T Qwhere G acts bythe

cotangent lift of some Gaction on Q Let T Q Q denote the pro jection The

moment map is given by

hmi hm q i

where q m This shows that the zero levelsetistheunionofcovectors orthogonal

to orbits

ann T G q

q Q

Since contains the zero section Q it is clear that the Gaction on is

lo cally free if and only if the action on Q is lo cally free If the action is free wehave

T QG T QG To see this at least settheoretically note that

T QG T QT G q G

q q

so that

T QG ann T G q G

To identify the symplectic forms one has to identify the reduced canonical form with

the canonical form on T QG weleave this as an exercise

For an arbitrary Gspace Q the singular reduced space T QG may b e viewed as

a cotangent bundle for the singular space QG

Example Returning to our d gauge theory example the reduction

AG is the mo duli space of at connections on

Reduced Hamiltonians Let G b e a compact Lie group Supp ose M

is a Hamiltonian Gspace that g is a regular value of the moment map and that

the action of G on the level set is free Then every invariant Hamiltonian

H C M descends to a unique function H C M with H H Passing

to the reduced Hamiltonian H is often a rst step in solving the equations of motion

for H From H invariance of X it follows that the restriction X X

H H

is related to X XM that is its ow pro jects down to the owonM After

one has solved the reduced system ie determined its ow F t it is a second step to

lift F tuptothelevel set

SYMPLECTIC REDUCTION

Example Consider the motion of a particle on R in a p otential V q It is

describ ed by the Hamiltonian on T R

jjpjj

H q p V q

Supp ose the p otential has rotational symmetry ie that it dep ends only on r jjq jj

Then H is invariant under the cotangent lift of the rotation action of G S We had

seen that the moment map for this action is angular momentumq p p q q p

In p olar co ordinates ronR and corresp onding cotangent co ordinates on T R

p p V r H rp p

and p The symplectic form on T R is dr p d p Every value

is a regular value of since S acts freely on the set where p r On

T R the second term disapp ears ie dr p It follows that

M T R symplectically and the reduced Hamiltonian is

H rp p V r

r ef f

with the eective p otential

V r V r

ef f

Using conservation of energy

r p

V r V r E

ef f ef f

ier E V r one obtains the solution in implicit form

ef f

t t

E V r

ef f

Using r p one also obtains a dierential equation for the tra jectories

r r

E V r

ef f

with solutions

r E V r

ef f

Kepler problem this integral can b e solved and leads to In the sp ecial case V r

conic sections see any textb o ok on classical mechanics

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Reduction in stages As a sp ecial case of reduced Hamiltonian one some

times has a reduced moment map For the simplest situation supp ose G H are compact

Lie groups and M is a Hamiltonian G H space with moment map Since

the two actions commute h g forall g h In particular is H

invariantandisGinvariant Let b e a regular value of so that the reduced space

M is dened Since is Ginvariant it descends to a map M h Itisthe

moment map for the H action on M induced from the H action on M

Lemma Reduction in Stages Suppose is a regular value of and a

regular value for Then isaregular value for IfG acts freely on

and G H acts freely on then H acts freely on and there

is a natural symplectomorphism

M M

Proof ClearlyifG acts with nite resp trivial stabilizers on andG

H acts with nite resp trivial stabilizers on the same is true for the

H action on This proves the rst part since a level set having nite stabilizers

is equivalent to the level b eing a regular value The second part follows b ecause the

natural identications

H G H M M

all preserve forms

The cotangent bundle of a Lie group Let G b e a compact Lie group For

all g the left and rightinvariantvector elds are related by the adjoint action as

follows

L R

g Ad g

The map

G g TG g g

is a vector bundle isomorphism called left trivialization of TG In left trivialization

b ecomes the constantvector eld and b ecomes the vector eld Ad Dual to

the lefttrivialization of TG there is the trivialization T G G g byleftinvariant

forms

Exercise Show that in left trivialization of TG the tangent maps to inversion

Inv G G g g left action L G G g hg and rightactionR G

G g gh are given by

Inv g g Ad

L g hg

R g gh Ad

h h

NORMAL FORMS AND THE DUISTERMAATHECKMAN THEOREM

Show that the resp ective cotangent lifts are

Inv g g Ad

L g hg

R g gh Ad

h h

R L

Since the generating vector elds for the leftand right action are resp ectively

we nd that the moment maps for these actions are

g Ad g

L R

Note that the cotangent lifts of b oth the left action and the right action are free In

particular every g is a regular value for b oth moment maps

Theorem The symplectic reduction T G by the right action with Gaction

inheritedfrom the leftaction is the coadjoint orbit G

is free and Proof It suces to note that the left action of G on the level set

transitive and the action on the quotient has stabilizer conjugate to G The moment

map induced by identies gives a symplectomorphism onto G

Of course the reduced spaces with resp ect to the left action are coadjoint orbits G

as well the cotangent lift of the inversion map G G g g exchanges the roles of

the left and right action

Theorem Let M be a Hamiltonian Gspace Let G act diagonal ly on

T G M where the action on T G is the right action Consider the reducedspaceat

as a Hamiltonian Gspace with Gaction inducedfrom the leftGaction on T G Then

there isacanonical isomorphism of Hamiltonian Gspaces

M T G M G

Proof Use left trivialization T G G g The map

T G M T G M g m g gm

is symplectic and takes the diagonal action to the rightaction on the rst factor and

the leftaction b ecomes a diagonal action Hence it induces a symplectomorphism

T G M G T GG M M

Normal forms and the DuistermaatHeckman theorem

Let M b e a Hamiltonian Gspace where G is a compact Lie group If is a

regular value of the moment map then so are nearbyvalues g What is the relation

between M and reduced spaces M at nearbyvalues

Toinvestigate this question we describ e the reduction pro cess in terms of a normal form

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

Let Z b e the zero level set Z M the inclusion and Z M the

pro jection Since the Gaction on has nite stabilizers there exists a connection

form

Z g

that is satises the two conditions

g Ad

Such a form can b e constructed as follows Consider the embedding Z g

TZ m masa vector subbundle Cho ose a Ginvariant Riemannian met

ric on Z and let p TZ Z g b e the orthogonal pro jection with resp ect to that

metric The form is dened by pv m v for v T M

Let pr pr b e the pro jections from Z g to the rst and second factor Dene a

form on the pro duct X Z g by

pr dhpr i

let Gaction act diagonally using the coadjoint action on g

Theorem Lo cal normal form near the zero level set The form is non

degenerate ie symplectic on some neighborhoodof Z and satises

dhpr i

for al l g that is pr is a moment map There exists an equivariant symplecto

morphism between neighborhoods of Z in M and in X intertwining the two moment

maps

Proof Notice that the form

pr hdpr i

is nondegenerate Since diers from this form by a term hpr di whichvanishes

along Z itfollows that is nondegenerate near Z For the second part notice that the

tangent bundle to the null foliation ker TZ is a trivial bundle Z g Z where G

acts on g by the adjoint action By the equivariantversion of the coisotropic emb edding

theorem it follows that neighborhoods of Z in M and of Z in X are equivariantly

symplectomorphic Since b oth momentmapsvanish on Z it is automatic that the

symplectomorphism intertwines the momentmaps

WecanviewZ g also as a quotientZ T GG using left trivialization to

identify T G G g theGaction on Z g is induced from the cotangentliftofthe

right Gaction

It follows that the reduced space at close to is symplectomorphic to Z G

Gthatis

Corollary The reducedspace M for close to ber over M withbers

coadjoint orbits

M Z G G

THE SYMPLECTIC SLICE THEOREM

Letting G g be the embedding the pul lback of the form to Z G

dhi

Consider in particular the case that G is a torus T RZ Then the coadjoint

action is trivial and the reduced spaces at and at nearbyvalues are dieomorphic

Notice that d descends to a form M which is just the curvature form F M t

of the torus bundle M As a consequence we nd that the symplectic form

changes according to

h F i

In particular this change is linear in This result dep ends on our identication of

M M This identication is not natural but anytwo identications are related by

an isotopyof M Since cohomology classes are stable under isotopies it makes sense to

compare cohomology classes and the ab ove discussion proves

Theorem DuistermaatHeckman The cohomology class of the symplectic

form changes according to

h ci

where c H M t is the rst Chern class of the torus bund le M

In particular this change is linear in As an immediate consequence one has

Corollary Let M be a Hamiltonian T space and let U beaconnected

component of the set of regular values of Then the volume function U R

VolM is given by a polynomial of degree at most dim M dim T

Here the maximum degree is obtained as half the dimension of a reduced space

The symplectic slice theorem

The slice theorem for Gmanifolds Let G b e a compact Lie group and H

a closed Lie subgroup Then every Gequivariantvector bundle over the homogeneous

space GH is of the form

E G W G W H

where W is an H representation the quotientistaken by the H action hg w

gh hw and the Gaction is given by g g w g g w Indeed given E one

denes W E to b e the b er over the identity coset m eH and the map

G W E g w gw is easily seen to b e a welldened equivariantvector

bundle isomorphism

For example supp ose M is a Gmanifold m M andH G Let W

T MT Gm b e the socalled slicerepresentation of H Then E G W is the

m m H

normal bundle TMj T O to the orbit O Gm GH Recall now that by

O O

the tubular neighb orho o d theorem if N is any submanifold of M there is a dieomor

phism of neighborhoods of N in M and in The dieomorphism is constructed using N

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

geo desic ow with resp ect to a Riemannian metric on M If N is Ginvariant this

dieomorphism can b e chosen Gequivariant Just take the Riemannian metric to b e

Ginvariant We conclude

Theorem Slice theorem Let M beaGmanifold and m M apoint with

stabilizer H Gm and slicerepresentation W T MT Gm There exists a G

m m

equivariant dieomorphism from an invariant open neighborhood of the orbit O Gm

to a neighborhood of the zerosection of E G W

Corollary There exists a neighborhood U of O Gm with the property that

al l stabilizer groups G x U areconjugate to subgroups of H GmInparticular if

M is compact thereare only nitely many conjugacy classes of stabilizer groups

Proof Identify some neighb orho o d of the orbit with the mo del E G W Let

x gy with y W ThenG Ad G But G is a subgroup of H since it preserves

x g y y

the b er W E

Corollary If G is compact abelian and M is compact the number of stabilizer

subgroups fH Gj H G for some m M g is nite

Proof For an ab elian group conjugation is trivial

Definition For any subgroup H of G one denotes its conjugacy class byH

and calls the Ginvariant subset

M fm M j G is Gconjugate to H g

the p oints of orbit type H One also denes

M fm M j G H g M fm M j G H g

m H m

Proposition The connectedcomponents of M M and M aresmooth sub

manifolds of M

Proof Any orbit OM contains a p oint m M with G H In the mo del

H m

E G W near O wehave

H H

E G W GH W

since W is a vector subspace of W this is clearly a smo oth subbundle of E The

H H

connected comp onents of M are smo oth submanifolds since for all m M a neigh

b orho o d is H equivariantly mo deled by the H action on T M and T M is a vector

m m

subspace The closure M is a union of connected comp onents of M Since M is

op en in its closure it is in particular a submanifold

The decomp osition M M is called the orbit typ e stratication of M

Using the lo cal mo del near orbits one can show that it is indeed a stratication in

the technical sense Note that since each M G is a union of smo oth manifolds it

induces a decomp osition in fact stratication of the orbit space MG

THE SYMPLECTIC SLICE THEOREM

The slice theorem for Hamiltonian Gmanifolds In symplectic geometry

one can go one step further and try to equip the total space to the normal bundle

with a symplectic structure Thus let M b e a Hamiltonian Gspace Assume

that m is in the zero level set This implies that the orbit is an isotropic

submanifold Since vanishes on O wehave T Okerd on the other hand

m m

kerd T O The symplectic vector space V T O T O with the action of

m m m m

H G is called the symplectic slice representation at m

Definition Let H act on a symplectic vector space V by linear symplectic

transformations and let V h b e the unique momentmapvanishing at cf

V h hv i v v

One calls the symplectic quotient

E T G V H

the model dened by V Here H acts on T G V by the diagonal action where the

action on T G is given by the cotangent lift of the rightaction of H on G Welet

E g b e the moment map for the Gaction on E inherited from the cotangent

lift of the leftGaction on T G

The orbit O GH is naturally emb edded as an isotropic submanifold of E namely

as the zero section of T GH T GH Its symplectic normal bundle in E is an

asso ciated bundle G V

Remark The mo del can also b e written as an asso ciated bundle Identify

T G G g using left trivialization The zero level set for the H action consists

v Thus if wecho ose an H equivariant of p oints g v suchthat pr

complementtoannhing identifying g h ann h we see that the zero level

set consists of p oints g v v with annh and is therefore isomorphic to

G annh V Thus

E G ann h V

In this description the momentmap is given by

gv g v

E V

Note however that this identication dep ends on the choice of splitting

Theorem Let M be a Hamiltonian Gmanifold and O Gm an orbit

in the zero level set of There exists a Gequivariant symplectomorphism between

neighborhoods of O in M and in the model E dened by the symplectic slicerepresentation

V T O T O of H G intertwining the two moment maps

m m m

Proof This follows from equivariantversion of the constant rank emb edding

theorem The symplectic normal bundles of O in b oth spaces are G V H

SYMPLECTIC GROUP ACTIONS AND MOMENT MAPS

The symplectic slice theorem is extremely useful For example we obtain a mo del for

the singularities of MG in case is a singular value Indeed by reduction in stages we

have

T G VH G T G VGH VH

whichshows that the singularities are mo deled by symplectic reductions of unitary repre

sentations Since the moment map for a unitary representation is homogeneous the zero

is a cone and hence the singularities are conic singularities This discus level set

sion can b e carried much further see the pap er SjamaarLerman Stratied symplectic

spaces and reduction Ann of Math no

Proposition Let M be a Hamiltonian Gspace H G a closed sub

group The connectedcomponents of M and M are symplectic submanifolds of M

For every connectedopen subset U M the image U is an open subset of an ane

H H

subspace ann h g for some g

H H H

Proof For all m M the tangent space T M is equal to T M But

m m

for any symplectic representation V of a compact Lie group H the subspace V is

symplectic Pro of Cho ose an H invariant compatible complex structure Then V

is a complex hence symplectic subspace This shows that M and the op en subset

M M are symplectic

The second part follows from the lo cal mo del or alternatively follows Let Z

Z H b e the centralizer and K N H the normalizer of H in G resp ectivelyThus

G G

H H H H

Z K G and z k g Duallyidentify z k g Byequivariance of

H H

the momentmapM M g z The action of K G preserves M

H H

Its moment map M k is the restriction of followed by pro jection g k

but since it takes valuesink z it is actually just the restriction of Since

im d ann h annh is indep endentof m U we conclude that U isan

m k

op en neighb orho o d of minmannh

CHAPTER

Hamiltonian torus actions

The AtiyahGuilleminSternb erg convexity theorem

Motivation Asamotivating example which on rst sight seems quite unre

lated to symplectic geometry consider the following problem ab out selfadjoint matrices

Let R be an ntuple of real numb ers and let O b e the set of all

selfadjoint complex n nmatrices having eigenvalues Let O R

b e the pro jection to the diagonal

Theorem The image O is the convex hul l

hullf S g

where S is the permutation group

This and related results were proved bySchur and Horn later greatly generalized by

Kostant and Heckman

The relation to symplectic geometry is as follows First instead of selfadjoint matri

ces we can equivalently consider skewadjoint matrices ie the Lie algebra g of G Un

Since all matrices with given eigenvalues are conjugate O is an orbit for the action

of Un Using the inner pro duct

A B trAB

on g we can also view it as a coadjoint orbit

The pro jection is just orthogonal pro jection onto the diagonal matrices which are

a maximal commutative subalgebra t g Using the inner pro duct to identify t t it

b ecomes the moment map for the induced T G action

For this reason the SchurHorn theorem can b e viewedasaconvexity theorem for

Hamiltonian torus actions on coadjoint orbits of Un Nothing is sp ecial ab out Un

analogous results hold for arbitrary compact groups

Lo cal convexity Our discussion of the convexity theorem will follow the ex

p osition in GuilleminSternb erg Symplectic Techniques in Physics In order to under

stand how images of moment maps for Hamiltonian T spaces lo ok like we rst haveto

understand how they lo ok like lo cally Wewillwork with the lo cal mo del E for the

T action near any orbit O Tm In that section we assumed Tm but this

can b e arranged by adding a constant to the moment map Letting H T be the

HAMILTONIAN TORUS ACTIONS

stabilizer wehave the mo del

E T T V H

where V is a unitary H representation Using the identication T T T t the

v and the moment map for the H action on T T V is t v pr

moment map for the T action on E is induced from the map t v The zero

level set condition for the H action reads pr v This shows

Lemma The image of the moment map is of the form

E pr V

E V

To understand the shap e of this set we need to describ e the moment map images of

unitary torus representations here for the identitycomponentofH acting on V

Let T b e a torus t the integral lattice Every dimensional unitary representa

tion

S RZ T U

denes a map t Rwhich restricts to a homomorphism of lattices

called the weight of the representation One calls

Hom Z t

Given one denes a dimensional representation C by

ih i

exp z e z

By Schurs Lemma any unitary T representation V splits into a sum of dimensional

representations ie is isomorphic to a representation of the form

C V

j j

where are called the weights of V Given a symplectic vector space V with a symplectic

T representation one cho oses an invariant compatible complex structure I which makes

V into a unitary T representation The weights for this representation are indep endent

of the choice of I since anytwo I s are deformation equivalent They are called the

weights of the symplectic T representation

Lemma Let V be a symplectic T representation with moment map

V V

h v i v v The image of the moment map is a convex rational polyhe

V V

dral cone V C spanned by the weight of the representation

V V j

C conef g

V n

For any open subset U V the image U is open in C inparticular any open

V V

neighborhood of the origin gets mappedontoanopen neighborhood of the tip of the cone

THE ATIYAHGUILLEMINSTERNBERG CONVEXITY THEOREM

Proof It is convenient to write the moment map in a slightly dierent form By

the ab ove discussion we can cho ose an equivariant symplectomorphism V C such

n n

that T acts on C by the homomorphism T S determined bytheweights The

n n

moment map for the standard S action on C is

z z jz j jz j jz j e

n n j j

n n

where e is the j th standard basis vector for R R Hence is a comp osition of

j V

with the map R t dual to the tangent map The latter map takes e to

j j

Thus

z z jz j

V n j j

The description of the image of is nowimmediate If U V is op en then U is

n n n

op en in the p ositive orthant R ie is the intersection of R with an op en set U R

n n n

This amounts to saying that the quotient top ology on R C S coincides with

the subset top ology Since is obtained from by comp osition with a linear map it

follows that is op en onto its image

Following Sjamaar we nd it useful to make the following denition

Definition Let M b e a Hamiltonian T space m M LetH T be

the stabilizer of m and H its identitycomponent Let C h b e the cone spanned by

the weights for the H action on T M The ane cone

C m pr C

is called the local moment cone at m M

Thus C is exactly the moment map image for the lo cal mo del at mWehaveshown

Theorem Lo cal convexity theorem Let M be a Hamiltonian T space

m M O T m Then for any suciently smal l T invariant neighborhood U of O

there is a neighborhood V of m C such that

U C V

Global convexity We now come to the key observation of Guillemin

Sternberg Given t consider the corresp onding comp onent hi of the mo

ment map A value s R is called a lo cal minimum for if there exists m M with

ms and s on some neighborhood

Lemma GuilleminSternb erg Let M beacompact connected Hamilton

ian T space Then al l bers of areconnected Moreover the function has a unique

local minimummaximum

We will prove this Lemma in the next section For any subset S t and t let

cone S f t j S g

b e the cone over S at

HAMILTONIAN TORUS ACTIONS

Theorem GuilleminSternb erg Atiyah Let M beacompact connected

Hamiltonian T space on which T acts eectively The image M is a convex

rational polytope of dimension dim T For al l m M one has

C cone

Proof Since lo cal convexity of a compact set implies global convexity it suces to

prove Equation Since a neighb orho o d of the tip of the cone C gets mapp ed into

it follows that cone C To see the opp osite inclusion we dene for all t

m m

the ane linear functional f hih i on t Wehavetoshow that for al

f j f j

But f onC means by the mo del that h i is a lo cal minimum for Bythe

Lemma this has to b e a global minimum or equivalently f on This proves

We note that was rst observed by Sjamaar who generalized it also to the non

ab elian case Since C is the moment map image for the lo cal mo del it shows that is

open as a map onto

We obtain the following description of the faces and the ne structure of Let

H T b e in the nite list of stabilizer groups and M the p oints with stabilizer

H Recall again that M is an op en subset of the symplectic submanifold M Each

connected comp onentof M is a Hamiltonian T space in its own right with H acting

trivially Thus its moment map image is a convex p olytop e of dimension dimTH

inside an ane subspace annh with the corresp onding comp onentofM mapping

to its interior That is the op en faces of corresp ond to orbit typ e strata and in

particular the vertices of corresp ond to xed p oints M That is

hullM

is the convex hull of the xed p ointsetNotehowever that some of the p olytop es M

get mapp ed to the interior of Thus gets sub divided into p olyhedral subregions

consisting of regular values of

Theorem Let M be a Hamiltonian T space with T acting eectively

and t its moment polytope For any closedface of of codimension d the

i i

preimage is symplectic and is a connectedcomponent of the xedpoint set for

some d dimensional stabilizer group H T where h is the subspace orthogonal to

i i i i

In particular the vertices of correspond to xedpoint manifolds

Proof We note that each M is closed and connected by connectedness

of the b ers of Hence it is a connected comp onent of some M where annh is

parallel to

In particular Hamiltonian torus actions on compact symplectic manifolds are never

xedpoint free This shows immediately that the standard ktorus action on itself cannot b e Hamiltonian

SOME BASIC MORSEBOTT THEORY

Exercise Let M b e a compact connected Hamiltonian T space where T

acts eectivelyLetM M b e the subset on which the action is free Show that M

feg

is connected and that its image M is precisely the interior of the moment p olytop e

DuistermaatHeckman measure Let us assume with no loss of generality

that the T action on M is eective The images of the xed p oint manifolds for

nontrivial stabilizer groups dene a sub divison of the p olytop e into chamb ers given

as the connected comp onents of the set of regular values of By the Duistermaat

Heckman theorem the volume function VolM is a p olynomial on eachofthese

connected comp onents

The volume function is equivalenttothe DuistermaatHeckman measure dened

as the pushforward

of the Liouville measure under the momentmapThus is the measure such that for

every continuous function on t

Z Z

t M

Let b e Leb esgue measure on t normalized in sucha way that t where is

Leb

the weight lattice has volume

Exercise Let M b e a compact Hamiltonian T space with T acting

eectivelyShow that at regular values of is smo oth with resp ect to the normalized

Leb esgue measure and

Leb

VolM

Leb

The pro of of this fact is left as an exercise Hint Use the lo cal mo del for reduction

near a regular level set The DuistermaatHeckman theorem maythus b e rephrased by

saying that is a piecewise polynomial measure

DuistermaatHeckman in their pap er use this fact to derive a remarkable exact

integration formula whichwe will in Section

Some basic MorseBott theory

The pro of of the fact that every comp onent f of the moment map has a unique

lo cal minimum relies on the idea of viewing f as a MorseBott function For any function

f C M R on a manifold M the set of critical p oints is the closed subset

C fmj df mg

For all m C there is a welldened symmetric bilinear form on T M called the Hessian

f mX Y L L f m d

m m X Y

for all X Y VectM In lo cal co ordinates the Hessian is simply given by the matrix

of second derivatives of f

HAMILTONIAN TORUS ACTIONS

The function f is called a Morse function if C is discrete and for all m C the Hessian

is nondegenerate More generally f is called MorseBott if the connected comp onents

j j

C of C fmj df mg are smo oth manifolds and for all m C wehave

kerd f m T C

Given a Riemannian metric on M consider the negative gradientowoff ie the

t j

ow F of the vector eld rf VectM For all connected comp onents C wecan

consider the sets

t j

W fm M lim F m C g

and

t j

W fm M lim F m C g

If f is MorseBott then all W are smo oth manifolds and one has natural nite decom

p ositions

j j

W M W

j j

is equal to the di resp W into unstablestable manifolds The dimension of W

mension of C plus the dimension of the negative eigenspace of Hessf denoted n

Thus

j j

n codimW

The number n is called the index of C

Proposition If none of the indices n is equal to there exists a unique

critical manifold of index ie a unique local minimum of f Ifmoreover al l n

then al l level sets f c areconnected

Proof The condition n means that all W of p ositive index have co dimension

at least so that their complement is connected Hence there is a unique stable manifold

W with n If in addition n the set M obtained from M by removing all

j j

j j j j

M with n and all M with n is op en dense and connected in M Notice

that M consists of all p oints whichow to the unique minimum of f for t and to

the unique maximum of f for t Ifminf cmaxf then every tra jectory

of the gradientow of a p ointinM intersects f c in a unique p oint Therefore the

map

f c M R M m t F m

is a dieomorphism and in particular f c M is connected To prove the prop osition

it suces to showthatf c M is dense in f c Let m f cand U a connected

op en neighb orho o d of m Since c is neither maximum or minimum U M meets b oth

the sets where fcand fc and since it is connected it meets f c

Returning to the symplectic geometry context weneedtoshow

SOME BASIC MORSEBOTT THEORY

Theorem Let M be a Hamiltonian Gspace g Then f is a

MorseBott function Moreover al l critical manifolds C are symplectic submanifolds of

M and the indices n are al l even

Proof Let H G b e the closure of the parameter subgroup generated by

Then H is a torus The critical set of f is given by the condition

dhim m

M m

Since is nondegenerate it is precisely the set of zero es of the vector eld or

equivalently the xed p oint set for the parameter subgroup fexpt j t Rg G Let

H fexpt j t Rg

then H is ab elian and connected hence is a torus and C is just the set of xed p oints

for this torus action Let m C and equip T M V with an H invariant compatible

complex structure As a unitary representation V is equivalentto V C where

are the weights for the action By the equivariant Darb ouxtheorem V serves as a

mo del for the H action near m In particular the xed p oint manifold C M gets

mo deled by the space of xed vectors V which is a complex hence also symplectic

subspace This shows that all C are symplectic manifolds Moreover the moment map

in this mo del is a constant plus

X X

z jz j q p

j j j

j j

in particular

X X

f jz j q p h i

j j j

j j

From this it is evident that f is MorseBott and that all indices are even

The fact that all indices are even has very strong implications in Morse theory It

implies that the socalled lacunary principle applies and the MorseBott p olynomial is

equal to the Poincare p olynomial Ie the Morse inequalities are equalities Morse

functions for which this is the case are called perfect This gives a p owerful to ol to

calculate the cohomology of Hamiltonian Gspaces in particular for isolated xed p oints

this gives

dim H M Qf critical p oints of index k g

in particular all cohomology sits in even degree if all indices are even

Corollary Suppose M admits a MorseBott function f such that the minimum

of f is an isolatedpoint and al l n Then M is simply connected

Proof Given any m X and a lo op X based at m one can always p erturb

so that it do es not meet the stable manifolds of index Applying the gradientow

to contracts to the minimum

HAMILTONIAN TORUS ACTIONS

Examples are coadjoint orbits of a compact Lie group the fact that coadjoint orbits are

compact submanifolds of a vector space allows one to show that for generic comp onents of

the moment map the minimum is isolated Thus coadjoint orbits are simply connected

We remark that this is not true in general for conjugacy classes Let GG be a

coadjoint orbit where G is compact connected View G as the b er over the identity

coset Given anytwopoints in G they can b e joined by a path in G The pro jection to

GG is a closed path hence can b e contracted Lifting the contraction to G pro duces

a path in G connecting the twopoints Thus all stabilizer groups for the coadjoint

action are connected

Lo calization formulas

Let M b e a compact Hamiltonian T space For simplicitywe assume that the

set M of xed p oints is nite This is for example the case for the action of a maximal

torus T G on a coadjointorbitO G Given p M let a p a p t

be the weights for the action on T M

Theorem DuistermaatHeckman Let t be such that ha pi for al l

p j Then one has the exact integration formula

n hp i

h i

n ha pi

One way of lo oking at this result is to say that the stationary phase approximation

ith i

e for the integral is exact

M n

Our pro of of the DHformula will follow an argument of BerlineVergne Notice rst

that the integrand is just the top form degree part of

h i h i

M e e

Consider the derivation

d M M d d

The dierential form hi is d closed ie killed byd

d hi dhi

h i

Moreover e is d closed as well BerlineVergne prove the following general

ization of the DHformula

Theorem Let M beacompact oriented T manifold with isolated xedpoint

let a p be the weights for the action on T M dened with respect set Given p M

j p

to some choiceofT invariant complex structureonT M Suppose on M nM

p M

LOCALIZATION FORMULAS

Then for al l forms M such that d one has the integration formula

dim M

ha pi

In the pro of we will use the useful notion of real blowups Consider rst the case of

a real vector space V Let

S V V nfgR

b e its sphere thought of as the space of rays based at Dene V as the subset of

V S V

V fv x V S V j v lies on the ray parametrized by xg

Then V is a manifold with b oundary In fact if one intro duces an inner pro duct on

V then V S V R There is a natural smo oth map V V whichisa

dieomorphism away from S V If M is a manifold and m M one can dene its

blowup M M by using a co ordinate chart based at m Just as in the complex

category one shows that this is indep endentofthechoice of chart although this is

actually not imp ortant for our purp oses

Supp ose now that M is a T spaceasabove Let M M b e the manifold with

b oundary obtained by real blowup at all the xed p oints M TheT action on M lifts

to a T action on M with no xedpoints In particular has no zero es Cho ose an

M and dene invariant Riemannian metric g on

M M

andd L Therefore Then satises

d d

HAMILTONIAN TORUS ACTIONS

isawelldened form satisfying d Thekey idea of BerlineVergne is to use this

form for partial integration

Z Z

M M

S T M

Thus to complete the pro of wehave to carry out the remaining integral over the sphere

We will do this by a trick dening a d closed form where we can actually compute

the integral byhand

C for a given p M Intro duce co ordi Consider the T action on T M

a p p

nates r t by z r e Given let C R beacuto

j j j j

function with r for r and for Dene a form

n n

Y Y

ha pi r dr dt d r dt

j j j j j j

j j

Note that this form is welldened even though the co ordinates are not globally well

dened compactly supp orted and d closed Its integral is equal to

r dr

j j

T M

On the other hand ha pi

Cho osing suciently small we can consider as a form on M vanishing at all the

other xed p oints Applying the lo calization formula wend

Z Z

ha pi

S T M M

thus

ha pi

S T M

p QED

FRANKELS THEOREM

The ab ove discussion extends to nonisolated xed p oints in this case the pro duct

ha pi is replaced by the equivariant Euler class of the normal bundle of the xed

point manifold

One often applies the DuistermaatHeckman theorem in order to compute Liouville

volumes of symplectic manifolds with Hamiltonian group action Consider for example

a Hamiltonian S RZaction with isolated xed p oints Identify LieS so that the

integral lattice and its dual are just Z ZLetH hi where corresp onds

to R By DuistermaatHeckman

n tH p

a p n t

Notice by the way that the individual terms on the right hand side are singular for t

This implies very subtle relationships b etween the weight for example one must have

H p

a p

for all knFor the volume one reads o

H p

VolM

n a p

Frankels theorem

As wehave seen Hamiltonian torus actions are very sp ecial in many resp ects In

particular they always have xed p oints It is a classical result of Frankel long b efore

moment maps were invented that on Kahler manifolds the converse is true

Theorem Let M bea compact Kahler manifold with Kahler form Con

sider a symplectic S action on M with at least one xedpoint Then the action is

Hamiltonian

Proof Let dim M nWe need one nontrivial result from complex geometry

which is a particular case of the hard Lefschetz theorem Wedge pro duct with

induces an isomorphism in cohomology

n n

H M H M

Let X VectM b e the vector eld corresp onding to R LieS We need to

show that is exact By hard Lefschetz this is equivalent to showing that is

X X

b e a xed p oint In a neighborhood of m wecanidentify M as exact Let m M

T M beaninvariant form supp orted in an ball a T space with T M Let

m m

R R

Cho osing suciently small we around T M normalized so that

T M M m

HAMILTONIAN TORUS ACTIONS

can view as a form on M Since and have the same integral it follows that

n n

d for some invariant form M Then

X X d L dX dX

showing that X isexactWethus need to show that X is exact This

however follows from the Poincare lemma since it is supp orted in a ball around mwhere

one can just apply the homotopy op erator

Delzant spaces

Definition A Hamiltonian T space M with prop er moment map is

called multiplicityfree if all reduced space M are either empty or dimensional We

call M a Delzantspace if in addition M is connected the moment map is prop er

and the numb er of orbit typ e strata is nite

Thus if T acts eectivelyM is Delzant if and only if dim M dimT

n n

Examples a M C with the standard action of T S The moment

n n

t More abstractlyifV is a R map image is the p ositive orthant R

Hermitian vector space the action of the maximal torus T U V onV os Delzant

b M C P n with the action of T S S quotientby diagonal subgroup

n n

coming from the action of S on C The moment map image is a simplex

t with the hyp erplane given as the intersection of the p ositive orthant R

More generallyifV is a Hermitian vector space the action of the maximal torus

T UV on the pro jectivization P V is Delzant

c M T T with the cotangent lift of the leftaction of T on itself The moment

map image is all of t We will call this from now on the standard T action on

T T

d Supp ose M is a Delzant T space and H T is a subgroup acting freely on

the level set of h Then the H reduced space M is Delzant The

moment map image M t is the intersection of M with the ane subspace

We can view M as a Delzant THspace after cho osing a moment map pr

for the THaction suchachoice amounts to cho osing a p ointinpr

The moment map images for Delzant spaces can b e characterized as follows Let

t be the integral lattice ie the kernel of exp t T Let t b e a rational

convex p olyhedral set of dimension d dimT with k b oundary hyp erplanes That is

is of the form

i i

where v are primitive lattice vectors and Rand

i i

H f t jh v i g

v i i

i i

The niteness assumption is not very imp ortant and is of course automatic if M is compact

DELZANT SPACES

For any subset I f kg let b e the set of all with h v i for i I We

I i i

set int

Definition The p olyhedral set t is called Delzant if for all I with

the vectors v i I are linearly indep endent and

span fv j i I g span fv j i I g

i i

Z R

Remark For compact p olyhedral sets that is p olytop es it is enough to check

the Delzant condition at the vertices The Delzant condition means in particular that

each v has to b e a primitive normal vector ie is not of the form v av where v

i i

and a Z

Example Let T S and identify t t R and Z The

p olytop e with vertices at is Delzant However the p olytop e with

vertices at is not Delzant Indeed for the vertex at the two

primitive normal vectors are v and v and they do not span the

lattice Z

The Delzant condition for says that s v s Z for all j I or

I j j j

j I

equivalently

exp s v s mod Z for all j I

j j j

j I

Thus if we dene a homomorphism

S T s s exp s v

k i i

and let

I k

S fs s S j s modZ for j I g

k j

b e the pro duct of S factors corresp onding to indices j I the Delzant condition is

equivalenttosaying that restricts to an inclusion S T The image

H S T is obtained by exp onentiating h span fv j j I gby denition

I I j

it is the subspace p erp endicular to t

Theorem Let M be a Delzant T space with eective T action Then

M is a Delzant polyhedron For al l open faces F the preimage F is

aconnectedcomponent of the orbit typestratum M M for H exph where h t

H F F

is the subspaceperpendicular to F Inparticular al l stabilizer groups areconnected

Proof Let F O Tm an orbit and H T the stabilizer group

We had seen that the cone is equal to the lo cal momentcone

C C pr

where C h is the cone spanned bytheweights h for the H action on

T O By dimension count k dim V the symplectic vector space V T O

m C m

HAMILTONIAN TORUS ACTIONS

dim M dimTH dim H It follows that are a basis of h Since annh t is

C it must coincide with the space the maximal linear subspace inside the cone pr

parallel to F Thatish h

The action of H on V must b e eective since the T action on E is eective Thus

H acts as a compact ab elian subgroup of UV of dimension dim H dim V Soits

identity comp onent H is a maximal torus But it is a wellknown fact from Lie group

theory that maximal tori are maximal ab elian so H H In particular wehave shown

that all p oints in F have the same stabilizer group

It follows that the map H S dened by the ro ots is an isomorphism This

means that are a basis for the weight lattice weight lattice h in h

Equivalently the dual basis w w h are a basis for hWehave

C conef g f h jh w ig

n i

which identies the fw w g with fv j i I g

k i

Delzantgave an explicit recip e for constructing a Delzant space with moment p oly

top e a given Delzant p olyhedron The following version of Delzants construction is due

to Eugene Lerman

Let S act on the cotangent bundle T T via the comp osition of with the

standard T action on T T In the left trivialization T T T t a moment map

for the T action is pro jection to t Hence

X X

t h v ie e

j j j j

j j

k k k

is a moment map for the action of S Let S act on C in the standard way with

jz j e momentmap

j j

Definition For any p olyhedron let D b e the symplectic quotient

k k

D T T C S

by the diagonal action with T action induced from the standard T action on T T

Theorem Suppose is a Delzant polyhedron Then the action of S on the

zero level set of T T C is free and the quotient D is a DelzantT space The

moment map image of D is exactly

Proof Let t z in the zero level set Thus

h v i jz j

i i i

If z then the ith factor of S acts freely at t z Thus we need only worry

ab out the set I of indices i with z For these indices h v i LetS be the

i i i

pro duct of copies of S corresp onding to these indices By the Delzant condition

I I

T Since T acts freely on T T so do es S This restricts to an embedding S

shows that the action is free and D is a smo oth symplectic manifold To identify the

DELZANT SPACES

image of the T moment map note that given t one can nd t z with t zis

in the zero level set if and only if h v i

i i

Definition Lerman Let b e a Delzant p olyhedron and M a Hamil

tonian T space The cut space dened by is the symplectic quotient

M M D T

with T action induced from the action on the rst factor

It is immediate that T T D In particular T S C Wewillnowuse

these two facts to prove

Theorem Delzant Every Delzant space M is determined by its mo

ment polyhedron M uptoequivariant symplectomorphism intertwining the

moment maps

Proof Usually this is proved using a Cech theoretic argument Belowwesketch

a more elementary approach The idea is to present M as a symplectic cut M

of a connected multiplicity free Hamiltonian T space M with free T action Since the

action of T on M is free the map is a Lagrangian bration over its image Thus

we can intro duce actionangle variables whichidenties M as an op en subset of T T

Therefore M M T T D

Wenow indicate how to construct such a space M Let i f kg be an

index suchthat and S the symplectic submanifold obtained as its

i i

preimage It is a connected comp onent of the xed p ointsetof H and has co dimension

Let TS b e its symplectic normal bundle After cho osing a compatible complex

structure it can b e viewed as a Hermitian line bundle Let Q b e the unit circle

bundle inside Q It is a T equivariant principal S bundle and Q C Let

Q S b e the pro jection map Let Q be a T invariant connection

form and consider the closed form

djz j

QC S C

It is easy to check that this form is basic for the S action so it descends to a closed

form

Furthermore is nondegenerate near S QS It follows that there exists an

equivariant symplectomorphism b etween op en neighborhoods of S in M and in Now

Q R cut with resp ect to the S action where Q R is equipp ed with the

form

QR S C

Wehave a natural dieomorphism b etween Q R and nS preserving forms We

can thus glue M nS with a small neighborhood of Q in Q R to obtain a new connected

HAMILTONIAN TORUS ACTIONS

multiplicity free Hamiltonian T space M with one orbit typ e stratum less The

original space is obtained from M by cutting

M M

where H is the ane halfspace h v i Continuing in this fashion construct

i i

spaces M M M M where n is the numb er of faces of Wehave

M M M M

n H H H

The nal space M M no longer has dimensional stabilizer groups so the T action

n is free as required