{f, g} = ’ .

‚xi ‚yi ‚yi ‚xi

i=1

™¦

4.4 Integrable Systems

De¬nition 4.7 A hamiltonian system is a triple (M, ω, H), where (M, ω) is

a symplectic manifold and H ∈ C ∞ (M ; R) is a function, called the hamiltonian

function.

Theorem 4.8 We have {f, H} = 0 if and only if f is constant along integral

curves of XH .

Proof. Let ρt be the ¬‚ow of XH . Then

d

= ρ— LXH f = ρ— ±XH df = ρ— ±XH ±Xf ω

—¦ ρt )

dt (f t t t

— —

= ρt ω(Xf , XH ) = ρt {f, H} = 0 .

A function f as in Theorem 4.8 is called an integral of motion (or a ¬rst

integral or a constant of motion). In general, hamiltonian systems do not admit

integrals of motion which are independent of the hamiltonian function. Functions

f1 , . . . , fn on M are said to be independent if their di¬erentials (df1 )p , . . . , (dfn )p

are linearly independent at all points p in some open dense subset of M . Loosely

speaking, a hamiltonian system is (completely) integrable if it has as many commut-

ing integrals of motion as possible. Commutativity is with respect to the Poisson

bracket. Notice that, if f1 , . . . , fn are commuting integrals of motion for a hamil-

tonian system (M, ω, H), then, at each p ∈ M , their hamiltonian vector ¬elds

generate an isotropic subspace of Tp M :

ω(Xfi , Xfj ) = {fi , fj } = 0 .

If f1 , . . . , fn are independent at p, then, by symplectic linear algebra, n can be at

most half the dimension of M .

54 LECTURE 4. HAMILTONIAN FIELDS

De¬nition 4.9 A hamiltonian system (M, ω, H) is (completely) integrable

if it possesses n = 1 dim M independent integrals of motion, f1 = H, f2 , . . . , fn ,

2

which are pairwise in involution with respect to the Poisson bracket, i.e., {f i , fj } =

0, for all i, j.

Examples.

1. The simple pendulum (discussed in the next section) and the harmonic oscil-

lator are trivially integrable systems “ any 2-dimensional hamiltonian system

(where the set of non-¬xed points is dense) is integrable.

2. A hamiltonian system (M, ω, H) where M is 4-dimensional is integrable if

there is an integral of motion independent of H (the commutativity condi-

tion is automatically satis¬ed). The next section shows that the spherical

pendulum is integrable.

™¦

For sophisticated examples of integrable systems, see [9, 28].

Let (M, ω, H) be an integrable system of dimension 2n with integrals of

motion f1 = H, f2 , . . . , fn . Let c ∈ Rn be a regular value of f := (f1 , . . . , fn ).

The corresponding level set, f ’1 (c), is a lagrangian submanifold, because it is

n-dimensional and its tangent bundle is isotropic.

Proposition 4.10 If the hamiltonian vector ¬elds Xf1 , . . . , Xfn are complete on

the level f ’1 (c), then the connected components of f ’1 (c) are homogeneous spaces

for Rn , i.e., are of the form Rn’k — Tk for some k, 0 ¤ k ¤ n, where Tk is a

k-dimensional torus.

Proof. Exercise (just follow the ¬‚ows).

Any compact component of f ’1 (c) must hence be a torus. These components,

when they exist, are called Liouville tori. (The easiest way to ensure that compact

components exist is to have one of the fi ™s proper.)

Theorem 4.11 (Arnold-Liouville [3]) Let (M, ω, H) be an integrable system

of dimension 2n with integrals of motion f1 = H, f2 , . . . , fn . Let c ∈ Rn be a

regular value of f := (f1 , . . . , fn ). The corresponding level f ’1 (c) is a lagrangian

submanifold of M .

(a) If the ¬‚ows of Xf1 , . . . , Xfn starting at a point p ∈ f ’1 (c) are complete, then

the connected component of f ’1 (c) containing p is a homogeneous space for

Rn . With respect to this a¬ne structure, that component has coordinates

•1 , . . . , •n , known as angle coordinates, in which the ¬‚ows of the vector

¬elds Xf1 , . . . , Xfn are linear.

55

4.5. PENDULUMS

(b) There are coordinates ψ1 , . . . , ψn , known as action coordinates, comple-

mentary to the angle coordinates such that the ψi ™s are integrals of motion

and •1 , . . . , •n , ψ1 , . . . , ψn form a Darboux chart.

Therefore, the dynamics of an integrable system is extremely simple and the

system has an explicit solution in action-angle coordinates. The proof of part (a)

“ the easy part “ of the Arnold-Liouville theorem is sketched above. For the proof

of part (b), see [3, 17].

Geometrically, part (a) of the Arnold-Liouville theorem says that, in a neigh-

borhood of the value c, the map f : M ’ Rn collecting the given integrals of

motion is a lagrangian ¬bration, i.e., it is locally trivial and its ¬bers are la-

grangian submanifolds. The coordinates along the ¬bers are the angle coordinates. 2

Part (b) of the theorem guarantees the existence of coordinates on Rn , the action

coordinates, which satisfy {•i , ψj } = δij with respect to the angle coordinates. No-

tice that, in general, the action coordinates are not the given integrals of motion

because •1 , . . . , •n , f1 , . . . , fn do not form a Darboux chart.

4.5 Pendulums

The simple pendulum is a mechanical system consisting of a massless rigid rod

of length , ¬xed at one end, whereas the other end has a plumb bob of mass m,

which may oscillate in the vertical plane. We assume that the force of gravity is

constant pointing vertically downwards, and that this is the only external force

acting on this one-particle system.

Let θ be the oriented angle between the rod (regarded as a point mass) and

the vertical direction. Let ξ be the coordinate along the ¬bers of T — S 1 induced by

the standard angle coordinate on S 1 . Then the function H : T — S 1 ’ R given by

ξ2

+ m (1 ’ cos θ) ,

H(θ, ξ) =

2m 2

V

K

is an appropriate hamiltonian function to describe the spherical pendulum. More

precisely, gravity corresponds to the potential energy V (θ) = m (1 ’ cos θ) (we

1

omit universal constants), and the kinetic energy is given by K(θ, ξ) = 2m 2 ξ 2 .

For simplicity, we assume that m = = 1.

Exercise 20

Show that there exists a number c such that for 0 < h < c the level curve

H = h in the (θ, ξ) plane is a disjoint union of closed curves. Show that the

projection of each of these curves onto the θ-axis is an interval of length less

than π.

Show that neither of these assertions is true if h > c.

What types of motion are described by these two types of curves?

What about the case H = c?

2 The name “angle coordinates” is used even if the ¬bers are not tori.

56 LECTURE 4. HAMILTONIAN FIELDS

Modulo 2π in θ, the function H has exactly two critical points: a critical

point s where H vanishes, and a critical point u where H equals c. These points

are called the stable and unstable points of H, respectively. This terminology is

justi¬ed by the fact that a trajectory of the hamiltonian vector ¬eld of H whose

initial point is close to s stays close to s forever, whereas this is not the case for

u. (What is happening physically?)

The spherical pendulum is a mechanical system consisting of a massless

rigid rod of length , ¬xed at one end, whereas the other end has a plumb bob

of mass m, which may oscillate freely in all directions. We assume that the force

of gravity is constant pointing vertically downwards, and that this is the only

external force acting on this one-particle system.

Let •, θ (0 < • < π, 0 < θ < 2π) be spherical coordinates for the bob. For

simplicity we take m = = 1.

Let ·, ξ be the coordinates along the ¬bers of T — S 2 induced by the spheri-

cal coordinates •, θ on S 2 . An appropriate hamiltonian function to describe the

spherical pendulum is H : T — S 2 ’ R given by

ξ2

1 2

H(•, θ, ·, ξ) = ·+ + cos • .

(sin •)2

2

On S 2 , the function H has exactly two critical points: s (where H has a

minimum) and u. These points are called the stable and unstable points of H,

respectively. A trajectory whose initial point is close to s stays close to s forever,

whereas this is not the case for u.

The group of rotations about the vertical axis is a group of symmetries of the

spherical pendulum. In the coordinates above, the integral of motion associated

with these symmetries is the function

J(•, θ, ·, ξ) = ξ .

Exercise 21

Give a more coordinate-independent description of J, one that makes sense

also on the cotangent ¬bers above the North and South poles.

We will locate all points p ∈ T — S 2 where dHp and dJp are linearly dependent:

• Clearly, the two critical points s and u belong to this set. These are the only

two points where dHp = dJp = 0.

• If x ∈ S 2 is in the southern hemisphere (x3 < 0), then there exist exactly

two points, p+ = (x, ·, ξ) and p’ = (x, ’·, ’ξ), in the cotangent ¬ber above

x where dHp and dJp are linearly dependent.

• Since dHp and dJp are linearly dependent along the trajectory of the hamil-

tonian vector ¬eld of H through p+ , this trajectory is also a trajectory of the

57

4.6. SYMPLECTIC AND HAMILTONIAN ACTIONS

hamiltonian vector ¬eld of J, and, hence, that its projection onto S 2 is a lat-

itudinal circle (of the form x3 = constant). The projection of the trajectory

through p’ is the same latitudinal circle traced in the opposite direction.

One can check that any nonzero value j is a regular value of J, and that S 1

acts freely on the level set J = j.

Exercise 22

What happens on the cotangent ¬bers above the North and South poles?

The integral curves of the original system on the level set J = j can be

obtained from those of the reduced system by “quadrature”, in other words, by a

simple integration.

The reduced system for j = 0 has exactly one equilibrium point. The corre-

sponding relative equilibrium for the original system is one of the horizontal curves

from above.

The energy-momentum map is the map (H, J) : T — S 2 ’ R2 . If j = 0, the

level set (H, J) = (h, j) of the energy-momentum map is either a circle (in which

case it is one of the horizontal curves above), or a two-torus. The projection onto

the con¬guration space of the two-torus is an annular region on S 2 .

4.6 Symplectic and Hamiltonian Actions

Let (M, ω) be a symplectic manifold, and G a Lie group. Let ψ : G ’’ Di¬(M )

be a (smooth) action.

De¬nition 4.12 The action ψ is a symplectic action if

ψ : G ’’ Sympl(M, ω) ‚ Di¬(M ) ,

i.e., G acts by symplectomorphisms.

In particular, symplectic actions of R on (M, ω) are in one-to-one correspon-

dence with complete symplectic vector ¬elds on M .

Examples.

1. On R2n with ω = ‚

dxi § dyi , let X = ’ ‚y1 . The orbits of the action

generated by X are lines parallel to the y1 -axis,

{(x1 , y1 ’ t, x2 , y2 , . . . , xn , yn ) | t ∈ R} .

Since X = Xx1 is hamiltonian (with hamiltonian function H = x1 ), this is

actually an example of a hamiltonian action of R.

58 LECTURE 4. HAMILTONIAN FIELDS

2. On the symplectic 2-torus (T2 , dθ1 § dθ2 ), the one-parameter groups of dif-

feomorphisms given by rotation around each circle, ψ1,t (θ1 , θ2 ) = (θ1 + t, θ2 )

(t ∈ R) and ψ2,t similarly de¬ned, are symplectic actions of S 1 .

3. On the symplectic 2-sphere (S 2 , dθ § dh) in cylindrical coordinates, the one-

parameter group of di¬eomorphisms given by rotation around the verti-

cal axis, ψt (θ, h) = (θ + t, h) (t ∈ R) is a symplectic action of the group

S1 R/ 2π , as it preserves the area form dθ § dh. Since the vector ¬eld

corresponding to ψ is hamiltonian (with hamiltonian function H = h), this

is an example of a hamiltonian action of S 1 .

™¦

De¬nition 4.13 A symplectic action ψ of S 1 or R on (M, ω) is hamiltonian if

the vector ¬eld generated by ψ is hamiltonian. Equivalently, an action ψ of S 1 or

R on (M, ω) is hamiltonian if there is H : M ’ R with dH = ±X ω, where X is

the vector ¬eld generated by ψ.

What is a “hamiltonian action” of an arbitrary Lie group?

For the case where G = Tn = S 1 — . . . — S 1 is an n-torus, an action ψ : G ’

Sympl(M, ω) should be called hamiltonian when each restriction

: S 1 ’’ Sympl(M, ω)

ψ i := ψ|ith S 1 factor

is hamiltonian in the previous sense with hamiltonian function preserved by the

action of the rest of G.

When G is not a product of S 1 ™s or R™s, the solution is to use an upgraded

hamiltonian function, known as a moment map. Up to an additive constant, a

moment map µ is determined by coordinate functions µi satisfying dµi = ±Xi ω for

a basis Xi of the Lie algebra of G. There are various ways to ¬x that constant,

and we can always choose µ equivariant, i.e., intertwining the action of G on M

with the coadjoint action of G on the dual of the Lie algebra (see Appendix B),

as de¬ned in the next section.

4.7 Moment Maps

Let

(M, ω) be a symplectic manifold,

G a Lie group, and

ψ : G ’ Sympl(M, ω) a symplectic action, i.e., a group homomorphism such

that the evaluation map evψ (g, p) := ψg (p) is smooth.

59

4.7. MOMENT MAPS

Case G = R:

We have the following bijective correspondence:

{symplectic actions of R on M } ←’ {complete symplectic vector ¬elds on M }

dψt (p)

’’

ψ Xp = dt

←’

ψ = exp tX X

“¬‚ow of X” “vector ¬eld generated by ψ”

The action ψ is hamiltonian if there exists a function H : M ’ R such that

dH = ±X ω where X is the vector ¬eld on M generated by ψ.

Case G = S 1 :

An action of S 1 is an action of R which is 2π-periodic: ψ2π = ψ0 . The S 1 -

action is called hamiltonian if the underlying R-action is hamiltonian.

General case:

Let

(M, ω) be a symplectic manifold,

G a Lie group,

the Lie algebra of G,

g

g— the dual vector space of g, and

ψ : G ’’ Sympl(M, ω) a symplectic action.

De¬nition 4.14 The action ψ is a hamiltonian action if there exists a map

µ : M ’’ g—

satisfying:

1. For each X ∈ g, let

• µX : M ’ R, µX (p) := µ(p), X , be the component of µ along X,

• X # be the vector ¬eld on M generated by the one-parameter subgroup

{exp tX | t ∈ R} ⊆ G.

Then

dµX = ±X # ω

i.e., µX is a hamiltonian function for the vector ¬eld X # .

60 LECTURE 4. HAMILTONIAN FIELDS

2. µ is equivariant with respect to the given action ψ of G on M and the coad-

joint action Ad— of G on g— :

µ —¦ ψg = Ad— —¦ µ , for all g ∈ G .

g

The vector (M, ω, G, µ) is then called a hamiltonian G-space and µ is a mo-

ment map.

This de¬nition matches the previous ones for the cases G = R, S1 , torus,

where equivariance becomes invariance since the coadjoint action is trivial.

Case G = S 1 (or R):

R, g— R. A moment map µ : M ’’ R satis¬es:

Here g

1. For the generator X = 1 of g, we have µX (p) = µ(p) · 1, i.e., µX = µ, and

X # is the standard vector ¬eld on M generated by S 1 . Then dµ = ±X # ω.

2. µ is invariant: LX # µ = ±X # dµ = 0.

Case G = Tn = n-torus:

Rn , g— Rn . A moment map µ : M ’’ Rn satis¬es:

Here g

1. For each basis vector Xi of Rn , µXi is a hamiltonian function for Xi# .

2. µ is invariant.

Atiyah, Guillemin and Sternberg [6, 26] showed that the image of the moment

map for a hamiltonian torus action on a compact connected symplectic manifold

is always a polytope3

Theorem 4.15 (Atiyah [6], Guillemin-Sternberg [26]) Let (M, ω) be a

compact connected symplectic manifold, and let T m be an m-torus. Suppose that

ψ : Tm ’ Sympl(M, ω) is a hamiltonian action with moment map µ : M ’ Rm .

Then:

(a) the levels of µ are connected;

(b) the image of µ is convex;

(c) the image of µ is the convex hull of the images of the ¬xed points of the

action.

3A

polytope in Rn is the convex hull of a ¬nite number of points in R n . A convex polyhe-

dron is a subset of Rn which is the intersection of a ¬nite number of a¬ne half-spaces. Hence,

polytopes coincide with bounded convex polyhedra.

61

4.7. MOMENT MAPS

The image µ(M ) of the moment map is called the moment polytope. A

proof of Theorem 4.15 can be found in [35].

An action of a group G on a manifold M is called e¬ective if each group

element g = e moves at least one p ∈ M , that is, ©p∈M Gp = {e}, where Gp =

{g ∈ G | g · p = p} is the stabilizer of p.

Exercise 23

Suppose that T m acts linearly on (Cn , ω0 ). Let »(1) , . . . , »(n) ∈ Z m be the

weights appearing in the corresponding weight space decomposition, that is,

n

n

V»(k) ,

C

k=1

(k) (k)

where, for »(k) = (»1 , . . . , »m ), T m acts on the complex line V»(k) by

(k)

i j »j tj v

(eit1 , . . . , eitm ) · v = e , ∀v ∈ V»(k) , ∀k = 1, . . . , n .

(a) Show that, if the action is e¬ective, then m ¤ n and the weights

»(1) , . . . , »(n) are part of a Z-basis of Z m.

(b) Show that, if the action is symplectic (hence, hamiltonian), then the

weight spaces V»(k) are symplectic subspaces.

(c) Show that, if the action is hamiltonian, then a moment map is given by

n

1

»(k) ||v»(k) ||2 ( + constant ) ,

µ(v) = ’ 2

k=1

where || · || is the standard norma and v = v»(1) + . . . + v»(n) is the

weight space decomposition. Cf. Example 1.

(d) Conclude that, if T n acts on Cn in a linear, e¬ective and hamiltonian

way, then any moment map µ is a submersion, i.e., each di¬erential

dµv : Cn ’ Rn (v ∈ Cn ) is surjective.

a Noticethat the standard inner product satis¬es (v, w) = ω0 (v, Jv) where

‚ ‚ ‚ ‚

J = i ‚z and J ‚ z = ’i ‚ z . In particular, the standard norm is invariant for

¯ ¯

‚z

a symplectic complex-linear action.

The following two results use the crucial fact that any e¬ective action T m ’

Di¬(M ) has orbits of dimension m; a proof may be found in [11].

Corollary 4.16 Under the conditions of the convexity theorem, if the T m -action

is e¬ective, then there must be at least m + 1 ¬xed points.

Proof. At a point p of an m-dimensional orbit the moment map is a submersion,

i.e., (dµ1 )p , . . . , (dµm )p are linearly independent. Hence, µ(p) is an interior point

of µ(M ), and µ(M ) is a nondegenerate polytope. Any nondegenerate polytope in

Rm must have at least m + 1 vertices. The vertices of µ(M ) are images of ¬xed

points.

62 LECTURE 4. HAMILTONIAN FIELDS

Proposition 4.17 Let (M, ω, Tm , µ) be a hamiltonian Tm -space. If the Tm -action

is e¬ective, then dim M ≥ 2m.

Proof. Since the moment map is constant on an orbit O, for p ∈ O the exterior

derivative

dµp : Tp M ’’ g—

maps Tp O to 0. Thus

Tp O ⊆ ker dµp = (Tp O)ω ,

where (Tp O)ω is the symplectic orthogonal of Tp O. This shows that orbits O of

a hamiltonian torus action are always isotropic submanifolds of M . In particular,

by symplectic linear algebra we have that dim O ¤ 1 dim M . Now consider an

2

m-dimensional orbit.

Examples.

1. Let Tn = {(t1 , . . . , tn ) ∈ Cn : |tj | = 1, for all j } be a torus acting on Cn by

(t1 , . . . , tn ) · (z1 , . . . , zn ) = (tk1 z1 , . . . , tkn zn ) ,

1 n

where k1 , . . . , kn ∈ Z are ¬xed. This action is hamiltonian with moment map

µ : Cn ’ (tn )— Rn given by

µ(z1 , . . . , zn ) = ’ 1 (k1 |z1 |2 , . . . , kn |zn |2 ) ( + constant ) .

2

2. Suppose that a Lie group G acts in a hamiltonian way on two symplectic

manifolds (Mj , ωj ), j = 1, 2, with moment maps µ : Mj ’ g— . Then the

diagonal action of G on M1 — M2 is hamiltonian with moment map µ :

M1 — M2 ’ g— given by

µ(p1 , p2 ) = µ1 (p1 ) + µ2 (p2 ) , for pj ∈ Mj .

3. The vector ¬eld X # generated by X ∈ g for the coadjoint representation of

a Lie group G on g— satis¬es Xξ , Y = ξ, [Y, X] , for any Y ∈ g. Equip

#

the coadjoint orbits with the canonical symplectic forms (Section 4.3). Then,

for each ξ ∈ g— , the coadjoint action on the orbit G · ξ is hamiltonian with

moment map the inclusion map:

µ : G · ξ ’ g— .

™¦

63

4.8. LANGUAGE FOR MECHANICS

Exercises 24

(a) Consider the natural action of U(n) on (Cn , ω0 ). Show that this action

is hamiltonian with moment map µ : Cn ’ u(n) given by

i

zz —

µ(z) = ,

2

where we identify the Lie algebra u(n) with its dual via the inner product

(A, B) = trace(A— B).

Hint: Denote the elements of U(n) in terms of real and imaginary

parts g = h + i k. Then g acts on R2n by the linear symplectomorphism

h ’k

. The Lie algebra u(n) is the set of skew-hermitian matrices

k h

X = V + i W where V = ’V t ∈ Rn—n and W = W t ∈ Rn—n .

Show that the in¬nitesimal action is generated by the hamiltonian

functions

µX (z) = ’ 2 (x, W x) + (y, V x) ’ 1 (y, W y)

1

2

where z = x + i y, x, y ∈ Rn and (·, ·) is the standard inner product.

Show that

µX (z) = 1 i z — Xz = 2 i trace(zz — X) .

1

2

Check that µ is equivariant.

(b) Consider the natural action of U(k) on the space (Ck—n , ω0 ) of complex

(k — n)-matrices. Identify the Lie algebra u(k) with its dual via the inner

product (A, B) = trace(A— B). Prove that a moment map for this action

is given by

µ(A) = 2 AA— + Id , for A ∈ Ck—n .

i

2i

Id

(The constant is just a choice.)

2i

Hint: Example 2 and Exercise (a).

2

(c) Consider the U(n)-action by conjugation on the space (Cn , ω0 ) of com-

plex (n — n)-matrices. Show that a moment map for this action is given

by

i

µ(A) = 2 [A, A— ] .

Hint: Previous exercise and its “transpose” version.

4.8 Language for Mechanics

Example.

Let G = SO(3) = {A ∈ GL(3; R) | A t A = Id and detA = 1}. Then g =

{A ∈ gl(3; R) | A + At = 0} is the space of 3 — 3 skew-symmetric matrices and

can be identi¬ed with R3 . The Lie bracket on g can be identi¬ed with the exterior

64 LECTURE 4. HAMILTONIAN FIELDS

product via

®

’a3

0 a2

’ = (a1 , a2 , a3 )

’

° a3 ’a1 » ’’

A= 0 a

’a2 a1 0

’ — ’.

’’

[A, B] = AB ’ BA ’’ a b

Exercise 25

Under the identi¬cations g, g— R3 , the adjoint and coadjoint actions are the

usual SO(3)-action on R3 by rotations.

Therefore, the coadjoint orbits are the spheres in R3 centered at the origin.

™¦

Section 4.3 shows how general coadjoint orbits are symplectic.

The name “moment map” comes from being the generalization of linear and

angular momenta in classical mechanics.

Translation: Consider R6 with coordinates x1 , x2 , x3 , y1 , y2 , y3 and symplectic

form ω = dxi § dyi . Let R3 act on R6 by translations:

’ ∈ R3 ’’ ψ’ ∈ Sympl(R6 , ω)

’ ’

a a

ψ’ (’, ’) = (’ + ’, ’) .

’’ ’ ’ ’’

axy x ay

+ a3 ‚x3 for X = ’, and

’

‚ ‚ ‚

Then X # = a1 ‚x1 + a2 ‚x2 a

µ(’, ’) = ’

’’ ’

µ : R6 ’’ R3 , xy y

is a moment map, with

’’ ’

’

µ a (’, ’) = µ(’, ’), ’ = ’ · ’ .

’’ ’ ’’

xy xy a ya

Classically, ’ is called the momentum vector corresponding to the position

’

y

’, and the map µ is called the linear momentum.

’

vector x

Rotation: The SO(3)-action on R3 by rotations lifts to a symplectic action ψ on

the cotangent bundle R6 . The in¬nitesimal version of this action is

’ ∈ R3 ’’ dψ(’) ∈ χsympl (R6 )

’ ’

a a

dψ(’)(’, ’) = (’ — ’, ’ — ’) .

’ ’’ ’ ’’ ’

a xy a xa y

Then

µ(’, ’) = ’ — ’

’’ ’’

µ : R6 ’’ R3 , xy x y

is a moment map, with

’’ ’

’

µ a (’, ’) = µ(’, ’), ’ = (’ — ’) · ’.

’’ ’ ’’’

xy xy a x y a

The map µ is called the angular momentum.

Let (M, ω, G, µ) be a hamiltonian G-space.

65

4.9. EXISTENCE AND UNIQUENESS OF MOMENT MAPS

Theorem 4.18 (Noether) If f : M ’ R is a G-invariant function, then µ is

constant on the trajectories of the hamiltonian vector ¬eld of f .

Proof. Let vf be the hamiltonian vector ¬eld of f . Let X ∈ g and µX = µ, X :

M ’ R. We have

Lvf µX = ±vf dµX = ±vf ±X # ω

= ’±X # ±vf ω = ’±X # df

= ’LX # f = 0

because f is G-invariant.

De¬nition 4.19 A G-invariant function f : M ’ R is called an integral of

motion of (M, ω, G, µ). If µ is constant on the trajectories of a hamiltonian vector

¬eld vf , then the corresponding one-parameter group of di¬eomorphisms {exp tv f |

t ∈ R} is called a symmetry of (M, ω, G, µ).

The Noether principle asserts that there is a one-to-one correspondence

between symmetries and integrals of motion.

4.9 Existence and Uniqueness of Moment Maps

Let g be a Lie algebra, and

:= Λk g— = k-cochains on g

Ck

= alternating k-linear maps g — . . . — g ’’ R .

k

De¬ne a linear operator δ : C k ’ C k+1 by

(’1)i+j c([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ) .

δc(X0 , . . . , Xk ) =

i<j

Exercise 26

Check that δ 2 = 0.

The Lie algebra cohomology groups (or Chevalley cohomology groups)

δ δ δ

of g are the cohomology groups of the complex 0 ’ C 0 ’ C 1 ’ . . .:

ker δ : C k ’’ C k+1

H k (g; R) := .

im δ : C k’1 ’’ C k

Theorem 4.20 If g is the Lie algebra of a compact connected Lie group G, then

H k (g; R) = HdeRham (G) .

k

66 LECTURE 4. HAMILTONIAN FIELDS

Proof. Exercise. Hint: by averaging show that the de Rham cohomology can be

computed from the subcomplex of G-invariant forms.

Meaning of H 1 (g; R) and H2 (g; R):

• An element of C 1 = g— is a linear functional on g. If c ∈ g— , then δc(X0 , X1 ) =

’c([X0 , X1 ]). The commutator ideal of g is

[g, g] := {linear combinations of [X, Y ] for any X, Y ∈ g} .

Since δc = 0 if and only if c vanishes on [g, g], we conclude that

H 1 (g; R) = [g, g]0

where [g, g]0 ⊆ g— is the annihilator of [g, g].

• An element of C 2 is an alternating bilinear map c : g — g ’ R.

δc(X0 , X1 , X2 ) = ’c([X0 , X1 ], X2 ) + c([X0 , X2 ], X1 ) ’ c([X1 , X2 ], X0 ) .

If c = δb for some b ∈ C 1 , then

c(X0 , X1 ) = (δb)(X0 , X1 ) = ’b([X0 , X1 ] ).

Theorem 4.21 If H 1 (g; R) = H2 (g, R) = 0, then any symplectic G-action is

hamiltonian.

Proof. Let ψ : G ’ Sympl(M, ω) be a symplectic action of G on a symplectic

manifold (M, ω). Since

H 1 (g; R) = 0 ⇐’ [g, g] = g

and since commutators of symplectic vector ¬elds are hamiltonian, we have

dψ : g = [g, g] ’’ χham (M ).

The action ψ is hamiltonian if and only if there is a Lie algebra homomorphism

µ— : g ’ C ∞ (M ) such that the following diagram commutes.

E C ∞ (M ) E χham (M )

R

d

s

d

d

?d dψ

d

g

We ¬rst take an arbitrary vector space lift „ : g ’ C ∞ (M ) making the diagram

commute, i.e., for each basis vector X ∈ g, we choose

„ (X) = „ X ∈ C ∞ (M ) such that v(„ X ) = dψ(X) .

67

4.9. EXISTENCE AND UNIQUENESS OF MOMENT MAPS

The map X ’ „ X may not be a Lie algebra homomorphism. By construction,

„ [X,Y ] is a hamiltonian function for [X, Y ]# , and (as computed in Section 4.3)

{„ X , „ Y } is a hamiltonian function for ’[X # , Y # ]. Since [X, Y ]# = ’[X #, Y # ],

the corresponding hamiltonian functions must di¬er by a constant:

„ [X,Y ] ’ {„ X , „ Y } = c(X, Y ) ∈ R .

By the Jacobi identity, δc = 0. Since H 2 (g; R) = 0, there is b ∈ g— satisfying

c = δb, c(X, Y ) = ’b([X, Y ]). We de¬ne

µ— : g ’’ C ∞ (M )

X ’’ µ— (X) = „ X + b(X) = µX .

Now µ— is a Lie algebra homomorphism:

µ— ([X, Y ]) = „ [X,Y ] + b([X, Y ]) = {„ X , „ Y } = {µX , µY } .

So when is H 1 (g; R) = H2 (g; R) = 0?

A compact Lie group G is semisimple if g = [g, g].

Examples. The unitary group U(n) is not semisimple because the multiples of

the identity, S 1 · Id, form a nontrivial center; at the level of the Lie algebra, this

corresponds to the 1-dimensional subspace R · Id of constant matrices which are

not commutators since they are not traceless.

Any direct product of the other compact classical groups SU(n), SO(n) and

Sp(n) is semisimple (n > 1). Any commutative Lie group is not semisimple. ™¦

Theorem 4.22 (Whitehead Lemmas) Let G be a compact Lie group.

H 1 (g; R) = H2 (g; R) = 0 .

⇐’

G is semisimple

A proof can be found in [30, pages 93-95].

Corollary 4.23 If G is semisimple, then any symplectic G-action is hamiltonian.

As for the question of uniqueness, let G be a compact Lie group.

Theorem 4.24 If H 1 (g; R) = 0, then moment maps for hamiltonian G-actions

are unique.

Proof. Suppose that µ— and µ— are two comoment maps for an action ψ:

1 2

E χham (M )

C ∞ (M )

d

s

d µ—

d2

µ— d dψ

1

d

g

68 LECTURE 4. HAMILTONIAN FIELDS

For each X ∈ g, µX and µX are both hamiltonian functions for X # , thus µX ’

1 2 1

µX = c(X) is locally constant. This de¬nes c ∈ g— , X ’ c(X).

2

Since µ— , µ— are Lie algebra homomorphisms, we have c([X, Y ]) = 0, ∀X, Y ∈

1 2

g, i.e., c ∈ [g, g]0 = {0}. Hence, µ— = µ— .

1 2

Corollary of this proof. In general, if µ : M ’ g— is a moment map, then

given any c ∈ [g, g]0 , µ1 = µ + c is another moment map.

In other words, moment maps are unique up to elements of the dual of the

Lie algebra which annihilate the commutator ideal.

The two extreme cases are:

G semisimple: any symplectic action is hamiltonian ,

moment maps are unique .

G commutative: symplectic actions may not be hamiltonian ,

moment maps are unique up to any constant c ∈ g— .

Example. The circle action on (T 2 , ω = dθ1 § dθ2 ) by rotations in the θ1 direction

‚

has vector ¬eld X # = ‚θ1 ; this is a symplectic action but is not hamiltonian. ™¦

Lecture 5

Symplectic Reduction

The phase space of a system of n particles is the space parametrizing the position

and momenta of the particles. The mathematical model for the phase space is a

symplectic manifold. Classical physicists realized that, whenever there is a sym-

metry group of dimension k acting on a mechanical system, then the number of

degrees of freedom for the position and momenta of the particles may be reduced

by 2k. Symplectic reduction formulates this feature mathematically.

5.1 Marsden-Weinstein-Meyer Theorem

i

dzi § d¯i = dxi § dyi = ri dri § dθi be the standard symplectic

Let ω = 2 z

1

form on C . Consider the following S -action on (Cn , ω):

n

t ∈ S 1 ’’ ψt = multiplication by eit .

The action ψ is hamiltonian with moment map

µ : Cn ’’ R

2

’’ ’ |z| + constant

z 2

since

1 2

dµ = ’ 2 d( ri )

‚ ‚ ‚

X# = + +... +

‚θ1 ‚θ2 ‚θn

1 2

=’ ri dri = ’ 2

±X # ω dri .

If we choose the constant to be 1 , then µ’1 (0) = S 2n’1 is the unit sphere. The

2

orbit space of the zero level of the moment map is

µ’1 (0)/S 1 = S 2n’1 /S 1 = C Pn’1

,

69

70 LECTURE 5. SYMPLECTIC REDUCTION

which is thus called a reduced space. This is a particular observation of the

major theorem Marsden-Weinstein-Meyer which shows that reduced spaces are

symplectic manifolds.

Theorem 5.1 (Marsden-Weinstein-Meyer [34, 36]) Let (M, ω, G, µ) be a

hamiltonian G-space for a compact Lie group G. Let i : µ’1 (0) ’ M be the

inclusion map. Assume that G acts freely on µ’1 (0). Then

• the orbit space Mred = µ’1 (0)/G is a manifold,

• π : µ’1 (0) ’ Mred is a principal G-bundle, and

• there is a symplectic form ωred on Mred satisfying i— ω = π — ωred .

De¬nition 5.2 The pair (Mred , ωred ) is the reduction of (M, ω) with respect to

G, µ, or the reduced space, or the symplectic quotient, or the Marsden-

Weinstein-Meyer quotient, etc.

Low-brow proof for the case G = S 1 and dim M = 4.

In this case the moment map is µ : M ’ R. Let p ∈ µ’1 (0). Choose local

coordinates:

• θ along the orbit through p,

• µ given by the moment map, and

• ·1 , ·2 pullback of coordinates on µ’1 (0)/S 1 .

Then the symplectic form can be written

ω = A dθ § dµ + Bj dθ § d·j + Cj dµ § d·j + D d·1 § d·2 .

‚

Since dµ = ± ω, we must have A = 1, Bj = 0. Hence,

‚θ

ω = dθ § dµ + Cj dµ § d·j + D d·1 § d·2 .

Since ω is symplectic, we must have D = 0. Therefore, i— ω = D d·1 § d·2 is the

pullback of a symplectic form on Mred .

Examples.

1. For the natural action of U(k) on Ck—n with moment map computed in

Section 4.7, we have µ’1 (0) = {A ∈ Ck—n | AA— = Id}. Then the quotient

manifold

µ’1 (0)/U(k) = G (k, n)

is the grassmannian of k-planes in Cn .

71

5.2. INGREDIENTS

2. Consider the S 1 -action on (R2n+2 , ω0 ) which, under the usual identi¬cation

of R2n+2 with Cn+1 , corresponds to multiplication by eit . This action is

hamiltonian with a moment map µ : Cn+1 ’ R given by

1 1

µ(z) = ’ 2 |z|2 + .

2

Then the reduction µ’1 (0)/S 1 is C P with the Fubini-Study symplectic form

n

ωred = ωFS . To prove this assertion, let pr : Cn+1 \ {0} ’ C P denote the

n

standard projection, and check that

i¯

pr— ωFS = 2 ‚ ‚ log(|z|2 ) .

This form has the same restriction to S 2n+1 as ωred .

™¦

Exercise 27

The natural actions of T n+1 and U(n + 1) on (C Pn , ωFS ) are hamiltonian, and

¬nd formulas for their moment maps.

Hint: Previous example and Section 4.7.

5.2 Ingredients

The actual proof of the Marsden-Weinstein-Meyer theorem requires the following

ingredients.

1. Let gp be the Lie algebra of the stabilizer of p ∈ M . Then dµp : Tp M ’ g—

has

ker dµp = (Tp Op )ωp

im dµp = g0 p

where Op is the G-orbit through p, and g0 = {ξ ∈ g— | ξ, X = 0, ∀X ∈ gp }

p

is the annihilator of gp .

#

Proof. Stare at the expression ωp (Xp , v) = dµp (v), X , for all v ∈ Tp M

and all X ∈ g, and count dimensions.

Consequences:

• The action is locally free at p

⇐’ gp = {0}

⇐’ dµp is surjective

⇐’ p is a regular point of µ.

72 LECTURE 5. SYMPLECTIC REDUCTION

• G acts freely on µ’1 (0)

=’ 0 is a regular value of µ

=’ µ’1 (0) is a closed submanifold of M

of codimension equal to dim G.

• G acts freely on µ’1 (0)

=’ Tp µ’1 (0) = ker dµp (for p ∈ µ’1 (0))

=’ Tp µ’1 (0) and Tp Op are symplectic orthocomplements in Tp M .

In particular, the tangent space to the orbit through p ∈ µ’1 (0) is an

isotropic subspace of Tp M . Hence, orbits in µ’1 (0) are isotropic.

Since any tangent vector to the orbit is the value of a vector ¬eld generated

by the group, we can con¬rm that orbits are isotropic directly by computing,

for any X, Y ∈ g and any p ∈ µ’1 (0),

ωp (Xp , Yp# ) = hamiltonian function for [Y # , X # ] at p

#

= hamiltonian function for [Y, X]# at p

= µ[Y,X] (p) = 0 .

2. Lemma 5.3 Let (V, ω) be a symplectic vector space. Suppose that I is an

isotropic subspace, that is, ω|I ≡ 0. Then ω induces a canonical symplectic

form „¦ on I ω /I.

Proof. Let u, v ∈ I ω , and [u], [v] ∈ I ω /I. De¬ne „¦([u], [v]) = ω(u, v).

• „¦ is well-de¬ned:

∀i, j ∈ I .

ω(u + i, v + j) = ω(u, v) + ω(u, j) + ω(i, v) + ω(i, j) ,

0 0 0

• „¦ is nondegenerate:

Suppose that u ∈ I ω has ω(u, v) = 0, for all v ∈ I ω .

Then u ∈ (I ω )ω = I, i.e., [u] = 0.

3. Proposition 5.4 If a compact Lie group G acts freely on a manifold M ,

then M/G is a manifold and the map π : M ’ M/G is a principal G-

bundle.

Proof. We will ¬rst show that, for any p ∈ M , the G-orbit through p is a

compact embedded submanifold of M di¬eomorphic to G.

Since the action is smooth, the evaluation map ev : G — M ’ M , ev(g, p) =

g · p, is smooth. Let evp : G ’ M be de¬ned by evp (g) = g · p. The map evp

provides the embedding we seek:

73

5.2. INGREDIENTS

The image of evp is the G-orbit through p. Injectivity of evp follows from the

action of G being free. The map evp is proper because, if A is a compact,

hence closed, subset of M , then its inverse image (ev p )’1 (A), being a closed

subset of the compact Lie group G, is also compact. It remains to show that

evp is an immersion. For X ∈ g Te G, we have

#

d(evp )e (X) = 0 ⇐’ Xp = 0 ⇐’ X = 0 ,

as the action is free. We conclude that d(evp )e is injective. At any other point

g ∈ G, for X ∈ Tg G, we have

d(evp )g (X) = 0 ⇐’ d(evp —¦ Rg )e —¦ (dRg’1 )g (X) = 0 ,

where Rg : G ’ G is right multiplication by g. But evp —¦ Rg = evg·p has an

injective di¬erential at e, and (dRg’1 )g is an isomorphism. It follows that

d(evp )g is always injective.

Exercise 28

Show that, even if the action is not free, the G-orbit through p is a compact

embedded submanifold of M . In that case, the orbit is di¬eomorphic to the

quotient of G by the isotropy of p: Op G/Gp .

Let S be a transverse section to Op at p; this is called a slice. Choose a

coordinate system x1 , . . . , xn centered at p such that

Op G : x1 = . . . = xk =0

S : xk+1 = . . . = xn = 0.

Let Sµ = S © Bµ (0, Rn ) where Bµ (0, Rn ) is the ball of radius µ centered at 0

in Rn . Let · : G — S ’ M , ·(g, s) = g · s. Apply the following equivariant

tubular neighborhood theorem.

Theorem 5.5 (Slice Theorem) Let G be a compact Lie group acting

on a manifold M such that G acts freely at p ∈ M . For su¬ciently small

µ, · : G — Sµ ’ M maps G — Sµ di¬eomorphically onto a G-invariant

neighborhood U of the G-orbit through p.

The proof of this slice theorem is sketched further below.

Corollary 5.6 If the action of G is free at p, then the action is free on U.

Corollary 5.7 The set of points where G acts freely is open.

Corollary 5.8 The set G — Sµ U is G-invariant. Hence, the quotient

U/G Sµ is smooth.

74 LECTURE 5. SYMPLECTIC REDUCTION

Conclusion of the proof that M/G is a manifold and π : M ’ M/G is a

smooth ¬ber map.

For p ∈ M , let q = π(p) ∈ M/G. Choose a G-invariant neighborhood U of

p as in the slice theorem: U G — S (where S = Sµ for an appropriate µ).

Then π(U) = U/G =: V is an open neighborhood of q in M/G. By the slice

theorem, S ’ V is a homeomorphism. We will use such neighborhoods V

as charts on M/G. To show that the transition functions associated with

these charts are smooth, consider two G-invariant open sets U1 , U2 in M and

corresponding slices S1 , S2 of the G-action. Then S12 = S1 ©U2 , S21 = S2 ©U1

are both slices for the G-action on U1 © U2 . To compute the transition map

S12 ’ S21 , consider the diagram

’’ id — S12 ’ G — S12

S12

U1 © U 2 .

’’ id — S21 ’ G — S21

S21

Then the composition

pr

S12 ’ U1 © U2 ’’ G — S21 ’’ S21

is smooth.

Finally, we need to show that π : M ’ M/G is a smooth ¬ber map. For

p ∈ M , q = π(p), choose a G-invariant neighborhood U of the G-orbit

through p of the form · : G—S ’ U. Then V = U/G S is the corresponding

neighborhood of q in M/G:

·

M⊇ U G—S G—V

“π “

M/G ⊇ V V

=

Since the projection on the right is smooth, π is smooth.

Exercise 29

Check that the transition functions for the bundle de¬ned by π are smooth.

Sketch for the proof of the slice theorem. We need to show that, for

µ su¬ciently small, · : G — Sµ ’ U is a di¬eomorphism where U ⊆ M is a

G-invariant neighborhood of the G-orbit through p. Show that:

(a) d·(id,p) is bijective.

75

5.3. PROOF OF THE REDUCTION THEOREM

(b) Let G act on G — S by the product of its left action on G and trivial

action on S. Then · : G — S ’ M is G-equivariant.

(c) d· is bijective at all points of G — {p}. This follows from (a) and (b).

(d) The set G — {p} is compact, and · : G — S ’ M is injective on G — {p}

with d· bijective at all these points. By the implicit function theorem,

there is a neighborhood U0 of G — {p} in G — S such that · maps U0

di¬eomorphically onto a neighborhood U of the G-orbit through p.

(e) The sets G — Sµ , varying µ, form a neighborhood base for G — {p} in

G — S. So in (d) we may take U0 = G — Sµ .

5.3 Proof of the Reduction Theorem

Since

G acts freely on µ’1 (0) =’ dµp is surjective for all p ∈ µ’1 (0)

=’ 0 is a regular value

=’ µ’1 (0) is a submanifold of codimension = dim G

for the ¬rst two parts of the Marsden-Weinstein-Meyer theorem it is enough to

apply the third ingredient from Section 5.2 to the free action of G on µ’1 (0).

At p ∈ µ’1 (0) the tangent space to the orbit Tp Op is an isotropic subspace

of the symplectic vector space (Tp M, ωp ), i.e., Tp Op ⊆ (Tp Op )ω .

(Tp Op )ω = ker dµp = Tp µ’1 (0) .

The lemma (second ingredient) gives a canonical symplectic structure on the

quotient Tp µ’1 (0)/Tp Op . The point [p] ∈ Mred = µ’1 (0)/G has tangent space

T[p] Mred Tp µ’1 (0)/Tp Op . Thus the lemma de¬nes a nondegenerate 2-form ωred

on Mred . This is well-de¬ned because ω is G-invariant.

By construction i— ω = π — ωred where

i

µ’1 (0) ’M

“π

Mred

Hence, π — dωred = dπ — ωred = d±— ω = ±— dω = 0. The closedness of ωred follows from

the injectivity of π — .

Remark. Suppose that another Lie group H acts on (M, ω) in a hamiltonian way

with moment map φ : M ’ h— . If the H-action commutes with the G-action, and

if φ is G-invariant, then Mred inherits a hamiltonian action of H, with moment

map φred : Mred ’ h— satisfying φred —¦ π = φ —¦ i. ™¦

76 LECTURE 5. SYMPLECTIC REDUCTION

5.4 Elementary Theory of Reduction

Finding a symmetry for a 2n-dimensional mechanical problem may reduce it to

a (2n ’ 2)-dimensional problem as follows: an integral of motion f for a 2n-

dimensional hamiltonian system (M, ω, H) may enable us understand the tra-

jectories of this system in terms of the trajectories of a (2n ’ 2)-dimensional

hamiltonian system (Mred , ωred , Hred ). To make this precise, we will describe this

process locally. Suppose that U is an open set in M with Darboux coordinates

x1 , . . . , xn , ξ1 , . . . , ξn such that f = ξn for this chart, and write H in these coordi-

nates: H = H(x1 , . . . , xn , ξ1 , . . . , ξn ). Then

±

the trajectories of vH lie on the

hyperplane ξ = constant

n

ξn is an integral of motion =’ ‚H

{ξn , H} = 0 = ’ ‚xn

=’ H = H(x1 , . . . , xn’1 , ξ1 , . . . , ξn ) .

If we set ξn = c, the motion of the system on this hyperplane is described by

the following Hamilton equations:

±

dx1 ‚H

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚ξ1

.

.

.

dxn’1 ‚H

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚ξn’1

dξ

‚H

1

’

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚x1

.

.

.

dξn’1 ‚H

=’ (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚xn’1

dxn ‚H

=

dt ‚ξn

dξn ‚H

=’ =0.

dt ‚xn

The reduced phase space is

Ured = {(x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 ) ∈ R2n’2 |

(x1 , . . . , xn’1 , a, ξ1 , . . . , ξn’1 , c) ∈ U for some a} .

The reduced hamiltonian is

Hred : Ured ’’ R ,

Hred (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 ) = H(x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c) .

77

5.5. REDUCTION FOR PRODUCT GROUPS

In order to ¬nd the trajectories of the original system on the hypersurface

ξn = c, we look for the trajectories

x1 (t), . . . , xn’1 (t), ξ1 (t), . . . , ξn’1 (t)

of the reduced system on Ured . We integrate the equation

dxn ‚H

(t) = (x1 (t), . . . , xn’1 (t), ξ1 (t), . . . , ξn’1 (t), c)

dt ‚ξn

to obtain the original trajectories

t ‚H

xn (t) = xn (0) + 0 ‚ξn (. . .)dt

ξn (t) = c.

5.5 Reduction for Product Groups

Let G1 and G2 be compact connected Lie groups and let G = G1 — G2 . Then

g— = g— • g— .

g = g 1 • g2 and 1 2

Suppose that (M, ω, G, ψ) is a hamiltonian G-space with moment map

ψ : M ’’ g— • g— .

1 2

Write ψ = (ψ1 , ψ2 ) where ψi : M ’ g— for i = 1, 2. The fact that ψ is equivariant

i

implies that ψ1 is invariant under G2 and ψ2 is invariant under G1 . Now reduce

(M, ω) with respect to the G1 -action. Let

’1

Z1 = ψ1 (0) .

Assume that G1 acts freely on Z1 . Let M1 = Z1 /G1 be the reduced space and let ω1

be the corresponding reduced symplectic form. The action of G2 on Z1 commutes

with the G1 -action. Since G2 preserves ω, it follows that G2 acts symplectically

on (M1 , ω1 ). Since G1 preserves ψ2 , G1 also preserves ψ2 —¦ ι1 : Z1 ’ g— , where

2

p1

ι1 : Z1 ’ M is inclusion. Thus ψ2 —¦ ι is constant on ¬bers of Z1 ’ M1 . We

conclude that there exists a smooth map µ2 : M1 ’ g— such that µ2 —¦ p = ψ2 —¦ i.

2

Exercise 30

Show that:

(a) the map µ2 is a moment map for the action of G 2 on (M1 , ω1 ), and

(b) if G acts freely on ψ ’1 (0, 0), then G2 acts freely on µ’1 (0), and there

2

is a natural symplectomorphism

µ’1 (0)/G2 ψ ’1 (0, 0)/G .

2

This technique of performing reduction with respect to one factor of a product

group at a time is called reduction in stages. It may be extended to reduction

by a normal subgroup H ‚ G and by the corresponding quotient group G/H.

78 LECTURE 5. SYMPLECTIC REDUCTION

5.6 Reduction at Other Levels

Suppose that a compact Lie group G acts on a symplectic manifold (M, ω) in a

hamiltonian way with moment map µ : M ’ g— . Let ξ ∈ g— .

To reduce at the level ξ of µ, we need µ’1 (ξ) to be preserved by G, or else

take the G-orbit of µ’1 (ξ), or else take the quotient by the maximal subgroup of

G which preserves µ’1 (ξ).

Since µ is equivariant,

G preserves µ’1 (ξ) ⇐’ G preserves ξ

Ad— ξ = ξ, ∀g ∈ G .

⇐’ g

Of course the level 0 is always preserved. Also, when G is a torus, any level

is preserved and reduction at ξ for the moment map µ, is equivalent to reduction

at 0 for a shifted moment map φ : M ’ g— , φ(p) := µ(p) ’ ξ.

Let O be a coadjoint orbit in g— equipped with the canonical symplectic

form (also know as the Kostant-Kirillov symplectic form or the Lie-Poisson

symplectic form) ω O de¬ned in Section 4.3. Let O ’ be the orbit O equipped with

’ωO . The natural product action of G on M — O ’ is hamiltonian with moment

map µO (p, ξ) = µ(p) ’ ξ. If the Marsden-Weinstein-Meyer hypothesis is satis¬ed

for M — O’ , then one obtains a reduced space with respect to the coadjoint

orbit O.

5.7 Orbifolds

Example. Let G = T n be an n-torus. For any ξ ∈ (tn )— , µ’1 (ξ) is preserved by

the Tn -action. Suppose that ξ is a regular value of µ. (By Sard™s theorem, the

singular values of µ form a set of measure zero.) Then µ’1 (ξ) is a submanifold of

codimension n. Note that

dµp is surjective at all p ∈ µ’1 (ξ)

ξ regular =’

gp = 0 for all p ∈ µ’1 (ξ)

=’

the stabilizers on µ’1 (ξ) are ¬nite

=’

µ’1 (ξ)/G is an orbifold [38, 39] .

=’

Let Gp be the stabilizer of p. By the slice theorem (Theorem 5.5), µ’1 (ξ)/G

is modeled by S/Gp , where S is a Gp -invariant disk in µ’1 (ξ) through p and

transverse to Op . Hence, locally µ’1 (ξ)/G looks indeed like Rn divided by a ¬nite

™¦

group action.

Example. Consider the S 1 -action on C2 given by eiθ · (z1 , z2 ) = (eikθ z1 , eiθ z2 ) for

some ¬xed integer k ≥ 2. This is hamiltonian with moment map

C2 ’’ R

µ:

1

(z1 , z2 ) ’’ ’ 2 (k|z1 |2 + |z2 |2 ) .

79

5.8. SYMPLECTIC TORIC MANIFOLDS

Any ξ < 0 is a regular value and µ’1 (ξ) is a 3-dimensional ellipsoid. The stabilizer

2π

of (z1 , z2 ) ∈ µ’1 (ξ) is {1} if z2 = 0, and is Z k = ei | = 0, 1, . . . , k ’ 1 if

k

z2 = 0. The reduced space µ’1 (ξ)/S 1 is called a teardrop orbifold or conehead;

it has one cone (also known as a dunce cap) singularity of type k (with cone

angle 2π ). ™¦

k

Example. Let S 1 act on C2 by eiθ · (z1 , z2 ) = (eikθ z1 , ei θ z2 ) for some integers

k, ≥ 2. Suppose that k and are relatively prime. Then

(z1 , 0) has stabilizer Z k (for z1 = 0) ,

(0, z2 ) has stabilizer Z (for z2 = 0) ,

(z1 , z2 ) has stabilizer {1} (for z1 , z2 = 0) .

The quotient µ’1 (ξ)/S 1 is called a football orbifold. It has two cone singularities,

™¦

one of type k and another of type .

Example. More generally, the reduced spaces of S 1 acting on Cn by

eiθ · (z1 , . . . , zn ) = (eik1 θ z1 , . . . , eikn θ zn ) ,

™¦

are called weighted (or twisted) projective spaces.

5.8 Symplectic Toric Manifolds

De¬nition 5.9 A symplectic toric manifold is a compact connected symplectic

manifold (M, ω) equipped with an e¬ective hamiltonian action of a torus T of

dimension equal to half the dimension of the manifold,

1

dim T = dim M ,

2

and with a choice of a corresponding moment map µ.

Exercise 31

Show that an e¬ective hamiltonian action of a torus T n on a 2n-dimensional

symplectic manifold gives rise to an integrable system.

Hint: The coordinates of the moment map are commuting integrals of motion.

De¬nition 5.10 Two symplectic toric manifolds, (Mi , ωi , Ti , µi ), i = 1, 2, are

equivalent if there exists an isomorphism » : T 1 ’ T2 and a »-equivariant

symplectomorphism • : M1 ’ M2 such that µ1 = µ2 —¦ •.

80 LECTURE 5. SYMPLECTIC REDUCTION

Equivalent symplectic toric manifolds are often undistinguished.

Examples of symplectic toric manifolds.

1. The circle S 1 acts on the 2-sphere (S 2 , ωstandard = dθ § dh) by rotations

eiν · (θ, h) = (θ + ν, h)

with moment map µ = h equal to the height function and moment polytope

[’1, 1].

Equivalently, the circle S 1 acts on P1 = C2 ’ 0/ ∼ with the Fubini-Study

1

form ωFS = 4 ωstandard , by eiθ · [z0 : z1 ] = [z0 : eiθ z1 ]. This is hamiltonian

2

with moment map µ[z0 : z1 ] = ’ 2 · |z0 ||z+|z1 |2 , and moment polytope ’ 2 , 0 .

1|

1 1

2

t t1

'$

µ=h

E