25.2 Connection and Curvature Forms

A connection on a principal bundle P may be equivalently described in terms

of 1-forms.

De¬nition 25.3 A connection form on a principal bundle P is a Lie-algebra-

valued 1-form

k

∈ „¦1 (P ) — g

Ai — X i

A=

i=1

such that:

157

25.2 Connection and Curvature Forms

(a) A is G-invariant, with respect to the product action of G on „¦1 (P ) (in-

duced by the action on P ) and on g (the adjoint representation), and

(b) A is vertical, in the sense that ±X # A = X for any X ∈ g.

Exercise. Show that a connection T P = V • H determines a connection form

A and vice-versa by the formula

H = ker A = {v ∈ T P | ±v A = 0} .

™¦

Given a connection on P , the splitting T P = V • H induces the following

splittings for bundles:

T —P = V — • H—

§2 T — P = (§2 V — ) • (V — § H — ) • (§2 H — )

.

.

.

and for their sections:

„¦1 (P ) = „¦1 (P ) • „¦1 (P )

vert horiz

„¦2 (P ) = „¦2 (P ) • „¦2 (P ) • „¦2 (P )

vert mix horiz

.

.

.

The corresponding connection form A is in „¦1 — g. Its exterior derivative dA

vert

is in

„¦2 (P ) — g = „¦2 • „¦2 • „¦2 horiz — g ,

vert mix

and thus decomposes into three components,

dA = (dA)vert + (dA)mix + (dA)horiz .

Exercise. Check that:

1

ci m A § Am — Xi , where

(a) (dA)vert (X, Y ) = [X, Y ], i.e., (dA)vert = 2

i, ,m

i

the c m ™s are the structure constants of the Lie algebra with respect to

c i m Xi ;

the chosen basis, and de¬ned by [X , Xm ] =

i, ,m

(b) (dA)mix = 0.

™¦

According to the previous exercise, the relevance of dA may come only from

its horizontal component.

De¬nition 25.4 The curvature form of a connection is the horizontal com-

ponent of its connection form. I.e., if A is the connection form, then

∈ „¦2

horiz — g .

curv A = (dA)horiz

De¬nition 25.5 A connection is called ¬‚at if its curvature is zero.

158 25 MOMENT MAP IN GAUGE THEORY

25.3 Symplectic Structure on the Space of Connections

Let P be a principal G-bundle over B. If A is a connection form on P , and if

a ∈ „¦1horiz — g is G-invariant for the product action, then it is easy to check that

A + a is also a connection form on P . Reciprocally, any two connection forms on

P di¬er by an a ∈ („¦1 G

horiz — g) . We conclude that the set A of all connections

on the principal G-bundle P is an a¬ne space modeled on the linear space

a = („¦1 G

horiz — g) .

Now let P be a principal G-bundle over a compact oriented 2-dimensional

riemannian manifold B (for instance, B is a Riemann surface). Suppose that

the group G is compact or semisimple. Atiyah and Bott [7] noticed that the cor-

responding space A of all connections may be treated as an in¬nite-dimensional

symplectic manifold. This will require choosing a G-invariant inner product

·, · on g, which always exists, either by averaging any inner product when G

is compact, or by using the Killing form on semisimple groups.

Since A is an a¬ne space, its tangent space at any point A is identi¬ed with

the model linear space a. With respect to a basis X1 , . . . , Xk for the Lie algebra

g, elements a, b ∈ a14 are written

ai — Xi bi — X i .

a= and b=

If we wedge a and b, and then integrate over B using the riemannian volume,

we obtain a real number:

G

„¦2 (P ) „¦2 (B)

a — a ’’ ’’ R

ω: horiz

(a, b) ’’ a i § b j X i , Xj ’’ a i § b j X i , Xj .

i,j B i,j

We have used that the pullback π — : „¦2 (B) ’ „¦2 (P ) is an isomorphism onto

G

its image „¦2 (P ) .

horiz

Exercise. Show that if w(a, b) = 0 for all b ∈ a, then a must be zero. ™¦

The map ω is nondegenerate, skew-symmetric, bilinear and constant in the

sense that it does not depend on the base point A. Therefore, it has the right to

be called a symplectic form on A, so the pair (A, ω) is an in¬nite-dimensional

symplectic manifold.

25.4 Action of the Gauge Group

Let P be a principal G-bundle over B. A di¬eomorphism f : P ’ P commuting

with the G-action determines a di¬eomorphism fbasic : B ’ B by projection.

14 The choice of symbols is in honor of Atiyah and Bott!

159

25.5 Case of Circle Bundles

De¬nition 25.6 A di¬eomorphism f : P ’ P commuting with the G-action

is a gauge transformation if the induced fbasic is the identity. The gauge

group of P is the group G of all gauge transformations of P .

The derivative of an f ∈ G takes a connection T P = V • H to another

connection T P = V • Hf , and thus induces an action of G in the space A of

all connections. Recall that A has a symplectic form ω. Atiyah and Bott [7]

noticed that the action of G on (A, ω) is hamiltonian, where the moment map

(appropriately interpreted) is the map

G

„¦2 (P ) — g

µ: A ’’

’’ curv A ,

A

i.e., the moment map “is” the curvature! We will describe this construction in

detail for the case of circle bundles in the next section.

Remark. The reduced space at level zero

M = µ’1 (0)/G

is the space of ¬‚at connections modulo gauge equivalence, known as the mod-

uli space of ¬‚at connections. It turns out that M is a ¬nite-dimensional

™¦

symplectic orbifold.

25.5 Case of Circle Bundles

What does the Atiyah-Bott construction of the previous section look like for the

case when G = S 1 ?

EP

S1 ‚

π

c

B

1

Let v be the generator of the S -action on P , corresponding to the basis 1 of

g R. A connection form on P is a usual 1-form A ∈ „¦1 (P ) such that

Lv A = 0 and ±v A = 1 .

If we ¬x one particular connection A0 , then any other connection is of the form

G

A = A0 + a for some a ∈ a = „¦1 (P ) = „¦1 (B). The symplectic form on

horiz

a = „¦1 (B) is simply

ω : a — a ’’ R

(a, b) ’’ a§b .

B

∈„¦2 (B)

160 25 MOMENT MAP IN GAUGE THEORY

The gauge group is G = Maps(B, S 1 ), because a gauge transformation is multi-

plication by some element of S 1 over each point in B:

G ’’ Di¬(P )

ψ:

h : B ’ S1 ψh : P ’ P

’’

p ’ h(π(p)) · p

The Lie algebra of G is

Lie G = Maps(B, R) = C ∞ (B) .

Its dual space is

—

(Lie G) = „¦2 (B) ,

where the duality is provided by integration over B

C ∞ (B) — „¦2 (B) ’’ R

(h, β) ’’ hβ .

B

(it is topological or smooth duality, as opposed to algebraic duality) .

The gauge group acts on the space of all connections by

G ’’ Di¬(A)

’’ (A ’ A ’π — dθ)

h(x) = eiθ(x)

∈a

Exercise. Check the previous assertion about the action on connections.

Hint: First deal with the case where P = S 1 — B is a trivial bundle, in which

case h ∈ G acts on P by

ψh : (t, x) ’’ (t + θ(x), x) ,

and where every connection can be written A = dt + β, with β ∈ „¦1 (B). A

gauge transformation h ∈ G acts on A by

—

A ’’ ψh’1 (A) .

™¦

The in¬nitesimal action of G on A is

dψ : Lie G ’’ χ(A)

’’ X # = vector ¬eld described by the transformation

X

(A ’ A ’dX )

∈„¦1 (B)=a

so that X # = ’dX.

161

25.5 Case of Circle Bundles

Finally, we will check that

µ : A ’’ (Lie G)— = „¦2 (B)

A ’’ curv A

is indeed a moment map for the action of the gauge group on A.

Exercise. Check that in this case:

G

„¦2 (P ) = „¦2 (B) ,

∈

(a) curv A = dA horiz

(b) µ is G-invariant.

™¦

The previous exercise takes care of the equivariance condition, since the

action of G on „¦2 (B) is trivial.

Take any X ∈ Lie G = C ∞ (B). We need to check that

dµX (a) = ω(X # , a) , ∀a ∈ „¦1 (B) . ()

As for the left-hand side of ( ), the map µX ,

µX : A ’’ R

A ’’ X · dA ,

X , dA =

B

∈C ∞ (B) ∈„¦2 (B)

is linear in A. Consequently,

dµX : a ’’ R

a ’’ X · da .

B

As for the right-hand side of ( ), by de¬nition of ω, we have

ω(X # , a) = X# · a = ’ dX · a .

B B

But, by Stokes theorem,, the last integral is

’ dX · a = X · da ,

B B

so we are done in proving that µ is the moment map.

Homework 19: Examples of Moment Maps

1. 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— . Prove that

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 .

2. 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. Show that this action is hamiltonian with

moment map µ : Cn ’ (tn )— Rn given by

1

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

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. Show

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

with moment map the inclusion map:

µ : G · ξ ’ g— .

4. 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

µ(z) = 2 zz — ,

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

(A, B) = trace(A— B).

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

Hint:

h ’k

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

k h

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

1 1

µX (z) = i z — Xz i trace(zz — X)

= .

2 2

Check that µ is equivariant.

162

163

HOMEWORK 19

5. 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 choice of the constant is for convenience in Homework 20.)

2i

Exercises 1 and 4.

Hint:

2

6. Consider the U(n)-action by conjugation on the space (Cn , ω0 ) of complex

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

µ(A) = 2 [A, A— ] .

i

Previous exercise and its “transpose” version.

Hint:

26 Existence and Uniqueness of Moment Maps

26.1 Lie Algebras of Vector Fields

Let (M, ω) be a symplectic manifold and v ∈ χ(M ) a vector ¬eld on M .

⇐’

v is symplectic ±v ω is closed ,

⇐’

v is hamiltonian ±v ω is exact .

The spaces

χsympl (M ) = symplectic vector ¬elds on M ,

χham (M ) = hamiltonian vector ¬elds on M .

are Lie algebras for the Lie bracket of vector ¬elds. C ∞ (M ) is a Lie algebra for

the Poisson bracket, {f, g} = ω(vf , vg ). H 1 (M ; R) and R are regarded as Lie

algebras for the trivial bracket. We have two exact sequences of Lie algebras:

0 ’’ χham (M ) ’ χsympl (M ) ’’ H 1 (M ; R) ’’ 0

v ’’ [±v ω]

C ∞ (M ) ’’ χham (M )

0 ’’ ’ ’’ 0

R

f ’’ vf .

In particular, if H 1 (M ; R) = 0, then χham (M ) = χsympl (M ).

Let G be a connected Lie group. A symplectic action ψ : G ’ Sympl(M, ω)

induces an in¬nitesimal action

g ’’ χsympl (M )

dψ :

X ’’ X # = vector ¬eld generated by the

one-parameter group {exp tX(e) | t ∈ R} .

™¦

Exercise. Check that the map dψ is a Lie algebra anti-homomorphism.

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

morphism µ— : g ’ C ∞ (M ) lifting dψ, i.e., making the following diagram

commute.

E χsympl (M )

C ∞ (M )

d

s

d

d

µ— d dψ

d

g

—

The map µ is then called a comoment map (de¬ned in Lecture 22).

Existence of µ— ⇐’ Existence of µ

comoment map moment map

←’

Lie algebra homomorphism equivariance

164

165

26.2 Lie Algebra Cohomology

26.2 Lie Algebra Cohomology

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. 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 26.1 If g is the Lie algebra of a compact connected Lie group G,

then

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

k

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 H 2 (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 ] ).

166 26 EXISTENCE AND UNIQUENESS OF MOMENT MAPS

26.3 Existence of Moment Maps

Theorem 26.2 If H 1 (g; R) = H 2 (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) .

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 Lecture 16)

{„ 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) = H 2 (g; R) = 0?

167

26.4 Uniqueness of Moment Maps

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

H 1 (g; R) = H 2 (g; R) = 0 .

⇐’

G is semisimple

A proof can be found in [66, pages 93-95].

Corollary 26.4 If G is semisimple, then any symplectic G-action is hamilto-

nian.

26.4 Uniqueness of Moment Maps

Let G be a compact Lie group.

Theorem 26.5 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

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

1 2

µ1 ’ µ2 = c(X) is locally constant. This de¬nes c ∈ g— , X ’ c(X).

X X

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— .

168 26 EXISTENCE AND UNIQUENESS OF MOMENT MAPS

Example. The circle action on (T2 , ω = dθ1 § dθ2 ) by rotations in the θ1

‚

direction has vector ¬eld X # = ‚θ1 ; this is a symplectic action but is not

™¦

hamiltonian.

Homework 20: Examples of Reduction

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

exercise 5 of Homework 19, we have µ’1 (0) = {A ∈ Ck—n | AA— = Id}.

Show that the quotient

µ’1 (0)/U(k) = G(k, n)

is the grassmannian of k-planes in Cn .

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

tion 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

µ(z) = ’ 1 |z|2 + 1

.

2 2

Prove that the reduction µ’1 (0)/S 1 is CPn with the Fubini-Study sym-

plectic form ωred = ωFS .

Let pr : Cn+1 \ {0} ’ CPn denote the standard projection. Check

Hint:

that

i¯

pr— ωFS = ‚ ‚ log(|z|2 ) .

2

Prove that this form has the same restriction to S 2n+1 as ωred .

3. Show that the natural actions of Tn+1 and U(n + 1) on (CPn , ωFS ) are

hamiltonian, and ¬nd formulas for their moment maps.

Previous exercise and exercises 2 and 4 of Homework 19.

Hint:

169

27 Convexity

27.1 Convexity Theorem

From now on, we will concentrate on actions of a torus G = Tm = Rm /Zm .

Theorem 27.1 (Atiyah [6], Guillemin-Sternberg [56])

Let (M, ω) be a compact connected symplectic manifold, and let Tm be an

m-torus. Suppose that ψ : Tm ’ Sympl(M, ω) is a hamiltonian action with

moment map µ : M ’ Rm . Then:

1. the levels of µ are connected;

2. the image of µ is convex;

3. the image of µ is the convex hull of the images of the ¬xed points of the

action.

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

Proof. This proof (due to Atiyah) involves induction over m = dim Tm . Con-

sider the statements:

Am : “the levels of µ are connected, for any Tm -action;”

Bm : “the image of µ is convex, for any Tm -action.”

Then

⇐’

(1) Am holds for all m ,

⇐’

(2) Bm holds for all m .

• A1 is a non-trivial result in Morse theory.

• Am’1 =’ Am (induction step) is in Homework 21.

• B1 is trivial because in R connectedness is convexity.

• Am’1 =’ Bm is proved below.

Choose an injective matrix A ∈ Zm—(m’1) . Consider the action of an (m’1)-

subtorus

ψA : Tm’1 ’’ Sympl(M, ω)

θ ’’ ψAθ .

Exercise. The action ψA is hamiltonian with moment map µA = At µ : M ’

Rm’1 . ™¦

Given any p0 ∈ µ’1 (ξ),

A

p ∈ µ’1 (ξ) ⇐’ At µ(p) = ξ = At µ(p0 )

A

170

171

27.2 E¬ective Actions

so that

µ’1 (ξ) = {p ∈ M | µ(p) ’ µ(p0 ) ∈ ker At } .

A

By the ¬rst part (statement Am’1 ), µ’1 (ξ) is connected. Therefore, if we

A

’1

connect p0 to p1 by a path pt in µA (ξ), we obtain a path µ(pt ) ’ µ(p0 ) in

ker At . But ker At is 1-dimensional. Hence, µ(pt ) must go through any convex

combination of µ(p0 ) and µ(p1 ), which shows that any point on the line segment

from µ(p0 ) to µ(p1 ) must be in µ(M ):

(1 ’ t)µ(p0 ) + tµ(p1 ) ∈ µ(M ) , 0¤t¤1.

Any p0 , p1 ∈ M can be approximated arbitrarily closely by points p0 and p1

with µ(p1 ) ’ µ(p0 ) ∈ ker At for some injective matrix A ∈ Zm—(m’1) . Taking

limits p0 ’ p0 , p1 ’ p1 , we obtain that µ(M ) is convex.15

To prove part 3, consider the ¬xed point set C of ψ. Homework 21 shows

that C is a ¬nite union of connected symplectic submanifolds, C = C1 ∪. . .∪CN .

The moment map is constant on each Cj , µ(Cj ) = ·j ∈ Rm , j = 1, . . . , N . By

the second part, the convex hull of {·1 , . . . , ·N } is contained in µ(M ).

For the converse, suppose that ξ ∈ Rm and ξ ∈ convex hull of {·1 , . . . , ·N }.

/

m

Choose X ∈ R with rationally independent components and satisfying

ξ, X > ·j , X , for all j .

By the irrationality of X, the set {exp tX(e) | t ∈ R} is dense in Tm , hence the

zeros of the vector ¬eld X # on M are the ¬xed points of the Tm -action. Since

µX = µ, X attains its maximum on one of the sets Cj , this implies

ξ, X > sup µ(p), X ,

p∈M

hence ξ ∈ µ(M ). Therefore,

/

µ(M ) = convex hull of {·1 , . . . , ·N } .

27.2 E¬ective Actions

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,

Gp = {e} ,

p∈M

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

Corollary 27.2 Under the conditions of the convexity theorem, if the Tm -

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

15 Clearly µ(M ) is closed because it is compact.

172 27 CONVEXITY

Proof. If the Tm -action is e¬ective, there must be a point p where 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 convex polytope.

Any nondegenerate convex polytope in Rm must have at least m + 1 vertices.

The vertices of µ(M ) are images of ¬xed points.

Theorem 27.3 Let (M, ω, Tm , µ) be a hamiltonian Tm -space. If the Tm -action

is e¬ective, then dim M ≥ 2m.

Proof. Fact: If ψ : Tm ’ Di¬(M ) is an e¬ective action, then it has orbits of

dimension m; a proof may be found in [17].

On an m-dimensional orbit O, the moment map µ(O) = ξ is constant. For

p ∈ O, the exterior derivative

dµp : Tp M ’’ g—

maps Tp O to 0. Thus

Tp O ⊆ ker dµp = (Tp O)ω ,

which shows that orbits O of a hamiltonian torus action are always isotropic

submanifolds of M . In particular, dim O = m ¤ 1 dim M .

2

De¬nition 27.4 A (symplectic) toric manifold16 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. Show that an e¬ective hamiltonian action of a torus Tn on a 2n-

dimensional symplectic manifold gives rise to an integrable system.

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

™¦

27.3 Examples

1. The circle S1 acts on the 2-sphere (S 2 , ωstandard = dθ § dh) by rota-

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

polytope [’1, 1].

16 In these notes, a toric manifold is always a symplectic toric manifold.

173

27.3 Examples

t t1

'$

µ=h

E

t t’1

&%

1.™ The circle S1 acts on CP1 = C2 ’ 0/ ∼ with the Fubini-Study form

ωFS = 1 ωstandard , by eiθ · [z0 , z1 ] = [z0 , eiθ z1 ]. This is hamiltonian with

4

|z1 |2

1 1

moment map µ[z0 , z1 ] = ’ 2 · and moment polytope ’ 2 , 0 .

|z0 |2 +|z1 |2 ,

2. The T2 -action on CP2 by

(eiθ1 , eiθ2 ) · [z0 , z1 , z2 ] = [z0 , eiθ1 z1 , eiθ2 z2 ]

has moment map

|z1 |2 |z2 |2

1

µ[z0 , z1 , z2 ] = ’ , .

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

2

The ¬xed points get mapped as

[1, 0, 0] ’’ (0, 0)

1

[0, 1, 0] ’’ ’ 2 , 0

1

[0, 0, 1] ’’ 0, ’ 2

Notice that the stabilizer of a preimage of the edges is S 1 , while the action

is free at preimages of interior points of the moment polytope.

174 27 CONVEXITY

T

1

(’ 2 , 0)

t(0, 0) E

t

d

d

d

d

d

dt 1

(0, ’ 2 )

Exercise. What is the moment polytope for the T3 -action on CP3 as

(eiθ1 , eiθ2 , eiθ3 ) · [z0 , z1 , z2 , z3 ] = [z0 , eiθ1 z1 , eiθ2 z2 , eiθ3 z3 ] ?

™¦

Exercise. What is the moment polytope for the T2 -action on CP1 — CP1 as

(eiθ , ei· ) · ([z0 , z1 ], [w0 , w1 ]) = ([z0 , eiθ z1 ], [w0 , ei· w1 ]) ?

™¦

Homework 21: Connectedness

Consider a hamiltonian action ψ : Tm ’ Sympl (M, ω), θ ’ ψθ , of an m-

dimensional torus on a 2n-dimensional compact connected symplectic manifold

(M, ω). If we identify the Lie algebra of Tm with Rm by viewing Tm = Rm /Zm ,

and we identify the Lie algebra with its dual via the standard inner product,

then the moment map for ψ is µ : M ’ Rm .

1. Show that there exists a compatible almost complex structure J on (M, ω)

— —

which is invariant under the Tm -action, that is, ψθ J = Jψθ , for all θ ∈ Tm .

Hint: We cannot average almost complex structures, but we can average

riemannian metrics (why?). Given a riemannian metric g0 on M , its Tm -

—

average g = Tm ψθ g0 dθ is Tm -invariant.

2. Show that, for any subgroup G ⊆ Tm , the ¬xed-point set for G ,

Fix (G) = Fix (ψθ ) ,

θ∈G

is a symplectic submanifold of M .

For each p ∈ Fix (G) and each θ ∈ G, the di¬erential of ψθ at p,

Hint:

dψθ (p) : Tp M ’’ Tp M ,

preserves the complex structure Jp on Tp M . Consider the exponential map

expp : Tp M ’ M with respect to the invariant riemannian metric g(·, ·) =

ω(·, J·). Show that, by uniqueness of geodesics, exp p is equivariant, i.e.,

expp (dψθ (p)v) = ψθ (exp p v)

for any θ ∈ G, v ∈ Tp M . Conclude that the ¬xed points of ψθ near p correspond

to the ¬xed points of dψθ (p) on Tp M , that is

Tp Fix (G) = ker(Id ’ dψθ (p)) .

θ∈G

Since dψθ (p) —¦ Jp = Jp —¦ dψθ (p), the eigenspace with eigenvalue 1 is invariant

under Jp , and is therefore a symplectic subspace.

3. A smooth function f : M ’ R on a compact riemannian manifold M

is called a Morse-Bott function if its critical set Crit (f ) = {p ∈

M | df (p) = 0} is a submanifold of M and for every p ∈ Crit (f ), Tp Crit (f ) =

ker 2 f (p) where 2 f (p) : Tp M ’ Tp M denotes the linear operator ob-

tained from the hessian via the riemannian metric. This is the natural

generalization of the notion of Morse function to the case where the crit-

ical set is not just isolated points. If f is a Morse-Bott function, then

Crit (f ) decomposes into ¬nitely many connected critical manifolds C.

The tangent space Tp M at p ∈ C decomposes as a direct sum

+ ’

Tp M = T p C • E p • E p

+ ’

where Ep and Ep are spanned by the positive and negative eigenspaces of

f (p). The index of a connected critical submanifold C is n’ = dim Ep ,

2 ’

C

for any p ∈ C, whereas the coindex of C is n+ = dim Ep . +

C

175

176 HOMEWORK 21

For each X ∈ Rm , let µX = µ, X : M ’ R be the component of µ along

X. Show that µX is a Morse-Bott function with even-dimensional critical

manifolds of even index. Moreover, show that the critical set

Crit (µX ) = Fix (ψθ )

θ∈TX

is a symplectic manifold, where TX is the closure of the subgroup of Tm

generated by X.

Hint: Assume ¬rst that X has components independent over Q, so that TX =

Tm and Crit (µX ) = Fix (Tm ). Apply exercise 2. To prove that Tp Crit (µX ) =

ker 2 µX (p), show that ker 2 µX (p) = ©θ∈Tm ker(Id ’ dψθ (p)). To see

this, notice that the 1-parameter group of matrices (dψexp tX )p coincides with

exp(tvp ), where vp = ’Jp 2 µX (p) : Tp M ’ Tp M is a vector ¬eld on Tp M .

The kernel of 2 µX (p) corresponds to the ¬xed points of dψtX (p), and since X

has rationally independent components, these are the common ¬xed points of

all dψθ (p), θ ∈ Tm . The eigenspaces of 2 µX (p) are even-dimensional because

they are invariant under Jp .

4. The moment map µ = (µ1 , . . . , µm ) is called e¬ective if the 1-forms

dµ1 , . . . , dµm of its components are linearly independent. Show that, if µ

is not e¬ective, then the action reduces to that of an (m ’ 1)-subtorus.

Hint: If µ is not e¬ective, then the function µX = µ, X is constant for

some nonzero X ∈ Rm . Show that we can neglect the direction of X.

5. Prove that the level set µ’1 (ξ) is connected for every regular value ξ ∈ Rm .

Hint: Prove by induction over m = dim Tm . For the case m = 1, use the

lemma that all level sets f ’1 (c) of a Morse-Bott function f : M ’ R on a com-

pact manifold M are necessarily connected, if the critical manifolds all have

index and coindex = 1 (see [82, p.178-179]). For the induction step, you can as-

sume that ψ is e¬ective. Then, for every 0 = X ∈ Rm , the function µX : M ’

R is not constant. Show that C := ∪X=0 Crit µX = ∪0=X∈Zm Crit µX where

each Crit µX is an even-dimensional proper submanifold, so the complement

M \ C must be dense in M . Show that M \ C is open. Hence, by continuity, to

show that µ’1 (ξ) is connected for every regular value ξ = (ξ1 , . . . , ξm ) ∈ Rm , it

su¬ces to show that µ’1 (ξ) is connected whenever (ξ1 , . . . , ξm’1 ) is a regular

value for a reduced moment map (µ1 , . . . , µm’1 ). By the induction hypoth-

esis, the manifold Q = ©j=1 µ’1 (ξj ) is connected whenever (ξ1 , . . . , ξm’1 )

m’1

j

is a regular value for (µ1 , . . . , µm’1 ). It su¬ces to show that the function

µm : Q ’ R has only critical manifolds of even index and coindex (see [82,

p.183]), because then, by the lemma, the level sets µ’1 (ξ) = Q © µ’1 (ξm ) are

m

connected for every ξm .

Part XI

Symplectic Toric Manifolds

Native to algebraic geometry, toric manifolds have been studied by symplec-

tic geometers as examples of extremely symmetric hamiltonian spaces, and as

guinea pigs for new theorems. Delzant showed that symplectic toric manifolds

are classi¬ed (as hamiltonian spaces) by a set of special polytopes.

28 Classi¬cation of Symplectic Toric Manifolds

28.1 Delzant Polytopes

A 2n-dimensional (symplectic) toric manifold is a compact connected sym-

plectic manifold (M 2n , ω) equipped with an e¬ective hamiltonian action of an

n-torus Tn and with a corresponding moment map µ : M ’ Rn .

De¬nition 28.1 A Delzant polytope ∆ in Rn is a convex polytope satisfying:

• it is simple, i.e., there are n edges meeting at each vertex;

• it is rational, i.e., the edges meeting at the vertex p are rational in the

sense that each edge is of the form p + tui , 0 ¤ t < ∞, where ui ∈ Zn ;

• it is smooth, i.e., these u1 , . . . , un can be chosen to be a basis of Zn .

Remark. The Delzant polytopes are the simple rational smooth polytopes.

These are closely related to the Newton polytopes (which are the nonsingular

n-valent polytopes), except that the vertices of a Newton polytope are required

™¦

to lie on the integer lattice and for a Delzant polytope they are not.

Examples of Delzant polytopes:

d

d

d

d

¢d

d ¢ d

d ¢ d

d ¢ d

d ¢

¢

177

178 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

The dotted vertical line in the trapezoildal example means nothing, except that

it™s a picture of a rectangle plus an isosceles triangle. For “taller” triangles,

smoothness would be violated. “Wider” triangles (with integral slope) may

still be Delzant. The family of the Delzant trapezoids of this type, starting

with the rectangle, correspond, under the Delzant construction, to Hirzebruch

surfaces; see Homework 22.

Examples of polytopes which are not Delzant:

„

d

r

rr „d

rr „d

rr „d

r „

The picture on the left fails the smoothness condition, whereas the picture

on the right fails the simplicity condition.

Algebraic description of Delzant polytopes:

A facet of a polytope is a (n ’ 1)-dimensional face.

Let ∆ be a Delzant polytope with n = dim ∆ and d = number of facets.

A lattice vector v ∈ Zn is primitive if it cannot be written as v = ku with

u ∈ Zn , k ∈ Z and |k| > 1; for instance, (1, 1), (4, 3), (1, 0) are primitive, but

(2, 2), (4, 6) are not.

Let vi ∈ Zn , i = 1, . . . , d, be the primitive outward-pointing normal vectors

to the facets.

r

¡ rr !

¡

¡ rr ¡ n=2

r

rr ¡ rr ¡

r¡ r¡ d=3

r

¡ rr

¡ rr

¡ rr

¡ r

c

Then we can describe ∆ as an intersection of halfspaces

∆ = {x ∈ (Rn )— | x, vi ¤ »i , i = 1, . . . , d} for some »i ∈ R .

179

28.2 Delzant Theorem

Example. For the picture below, we have

∆ = {x ∈ (R2 )— | x1 ≥ 0, x2 ≥ 0, x1 + x2 ¤ 1}

= {x ∈ (R2 )— | x, (’1, 0) ¤ 0 , x, (0, ’1) ¤ 0 , x, (1, 1) ¤ 1} .

(0, 1)

r

d

d

d

' d

d

d

d

dr

r

(0, 0) (1, 0)

c

™¦

28.2 Delzant Theorem

We do not have a classi¬cation of symplectic manifolds, but we do have a clas-

si¬cation of toric manifolds in terms of combinatorial data. This is the content

of the Delzant theorem.

Theorem 28.2 (Delzant [23]) Toric manifolds are classi¬ed by Delzant

polytopes. More speci¬cally, there is the following one-to-one correspondence

1’1

{toric manifolds} ’’ {Delzant polytopes}

(M 2n , ω, Tn , µ) ’’ µ(M ).

We will prove the existence part (or surjectivity) in the Delzant theorem

following [53]. Given a Delzant polytope, what is the corresponding toric man-

ifold?

?

(M∆ , ω∆ , Tn , µ) ←’ ∆n

180 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

28.3 Sketch of Delzant Construction

Let ∆ be a Delzant polytope with d facets. Let vi ∈ Zn , i = 1, . . . , d, be the

primitive outward-pointing normal vectors to the facets. For some »i ∈ R,

∆ = {x ∈ (Rn )— | x, vi ¤ »i , i = 1, . . . , d} .

Let e1 = (1, 0, . . . , 0), . . . , ed = (0, . . . , 0, 1) be the standard basis of Rd . Con-

sider

π : Rd ’’ Rn

ei ’’ vi .

Claim. The map π is onto and maps Zd onto Zn .

Proof. The set {e1 , . . . , ed } is a basis of Zd . The set {v1 , . . . , vd } spans Zn for

the following reason. At a vertex p, the edge vectors u1 , . . . , un ∈ (Rn )— , form a

basis for (Zn )— which, without loss of generality, we may assume is the standard

basis. Then the corresponding primitive normal vectors to the facets meeting at

p are symmetric (in the sense of multiplication by ’1) to the ui ™s, hence form

a basis of Zn .

Therefore, π induces a surjective map, still called π, between tori:

π

Rd /Zd ’’ Rn /Zn

Td Tn

’’ ’’ 0 .

Let

(N is a Lie subgroup of Td )

N = kernel of π

n = Lie algebra of N

Rd = Lie algebra of Td

Rn = Lie algebra of Tn .

The exact sequence of tori

i π

0 ’’ N ’’ Td ’’ Tn ’’ 0

induces an exact sequence of Lie algebras

i π

0 ’’ n ’’ Rd ’’ Rn ’’ 0

with dual exact sequence

π— i—

0 ’’ (Rn )— ’’ (Rd )— ’’ n— ’’ 0 .

i

Now consider Cd with symplectic form ω0 = dzk § d¯k , and standard

z

2

hamiltonian action of Td

(e2πit1 , . . . , e2πitd ) · (z1 , . . . , zd ) = (e2πit1 z1 , . . . , e2πitd zd ) .

181

28.3 Sketch of Delzant Construction

The moment map is φ : Cd ’’ (Rd )—

φ(z1 , . . . , zd ) = ’π(|z1 |2 , . . . , |zd |2 ) + constant ,

where we choose the constant to be (»1 , . . . , »d ). What is the moment map for

the action restricted to the subgroup N ?

Exercise. Let G be any compact Lie group and H a closed subgroup of G,

with g and h the respective Lie algebras. The inclusion i : h ’ g is dual to the

projection i— : g— ’ h— . Suppose that (M, ω, G, φ) is a hamiltonian G-space.

Show that the restriction of the G-action to H is hamiltonian with moment map

i— —¦ φ : M ’’ h— .

™¦

The subtorus N acts on Cd in a hamiltonian way with moment map

i— —¦ φ : Cd ’’ n— .

Let Z = (i— —¦ φ)’1 (0) be the zero-level set.

Claim. The set Z is compact and N acts freely on Z.

This claim will be proved in the next lecture.

By the ¬rst claim, 0 ∈ n— is a regular value of i— —¦ φ. Hence, Z is a compact

submanifold of Cd of dimension

dimR Z = 2d ’ (d ’ n) = d + n .

dim n—

The orbit space M∆ = Z/N is a compact manifold of dimension

dimR M∆ = d + n ’ (d ’ n) = 2n .

dim N

The point-orbit map p : Z ’ M∆ is a principal N -bundle over M∆ .

Consider the diagram

j

Cd

’

Z

p“

M∆

where j : Z ’ Cd is inclusion. The Marsden-Weinstein-Meyer theorem guaran-

tees the existence of a symplectic form ω∆ on M∆ satisfying

p — ω∆ = j — ω0 .

Exercise. Work out all details in the following simple example.

Let ∆ = [0, a] ‚ R— (n = 1, d = 2). Let v(= 1) be the standard basis vector

in R. Then

∆ : x, v1 ¤ 0 v1 = ’v

x, v2 ¤ a v2 = v .

182 28 CLASSIFICATION OF SYMPLECTIC TORIC MANIFOLDS

The projection

π

R2 ’’ R

’’ ’v

e1

’’ v

e2

has kernel equal to the span of (e1 + e2 ), so that N is the diagonal subgroup of

T2 = S 1 — S 1 . The exact sequences become

i π

T2 S1

0 ’’ ’’ ’’ ’’ 0

N

— —

π i

0 ’’ R— (R2 )— n—

’’ ’’ ’’ 0

’’

(x1 , x2 ) x 1 + x2 .

The action of the diagonal subgroup N = {(e2πit , e2πit ) ∈ S 1 — S 1 } on C2 ,

(e2πit , e2πit ) · (z1 , z2 ) = (e2πit z1 , e2πit z2 ) ,

has moment map

(i— —¦ φ)(z1 , z2 ) = ’π(|z1 |2 + |z2 |2 ) + a ,

with zero-level set

a

(i— —¦ φ)’1 (0) = {(z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 = }.

π

Hence, the reduced space is

(i— —¦ φ)’1 (0)/N = CP1 projective space!

™¦

29 Delzant Construction

29.1 Algebraic Set-Up

Let ∆ be a Delzant polytope with d facets. We can write ∆ as

∆ = {x ∈ (Rn )— | x, vi ¤ »i , i = 1, . . . , d} ,

for some »i ∈ R. Recall the exact sequences from the previous lecture

i π

’’ Td ’’ Tn

0 ’’ N ’’ 0

i π

’’ Rd ’’ Rn

0 ’’ n ’’ 0

’’ vi

ei

and the dual sequence

π— i—

0 ’’ (Rn )— ’’ (Rd )— ’’ n— ’’ 0 .

The standard hamiltonian action of Td on Cd

(e2πit1 , . . . , e2πitd ) · (z1 , . . . , zd ) = (e2πit1 z1 , . . . , e2πitd zd )

has moment map φ : Cd ’ (Rd )— given by

φ(z1 , . . . , zd ) = ’π(|z1 |2 , . . . , |zd |2 ) + (»1 , . . . , »d ) .

The restriction of this action to N has moment map

i— —¦ φ : Cd ’’ n— .

29.2 The Zero-Level

Let Z = (i— —¦ φ)’1 (0).

Theorem 29.1 The level Z is compact and N acts freely on Z.

Proof. Let ∆ be the image of ∆ by π — . We will show that φ(Z) = ∆ . Since

φ is a proper map and ∆ is compact, it will follow that Z is compact.

Lemma 29.2 Let y ∈ (Rd )— . Then:

y∈∆ ⇐’ y is in the image of Z by φ .

Proof of the lemma. The value y is in the image of Z by φ if and only if

both of the following conditions hold:

1. y is in the image of φ;

183

184 29 DELZANT CONSTRUCTION

2. i— y = 0.

Using the expression for φ and the third exact sequence, we see that these

conditions are equivalent to:

1. y, ei ¤ »i for i = 1, . . . , d.

2. y = π — (x) for some x ∈ (Rn )— .

Suppose that the second condition holds, so that y = π — (x). Then

π — (x), ei ¤ »i , ∀i

y, ei ¤ »i , ∀i ⇐’

⇐’ x, π(ei ) ¤ »i , ∀i

⇐’ x ∈ ∆.

Thus, y ∈ φ(z) ⇐’ y ∈ π — (∆) = ∆ .

Hence, we have a surjective proper map φ : Z ’ ∆ . Since ∆ is compact,

we conclude that Z is compact. It remains to show that N acts freely on Z.

We de¬ne a strati¬cation of Z with three equivalent descriptions:

• De¬ne a strati¬cation on ∆ whose ith stratum is the closure of the union

of the i-dimensional faces of ∆ . Pull this strati¬cation back to Z by φ.

We can obtain a more explicit description of the strati¬cation on Z:

• Let F be a face of ∆ with dim F = n ’ r. Then F is characterized (as a

subset of ∆ ) by r equations

y, ei = »i , i = i 1 , . . . , ir .

We write F = FI where I = (i1 , . . . , ir ) has 1 ¤ i1 < i2 . . . < ir ¤ d.

Let z = (z1 , . . . , zd ) ∈ Z.

z ∈ φ’1 (FI ) ⇐’ φ(z) ∈ FI