3Hamiltonian vector fields
III Symplectic Geometry
3.6 The convexity theorem
We focus on the case
G
=
T
n
. It turns out the image of the moment map
µ
has
a very rigid structure.
Theorem
(Convexity theorem (Atiyah, Guillemin–Sternberg))
.
Let (
M, ω
) be
a compact connected symplectic manifold, and
µ : M → R
n
a moment map for a Hamiltonian torus action. Then
(i) The levels µ
−1
(c) are connected for all c
(ii) The image µ(M) is convex.
(iii) The image µ(M) is in fact the convex hull of µ(M
G
).
We call µ(M) the moment polytope.
Here we identify
G
∼
=
T
n
' R
n
/Z
n
, which gives us an identification
g
∼
=
R
n
and g
∗
∼
=
(R
n
)
∗
∼
=
R
n
.
Example.
Consider (
M
=
CP
n
, ω
F S
).
T
n
acts by letting
t
= (
t
1
, . . . , t
n
)
∈
T
n
∼
=
U(1)
n
send
ψ
t
([z
0
: ···z
n
]) = [z
0
: t
1
z
1
: ··· : t
n
z
n
].
This has moment map
µ([z
0
: ···z
n
]) = −
1
2
(|z
1
|
2
, . . . , |z
n
|
2
)
|z
0
|
2
+ ··· + |z
n
|
2
.
The image of this map is
µ(M) =
x ∈ R
n
: x
k
≤ 0, x
1
+ ··· + x
n
≥ −
1
2
.
For example, in the CP
2
case, the moment image lives in R
2
, and is just
−
1
2
−
1
2
The three vertices are µ([0 : 0 : 1]), µ([0 : 1 : 0]) and µ([1 : 0 : 0]).
We now want to prove the convexity theorem. We first look at (iii). While it
seems like a very strong claim, it actually follows formally from (ii).
Lemma. (ii) implies (iii).
Proof.
Suppose the fixed point set of the action has
k
connected components
Z
=
Z
1
∪···∪Z
k
. Then
µ
is constant on each
Z
j
, since
X
#
|
Z
j
= 0 for all
X
. Let
µ
(
Z
j
) =
η
j
∈ R
n
. By (ii), we know the convex hull of
{η
1
, . . . η
k
}
is contained in
µ(M).
To see that
µ
(
M
) is exactly the convex hull, observe that if
X ∈ R
n
has
rationally independent components, so that
X
topologically generates
T
, then
p
is fixed by
T
iff
X
#
p
= 0, iff d
µ
X
p
= 0. Thus,
µ
X
attains its maximum on one of
the Z
j
.
Now if
ξ
is not in the convex hull of
{η
j
}
, then we can pick an
X ∈ R
n
with
rationally independent components such that
hξ, Xi > hη
j
, Xi
for all
j
, since
the space of such X is open and non-empty. Then
hξ, Xi > sup
p∈
S
Z
j
hµ(p), Xi = sup
p∈M
hµ(p), Xi.
So ξ 6∈ µ(M).
With a bit more work, (i) also implies (ii).
Lemma. (i) implies (ii).
Proof.
The case
n
= 1 is immediate, since
µ
(
M
) is compact and connected,
hence a closed interval.
In general, to show that
µ
(
M
) is convex, we want to show that the intersection
of
µ
(
M
) with any line is connected. In other words, if
π
:
R
n+1
→ R
n
is any
projection and ν = π ◦ µ, then
π
−1
(c) ∩ µ(M) = µ(ν
−1
(c))
is connected. This would follow if we knew
ν
−1
(
c
) were connected, which would
follow from (i) if
ν
were a moment map of an
T
n
action. Unfortunately, most of
the time, it is just the moment map of an
R
n
action. For it to come from a
T
n
action, we need π to be represented by an integer matrix. Then
T = {π
T
t : t ∈ T
n
= R
n
/Z
n
} ⊆ T
n+1
is a subtorus, and one readily checks that
ν
is the moment map for the
T
action.
Now for any
p
0
, p
1
∈ M
, we can find
p
0
0
, p
0
1
arbitrarily close to
p
0
, p
1
and
a line of the form
π
−1
(
c
) with
π
integral. Then the line between
p
0
0
and
p
0
1
is contained in
µ
(
M
) by the above argument, and we are done since
µ
(
M
) is
compact, hence closed.
It thus remains to prove (i), where we have to put in some genuine work.
This requires Morse–Bott theory.
Let M be a manifold, dim M = m, and f : M → R a smooth map. Let
Crit(f) = {p ∈ M : df
p
= 0}
be the set of critical points. For
p ∈ Crit
(
f
) and (
U, x
1
, . . . , x
m
) a coordinate
chart around p, we have a Hessian matrix
H
p
f =
∂
2
f
∂x
i
∂x
j
Definition
(Morse(-Bott) function)
. f
is a Morse function if at each
p ∈ Crit
(
f
),
H
p
f is non-degenerate.
f
is a Morse–Bott function if the connected components of
Crit
(
f
) are
submanifolds and for all p ∈ Crit(p), T
p
Crit(f) = ker(H
p
f).
If
f
is Morse, then the critical points are isolated. If
f
is Morse–Bott, then
the Hessian is non-degenerate in the normal bundle to Crit(f).
If
M
is compact, then there is a finite number of connected components of
Crit(f). So we have
Crit(f) = Z
1
∪ ··· ∪ Z
k
,
and the Z
i
are called the critical submanifolds.
For p ∈ Z
i
, the Hessian H
p
f is a quadratic form and we can write
T
p
M = E
−
p
⊕ T
p
Z
i
⊕ E
+
p
,
where
E
±
p
are the positive and negative eigenspaces of
H
p
f
respectively. Note
that
dim E
±
p
are locally constant, hence constant along
Z
i
. So we can define the
index of Z
i
to be dim E
−
p
, and the coindex to be dim E
+
p
.
We can then define a vector bundle
E
−
→ Z
i
, called the negative normal
bundle.
Morse theory tells us how the topology of
M
−
c
=
{p ∈ M
:
f
(
p
)
≤ c}
changes
with c ∈ R.
Theorem (Morse theory).
(i)
If
f
−1
([
c
1
, c
2
]) does not contain any critical point. Then
f
−1
(
c
1
)
∼
=
f
−1
(
c
2
)
and M
c
1
∼
=
M
c
2
(where
∼
=
means diffeomorphic).
(ii)
If
f
−1
([
c
1
, c
2
]) contains one critical manifold
Z
, then
M
−
c
2
' M
−
c
1
∪D
(
E
−
),
where D(E
−
) is the disk bundle of E
−
.
In particular, if
Z
is an isolated point,
M
−
c
2
is, up to homotopy equivalence,
obtained by adding a dim E
−
p
-cell to M
−
c
1
.
The key lemma in this proof is the following result:
Lemma.
Let
M
be a compact connected manifold, and
f
:
M → R
a Morse–Bott
function with no critical submanifold of index or coindex 1. Then
(i) f has a unique local maximum and local minimum
(ii) All level sets of f
−1
(c) are connected.
Proof sketch.
There is always a global minimum since
f
is compact. If there is
another local minimum at c, then the disk bundle is trivial, and so in
M
−
c+ε
' M
−
c−ε
∪ D(E
−
)
for
ε
small enough, the union is a disjoint union. So
M
c+ε
has two components.
Different connected components can only merge by crossing a level of index 1,
so this cannot happen. To handle the maxima, consider −f .
More generally, the same argument shows that a change in connectedness
must happen by passing through a index or coindex 1 critical submanifold.
To apply this to prove the convexity theorem, we will show that for any
X
,
µ
X
is a Morse–Bott function where all the critical submanifolds are symplectic.
In particular, they are of even index and coindex.
Lemma.
For any
X ∈ R
n
,
µ
X
is a Morse–Bott function where all critical
submanifolds are symplectic.
Proof sketch.
Note that
p
is a fixed point iff
X
#
p
= 0 iff d
µ
X
p
= 0 iff
p
is a critical
point. So the critical points are exactly the fixed points of X.
(
T
p
M, ω
p
) models (
M, ω
) in a neighbourhood of
p
by Darboux theorem. Near
a fixed point
T
n
, an equivariant version of the Darboux theorem tells us there is
a coordinate chart U where (M, ω, µ) looks like
ω|
U
=
X
dx
i
∧ dy
i
µ|
U
= µ(p) −
1
2
X
i
(x
2
i
+ y
2
i
)α
i
,
where α
i
∈ Z are weights.
Then the critical submanifolds of µ are given by
{x
i
= y
i
= 0 : α
i
6= 0},
which is locally a symplectic manifold and has even index and coindex.
Finally, we can prove the theorem.
Lemma. (i) holds.
Proof.
The
n
= 1 case follows from the previous lemmas. We then induct on
n
.
Suppose the theorem holds for
n
, and let
µ
= (
µ
1
, . . . , µ
n
) :
M → R
n+1
be
a moment map for a Hamiltonian
T
n+1
-action. We want to show that for all
c = (c
1
, . . . , c
n
) ∈ R
n+1
, the set
µ
−1
(c) = µ
−1
1
(c
1
) ∩ ··· ∩ µ
−1
n+1
(c
n+1
)
is connected.
The idea is to set
N = µ
−1
1
(c
1
) ∩ ···µ
−1
1
(c
n
),
and then show that
µ
n+1
|
N
:
N → R
is a Morse–Bott function with no critical
submanifolds of index or coindex 1.
We may assume that d
µ
1
, . . . ,
d
µ
n
are linearly independent, or equivalently,
dµ
X
6= 0 for all X ∈ R
n
. Otherwise, this reduces to the case of an n-torus.
To make sense of
N
, we must pick
c
to be a regular value. Density arguments
imply that
C =
[
X6=0
Crit(µ
X
) =
[
X∈Z
n+1
\{0}
Crit µ
X
.
Since
Crit µ
X
is a union of codimension
≥
2 submanifolds, its complement is
dense. Hence by the Baire category theorem,
C
has dense complement. Then a
continuity argument shows that we only have to consider the case when
c
is a
regular value of µ, hence N is a genuine submanifold of codimension n.
By the induction hypothesis,
N
is connected. We now show that
µ
n+1
|
N
:
N → R is Morse–Bott with no critical submanifolds of index or coindex 1.
Let
x
be a critical point. Then the theory of Lagrange multipliers tells us
there are some λ
i
∈ R such that
"
dµ
n+1
+
n
X
n=1
λ
i
dµ
i
#
x
= 0
Thus, µ is critical in M for the function
µ
Y
= µ
n+1
+
n
X
i=1
λ
i
µ
i
,
where
Y
= (
λ
1
, . . . , λ
n
,
1)
∈ R
n+1
. So by the claim,
µ
Y
is Morse–Bott with only
even indices and coindices. Let
W
be a critical submanifold of
µ
Y
containing
x
.
Claim. W intersects N transversely at x.
If this were true, then
µ
Y
|
N
has
W ∩ N
as a non-degenerate critical sub-
manifold of even index and coindex, since the coindex doesn’t change and
W
is even-dimensional. Moreover, when restricted to
N
,
P
λ
i
µ
i
is a constant. So
µ
n+1
|
N
satisfies the same properties.
To prove the claim, note that
T
x
N = ker dµ
1
|
x
∩ ··· ∩ ker dµ
n
|
x
.
With a moments thought, we see that it suffices to show that d
µ
1
, . . . ,
d
µ
n
remain linearly independent when restricted to
T
x
W
. Now observe that the
Hamiltonian vector fields
X
#
1
|
x
, . . . , X
#
n
|
x
are independent since d
µ
1
|
x
, . . .
d
µ
n
|
x
are, and they live in T
x
W since their flows preserve W .
Since
W
is symplectic (by the claim), for all
k
= (
k
1
, . . . k
n
), there exists
v ∈ T
x
W such that
ω
X
k
i
X
#
i
|
x
, v
6= 0.
In other words,
X
k
i
dµ
i
(v) 6= 0.
It is natural to seek a non-abelian generalization of this, and it indeed
exists. Let (
M, ω
) be a symplectic manifold, and
G
a compact Lie group with
a Hamiltonian action on
M
with moment map
µ
:
M → g
∗
. From Lie group
theory, there is a maximal torus
T ⊆ G
with Lie algebra
t
, and the Weyl group
W = N(T)/T is finite (where N(T ) is the normalizer of T ).
Then under the coadjoint action, we have
g
∗
/G
∼
=
t
∗
/W,
Pick a Weyl chamber t
∗
+
of t
∗
, i.e. a fundamental domain of t
∗
under W . Then
µ
induces a moment map
µ
+
:
M → t
∗
+
, and the non-abelian convexity theorem
says
Theorem (Kirwan, 1984). µ
+
(M) ⊆ t
∗
+
is a convex polytope.
We shall end with an application of the convexity theorem to linear algebra.
Let
λ
= (
λ
1
, λ
2
)
∈ R
2
and
λ
1
≥ λ
2
, and
H
2
λ
the set of all 2
×
2 Hermitian
matrices with eigenvalues (
λ
1
, λ
2
). What can the diagonal entries of
A ∈ H
2
λ
be?
We can definitely solve this by brute force, since any entry in H looks like
A =
a z
¯z b
where a, b ∈ R and z ∈ C. We know
tr a = a + b = λ
1
+ λ
2
det a = ab − |z|
2
= λ
1
λ
2
.
The first implies
b
=
λ
1
+
λ
2
− a
, and all the second condition gives is that
ab > λ
1
λ
2
.
a
b
This completely determines the geometry of
H
2
λ
, since for each allowed value
of
a
, there is a unique value of
b
, which in turn determines the norm of
z
.
Topologically, this is a sphere, since there is a
S
1
’s worth of choices of
z
except
at the two end points ab = λ
1
λ
2
.
What has this got to do with Hamiltonian actions? Observe that U(2) acts
transitively on H
2
λ
by conjugation, and
T
2
=
e
iθ
1
0
0 e
iθ
2
⊆ U(2).
This contains a copy of S
1
given by
S
1
=
e
iθ
1
0
0 e
−iθ
1
⊆ T
2
.
We can check that
e
iθ
0
0 e
−iθ
a z
¯z b
e
iθ
0
0 e
−iθ
−1
=
a e
iθ
z
e
iθ
z b
Thus, if ϕ : H
2
λ
→ R
2
is the map that selects the diagonal elements, then
ϕ
−1
(a, b) =
a |z|e
iθ
|z|e
−iθ
b
is one
S
1
-orbit. This is reminiscent of the
S
1
action of
S
2
quotienting out to a
line segment.
We can think more generally about H
n
λ
, the n × n Hermitian matrices with
eigenvalues λ
1
≥ ··· ≥ λ
n
, and ask what the diagonal elements can be.
Take
ϕ
:
H
n
λ
→ R
n
be the map that selects the diagonal entries. Then the
image lies on the plane
tr A
=
P
λ
i
. This certainly contain the
n
! points whose
coordinates are all possible permutation of the
λ
i
, given by the diagonal matrices.
Theorem
(Schur–Horn theorem)
. ϕ
(
H
n
λ
) is the convex hull of the
n
! points
from the diagonal matrices.
To view this from a symplectic perspective, let
M
=
H
n
λ
, and U(
n
) acts
transitively by conjugation. For A ⊆ M, let G
A
be the stabilizer of A. Then
H
n
λ
= U(n)/G
A
.
Thus,
T
A
H
n
λ
∼
=
iH
n
g
A
where
H
n
is the Hermitian matrices. The point of this is to define a symplectic
form. We define
ω
A
: iH
n
× iH
n
→ R
(X, Y ) 7→ i tr([X, Y ]A) = i tr(X(Y A − AY ))
So
ker ω
A
= {Y : [A, Y ] = 0} = g
A
.
So
ω
A
induces a non-degenerate form on
T
A
H
n
λ
. In fact, this gives a symplectic
form on H
n
λ
.
Let
T
n
⊆
U(
n
) be the maximal torus consisting of diagonal entries. We can
show that the
T
n
action is Hamiltonian with moment map
ϕ
. Since
T
n
fixes
exactly the diagonal matrices, we are done.