3Hamiltonian vector fields
III Symplectic Geometry
3.5 Symplectic reduction
Given a Lie group action of
G
on
M
, it is natural to ask what the “quotient” of
M
looks like. What we will study is not quite the quotient, but a symplectic
reduction, which is a natural, well-behaved subspace of the quotient that is in
particular symplectic.
We first introduce some words. Let ψ : G → Diff(M) be a smooth action.
Definition (Orbit). If p ∈ M, the orbit of p under G is
O
p
= {ψ
g
(p) : g ∈ G}.
Definition
(Stabilizer)
.
The stabilizer or isotropy group of
p ∈ M
is the closed
subgroup
G
p
= {g ∈ G : ψ
g
(p) = p}.
We write g
p
for the Lie algebra of G
p
.
Definition (Transitive action). We say G acts transitively if M is one orbit.
Definition (Free action). We say G acts freely if G
p
= {e} for all p.
Definition
(Locally free action)
.
We say
G
acts locally freely if
g
p
=
{
0
}
, i.e.
G
p
is discrete.
Definition
(Orbit space)
.
The orbit space is
M/G
, and we write
π
:
M → M/G
for the orbit projection. We equip M/G with the quotient topology.
The main theorem is the following:
Theorem
(Marsden–Weinstein, Meyer)
.
Let
G
be a compact Lie group, and
(
M, ω
) a symplectic manifold with a Hamiltonian
G
-action with moment map
µ
:
M → g
∗
. Write
i
:
µ
−1
(0)
→ M
for the inclusion. Suppose
G
acts freely on
µ
−1
(0). Then
(i) M
red
= µ
−1
(0)/G is a manifold;
(ii) π : µ
−1
(0) → M
red
is a principal G-bundle; and
(iii) There exists a symplectic form ω
red
on M
red
such that i
∗
ω = π
∗
ω
red
.
Definition
(Symplectic quotient)
.
We call
M
red
the symplectic quotient/reduced
space/symplectic reduction of (
M, ω
) with respect to the given
G
-action and
moment map.
What happens if we do reduction at other levels? In other words, what can
we say about µ
−1
(ξ)/G for other ξ ∈ g
∗
?
If we want to make sense of this, we need
ξ
to be preserved under the coadjoint
action of G. This is automatically satisfied if G is abelian, and in this case, we
simply have
µ
−1
(
ξ
) =
ϕ
−1
(0), where
ϕ
=
µ −ξ
is another moment map. So this
is not more general.
If
ξ
is not preserved by
G
, then we can instead consider
µ
−1
(
ξ
)
/G
ξ
, or
equivalently take µ
−1
(O
ξ
)/G. We check that
µ
−1
(ξ)/G
ξ
∼
=
µ
−1
(O
ξ
)/G
∼
=
ϕ
−1
(0)/G,
where
ϕ : M × O
ξ
→ g
∗
(ρ, η) 7→ µ(p) − η
is a moment map for the product action of G on (M × O
ξ
, ω × ω
ξ
).
So in fact there is no loss in generality for considering just µ
−1
(0).
Proof.
We first show that
µ
−1
(0) is a manifold. This follows from the following
claim:
Claim. G acts locally freely at p iff p is a regular point of µ.
We compute the dimension of
im
d
µ
p
using the rank-nullity theorem. We
know dµ
p
v = 0 iff hdµ
p
(v), Xi = 0 for all X ∈ g. We can compute
hdµ
p
(v), Xi = (dµ
X
)
p
(v) = (ι
X
#
p
ω)(v) = ω
p
(X
#
p
, v).
Moreover, the span of the X
#
p
is exactly T
p
O
p
. So
ker dµ
p
= (T
p
O
p
)
ω
.
Thus,
dim(im dµ
p
) = dim O
p
= dim G − dim G
p
.
In particular, dµ
p
is surjective iff G
p
= 0.
Then (i) and (ii) follow from the following theorem:
Theorem.
Let
G
be a compact Lie group and
Z
a manifold, and
G
acts freely
on Z. Then Z/G is a manifold and Z → Z/G is a principal G-bundle.
Note that if
G
does not act freely on
µ
−1
(0), then by Sard’s theorem,
generically,
ξ
is a regular value of
µ
, and so
µ
−1
(
ξ
) is a manifold, and
G
acts
locally freely on
µ
−1
(
ξ
). If
µ
−1
(
ξ
) is preserved by
G
, then
µ
−1
(
ξ
)
/G
is a
symplectic orbifold.
It now remains to construct the symplectic structure. Observe that if
p ∈
µ
−1
(0), then
T
p
O
p
⊆ T
p
µ
−1
(0) = ker dµ
p
= (T
p
O
p
)
ω
.
So
T
p
O
p
is an isotropic subspace of (
T
p
M, ω
). We then observe the following
straightforward linear algebraic result:
Lemma.
Let (
V,
Ω) be a symplectic vector space and
I
an isotropic sub-
space. Then Ω induces a canonical symplectic structure Ω
red
on
I
Ω
/I
, given by
Ω
red
([u], [v]) = Ω(u, v).
Applying this, we get a canonical symplectic structure on
(T
p
O
p
)
ω
T
p
O
p
=
T
p
µ
−1
(0)
T
p
O
p
= T
[p]
M
red
.
This defines
ω
red
on
M
red
, which is well-defined because
ω
is
G
-invariant, and is
smooth by local triviality and canonicity.
It remains to show that dω
red
= 0. By construction, i
∗
ω = π
∗
ω
red
. So
π
∗
(dω
red
) = dπ
∗
ω
red
= di
∗
ω = i
∗
dω = 0
Since π
∗
is injective, we are done.
Example. Take
(M, ω) =
C
n
, ω
0
=
i
2
X
dz
k
∧ d¯z
k
=
X
dx
k
∧ dy
k
=
X
r
k
dr
k
∧ dθ
k
.
We let G = S
1
act by multiplication
e
it
· (z
1
, . . . , z
n
) = (e
it
z
1
, . . . , e
it
z
n
).
This action is Hamiltonian with moment map
µ : C
n
→ R
z 7→ −
|z|
2
2
+
1
2
.
The +
1
2
is useful to make the inverse image of 0 non-degenerate. To check this
is a moment map, we compute
d
1
2
|z|
2
= d
−
1
2
X
r
2
k
= −
X
r
k
dr
k
.
On the other hand, if
X = a ∈ g
∼
=
R,
then we have
X
#
= a
∂
∂θ
1
+ ··· +
∂
∂θ
n
.
So we have
ι
X
#
ω = −a
X
r
k
dr
k
= dµ
X
.
It is also clear that µ is S
1
-invariant. We then have
µ
−1
(0) = {z ∈ C
n
: |z|
2
= 1} = S
2n−1
.
Then we have
M
red
= µ
−1
(0)/S
1
= CP
n−1
.
One can check that this is in fact the Fubini–Study form. So (
CP
n−1
, ω
F S
) is
the symplectic quotient of (
C
n
, ω
0
) with respect to the diagonal
S
1
action and
the moment map µ.
Example. Fix k, ∈ Z relatively prime. Then S
1
acts on C
2
by
e
it
(z
1
, z
2
) = (e
ikt
z
1
, e
i`t
z
2
).
This action is Hamiltonian with moment map
µ : C
2
→ R
(z
1
, z
2
) 7→ −
1
2
(k|z
1
|
2
+ |z
2
|
2
).
There is no level set of µ where the action is free, since
– (z, 0) has stabilizer Z/kZ
– (0, z) has stabilizer Z/Z
– (z
1
, z
2
) has trivial stabilizer if z
1
, z
2
6= 0.
On the other hand, the action is still locally free on
C
2
\ {
(0
,
0)
}
since the
stabilizers are discrete.
The reduced spaces
µ
−1
(
ξ
)
/S
1
for
ξ 6
= 0 are orbifolds, called weighted or
twisted projective spaces.
The final example is an infinite dimensional one by Atiyah and Bott. We
will not be able to prove the result in full detail, or any detail at all, but we
will build up to the statement. The summary of the result is that performing
symplectic reduction on the space of all connections gives the moduli space of
flat connections.
Let
G → P
π
→ B
be a principal
G
-bundle, and
ψ
:
G → Diff
(
P
) the associated
free action. Let
dψ : g → χ(P )
X 7→ X
#
be the associated infinitesimal action. Let
X
1
, . . . , X
k
be a basis of the Lie
algebra
g
. Then since
ψ
is a free action,
X
#
1
, . . . , X
#
k
are all linearly independent
at each p ∈ P .
Define the vertical tangent space
V
p
= span{(X
#
1
)
p
, . . . , (X
#
k
)
p
} ⊆ T
p
P = ker(dπ
p
)
We can then put these together to get V ⊆ T P , the vertical tangent bundle.
Definition
(Ehresmann Connection)
.
An (Ehresmann) connection on
P
is a
choice of subbundle H ⊆ T P such that
(i) P = V ⊕H
(ii) H is G-invariant.
Such an H is called a horizontal bundle.
There is another way of describing a connection. A connection form on
P
is
a g-valued 1-form A ∈ Ω
1
(P ) ⊗ g such that
(i) ι
X
#
A = X for all X ∈ g
(ii) A is G-invariant for the action
g · (A
i
⊗ X
i
) = (ψ
g
−1
)
∗
A
i
⊗ Ad
g
X
i
.
Lemma.
Giving an Ehresmann connection is the same as giving a connection
1-form.
Proof. Given an Ehresmann connection H, we define
A
p
(v) = X,
where v = X
#
p
+ h
p
∈ V ⊕H.
Conversely, given an A, we define
H
p
= ker A
p
= {v ∈ T
p
P : i
v
A
p
= 0}.
We next want to define the notion of curvature. We will be interested in flat
connections, i.e. connections with zero curvature.
To understand curvature, if we have a connection
T P
=
V ⊕H
, then we get
further decompositions
T
∗
P = V
∗
⊕ H
∗
, Λ
2
(T
∗
P ) = (
V
2
V
∗
) ⊕ (V
∗
∧ H
∗
) ⊕ (
V
2
H
∗
).
So we end up having
Ω
1
(P ) = Ω
1
vert
(P ) ⊕ Ω
1
hor
(P )
Ω
2
(P ) = Ω
2
vert
⊕ Ω
mixed
⊕ Ω
2
hor
If A =
P
k
i=1
A
i
⊗ X
i
∈ Ω
1
⊗ g, then
dA ∈ Ω
2
⊗ g.
We can then decompose this as
dA = (dA)
vert
+ (dA)
mix
+ (dA)
hor
.
The first part is uninteresting, in the sense that it is always given by
(dA)
vert
(X, Y ) = [X, Y ],
the second part always vanishes, and the last is the curvature form
curv A ∈ Ω
2
hor
⊗ g.
Definition (Flat connection). A connection A is flat if curv A = 0.
We write
A
for the space of all connections on
P
. Observe that if
A
1
, A
0
∈ A
,
then A
1
− A
0
is G-invariant and
ι
X
#
(A
1
− A
0
) = X − X = 0
for all X ∈ g. So A
1
− A
0
∈ (Ω
1
hor
⊗ g)
G
. So
A = A
0
+ (Ω
1
hor
⊗ g)
G
,
and T
A
0
A = (Ω
1
hor
⊗ g)
G
.
Suppose
B
is a compact Riemann surface, and
G
a compact or semisimple
Lie group. Then there exists an Ad-invariant inner product on g. We can then
define
ω : (Ω
1
hor
⊗ g)
G
× (Ω
1
hor
⊗ g)
G
→ R,
sending
X
a
i
X
i
,
X
b
i
X
i
7→
Z
B
X
i,j
a
i
∧ b
j
(X
i
, X
j
).
This is easily seen to be bilinear, anti-symmetric and non-degenerate. It is also
closed, if suitably interpreted, since it is effectively constant across the affine
space A. Thus, A is an “infinite-dimensional symplectic manifold”.
To perform symplectic reduction, let
G
be the gauge group, i.e. the group of
G
-equivariant diffeomorphisms
f
:
P → P
covering the identity.
G
acts on
A
by
V ⊕H 7→ V ⊕H
f
,
where
H
f
is the image of
H
by d
f
. Atiyah and Bott proved that the action of
G
on (A, ω) is Hamiltonian with moment map
µ : A 7→ Lie(G)
∗
= (Ω
2
hor
⊗ g)
G
A 7→ curv A
Performing symplectic reduction, we get
M = µ
−1
(0)/G,
the moduli space of flat connections, which has a symplectic structure. It turns
out this is in fact a finite-dimensional symplectic orbifold.