L12: Circle actions
So far we have constructed symplectic manifolds by surgery, but we can also construct
them from symmetry. We need to be careful: if S 1 acts on a symplectic manifold M ,
then M/S 1 is of odd dimension; so the quotient construction is more subtle.
Definition: A circle action S 1 → Symp(M ) is
called Hamiltonian if it’s generated by a (periodic) Hamiltonian vector field XH . The function H : M → R is called the moment map.
Theorem: If t is regular for H, and if S 1 acts
freely on H −1(t), the quotient Qt = H −1(t)/S 1
carries a natural symplectic structure ωt; the
form ωt on the quotient is uniquely characterised by saying that, for π : H −1(t) → Qt
π ∗ωt = ωM |H −1(t)
Proof: Write Q̃t = H −1(t). If p ∈ Q̃t, then the
space (T Q̃t)⊥
p is one-dimensional, generated by
the tangent line to the S 1-action (since XH is
non-zero, by freeness, and does lie inside this
space). From q.sheet 1, if U ⊂ V is coisotropic
in a symplectic vector space, U/U ⊥ is naturally
symplectic. Hence TpQ̃t/(TpQ̃t)⊥ is naturally
symplectic. But T[p]Qt is canonically equal to
this quotient; and since S 1 preserves ω, the
choice of p ∈ [p] is immaterial.
This shows the quotient has a canonical nondegenerate 2-form ωt, and π ∗ωt = ωM |Q̃ by
t
construction. We also need this form to be
closed. But π : Q̃t → Qt is an S 1-bundle,
so (dπ)∗ : T Qt → T Q̃t is injective; hence dωt
is completely determined by π ∗dωt, but this is
dωM |Q̃ by the above, and dωM = 0. ¥
t
Example: if H 1(M ; R ) = 0 every circle action
is Hamiltonian; since it’s generated by a vector
field X = µω (φ) for some φ = dH ∈ Ω1(M ), so
X = XH . Rotation on T 2 is not Hamiltonian.
Example: Let S 1 act on C n+1 by t 7→ e2iπt
(i.e. the usual diagonal action), for the standard form ω0 on C n+1. This has a Hamiltonian function H : C n+1 → R , namely z 7→
−π|z|2. (To see this, check XH = −i ∇H for
functions H on C k .) Then H −1(−π)/S 1 =
C Pn; so Pn carries a symplectic form uniquely
characterised by the fact that its pullback to
(the unit) S 2n+1 ⊂ C n+1 is the restriction of
ω0. This is just our earlier Fubini-Study Kähler
form: note they are both U (n+1) invariant and
both give a line area π.
Proposition: Suppose (M, ω) is closed and
∧ [ω]n−1 : H 1(M ; R ) → H 2n−1(M ; R )
is an isomorphism (e.g. M is Kähler). Then
an S 1-action S 1 → Symp(M ) is Hamiltonian if
and only if it has fixed points.
Proof: If the action is Hamiltonian then a critical point of H : M → R gives a zero of XH ,
hence a fixed point of the flow and the action.
The converse is more interesting. Suppose the
action is not Hamiltonian, i.e. its generated by
a vector field X s.t. ιX ω is not exact. Exercise:
all the orbits of the S 1-action represent the
same class in H1(M ; Z). So it is enough to
prove that the homology class of some nonconstant orbit γ is non-zero. Pick a volume
form σ ∈ Ω2n(M ) supported near γ.
Claim: [ιX σ] = PD[γ] ∈ H 2n−1(M ; R ).
For PD[γ] is characterised as being represented
by a form supported near γ and with integral 1
over each fibre of the normal bundle νγ/M . In
general, a form φ supported near a submanifold
Y k ⊂ X d represents PD[Y ] ∈ H d−k (X) if and
d−k
∼ R at each y ∈ Y .
(νy ) =
only if φ|νy 6= 0 ∈ Hct
Now ιX σ has the required property, for γ, since
X is tangent to γ and σx 6= 0 ∈ Λ2nTx∗M for
each x ∈ γ, as σ is a volume form.
Finally, recall [σ] = [ω]n ∈ H 2n(M ) so [ιX σ] =
[ω]n−1 ∧ [ιX ω]. The RHS is non-zero by assumption, hence [γ] 6= 0 as required. ¥
Fixed points actually play a large role in the
theory. Fix a circle action φ : S 1 → Symp(M, ω).
Lemma: ∃ a φ-invariant J compatible with ω.
Proof: pick some metric g. By averaging φ∗θ g
over the compact set of θ ∈ S 1, we obtain a
φ-invariant metric g 0. Now take the J canonically associated to g 0 in our construction of
compatible almost complex structures. ¥
Lemma: F ix(S 1) is a symplectic submanifold.
Proof: Let p ∈ F ix(S 1). For each θ ∈ S 1
we have (dφθ )p : TpM → TpM ; this defines
a unitary action of S 1 on (TpM, J, ω). The
tangent space to the fixed point set is just
the eigenspace with eigenvalue 1 for (dφθ )p;
since the action is unitary, this eigenspace is
J-invariant, hence symplectic. ¥
Note: the proof shows if 1 is an e-value of a
symplectic matrix, it occurs with even multiplicity.
Now turn to the Hamiltonian case. Suppose
H : M 2n → R is actually a Morse function,
i.e. the critical points p of H are isolated and
non-degenerate: det(Hessf (p)) 6= 0. Then H
is locally a non-degenerate quadratic form.
For any closed manifold, Morse functions are
dense in C ∞(M ; R ). These are “generic functions”. The critical points of H are the fixed
points of the associated circle action. Now a
unitary action of S 1 on a complex vector space
C n, after diagonalising, becomes
θ 7→ diag(e2πik1 , e2πik2 , . . . , e2πikn )
for some k1, . . . , kq
n ∈ Z. Write e = i ki . In the
Morse case, e = det(Hessf (p)) 6= 0.
Q
Theorem: [Duistermaat-Heckmann localisation]
If the moment map H of a Hamiltonian S 1action is Morse, then for all ~ ∈ C
Z
ωn
i
~
H
e
=
n!
M
µ ¶n
i
~
ei~H(p)
e(p)
p∈Crit(H)
X
Expanding in powers of ~ this gives identities,
for instance:
Z
X H(p)n
n
ω =
vol(M ) =
e(p)
M
p
and for each k = 0, 1, . . . , n − 1:
X H(p)k
p
e(p)
= 0
Example: the rotation circle action on S 2 ⊂ R 3
is generated by H = 2πz : S 2 → R . The poles
(0, 0, ±1) are fixed, with weights e = ±1.
Remark: stationary phase approximation gives
a way Rto estimate oscillatory integrals, of the
shape M a(x)eitf (x)volM , by contributions from
critical points of f : the oscillation makes everything else, where f is varying a lot, cancels
out. The above theorem gives a rare example
in which the approximation is exact. Usually,
for weights w̃f,p, dim(M ) = d, and real t À 0
d
t2
Z
M
a(x)eitf (x)vol
M =
1
w̃f,p a(p)eitf (p)+O( )
t
p
X