7
Symplectic Quotients
M symplectic, equipped with Hamiltonian action of G
Definition 7.1 The symplectic quotient of M is
M = Φ−1 (0)/G.
In general let c ∈ Z(g). Mc = Φ−1 (c)/G
Theorem The annihilator of Im(dΦ)m in g∗ is Lie(Stab(m)). Proof: η ∈ g implies
ωm (η # , Ym ) = (dΦm )(Ym )(η). Ym ∈ Tm M (for fixed m) So η ∈ g annihilates all
#
#
dΦm (Ym ) ∈ g∗ iff ωm (ηm
, Ym ) = 0 for all Ym ∈ Tm M iff ηm
= 0 (since ω is nondegenerate).
#
Corollary: dΦm : Tm M → g∗ is surjective iff ηm
= 0 for all η ∈ g, iff Stab(m) is
a finite group (the only Lie groups with Lie algebra 0 are finite groups).
Corollary: 0 is a regular value for Φ iff dΦm is surjective for all m ∈ Φ−1 (0) iff
Stab(m) is finite iff G acts locally freely on Φ−1 (0).
Definition 7.2 A group G acts locally freely on a manifold M if ∀m ∈ M , Stab(m)
is a finite subgroup of G.
Theorem 7.3 (Regular Value Theorem)
Let M be a smooth manifold and Ψ : M → N a smooth map. If n ∈ N is a regular
value of Ψ (in other words dΨm is surjective for all m ∈ Ψ−1 (n)) then Ψ−1 (n) is a
smooth manifold.
Theorem 7.4 Let M be a smooth manifold and G a (compact) group acting locally
freely on M . Then M/G is an orbifold.
Definition 7.5 Orbifold or V-manifold: a space of dimension n which has a cover
by open sets Uα s.t. Uα = U˜α /Γα where U˜α is an open set in Rn and Γα is a finite
group.
(References: Audin III.2, III.17; Henriques-Metzler [23].)
Theorem 7.6 Suppose G acts freely on Φ−1 (0). Then 0 is a regular value of Φ and
M0 = Φ−1 (0)/G is a smooth manifold.
Theorem 7.7 In the notation of the previous theorem, M0 is symplectic.
Proof: Define
Tm̄ M0 = Tm Φ−1 (0)/(g · m = Tm (G ◦ m)).
Let η¯1 , η¯2 ∈ Tm Φ−1 (0). The quotient by Tm (G · m). gives equivariant classes [η1 ], [η2 ]
∈ Tm̄ M0 . Define ω̄m̄ ([η1 ], [η2 ]) = ωm (η1 , η2 ).
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On any sufficiently small chart U in Φ−1 (0)/G there is a section σ : U → π −1 (U )
(where π : Φ−1 (0) → Φ−1 (0)/G is the projection) for which ω̄|U = σ ∗ ωΦ−1 (0) . Thus ω̄
is closed (as dω̄ = d(σ ∗ ω) = σ ∗ (dω) = 0).
To show that ω̄ is nondegenerate, suppose X ∈ Tm Φ−1 (0) and suppose Tm M =
g ⊕ Tm Φ−1 (0).
iηm ω = dΦη .
So
g · m ⊂ Ker(ω|Φ−1 (0) ).
But on dimensional grounds Ker(ω|Φ−1 (0) ) has dimension ≤ d = dim(G) so Ker(ω|Φ−1 (0) ) =
g · m so ω̄|Tm M0 is nondegenerate.
Let V be a symplectic vector space, and let F be a subspace of V .
Definition 7.8 F 0 is the annihilator or symplectic orthocomplement of F (defined
as F = {X ∈ V : ω(X, Y ) = 0∀Y ∈ F }).
Definition 7.9 F is isotropic if F ⊂ F 0
F is coisotropic if F 0 ⊂ F
F is Lagrangian if F 0 = F
Lemma 7.10 (F 0 )0 = F
Lemma 7.11 If F is coisotropic, then ω induces a symplectic form on F/F 0 .
Proof: If X ∈ F and ω(X, Y ) = 0∀Y ∈ F , then X ∈ F ⊥ implies [X] = 0 in F/F ⊥ .
Lemma 7.12 If F = Ker(dΦm ) then F ⊥ = Tm (G · m) (so Ker(dΦm ) is coisotropic).
Proof: dimKer(dΦm ) = dimM − dimG (so dim(Im(dΦm )) = dim(G) − dimStabm )
But Tm (G · m) ⊂ F 0 and dimTm (G · m) = dimG So Tm (G · m) = F 0 .
Hence in this situation Ker(dΦm )/Tm (G · m) inherits a symplectic structure from
ω.
Theorem 7.13 (“Reduction in stages”) Suppose G acts on M in a Hamiltonian
fashion. Suppose H is a normal subgroup of G.1 Then G/H is a Lie group with Lie
algebra g/h. Also Φ−1
H (0)/H is acted on by G/H in a Hamiltonian fashion and the
moment map is
P
(g/h)∗ → g∗ → h∗ → · · ·
∗
ΦG |Φ−1 (0) : Φ−1
H (0) → Ker(P ) ⊂ g
Further, the symplectic quotient of Φ−1
H (0)/H by G/H is isomorphic to the symplectic
quotient of M by G.
1
For example, this is true if G and H are tori.
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Proof: If ζ ∈ Tm (Φ−1
H (0)) then
ΦH (ζ) = P ◦ ΦG (ζ) = 0.
So
dΦG (ζ) ∈ (Ker(P ) ⊂ g∗ ) = g∗ /h∗ .
−1
−1
Then G/H acts on Φ−1
H (0)/H and ΦG/H (0)/(G/H) = ΦG (0)/G.
Example 7.14 CP n is the symplectic quotient of Cn+1 by the diagonal action of
U (1). The action of U (n + 1) on CP n descends from the action on Cn+1 according to
the previous theorem.
Shifting trick
Suppose M is acted on in a Hamiltonian fashion by a nonabelian group G. Then
−1
Φ (λ) is preserved by the subgroup Zλ ⊂ G but not by G. Hence we may define the
symplectic quotient as Φ−1 (λ)/G.
We may define a symplectic quotient
Mξ = Φ−1 (ξ)/Zξ = Φ−1 (Oξ )/G.
Shifting trick: Define a symplectic structure on M × Oξ by
Ω = ωM − ωOξ .
Then for the moment map with respect to the induced action of G on M × Oξ we
have
Mξ = (M × Oξ )0
so that reduction at a coadjoint orbit is a special case of reduction at 0.
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