# Moment Maps

# Symplectic Cutting

Let (M,ω)(M,\omega) be symplectic with a Hamiltonian S1S^{1}-action whose moment map is μ:MR\mu: M \rightarrow \R. Consider now the product M×CM \times \mathbb{C} equipped with the product symplectic structure and the S1S^{1}-action

eiθ(m,ξ):=(eiθm,eiθξ),e^{i \theta} \cdot (m,\xi) := (e^{i \theta} \cdot m, e^{-i \theta} \xi),

whose moment map is

Φ:M×CR,Φ(m,ξ)=μ(m)ξ2.\Phi: M \times \mathbb{C} ⟶ \R,\qquad \Phi(m,\xi) = \mu(m) - |\xi|^{2}.

The symplectic cut Mcut:=Φ1(ϵ)/S1M_{cut} := \Phi^{-1}(ϵ)/S^{1} is the symplectic quotient of M×CM \times \mathbb{C} at level ϵϵ.


The set Φ1(ϵ)\Phi^{-1}(ϵ) decomposes into the disjoint union Φ1(ϵ)=Σ1Σ2\Phi^{-1}(ϵ) = \Sigma_{1} ⊔ \Sigma_{2}, where

Σ1:={(m,ξ)M×C:μ(m)>ϵ,z=+μ(m)ϵ},Σ2:={(m,0)M×C:μ(m)=ϵ}.\Sigma_{1} := \{ (m,\xi) \in M \times \mathbb{C} : \mu(m) > ϵ,\ |z| = + √{\mu(m) - ϵ} \},\quad \Sigma_{2} := \{ (m,0) \in M \times \mathbb{C} : \mu(m) = ϵ \}.

On Σ1\Sigma_{1}, each orbit of the S1S^{1}-action contains a unique choice of (m,z)(m,z), where zz is in fact real and positive. Thus Σ1/S1{mM:μ(m)>ϵ}=:M>ϵ\Sigma_{1}/S^{1} \cong \{m \in M : \mu(m) > ϵ \} =: M_{>ϵ}. On the other hand, Σ2/S1\Sigma_{2}/S^{1} is just the usual symplectic quotient μ1(ϵ)/S1Σ2/S1\mu^{-1}(ϵ)/S^{1} \cong \Sigma_{2}/S^{1}.

So the symplectic cut Mcut=Φ1(ϵ)/S1M>ϵ(μ1(ϵ)/S1)M_{cut} = \Phi^{-1}(ϵ)/S^{1} \cong M_{>ϵ} \sqcup (\mu^{-1}(ϵ)/S^{1}) can be viewed as {mM:μ(m)ϵ}\{ m \in M : \mu(m) \geq ϵ \}, with the S1S^{1}-action on the boundary being factored out.