Hamiltonian Local Normal Form

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This summarizes what I learned from Professor Karshon's lecture notes for [MAT1347HS, http://www.math.toronto.edu/~karshon/grad/2009-10/index-archived.html]. (This page is still under construction.)

Arthur Huang

Contents

Hamiltonian Manifolds

A compact Hamiltonian $G$-manifold (M,ω,Φ) is defined as following:

- M is a compact smooth manifold of real dimension $2n$;

- ω is a $G$-invariang (i.e. \forall g\in G, (g.)^{\ast}\omega=\omega)symplectic form (non-degenerate, closed $2$-form) on $M$;

- \Phi : M \rightarrow g^{\ast} is a momentum map.

Note that all compact Hamiltonian $G$-manifolds together with $G$-equivariant symplectomorphisms form a category $Ham$.

Now we fix some torus $G$ of rank $k$.

Given any Hamiltonian $G$-manifold (M,ω,Φ), and any x\in M, we have a normal form around its orbit $G.x$ in the category $Ham$:

Denote H = StabG(x), then there exists a morphism $f$ in $Ham$, which is an isomorphism whose source is (G\times_H[(g/h)^{\ast}\times D],\Omega) (to be defined clearly later) and target is $(U_x,\omega)$, some closed neighborhood of $G.x$.


Basically speaking, on the one hand, we have a new version of the slice theorem respecting the linear symplectic structure at the point $x$, and on the other hand we have some canonical symplectic structure on the normal form; we put them together by a uniqueness theorem.


The Generalized Slice Theorem

Set $H=Stab_G(x)$, then T_x(G.x)\cong g/h.

1. The linear structure at $x$:

$G$ is a torus \Rightarrow N = g / h is an isotropic subspace of $T_xM$, then T_xM=N\oplus N^{\ast}\oplus V, where V=N^{\omega_x}/N is a symplectic subspace of $T_xM$ with $\omega_V$ the restriction of $\omega_x$ to $V$. Then the symplectic action of $G$ on $M$ induces an $H$ symplectic linear action on $V$; call H\rightarrow Sp(V,\omega_V) the symplectic slice representation of $H$ on $(V,\omega_V)$.


2. The symplectic form at $x$:

$H$ fixes $x$, so by the local linearization theorem we have an $H$-equivariant diffeomorphism E: W\rightarrow U_0 from some nbhd $W$ of 0\in T_xM to some open neighborhood $U_0$ around x\in M, the symplectic form $\omega$, pulled back under $E$ at the point $x$, is exactly \left[ \begin{array}{cc} A & 0 \\ 0 & \omega_V \end{array}\right], where $A$ is the standard pairing of N\oplus N^{\ast}: A=\left[ \begin{array}{cc} 0 & I \\ -I & 0 \end{array}\right].


3. Sweeping:

Now let D\subset V be the disc centred at the origin with sufficiently small radius s.t. \{0\}\oplus D \subset W ($0$ being the origin of N\oplus N^{\ast}), then f: G\times_H N^{\ast}\oplus D \rightarrow U_x where $U_x$ is a suitable neighborhood of $G.x$ (s.t. f(\{0\}\oplus D)\subset U_0) and $f([g,(\eta, v)])=g.E(\eta+v)$.

The Natural Symplectic Structure on the Normal Form with Given Moment Map

We already have linear symplectic representation H\rightarrow Sp(V,\Omega_V), and are given a momentum map \Phi_V: V\rightarrow h^{\ast}, where $H$ and $(V,\Omega_V)$ are as in the last section.

1. The symplectic structure on G\times g^{\ast} and its exact momentum map $\Phi_G$:


- T^{\ast}G has a trivialization by F:G\times g^{\ast}\rightarrow T^{\ast}G with (h,\xi)\mapsto (h, \xi\circ (h^{-1}.)_{\ast}).


- $G$ acts on $G\times g^{\ast}$ by left multiplication on the first factor: a.(h,ξ) = (ah,ξ); $G$ acts on $T^{\ast}G$ by a.(h, \xi)=(ah, \xi\circ (a^{-1}.)_{\ast}). Then $F$ is equivariant (Check1N).


- T^{\ast}G has a natural symplectic structure given by the tautological 1-form \tau_{h,\beta}=\beta\circ T_{(h,\beta)}\Pi_G, where $T_{(h,\beta)}\Pi_G$ is the differential at $(h,\beta)$ of the natural projection \Pi:T^{\ast}G\rightarrow G, and $\Omega_G=-d\tau$. Note that $\tau$ is invariant under the above $G$ action.(Check2N)


- Then $F^{\ast}\Omega_G=-dF^{\ast}\tau$ is a symplectic form on $G\times g^{\ast}$; also $F^{\ast}\tau$ is invariant under the $G$ action: (a.)^{\ast}F^{\ast}\tau= (F\circ a.)^{\ast}\tau=(a.\circ F)^{\ast}\tau=F^{\ast}(a.)^{\ast}\tau=F^{\ast}\tau. We then have an exact momentum map for F^{\ast}\tau: \Phi_G: G\times g^{\ast}\rightarrow g^{\ast} such that ΦG(h,β) = Ad(h)β.(Check3N)


2. The symplectic structure on G\times g^{\ast}\times V and its momentum map:


- Since $(V,\Omega_V)$ is already a symplectic manifold, we have the symplectic structure on G\times g^{\ast}\times V given by F^{\ast}\Omega_G\oplus \Omega_V.


- $H$ acts on G\times g^{\ast}\times V by h.(a,\beta, z)=(ha^{-1},Ad^{\ast}(a)\beta, h.z), where $h.z$ is the given symplectic $H$-action on $(V,\Omega_V)$ coming from the linear symplectic representation H\rightarrow Sp(V,\Omega_V) with given momentum map \Phi_V: V\rightarrow h^{\ast}.


- The momentum map \Phi_H:G\times g^{\ast}\times V\rightarrow h^{\ast} for the $H$-action is $\Phi_H(h,\beta,z)=-\beta|_h+\Phi_V(z)$.(Check4N)


3. The zero level set of $\Phi_H$:


- We have \Phi_H^{-1}(0)=\{(a,\beta,z)\in G\times g^{\ast} \times V: \beta|_h=\Phi_V(z)\}. Note that we have a decomposititon g^{\ast}= (g/h)^{\ast}\oplus h^{\ast}, by choosing some fixed $G$-invraiant inner product, then \beta=\nu\oplus \beta|_h for some unique \nu\in (g/h)^{\ast}. Hence the map \Phi^{-1}_H(0)\rightarrow G\times (g/h)^{\ast} \times V such that (a,\beta,z)\mapsto (a, \beta-\Phi_V(z),z) is an $(G\times H)$-quivariant diffeomorphism.


4. Sweeping:


- Now we quotient both sides of the above equation by the $H$-action, and get: \Phi_H^{-1}(0)/H\rightarrow G\times_H[(g/h)^{\ast}\times V] being a $G$-quivariant diffeomorphism.


- The zero level set carries the symplectic structure on G\times g^{\ast}\times V, and the symplectic reduction theorem for the zero level set of momentum maps says that F^{\ast}\Omega_G\oplus \Omega_V descends to the orbit \Phi_H^{\ast}(0)/H as a symplectic form; and the symplectic form $\Omega$ on G\times_H[(g/h)^{\ast}\times V] comes by the above diffeomorphism.


Now we have given G\times_H[(g/h)^{\ast}\times V] a symplectic form $\Omega$.

Connecting the two sides

Now we have f: G\times_H[(g/h)^{\ast}\times D]\rightarrow U_x\subset M which is an equivariant diffeomorphism such that (f^{\ast}\omega)_{[e,0,0]}=\Omega_{[e,0,0]}. (D\subset V is the disc given above.)

-- Note that the pull back of $\Omega$ at $[e,0,0]=f^{-1}(x)$ by $f$ coincides with $\omega$ at that point, both being \left[ \begin{array}{cc} A & 0 \\ 0 & \omega_V \end{array}\right], where $A$ is the standard pairing of N\oplus N^{\ast}: A=\left[ \begin{array}{cc} 0 & I \\ -I & 0 \end{array}\right]. Then the $G$-equivariance of $f$ and the $G$-invariance of $\omega$ and $\Omega$ shows that they coincides on the zero section. (Check1C)


-- Now we use Moser's method to show that the coincidence of f^{\ast}\omega with Ω on the zero section extends to the whole bundle.

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