# Hamiltonian Local Normal Form

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.)

# 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.(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.