Action-angle coordinates. [04HY]

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Action-angle coordinates.

We review here some standard facts about Lagrangian fibrations which we will use in the next Sections. For details we refer to Duistermaat [4]. Assume we are given a $2n$ -dimensional symplectic manifold $X$ with symplectic form $\omega$ , a smooth $n$ -dimensional manifold $B$ and a proper smooth submersion $f:X\rightarrow B$ whose fibres are connected Lagrangian submanifolds. For every $b\in B$ , denote by $F_{b}$ the fibre of $f$ at $b$ .

Proposition 3.2 (Arnold-Liouville).

In the above situation, for every $b\in B$ , $T^{\ast}_{b}B$ acts transitively on $F_{b}$ . In particular there exists a maximal sub-lattice $\Lambda_{b}$ of $T^{\ast}_{b}B$ such that $F_{b}$ is naturally diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$ , therefore $F_{b}$ is an $n$ -torus.

Proof.

To every $\alpha\in T^{\ast}_{b}B$ we can associate a vector field $v_{\alpha}$ on $F_{b}$ by

\iota_{v_{\alpha}}\omega=f^{\ast}\alpha.

Let $\phi_{\alpha}^{t}$ be the flow of $v_{\alpha}$ with time $t\in\mathbb{R}$ . Then we define the action $\theta_{\alpha}$ of $\alpha$ on $F_{b}$ by

\theta_{\alpha}(p)=\phi_{\alpha}^{1}(p),

where $p\in F_{b}$ . One can check that such an action is well defined and transitive. Then, $\Lambda_{b}$ defined as

\Lambda_{b}=\{\lambda\in T^{\ast}_{b}B\ |\ \theta_{\lambda}(p)=p,\ \text{for all}\ p\in F_{b}\}

is a closed discrete subgroup of $T^{\ast}_{b}B$ , i.e. a lattice. From the properness of $F_{b}$ it follows that $\Lambda_{b}$ is maximal (in particular homomorphic to $\mathbb{Z}^{n}$ ) and that $F_{b}$ is diffeomorphic to $T^{\ast}_{b}B/\Lambda_{b}$ . ∎

We denote $\Lambda=\cup_{b\in B}\Lambda_{b}$ . Given the presheaf on $B$ defined by $U\mapsto H_{1}(f^{-1}(U),\mathbb{Z})$ , the associated sheaf is a locally constant sheaf. We can identify it with $\Lambda$ as follows. Let $U\subseteq B$ be a contractible open set. For every $b\in U$ , $H_{1}(F_{b},\mathbb{Z})$ can be naturally identified with $H_{1}(f^{-1}(U),\mathbb{Z})$ . To every $\gamma\in H_{1}(f^{-1}(U),\mathbb{Z})$ , we can associate a $1$ -form $\lambda$ on $U$ as follows. For every vector field $v$ on $U$ , if we denote by $\tilde{v}$ a lift, define

\lambda(v)=-\int_{\gamma}\iota_{\tilde{v}}\omega.

(4)

It turns out that this identifies the above sheaf with $\Lambda\subset T^{\ast}_{B}$ . If $\gamma_{1},\ldots,\gamma_{n}$ are a basis for $H_{1}(F_{b},\mathbb{Z})$ , then (4) gives us a $\mathbb{Z}$ -basis $\lambda_{1},\ldots,\lambda_{n}$ of $\Lambda$ over a contractible open neighborhood $U$ of $b$ .

In particular, one can read the monodromy of $f:X\rightarrow B$ from the monodromy of $\Lambda$ . We state now the fundamental theorem of smooth proper Lagrangian submersions:

Theorem 3.3 (Duistermaat).

Given a basis $\{\gamma_{1},\ldots,\gamma_{n}\}$ of $H_{1}(F_{b},\mathbb{Z})$ , then the corresponding 1-forms $\lambda_{1},\ldots,\lambda_{n}$ defined on a contractible open neighborhood $U$ of $b$ are closed and locally generate $\Lambda$ . In particular, $\Lambda$ is Lagrangian with respect to the standard symplectic structure in $T^{\ast}_{B}$ . A choice of functions $a_{j}$ such that $\lambda_{j}=da_{j}$ defines coordinates $a=(a_{1},\ldots,a_{n})$ called action coordinates. A covering $\{U_{i}\}$ of $B$ by contractible open sets and a choice of action coordinates on each $U_{i}$ defines an integral affine structure $\mathscr{A}$ on $B$ . Moreover, if $f$ has a Lagrangian section $\sigma:U\rightarrow X$ over an open set $U\subseteq B$ , then there is a natural symplectomorphism

\Theta:T^{\ast}_{U}/\Lambda\rightarrow f^{-1}(U).

(5)

If $\sigma$ is a global section then $X(B,\mathscr{A})$ is symplectically conjugate to $X$ . If in addition the monodromy of $\Lambda$ is trivial $X$ is symplectically conjugate to $B\times T^{n}$ . The map $\Theta$ is called the period map or action-angle coordinates map.

Proof.

We just give a sketch of the proof. Using the Weinstein neighborhood theorem one can show that in a sufficiently small tubular neighborhood of a fibre $F_{b}$ , the symplectic form is exact, i.e $\omega=-d\eta$ for some 1-form $\eta$ . Notice that $\eta|_{F_{b}}$ is a closed 1-form. Define functions $a_{j}$ on $U$ by

a_{j}=\int_{\gamma_{j}}\eta.

One can show that

\lambda_{j}=da_{j}

and therefore $\lambda_{j}$ is closed. It is clear that the coordinates $a=(a_{1},...,a_{n})$ are well defined up to an integral affine transformation and therefore they define an integral affine structure on $B$ inducing the lattice $\Lambda$ in $T^{\ast}_{B}$ . Finally, notice that given a section $\sigma:U\rightarrow X$ we have a covering map

\begin{array}[]{rcl}T^{\ast}_{U}&\rightarrow&f^{-1}(U)\\ \alpha&\mapsto&\theta_{\alpha}(\sigma(\pi(\alpha)))\end{array}

This map induces a diffeomorphism between $T^{\ast}_{U}/\Lambda$ and $f^{-1}(U)$ . One can check that in the case $\sigma$ is Lagrangian this map is a symplectomorphism. For the proof of the last statement we refer the reader to [4]. ∎

Corollary 3.4.

Let $\mathcal{F}=(X,f,B)$ and $\mathcal{F}^{\prime}=(X^{\prime},f^{\prime},B^{\prime})$ be smooth proper Lagrangian fibrations inducing integral affine structures $\mathscr{A}$ and $\mathscr{A}^{\prime}$ on $B$ and $B^{\prime}$ respectively. Assume there exist Lagrangian sections $\sigma$ and $\sigma^{\prime}$ of $f$ and $f^{\prime}$ respectively. Then an integral affine isomorphism $\phi$ between $B$ and $B^{\prime}$ induces a symplectic $(\psi,\phi)$ -conjugation between $\mathcal{F}$ and $\mathcal{F}^{\prime}$ such that $\psi\circ\sigma=\sigma^{\prime}\circ\phi$ .

Proof.

Let $\Lambda\subset T^{\ast}_{B}$ and $\Lambda^{\prime}\subset T^{\ast}_{B^{\prime}}$ be the lattices induced from the integral affine structures on $B$ and $B^{\prime}$ , respectively. From Theorem 3.3 it follows that $X$ and $X^{\prime}$ are symplectomorphic to $T^{\ast}_{B}/\Lambda$ and $T^{\ast}_{B^{\prime}}/\Lambda^{\prime}$ , respectively. Given an integral affine isomorphism $\phi$ between $B$ and $B^{\prime}$ , clearly $\phi^{\ast}$ is a symplectomorphism between $T^{\ast}_{B^{\prime}}$ and $T^{\ast}_{B}$ inducing an isomorphism between $\Lambda^{\prime}$ and $\Lambda$ . Therefore $\phi^{\ast}$ descends to a symplectomorphism $\tilde{\psi}$ between $T^{\ast}_{B^{\prime}}/\Lambda^{\prime}$ and $T^{\ast}_{B}/\Lambda$ . Defining $\psi=\Theta^{\prime}\circ(\tilde{\psi})^{-1}\circ\Theta^{-1}$ the claim follows. ∎

The following is an easy but important consequence of Arnold-Liouville-Duistermaat theorem:

Corollary 3.5.

Proper Lagrangian submersions do not have semi-global symplectic invariants. In other words, all such fibrations are symplectically conjugate to $U\times T^{n}$ when restricted to a small enough neighborhood $U$ of a base point.

It is clear that smoothness of the fibration map plays a crucial role in the above result. Semi-global invariants do arise for certain piecewise $C^{\infty}$ Lagrangian fibrations [2]. We say more about this in §6.