I’ve recently been reading Torus actions on symplectic manifolds [Aud04] by Michèle Audin. I’d like to write down some of the facts I learned from this book.
The main object the book covers is a symplectic manifold with a torus action. Why look at torus actions, not Lie group actions? This is not a rhetoric question, unfortunately. Torus actions seem to be quite important in various places, e.g., the classification of semisimple Lie algebras (Cartan subalgebras), but I have no idea why they are important. These two work well, because if there is a general compact Lie group action, we can look at the maximal subtorus of the algebra and only consider its action.
One other aspect that I don’t understand well is that we look at Hamiltonian group actions on symplectic manifolds. At the end, everything turns out very well; Hamiltonian torus actions on symplectic manifolds correspond to a special class of polytopes.
1. Crash course on symplectic manifolds#
Hamiltonian group actions are defined on symplectic manifolds.
Definition 1. A symplectic form is a closed \( 2 \)-form that is pointwise non-degenerate. A symplectic manifold is a manifold along with a symplectic form.
A skew-symmetric bilinear form on an odd-dimensional vector space is necessarily degenerate. Thus every symplectic manifold is \( 2n \)-dimensional for some \( n \).
Example 1. For any manifold \( M \), its cotangent bundle \( T^\ast M \) has a canonical symplectic form \[ \displaystyle \omega = dx _ 1 \wedge d\xi _ 1 + dx _ 2 \wedge d\xi _ 2 + \cdots + dx _ n \wedge d\xi _ n, \] where \( (x _ 1, \ldots, x _ n) \) is a coordinate chart on \( M \) and \( (x _ 1, \ldots, \xi _ n) \) is the induced coordinate chart on \( T^\ast M \). This \( 2 \)-form is in fact exact.
Take a symplectic manifold \( M \) with symplectic form \( \omega \). This \( \omega \) determines an isomorphism \( T _ p M \cong T _ p^\ast M \). So here is a construction we can do. Take a smooth function \( H : M \rightarrow \mathbb{R} \), and then its derivative \( dH \) is a global \( 1 \)-form on \( M \). Using the isomorphism \( T _ p M \cong T _ p^\ast M \) given by \( \omega \), we can translate it into a vector field \( X _ H \). In terms of the equation, we will be able to write \[ \displaystyle i _ {X _ H} \omega = -dH, \] where \( i \) means putting it in the first component. This is called the Hamiltonian vector field associated to Hamiltonian function \( H : M \rightarrow \mathbb{R} \).
Example 2. The physical interpretation is really interesting. In Hamiltonian mechanics, the Hamiltonian equations are given by \[ \displaystyle \frac{\partial H}{\partial p _ i} = \dot{q} _ i, \quad \frac{\partial H}{\partial q _ i} = -\dot{p} _ i. \] Give the phase space \( (p, q) \) the symplectic form \( \omega = \sum _ {i}^{} dp _ i \wedge dq _ i \). Then the equations are saying that the motion of physical system is exactly the flow of the Hamiltonian vector field associated to \( H \). (Maybe there is a sign difference, but I don’t care.)
Proposition 2. Let \( (M, \omega) \) be a symplectic manifold. For any \( H : M \rightarrow \mathbb{R} \), the flow generated by \( X _ H \) preserves the symplectic form \( \omega \), i.e., \( \mathcal{L} _ {X _ H} \omega = 0 \).
Proof. We can just compute this. We have \[ \displaystyle \mathcal{L} _ {X _ H} \omega = d(i _ {X _ H} \omega) + i _ {X _ H} (d\omega) = d(-dH) + i _ {X _ H}(0) = 0 \] using Cartan’s magic formula. ▨
It is clear that the zeros of \( X _ H \) are the critical points of \( H \). We can also define the Poisson bracket of \( H, K : M \rightarrow \mathbb{R} \) as \[ \displaystyle \{ H, K \} = X _ H(K) = \omega(X _ H, X _ K) = -\{K, H\}. \] This is in fact a Lie algebra structure on \( C^\infty(M) \). Verify the Jacobi identity if you feel energetic.
Let us now look at group actions on symplectic manifolds. Take a symplectic manifold \( M \) with an smooth action by a Lie group \( G \). Obviously, we should require that any \( g : M \rightarrow M \) is a symplectomorphism, i.e., preserves \( \omega \). But we are going to require more.
For each \( X \in \mathfrak{g} \), we get a global vector field \( \underline{X} \) on \( M \). This is defined so that \[ \displaystyle (1 + \epsilon X) \cdot p = p + \epsilon \underline{X} _ p, \] infinitesimally. We want this to be a Hamiltonian vector field of some function. Then we get a linear map \[ \displaystyle \tilde{\mu} : \mathfrak{g} \rightarrow C^\infty(M); \quad \underline{X} = X _ {\tilde{\mu} _ X} \text{ where } \tilde{\mu} _ X = \tilde{\mu}(X). \] The map \( \tilde{\mu} : \mathfrak{g} \rightarrow C^\infty(M) \) is called the comoment map.
Definition 3. A Hamiltonian group action is an action of \( G \) on \( M \) such that the comoment map \( \tilde{\mu} : \mathfrak{g} \rightarrow C^\infty(M) \) is a Lie algebra morphism.
It is always true that \( \{\tilde{\mu} _ X, \tilde{\mu} _ Y\} - \tilde{\mu} _ {[X, Y]} \) is locally constant; we are just requiring that this is zero.
There is an alternative formulation. The comoment map \( \tilde{\mu} : \mathfrak{g} \rightarrow C^\infty(M) \) can be considered as a map \( \mathfrak{g} \times M \rightarrow \mathbb{R} \) that is linear on \( \mathfrak{g} \) and just smooth on \( M \). So this has the same data as a smooth map \( \mu : M \rightarrow \mathfrak{g}^\ast \), called the moment map.
Definition 4. A Hamiltonian group action is an action of \( G \) on \( M \) such that the moment map \( \mu : M \rightarrow \mathfrak{g}^\ast \) is \( G \)-equivariant. (Here, \( \mathfrak{g}^\ast \) has the coadjoint action, i.e., the dual of the adjoint action.)
The two definitions can be shown to be equivalent.
Example 3. Consider the sphere \( S^2 \) with the area form as \( \omega \). (This is necessarily closed because it is of top dimension.) Consider a \( S^1 = T^1 \) action on \( S^2 \) by rotations around the \( z \)-axis. This is a Hamiltonian torus action with momentum map being the projection to the \( z \)-axis, as in Figure 1.
Figure 1: The moment map of the stnadard circle action on the sphere
2. Morse theoretic preliminaries#
We’re now interested in Hamiltonian actions of \( T^k \) on a symplectic manifold \( M \). As I’ve said before, I don’t know why the torus is so important. But if we focus on torus actions, there is a remarkable theorem that was proved first by Guillemin–Sternberg [GS82] and later by Atiyah [Ati82] in a simpler way. Proving it is the goal of this section.
Theorem 5. Let \( (M, \omega) \) be a compact connected symplectic manifold and \( \mu : M \rightarrow \mathfrak{t}^{\ast k} \) be a moment map for a Hamiltonian torus action. Let \( Z _ 1, \ldots, Z _ m \) be the connected components of the fixed points of the torus action. Then each \( \mu(Z _ j) = c _ j \) is a point and the image of \( \mu \) is the convex hull of the points \( c _ j \).
It’s helpful to look at the example of complex projective space.
Example 4. There is a canonical symplectic structure on \( \mathbb{C}P^{n-1} \), called the Fubini–Study structure. (The Example 3 is the case \( n = 2 \).) There is also an easy torus action given by \[ \displaystyle (x _ 1, \ldots, x _ n) \cdot [z _ 1, \ldots, z _ n] = [e^{ix _ 1} z _ 1, \ldots, e^{ix _ n} z _ n]. \] The moment map for this action may be given as \[ \displaystyle \mu : [z _ 1, \ldots, z _ n] \mapsto \biggl( \frac{\lvert z _ 1 \rvert^2}{\sum _ {i}^{} \lvert z _ i \rvert^2}, \ldots, \frac{\lvert z _ n \rvert^2}{\sum _ {i}^{} \lvert z _ i \rvert^2} \biggr), \] and its image is an \( (n-1) \)-dimensional simplex. Notice that the vertices correspond to fixed points of the action, which are \( [0, \ldots, 0, 1, 0, \ldots, 0] \).
Moment map for a torus action on complex projective space
The main tool we are going to use in the proof is Morse theory. Here is a short recap of the main theorem.
Theorem 6. Let \( M \) be a compact manifold and \( f : M \rightarrow \mathbb{R} \) be a Morse–Bott function. Assume that \( Z _ 1, \ldots, Z _ m \) are the critical components with critical values \( c _ 1 < \cdots < c _ m \), and let \( i _ 1, \ldots, i _ m \) be the (descending) indices of \( Z _ 1, \ldots, Z _ m \). Then \( f^{-1}(-\infty, c _ j + \epsilon] \) and \( f^{-1}(-\infty, c _ {j+1}-\epsilon] \) are homotopic, and \( f^{-1}(-\infty, c _ j + \epsilon] \) is homotopic to \( f^{-1}(-\infty, c _ j - \epsilon] \) with a \( D^{i _ j} \)-bundle over \( Z _ j \) attached along its boundary.
Corollary 7. Let \( M \) be a connected compact manifold and \( f : M \rightarrow \mathbb{R} \) be a Morse–Bott function such that none of the descending indices nor the ascending indices are \( 1 \). Then \( f \) has a unique local minimum component and a local maximum component. Moreover, all its nonempty levels are connected.
Proof. When you pass a critical value and go from one level set to the next level set, you’re sort of doing a surgery. Suppose \( Z \) is the critical component, \( i _ d \) and \( i _ a \) are the descending and ascending indices (so that \( \dim Z + i _ d + i _ a = \dim M \)). Then roughly \( f^{-1}(c+\epsilon) \) can be obtained from \( f^{-1}(c-\epsilon) \) by taking a \( S^{i _ d - 1} \times D^{i _ a} \)-bundle over \( Z \) inside \( M \), and then replacing this by a \( D^{i _ d} \times S^{i _ a - 1} \)-bundle. In this process, nothing can be connected or disconnected. ▨
In the case of a Hamiltonian torus action, this index condition is satisfied. Thus we are allowed to directly apply it.
Proposition 8. Let \( \tilde{\mu} : \mathfrak{t}^k \rightarrow C^\infty(M) \) be a comoment map for a Hamiltonian torus action. Then for any vector \( X \in \mathfrak{t}^n \), the function \( \tilde{\mu}(X) \) is a Morse–Bott function, and \( \dim Z \), \( i _ d \), \( i _ a \) are all even for all critical components \( Z \). Moreover, \( Z \) is a symplectic submanifold of \( M \).
Proof. Let \( z \in Z \) be a critical point of \( H = \tilde{\mu}(X) \). We may assume that \( X \) generates a dense subset of the whole torus \( T^m \), since we can restrict attention to the closure of \( \exp(tX) \) otherwise. Since the Hamiltonian vector field associated to \( H \) vanishes at \( z \), it follows that \( z \in M \) is a fixed point of the \( T^m \)-action.
Now there is some "equivariant" version of Darboux’s theorem, which gives a good coordinate chart \( (x _ 1, \ldots, x _ n, y _ 1, \ldots, y _ n) \) near \( z \), such that
- the symplectic form is given by \( \omega = dx _ 1 \wedge dy _ 1 + \cdots + dx _ n \wedge dy _ n \),
- each function in the image of \( \tilde{\mu} \) takes the form of \( c + a _ 1 \lvert z _ 1 \rvert^2 + \cdots + a _ n \lvert z _ n \rvert^2 \), where \( z _ j = x _ j + i y _ j \).
The negative \( a _ i \) contribute to the descending index, positive \( a _ i \) to the ascending index, and \( a _ i = 0 \) to the dimension of \( Z \). This immediately implies that \( i _ d \), \( i _ a \), and \( \dim Z \) are all even. It further shows that \( Z \) is a symplectic submanifold of \( M \). ▨
Corollary 9. Let \( (M, \omega) \) be a compact connected manifold and \( \tilde{\mu} : \mathfrak{t}^{\ast k} \rightarrow C^\infty(M) \) be a moment map for a Hamiltonian torus action. Then for any \( X \in \mathfrak{t}^{\ast k} \), all nonempty level sets of \( H = \tilde{\mu}(X) \) are connected.
Now we are ready to prove the convexity theorem.
3. Convexity theorem of Atiyah and Gullemin–Sternberg#
Let me state the convexity theorem again.
Theorem 10. Let \( (M, \omega) \) be a compact connected symplectic manifold and \( \mu : M \rightarrow \mathfrak{t}^{\ast k} \) be a moment map for a Hamiltonian torus action. Let \( Z _ 1, \ldots, Z _ m \) be the connected components of the fixed points of the torus action. Then each \( \mu(Z _ j) = c _ j \) is a point and the image of \( \mu \) is the convex hull of the points \( c _ j \).
In my opinion, the proof given in [Aud04] is quite technical from now on. Let’s start by proving another theorem that doesn’t look related too much.
Theorem 11. Let \( (M, \omega) \) be a compact connected symplectic manifold with a comoment map \( \tilde{\mu} : \mathfrak{t}^{\ast l} \rightarrow C^\infty(M) \) of a Hamiltonian torus action. Take arbitrary \( X _ 1, \ldots, X _ k \in \mathfrak{t}^{\ast l} \) and define functions \( f _ j = \tilde{\mu}(X _ j) \). Then for every \( \xi \in \mathbb{R}^k \), the level set \( f^{-1}(\xi) \subseteq M \) is empty or connected.
Proof. The case \( k = 1 \) is exactly Corollary 9. We proceed by induction.
Assume that the statement is true for some \( k \), and now consider \( k+1 \) functions \( f _ 1, \ldots, f _ {k+1} \). Roughly the argument goes like this. If there is no regular value of \( f \) that is also in the image of \( f \), then this means that \( df _ 1, \ldots, df _ {k+1} \) are linearly dependent. In this case, one can just apply the inductive hypothesis. Consider now the case when there is a regular value of \( f \) that is in the image. Then the set of regular values is dense in the image of \( \mu \). Since "having connected fiber" is a closed condition, it suffices to show that \( f^{-1}(\xi) \) is connected for all regular values \( \xi \). (We can take the limit of a sequence of regular values converging to the given \( \xi \).)
Thus we are assuming that \( \xi \) a regular value for \( f \) and is contained in the image of \( f \). By a similar argument we can also assume that \( (\xi _ 1, \ldots, \xi _ k) \) is a regular value for \( (f _ 1, \ldots, f _ k) \). Then \[ \displaystyle N = f _ 1^{-1}(\xi _ 1) \cap \cdots \cap f _ k^{-1}(\xi _ k) \] is a connected submanifold of \( M \) by the induction hypothesis. We will be happy if we can apply Corollary 7 to \( N \) and \( f _ {k+1} \).
Consider the critical points of \( f _ {k+1} \vert _ N \) in \( N \). These are the points \( x \in N \) where there is a representation \[ \displaystyle df _ {k+1}(x) = \lambda _ 1(x) df _ 1 + \cdots + \lambda _ k(x) df _ k(x) \in T _ x^\ast M. \] Here, \( \lambda _ j(x) \) are uniquely determined since \( df _ 1, \ldots, df _ k \) are linearly independent by regularity of \( (\xi _ 1, \ldots, \xi _ k) \). It can be proven that \( \lambda _ j(x) \) are locally constant on the set of critical points of \( f _ {k+1} \vert _ N \). (This is hard and uses the fact that the functions come from a Hamiltonian action.)
Anyways, given a critical point \( x \) of \( f _ {k+1} \vert _ N \), consider the function \( F = f _ {k+1} - \lambda _ 1(x) f _ 1 - \cdots - \lambda _ k(x) f _ k \) so that \( dF(x) = 0 \). This function \( F \) also is in the image of \( \tilde{\mu} \). Let \( Z \) be the component of the critical points of \( F \) that contains \( x \). With this definition, note that \( Z \cap N \) is (near \( x \)) the critical points of \( f _ {k+1} \vert N \).
We claim that \( Z \) and \( N \) transverse at \( x \). This is equivalent to showing that \( df _ 1 \vert _ Z(x), \ldots, df _ k \vert _ Z(x) \) are linearly independent. Since the Hamiltonian vector flows for \( f _ j \) and \( F \) commute, the set \( Z \) is preserved under the Hamiltonian vector flow of \( f _ j \). That is, the vectors \( X _ {f _ j}(x) \) are tangent to \( Z \) at \( x \). Since \( df _ 1(x), \ldots, df _ k(x) \) are linearly independent, the vectors \( X _ {f _ 1}(x), \ldots, X _ {f _ k}(x) \) are linearly independent. Because \( Z \) is a symplectic submanifold, it follows that \( df _ 1 \vert _ Z(x), \ldots, df _ k \vert _ Z(x) \) are linearly independent. Thus \( Z \) and \( N \) are transversal at \( x \).
This implies that the descending index of \( Z \cap N \) for \( f _ {k+1} \vert _ N \) is the same as the descending index of \( Z \) for \( f _ {k+1} \). Since \( f _ {k+1} \) is in the image of \( \tilde{\mu} \), this index is even. Likewise, the ascending index of \( Z \cap N \) for \( f _ {k+1} \vert _ N \) is even. We can now apply Corollary 7 to conclude that \( f^{-1}(\xi) = f _ {k+1}^{-1}(\xi _ {k+1}) \subseteq N \) is connected. ▨
From this, it is not hard to deduce the convexity theorem.
Corollary 12. If \( (M, \omega) \) is a connected compact symplectic manifold and \( \mu : M \rightarrow \mathfrak{t}^{k} \) is a moment map of a Hamiltonian torus action, then the image of \( \mu \) is convex.
Proof. It suffices to show that the intersection of \( \mathrm{im}(\mu) \) and an arbitrary line \( \ell \) is connected. But note that the inverse image \( \mu^{-1}(\ell) \) can be represented as level set of \( k-1 \) functions. Thus by Theorem 11, the set \( \mu^{-1}(\ell) \) is connected. Its image under \( \mu \) is \( \mathrm{im}(\mu) \cap \ell \), and it also has to be connected. ▨
Proof. } Since \( Z _ j \) are the components of the fixed points of the torus action, it is clear that \( f(Z _ j) = c _ j \) is a single point. A generic vector \( X \in \mathfrak{t}^k \) generates a dense subgroup of \( T^k \). Then the critical points of \( \tilde{\mu}(X) \) are exactly the fixed point of the torus action, which is \( \bigcup _ j Z _ j \). By Corollary 9 the minimum of \( \tilde{\mu}(X) = \langle \mu(-), X \rangle \) is obtained at some unique component \( Z _ j \). Since this is true for generic \( X \), it follows that \( \mathrm{im}(\mu) \) is a convex polytope with \( c _ j \) as vertices. But \( \mathrm{im}(\mu) \) is a convex set containing all of \( c _ j \). Therefore \( \mathrm{im}(\mu) \) is the convex hull of \( c _ j \). ▨
We would like to see some more examples. For instance, what polytopes arise as an image of a moment map? Or given the image of \( \mu \), what information of \( M \) can be distill out of it? To answer them, we need an apparatus for constructing or manipulating symplectic manifolds. This is going to be the topic of the sequel.
Before ending this post, I would like to point out one thing. Recall that the "equivariant" version of Darboux’s theorem gave a full description of the torus action near a fixed point. If \( p \) is fixed by \( T^k \), then there is a good coordinate chart \( (z _ 1, \ldots, z _ n) \) near \( x \) such that
- \( p \) has the coordinate \( (0, \ldots, 0) \),
- the symplectic form is \( \omega = dx _ 1 \wedge dy _ 1 + \cdots + dx _ n \wedge dy _ n \),
- the action is given by \[ \displaystyle (\theta _ 1, \ldots, \theta _ k) \cdot z = (e^{i(a _ {11} \theta _ 1 + \cdots + a _ {1k} \theta _ k)} z _ 1, \ldots, e^{i(a _ {n1} \theta _ 1 + \cdots + a _ {nk} \theta _ k)} z _ n). \]
Then the \( j \)-th component of the moment map is (possible with some sign) \[ \displaystyle \mu _ j(z) = \mu _ j(0) + a _ {1j} \lvert z _ 1 \rvert^2 + a _ {2j} \lvert z _ 2 \rvert^2 + \cdots + a _ {nj} \lvert z _ n \rvert^2. \] This means that the image of \( \mu \) near \( p \) is going to look like the image of \( \{ x : x _ j \ge 0 \} \subseteq \mathbb{R}^n \) under a linear map \( \mathbb{R}^n \rightarrow \mathbb{R}^k \).
References#
[Ati82] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416
[Aud04] Michèle Audin, Torus actions on symplectic manifolds, revised ed., Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 2004. MR 2091310
[GS82] V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117