Hamiltonian torus action

Definition

Let $T$ be a compact torus with Lie algebra $\mathfrak t$ , acting on a symplectic manifold $(M,\omega)$ . A Hamiltonian torus action is a Hamiltonian Lie group action with a map $\mu:M\to\mathfrak t^*$ satisfying

d\langle\mu,\xi\rangle=\iota_{\xi_M}\omega

for every $\xi\in\mathfrak t$ , and $\mu(tx)=\mu(x)$ for $t\in T$ . The second condition is equivariance: because $T$ is abelian, its coadjoint action on $\mathfrak t^*$ is trivial. The integral lattice $\ker(\exp:\mathfrak t\to T)$ records which infinitesimal generators integrate to periodic circle actions.

Components and normalization

Each $\xi\in\mathfrak t$ determines a Hamiltonian function

\mu^\xi=\langle\mu,\xi\rangle

whose Hamiltonian vector field is the infinitesimal generator $\xi_M$ . The moment map is determined only up to addition of a constant element of $\mathfrak t^*$ . Translating $\mu$ changes neither the action nor its component vector fields.

Lattice conventions vary: some authors define $T=\mathfrak t/\Lambda$ , while others take the exponential kernel to be $2\pi\Lambda$ . Statements involving integral weights or polytope labels must retain the chosen convention.

Convexity and toric actions

If $M$ is compact and connected, the Atiyah–Guillemin–Sternberg convexity theorem says that $\mu(M)$ is the convex hull of the images of the $T$ -fixed-point components; in particular it is a convex polytope Guillemin, chapter 1. This special convexity is one reason torus actions are more rigid than general Hamiltonian group actions.

When the action is effective and $\dim T=\frac12\dim M$ , the Hamiltonian $T$ -space is called symplectic toric. Compact connected symplectic toric manifolds are classified, up to the appropriate equivariant symplectomorphism, by Delzant polytopes.

Standard example

The coordinatewise action of $T^n$ on $\mathbb C^n$ ,

(e^{it_1},\ldots,e^{it_n})\cdot(z_1,\ldots,z_n) =(e^{it_1}z_1,\ldots,e^{it_n}z_n),

is Hamiltonian. For $\omega=\sum_j dx_j\wedge dy_j$ and the sign convention in the core, one moment map is

\mu(z)=-\frac12\bigl(|z_1|^2,\ldots,|z_n|^2\bigr).

Changing the convention for Hamiltonian vector fields reverses this sign.

References

Victor Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$ -Spaces, Birkhäuser, 1994. DOI record. Relevant: chapter 1, Hamiltonian torus actions, moment polytopes, and Delzant spaces.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: chapters on moment maps, symplectic reduction, and symplectic toric manifolds.

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