Moment polytope - Knowlpedia

Let a compact torus $T$ , with Lie algebra $\mathfrak t$ , act in a Hamiltonian fashion on a compact connected symplectic manifold $(M,\omega)$ , and let $\mu:M\to\mathfrak t^*$ be a moment map. The moment polytope is

\Delta(M,\omega,\mu)=\mu(M)\subseteq\mathfrak t^*.

The Atiyah–Guillemin–Sternberg convexity theorem guarantees that this image is a convex polytope. Because a torus moment map is determined only up to addition of a constant in $\mathfrak t^*$ , the polytope is likewise defined only up to translation unless a normalization of $\mu$ is chosen.

Fixed points and lattice structure

The polytope is the convex hull of the moment-map values on the fixed-point set $M^T$ . Every component of $M^T$ maps to a single point. Its faces and edge directions reflect the isotropy weights of the torus action, hence are rational relative to the weight lattice determined by $\ker(\exp:\mathfrak t\to T)$ .

Translation does not change face directions or the normal fan, but it does change the numerical coordinates of vertices. Integrality of the vertices is an extra prequantization condition, not a consequence of Hamiltonianity alone.

Standard examples

For the standard $T^n$ -action on complex projective space $\mathbb{CP}^n$ , with a normalized Fubini–Study form, the moment polytope is an $n$ -simplex. Rescaling the symplectic form rescales the simplex, and changing the additive normalization translates it.

For a compact connected symplectic toric manifold, the polytope has dimension $n=\frac12\dim M$ and satisfies the Delzant smoothness condition. For a noneffective action, the polytope lies in a proper affine subspace corresponding to the effective quotient torus.

Conventions and scope

Some authors say “moment polytope” for the intersection of a nonabelian moment-map image with a chosen positive Weyl chamber. That is a related nonabelian construction, not the torus image defined here.

Compactness and connectedness matter. For a noncompact Hamiltonian $T$ -space, the image may be unbounded or fail to be a polytope without properness and convexity hypotheses. A general smooth map into $\mathfrak t^*$ is not a moment map, so its image does not acquire this structure merely from being compact.

References

V. Guillemin, Moment Maps and Combinatorial Invariants of Hamiltonian $T^n$ -Spaces, Birkhäuser, 1994. DOI record. Relevant: Chapter 1, moment images, isotropy weights, and rational polytopes.
M. Audin, Torus Actions on Symplectic Manifolds, 2nd revised ed., Birkhäuser, 2004. DOI record. Relevant: Chapter IV, convexity and moment polytopes.