A Hamiltonian action of a torus $T = (S^1)^k$ on a compact connected symplectic manifold $(M^{2n}, \omega)$ with moment map $\mu: M \to \mathfrak{t}^* \cong \mathbb{R}^k$ is called a Hamiltonian torus action. The fixed point set $M^T = \{x \in M : t \cdot x = x, \forall t \in T\}$ is a union of symplectic submanifolds.

The moment polytope $\Delta = \mu(M) \subseteq \mathfrak{t}^* \cong \mathbb{R}^k$ is the image of the moment map. By the Atiyah-Guillemin-Sternberg theorem, $\Delta$ is a convex polytope whose vertices are exactly the images $\mu(F)$ of the connected components $F$ of the fixed point set $M^T$.

A Delzant polytope is a convex polytope in $\mathbb{R}^n$ satisfying: (1) $n$ edges meet at each vertex; (2) the edge directions at each vertex form a $\mathbb{Z}$-basis of $\mathbb{Z}^n$; (3) each facet is defined by $\langle x, v_i \rangle \geq \lambda_i$ with $v_i$ primitive in $\mathbb{Z}^n$. Delzant's theorem: the moment polytope of a toric manifold is Delzant, and conversely, every Delzant polytope arises from a unique (up to equivariant symplectomorphism) toric manifold.