Action-Angle and Complex Coordinates on Toric Manifolds

Abstract

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds (M2n, ω) equipped with an effective Hamiltonian \(\mathbb {T}^n\cong (S^1)^n\)-action. We summarize the construction of M as a symplectic quotient of \(\mathbb {C}^d\), the \(\mathbb {T}^n\)-actions on M and their moment maps, and Guillemin’s Kähler potential on M. While the theories presented in this paper are for compact toric manifolds, they do carry over for some noncompact examples as well, such as the canonical line bundle KM, which is one of our main running examples, along with the complex projective space \(\mathbb {P}^n\) and its canonical bundle \(K_{\mathbb {P}^n}\). One main topic explored in this article is how to write the moment map in terms of the complex homogeneous coordinates \(z\in \mathbb {C}^d\), or equivalently, the relationship between the action-angle coordinates and the complex toric coordinates. We end with a brief review of homological mirror symmetry for toric geometries, where the main connection with the rest of the paper is that KM provides a prototypical class of examples of a Calabi-Yau toric manifold Y which serves as the total space of a symplectic fibration \(W: Y \to \mathbb {C}\) with a singular fiber above 0, known as a Landau-Ginzburg model in mirror symmetry. Here we write W in terms of the action-angle coordinates, which will prove to be useful in understanding the geometry of the fibration in our forthcoming work (Azam et al., Global homological mirror symmetry for genus two curves, in preparation).