Symmetry of a symplectic toric manifold

TL;DR

This paper characterizes the symmetry group of a symplectic toric manifold using a root system derived from its moment polytope, revealing that the group's structure is constrained by the polytope's combinatorial properties.

Contribution

It introduces a root system associated with the moment polytope and describes the structure of maximal compact Lie subgroups of symplectomorphisms containing the torus.

Findings

Any irreducible subsystem of R(P) is of type A.

The root system of G is a subsystem of R(P).

A maximal compact Lie subgroup G_max containing T is identified.

Abstract

The action of a torus group $T$ on a symplectic toric manifold $(M,\omega)$ often extends to an effective action of a (non-abelian) compact Lie group $G$. We may think of $T$ and $G$ as compact Lie subgroups of the symplectomorphism group $Symp(M,\omega)$ of $(M,\omega)$. On the other hand, $(M,\omega)$ is determined by the associated moment polytope $P$ by the result of Delzant. Therefore, the group $G$ should be estimated in terms of $P$ or we may say that a maximal compact Lie subgroup of $Symp(M,\omega)$ containing the torus $T$ should be described in terms of $P$. In this paper, we introduce a root system $R(P)$ associated to $P$ and prove that any irreducible subsystem of $R(P)$ is of type A and the root system $\Delta(G)$ of the group $G$ is a subsystem of $R(P)$ (so that $R(P)$ gives an upper bound for the identity component of $G$ and any irreducible factor of $\Delta(G)$ is of type A). We also introduce a homomorphism from the normalizer of $T$ in $G$ to an automorphism group $Aut(P)$ of $P$, which detects the connected components of $G$. Finally we find a maximal compact Lie subgroup $G_{\max}$ of $Symp(M,\omega)$ containing the torus $T$.