Root systems and symmetries of torus manifolds

TL;DR

This paper establishes a connection between root systems and torus manifolds, showing that symmetries extend to Lie groups of specific types, and provides an alternative proof to a known theorem with implications for complex structures.

Contribution

It introduces a root system framework for torus manifolds and characterizes the types of Lie groups acting on them, offering a new proof of Wiemeler's theorem.

Findings

Irreducible root subsystems are of type A, B, or D.

If a torus action extends to a Lie group, its simple factors are of type A, B, or D.

For invariant stably complex structures, only type A appears.

Abstract

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected closed smooth manifold with a smooth effective action of an $n$-dimensional compact torus having a fixed point, and show that if the torus action extends to a smooth action of a connected compact Lie group $G$, then a simple factor of the Lie algebra of $G$ is of type A, B or D. This gives an alternative proof to Wiemeler's theorem. We also discuss a similar problem for a torus manifold with an invariant stably complex structure. In this case only type A appears.