An invitation to circle actions

Abstract

The problem of determining whether a manifold admits symmetries has been widely studied in mathematics and physics. It is in general hard to determine whether, given a Lie group $G$ and a manifold $M$, there exists a nontrivial action of $G$ on $M$ that preserves a prescribed structure. When $M$ is symplectic, for instance when $M$ is the phase space of a particle, having one conserved quantity whose associated (Hamiltonian) flow on the manifold is periodic, is equivalent to having a Hamiltonian circle action. The following questions are therefore natural: Which symplectic manifolds admit symplectic circle actions? What are their topological properties? Here we discuss these problems when the fixed point set is discrete. In particular we review some of the consequences of the fact that the Chern number $c_1{c}_{n-1}[†M]$‡ is completely determined by the fixed point data of the action.

For example, this allows us to construct an algorithm which, in some cases where the fixed point set is minimal, determines the possible representations at the fixed points of Hamiltonian circle actions with discrete fixed point sets. In general it can be applied towards proving the symplectic Petrie conjecture on these actions. Another application is the obtainment of lower bounds and divisibility results for the number of fixed points on the broader category of almost complex manifolds. This lower bound problem is related to the Kosniowski conjecture, which has been open since 1979. Finally, we show how to extend the 12 and 24 Theorems on the number of lattice points of reflexive polytopes of dimensions 2 and 3 to Delzant reflexive polytopes of any dimension.