Topological properties of Hamiltonian circle actions

By D. Mcduff, S. Tolman. 2004

114 citations recorded by the source corpus.

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,ω). Our main tool is the Seidel representation of π1(Ham(M,ω)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π1(Ham(M,ω)). Further, if the isotropy has order at most two and the circle contracts in Ham(M,ω) then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval [−a, a]. If the action is semifree (i.e. the isotropy weights are 0 or ±1) then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of M to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov–Witten invariants on symplectic manifolds with S1-actions.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa4fed2ce5298a2a203da75a1dec52e62a55798d

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Citations in the corpus

Bounds on spectral norms and barcodes (2018, methodology): “s of Proposition 15. For use in this section, we recall that an S1-action on a space Xis called semi-free if the stabilizer of each point x2Xis either 0 or S1:The main references for this section are [76] and [61]. 3.1. CPn. We start with Case (2) of Theorem G, and show that it satises the condition”
Lagrangian Floer theory on compact toric manifolds II: bulk deformations (2008, methodology): “roof of Proposition 6.7 besides the ﬁxed point localization. One is written by Cieliebak and Salamon [CS] which uses vortex equations (gauged sigma model) and the other is written by McDuﬀ and Tolman [MT] which uses Seidel’s result [Se1]. 7. Localizationofquantumcohomologyringat momentpolytope In th”
QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, methodology): “Related computations can be found in McDuff-Tolman [22] and Iritani [19].”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, background): “Legendre for bringing to my attention the work [69], and to G.”