Hamiltonian Classification of toric fibres and symmetric probes

TL;DR

This paper investigates when two toric fibres in a symplectic manifold are Hamiltonian equivalent, introduces a new symmetric probe method for constructing such equivalences, and determines the Hamiltonian monodromy groups for key examples.

Contribution

It introduces symmetric McDuff's probes for classifying toric fibres and proves a conjecture that these generate all equivalences in several important cases.

Findings

New symmetric probe method for toric fibre equivalences

Conjecture that symmetric probes generate all equivalences, proven in key cases

Determination of Hamiltonian monodromy groups for several examples

Abstract

In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On the construction side of this question, we introduce a new method of constructing equivalences of toric fibres by using a symmetric version of McDuff's probes (see arXiv:0904.1686 and arXiv:1203.1074). On the other hand, we derive some obstructions to such equivalence by using Chekanov's classification of product tori together with a lifting trick from toric geometry. Furthermore, we conjecture that (iterated) symmetric probes yield all possible equivalences and prove this conjecture for $\mathbb{C}^n,\mathbb{C}P^2, \mathbb{C} \times S^2, \mathbb{C}^2 \times T^*S^1, T^*S^1 \times S^2$ and monotone $S^2 \times S^2$. This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, i.e. determining which automorphisms of the homology of the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the toric fibre in question to itself. For the above list of examples, we determine the Hamiltonian monodromy group for all toric fibres.