2023/02/01 by Joé Brendel, Brendel, Joé · 1 citation
Mathematics · Physics and Astronomy · #Geometric and Algebraic Topology #Geometry and complex manifolds #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2302.00334
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 ℂn,ℂP2, ℂ × S2, ℂ2 × T^*S1, T^*S1 × S2 and monotone S2 × S2. 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.