Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$

Symmetry Integrability and Geometry-Methods and Applications

article

Daniel M. Kane¹  Joseph Palmer¹  Alvaro Pelayo¹ 

[1] University of California, Department of Mathematics

关键词: symplectic geometry;    integrable system;    semitoric integrable systems;    toric integrable systems;    focus-focus singularities;    SL2(Z);   

DOI  :  10.3842/SIGMA.2018.016

来源: National Academy of Science of Ukraine

【 摘 要 】

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points and which are in the same twisting index class. In particular, we show that any semitoric system with precisely one focus-focus singular point can be continuously deformed into a system in the same isomorphism class as the Jaynes-Cummings model from optics.

【 授权许可】

Unknown