Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z)

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2018
• Main Authors: Kane, D.M., Palmer, J., Pelayo, Á.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2018
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/209448
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Classifying Toric and Semitoric Fans by Lifting Equations from SL₂(Z) / D.M. Kane, J. Palmer, Á. Pelayo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 48 назв. — англ.

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL₂(Z) to its preimage in the universal cover of SL₂(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.