Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 016, 43 pages      arXiv:1502.07698

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

Daniel M. Kane, Joseph Palmer and Álvaro Pelayo
University of California, San Diego, Department of Mathematics, 9500 Gilman Drive #0112, La Jolla, CA 92093-0112, USA

Received April 17, 2017, in final form February 13, 2018; Published online February 22, 2018

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.

Key words: symplectic geometry; integrable system; semitoric integrable systems; toric integrable systems; focus-focus singularities; ${\rm SL}_2({\mathbb Z})$.

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