This article is cited in 3 papers

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

Daniel M. Kane, Joseph Palmer, Álvaro Pelayo

University of California, San Diego, Department of Mathematics, 9500 Gilman Drive #0112, La Jolla, CA 92093-0112, USA

Abstract: We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $\mathrm{SL}_2(\mathbb{Z})$ to its preimage in the universal cover of $\mathrm{SL}_2(\mathbb{R})$. 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.

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

MSC: 52B20; 15B36; 53D05

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

Language: English

DOI: 10.3842/SIGMA.2018.016

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