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

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.