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

SIGMA 14 (2018), 084, 66 pages      arXiv:1706.09935      https://doi.org/10.3842/SIGMA.2018.084

Faithful Semitoric Systems

Sonja Hohloch a, Silvia Sabatini b, Daniele Sepe c and Margaret Symington d
a) Department of Mathematics - Computer Science, University of Antwerpen, Campus Middelheim, Building G, M.G.211, Middelheimlaan 1, 2020 Antwerpen, Belgium
b) Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931 Köln, Germany
c) Universidade Federal Fluminense, Instituto de Matemática, Departamento de Matemática Aplicada, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H, Campus do Gragoatá, CEP 24210-201, Niterói, RJ, Brazil
d) Department of Mathematics, Mercer University, 1501 Mercer University Drive, Macon, GA 31207, USA

Received July 07, 2017, in final form July 30, 2018; Published online August 16, 2018

Abstract
This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vũ Ngoc and classified by Pelayo and Vũ Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

Key words: completely integrable Hamiltonian systems; almost toric systems; semitoric systems; integral affine geometry; focus-focus singularities.

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