On the affine invariant of simple hypersemitoric systems

TL;DR

This paper introduces an affine invariant for hypersemitoric systems, extending known invariants from toric and semitoric systems, with computations for various examples.

Contribution

It defines a new affine invariant for hypersemitoric systems, generalizing existing invariants and providing explicit computations for complex cases.

Findings

The affine invariant generalizes Delzant and semitoric invariants.

Explicit computation and visualization of the invariant for multiple examples.

The invariant captures the structure of hypersemitoric systems.

Abstract

Hypersemitoric systems are a class of integrable systems on $4$-dimensional symplectic manifolds which only have mildly degenerate singularities and where one of the integrals induces an effective Hamiltonian $S^1$-action and is proper. We introduce the affine invariant of hypersemitoric systems, which is a generalization of the Delzant polytope of toric systems and the polytope invariant of semitoric systems. Along the way, we compute and plot this invariant for meaningful and more and more complicated examples.