The affine invariant of generalized semitoric systems

TL;DR

This paper introduces a new symplectic invariant called the affine invariant for generalized semitoric systems, capturing their affine structure and singularities, extending previous constructions to broader classes of integrable systems.

Contribution

It constructs a generalized affine invariant for semitoric systems with focus-focus singularities, broadening the scope of previous invariants and providing a detailed geometric description.

Findings

The invariant encodes the affine structure of the system's image.

The invariant can be described as a union of regions with specific geometric properties.

For toric systems, the invariant reduces to a convex polygon.

Abstract

A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not necessarily proper. These systems can exhibit focus-focus singularities, which correspond to fibers of F which are topologically multipinched tori. The image F(M) is a singular affine manifold which contains a distinguished set of isolated points in its interior: the focus-focus values {(x_i,y_i)} of F. By performing a vertical cutting procedure along the lines {x:=x_i}, we construct a homeomorphism f : F(M) --> f(F(M)), which restricts to an affine diffeomorphism away from these vertical lines, and generalizes a construction of Vu Ngoc. The set \Delta:=f(F(M)) in R^2 is a symplectic invariant of (M,\omega,F), which encodes the affine structure of F. Moreover, \Delta may be described as a countable union of planar regions of four distinct types, where each type is defined as the region bounded between the graphs of two functions with various properties (piecewise linear, continuous, convex, etc). If F is a toric system, \Delta is a convex polygon (as proven by Atiyah and Guillemin-Sternberg) and f is the identity.