So recently I have been studying semi-toric systems which are a generalization of toric symplectic manifolds and allow for the presence of focus-focus fibers. These were proved to be classified by $5$ invariants, one of them is the polygon invariant which is an adaptation from the standard momentum map invariant coming from toric systems. In particular this polygon invariant is a rational convex polygon.
Now from what I understand, much like toric symplectic manifolds, one has for toric symplectic orbifolds a classification theorem analogous to Delzant's theorem for toric manifolds. It says that these are classified by rational convex polygons. Therefore I was wondering if anyone knows of a link between toric symplectic orbifolds and semi-toric systems, making use of the polygon invariant and the classification results of toric symplectic orbifolds, for example if somehow one can relate the focus-focus fiber of the semi-toric system with some fiber of symplectic toric orbifold ?
I am no expert in this, so any insight is appreciated, thanks in advance.
