Asymptotics of action variables near semi-toric singularities

TL;DR

This paper investigates the local behavior of action variables near focus-focus singularities in semi-toric integrable systems, revealing that their multi-valuedness is characterized by a complex logarithm and computing the associated monodromy.

Contribution

It generalizes Vu Ngoc's result to higher dimensions, providing a detailed description of the singular behavior and monodromy in semi-toric systems.

Findings

Action variables exhibit multi-valuedness described by a complex logarithm.

Monodromy matrix for semi-toric systems is explicitly calculated.

The results extend understanding of singularities in integrable Hamiltonian systems.

Abstract

The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the Liouville-Arnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behaviour and the multi-valuedness of the Action integrals is given by a complex logarithm. This extends a previous result by Vu Ngoc to any dimension. We also calculate the monodromy matrix for these systems.