Parallel transport along Seifert manifolds and fractional monodromy

TL;DR

This paper develops a method to compute fractional monodromy in integrable Hamiltonian systems using parallel transport along Seifert manifolds, linking non-triviality to fixed points of circle actions.

Contribution

It introduces a general approach for calculating fractional monodromy in systems with circle symmetry, connecting it to fixed points via Seifert manifold transport.

Findings

Fractional monodromy relates to fixed points of circle actions.

Parallel transport along Seifert manifolds enables monodromy computation.

The method applies to various integrable Hamiltonian systems.

Abstract

The notion of fractional monodromy was introduced by Nekhoroshev, Sadovski\'{i} and Zhilinski\'{i} as a generalization of standard (`integer') monodromy in the sense of Duistermaat from torus bundles to singular torus fibrations. In the present paper we prove a general result that allows to compute fractional monodromy in various integrable Hamiltonian systems. In particular, we show that the non-triviality of fractional monodromy in 2 degrees of freedom systems with a Hamiltonian circle action is related only to the fixed points of the circle action. Our approach is based on the study of a specific notion of parallel transport along Seifert manifolds.