Parallel Transport Along Seifert Manifolds and Fractional Monodromy

N Martynchuk,K Efstathiou

doi:10.1007/s00220-017-2988-5

Abstract

The notion of fractional monodromy was introduced by Nekhoroshev, Sadovskií and Zhilinskií 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 one 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.

Highlights

A fundamental notion in classical mechanics is the notion of Liouville integrability
We note that fractional monodromy is not a complete invariant of such fibrations—it contains less information than the marked molecule in Fomenko– Zieschang theory [6,7,19]—but it is important for applications and appears, for instance, in the so-called m:(−n) resonant systems [15,27,28,30,31]; see Sect. 4.1 for details. It was observed by Bolsinov et al [6] that in m:(−n) resonant systems the circle action defines a Seifert fibration on a small 3-sphere around the equilibrium point and that the Euler number of this fibration equals the number appearing in the matrix of fractional monodromy, cf
In [17] we have shown that if the circle action is free outside isolated fixed points standard monodromy can be completely determined by the weights 1:(±1) of the circle action at those points

Summary

Introduction

A fundamental notion in classical mechanics is the notion of Liouville integrability. It was observed by Bolsinov et al [6] that in m:(−n) resonant systems the circle action defines a Seifert fibration on a small 3-sphere around the equilibrium point and that the Euler number of this fibration equals the number appearing in the matrix of fractional monodromy, cf Remark 1. In the case when a Seifert fibration admits an equivariant filling, the Euler number is given by the fixed points of the circle action inside the filling manifold; see Theorem 4. These theorems specify the subgroup of homology cycles that admit parallel transport, and give a formula for the computation of the fractional monodromy These results, demonstrate that for standard and fractional monodromy the circle action is more important than the precise form of the integral map.

Parallel Transport along Seifert Manifolds

Monodromy in Integrable Systems

Examples

Proof of Theorem 3

Discussion