Abstract
The notion of monodromy was introduced by J.J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalizations. In particular, we will discuss the monodromy around a focus–focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.
Highlights
In the context of finite-dimensional integrable Hamiltonian systems, the notion of monodromy was introduced by Duistermaat in his seminal paper [31] published in 1980
The notion of fractional monodromy was introduced in the paper [65] as a generalization of the usual Duistermaat’s monodromy to the case of singular fibrations; it naturally appears in integrable systems with hyperbolic singularities
We briefly review a construction of these coordinates here and explain the relation to the definition of Hamiltonian monodromy given in the Introduction
Summary
In the context of finite-dimensional integrable Hamiltonian systems, the notion of monodromy was introduced by Duistermaat in his seminal paper [31] published in 1980. We shall come back to case of focus-focus singularities later in this paper, in connection with the classical Morse theory and principal circle bundles; this is the content of the recent topological theory of monodromy developed in [55] Another breakthrough in the monodromy theory was the quantum formulation of this invariant; first, for the quantum spherical pendulum [24,43] and later, in more generality, by S. The notion of fractional monodromy was introduced in the paper [65] as a generalization of the usual Duistermaat’s monodromy (sometimes referred to as Hamiltonian monodromy) to the case of singular fibrations; it naturally appears in integrable systems with hyperbolic singularities. Several parts of this work appeared in a more extended form in [55]
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