Fractional Hamiltonian Monodromy

Abstract

We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy-momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:(−2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems.

Fractional Hamiltonian monodromy from a Gauss–Manin monodromy

“…The monodromy matrix, which is an automorphism of H 1 , is the holonomy of this connection. This construction can be generalized to fractional monodromy but only a subgroup of H 1 (T 2 (h, j), Z) can be transported continuously across the line C [NSZ06,ECS07]. These remarks can be understood by the following construction.…”

“…2.5) were formulated as conjectures in Refs. [NSZ06,Efs04]. They are based on the analysis of the quantum joint spectrum of the energy-momentum maps.…”

“…The non-trivial monodromy in the spherical pendulum is due to this singularity [Dui80,CB97]. Recently, the definition of Hamiltonian monodromy has been extended to characterize not only isolated singularities but also some types of non-isolated singularities, leading to the concept of Fractional Hamiltonian Monodromy [NSZ02,NSZ06,Efs04,ECS07]. More precisely, one considers an energy-momentum map with a 1-dimensional set C of weak critical values defined by the property that each point of this set lifts to a particular type of singular torus, a curled torus, i.e.…”

“…The formal extension of this action to the whole group leads to monodromy matrices with fractional coefficients and to the denomination fractional monodromy. One of the main motivations for the introduction of this new concept is given by the quantum manifestation of monodromy in the discrete joint spectrum of the energy-momentum map [Ngoc99,NSZ06]. This spectrum can be represented as a lattice of points in R 2 .…”

“…Using this construction, we recover the results of the real approach obtained in Refs. [NSZ02,NSZ06,Efs04,ECS07] for the 1:-2 resonance and Ref. [Nek07] for m : −n resonance.…”