Uncovering fractional monodromy

K. Efstathiou and H.W. Broer
Communications in Mathematical Physics, 324(2), pp. 549-588, 2013
DOI: 10.1007/s00220-013-1816-9

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The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for \(n_1{:}(-n_2)\) resonant Hamiltonian systems with \(n_1\), \(n_2\) coprime natural numbers. We consider, in particular, systems that for \(n_1, n_2 > 1\) contain one-parameter families of singular fibres which are ‘curled tori’. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of \(n_1\), \(n_2\). Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.