Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

The n-particle periodic Toda chain is a well known example of an integrable but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold singularities of the Toda chain, ie points where there exist k independent linear relations amongst the gradients of the integrals of motion, coincide with points where there are k (doubly) degenerate eigenvalues of representatives L and Lbar of the two inequivalent classes of Lax matrices (corresponding to degenerate periodic or antiperiodic solutions of the associated second-order difference equation). The singularities are shown to be nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold. Sigma_k is shown to be of elliptic type, and the frequencies of transverse oscillations under Hamiltonians which fix Sigma_k are computed in terms of spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is given by the product of the holonomies (equal to +/- 1) of the even- (or odd-) indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio

Uncovering Fractional Monodromy

<p>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.</p>