Abstract
We describe nonautonomous Hamiltonian systems derived from the Hitchin integrable systems. The Hitchin integrals of motion depend on W-structures of the basic curve. The parameters of the W-structures play the role of times. In particular, the quadratic integrals depend on the complex structure (the W2-structure) of the basic curve, and the times are coordinates in the Teichmüller space. The corresponding flows are the monodromy-preserving equations such as the Schlesinger equations, the Painlevé VI equation, and their generalizations. The equations corresponding to the higher integrals are the monodromypreserving conditions with respect to changing the Wk-structures (k>2). They are derived by the sympletic reduction of a gauge field theory on the basic curve interacting with the Wk-gravity. As a by-product, we obtain the classical Ward identities in this theory.
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In memory of Mikhail Saveliev
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 237–263, May, 2000.
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Levin, A.M., Olshanetsky, M.A. Nonautonomous Hamiltonian systems related to higher Hitchin integrals. Theor Math Phys 123, 609–632 (2000). https://doi.org/10.1007/BF02551395
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DOI: https://doi.org/10.1007/BF02551395