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 TMF, 2000, Volume 123, Number 2, Pages 237–263 (Mi tmf600)

Nonautonomous Hamiltonian systems related to higher Hitchin integrals

A. M. Levinab, M. A. Olshanetskyca

a Max Planck Institute for Mathematics
b P. P. Shirshov institute of Oceanology of RAS
c Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We describe nonautonomous Hamiltonian systems derived from the Hitchin integrable systems. The Hitchin integrals of motion depend on $\mathcal W$-structures of the basic curve. The parameters of the $\mathcal W$-structures play the role of times. In particular, the quadratic integrals depend on the complex structure (the $\mathcal W_2$-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 monodromy-preserving conditions with respect to changing the $\mathcal W_k$-structures $(k>2)$. They are derived by the symplectic reduction of a gauge field theory on the basic curve interacting with the $\mathcal W_k$-gravity. As a by-product, we obtain the classical Ward identities in this theory.

DOI: https://doi.org/10.4213/tmf600

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English version:
Theoretical and Mathematical Physics, 2000, 123:2, 609–632

Bibliographic databases:

Citation: A. M. Levin, M. A. Olshanetsky, “Nonautonomous Hamiltonian systems related to higher Hitchin integrals”, TMF, 123:2 (2000), 237–263; Theoret. and Math. Phys., 123:2 (2000), 609–632

Citation in format AMSBIB
\Bibitem{LevOls00} \by A.~M.~Levin, M.~A.~Olshanetsky \paper Nonautonomous Hamiltonian systems related to higher Hitchin integrals \jour TMF \yr 2000 \vol 123 \issue 2 \pages 237--263 \mathnet{http://mi.mathnet.ru/tmf600} \crossref{https://doi.org/10.4213/tmf600} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1794158} \zmath{https://zbmath.org/?q=an:0970.37044} \transl \jour Theoret. and Math. Phys. \yr 2000 \vol 123 \issue 2 \pages 609--632 \crossref{https://doi.org/10.1007/BF02551395} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000165897000006} 

• http://mi.mathnet.ru/eng/tmf600
• https://doi.org/10.4213/tmf600
• http://mi.mathnet.ru/eng/tmf/v123/i2/p237

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This publication is cited in the following articles:
1. Levin, AM, “Painlevé VI, rigid tops and reflection equation”, Communications in Mathematical Physics, 268:1 (2006), 67
2. Hurtubise, J, “On the geometry of isomonodromic deformations”, Journal of Geometry and Physics, 58:10 (2008), 1394
3. A. V. Zotov, A. V. Smirnov, “Modifications of bundles, elliptic integrable systems, and related problems”, Theoret. and Math. Phys., 177:1 (2013), 1281–1338
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