Geodesic
Nonautonomous Hamiltonian systems related to higher Hitchin integrals
Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 237-263
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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 Teichmьller 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. 
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A. M. Levin; M. A. Olshanetsky. Nonautonomous Hamiltonian systems related to higher Hitchin integrals. Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 237-263. http://geodesic.mathdoc.fr/item/TMF_2000_123_2_a6/

[1] C. M. Hull, Lectures on ${\mathcal W}$-gravity, ${\mathcal W}$-geometry and ${\mathcal W}$-strings, E-print hep-th/9302110

[2] M. A. Gabeskeria, M. V. Saveliev, Lax type representation for the embeddings of Riemann manifolds, Preprint IHEP, IHEP, Protvino, 1983

[3] M. Ugaglia, On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEs, E-print solv-int/9902006 | MR

[4] N. Hitchin, Duke Math. J., 54 (1987), 91–114 | DOI | MR | Zbl

[5] B. van Geemen, E. Previato, On the Hitchin System, E-print alg-geom/9410015 | MR

[6] K. Gawedzki, P. Tran-Ngog-Bich, Hitchin systems at low genera, E-print hep-th/9803101 | MR

[7] N. Nekrasov, Commun. Math. Phys., 180 (1996), 587–604 ; E-print hep-th/9503157 | DOI | MR

[8] A. Levin, M. Olshanetsky, Am. Math. Soc. Transl. 2, 191 (1999) | Zbl

[9] A. Beilinson, V. Drinfeld, Opers. Preprint, 1994

[10] V. Fock, Towards the geometrical sense of operator expansions for chiral currents and $W$-algebras, Preprint ITEP, ITEP, Moscow, 1990; A. Bilal, V. Fock, Ia Kogan, Nucl. Phys. B, 359 (1991), 635–672 | DOI | MR

[11] A. Gerasimov, A. Levin, A. Marshakov, Nucl. Phys. B, 360 (1991), 537–558 | DOI | MR

[12] I. Krichever, Commun. Pure Appl. Math., 47:4 (1994), 437–475 | DOI | MR | Zbl

[13] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1974 | MR