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This article is cited in 18 scientific papers (total in 18 papers)
Classification of isomonodromy problems on elliptic curves
A. M. Levinab, M. A. Olshanetskyac, A. V. Zotovdac a Institute for Theoretical and Experimental Physics
b Laboratory of Algebraic Geometry, National Research University "Higher School of Economics"
c Moscow Institute of Physics and Technology
d Steklov Mathematical Institute of Russian Academy of Sciences
Abstract:
This paper describes isomonodromy problems in terms of flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ and connections with regular singularities at marked points. The bundles are classified by their characteristic classes, which are elements of the second cohomology group $H^2(\Sigma_\tau,{\mathscr Z}(G))$, where ${\mathscr Z}(G)$ is the centre of $G$. For any complex simple Lie group $G$ and any characteristic class the moduli space of flat connections is defined, and for them the monodromy-preserving deformation equations are given in Hamiltonian form together with the corresponding Lax representation. In particular, they include the Painlevé VI equation, its multicomponent generalizations, and the elliptic Schlesinger equations. The general construction is described for punctured complex curves of arbitrary genus. The Drinfeld–Simpson (double coset) description of the moduli space of Higgs bundles is generalized to the case of the space of flat connections. This local description makes it possible to establish the Symplectic Hecke Correspondence for a wide class of monodromy-preserving problems classified by the characteristic classes of the underlying bundles. In particular, the Painlevé VI equation can be described in terms of $\operatorname{SL}(2,{\mathbb C})$-bundles. Since ${\mathscr Z}(\operatorname{SL}(2,{\mathbb C}))={\mathbb Z}_2$, the Painlevé VI equation has two representations related by the Hecke transformation: 1) as the well-known elliptic form of the Painlevé VI equation (for trivial bundles); 2) as the non-autonomous Zhukovsky–Volterra gyrostat (for non-trivial bundles).
Bibliography: 123 titles.
Keywords:
monodromy-preserving deformations, Painlevé equations, flat connections, Schlesinger systems, Higgs bundles.
Received: 15.11.2013
Citation:
A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Classification of isomonodromy problems on elliptic curves”, Uspekhi Mat. Nauk, 69:1(415) (2014), 39–124; Russian Math. Surveys, 69:1 (2014), 35–118
Linking options:
https://www.mathnet.ru/eng/rm9576https://doi.org/10.1070/RM2014v069n01ABEH004878 https://www.mathnet.ru/eng/rm/v69/i1/p39
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