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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 1(415), Pages 39–124 (Mi umn9576)

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 12-02-0059412-01-33071_ìîë_à_âåä Ministry of Education and Science of the Russian Federation ÍØ-4724.2014.211.G34.31.0023 Dynasty Foundation This work was supported by the Russian Foundation for Basic Research (grant no. 12-02-00594 and grant no. 12-01-33071-ìîë-à-âåä for young researchers) and by the Programme "Leading Scientific Schools" (grant no. ÍØ-4724.2014.2). The first author was also supported by the Laboratory of Algebraic Geometry and its Applications at the National Research University "Higher School of Economics" (Agreement 11.G34.31.0023 of the Government of the Russian Federation). The third author was also supported by Dmitrii Zimin's "Dynasty" Foundation.

DOI: https://doi.org/10.4213/rm9576

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English version:
Russian Mathematical Surveys, 2014, 69:1, 35–118

Bibliographic databases:

ArXiv: 1311.4498
UDC: 514.7+514.8+517.923
MSC: Primary 34M56, 14H60; Secondary 14H70, 17B80

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

Citation in format AMSBIB
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