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

This article is cited in 10 scientific papers (total in 10 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.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-02-00594
12-01-33071_мол_а_вед
Ministry of Education and Science of the Russian Federation НШ-4724.2014.2
11.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-мол-а-вед for young researchers) and by the Programme "Leading Scientific Schools" (grant no. НШ-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

Full text: PDF file (1369 kB)
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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
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

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    This publication is cited in the following articles:
    1. G. Aminov, S. Arthamonov, A. Smirnov, A. Zotov, “Rational top and its classicalr-matrix”, J. Phys. A, 47:30 (2014), 305207, 19 pp.  crossref  mathscinet  zmath  isi  scopus
    2. A. Levin, M. Olshanetsky, A. Zotov, “Classical integrable systems and soliton equations related to eleven-vertex $R$-matrix”, Nuclear Phys. B, 887 (2014), 400–422  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. A. Levin, M. Olshanetsky, A. Zotov, “Planck constant as spectral parameter in integrable systems and KZB equations”, J. High Energ. Phys., 2014:10 (2014), 109  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. A. M. Levin, M. A. Olshanetsky, A. V. Zotov, “Quantum Baxter–Belavin $R$-matrices and multidimensional Lax pairs for Painlevé VI”, Theoret. and Math. Phys., 184:1 (2015), 924–939  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    5. JETP Letters, 101:9 (2015), 648–655  mathnet  crossref  crossref  isi  elib
    6. D. P. Novikov, B. I. Suleimanov, ““Quantization” of an isomonodromic Hamiltonian Garnier system with two degrees of freedom”, Theoret. and Math. Phys., 187:1 (2016), 479–496  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    7. B. I. Suleimanov, “Quantum aspects of the integrability of the third Painlevé equation and a non-stationary time Schrödinger equation with the Morse potential”, Ufa Math. J., 8:3 (2016), 136–154  mathnet  crossref  mathscinet  isi  elib
    8. V. A. Pavlenko, B. I. Suleimanov, ““Quantizations” of isomonodromic Hamilton system $H^{\frac{7}{2}+1}$”, Ufa Math. J., 9:4 (2017), 97–107  mathnet  crossref  isi  elib
    9. V. A. Pavlenko, B. I. Suleimanov, “Solutions to analogues of non-stationary Schrödinger equations defined by isomonodromic Hamilton system $H^{2+1+1+1}$”, Ufa Math. J., 10:4 (2018), 92–102  mathnet  crossref  isi
    10. Vasilyev M., Zotov A., “On Factorized Lax Pairs For Classical Many-Body Integrable Systems”, Rev. Math. Phys., 31:6 (2019), 1930002  crossref  isi
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