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Cluster Toda Chains and Nekrasov Functions

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Abstract

We extend the relation between cluster integrable systems and q-difference equations beyond the Painlev´e case. We consider the class of hyperelliptic curves where the Newton polygons contain only four boundary points. We present the corresponding cluster integrable Toda systems and identify their discrete automorphisms with certain reductions of the Hirota difference equation. We also construct nonautonomous versions of these equations and find that their solutions are expressed in terms of five-dimensional Nekrasov functions with Chern–Simons contributions, while these equations in the autonomous case are solved in terms of Riemann theta functions.

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Author information

Authors and Affiliations

  1. Landau Institute for Theoretical Physics, RAS, Moscow Oblast, Chernogolovka, Russia

    M. A. Bershtein, P. G. Gavrylenko & A. V. Marshakov

  2. Laboratory for Representation Theory and Mathematical Physics, Mathematics Faculty, National Research University Higher School of Economics, Moscow, Russia

    M. A. Bershtein, P. G. Gavrylenko & A. V. Marshakov

  3. Center for Advanced Studies, Skoltech, Moscow, Russia

    M. A. Bershtein

  4. Independent University of Moscow, Moscow, Russia

    M. A. Bershtein

  5. Institute for Information Transmission Problems, RAS, Moscow, Russia

    M. A. Bershtein

  6. Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

    P. G. Gavrylenko

  7. Institute for Theoretical and Experimental Physics, Moscow, Russia

    A. V. Marshakov

  8. Theory Department, Lebedev Physical Institute, RAS, Moscow, Russia

    A. V. Marshakov

Authors
  1. M. A. Bershtein
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  2. P. G. Gavrylenko
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  3. A. V. Marshakov
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Corresponding author

Correspondence to M. A. Bershtein.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 2, pp. 179–214, February, 2019.

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Bershtein, M.A., Gavrylenko, P.G. & Marshakov, A.V. Cluster Toda Chains and Nekrasov Functions. Theor Math Phys 198, 157–188 (2019). https://doi.org/10.1134/S0040577919020016

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  • DOI: https://doi.org/10.1134/S0040577919020016

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