Theoretical and Mathematical Physics

, Volume 198, Issue 2, pp 157–188 | Cite as

Cluster Toda Chains and Nekrasov Functions

  • M. A. BershteinEmail author
  • P. G. Gavrylenko
  • A. V. Marshakov


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.


integrable system topological string cluster algebra supersymmetric gauge theory 


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • M. A. Bershtein
    • 1
    • 2
    • 3
    • 4
    • 5
    Email author
  • P. G. Gavrylenko
    • 1
    • 2
    • 6
  • A. V. Marshakov
    • 1
    • 2
    • 7
    • 8
  1. 1.Landau Institute for Theoretical Physics, RASMoscow Oblast, ChernogolovkaRussia
  2. 2.Laboratory for Representation Theory and Mathematical Physics, Mathematics FacultyNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Center for Advanced Studies, SkoltechMoscowRussia
  4. 4.Independent University of MoscowMoscowRussia
  5. 5.Institute for Information Transmission Problems, RASMoscowRussia
  6. 6.Bogolyubov Institute for Theoretical PhysicsKievUkraine
  7. 7.Institute for Theoretical and Experimental PhysicsMoscowRussia
  8. 8.Theory Department, Lebedev Physical Institute, RASMoscowRussia

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