Seiberg-Witten curves and double-elliptic integrable systems

  • G. Aminov
  • H. W. Braden
  • A. Mironov
  • A. Morozov
  • A. Zotov
Open Access
Regular Article - Theoretical Physics


An old conjecture claims that commuting Hamiltonians of the double-elliptic integrable system are constructed from the theta-functions associated with Riemann surfaces from the Seiberg-Witten family, with moduli treated as dynamical variables and the Seiberg-Witten differential providing the pre-symplectic structure. We describe a number of theta-constant equations needed to prove this conjecture for the N-particle system. These equations provide an alternative method to derive the Seiberg-Witten prepotential and we illustrate this by calculating the perturbative contribution. We provide evidence that the solutions to the commutativity equations are exhausted by the double-elliptic system and its degenerations (Calogero and Ruijsenaars systems). Further, the theta-function identities that lie behind the Poisson commutativity of the three-particle Hamiltonians are proven.


Supersymmetric gauge theory Integrable Hierarchies Integrable Equations in Physics 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2015

Authors and Affiliations

  • G. Aminov
    • 1
    • 2
  • H. W. Braden
    • 3
  • A. Mironov
    • 1
    • 4
    • 5
  • A. Morozov
    • 1
    • 5
  • A. Zotov
    • 1
    • 2
    • 6
  1. 1.ITEPMoscowRussia
  2. 2.Moscow Institute of Physics and Technology, Institute al.DolgoprudnyRussia
  3. 3.School of Mathematics, The King’s BuildingsUniversity of EdinburghEdinburghScotland
  4. 4.Theory DepartmentLebedev Physics InstituteMoscowRussia
  5. 5.Moscow Physical Engineering InstituteMoscowRussia
  6. 6.Steklov Mathematical InstituteRASMoscowRussia

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