Seiberg-Witten curves and double-elliptic integrable systems
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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.
KeywordsSupersymmetric gauge theory Integrable Hierarchies Integrable Equations in Physics
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- P. Etingoff and A. Varchenko, Traces of intertwiners for quantum groups and difference equations, I, math/9907181.
- V. Tarasov and A. Varchenko, Difference equations compatible with trigonometric KZ differential equations, math/0002132.
- S. Mironov, A. Morozov and Y. Zenkevich, Generalized MacDonald polynomials and five dimensional AGT conjecture, to appear.Google Scholar
- R. Donagi and E. Markman, Cubics, integrable systems, and Calabi-Yau threefolds, in the proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, May 2–7, Ramat Gan, Israel (1996).Google Scholar
- J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics volume 352. Springer-Verlag, Berlin Germany (1973).Google Scholar