Seiberg-Witten curves and double-elliptic integrable systems
Abstract
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 -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.
Cite
@article{arxiv.1410.0698,
title = {Seiberg-Witten curves and double-elliptic integrable systems},
author = {G. Aminov and H. W. Braden and A. Mironov and A. Morozov and A. Zotov},
journal= {arXiv preprint arXiv:1410.0698},
year = {2015}
}