Threeparticle integrable systems with elliptic dependence on momenta and theta function identities
Abstract
We claim that some nontrivial thetafunction identities at higher genus can stand behind the Poisson commutativity of the Hamiltonians of elliptic integrable systems, which were introduced in [1,2] and are made from the thetafunctions on Jacobians of the SeibergWitten curves. For the case of threeparticle systems the genus2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus2 thetafunction identities provide the direct way to construct the Poisson structure in terms of the coordinates on the Jacobian of the spectral curve and the elements of its period matrix. The Lax representations for the twoparticle systems are also obtained.
 Publication:

Physics Letters B
 Pub Date:
 November 2013
 DOI:
 10.1016/j.physletb.2013.09.004
 arXiv:
 arXiv:1307.1465
 Bibcode:
 2013PhLB..726..802A
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 Physics Letters B726 (2013) 802808