Phys. Lett. B, 2013, Volume 726, Issue 4, Pages 802–808
Three-particle integrable systems with elliptic dependence on momenta and theta function identities
G. Aminovab, A. Mironovac, A. Morozova, A. Zotovabd
a Institute of Theoretical and Experimental Physics, Moscow, Russia
b Moscow Institute of Physics and Technology, Dolgoprudny, Russia
c Theory Department, Lebedev Physics Institute, Moscow, Russia
d Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
We claim that some non-trivial theta-function 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 theta-functions on Jacobians of the Seiberg–Witten curves. For the case of three-particle systems the genus-2 identities are found and presented in the Letter. The connection with the Macdonald identities is established. The genus-2 theta-function 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 two-particle systems are also obtained.
|Ministry of Education and Science of the Russian Federation
|National Council for Scientific and Technological Development (CNPq)
|Russian Foundation for Basic Research
|The work was partly supported by Ministry of Education and Science of the Russian Federation under contract 8207 (A.Mir., A.Mor., A.Z.), the Brazil National Counsel of Scientific and Technological Development (A.Mor.), by NSh-3349.2012.2 (A.Mir., A.Mor.), by RFBR grants 13-02-00457 (A.Mir.), 13-02-00478 (A.Mor.) and 12-01-00482 (G.A., A.Z.), by joint grants 12-02-92108-Yaf (A.Mir., A.Mor.), 13-02-91371-ST (A.Mir., A.Mor.), 14-01-93004-Viet (A.Mir., A.Mor.), by leading young scientific groups RFBR 12-01-33071 mol_a_ved (G.A., A.Z.) and by D. Zimin's fund "Dynasty" (A.Z.).
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