Three-particle Integrable Systems with Elliptic Dependence on Momenta and Theta Function Identities

TL;DR

This paper demonstrates how genus-2 theta-function identities underpin the Poisson commutativity of three-particle elliptic integrable systems, linking advanced theta identities with Hamiltonian structures and Lax representations.

Contribution

It derives new genus-2 theta-function identities that establish the Poisson structure and Lax representations for three-particle elliptic integrable systems.

Findings

Identified genus-2 theta-function identities related to integrable systems

Connected theta identities with Poisson commutativity of Hamiltonians

Constructed Lax representations for two-particle systems

Abstract

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 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 paper. 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.