An integrable system on the moduli space of rational functions and its variants

Abstract

We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action-angle variables and the separated variables (in Sklyanin's sense) are related via a canonical transformation, the generating function of which is the Abel-Jacobi type integral of the Seiberg-Witten differential over the spectral curve.

Integrable systems whose spectral curves are the graph of a function

“…In this quite general setup, one can use a trick based on the residue theorem [14] to derive the identity…”

“…Apart from the CM and RS systems, the foregoing examples are associated with a pair of polynomials A(λ) and B(λ). The main result of the joint work with Takebe [14] is to construct the following analogues defined on a cylinder and a torus:…”

“…where g 2 (λ) and g 3 (λ) are functions (e.g., polynomials) of λ. Geometrically, this is the structure called an "elliptic fibration". Integrable systems of this type, too, are considered in the joint work with Takebe [14].…”

“…This article is a supplement to the joint work with Takebe [14] on SOV of integrable systems. This work was motivated by a rather naive question -what will be the simplest nontrivial examples of SOV of integrable systems?…”