Seiberg-Witten curves and double-elliptic integrable systems

TL;DR

This paper investigates the connection between Seiberg-Witten curves and double-elliptic integrable systems, providing new equations and evidence for their relationship, and deriving the prepotential through theta-constant equations.

Contribution

It introduces theta-constant equations to prove the conjecture relating Seiberg-Witten curves to double-elliptic systems and offers an alternative method to derive the prepotential.

Findings

Derived theta-constant equations for the N-particle system

Provided evidence that solutions are exhausted by the double-elliptic system and degenerations

Proven theta-function identities ensuring Poisson commutativity

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 $N$-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.