New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential

TL;DR

This paper introduces new non-linear equations and modular form expansions for the double-elliptic Seiberg-Witten prepotential, extending the understanding of integrable systems beyond the WDVV framework.

Contribution

It derives novel non-linear equations for the perturbative prepotential that include non-perturbative corrections, with solutions expressed as modular expansions.

Findings

New non-linear equations for double-elliptic systems

Explicit non-perturbative corrections for N=3 case

Modular form expansions of solutions

Abstract

Integrable N-particle systems have an important property that the associated Seiberg-Witten prepotentials satisfy the WDVV equations. However, this does not apply to the most interesting class of elliptic and double-elliptic systems. Studying the commutativity conjecture for theta-functions on the families of associated spectral curves, we derive some other non-linear equations for the perturbative Seiberg-Witten prepotential, which turn out to have exactly the double-elliptic system as their generic solution. In contrast with the WDVV equations, the new equations acquire non-perturbative corrections which are straightforwardly deducible from the commutativity conditions. We obtain such corrections in the first non-trivial case of N=3 and describe the structure of non-perturbative solutions as expansions in powers of the flat moduli with coefficients that are (quasi)modular forms of the elliptic parameter.