https://doi.org/10.1140/epjc/s10052-016-4267-6
Regular Article - Theoretical Physics
New non-linear equations and modular form expansion for double-elliptic Seiberg–Witten prepotential
1
ITEP, Moscow, 117218, Russia
2
Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Russia
3
Lebedev Physics Institute, Moscow, 119991, Russia
4
National Research Nuclear University MEPhI, Moscow, 115409, Russia
5
Institute for Information Transmission Problems, Moscow, 127994, Russia
* e-mail: mironov@itep.ru
** e-mail: mironov@lpi.ru
Received:
23
June
2016
Accepted:
13
July
2016
Published online:
2
August
2016
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 
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.
© The Author(s), 2016
