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arXiv · 1606.05274

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

G. Aminov·A. Mironov·A. Morozov
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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.

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G. Aminov, A. Mironov, A. Morozov. 2016-06-26. New non-linear equations and modular form expansion for double-elliptic Seiberg-Witten prepotential. https://doi.org/10.1140/epjc%2Fs10052-016-4267-6

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