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.
Highlights
Seiberg–Witten theory interprets the eigenvalues of Lax operator as a 1-form on the spectral curve and treats integrals along the A-cycles as flat moduli aI, while those along the B-cycles as the gradient ∂F /∂aI of a function F(a) known as Seiberg–Witten prepotential
We present the calculations for the three-particle (N = 3) elliptic integrable systems associated with the low-energy limit of N = 2 SUSY gauge theories with adjoint matter hypermultiplets, the presentation being performed in the form allowing an immediate extension to an arbitrary number of particles N
We proposed new non-linear equations that allow one to effectively describe the instanton expansions for the Seiberg– Witten prepotentials associated with the N = 3 elliptic Calogero–Moser system (4d case), the N = 3 elliptic Ruijsenaars system (5d case) and the N = 3 double-elliptic integrable system (6d case)
Summary
Seiberg–Witten theory interprets the eigenvalues of Lax operator as a 1-form on the spectral curve and treats integrals along the A-cycles as flat moduli aI , while those along the B-cycles as the gradient ∂F /∂aI of a function F(a) known as Seiberg–Witten prepotential. We present the calculations for the three-particle (N = 3) elliptic integrable systems associated with the low-energy limit of N = 2 SUSY gauge theories with adjoint matter hypermultiplets, the presentation being performed in the form allowing an immediate extension to an arbitrary number of particles N. We demonstrate that the non-perturbative prepotential is a series in flat moduli with the coefficients being modular forms This fact is in a complete agreement with modular properties of the spectral curves of the corresponding integrable systems, and part of the behavior (the dependence on the quasimodular form E2) is described by the modular anomaly equation [39] in the Calogero and the Ruijsenaars cases. In the double-elliptic case, it is more involved and will be discussed elsewhere [40]
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