Experiments with the WDVV equations for the gluino-condensate prepotential: the cubic (two-cut) case

We demonstrate by explicit calculation that the first two terms in the CIV-DV prepotential for the two-cut case satisfy the generalized WDVV equations, just as in all other known examples of hyper-elliptic Seiberg–Witten models. The WDVV equations are non-trivial in this situation, provided the set of moduli is extended as compared to the Dijkgraaf–Vafa suggestion and includes also moduli, associated with the positions of the cuts (not only with their lengths). Expression for the extra modulus dictated by WDVV equation, however, appears different from a naive expectation implied by the Whitham theory. Moreover, for every value of the “quantum-deformation parameter” 1/g3, we actually find an entire one-parameter family of solutions to the WDVV equations, of which the conventional prepotential is just a single point.