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

TL;DR

This paper explicitly verifies that the first two terms of the CIV-DV prepotential in the two-cut case satisfy the generalized WDVV equations, revealing new insights into the moduli structure and solution space of these equations.

Contribution

It demonstrates that the WDVV equations hold for the two-cut case with an extended set of moduli, and uncovers a family of solutions parameterized by the quantum-deformation parameter.

Findings

WDVV equations satisfied in the two-cut case with extended moduli

Extra modulus expression differs from naive Whitham theory expectations

Existence of a one-parameter family of solutions to WDVV equations

Abstract

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 hyperelliptic 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/g_3, we actually find an entire one-parameter family of solutions to the WDVV equations, of which the conventional prepotential is just a single point.