On the WDVV equations

Abstract:
The original Witten-Dijkgraaf-Verlinde-Verlinde or WDVV equations form a system of third order nonlinear partial differential equations playing an important role in such areas as topological field theory, Frobenius manifolds, quantum cohomology and integrable systems.

In the context of N = 2 supersymmetric Yang-Mills theory it was realized by Marshakov, Mironov and Morozov that a suitable generalization of the WDVV equations is necessary to identify the so-called prepotential of the Yang-Mills theory as a solution to these equations. Typical solutions (which are not solutions to the original equations) involve logarithmic dependence on the variables.

We will introduce the original as well as the generalized WDVV equations and present some exact solutions. In particular, inspired by N = 2 supersymmetric Yang-Mills theory in five dimensions we obtain a set of functions with trilogarithmic dependence which are surprisingly solutions to the original (and not just the generalized) WDVV system.