Legendre transformations for special solutions of the WDVV equations

The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations admit many interrelated classes of exact solutions, some of which are associated with the richer structure of a Frobenius manifold. This thesis explores a symmetry of the WDVV equations known as a Legendre transformation. Remarkably, Legendre transformations have previously been shown to connect rational and trigonometric solutions in limited examples. Legendre transformations are generated by a choice of vector field, which must satisfy a geometric requirement called the Legendre field condition. In the first part of the thesis, we analyse the Legendre field condition in the setting of each of the two-dimensional Frobenius manifolds. We explicitly describe all homogeneous Legendre fields for these manifolds. We also consider so-called twisted Legendre fields for the Frobenius manifold associated with the Coxeter root system A₂. Twisted fields arise from almost duality for Frobenius manifolds, which provides a mapping between certain polynomial and rational solutions of the WDVV equations. In the second part of the thesis, we investigate particular examples of Legendre transformations applied to multi-parameter families of rational solutions. The solutions we consider are associated with deformations of the Aₙ and Bₙ root systems; for special values of the parameters, these are almost dual to the corresponding polynomial Frobenius manifold. In all cases that we consider, Legendre transformations of the rational solutions produce trigonometric solutions which are also related to root systems. The Bₙ₋type rational solutions are mapped to a sub-class of a family of BCₙ₋₁-type trigonometric solutions already known in the literature. Motivated by results in the type A case, we introduce a new multi-parameter family of A-type trigonometric solutions. We show that this family generalises and relates some previously known trigonometric solutions of the same form. We find that two different types of Legendre transformations applied to Aₙ₋type rational solutions both produce sub-families of the new Aₙ₋₁-type trigonometric solutions.