This thesis studies how a symmetry defined on the solution space to the WDVV equations, called the inversion symmetry, singles out a special class of solution: those that lie at its fixed points. We will learn how demanding invariance of the solution under this symmetry forces it to take on a very rich form: that of a quasi-modular function. We will also study how the corresponding principal hierarchies inherit this symmetry from the solution to the WDVV equations.
More specifically, Chapter 1 is introductory material on the theory of Frobenius manifolds. We provide motivation, basic examples and tools that will be useful to us later. As such it contains no original material. The references [1,16,17,18,19,35,37] were extremely useful in the preparation of this chapter.
Chapter 2 is based on [45], which was written in collaboration with Professor Ian Strachan. It appeared in Physica D: Nonlinear Phenomena. We study those solutions to WDVV that lie at the fixed points the inversion symmetry. By studying the transformation and homogeneity properties that such solutions must have, we set out a program for classification, with complete results presented for dimensions three and four, together with partial results for dimension five. We show how various examples that have appeared in the literature fit into our framework. Any material here that is not original is clearly referenced.
Chapter 3 is background material on integrable systems. It begins with a little history of the KdV equation, and explores some of the key properties that make it interesting [15,13]. We move on to the construction of Poisson brackets on loop spaces [48,24], and more specifically Poisson brackets of hydrodynamic type. We then sketch Dubrovin's construction [16] of the principal hierarchy from the geometry of the Frobenius manifold.
Chapter 4 is based on [44], a joint work with Professor Ian Strachan which appeared in International Mathematics Research Notices. We study how the inversion symmetry defined on the solution space to the WDVV equations lifts to the principal hierarchy. It turns out that the action is an example of a so-called reciprocal transformation, introduced by Rogers [52]. These results were obtained independently in [41], and more recently in [26]. There is also some background material which has been tailored from Dubrovin's work [10] on so-called almost dual solutions to WDVV. We give a definition of the inversion symmetry for these solutions and study how the inversion symmetry acts on the associated hydrodynamic systems.
Chapter 5 is a very natural continuation of the results obtained in Chapter 4. Motivated by the Witten-Kontsevich theorem, Dubrovin & Zhang [23] showed how the inclusion of the elliptic Gromov-Witten invariants into the tau-function perturbs the equations of the hierarchy. The results of Chapter 5 show how to extend the symmetry found at the level of the hydrodynamic equations to these first order, (or genus one), deformations
