Deformations, extensions and symmetries of solutions to the WDVV equations

Stedman, Richard James (2017) Deformations, extensions and symmetries of solutions to the WDVV equations. PhD thesis, University of Glasgow.

Abstract

We investigate almost-dual-like solutions of the WDVV equations for which the metric, under the standard definition, is degenerate. Such solutions have previously been considered in [21] as complex Euclidean v-systems with zero canonical form but were not regarded as solutions since a non-degenerate metric is required for a solution. We have found that, in every case we considered, we can impose a metric and hence recover a solution. We also found that for the deformed A_n(c) family (first appearing in [8]) with the choice of parameters that renders the metric singular we can also recover a solution. The generalised root system A(n-1,n) (as it appears in our notation) has zero canonical form but we found that by restricting the covectors we can again recover a solution which we generalise to a family with (n+1) parameters which we denote as P_n.

We next look at extended v-systems. These are root-systems which possess the small orbit property (as defined in [36]) which we then extend into a dimension perpendicular to the original system. We then impose the v-conditions onto these systems and obtain 1-parameter infinite families of v-systems. We also find that for the B_n family we can extend into two perpendicular directions.

We then go on to look at a generalisation of the Legendre transformations (which originally appeared in [13])which map solutions to WDVV to other solutions. We find that such transformations are generated not only by constant vector fields but by functional vector fields too and we find a very simple rule which such vector fields must obey. Finally we link our work on extended v-systems and on generalised Legendre transformations to that on extended affine Weyl groups found in [16] and [17].

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: WDVV equations, Frobenius manifolds.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Strachan, Prof. I.A.B.
• Date of Award: 2017
• Depositing User: Mr Richard Stedman
• Unique ID: glathesis:2017-8011
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 24 Mar 2017 14:45
• Last Modified: 27 Apr 2017 07:35
• URI: https://theses.gla.ac.uk/id/eprint/8011