Stedman, Richard James
(2017)
Deformations, extensions and symmetries of solutions to the WDVV equations.
PhD thesis, University of Glasgow.
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Abstract
We investigate almostduallike 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 vsystems with zero canonical form but were not regarded as solutions since a nondegenerate 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(n1,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 vsystems. These are rootsystems 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 vconditions onto these systems and obtain 1parameter infinite families of vsystems. 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 vsystems and on generalised Legendre transformations to that on extended affine Weyl groups found in [16] and [17].
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