Deformations, extensions and symmetries of solutions to the WDVV equations

[摘要] 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 welink our work on extended v-systems and on generalised Legendre transformations to that on extended affine Weyl groups found in [16] and [17].

[发布机构] University:University of Glasgow;Department:School of Mathematics and Statistics

[关键词] WDVV equations, Frobenius manifolds.