Abstract
The Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class—the so-called Legendre transformations—were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.
Highlights
The Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity∂3F (t) ∂ tα∂ tβ ∂ tν ηνμ∂3F (t) ∂ tμ ∂ tγ ∂ tλ = ∂∂3F (t) tβ∂tγ ∂tν ηνμ ∂3F (t) tα∂tμ∂tλ, have been much studied from a variety of different points of view, amongst them, topological quantum field theories, Seiberg-Witten theory, singularity theory and integrable systems
The aim of this paper is to study more general Legendre-type transformations where the vector field ∂ is replaced by a field satisfying the condition
Similar results may be proved for the homogeneity of ∇e∂ and related objects. Before applying these ideas to generate new solutions of the WDVV equations from old, we develop a submanifold theory for Legendre fields
Summary
Have been much studied from a variety of different points of view, amongst them, topological quantum field theories, Seiberg-Witten theory, singularity theory and integrable systems. In what follows we will apply Legendre fields to map solutions of the WDVV equations to new solutions, but the ideas may be applied more generally to situations where one has torsion free connections and non-metric connections, or curvature, such as Riemannian F -manifolds [1, 10]. Before doing this we derive certain basic properties of such Legendre fields. Similar results may be proved for the homogeneity of ∇e∂ and related objects Before applying these ideas to generate new solutions of the WDVV equations from old, we develop a submanifold theory for Legendre fields
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