Generalized Legendre transformations and symmetries of the WDVV equations

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.

23 pages

References in corpus (1)

• Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

Cited by in corpus (6)

• A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$
• WDVV equations and invariant bi-Hamiltonian formalism
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• Legendre transforms for type $A_n$ and $B_n$ $\vee$-systems
• On Equivariant Gromov--Witten Invariants of Resolved Conifold with Diagonal and Anti-Diagonal Actions
• Diagonal invariants and genus-zero Hurwitz Frobenius manifolds