Legendre transforms for type $A_n$ and $B_n$ $\vee$-systems

Abstract

The Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to $A_n$- and $B_n$-type multi-parameter rational solutions, relating them to known and new trigonometric solutions.

28 pages

References in corpus (8)

• Deformations of the root systems and new solutions to generalised WDVV equations
• Locus configurations and $\vee$-systems
• Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems
• A note on the relationship between rational and trigonometric solutions of the WDVV equations
• Generalized Legendre transformations and symmetries of the WDVV equations
• On the WDVV equations in five-dimensional gauge theories
• Explicit solutions of the WDVV equation determined by the "flat" hydrodynamic reductions of the Egorov hydrodynamic chains
• Trigonometric $\vee$-systems and solutions of WDVV equations