Extended V-systems and almost-duality for extended affine Weyl orbit spaces

Abstract

Rational solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations of associativity are given in terms a configurations of vectors which satisfy certain algebraic conditions known as $\bigvee$-conditions. The simplest examples of such configuration are the root systems of finite Coxeter groups. In this paper conditions are derived which ensure that an extended configuration - a configuration in a space one-dimension higher -satisfy these $\bigvee$-conditions. Such a construction utilizes the notion of a small-orbit, as defined by Serganova. Symmetries of such resulting solutions to the WDVV-equations are studied; in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions and, for certain values of the free data, one obtains a transformation from extended $\bigvee$-systems to the trigonometric almost dual solutions corresponding to the classical extended affine Weyl groups.

26 pages, 7 figures

References in corpus (5)

• Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials
• Complex reflection groups, logarithmic connections and bi-flat F-manifolds
• Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems
• N=4 Multi-Particle Mechanics, WDVV Equation and Roots
• Generalized Legendre transformations and symmetries of the WDVV equations

Cited by in corpus (2)

• WDVV equations and invariant bi-Hamiltonian formalism
• Five-dimensional gauge theories and the local B-model