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

TL;DR

This paper extends the theory of V-systems and explores their almost-duality in the context of extended affine Weyl orbit spaces, revealing new symmetries and transformations between rational and trigonometric solutions of the WDVV equations.

Contribution

It introduces conditions for extended configurations to satisfy $igvee$-conditions and studies their symmetries, including Legendre transformations, linking rational and trigonometric solutions.

Findings

Extended configurations satisfy $igvee$-conditions under new criteria.

Legendre transformations map rational solutions to trigonometric solutions.

Connections established between extended V-systems and classical affine Weyl groups.

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.