2016/12/12 by Richard Stedman, Stedman, Richard, Ian A. B. Strachan +1 · 1 citation
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.1612.03667
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.