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arXiv · 1612.03667

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

Richard Stedman·Ian A. B. Strachan
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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.

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Richard Stedman, Ian A. B. Strachan. 2016-12-12. Extended V-systems and almost-duality for extended affine Weyl orbit spaces. https://doi.org/10.1063/5.0024108

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