Extended ⋁-systems and trigonometric solutions to the WDVV equations
Richard Stedman, Ian A. B. Strachan
Source abstract
Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration—a configuration in a space one-dimension higher—satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281–4299 (1996)]. 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 ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.
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