Extended ⋁-systems and trigonometric solutions to the WDVV equations

Stedman, R. and Strachan, I. A.B. (2021) Extended ⋁-systems and trigonometric solutions to the WDVV equations. Journal of Mathematical Physics, 62(2), 022301. (doi: 10.1063/5.0024108)

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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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Stedman, Richard James and Strachan, Professor Ian
Authors: Stedman, R., and Strachan, I. A.B.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Mathematical Physics
Publisher:AIP Publishing
ISSN (Online):1089-7658
Published Online:01 February 2021
Copyright Holders:Copyright © 2021 AIP Publishing
First Published:First published in Journal of Mathematical Physics 62(2): 022301
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
190490Engineering and Physical Sciences Doctoral Training Grant 2012-16Mary Beth KneafseyEngineering and Physical Sciences Research Council (EPSRC)EP/K503058/1Research and Innovation Services