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)

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.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• 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: 0022-2488
• 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

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
190490		Engineering and Physical Sciences Doctoral Training Grant 2012-16	Mary Beth Kneafsey	Engineering and Physical Sciences Research Council (EPSRC)	EP/K503058/1	Research and Innovation Services