Deformations of the root systems and new solutions to generalised WDVV equations

TL;DR

This paper explores special solutions to the generalized WDVV equations linked to root systems and their deformations, establishing geometric conditions that ensure these solutions and discovering new solutions related to Calogero-Moser systems.

Contribution

It introduces geometric conditions on covector sets that guarantee solutions to the WDVV equations, including all root systems and their deformations, leading to new solutions.

Findings

Geometric check-conditions for WDVV solutions

Verification that root systems satisfy these conditions

Discovery of new solutions via deformations of root systems

Abstract

A special class of solutions to the generalised WDVV equations related to a finite set of covectors is investigated. Some geometric conditions on such a set which guarantee that the corresponding function satisfies WDVV equations are found (check-conditions). These conditions are satisfied for all root systems and their special deformations discovered in the theory of the Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the new solutions for the generalized WDVV equations.