Commutativity equations and their trigonometric solutions

TL;DR

This paper explores solutions to commutativity equations related to Frobenius algebras, establishing conditions under which these solutions satisfy WDVV equations and providing explicit formulas for trigonometric solutions linked to root systems.

Contribution

It demonstrates that under certain conditions, solutions to commutativity equations satisfy WDVV equations and provides explicit formulas for trigonometric solutions associated with root systems.

Findings

Solutions satisfy WDVV equations under non-degeneracy conditions

Explicit formulas for identity element in trigonometric solutions

Connections to non-simply laced root systems

Abstract

We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1, \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$. We show that under certain non-degeneracy conditions a solution $F$ satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field $e$. We also study trigonometric solutions $F$ determined by a finite collection of vectors with multiplicities, and we give an explicit formula for $e$ for all the known such solutions. The corresponding collections of vectors are given by non-simply laced root systems or are related to their projections to the intersection of mirrors.