Commutativity equations and their trigonometric solutions

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.

26 pages