Flat coordinates of algebraic Frobenius manifolds in small dimensions

TL;DR

This paper explicitly relates flat coordinates of Frobenius manifolds derived from Coxeter groups to algebraic functions, focusing on small dimensions up to four, enhancing understanding of their structure.

Contribution

It provides explicit relations between flat coordinates of the Frobenius metric and the intersection form for algebraic Frobenius manifolds up to dimension four.

Findings

Flat coordinates are algebraic functions on orbit spaces.

Explicit relations are found for most known examples up to dimension 4.

Flat coordinates of the metric η relate to basic invariants of Coxeter groups.

Abstract

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide polynomial Frobenius manifolds. Flat coordinates of the Frobenius metric $\eta$ are Saito polynomials which are distinguished basic invariants of the Coxeter group. Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We find explicit relations between flat coordinates of the Frobenius metric $\eta$ and flat coordinates of the intersection form $g$ for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric $\eta$ appear to be algebraic functions on the orbit space of the Coxeter group.