Diagonal invariants and genus-zero Hurwitz Frobenius manifolds

TL;DR

This paper constructs Frobenius manifolds on rational functions with simple poles, revealing polynomial diagonal invariants and exploring their relation to Coxeter and affine-Weyl groups, advancing invariant theory in this context.

Contribution

It introduces a Frobenius manifold structure on rational functions, studies the polynomial diagonal invariants, and extends invariant theory for Coxeter and affine-Weyl groups.

Findings

Diagonal invariants are polynomial functions.

Explicit Frobenius manifold structures are constructed.

Invariant theory is developed for specific group classes.

Abstract

The Frobenius manifold structure on the space of rational functions with multiple simple poles is constructed. In particular, the dependence of the Saito-flat coordinates on the flat coordinates of the intersection form is studied. While some of the individual flat coordinates are complicated rational functions, they appear in the prepotential in certain combinations known as diagonal invariants, which turn out to be polynomial. Two classes are studied in more detail. These are generalisations of the Coxeter and extended-affine-Weyl orbits space for the group $W=W(A_\ell)\,.$ An invariant theory is also developed.