Complex reflection groups, logarithmic connections and bi-flat F-manifolds

TL;DR

This paper explores the geometric structures of bi-flat F-manifolds and extends duality concepts from Frobenius manifolds to complex reflection groups, revealing new invariants and connections in the theory.

Contribution

It introduces a new interpretation of bi-flat F-manifolds, extends duality to complex reflection groups, and generalizes Veselov's 8$-systems, uncovering previously unknown invariants.

Findings

Bi-flat F-manifolds encode almost duality for Frobenius manifolds without metric.

Extended duality to orbit spaces of exceptional complex reflection groups.

Discovered non-uniqueness of basic flat invariants in some examples.

Abstract

We show that bi-flat $F$-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's $\vee$-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank $2$ and $3$. On the Veselov's $\vee$-systems side, we provide a generalization of the notion of $\vee$-systems that gives rise to a dual connection which coincides with a Dunkl-Kohno-type connection associated with such groups. In particular, this allows us to treat on the same ground several different examples including Coxeter and Shephard groups. Remarkably, as a byproduct of our results, we prove that in some examples basic flat invariants are not uniquely defined. As far as we know, such a phenomenon has never been pointed out before.