Regular non-semisimple Dubrovin-Frobenius manifolds

TL;DR

This paper investigates specific classes of regular non-semisimple Dubrovin-Frobenius manifolds in low dimensions, focusing on cases with a single Jordan block, and utilizes special local coordinates to analyze their invariant metrics.

Contribution

It introduces a detailed study of regular non-semisimple Dubrovin-Frobenius manifolds with a single Jordan block, leveraging special local coordinates for their analysis.

Findings

Characterization of invariant metrics in special coordinates

Explicit descriptions in dimensions 2, 3, 4

Insights into the structure of non-semisimple manifolds

Abstract

We study regular non-semisimple Dubrovin-Frobenius manifolds in dimensions 2,3,4. We focus on the case where the Jordan canonical form of the operator of multiplication by the Euler vector field has a single Jordan block. Our results rely on the existence of special local coordinates introduced in [4] for regular flat F-manifolds with Euler vector field. In such coordinates the invariant metric of the Dubrovin-Frobenius manifold takes a special form which is the starting point of our construction.