A Frobenius manifold for $\ell$-Kronecker quiver

TL;DR

This paper constructs a Frobenius structure linked to the $ ext{Kronecker}$ quiver's Cartan matrix, providing a geometric framework on the space of stability conditions for its derived category.

Contribution

It introduces a Frobenius manifold structure that aligns with the $ ext{Kronecker}$ quiver's algebraic data and stability conditions.

Findings

Frobenius structure matches the generalized Cartan matrix.

Complex manifold is isomorphic to the stability condition space.

Provides geometric insight into the $ ext{Kronecker}$ quiver's representation theory.

Abstract

We construct a Frobenius structure whose intersection form coincides with the generalized Cartan matrix of the $\ell$-Kronecker quiver $K_{\ell}$ and underlying complex manifold is isomorphic to the space of stability conditions for the bounded derived category of finitely generated modules over the path algebra $\mathbb{C} K_{\ell}$.