The Frobenius-manifold and stability-condition isomorphism conjecture for the -Kronecker quiver

Let $K_{\ell}$ be the $\ell$-Kronecker quiver, let ${\mathcal D}^{b}(K_{\ell})$ be its bounded derived category, and let $\widetilde{X}/W$ be the quotient Frobenius manifold constructed in the paper, with first structure connection $\widehat{\nabla}$ and deformed flat coordinates $\widetilde{t}^{1},\widetilde{t}^{2}$. Let $\mathcal Z$ be the central charge map, and let $d$ be the trivial connection on

$$T\operatorname{Stab}({\mathcal D}^{b}(K_{\ell}))\cong \operatorname{Stab}({\mathcal D}^{b}(K_{\ell}))\times \operatorname{Hom}_{\mathbb Z}(K_{0}({\mathcal D}^{b}(K_{\ell})),{\mathbb C}).$$

Frobenius-manifold and stability-condition isomorphism conjecture. There should exist an isomorphism

$$\varphi:\widetilde{X}/W\longrightarrow \operatorname{Stab}({\mathcal D}^{b}(K_{\ell}))$$

compatible with deformed flat coordinates and the central charge map, so that the diagram with maps $(\widetilde{t}^{1}|_{u=1},\widetilde{t}^{2}|_{u=1})$ and $\mathcal Z$ commutes. In particular,

$$\widehat{\nabla}|_{u=1}=\varphi^{*}d,$$

where

$$\widehat{\nabla}_{\delta}\delta':=\nabla_{\delta'}\delta+\dfrac{1}{u}\delta\circ\delta'.$$

In type $A_{2}$, an analogous isomorphism is known; the conjecture proposes the corresponding identification for the $\ell$-Kronecker quiver, linking the constructed Frobenius manifold to the stability-condition space and its central charge.