$q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories

Categorically, we introduce the Calabi-Yau-$\mathbb{X}$ categories ${\mathcal{D}}_{\mathbb{X}}$ of a graded marked surface ${\mathbf{S}}^{\lambda }$, as a $q$-deformation of the topological Fukaya category ${\mathcal{D}}_{\mathrm{\infty }}$ of ${\mathbf{S}}^{\lambda }$. We show that ${\mathcal{D}}_{\mathrm{\infty }}$ can be identified with the cluster-$\mathbb{X}$ category associated to ${\mathcal{D}}_{\mathbb{X}}$. Geometrically, we construct and identify the space of $q$-quadratic differentials on the logarithm surface $\operatorname{log}{$\mathbf{c}$} $\mathbf{S}$\vartriangle^\lambda$withthespaceofinduced$q$-stability conditions on ${\mathcal{D}}_{\mathbb{X}}$, for a complex parameter $s$ satisfying $\mathrm{Re}(s)\gg 1$. When $s=N$ is an integer, the result gives an $N$-analogue of Bridgeland-Smith’s result for realizing stability conditions on the orbit Calabi-Yau-$N$ category ${\mathcal{D}}_{\mathbb{X}}[\mathbb{X}-N]$ via CY-$N$ type quadratic differentials. When the genus of $\mathbf{S}$ is zero, the space of $q$-quadratic differentials can be also identified with framed Hurwitz spaces. As a byproduct, the result confirms the conjectural almost Frobenius structure on spaces of $q$-stability conditions for type $A$.