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

Abstract

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 ${\mathcal{D}}_{\infty }$ of ${\mathbf{S}}^{\lambda }$. We show that ${\mathcal{D}}_{\infty }$ 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 ${\log }_{\mathbf{c}}{\mathbf{S}}_{△}^{\lambda }$ with the space of induced $q$-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 $A$.

Memoirs of Amer. Math. Soc. to appear