Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras

Let $\mathbf{S}$ be a graded marked surface. We construct a string model for Calabi-Yau-$\mathbb{X}$ category ${\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{△})$, via the graded DMS (=decorated marked surface) ${\mathbf{S}}_{\mathrm{\Delta }}$. We prove an isomorphism between the braid twist group of ${\mathbf{S}}_{△}$ and the spherical twist group of ${\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{△})$, and $\mathbf{q}$-intersection formulas. We also give a topological realization of the Lagrangian immersion ${\mathcal{D}}_{\mathrm{\infty }}(\mathbf{S})\to {\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{△})$, where ${\mathcal{D}}_{\mathrm{\infty }}(\mathbf{S})$ is the topological Fukaya category associated to $\mathbf{S}$, that is triangle equivalent to the bounded derived category of some graded gentle algebra. This generalizes previous works of [Qiu, Qiu-Zhou] in the Calabi-Yau-3 case and and also unifies the Calabi-Yau-$\mathrm{\infty }$ case ${\mathcal{D}}_{\mathrm{\infty }}(\mathbf{S})$ (cf. [Haiden-Katzarkov-Kontsevich, Opper-Plamondon-Schroll]).