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 $\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, via the graded DMS (=decorated marked surface) $\mathbf{S}_\Delta$. We prove an isomorphism between the braid twist group of $\mathbf{S}_\bigtriangleup$ and the spherical twist group of $\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, and $\mathbf{q}$-intersection formulas. We also give a topological realization of the Lagrangian immersion $\mathcal{D}_\infty(\mathbf{S})\to\mathcal{D}_\mathbb{X}(\mathbf{S}_\bigtriangleup)$, where $\mathcal{D}_\infty(\mathbf{S})$ 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-$\infty$ case $\mathcal{D}_\infty(\mathbf{S})$ (cf. [Haiden-Katzarkov-Kontsevich, Opper-Plamondon-Schroll]).