  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]).
