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

Abstract

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) ${\mathbf{S}}_{\Delta }$. 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 ${\mathcal{D}}_{\infty }(\mathbf{S})\to {\mathcal{D}}_{\mathbb{X}}({\mathbf{S}}_{△})$, 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]).

arXiv commentTable of notations added and typos fixed

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• Flat surfaces and stability structures
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Cited by in corpus (1)

• Contractible flow of stability conditions via global dimension function