Cluster theory of topological Fukaya categories

We establish a novel relation between the cluster categories associated with marked surfaces and the topological Fukaya categories of the surfaces. We consider a generalization of the triangulated cluster category of the surface by a $2$-Calabi-Yau extriangulated/exact $\infty$-category, which arises via Amiot's construction from the relative Ginzburg algebra of the triangulated surface. This category is shown to be equivalent to the $1$-periodic version of the topological Fukaya category of the marked surface, as well as to Wu's Higgs category. We classify the cluster tilting objects in this extriangulated cluster category and describe a cluster character to the upper cluster algebra of the marked surface with coefficients in the boundary arcs. We furthermore give a general construction of $2$-Calabi-Yau Frobenius extriangulated structures/exact $\infty$-structures on stable $\infty$-categories equipped with a relative right $2$-Calabi-Yau structure in the sense of Brav-Dyckerhoff, that may be of independent interest.