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Cluster theory of topological Fukaya categories

Merlin Christ

Abstract

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.

Cluster theory of topological Fukaya categories

Abstract

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 -Calabi-Yau extriangulated/exact -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 -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 -Calabi-Yau Frobenius extriangulated structures/exact -structures on stable -categories equipped with a relative right -Calabi-Yau structure in the sense of Brav-Dyckerhoff, that may be of independent interest.
Paper Structure (40 sections, 53 theorems, 87 equations, 3 figures)

This paper contains 40 sections, 53 theorems, 87 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on cluster categories
  3. 1-periodic topological Fukaya categories
  4. The generalized cluster categories of marked surfaces and perverse schobers
  5. The exact infinity-structure and the categorification of the cluster algebra
  6. Relation with other types of cluster categories of surfaces
  7. Acknowledgements
  8. Notations and conventions
  9. Preliminaries
  10. infinity-categories of infinity-categories
  11. Semiorthogonal decompositions and Verdier quotients
  12. Monadicity and derived categories of graded Laurent algebras
  13. Relative Calabi--Yau structures
  14. Perverse schobers on surfaces
  15. Marked surfaces and ribbon graphs
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let ${\bf S}$ be a marked surface and $\mathcal{C}_{\bf S}$ the associated $1$-periodic topological Fukaya category. Every compact object $X\in \mathcal{C}_{\bf S}$ splits uniquely into the direct sum of indecomposable objects associated to matching curves.

Figures (3)

  • Figure 1: A crossing and a direct endpoint intersection of two matching curves. The boundary of the surface is depicted in green. The magenta arrow indicates the clockwise direction. The orange vertex describes a marked point.
  • Figure 2: On the left: a crossing of two matching curves $\gamma,\gamma$ (in blue) and the two possible smoothings $\gamma_1,\gamma_2$ and $\gamma_3,\gamma_4$. On the right: a directed boundary intersection of two matching curves $\gamma,\gamma'$ and the corresponding smoothed composite $\gamma_1$. The boundary of ${\bf S}$ is depicted in green. Outside of the depicted parts of ${\bf S}$, the matching curves continue identically.
  • Figure 3: One of the three possible segments (in blue) at a vertex of $\mathcal{T}$.

Theorems & Definitions (171)

  • Theorem 1.1: \ref{['thm:geom']}
  • Theorem 1.2: \ref{['thm:clstcat']}, \ref{['prop:indfin']}
  • Proposition 1.3
  • Definition 1.4: \ref{['def:cyfun']} and \ref{['lem:cyfun']}
  • Proposition 1.5: \ref{['prop:cydual']}
  • Theorem 1.6: \ref{['cor:bmrcluster']}
  • Theorem 1.7: \ref{['thm:char']}
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2: $\!\!$HA
  • ...and 161 more