Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cluster theory of topological Fukaya categories. Part II: Higher Teichmüller theory

Merlin Christ

TL;DR

The work constructs relative $3$-Calabi--Yau categories attached to marked surfaces and Dynkin quivers, and provides an additive categorification of higher Teichmüller cluster algebras via gluing along triangulations implemented by perverse schobers. It identifies a canonical equivalence between the cosingularity category, the Higgs category, and a $1$-Calabi--Yau cluster category–valued topological Fukaya category, with the latter interpreted as the global sections of a quotient perverse schober. The cosingularity category is shown to be governed by a $2$-periodic topological Fukaya theory with coefficients in $ ext{C}_I= ext{CoSing}( ext{Pi}_2(I))$, and the Higgs category provides a coefficient-enhanced triangulated framework carrying cluster tilting objects. The paper gives a detailed construction via perverse schobers, describes the amalgamation of ice quivers with potentials along triangulations, and proves that the global sections are independent of triangulation, yielding a robust additive categorification of higher Teichmüller cluster algebras with explicit ties to topological Fukaya categories. This advances the understanding of higher rank cluster structures on surfaces and their categorical realizations, bridging dg/categorical, topological, and representation-theoretic viewpoints.

Abstract

We construct relative $3$-Calabi--Yau categories related with higher Teichmüller theory. We further study their corresponding cosingularity categories and the additive categorification of the corresponding cluster algebras. The input for our constructions is a marked surface with boundary and a Dynkin quiver $I$. In the case of the triangle, these categories have been described in recent work of Keller--Liu. For general surfaces, the categories are constructed via gluing along a perverse schober, categorifying the amalgamation of cluster varieties. The case $I=A_1$ was subject of the prequel paper. We show that the cosingularity category is equivalent to the corresponding Higgs category and to the topological Fukaya category of the marked surface valued in the $1$-Calabi--Yau cluster category of type $I$.

Cluster theory of topological Fukaya categories. Part II: Higher Teichmüller theory

TL;DR

The work constructs relative -Calabi--Yau categories attached to marked surfaces and Dynkin quivers, and provides an additive categorification of higher Teichmüller cluster algebras via gluing along triangulations implemented by perverse schobers. It identifies a canonical equivalence between the cosingularity category, the Higgs category, and a -Calabi--Yau cluster category–valued topological Fukaya category, with the latter interpreted as the global sections of a quotient perverse schober. The cosingularity category is shown to be governed by a -periodic topological Fukaya theory with coefficients in , and the Higgs category provides a coefficient-enhanced triangulated framework carrying cluster tilting objects. The paper gives a detailed construction via perverse schobers, describes the amalgamation of ice quivers with potentials along triangulations, and proves that the global sections are independent of triangulation, yielding a robust additive categorification of higher Teichmüller cluster algebras with explicit ties to topological Fukaya categories. This advances the understanding of higher rank cluster structures on surfaces and their categorical realizations, bridging dg/categorical, topological, and representation-theoretic viewpoints.

Abstract

We construct relative -Calabi--Yau categories related with higher Teichmüller theory. We further study their corresponding cosingularity categories and the additive categorification of the corresponding cluster algebras. The input for our constructions is a marked surface with boundary and a Dynkin quiver . In the case of the triangle, these categories have been described in recent work of Keller--Liu. For general surfaces, the categories are constructed via gluing along a perverse schober, categorifying the amalgamation of cluster varieties. The case was subject of the prequel paper. We show that the cosingularity category is equivalent to the corresponding Higgs category and to the topological Fukaya category of the marked surface valued in the -Calabi--Yau cluster category of type .

Paper Structure

This paper contains 28 sections, 57 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background on higher Teichmüller theory and cluster varieties
  3. Higher rank 3-Calabi--Yau categories of marked surfaces
  4. Cosingularity categories
  5. The Higgs category and cluster tilting theory
  6. Higher categorical preliminaries
  7. Linear infinity-categories
  8. Exact infinity-categories and cluster tilting objects
  9. infinity-categorical group actions
  10. Categorical compactifications and inverse Serre functors
  11. The 1-cluster categories of Dynkin type
  12. Orbit categories
  13. Fiber and cofiber sequences from squares in the Auslander--Reiten quiver
  14. Description of the suspension functor
  15. 3-Calabi--Yau perverse schobers for higher Teichmüller theory
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let ${\bf S}$ be a marked surface and $I$ a Dynkin quiver. ${\bf S}$ is assumed to have non-empty boundary and no punctures. With this, we associate a relative $3$-Calabi--Yau category $\mathcal{D}^{\operatorname{perf}}(\mathscr{G}_{{\bf G},I})$, see def:GS and cor:global_sections_independence. Ther

Figures (4)

  • Figure 1: The ice quiver of the basic triangle for $G=\operatorname{SL}_4$.
  • Figure 2: The amalgamation ice quiver of the $4$-gon for $G=\operatorname{SL}_4$.
  • Figure 3: The perverse schober $\mathcal{F}_{\Delta,I}$ on the basic triangle, parametrized by the $3$-spider. The distinguished bottom edge is dashed.
  • Figure 4: The the $1$-Calabi--Yau category $\mathcal{C}_I=\operatorname{CoSing}(\Pi_2(I))$ arises both as the cosingularity category of an orbit category and as an orbit category of the cosingularity category of $\mathcal{D}^{\operatorname{perf}}(\widetilde{\Pi_2(I)})$. The diagram commutes by the fact that colimits commute with colimit. Depicted are also compatible automorphisms, induced by $\mathcal{D}^{\operatorname{perf}}(\tilde{\sigma})$.

Theorems & Definitions (160)

  • Theorem 1.1: \ref{['thm:Cosing_Fukaya', 'thm:GS_cluster_tilting', 'thm:equivHiggsCosing']}
  • Example 1.2
  • Theorem 1.3: \ref{['thm:3gonschober']}
  • Theorem 1.4: \ref{['cor:global_sections_independence']} and \ref{['prop:global_sections_GS', 'prop:rel3CY']}
  • Proposition 1.5: \ref{['prop:Ginzburg_alg']}
  • Proposition 1.6: \ref{['prop:shift']}
  • Theorem 1.7: \ref{['thm:global_sections_=_Fukaya']}
  • Theorem 1.8: \ref{['thm:GS_cluster_tilting']}
  • Definition 2.1: $\!\!$Bar15
  • Definition 2.2
  • ...and 150 more