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Autoequivalences of Fukaya categories of surfaces and graded gentle algebras

Sebastian Opper

TL;DR

The paper determines the derived Picard groups of partially wrapped Fukaya categories of graded surfaces and their graded gentle algebras, unifying geometric and algebraic viewpoints. It employs an exponential integration map from Hochschild cohomology to the derived Picard group in characteristic zero, and leverages formality results in positive characteristic to extend the classification. A key outcome is that the surface with its decorations is a complete derived invariant for these categories, enabling a derived equivalence classification via geometrisation maps and mapping class group data. The work also establishes explicit group actions of mapping class groups on Fukaya categories, constructs string-complex–based functors between arc-system models, and analyzes kernels and kernels' structure to obtain a detailed description of the derived Picard groups across both wrapped and non-wrapped (proper) settings, including punctured cases.

Abstract

We compute the derived Picard groups of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich and the related graded gentle algebras. An important ingredient in characteristic zero is the exponential map from Hochschild cohomology to the derived Picard group introduced in recent work by the author. To prove our results in positive characteristic, we combine deformation theory and formality results for Hochschild complexes. Along the way we show that the surface together with its decorations forms a complete derived invariant of partially wrapped Fukaya categories and we prove analogous results for graded gentle algebras. This removes all previous restrictions from earlier results of this kind.

Autoequivalences of Fukaya categories of surfaces and graded gentle algebras

TL;DR

The paper determines the derived Picard groups of partially wrapped Fukaya categories of graded surfaces and their graded gentle algebras, unifying geometric and algebraic viewpoints. It employs an exponential integration map from Hochschild cohomology to the derived Picard group in characteristic zero, and leverages formality results in positive characteristic to extend the classification. A key outcome is that the surface with its decorations is a complete derived invariant for these categories, enabling a derived equivalence classification via geometrisation maps and mapping class group data. The work also establishes explicit group actions of mapping class groups on Fukaya categories, constructs string-complex–based functors between arc-system models, and analyzes kernels and kernels' structure to obtain a detailed description of the derived Picard groups across both wrapped and non-wrapped (proper) settings, including punctured cases.

Abstract

We compute the derived Picard groups of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich and the related graded gentle algebras. An important ingredient in characteristic zero is the exponential map from Hochschild cohomology to the derived Picard group introduced in recent work by the author. To prove our results in positive characteristic, we combine deformation theory and formality results for Hochschild complexes. Along the way we show that the surface together with its decorations forms a complete derived invariant of partially wrapped Fukaya categories and we prove analogous results for graded gentle algebras. This removes all previous restrictions from earlier results of this kind.

Paper Structure

This paper contains 51 sections, 65 theorems, 55 equations, 1 figure.

Table of Contents

  1. Reminder on A-categories and their Hochschild cohomology
  2. Hochschild spaces and brace operations
  3. $A_\infty$-categories and $A_\infty$-functors
  4. Twisted complexes and (perfect) derived categories
  5. The Hochschild $A_\infty$-algebra of an $A_\infty$-category
  6. Topological Fukaya categories after Haiden-Katzarkov-Kontsevich
  7. Marked surfaces
  8. Gradings and winding numbers
  9. Marked intervals versus marked points
  10. Topological Fukaya categories
  11. Graded gentle algebras
  12. Koszul duality: smooth versus proper
  13. Geometric model for derived categories of gentle algebras
  14. Derived Picard groups
  15. The homotopy category of dg categories and derived Picard groups
  16. ...and 36 more sections

Key Result

Theorem A

Let $\Sigma$ be a graded marked surface and let $\operatorname{Fuk}(\Sigma)$ denote the associated partially wrapped Fukaya category. If $\operatorname{char} \Bbbk=0$ or $\operatorname{Fuk}(\Sigma)$ is homologically smooth and proper, there exists an isomorphism where $\mathop{\mathrm{\operatorname{MCG}}}\nolimits_{\operatorname{gr}}(\Sigma)$ denotes the graded mapping class group of $\Sigma$ and

Figures (1)

  • Figure 1: A diagrammatic representation of a graph map (left) with identities indicated by double lines and a boundary singleton single map (right).

Theorems & Definitions (127)

  • Theorem A: \ref{['cor: Theorem A']} & \ref{['cor: Theorem A (2)']}
  • Theorem B: \ref{['thm: derived invariant Fukaya categories']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5: cf. SeidelFormalGroups, OpperIntegration
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • ...and 117 more