Autoequivalences of Fukaya categories of surfaces and graded gentle algebras

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. This includes the wrapped cases as introduced by Bocklandt. An important ingredient for our proof in characteristic zero is the exponential map from Hochschild cohomology to the derived Picard group introduced in recent work by the author. In positive characteristics, we combine deformation theory and formality results for Hochschild complexes to prove our results. 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.

💡 Research Summary

The paper determines the derived Picard groups (DPic) of partially wrapped Fukaya categories of surfaces, as introduced by Haiden‑Katzarkov‑Kontsevich, and of the associated graded gentle algebras. The authors treat both the wrapped case of Bocklandt and the more general partially wrapped situation, covering proper and homologically smooth algebras.

The main result (Theorem A) states that, when the base field k has characteristic zero or when the Fukaya category Fuk(Σ) is both homologically smooth and proper, the derived Picard group fits into a split semi‑direct product
 DPic(Fuk(Σ)) ≅ N(Σ) ⋊ MCG_gr(Σ).
Here MCG_gr(Σ) is the graded mapping‑class group of the marked surface Σ (preserving the set of stops and the homotopy class of the line field), and N(Σ) is an explicit abelian group read off from the topology of Σ. For punctured surfaces N(Σ) ≅ H₁(Σ, k^×) (with a single exceptional case); for other surfaces N(Σ) is described in detail in Propositions 10.6 and 11.4. The shift functor provides a central ℤ‑extension inside MCG_gr(Σ).

A key technical tool is the exponential map from Hochschild cohomology to the derived Picard group, constructed by the author in earlier work. In characteristic zero this map integrates a portion of HH¹(Fuk(Σ)) into actual auto‑equivalences, with the Baker‑Campbell‑Hausdorff formula and the Gerstenhaber bracket governing the group law. The authors compute HH⁎ of graded gentle algebras (building on their own previous calculations and earlier literature) and use these results to identify N(Σ) as the neutral component of DPic.

In positive characteristic the exponential map is unavailable. The authors instead exploit the formality of the Hochschild dg‑algebra for homologically smooth proper graded gentle algebras. Formality implies that deformations of the identity functor are controlled solely by HH¹, allowing the same description of N(Σ) and the semi‑direct product structure to hold.

The paper also proves a derived‑equivalence classification (Theorem C): two partially wrapped Fukaya categories are equivalent if and only if their underlying graded marked surfaces are isomorphic; similarly, two graded gentle algebras (proper or homologically smooth) have equivalent derived categories precisely when the associated surfaces coincide. This gives an effective derived invariant: the surface together with its decorations can be recovered from the compact subcategory of the derived category. The result generalises earlier work for proper degree‑zero algebras and for graded homologically smooth algebras, and it includes new cases such as punctured surfaces.

As an application, the authors consider non‑commutative nodal projective curves of gentle type (e.g., Kodaira cycles, stacky chains, and cycles). Using known equivalences between their Auslander curves and partially wrapped Fukaya categories, they deduce that the derived Picard groups of the corresponding derived categories are again of the form N(Σ) ⋊ MCG_gr(Σ). This recovers known auto‑equivalence groups from earlier literature and extends them to a broader class of non‑commutative curves.

The paper is organized as follows: Sections 1–5 review A∞‑categories, Hochschild cohomology, partially wrapped Fukaya categories, and graded gentle algebras. Section 6 constructs the geometrisation homomorphism DPic → MCG_gr and proves part of Theorem C. Section 7 establishes the existence of a section (the action of the mapping‑class group on the Fukaya category). Sections 8–9 recall the exponential map and compute Hochschild cohomology for graded gentle algebras. Section 10 describes the group N(Σ); Section 11 treats the positive‑characteristic case; Section 12 completes the proof for punctured surfaces.

Overall, the work provides a unified, algebraic‑geometric description of the auto‑equivalence groups of a large class of Fukaya‑type categories and their algebraic models, linking symplectic topology, homological algebra, and non‑commutative geometry through Hochschild cohomology and derived Picard theory.